diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 5f95802a..50372f5c 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -54,7 +54,8 @@ "K0106003B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-02_常用逻辑用语" ], "genre": "填空题", "ans": "(1) 必要非充分; (2) 充分非必要", @@ -223,7 +224,8 @@ "K0106001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-02_常用逻辑用语" ], "genre": "选择题", "ans": "A", @@ -455,7 +457,8 @@ "K0107003B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-02_常用逻辑用语" ], "genre": "解答题", "ans": "证明略", @@ -508,7 +511,8 @@ "K0109004B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-04_方程与不等式的求解" ], "genre": "解答题", "ans": "(1) $\\dfrac{5}{2}$; (2) $-\\dfrac{35}{4}$.", @@ -560,7 +564,8 @@ "K0111003B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-03_等式与不等式的性质及证明" ], "genre": "解答题", "ans": "证明略", @@ -589,7 +594,9 @@ "K0112001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-04_方程与不等式的求解" ], "genre": "解答题", "ans": "$(-\\infty,\\dfrac 53)$", @@ -837,7 +844,9 @@ "K0109004B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-04_方程与不等式的求解" ], "genre": "解答题", "ans": "$-1$", @@ -887,7 +896,9 @@ "K0112001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-04_方程与不等式的求解" ], "genre": "解答题", "ans": "$(-\\infty,-1)$", @@ -1166,7 +1177,8 @@ "K0118003B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-03_等式与不等式的性质及证明" ], "genre": "解答题", "ans": "证明略", @@ -1222,7 +1234,9 @@ "K0206002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-05_幂指数与对数" ], "genre": "填空题", "ans": "(1) $\\sqrt[3]{5}$, $\\log_3 5$; (2) $a^{\\frac 13}$; (3) $\\dfrac 14$; (4) $125$", @@ -1249,7 +1263,9 @@ "K0202002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-05_幂指数与对数" ], "genre": "选择题", "ans": "(1) B (2) D", @@ -1350,7 +1366,8 @@ "K0204003B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第一轮复习讲义-05_幂指数与对数" ], "genre": "解答题", "ans": "$5$", @@ -1377,7 +1394,8 @@ "K0205002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第一轮复习讲义-05_幂指数与对数" ], "genre": "填空题", "ans": "(1) $-3$; (2) $64$; (3) $1$", @@ -1480,7 +1498,8 @@ "K0203004B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第一轮复习讲义-05_幂指数与对数" ], "genre": "解答题", "ans": "证明略", @@ -1531,7 +1550,9 @@ "K0206002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-05_幂指数与对数" ], "genre": "解答题", "ans": "证明略", @@ -1561,7 +1582,9 @@ "K0210002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-06_幂指对函数" ], "genre": "填空题", "ans": "(1) $y=x^{\\frac 12}$, $y=(\\sqrt[4]{2})^x$, $y=\\log_{\\sqrt[4]{2}}x$; (2) $(-\\infty,0)$; (3) $(2,2)$", @@ -1770,7 +1793,9 @@ "K0210002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-06_幂指对函数" ], "genre": "填空题", "ans": "(1) $1$或$3$; (2) 一", @@ -1900,7 +1925,9 @@ "K0107003B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-较难题", + "2023届高三-第一轮复习讲义-05_幂指数与对数" ], "genre": "解答题", "ans": "不是, 证明略", @@ -2256,7 +2283,9 @@ "K0221002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-09_函数的零点与最值" ], "genre": "解答题", "ans": "$-1$或$2$", @@ -2405,7 +2434,9 @@ "K0220001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-08_函数的单调性" ], "genre": "解答题", "ans": "在$(-\\infty,-1]$及$[0,1]$上分别严格增, 在$[-1,0]$及$[1,+\\infty)$上分别严格减; 证明略; 图像如下:\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1.7:1.7] plot (\\x,{pow((\\x * \\x -1),2)-1});\n\\end{tikzpicture}\n\\end{center}", @@ -2655,7 +2686,9 @@ "K0314004B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-13_解三角形" ], "genre": "解答题", "ans": "$\\dfrac 58$", @@ -2986,7 +3019,8 @@ "K0310002B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-13_解三角形" ], "genre": "解答题", "ans": "(1) 等腰三角形($a=b$)或直角三角形($C=90^\\circ$); (2) 等腰三角形($a=b$)或直角三角形($C=90^\\circ$).", @@ -3072,7 +3106,8 @@ "K0315004B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-13_解三角形" ], "genre": "解答题", "ans": "(1) \\textcircled{1} $B$不存在, \\textcircled{2} $B=90^\\circ$, \\textcircled{3} $B=\\arcsin\\dfrac 9{13}$或$\\pi-\\arcsin \\dfrac 9{13}$, \\textcircled{4} $B=30^\\circ$, \\textcircled{5} $B=\\arcsin\\dfrac 9{44}$; (2) 当$0=latex,xscale = {8/15}, yscale = {5/0.25}]\n\\draw [->] (0,0) -- (0.25,0) -- (0.5,0.01) -- (0.75,-0.01) -- (1,0) -- (15,0) node [below] {距离};\n\\draw [->] (0,0) -- (0,0.25) node [left] {频率/组距};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j/\\n in {1.5/{1/30}/11.5,3.5/{4/30}/13.5,5.5/{11/60}/15.5,7.5/{5/60}/17.5,9.5/{4/60}/19.5} {\\draw (\\i,0) --++ (0,\\j) --++ (2,0) --++ (0,{-\\j}); \\draw (\\i,0) node [below] {$\\n$};};\n\\draw (11.5,0) node [below] {$21.5$};\n\\draw [dashed] (0,{2/60}) node [left] {$1/30$} --++ (1.5,0);\n\\draw [dashed] (0,{2/15}) node [left] {$2/15$} --++ (3.5,0);\n\\draw [dashed] (0,{11/60}) node [left] {$11/60$} --++ (5.5,0);\n\\draw [dashed] (0,{1/12}) node [left] {$1/12$} --++ (7.5,0);\n\\draw [dashed] (0,{1/15}) node [left] {$1/15$} --++ (9.5,0);\n\\end{tikzpicture}", @@ -6504,7 +6592,8 @@ "K0905001B" ], "tags": [ - "第九单元" + "第九单元", + "2023届高三-第一轮复习讲义-27_统计初步中的术语" ], "genre": "解答题", "ans": "\\begin{tabular}{c|cccccccccccc}\n$6$ & $4$ & $7$\\\\\n$7$ & $0$ & $2$ & $4$ & $6$ & $6$ & $9$ \\\\\n$8$ & $0$ & $1$ & $2$ & $2$ & $3$ & $5$ & $6$ & $8$ \\\\\n$9$ & $1$ & $1$ & $2$ & $3$ & $3$ & $3$ & $5$ & $6$ & $6$ & $7$ & $7$ & $9$ \\\\\n$10$ & $0$ & $0$ & $2$ & $4$ & $6$ & $6$ & $7$ & $8$ & $8$ \\\\\n$11$ & $2$ & $2$ & $4$ & $6$ & $8$ & $9$ & $9$ \\\\\n$12$ & $2$ & $3$ & $5$ & $6$ & $8$\\\\\n$13$ & $3$\n\\end{tabular}", @@ -6814,7 +6903,9 @@ ], "tags": [ "第七单元", - "直线" + "直线", + "2023届高三-四月错题重做-04_解析几何", + "2023届高三-第一轮复习讲义-34_直线及其方程" ], "genre": "解答题", "ans": "$y=-\\dfrac 23 x$或$y=-x+1$", @@ -7089,7 +7180,9 @@ ], "tags": [ "第七单元", - "直线" + "直线", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-34_直线及其方程" ], "genre": "解答题", "ans": "$y=\\dfrac 23 x$", @@ -7233,7 +7326,8 @@ "K0704003X" ], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第一轮复习讲义-34_直线及其方程" ], "genre": "解答题", "ans": "图形略, 面积为$2$", @@ -7347,7 +7441,8 @@ "K0721002X" ], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第一轮复习讲义-35_圆及曲线方程" ], "genre": "解答题", "ans": "(1) 不正确, 如$(1,-1)$; (2) 正确, 理由略; (3) 不正确, 如$(1,0)$", @@ -7469,7 +7564,8 @@ ], "tags": [ "第七单元", - "椭圆" + "椭圆", + "2023届高三-第一轮复习讲义-36_椭圆与双曲线的概念及性质" ], "genre": "解答题", "ans": "$\\dfrac{\\sqrt{5}-1}2$", @@ -7560,7 +7656,8 @@ ], "tags": [ "第七单元", - "圆" + "圆", + "2023届高三-第一轮复习讲义-35_圆及曲线方程" ], "genre": "解答题", "ans": "$x^2+9y^2=4$", @@ -7794,7 +7891,9 @@ "K0720003X" ], "tags": [ - "第七单元" + "第七单元", + "2023届高三-四月错题重做-04_解析几何", + "2023届高三-第一轮复习讲义-37_抛物线的概念及性质" ], "genre": "解答题", "ans": "$a=0$或$-\\dfrac 12$", @@ -7941,7 +8040,8 @@ "K0721002X" ], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第一轮复习讲义-35_圆及曲线方程" ], "genre": "解答题", "ans": "$y^2=k(x^2-36)$($y\\ne 0$且$k\\ne \\pm 6$)", @@ -8114,7 +8214,8 @@ ], "tags": [ "第六单元", - "空间向量" + "空间向量", + "2023届高三-第一轮复习讲义-33_立体几何中的定量计算" ], "genre": "解答题", "ans": "(1) $(\\dfrac 73,\\dfrac 76,\\dfrac 76)$, $(0,0,0)$, $(\\dfrac 72,0,\\dfrac 72)$; (2) $\\dfrac{5\\sqrt{3}}3$", @@ -8153,7 +8254,8 @@ ], "tags": [ "第六单元", - "空间向量" + "空间向量", + "2023届高三-第一轮复习讲义-33_立体几何中的定量计算" ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\arcsin \\dfrac{\\sqrt{10}}5$", @@ -8218,7 +8320,8 @@ ], "tags": [ "第六单元", - "空间向量" + "空间向量", + "2023届高三-第一轮复习讲义-33_立体几何中的定量计算" ], "genre": "解答题", "ans": "(1) 证明略; (2) $2$", @@ -8281,7 +8384,8 @@ "K0619003B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期周末卷-周末卷05" ], "genre": "解答题", "ans": "(1) $\\dfrac\\pi 3$; (2) $\\dfrac 56$", @@ -8314,7 +8418,8 @@ ], "tags": [ "第六单元", - "空间向量" + "空间向量", + "2023届高三-第一轮复习讲义-33_立体几何中的定量计算" ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\arccos \\dfrac 23$", @@ -8351,7 +8456,8 @@ "K0610004B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-21_空间直线与平面的位置关系" ], "genre": "解答题", "ans": "$\\dfrac{\\pi}6$", @@ -8438,7 +8544,8 @@ "K0613003B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-22_空间平面与平面的位置关系" ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\arctan \\dfrac{\\sqrt{5}}2$与$\\pi-\\arctan \\dfrac{\\sqrt{5}}2$", @@ -8469,7 +8576,8 @@ ], "tags": [ "第六单元", - "空间向量" + "空间向量", + "2023届高三-第一轮复习讲义-33_立体几何中的定量计算" ], "genre": "解答题", "ans": "(1) 证明略; (2) $1$; (3) $(\\dfrac 13,1)$", @@ -8506,7 +8614,8 @@ ], "tags": [ "第六单元", - "空间向量" + "空间向量", + "2023届高三-第一轮复习讲义-33_立体几何中的定量计算" ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\dfrac\\pi 3$与$\\dfrac{2\\pi}3$", @@ -8697,7 +8806,9 @@ "K0402006X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-30_等差数列与等比数列" ], "genre": "解答题", "ans": "$15$项, $225$", @@ -8913,7 +9024,8 @@ "K0406004X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第一轮复习讲义-31_数列的递推与通项及数学归纳法" ], "genre": "解答题", "ans": "(1) 证明略; (2) 最大项为$b_1=2$, 最小项为$b_2=\\dfrac 25$", @@ -8947,7 +9059,10 @@ "K0402002X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-四月错题重做-03_数列", + "2023届高三-四月错题重做-03_易错题-数列", + "2023届高三-第一轮复习讲义-30_等差数列与等比数列" ], "genre": "解答题", "ans": "证明略", @@ -8991,7 +9106,9 @@ "K0408003X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-31_数列的递推与通项及数学归纳法" ], "genre": "解答题", "ans": "证明略", @@ -9033,7 +9150,8 @@ "K0408003X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第一轮复习讲义-31_数列的递推与通项及数学归纳法" ], "genre": "解答题", "ans": "存在, $a=\\dfrac 14$, $b=0$, $c=-\\dfrac 14$, 证明略", @@ -9118,7 +9236,9 @@ "K0105002B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-容易题", + "2023届高三-赋能-赋能01" ], "genre": "填空题", "ans": "真", @@ -9156,7 +9276,8 @@ "K0104003B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-赋能-赋能01" ], "genre": "填空题", "ans": "$a\\le 0$", @@ -9195,7 +9316,8 @@ "K0514001B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-赋能-赋能01" ], "genre": "填空题", "ans": "$5$", @@ -9231,7 +9353,8 @@ "content": "若$\\triangle ABC$中, $a+b=4$, $\\angle C=30^\\circ$, 则$\\triangle ABC$面积的最大值是\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-赋能-赋能01" ], "genre": "填空题", "ans": "$1$", @@ -9317,7 +9440,8 @@ "content": "若半径为2的球$O$表面上一点$A$作球$O$的截面, 若$OA$与该截面所成的角是$60^\\circ$, 则该\n截面的面积是\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-赋能-赋能01" ], "genre": "填空题", "ans": "$\\pi$", @@ -9359,7 +9483,11 @@ "tags": [ "第八单元", "概率", - "加法原理与乘法原理" + "加法原理与乘法原理", + "2023届高三-四月错题重做-05_易错题-概率与统计", + "2023届高三-四月错题重做-05_概率与统计", + "2023届高三-第一轮复习讲义-25_概率的概念及性质", + "2023届高三-赋能-赋能01" ], "genre": "填空题", "ans": "$\\dfrac 1{108}$", @@ -9477,7 +9605,9 @@ ], "tags": [ "第七单元", - "双曲线" + "双曲线", + "2023届高三-寒假作业-较难题", + "2023届高三-赋能-赋能01" ], "genre": "填空题", "ans": "$16y^2-4x^2=1$", @@ -9557,7 +9687,8 @@ ], "tags": [ "第七单元", - "抛物线" + "抛物线", + "2023届高三-赋能-赋能02" ], "genre": "填空题", "ans": "$\\dfrac 92$", @@ -9625,7 +9756,8 @@ "K0514004B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-赋能-赋能02" ], "genre": "填空题", "ans": "$2$", @@ -9666,7 +9798,10 @@ ], "tags": [ "第八单元", - "二项式定理" + "二项式定理", + "2023届高三-四月错题重做-05_易错题-概率与统计", + "2023届高三-四月错题重做-05_概率与统计", + "2023届高三-赋能-赋能02" ], "genre": "填空题", "ans": "$160$", @@ -9709,7 +9844,8 @@ "content": "在长方体$ABCD-A_1B_1C_1D_1$中, 若$AB=BC=1$, $AA_1=\\sqrt{2}$, 则异面直线$BD_1$与$CC_1$所成角的大小为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-赋能-赋能02" ], "genre": "填空题", "ans": "$\\frac{\\pi }4$", @@ -9750,7 +9886,11 @@ "K0215005B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-四月错题重做-02_函数二", + "2023届高三-四月错题重做-02_易错题-函数2", + "2023届高三-第一轮复习讲义-07_函数的概念与奇偶性", + "2023届高三-赋能-赋能02" ], "genre": "填空题", "ans": "$0=latex]\n \\draw (0,0) node [left] {$B$} -- (3,0) node [right] {$C$} -- (1.5,{1.5*sqrt(3)}) node [above] {$A$} coordinate (A) -- cycle;\n \\draw [->] (A) -- (1,0) node [below] {$D$};\n \\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-赋能-赋能02" ], "genre": "填空题", "ans": "$-\\frac 32$", @@ -9839,7 +9980,8 @@ "K0223001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-赋能-赋能02" ], "genre": "填空题", "ans": "$4$", @@ -9876,7 +10018,8 @@ "objs": [], "tags": [ "第八单元", - "排列" + "排列", + "2023届高三-赋能-赋能02" ], "genre": "填空题", "ans": "$40320$", @@ -9914,7 +10057,8 @@ "K0104001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-赋能-赋能03" ], "genre": "填空题", "ans": "$\\{2\\}$", @@ -9950,7 +10094,9 @@ "content": "函数$y=\\sin (\\omega x-\\dfrac{\\pi}{3})$($\\omega >0$)的最小正周期是$\\pi$, 则$\\omega =$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-寒假作业-容易题", + "2023届高三-赋能-赋能03" ], "genre": "填空题", "ans": "$2$", @@ -10004,7 +10150,8 @@ "K0514002B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-赋能-赋能03" ], "genre": "填空题", "ans": "$\\frac 35$", @@ -10042,7 +10189,9 @@ "K0226004B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-容易题", + "2023届高三-赋能-赋能03" ], "genre": "填空题", "ans": "$3$", @@ -10110,7 +10259,8 @@ "objs": [], "tags": [ "第八单元", - "排列" + "排列", + "2023届高三-赋能-赋能03" ], "genre": "填空题", "ans": "$60$", @@ -10162,7 +10312,8 @@ "K0620002B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-赋能-赋能03" ], "genre": "填空题", "ans": "$\\dfrac{3\\sqrt 7}8\\pi$", @@ -10226,7 +10377,8 @@ "content": "如图, 在$\\triangle ABC$中, $\\angle B=45^\\circ$, $D$是$BC$边上的一点, $AD=5$, $AC=7$, $DC=3$, 则$AB$的长为\\blank{50}.\n\\begin{center}\n \\begin{tikzpicture}[scale = 0.5]\n \\draw (-6.830127018922193,0.)-- (3.,0.) node [below right] {$C$};\n \\draw (3.,0.)-- (-2.5,4.330127018922193) node [above] {$A$};\n \\draw (-2.5,4.330127018922193)-- (-6.830127018922193,0.) node [below left] {$B$};\n \\draw (-2.5,4.330127018922193)-- (0.,0.) node [below] {$D$};\n \\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-赋能-赋能03" ], "genre": "填空题", "ans": "$\\frac{5\\sqrt 6}2$", @@ -10311,7 +10463,8 @@ "K0104001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-赋能-赋能04" ], "genre": "填空题", "ans": "$[0,1]$", @@ -10347,7 +10500,8 @@ "content": "若$-\\dfrac{\\pi}{2}<\\alpha <\\dfrac{\\pi}{2}$, $\\sin \\alpha =\\dfrac{3}{5}$, 则$\\cot 2\\alpha =$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-赋能-赋能04" ], "genre": "填空题", "ans": "$\\frac 7{24}$", @@ -10416,7 +10570,8 @@ ], "tags": [ "第八单元", - "二项式定理" + "二项式定理", + "2023届高三-赋能-赋能04" ], "genre": "填空题", "ans": "$31$", @@ -10456,7 +10611,9 @@ ], "tags": [ "第七单元", - "双曲线" + "双曲线", + "2023届高三-寒假作业-中档题", + "2023届高三-赋能-赋能04" ], "genre": "填空题", "ans": "$(\\sqrt 2,+\\infty)$", @@ -10495,7 +10652,8 @@ "K0218001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-赋能-赋能04" ], "genre": "填空题", "ans": "$[0,+\\infty)$", @@ -10533,7 +10691,9 @@ "K0213008B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第一轮复习讲义-06_幂指对函数", + "2023届高三-赋能-赋能04" ], "genre": "填空题", "ans": "$x=1$", @@ -10577,7 +10737,10 @@ ], "tags": [ "第七单元", - "圆" + "圆", + "2023届高三-四月错题重做-04_易错题-解析几何", + "2023届高三-四月错题重做-04_解析几何", + "2023届高三-赋能-赋能04" ], "genre": "填空题", "ans": "$k<-2$或$k>0$", @@ -10628,7 +10791,8 @@ "K0619003B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-赋能-赋能04" ], "genre": "填空题", "ans": "$\\frac{\\sqrt 2}6$", @@ -10696,7 +10860,9 @@ "K0512002B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-寒假作业-容易题", + "2023届高三-赋能-赋能05" ], "genre": "填空题", "ans": "$2$", @@ -10737,7 +10903,9 @@ "K0204001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-容易题", + "2023届高三-赋能-赋能05" ], "genre": "填空题", "ans": "$-2$", @@ -10776,7 +10944,9 @@ "K0116002B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-容易题", + "2023届高三-赋能-赋能05" ], "genre": "填空题", "ans": "$[-1,1]$", @@ -10815,7 +10985,8 @@ ], "tags": [ "第七单元", - "抛物线" + "抛物线", + "2023届高三-赋能-赋能05" ], "genre": "填空题", "ans": "$\\frac 34$", @@ -10884,7 +11055,9 @@ "K0119001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-03_等式与不等式的性质及证明", + "2023届高三-赋能-赋能05" ], "genre": "填空题", "ans": "$\\frac 18$", @@ -10926,7 +11099,8 @@ "K0620004B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-赋能-赋能05" ], "genre": "填空题", "ans": "$75\\pi$", @@ -10965,7 +11139,8 @@ ], "tags": [ "第八单元", - "二项式定理" + "二项式定理", + "2023届高三-赋能-赋能05" ], "genre": "填空题", "ans": "$12$", @@ -11001,7 +11176,8 @@ "content": "已知$A,B$分别是函数$f(x)=2\\sin \\omega x$($\\omega >0$)在$y$轴右侧图像上的第一个最高点和第一个最低点, 且$\\angle AOB=\\dfrac\\pi 2$, 则该函数的最小正周期是\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-赋能-赋能05" ], "genre": "填空题", "ans": "$\\frac{8\\sqrt 3}3$", @@ -11041,7 +11217,8 @@ "tags": [ "第八单元", "组合", - "排列" + "排列", + "2023届高三-赋能-赋能05" ], "genre": "填空题", "ans": "$96$", @@ -11123,7 +11300,9 @@ "K0104006B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-容易题", + "2023届高三-赋能-赋能06" ], "genre": "填空题", "ans": "$\\{-1,0,1\\}$", @@ -11166,7 +11345,9 @@ "K0116002B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-容易题", + "2023届高三-赋能-赋能06" ], "genre": "填空题", "ans": "$(-2,-1)$", @@ -11291,7 +11472,9 @@ "content": "已知向量$\\overrightarrow{a}=(1,2)$, $\\overrightarrow{b}=(0,3)$, 则$\\overrightarrow{b}$在$\\overrightarrow{a}$的方向上的数量投影为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期测验卷-线上测验01", + "2023届高三-赋能-赋能06" ], "genre": "填空题", "ans": "$\\frac{6\\sqrt 5}5$", @@ -11337,7 +11520,9 @@ "content": "已知一个底面置于水平面上的圆锥, 其轴截面是边长为$6$的正三角形, 则该圆锥的侧面积为\\blank{50}.", "objs": [], "tags": [ - "暂无对应" + "暂无对应", + "2023届高三-寒假作业-容易题", + "2023届高三-赋能-赋能06" ], "genre": "填空题", "ans": "$18\\pi$", @@ -11379,7 +11564,9 @@ "tags": [ "第八单元", "概率", - "组合" + "组合", + "2023届高三-寒假作业-容易题", + "2023届高三-赋能-赋能06" ], "genre": "填空题", "ans": "$\\frac 57$", @@ -11433,7 +11620,8 @@ ], "tags": [ "第八单元", - "二项式定理" + "二项式定理", + "2023届高三-赋能-赋能06" ], "genre": "填空题", "ans": "$2$", @@ -11472,7 +11660,8 @@ "K0104001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-赋能-赋能07" ], "genre": "填空题", "ans": "$\\{1\\}$", @@ -11513,7 +11702,9 @@ "K0511005B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第一轮复习讲义-18_复数的代数运算与性质", + "2023届高三-赋能-赋能07" ], "genre": "填空题", "ans": "$3-4\\mathrm{i}$", @@ -11562,7 +11753,8 @@ "K0225004B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-赋能-赋能07" ], "genre": "填空题", "ans": "$2$", @@ -11600,7 +11792,9 @@ "K0117002B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-04_方程与不等式的求解", + "2023届高三-赋能-赋能07" ], "genre": "填空题", "ans": "$(0,1)\\cup (1,+\\infty)$", @@ -11638,7 +11832,8 @@ "objs": [], "tags": [ "第三单元", - "第五单元" + "第五单元", + "2023届高三-赋能-赋能07" ], "genre": "填空题", "ans": "$\\pi$", @@ -11678,7 +11873,8 @@ "tags": [ "第八单元", "概率", - "排列" + "排列", + "2023届高三-赋能-赋能07" ], "genre": "填空题", "ans": "$\\frac 14$", @@ -11714,7 +11910,8 @@ "content": "如图, 在棱长为$1$的正方体$ABCD-A_1B_1C_1D_1$中, 点$P$在截面$A_1DB$上, 则线段$AP$的最小值为\\blank{50}.\n\\begin{center}\n \\begin{tikzpicture}\n \\draw (0,0) node [below left] {$A$} coordinate (A) --++ (2,0) node [below right] {$B$} coordinate (B) --++ (45:{2/2}) node [right] {$C$} coordinate (C)\n --++ (0,2) node [above right] {$C_1$} coordinate (C1)\n --++ (-2,0) node [above left] {$D_1$} coordinate (D1) --++ (225:{2/2}) node [left] {$A_1$} coordinate (A1) -- cycle;\n \\draw (A) ++ (2,2) node [right] {$B_1$} coordinate (B1) -- (B) (B1) --++ (45:{2/2}) (B1) --++ (-2,0);\n \\draw [dashed] (A) --++ (45:{2/2}) node [left] {$D$} coordinate (D) --++ (2,0) (D) --++ (0,2);\n \\draw (A1) -- (B);\n \\draw [dashed] (B) -- (D) -- (A1);\n \\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-赋能-赋能07" ], "genre": "填空题", "ans": "$\\frac{\\sqrt 3}3$", @@ -11753,7 +11950,8 @@ ], "tags": [ "第八单元", - "二项式定理" + "二项式定理", + "2023届高三-赋能-赋能07" ], "genre": "填空题", "ans": "$11$", @@ -11793,7 +11991,8 @@ "K0623002B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-赋能-赋能07" ], "genre": "填空题", "ans": "$\\sqrt{17}\\pi$", @@ -11829,7 +12028,8 @@ "content": "设$P(x,y)$是曲线$C:\\sqrt{\\dfrac{x^2}{25}}+\\sqrt{\\dfrac{y^2}9}=1$上的点, $F_1(-4,0)$, $F_2(4,0)$, 则$|PF_1|+|PF_2|$的最大值为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-赋能-赋能07" ], "genre": "填空题", "ans": "$10$", @@ -11868,7 +12068,8 @@ "K0512005B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-赋能-赋能08" ], "genre": "填空题", "ans": "$3-4\\mathrm{i}$", @@ -11914,7 +12115,8 @@ "K0212002B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-赋能-赋能08" ], "genre": "填空题", "ans": "$[-1,3)$", @@ -11953,7 +12155,8 @@ ], "tags": [ "第八单元", - "二项式定理" + "二项式定理", + "2023届高三-赋能-赋能08" ], "genre": "填空题", "ans": "$160$", @@ -12013,7 +12216,8 @@ ], "tags": [ "第七单元", - "双曲线" + "双曲线", + "2023届高三-赋能-赋能08" ], "genre": "填空题", "ans": "$4$", @@ -12078,7 +12282,11 @@ ], "tags": [ "第五单元", - "第八单元" + "第八单元", + "2023届高三-四月错题重做-05_易错题-概率与统计", + "2023届高三-四月错题重做-05_概率与统计", + "2023届高三-第一轮复习讲义-25_概率的概念及性质", + "2023届高三-赋能-赋能08" ], "genre": "填空题", "ans": "$\\frac{13}{28}$", @@ -12135,7 +12343,8 @@ "K0620004B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-赋能-赋能08" ], "genre": "填空题", "ans": "$16\\pi$", @@ -12175,7 +12384,11 @@ "K0406004X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-四月错题重做-03_数列", + "2023届高三-四月错题重做-03_易错题-数列", + "2023届高三-第一轮复习讲义-31_数列的递推与通项及数学归纳法", + "2023届高三-赋能-赋能08" ], "genre": "填空题", "ans": "$b>-3$", @@ -12235,7 +12448,8 @@ "content": "将边长为$10$的正三角形$ABC$, 按``斜二测''画法在水平放置的平面上画出为$\\triangle A'B'C'$, 则$\\triangle A'B'C'$中最短边的边长为\\blank{50}(精确到0.01).", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-赋能-赋能08" ], "genre": "填空题", "ans": "$3.62$", @@ -12271,7 +12485,8 @@ "content": "已知点$A$是圆$O: x^2+y^2=4$上的一个定点, 点$B$是圆$O$上的一个动点, 若满足$|\\overrightarrow{AO}+\\overrightarrow{BO}|=|\\overrightarrow{AO}-\\overrightarrow{BO}|$, 则$\\overrightarrow{AO}\\cdot \\overrightarrow{AB}=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-赋能-赋能08" ], "genre": "填空题", "ans": "$4$", @@ -12309,7 +12524,9 @@ "K0204004B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-容易题", + "2023届高三-赋能-赋能09" ], "genre": "填空题", "ans": "$2$", @@ -12350,7 +12567,9 @@ "K0116002B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-容易题", + "2023届高三-赋能-赋能09" ], "genre": "填空题", "ans": "$5$", @@ -12388,7 +12607,8 @@ "K0402005X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-赋能-赋能09" ], "genre": "填空题", "ans": "$a_n=2^{n-1}$", @@ -12455,7 +12675,8 @@ ], "tags": [ "第八单元", - "二项式定理" + "二项式定理", + "2023届高三-赋能-赋能09" ], "genre": "填空题", "ans": "$160$", @@ -12496,7 +12717,9 @@ "K0619003B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-24_体积及表面积的计算", + "2023届高三-赋能-赋能09" ], "genre": "填空题", "ans": "$\\frac 43$", @@ -12537,7 +12760,10 @@ "objs": [], "tags": [ "第八单元", - "排列" + "排列", + "2023届高三-四月错题重做-05_易错题-概率与统计", + "2023届高三-四月错题重做-05_概率与统计", + "2023届高三-赋能-赋能09" ], "genre": "填空题", "ans": "$240$", @@ -12580,7 +12806,9 @@ "content": "集合$\\{x|\\cos (\\pi \\cos x)=0,x\\in [0,\\pi]\\}=$\\blank{50}(用列举法表示).", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-寒假作业-中档题", + "2023届高三-赋能-赋能09" ], "genre": "填空题", "ans": "$\\{\\frac{\\pi }3,\\frac{2\\pi }3\\}$", @@ -12621,7 +12849,9 @@ "K0504007B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第一轮复习讲义-17_平面向量的投影及数量积", + "2023届高三-赋能-赋能09" ], "genre": "填空题", "ans": "$[-\\frac 12,\\frac 12]$", @@ -12671,7 +12901,8 @@ "K0215005B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-赋能-赋能09" ], "genre": "填空题", "ans": "$[\\frac 12,+\\infty)$", @@ -13038,7 +13269,8 @@ "K0104001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-赋能-赋能10" ], "genre": "填空题", "ans": "$\\{2,4,8\\}$", @@ -13079,7 +13311,8 @@ "K0512005B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-赋能-赋能10" ], "genre": "填空题", "ans": "$1$", @@ -13117,7 +13350,9 @@ "content": "设函数$f(x)=\\sin x-\\cos x$, 且$f(a)=1$, 则$\\sin 2a=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-寒假作业-容易题", + "2023届高三-赋能-赋能10" ], "genre": "填空题", "ans": "$0$", @@ -13222,7 +13457,10 @@ ], "tags": [ "第一单元", - "第三单元" + "第三单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-02_常用逻辑用语", + "2023届高三-赋能-赋能10" ], "genre": "填空题", "ans": "充分非必要", @@ -13267,7 +13505,8 @@ ], "tags": [ "第七单元", - "双曲线" + "双曲线", + "2023届高三-赋能-赋能10" ], "genre": "填空题", "ans": "$6$", @@ -13309,7 +13548,8 @@ ], "tags": [ "第一单元", - "第四单元" + "第四单元", + "2023届高三-赋能-赋能10" ], "genre": "填空题", "ans": "$2$", @@ -13347,7 +13587,8 @@ "K0321003B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-赋能-赋能10" ], "genre": "填空题", "ans": "$00)$的最小正周期是$\\pi$, 则$\\omega=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-赋能-赋能14" ], "genre": "填空题", "ans": "$2$", @@ -14834,7 +15109,8 @@ "K0226004B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-赋能-赋能14" ], "genre": "填空题", "ans": "$\\frac 12$", @@ -14869,7 +15145,8 @@ "content": "将一个正方形绕着它的一边所在的直线旋转一周, 所得几何体的体积为$27\\pi\\text{cm}^3$, 则该几何体的侧面积为\\blank{50}$\\text{cm}^3$.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-赋能-赋能14" ], "genre": "填空题", "ans": "$18\\pi$", @@ -14907,7 +15184,10 @@ "K0215001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-07_函数的概念与奇偶性", + "2023届高三-赋能-赋能14" ], "genre": "填空题", "ans": "$-\\frac 98$", @@ -14945,7 +15225,8 @@ "KNONE" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-赋能-赋能14" ], "genre": "填空题", "ans": "$2$", @@ -14983,7 +15264,8 @@ "K0104006B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-赋能-赋能15" ], "genre": "填空题", "ans": "$\\{5\\}$", @@ -15020,7 +15302,9 @@ "K0512002B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第一轮复习讲义-18_复数的代数运算与性质", + "2023届高三-赋能-赋能15" ], "genre": "填空题", "ans": "$-1$", @@ -15066,7 +15350,8 @@ "objs": [], "tags": [ "第八单元", - "排列" + "排列", + "2023届高三-赋能-赋能15" ], "genre": "填空题", "ans": "$60$", @@ -15099,7 +15384,8 @@ "content": "已知$\\tan \\theta =-2$, 且$\\theta \\in (\\dfrac\\pi 2,\\pi)$, 则$\\cos\\theta=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-赋能-赋能15" ], "genre": "填空题", "ans": "$-\\dfrac{\\sqrt 5}5$", @@ -15132,7 +15418,8 @@ "content": "圆锥的底面半径为$1$, 母线长为$3$, 则圆锥的侧面积等于\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-赋能-赋能15" ], "genre": "填空题", "ans": "$3\\pi$", @@ -15171,7 +15458,10 @@ "K0504007B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-17_平面向量的投影及数量积", + "2023届高三-赋能-赋能15" ], "genre": "填空题", "ans": "$\\sqrt 3$", @@ -15216,7 +15506,8 @@ "content": "已知球的大圆面积等于$9\\pi$, 则该球的体积为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-赋能-赋能15" ], "genre": "填空题", "ans": "$36\\pi$", @@ -15256,7 +15547,8 @@ ], "tags": [ "第八单元", - "二项式定理" + "二项式定理", + "2023届高三-赋能-赋能15" ], "genre": "填空题", "ans": "$84$", @@ -15296,7 +15588,8 @@ "content": "已知$A(2,0)$, $B(4,0)$, 动点$P$满足$|PA|=\\dfrac{\\sqrt2} 2|PB|$, 则$P$到原点的距离为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-赋能-赋能15" ], "genre": "填空题", "ans": "$2\\sqrt 2$", @@ -15333,7 +15626,8 @@ ], "tags": [ "第七单元", - "椭圆" + "椭圆", + "2023届高三-赋能-赋能15" ], "genre": "填空题", "ans": "$\\dfrac 32$", @@ -15368,7 +15662,9 @@ "K0212001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验02", + "2023届高三-赋能-赋能16" ], "genre": "填空题", "ans": "$(-\\infty ,2)$", @@ -15408,7 +15704,8 @@ "K0217003B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-赋能-赋能16" ], "genre": "填空题", "ans": "$0$", @@ -15445,7 +15742,8 @@ "K0405003X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-赋能-赋能16" ], "genre": "填空题", "ans": "$1$", @@ -15479,7 +15777,8 @@ "content": "在$\\triangle ABC$中, $\\angle A,\\angle B,\\angle C$所对的边分别是$a,b,c$, 若$a:b:c=2:3:4 $, 则$\\cos C=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-赋能-赋能16" ], "genre": "填空题", "ans": "$-\\dfrac 14$", @@ -15518,7 +15817,8 @@ ], "tags": [ "第一单元", - "第五单元" + "第五单元", + "2023届高三-赋能-赋能16" ], "genre": "填空题", "ans": "$[ -\\dfrac 12, \\dfrac 12 ]$", @@ -15558,7 +15858,8 @@ "tags": [ "第八单元", "组合", - "加法原理与乘法原理" + "加法原理与乘法原理", + "2023届高三-赋能-赋能16" ], "genre": "填空题", "ans": "$18$", @@ -15592,7 +15893,8 @@ "content": "已知$M$、$N$是三棱锥$P-ABC$的棱$AB$, $PC$的中点, 记三棱锥$P-ABC$的体积为$V_1$, 三棱锥$N-MBC$的体积为$V_2$, 则$\\dfrac{V_2}{V_1}$等于\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-赋能-赋能16" ], "genre": "填空题", "ans": "$\\dfrac 14$", @@ -15629,7 +15931,8 @@ ], "tags": [ "第七单元", - "双曲线" + "双曲线", + "2023届高三-赋能-赋能16" ], "genre": "填空题", "ans": "$y=\\pm \\frac 13x$", @@ -15663,7 +15966,8 @@ "content": "已知$y=\\sin x$和$y=\\cos x$的图像的连续的三个交点$A$、$B$、$C$构成三角形$\\triangle ABC$, 则$\\triangle ABC$的面积等于\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-赋能-赋能16" ], "genre": "填空题", "ans": "$\\sqrt 2\\pi$", @@ -15701,7 +16005,8 @@ "K0319002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-赋能-赋能16" ], "genre": "填空题", "ans": "$\\frac{3025}2$", @@ -15742,7 +16047,8 @@ "K0117002B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-赋能-赋能17" ], "genre": "填空题", "ans": "$[0,2]$", @@ -15781,7 +16087,8 @@ "content": "已知角$\\theta$的顶点在坐标原点, 始边与$x$轴的正半轴重合, 若角$\\theta$的终边落在第三象限内, 且$\\cos(\\dfrac\\pi 2+\\theta)=\\dfrac35$, 则$\\cos 2\\theta=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-赋能-赋能17" ], "genre": "填空题", "ans": "$\\dfrac 7{25}$", @@ -15819,7 +16126,9 @@ "K0207001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验02", + "2023届高三-赋能-赋能17" ], "genre": "填空题", "ans": "$(-\\infty ,0)$", @@ -15885,7 +16194,8 @@ "content": "某圆锥体的底面圆的半径长为$\\sqrt2$, 其侧面展开图是圆心角为$\\dfrac23\\pi$的扇形, 则该圆锥体的体积是\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-赋能-赋能17" ], "genre": "填空题", "ans": "$\\frac 83\\pi$", @@ -15926,7 +16236,9 @@ ], "tags": [ "第七单元", - "圆" + "圆", + "2023届高三-第一轮复习讲义-35_圆及曲线方程", + "2023届高三-赋能-赋能17" ], "genre": "填空题", "ans": "$-2\\cdot (x+2)+1\\cdot (y-1)=0$", @@ -15982,7 +16294,8 @@ "tags": [ "第五单元", "第八单元", - "二项式定理" + "二项式定理", + "2023届高三-赋能-赋能17" ], "genre": "填空题", "ans": "$5\\sqrt 2$", @@ -16019,7 +16332,8 @@ "objs": [], "tags": [ "第八单元", - "组合" + "组合", + "2023届高三-赋能-赋能17" ], "genre": "填空题", "ans": "$25$", @@ -16057,7 +16371,8 @@ "content": "已知$\\triangle ABC$的三个内角$A,B,C$所对边长分别为$a,b,c$, 记$\\triangle ABC$的面积为$S$, 若$S=a^2-(b-c)^2$, 则内角$A=$\\blank{50}(结果用反三角函数值表示).", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-赋能-赋能17" ], "genre": "填空题", "ans": "$\\arccos \\dfrac{15}{17}$", @@ -16096,7 +16411,8 @@ "K0223004B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-赋能-赋能17" ], "genre": "填空题", "ans": "$\\begin{cases} b+c=-1, \\\\ b<-2. \\end{cases}$", @@ -16135,7 +16451,8 @@ "K0104006B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-赋能-赋能18" ], "genre": "填空题", "ans": "$A=\\{x|00$, 若函数$g(x)=f(x+\\alpha)$为 奇函数, 则$\\alpha$的值为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-赋能-赋能25" ], "genre": "填空题", "ans": "$\\alpha =\\dfrac{k\\pi }2-\\frac{\\pi }6 \\ (k\\in \\mathbf{N}^*)$", @@ -19034,7 +19430,8 @@ "K0113001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-赋能-赋能26" ], "genre": "填空题", "ans": "$(-1,0]$", @@ -19073,7 +19470,8 @@ "content": "已知$\\sin\\alpha=\\dfrac45$, 则$\\cos(\\alpha+\\dfrac{\\pi}2)=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-赋能-赋能26" ], "genre": "填空题", "ans": "$-\\dfrac 45$", @@ -19145,7 +19543,8 @@ "content": "已知球的表面积为$16\\pi$, 则该球的体积为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-赋能-赋能26" ], "genre": "填空题", "ans": "$\\dfrac{32\\pi }3$", @@ -19186,7 +19585,8 @@ "K0226004B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-赋能-赋能26" ], "genre": "填空题", "ans": "$4$", @@ -19221,7 +19621,8 @@ "KNONE" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-赋能-赋能26" ], "genre": "填空题", "ans": "$18$", @@ -19254,7 +19655,8 @@ "content": "在$\\triangle ABC$中, 角$A$、$B$、$C$所对的边分别为$a$、$b$、$c$, 若$(a+b+c)(a-b+c)=ac$, 则$B=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-赋能-赋能26" ], "genre": "填空题", "ans": "$\\dfrac{2\\pi }3$", @@ -19290,7 +19692,8 @@ ], "tags": [ "第八单元", - "二项式定理" + "二项式定理", + "2023届高三-赋能-赋能26" ], "genre": "填空题", "ans": "$1120$", @@ -19325,7 +19728,8 @@ "K0319003B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-赋能-赋能26" ], "genre": "填空题", "ans": "$\\dfrac 12$", @@ -19361,7 +19765,8 @@ "K0407002X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-赋能-赋能26" ], "genre": "填空题", "ans": "$-1+\\dfrac{{(-1)}^n}{n+1}$", @@ -19397,7 +19802,8 @@ "K0104006B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-赋能-赋能27" ], "genre": "填空题", "ans": "$\\{1,4\\}$", @@ -19430,7 +19836,8 @@ "objs": [], "tags": [ "第七单元", - "参数方程" + "参数方程", + "2023届高三-赋能-赋能27" ], "genre": "填空题", "ans": "$(1,0)$", @@ -19464,7 +19871,8 @@ "K0514007B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-赋能-赋能27" ], "genre": "填空题", "ans": "$[1,3]$", @@ -19498,7 +19906,8 @@ "KNONE" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-赋能-赋能27" ], "genre": "填空题", "ans": "$1$", @@ -19535,7 +19944,8 @@ "tags": [ "第四单元", "第八单元", - "二项式定理" + "二项式定理", + "2023届高三-赋能-赋能27" ], "genre": "填空题", "ans": "$8$", @@ -19570,7 +19980,8 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-赋能-赋能27" ], "genre": "填空题", "ans": "$\\frac 7{10}$", @@ -19661,7 +20072,8 @@ "K0223004B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-赋能-赋能27" ], "genre": "填空题", "ans": "$(\\frac 59,1)$", @@ -19693,7 +20105,8 @@ "content": "某部门有$8$位员工, 其中$6$位员工的月工资分别为$8200$, $8300$, $8500$, $9100$, $9500$, $9600$(单位: 元), 另两位员工的月工资数据不清楚, 但两人的月工资和为$17000$元, 则这$8$位员工月工资的中位数可能的最大值为\\blank{50}元.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-赋能-赋能27" ], "genre": "填空题", "ans": "$8800$", @@ -19761,7 +20174,8 @@ "K0215003B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-赋能-赋能28" ], "genre": "填空题", "ans": "$(-\\infty ,0)\\cup (1,+\\infty)$", @@ -19792,7 +20206,8 @@ "content": "若$\\dfrac{\\pi}2<\\alpha<\\pi$, $\\sin\\alpha=\\dfrac35$, 则$\\tan\\dfrac{\\alpha}2=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-赋能-赋能28" ], "genre": "填空题", "ans": "$3$", @@ -19824,7 +20239,8 @@ "K0512005B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-赋能-赋能28" ], "genre": "填空题", "ans": "$-1+\\mathrm{i}$", @@ -19852,7 +20268,8 @@ "objs": [], "tags": [ "第七单元", - "参数方程" + "参数方程", + "2023届高三-赋能-赋能28" ], "genre": "填空题", "ans": "$2$", @@ -19882,7 +20299,8 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-赋能-赋能28" ], "genre": "填空题", "ans": "$\\frac 1{169}$", @@ -19909,7 +20327,8 @@ "content": "若关于$x$的方程$\\sin x+\\cos x-m=0$在区间$[0,\\dfrac{\\pi}2]$上有解, 则实数$m$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-赋能-赋能28" ], "genre": "填空题", "ans": "$1\\le m\\le \\sqrt 2$.", @@ -19936,7 +20355,8 @@ "content": "若一个圆锥的母线与底面所成的角为$\\dfrac{\\pi}6$,体积为$125\\pi$,则此圆锥的高为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-赋能-赋能28" ], "genre": "填空题", "ans": "$5$", @@ -19967,7 +20387,8 @@ "K0225004B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-赋能-赋能28" ], "genre": "填空题", "ans": "$4$", @@ -19994,7 +20415,8 @@ "content": "若三棱锥$S-ABC$的所有的顶点都在球$O$的球面上, $SA\\perp$平面$ABC$, $SA=AB=2$, $AC=4$, $\\angle BAC=\\dfrac{\\pi}3$, 则球$O$的表面积为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-赋能-赋能28" ], "genre": "填空题", "ans": "$20\\pi$", @@ -20023,7 +20445,8 @@ "K0204004B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-赋能-赋能29" ], "genre": "填空题", "ans": "$x=4$", @@ -20069,7 +20492,8 @@ "K0104001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-赋能-赋能29" ], "genre": "填空题", "ans": "$\\{-1,0\\}$", @@ -20112,7 +20536,8 @@ "K0512003B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-赋能-赋能29" ], "genre": "填空题", "ans": "$1$", @@ -20158,7 +20583,8 @@ "tags": [ "第七单元", "直线", - "参数方程" + "参数方程", + "2023届高三-赋能-赋能29" ], "genre": "填空题", "ans": "$x+y-1=0$", @@ -20193,7 +20619,8 @@ ], "tags": [ "第八单元", - "二项式定理" + "二项式定理", + "2023届高三-赋能-赋能29" ], "genre": "填空题", "ans": "$16$", @@ -20257,7 +20684,10 @@ "K0223004B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-09_函数的零点与最值", + "2023届高三-赋能-赋能29" ], "genre": "填空题", "ans": "$[-\\frac 12,1]$", @@ -20330,7 +20760,8 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-赋能-赋能29" ], "genre": "填空题", "ans": "$\\frac 29$", @@ -20373,7 +20804,11 @@ ], "tags": [ "第七单元", - "椭圆" + "椭圆", + "2023届高三-四月错题重做-04_易错题-解析几何", + "2023届高三-四月错题重做-04_解析几何", + "2023届高三-第一轮复习讲义-36_椭圆与双曲线的概念及性质", + "2023届高三-赋能-赋能29" ], "genre": "填空题", "ans": "$\\frac{\\sqrt 3}2$", @@ -20430,7 +20865,8 @@ "content": "函数$y=1-2\\sin^2(2x)$的最小正周期是\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-赋能-赋能30" ], "genre": "填空题", "ans": "$\\frac{\\pi}2$", @@ -20467,7 +20903,8 @@ "K0104006B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-赋能-赋能30" ], "genre": "填空题", "ans": "$[0,1)$", @@ -20502,7 +20939,8 @@ "K0514001B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-赋能-赋能30" ], "genre": "填空题", "ans": "$\\sqrt{10}$", @@ -20535,7 +20973,8 @@ ], "tags": [ "第七单元", - "双曲线" + "双曲线", + "2023届高三-赋能-赋能30" ], "genre": "填空题", "ans": "$16$", @@ -20568,7 +21007,8 @@ "content": "已知正四棱锥的底面边长是$2$, 侧棱长是$\\sqrt3$, 则该正四棱锥的体积为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-赋能-赋能30" ], "genre": "填空题", "ans": "$\\frac 43$", @@ -20631,7 +21071,8 @@ ], "tags": [ "第八单元", - "二项式定理" + "二项式定理", + "2023届高三-赋能-赋能30" ], "genre": "填空题", "ans": "$15$", @@ -20663,7 +21104,8 @@ "K0405003X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-赋能-赋能30" ], "genre": "填空题", "ans": "$\\frac 83$", @@ -20697,7 +21139,8 @@ ], "tags": [ "第二单元", - "第七单元" + "第七单元", + "2023届高三-赋能-赋能30" ], "genre": "填空题", "ans": "$0$", @@ -20730,7 +21173,8 @@ "objs": [], "tags": [ "第八单元", - "排列" + "排列", + "2023届高三-赋能-赋能30" ], "genre": "填空题", "ans": "$64$", @@ -20764,7 +21208,8 @@ "K0104001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-赋能-赋能31" ], "genre": "填空题", "ans": "$\\{2,3,4\\}$", @@ -20810,7 +21255,9 @@ "K0513003B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第一轮复习讲义-19_复数的几何意义与实系数二次方程", + "2023届高三-赋能-赋能31" ], "genre": "填空题", "ans": "四", @@ -20900,7 +21347,8 @@ "K0706001X" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-赋能-赋能31" ], "genre": "填空题", "ans": "$\\pm 2$", @@ -20935,7 +21383,8 @@ ], "tags": [ "第八单元", - "二项式定理" + "二项式定理", + "2023届高三-赋能-赋能31" ], "genre": "填空题", "ans": "$1$", @@ -20968,7 +21417,8 @@ ], "tags": [ "第七单元", - "双曲线" + "双曲线", + "2023届高三-赋能-赋能31" ], "genre": "填空题", "ans": "$2$", @@ -20998,7 +21448,8 @@ "content": "在$\\triangle ABC$中, 三边长分别为$a=2$, $b=3$, $c=4$, 则$\\dfrac{\\sin 2A}{\\sin B}=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-赋能-赋能31" ], "genre": "填空题", "ans": "$\\frac 76$", @@ -21032,7 +21483,8 @@ ], "tags": [ "第七单元", - "直线" + "直线", + "2023届高三-赋能-赋能31" ], "genre": "填空题", "ans": "$[0,5]$", @@ -21064,7 +21516,8 @@ "K0223004B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-赋能-赋能31" ], "genre": "填空题", "ans": "$4$", @@ -21120,7 +21573,8 @@ "K0215003B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-赋能-赋能32" ], "genre": "填空题", "ans": "$[0,2]$", @@ -21157,7 +21611,8 @@ "K0706001X" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-赋能-赋能32" ], "genre": "填空题", "ans": "$2$", @@ -21189,7 +21644,8 @@ "K0106001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-赋能-赋能32" ], "genre": "填空题", "ans": "$-1$", @@ -21223,7 +21679,8 @@ "K0512005B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-赋能-赋能32" ], "genre": "填空题", "ans": "$\\frac 34$", @@ -21256,7 +21713,8 @@ "K0219003B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-赋能-赋能32" ], "genre": "填空题", "ans": "$[\\frac 23,1)$", @@ -21317,7 +21775,8 @@ ], "tags": [ "第七单元", - "圆" + "圆", + "2023届高三-赋能-赋能32" ], "genre": "填空题", "ans": "$[3,7]$", @@ -21349,7 +21808,8 @@ "content": "已知向量$\\overrightarrow a=(\\cos(\\dfrac{\\pi}3+\\alpha),1)$, $\\overrightarrow b=(1,4)$, 如果$\\overrightarrow a \\parallel \\overrightarrow b$, 那么$\\cos(\\dfrac{\\pi}3-2\\alpha)$的值为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-赋能-赋能32" ], "genre": "填空题", "ans": "$\\dfrac 78$", @@ -21382,7 +21842,11 @@ ], "tags": [ "第八单元", - "加法原理与乘法原理" + "加法原理与乘法原理", + "2023届高三-四月错题重做-05_易错题-概率与统计", + "2023届高三-四月错题重做-05_概率与统计", + "2023届高三-第一轮复习讲义-25_概率的概念及性质", + "2023届高三-赋能-赋能32" ], "genre": "填空题", "ans": "$\\dfrac 37$", @@ -21434,7 +21898,10 @@ "K0321001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷04", + "2023届高三-寒假作业-中档题", + "2023届高三-赋能-赋能32" ], "genre": "填空题", "ans": "$\\dfrac 32$", @@ -21475,7 +21942,9 @@ "K0104001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验02", + "2023届高三-赋能-赋能33" ], "genre": "填空题", "ans": "$(1,\\log_2 3)$", @@ -21534,7 +22003,8 @@ ], "tags": [ "第八单元", - "二项式定理" + "二项式定理", + "2023届高三-赋能-赋能33" ], "genre": "填空题", "ans": "$2$", @@ -21566,7 +22036,8 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-赋能-赋能33" ], "genre": "填空题", "ans": "$\\dfrac 35$", @@ -21597,7 +22068,8 @@ "K0218001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-赋能-赋能33" ], "genre": "填空题", "ans": "$-3$", @@ -21628,7 +22100,8 @@ "tags": [ "第七单元", "直线", - "参数方程" + "参数方程", + "2023届高三-赋能-赋能33" ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt 5}5$", @@ -21660,7 +22133,8 @@ ], "tags": [ "第四单元", - "第五单元" + "第五单元", + "2023届高三-赋能-赋能33" ], "genre": "填空题", "ans": "$(-\\infty ,-1)\\cup (0,+\\infty)$", @@ -21691,7 +22165,8 @@ "content": "已知正四棱锥$P-ABCD$的棱长都相等, 侧棱$PB$、$PD$的中点分别为$M$、$N$, 则截面$AMN$与底面$ABCD$所成的锐二面角的余弦值是\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-赋能-赋能33" ], "genre": "填空题", "ans": "$\\dfrac{2\\sqrt 5}5$", @@ -21724,7 +22199,8 @@ "K0221002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-赋能-赋能33" ], "genre": "填空题", "ans": "$(\\frac{\\sqrt 2}4,+\\infty)$", @@ -21756,7 +22232,8 @@ "K0223005B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-赋能-赋能33" ], "genre": "填空题", "ans": "$8$", @@ -21788,7 +22265,8 @@ "K0113001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-赋能-赋能34" ], "genre": "填空题", "ans": "$[2,4)$", @@ -21818,7 +22296,8 @@ "tags": [ "第七单元", "直线", - "参数方程" + "参数方程", + "2023届高三-赋能-赋能34" ], "genre": "填空题", "ans": "$1$", @@ -21851,7 +22330,8 @@ "content": "已知圆锥的母线长为$4$, 母线与旋转轴的夹角为$30^\\circ$, 则该圆锥的侧面积为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-赋能-赋能34" ], "genre": "填空题", "ans": "$8\\pi$", @@ -21882,7 +22362,9 @@ ], "tags": [ "第七单元", - "抛物线" + "抛物线", + "2023届高三-第一轮复习讲义-37_抛物线的概念及性质", + "2023届高三-赋能-赋能34" ], "genre": "填空题", "ans": "$2$", @@ -21953,7 +22435,8 @@ "content": "若三个数$a_1,a_2,a_3$的方差为$1$, 则$3a_1+2,3a_2+2,3a_3+2$的方差为\\blank{50}.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-赋能-赋能34" ], "genre": "填空题", "ans": "$9$", @@ -21984,7 +22467,10 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-25_概率的概念及性质", + "2023届高三-赋能-赋能34" ], "genre": "填空题", "ans": "$0.98$", @@ -22018,7 +22504,8 @@ "content": "函数$y=\\sin (\\dfrac{\\pi}6-x), \\ x\\in [0,\\dfrac32\\pi]$的单调递减区间是\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-赋能-赋能34" ], "genre": "填空题", "ans": "$[ 0,\\dfrac 23\\pi]$", @@ -22097,7 +22584,8 @@ "K0216003B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-赋能-赋能34" ], "genre": "填空题", "ans": "$6$", @@ -22140,7 +22628,8 @@ "content": "函数$y=2\\sin^2(2x)-1$的最小正周期是\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-赋能-赋能35" ], "genre": "填空题", "ans": "$\\frac{\\pi}2$", @@ -22176,7 +22665,8 @@ "K0514004B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-赋能-赋能35" ], "genre": "填空题", "ans": "$1$", @@ -22207,7 +22697,8 @@ "K0225004B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-赋能-赋能35" ], "genre": "填空题", "ans": "$1$", @@ -22272,7 +22763,8 @@ "content": "若圆锥的侧面积是底面积的$2$倍, 则其母线与轴所成角的大小是\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-赋能-赋能35" ], "genre": "填空题", "ans": "$\\frac{\\pi}6$", @@ -22305,7 +22797,8 @@ "K0401004X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-赋能-赋能35" ], "genre": "填空题", "ans": "$\\frac 52$", @@ -22336,7 +22829,8 @@ "tags": [ "第七单元", "直线", - "参数方程" + "参数方程", + "2023届高三-赋能-赋能35" ], "genre": "填空题", "ans": "$1$", @@ -22373,7 +22867,8 @@ ], "tags": [ "第七单元", - "双曲线" + "双曲线", + "2023届高三-赋能-赋能35" ], "genre": "填空题", "ans": "$x^2-\\dfrac{y^2}3=1$", @@ -22405,7 +22900,8 @@ "K0223005B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-赋能-赋能35" ], "genre": "填空题", "ans": "$(1,+\\infty)$", @@ -22442,7 +22938,9 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-第一轮复习讲义-26_大数定律及独立性", + "2023届高三-赋能-赋能35" ], "genre": "填空题", "ans": "$\\frac{13}{15}$", @@ -22483,7 +22981,8 @@ "K0104001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-赋能-赋能36" ], "genre": "填空题", "ans": "$(-1,2)$", @@ -22513,7 +23012,8 @@ "K0514004B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-赋能-赋能36" ], "genre": "填空题", "ans": "$1$", @@ -22542,7 +23042,8 @@ "content": "函数$f(x)=\\sin^2 x- 4\\cos^2 x$的最小正周期是\\blank{50}.", "objs": [], "tags": [ - "暂无对应" + "暂无对应", + "2023届高三-赋能-赋能36" ], "genre": "填空题", "ans": "$\\pi$", @@ -22575,7 +23076,8 @@ ], "tags": [ "第七单元", - "双曲线" + "双曲线", + "2023届高三-赋能-赋能36" ], "genre": "填空题", "ans": "$3$", @@ -22607,7 +23109,8 @@ "K0617002B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-赋能-赋能36" ], "genre": "填空题", "ans": "$5.1$", @@ -22667,7 +23170,8 @@ "tags": [ "第七单元", "直线", - "参数方程" + "参数方程", + "2023届高三-赋能-赋能36" ], "genre": "填空题", "ans": "$2$", @@ -22698,7 +23202,8 @@ "K0225004B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-赋能-赋能36" ], "genre": "填空题", "ans": "$-1$", @@ -22760,7 +23265,9 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-第一轮复习讲义-26_大数定律及独立性", + "2023届高三-赋能-赋能36" ], "genre": "填空题", "ans": "$0.03$", @@ -22824,7 +23331,8 @@ "content": "设实数$\\omega>0$, 若函数$f(x)=\\cos(\\omega x)+\\sin(\\omega x)$的最小正周期为$\\pi$, 则$\\omega=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-赋能-赋能37" ], "genre": "填空题", "ans": "$2$", @@ -22848,7 +23356,8 @@ "content": "已知圆锥的底面半径和高均为$1$, 则该圆锥的侧面积为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-赋能-赋能37" ], "genre": "填空题", "ans": "$\\sqrt 2\\pi$", @@ -22875,7 +23384,8 @@ "K0508003B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-赋能-赋能37" ], "genre": "填空题", "ans": "$(-\\infty,-4)$", @@ -22901,7 +23411,8 @@ "K0104003B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-赋能-赋能37" ], "genre": "填空题", "ans": "$2$", @@ -22925,7 +23436,8 @@ "content": "设$z_1,z_2$是方程$z^2+2z+3=0$的两根, 则$|z_1-z_2|=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-赋能-赋能37" ], "genre": "填空题", "ans": "$2\\sqrt 2$", @@ -22952,7 +23464,8 @@ "K0210006B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-赋能-赋能37" ], "genre": "填空题", "ans": "$(-\\infty,-3)$", @@ -23004,7 +23517,10 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-26_大数定律及独立性", + "2023届高三-赋能-赋能37" ], "genre": "填空题", "ans": "$\\dfrac 79$", @@ -23042,7 +23558,8 @@ ], "tags": [ "第七单元", - "椭圆" + "椭圆", + "2023届高三-赋能-赋能37" ], "genre": "填空题", "ans": "$(8,12)$", @@ -23069,7 +23586,8 @@ "K0104006B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-赋能-赋能38" ], "genre": "填空题", "ans": "$\\{2\\}$", @@ -23101,7 +23619,8 @@ ], "tags": [ "第七单元", - "抛物线" + "抛物线", + "2023届高三-赋能-赋能38" ], "genre": "填空题", "ans": "$y^2=4x$", @@ -23128,7 +23647,8 @@ "content": "某次体检, $8$位同学的身高(单位: 米)分别为. $1.68$, $1.71$, $1.73$, $1.63$, $1.81$, $1.74$, $1.66$, $1.78$, 则这组数据的中位数是\\blank{50}(米).", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-赋能-赋能38" ], "genre": "填空题", "ans": "$1.72$", @@ -23155,7 +23675,8 @@ "content": "函数$f(x)=2\\sin 4x \\cos 4x$的最小正周期为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-赋能-赋能38" ], "genre": "填空题", "ans": "$\\frac{\\pi }4$", @@ -23185,7 +23706,8 @@ "content": "已知球的俯视图面积为$\\pi$, 则该球的表面积为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-赋能-赋能38" ], "genre": "填空题", "ans": "$4\\pi$", @@ -23243,7 +23765,8 @@ "objs": [], "tags": [ "第八单元", - "组合" + "组合", + "2023届高三-赋能-赋能38" ], "genre": "填空题", "ans": "$1688$", @@ -23275,7 +23798,8 @@ "K0405003X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-赋能-赋能38" ], "genre": "填空题", "ans": "$\\frac{\\sqrt 5-1}2$", @@ -23307,7 +23831,8 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-赋能-赋能38" ], "genre": "填空题", "ans": "$\\frac 3{10}$", @@ -23338,7 +23863,8 @@ "K0221002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-赋能-赋能38" ], "genre": "填空题", "ans": "$[2,+\\infty)$", @@ -23368,7 +23894,8 @@ "K0104006B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-赋能-赋能39" ], "genre": "填空题", "ans": "$\\{1,3\\}$", @@ -23432,7 +23959,8 @@ "K0512003B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-赋能-赋能39" ], "genre": "填空题", "ans": "$-2$", @@ -23458,7 +23986,8 @@ "content": "若$2\\log_2 x-4=0$, 则$x=$\\blank{50}.", "objs": [], "tags": [ - "暂无对应" + "暂无对应", + "2023届高三-赋能-赋能39" ], "genre": "填空题", "ans": "$4$", @@ -23486,7 +24015,8 @@ "content": "我国古代数学名著《九章算术》有``米谷粒分''题: 粮仓开仓收粮, 有人送来米$1534$石, 验得米内夹谷, 抽样取米一把, 数得$254$粒内夹谷$28$粒, 则这批米内夹谷约为\\blank{50}石(精确到小数点后一位数字).", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-赋能-赋能39" ], "genre": "填空题", "ans": "$169.1$", @@ -23510,7 +24040,8 @@ "content": "已知圆锥的母线长为$5$, 侧面积为$15\\pi$, 则此圆锥的体积为\\blank{50}(结果保留$\\pi$).", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-赋能-赋能39" ], "genre": "填空题", "ans": "$12\\pi$", @@ -23545,7 +24076,8 @@ ], "tags": [ "第八单元", - "二项式定理" + "二项式定理", + "2023届高三-赋能-赋能39" ], "genre": "填空题", "ans": "$-\\frac 13$", @@ -23572,7 +24104,8 @@ ], "tags": [ "第七单元", - "椭圆" + "椭圆", + "2023届高三-赋能-赋能39" ], "genre": "填空题", "ans": "$\\sqrt 2$", @@ -23601,7 +24134,9 @@ "K0319003B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷04", + "2023届高三-赋能-赋能39" ], "genre": "填空题", "ans": "$f(x)=\\log_2(3-x)$", @@ -23661,7 +24196,8 @@ "K0116002B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-赋能-赋能40" ], "genre": "填空题", "ans": "$\\{1\\}$", @@ -23685,7 +24221,8 @@ "content": "已知半径为$2R$和$R$的两个球, 则大球和小球的体积比为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-赋能-赋能40" ], "genre": "填空题", "ans": "$8:1$", @@ -23712,7 +24249,8 @@ ], "tags": [ "第七单元", - "抛物线" + "抛物线", + "2023届高三-赋能-赋能40" ], "genre": "填空题", "ans": "$(0,\\frac 14)$", @@ -23773,7 +24311,8 @@ "content": "已知在$\\triangle ABC$中, $a$, $b$, $c$分别为$\\angle A$, $\\angle B$, $\\angle C$所对的边. 若$b^2+c^2-a^2=\\sqrt{2}bc$, 则$\\angle A=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-赋能-赋能40" ], "genre": "填空题", "ans": "$\\frac{\\pi}4$", @@ -23826,7 +24365,8 @@ "K0511009B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-赋能-赋能40" ], "genre": "填空题", "ans": "$4$", @@ -23852,7 +24392,8 @@ "K0405003X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-赋能-赋能40" ], "genre": "填空题", "ans": "$\\frac{\\pi}6$或$\\frac{5\\pi}6$", @@ -23880,7 +24421,8 @@ "K0803002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第一轮复习讲义-07_函数的概念与奇偶性" ], "genre": "填空题", "ans": "$\\frac 37$", @@ -23911,7 +24453,8 @@ ], "tags": [ "第八单元", - "二项式定理" + "二项式定理", + "2023届高三-赋能-赋能40" ], "genre": "填空题", "ans": "$3$", @@ -23939,7 +24482,8 @@ "K0114001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-赋能-赋能41" ], "genre": "填空题", "ans": "$[-1,3]$", @@ -23967,7 +24511,8 @@ ], "tags": [ "第八单元", - "二项式定理" + "二项式定理", + "2023届高三-赋能-赋能41" ], "genre": "填空题", "ans": "$20$", @@ -24003,7 +24548,10 @@ "K0212002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-06_幂指对函数", + "2023届高三-赋能-赋能41" ], "genre": "填空题", "ans": "$(0,+\\infty)$", @@ -24031,7 +24579,8 @@ ], "tags": [ "第七单元", - "抛物线" + "抛物线", + "2023届高三-赋能-赋能41" ], "genre": "填空题", "ans": "$1$", @@ -24055,7 +24604,8 @@ "content": "若一个球的体积为$\\dfrac{32\\pi}3$, 则该球的表面积为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-赋能-赋能41" ], "genre": "填空题", "ans": "$16\\pi$", @@ -24109,7 +24659,8 @@ "content": "函数$f(x)=(\\sin x+\\cos x)^2+1$的最小正周期是\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-赋能-赋能41" ], "genre": "填空题", "ans": "$\\pi$", @@ -24134,7 +24685,8 @@ "content": "若一圆锥的底面半径为$3$, 体积是$12\\pi$, 则该圆锥的侧面积等于\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-赋能-赋能41" ], "genre": "填空题", "ans": "$15\\pi$", @@ -24161,7 +24713,8 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-赋能-赋能41" ], "genre": "填空题", "ans": "$\\frac 16$", @@ -24189,7 +24742,8 @@ "tags": [ "第七单元", "直线", - "圆" + "圆", + "2023届高三-赋能-赋能41" ], "genre": "填空题", "ans": "$x^2+y^2-2x-y=0$", @@ -24215,7 +24769,8 @@ "K0104001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-赋能-赋能42" ], "genre": "填空题", "ans": "$a\\ge 1$", @@ -24243,7 +24798,8 @@ ], "tags": [ "第七单元", - "直线" + "直线", + "2023届高三-赋能-赋能42" ], "genre": "填空题", "ans": "$2$", @@ -24267,7 +24823,8 @@ "content": "已知$\\alpha \\in (0,\\pi)$, $\\cos\\alpha =-\\dfrac35$, 则$\\tan(\\alpha+\\dfrac{\\pi}4)=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-赋能-赋能42" ], "genre": "填空题", "ans": "$-\\frac 17$", @@ -24294,7 +24851,10 @@ "K0610004B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-21_空间直线与平面的位置关系", + "2023届高三-赋能-赋能42" ], "genre": "填空题", "ans": "$2$", @@ -24326,7 +24886,8 @@ "K0225004B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-赋能-赋能42" ], "genre": "填空题", "ans": "$-2$", @@ -24353,7 +24914,8 @@ ], "tags": [ "第七单元", - "双曲线" + "双曲线", + "2023届高三-赋能-赋能42" ], "genre": "填空题", "ans": "$\\frac 12$", @@ -24380,7 +24942,8 @@ "K0403002X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-赋能-赋能42" ], "genre": "填空题", "ans": "$1$或$-\\frac 12$", @@ -24407,7 +24970,8 @@ ], "tags": [ "第八单元", - "二项式定理" + "二项式定理", + "2023届高三-赋能-赋能42" ], "genre": "填空题", "ans": "$20$", @@ -24465,7 +25029,8 @@ ], "tags": [ "第七单元", - "椭圆" + "椭圆", + "2023届高三-赋能-赋能42" ], "genre": "填空题", "ans": "$\\frac 12mn$", @@ -24517,7 +25082,9 @@ "K0117001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-04_方程与不等式的求解" ], "genre": "填空题", "ans": "$(-\\infty,0)\\cup (2,+\\infty)$", @@ -24620,7 +25187,8 @@ "K0504007B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第一轮复习讲义-17_平面向量的投影及数量积" ], "genre": "填空题", "ans": "$-6$", @@ -25163,7 +25731,11 @@ "K0215003B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-四月错题重做-01_函数一", + "2023届高三-四月错题重做-01_易错题-函数1", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-07_函数的概念与奇偶性" ], "genre": "填空题", "ans": "$\\{x|x\\ge -1\\}$", @@ -25464,7 +26036,8 @@ "K0513003B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第一轮复习讲义-19_复数的几何意义与实系数二次方程" ], "genre": "填空题", "ans": "$-1$", @@ -25584,7 +26157,8 @@ "K0505005B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第一轮复习讲义-17_平面向量的投影及数量积" ], "genre": "填空题", "ans": "$3$", @@ -25669,7 +26243,9 @@ "K0214002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-06_幂指对函数" ], "genre": "填空题", "ans": "$(0,1)\\cup [2,+\\infty)$", @@ -25940,7 +26516,8 @@ ], "tags": [ "第七单元", - "抛物线" + "抛物线", + "2023届高三-第一轮复习讲义-37_抛物线的概念及性质" ], "genre": "填空题", "ans": "$4\\sqrt 6$", @@ -26512,7 +27089,8 @@ "K0221002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验02" ], "genre": "填空题", "ans": "$m\\ge -5$", @@ -27472,7 +28050,8 @@ "K0514004B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第一轮复习讲义-18_复数的代数运算与性质" ], "genre": "填空题", "ans": "$1$", @@ -27638,7 +28217,9 @@ "K0218001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-07_函数的概念与奇偶性" ], "genre": "填空题", "ans": "$(-\\infty, -2]\\cup [0,2]$", @@ -27887,7 +28468,8 @@ "K0504006B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第一轮复习讲义-17_平面向量的投影及数量积" ], "genre": "填空题", "ans": "$\\sqrt 7$", @@ -28247,7 +28829,8 @@ "K0503001B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第一轮复习讲义-16_平面向量的概念及线性运算" ], "genre": "填空题", "ans": "$2$", @@ -28285,7 +28868,9 @@ "K0221002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-09_函数的零点与最值" ], "genre": "填空题", "ans": "$\\frac{3\\sqrt 2}2$", @@ -28491,7 +29076,8 @@ "K0514004B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第一轮复习讲义-18_复数的代数运算与性质" ], "genre": "填空题", "ans": "$\\sqrt 2$", @@ -28554,7 +29140,9 @@ "K0504007B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-17_平面向量的投影及数量积" ], "genre": "填空题", "ans": "$2$", @@ -28592,7 +29180,8 @@ ], "tags": [ "第七单元", - "椭圆" + "椭圆", + "2023届高三-第一轮复习讲义-36_椭圆与双曲线的概念及性质" ], "genre": "填空题", "ans": "$3$", @@ -28793,7 +29382,8 @@ "K0514002B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第一轮复习讲义-19_复数的几何意义与实系数二次方程" ], "genre": "填空题", "ans": "$2$", @@ -28957,7 +29547,9 @@ ], "tags": [ "第七单元", - "椭圆" + "椭圆", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-36_椭圆与双曲线的概念及性质" ], "genre": "填空题", "ans": "$\\frac{\\sqrt 2}2$", @@ -29417,7 +30009,9 @@ "K0107003B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-03_等式与不等式的性质及证明" ], "genre": "填空题", "ans": "$3$", @@ -29827,7 +30421,9 @@ "K0105001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-03_等式与不等式的性质及证明" ], "genre": "填空题", "ans": "$4$", @@ -30252,7 +30848,8 @@ "K0211001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第一轮复习讲义-06_幂指对函数" ], "genre": "填空题", "ans": "$[0,+\\infty)$", @@ -30420,7 +31017,10 @@ ], "tags": [ "第七单元", - "直线" + "直线", + "2023届高三-四月错题重做-04_易错题-解析几何", + "2023届高三-四月错题重做-04_解析几何", + "2023届高三-第一轮复习讲义-34_直线及其方程" ], "genre": "填空题", "ans": "$\\frac{\\sqrt{10}}{10}$", @@ -30883,7 +31483,8 @@ "K0107001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-02_常用逻辑用语" ], "genre": "填空题", "ans": "$\\times$,\\checkmark,$\\times$,$\\times$,\\checkmark,$\\times$,\\checkmark,\\checkmark,$\\times$", @@ -30913,7 +31514,8 @@ "K0107002B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-02_常用逻辑用语" ], "genre": "填空题", "ans": "(1) $a\\ne 0$或$b\\ne 0$; (2) $-30$, 使得$|a|>a$; (3) 存在实数$x$满足$x^2-x=0$, 使得$x\\ne 1$且$x\\ne 0$; (4) 存在实数$x$满足$x^2-x<0$, 使得$x\\le 0$或$x\\ge 1$.", @@ -31135,7 +31738,8 @@ "K0106003B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-02_常用逻辑用语" ], "genre": "填空题", "ans": "\\textcircled{3},\\textcircled{2},\\textcircled{1},\\textcircled{2},\\textcircled{1},\\textcircled{2},\\textcircled{4},\\textcircled{3},\\textcircled{1},\\textcircled{3},\\textcircled{4},\\textcircled{1},\\textcircled{1}", @@ -31603,7 +32207,8 @@ "K0103005B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-01_集合" ], "genre": "解答题", "ans": "$\\{x|x\\ne 2, \\ x\\in \\mathbf{R}\\}$", @@ -31886,7 +32491,8 @@ "K0104002B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期测验卷-测验04" ], "genre": "填空题", "ans": "$\\{(\\dfrac{3+\\sqrt{5}}2,\\dfrac{5+\\sqrt{5}}2),(\\dfrac{3-\\sqrt{5}}2,\\dfrac{5-\\sqrt{5}}2)\\}$", @@ -31927,7 +32533,8 @@ "K0104001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-01_集合" ], "genre": "填空题", "ans": "$(-\\infty,4]$", @@ -31961,7 +32568,8 @@ "K0104003B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-01_集合" ], "genre": "解答题", "ans": "$p=2$, $q=1$, $r=-2$", @@ -31998,7 +32606,8 @@ "K0104001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-01_集合" ], "genre": "解答题", "ans": "$[\\dfrac 18,\\dfrac 13)$", @@ -32890,7 +33499,9 @@ "K0108003B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-04_方程与不等式的求解" ], "genre": "解答题", "ans": "当$a\\ne 1$时, 解为$\\dfrac{ab+1}{a-1}$; 当$a=1$且$b\\ne -1$时, 解集为$\\varnothing$; 当$a=1$且$b=-1$时, 解集为$\\mathbf{R}$.", @@ -32921,7 +33532,9 @@ "K0108003B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-04_方程与不等式的求解" ], "genre": "解答题", "ans": "当$m=2$时, 解集为$\\mathbf{R}$; 当$m=-3$时, 解集为$\\varnothing$; 当$m\\ne -3$且$m\\ne 2$时, 解为$x=\\dfrac{m-1}{m+3}$", @@ -33445,7 +34058,9 @@ "K0109004B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-04_方程与不等式的求解" ], "genre": "填空题", "ans": "$1$", @@ -33531,7 +34146,9 @@ "K0109001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-04_方程与不等式的求解" ], "genre": "解答题", "ans": "$\\begin{cases}p=3, \\\\ q=2\\end{cases}$或$\\begin{cases}p=-3, \\\\ q=2\\end{cases}$", @@ -33559,7 +34176,9 @@ "K0109004B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-04_方程与不等式的求解" ], "genre": "解答题", "ans": "$4$", @@ -33587,7 +34206,9 @@ "K0109004B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-04_方程与不等式的求解" ], "genre": "解答题", "ans": "$24x^2-2ax+4=0$", @@ -33887,7 +34508,8 @@ "K0111003B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-03_等式与不等式的性质及证明" ], "genre": "填空题", "ans": "(1) $\\times$ (2) $\\checkmark$ (3) $\\times$ (4) $\\times$ (5) $\\times$ (6) $\\times$ (7) $\\checkmark$ (8) $\\checkmark$ (9) $\\checkmark$ (10) $\\times$", @@ -33918,7 +34540,8 @@ "K0111003B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-03_等式与不等式的性质及证明" ], "genre": "填空题", "ans": "$\\dfrac ab$, $\\dfrac{a+m}{b+m}$, $\\dfrac{b+n}{a+n}$, $\\dfrac ba$", @@ -34192,7 +34815,8 @@ "K0120002B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-03_等式与不等式的性质及证明" ], "genre": "解答题", "ans": "证明略", @@ -34771,7 +35395,9 @@ "K0117002B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-较难题", + "2023届高三-第一轮复习讲义-04_方程与不等式的求解" ], "genre": "解答题", "ans": "$a=-\\dfrac 25$, $b=\\dfrac = 15$", @@ -34855,7 +35481,8 @@ "K0111003B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-03_等式与不等式的性质及证明" ], "genre": "填空题", "ans": "(1) $\\checkmark$ (2) $\\checkmark$ (3) $\\checkmark$ (4) $\\times$ (5) $\\times$ (6) $\\times$ (7) $\\checkmark$ (8) $\\times$ (9) $\\times$ (10) $\\checkmark$ (11) $\\times$", @@ -34940,7 +35567,8 @@ "K0118003B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-03_等式与不等式的性质及证明" ], "genre": "解答题", "ans": "$k=\\dfrac 13$", @@ -35050,7 +35678,8 @@ "K0119001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-03_等式与不等式的性质及证明" ], "genre": "解答题", "ans": "$9$", @@ -35079,7 +35708,8 @@ "K0119001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-03_等式与不等式的性质及证明" ], "genre": "解答题", "ans": "$3$", @@ -35136,7 +35766,8 @@ "K0119001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-03_等式与不等式的性质及证明" ], "genre": "解答题", "ans": "$4\\sqrt{2}$", @@ -35194,7 +35825,8 @@ "K0119001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-03_等式与不等式的性质及证明" ], "genre": "解答题", "ans": "最大面积为$2L^2$, 长与宽分别为$2L$与$L$时面积最大.", @@ -35250,7 +35882,8 @@ "K0111003B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-03_等式与不等式的性质及证明" ], "genre": "解答题", "ans": "证明略", @@ -35363,7 +35996,8 @@ "K0118002B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-03_等式与不等式的性质及证明" ], "genre": "解答题", "ans": "证明略", @@ -35394,7 +36028,8 @@ "K0110002B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-03_等式与不等式的性质及证明" ], "genre": "解答题", "ans": "证明略", @@ -36075,7 +36710,8 @@ "K0215003B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第一轮复习讲义-07_函数的概念与奇偶性" ], "genre": "解答题", "ans": "(1) $(-\\infty,1)\\cup (1,2)\\cup (2,+\\infty)$; (2) $(-\\infty,1)\\cup (1,+\\infty)$; (3) $(\\dfrac 23,+\\infty)$; (4) $\\{\\dfrac 12\\}$; (5) $(-\\infty,-2]\\cup [2,+\\infty)$; (6) $[-\\dfrac 12,3)\\cup (3,+\\infty)$", @@ -36187,7 +36823,8 @@ "K0222002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第一轮复习讲义-10_有关函数的应用问题" ], "genre": "解答题", "ans": "$y=12ax+\\dfrac{16000a}x+\\dfrac{8000a}3, \\ x\\in (0,+\\infty)$", @@ -36271,7 +36908,8 @@ "K0222002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第一轮复习讲义-10_有关函数的应用问题" ], "genre": "解答题", "ans": "$y=\\begin{cases} \\dfrac{24}5 x, & 0\\sqrt{2}$或$y<-\\sqrt{2}$); (2) $2x^2-y^2-4x+y=0$, 能取到每一个点", @@ -71483,7 +72425,8 @@ ], "tags": [ "第七单元", - "抛物线" + "抛物线", + "2023届高三-第一轮复习讲义-37_抛物线的概念及性质" ], "genre": "填空题", "ans": "$(1,2)$", @@ -71521,7 +72464,8 @@ ], "tags": [ "第七单元", - "抛物线" + "抛物线", + "2023届高三-第一轮复习讲义-37_抛物线的概念及性质" ], "genre": "填空题", "ans": "$-p^2$", @@ -71886,7 +72830,9 @@ ], "tags": [ "第七单元", - "抛物线" + "抛物线", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-37_抛物线的概念及性质" ], "genre": "填空题", "ans": "$4x+y+3=0$", @@ -71924,7 +72870,10 @@ ], "tags": [ "第七单元", - "抛物线" + "抛物线", + "2023届高三-四月错题重做-04_易错题-解析几何", + "2023届高三-四月错题重做-04_解析几何", + "2023届高三-第一轮复习讲义-37_抛物线的概念及性质" ], "genre": "填空题", "ans": "$x=\\dfrac 12$且$y>\\dfrac 12$", @@ -72096,7 +73045,9 @@ ], "tags": [ "第七单元", - "抛物线" + "抛物线", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-37_抛物线的概念及性质" ], "genre": "选择题", "ans": "C", @@ -72159,7 +73110,8 @@ ], "tags": [ "第七单元", - "抛物线" + "抛物线", + "2023届高三-第一轮复习讲义-37_抛物线的概念及性质" ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{11}}2-1$", @@ -72248,7 +73200,9 @@ ], "tags": [ "第七单元", - "抛物线" + "抛物线", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-37_抛物线的概念及性质" ], "genre": "解答题", "ans": "$3\\sqrt{2}$或$5\\sqrt{2}$", @@ -72395,7 +73349,9 @@ ], "tags": [ "第七单元", - "抛物线" + "抛物线", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-37_抛物线的概念及性质" ], "genre": "解答题", "ans": "(1) $y=2x-3$; (2) $y=2x-3$或$y=-\\dfrac 23 x-\\dfrac 13$", @@ -72436,7 +73392,8 @@ ], "tags": [ "第七单元", - "双曲线" + "双曲线", + "2023届高三-第一轮复习讲义-36_椭圆与双曲线的概念及性质" ], "genre": "解答题", "ans": "(1) $y=\\sqrt{3}x+1$; (2) $y=\\dfrac{1+\\sqrt{7}}2x+\\dfrac{1-\\sqrt{7}}2$或$y=\\dfrac{1-\\sqrt{7}}2x+\\dfrac{1+\\sqrt{7}}2$", @@ -75184,7 +76141,8 @@ "objs": [], "tags": [ "第八单元", - "排列" + "排列", + "2023届高三-第一轮复习讲义-38_计数原理与排列组合" ], "genre": "填空题", "ans": "$3720$", @@ -75540,7 +76498,8 @@ "objs": [], "tags": [ "第八单元", - "排列" + "排列", + "2023届高三-第一轮复习讲义-38_计数原理与排列组合" ], "genre": "填空题", "ans": "$5$", @@ -75918,7 +76877,8 @@ "objs": [], "tags": [ "第八单元", - "排列" + "排列", + "2023届高三-第一轮复习讲义-38_计数原理与排列组合" ], "genre": "填空题", "ans": "$1-\\dfrac{1}{(n+1)!}$", @@ -76595,7 +77555,10 @@ "tags": [ "第八单元", "组合", - "加法原理与乘法原理" + "加法原理与乘法原理", + "2023届高三-四月错题重做-05_易错题-概率与统计", + "2023届高三-四月错题重做-05_概率与统计", + "2023届高三-第一轮复习讲义-38_计数原理与排列组合" ], "genre": "填空题", "ans": "$45213$, $88$", @@ -77042,7 +78005,8 @@ "objs": [], "tags": [ "第八单元", - "排列" + "排列", + "2023届高三-第一轮复习讲义-38_计数原理与排列组合" ], "genre": "填空题", "ans": "$60$", @@ -77135,7 +78099,10 @@ "tags": [ "第八单元", "排列", - "加法原理与乘法原理" + "加法原理与乘法原理", + "2023届高三-四月错题重做-05_易错题-概率与统计", + "2023届高三-四月错题重做-05_概率与统计", + "2023届高三-第一轮复习讲义-38_计数原理与排列组合" ], "genre": "填空题", "ans": "$21$", @@ -77675,7 +78642,8 @@ ], "tags": [ "第八单元", - "二项式定理" + "二项式定理", + "2023届高三-第一轮复习讲义-39_二项式定理" ], "genre": "填空题", "ans": "$7$", @@ -78797,7 +79765,9 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-25_概率的概念及性质" ], "genre": "填空题", "ans": "$\\dfrac 38$", @@ -78859,7 +79829,10 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-四月错题重做-05_易错题-概率与统计", + "2023届高三-四月错题重做-05_概率与统计", + "2023届高三-第一轮复习讲义-25_概率的概念及性质" ], "genre": "填空题", "ans": "$\\dfrac{16}{33}$", @@ -79153,7 +80126,8 @@ "content": "一组数据两两不同, 由小到大排列, 如果第$99$个数和第$100$个数的平均数是这组数据的中位数, 那么该组数据有\\blank{80}个.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-赋能-赋能02" ], "genre": "填空题", "ans": "$198$", @@ -79524,7 +80498,8 @@ ], "tags": [ "第七单元", - "抛物线" + "抛物线", + "2023届高三-第一轮复习讲义-37_抛物线的概念及性质" ], "genre": "解答题", "ans": "证明略", @@ -79635,7 +80610,8 @@ "K0103001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-01_集合" ], "genre": "填空题", "ans": "\\textcircled{4}\\textcircled{5}", @@ -79743,7 +80719,8 @@ "K0104006B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-01_集合" ], "genre": "填空题", "ans": "$2$", @@ -79826,7 +80803,8 @@ "K0103003B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-01_集合" ], "genre": "解答题", "ans": "$C\\subset D$, 证明略", @@ -79882,7 +80860,8 @@ "K0103005B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-01_集合" ], "genre": "解答题", "ans": "(1) $[1,2]$; (2) $[\\dfrac 12, 3]$.", @@ -79915,7 +80894,8 @@ "K0104007B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-01_集合" ], "genre": "填空题", "ans": "$A\\cap B\\cap \\overline C$", @@ -79946,7 +80926,8 @@ "K0102003B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-01_集合" ], "genre": "填空题", "ans": "(1) $[0,4]$; (2) $\\{(-\\sqrt{2},2),(\\sqrt{2},2)\\}$", @@ -80102,7 +81083,8 @@ "K0104007B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-01_集合" ], "genre": "选择题", "ans": "C", @@ -80163,7 +81145,8 @@ "K0104004B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-01_集合" ], "genre": "填空题", "ans": "$2$", @@ -80266,7 +81249,8 @@ "K0101001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-01_集合" ], "genre": "选择题", "ans": "B", @@ -80578,7 +81562,8 @@ "K0101004B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-01_集合" ], "genre": "填空题", "ans": "$-1$", @@ -80659,7 +81644,8 @@ "K0107002B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-02_常用逻辑用语" ], "genre": "填空题", "ans": "(1) $m>0$且$n\\le 0$; (2) 空间三条直线$l,m,n$中至少有两条不相交; (3) 复数$z_1,z_2,z_3$中至少有两个纯虚数", @@ -80711,7 +81697,8 @@ "K0105002B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-02_常用逻辑用语" ], "genre": "选择题", "ans": "B", @@ -80739,7 +81726,8 @@ "K0107003B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-02_常用逻辑用语" ], "genre": "填空题", "ans": "充分非必要", @@ -80815,7 +81803,8 @@ "K0106001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-02_常用逻辑用语" ], "genre": "选择题", "ans": "D", @@ -80893,7 +81882,8 @@ "K0106003B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-02_常用逻辑用语" ], "genre": "解答题", "ans": "(1) 存在, 如$m=100$; (2) 不存在, 反例只需既大于$3$又大于$-\\dfrac m2$, 如$|\\dfrac m2|+4$.", @@ -80923,7 +81913,8 @@ "K0112001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-04_方程与不等式的求解" ], "genre": "解答题", "ans": "(1) $\\begin{cases}b<0, \\\\ c=0;\\end{cases}$ (2) $\\begin{cases} b^2-4ac \\ge 0, \\\\ 4a+b>0, \\\\ 4a+2b+c>0.\\end{cases}$", @@ -81021,7 +82012,8 @@ "K0106001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期周末卷-周末卷03_暂未使用" ], "genre": "选择题", "ans": "C", @@ -81047,7 +82039,8 @@ "K0106001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-02_常用逻辑用语" ], "genre": "填空题", "ans": "$(-\\infty,0]$", @@ -81075,7 +82068,8 @@ "K0107002B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-02_常用逻辑用语" ], "genre": "解答题", "ans": "$(0,\\dfrac 1{16}]\\cup [\\dfrac 14,+\\infty)$", @@ -81155,7 +82149,9 @@ "K0111003B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-03_等式与不等式的性质及证明" ], "genre": "填空题", "ans": "\\textcircled{2}\\textcircled{4}\\textcircled{5}\\textcircled{7}", @@ -81247,7 +82243,9 @@ "K0119001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-较难题", + "2023届高三-第一轮复习讲义-03_等式与不等式的性质及证明" ], "genre": "填空题", "ans": "\\textcircled{1}", @@ -81298,7 +82296,8 @@ "K0119001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-03_等式与不等式的性质及证明" ], "genre": "选择题", "ans": "C", @@ -81452,7 +82451,8 @@ "K0111003B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-03_等式与不等式的性质及证明" ], "genre": "选择题", "ans": "C", @@ -81630,7 +82630,8 @@ "K0119001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-03_等式与不等式的性质及证明" ], "genre": "填空题", "ans": "(1) 大, $-4$, $0$; (2) $\\sqrt{2}$, $1$", @@ -81732,7 +82733,9 @@ "K0115002B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-04_方程与不等式的求解" ], "genre": "填空题", "ans": "$-3$", @@ -81758,7 +82761,9 @@ "K0114001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-04_方程与不等式的求解" ], "genre": "填空题", "ans": "$[-3,-2)\\cup (0,1]$", @@ -81808,7 +82813,9 @@ "K0115002B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-04_方程与不等式的求解" ], "genre": "填空题", "ans": "\\textcircled{3}\\textcircled{5}", @@ -81884,7 +82891,9 @@ "K0115002B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-04_方程与不等式的求解" ], "genre": "解答题", "ans": "$(-\\infty,\\dfrac 14)\\cup (\\dfrac 12,+\\infty)$", @@ -82033,7 +83042,9 @@ "K0115002B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-04_方程与不等式的求解" ], "genre": "解答题", "ans": "$(0,3)$", @@ -82059,7 +83070,9 @@ "K0115001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-04_方程与不等式的求解" ], "genre": "解答题", "ans": "$[-2,\\dfrac 65)$", @@ -82184,7 +83197,9 @@ "K0116002B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-04_方程与不等式的求解" ], "genre": "填空题", "ans": "$[\\dfrac{26}9,5)$", @@ -82213,7 +83228,9 @@ "K0116002B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-04_方程与不等式的求解" ], "genre": "填空题", "ans": "$-5$", @@ -82240,7 +83257,9 @@ "K0116001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-较难题", + "2023届高三-第一轮复习讲义-04_方程与不等式的求解" ], "genre": "填空题", "ans": "$(-\\infty,-1]\\cup \\{1\\}\\cup [2,+\\infty)$", @@ -82269,7 +83288,9 @@ "K0117001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-04_方程与不等式的求解" ], "genre": "填空题", "ans": "$(-4,-3)\\cup (1,2)$", @@ -82302,7 +83323,9 @@ "K0117002B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-04_方程与不等式的求解" ], "genre": "填空题", "ans": "$(-\\infty,\\dfrac 15)$", @@ -82407,7 +83430,8 @@ "K0117002B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-04_方程与不等式的求解" ], "genre": "解答题", "ans": "(1) $(-\\infty,-2)\\cup (1,+\\infty)$; (2) $(-\\infty,-1)\\cup (0,+\\infty)$", @@ -82460,7 +83484,9 @@ "K0117002B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-04_方程与不等式的求解" ], "genre": "填空题", "ans": "$(-1,0)$", @@ -82662,7 +83688,8 @@ "K0116001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-04_方程与不等式的求解" ], "genre": "解答题", "ans": "(1) $(-\\infty,-\\sqrt{5})\\cup (1,\\sqrt{5})$; (2) $[1,\\dfrac 52)\\cup (4,25]$", @@ -82790,7 +83817,8 @@ "K0118003B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-03_等式与不等式的性质及证明" ], "genre": "解答题", "ans": "证明略", @@ -83185,7 +84213,8 @@ "K0216005B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第一轮复习讲义-10_有关函数的应用问题" ], "genre": "解答题", "ans": "$y=\\begin{cases}\n -5x^2+20x, & 1\\le x<4,\\\\ \\dfrac{125}2-\\dfrac{90}x-10x, & x\\ge 4.\n\\end{cases}$", @@ -83234,7 +84263,8 @@ "K0222001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第一轮复习讲义-10_有关函数的应用问题" ], "genre": "解答题", "ans": "$y=-(\\dfrac \\pi 2+2)x^2+lx, \\ x\\in (0,\\dfrac l{\\pi+2})$", @@ -83261,7 +84291,11 @@ "K0215005B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-四月错题重做-02_函数二", + "2023届高三-四月错题重做-02_易错题-函数2", + "2023届高三-寒假作业-较难题", + "2023届高三-第一轮复习讲义-07_函数的概念与奇偶性" ], "genre": "解答题", "ans": "(1) $[\\dfrac 14,+\\infty)$; (2) $[0,\\dfrac 14]$", @@ -83323,7 +84357,9 @@ "K0215003B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-07_函数的概念与奇偶性" ], "genre": "填空题", "ans": "$[\\dfrac 12,2)$", @@ -83448,7 +84484,9 @@ "K0215001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-07_函数的概念与奇偶性" ], "genre": "选择题", "ans": "B", @@ -83678,7 +84716,8 @@ "K0218001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第一轮复习讲义-07_函数的概念与奇偶性" ], "genre": "填空题", "ans": "(1) $-x^2-2x$; (2) $x^2+2x$", @@ -83778,7 +84817,11 @@ "K0217004B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-四月错题重做-02_函数二", + "2023届高三-四月错题重做-02_易错题-函数2", + "2023届高三-寒假作业-较难题", + "2023届高三-第一轮复习讲义-07_函数的概念与奇偶性" ], "genre": "解答题", "ans": "(1) 证明略; (2) $a=0$", @@ -83840,7 +84883,8 @@ "K0218001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验02" ], "genre": "选择题", "ans": "D", @@ -83912,7 +84956,8 @@ "K0217004B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第一轮复习讲义-07_函数的概念与奇偶性" ], "genre": "解答题", "ans": "(1) 奇函数; (2) 奇函数", @@ -83986,7 +85031,11 @@ "K0218001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验01", + "2023届高三-四月错题重做-02_函数二", + "2023届高三-四月错题重做-02_易错题-函数2", + "2023届高三-第一轮复习讲义-07_函数的概念与奇偶性" ], "genre": "解答题", "ans": "$b=1$", @@ -84111,7 +85160,9 @@ "K0217003B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-07_函数的概念与奇偶性" ], "genre": "选择题", "ans": "C", @@ -84336,7 +85387,9 @@ "K0213001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-06_幂指对函数" ], "genre": "填空题", "ans": "$-4$", @@ -84413,7 +85466,9 @@ "K0213001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷02", + "2023届高三-第一轮复习讲义-06_幂指对函数" ], "genre": "选择题", "ans": "A", @@ -84524,7 +85579,9 @@ "K0213001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-06_幂指对函数" ], "genre": "填空题", "ans": "$2$, $\\log_2(5-a)$", @@ -84609,7 +85666,8 @@ "K0218001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验02" ], "genre": "解答题", "ans": "(1) 证明略; (2) $f(6)=-2a$, $f(300)=-100a$.", @@ -84682,7 +85740,9 @@ "K0219003B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-08_函数的单调性" ], "genre": "填空题", "ans": "\\textcircled{3}\\textcircled{4}", @@ -84786,7 +85846,9 @@ "K0220002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-08_函数的单调性" ], "genre": "填空题", "ans": "$(-\\infty,-3)$", @@ -84917,7 +85979,9 @@ "KNONE" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-08_函数的单调性" ], "genre": "填空题", "ans": "$(-\\infty,0]$", @@ -84943,7 +86007,9 @@ "K0219003B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-08_函数的单调性" ], "genre": "解答题", "ans": "(1) 证明略; (2) 证明略", @@ -84969,7 +86035,9 @@ "K0219001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-08_函数的单调性" ], "genre": "解答题", "ans": "$(-3,-2]$", @@ -85047,7 +86115,9 @@ "K0213004B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-06_幂指对函数" ], "genre": "填空题", "ans": "$(2,+\\infty)$", @@ -85187,7 +86257,8 @@ "K0219003B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验02" ], "genre": "解答题", "ans": "(1) $a=4$, $b=0$; (2) 证明略.", @@ -85236,7 +86307,11 @@ "K0213004B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-四月错题重做-02_函数二", + "2023届高三-四月错题重做-02_易错题-函数2", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-06_幂指对函数" ], "genre": "解答题", "ans": "$(1,2)$", @@ -85359,7 +86434,9 @@ "K0207003B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-06_幂指对函数" ], "genre": "选择题", "ans": "B", @@ -85410,7 +86487,9 @@ "K0208002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-06_幂指对函数" ], "genre": "填空题", "ans": "$-1$", @@ -85496,7 +86575,9 @@ "K0208002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-06_幂指对函数" ], "genre": "填空题", "ans": "$-1$", @@ -85595,7 +86676,11 @@ "K0208002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-四月错题重做-01_函数一", + "2023届高三-四月错题重做-01_易错题-函数1", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-06_幂指对函数" ], "genre": "解答题", "ans": "$t=1$, 图像如下:\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0:2,samples = 50] plot (\\x,{pow(\\x,2/3)});\n\\draw [domain = 0:2,samples = 50] plot (-\\x,{pow(\\x,2/3)});\n\\end{tikzpicture}\n\\end{center}", @@ -85783,7 +86868,9 @@ "K0207004B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-06_幂指对函数" ], "genre": "选择题", "ans": "B", @@ -86528,7 +87615,9 @@ "K0211002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-09_函数的零点与最值" ], "genre": "填空题", "ans": "$\\dfrac{1+\\sqrt{5}}2$", @@ -86631,7 +87720,11 @@ "K0215005B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-四月错题重做-02_函数二", + "2023届高三-四月错题重做-02_易错题-函数2", + "2023届高三-寒假作业-较难题", + "2023届高三-第一轮复习讲义-09_函数的零点与最值" ], "genre": "解答题", "ans": "$g(a)=\\begin{cases} \\dfrac 32a-\\dfrac 34, & a\\ge \\dfrac 12,\\\\ 0, & a\\le \\dfrac 12. \\end{cases}$", @@ -86825,7 +87918,9 @@ "K0214002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-09_函数的零点与最值" ], "genre": "填空题", "ans": "$6$", @@ -86877,7 +87972,11 @@ "K0212001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-四月错题重做-01_函数一", + "2023届高三-四月错题重做-01_易错题-函数1", + "2023届高三-寒假作业-较难题", + "2023届高三-第一轮复习讲义-07_函数的概念与奇偶性" ], "genre": "解答题", "ans": "(1) $[-3\\sqrt{3},-\\sqrt{3}]\\cup [\\sqrt{3},3\\sqrt{3}]$; (2) $[13,\\dfrac {69}4]$", @@ -86941,7 +88040,8 @@ "K0221002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验02" ], "genre": "解答题", "ans": "(1) $(3,+\\infty)$; (2) $[-5,1]$.", @@ -87063,7 +88163,9 @@ "K0221002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-较难题", + "2023届高三-第一轮复习讲义-09_函数的零点与最值" ], "genre": "填空题", "ans": "$\\begin{cases} 0, & a\\ge \\dfrac 12, \\\\ 2a-1, & a\\le \\dfrac 12. \\end{cases}$", @@ -87548,7 +88650,9 @@ "K0215005B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-07_函数的概念与奇偶性" ], "genre": "填空题", "ans": "$[-2,+\\infty)$", @@ -88049,7 +89153,9 @@ "K0224001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-较难题", + "2023届高三-第一轮复习讲义-09_函数的零点与最值" ], "genre": "填空题", "ans": "$(-\\infty,-1]\\cup [\\dfrac 15,+\\infty)$", @@ -88751,7 +89857,9 @@ "K0212003B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-06_幂指对函数" ], "genre": "填空题", "ans": "\\textcircled{3}\\textcircled{4}", @@ -89121,7 +90229,8 @@ "K0301004B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-11_三角比的定义及直接性质" ], "genre": "填空题", "ans": "$k\\pi-\\dfrac\\pi 6, \\ k\\in \\mathbf{Z}$", @@ -89294,7 +90403,8 @@ "K0302003B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-11_三角比的定义及直接性质" ], "genre": "解答题", "ans": "$2\\alpha$的终边在第三、四象限或$y$轴负半轴, $\\dfrac{\\alpha}2$的终边在第一、三象限, $\\dfrac{\\alpha}3$的终边在第一、二、四象限", @@ -89331,7 +90441,8 @@ "K0304001B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-11_三角比的定义及直接性质" ], "genre": "解答题", "ans": "(1) 图略, $\\{\\alpha|\\dfrac\\pi 3+2k\\pi <\\alpha<\\dfrac{2\\pi}3+2k\\pi, \\ k\\in \\mathbf{Z}\\}$; (2) 图略, $\\{\\alpha|\\dfrac{2\\pi}3+2k\\pi<\\alpha<\\dfrac{4\\pi}3+2k\\pi, \\ k\\in \\mathbf{Z}\\}$; (3) 图略, $\\{\\alpha|\\dfrac\\pi 2+k\\pi<\\alpha<\\dfrac{3\\pi}4+k\\pi, \\ k\\in \\mathbf{Z}\\}$.", @@ -89984,7 +91095,9 @@ "K0309003B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-12_和差倍角公式" ], "genre": "填空题", "ans": "$\\dfrac 12$", @@ -90021,7 +91134,9 @@ "K0311002B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-寒假作业-较难题", + "2023届高三-第一轮复习讲义-12_和差倍角公式" ], "genre": "填空题", "ans": "$2\\sin(\\alpha+\\dfrac{5\\pi}3)$", @@ -90058,7 +91173,8 @@ "K0309003B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-12_和差倍角公式" ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{15}}4$", @@ -90095,7 +91211,9 @@ "K0310002B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-12_和差倍角公式" ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{2}}{10}$", @@ -90132,7 +91250,8 @@ "K0310002B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-周末卷03" ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{35}+\\sqrt{3}}{12}$", @@ -90169,7 +91288,8 @@ "K0310002B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-12_和差倍角公式" ], "genre": "填空题", "ans": "$\\dfrac 5{14}$", @@ -90256,7 +91376,9 @@ "K0310002B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-寒假作业-较难题", + "2023届高三-第一轮复习讲义-12_和差倍角公式" ], "genre": "解答题", "ans": "$\\dfrac{11\\pi}4$", @@ -90317,7 +91439,8 @@ "K0309003B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-12_和差倍角公式" ], "genre": "填空题", "ans": "$\\dfrac{59}{72}$", @@ -90408,7 +91531,8 @@ "K0310002B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-12_和差倍角公式" ], "genre": "填空题", "ans": "$\\dfrac{7\\pi}4$", @@ -90466,7 +91590,9 @@ "K0310002B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-寒假作业-较难题", + "2023届高三-第一轮复习讲义-12_和差倍角公式" ], "genre": "解答题", "ans": "$(-\\dfrac 34,\\dfrac 32)\\cup (\\dfrac 32,+\\infty)$", @@ -90618,7 +91744,8 @@ "K0312003B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-12_和差倍角公式" ], "genre": "填空题", "ans": "$-\\dfrac{24}{25}$", @@ -90655,7 +91782,8 @@ "K0312003B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-12_和差倍角公式" ], "genre": "填空题", "ans": "$\\dfrac 14$", @@ -90694,7 +91822,8 @@ "K0312003B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-周末卷03" ], "genre": "填空题", "ans": "$\\dfrac 45$", @@ -90931,7 +92060,8 @@ "K0312003B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-12_和差倍角公式" ], "genre": "解答题", "ans": "证明略", @@ -91017,7 +92147,8 @@ "K0314006B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-13_解三角形" ], "genre": "填空题", "ans": "(1) $2R^2\\sin A\\sin B\\sin C$; (2) $\\dfrac{abc}{4R}$; (3) $pr$.", @@ -91076,7 +92207,8 @@ "K0315003B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-13_解三角形" ], "genre": "填空题", "ans": "$\\arccos \\dfrac 14$", @@ -91135,7 +92267,8 @@ "K0314004B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-13_解三角形" ], "genre": "填空题", "ans": "(1) $\\dfrac{5\\pi}{6}$; (2) $\\dfrac{5\\pi}{6}$或$\\dfrac{\\pi}6$.", @@ -91196,7 +92329,8 @@ "K0315002B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-13_解三角形" ], "genre": "填空题", "ans": "$\\sqrt{3}$", @@ -91234,7 +92368,8 @@ "K0315004B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-13_解三角形" ], "genre": "填空题", "ans": "$(\\sqrt{3},\\sqrt{5})$", @@ -91293,7 +92428,8 @@ "K0316002B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-13_解三角形" ], "genre": "解答题", "ans": "(1) $a=b$的等腰三角形; (2) $\\angle A=90^\\circ$的直角三角形.", @@ -91330,7 +92466,9 @@ "K0317002B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-13_解三角形" ], "genre": "解答题", "ans": "长约为$445$米.", @@ -91367,7 +92505,8 @@ "K0315004B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-13_解三角形" ], "genre": "填空题", "ans": "$3$", @@ -91404,7 +92543,8 @@ "K0314004B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-13_解三角形" ], "genre": "填空题", "ans": "$\\dfrac{2\\sqrt{3}}3$", @@ -91442,7 +92582,8 @@ "K0315003B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-13_解三角形" ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{3}}3$", @@ -91479,7 +92620,8 @@ "K0316002B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-13_解三角形" ], "genre": "选择题", "ans": "B", @@ -91516,7 +92658,8 @@ "K0315002B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-13_解三角形" ], "genre": "填空题", "ans": "$\\dfrac{15}2$", @@ -91575,7 +92718,8 @@ "K0317002B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-13_解三角形" ], "genre": "填空题", "ans": "$\\dfrac{a\\sin\\alpha\\sin\\beta}{\\sin(\\alpha-\\beta)}$", @@ -91660,7 +92804,8 @@ "K0322002B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-14_正弦函数及正弦型函数" ], "genre": "填空题", "ans": "$\\{x|\\dfrac\\pi 2+2k\\pi\\le x\\le \\dfrac{3\\pi}2+2k\\pi, \\ k\\in \\mathbf{Z}\\}$", @@ -91706,7 +92851,8 @@ "K0320002B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-14_正弦函数及正弦型函数" ], "genre": "填空题", "ans": "$[-1,2]$", @@ -91787,7 +92933,8 @@ "K0324005B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-15_周期性与其他三角函数" ], "genre": "填空题", "ans": "$-5$", @@ -91824,7 +92971,8 @@ "K0320001B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-14_正弦函数及正弦型函数" ], "genre": "选择题", "ans": "C", @@ -91861,7 +93009,8 @@ "K0319003B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-15_周期性与其他三角函数" ], "genre": "选择题", "ans": "B", @@ -91899,7 +93048,8 @@ "K0322002B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-14_正弦函数及正弦型函数" ], "genre": "解答题", "ans": "(1) $\\{x|2k\\pi=latex, line cap = round, line join = round, scale = 1.5]\n \\draw (0,0) -- (3,0) (0,1) node [above left] {$A$} -- (4,1) (1,-1) -- (2,-1) (2,2) node [above left] {$C$} -- (4,2);\n \\draw (0,0) -- (0,1) (1,-1) -- (1,1) (2,-1) -- (2,2) (3,0) -- (3,2) (4,1) -- (4,2);\n \\draw (0,1) -- (1,0) node [below right] {$B$} coordinate (B) (2,2) -- (3,1) node [below right] {$D$};\n \\draw [dashed] (B) --++ (45:1/2) --++ (1,0) --++ (225:1/2);\n \\draw [dashed] (B) ++ (0,1) --++ (45:1/2) --++ (1,0) --++ (225:1/2);\n \\draw [dashed] (B) ++ (45:1/2) --++ (0,1) ++ (1,0) --++ (0,-1);\n \\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期周末卷-周末卷05", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$\\dfrac\\pi 3$", @@ -100347,7 +101595,8 @@ "content": "如图, $P$为三角形$ABC$所在平面外一点, $PA\\perp$平面$ABC$, $\\angle ABC=90^\\circ$, $AE\\perp PB$于$E$, $AF\\perp PC$于$F$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, line cap = round, line join = round]\n \\draw (0,0) coordinate (B) (3,0) coordinate (C) (1,1) coordinate (A) node [below] {$A$} (1,4) coordinate (P);\n \\draw (B) node [below left] {$B$} -- (C) node [below right] {$C$} -- (P) node [above] {$P$} -- cycle;\n \\draw ($(B)!0.6!0:(P)$) coordinate (E) node [left] {$E$} ($(C)!0.65!0:(P)$) coordinate (F) node [right] {$F$};\n \\draw (E) -- (F);\n \\draw [dashed] (A) -- (P) (A) -- (B) (A) -- (C) (A) -- (E) (A) -- (F); \n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $BC\\perp$平面$PAB$, $AE\\perp$平面$PBC$, $PC\\perp$平面$AEF$;\\\\\n(2) 若$AP=AC=2$, $\\angle BPC=\\theta$, 当$\\theta$为何值时, 三角形$AEF$的面积$S$取到最大值? 最大值是多少?", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期周末卷-周末卷05" ], "genre": "解答题", "ans": "(1) 证明略; (2) 当且仅当$\\theta=\\arcsin\\dfrac{\\sqrt{3}}3$时, 三角形$AEF$的面积最大, 最大值为$\\dfrac 12$", @@ -100441,7 +101690,8 @@ "content": "对于分别与两条异面直线都相交的两条直线, 下列结论中, 真命题有\\blank{50}(填入序号).\\\\\n\\textcircled{1} 一定是异面直线; \\textcircled{2} 不可能是平行直线; \\textcircled{3} 不可能是相交直线.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-20_描述空间位置关系的公理" ], "genre": "填空题", "ans": "\\textcircled{2}", @@ -100648,7 +101898,8 @@ "K0617006B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-24_体积及表面积的计算" ], "genre": "填空题", "ans": "$1+\\dfrac 1{2\\pi}$", @@ -101132,7 +102383,8 @@ "content": "如图, 在直三棱柱$ABC-A_1B_1C_1$中, $AA_1=BC=AB=2$, $AB\\perp BC$.\n\\begin{center}\n \\begin{tikzpicture}[>=latex, line cap = round, line join = round, scale = 0.6]\n \\draw (0,0) node [below left] {$A$} coordinate (A) ++ (45:2) node [below right] {$B$} coordinate (B) \n ++ (4,0) node [below right] {$C$} coordinate (C);\n \\draw (A) ++ (0,4) coordinate (A1) node [left] {$A_1$};\n \\draw (B) ++ (0,4) coordinate (B1) node [above] {$B_1$};\n \\draw (C) ++ (0,4) coordinate (C1) node [above right] {$C_1$};\n \\draw (A1) -- (B1) -- (C1) -- cycle;\n \\draw (C1) -- (C) -- (A) -- (A1) -- (C);\n \\draw [dashed] (A) -- (B) -- (C) -- (B1)-- (B) -- (A1);\n \\end{tikzpicture}\n\\end{center}\n(1) 求直线$A_1C$和直线$B_1C_1$所成的角的大小;\\\\\n(2) 求二面角$C-A_1B_1-C_1$的大小;\\\\\n(3) 求点$A$到平面$A_1BC$的距离.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期周末卷-周末卷05" ], "genre": "解答题", "ans": "(1) $\\arctan\\sqrt{2}$; (2) $\\dfrac\\pi 4$; (3) $\\sqrt{2}$", @@ -101228,7 +102480,8 @@ "K0612006B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-22_空间平面与平面的位置关系" ], "genre": "填空题", "ans": "(1) $\\sqrt{2\\sqrt{3}}3$; (2) $\\sqrt{2\\sqrt{3}}3$; (3) $\\sqrt{2\\sqrt{3}}3$; (4) $\\sqrt{2}$", @@ -101255,7 +102508,8 @@ "content": "在四棱锥$P-ABCD$中, 底面是边长为$2$的菱形. $\\angle DAB=60^\\circ$, 对角线$AC$与$BD$相交于点$O$, $PO\\perp$平面$ABCD$, $PB$与平面$ABCD$所成的角的大小为$60^\\circ$.\n\\begin{center}\n \\begin{tikzpicture}[>=latex, line cap = round, line join = round, scale = 1.3]\n \\draw ({-sqrt(3)},0) coordinate (A) node [left] {$A$};\n \\draw ({sqrt(3)},0) coordinate (C) node [right] {$C$};\n \\draw (45:0.5) coordinate (D) node [above right] {$D$};\n \\draw (225:0.5) coordinate (B) node [below left] {$B$};\n \\draw (0,{sqrt(3)}) coordinate (P) node [above] {$P$};\n \\draw ($(P)!0.5!(B)$) coordinate (E) node [left] {$E$};\n \\draw (A) -- (B) -- (C) -- (P) -- cycle (P) -- (B);\n \\draw [dashed] (D) -- (A) (D) -- (E) (D) -- (C) (D) -- (P);\n \\end{tikzpicture}\n\\end{center}\n(1) 求四棱锥$P-ABCD$的体积;\\\\\n(2) 若$E$是$PB$的中点, 求异面直线$DE$与$PA$所成的角的大小.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期周末卷-周末卷05" ], "genre": "解答题", "ans": "(1) $2$; (2) $\\arccos \\dfrac{\\sqrt{2}}4$", @@ -101285,7 +102539,8 @@ "K0101003B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第一轮复习讲义-01_集合" ], "genre": "填空题", "ans": "$\\mathbf{Z}\\subseteq \\mathbf{Q}\\subseteq \\mathbf{R}\\subseteq \\mathbf{C}$", @@ -101315,7 +102570,8 @@ "K0512005B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第一轮复习讲义-18_复数的代数运算与性质" ], "genre": "填空题", "ans": "$-1-4\\mathrm{i}$", @@ -101403,7 +102659,8 @@ "K0514001B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第一轮复习讲义-18_复数的代数运算与性质" ], "genre": "填空题", "ans": "$-\\mathrm{i}$", @@ -101492,7 +102749,8 @@ "K0514001B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第一轮复习讲义-18_复数的代数运算与性质" ], "genre": "填空题", "ans": "(1) 真; (2) 假, $z=0$; (3) 真; (4) 假, $1+\\sqrt{3}\\mathrm{i}$", @@ -101555,7 +102813,8 @@ "K0514001B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第一轮复习讲义-18_复数的代数运算与性质" ], "genre": "解答题", "ans": "$\\sqrt{10}$", @@ -101687,7 +102946,8 @@ "K0512002B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第一轮复习讲义-18_复数的代数运算与性质" ], "genre": "填空题", "ans": "$-2$", @@ -101777,7 +103037,8 @@ "K0513002B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第一轮复习讲义-18_复数的代数运算与性质" ], "genre": "解答题", "ans": "(1) $|z|=2$; (2) $(-4,4)$; (3) $2\\mathrm{i}$", @@ -101836,7 +103097,8 @@ "K0513003B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第一轮复习讲义-19_复数的几何意义与实系数二次方程" ], "genre": "选择题", "ans": "B", @@ -101871,7 +103133,8 @@ "K0513003B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第一轮复习讲义-19_复数的几何意义与实系数二次方程" ], "genre": "选择题", "ans": "A", @@ -101908,7 +103171,8 @@ "K0514007B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第一轮复习讲义-19_复数的几何意义与实系数二次方程" ], "genre": "填空题", "ans": "$\\dfrac\\pi 2$", @@ -101945,7 +103209,8 @@ "K0513004B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第一轮复习讲义-19_复数的几何意义与实系数二次方程" ], "genre": "填空题", "ans": "$-2+\\mathrm{i}$", @@ -101982,7 +103247,8 @@ "K0513003B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第一轮复习讲义-19_复数的几何意义与实系数二次方程" ], "genre": "填空题", "ans": "$a+b\\mathrm{i}$", @@ -102017,7 +103283,8 @@ "K0514007B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第一轮复习讲义-19_复数的几何意义与实系数二次方程" ], "genre": "填空题", "ans": "(1) 直线$y=0$; (2) 圆$(x+1)^2+y^2=1$; (3) 椭圆$\\dfrac{x^2}{36}+\\dfrac{y^2}{11}=1$; (4) 线段$y=0 \\ (x\\in [-1,1])$; (5) 双曲线的一支: $y^2-\\dfrac{x^2}{3}=1 \\ (y<0)$", @@ -102125,7 +103392,8 @@ "K0514007B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第一轮复习讲义-19_复数的几何意义与实系数二次方程" ], "genre": "解答题", "ans": "以$(-3,7)$为圆心, $12$为半径的圆", @@ -102210,7 +103478,8 @@ "K0513003B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期周末卷-周末卷05" ], "genre": "填空题", "ans": "以$(1,0)$为圆心, $3$为半径的圆", @@ -102264,7 +103533,9 @@ "K0514007B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期周末卷-周末卷05", + "2023届高三-寒假作业-较难题" ], "genre": "填空题", "ans": "$\\{4\\mathrm{i}\\}$", @@ -102294,7 +103565,8 @@ "K0514007B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第一轮复习讲义-19_复数的几何意义与实系数二次方程" ], "genre": "解答题", "ans": "(1) $[4\\sqrt{2}-2,4\\sqrt{2}+2]$; (2) $(3+\\sqrt{2})-(4+\\sqrt{2})\\mathrm{i}$", @@ -102332,7 +103604,8 @@ "K0513005B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第一轮复习讲义-19_复数的几何意义与实系数二次方程" ], "genre": "解答题", "ans": "以$(1,0)$为圆心, $\\dfrac 12$为半径的圆", @@ -102420,7 +103693,8 @@ "K0511009B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第一轮复习讲义-18_复数的代数运算与性质" ], "genre": "填空题", "ans": "$1$", @@ -102538,7 +103812,9 @@ "K0515006B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-寒假作业-较难题", + "2023届高三-第一轮复习讲义-19_复数的几何意义与实系数二次方程" ], "genre": "填空题", "ans": "\\textcircled{2}\\textcircled{5}", @@ -102599,7 +103875,8 @@ "K0515007B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第一轮复习讲义-19_复数的几何意义与实系数二次方程" ], "genre": "解答题", "ans": "$p=12$, $q=26$", @@ -102765,7 +104042,8 @@ "K0511009B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第一轮复习讲义-18_复数的代数运算与性质" ], "genre": "填空题", "ans": "$-1$", @@ -102805,7 +104083,8 @@ "K0515007B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第一轮复习讲义-19_复数的几何意义与实系数二次方程" ], "genre": "解答题", "ans": "(1) $-\\dfrac 14$或$\\dfrac{17}{4}$; (2) $-\\dfrac 14$或$\\dfrac 94$", @@ -103202,7 +104481,9 @@ "objs": [], "tags": [ "第八单元", - "排列" + "排列", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-38_计数原理与排列组合" ], "genre": "填空题", "ans": "$\\mathrm{P}_{34-m}^8$", @@ -103382,7 +104663,10 @@ "objs": [], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-四月错题重做-05_易错题-概率与统计", + "2023届高三-四月错题重做-05_概率与统计", + "2023届高三-第一轮复习讲义-38_计数原理与排列组合" ], "genre": "填空题", "ans": "$0.985$", @@ -103572,7 +104856,8 @@ "objs": [], "tags": [ "第八单元", - "排列" + "排列", + "2023届高三-第一轮复习讲义-38_计数原理与排列组合" ], "genre": "填空题", "ans": "$407$", @@ -103683,7 +104968,11 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-四月错题重做-05_易错题-概率与统计", + "2023届高三-四月错题重做-05_概率与统计", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-25_概率的概念及性质" ], "genre": "填空题", "ans": "$\\dfrac 3{10}$", @@ -104226,7 +105515,8 @@ "K0315004B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-13_解三角形" ], "genre": "解答题", "ans": "(1) $\\dfrac{9\\sqrt{3}}{14}$; (2) $3+4\\sqrt{2}\\pm \\sqrt{5}$.", @@ -104508,7 +105798,8 @@ "K0401003X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第一轮复习讲义-30_等差数列与等比数列" ], "genre": "填空题", "ans": "$\\dfrac{27}{8}$", @@ -104640,7 +105931,8 @@ "content": "已知$\\overrightarrow{a_1}, \\overrightarrow{a_2}, \\overrightarrow{b_1}, \\overrightarrow{b_2},\\cdots,\\overrightarrow{b_k}$($k\\in \\mathbf{N}$, $k\\ge 1$)是平面内两两互不相等的向量, 满足$|\\overrightarrow{a_1}-\\overrightarrow{a_2}|=1$, 且$|\\overrightarrow{a_i}-\\overrightarrow{b_j}|\\in \\{1,2\\}$(其中$i=1,2$, $j=1,2,\\cdots,k$), 则$k$的最大值为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期周末卷-周末卷04" ], "genre": "填空题", "ans": "$6$", @@ -104798,7 +106090,8 @@ "content": "已知$f(x)=\\sin\\omega x$($\\omega>0$).\\\\\n(1) $f(x)$的最小正周期是$4\\pi$, 求$\\omega$, 并求此时$f(x)=\\dfrac12$的解集;\\\\\n(2) 已知$\\omega=1$, $g(x)=f^2(x)+\\sqrt3f(-x)f(\\dfrac{\\pi}2-x)$, $x\\in [0,\\dfrac{\\pi}4]$, 求$g(x)$的值域.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-测验03" ], "genre": "解答题", "ans": "(1) $\\omega = \\dfrac 12$; (2) $[-\\dfrac 12,0]$", @@ -104862,7 +106155,8 @@ ], "tags": [ "第七单元", - "双曲线" + "双曲线", + "2023届高三-第二轮复习讲义-12_圆锥曲线" ], "genre": "解答题", "ans": "(1) $2$; (2) $\\arccos\\dfrac{11}{16}$; (3) $\\overrightarrow{OM}\\cdot \\overrightarrow{ON}=4+b^2$, 取值范围为$(6+2\\sqrt{5},+\\infty)$", @@ -104949,7 +106243,9 @@ "content": "已知$z\\in \\mathbf{C}$. 若$\\dfrac{1}{z-5}=\\mathrm{i}$($\\mathrm{i}$为虚数单位), 则$z=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期测验卷-测验05", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$5-\\mathrm{i}$", @@ -105062,7 +106358,9 @@ "K0319002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验05", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$-1$", @@ -105170,7 +106468,10 @@ "objs": [], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-四月错题重做-05_易错题-概率与统计", + "2023届高三-四月错题重做-05_概率与统计", + "2023届高三-第一轮复习讲义-38_计数原理与排列组合" ], "genre": "填空题", "ans": "$\\dfrac{27}{100}$", @@ -105317,7 +106618,8 @@ "content": "已知$\\omega\\in \\mathbf{R}$, 函数$f(x)=(x-6)^2\\cdot \\sin (\\omega x)$. 若存在常数$a\\in \\mathbf{R}$, 使得$f(x+a)$为偶函数, 则$\\omega$的值可能为\\bracket{15}.\n\\fourch{$\\dfrac{\\pi}{2}$}{$\\dfrac{\\pi}{3}$}{$\\dfrac{\\pi}{4}$}{$\\dfrac{\\pi}{5}$}", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-测验04" ], "genre": "选择题", "ans": "C", @@ -105407,7 +106709,11 @@ "K0223004B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-四月错题重做-02_函数二", + "2023届高三-四月错题重做-02_易错题-函数2", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-09_函数的零点与最值" ], "genre": "解答题", "ans": "(1) $(-2,-1)$; (2) $[-\\dfrac 12,-\\dfrac 16]$", @@ -105675,7 +106981,9 @@ "K0217004B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验05", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$-1$", @@ -105737,7 +107045,8 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-第一轮复习讲义-25_概率的概念及性质" ], "genre": "填空题", "ans": "$\\dfrac 15$", @@ -105796,7 +107105,9 @@ "K0203005B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-05_幂指数与对数" ], "genre": "填空题", "ans": "$6$", @@ -105914,7 +107225,8 @@ "objs": [], "tags": [ "第八单元", - "加法原理与乘法原理" + "加法原理与乘法原理", + "2023届高三-第一轮复习讲义-38_计数原理与排列组合" ], "genre": "选择题", "ans": "D", @@ -106026,7 +107338,8 @@ "K0222001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第一轮复习讲义-10_有关函数的应用问题" ], "genre": "解答题", "ans": "(1) $(45,100)$; (2) $g(x)=\\begin{cases} 40-\\dfrac x{10}, & 08736$, 故第$42$个月底单车保有量超过了容纳量", @@ -106911,7 +108227,8 @@ "content": "设$\\overrightarrow a,\\overrightarrow b,\\overrightarrow c$是平面上的向量,$|\\overrightarrow a| =1,|\\overrightarrow b| =3,|\\overrightarrow c|=4$, 且$\\overrightarrow b\\cdot \\overrightarrow c=0$, 实数$\\lambda$满足$0 \\le \\lambda \\le 1$. 若$\\overrightarrow a,\\overrightarrow b,\\overrightarrow c$及$\\lambda$, 使得$s=|\\overrightarrow a-\\lambda \\overrightarrow b-(1-\\lambda)\\overrightarrow c|$是正整数, 则$s$的值的集合是\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期测验卷-线上测验01" ], "genre": "填空题", "ans": "$\\{2,3,4,5\\}$", @@ -106941,7 +108258,8 @@ "content": "如图, 在平面内, $l_1,l_2$是两条平行直线, 它们之间的距离为$2$, 点$P$位于$l_1,l_2$的下方, 动点$N,M$分别在$l_1,l_2$上, 满足$|\\overrightarrow{PM}+\\overrightarrow{PN}|=6$, 则$\\overrightarrow{PM}\\cdot \\overrightarrow{PN}$的最大值为\\bracket{20}.\n\\fourch{$6$}{$8$}{$12$}{$15$}\n\\begin{center}\n \\begin{tikzpicture}[>=latex]\n \\draw (0,0) -- (5,0) node [right] {$l_1$} (0,2) -- (5,2) node [right] {$l_2$};\n \\draw [->] (4.5,-1) node [below right] {$P$} -- (2,0) node [below] {$N$};\n \\draw [->] (4.5,-1) -- (2.5,2) node [below] {$M$};\n \\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期周末卷-周末卷04" ], "genre": "选择题", "ans": "B", @@ -107071,7 +108389,8 @@ "K0105001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-02_常用逻辑用语" ], "genre": "填空题", "ans": "$[0,1]$", @@ -108038,7 +109357,11 @@ "K0214002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-四月错题重做-01_函数一", + "2023届高三-四月错题重做-01_易错题-函数1", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-06_幂指对函数" ], "genre": "填空题", "ans": "$(0,\\dfrac 23)\\cup (1,+\\infty)$", @@ -108325,7 +109648,8 @@ ], "tags": [ "第一单元", - "第五单元" + "第五单元", + "2023届高三-第一轮复习讲义-19_复数的几何意义与实系数二次方程" ], "genre": "解答题", "ans": "不存在推出关系, 因为$P$成立当且仅当$a\\in (-2\\sqrt{2},2\\sqrt{2})$, $Q$成立当且仅当$a\\in [-3,-1]\\cup [1,3]$, $a=0$时$P$成立但$Q$不成立, $a=3$时$Q$成立但$P$不成立", @@ -108483,7 +109807,9 @@ ], "tags": [ "第八单元", - "二项式定理" + "二项式定理", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-39_二项式定理" ], "genre": "填空题", "ans": "$-2$", @@ -108821,7 +110147,9 @@ "K0108003B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-04_方程与不等式的求解" ], "genre": "填空题", "ans": "$(1,+\\infty)$", @@ -108917,7 +110245,9 @@ ], "tags": [ "第七单元", - "抛物线" + "抛物线", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-37_抛物线的概念及性质" ], "genre": "填空题", "ans": "$2$", @@ -109073,7 +110403,9 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-25_概率的概念及性质" ], "genre": "选择题", "ans": "B", @@ -109342,7 +110674,8 @@ "content": "三角形的三内角$A,B,C$所对边的长分别为$a,b,c$, 设向量$\\overrightarrow{m}=(a-b,a-c)$, $\\overrightarrow{n}=(c,a+b)$. 若$\\overrightarrow{m}\\parallel \\overrightarrow{n}$, 则角$B$的大小为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期测验卷-测验04" ], "genre": "填空题", "ans": "$\\dfrac\\pi 3$", @@ -109755,7 +111088,9 @@ "K0213002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-06_幂指对函数" ], "genre": "选择题", "ans": "D", @@ -110060,7 +111395,9 @@ ], "tags": [ "第一单元", - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-05_幂指数与对数" ], "genre": "填空题", "ans": "$3+2\\sqrt{2}$", @@ -110272,7 +111609,8 @@ ], "tags": [ "第七单元", - "抛物线" + "抛物线", + "2023届高三-第一轮复习讲义-37_抛物线的概念及性质" ], "genre": "填空题", "ans": "$4$", @@ -110331,7 +111669,8 @@ "content": "方程$\\dfrac{\\sin x}{1+\\cos x}=\\dfrac{1-\\cos x}{\\sin x}$的解集为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-赋能-赋能27" ], "genre": "填空题", "ans": "$\\{x|x\\ne k\\pi, \\ k\\in \\mathbf{Z}\\}$", @@ -110389,7 +111728,8 @@ "content": "如图, $M$是平行四边形$ABCD$的边$AB$的中点, 直线$l$过点$M$分别交$AD,AC$于点$E,F$. 若$\\overrightarrow{AD}=3\\overrightarrow{AE}$, 则$AF:FC=$\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}\n\\draw (0,0) node [left] {$A$}--(3,0) node [right] {$B$}--(4,2) node [right] {$C$}--(1,2) node [left] {$D$}--cycle;\n\\draw (0,0)--(4,2);\n\\draw (0,{7/6})--(2,{-2/7}) node [below right] {$l$};\n\\draw ({3/2},0) node [below] {$M$} ({1/3},{2/3}) node [left] {$E$} (1,0.5) node [above] {$F$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期测验卷-线上测验01" ], "genre": "填空题", "ans": "$\\dfrac 14$", @@ -111196,7 +112536,9 @@ "K0607004B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-20_描述空间位置关系的公理" ], "genre": "选择题", "ans": "D", @@ -111414,7 +112756,9 @@ "K0215001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-07_函数的概念与奇偶性" ], "genre": "填空题", "ans": "\\textcircled{4}", @@ -111580,7 +112924,8 @@ "K0608001B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期周末卷-周末卷05" ], "genre": "选择题", "ans": "D", @@ -111637,7 +112982,8 @@ ], "tags": [ "第七单元", - "圆" + "圆", + "2023届高三-第一轮复习讲义-35_圆及曲线方程" ], "genre": "解答题", "ans": "(1) $y=x$或$y=7x$; (2) $y=2x+\\dfrac{5\\sqrt{3}}2$或$y=2x-\\dfrac{5\\sqrt{3}}2$", @@ -112673,7 +114019,9 @@ "K0216003B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-07_函数的概念与奇偶性" ], "genre": "选择题", "ans": "A", @@ -112837,7 +114185,8 @@ "K0315004B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-13_解三角形" ], "genre": "填空题", "ans": "$\\dfrac \\pi 6$", @@ -113243,7 +114592,9 @@ "K0304001B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-11_三角比的定义及直接性质" ], "genre": "填空题", "ans": "$\\dfrac{11\\pi}6$", @@ -113806,7 +115157,8 @@ "K0622005B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-23_多面体及旋转体的概念与性质" ], "genre": "选择题", "ans": "D", @@ -113884,7 +115236,9 @@ "content": "在正方体$ABCD-A_1B_1C_1D_1$中, $E,F,G,H$分别为$AB_1,AB,BB_1,B_1C_1$的中点, 则异面直线$EF,GH$所成角的大小为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-20_描述空间位置关系的公理" ], "genre": "填空题", "ans": "$\\dfrac \\pi 4$", @@ -113912,7 +115266,8 @@ "content": "若一个球的体积为$4\\sqrt{3}\\pi$, 则它的表面积为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期测验卷-测验05" ], "genre": "填空题", "ans": "$12\\pi$", @@ -114160,7 +115515,8 @@ "K0622004B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-23_多面体及旋转体的概念与性质" ], "genre": "选择题", "ans": "B", @@ -114463,7 +115819,9 @@ "K0228004X" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-28_导数的概念及常用公式" ], "genre": "解答题", "ans": "$1$或$\\dfrac{13}4$", @@ -114493,7 +115851,10 @@ "K0233003X" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-四月错题重做-01_函数一", + "2023届高三-四月错题重做-01_易错题-函数1", + "2023届高三-第一轮复习讲义-29_导数的应用" ], "genre": "解答题", "ans": "$y=x^3-3x^2$, 单调减区间为$[0,2]$", @@ -115050,7 +116411,8 @@ ], "tags": [ "第八单元", - "二项式定理" + "二项式定理", + "2023届高三-第一轮复习讲义-39_二项式定理" ], "genre": "解答题", "ans": "证明略", @@ -115816,7 +117178,8 @@ "K0103001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-下学期测验卷-高三下学期测验01" ], "genre": "填空题", "ans": "$1$", @@ -115889,7 +117252,8 @@ "K0616003B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期测验卷-高三下学期测验01" ], "genre": "填空题", "ans": "$\\sqrt{5}$", @@ -115926,7 +117290,8 @@ "objs": [], "tags": [ "第八单元", - "组合" + "组合", + "2023届高三-下学期测验卷-高三下学期测验01" ], "genre": "填空题", "ans": "$220$", @@ -115991,7 +117356,8 @@ ], "tags": [ "第七单元", - "双曲线" + "双曲线", + "2023届高三-第一轮复习讲义-36_椭圆与双曲线的概念及性质" ], "genre": "填空题", "ans": "$\\dfrac{y^2}{5}-\\dfrac{x^2}{4}=1$", @@ -116069,7 +117435,8 @@ "content": "函数$y=\\sin (\\dfrac{\\pi}6-x)$, $x\\in [0,\\dfrac{3\\pi }2]$的单调递减区间是\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期测验卷-高三下学期测验01" ], "genre": "填空题", "ans": "$[0,\\dfrac{2\\pi}{3}]$", @@ -116116,7 +117483,8 @@ "K0223004B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期测验卷-高三下学期测验03" ], "genre": "填空题", "ans": "$6$", @@ -116159,7 +117527,8 @@ "content": "已知$6$个正整数, 它们的平均数是$5$, 中位数是$4$, 唯一众数是$3$, 则这$6$个数方差的最大值为\\blank{50}(精确到小数点后一位).", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-下学期测验卷-高三下学期测验01" ], "genre": "填空题", "ans": "$12.3$", @@ -116195,7 +117564,8 @@ "content": "已知正方形$ABCD$边长为$8$, $\\overrightarrow{BE}=\\overrightarrow{EC}$, $\\overrightarrow{DF}=3\\overrightarrow{FA}$, 若在正方形边上恰有$6$个不同的点$P$, 使$\\overrightarrow{PE}\\cdot \\overrightarrow{PF}=\\lambda$, 则$\\lambda$的取值范围为\\blank{50}", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-下学期测验卷-高三下学期测验01" ], "genre": "填空题", "ans": "$(-1,8)$", @@ -116282,7 +117652,8 @@ "content": "已知$z=x+y\\mathrm{i}$, $x,y\\in \\mathbf{R}$, $\\mathrm{i}$是虚数单位.若复数$\\dfrac z{1+\\mathrm{i}}+\\mathrm{i}$是实数, 则$|z|$的最小值为\\bracket{20}.\n\\fourch{$0$}{$\\dfrac 52$}{5}{$\\sqrt 2$}", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-下学期测验卷-高三下学期测验01" ], "genre": "选择题", "ans": "D", @@ -116346,7 +117717,8 @@ "tags": [ "第二单元", "第七单元", - "直线" + "直线", + "2023届高三-下学期测验卷-高三下学期测验01" ], "genre": "选择题", "ans": "A", @@ -116522,7 +117894,9 @@ "K0113001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期测验卷-月考01", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$\\{2,3,4\\}$", @@ -116565,7 +117939,9 @@ "content": "复数$z=\\dfrac{2-\\mathrm{i}}{1+\\mathrm{i}}$所对应的点在复平面内位于第\\blank{50}象限.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期测验卷-测验05", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "四", @@ -116655,7 +118031,8 @@ ], "tags": [ "第七单元", - "双曲线" + "双曲线", + "2023届高三-下学期测验卷-高三下学期测验03" ], "genre": "填空题", "ans": "$3$", @@ -116697,7 +118074,9 @@ "K0617002B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-24_体积及表面积的计算" ], "genre": "填空题", "ans": "$\\dfrac{16}\\pi$", @@ -116855,7 +118234,8 @@ "K0223004B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期测验卷-高三下学期测验03" ], "genre": "填空题", "ans": "$-1$与$\\dfrac 12$", @@ -116920,7 +118300,9 @@ "K0223004B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-月考01", + "2023届高三-寒假作业-较难题" ], "genre": "填空题", "ans": "$\\{2\\}$", @@ -116962,7 +118344,8 @@ ], "tags": [ "第一单元", - "第六单元" + "第六单元", + "2023届高三-上学期周末卷-周末卷05" ], "genre": "选择题", "ans": "B", @@ -117076,7 +118459,8 @@ ], "tags": [ "第六单元", - "空间向量" + "空间向量", + "2023届高三-第一轮复习讲义-33_立体几何中的定量计算" ], "genre": "解答题", "ans": "(1) $\\arcsin \\dfrac{\\sqrt{15}}5$; (2) $\\arccos \\dfrac{\\sqrt{10}}5$", @@ -117140,7 +118524,8 @@ "K0317002B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-测验04" ], "genre": "解答题", "ans": "(1) $AC=\\sqrt{7}\\text{km}$, $S_{ABCD}=2\\sqrt{3}\\text{km}^2$; (2) 当且仅当$P$位于弧$\\overset\\frown{ABC}$的中点时, 改造后的新建筑用地面积最大, 最大面积为$\\dfrac{9\\sqrt{3}}4\\text{km}^2$", @@ -117359,7 +118744,8 @@ "K0206002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验01" ], "genre": "填空题", "ans": "$2$", @@ -117418,7 +118804,8 @@ "K0218001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验01" ], "genre": "填空题", "ans": "$-15$", @@ -117505,7 +118892,8 @@ "content": "函数$f(x)=\\sin(\\omega x)+\\sqrt 3\\cos (\\omega x)$($\\omega >0$), 若恰有两个实数$m$满足: \\textcircled{1} $0\\le m\\le \\dfrac{\\pi}2$; \\textcircled{2} 直线$x=m$是函数图像的一条对称轴, 则$\\omega$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-测验04" ], "genre": "填空题", "ans": "$[\\dfrac 73,\\dfrac{13}3)$", @@ -117656,7 +119044,8 @@ ], "tags": [ "第二单元", - "第五单元" + "第五单元", + "2023届高三-上学期测验卷-测验04" ], "genre": "选择题", "ans": "A", @@ -117820,7 +119209,8 @@ "K0306002B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-月考01" ], "genre": "填空题", "ans": "$-\\dfrac 14$", @@ -117886,7 +119276,9 @@ "content": "已知圆锥的底面半径为$1$, 母线长为$2$, 则该圆锥的体积为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期测验卷-测验05", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{3}}3\\pi$", @@ -117925,7 +119317,9 @@ "K0116002B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期测验卷-测验04", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$(0,1)$", @@ -118020,7 +119414,8 @@ "K0401004X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-下学期测验卷-高三下学期测验02" ], "genre": "填空题", "ans": "$2$", @@ -118058,7 +119453,8 @@ ], "tags": [ "第七单元", - "椭圆" + "椭圆", + "2023届高三-下学期测验卷-高三下学期测验02" ], "genre": "填空题", "ans": "$\\sqrt{3}$", @@ -118098,7 +119494,8 @@ "K0218001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期测验卷-高三下学期测验02" ], "genre": "填空题", "ans": "$-3$", @@ -118164,7 +119561,8 @@ "content": "已知定点$A(1,0)$, 圆$\\omega:x^2+y^2=4$, $M,N$为$\\omega$上的动点, 满足$|MN|=2\\sqrt{3}$, 则$\\overrightarrow{AM}\\cdot \\overrightarrow{AN}$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-下学期测验卷-高三下学期月考01" ], "genre": "填空题", "ans": "$[-3,1]$", @@ -118250,7 +119648,8 @@ "content": "函数$f(x)=\\sin(2x+\\dfrac\\pi 4)$的图像关于\\bracket{20}对称.\n\\fourch{直线$x=\\dfrac\\pi 4$}{直线$x=\\dfrac{3\\pi}8$}{点$(\\dfrac\\pi 4,0)$}{点$(\\dfrac{3\\pi}8,0)$}", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期测验卷-高三下学期测验03" ], "genre": "选择题", "ans": "D", @@ -118289,7 +119688,8 @@ ], "tags": [ "第一单元", - "第六单元" + "第六单元", + "2023届高三-下学期测验卷-高三下学期测验03" ], "genre": "选择题", "ans": "C", @@ -118423,7 +119823,8 @@ "K0317002B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-13_解三角形" ], "genre": "解答题", "ans": "(1) $S=1200\\sqrt{3}\\sin(2\\theta+\\dfrac\\pi 6)-600\\sqrt{3}, \\ \\theta \\in (0,\\dfrac \\pi 3)$; (2) $S$的最大值为$600\\sqrt{3}$(平方米), 相应的$\\theta$的值为$\\dfrac{\\pi}6$.", @@ -118519,7 +119920,8 @@ "K0205002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期测验卷-高三下学期测验02" ], "genre": "填空题", "ans": "$4$", @@ -118564,7 +119966,9 @@ "K0104001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期测验卷-测验05", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$\\{-1,0\\}$", @@ -118604,7 +120008,8 @@ "content": "若复数$z_1=a+2\\mathrm{i}$, $z_2=2+\\mathrm{i}$($\\mathrm{i}$是虚数单位), 且$z_1z_2$为纯虚数, 则实数$a=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-下学期测验卷-高三下学期测验03" ], "genre": "填空题", "ans": "$1$", @@ -118745,7 +120150,8 @@ "K0219003B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期测验卷-高三下学期测验03" ], "genre": "填空题", "ans": "$[-\\dfrac 12,1]$", @@ -118794,7 +120200,8 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-第一轮复习讲义-26_大数定律及独立性" ], "genre": "填空题", "ans": "$\\dfrac 29$", @@ -118921,7 +120328,10 @@ "K0221002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验05", + "2023届高三-四月错题重做-02_函数二", + "2023届高三-四月错题重做-02_易错题-函数2" ], "genre": "填空题", "ans": "$[1-\\sqrt{2},0]$", @@ -118966,7 +120376,8 @@ "content": "已知函数$f(x)=|\\sin x|+|\\cos x|-4\\sin x\\cos x-k$, 若函数$f(x)$在区间$(0,\\pi)$内恰好有奇数个零点, 则实数$k$的所有取值之和为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-测验05" ], "genre": "填空题", "ans": "$1+2\\sqrt{2}$", @@ -119052,7 +120463,10 @@ "K0118002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-月考01", + "2023届高三-四月错题重做-01_函数一", + "2023届高三-四月错题重做-01_易错题-函数1" ], "genre": "选择题", "ans": "A", @@ -119124,7 +120538,8 @@ "content": "如图, 在四棱锥$M-ABCD$中, 已知$AM\\perp\\text{平面}ABCD$, $AB\\perp AD$, $AB\\parallel CD$, $AB=2CD$, 且$AB=AM=AD=2$.\n\\begin{center}\n \\begin{tikzpicture}\n \\draw [dashed] (0,0) node [left] {$A$} coordinate (A) -- (2,0) node [right] {$D$} coordinate (D) (A) -- (0,2) node [above] {$M$} coordinate (M) (A) -- (225:1) node [left] {$B$} coordinate (B);\n \\draw (D) --++ (225:0.5) node [right] {$C$} coordinate (C);\n \\draw (M) -- (B) -- (C) -- (D) -- cycle;\n \\draw (C) -- (M);\n \\end{tikzpicture}\n\\end{center}\n(1) 求四棱锥$M-ABCD$的体积;\\\\\n(2) 求直线$MC$与平面$ADM$所成的角.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期测验卷-高三下学期测验03" ], "genre": "解答题", "ans": "(1) $2$; (2) $\\arctan\\dfrac{\\sqrt{2}}4$", @@ -119189,7 +120604,8 @@ "K0221002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第一轮复习讲义-10_有关函数的应用问题" ], "genre": "解答题", "ans": "(1) $y=225x+\\dfrac{129600}x+2000, \\ x\\in (2,+\\infty)$; (2) 当且仅当$x=24$时, 修建此矩形场地围墙的总费用最少, 最少总费用为$12800$元.", @@ -119271,7 +120687,8 @@ "K0104004B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-01_集合" ], "genre": "填空题", "ans": "$(-1,2)$", @@ -119365,7 +120782,8 @@ "K0515005B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第一轮复习讲义-19_复数的几何意义与实系数二次方程" ], "genre": "填空题", "ans": "$2\\sqrt{3}$", @@ -119426,7 +120844,8 @@ ], "tags": [ "第七单元", - "双曲线" + "双曲线", + "2023届高三-下学期测验卷-高三下学期测验02" ], "genre": "填空题", "ans": "$3$", @@ -119517,7 +120936,8 @@ "content": "在$\\triangle ABC$中, 已知$AB=1$, $BC=2$, 若$y=\\cos^2 C-\\sin^2 C$, 则$y$的最小值是\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期测验卷-高三下学期测验03" ], "genre": "填空题", "ans": "$\\dfrac 12$", @@ -119584,7 +121004,8 @@ "objs": [], "tags": [ "第三单元", - "第八单元" + "第八单元", + "2023届高三-下学期测验卷-高三下学期测验02" ], "genre": "填空题", "ans": "$2^{2021}$", @@ -119747,7 +121168,8 @@ "content": "如图, 在直三棱柱$ABC-A_1B_1C_1$中, $BA\\perp BC$, $BA=BC=BB_1=2$.\n\\begin{center}\n \\begin{tikzpicture}\n \\draw [dashed] (0,0) node [left] {$B$} coordinate (B)-- (2,0) node [right] {$B_1$} coordinate (B1) (B) -- (225:1) node [left] {$A$} coordinate (A) (B) -- (0,2) node [above] {$C$} coordinate (C);\n \\draw (A) -- (C) --++ (2,0) node [above] {$C_1$} coordinate (C1) -- (B1) --++ (225:1) node [right] {$A_1$} coordinate (A1) -- (A) (A1) -- (C1) (A1) -- (C);\n \\draw [dashed] (A) -- (B1) (C) -- (B1);\n \\filldraw (0,1) circle (0.03) node [right] {$M$};\n \\end{tikzpicture}\n\\end{center}\n(1) 求异面直线$AB_1$与$A_1C_1$所成角的大小;\\\\\n(2) 若$M$是棱$BC$的中点.求点$M$到平面$A_1{B_1}C$的距离.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期周末卷-周末卷05" ], "genre": "解答题", "ans": "(1) $\\dfrac\\pi 3$; (2) $\\dfrac{\\sqrt{2}}2$", @@ -119778,7 +121200,8 @@ "K0317002B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-13_解三角形" ], "genre": "解答题", "ans": "(1) $\\pi\\text{km}$; (2) 约$1.39\\text{km}$.", @@ -119873,7 +121296,8 @@ "K0221002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验02" ], "genre": "解答题", "ans": "(1) $\\log_2 \\dfrac 43$; (2) $[3-\\sqrt{5},3+\\sqrt{5}]$; (3) $1$.", @@ -120005,7 +121429,8 @@ "K0405003X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-下学期测验卷-高三下学期测验02" ], "genre": "填空题", "ans": "$-\\dfrac 12$", @@ -120040,7 +121465,8 @@ "content": "一个圆锥的表面积为$\\pi$, 母线长为$\\dfrac 56$, 则其底面半径为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期测验卷-高三下学期测验02" ], "genre": "填空题", "ans": "$\\dfrac 23$", @@ -120207,7 +121633,8 @@ ], "tags": [ "第六单元", - "第七单元" + "第七单元", + "2023届高三-第一轮复习讲义-24_体积及表面积的计算" ], "genre": "填空题", "ans": "$\\dfrac{80\\pi}3$", @@ -120658,7 +122085,8 @@ ], "tags": [ "第七单元", - "直线" + "直线", + "2023届高三-第一轮复习讲义-34_直线及其方程" ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{34}}{85}$", @@ -120749,7 +122177,8 @@ "K0409001X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-下学期测验卷-高三下学期测验03" ], "genre": "填空题", "ans": "$4$", @@ -120787,7 +122216,8 @@ "K0215001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期测验卷-高三下学期月考01" ], "genre": "填空题", "ans": "$36$", @@ -120985,7 +122415,8 @@ "K0215005B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-月考01" ], "genre": "解答题", "ans": "(1) $0$; (2) $(-\\infty,-\\dfrac 12]\\cup [\\dfrac 72,+\\infty)$", @@ -121221,7 +122652,8 @@ "objs": [], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-下学期测验卷-高三下学期测验03" ], "genre": "填空题", "ans": "$\\dfrac{34}{35}$", @@ -121257,7 +122689,8 @@ "content": "在复平面内, 三点$A$、$B$、$C$分别对应复数$z_A$、$z_B$、$z_C$, 若$\\dfrac{z_B-z_A}{z_C-z_A}=1+\\dfrac 43\\mathrm{i}$, 则$\\triangle ABC$的三边长之比为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-下学期测验卷-高三下学期测验02" ], "genre": "填空题", "ans": "$3:4:5$(顺序不考虑)", @@ -121444,7 +122877,8 @@ "content": "已知函数$f(x)=\\cos (3x+\\varphi)$满足$f(x)\\le f(1)$恒成立, 则\\bracket{20}.\n\\twoch{函数$f(x-1)$一定是奇函数}{函数$f(x+1)$一定是奇函数}{函数$f(x-1)$一定是偶函数}{函数$f(x+1)$一定是偶函数}", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-测验04" ], "genre": "选择题", "ans": "D", @@ -121536,7 +122970,9 @@ ], "tags": [ "第六单元", - "空间向量" + "空间向量", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-33_立体几何中的定量计算" ], "genre": "解答题", "ans": "(1) $\\dfrac{\\sqrt{21}}7$; (2) $\\sqrt{2}$", @@ -121622,7 +123058,8 @@ "content": "已知曲线$\\Gamma:F(x,y)=0$, 对坐标平面上任意一点$P(x,y)$, 定义$F[P]=F(x,y)$. 若两点$P$、$Q$, 满足$F[P]\\cdot F[Q]>0$, 称点$P$、$Q$在曲线$\\Gamma$同侧; 若$F[P]\\cdot F[Q]<0$, 称点$P$、$Q$在曲线$\\Gamma$两侧.\\\\\n(1) 直线$l:kx-y=0$过原点, 线段$AB$上所有点都在直线$l$同侧, 其中$A(-1,1)$、$B(2,3)$, 求直线$l$的倾斜角的取值范围;\\\\\n(2)已知曲线$F(x,y)=(3x+4y-5)\\cdot \\sqrt{4-x^2-y^2}=0$, $O$为坐标原点, 求点集$S=\\{P|F[P]\\cdot F[O]>0\\}$的面积;\\\\\n(3)记到点$(0,1)$与到$x$轴距离和为$5$的点的轨迹为曲线$C$, 曲线$\\Gamma :F(x,y)=x^2+y^2-y-a=0$, 若曲线$C$上总存在两点$M$、$N$在曲线$\\Gamma$两侧, 求曲线$C$的方程与实数$a$的取值范围.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期测验卷-高三下学期测验03" ], "genre": "解答题", "ans": "(1) $[0,\\arctan\\dfrac 32)\\cup (\\dfrac{3\\pi}4,\\pi)$; (2) $\\dfrac 83\\pi+\\sqrt{3}$; (3) 曲线$C$的方程为$x^2=\\begin{cases} 24-8y, & y\\ge 0, \\\\ 24+12y, & y\\le 0,\\end{cases}$ $a$的取值范围为$(6,24)$", @@ -121662,7 +123099,8 @@ "K0221002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-月考01" ], "genre": "解答题", "ans": "(1) $[-2,+\\infty)$; (2) 具有性质$P$; (3) 证明略", @@ -121698,7 +123136,9 @@ "content": "已知$\\tan \\alpha =\\dfrac 12$, 则$\\tan 2\\alpha =$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-测验03", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$\\dfrac 43$", @@ -121846,7 +123286,10 @@ ], "tags": [ "第七单元", - "圆" + "圆", + "2023届高三-下学期测验卷-高三下学期月考01", + "2023届高三-下学期测验卷-高三下学期测验01", + "2023届高三-赋能-赋能40" ], "genre": "填空题", "ans": "$3$", @@ -121924,7 +123367,8 @@ ], "tags": [ "第七单元", - "直线" + "直线", + "2023届高三-下学期测验卷-高三下学期测验03" ], "genre": "填空题", "ans": "$3$", @@ -121987,7 +123431,8 @@ "K0225002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期测验卷-高三下学期月考01" ], "genre": "填空题", "ans": "$(-\\infty,-1]\\cup [0,+\\infty)$", @@ -122024,7 +123469,8 @@ "objs": [], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-上学期测验卷-测验05" ], "genre": "填空题", "ans": "$\\dfrac{29}{32}$", @@ -122237,7 +123683,8 @@ "K0221002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第一轮复习讲义-08_函数的单调性" ], "genre": "解答题", "ans": "(1) 在$(-\\infty,1]$上严格增, 在$[1,2]$上严格减, 在$[2,+\\infty)$上严格增; (2) $[\\dfrac 12,+\\infty)$", @@ -122362,7 +123809,8 @@ "K0215003B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期测验卷-高三下学期测验03" ], "genre": "填空题", "ans": "$(-3,1]$", @@ -122530,7 +123978,8 @@ "K0218001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-月考01" ], "genre": "填空题", "ans": "$-\\dfrac 12$", @@ -122590,7 +124039,8 @@ "content": "已知函数$f(x)=\\cos (2x-\\dfrac \\pi 6)$, 若对于任意的$x_1\\in [-\\dfrac \\pi 4,\\dfrac\\pi 4]$, 总存在$x_2\\in [m,n]$, 使得$f(x_1)+f(x_2)=0$, 则$|m-n|$的最小值为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-测验05" ], "genre": "填空题", "ans": "$\\dfrac\\pi 3$", @@ -122704,7 +124154,8 @@ "K0105001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-02_常用逻辑用语" ], "genre": "选择题", "ans": "A", @@ -122733,7 +124184,9 @@ "K0610004B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-21_空间直线与平面的位置关系" ], "genre": "选择题", "ans": "C", @@ -122990,7 +124443,8 @@ "K0104001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-下学期测验卷-高三下学期测验02" ], "genre": "填空题", "ans": "$(2,3)$", @@ -123548,7 +125002,9 @@ "K0117001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期测验卷-月考01", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$(-\\infty,0)\\cup (2,+\\infty)$", @@ -123596,7 +125052,8 @@ "K0218001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验05" ], "genre": "填空题", "ans": "$[-2,2]$", @@ -123805,7 +125262,10 @@ "K0220001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-月考01", + "2023届高三-四月错题重做-02_函数二", + "2023届高三-四月错题重做-02_易错题-函数2" ], "genre": "填空题", "ans": "$[0,1]$", @@ -123879,7 +125339,8 @@ "content": "设$\\varphi \\in (0,\\pi)$. 若存在实数$a,b$使得关于$x$的方程$a\\sin (2x+\\varphi)+b=0$在$[0,2\\pi]$时恰有$5$个解, 且解的和为$\\dfrac{63}{11}\\pi$, 则$\\varphi =$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-测验03" ], "genre": "填空题", "ans": "$\\dfrac{7\\pi}{11}$", @@ -123945,7 +125406,9 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-26_大数定律及独立性" ], "genre": "选择题", "ans": "D", @@ -124053,7 +125516,8 @@ "K0222002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-月考01" ], "genre": "解答题", "ans": "(1) 年存储成本费为$68000$元; (2) 每次订购$500$吨甲醇, 可使化工厂年存储成本费最少, 最少费用为$60000$元", @@ -124227,7 +125691,8 @@ "K0402001X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-下学期测验卷-高三下学期测验01" ], "genre": "填空题", "ans": "$150$", @@ -124446,7 +125911,8 @@ "tags": [ "第四单元", "第八单元", - "二项式定理" + "二项式定理", + "2023届高三-第一轮复习讲义-39_二项式定理" ], "genre": "填空题", "ans": "$(8,+\\infty)$", @@ -124612,7 +126078,9 @@ ], "tags": [ "第六单元", - "空间向量" + "空间向量", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-33_立体几何中的定量计算" ], "genre": "解答题", "ans": "(1) $\\arccos\\dfrac{\\sqrt{10}}{10}$; (2) $\\dfrac{2\\sqrt{33}}{11}$", @@ -124727,7 +126195,8 @@ "K0401002X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-下学期测验卷-高三下学期测验03" ], "genre": "解答题", "ans": "(1) $-5,-3,-1,1,3,5,7$; (2) $a_n=\\begin{cases}\\dfrac {n+1}2, & n=2k-1,\\\\ 1-\\dfrac n 2, & n = 2k \\end{cases}$($k$为正整数); (3) 证明略", @@ -124765,7 +126234,8 @@ "K0104006B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-下学期测验卷-高三下学期测验03" ], "genre": "填空题", "ans": "$[1,2)$", @@ -124857,7 +126327,8 @@ "K0216005B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-月考01" ], "genre": "填空题", "ans": "$(-\\infty,-1)\\cup (1,+\\infty)$", @@ -124949,7 +126420,10 @@ "K0221002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-月考01", + "2023届高三-四月错题重做-02_函数二", + "2023届高三-四月错题重做-02_易错题-函数2" ], "genre": "填空题", "ans": "$(-\\dfrac 12,4)$", @@ -125163,7 +126637,9 @@ "K0106001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期测验卷-测验05", + "2023届高三-寒假作业-容易题" ], "genre": "选择题", "ans": "B", @@ -125320,7 +126796,8 @@ "K0219003B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-月考01" ], "genre": "解答题", "ans": "(1) $(0,1]$; (2) $(0,81]$", @@ -125758,7 +127235,8 @@ "K0210005B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验02" ], "genre": "选择题", "ans": "C", @@ -126033,7 +127511,8 @@ "K0218001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期测验卷-高三下学期测验01" ], "genre": "填空题", "ans": "$(-\\infty,-1)\\cup (1,+\\infty)$", @@ -126186,7 +127665,8 @@ "K0111003B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期测验卷-月考01" ], "genre": "选择题", "ans": "D", @@ -126307,7 +127787,8 @@ "content": "如图所示, 在直三棱柱$ABC-A_1B_1C_1$中, 底面是等腰直角三角形, $\\angle ACB=90^\\circ$, $CA=CB=CC_1=2$. 点$D,D_1$分别是棱$AC,A_1C_1$的中点.\n\\begin{center}\n \\begin{tikzpicture}[scale = 1.2]\n \\draw [dashed] (0,0) node [left] {$C$} coordinate (C) -- (2,0) node [right] {$B$} coordinate (B) (C) -- (0,2) node [above] {$C_1$} coordinate (C1) (C) -- (225:1) node [left] {$A$} coordinate (A);\n \\draw (A) --++ (0,2) node [left] {$A_1$} coordinate (A1) (B) --++ (0,2) node [right] {$B_1$} coordinate (B1);\n \\draw (A) -- (B) (A1) -- (B1) -- (C1) -- cycle;\n \\draw ($(A)!0.5!(C)$) node [left] {$D$} coordinate (D) ++ (0,2) node [left] {$D_1$} coordinate (D1);\n \\draw (B1) -- (D1);\n \\draw [dashed] (B) -- (D) -- (D1) (C1) -- (B);\n \\end{tikzpicture}\n\\end{center}\n(1)\t求四棱锥$C-AA_1B_1B$的体积;\\\\\n(2)\t求直线$BC_1$与平面$DBB_1D_1$所成角的大小.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期测验卷-高三下学期测验01" ], "genre": "解答题", "ans": "(1) $\\dfrac 43$; (2) $\\arcsin \\dfrac{\\sqrt{10}}{10}$", @@ -126343,7 +127824,8 @@ "content": "设常数$k\\in \\mathbf{R}$, $f(x)=k\\cos^2x+\\sqrt 3\\sin x\\cos x$, $x\\in \\mathbf{R}$.\\\\\n(1) 若$\\tan \\alpha =2$且$f(\\alpha)=\\sqrt 3$, 求实数$k$的值;\\\\\n(2) 设$k=1$, $\\triangle ABC$中, 内角$A,B,C$的对边分别为$a,b,c$. 若$f(A)=1$, $a=\\sqrt 7$, $b=3$, 求$\\triangle ABC$的面积$S$.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期测验卷-高三下学期测验01" ], "genre": "解答题", "ans": "(1) $3\\sqrt{3}$; (2) $\\dfrac{3\\sqrt{3}}2$或$\\dfrac{3\\sqrt{3}}4$", @@ -126458,7 +127940,9 @@ "K0116001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-04_方程与不等式的求解" ], "genre": "填空题", "ans": "$(-\\infty,0)\\cup [\\dfrac 13,+\\infty)$", @@ -126543,7 +128027,8 @@ "K0215005B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验02" ], "genre": "填空题", "ans": "$(-\\infty,0]\\cup [4,+\\infty)$", @@ -126600,7 +128085,8 @@ "K0117001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期测验卷-测验04" ], "genre": "填空题", "ans": "$(3,+\\infty)$", @@ -126638,7 +128124,8 @@ "K0215005B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-下学期测验卷-高三下学期测验03" ], "genre": "填空题", "ans": "$1$", @@ -126778,7 +128265,9 @@ "content": "为了得到函数$y=\\sin (2x+\\dfrac{\\pi}3)$的图像, 可将函数$y=\\sin 2x$的图像\\bracket{20}.\n\\fourch{左移$\\dfrac{\\pi}3$个长度}{右移$\\dfrac{\\pi}3$个长度}{左移$\\dfrac{\\pi}6$个长度}{右移$\\dfrac{\\pi}6$个长度}", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-测验05", + "2023届高三-寒假作业-容易题" ], "genre": "选择题", "ans": "C", @@ -126895,7 +128384,8 @@ "KNONE" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验04" ], "genre": "解答题", "ans": "(1) $(3,4]$; (2) $(0,1]$上的值域为$\\{1\\}$, $(1,2]$上的值域为$\\{3,4\\}$, $(2,3]$上的值域为$\\{7,8,9\\}$, $(0,n]$上的值域中含有的元素个数为$\\dfrac{n(n+1)}2$; (3) $(3,+\\infty)$", @@ -127251,7 +128741,10 @@ ], "tags": [ "第一单元", - "第二单元" + "第二单元", + "2023届高三-下学期测验卷-高三下学期测验01", + "2023届高三-四月错题重做-01_函数一", + "2023届高三-四月错题重做-01_易错题-函数1" ], "genre": "填空题", "ans": "$(-\\infty,-\\dfrac 23]$", @@ -127325,7 +128818,8 @@ "content": "若空间中三条不同的直线$l_1$、$l_2$、$l_3$, 满足$l_1\\perp l_2$, $l_2\\parallel l_3$, 则下列结论一定正确的是\\bracket{20}.\n\\twoch{$l_1\\perp l_3$}{$l_1\\parallel l_3$}{$l_1$、$l_3$既不平行也不垂直}{$l_1$、$l_3$相交且垂直}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期测验卷-高三下学期测验01" ], "genre": "选择题", "ans": "A", @@ -127389,7 +128883,9 @@ "K0221002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-09_函数的零点与最值" ], "genre": "选择题", "ans": "C", @@ -127416,7 +128912,8 @@ "K0215005B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-月考01" ], "genre": "选择题", "ans": "C", @@ -127478,7 +128975,8 @@ "content": "已知函数$f(x)=\\dfrac 32\\sin \\omega x+\\dfrac{\\sqrt 3}2\\cos \\omega x$(其中$\\omega >0$).\\\\\n(1)\t若$\\omega =2$, $0<\\alpha <\\pi$, 且$f(\\alpha)=\\dfrac 32$, 求$\\alpha$的值;\\\\\n(2)\t若函数$f(x)$的最小正周期为$3\\pi$, 求$\\omega$的值, 并求函数$f(x)$在$[0,\\pi]$上的单调递增区间.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期测验卷-高三下学期测验03" ], "genre": "解答题", "ans": "(1) $\\dfrac{\\pi}{12}$或$\\dfrac\\pi 4$; (2) $\\omega = \\dfrac 23$, 单调递增区间为$[0,\\dfrac\\pi 2]$", @@ -127542,7 +129040,8 @@ "K0214002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-月考01" ], "genre": "解答题", "ans": "(1) $7$; (2) $\\dfrac{2\\sqrt{3}}3$; (3) $(0,\\dfrac 2{27}]$", @@ -127607,7 +129106,8 @@ "K0215003B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷07" ], "genre": "填空题", "ans": "$[-2,+\\infty)$", @@ -127648,7 +129148,9 @@ "KNONE" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷07", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$3$", @@ -127681,7 +129183,9 @@ "content": "在正方体$ABCD-A_1B_1C_1D_1$中, 直线$BC_1$与平面$BB_1D_1D$所成角的大小等于\\blank{50}.\n\\begin{center}\n \\begin{tikzpicture}\n \\draw (0,0) node [below left] {$A$} coordinate (A) --++ (2,0) node [below right] {$B$} coordinate (B) --++ (45:{2/2}) node [right] {$C$} coordinate (C)\n --++ (0,2) node [above right] {$C_1$} coordinate (C1)\n --++ (-2,0) node [above left] {$D_1$} coordinate (D1) --++ (225:{2/2}) node [left] {$A_1$} coordinate (A1) -- cycle;\n \\draw (A) ++ (2,2) node [right] {$B_1$} coordinate (B1) --++ (45:{2/2}) (B1) ++ (-2,0);\n \\draw [dashed] (A) --++ (45:{2/2}) node [left] {$D$} coordinate (D) ++ (2,0) (D) ++ (0,2);\n \\filldraw [gray!30] (B) -- (D) -- (D1) -- (B1) -- cycle;\n \\draw (B) -- (C1) (D1) -- (B1) -- (B) (B1) -- (A1);\n \\draw [dashed] (B) -- (D) -- (D1) (D) -- (C);\n \\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期周末卷-周末卷07", + "2023届高三-寒假作业-中档题" ], "genre": "填空题", "ans": "$\\dfrac\\pi 6$", @@ -127710,7 +129214,9 @@ "content": "已知角$\\alpha$的终边经过点$P(-1,2)$(始边为$x$轴正半轴), 则$\\sin 2\\alpha=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-周末卷07", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$-\\dfrac 45$", @@ -127742,7 +129248,8 @@ ], "tags": [ "第八单元", - "二项式定理" + "二项式定理", + "2023届高三-上学期周末卷-周末卷07" ], "genre": "填空题", "ans": "$252$", @@ -127776,7 +129283,9 @@ "K0111002B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期周末卷-周末卷07", + "2023届高三-上学期测验卷-月考01" ], "genre": "填空题", "ans": "$\\dfrac 18$", @@ -127845,7 +129354,9 @@ "tags": [ "第八单元", "组合", - "加法原理与乘法原理" + "加法原理与乘法原理", + "2023届高三-上学期周末卷-周末卷07", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$30$", @@ -127874,7 +129385,9 @@ "content": "已知圆锥的侧面展开图是一个扇形, 若此扇形的圆心角为$\\dfrac{6\\pi}5$, 面积为$15\\pi$, 则该圆锥的体积为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期周末卷-周末卷07", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$12\\pi$", @@ -127903,7 +129416,8 @@ "content": "在$\\triangle ABC$中, 内角$A,B,C$的对边分别为$a,b,c$, 若$b=2$, $\\dfrac{\\sin A}{a}=\\dfrac{\\sqrt{3}\\cos B}{b}$. 则$\\triangle ABC$的面积的最大值等于\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-周末卷07" ], "genre": "填空题", "ans": "$\\sqrt{3}$", @@ -127932,7 +129446,8 @@ "content": "在高中阶段, 我们学习过函数的概念、性质和图像, 以下两个结论是正确的: \\textcircled{1} 偶函数$f(x)$在区间$[a,b]$($a0$, $\\omega>0$, $0\\le \\varphi\\le \\pi$)一个周期内的图像. 将$f(x)$图像上所有点的横坐标伸长为原来的$2$倍, 纵坐标不变, 再把所得图像向右平移$\\dfrac\\pi 2$个单位长度, 得到函数$g(x)$的图像.\n\\begin{center}\n \\begin{tikzpicture}[>=latex]\n \\draw [->] (-1,0) -- (3,0) node [below] {$x$};\n \\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\draw [domain = -45:135, samples = 200] plot ({\\x/180*pi},{2*sin(2*\\x+90)});\n \\filldraw ({-pi/4},0) circle (0.03) node [above left] {$-\\frac\\pi 4$} ({pi/4},0) circle (0.03) node [above right] {$\\frac\\pi 4$};\n \\draw (0,2) node [above left] {$2$};\n \\draw [dashed] (0,-2) node [left] {$-2$} -- ({pi/2},-2) -- ({pi/2},0);\n \\end{tikzpicture}\n\\end{center}\n(1) 求函数$f(x)$和$g(x)$的解析式;\\\\\n(2) 若$f(x_0)=g(x_0)$, 求$\\sin (x_0-\\dfrac\\pi 3)$的所有可能的值;\\\\\n(3) 求函数$F(x)=f(x)+ag(x)$($a$为正常数)在区间$(0,19\\pi)$内的所有零点之和.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-测验05" ], "genre": "解答题", "ans": "(1) $f(x)=2\\sin(2x+\\dfrac\\pi 2)$, $g(x)=2\\sin x$; (2) $-\\dfrac 12$或$1$; (3) 当$a>1$时, 和为$171\\pi$, 当$a=1$时, 和为$266\\pi$, 当$0f(\\text{t})>f(\\text{i})>f(\\text{a})$; (2) $\\begin{pmatrix} 2 & 3 & 4 & 5 \\\\ \\dfrac 29 & \\dfrac 49 & \\dfrac 29 & \\dfrac 19 \\end{pmatrix}$; (3) $\\dfrac 5{18}$.", @@ -132538,7 +134131,8 @@ "objs": [], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-暑假概率初步续-04期望" ], "genre": "填空题", "ans": "$-\\dfrac 13$, $\\dfrac{13}6$", @@ -132575,7 +134169,8 @@ "objs": [], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-暑假概率初步续-04期望" ], "genre": "解答题", "ans": "(1) $26.5$; (2) 分布列为$\\begin{pmatrix} 0 & 1 & 2 & 3 & 4 \\\\ \\dfrac 1{16} & \\dfrac 4{16} & \\dfrac 6{16} & \\dfrac 4{16} & \\dfrac 1{16}\\end{pmatrix}$, 期望为$2$", @@ -132610,7 +134205,8 @@ "objs": [], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-暑假概率初步续-04期望" ], "genre": "解答题", "ans": "(1) 调整前的平均利润为$5000$元每天, 调整后的平均利润为$15000$元每天, 因此调整后的平均利润比调整前更多; (2) 应定价为每张$13$元", @@ -132645,7 +134241,8 @@ "objs": [], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-暑假概率初步续-04期望" ], "genre": "解答题", "ans": "(1) $\\begin{pmatrix} 0 & 1 & 2 & 3 & 4 & 5 & 6\\\\ 0.01 & 0.04 & 0.12 & 0.22 & 0.28 & 0.24 & 0.09 \\end{pmatrix}$; (2) 方案一所需费用的期望为$10720$元, 方案二所需费用的期望为$10420$元, 因此选择第二种延保方案更合算", @@ -132680,7 +134277,8 @@ "objs": [], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-暑假概率初步续-05方差" ], "genre": "填空题", "ans": "$\\dfrac{5}{9}$, $\\dfrac{5}{36}$", @@ -132716,7 +134314,8 @@ "objs": [], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-暑假概率初步续-05方差" ], "genre": "解答题", "ans": "(1) $\\dfrac 35$; (2) 分布列为$\\begin{pmatrix} 0 & 1 & 2 \\\\ \\dfrac 25 & \\dfrac 25 & \\dfrac 15\\end{pmatrix}$, $E[X]=\\dfrac 45$, $D[X]=\\dfrac{14}{25}$", @@ -132750,7 +134349,8 @@ "objs": [], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-暑假概率初步续-05方差" ], "genre": "解答题", "ans": "(1) $\\dfrac 5{12}$; (2) 分布列为$\\begin{pmatrix} 0 & 40 & 80 & 120 & 160 \\\\ \\dfrac 1{24} & \\dfrac 14 & \\dfrac 5{12} & \\dfrac 14 & \\dfrac 1{24}\\end{pmatrix}$, $E[X]=80$, $D[X]=\\dfrac{4000}3$", @@ -132784,7 +134384,8 @@ "objs": [], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-暑假概率初步续-06二项分布" ], "genre": "填空题", "ans": "$\\dfrac 2{27}$", @@ -132809,7 +134410,8 @@ "objs": [], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-暑假概率初步续-06二项分布" ], "genre": "填空题", "ans": "$6$", @@ -132834,7 +134436,8 @@ "objs": [], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-暑假概率初步续-06二项分布" ], "genre": "填空题", "ans": "$\\dfrac{20}{243}$", @@ -132859,7 +134462,8 @@ "objs": [], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-暑假概率初步续-06二项分布" ], "genre": "解答题", "ans": "(1) 分布列为$\\begin{pmatrix} 0 & 1 & 2 & 3 \\\\ 0.2p^2-0.4p+0.2 & 0.4p^2-1.2p+0.8 & -1.4p^2+1.6p & 0.8p^2\\end{pmatrix}$, $E[X]=2p+0.8$; (2) $0.96$, $700$棵", @@ -132884,7 +134488,8 @@ "objs": [], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-暑假概率初步续-06二项分布" ], "genre": "解答题", "ans": "(1) $\\begin{pmatrix} 0 & 1 & 2 & 3 \\\\ \\dfrac{729}{1000} & \\dfrac{243}{1000} & \\dfrac{27}{1000} & \\dfrac 1{1000}\\end{pmatrix}$; (2) 一轮游戏获得的分数$Y$的期望$E[Y]=-1.69<0$, 所以许多人的分数没有增加反而减少了", @@ -132909,7 +134514,8 @@ "objs": [], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-暑假概率初步续-06二项分布" ], "genre": "解答题", "ans": "(1) 当$n=5$或$6$时, 有$3$个坑需要补种的概率最大, 最大概率为$\\dfrac 5{16}$; (2) 分布列为$\\begin{pmatrix}0 & 1 & 2 & 3 & 4 \\\\ \\dfrac 1{16} & \\dfrac 14 & \\dfrac 38 & \\dfrac 14 & \\dfrac 1{16}\\end{pmatrix}$, $E[X]=2$", @@ -132934,7 +134540,8 @@ "objs": [], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-暑假概率初步续-06二项分布" ], "genre": "解答题", "ans": "", @@ -132957,7 +134564,8 @@ "objs": [], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-暑假概率初步续-06二项分布" ], "genre": "解答题", "ans": "", @@ -132980,7 +134588,8 @@ "objs": [], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-暑假概率初步续-06二项分布" ], "genre": "解答题", "ans": "(1) $20p$; (2) $3.2p-1.2$; (3) 当$p\\in (0,\\dfrac 34]$时, 应选择第一个项目(期望更高, 或者期望相同的情况下方差更低), 当$p\\in (\\dfrac 34,1)$时, 应选择第二个项目", @@ -133007,7 +134616,8 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-暑假概率初步续-07超几何分布" ], "genre": "填空题", "ans": "$\\dfrac {43}{138}$", @@ -133032,7 +134642,8 @@ "objs": [], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-暑假概率初步续-07超几何分布" ], "genre": "填空题", "ans": "$\\dfrac{56}{165}$", @@ -133057,7 +134668,8 @@ "objs": [], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-暑假概率初步续-07超几何分布" ], "genre": "填空题", "ans": "$0.042$", @@ -133082,7 +134694,8 @@ "objs": [], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-暑假概率初步续-07超几何分布" ], "genre": "解答题", "ans": "(1) $0.191$; (2) $\\dfrac 53$", @@ -133107,7 +134720,8 @@ "objs": [], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-暑假概率初步续-07超几何分布" ], "genre": "解答题", "ans": "(1) $48$; (2) 分布列为$\\begin{pmatrix}0 & 1 & 2 \\\\ \\dfrac{12}{19} & \\dfrac{32}{95} & \\dfrac 3{95}\\end{pmatrix}$, $E[X]=\\dfrac 25$; (3) $S=0.012<0.05$, 故本次测试对难度的预估是合理的", @@ -133132,7 +134746,8 @@ "objs": [], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-暑假概率初步续-07超几何分布" ], "genre": "解答题", "ans": "(1) $(a,b,c)=(9,6,6)$; (2) $\\begin{pmatrix}0 & 1 & 2 \\\\ \\dfrac 17 & \\dfrac 47 & \\dfrac 27\\end{pmatrix}$", @@ -133157,7 +134772,8 @@ "objs": [], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-暑假概率初步续-07超几何分布" ], "genre": "解答题", "ans": "(1) 约$400$名; (2) $0.49$; (3) 分布列为$\\begin{pmatrix}0 & 1 & 2 & 3 \\\\ \\dfrac 1{20} & \\dfrac 9{20} & \\dfrac 9{20} & \\dfrac 1{20}\\end{pmatrix}$, $E[X]=\\dfrac 32$", @@ -133265,7 +134881,8 @@ "KNONE" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验02" ], "genre": "填空题", "ans": "$-x^2+x$", @@ -133648,7 +135265,9 @@ ], "tags": [ "第一单元", - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验04", + "2023届高三-寒假作业-中档题" ], "genre": "解答题", "ans": "(1) 当$a>2$时, 解集为$(-\\infty,-2)\\cup (0,+\\infty)$; 当$a=2$时, 解集为$\\varnothing$; 当$a<2$时, 解集为$(-2,0)$; (2) $(-\\infty,\\dfrac {34}{15})$", @@ -133827,7 +135446,8 @@ "K0307003B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-11_三角比的定义及直接性质" ], "genre": "填空题", "ans": "$-\\dfrac 45$", @@ -133975,7 +135595,9 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-25_概率的概念及性质" ], "genre": "填空题", "ans": "$\\dfrac 34$", @@ -134009,7 +135631,8 @@ ], "tags": [ "第七单元", - "圆" + "圆", + "2023届高三-第一轮复习讲义-35_圆及曲线方程" ], "genre": "填空题", "ans": "$\\dfrac 43$", @@ -134273,7 +135896,8 @@ "content": "在$\\triangle ABC$中, $a$, $b$, $c$分别是角$A$, $B$, $C$的对边, 且$8\\sin^2\\dfrac{B+C}2-2\\cos 2A=7$.\\\\\n(1) 求角$A$的大小;\\\\\n(2) 若$a=\\sqrt 3$, $b+c=3$, 求$b$和$c$的值.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期测验卷-高三下学期测验02" ], "genre": "解答题", "ans": "(1) $A=\\dfrac\\pi 3$; (2)$(b,c)=(1,2)$或$(2,1)$", @@ -134791,7 +136415,8 @@ "K0315003B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-13_解三角形" ], "genre": "填空题", "ans": "$\\dfrac{3\\sqrt{5}}5+3$", @@ -135043,7 +136668,8 @@ "K0221002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验02" ], "genre": "解答题", "ans": "(1) $-1$; (2) $1$; (3) $[8,9)$.", @@ -135471,7 +137097,8 @@ "K0611002B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-21_空间直线与平面的位置关系" ], "genre": "选择题", "ans": "C", @@ -135532,7 +137159,8 @@ ], "tags": [ "第六单元", - "空间向量" + "空间向量", + "2023届高三-第一轮复习讲义-33_立体几何中的定量计算" ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\arcsin \\dfrac{4\\sqrt{5}}{25}$", @@ -135648,7 +137276,8 @@ "K0319003B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷04" ], "genre": "解答题", "ans": "(1) 具有, 不具有; (2) 证明略; (3) $1+2\\sqrt{2}$", @@ -135743,7 +137372,8 @@ ], "tags": [ "第七单元", - "直线" + "直线", + "2023届高三-赋能-赋能35" ], "genre": "填空题", "ans": "$6$", @@ -136062,7 +137692,8 @@ ], "tags": [ "第七单元", - "椭圆" + "椭圆", + "2023届高三-第一轮复习讲义-36_椭圆与双曲线的概念及性质" ], "genre": "选择题", "ans": "B", @@ -136150,7 +137781,8 @@ "K0221002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第一轮复习讲义-09_函数的零点与最值" ], "genre": "解答题", "ans": "(1) $-1$; (2) $(-\\infty,-4)\\cup \\{-2\\sqrt{3}\\}$", @@ -136179,7 +137811,8 @@ "K0221002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期测验卷-高三下学期测验03" ], "genre": "解答题", "ans": "(1) $8\\sqrt{3}-8$; (2) 当$A$在弧$\\overset\\frown{MN}$的四等分点(更靠近$M$)处时, 矩形$ABCD$的面积最大, 最大面积为$16\\sqrt{2}-16$", @@ -136300,7 +137933,8 @@ "content": "复数$z=2-\\mathrm{i}$, 则$|z|=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-赋能-赋能37" ], "genre": "填空题", "ans": "$\\sqrt{5}$", @@ -136471,7 +138105,8 @@ ], "tags": [ "第二单元", - "第八单元" + "第八单元", + "2023届高三-赋能-赋能36" ], "genre": "填空题", "ans": "$\\dfrac{3}{8}$", @@ -137097,7 +138732,8 @@ "content": "如图所示在$\\triangle ABC$中, $BC$边上的中垂线分别交$BC$、$AC$于点$D$、$E$, 若$\\overrightarrow{AE}\\cdot \\overrightarrow{BC}=6$, $|\\overrightarrow{AB}|=2$, 则$|\\overrightarrow{AC}|=$\\blank{50}.\n\\begin{center}\n \\begin{tikzpicture}\n \\draw (0,0) ++ ({-sqrt(2)},0) node [left] {$B$} coordinate (B);\n \\draw (0,0) ++ ({sqrt(14)},0) node [right] {$C$} coordinate (C);\n \\draw (0,{sqrt(2)}) node [above] {$A$} coordinate (A);\n \\draw (A) -- (B) -- (C) -- cycle;\n \\path [name path = AC] (A) -- (C);\n \\path [name path = perp] ($(B)!0.5!(C)$) node [below] {$D$} coordinate (D) --++ (0,{sqrt(2)});\n \\path [name intersections = {of = AC and perp, by = E}];\n \\draw (E) node [above] {$E$} -- (D);\n \\draw (D) ++ (0,0.2) --++ (0.2,0) --++ (0,-0.2);\n \\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-赋能-赋能37" ], "genre": "填空题", "ans": "$4$", @@ -137331,7 +138967,8 @@ "K0222002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验05" ], "genre": "解答题", "ans": "(1) $y=(180k-20)-x-\\dfrac{360k}{x+4}, \\ x\\in [0,10]$; (2) $[\\dfrac{35}{54},1]$", @@ -137473,7 +139110,8 @@ "K0104007B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-01_集合" ], "genre": "解答题", "ans": "$A=\\{2,5,13,17,23\\}$, $B=\\{2,11,17,19,29\\}$", @@ -137533,7 +139171,9 @@ "K0221002B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-01_集合", + "2023届高三-第一轮复习讲义-09_函数的零点与最值" ], "genre": "解答题", "ans": "$(-1,\\dfrac{18}7]$", @@ -137808,7 +139448,8 @@ "K0102001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-01_集合" ], "genre": "解答题", "ans": "$\\{-7,-1,1,2,3,4\\}$", @@ -138132,7 +139773,8 @@ "K0101001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-01_集合" ], "genre": "选择题", "ans": "C", @@ -138524,7 +140166,8 @@ "K0104001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期测验卷-测验01" ], "genre": "填空题", "ans": "$[0,1]$", @@ -139449,7 +141092,8 @@ "K0105002B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-02_常用逻辑用语" ], "genre": "解答题", "ans": "(1) 不正确, 如$x=1$, $y=1$; (2) 正确, 用反证法, 假设结论不成立, 即$x=y$且$x=-y$, 则$x=y$, 故$x^2=y^2$.", @@ -139614,7 +141258,8 @@ "K0107001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-02_常用逻辑用语" ], "genre": "选择题", "ans": "B", @@ -139849,7 +141494,8 @@ "K0107003B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-02_常用逻辑用语" ], "genre": "解答题", "ans": "证明略", @@ -140083,7 +141729,8 @@ "K0106003B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-02_常用逻辑用语" ], "genre": "解答题", "ans": "证明略", @@ -140136,7 +141783,8 @@ "K0107003B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-02_常用逻辑用语" ], "genre": "选择题", "ans": "B", @@ -140167,7 +141815,8 @@ "K0107003B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-02_常用逻辑用语" ], "genre": "选择题", "ans": "B", @@ -140219,7 +141868,8 @@ "K0107001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-02_常用逻辑用语" ], "genre": "选择题", "ans": "D", @@ -140429,7 +142079,8 @@ "K0106003B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-02_常用逻辑用语" ], "genre": "解答题", "ans": "(1) $p$是$q$的充分非必要条件; (2) $p$是$q$的必要非充分条件; (3) $p$是$q$的充要条件; (4) $p$是$q$的既非充分又非必要条件", @@ -140630,7 +142281,8 @@ "K0106003B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-02_常用逻辑用语" ], "genre": "解答题", "ans": "证明略", @@ -141488,7 +143140,8 @@ "K0114001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-04_方程与不等式的求解" ], "genre": "填空题", "ans": "(1) $(-\\infty,1)\\cup (1,+\\infty)$; (2) $(-\\dfrac 13,2)$; (3) $(-1,\\dfrac 13)$; (4) $[-1-\\sqrt{2},-1+\\sqrt{2}]$; (5) $[0,4)$.", @@ -141515,7 +143168,8 @@ "K0117002B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-04_方程与不等式的求解" ], "genre": "填空题", "ans": "(1) $(-\\infty,-\\dfrac 43]\\cup (2,+\\infty)$; (2) $(-\\dfrac 13,2)$; (3) $(-\\infty,-1)\\cup (0,1)$; (4) $(-\\infty,-3)\\cup (3,+\\infty)$; (5) $(-\\infty,-3)\\cup (4,+\\infty)$.", @@ -142000,7 +143654,8 @@ "K0109003B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期测验卷-测验01" ], "genre": "填空题", "ans": "$10+4\\sqrt{6}$", @@ -143128,7 +144783,8 @@ ], "tags": [ "第一单元", - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-24_体积及表面积的计算" ], "genre": "解答题", "ans": "$3\\sqrt[3]{2\\pi V^2}$", @@ -143607,7 +145263,8 @@ ], "tags": [ "第一单元", - "第二单元" + "第二单元", + "2023届高三-第一轮复习讲义-05_幂指数与对数" ], "genre": "填空题", "ans": "$2$", @@ -143688,7 +145345,9 @@ ], "tags": [ "第一单元", - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-05_幂指数与对数" ], "genre": "选择题", "ans": "B", @@ -145437,7 +147096,8 @@ ], "tags": [ "第二单元", - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-12_和差倍角公式" ], "genre": "解答题", "ans": "证明略", @@ -146374,7 +148034,11 @@ "K0206002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-四月错题重做-01_函数一", + "2023届高三-四月错题重做-01_易错题-函数1", + "2023届高三-寒假作业-较难题", + "2023届高三-第一轮复习讲义-05_幂指数与对数" ], "genre": "解答题", "ans": "$\\sqrt{\\log_2 10}$", @@ -147062,7 +148726,9 @@ "K0116001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-04_方程与不等式的求解" ], "genre": "选择题", "ans": "B", @@ -148226,7 +149892,8 @@ "K0214002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验02" ], "genre": "选择题", "ans": "D", @@ -148298,7 +149965,9 @@ "K0214002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-06_幂指对函数" ], "genre": "解答题", "ans": "$(\\dfrac 23,\\dfrac 32)$", @@ -148963,7 +150632,9 @@ "K0120001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-03_等式与不等式的性质及证明" ], "genre": "选择题", "ans": "C", @@ -149089,7 +150760,8 @@ "K0117002B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期测验卷-测验01" ], "genre": "解答题", "ans": "$(2,+\\infty)$", @@ -149256,7 +150928,9 @@ "K0223005B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-09_函数的零点与最值" ], "genre": "解答题", "ans": "$(-\\infty,\\dfrac 52)$", @@ -149333,7 +151007,9 @@ "K0120003B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-03_等式与不等式的性质及证明" ], "genre": "解答题", "ans": "$(1,+\\infty)$", @@ -150853,7 +152529,9 @@ "K0215004B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-07_函数的概念与奇偶性" ], "genre": "选择题", "ans": "B", @@ -150952,7 +152630,9 @@ "K0215003B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验07", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$[-1,1]$", @@ -151900,7 +153580,8 @@ "K0709001X" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第一轮复习讲义-35_圆及曲线方程" ], "genre": "解答题", "ans": "$x$的范围为$[0,2]$, $x^2+y^2$的范围为$[0,4]$", @@ -152391,7 +154072,8 @@ "K0221002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第一轮复习讲义-10_有关函数的应用问题" ], "genre": "解答题", "ans": "$y=a(\\dfrac{4900}v+0.01v), \\ v\\in (0,+\\infty)$; 总耗费最小时, 飞行速度为$700$千米$/$时.", @@ -155080,7 +156762,9 @@ "K0208003B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-06_幂指对函数" ], "genre": "填空题", "ans": "$(1,1)$, $pq=1$", @@ -155106,7 +156790,9 @@ "K0208004B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-06_幂指对函数" ], "genre": "解答题", "ans": "$(-\\infty,0)$", @@ -156193,7 +157879,9 @@ "K0217001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-07_函数的概念与奇偶性" ], "genre": "选择题", "ans": "C", @@ -157663,7 +159351,9 @@ "K0210006B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-06_幂指对函数" ], "genre": "选择题", "ans": "C", @@ -157690,7 +159380,9 @@ "K0210005B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-06_幂指对函数" ], "genre": "选择题", "ans": "C", @@ -157813,7 +159505,8 @@ "K0210005B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验02" ], "genre": "选择题", "ans": "D", @@ -158064,7 +159757,8 @@ "K0220002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验02" ], "genre": "填空题", "ans": "$(-\\infty,-1)\\cup (1,+\\infty)$", @@ -158264,7 +159958,9 @@ "K0210006B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-较难题", + "2023届高三-第一轮复习讲义-06_幂指对函数" ], "genre": "填空题", "ans": "$a=latex]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -2.7:2.7] plot (\\x,{pow(\\x,2)-2*abs(\\x)});\n\\end{tikzpicture}\n\\end{center}", @@ -215129,7 +216953,11 @@ "K0219001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-四月错题重做-01_函数一", + "2023届高三-四月错题重做-01_易错题-函数1", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-08_函数的单调性" ], "genre": "解答题", "ans": "$(0,1)$", @@ -215196,7 +217024,9 @@ "K0219001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-08_函数的单调性" ], "genre": "解答题", "ans": "如$y=1-4|x|$等", @@ -215423,7 +217253,9 @@ "K0219001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-08_函数的单调性" ], "genre": "解答题", "ans": "存在, 如$a=0$(小于$\\dfrac 12$的整数都可以)", @@ -215647,7 +217479,9 @@ "K0211001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验01", + "2023届高三-寒假作业-容易题" ], "genre": "选择题", "ans": "A", @@ -216289,7 +218123,8 @@ "K0215001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第一轮复习讲义-07_函数的概念与奇偶性" ], "genre": "选择题", "ans": "C", @@ -216475,7 +218310,9 @@ "K0115001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-较难题", + "2023届高三-第一轮复习讲义-04_方程与不等式的求解" ], "genre": "解答题", "ans": "$(-\\dfrac 89,0]$", @@ -216681,7 +218518,8 @@ "K0215001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第一轮复习讲义-10_有关函数的应用问题" ], "genre": "解答题", "ans": "(1) $S=40+8t-32\\sqrt{t}, \\ t\\in [0,+\\infty)$; (2) $2.25$小时后开始出现供水紧张, 这一天内供水紧张的有$4$小时.", @@ -218735,7 +220573,8 @@ "K0226004B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-赋能-赋能06" ], "genre": "填空题", "ans": "$(-8,57)$", @@ -219273,7 +221112,8 @@ "K0301002B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-月考01" ], "genre": "选择题", "ans": "C", @@ -219404,7 +221244,8 @@ "K0302002B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-11_三角比的定义及直接性质" ], "genre": "解答题", "ans": "$S=\\dfrac{5\\pi}{3}$, $\\alpha=\\dfrac{5\\pi}6$", @@ -219444,7 +221285,8 @@ "K0302001B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-11_三角比的定义及直接性质" ], "genre": "解答题", "ans": "角度制: $\\{\\alpha|\\alpha=k\\cdot 90^\\circ+45^\\circ, \\ k\\in \\mathbf{Z}\\}$, 弧度制: $\\{\\alpha|\\alpha=\\dfrac{k\\pi}2+\\dfrac\\pi 4, \\ k\\in \\mathbf{Z}\\}$", @@ -220323,7 +222165,8 @@ "K0304003B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-11_三角比的定义及直接性质" ], "genre": "解答题", "ans": "$(\\sin\\alpha,\\cos\\alpha,\\tan\\alpha)=(\\dfrac{2\\sqrt{5}}5,-\\dfrac{\\sqrt{5}}5,-2)$或$(-\\dfrac{2\\sqrt{5}}5,\\dfrac{\\sqrt{5}}5,-2)$", @@ -220894,7 +222737,8 @@ "K0313002B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-12_和差倍角公式" ], "genre": "解答题", "ans": "证明略", @@ -220955,7 +222799,8 @@ "K0311002B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-12_和差倍角公式" ], "genre": "解答题", "ans": "$13\\sin(\\alpha-\\arcsin\\dfrac{12}{13})$", @@ -221710,7 +223555,8 @@ "K0316001B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-13_解三角形" ], "genre": "解答题", "ans": "证明略", @@ -222083,7 +223929,8 @@ "K0301001B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-11_三角比的定义及直接性质" ], "genre": "选择题", "ans": "C", @@ -222358,7 +224205,8 @@ "K0305001B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-11_三角比的定义及直接性质" ], "genre": "解答题", "ans": "$\\sin\\alpha=\\dfrac 13+\\dfrac{\\sqrt{14}}6$, $\\cos\\alpha=\\dfrac 13-\\dfrac{\\sqrt{14}}6$", @@ -223810,7 +225658,8 @@ "K0323003B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-14_正弦函数及正弦型函数" ], "genre": "解答题", "ans": "\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-1,0) -- (15,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0:720] plot ({\\x/180*pi},{2*sin(\\x/2)});\n\\draw [dashed] (pi,0) node [below] {$\\pi$} -- (pi,2) -- (0,2) node [left] {$2$};\n\\draw [dashed] ({3*pi},0) node [above] {$3\\pi$} -- ({3*pi},-2) -- (0,-2) node [left] {$-2$};\n\\draw ({4*pi},0) node [above] {$4\\pi$} ({2*pi},0) node [below left] {$2\\pi$};\n\\end{tikzpicture}", @@ -224421,7 +226270,8 @@ "K0308002B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-11_三角比的定义及直接性质" ], "genre": "填空题", "ans": "$\\{x|x=-\\dfrac \\pi 4+k\\pi, \\ k\\in \\mathbf{Z}\\}$", @@ -224929,7 +226779,8 @@ "content": "函数$y=2\\cos ^2(2x+\\dfrac{\\pi}3)$的最小正周期是\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-赋能-赋能06" ], "genre": "填空题", "ans": "$\\dfrac\\pi 2$", @@ -225018,7 +226869,8 @@ "K0323003B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-14_正弦函数及正弦型函数" ], "genre": "解答题", "ans": "\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,3.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -pi:pi,samples = 200] plot (\\x,{2*sin(\\x/pi*180-45)+1});\n\\draw [dashed] (0,-1) -- (-pi/4,-1) -- (-pi/4,0) (-pi,0) -- (-pi,{1+sqrt(2)}) (3*pi/4,0) -- (3*pi/4,3) -- (0,3) (pi,0) -- (pi,{1+sqrt(2)});\n\\draw (0,-1) node [right] {$-1$} (-pi/4,0) node [above] {$-\\dfrac \\pi 4$} (-pi,0) node [below] {$-\\pi$} (pi,0) node [below] {$\\pi$} (3*pi/4,0) node [below] {$\\dfrac {3\\pi}4$} (0,3) node [left] {$3$};\n\\end{tikzpicture}\n振幅: $2$; 周期: $2\\pi$; 频率: $\\dfrac 1{2\\pi}$; 初相: $-\\dfrac\\pi 4$; 单调区间: $[-\\dfrac \\pi 4,\\dfrac{3\\pi}{4}]$(严格增), $[-\\pi,-\\dfrac \\pi 4]$(严格减), $[\\dfrac{3\\pi}4,\\pi]$(严格减); 值域: $[-1,3]$", @@ -225451,7 +227303,8 @@ "content": "函数$y=1+\\sin x, \\ x\\in [-\\pi ,2\\pi]$的图像与直线$y=\\dfrac 32$的交点个数是\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-赋能-赋能06" ], "genre": "填空题", "ans": "$2$", @@ -226331,7 +228184,9 @@ "K0213008B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-08_函数的单调性" ], "genre": "填空题", "ans": "$(0,\\dfrac 12)\\cup (2,+\\infty)$", @@ -226951,7 +228806,8 @@ "K0401006X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第一轮复习讲义-30_等差数列与等比数列" ], "genre": "解答题", "ans": "证明略", @@ -227967,7 +229823,8 @@ "K0403005X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第一轮复习讲义-30_等差数列与等比数列" ], "genre": "解答题", "ans": "第$6$年底浮萍面积最大", @@ -235261,7 +237118,8 @@ ], "tags": [ "第七单元", - "直线" + "直线", + "2023届高三-第一轮复习讲义-34_直线及其方程" ], "genre": "解答题", "ans": "$l_{BC}:2x+y-5=0$, $l_{AC}:4x-5y+13=0$", @@ -235351,7 +237209,8 @@ ], "tags": [ "第七单元", - "直线" + "直线", + "2023届高三-第一轮复习讲义-34_直线及其方程" ], "genre": "解答题", "ans": "$y=\\dfrac 12 x+\\dfrac 52$或$y=\\dfrac 23 x+2$", @@ -235394,7 +237253,8 @@ ], "tags": [ "第七单元", - "直线" + "直线", + "2023届高三-第一轮复习讲义-34_直线及其方程" ], "genre": "解答题", "ans": "$a=7$", @@ -236551,7 +238411,8 @@ ], "tags": [ "第七单元", - "直线" + "直线", + "2023届高三-第一轮复习讲义-34_直线及其方程" ], "genre": "解答题", "ans": "$y=-\\dfrac 56 x$或$y=-\\dfrac{11}2 x$", @@ -236720,7 +238581,9 @@ ], "tags": [ "第七单元", - "直线" + "直线", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-34_直线及其方程" ], "genre": "解答题", "ans": "$19x-4y-22=0$", @@ -237230,7 +239093,8 @@ ], "tags": [ "第七单元", - "直线" + "直线", + "2023届高三-第一轮复习讲义-34_直线及其方程" ], "genre": "填空题", "ans": "$[0,\\dfrac\\pi 4]\\cup [\\dfrac{3\\pi}4,\\pi)$", @@ -237268,7 +239132,8 @@ ], "tags": [ "第七单元", - "直线" + "直线", + "2023届高三-第一轮复习讲义-34_直线及其方程" ], "genre": "填空题", "ans": "$x+2y+2=0$", @@ -237306,7 +239171,8 @@ ], "tags": [ "第七单元", - "直线" + "直线", + "2023届高三-第一轮复习讲义-34_直线及其方程" ], "genre": "解答题", "ans": "$3x+y-13=0$", @@ -237849,7 +239715,8 @@ ], "tags": [ "第七单元", - "圆" + "圆", + "2023届高三-第一轮复习讲义-35_圆及曲线方程" ], "genre": "解答题", "ans": "(1) $(x+\\dfrac 32)^2+(y-3)^2=3$; (2) $(x-\\sqrt{2})^2+(y-1)^2=6$; (3) $(x-3)^2+(y-1)^2=5$或$(x-3)^2+(y+1)^2=5$", @@ -237916,7 +239783,8 @@ ], "tags": [ "第七单元", - "圆" + "圆", + "2023届高三-第一轮复习讲义-35_圆及曲线方程" ], "genre": "解答题", "ans": "$4x-3y-35=0$", @@ -238116,7 +239984,9 @@ ], "tags": [ "第七单元", - "圆" + "圆", + "2023届高三-寒假作业-较难题", + "2023届高三-第一轮复习讲义-35_圆及曲线方程" ], "genre": "解答题", "ans": "$x^2+(y-\\dfrac 12)^2=\\dfrac 14$且$x\\ne 0$", @@ -238181,7 +240051,8 @@ ], "tags": [ "第七单元", - "圆" + "圆", + "2023届高三-第一轮复习讲义-35_圆及曲线方程" ], "genre": "解答题", "ans": "$3x+4y+20=0$或$3x+4y-10=0$", @@ -238219,7 +240090,10 @@ ], "tags": [ "第七单元", - "圆" + "圆", + "2023届高三-四月错题重做-04_易错题-解析几何", + "2023届高三-四月错题重做-04_解析几何", + "2023届高三-第一轮复习讲义-35_圆及曲线方程" ], "genre": "解答题", "ans": "$(x-\\dfrac {24}5)^2+(y+\\dfrac{18}5)^2=1$", @@ -238442,7 +240316,8 @@ ], "tags": [ "第七单元", - "椭圆" + "椭圆", + "2023届高三-第一轮复习讲义-36_椭圆与双曲线的概念及性质" ], "genre": "解答题", "ans": "(1) $\\dfrac{x^2}{100}+\\dfrac{y^2}{64}=1$; (2) $\\dfrac{x^2}{6}+\\dfrac{y^2}{4}=1$; (3) $\\dfrac{x^2}8+\\dfrac{y^2}4=1$或$\\dfrac{y^2}8+\\dfrac{x^2}4=1$", @@ -238981,7 +240856,8 @@ ], "tags": [ "第七单元", - "椭圆" + "椭圆", + "2023届高三-第一轮复习讲义-36_椭圆与双曲线的概念及性质" ], "genre": "填空题", "ans": "$[1,5)\\cup (5,+\\infty)$", @@ -239413,7 +241289,8 @@ ], "tags": [ "第七单元", - "双曲线" + "双曲线", + "2023届高三-第一轮复习讲义-36_椭圆与双曲线的概念及性质" ], "genre": "选择题", "ans": "B", @@ -239474,7 +241351,10 @@ ], "tags": [ "第七单元", - "双曲线" + "双曲线", + "2023届高三-四月错题重做-04_解析几何", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-36_椭圆与双曲线的概念及性质" ], "genre": "解答题", "ans": "$x^2-\\dfrac{y^2}9=1$或$\\dfrac{y^2}{81}-\\dfrac{x^2}9=1$", @@ -239769,7 +241649,9 @@ ], "tags": [ "第七单元", - "抛物线" + "抛物线", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-37_抛物线的概念及性质" ], "genre": "解答题", "ans": "$2$", @@ -239877,7 +241759,9 @@ ], "tags": [ "第七单元", - "抛物线" + "抛物线", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-37_抛物线的概念及性质" ], "genre": "填空题", "ans": "$(0,-8)$, $y=8$", @@ -240173,7 +242057,8 @@ ], "tags": [ "第七单元", - "抛物线" + "抛物线", + "2023届高三-第一轮复习讲义-37_抛物线的概念及性质" ], "genre": "解答题", "ans": "$x^2=40y$, $x\\in [-\\dfrac{37}2,\\dfrac{37}2]$", @@ -240260,7 +242145,8 @@ ], "tags": [ "第七单元", - "圆" + "圆", + "2023届高三-第一轮复习讲义-35_圆及曲线方程" ], "genre": "解答题", "ans": "(1) $(x-3)^2+(y+3)^2=10$; (2) $(x-1)^2+(y+2)^2=2$", @@ -240299,7 +242185,8 @@ ], "tags": [ "第七单元", - "圆" + "圆", + "2023届高三-第一轮复习讲义-35_圆及曲线方程" ], "genre": "解答题", "ans": "(1) $y=x+\\sqrt{2}$或$y=x-\\sqrt{2}$; (2) $y=x+\\sqrt{2}$或$y=-x+\\sqrt{2}$", @@ -240589,7 +242476,8 @@ ], "tags": [ "第七单元", - "双曲线" + "双曲线", + "2023届高三-第一轮复习讲义-36_椭圆与双曲线的概念及性质" ], "genre": "解答题", "ans": "斜率为$2$, 但是判别式小于零, 故直线不存在.", @@ -240704,7 +242592,9 @@ "tags": [ "第七单元", "双曲线", - "抛物线" + "抛物线", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-37_抛物线的概念及性质" ], "genre": "解答题", "ans": "$y^2=-12x$", @@ -241110,7 +243000,8 @@ "content": "已知复数$z_1=a+b\\mathrm{i}$($a,b\\in \\mathbf{R}$)和复数$z_2=c+d\\mathrm{i}$($c,d\\in \\mathbf{R}$). ``$a=c$''是``$z_1=z_2$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-下学期测验卷-高三下学期月考01" ], "genre": "选择题", "ans": "B", @@ -242493,7 +244384,8 @@ "content": "在复数集中分解因式:\\\\\n(1) $x^2+5y^2$;\\\\ \n(2) $2x^2-6x+5$.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第一轮复习讲义-19_复数的几何意义与实系数二次方程" ], "genre": "解答题", "ans": "(1) $(x+\\sqrt{5}\\mathrm{i}y)(x-\\sqrt{5}\\mathrm{i}y)$; (2) $2(x-\\dfrac{3+\\mathrm{i}}2)(x-\\dfrac{3-\\mathrm{i}}2)$", @@ -243489,7 +245381,8 @@ ], "tags": [ "第七单元", - "圆" + "圆", + "2023届高三-第一轮复习讲义-35_圆及曲线方程" ], "genre": "填空题", "ans": "$(0,-1)$", @@ -243621,7 +245514,9 @@ ], "tags": [ "第七单元", - "抛物线" + "抛物线", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-37_抛物线的概念及性质" ], "genre": "选择题", "ans": "D", @@ -243656,7 +245551,8 @@ "content": "如果实数$x,y$满足$(x-2)^2+y^2=3$, 那么$\\dfrac yx$的最大值是\\bracket{20}.\n\\fourch{$\\dfrac 12$}{$\\dfrac{\\sqrt 3}3$}{$\\dfrac{\\sqrt 3}2$}{$\\sqrt 3$}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期测验卷-高三下学期月考01" ], "genre": "选择题", "ans": "D", @@ -244969,7 +246865,8 @@ "K0610002B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-21_空间直线与平面的位置关系" ], "genre": "填空题", "ans": "(1) $\\times$; (2) $\\times$; (3) $\\times$; (4) $\\times$", @@ -245137,7 +247034,8 @@ "K0608004B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-21_空间直线与平面的位置关系" ], "genre": "解答题", "ans": "证明略", @@ -245371,7 +247269,8 @@ "K0613003B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-22_空间平面与平面的位置关系" ], "genre": "解答题", "ans": "(1) $\\arctan 2$; (2) $\\pi - \\arctan\\sqrt{2}$", @@ -245467,7 +247366,8 @@ "K0612002B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-22_空间平面与平面的位置关系" ], "genre": "选择题", "ans": "D", @@ -245991,7 +247891,8 @@ "K0610004B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-21_空间直线与平面的位置关系" ], "genre": "填空题", "ans": "$1$", @@ -246040,7 +247941,8 @@ "content": "在空间四边形$ABCD$中, $AB=BC=CD=DA=AC=BD=a$, $F$是$BC$的中点, 求异面直线$AC$与$DF$所成的角的大小.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-20_描述空间位置关系的公理" ], "genre": "解答题", "ans": "$\\arccos \\dfrac{\\sqrt{3}}6$", @@ -246114,7 +248016,8 @@ "K0608002B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-21_空间直线与平面的位置关系" ], "genre": "解答题", "ans": "证明略", @@ -246327,7 +248230,9 @@ "K0604003B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-寒假作业-较难题", + "2023届高三-第一轮复习讲义-20_描述空间位置关系的公理" ], "genre": "解答题", "ans": "图略", @@ -246446,7 +248351,9 @@ "K0618002B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-23_多面体及旋转体的概念与性质" ], "genre": "解答题", "ans": "图略", @@ -246500,7 +248407,8 @@ "K0618006B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-23_多面体及旋转体的概念与性质" ], "genre": "解答题", "ans": "$\\pi (R^2-d^2)$", @@ -246700,7 +248608,8 @@ "K0617006B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-24_体积及表面积的计算" ], "genre": "解答题", "ans": "(1) $S_{\\text{侧}}=-2\\pi x^2+4\\pi x, \\ x\\in (0,2)$; (2) 当且仅当$x=1$时圆柱的侧面积最大", @@ -247115,7 +249024,8 @@ "K0618004B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-23_多面体及旋转体的概念与性质" ], "genre": "选择题", "ans": "D", @@ -247749,7 +249659,8 @@ "objs": [], "tags": [ "第八单元", - "加法原理与乘法原理" + "加法原理与乘法原理", + "2023届高三-第一轮复习讲义-38_计数原理与排列组合" ], "genre": "解答题", "ans": "$100$", @@ -247952,7 +249863,8 @@ "objs": [], "tags": [ "第八单元", - "排列" + "排列", + "2023届高三-第一轮复习讲义-38_计数原理与排列组合" ], "genre": "解答题", "ans": "(1) $6$; (2) $5$; (3) $4$", @@ -249690,7 +251602,9 @@ "objs": [], "tags": [ "第八单元", - "组合" + "组合", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-38_计数原理与排列组合" ], "genre": "解答题", "ans": "$375$", @@ -250084,7 +251998,9 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-25_概率的概念及性质" ], "genre": "解答题", "ans": "$\\dfrac 1{28}$, $\\dfrac 37$", @@ -250142,7 +252058,8 @@ "objs": [], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-第一轮复习讲义-38_计数原理与排列组合" ], "genre": "解答题", "ans": "$\\dfrac{11}{21}$", @@ -250256,7 +252173,8 @@ "objs": [], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-第一轮复习讲义-38_计数原理与排列组合" ], "genre": "解答题", "ans": "$\\dfrac{3439}{10000}$", @@ -250323,7 +252241,8 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-第一轮复习讲义-26_大数定律及独立性" ], "genre": "解答题", "ans": "$0.97\\dot{3}$", @@ -252433,7 +254352,9 @@ "K0108003B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-较难题", + "2023届高三-第一轮复习讲义-04_方程与不等式的求解" ], "genre": "解答题", "ans": "当$k=0$时, 解集为$\\varnothing$; 当$k\\ne 0$时, 解集为$\\{(-\\dfrac 2k,-1)\\}$", @@ -252977,7 +254898,9 @@ "K0120002B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-03_等式与不等式的性质及证明" ], "genre": "解答题", "ans": "证明略", @@ -253471,7 +255394,9 @@ "K0208005B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-06_幂指对函数" ], "genre": "解答题", "ans": "(1) $y=(x-1)^{\\frac 23}$的图像由$y=x^{\\frac 23}$的图像向右平移一个单位长度而得到.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0:2,samples = 50] plot (\\x,{pow(\\x,2/3)}) node [above right] {$y=x^{\\frac 23}$};\n\\draw [domain = 0:2,samples = 50] plot (-\\x,{pow(\\x,2/3)});\n\\draw [domain = 0:2,samples = 50] plot ({\\x+1},{pow((\\x,2/3)}) node [right] {$y=(x-1)^{\\frac 23}$};\n\\draw [domain = 0:2,samples = 50] plot ({-\\x+1},{pow(\\x,2/3)});\n\\end{tikzpicture}\n\\end{center}\n(2) $y=x^{\\frac 23}+1$的图像由$y=x^{\\frac 23}$的图像向上平移一个单位长度而得到.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0:2,samples = 50] plot (\\x,{pow(\\x,2/3)}) node [above right] {$y=x^{\\frac 23}$};\n\\draw [domain = 0:2,samples = 50] plot (-\\x,{pow(\\x,2/3)});\n\\draw [domain = 0:2,samples = 50] plot (\\x,{pow(\\x,2/3)+1}) node [right] {$y=x^{\\frac 23}+1$};\n\\draw [domain = 0:2,samples = 50] plot (-\\x,{pow(\\x,2/3)+1});\n\\end{tikzpicture}\n\\end{center}", @@ -253521,7 +255446,9 @@ "K0208005B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-06_幂指对函数" ], "genre": "解答题", "ans": "图像如下:\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [above right] {$O$};\n\\draw [domain = -4:{-7/3}] plot (\\x,{(-\\x-1)/(\\x+2)});\n\\draw [domain = {-9/5}:4] plot (\\x,{(-\\x-1)/(\\x+2)});\n\\draw [dashed] (-4,-1) -- (4,-1) (-2,-4) -- (-2,4);\n\\end{tikzpicture}\n\\end{center}", @@ -253969,7 +255896,8 @@ "K0215004B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第一轮复习讲义-07_函数的概念与奇偶性" ], "genre": "选择题", "ans": "C", @@ -254043,7 +255971,9 @@ "K0216005B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-07_函数的概念与奇偶性" ], "genre": "解答题", "ans": "$y=\\begin{cases} 3x+10, & x\\in [-3,-2],\\\\ 4, & x\\in [-2,2],\\\\ 10-3x, & x\\in [2,3]. \\end{cases}$", @@ -254189,7 +256119,9 @@ "K0219001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-08_函数的单调性" ], "genre": "解答题", "ans": "不正确, 如$y=\\begin{cases} 0, & x=0, \\\\ \\dfrac 1x, & x>0.\\end{cases}$", @@ -254216,7 +256148,9 @@ "K0219003B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-08_函数的单调性" ], "genre": "解答题", "ans": "证明略", @@ -254315,7 +256249,9 @@ "K0220003B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-较难题", + "2023届高三-第一轮复习讲义-08_函数的单调性" ], "genre": "解答题", "ans": "(1) 证明略; (2) 是的, 证明略", @@ -254416,7 +256352,8 @@ "K0222002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第一轮复习讲义-10_有关函数的应用问题" ], "genre": "解答题", "ans": "$y=6-\\dfrac x2, \\ x\\in (0,6)$", @@ -254512,7 +256449,9 @@ "K0223005B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-09_函数的零点与最值" ], "genre": "解答题", "ans": "$(1,+\\infty)$", @@ -254538,7 +256477,9 @@ "K0224001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-09_函数的零点与最值" ], "genre": "解答题", "ans": "不是, 例如$f(x)=x^2-1$, $a=-10$, $b=10$等", @@ -254564,7 +256505,9 @@ "K0224002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-09_函数的零点与最值" ], "genre": "解答题", "ans": "$1.6$", @@ -254924,7 +256867,8 @@ "K0304001B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-11_三角比的定义及直接性质" ], "genre": "选择题", "ans": "C", @@ -255262,7 +257206,8 @@ "K0307004B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-11_三角比的定义及直接性质" ], "genre": "解答题", "ans": "$(4,-3)$", @@ -255302,7 +257247,8 @@ "K0308002B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-11_三角比的定义及直接性质" ], "genre": "解答题", "ans": "(1) $x=2k\\pi-\\dfrac \\pi 3$或$2k\\pi-\\dfrac{2\\pi}3, \\ k\\in \\mathbf{Z}$; (2) $x=2k\\pi\\pm \\dfrac{2\\pi}3, \\ k\\in \\mathbf{Z}$; (3) $x=k\\pi-\\dfrac \\pi 3, \\ k\\in \\mathbf{Z}$.", @@ -256110,7 +258056,8 @@ "K0319002B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-15_周期性与其他三角函数" ], "genre": "解答题", "ans": "如$760\\text{ms}$等", @@ -256455,7 +258402,8 @@ "K0324006B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-15_周期性与其他三角函数" ], "genre": "解答题", "ans": "定义域为$\\{x|x\\ne \\dfrac \\pi{12}+\\dfrac{k\\pi}3, \\ k\\in \\mathbf{Z}\\}$; 单调区间为$(\\dfrac\\pi{12}+\\dfrac{k\\pi}3,\\dfrac{5\\pi}{12}+\\dfrac{k\\pi}3) \\ (k\\in \\mathbf{Z})$, 在这些区间上是严格增函数", @@ -257778,7 +259726,8 @@ "content": "如图, 在长方体$ABCD-A_1B_1C_1D_1$中,\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw (0,0) node [below left] {$A$} coordinate (A) --++ (3,0) node [below right] {$B$} coordinate (B) --++ (45:{3/2}) node [right] {$C$} coordinate (C)\n--++ (0,2) node [above right] {$C_1$} coordinate (C1)\n--++ (-3,0) node [above left] {$D_1$} coordinate (D1) --++ (225:{3/2}) node [left] {$A_1$} coordinate (A1) -- cycle;\n\\draw (A) ++ (3,2) node [right] {$B_1$} coordinate (B1) -- (B) (B1) --++ (45:{3/2}) (B1) --++ (-3,0);\n\\draw [dashed] (A) --++ (45:{3/2}) node [left] {$D$} coordinate (D) --++ (3,0) (D) --++ (0,2);\n\\end{tikzpicture}\n\\end{center}\n(1) 设$AC$与$BD$的交点为$O$, $O$必为平面\\blank{50}与平面\\blank{50}的公共点(答案不唯一);\\\\\n(2) 画出平面$A_1BCD_1$与平面$B_1BDD_1$的交线.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-20_描述空间位置关系的公理" ], "genre": "填空题", "ans": "(1) $ACB_1$, $BDD_1B_1$(不唯一); (2) 图略(直线$BD_1$)", @@ -257830,7 +259779,8 @@ "K0604002B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-20_描述空间位置关系的公理" ], "genre": "解答题", "ans": "图略", @@ -258134,7 +260084,9 @@ "K0608004B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-寒假作业-较难题", + "2023届高三-第一轮复习讲义-21_空间直线与平面的位置关系" ], "genre": "解答题", "ans": "证明略", @@ -258230,7 +260182,8 @@ "K0609003B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-21_空间直线与平面的位置关系" ], "genre": "解答题", "ans": "证明略", @@ -258332,7 +260285,8 @@ "K0610002B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-21_空间直线与平面的位置关系" ], "genre": "填空题", "ans": "(1) 外; (2) 中; (3) 垂", @@ -258386,7 +260340,8 @@ "K0611003B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-21_空间直线与平面的位置关系" ], "genre": "解答题", "ans": "(1) 证明略; (2) 证明略", @@ -258444,7 +260399,9 @@ "K0609008B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-22_空间平面与平面的位置关系" ], "genre": "解答题", "ans": "(1) $\\sqrt{2}a$; (2) $a$; (3) $a$; (4) $a$", @@ -258876,7 +260833,8 @@ "K0618005B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-23_多面体及旋转体的概念与性质" ], "genre": "解答题", "ans": "(1) 是三角形, 当底面半径$r\\ge \\dfrac{\\sqrt{2}}2l$($l$为母线长)时, 底面圆中的弦长为$\\sqrt{2}l$时截面面积最大; 当底面半径$r<\\dfrac{\\sqrt{2}}2l$时, 轴截面面积最大; (2) 不可能, 证明略", @@ -258973,7 +260931,8 @@ "K0619005B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-24_体积及表面积的计算" ], "genre": "解答题", "ans": "证明略", @@ -259362,7 +261321,8 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-第一轮复习讲义-25_概率的概念及性质" ], "genre": "解答题", "ans": "$\\Omega = \\{1,2,3,4,5,6\\}$, $A=\\{2,3,4,5,6\\}$, $B=\\{1,3,5\\}$, $C=\\{3,4,5,6\\}$", @@ -259497,7 +261457,9 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-25_概率的概念及性质" ], "genre": "解答题", "ans": "$P(A_0)=\\dfrac 13$, $P(A_1)=\\dfrac 12$, $P(A_2)=0$, $P(A_3)=\\dfrac 16$", @@ -259577,7 +261539,9 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-25_概率的概念及性质" ], "genre": "解答题", "ans": "证明略", @@ -259609,7 +261573,8 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-第一轮复习讲义-25_概率的概念及性质" ], "genre": "解答题", "ans": "证明略", @@ -259716,7 +261681,8 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-第一轮复习讲义-26_大数定律及独立性" ], "genre": "解答题", "ans": "约$6.7\\times 10^4$条", @@ -259779,7 +261745,9 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-26_大数定律及独立性" ], "genre": "填空题", "ans": "$0.18$, $0.12$", @@ -259810,7 +261778,8 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-第一轮复习讲义-26_大数定律及独立性" ], "genre": "解答题", "ans": "(1) 证明略; (2) 证明略", @@ -259848,7 +261817,8 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-第一轮复习讲义-26_大数定律及独立性" ], "genre": "解答题", "ans": "(1) $pq$; (2) $1-p-q+pq$; (3) $p+q-pq$; (4) $1-pq$", @@ -260568,7 +262538,9 @@ ], "tags": [ "第七单元", - "直线" + "直线", + "2023届高三-四月错题重做-04_解析几何", + "2023届高三-第一轮复习讲义-34_直线及其方程" ], "genre": "解答题", "ans": "$y=-\\dfrac{\\sqrt 3}{3}(x+1)$", @@ -260761,7 +262733,8 @@ ], "tags": [ "第七单元", - "直线" + "直线", + "2023届高三-第一轮复习讲义-34_直线及其方程" ], "genre": "解答题", "ans": "总经过定点$(-1,0)$, 证明略", @@ -260849,7 +262822,8 @@ ], "tags": [ "第七单元", - "直线" + "直线", + "2023届高三-第一轮复习讲义-34_直线及其方程" ], "genre": "解答题", "ans": "$-\\dfrac 94$", @@ -261042,7 +263016,8 @@ ], "tags": [ "第七单元", - "直线" + "直线", + "2023届高三-第一轮复习讲义-34_直线及其方程" ], "genre": "解答题", "ans": "(1) $\\arccos\\dfrac{\\sqrt{170}}{170}$; (2) $\\arccos \\dfrac{7\\sqrt{65}}{65}$", @@ -261106,7 +263081,8 @@ ], "tags": [ "第七单元", - "直线" + "直线", + "2023届高三-第一轮复习讲义-34_直线及其方程" ], "genre": "解答题", "ans": "$y+3=(2+\\sqrt{3})(x-4)$或$y+3=(2-\\sqrt{3})(x-4)$", @@ -261245,7 +263221,8 @@ ], "tags": [ "第七单元", - "圆" + "圆", + "2023届高三-第一轮复习讲义-35_圆及曲线方程" ], "genre": "解答题", "ans": "$(x-5)^2+(y-4)^2=25$", @@ -261282,7 +263259,8 @@ ], "tags": [ "第七单元", - "圆" + "圆", + "2023届高三-第一轮复习讲义-35_圆及曲线方程" ], "genre": "解答题", "ans": "$x^2+y^2-5x-y+4=0$", @@ -261670,7 +263648,8 @@ ], "tags": [ "第七单元", - "椭圆" + "椭圆", + "2023届高三-第一轮复习讲义-36_椭圆与双曲线的概念及性质" ], "genre": "解答题", "ans": "最大值为$10$, 最小值为$2$", @@ -261856,7 +263835,8 @@ ], "tags": [ "第七单元", - "双曲线" + "双曲线", + "2023届高三-第一轮复习讲义-36_椭圆与双曲线的概念及性质" ], "genre": "选择题", "ans": "A", @@ -261922,7 +263902,8 @@ ], "tags": [ "第七单元", - "双曲线" + "双曲线", + "2023届高三-第一轮复习讲义-36_椭圆与双曲线的概念及性质" ], "genre": "解答题", "ans": "(1) $\\dfrac{x^2}{25}-\\dfrac{y^2}{3}=1$; (2) $\\dfrac{x^2}{16}-\\dfrac{y^2}9=1$", @@ -262561,7 +264542,9 @@ ], "tags": [ "第六单元", - "空间向量" + "空间向量", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-32_空间向量的概念与性质及立体几何中的证明问题" ], "genre": "解答题", "ans": "$\\dfrac 12 a^2$, $-\\dfrac 14 a^2$", @@ -262624,7 +264607,9 @@ ], "tags": [ "第六单元", - "空间向量" + "空间向量", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-32_空间向量的概念与性质及立体几何中的证明问题" ], "genre": "解答题", "ans": "(1) 假命题, 理由略; (2) 假命题, 理由略; (3) 真命题, 理由略", @@ -262833,7 +264818,8 @@ ], "tags": [ "第六单元", - "空间向量" + "空间向量", + "2023届高三-第一轮复习讲义-24_体积及表面积的计算" ], "genre": "解答题", "ans": "$\\dfrac 67$", @@ -263032,7 +265018,8 @@ "K0401004X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第一轮复习讲义-30_等差数列与等比数列" ], "genre": "解答题", "ans": "(1) $27$; (2) $86$; (3) $13$", @@ -263318,7 +265305,9 @@ "K0405002X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-30_等差数列与等比数列" ], "genre": "解答题", "ans": "$\\dfrac 12$", @@ -263796,7 +265785,8 @@ "K0227004X" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第一轮复习讲义-28_导数的概念及常用公式" ], "genre": "解答题", "ans": "(1) $2\\pi$; (2) $2\\pi$", @@ -263958,7 +265948,9 @@ "K0229005X" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-28_导数的概念及常用公式" ], "genre": "解答题", "ans": "证明略", @@ -264217,7 +266209,8 @@ "K0234001X" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第一轮复习讲义-29_导数的应用" ], "genre": "解答题", "ans": "(1) 错误, 理由略; (2) 正确, 由最大(小)值的定义; (3) 错误, 理由略; (4) 错误, 理由略", @@ -264335,7 +266328,8 @@ "K0235001X" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第一轮复习讲义-29_导数的应用" ], "genre": "解答题", "ans": "$\\dfrac{\\sqrt{3}}3R$", @@ -264555,7 +266549,8 @@ "objs": [], "tags": [ "第八单元", - "排列" + "排列", + "2023届高三-第一轮复习讲义-38_计数原理与排列组合" ], "genre": "解答题", "ans": "$15$", @@ -264877,7 +266872,8 @@ ], "tags": [ "第八单元", - "二项式定理" + "二项式定理", + "2023届高三-第一轮复习讲义-39_二项式定理" ], "genre": "解答题", "ans": "(1) 系数最大的项为$672x^5$; (2) 系数最大的项为$560x^4$", @@ -265732,7 +267728,9 @@ ], "tags": [ "第七单元", - "双曲线" + "双曲线", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-36_椭圆与双曲线的概念及性质" ], "genre": "填空题", "ans": "$6$", @@ -266059,7 +268057,9 @@ "content": "如图, 正方体$ABCD-A_1B_1C_1D_1$中, $P,Q,R,S$分别为棱$AB,BC,BB_1,CD$的中点, 联结$A_1S,B_1D$. 空间任意两点$M,N$, 若线段$MN$上不存在点在线段$A_1S,B_1D$上, 则称$M,N$两点可视, 则下列选项中, 与点$D_1$可视的为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$A$} coordinate (A) --++ (2,0) node [below right] {$B$} coordinate (B) --++ (45:{2/2}) node [right] {$C$} coordinate (C)\n--++ (0,2) node [above right] {$C_1$} coordinate (C1)\n--++ (-2,0) node [above left] {$D_1$} coordinate (D1) --++ (225:{2/2}) node [left] {$A_1$} coordinate (A1) -- cycle;\n\\draw (A) ++ (2,2) node [right] {$B_1$} coordinate (B1) -- (B) (B1) --++ (45:{2/2}) (B1) --++ (-2,0);\n\\draw [dashed] (A) --++ (45:{2/2}) node [left] {$D$} coordinate (D) --++ (2,0) (D) --++ (0,2);\n\\filldraw ($(A)!0.5!(B)$) circle (0.05) node [below] {$P$};\n\\filldraw ($(C)!0.5!(B)$) circle (0.05) node [below right] {$Q$};\n\\filldraw ($(B1)!0.5!(B)$) circle (0.05) node [above right] {$R$};\n\\filldraw ($(C)!0.5!(D)$) circle (0.05) node [above] {$S$} coordinate (S);\n\\draw [dashed] (A1) -- (S) (B1) -- (D);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{点$P$}{点$B$}{点$R$}{点$Q$}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期周末卷-周末卷06", + "2023届高三-寒假作业-容易题" ], "genre": "选择题", "ans": "D", @@ -266107,7 +268107,9 @@ "content": "设$ABC$是等边三角形, $O$为边$AC$的中点, $PO\\perp$平面$ABC$, $PA=AC=2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.8]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (1,0,{sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw (1,{sqrt(3)},0) node [above] {$P$} coordinate (P);\n\\draw (1,0,0) node [below] {$O$} coordinate (O);\n\\draw (A) -- (B) -- (C) -- (P) -- cycle (P) -- (B);\n\\draw [dashed] (O) -- (P) (A) -- (C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求三棱锥$P-ABC$的体积;\\\\\n(2) 若$M$为$BC$的中点, 求$PM$与平面$PAC$所成角的大小.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期周末卷-周末卷06", + "2023届高三-下学期测验卷-高三下学期月考01" ], "genre": "解答题", "ans": "(1) $1$; (2) $\\arctan\\dfrac{\\sqrt{39}}{13}$", @@ -266774,7 +268776,8 @@ "K0103005B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-01_集合" ], "genre": "解答题", "ans": "$2$", @@ -266905,7 +268908,8 @@ "K0101001B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-01_集合" ], "genre": "解答题", "ans": "$0$或$-1$", @@ -266937,7 +268941,8 @@ "K0103003B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-01_集合" ], "genre": "解答题", "ans": "$B\\subset A$, 证明略", @@ -267021,7 +269026,8 @@ "K0105002B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-02_常用逻辑用语" ], "genre": "解答题", "ans": "是, 是, 不是, 是, 是", @@ -267767,7 +269773,9 @@ "K0109003B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-04_方程与不等式的求解" ], "genre": "解答题", "ans": "证明略", @@ -268864,7 +270872,8 @@ "K0120002B" ], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第一轮复习讲义-03_等式与不等式的性质及证明" ], "genre": "解答题", "ans": "证明略, 等号成立时$x$的取值范围为$(-\\infty,-2]$", @@ -269015,7 +271024,8 @@ "K0202002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第一轮复习讲义-05_幂指数与对数" ], "genre": "解答题", "ans": "(1) $a^\\frac{7}{12}$; (2) $a^{\\frac 12}$; (3) $a^2b^{-3}$; (4) $ab^{-\\frac 76}$", @@ -269119,7 +271129,9 @@ "K0203005B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-05_幂指数与对数" ], "genre": "解答题", "ans": "证明略", @@ -269390,7 +271402,9 @@ "K0205002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-05_幂指数与对数" ], "genre": "解答题", "ans": "约等于$22$($10^{1.35}$)", @@ -269593,7 +271607,8 @@ "K0207004B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验02" ], "genre": "填空题", "ans": "\\textcircled{2}", @@ -269669,7 +271684,8 @@ "K0211001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验02" ], "genre": "填空题", "ans": "$(-\\infty,1)$", @@ -269700,7 +271716,9 @@ "K0208003B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-06_幂指对函数" ], "genre": "选择题", "ans": "D", @@ -270726,7 +272744,9 @@ "K0219003B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-08_函数的单调性" ], "genre": "解答题", "ans": "证明略", @@ -270976,7 +272996,8 @@ "K0219001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第一轮复习讲义-08_函数的单调性" ], "genre": "填空题", "ans": "$[2,+\\infty)$", @@ -271115,7 +273136,9 @@ "K0224002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-较难题", + "2023届高三-第一轮复习讲义-09_函数的零点与最值" ], "genre": "解答题", "ans": "证明略, 近似值为$0.5$", @@ -271220,7 +273243,9 @@ "K0223004B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-09_函数的零点与最值" ], "genre": "解答题", "ans": "证明略", @@ -271251,7 +273276,11 @@ "K0223005B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-四月错题重做-01_函数一", + "2023届高三-四月错题重做-01_易错题-函数1", + "2023届高三-寒假作业-较难题", + "2023届高三-第一轮复习讲义-09_函数的零点与最值" ], "genre": "解答题", "ans": "$(-\\infty,0)\\cup (0,1]$", @@ -271620,7 +273649,8 @@ "K0303002B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-11_三角比的定义及直接性质" ], "genre": "解答题", "ans": "(1) $-$; (2) $-$; (3) $+$.", @@ -271725,7 +273755,8 @@ "K0304003B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-11_三角比的定义及直接性质" ], "genre": "解答题", "ans": "$(\\sin\\alpha,\\cos\\alpha)=(\\dfrac{\\sqrt{5}}5,-\\dfrac{2\\sqrt{5}}5)$或$(-\\dfrac{\\sqrt{5}}5,\\dfrac{2\\sqrt{5}}5)$", @@ -271883,7 +273914,8 @@ "K0307003B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-11_三角比的定义及直接性质" ], "genre": "解答题", "ans": "(1) $0$; (2) $\\sin\\alpha$; (3) $\\cot\\alpha$; (4) $-\\tan\\alpha$", @@ -271945,7 +273977,8 @@ "K0302003B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-11_三角比的定义及直接性质" ], "genre": "解答题", "ans": "$60^\\circ$或$150^\\circ$", @@ -272053,7 +274086,8 @@ "K0301004B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-11_三角比的定义及直接性质" ], "genre": "填空题", "ans": "$\\alpha+\\beta=2k\\pi, \\ k\\in \\mathbf{Z}$", @@ -272182,7 +274216,8 @@ "content": "已知$\\alpha$是第二象限的角, 化简: $\\sqrt{\\dfrac{1+\\sin \\alpha}{1-\\sin \\alpha}}+\\sqrt{\\dfrac{1-\\sin\\alpha}{1+\\sin\\alpha}}$.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-11_三角比的定义及直接性质" ], "genre": "解答题", "ans": "$-\\dfrac{2}{\\cos\\alpha}$", @@ -272221,7 +274256,8 @@ "K0305001B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-11_三角比的定义及直接性质" ], "genre": "解答题", "ans": "$\\sin\\alpha=\\dfrac 45$, $\\cos\\alpha=-\\dfrac 35$.", @@ -273091,7 +275127,8 @@ "K0308005B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-11_三角比的定义及直接性质" ], "genre": "解答题", "ans": "(1) $x=\\arcsin \\dfrac 25$或$\\pi-\\arcsin \\dfrac 25$; (2) $x=\\pi\\pm \\arccos\\dfrac 23$; (3) $x=-\\arctan \\dfrac 12+k\\pi, \\ k\\in \\mathbf{Z}$", @@ -276867,7 +278904,8 @@ "K0603004B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-20_描述空间位置关系的公理" ], "genre": "解答题", "ans": "证明略", @@ -277271,7 +279309,9 @@ "K0606002B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-20_描述空间位置关系的公理" ], "genre": "解答题", "ans": "证明略", @@ -277301,7 +279341,9 @@ "K0607001B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-20_描述空间位置关系的公理" ], "genre": "解答题", "ans": "$5$", @@ -277661,7 +279703,9 @@ "content": "如图, 在$\\triangle ABC$中, $\\angle ACB=90^\\circ$, 且$DA$垂直于$\\triangle ABC$所在的平面$\\alpha$, $M$、$N$分别是边$AC$、$DB$的中点.求证: $MN\\perp AC$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$C$} coordinate (C) -- (2,0) node [right] {$B$} coordinate (B);\n\\draw (0.8,0.8) node [left] {$A$} coordinate (A) --++ (0,1.5) node [left] {$D$} coordinate (D);\n\\draw (A) -- (C) (A) -- (B) (B) -- (D);\n\\draw ($(A)!0.5!(C)$) node [left] {$M$} coordinate (M);\n\\draw ($(B)!0.5!(D)$) node [right] {$N$} coordinate (N);\n\\draw (M) -- (N);\n\\draw (-0.9,-0.3) --++ (3.3,0) --++ (1.3,1.3) coordinate (R) ++ (-3.3,0) coordinate (P) --++ (-1.3,-1.3);\n\\path [name path = rear] (R) --++ (-3.3,0);\n\\path [name path = BD] (B) -- (D);\n\\path [name path = AD] (A) -- (D);\n\\path [name intersections = {of = AD and rear, by = S}];\n\\path [name intersections = {of = BD and rear, by = T}];\n\\draw (S) -- (P) (T) -- (R);\n\\draw [dashed] (S) -- (T);\n\\draw (R) ++ (-0.5,0) node [below] {$\\alpha$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期周末卷-周末卷06", + "2023届高三-寒假作业-中档题" ], "genre": "解答题", "ans": "证明略", @@ -278039,7 +280083,9 @@ "content": "已知一平面平行于两条异面直线, 一直线与两异面直线都垂直, 那么这个平面与这条直线的位置关系是\\bracket{20}.\n\\fourch{平行}{垂直}{斜交}{不能确定}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期周末卷-周末卷06", + "2023届高三-寒假作业-容易题" ], "genre": "选择题", "ans": "B", @@ -278131,7 +280177,8 @@ "content": "若两异面直线$a$、$b$所成的角为$70^\\circ$, 过空间内一点$P$作与直线$a$、$b$所成角均是$70^\\circ$的直线$l$, 则所作直线$l$的条数为\\bracket{20}.\n\\fourch{$1$}{$2$}{$3$}{$4$}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期周末卷-周末卷06" ], "genre": "选择题", "ans": "D", @@ -278272,7 +280319,9 @@ "K0617006B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期周末卷-周末卷06", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$4\\pi$", @@ -278300,7 +280349,8 @@ "K0616003B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-24_体积及表面积的计算" ], "genre": "填空题", "ans": "$24\\sqrt{3}$", @@ -278329,7 +280379,8 @@ "K0617004B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-24_体积及表面积的计算" ], "genre": "填空题", "ans": "$12a^2$", @@ -278358,7 +280409,9 @@ "K0617004B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-24_体积及表面积的计算" ], "genre": "填空题", "ans": "$40$", @@ -278386,7 +280439,9 @@ "content": "在正四棱柱$ABCD-A_1B_1C_1D_1$中, 若$AA_1=2AB$, 则异面直线$CD$与$AC_1$所成角的大小为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期周末卷-周末卷06", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$\\arctan \\sqrt{5}$", @@ -278550,7 +280605,9 @@ "content": "如图, 已知三棱锥$P-ABC$中, $PA$垂直于平面$ABC$,$ AB\\perp BC$, $PA=4$, $AB=3$, $AC=5$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (A) ++ (0,4,0) node [left] {$P$} coordinate (P);\n\\draw (A) ++ (3,0,0) node [right] {$B$} coordinate (B);\n\\draw (B) ++ (0,0,-4) node [right] {$C$} coordinate (C);\n\\draw (A) -- (B) -- (C) (P) -- (A) (P) -- (C) (P) -- (B);\n\\draw [dashed] (A) -- (C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求点$A$到平面$PBC$的距离;\\\\\n(2) 求三棱锥$P-ABC$的表面积.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期周末卷-周末卷06", + "2023届高三-寒假作业-容易题" ], "genre": "解答题", "ans": "(1) $\\dfrac {12}5$; (2) $32$", @@ -278600,7 +280657,8 @@ "K0618005B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-23_多面体及旋转体的概念与性质" ], "genre": "解答题", "ans": "证明略", @@ -278730,7 +280788,8 @@ "K0619004B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-24_体积及表面积的计算" ], "genre": "解答题", "ans": "$h=\\dfrac 12$, $V=\\dfrac{\\sqrt{2}}{12}$", @@ -278890,7 +280949,8 @@ "K0617007B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-24_体积及表面积的计算" ], "genre": "解答题", "ans": "表面积为$8+8\\sqrt{3}$, 体积为$\\dfrac{10}3\\sqrt{2}$", @@ -278946,7 +281006,9 @@ "K0621001B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-23_多面体及旋转体的概念与性质" ], "genre": "填空题", "ans": "\\textcircled{1}\\textcircled{3}\\textcircled{5}\\textcircled{7}", @@ -278979,7 +281041,8 @@ "K0619003B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-24_体积及表面积的计算" ], "genre": "解答题", "ans": "棱柱的体积更大($3:2\\sqrt{2}$)", @@ -279007,7 +281070,8 @@ "content": "若一个球的体积是$\\dfrac 43\\pi$, 则这个球的表面积是\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-赋能-赋能29" ], "genre": "填空题", "ans": "$4\\pi$", @@ -279108,7 +281172,8 @@ "K0622005B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-23_多面体及旋转体的概念与性质" ], "genre": "填空题", "ans": "$\\dfrac{9\\pi}4$", @@ -279161,7 +281226,8 @@ "K0622002B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-23_多面体及旋转体的概念与性质" ], "genre": "解答题", "ans": "$\\dfrac{13}2$", @@ -279212,7 +281278,8 @@ "content": "如图, 半径为$R$的球$O$中有一内接圆柱, 当圆柱的侧面积最大时, 求球的表面积与该圆柱的侧面积之差.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw (0,0) circle (2);\n\\filldraw (0,0) circle (0.03) node [below] {$O$} coordinate (O);\n\\draw [dashed] (1.6,1.2) coordinate (P) -- (O);\n\\draw (-2,0) arc (180:360:2 and 0.5);\n\\draw (0.8,0.6) node [below] {$R$};\n\\draw [dashed] (-2,0) arc (180:0:2 and 0.5);\n\\draw [dashed] (P) -- (1.6,-1.2) (-1.6,1.2) -- (-1.6,-1.2);\n\\draw (P) arc (0:-180:1.6 and 0.4);\n\\draw (P) ++ (0,-2.4) arc (0:-180:1.6 and 0.4);\n\\draw [dashed] (P) arc (0:180:1.6 and 0.4);\n\\draw [dashed] (P) ++ (0,-2.4) arc (0:180:1.6 and 0.4);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期周末卷-周末卷06" ], "genre": "解答题", "ans": "$2\\pi R^2$", @@ -279266,7 +281333,9 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-25_概率的概念及性质" ], "genre": "解答题", "ans": "(1) 错误; (2) 错误; (3) 正确", @@ -279374,7 +281443,8 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-第一轮复习讲义-25_概率的概念及性质" ], "genre": "解答题", "ans": "$\\Omega=\\{W_1B_1,W_1B_2,W_1R_1,W_1R_2,W_1R_3,B_1B_2,B_1R_1,B_1R_2,B_1R_3,B_2R_1,B_2R_2,B_2R_3,R_1R_2,R_1R_3,R_2R_3\\}$, $A=\\{W_1B_1,W_1B_2,W_1R_1,W_1R_2,W_1R_3\\}$, $B=\\{W_1B_1,W_1B_2,B_1B_2,B_1R_1,B_1R_2,B_1R_3,B_2R_1,B_2R_2,B_2R_3\\}$. 或者$\\Omega' = \\{WB,WR,BB,BR,RR\\}$, $A'=\\{WB,WR\\}$, $B'=\\{WB,BB,BR\\}$", @@ -279407,7 +281477,9 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-25_概率的概念及性质" ], "genre": "解答题", "ans": "(1) $\\Omega = \\{BBGG,BGBG,BGGB,GGBB,GBGB,GBBG\\}$等; (2) $A=\\{BGBG,GBGB\\}$等; (3) $B=\\{BGBG,BGGB,GBGB,GBBG\\}$等", @@ -279514,7 +281586,8 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-第一轮复习讲义-25_概率的概念及性质" ], "genre": "选择题", "ans": "B", @@ -279546,7 +281619,9 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-25_概率的概念及性质" ], "genre": "解答题", "ans": "$0.10$", @@ -279680,7 +281755,8 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-第一轮复习讲义-25_概率的概念及性质" ], "genre": "解答题", "ans": "$\\dfrac 35$", @@ -279763,7 +281839,9 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-26_大数定律及独立性" ], "genre": "解答题", "ans": "(1) $0.75$; (2) $15$", @@ -279800,7 +281878,8 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-第一轮复习讲义-26_大数定律及独立性" ], "genre": "解答题", "ans": "(1) $\\dfrac 79$; (2) 作``取两个球''的操作$n$次($n$充分大), 记录颜色不同的次数$S_n$, 计算$\\dfrac{S_n}n$", @@ -279831,7 +281910,8 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-第一轮复习讲义-26_大数定律及独立性" ], "genre": "解答题", "ans": "$0.75$", @@ -279889,7 +281969,8 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-第一轮复习讲义-26_大数定律及独立性" ], "genre": "选择题", "ans": "C", @@ -279922,7 +282003,8 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-第一轮复习讲义-26_大数定律及独立性" ], "genre": "解答题", "ans": "(1) $\\dfrac 56$; (2) $\\dfrac 16$; (3) $\\dfrac 23$; (4) $\\dfrac 12$", @@ -280751,7 +282833,8 @@ ], "tags": [ "第七单元", - "直线" + "直线", + "2023届高三-第一轮复习讲义-34_直线及其方程" ], "genre": "解答题", "ans": "$a=-\\dfrac 57$, 斜率为$-\\dfrac 16$", @@ -280816,7 +282899,8 @@ ], "tags": [ "第七单元", - "直线" + "直线", + "2023届高三-第一轮复习讲义-34_直线及其方程" ], "genre": "解答题", "ans": "$\\begin{cases}\\dfrac\\pi 2-\\theta, & 0\\le \\theta\\le \\dfrac \\pi 2, \\\\ \\dfrac{3\\pi}2-\\theta, & \\dfrac\\pi 2<\\theta<\\pi.\\end{cases}$", @@ -280933,7 +283017,8 @@ ], "tags": [ "第七单元", - "直线" + "直线", + "2023届高三-第一轮复习讲义-34_直线及其方程" ], "genre": "解答题", "ans": "$y=\\dfrac 34x+4$或$y=-\\dfrac 34 x+4$", @@ -280998,7 +283083,8 @@ ], "tags": [ "第七单元", - "直线" + "直线", + "2023届高三-第一轮复习讲义-34_直线及其方程" ], "genre": "解答题", "ans": "(1) $\\pi-\\arctan 5$; (2) $\\dfrac 25$", @@ -281037,7 +283123,8 @@ ], "tags": [ "第七单元", - "直线" + "直线", + "2023届高三-第一轮复习讲义-34_直线及其方程" ], "genre": "解答题", "ans": "$y-1=\\dfrac{5+\\sqrt{33}}4(x+2)$或$y-1=\\dfrac{5-\\sqrt{33}}4(x+2)$", @@ -281530,7 +283617,8 @@ ], "tags": [ "第七单元", - "直线" + "直线", + "2023届高三-第一轮复习讲义-34_直线及其方程" ], "genre": "解答题", "ans": "(1) $(-\\infty,-5)\\cup (-5,8)\\cup (8,+\\infty)$; (2) $\\{-5\\}$; (3) $\\{8\\}$", @@ -281745,7 +283833,8 @@ ], "tags": [ "第七单元", - "直线" + "直线", + "2023届高三-第一轮复习讲义-34_直线及其方程" ], "genre": "解答题", "ans": "$0$", @@ -281784,7 +283873,9 @@ ], "tags": [ "第七单元", - "直线" + "直线", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-34_直线及其方程" ], "genre": "解答题", "ans": "$x+3y-6=0$或$3x-y+6=0$", @@ -282295,7 +284386,8 @@ ], "tags": [ "第七单元", - "圆" + "圆", + "2023届高三-第一轮复习讲义-35_圆及曲线方程" ], "genre": "解答题", "ans": "$x-y+1=0$", @@ -282410,7 +284502,8 @@ ], "tags": [ "第七单元", - "圆" + "圆", + "2023届高三-第一轮复习讲义-35_圆及曲线方程" ], "genre": "解答题", "ans": "$(x-\\dfrac{12}5)^2+(y+\\dfrac{16}5)^2=1$", @@ -282914,7 +285007,8 @@ ], "tags": [ "第七单元", - "椭圆" + "椭圆", + "2023届高三-第一轮复习讲义-36_椭圆与双曲线的概念及性质" ], "genre": "解答题", "ans": "$\\dfrac{x^2}{5.9^2}+\\dfrac{y^2}{5.8^2}=1$(以$1\\times 10^8\\text{km}$为单位)", @@ -283975,7 +286069,8 @@ ], "tags": [ "第六单元", - "空间向量" + "空间向量", + "2023届高三-第一轮复习讲义-32_空间向量的概念与性质及立体几何中的证明问题" ], "genre": "解答题", "ans": "证明略", @@ -284009,7 +286104,9 @@ ], "tags": [ "第六单元", - "空间向量" + "空间向量", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-32_空间向量的概念与性质及立体几何中的证明问题" ], "genre": "解答题", "ans": "$\\dfrac 12 \\overrightarrow{a}+\\dfrac 12\\overrightarrow{b}+\\dfrac 12 \\overrightarrow {c}$", @@ -284173,7 +286270,9 @@ ], "tags": [ "第六单元", - "空间向量" + "空间向量", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-33_立体几何中的定量计算" ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\dfrac{\\sqrt{51}}{17}$; (3) $\\dfrac{\\sqrt{41}}8$", @@ -284408,7 +286507,9 @@ ], "tags": [ "第六单元", - "空间向量" + "空间向量", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-33_立体几何中的定量计算" ], "genre": "解答题", "ans": "$\\arccos \\dfrac{3\\sqrt{10}}{10}$", @@ -284495,7 +286596,9 @@ ], "tags": [ "第六单元", - "空间向量" + "空间向量", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-32_空间向量的概念与性质及立体几何中的证明问题" ], "genre": "解答题", "ans": "证明略", @@ -285411,7 +287514,9 @@ "K0406002X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-31_数列的递推与通项及数学归纳法" ], "genre": "解答题", "ans": "(1) $-4,-6,-6,-4$; (2) $-\\dfrac 12,\\dfrac 12,-\\dfrac 12,\\dfrac 12$", @@ -285497,7 +287602,8 @@ "K0406005X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第一轮复习讲义-31_数列的递推与通项及数学归纳法" ], "genre": "解答题", "ans": "有最大项$a_3$, 无最小项", @@ -285608,7 +287714,9 @@ "K0406005X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-四月错题重做-03_数列", + "2023届高三-第一轮复习讲义-31_数列的递推与通项及数学归纳法" ], "genre": "解答题", "ans": "第$10$项最大, 第$9$项最小", @@ -285877,7 +287985,8 @@ "K0408003X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第一轮复习讲义-31_数列的递推与通项及数学归纳法" ], "genre": "解答题", "ans": "证明略", @@ -286130,7 +288239,9 @@ "K0227005X" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-较难题", + "2023届高三-第一轮复习讲义-28_导数的概念及常用公式" ], "genre": "解答题", "ans": "(1) $4\\pi \\text{rad}/\\text{s}$; (2) $12\\pi\\text{rad}/\\text{s}$", @@ -286159,7 +288270,8 @@ "K0228002X" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第一轮复习讲义-28_导数的概念及常用公式" ], "genre": "解答题", "ans": "(1) $-\\dfrac{\\sqrt{3}}3$; (2) $0$; (3) $\\dfrac{\\sqrt{3}}3$", @@ -286188,7 +288300,9 @@ "K0228004X" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-28_导数的概念及常用公式" ], "genre": "解答题", "ans": "$f(1)=1$, $f'(1)=4$", @@ -286423,7 +288537,8 @@ "K0230003X" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第一轮复习讲义-28_导数的概念及常用公式" ], "genre": "解答题", "ans": "(1) $h(1)=\\dfrac{22}3$, $h'(1)=9$; (2) $h(1)=\\dfrac{47}3$, $h'(1)=44$; (3) $h(1)=\\dfrac 76$, $h'(1)=-\\dfrac 1{12}$;", @@ -286577,7 +288692,8 @@ "K0228001X" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第一轮复习讲义-28_导数的概念及常用公式" ], "genre": "解答题", "ans": "(1) 不是, 导数$\\dfrac 1x$不会等于$-1$; (2) $b=2$或$-2$", @@ -286714,7 +288830,8 @@ "K0230004X" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第一轮复习讲义-28_导数的概念及常用公式" ], "genre": "解答题", "ans": "(1) $f'(x)=2x\\sin 3x+3x^2\\cos 3x+\\dfrac 1{\\sqrt{x^3}}$; (2) $f'(x)=\\dfrac{4}{\\mathrm{e}^{2x}+\\mathrm{e}^{-2x}+2}$", @@ -287734,7 +289851,8 @@ "objs": [], "tags": [ "第八单元", - "组合" + "组合", + "2023届高三-第一轮复习讲义-38_计数原理与排列组合" ], "genre": "解答题", "ans": "$24$", @@ -287906,7 +290024,11 @@ "objs": [], "tags": [ "第八单元", - "组合" + "组合", + "2023届高三-四月错题重做-05_易错题-概率与统计", + "2023届高三-四月错题重做-05_概率与统计", + "2023届高三-寒假作业-较难题", + "2023届高三-第一轮复习讲义-38_计数原理与排列组合" ], "genre": "解答题", "ans": "$2174$", @@ -288754,7 +290876,8 @@ "objs": [], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "解答题", "ans": "$0.2$", @@ -288806,7 +290929,8 @@ "content": "$A$校$66$名高一年级学生身高(单位: $\\text{cm}$)与体重(单位: $\\text{cm}$)的数据, 见下表. \n\\begin{center}\n\\begin{longtable}{|c|c|c|c|c|c|c|c|c|}\n\\hline\n性别 & 身高/$\\text{cm}$ & 体重/$\\text{kg}$ & 性别 & 身高/$\\text{cm}$ & 体重/$\\text{kg}$ & 性别 & 身高/$\\text{cm}$ & 体重/$\\text{kg}$ \\\\ \\hline\n\\endhead\n女 & $152$ & $46$ & 女 & $164$ & $52$ & 男 & $172$ & $92$ \\\\ \\hline\n女 & $153$ & $47$ & 男 & $165$ & $54$ & 男 & $172$ & $64$ \\\\ \\hline\n女 & $154$ & $63$ & 男 & $165$ & $60$ & 女 & $172$ & $69$ \\\\ \\hline\n女 & $155$ & $50$ & 男 & $165$ & $48$ & 男 & $173$ & $75$ \\\\ \\hline\n女 & $156$ & $48$ & 女 & $165$ & $51$ & 男 & $173$ & $72$ \\\\ \\hline\n女 & $156$ & $50$ & 女 & $165$ & $55$ & 男 & $174$ & $55$ \\\\ \\hline\n女 & $156$ & $51$ & 女 & $165$ & $58$ & 男 & $174$ & $56$ \\\\ \\hline\n女 & $157$ & $51$ & 女 & $165$ & $63$ & 男 & $174$ & $63$ \\\\ \\hline\n女 & $157$ & $50$ & 男 & $166$ & $64$ & 男 & $174$ & $74$ \\\\ \\hline\n女 & $159$ & $49$ & 男 & $167$ & $54$ & 男 & $175$ & $53$ \\\\ \\hline\n女 & $159$ & $51$ & 男 & $167$ & $52$ & 男 & $176$ & $64$ \\\\ \\hline\n女 & $160$ & $47$ & 男 & $167$ & $53$ & 男 & $176$ & $60$ \\\\ \\hline\n女 & $160$ & $62$ & 女 & $167$ & $69$ & 男 & $177$ & $63$ \\\\ \\hline\n女 & $160$ & $50$ & 女 & $167$ & $61$ & 男 & $177$ & $75$ \\\\ \\hline\n女 & $160$ & $63$ & 男 & $168$ & $97$ & 男 & $178$ & $62$ \\\\ \\hline\n女 & $161$ & $53$ & 女 & $168$ & $60$ & 男 & $178$ & $60$ \\\\ \\hline\n女 & $162$ & $84$ & 女 & $168$ & $44$ & 男 & $178$ & $73$ \\\\ \\hline\n女 & $163$ & $66$ & 男 & $170$ & $53$ & 男 & $178$ & $68$ \\\\ \\hline\n女 & $163$ & $53$ & 男 & $170$ & $54$ & 男 & $179$ & $78$ \\\\ \\hline\n女 & $164$ & $63$ & 男 & $170$ & $57$ & 男 & $181$ & $80$ \\\\ \\hline\n女 & $164$ & $68$ & 男 & $170$ & $47$ & 男 & $182$ & $92$ \\\\ \\hline\n女 & $164$ & $52$ & 男 & $170$ & $69$ & 男 & $184$ & $78$ \\\\ \\hline\n\\end{longtable}\n\\end{center}\n试计算它们的相关系数.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "解答题", "ans": "$0.561$", @@ -288834,7 +290958,8 @@ "content": "某公司为研究工人操作熟练程度对产品合格率的影响, 随机抽取$15$名工人进行调查, 得到如下数据:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n\\hline\n工人编号 & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$ \\\\ \\hline\n操作熟练程度/$\\%$ & $7.6$ & $15.2$ & $37.9$ & $45.5$ & $7.6$ & $0.0$ & $15.2$ & $75.8$ \\\\ \\hline\n产品合格率/$\\%$ & $50$ & $55$ & $68$ & $75$ & $52$ & $30$ & $55$ & $90$ \\\\ \\hline\n工人编号 & $9$ & $10$ & $11$ & $12$ & $13$ & $14$ & $15$ & / \\\\ \\hline\n操作熟练程度/$\\%$ & $90.9$ & $60.6$ & $7.6$ & $15.2$ & $37.9$ & $45.5$ & $98.5$ & / \\\\ \\hline\n产品合格率/$\\%$ & $92$ & $80$ & $58$ & $60$ & $70$ & $80$ & $95$ & / \\\\ \\hline\n\\end{tabular}\n\\end{center}\n试计算工人操作熟练程度与产品合格率的相关系数.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "解答题", "ans": "$0.945$", @@ -288884,7 +291009,8 @@ "content": "如果两种证券在一段时间内收益数据的相关系数为正数, 那么表明\\bracket{20}.\n\\onech{两种证券的收益之间存在完全同向的联动关系, 即同时涨或同时跌}{两种证券的收益之间存在完全反向的联动关系, 即涨或跌是相反的}{两种证券的收益有同向变动的倾向}{两种证券的收益有反向变动的倾向}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "选择题", "ans": "C", @@ -288956,7 +291082,8 @@ "content": "下表中是某家庭$2009$年至$2018$年电费开支的情况, 设年电费开支为$y$(单位: 元), 试建立年份$x$与$y$的回归方程.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline\n年份$x$ & $2009$ & $2010$ & $2011$ & $2012$ & $2013$ & $2014$ & $2015$ & $2016$ & $2017$ & $2018$ \\\\ \\hline\n电费$y$/元 & $1323$ & $1552$ & $1679$ & $1852$ & $1975$ & $2129$ & $2327$ & $2494$ & $2667$ & $2791$ \\\\ \\hline\n\\end{tabular}\n\\end{center}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "解答题", "ans": "$y=161.64x-323388$", @@ -288984,7 +291111,8 @@ "content": "随机抽取$8$对成年母女的身高数据(单位: $\\text{cm}$), 试据此建立母亲身高与女儿身高的回归方程.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n\\hline\n母亲身高$x$/$\\text{cm}$ & $154$ & $157$ & $158$ & $159$ & $160$ & $161$ & $162$ & $163$ \\\\ \\hline\n女儿身高$y$/$\\text{cm}$ & $155$ & $156$ & $159$ & $162$ & $161$ & $164$ & $165$ & $166$ \\\\ \\hline\n\\end{tabular}\n\\end{center}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "解答题", "ans": "$y=1.34x-53.12$", @@ -289012,7 +291140,8 @@ "content": "某生物学家对白鲸游泳速度与其摆尾频率之间的关系进行了研究. 研究的样本为$19$头白鲸, 测量其游泳速度和摆尾频率. 白鲸游泳速度的测量单位为每秒向前移动的身长数($1.0$代表每秒向前移动一个身长), 而摆尾频率的测量单位是赫兹($1.0$代表每秒摆尾$1$个来回). 测量数据如下表所示. \n\\begin{center}\n\\begin{longtable}{|c|c|c|c|c|c|}\n\\hline\n白鲸编号 & 游泳速度/($\\text{L}$/$\\text{s}$) & 摆尾频率/$\\text{Hz}$ & 白鲸编号 & 游泳速度/($\\text{L}$/$\\text{s}$) & 摆尾频率/$\\text{Hz}$ \\\\ \\hline\n\\endhead\n$1$ & $0.37$ & $0.62$ & $2$ & $0.50$ & $0.68$ \\\\ \\hline\n$3$ & $0.35$ & $0.68$ & $4$ & $0.34$ & $0.71$ \\\\ \\hline\n$5$ & $0.46$ & $0.80$ & $6$ & $0.44$ & $0.88$ \\\\ \\hline\n$7$ & $0.51$ & $0.88$ & $8$ & $0.68$ & $0.92$ \\\\ \\hline\n$9$ & $0.51$ & $1.08$ & $10$ & $0.67$ & $1.14$ \\\\ \\hline\n$11$ & $0.68$ & $1.20$ & $12$ & $0.86$ & $1.38$ \\\\ \\hline\n$13$ & $0.68$ & $1.41$ & $14$ & $0.73$ & $1.44$ \\\\ \\hline\n$15$ & $0.95$ & $1.49$ & $16$ & $0.79$ & $1.50$ \\\\ \\hline\n$17$ & $0.84$ & $1.50$ & $18$ & $1.06$ & $1.56$ \\\\ \\hline\n$19$ & $1.04$ & $1.67$ & / & / & / \\\\ \\hline\n\\end{longtable}\n\\end{center}\n生物学家聚焦的研究问题是``白鲸的摆尾频率依赖于其游泳速度吗'', 这里的因变量$y$是摆尾频率, 自变量$x$是游泳速度.\\\\\n(1) 绘制数据散点图;\\\\\n(2) 建立$x$与$y$的回归方程.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "解答题", "ans": "(1) 散点图略; (2) $y=1.44x+0.19$", @@ -289062,7 +291191,8 @@ "content": "某工厂生产某种产品的月产量(单位: 千件)与单位成本(单位: 元/件)的数据如下:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\n月份 & 产量$x$/千件 & 单位成本$y$/(元/件) & 月份 & 产量$x$/千件 & 单位成本$y$/(元/件) \\\\ \\hline\n$1$ & $2$ & $73$ & $2$ & $3$ & $72$ \\\\ \\hline\n$3$ & $4$ & $71$ & $4$ & $3$ & $73$ \\\\ \\hline\n$5$ & $4$ & $69$ & $6$ & $5$ & $68$ \\\\ \\hline\n\\end{tabular}\n\\end{center}\n(1) 计算产量与单位成本的相关系数;\\\\\n(2) 建立产量与单位成本的回归方程;\\\\\n(3) 若该工厂计划$7$月份生产$7$千件该产品, 则单位成本预计是多少?", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "解答题", "ans": "(1) $r\\approx -0.909$; (2) $y=-1.818 x+77.364$; (3) $64.6$元/件", @@ -289090,7 +291220,8 @@ "content": "为了解大学校园附近餐馆的月营业收入(单位: 千元)和该店周围的大学生人数(单位: 千人)之间的关系, 抽取了$10$所大学附近餐馆的有关数据, 如下表所示.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline\n学生人数$x$/千人 & $2$ & $6$ & $8$ & $8$ & $12$ & $16$ & $20$ & $20$ & $22$ & $26$ \\\\ \\hline\n月营业收入$y$/千元 & $58$ & $105$ & $88$ & $118$ & $117$ & $137$ & $157$ & $169$ & $149$ & $202$ \\\\ \\hline\n\\end{tabular}\n\\end{center}\n(1) 根据以上数据, 建立月营业收入$y$与该店周围的大学生人数$x$的回归方程;\\\\\n(2) 已知某餐馆周围的大学生人数为$10000$人, 试对该店月营业收入作出预测.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "解答题", "ans": "(1) $y=5x+60$; (2) $110$千元", @@ -289118,7 +291249,8 @@ "content": "某运动生理学家在一项健身活动中选择了$19$位参与者, 以他们的皮下脂肪厚度来估计身体的脂肪含量, 其中脂肪含量以占体重(单位: $\\text{kg}$)的百分比表示. 得到脂肪含量和体重的数据如下表所示. 其中, 参与者$1-10$为男性, $11-19$为女性.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\n参与者编号 & 体重$x$/$\\text{kg}$ & 脂肪含量$y$/$\\%$ & 参与者编号 & 体重$x$/$\\text{kg}$ & 脂肪含量$y$/$\\%$ \\\\ \\hline\n$1$ & $89$ & $28$ & $2$ & $88$ & $27$ \\\\ \\hline\n$3$ & $66$ & $24$ & $4$ & $59$ & $23$ \\\\ \\hline\n$5$ & $93$ & $29$ & $6$ & $73$ & $25$ \\\\ \\hline\n$7$ & $82$ & $29$ & $8$ & $77$ & $25$ \\\\ \\hline\n$9$ & $100$ & $30$ & $10$ & $67$ & $23$ \\\\ \\hline\n$11$ & $57$ & $29$ & $12$ & $68$ & $32$ \\\\ \\hline\n$13$ & $69$ & $35$ & $14$ & $59$ & $31$ \\\\ \\hline\n$15$ & $62$ & $29$ & $16$ & $59$ & $26$ \\\\ \\hline\n$17$ & $56$ & $28$ & $18$ & $66$ & $33$ \\\\ \\hline\n$19$ & $72$ & $33$ & / & / & / \\\\ \\hline\n\\end{tabular}\n\\end{center}\n(1) 分别建立男性和女性体重与脂肪含量的回归方程;\\\\\n(2) 男性和女性合在一起所构成的样本的回归方程为$y=0.021x+26.88$, 其斜率与(1)中所计算的斜率有差异吗? 能否对这种差异进行解释?\\\\\n(3) 计算下列情况下体重与脂肪含量的相关系数: \\textcircled{1} 男性; \\textcircled{2} 女性; \\textcircled{3} 男女合计. 这些值与(2)中所反映的信息是否一致?", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "解答题", "ans": "(1) 男性: $y=0.186x+11.571$; 女性: $y=0.403x+5.240$; (2) 有差异, 男女身体构造不同; (3) $r_{男性}\\approx 0.935$, $r_{女性}\\approx 0.813$, $r_{合计}\\approx 0.078$, 与(2)中反映的信息一致", @@ -289146,7 +291278,8 @@ "content": "我国$1999$年至$2018$年国内游客数量与年份关系如下表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline\n年份($Y$) & $1999$ & $2000$ & $2001$ & $2002$ & $2003$ & $2004$ & $2005$ \\\\ \\hline\n$N$ & $71900$ & $74400$ & $78400$ & $87800$ & $87000$ & $110200$ & $121200$ \\\\ \\hline\n$\\ln N$ & $11.18$ & $11.22$ & $11.27$ & $11.38$ & $11.37$ & $11.61$ & $11.71$ \\\\ \\hline\n年份($Y$) & $2006$ & $2007$ & $2008$ & $2009$ & $2010$ & $2011$ & $2012$ \\\\ \\hline\n$N$ & $139400$ & $161000$ & $171200$ & $190200$ & $210300$ & $264100$ & $295700$ \\\\ \\hline\n$\\ln N$ & $11.85$ & $11.99$ & $12.05$ & $12.16$ & $12.26$ & $12.48$ & $12.60$ \\\\ \\hline\n年份($Y$) & $2013$ & $2014$ & $2015$ & $2016$ & $2017$ & $2018$ & / \\\\ \\hline\n$N$ & $326200$ & $361100$ & $400000$ & $444000$ & $500000$ & $553900$ & / \\\\ \\hline\n$\\ln N$ & $12.70$ & $12.80$ & $12.90$ & $13.00$ & $13.12$ & $13.22$ & / \\\\ \\hline\n\\end{tabular}\n\\end{center}\n(1) 完成回归模型, 并据此模型预测$2021$年我国国内的游客数量;\\\\\n(2) 查阅$2021$年我国国内实际游客数量, 与上述模型预测数据进行比较, 并讨论数据出现偏差的原因.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "解答题", "ans": "(1) $\\ln N=0.114744Y-218.321$, 预测约$788000$万; (2) 实际仅有$324600$万人次, 可能受到疫情影响", @@ -289196,7 +291329,8 @@ "content": "慢性气管炎是一种常见的呼吸道疾病. 医药研究人员对甲、乙两种中草药治疗慢性气管炎的效果进行了对比, 所得数据如下表所示.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n & 有效 & 无效 & 总计 \\\\ \\hline\n甲药 & $184$ & $61$ & $245$ \\\\ \\hline\n乙药 & $91$ & $9$ & $100$ \\\\ \\hline\n总计 & $275$ & $70$ & $345$ \\\\ \\hline\n\\end{tabular}\n\\end{center}\n根据表中的数据回答:甲、乙两种中草药的疗效有无显著差异?", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "解答题", "ans": "有显著差异($\\chi^2=11.10$)", @@ -289245,7 +291379,8 @@ "content": "证明本节中的公式: $\\chi^2=\\dfrac{n(ad-bc)^2}{(a+b)(c+d)(a+c)(b+d)}$.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "解答题", "ans": "证明略", @@ -289272,7 +291407,8 @@ "content": "已知集合$A=\\{1,2,3,4\\}$, $B=\\{2,4,6\\}$, 则$A\\cup B=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期周末卷-周末卷01" ], "genre": "填空题", "ans": "$\\{1,2,3,4,6\\}$", @@ -289311,7 +291447,9 @@ "content": "不等式$\\dfrac{x-2}{x+1}<0$的解集为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期周末卷-周末卷01", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$(-1,2)$", @@ -289352,7 +291490,9 @@ "content": "函数$y=\\lg (x-1)+\\dfrac 1{\\sqrt {2-x}}$的定义域是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷01", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$(1,2)$", @@ -289389,7 +291529,9 @@ "content": "函数$y=\\sin( \\omega x-\\dfrac{\\pi}{3})$($\\omega >0$)的最小正周期是$\\pi$, 则$\\omega =$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-周末卷01", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$2$", @@ -289480,7 +291622,8 @@ "content": "已知幂函数$f(x)={x^\\alpha}$的图像过点$(2,\\dfrac{\\sqrt 2}2)$, 则$f(x)$的定义域为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷01" ], "genre": "填空题", "ans": "$(0,+\\infty)$", @@ -289515,7 +291658,8 @@ "objs": [], "tags": [ "第八单元", - "加法原理与乘法原理" + "加法原理与乘法原理", + "2023届高三-上学期周末卷-周末卷01" ], "genre": "填空题", "ans": "$60$", @@ -289564,7 +291708,9 @@ "content": "设集合$M=\\{x|x^2\\le 1\\}$, $N=\\{b\\}$, 若$M\\cup N=M$, 则实数$b$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期周末卷-周末卷01", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$[-1,1]$", @@ -289634,7 +291780,8 @@ "content": "如图, 在$\\triangle ABC$中, $\\angle B=45^\\circ$, $D$是$BC$边上的一点, $AD=5$, $AC=7$, $DC=3$, 则$AB$的长为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.25]\n\\draw (0,0) node [below] {$D$} coordinate (D);\n\\draw (3,0) node [below] {$C$} coordinate (C);\n\\draw (-2.5,{5*sqrt(3)/2}) node [above] {$A$} coordinate (A);\n\\draw ({-2.5-5*sqrt(3)/2},0) node [below] {$B$} coordinate (B);\n\\draw (B) -- (C) -- (A) -- cycle (A) -- (D);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-周末卷01" ], "genre": "填空题", "ans": "$\\dfrac{5\\sqrt{6}}2$", @@ -289683,7 +291830,8 @@ "content": "若函数$f(x)$满足: \\textcircled{1} 在定义域$D$内是严格单调函数; \\textcircled{2} 存在$[a, b]\\subseteq D$($a=latex,scale = 0.5]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (1,0,{sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw (A) -- (B) -- (C);\n\\draw (A) --++ (0,5) node [left] {$A_1$} coordinate (A1);\n\\draw (B) --++ (0,5) node [below right] {$B_1$} coordinate (B1);\n\\draw (C) --++ (0,5) node [right] {$C_1$} coordinate (C1);\n\\draw (A1) -- (B1) -- (C1) -- cycle;\n\\draw [dashed] (A) -- (C);\n\\draw (A) -- ($(B)!{1/6}!(B1)$) -- ($(C)!{1/3}!(C1)$) ($(A)!0.5!(A1)$) -- ($(B)!{2/3}!(B1)$) -- ($(C)!{5/6}!(C1)$);\n\\draw [dashed] ($(C)!{1/3}!(C1)$) -- ($(A)!0.5!(A1)$) ($(C)!{5/6}!(C1)$) -- (A1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期周末卷-周末卷01" ], "genre": "填空题", "ans": "$13$", @@ -289751,7 +291900,8 @@ "content": "``$x<2$''是``$x^2<4$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充分必要条件}{既非充分又非必要条件}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期周末卷-周末卷01" ], "genre": "选择题", "ans": "B", @@ -289785,7 +291935,8 @@ "content": "对任意向量$\\overrightarrow a$、$\\overrightarrow b$, 下列关系式中不恒成立的是\\bracket{20}.\n\\twoch{$(\\overrightarrow a+\\overrightarrow b)^2=|\\overrightarrow a+\\overrightarrow b|^2$}{$(\\overrightarrow a+\\overrightarrow b)\\cdot (\\overrightarrow a-\\overrightarrow b)=\\overrightarrow a^2-\\overrightarrow b^2$}{$|\\overrightarrow a\\cdot \\overrightarrow b|\\le|\\overrightarrow a|\\cdot|\\overrightarrow b|$}{$|\\overrightarrow a-\\overrightarrow b|\\le||\\overrightarrow a|-|\\overrightarrow b||$}", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期周末卷-周末卷01" ], "genre": "选择题", "ans": "D", @@ -289819,7 +291970,8 @@ "content": "设$m$、$n$为两条直线, $\\alpha$、$\\beta$为两个平面, 则下列命题中假命题是\\bracket{20}.\n\\twoch{若$m\\perp n$, $m\\perp \\alpha$, $n\\perp \\beta$, 则$\\alpha \\perp \\beta$}{若$m\\parallel n$, $m\\perp \\alpha$, $n\\parallel \\beta$, 则$\\alpha \\perp \\beta$}{若$m\\perp n$, $m\\parallel \\alpha$, $n\\parallel \\beta$, 则$\\alpha \\parallel \\beta$}{若$m\\parallel n$, $m\\perp \\alpha$, $n\\perp \\beta$, 则$\\alpha \\parallel \\beta$}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期周末卷-周末卷01" ], "genre": "选择题", "ans": "C", @@ -289853,7 +292005,10 @@ "content": "已知函数$y=f(x)$的定义域为$D$, $x_1,x_2\\in D$. 关于$y=f(x)$ 的两个命题:\\\\\n命题\\textcircled{1}: 若当$f(x_1)+f(x_2)=0$时, 都有$x_1+x_2=0$, 则函数$y=f(x)$是$D$上的奇函数.\\\\\n命题\\textcircled{2}: 若当$f(x_1)=latex,scale = 0.8]\n\\draw (0,0,0) node [left] {$B$} coordinate (B);\n\\draw (0,2,0) node [above] {$A$} coordinate (A);\n\\draw (0,0,2) node [below] {$C$} coordinate (C);\n\\draw ({2*sqrt(2)},0,2) node [right] {$D$} coordinate (D);\n\\draw ($(B)!0.5!(D)$) node [above] {$M$} coordinate (M);\n\\draw [dashed] (C) -- (B) -- (D) (A) -- (B) (C) -- (M);\n\\draw (A) -- (C) -- (D) -- cycle;\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期周末卷-周末卷01" ], "genre": "解答题", "ans": "(1) $\\dfrac{4\\sqrt{2}}3$; (2) $\\arccos\\dfrac{\\sqrt{3}}6$.", @@ -289933,7 +292089,8 @@ "content": "已知$x\\in \\mathbf{R}$, 设$\\overrightarrow m=(2\\cos x, \\sin x+\\cos x)$, $\\overrightarrow n=(\\sqrt 3\\sin x, \\sin x-\\cos x)$, 记函数$f(x)=\\overrightarrow m\\cdot \\overrightarrow n$.\\\\\n(1) 求函数$f(x)$取最小值时$x$的取值范围;\\\\\n(2) 设$\\triangle ABC$的角$A$, $B$, $C$所对的边分别为$a$, $b$, $c$, 若$f(C)=2$, $c=\\sqrt 3$, 求$\\triangle ABC$的面积$S$的最大值.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-周末卷01" ], "genre": "解答题", "ans": "(1) $\\{x|x=k\\pi-\\dfrac\\pi 6, \\ k\\in \\mathbf{Z}\\}$; (2) $\\dfrac{3\\sqrt{3}}4$.", @@ -289967,7 +292124,8 @@ "content": "如图, 某城市有一矩形街心广场$ABCD$, 如图. 其中$AB=4$百米, $BC=3$百米.现将在其内部挖掘一个三角形水池$DMN$种植荷花, 其中点$M$在边$BC$上, 点$N$在边$AB$上, 要求$\\angle MDN=\\dfrac{\\pi}4$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw (0,0) node [below] {$A$} coordinate (A) -- (4,0) node [below] {$B$} coordinate (B) -- (4,3) node [above] {$C$} coordinate (C) -- (0,3) node [above] {$D$} coordinate (D) -- cycle;\n\\draw (2,0) node [below] {$N$} coordinate (N) -- (4,1) node [right] {$M$} coordinate (M) -- (D) -- cycle;\n\\end{tikzpicture}\n\\end{center}\n(1) 若$AN=CM=2$百米, 判断$\\triangle DMN$是否符合要求, 并说明理由;\\\\\n(2) 设$\\angle CDM=\\theta$, 写出$\\triangle DMN$面积的$S$关于$\\theta$的表达式, 并求$S$的最小值.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-周末卷01" ], "genre": "解答题", "ans": "(1) 不符合要求, 证明略; (2) $S=\\dfrac{12}{1+\\sin(2\\theta+\\dfrac \\pi 4)}, \\ \\theta\\in [0,\\arctan\\dfrac 34]$, $S$的最小值为$12\\sqrt{2}-12$.", @@ -290001,7 +292159,10 @@ "content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n$, 且$a_1=1$, $a_2=a$.\\\\\n(1) 若数列$\\{a_n\\}$是等差数列, 且$a_8=15$, 求实数$a$的值;\\\\\n(2) 若数列$\\{a_n\\}$满足$a_{n+2}-a_n=2$($n\\in \\mathbf{N}$且$n\\ge 1$), 且$S_{19}=19a_{10}$, 求证: 数列$\\{a_n\\}$是等差数列;", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-上学期周末卷-周末卷01", + "2023届高三-四月错题重做-03_数列", + "2023届高三-四月错题重做-03_易错题-数列" ], "genre": "解答题", "ans": "(1) $3$; (2) 证明略.", @@ -290046,7 +292207,8 @@ "content": "已知函数$f(x)=2^x+k\\cdot 2^{-x}$($x\\in \\mathbf{R}$).\\\\\n(1) 判断函数$f(x)$的奇偶性, 并说明理由;\\\\\n(2) 设$k>0$, 问函数$f(x)$的图像是否关于某直线$x=m$成轴对称图形, 如果是, 求出$m$的值; 如果不是, 请说明理由; (可利用真命题: ``函数$g(x)$的图像关于某直线$x=m$成轴对称图形''的充要条件为``函数$g(m+x)$是偶函数'')\\\\\n(3) 设$k=-1$, 函数$h(x)=a\\cdot 2^x-2^{1-x}-\\dfrac 43a$, 若函数$f(x)$与$h(x)$的图像有且只有一个公共点, 求实数$a$的取值范围.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷01" ], "genre": "解答题", "ans": "(1) 当$k=1$时, $y=f(x)$是偶函数; 当$k=-1$时, $y=f(x)$是奇函数; 当$k\\ne 1$且$k \\ne -1$时, $y=f(x)$既不是奇函数, 又不是偶函数; (2) $m=\\log_4 k$; (3) $\\{-3\\}\\cup (1,+\\infty)$.", @@ -290080,7 +292242,8 @@ "content": "设集合$A=\\{x\\in\\mathbf{R}|0\\le x\\le 1\\}$, $B=\\{x\\in \\mathbf{R}|(x-1)(x-2)\\le 0\\}$, 则$A\\cup B=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期周末卷-周末卷02" ], "genre": "填空题", "ans": "$[0,2]$", @@ -290144,7 +292307,8 @@ "content": "若$0<\\alpha<\\pi$, $\\cos\\alpha=-\\dfrac 13$, 则$\\tan\\alpha=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-周末卷02" ], "genre": "填空题", "ans": "$-2\\sqrt{2}$", @@ -290179,7 +292343,9 @@ "content": "复数$\\dfrac{2+4\\mathrm{i}}{1+\\mathrm{i}}$的虚部为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期周末卷-周末卷02", + "2023届高三-寒假作业-中档题" ], "genre": "填空题", "ans": "$1$", @@ -290214,7 +292380,8 @@ "content": "若正方体的棱长为$1$, 则其外接球的体积为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期周末卷-周末卷02" ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{3}}2\\pi$", @@ -290249,7 +292416,8 @@ "content": "已知函数$f(x)=\\sin(3x+\\varphi)$($-\\dfrac\\pi 2<\\varphi<\\dfrac\\pi 2$)的图像关于直线$x=\\dfrac \\pi 4$对称, 则$\\varphi=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-周末卷02" ], "genre": "填空题", "ans": "$-\\dfrac\\pi 4$", @@ -290285,7 +292453,8 @@ "objs": [], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-上学期周末卷-周末卷02" ], "genre": "填空题", "ans": "$\\dfrac 12$", @@ -290320,7 +292489,10 @@ "content": "若抛物线$y^2=8x$的准线与曲线$\\dfrac{x^2}a+\\dfrac{y^2}4=1$($a>0$)只有一个公共点, 则$a$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-上学期周末卷-周末卷02", + "2023届高三-四月错题重做-04_易错题-解析几何", + "2023届高三-四月错题重做-04_解析几何" ], "genre": "填空题", "ans": "$\\{4\\}$", @@ -290367,7 +292539,8 @@ "content": "设函数$f(x)=\\dfrac 1x-\\lg x$, 则不等式$f(\\dfrac 1x-1)<1$的解集为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷02" ], "genre": "填空题", "ans": "$(0,\\dfrac 12)$", @@ -290403,7 +292576,8 @@ "objs": [], "tags": [ "第二单元", - "第一单元" + "第一单元", + "2023届高三-上学期周末卷-周末卷02" ], "genre": "填空题", "ans": "$\\mathrm{e}^{\\mathrm{e}}$", @@ -290438,7 +292612,8 @@ "content": "正方形$ABCD$的边长为$4$, $O$是正方形$ABCD$的中心, 过中心$O$的直线$l$与边$AB$交于点$M$, 与边$CD$交于点$N$. $P$为平面上一点, 满足$2\n\\overrightarrow{OP}=\\lambda \\overrightarrow{OB}+(1-\\lambda)\\overrightarrow{OC}$, 则$\\overrightarrow{PM}\\cdot \\overrightarrow{PN}$的最小值为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期周末卷-周末卷02" ], "genre": "填空题", "ans": "$-7$", @@ -290475,7 +292650,8 @@ "content": "已知常数$b,c\\in \\mathbf{R}$, 若函数$f(x)=x+\\dfrac bx+c$在区间$[1,+\\infty)$上存在零点, 则$b^2+c^2$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷02" ], "genre": "填空题", "ans": "$[\\dfrac 12,+\\infty)$", @@ -290510,7 +292686,8 @@ "content": "曲线$y^2=8x$的准线方程是\\bracket{20}.\n\\fourch{$x=4$}{$x=2$}{$x=-2$}{$x=-4$}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-上学期周末卷-周末卷02" ], "genre": "选择题", "ans": "C", @@ -290568,7 +292745,10 @@ "objs": [], "tags": [ "第六单元", - "第七单元" + "第七单元", + "2023届高三-上学期周末卷-周末卷02", + "2023届高三-四月错题重做-04_易错题-解析几何", + "2023届高三-四月错题重做-04_解析几何" ], "genre": "选择题", "ans": "B", @@ -290612,7 +292792,8 @@ "content": "已知常数$b,c\\in \\mathbf{R}$, 关于$x$的方程$x^2+b|x|+c=0$在复数集$\\mathbf{C}$上给出下列两个结论: \\textcircled{1} 存在$b,c$, 使得该方程有且只有两个根, 且这两个根互为共轭虚根; \\textcircled{2} 存在$b,c$, 使得该方程有且只有$6$个互不相等的根, 则\\bracket{20}.\n\\fourch{\\textcircled{1}与\\textcircled{2}均正确}{\\textcircled{1}正确, \\textcircled{2}不正确}{\\textcircled{1}不正确, \\textcircled{2}正确}{\\textcircled{1}与\\textcircled{2}均不正确}", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期周末卷-周末卷02" ], "genre": "选择题", "ans": "A", @@ -290647,7 +292828,8 @@ "content": "设常数$a\\in \\mathbf{R}$, 函数$f(x)=a\\sin 2x+\\cos(2\\pi-2x)+1$.\\\\\n(1) 若$a=\\sqrt{3}$, 求$f(x)$的单调递增区间;\\\\\n(2) 若$f(x)$为偶函数, 求$f(x)$的值域.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-周末卷02" ], "genre": "解答题", "ans": "(1) $[k\\pi-\\dfrac \\pi 3,k\\pi+\\dfrac \\pi 6], \\ k\\in \\mathbf{Z}$; (2) $[0,2]$.", @@ -290682,7 +292864,8 @@ "content": "设$\\triangle ABC$的内角$A,B,C$所对的边分别为$a,b,c$, 且满足$\\sin A=\\sqrt{3}\\sin B$, $C=\\dfrac\\pi 6$.\\\\\n(1) 若$ac=\\sqrt{3}$, 求$\\triangle ABC$的面积;\\\\\n(2) 能否将$\\triangle ABC$的边长按某种顺序排列为一个等比数列? 说明理由.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-周末卷02" ], "genre": "解答题", "ans": "(1) $\\dfrac{\\sqrt{3}}4$; (2) 不能, 证明略.", @@ -290717,7 +292900,8 @@ "content": "某商场共有三层楼, 在其圆柱形空间内安装两部等长的扶梯I和II供顾客乘用. 如图, 一顾客自一楼点$A$处乘I到达二楼的点$B$处后, 沿着二楼面上的圆弧$BM$逆时针步行至点$C$处, 且$C$为圆弧$BM$的中点, 再乘II到达三楼的点$D$处. 设圆柱形空间三个楼面圆的中心分别为$O,O_1,O_2$, 半径为$8\\text{m}$, 相邻楼层的间距$AM=4\\text{m}$, 两部扶梯与楼面所成角的大小均为$\\arcsin\\dfrac 13$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.2]\n\\filldraw (0,8) circle (0.1) node [left] {$O_2$};\n\\filldraw (0,4) circle (0.1) node [left] {$O_1$};\n\\filldraw (0,0) circle (0.1) node [left] {$O$};\n\\draw (0,8) ellipse (8 and 2);\n\\draw (8,4) arc (0:-180:8 and 2) (8,0) arc (0:-180:8 and 2);\n\\draw [dashed] (8,4) arc (0:180:8 and 2) (8,0) arc (0:180:8 and 2);\n\\draw (8,0) node [right] {$A$} coordinate (A) -- (8,4) node [right] {$M$} coordinate (M) -- (8,8);\n\\draw (-8,0) -- (-8,8);\n\\draw ({8*cos(-120)},{4+2*sin(-120)}) node [below] {$B$} coordinate (B);\n\\draw ({8*cos(-55)},{4+2*sin(-55)}) node [below] {$C$} coordinate (C);\n\\draw ({8*cos(60)},{8+2*sin(60)}) node [above] {$D$} coordinate (D);\n\\draw [dashed] (A) -- (B) node [midway, below] {I};\n\\draw [dashed] (C) -- (D) node [midway, below left] {II};\n\\end{tikzpicture}\n\\end{center}\n(1) 求此顾客在二楼面上步行的路程;\\\\\n(2) 求异面直线$AB$与$CD$所成角的大小.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期周末卷-周末卷02" ], "genre": "解答题", "ans": "(1) $2\\pi$米; (2) $\\arccos\\dfrac{4\\sqrt{2}-1}9$.", @@ -290752,7 +292936,8 @@ "content": "已知曲线$\\Gamma:x^2-y|y|=1$与$x$轴分别相交于$A,B$两点($A$在$B$的左侧), $\\Gamma$与$y$轴相交于点$C$. 已知$F_1(-c,0)$, $F_2(c,0)$, $c>0$, $\\triangle BCF_1$的面积为$\\dfrac{1+\\sqrt{2}}{2}$.\\\\\n(1) 若过$F_2$的直线$l$与$\\Gamma$有且仅有一个公共点, 直接写出$l$倾斜角的取值范围;\\\\\n(2) 过点$B$作斜率存在的直线$m$交$\\Gamma$于$P,Q$两点(异于点$B$), 且点$P$在第一象限, 求证: $P,Q$的横坐标之积为定值, 并求该定值;\\\\\n(3) 在(2)的条件下, 当$\\overrightarrow{F_1P}\\cdot \\overrightarrow{F_1Q}=3+2\\sqrt{2}$时, 求$\\dfrac{|AP|}{|AQ|}$的值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-上学期周末卷-周末卷02" ], "genre": "解答题", "ans": "(1) $[\\dfrac\\pi 4,\\dfrac{3\\pi}4]$; (2) 定值为$1$; (3) $3+2\\sqrt{2}$.", @@ -290787,7 +292972,8 @@ "content": "已知数列$\\{a_n\\}$满足$a_n\\ne 0$恒成立.\\\\\n(1) 若$a_na_{n+2}=ka_{n+1}^2$且$a_n>0$, 当$\\{\\lg a_n\\}$成等差数列时, 求$k$的值;\\\\\n(2) 若$a_na_{n+2}=2a_{n+1}^2$且$a_n>0$, 当$a_1=1$, $a_4=16\\sqrt{2}$时, 求$\\{a_n\\}$的通项公式;\\\\\n(3) 若$a_na_{n+2}=-\\dfrac 12a_{n+1}a_{n+3}$, $a_1=-1$, $a_3\\in [4, 8]$, $a_{2020}<0$, 求$a_1+a_2+\\cdots+a_{2020}$的最大值.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-上学期周末卷-周末卷02" ], "genre": "解答题", "ans": "(1) $1$; (2) $a_n=2^{\\frac 12(n-1)^2}$; (3) $\\dfrac{1-4^{505}}{3}$.", @@ -290822,7 +293008,8 @@ "content": "已知全集$U=\\mathbf{R}$, 集合$A=\\{x||x-1|>1\\}$, $B=\\{x|\\dfrac{x-3}{x+1}<0\\}$, 则$\\overline{A}\\cap B=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期周末卷-周末卷03_暂未使用" ], "genre": "填空题", "ans": "$[0,2]$", @@ -290860,7 +293047,8 @@ "content": "已知幂函数的图像过点$(2,\\dfrac 14)$, 则该幂函数的单调递增区间是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷03_暂未使用" ], "genre": "填空题", "ans": "$(-\\infty,0)$", @@ -290926,7 +293114,8 @@ "content": "某圆锥体的底面圆的半径长为$\\sqrt 2$, 其侧面展开图是圆心角为$\\dfrac 23\\pi$的扇形, 则该圆锥体的体积是\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期周末卷-周末卷03_暂未使用" ], "genre": "填空题", "ans": "$\\dfrac{8\\pi}3$", @@ -290964,7 +293153,8 @@ "content": "过点$P(-2,1)$作圆$x^2+y^2=5$的切线, 则该切线的点法向式方程是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-上学期周末卷-周末卷03_暂未使用" ], "genre": "填空题", "ans": "$2(x+2)-(y-1)=0$", @@ -291015,7 +293205,8 @@ "content": "函数$f(x)=\\sqrt 3\\sin x\\cos x+\\cos ^2x$的最大值为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-周末卷03_暂未使用" ], "genre": "填空题", "ans": "$\\dfrac 32$", @@ -291078,7 +293269,8 @@ "objs": [], "tags": [ "第八单元", - "组合" + "组合", + "2023届高三-上学期周末卷-周末卷03_暂未使用" ], "genre": "填空题", "ans": "$25$", @@ -291140,7 +293332,8 @@ "content": "已知函数$f(x)=\\begin{cases} \\log_2(x+a), & -a0 \\end{cases}$有三个不同的零点, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷03_暂未使用" ], "genre": "填空题", "ans": "$[1,+\\infty)$", @@ -291176,7 +293369,8 @@ "content": "在边长为$1$的正六边形$ABCDEF$中, 记以$A$为起点, 其余顶点为终点的向量分别为$\\overrightarrow{a_1}$, $\\overrightarrow{a_2}$, $\\overrightarrow{a_3}$, $\\overrightarrow{a_4}$, $\\overrightarrow{a_5}$, 若$\\overrightarrow{a_i}$与$\\overrightarrow{a_j}$的夹角记为$\\theta _{ij}$, 其中$i,j\\in \\{1,2,3,4,5\\}$, 且$i\\ne j$, 则$|\\overrightarrow{a_i}|\\cdot \\cos \\theta _{ij}$的最大值为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期周末卷-周末卷03_暂未使用" ], "genre": "填空题", "ans": "$\\sqrt{3}$", @@ -291198,7 +293392,8 @@ "content": "设$l_1$、$l_2$是平面上过点$M$, 夹角为$\\dfrac{\\pi }3$的两条直线, 且与圆心为$O$, 半径长为$1$的圆均相切(圆心在两直线所夹的锐角中), 设圆周上一点$P$到$l_1$、$l_2$的距离分别为$d_1$、$d_2$, 那么$2d_1+d_2$的最小值为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-上学期周末卷-周末卷03_暂未使用" ], "genre": "填空题", "ans": "$3-\\sqrt{3}$", @@ -291220,7 +293415,8 @@ "content": "设函数$y=f(x)$, ``该函数的图像过点$(1,1)$''是``该函数为幂函数''的\t\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷03_暂未使用" ], "genre": "选择题", "ans": "B", @@ -291264,7 +293460,8 @@ "content": "如图, 平面直角坐标系中, 曲线(实线部分)的方程可以是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.8]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -{sqrt(3)}:{sqrt(3)},samples = 100] plot ({sqrt(\\x*\\x+1)},\\x);\n\\draw [domain = -{sqrt(3)}:{sqrt(3)},samples = 100] plot ({-sqrt(\\x*\\x+1)},\\x);\n\\draw (-1,0) node [below left] {$-1$} -- (0,-1) node [below left] {$-1$} -- (1,0) node [below right] {$1$};\n\\draw [dashed] (-2,1) -- (-1,0) (1,0) -- (2,1);\n\\end{tikzpicture}\n\\end{center}\n\\twoch{$(|x|-y-1)\\cdot (1-x^2+y^2)=0$}{$\\sqrt {|x|-y-1}\\cdot (1-x^2+y^2)=0$}{$(|x|-y-1)\\cdot \\sqrt {1-x^2+y^2}=0$}{$\\sqrt {|x|-y-1}\\cdot \\sqrt {1-x^2+y^2}=0$}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-上学期周末卷-周末卷03_暂未使用" ], "genre": "选择题", "ans": "C", @@ -291286,7 +293483,8 @@ "content": "在正方体$ABCD-A_1B_1C_1D_1$的八个顶点中任取两个点作直线, 与直线$A_1B$异面且夹角成$60^{\\circ}$的直线的条数为\\bracket{20}.\n\\fourch{$3$}{$4$}{$5$}{$6$}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期周末卷-周末卷03_暂未使用" ], "genre": "选择题", "ans": "B", @@ -291308,7 +293506,8 @@ "content": "已知正方体$ABCD-A_1B_1C_1D_1$的棱长为$2$, 点$E$、$F$分别是所在棱$A_1B_1$、$AB$的中点, 点$O_1$是面$A_1B_1C_1D_1$的中心, 如图所示.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$A$} coordinate (A) --++ (2,0) node [below right] {$B$} coordinate (B) --++ (45:{2/2}) node [right] {$C$} coordinate (C)\n--++ (0,2) node [above right] {$C_1$} coordinate (C1)\n--++ (-2,0) node [above left] {$D_1$} coordinate (D1) --++ (225:{2/2}) node [left] {$A_1$} coordinate (A1) -- cycle;\n\\draw (A) ++ (2,2) node [right] {$B_1$} coordinate (B1) -- (B) (B1) --++ (45:{2/2}) (B1) --++ (-2,0);\n\\draw [dashed] (A) --++ (45:{2/2}) node [left] {$D$} coordinate (D) --++ (2,0) (D) --++ (0,2);\n\\draw (A1) -- ($(A)!0.5!(B)$) node [below] {$F$};\n\\draw [dashed] (C) -- ($(A1)!0.5!(B1)$) node [above] {$E$};\n\\filldraw ($(A1)!0.5!(C1)$) circle (0.03) node [right] {$O_1$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求三棱锥$O_1-FBC$的体积$V_{O_1-FBC}$;\\\\\n(2) 求异面直线$A_1F$与$CE$所成角的大小.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期周末卷-周末卷03_暂未使用" ], "genre": "解答题", "ans": "(1) $\\dfrac 23$; (2) $\\arctan {2\\sqrt{5}}5$", @@ -291330,7 +293529,8 @@ "content": "已知函数$f(x)=\\dfrac a{2^x-1}+b$, 其中$a$、$b\\in \\mathbf{R}$.\\\\\n(1) 当$a=6$, $b=0$时, 求满足$f(|x|)=2^x$的$x$的值;\\\\\n(2) 若$f(x)$为奇函数且非偶函数, 求$a$与$b$的关系式.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷03_暂未使用" ], "genre": "解答题", "ans": "(1) $\\log_2 3$; (2) $a=2b\\ne 0$", @@ -291353,7 +293553,8 @@ "objs": [], "tags": [ "第三单元", - "第七单元" + "第七单元", + "2023届高三-上学期周末卷-周末卷03_暂未使用" ], "genre": "解答题", "ans": "(1) $\\sqrt{7}$千米; (2) 有$\\dfrac{8-\\sqrt{15}}7$小时, 两人不能通话", @@ -291375,7 +293576,8 @@ "content": "已知椭圆$\\Gamma:\\dfrac{x^2}9+\\dfrac{y^2}4=1$.\\\\\n(1) 若抛物线$C$的焦点与$\\Gamma$的焦点重合, 求$C$的标准方程;\\\\\n(2) 若$\\Gamma$的上顶点$A$、右焦点$F$及$x$轴上一点$M$构成直角三角形, 求点$M$的坐标;\\\\\n(3) 若$O$为$\\Gamma$的中心, $P$为$\\Gamma$上一点(非$\\Gamma$的顶点), 过$\\Gamma$的左顶点$B$, 作$BQ\\parallel OP$, $BQ$交$y$轴于点$Q$, 交$\\Gamma$于点$N$, 求证: $\\overrightarrow{BN}\\cdot \\overrightarrow{BQ}=2\\overrightarrow{OP}^2$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-上学期周末卷-周末卷03_暂未使用" ], "genre": "解答题", "ans": "(1) $y^2=4\\sqrt{5} x$或$y^2=-4\\sqrt{5} x$; (2) $M$的坐标为$(0,0)$或$(-\\dfrac{4\\sqrt{5}}5,0)$; (3) 证明略", @@ -291397,7 +293599,8 @@ "content": "给定整数$n$($n\\ge 4$), 设集合$A=\\{a_1,a_2,\\cdots ,a_n\\}$, $B=\\{a_i+a_j|a_i,a_j\\in A,\\ 1\\le i\\le j\\le n\\}$.\\\\\n(1) 若$A=\\{-3,0,1,2\\}$, 求集合$B$;\\\\\n(2) 若$a_1,a_2,\\cdots ,a_n$构成以$a_1$为首项, $d$($d>0$)为公差的等差数列, 求证: 集合$B$中的元素个数为$2n-1$;\\\\\n(3) 若$a_1,a_2,\\cdots ,a_n$构成以$3$为首项, $3$为公比的等比数列, 求集合$B$中元素的个数及所有元素之和.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-上学期周末卷-周末卷03_暂未使用" ], "genre": "解答题", "ans": "(1) $\\{-6,-3,-2,-1,0,1,2,3,4\\}$; (2) 证明略; (3) 元素个数为$\\dfrac 12 n(n+1)$; 元素之和为$\\dfrac{n+1}2(3^{n+1}-3)$", @@ -292027,7 +294230,9 @@ "content": "若关于$x$、$y$的方程组$\\begin{cases} 2x+3y=1 \\\\ ax-y=2 \\end{cases}$无解, 则实数$a=$\\blank{50}.", "objs": [], "tags": [ - "暂无对应" + "暂无对应", + "2023届高三-赋能-赋能08", + "2023届高三-赋能-赋能10" ], "genre": "填空题", "ans": "$-\\dfrac 23$", @@ -292924,7 +295129,9 @@ "content": "已知集合$A=(-\\infty ,-3)$, $B=(-4,+\\infty)$, 则$A\\cap B=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期周末卷-周末卷09", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$(-4,-3)$", @@ -292987,7 +295194,9 @@ "content": "已知复数$z$满足$\\dfrac 1{z-1}=\\mathrm{i}$($\\mathrm{i}$为虚数单位), 则$z=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期周末卷-周末卷09", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$1-\\mathrm{i}$", @@ -293023,7 +295232,9 @@ "content": "函数$f(x)=\\log_2(2x+4)$的反函数为$f^{-1}(x)$, 则$f^{-1}(4)=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷09", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$6$", @@ -293060,7 +295271,8 @@ "objs": [], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-上学期周末卷-周末卷09" ], "genre": "填空题", "ans": "$\\dfrac 16$", @@ -293097,7 +295309,9 @@ ], "tags": [ "第八单元", - "二项式定理" + "二项式定理", + "2023届高三-上学期周末卷-周末卷09", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$160$", @@ -293171,7 +295385,9 @@ "content": "已知圆锥的底面半径为$1$, 高为$\\sqrt 3$, 则该圆锥的侧面展开图的圆心角$\\theta$的大小为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期周末卷-周末卷09", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$\\pi$", @@ -293207,7 +295423,9 @@ "content": "已知$\\alpha \\in (0,\\pi)$, 且有$1-2\\sin 2\\alpha =\\cos 2\\alpha$, 则$\\cos \\alpha =$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-周末卷09", + "2023届高三-寒假作业-中档题" ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{5}}5$", @@ -293265,7 +295483,8 @@ "content": "若$a,b$分别是正数$p$, $q$的算术平均数和几何平均数, 且$a,b,-2$ 这三个数可适当排序后成等差数列, 也可适当排序后成等比数列, 则$p+q+pq$的值形成的集合是\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-上学期周末卷-周末卷09" ], "genre": "填空题", "ans": "$\\{9\\}$", @@ -293301,7 +295520,8 @@ "content": "设$f(x)=x^2+2a\\cdot x+b\\cdot 2^x$, 其中$a,b\\in \\mathbf{N}$, $x\\in \\mathbf{R}$, 如果函数$y=f(x)$与函数$y=f(f(x))$都有零点且它们的零点完全相同, 则有序数对$(a,b)$为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷09" ], "genre": "填空题", "ans": "$(0,0)$或$(1,0)$", @@ -293359,7 +295579,8 @@ "content": "在$\\triangle ABC$中, 若$\\overrightarrow{AB}\\cdot \\overrightarrow{BC}+\\overrightarrow{AB}^2=0$, 则$\\triangle ABC$的形状一定是\\bracket{20}.\n\\fourch{等边三角形}{直角三角形}{等腰三角形}{等腰直角三角形}", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-周末卷09" ], "genre": "选择题", "ans": "B", @@ -293393,7 +295614,8 @@ "content": "已知函数$f(x)=A\\sin (\\omega x+\\varphi)$($A>0$, $\\omega >0$)的图像与直线$y=b$($09$成立}{存在实数$x$、$y$满足$\\begin{cases}|x|\\le 1, \\\\|x+y|\\le 1, \\end{cases}$ 并使得$4(x+1)(y+1)>7$成立}{满足$\\begin{cases}|x|\\le 1, \\\\|x+y|\\le 1, \\end{cases}$ 且使得$4(x+1)(y+1)=-9$的实数$x$、$y$不存在}{满足$\\begin{cases}|x|\\le 1, \\\\|x+y|\\le 1, \\end{cases}$ 且使得$4(x+1)(y+1)<-9$的实数$x$、$y$不存在}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期周末卷-周末卷09" ], "genre": "选择题", "ans": "A", @@ -293465,7 +295688,8 @@ "content": "如图在三棱锥$P-ABC$中, 棱$AB$、$AC$、$AP$两两垂直, $AB=AC=AP=3$, 点$M$在$AP$上, 且$AM=1$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,2) node [left] {$B$} coordinate (B);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw ($(A)!0.5!(P)$) node [right] {$M$} coordinate (M);\n\\draw (P) -- (B) -- (C) -- cycle;\n\\draw [dashed] (A) -- (B) (A) -- (C) (A) -- (P) (M) -- (B) (M) -- (C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求异面直线$BM$和$PC$所成的角的大小;\\\\\n(2) 求三棱锥$P-BMC$的体积.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期周末卷-周末卷09" ], "genre": "解答题", "ans": "(1) $\\arccos\\dfrac{\\sqrt{5}}{10}$; (2) $3$", @@ -293501,7 +295725,8 @@ "content": "已知函数$f(x)=\\sin x\\cos (\\dfrac{\\pi }2+x)+\\sqrt 3\\sin x\\cos x$.\\\\\n(1) 求函数$f(x)$的最小正周期及对称中心;\\\\\n(2) 若$f(x)=a$在区间$[0,\\dfrac{\\pi }2]$上有两个解$x_1,x_2$, 求$a$的取值范围及$x_1+x_2$的值.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷09" ], "genre": "解答题", "ans": "(1) 最小正周期为$\\pi$, 对称中心为$(\\dfrac{k\\pi}2-\\dfrac\\pi{12},-\\dfrac 12)$($k\\in \\mathbf{Z}$); (2) $a$的取值范围为$[0,\\dfrac 12)$, $x_1+x_2=\\dfrac\\pi 3$", @@ -293535,7 +295760,8 @@ "content": "某企业接到生产$3000$台某产品的甲、乙、丙$3$种部件的订单, 每台产品需要这$3$种部件的数量分别为$2, 2, 1$(单位: 件). 已知每个工人每天可生产甲部件$6$件, 或乙部件$3$件, 或丙部件$2$件. 该企业计划安排$200$名工人分成三组分别生产这$3$种部件, 生产乙部件的人数与生产甲部件的人数成正比例, 比例系数为$k$($k\\ge 2$为正整数).\\\\\n(1) 设生产甲部件的人数为$x$, 分别写出完成甲、乙、丙$3$种部件生产需要的时间;\\\\\n(2) 假设这$3$种部件的生产同时开工, 试确定正整数$k$的值, 使完成订单任务的时间最短, 并给出时间最短时具体的人数分组方案.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷09" ], "genre": "解答题", "ans": "(1) 完成三种部件依次需要$\\dfrac{1000}{x}$天, $\\dfrac{2000}{kx}$天, $\\dfrac{1500}{200-(k+1)x}$天; (2) $k=2$时能使完成订单任务的时间最短, 具体人数分组方案为甲$44$人, 乙$88$人, 丙$68$人", @@ -293613,7 +295839,8 @@ "content": "函数$f(x)=x^{- \\frac 23}$的定义域为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷10" ], "genre": "填空题", "ans": "$(-\\infty,0)\\cup (0,+\\infty)$", @@ -293674,7 +295901,9 @@ "content": "设$a\\in \\mathbf{R}$, $a^2-a-2+(a+1)\\mathrm{i}$为纯虚数($\\mathrm{i}$为虚数单位), 则$a=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期周末卷-周末卷10", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$2$", @@ -293706,7 +295935,8 @@ "content": "若$\\triangle ABC$中, $a+b=4$, $\\angle C=30^\\circ$, 则$\\triangle ABC$面积的最大值是\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-周末卷10" ], "genre": "填空题", "ans": "$1$", @@ -293753,7 +295983,8 @@ "content": "若函数$f(x)=\\log_2\\dfrac{x-a}{x+1}$的反函数的图像过点$(-2,3)$, 则$a=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷10" ], "genre": "填空题", "ans": "$2$", @@ -293788,7 +296019,8 @@ "content": "椭圆$\\dfrac{x^2}9+\\dfrac{y^2}4=1$的焦点为$F_1,F_2$, $P$为椭圆上一点, 若$|PF_1|=5$, 则$\\cos \\angle F_1PF_2=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-上学期周末卷-周末卷10" ], "genre": "填空题", "ans": "$\\dfrac 35$", @@ -293820,7 +296052,8 @@ "content": "已知数列$\\{a_n\\}$的通项公式为$a_n=\\begin{cases} n, & n\\le 2, \\\\ (\\dfrac 12)^{n-1}, & n\\ge 3 \\end{cases}$($n\\in \\mathbf{N}$, $n\\ge 1$). $S_n$是数列$\\{a_n\\}$的前$n$项和. 则$\\displaystyle\\lim_{n\\to \\infty}S_n=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-上学期周末卷-周末卷10" ], "genre": "填空题", "ans": "$\\dfrac 72$", @@ -293874,7 +296107,9 @@ "content": "在直角坐标平面$xOy$中, $A(-2,0)$, $B(0,1)$, 动点$P$在圆$C:x^2+y^2=2$上, 则$\\overrightarrow{PA}\\cdot \\overrightarrow{PB}$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期周末卷-周末卷10", + "2023届高三-寒假作业-中档题" ], "genre": "填空题", "ans": "$[2-\\sqrt{10},2+\\sqrt{10}]$", @@ -293906,7 +296141,10 @@ "content": "已知$f(x)=\\begin{cases} x^2-4x+3, & x\\le 0, \\\\ -x^2-2x+3, & x>0, \\end{cases}$ 当$x\\in [a,a+1]$时, 不等式$f(x+a)\\ge f(2a-x)$恒成立, 则实数$a$的最大值是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷10", + "2023届高三-四月错题重做-02_函数二", + "2023届高三-四月错题重做-02_易错题-函数2" ], "genre": "填空题", "ans": "$-2$", @@ -293972,7 +296210,8 @@ "content": "如图所示, 已知函数$y=\\dfrac{1+x}x$($x>0$)图像上的点$A$, 和函数$y=\\dfrac{1-x}x$($x>0$)上的两点$B$、$C$, 且线段$AB$平行于$y$轴, 当三角形$ABC$为正三角形时, 点$C$的坐标为$(p,q)$, 则$\\dfrac pq$的值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.5,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = {2/7}:5, samples = 100] plot (\\x,{(1-\\x)/\\x}) node [right] {$y=\\dfrac{1-x}{x}$($x>0$)};\n\\draw [domain = {2/3}:5, samples = 100] plot (\\x,{(1+\\x)/\\x}) node [right] {$y=\\dfrac{1+x}{x}$($x>0$)};\n\\draw ({1/2*(sqrt(3)+sqrt(3+4*sqrt(3)))},{1/(1/2*(sqrt(3)+sqrt(3+4*sqrt(3))))}) ++ (0,1) node [above] {$A$} coordinate (A);\n\\draw (A) ++ (0,-2) node [below] {$B$} coordinate (B);\n\\draw (A) ++ (210:2) node [left] {$C$} coordinate (C);\n\\draw (A) -- (B) -- (C) -- cycle;\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷10" ], "genre": "填空题", "ans": "$\\sqrt{3}$", @@ -294004,7 +296243,8 @@ "content": "若$\\overrightarrow a$与$\\overrightarrow b-\\overrightarrow c$都是非零向量, 则``$\\overrightarrow a\\cdot \\overrightarrow b=\\overrightarrow a\\cdot \\overrightarrow c$''是``$\\overrightarrow a\\perp (\\overrightarrow b-\\overrightarrow c)$''的\\bracket{20}条件.\n\\fourch{充分不必要}{必要不充分}{充分必要}{既不充分也不必要}", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期周末卷-周末卷10" ], "genre": "选择题", "ans": "C", @@ -294036,7 +296276,9 @@ "content": "一个公司有$8$名员工, 其中$6$位员工的月工资分别为$5200$、$5300$、$5500$、$6100$、$6500$、$6600$, 另两位员工数据不清楚, 那么$8$位员工月工资的中位数不可能是\\bracket{20}.\n\\fourch{$5800$}{$6000$}{$6200$}{$6400$}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-上学期周末卷-周末卷10", + "2023届高三-寒假作业-容易题" ], "genre": "选择题", "ans": "D", @@ -294068,7 +296310,8 @@ "content": "设$z_1,z_2$为复数, 则下列命题中一定成立的是\\bracket{20}.\n\\twoch{如果$z_1-z_2>0$, 那么$z_1>z_2$}{如果$|z_1|=|z_2|$, 那么$z_1=\\pm z_2$}{如果$|\\dfrac{z_1}{z_2}|>1$, 那么$|z_1|>|z_2|$}{如果$z_1^2+z_2^2=0$, 那么$z_1=z_2=0$}", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期周末卷-周末卷10" ], "genre": "选择题", "ans": "C", @@ -294122,7 +296365,8 @@ "content": "如图, 正四棱柱$ABCD-A_1B_1C_1D_1$的底面边长为$1$, 异面直线$AD$与$BC_1$所成角的大小为$60^\\circ$, 求:\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$A$} coordinate (A) --++ (2,0) node [below right] {$B$} coordinate (B) --++ (45:{2/2}) node [right] {$C$} coordinate (C)\n--++ (0,{2*sqrt(3)}) node [above right] {$C_1$} coordinate (C1)\n--++ (-2,0) node [above left] {$D_1$} coordinate (D1) --++ (225:{2/2}) node [left] {$A_1$} coordinate (A1) -- cycle;\n\\draw (A) ++ (2,{2*sqrt(3)}) node [above] {$B_1$} coordinate (B1) -- (B) (B1) --++ (45:{2/2}) (B1) --++ (-2,0);\n\\draw [dashed] (A) --++ (45:{2/2}) node [below] {$D$} coordinate (D) --++ (2,0) (D) --++ (0,{2*sqrt(3)});\n\\draw (B) -- (C1) (A) -- (B1) -- (D1);\n\\draw [dashed] (A) -- (D1) (D) -- (C1);\n\\end{tikzpicture}\n\\end{center}\n(1) 线段$A_1B_1$到底面$ABCD$的距离;\\\\\n(2) 三棱椎$B_1-ABC_1$的体积.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期周末卷-周末卷10" ], "genre": "解答题", "ans": "(1) $\\sqrt{3}$; (2) $\\dfrac{\\sqrt{3}}6$", @@ -294154,7 +296398,8 @@ "content": "已知函数$f(x)=2^x+\\dfrac a{2^x}$, 其中$a$为实常数.\\\\\n(1) 若$f(0)=7$, 解关于$x$的方程$f(x)=5$;\\\\\n(2) 若对于任意的$x\\in \\mathbf{R}$, 恒有$f(x)\\ge a$成立, 求$a$的取值范围.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷10" ], "genre": "解答题", "ans": "(1) $\\{1,\\log_2 3\\}$; (2) $[0,4]$", @@ -294187,7 +296432,8 @@ "objs": [], "tags": [ "第九单元", - "第八单元" + "第八单元", + "2023届高三-上学期周末卷-周末卷10" ], "genre": "解答题", "ans": "(1) $n=400$; (2) $|x-y|=4$; (3) $\\dfrac{7}{10}$", @@ -294241,7 +296487,8 @@ "content": "已知无穷数列$\\{a_n\\}$的前$n$项和为$S_n$, 若对于任意的正整数$n$, 均有$S_{2n-1}\\ge 0$, $S_{2n}\\le 0$, 则称数列$\\{a_n\\}$具有性质$P$.\\\\\n(1) 判断首项为$1$, 公比为$-2$的无穷等比数列$\\{a_n\\}$是否具有性质$P$, 并说明理由;\\\\\n(2) 已知无穷数列$\\{a_n\\}$具有性质$P$, 且任意相邻四项之和都相等, 求证: $S_4=0$;\\\\\n(3) 已知$b_n=2n-1$($n\\in \\mathbf{N}$, $n\\ge 1$), 数列$\\{c_n\\}$是等差数列, $a_n=\\begin{cases} b_{\\frac{n+1}2}, & n\\text{为奇数}, \\\\c_{\\frac n2}, & n\\text{为偶数}. \\end{cases}$ 若无穷数列$\\{a_n\\}$具有性质$P$, 求$c_{2021}$的取值范围.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-上学期周末卷-周末卷10" ], "genre": "解答题", "ans": "(1) 具有性质$P$, 理由略; (2) 证明略; (3) $[-4043,-4041]$", @@ -294273,7 +296520,9 @@ "content": "不等式$\\dfrac 1x<1$的解集为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期周末卷-周末卷11", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$(-\\infty,0)\\cup (1,+\\infty)$", @@ -294316,7 +296565,8 @@ "content": "抛物线$y^2=2x$的焦点坐标为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-上学期周末卷-周末卷11" ], "genre": "填空题", "ans": "$(\\dfrac 12,0)$", @@ -294387,7 +296637,8 @@ "content": "已知向量$\\overrightarrow a=(1,-2)$, $\\overrightarrow b=(3,4)$, 则向量$\\overrightarrow a$在向量$\\overrightarrow b$的方向上的投影为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期周末卷-周末卷11" ], "genre": "填空题", "ans": "$(-\\dfrac 35,-\\dfrac 45)$", @@ -294422,7 +296673,9 @@ "content": "已知数列$\\{a_n\\}$为等差数列, 其前$n$项和为$S_n$.若$S_9=36$, 则$a_3+a_4+a_8=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-上学期周末卷-周末卷11", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$12$", @@ -294455,7 +296708,8 @@ "content": "已知直线$l:x-y+b=0$被圆$C:x^2+y^2=25$所截得的弦长为$6$, 则$b=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-上学期周末卷-周末卷11" ], "genre": "填空题", "ans": "$\\pm 4\\sqrt{2}$", @@ -294491,7 +296745,8 @@ "content": "已知函数$y=f(x)$是定义在$\\mathbf{R}$上的偶函数, 且在$[0,+\\infty)$上是严格增函数, 若$f(a+1)\\le f(4)$, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷11" ], "genre": "填空题", "ans": "$[-5,3]$", @@ -294525,7 +296780,9 @@ "content": "函数$f(x)=(\\sqrt 3\\sin x+\\cos x)(\\sqrt 3\\cos x-\\sin x)$的最小正周期为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷11", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$\\pi$", @@ -294561,7 +296818,8 @@ "content": "过双曲线$C:\\dfrac{x^2}{a^2}-\\dfrac{y^2}4=1$的右焦点$F$作一条垂直于$x$轴的垂线交双曲线$C$的两条渐近线于$A$、$B$两点, $O$为坐标原点, 则$\\triangle OAB$的面积的最小值为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-上学期周末卷-周末卷11" ], "genre": "填空题", "ans": "$8$", @@ -294597,7 +296855,9 @@ "content": "若关于$x$的不等式$|2^x-m|-\\dfrac 1{2^x}<0$在区间$[0,1]$内恒成立, 则实数$m$的范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷11", + "2023届高三-寒假作业-中档题" ], "genre": "填空题", "ans": "$(\\dfrac 32,2)$", @@ -294655,7 +296915,8 @@ "content": "已知函数$f(x)=\\begin{cases} \\dfrac x{4x^2+16}, & x\\ge 2, \\\\ (\\dfrac 12)^{|x-a|}, & x<2, \\end{cases}$ 若对任意的$x_1\\in [2,+\\infty)$, 都存在唯一的$x_2\\in (-\\infty ,2)$, 满足$f(x_1)=f(x_2)$, 则实数$a$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷11" ], "genre": "填空题", "ans": "$[-2,6)$", @@ -294688,7 +296949,8 @@ "content": "若实数$x,y\\in \\mathbf{R}$, 则陈述句甲``$\\begin{cases} x+y>4, \\\\ xy>4 \\end{cases}$''是陈述句乙``$\\begin{cases} x>2, \\\\ y>2 \\end{cases}$''的\\bracket{20}条件.\n\\fourch{充分非必要}{必要非充分}{充要}{既非充分又非必要}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期周末卷-周末卷11" ], "genre": "选择题", "ans": "B", @@ -294721,7 +296983,8 @@ "content": "已知$\\triangle ABC$中, $\\angle A=\\dfrac{\\pi }2$, $AB=AC=1$, 点$P$是$AB$边上的动点, 点$Q$是$AC$边上的动点, 则$\\overrightarrow{BQ}\\cdot \\overrightarrow{CP}$的最小值为\\bracket{20}.\n\\fourch{$-4$}{$-2$}{$-1$}{$0$}", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期周末卷-周末卷11" ], "genre": "选择题", "ans": "B", @@ -294754,7 +297017,8 @@ "content": "设$\\{a_n\\}$是等差数列, 下列命题中正确的是\\bracket{20}.\n\\twoch{若$a_1+a_2>0$, 则$a_2+a_3>0$}{若$a_1+a_3<0$, 则$a_1+a_2<0$}{若$0\\sqrt {a_1a_3}$}{若$a_1<0$, 则$(a_2-a_1)(a_2-a_3)>0$}", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-上学期周末卷-周末卷11" ], "genre": "选择题", "ans": "C", @@ -294788,7 +297052,8 @@ "objs": [], "tags": [ "第五单元", - "第七单元" + "第七单元", + "2023届高三-上学期周末卷-周末卷11" ], "genre": "选择题", "ans": "A", @@ -294821,7 +297086,8 @@ "content": "如图, 四棱锥$S-ABCD$的底面是正方形, $SD \\perp$平面$ABCD$, $SD=AD=2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$D$} coordinate (D);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (2,0,2) node [right] {$B$} coordinate (B);\n\\draw (0,0,2) node [left] {$A$} coordinate (A);\n\\draw (0,2,0) node [above] {$S$} coordinate (S);\n\\draw (S) -- (A) -- (B) -- (C) -- cycle (S) -- (B);\n\\draw [dashed] (A) -- (D) -- (C) (S) -- (D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $AC\\perp SB$;\\\\\n(2) 求二面角$C-SA-D$的大小.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期周末卷-周末卷11" ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\arctan \\sqrt{2}$", @@ -294854,7 +297120,8 @@ "content": "已知函数$f(x)=2\\sqrt 3\\sin x\\cos x-2\\sin ^2x$.\\\\\n(1) 若角$\\alpha$的终边与单位圆交于点$P(\\dfrac 35,\\dfrac 45)$, 求$f(\\alpha)$的值;\\\\\n(2) 当$x\\in [-\\dfrac{\\pi }6, \\dfrac{\\pi }3]$时, 求$f(x)$的单调递增区间和值域.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷11" ], "genre": "解答题", "ans": "(1) $\\dfrac{24\\sqrt{3}-32}{25}$; (2) 单调递增区间为$[-\\dfrac\\pi 6,\\dfrac\\pi 6]$, 值域为$[-2,1]$", @@ -294887,7 +297154,8 @@ "content": "设数列$\\{a_n\\}$满足$a_{n+1}=2a_n+n^2-4n+1$, $b_n=a_n+n^2-2n$.\\\\\n(1) 若$a_1=2$, 求证: 数列$\\{b_n\\}$为等比数列;\\\\\n(2) 在(1)的条件下, 对于正整数$2$、$q$、$r$($20$, $b>0$)的左、右焦点分别是 $F_1$、$F_2$, 左、右两顶点分别是 $A_1$、$A_2$, 弦$AB$和$CD$所在直线分别平行于$x$轴与$y$轴, 线段$BA$的延长线与线段$CD$相交于点$P$(如图).\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,{-2*sqrt(3)}) -- (0,{2*sqrt(3)}) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -3:3] plot ({2*sqrt(pow(\\x,2)/3+1)},\\x);\n\\draw [domain = -3:3] plot ({-2*sqrt(pow(\\x,2)/3+1)},\\x);\n\\draw [dashed] (-4,{-2*sqrt(3)}) -- (4,{2*sqrt(3)}) (-4,{2*sqrt(3)}) -- (4,{-2*sqrt(3)});\n\\draw (1,{-2*sqrt(3)}) -- (1,{2*sqrt(3)}) node [right] {$l$};\n\\filldraw ({-sqrt(7)},0) circle (0.03) node [below] {$F_1$} ({sqrt(7)},0) circle (0.03) node [below] {$F_2$};\n\\filldraw (-2,0) circle (0.03) node [below right] {$A_1$} coordinate (A1) (2,0) circle (0.03) node [above left] {$A_2$} coordinate (A2);\n\\draw ({-4/sqrt(3)},1) node [left] {$B$} coordinate (B) -- ({4/sqrt(3)},1) node [above left] {$A$} coordinate (A);\n\\draw [dashed] (A) -- (2.5,1) node [right] {$P$} coordinate (P);\n\\draw (2.5,{-3*sqrt(3)/4}) node [right] {$D$} coordinate (D) -- (2.5,{3*sqrt(3)/4}) node [right] {$C$} coordinate (C);\n\\draw (C) -- (A1) (C) -- ($(C)!3!(A2)$) node [left] {$N$} coordinate (N) ($(C)!{1.5/4.5}!(A1)$) node [above left] {$M$} coordinate (M);\n\\filldraw (C) circle (0.03) (D) circle (0.03) (M) circle (0.03) (N) circle (0.03) (A) circle (0.03) (B) circle (0.03) (P) circle (0.03);\n\\end{tikzpicture}\n\\end{center}\n(1) 若$\\overrightarrow d=(2,\\sqrt 3)$是$\\Gamma$的一条渐近线的一个方向向量, 试求$\\Gamma$的两渐近线的夹角$\\theta$;\\\\\n(2) 若$|PA|=1$, $|PB|=5$ , $|PC|=2$, $|PD|=4$, 试求双曲线$\\Gamma$的方程;\\\\\n(3) 在(1)的条件下, 且$|A_1A_2|=4$, 点$C$与双曲线的顶点不重合, 直线$CA_1$和直线$CA_2$与直线$l:x=1$分别相交于点$M$和$N$, 试问: 以线段$MN$为直径的圆是否恒经过定点? 若是, 请求出定点的坐标; 若不是, 试说明理由.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-上学期周末卷-周末卷11" ], "genre": "解答题", "ans": "(1) $\\arccos\\dfrac 17$; (2) $\\dfrac{x^2}{27/8}-\\dfrac{y^2}{27/5}=1$; (3) 恒经过两个定点$(-\\dfrac 12,0)$与$(\\dfrac 52,0)$", @@ -295412,7 +297681,8 @@ "content": "已知实数$a$是常数, 函数$f(x)=\\log_2(a x^2+2x-a)$.\\\\\n(1) 当$a=-1$时, 求该函数的定义域;\\\\\n(2) 当$a\\le 0$时, 如果$f(x)\\ge 1$对任何$x\\in [2,3]$都成立, 求实数$a$的取值范围;\\\\\n(3) 若$a<0$, 将函数$f(x)$的图像沿$x$轴或其相反方向平移, 得到一个偶函数$g(x)$的图像, 设函数$g(x)$的最大值为$h(a)$, 求$h(a)$的最小值.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验01" ], "genre": "解答题", "ans": "(1) $(1-\\sqrt{2},1+\\sqrt{2})$; (2) $[-\\dfrac 12,+\\infty)$; (3) $1$", @@ -295470,7 +297740,9 @@ "content": "函数$f(x)=\\log_2(x-1)$的定义域为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷08", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$(1,+\\infty)$", @@ -295506,7 +297778,9 @@ "content": "已知集合$A=\\{1,2,3,4\\}$, 集合$B=\\{4,5\\}$, 则$A\\cap B=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期周末卷-周末卷08", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$\\{4\\}$", @@ -295544,7 +297818,9 @@ "content": "函数$y=2\\cos ^2x-1$的最小正周期为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-周末卷08", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$\\pi$", @@ -295581,7 +297857,8 @@ "content": "已知球的体积为$36\\pi$, 则该球大圆的面积等于\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期周末卷-周末卷08" ], "genre": "填空题", "ans": "$9\\pi$", @@ -295619,7 +297896,8 @@ ], "tags": [ "第八单元", - "二项式定理" + "二项式定理", + "2023届高三-上学期周末卷-周末卷08" ], "genre": "填空题", "ans": "$-20$", @@ -295654,7 +297932,8 @@ "content": "若圆锥的母线长$l=5(\\text{cm})$, 高$h=4(\\text{cm})$, 则这个圆锥的体积为\\blank{50}$(\\text{cm}^3)$.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期周末卷-周末卷08" ], "genre": "填空题", "ans": "$12\\pi$", @@ -295686,7 +297965,8 @@ "content": "已知函数$f(x)=a^{x+1}-2$($a>0$且$a\\ne 1$), 设$f^{-1}(x)$是$f(x)$的反函数. 若$y=f^{-1}(x)$的图像不经过第二象限, 则$a$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷08" ], "genre": "填空题", "ans": "$[2,+\\infty)$", @@ -295718,7 +297998,8 @@ "content": "函数$f(x)=2\\sin (\\omega x+\\varphi)$($\\omega >0$)的部分图像, 如图所示, 若$|AB|=5$, 则$\\omega$的值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -2:3] plot (\\x,{2*sin(pi/3*\\x/pi*180+150)});\n\\draw [dashed] (-1,0) -- (-1,2) node [above] {$A$} coordinate (A) -- (0,2) node [right] {$2$};\n\\draw [dashed] (2,0) -- (2,-2) node [below] {$B$} coordinate (B) -- (0,-2) node [left] {$-2$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-周末卷08" ], "genre": "填空题", "ans": "$\\dfrac\\pi 3$", @@ -295751,7 +298032,8 @@ "objs": [], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-上学期周末卷-周末卷08" ], "genre": "填空题", "ans": "$0.30$", @@ -295783,7 +298065,9 @@ "content": "已知函数$f(x)=\\log_a(x+b)$($a>0$, $a\\ne 1$, $b\\in \\mathbf{R}$)的图像, 如图所示, 则$a+b$的值是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -3.5:4, samples = 100] plot (\\x,{ln(\\x+4)/ln(0.5)});\n\\draw (-3,0) node [below left] {$-3$} (0,-2) node [below left] {$-2$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷08", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$\\dfrac 92$", @@ -295815,7 +298099,8 @@ "content": "函数$F(x)=\\lg x-\\sin x$零点的个数是\\blank{50}个.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷08" ], "genre": "填空题", "ans": "$3$", @@ -295847,7 +298132,10 @@ "content": "设函数$f(x)$和$g(x)$都是定义在集合$M$上的函数, 若对于任意$x\\in M$, 都有$f(g(x))=g(f(x))$成立, 就称函数$f(x)$与$g(x)$在$M$上互为``互换函数''.若存在非空集合$M$, 使得函数$f(x)=a^x$($a>0$, $a\\ne 1$)与$g(x)=x+1$在集合$M$上互为``互换函数'', 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷08", + "2023届高三-四月错题重做-02_函数二", + "2023届高三-四月错题重做-02_易错题-函数2" ], "genre": "填空题", "ans": "$(1,+\\infty)$", @@ -295891,7 +298179,8 @@ "content": "函数$f(x)=2^x-\\dfrac 1{2^x}$的图像关于\\bracket{20}.\n\\fourch{原点对称}{直线$y=x$对称}{直线$y=-x$对称}{$y$轴对称}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷08" ], "genre": "选择题", "ans": "A", @@ -295923,7 +298212,8 @@ "content": "三国时期赵爽在《勾股方圆图注》中, 对勾股定理的证明可用现代数学表述为如图所示, 可以利用该图作为几何解释的不等式性质是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$A$} coordinate (A) -- (2,0) node [right] {$E$};\n\\draw (1,0) node [below] {$H$} -- (1,2) node [above] {$D$} coordinate (D);\n\\draw (1,1) node [left] {$G$} -- (3,1) node [right] {$C$} coordinate (C);\n\\draw (2,1) node [above] {$F$} -- (2,-1) node [below] {$B$} coordinate (B);\n\\draw (A) -- (D) -- (C) -- (B) -- cycle;\n\\draw (0.5,0) node [above] {$b$} (1,1) node [below right] {$a$} ($(A)!0.5!(D)$) node [above left] {$c$};\n\\end{tikzpicture}\n\\end{center}\n\\onech{如果$a>b$, $b>c$, 那么$a>c$}{如果$a>b>0$, 那么$a^2>b^2$}{对任意正实数$a$和$b$, 有$a^2+b^2\\ge 2ab$, 当且仅当$a=b$时等号成立}{如果$a>b$, $c>0$, 那么$ac>bc$}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期周末卷-周末卷08" ], "genre": "选择题", "ans": "C", @@ -295955,7 +298245,8 @@ "content": "若函数$f(x)=\\begin{cases} \\log_2x, & x\\ge 1, \\\\ x+c, & x<1. \\end{cases}$ 则``$c=-1$''是``$y=f(x)$是$\\mathbf{R}$上的单调增函数''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分也非必要条件}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷08" ], "genre": "选择题", "ans": "A", @@ -295988,7 +298279,10 @@ "objs": [], "tags": [ "第二单元", - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-周末卷08", + "2023届高三-四月错题重做-01_函数一", + "2023届高三-四月错题重做-01_易错题-函数1" ], "genre": "选择题", "ans": "D", @@ -296032,7 +298326,8 @@ "content": "如图, 正四棱柱$ABCD-A_1B_1C_1D_1$的底面边长$AB=2$, 异面直线$A_1A$与$B_1C$所成角的大小为$\\arctan \\dfrac 12$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw (0,0) node [below left] {$A$} coordinate (A) --++ (2,0) node [below right] {$B$} coordinate (B) --++ (45:{2/2}) node [right] {$C$} coordinate (C)\n--++ (0,4) node [above right] {$C_1$} coordinate (C1)\n--++ (-2,0) node [above left] {$D_1$} coordinate (D1) --++ (225:{2/2}) node [left] {$A_1$} coordinate (A1) -- cycle;\n\\draw (A) ++ (2,4) node [above] {$B_1$} coordinate (B1) -- (B) (B1) --++ (45:{2/2}) (B1) --++ (-2,0);\n\\draw [dashed] (A) --++ (45:{2/2}) node [left] {$D$} coordinate (D) --++ (2,0) (D) --++ (0,4);\n\\draw (C) -- (B1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求$BD_1$与底面$ABCD$所成角的正切值;\\\\\n(2) 求正四棱柱$ABCD-A_1B_1C_1D_1$的体积.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期周末卷-周末卷08" ], "genre": "解答题", "ans": "(1) $\\sqrt{2}$; (2) $16$", @@ -296064,7 +298359,8 @@ "content": "在$\\triangle ABC$中, 内角$A,B,C$所对的边长分别是$a,b,c$.\\\\\n(1) 若$c=2$, $C=\\dfrac{\\pi }3$, 且$\\triangle ABC$的面积$S=\\sqrt 3$, 求$a,b$的值;\\\\\n(2) 若$\\sin (A+B)+\\sin (B-A)=\\sin 2A$, 试判断$\\triangle ABC$的形状.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-周末卷08" ], "genre": "解答题", "ans": "(1) $a=b=2$; (2) 直角三角形($A$为直角顶点)或等腰三角形($C$为顶角)", @@ -296096,7 +298392,8 @@ "content": "如图, 某班级墙上有一壁画, 最高点$A$离地面$4$米, 最低点$B$离地面$2$米, 某同学从距离墙$x$($x>1$)米, 离地面高$a$($1\\le a\\le 2$)米的$C$处观赏该壁画, 设观赏视角$\\angle ACB=\\theta$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw (0,0) -- (0,1.5) node [above left] {$C$} coordinate (C) -- (4,2) node [above left] {$B$} coordinate (B) ++ (0,2) node [above] {$A$} coordinate (A);\n\\draw [very thick] (B) -- (A);\n\\draw (C) -- (A) (B) --++ (0,-2);\n\\draw (C) pic [\"$\\theta$\", draw, angle eccentricity = 1.5] {angle = B--C--A};\n\\filldraw [gray!50] (-1,0) rectangle (5.5,-0.2);\n\\draw (-0.1,1.5) -- (-0.3,1.5);\n\\draw [->] (-0.2,1.1) -- (-0.2,1.5);\n\\draw [->] (-0.2,0.4) -- (-0.2,0);\n\\draw (-0.2,0.75) node {$a$};\n\\draw [->] (1.6,0.2) -- (0,0.2);\n\\draw [->] (2.4,0.2) -- (4,0.2);\n\\draw (2,0.2) node {$x$};\n\\draw (4.1,2) -- (4.3,2);\n\\draw [->] (4.2,0.6) -- (4.2,0);\n\\draw [->] (4.2,1.4) -- (4.2,2);\n\\draw (4.2,1) node {$2$};\n\\draw (4.1,4) -- (4.9,4);\n\\draw [->] (4.6,1.6) -- (4.6,0);\n\\draw [->] (4.6,2.4) -- (4.6,4);\n\\draw (4.6,2) node {$4$};\n\\end{tikzpicture}\n\\end{center}\n(1) 若$a=1.5$米, 问该同学离墙多远时, 视角$\\theta$最大;\\\\\n(2) 若$\\tan \\theta =\\dfrac 12$, 当$a$变化时, 求$x$的取值范围.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-周末卷08" ], "genre": "解答题", "ans": "(1) 约$1.12$米; (2) $[3,4]$(单位: 米)", @@ -296128,7 +298425,8 @@ "content": "已知函数$f(x)$的定义域是$\\{x|x\\in \\mathbf{R},\\ x\\ne \\dfrac k2, \\ k\\in \\mathbf{Z}\\}$, 且$f(x)+f(2-x)=0$, $f(x+1)=-\\dfrac 1{f(x)}$, 当$0x^2-k-1$有解? 证明你的结论.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷08" ], "genre": "解答题", "ans": "(1) 奇函数, 理由略; (2) $f(x)=3^{x-2k-1}$; (3) 存在, $k=0$", @@ -296160,7 +298458,8 @@ "content": "已知函数$f(x)=\\ln (x^{-1}+a)$.\\\\\n(1) 设$f^{-1}(x)$是$f(x)$的反函数. 当$a=1$时, 解不等式$f^{-1}(x)>0$;\\\\\n(2) 若关于$x$的方程$f(x)+\\ln (x^2)=0$的解集中恰好有一个元素, 求实数$a$的值;\\\\\n(3) 设$a>0$, 若对任意$t\\in [\\dfrac 12,1]$, 函数$f(x)$在区间$[t,t+1]$上的最大值与最小值的差不超过$\\ln 2$, 求$a$的取值范围.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷08" ], "genre": "解答题", "ans": "(1) $(0,+\\infty)$; (2) $a=-\\dfrac 14$或$0$; (3) $[\\dfrac 23,+\\infty)$", @@ -297287,7 +299586,8 @@ "content": "函数$y=\\ln (-x^2+2x+3)$的单调递减区间是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-赋能-赋能38" ], "genre": "填空题", "ans": "$[1,3)$", @@ -297377,7 +299677,8 @@ "content": "不等式$x^2-3>ax-a$对一切$3\\le x\\le 4$恒成立, 则符合要求的自然数$a$有\\blank{50}个.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-赋能-赋能39" ], "genre": "填空题", "ans": "$3$", @@ -297622,7 +299923,8 @@ "content": "市场上有一种新型的强力洗衣液, 特点是去污速度快. 已知每投放$a$($1\\le a\\le 4$, 且$a\\in \\mathbf{R}$)个单位的洗衣液在一定量水的洗衣机中, 它在水中释放的浓度$y$(克/升)随着时间$x$(分钟)变化的函数关系式近似为$y=a\\cdot f(x)$, 其中$f(x)=\\begin{cases} \\dfrac{16}{8-x}-1, & 0\\le x\\le 4 \\\\ 5-\\dfrac 12x, & 40$)的左、右焦点, 直线$l$与椭圆交于不同的两点$A$, $B$, 且$|AF_1|+|AF_2|=2\\sqrt 2$.\\\\\n(1) 求椭圆$\\Gamma$的方程;\\\\\n(2) 已知直线$l$经过椭圆的右焦点$F_2$, $P,Q$是椭圆上两点, 四边形$ABPQ$是菱形, 求直线$l$的方程;\\\\\n(3) 已知直线$l$不经过椭圆的右焦点$F_2$, 直线$AF_2$, $l$, $BF_2$的斜率依次成等差数列, 求直线$l$在$y$轴上截距的取值范围.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期测验卷-高三下学期测验01" ], "genre": "解答题", "ans": "(1) $\\dfrac{x^2}2+y^2=1$; (2) $x=\\dfrac{\\sqrt{2}}2y+1$或$x=-\\dfrac{\\sqrt{2}}2y+1$; (3) $(-\\infty,-\\sqrt{2})\\cup (\\sqrt{2},+\\infty)$", @@ -298682,7 +300987,8 @@ "content": "已知集合$A=\\{y|y=10^x, \\ x\\in \\mathbf{R}\\}$, $B=\\{y|y=x^2, \\ 1\\le x\\le 2\\}$, 则$A\\cap B=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-赋能-赋能19" ], "genre": "填空题", "ans": "$[1,4]$", @@ -298745,7 +301051,8 @@ "content": "若关于$x, y$的方程组$\\begin{cases}x+y=m, \\\\ x+ny=1\\end{cases}$有无穷多组解, 则$m+n$的值为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-赋能-赋能18" ], "genre": "填空题", "ans": "$2$", @@ -298779,7 +301086,8 @@ "content": "若$-1+2\\mathrm{i}$($\\mathrm{i}$为虚数单位)是方程$x^2+bx+c=0$($b$、$c\\in \\mathbf{R}$)的一个根, 则$c-b=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-赋能-赋能19" ], "genre": "填空题", "ans": "$3$", @@ -299174,7 +301482,8 @@ "content": "若数列$\\{a_n\\}$满足``对任意正整数$i$, $j$, $i\\ne j$, 都存在正整数$k$, 使得$a_k=a_i\\cdot a_j$'', 则称数列$\\{a_n\\}$具有``性质$P$''.\\\\\n(1) 判断各项均等于$a$的常数列是否具有``性质$P$'', 并说明理由;\\\\\n(2) 若公比为$2$的无穷等比数列$\\{a_n\\}$具有``性质$P$'', 求首项$a_1$的值;\\\\\n(3) 若首项$a_1=2$的无穷等差数列$\\{a_n\\}$具有``性质$P$'', 求公差$d$的值.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-下学期测验卷-高三下学期测验01" ], "genre": "解答题", "ans": "(1) 当$a=0$或$a=1$时, 该数列具有``性质$P$''; 当$a\\not\\in \\{0,1\\}$时, 该数列不具有``性质$P$''; (2) $2^t$($t\\ge -1$, $t\\in \\mathbf{Z}$); (3) $d=1$或$2$", @@ -299600,7 +301909,8 @@ "content": "已知三角形$ABC$中, 三个内角$ABC$的对应边分别为$a$、$b$、$c$, 且$a=5$, $b=7$.\\\\\n(1) 若$B=\\dfrac{\\pi}3$, 求$c$;\\\\\n(2) 设点$M$是边$AB$的中点, 若$CM=3$, 求三角形$ABC$的面积.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期测验卷-高三下学期月考01" ], "genre": "解答题", "ans": "(1) $c=8$; (2) $6\\sqrt{6}$", @@ -300344,7 +302654,9 @@ "K0623002B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-24_体积及表面积的计算" ], "genre": "填空题", "ans": "$\\dfrac 32$", @@ -301212,7 +303524,8 @@ ], "tags": [ "第八单元", - "二项式定理" + "二项式定理", + "2023届高三-第一轮复习讲义-39_二项式定理" ], "genre": "填空题", "ans": "$4$", @@ -301716,7 +304029,8 @@ "content": "给定数列$\\{a_n\\}$, 若满足$a_1=a$($a>0$且$a\\ne 1$), 对于任意正整数$n, m$, 都有$a_{n+m}=a_n\\cdot a_m$, 则称数列$\\{a_n\\}$为指数数列.\\\\\n(1) 已知数列$\\{a_n\\}$, $\\{b_n\\}$的通项公式分别为$a_n=3\\cdot 2^{n-1}$, $b_n=3^n$, 试判断$\\{a_n\\}$, $\\{b_n\\}$是不是指数数列(需说明理由);\\\\\n(2) 若数列$\\{a_n\\}$满足: $a_1=2$, $a_2=4$, $a_{n+2}=3a_{n+1}-2a_n$, 证明: $\\{a_n\\}$是指数数列;\\\\\n(3) 若数列$\\{a_n\\}$是指数数列, $a_1=\\dfrac{t+3}{t+4}$($t\\in \\mathbf{N}$, $t\\ge 1$), 证明: 数列$\\{a_n\\}$中任意三项都不能构成等差数列.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-下学期测验卷-高三下学期测验02" ], "genre": "解答题", "ans": "(1) $\\{a_n\\}$不是指数数列, $\\{b_n\\}$是指数数列; (2) 证明略; (3) 证明略", @@ -303232,7 +305546,8 @@ "content": "已知动直线$l$与椭圆$C: x^2+\\dfrac{y^2}2=1$交于$P(x_1,y_1)$、$Q(x_2,y_2)$两不同点, 且$\\triangle OPQ$的面积$S_{\\triangle OPQ}=\\dfrac{\\sqrt 2}2$, 其中$O$为坐标原点.\\\\\n(1) 若动直线$l$垂直于$x$轴, 求直线$l$的方程;\\\\\n(2) 证明$x_1^2+x_2^2$和$y_1^2+y_2^2$均为定值;\\\\\n(3) 椭圆$C$上是否存在点$D,E,G$, 使得三角形面积$S_{\\triangle ODE}=S_{\\triangle ODG}=S_{\\triangle OEG}=\\dfrac{\\sqrt 2}2$? 若存在, 判断$\\triangle DEG$的形状; 若不存在, 请说明理由.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期测验卷-高三下学期月考01" ], "genre": "解答题", "ans": "(1) $x=\\dfrac{\\sqrt{2}}2$或$x=-\\dfrac{\\sqrt{2}}2$; (2) $x_1^2+x_2^2=1$, $y_1^2+y_2^2=2$; (3) 不存在, 证明略.", @@ -306767,7 +309082,8 @@ "content": "对任意不等于$1$的正数$a$, 函数$f(x)=\\log_a(x+3)$的反函数的图像都经过点$P$, 则点$P$的坐标是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-赋能-赋能09" ], "genre": "填空题", "ans": "$(0,-2)$", @@ -309050,7 +311366,9 @@ "objs": [], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-38_计数原理与排列组合" ], "genre": "填空题", "ans": "$\\dfrac 1{15}$", @@ -316077,7 +318395,8 @@ "content": "已知集合$A=(-\\infty ,a]$, $B=[2,3]$且$A\\cap B$非空, 则实数$a$的取值范围\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$[2,+\\infty)$", @@ -316105,7 +318424,8 @@ "content": "若函数$y=\\cos (x+\\varphi)$为奇函数, 则最小的正数$\\varphi =$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$\\dfrac\\pi 2$", @@ -316131,7 +318451,8 @@ "content": "已知长方体的长、宽、高分别为$3$、$4$、$12$, 则长方体的一条对角线长为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$13$", @@ -316157,7 +318478,8 @@ "content": "幂函数$f(x)$的图像过点$(4,2)$, 其反函数为$f^{-1}(x)$, 则$f^{-1}(3)=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$9$", @@ -316186,7 +318508,10 @@ ], "tags": [ "第八单元", - "二项式定理" + "二项式定理", + "2023届高三-四月错题重做-05_易错题-概率与统计", + "2023届高三-四月错题重做-05_概率与统计", + "2023届高三-寒假作业-中档题" ], "genre": "填空题", "ans": "$20x^{19}$", @@ -316324,7 +318649,8 @@ "content": "函数$y=f(x)$的定义域$D$和值域$A$都是集合$\\{1,2,3\\}$的非空真子集, 如果对于$D$内任意的$x$, 总有$x+f(x)+xf(x)$的值是奇数, 则满足条件的函数$y=f(x)$的个数是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-较难题" ], "genre": "填空题", "ans": "$29$", @@ -316350,7 +318676,9 @@ "content": "若分段函数$\\begin{cases} 3\\sin2x, & x\\le 0,\\\\ 2^x-3, & x>0, \\end{cases}$ 将函数$y=|f(x)-f(a)|$, $x\\in [m,n]$的最大值记作\n$Z_a[m,n]$, 那么当$-2\\le m\\le 2$时, $Z_2[m,m+4]$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-四月错题重做-02_函数二", + "2023届高三-四月错题重做-02_易错题-函数2" ], "genre": "填空题", "ans": "$[4,60]$", @@ -316416,7 +318744,8 @@ "objs": [], "tags": [ "第一单元", - "第二单元" + "第二单元", + "2023届高三-寒假作业-容易题" ], "genre": "选择题", "ans": "B", @@ -316442,7 +318771,8 @@ "content": "申辉中学从$4$名有数学特长的同学A、B、C、D中挑$1$人去参加中学生数学联赛, 4名同学各自对结果的估计如下: A: ``参赛的是A''; B: ``参赛的是B''; C: ``参赛的是A或B''; D: ``参赛的既不是A也不是C''; 已知其中有且只有$2$人的估计是正确的, 则参加联赛的是\\bracket{20}.\n\\fourch{A同学}{B同学}{C同学}{D同学}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-容易题" ], "genre": "选择题", "ans": "A", @@ -316494,7 +318824,8 @@ "content": "已知圆锥的体积为$\\pi$, 底面半径$OA$与$OB$互相垂直, 且$OA=\\sqrt 3$, $P$是母线$BS$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (2,0) node [right] {$B$} coordinate (B);\n\\draw (0,2) node [above] {$S$} coordinate (S);\n\\draw ($(B)!0.5!(S)$) node [above right] {$P$} coordinate (P);\n\\draw ({2*cos(-135)},{0.5*cos(-135)}) node [below left] {$A$} coordinate (A);\n\\draw (B) arc (0:-180:2 and 0.5) (B) -- (S) -- (-2,0);\n\\draw [dashed] (B) arc (0:180:2 and 0.5) (O) -- (S) (O) -- (B) (A) -- (P) (O) -- (A);\n\\end{tikzpicture}\n\\end{center}\n(1) 求圆锥的表面积;\\\\\n(2) 求异面直线$SO$与$PA$所成角的大小(结果用反三角函数表示).", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-寒假作业-中档题" ], "genre": "解答题", "ans": "(1) $(3+2\\sqrt{3})\\pi$; (2) $\\arctan \\sqrt{15}$", @@ -316617,7 +318948,8 @@ "content": "函数$f(x)=x^{-\\frac 12}$的定义域是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$(0,+\\infty)$", @@ -316650,7 +318982,8 @@ "content": "集合$A=\\{-1, 2m-1\\}$, $B=\\{m^2\\}$, 若$B\\subseteq A$, 则实数$m=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$1$", @@ -316749,7 +319082,8 @@ "content": "方程$\\lg (x+2)=2\\lg x$的解为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$2$", @@ -316813,7 +319147,8 @@ "content": "若函数$f(x)=\\sqrt {2x+1}$的反函数为$g(x)$, 则函数$g(x)$的零点为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-中档题" ], "genre": "填空题", "ans": "$1$", @@ -316845,7 +319180,8 @@ "content": "已知函数$y=\\sin (\\omega x-\\dfrac{\\pi }6) \\ (\\omega >0)$图像的一条对称轴为$x=\\dfrac{\\pi }6$, 则$\\omega$的最小值为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$4$", @@ -317003,7 +319339,8 @@ "content": "下列是``$a>b$''的充分不必要条件的是\\bracket{20}.\n\\fourch{$a>b+1$}{$\\dfrac ab>1$}{$a^2>b^2$}{$a^3>b^3$}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-下学期测验卷-高三下学期测验02" ], "genre": "选择题", "ans": "A", @@ -317333,7 +319670,8 @@ "content": "集合$A=\\{x|x\\ge 0\\}$, $B=\\{x|x\\ge a\\}$, 若$A\\subseteq B$, 则实数$a$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期测验卷-基础考" ], "genre": "填空题", "ans": "$(-\\infty,0]$", @@ -317368,7 +319706,8 @@ "content": "函数$y=\\lg(2-x)$的定义域为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-基础考" ], "genre": "填空题", "ans": "$(-\\infty,2)$", @@ -317403,7 +319742,8 @@ "content": "陈述句``$a\\ge 1$且$a\\le 3$''的否定形式为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期测验卷-基础考" ], "genre": "填空题", "ans": "$a<1$或$a>3$", @@ -317439,7 +319779,8 @@ "objs": [], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-上学期测验卷-基础考" ], "genre": "填空题", "ans": "$0.15$", @@ -317474,7 +319815,8 @@ "content": "若圆锥的轴截面是边长为$1$的正三角形, 则圆锥的侧面积是\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期测验卷-基础考" ], "genre": "填空题", "ans": "$\\dfrac\\pi 2$", @@ -317509,7 +319851,8 @@ "content": "若$z=\\dfrac{1-a\\mathrm{i}}{2+\\mathrm{i}}$($\\mathrm{i}$为虚数单位)为纯虚数, 则实数$a$的值为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期测验卷-基础考" ], "genre": "填空题", "ans": "$2$", @@ -317544,7 +319887,8 @@ "content": "已知$\\overrightarrow{a}=(2,1)$, $\\overrightarrow{b}$在$\\overrightarrow{a}$上的投影为$-2\\overrightarrow{a}$, 则$\\overrightarrow{a}\\cdot\\overrightarrow{b}=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期测验卷-基础考" ], "genre": "填空题", "ans": "$-10$", @@ -317579,7 +319923,8 @@ "content": "如果幂函数$y=f(x)$的图像经过点$(2,\\dfrac 12)$, 那么$y=f(x)$的单调减区间是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-基础考" ], "genre": "填空题", "ans": "$(-\\infty,0)$与$(0,+\\infty)$", @@ -317614,7 +319959,8 @@ "content": "某医院对某学校高三年级的$600$名学生进行身体健康调查, 采用男女分层抽样法抽取一个容量为$50$的样本, 已知女生比男生少抽了$10$人, 则该年级的女生人数是\\blank{50}.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-上学期测验卷-基础考" ], "genre": "填空题", "ans": "$240$", @@ -317649,7 +319995,8 @@ "content": "偶函数$y=f(x)$在区间$[0,+\\infty)$上是严格减函数, 若$f(1)=0$, 则关于$x$的不等式$f(x)-x^2>-1$的解集是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-基础考" ], "genre": "填空题", "ans": "$(-1,1)$", @@ -317684,7 +320031,8 @@ "content": "已知$a,b\\in \\mathbf{R}$且$a\\ne 0$, 则$|a+b|+|\\dfrac 4a-b|$的最小值是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期测验卷-基础考" ], "genre": "填空题", "ans": "$4$", @@ -317719,7 +320067,8 @@ "content": "已知函数$y=\\sin x+\\sin 2x$在$(-a,a)$上恰有$5$个零点, 则实数$a$的最大值为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-基础考" ], "genre": "填空题", "ans": "$\\dfrac{4\\pi}3$", @@ -317754,7 +320103,8 @@ "content": "设$x\\in \\mathbf{R}$, 则``$x<1$''是``$x^3<1$''的\\bracket{20}.\n\\twoch{充分而不必要条件}{必要而不充分条件}{充要条件}{既不充分也不必要条件}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期测验卷-基础考" ], "genre": "选择题", "ans": "C", @@ -317790,7 +320140,8 @@ "objs": [], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-上学期测验卷-基础考" ], "genre": "选择题", "ans": "D", @@ -317825,7 +320176,8 @@ "content": "已知某射击爱好者打靶成绩(单位:环)的茎叶图如图所示, 其中整数部分为``茎'', 小数部分为``叶'', 则这组数据的标准差为(精确到$0.01$)\\bracket{20}.\n\\begin{center}\n\\begin{tabular}{c|cccc}\n$5$ & $7$ & $9$ \\\\\n$6$ & $1$ & $2$ & $7$ & $7$ \\\\\n$7$ & $2$ & $5$\n\\end{tabular}\n\\end{center}\n\\fourch{$0.35$}{$0.59$}{$0.40$}{$0.63$}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-上学期测验卷-基础考" ], "genre": "选择题", "ans": "B", @@ -317860,7 +320212,8 @@ "content": "如图所示, 图中多面体是由两个底面相同的正四棱锥所拼接而成, 且这六个顶点在同一个球面上. 若二面角$M-AB-C$的正切值为$1$, 则二面角$N-AB-C$的正切值为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = (-120:0.5)]\n\\draw (-1,0,1) node [left] {$A$} coordinate (A);\n\\draw (1,0,1) node [right] {$B$} coordinate (B);\n\\draw (1,0,-1) node [right] {$C$} coordinate (C);\n\\draw (-1,0,-1) node [left] {$D$} coordinate (D);\n\\draw (0,1,0) node [above] {$M$} coordinate (M);\n\\draw (0,-2,0) node [below] {$N$} coordinate (N);\n\\draw (A) -- (B) -- (C) (A) -- (N) (B) -- (N) (C) -- (N) (M) -- (D) (M) -- (A) (M) -- (B) (M) -- (C) (A) -- (D);\n\\draw [dashed] (D) -- (C) (D) -- (N);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$1$}{$\\sqrt{2}$}{$2$}{$2\\sqrt{2}$}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期测验卷-基础考" ], "genre": "选择题", "ans": "C", @@ -317895,7 +320248,8 @@ "content": "已知$O$为坐标原点, $\\overrightarrow{OA}=(2,3)$, $\\overrightarrow{OB}=(4,2)$, $\\overrightarrow{OC}=(x,3)$.\\\\\n(1) 若$A,B,C$三点共线, 求$x$的值;\\\\\n(2) 若$\\overrightarrow{AB}$与$\\overrightarrow{OC}$夹角为钝角, 求$x$的取值范围.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期测验卷-基础考" ], "genre": "解答题", "ans": "(1) $2$; (2) $(-\\infty,-6)\\cup (-6,\\dfrac 32)$", @@ -317930,7 +320284,8 @@ "content": "已知函数$f(x)=ax^2+x-1$. ($a>0$)\\\\\n(1) 若关于$x$的不等式$f(x)<0$的解集为$(-1,b)$, 求实数$a$和$b$的值;\\\\\n(2) 若函数$y=f(x)$在$[-3,-1]$上的最大值为$2$, 求实数$a$的值.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-基础考" ], "genre": "解答题", "ans": "(1) $a=2$, $b=\\dfrac 12$; (2) $\\dfrac 23$", @@ -317966,7 +320321,8 @@ "objs": [], "tags": [ "第三单元", - "第六单元" + "第六单元", + "2023届高三-上学期测验卷-基础考" ], "genre": "解答题", "ans": "(1) 约为$1.58\\text{km}$; (2) 约为$49^\\circ$", @@ -318001,7 +320357,8 @@ "content": "如图, 三棱柱$ABC-A_1B_1C_1$中, $\\angle CAB=90^\\circ$, $AB=AC=A_1B=A_1C=2\\sqrt{2}$, $AA_1=2$, 点$M,F$分别为$BC,A_1B_1$的中点, 点$E$为$AM$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = (-45:0.5),scale = 1.25]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (0,0,{2*sqrt(2)}) node [below] {$B$} coordinate (B);\n\\draw ({2*sqrt(2)},0,0) node [right] {$C$} coordinate (C);\n\\draw (A) ++ ({sqrt(2)/2},{sqrt(3)},{sqrt(2)/2}) node [above left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ ({sqrt(2)/2},{sqrt(3)},{sqrt(2)/2}) node [below right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ ({sqrt(2)/2},{sqrt(3)},{sqrt(2)/2}) node [above right] {$C_1$} coordinate (C_1);\n\\draw ($(B)!0.5!(C)$) node [right] {$M$} coordinate (M);\n\\draw ($(A_1)!0.5!(B_1)$) node [above] {$F$} coordinate (F);\n\\draw ($(A)!0.5!(M)$) node [below left] {$E$} coordinate (E);\n\\draw (A) -- (B) -- (C) (A) -- (A_1) (B) -- (B_1) (C) -- (C_1) (A_1) -- (B_1) -- (C_1) (A_1) -- (C_1) (A_1) -- (B);\n\\draw [dashed] (A) -- (M) (E) -- (F) (A_1) -- (M) (A_1) -- (C) (A) -- (C);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $AA_1\\perp BC$;\\\\\n(2) 证明: $EF\\parallel$平面$BCC_1B_1$;\\\\\n(3) 求直线$EF$与平面$A_1BC$所成角的正弦值.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期测验卷-基础考" ], "genre": "解答题", "ans": "(1) 证明略; (2) 证明略; (3) $\\dfrac{\\sqrt{15}}5$", @@ -318037,7 +320394,8 @@ "objs": [], "tags": [ "第三单元", - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-基础考" ], "genre": "解答题", "ans": "(1) $\\begin{cases}0, & n=4k+1,4k+3,\\\\ 1, & n=4k+4, \\\\ -1, & n=4k+2,\\end{cases}$($k\\in \\mathbf{N}$); (2) 证明略; (3) 证明略", @@ -318072,7 +320430,9 @@ "content": "若集合$A=\\{2,a^2-a+1\\}$, $B=\\{3,a+3\\}$, 且$A\\cap B=\\{3\\}$, 则实数$a=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期测验卷-测验08", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$2$", @@ -318107,7 +320467,9 @@ "content": "若复数$z=(1+m\\mathrm{i})(2-\\mathrm{i})$($\\mathrm{i}$是虚数单位)是纯虚数, 则实数$m$的值为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期测验卷-测验08", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$-2$", @@ -318142,7 +320504,8 @@ "content": "已知全集$U=\\mathbf{R}$, 集合$A=\\{x|\\dfrac{x+1}{x-2}\\le 0\\}$, 则集合$\\overline{A}=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期测验卷-测验08" ], "genre": "填空题", "ans": "$(-\\infty,-1)\\cup [2,+\\infty)$", @@ -318177,7 +320540,9 @@ "content": "已知$\\{a_n\\}$为等差数列, 其前$n$项和为$S_n$, 若$a_1=1$, $a_3=5$, $S_n=64$, 则$n=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-上学期测验卷-测验08", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$8$", @@ -318212,7 +320577,9 @@ "content": "已知复数$z_0=3+\\mathrm{i}$($\\mathrm{i}$为虚数单位), 复数$z$满足$z\\cdot z_0=3z+z_0$, 则$|z|=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期测验卷-测验08", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$\\sqrt{10}$", @@ -318247,7 +320614,8 @@ "content": "已知$\\tan \\theta =3$, 则$\\sin 2\\theta -2\\cos ^2\\theta$的值为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-测验08" ], "genre": "填空题", "ans": "$\\dfrac 25$", @@ -318282,7 +320650,8 @@ "content": "已知$\\{a_n\\}$是各项均为正数的等比数列,\n且$a_6=2$, 则$\\log_2(a_1\\cdot a_2\\cdot a_3\\cdot \\cdots \\cdot a_{11})=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-上学期测验卷-测验08" ], "genre": "填空题", "ans": "$11$", @@ -318317,7 +320686,8 @@ "content": "已知函数$f(x)=A\\sin (\\omega x+\\varphi)(A,\\omega ,\\varphi$为常数且$A>0,\\omega >0$)的部分图像如图所示, 则$f(0)$的值是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = {-pi/6}:{11*pi/12}, samples = 100] plot (\\x,{sqrt(2)*sin(2*\\x/pi*180+60)});\n\\draw [dashed] ({pi/12},0) -- ({pi/12},{sqrt(2)}) -- (0,{sqrt(2)});\n\\draw [dashed] ({7*pi/12},0) -- ({7*pi/12},{-sqrt(2)}) -- (0,{-sqrt(2)});\n\\draw (0,{-sqrt(2)}) node [left] {$-\\sqrt{2}$};\n\\draw ({pi/3},0) node [below] {$\\dfrac{\\pi}{3}$};\n\\draw ({7*pi/12},0) node [above] {$\\dfrac{7\\pi}{12}$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-测验08" ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{6}}2$", @@ -318352,7 +320722,8 @@ "content": "设$f(x)$是$\\mathbf{R}$上的奇函数, $g(x)$是$\\mathbf{R}$上的偶函数, 若函数$f(x)+g(x)$的值域为$[-1,4]$, 则$f(x)-g(x)$的值域为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验08" ], "genre": "填空题", "ans": "$[-4,1]$", @@ -318387,7 +320758,10 @@ "content": "若函数$y=\\log_a(x^2-ax+1)$有最小值, 则$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验08", + "2023届高三-四月错题重做-02_函数二", + "2023届高三-四月错题重做-02_易错题-函数2" ], "genre": "填空题", "ans": "$(1,2)$", @@ -318434,7 +320808,10 @@ "content": "已知关于$x$的方程$|x+a^2|+|x-a^2|=-x^2+2x-1+2a^2$有解, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验08", + "2023届高三-四月错题重做-02_函数二", + "2023届高三-四月错题重做-02_易错题-函数2" ], "genre": "填空题", "ans": "$(-\\infty,-1]\\cup [1,+\\infty)$", @@ -318481,7 +320858,9 @@ "content": "如果数列$\\{a_n\\}$满足: $a_1=1$,$ a_{2021}=2017$, 且对于任意$n\\in \\mathbf{N}$, $n\\ge 1$, 存在实数$a$使得$a_n$, $a_{n+1}$是方程$x^2-(2a+1)x+a^2+a=0$的两个根, 则$a_{100}$的所有可能值构成的集合是\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-上学期测验卷-测验08", + "2023届高三-寒假作业-较难题" ], "genre": "填空题", "ans": "$\\{96,98,100\\}$", @@ -318516,7 +320895,8 @@ "content": "若$\\cos \\theta >0$, 且$\\sin 2\\theta <0$, 则角$\\theta$的终边所在象限是\\bracket{20}.\n\\fourch{第一象限}{第二象限}{第三象限}{第四象限}", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-测验08" ], "genre": "选择题", "ans": "D", @@ -318551,7 +320931,10 @@ "content": "记$S_n$为数列$\\{a_n\\}$的前$n$项和, ``$\\{a_n\\}$是递增数列''是``$\\{S_n\\}$是递增数列''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分也非必要条件}", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-上学期测验卷-测验08", + "2023届高三-四月错题重做-03_数列", + "2023届高三-四月错题重做-03_易错题-数列" ], "genre": "选择题", "ans": "D", @@ -318598,7 +320981,8 @@ "content": "有四个命题:\n\\textcircled{1} 若$0>a>b$, 则$\\dfrac 1a<\\dfrac 1b$; \\textcircled{2} 若$ab^2$; \\textcircled{3} 若$\\dfrac 1a>1$, 则$1>a$; \\textcircled{4} 若$10$时, 求解该不等式.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期测验卷-测验08" ], "genre": "解答题", "ans": "(1) $(-\\infty,\\dfrac 12)\\cup (2,+\\infty)$; (2) 当$a\\in (0,1)$时, 解集为$(a,\\dfrac 1a)$, 当$a=1$时, 解集为$\\varnothing$, 当$a>1$时, 解集为$(\\dfrac 1a,a)$", @@ -318703,7 +321089,8 @@ "content": "在$\\triangle ABC$中, 角$A$、$B$、$C$所对的边长分别为$a$、$b$、$c$,\n且$2\\sqrt 3\\sin B\\cos B-2\\cos ^2B=1$.\\\\\n(1) 求角$B$的大小;\\\\\n(2) 若$b=2$, 求$\\triangle ABC$的面积的最大值.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-测验08" ], "genre": "解答题", "ans": "(1) $\\dfrac \\pi 3$; (2) $\\sqrt{3}$", @@ -318738,7 +321125,8 @@ "content": "某地博物馆整体理念是将生态自然与人文历史有机的融合, 与周边环境自然过渡连接. 为了减少能源损耗, 馆顶和外墙需要建造隔热层. 博物馆每年节省的能源费用$h$(单位: 万元)与隔热层厚度$x$(单位: $\\text{cm}$)满足关系: $h(x)=32-\\dfrac{32}{x+k}$($0\\le x\\le 20$). 当不建造隔热层时, 每年节省费用为$0$, 但是隔热层自身需要消耗能源, 每年隔热层自身消耗的能源费用$g$(单位: 万元)与隔热层厚度$x$(单位: $\\text{cm}$)满足关系: $g(x)=2x$.\\\\\n(1) \\textcircled{1} 求$k$的值; \\textcircled{2} 为了使得每年隔热层节省的能源费用不低于隔热层自身消耗的能源费用, 隔热层建造的厚度$x$应该满足什么条件?\\\\\n(2) 在建造厚度为$x$的隔热层后, 每年博物馆真正节省的能源费用$f(x)=h(x)-g(x)$, 求每年博物馆真正节省的能源费用的最大值.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验08" ], "genre": "解答题", "ans": "(1) $0\\le x\\le 15$(单位: $\\text{cm}$); (2) $18$万元", @@ -318773,7 +321161,8 @@ "content": "设函数$f(x)=\\dfrac{2x+1}x$($x>0$), 数列$\\{a_n\\}$满足$a_1=1$, $a_n=f(\\dfrac 1{a_{n-1}})$($n\\in \\mathbf{N}$, $n\\ge 2$).\\\\\n(1) 求数列$\\{a_n\\}$的通项公式;\\\\\n(2) 设$T_n=a_1a_2-a_2a_3+a_3a_4-a_4a_5+\\cdots -a_{2n}a_{2n+1}$, 若$T_n\\ge tn^2$对$n\\in \\mathbf{N}$, $n\\ge 1$恒成立, 求实数$t$的取值范围;\\\\\n(3) 是否存在以$1$为首项, 公比为$q$($0g(x)$恒成立, 则称该函数满足性质$M$.\\\\\n(1) 判断函数$f_1(x)=\\sin x$, $f_2(x)=x^2$是否满足性质$M$(无需说明理由);\\\\\n(2) 若函数$f(x)$满足性质$M$, 求证: $f(x)$不是奇函数;\\\\\n(3) 若函数$f(x)$满足性质$M$, 求证: 当$\\lambda >0$, $x_1\\ne x_2$时, 不等式\n$\\dfrac{f(x_1)+\\lambda f(x_2)}{1+\\lambda }>f(\\dfrac{x_1+\\lambda x_2}{1+\\lambda })$恒成立,\n并求函数$h(x)=f(x)+f(2021-x),x\\in [1,2020]$的最大值.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验08" ], "genre": "解答题", "ans": "(1) $f_1(x)=\\sin x$不满足性质$M$, $f_2(x)=x^2$满足性质$M$; (2) 证明略; (3) 证明略, 最大值为$f(1)+f(2020)$", @@ -318843,7 +321233,9 @@ "content": "设集合$A=\\{x|(x-1)(x-4)<0\\}$, 集合$B=\\mathbf{Z}$, 则$A\\cap B=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期测验卷-测验09", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$\\{2,3\\}$", @@ -318878,7 +321270,8 @@ "content": "已知$\\mathrm{i}$为虚数单位, 则复数$z=\\dfrac{3+\\mathrm{i}}{2+\\mathrm{i}}$的模$|z|=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期测验卷-测验09" ], "genre": "填空题", "ans": "$\\sqrt{2}$", @@ -318913,7 +321306,9 @@ "content": "方程$\\log_2(3x+4)=3$的解为$x=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验09", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$\\dfrac 43$", @@ -318948,7 +321343,8 @@ "content": "在二项式$(x+\\dfrac2x)^6$的展开式中, 常数项是\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-上学期测验卷-测验09" ], "genre": "填空题", "ans": "$160$", @@ -318997,7 +321393,8 @@ "content": "若圆锥底面积为$20\\pi$, 且母线与底面所成角为$\\arccos \\dfrac 4\n5$, 则该圆锥的侧面积为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期测验卷-测验09" ], "genre": "填空题", "ans": "$25\\pi$", @@ -319032,7 +321429,8 @@ "content": "设点$H(2,3)$, 若直线$l$经过点$H$, 且与直线$OH$垂直($O$为坐标原点), 则直线$l$的方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-上学期测验卷-测验09" ], "genre": "填空题", "ans": "$2x+3y-13=0$", @@ -319067,7 +321465,8 @@ "content": "函数$y=2\\cos\\left(x+\\dfrac{\\pi}{4}\\right)\\cos\\left(x-\\dfrac{\\pi}{4}\\right)+\\sqrt{3}\\sin 2x$的值域为\\blank{80}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-测验09" ], "genre": "填空题", "ans": "$[-2,2]$", @@ -319106,7 +321505,8 @@ "content": "函数$y=\\dfrac{x}{x+1}$的图像是一个中心对称图形, 其对称中心的坐标为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验09" ], "genre": "填空题", "ans": "$(-1,1)$", @@ -319141,7 +321541,8 @@ "content": "已知随机变量$X$的分布列为$\\begin{pmatrix}\n -1 & 0 & 1 \\\\\n \\dfrac 12 & \\dfrac 13 & \\dfrac 16 \n \\end{pmatrix}$, 另一个随机变量$Y$满足$X+2Y=4$, 则$Y$的期望$E[Y]=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-上学期测验卷-测验09" ], "genre": "填空题", "ans": "$\\dfrac{13}6$", @@ -319176,7 +321577,8 @@ "content": "在由数字$1, 2, 3, 4, 5$组成的数字不重复的五位数中, 小于$50000$的奇数有\\blank{50}个.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-上学期测验卷-测验09" ], "genre": "填空题", "ans": "$60$", @@ -319215,7 +321617,8 @@ "content": "平面上的三个单位向量$\\overrightarrow{a}$, $\\overrightarrow{b}$, $\\overrightarrow{c}$满足$2\\overrightarrow{c}=3\\overrightarrow{a}+4\\overrightarrow{b}$, 则$\\overrightarrow{a}$, $\\overrightarrow{b}$, $\\overrightarrow{c}$两两间的夹角中, 最小的角的大小为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期测验卷-测验09" ], "genre": "填空题", "ans": "$\\arccos \\dfrac{11}{16}$", @@ -319251,7 +321654,9 @@ "objs": [], "tags": [ "第一单元", - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验09", + "2023届高三-寒假作业-较难题" ], "genre": "填空题", "ans": "$\\dfrac 59$", @@ -319288,7 +321693,9 @@ "content": "已知$a,b$是实数, 则``$a>b$''是``$a^3+1>b^3+1$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期测验卷-测验09", + "2023届高三-寒假作业-容易题" ], "genre": "选择题", "ans": "C", @@ -319323,7 +321730,9 @@ "content": "演讲比赛共有$9$位评委分别给出某选手的原始评分, 评定该选手的成绩时, 从$9$个原始评分中去掉$1$个最高分、$1$个最低分, 得到$7$个有效评分. $7$个有效评分与$9$个原始评分相比, 不变的数字特征是\\bracket{20}.\n\\fourch{中位数}{平均数}{方差}{极差}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-上学期测验卷-测验09", + "2023届高三-寒假作业-容易题" ], "genre": "选择题", "ans": "A", @@ -319358,7 +321767,8 @@ "content": "已知$\\omega$是常数, 若函数$y=|\\sin (\\omega x+\\dfrac \\pi 3)|$图像的一条对称轴是直线$x=\\dfrac\\pi 6$. 则$\\omega$的值不可能在区间\\bracket{20}中.\n\\fourch{$(0,2]$}{$(2,4]$}{$(4,6]$}{$(6,8]$}", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-测验09" ], "genre": "选择题", "ans": "C", @@ -319393,7 +321803,9 @@ "content": "对于两个定义在$\\mathbf{R}$上的函数$y=f(x)$与$y=g(x)$, 构造新的函数$y=h(x)$如下: 对任意$x_0\\in \\mathbf{R}$, $h(x_0)=f(x_0)+g(x_0)$. 现已知$y=h(x)$是严格增函数, 对于以下两个命题: \n\\textcircled{1} $y=f(x)$与$y=g(x)$中至少有一个是严格增函数;\n\\textcircled{2} $y=f(x)$与$y=g(x)$中至少有一个无最大值.\n其中\\bracket{20}.\n\\fourch{\\textcircled{1}和\\textcircled{2}都是真命题}{只有\\textcircled{1}是真命题}{只有\\textcircled{2}是真命题}{没有真命题}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验09", + "2023届高三-寒假作业-较难题" ], "genre": "选择题", "ans": "D", @@ -319428,7 +321840,8 @@ "content": "如图, 设$P-ABCD$是底面为矩形的四棱锥, $PA\\perp$平面$ABCD$. $PA=AB=2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (0,0,2) node [left] {$B$} coordinate (B);\n\\draw (3,0,2) node [right] {$C$} coordinate (C);\n\\draw (3,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,2,0) node [left] {$P$} coordinate (P);\n\\draw (P) -- (B) -- (C) -- (D) -- (P) (P) -- (C);\n\\draw [dashed] (A) -- (P) (A) -- (B) (A) -- (D);\n\\end{tikzpicture}\n\\end{center}\n(1) 若$PC\\perp BD$, 求四棱锥$P-ABCD$的体积;\\\\\n(2) 若直线$PD$与平面$PAB$所成的角的大小为$\\arctan 2$, 求直线$PC$与平面$ABCD$所成的角的大小.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期测验卷-测验09" ], "genre": "解答题", "ans": "(1) $\\dfrac 83$; (2) $\\arctan \\dfrac{\\sqrt{5}}5$", @@ -319463,7 +321876,8 @@ "content": "设$a$是实常数, 并记$f(x)=x^3+ax^2+2x$.\\\\\n(1) 当$a=-\\dfrac{5}{2}$时, 求函数$y=f(x)$的单调减区间;\\\\ \n(2) 是否存在$a$, 使得函数$y=f(x)$在实数范围内有且仅有三个零点, 且三个零点可按某种顺序排列后成等差数列? 若存在, 求所有满足条件的$a$的值; 若不存在, 说明理由.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验09" ], "genre": "解答题", "ans": "(1) $[\\dfrac 23,1]$; (2) 存在, $a=-3$或$3$", @@ -319498,7 +321912,8 @@ "content": "如图, 某市郊外景区内一条笔直的公路$a$经过三个景点$A$、$B$、$C$. 景区管委会又开发了风景优美的景点$D$.经测量景点$D$位于景点$A$的北偏东$30^\\circ$方向$8$千米处, 且位于景点$B$的正北方向, 还位于景点$C$的北偏西$75^\\circ$方向上.已知$AB=5$千米.\n\\begin{center}\n \\begin{tikzpicture}[>=latex,scale = 0.25]\n \\draw [->] (6,8) -- (10,8) node [right] {东};\n \\draw [->] (6,8) -- (6,12) node [above] {北};\n \\draw [->] (0,0) node [below] {$A$} coordinate (A) -- (0,8) node [left] {$N$} coordinate (N);\n \\draw (A) --++ (60:8) node [above] {$D$} coordinate (D);\n \\draw (4,3) node [below] {$B$} coordinate (B) -- (D);\n \\draw [name path = linea] (A) -- ($(A)!2.2!(B)$) node [right] {$a$} coordinate (a);\n \\path [name path = DC] (D) --++ (-15:4);\n \\path [name intersections = {of = linea and DC, by = C}];\n \\draw (D) -- (C) node [below] {$C$};\n \\draw (60:2) arc (60:90:2);\n \\draw (75:4) node {$30^\\circ$};\n \\end{tikzpicture}\n\\end{center}\n(1) 景区管委会准备由景点$D$向景点$B$修一条笔直的公路, 不考虑其他因素, 求出这条公路的长(结果精确到$0.1$千米);\\\\\n(2) 求景点$C$与景点$D$之间的距离(结果精确到$0.1$千米).", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-测验09" ], "genre": "解答题", "ans": "(1) 约为$3.9\\text{km}$; (2) 约为$4.0\\text{km}$", @@ -319533,7 +321948,8 @@ "content": "已知数列$\\{a_n\\}$的通项公式为$a_n=2^n+\\lambda n$, 其中常数$\\lambda\\in \\mathbf{R}$.\\\\\n(1) 若$a_3=4a_2$, 求$\\lambda$的值;\\\\\n(2) 若$\\{a_n\\}$前$10$项的和为$1551$, 试分析$\\{a_n\\}$的单调性;\\\\\n(3) 对于常数$t$, 记集合$C_t=\\{n|a_n=t\\}$, 试求当$\\lambda$与$t$变化时, 集合$C_t$中元素个数的最大值.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-上学期测验卷-测验09" ], "genre": "解答题", "ans": "(1) $-\\dfrac 85$; (2) 前四项严格递减, 从第四项起严格递增; (3) 元素个数的最大值为$2$", @@ -319568,7 +321984,8 @@ "content": "已知椭圆$E$的方程为$\\dfrac{x^2}{12}+\\dfrac{y^2}{4}=1$, $F_1(-2\\sqrt{2},0)$与$F_2(2\\sqrt{2},0)$是$E$的两个焦点, $A(0,-2)$是$E$的下顶点.\\\\\n(1) 设斜率为$1$的直线$l$过点$F_1$, 且与$E$交于$M,N$两点, 求弦$MN$的长;\\\\\n(2) 若$E$上一点$P$满足$|F_1P|=3|F_2P|$, 求$\\triangle F_1F_2P$的面积;\\\\\n(3) 是否存在椭圆$E$上, 且位于第一象限的点$Q$, 使得射线$QA$平分$\\angle F_1QF_2$? 若存在, 请写出一个满足条件的点$Q$的坐标并加以验证; 若不存在, 说明理由.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-上学期测验卷-测验09" ], "genre": "解答题", "ans": "(1) $2\\sqrt{3}$; (2) $2\\sqrt{5}$; (3) $Q$的坐标为$(3,1)$, 验证过程略", @@ -319603,7 +322020,8 @@ "content": "已知集合$A=\\{x|\\dfrac{2x}{x-1}\\le 1\\}$, $B=\\{-1,0,1,2\\}$, 则$A\\cap B=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期测验卷-测验10" ], "genre": "填空题", "ans": "$\\{-1,0\\}$", @@ -319638,7 +322056,8 @@ "content": "设$a\\in \\mathbf{R}$, $\\mathrm{i}$为虚数单位. 若$(a-\\mathrm{i})(1-2\\mathrm{i})$为纯虚数, 则$a=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期测验卷-测验10" ], "genre": "填空题", "ans": "$2$", @@ -319673,7 +322092,8 @@ "content": "在空间直角坐标系中, 点$A(1,2,-3)$关于$xOz$平面对称的点的坐标是$\\blank{50}$.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期测验卷-测验10" ], "genre": "填空题", "ans": "$(1,-2,-3)$", @@ -319708,7 +322128,8 @@ "content": "已知$m\\in \\mathbf{R}$, 直线$l_1:\\sqrt{3}x-y+7=0$, $l_2:mx+y-1=0$. 若$l_1\\parallel l_2$, 则$l_1$与$l_2$之间的距离为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-上学期测验卷-测验10" ], "genre": "填空题", "ans": "$3$", @@ -319743,7 +322164,8 @@ "content": "为了解某校高三年级男生的体重, 从该校高三年级男生中抽取$17$名, 测得他们的体重数据如下(按从小到大的顺序排列, 单位: kg): $56\\ 56 \\ 57\\ 58\\ 59\\ 59\\ 61\\ 63\\ 64\\ 65\\ 66\\ 68\\ 69\\ 70\\ 73\\ 74\\ 83$\\\\\n据此估计该校高三年级男生体重的第$75$百分位数为\\blank{50}kg.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-上学期测验卷-测验10" ], "genre": "填空题", "ans": "$69$", @@ -319778,7 +322200,9 @@ "content": "设$a,b$为实数. 若关于$x$的方程$x^2+abx+b=0$的解集为$\\{1,3\\}$, 则$a=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期测验卷-测验10", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$-\\dfrac 43$", @@ -319813,7 +322237,8 @@ "content": "已知常数$m>0$. 在$(x+\\dfrac mx)^6$的二项展开式中, $x^2$项的系数是$\\dfrac{1}{x^2}$项的系数的$4$倍, 则$m=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-上学期测验卷-测验10" ], "genre": "填空题", "ans": "$\\dfrac 12$", @@ -319848,7 +322273,8 @@ "content": "在平面上, 已知$\\overrightarrow{a}$, $\\overrightarrow{b}$为两个不平行的单位向量, $O$为定点, 集合$\\Omega = \\{P|\\overrightarrow{OP}=\\lambda \\overrightarrow{a}+\\mu \\overrightarrow{b}, \\ 0\\le \\lambda \\le 1, \\ 0\\le \\mu \\le 2\\}$. 若$\\Omega$中所有点构成图形的面积为$1$, 则$\\overrightarrow{a}$与$\\overrightarrow{b}$夹角的大小为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期测验卷-测验10" ], "genre": "填空题", "ans": "$\\dfrac\\pi 6$或$\\dfrac{5\\pi}6$", @@ -319883,7 +322309,9 @@ "content": "已知定义在$(-3,3)$上的奇函数$y=f(x)$的导函数是$f'(x)$, 当$x\\ge 0$时, $y=f(x)$的图像如图所示, 则关于$x$的不等式$\\dfrac{f'(x)}{x}>0$的解集为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=stealth, line cap = round, line join = round,scale = 0.6]\n\\draw [->] (-1,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0:3] plot (\\x,{1-pow(\\x-1,2)});\n\\draw [dashed] (1,0) node [below] {$1$} -- (1,1) (3,-3) -- (3,0) node [below right] {$3$};\n\\draw (2,0) node [below left] {$2$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验10", + "2023届高三-四月错题重做-01_函数一" ], "genre": "填空题", "ans": "$(-3,-1)\\cup (0,1)$", @@ -319920,7 +322348,8 @@ "content": "第$14$届国际数学教育大会(ICME-14)于$2021$年$7$月$12$日至$18$日在上海举办, 已知张老师和李老师都在$7$天中随机选择了连续的$3$天参会, 则两位老师所选的日期恰好都不相同的概率为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-上学期测验卷-测验10" ], "genre": "填空题", "ans": "$\\dfrac 6{25}$", @@ -319955,7 +322384,8 @@ "content": "已知$A$、$B$、$C$是半径为$1$的球面上的三点, 若$AB=AC=1$, 则$BC$的最大值为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期测验卷-测验10" ], "genre": "填空题", "ans": "$\\sqrt{3}$", @@ -319990,7 +322420,9 @@ "content": "设$a\\in \\mathbf{R}$, $m\\in \\mathbf{Z}$. 若存在唯一的$m$使得关于$x$的不等式组$\\dfrac 12 x^2-\\dfrac 12=stealth, line cap = round, line join = round]\n\\draw (0,0) node {$1$};\n\\draw (0,-0.5) node {$2$};\n\\draw (0,-1) node {$3$};\n\\draw (0.3,0) node {$6$};\n\\draw (0.6,0) node {$8$};\n\\draw (0.3,-0.5) node {$1$} (0.6,-0.5) node {$2$} (0.9,-0.5) node {$2$};\n\\draw (0.3,-1) node {$1$};\n\\draw (0.15,0.25) -- (0.15,-1.25);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$21$}{$21.5$}{$22$}{$22.5$}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-上学期测验卷-测验10", + "2023届高三-寒假作业-容易题" ], "genre": "选择题", "ans": "B", @@ -320060,7 +322494,8 @@ "content": "已知双曲线$\\Gamma_1:\\dfrac{x^2}{a_1^2}-\\dfrac{y^2}{b_1^2}=1$($a_1>0$, $b_1>0$)与$\\Gamma_2:\\dfrac{x^2}{a_2^2}-\\dfrac{y^2}{b_2^2}=1$($a_2>0$, $b_2>0$)有共同的渐近线, 则它们一定有相等的\\bracket{20}.\n\\fourch{实轴长}{虚轴长}{焦距}{离心率}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-上学期测验卷-测验10" ], "genre": "选择题", "ans": "D", @@ -320095,7 +322530,8 @@ "content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n$, 若$S_{2023}=a_{2023}$, 则$\\{a_n\\}$不可能是\\bracket{20}. \n\\twoch{公差大于$0$的等差数列}{公差小于$0$的等差数列}{公比大于$0$的等比数列}{公比小于$0$的等比数列}", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-上学期测验卷-测验10" ], "genre": "选择题", "ans": "C", @@ -320130,7 +322566,8 @@ "content": "已知$\\omega \\in \\mathbf{R}$, $\\varphi \\in [0, 2\\pi)$. 若对任意实数$x$均有$\\sin x\\ge \\cos (\\omega x+\\varphi)$, 则满足条件的有序实数对$(\\omega , \\varphi)$的个数为\\bracket{20}. \n\\fourch{$1$个}{$2$个}{$3$个}{无数个}", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-测验10" ], "genre": "选择题", "ans": "C", @@ -320165,7 +322602,10 @@ "content": "设等差数列$\\{a_n\\}$的前$n$项和为$S_n$, 且$a_4=10$.\\\\\n(1) 若$S_{20}=590$, 求$\\{a_n\\}$的公差;\\\\\n(2) 若$a_1\\in \\mathbf{Z}$, 且$S_7$是数列$\\{S_n\\}$中最大的项, 求$a_1$所有可能的值.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-上学期测验卷-测验10", + "2023届高三-四月错题重做-03_数列", + "2023届高三-四月错题重做-03_易错题-数列" ], "genre": "解答题", "ans": "(1) 公差为$3$; (2) $a_1$的所有可能的值为$18,19,20$", @@ -320212,7 +322652,8 @@ "content": "如图, 正四棱柱$ABCD-A_1B_1C_1D_1$的底面边长为$1$, 高为$2$, $AC$、$BD$相交于点$O$.\n\\begin{center}\n\\begin{tikzpicture}[>=stealth, line cap = round, line join = round, scale = 1.7]\n\\def\\l{1}\n\\def\\m{1}\n\\def\\n{2}\n\\draw (0,0,0) node [below left] {\\footnotesize $A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {\\footnotesize $B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {\\footnotesize $C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {\\footnotesize $D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {\\footnotesize $A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [above] {\\footnotesize $B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {\\footnotesize $C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {\\footnotesize $D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw (A1)-- (C1);\n\\draw [dashed] (A1) -- (D) (C1) -- (D) (A) -- (C) (B) -- (D);\n\\draw ($(A)!0.5!(C)$) node [below] {\\footnotesize $O$} coordinate (O);\n\\draw [dashed] (O) -- (B1);\n\\end{tikzpicture}\n\\end{center}\n(1)证明: 直线$B_1O$与平面$A_1C_1D$平行;\\\\\n(2)求三棱锥$O-A_1C_1D$的体积.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期测验卷-测验10" ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\dfrac 13$", @@ -320247,7 +322688,8 @@ "content": "闲置房出租是增加社会住房供给量, 满足人们居住需求的重要途径. 王先生有一套住房以每月$7000$元的价格出租, 但合同租期本月到期. 房客直接向王先生提出希望从下月起续租三年, 并愿意每月支付$8000$元的租金. 王先生通过中介公司了解到: 该房屋所在小区的类似住宅, 目前的租金为每月$8000$-$9000$元, 在委托中介公司后, 一般$2$-$4$周左右可以\n找到承租人, 同时每次租赁交易成功后, 中介公司向出租方和承租方各收取一个月租金的$50\\%$作为中介费. 对于是否同意房客续租, 王先生需要作出决策.\\\\\n(1) 除了上述了解到的情况, 还有哪些因素王先生可能需要考虑? 写出这些因素(不超过$5$个);\\\\ \n(2) 为了简化问题, 请对相关因素作出合情假设, 由此帮助王先生作出决策, 并说明理由.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验10" ], "genre": "解答题", "ans": "(1) (示例)王先生可能需要考虑的因素有:\\\\\n\\textcircled{1} 未来月租金的变化;\\\\\n\\textcircled{2} 找到承租人的时长的变化;\\\\\n\\textcircled{3} 未来租客的租期长短;\\\\\n\\textcircled{4} 房屋是否未来三年内可以用于出租;\\\\\n\\textcircled{5} 换租客的过程中是否需要重新装修;\n\\textcircled{6} 寻租过程中的时间、精力成本等.\\\\\n(2) 言之有理即可", @@ -320282,7 +322724,8 @@ "content": "已知椭圆$\\Gamma:\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的左、右焦点分别为$F_1,F_2$, 直线$l$的斜率为$k$, 在$y$轴上的截距为$m$.\\\\\n(1) 设$k=1$, 若$\\Gamma$的焦距为$2$, $l$过点$F_1$, 求$l$的方程;\\\\\n(2) 设$m=0$. 若$P(\\sqrt{3},\\dfrac 12)$是$\\Gamma$上的一点, 且$|\\overrightarrow{PF_1}|+|\\overrightarrow{PF_2}|=4$, $l$与$\\Gamma$交于不同的两点$A,B$, $Q$为$\\Gamma$的上顶点, 求$\\triangle ABQ$面积的最大值;\\\\\n(3) 设$\\overrightarrow{n}$是$l$的一个法向量, $M$是$l$上一点, 对于坐标平面内的点$N$, 定义$\\delta_N=\\dfrac{\\overrightarrow{n}\\cdot \\overrightarrow{MN}}{|\\overrightarrow{n}|}$. 用$a,b,k,m$表示$\\delta_{F_1}\\cdot \\delta_{F_2}$, 并利用$\\delta_{F_1}\\cdot \\delta_{F_2}$与$b^2$的大小关系, 提出一个关于$l$与$\\Gamma$位置关系的真命题, 给出该命题的证明.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-上学期测验卷-测验10" ], "genre": "解答题", "ans": "(1) $y=x+1$; (2) 面积的最大值为$2$, 此时$l:y=0$; (3) 当$\\delta_{F_1}\\cdot \\delta_{F_2}>b^2$($=b^2$, $0$''是``$\\dfrac 1a>0$''的\\bracket{20}条件.\n\\fourch{充分非必要}{必要非充分}{充要}{既非充分也非必要}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期周末卷-周末卷12" ], "genre": "选择题", "ans": "C", @@ -321733,7 +324192,8 @@ "content": "过正方体中心的平面截正方体所得的截面中, 不可能的图形是\\bracket{20}\n\\fourch{三角形}{长方形}{对角线不相等的菱形}{六边形}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期周末卷-周末卷12" ], "genre": "选择题", "ans": "A", @@ -321759,7 +324219,8 @@ "content": "如图所示, 正八边形$A_1A_2A_3A_4A_5A_6A_7A_8$的边长为$2$, 若$P$为该正八边形边上的动点, 则$\\overrightarrow{A_1A_3} \\cdot \\overrightarrow{A_1P}$的取值范围为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$A_1$} coordinate (A_1) --++ (0:1) node [below] {$A_2$} coordinate (A_2) --++ (45:1) node [right] {$A_3$} coordinate (A_3) --++ (90:1) node [right] {$A_4$} coordinate (A_4) --++ (135:1) node [above] {$A_5$} coordinate (A_5) --++ (180:1) node [above] {$A_6$} coordinate (A_6) --++ (225:1) node [left] {$A_7$} coordinate (A_7) --++ (270:1) node [left] {$A_8$} coordinate (A_8) -- cycle;\n\\draw [->] (A_1) -- (A_3);\n\\draw [->] (A_1) -- ($(A_6)!0.5!(A_7)$) node [above left] {$P$} coordinate (P);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$[0,8+6 \\sqrt 2]$}{$[-2 \\sqrt 2, 8+6 \\sqrt 2]$}{$[-8-6 \\sqrt 2, 2 \\sqrt 2]$}{$[-8-6 \\sqrt 2, 8+6 \\sqrt 2]$}", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期周末卷-周末卷12" ], "genre": "选择题", "ans": "B", @@ -321785,7 +324246,8 @@ "content": "如图, 长方体$ABCD-A_1B_1C_1D_1$中,$AB=BC=2$, $AA_1=3$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{3}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw (A1) -- (B);\n\\draw [dashed] (A1) -- (D) (A1) -- (C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求四棱锥$A_1-ABCD$的体积;\\\\\n(2) 求异面直线$A_1C$与$DD_1$所成角的大小.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期周末卷-周末卷12" ], "genre": "解答题", "ans": "(1) $4$; (2) $\\arctan \\dfrac{2\\sqrt{2}}3$", @@ -321811,7 +324273,10 @@ "content": "设$a \\in \\mathbf{R}$, 函数$f(x)=\\dfrac{2^x+a}{2^x+1}$.\\\\\n(1) 求$a$的值, 使得$f(x)$为奇函数;\\\\\n(2) 若$f(x)<\\dfrac{a+2}2$对任意$x \\in \\mathbf{R}$成立, 求$a$的取值范围.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷12", + "2023届高三-四月错题重做-02_函数二", + "2023届高三-四月错题重做-02_易错题-函数2" ], "genre": "解答题", "ans": "(1) $a=-1$; (2) $[0,2]$", @@ -321850,7 +324315,8 @@ "objs": [], "tags": [ "第三单元", - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷12" ], "genre": "解答题", "ans": "(1) $r_1\\approx 34.6$米, $r_2\\approx 16.1$米; (2) $M_1$的半径为$30$米, $M_2$的半径为$20$米时, 总造价最低, 此时总造价约为$263.9$千元", @@ -321876,7 +324342,8 @@ "content": "已知双曲线$\\Gamma: x^2-\\dfrac{y^2}{b^2}=1$($b>0$), 直线$l: y=k x+m$($km \\neq 0$), $l$与$\\Gamma$交于$P$、$Q$两点,$P'$为$P$关于$y$轴的对称点, 直线$P'Q$与$y$轴交于点$N(0,n)$.\\\\\n(1) 若点$(2,0)$是$\\Gamma$的一个焦点, 求$\\Gamma$的渐近线方程;\\\\\n(2) 若$b=1$, 点$P$的坐标为$(-1,0)$, 且$\\overrightarrow{NP'}=\\dfrac 32 \\overrightarrow{P'Q}$, 求$k$的值;\\\\\n(3) 若$m=2$, 求$n$关于$b$的表达式.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-上学期周末卷-周末卷12" ], "genre": "解答题", "ans": "(1) $y=\\sqrt{3}x$与$y=-\\sqrt{3} x$; (2) $\\pm \\dfrac 12$; (3) $n=-\\dfrac{b^2}2$", @@ -321904,7 +324371,8 @@ "tags": [ "第一单元", "第二单元", - "第四单元" + "第四单元", + "2023届高三-上学期周末卷-周末卷12" ], "genre": "解答题", "ans": "(1) 解集为$\\{\\dfrac 13\\}$; (2) 证明略; (3) $(-1,\\dfrac 13]$", @@ -321930,7 +324398,8 @@ "content": "不等式$|x|>1$的解集为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期周末卷-周末卷13" ], "genre": "填空题", "ans": "$(-\\infty,-1)\\cup (1,+\\infty)$", @@ -321980,7 +324449,8 @@ "content": "设集合$A=\\{x|0=latex, scale = 0.5]\n\\def\\l{3}\n\\def\\m{4}\n\\def\\n{5}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [above right] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(A1)!0.5!(C1)$) node [above left] {$O$} coordinate (O);\n\\draw (O) -- (B1) (A) -- (B1) (A1) -- (C1);\n\\draw [dashed] (O) -- (A);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期周末卷-周末卷13" ], "genre": "填空题", "ans": "$5$", @@ -322120,7 +324594,8 @@ "content": "某校组队参加辩论赛, 从$6$名学生中选出$4$人分别担任一、二、三、四辩, 若其中学生甲必须参赛且不担任四辩, 则不同的安排方法种数为\\blank{50}.(结果用数值表示)", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-上学期周末卷-周末卷13" ], "genre": "填空题", "ans": "$180$", @@ -322148,7 +324623,8 @@ "content": "设$a \\in \\mathbf{R}$, 若$(x^2+\\dfrac 2x)^9$与$(x+\\dfrac a{x^2})^9$的二项展开式中的常数项相等, 则$a=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-上学期周末卷-周末卷13" ], "genre": "填空题", "ans": "$4$", @@ -322176,7 +324652,8 @@ "content": "设$m \\in \\mathbf{R}$, 若$z$是关于$x$的方程$x^2+m x+m^2-1=0$的一个虚根, 则$|\\overline{z}|$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期周末卷-周末卷13" ], "genre": "填空题", "ans": "$(\\dfrac{\\sqrt{3}}3,+\\infty)$", @@ -322205,7 +324682,10 @@ "objs": [], "tags": [ "第二单元", - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-周末卷13", + "2023届高三-四月错题重做-02_函数二", + "2023届高三-四月错题重做-02_易错题-函数2" ], "genre": "填空题", "ans": "$(\\dfrac{11\\pi}6,\\dfrac{19\\pi}6]$", @@ -322245,7 +324725,8 @@ "content": "如图, 正方形$ABCD$的边长为$20$米, 圆$O$的半径为$1$米, 圆心是正方形的中心, 点$P$、$Q$分别在线段$AD$、$CB$上, 若线段$PQ$与圆$O$有公共点, 则称点$Q$在点$P$的 ``盲区'' 中, 已知点$P$以$1.5$米/秒的速度从$A$出发向$D$移动, 同时, 点$Q$以$1$米/秒的速度从$C$出发向$B$移动, 则在点$P$从$A$移动到$D$的过程中, 点$Q$在点$P$的盲区中的时长约为\\blank{50}秒.(精确到$0.1$秒)\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$A$} coordinate (A) --++ (2,0) node [below right] {$B$} coordinate (B) --++ (0,2) node [above right] {$C$} coordinate (C) --++ (-2,0) node [above left] {$D$} coordinate (D) -- cycle;\n\\filldraw (1,1) circle (0.01);\n\\draw (1,1) circle (0.1);\n\\draw (1.1,0.9) node [below right] {$O$};\n\\draw (0,0.45) node [left] {$P$} -- (2,1.7) node [right] {$Q$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-上学期周末卷-周末卷13" ], "genre": "填空题", "ans": "$4.4$", @@ -322274,7 +324755,8 @@ "content": "下列函数中, 为偶函数的是\\bracket{20}.\n\\fourch{$y=x^{-2}$}{$y=x^{\\frac 13}$}{$y=x^{-\\frac 12}$}{$y=x^3$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷13" ], "genre": "选择题", "ans": "A", @@ -322302,7 +324784,8 @@ "content": "如图, 在直三棱柱$ABC-A_1B_1C_1$的棱所在的直线中, 与直线$BC_1$异面的直线的条数为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (2,0,{sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw (A) --++ (0,2,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) --++ (0,2,0) node [below left] {$B_1$} coordinate (B_1);\n\\draw (C) --++ (0,2,0) node [right] {$C_1$} coordinate (C_1);\n\\draw (B) -- (C_1);\n\\draw (A) -- (B) -- (C) (A_1) -- (B_1) -- (C_1) (A_1) -- (C_1);\n\\draw [dashed] (A) -- (C);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{1}{2}{3}{4}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期周末卷-周末卷13" ], "genre": "选择题", "ans": "C", @@ -322355,7 +324838,8 @@ "objs": [], "tags": [ "第五单元", - "第七单元" + "第七单元", + "2023届高三-上学期周末卷-周末卷13" ], "genre": "选择题", "ans": "B", @@ -322384,7 +324868,8 @@ "content": "已知$f(x)=\\cos x$.\\\\\n(1) 若$f(\\alpha)=\\dfrac 13$, 且$\\alpha \\in[0, \\pi]$, 求$f(\\alpha-\\dfrac{\\pi}3)$的值;\\\\\n(2) 求函数$y=f(2x)-2 f(x)$的最小值.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-周末卷13" ], "genre": "解答题", "ans": "(1) $\\dfrac 16+\\dfrac{\\sqrt{6}}3$; (2) $-\\dfrac 32$", @@ -322413,7 +324898,10 @@ "content": "已知$a \\in \\mathbf{R}$, 双曲线$\\Gamma: \\dfrac{x^2}{a^2}-y^2=1$.\\\\\n(1) 若点$(2,1)$在$\\Gamma$上, 求$\\Gamma$的焦点坐标;\\\\\n(2) 若$a=1$, 直线$y=k x+1$与$\\Gamma$相交于$A$、$B$两点, 且线段$AB$中点的横坐标为$1$, 求实数$k$的值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-上学期周末卷-周末卷13", + "2023届高三-四月错题重做-04_易错题-解析几何", + "2023届高三-四月错题重做-04_解析几何" ], "genre": "解答题", "ans": "(1) $(\\pm \\sqrt{3},0)$; (2) $\\dfrac{\\sqrt{5}-1}2$", @@ -322454,7 +324942,8 @@ "objs": [], "tags": [ "第六单元", - "第七单元" + "第七单元", + "2023届高三-上学期周末卷-周末卷13" ], "genre": "解答题", "ans": "(1) $0.25$米; (2) $9.59^\\circ$", @@ -322506,7 +324995,8 @@ "content": "若$\\{c_n\\}$是严格递增数列, 数列$\\{a_n\\}$满足: 对任意$n \\in \\mathbf{N}$, $n\\ge 1$, 存在$m \\in \\mathbf{N}$, $m\\ge 1$, 使得$\\dfrac{a_m-c_n}{a_m-c_{n+1}} \\leq 0$, 则称$\\{a_n\\}$是$\\{c_n\\}$的``分隔数列''.\\\\\n(1) 设$c_n=2 n$, $a_n=n+1$, 证明: 数列$\\{a_n\\}$是$\\{c_n\\}$的分隔数列;\\\\\n(2) 设$c_n=n-4$, $S_n$是$\\{c_n\\}$的前$n$项和, $d_n=c_{3 n-2}$, 判断数列$\\{S_n\\}$是否是数列$\\{d_n\\}$的分隔数列, 并说明理由;\\\\\n(3) 设$c_n=a q^{n-1}$, $T_n$是$\\{c_n\\}$的前$n$项和, 若数列$\\{T_n\\}$是$\\{c_n\\}$的分隔数列, 求实数$a$、$q$的取值范围.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-上学期周末卷-周末卷13" ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\{S_n\\}$不是$\\{d_n\\}$的分隔数列, 例如$d_{10}$与$d_{11}$之间没有$\\{S_n\\}$中的项; (3) $a>0$且$q\\ge 2$", @@ -322534,7 +325024,9 @@ "content": "已知集合$A=\\{1,2,3,4,5\\}$, $B=\\{3,5,6\\}$, 则$A \\cap B=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期测验卷-测验11", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$\\{3,5\\}$", @@ -322598,7 +325090,8 @@ "content": "不等式$|x+1|<5$的解集为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期测验卷-测验11" ], "genre": "填空题", "ans": "$(-6,4)$", @@ -322658,7 +325151,9 @@ "content": "设$\\mathrm{i}$为虚数单位, $3 \\overline{z}-\\mathrm{i}=6+5 \\mathrm{i}$, 则$|z|$的值为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期测验卷-测验11", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$2\\sqrt{2}$", @@ -322696,7 +325191,9 @@ "content": "已知二元线性方程组$\\begin{cases}2 x+2 y=-1, \\\\4 x+a^2 y=a\\end{cases}$有无穷多解, 则实数$a=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期测验卷-测验11", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$-2$", @@ -322746,7 +325243,9 @@ "content": "在$(x+\\dfrac 1{\\sqrt x})^6$的二项展开式中, 常数项的值为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-上学期测验卷-测验11", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$15$", @@ -322784,7 +325283,9 @@ "content": "在$\\triangle ABC$中, $AC=3$, $3 \\sin A=2 \\sin B$, 且$\\cos C=\\dfrac 14$, 则$AB=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-测验11", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$\\sqrt{10}$", @@ -322824,7 +325325,8 @@ "content": "首届中国国际进口博览会在上海举行, 某高校拟派$4$人参与连续$5$天的志愿者活动, 其中甲连续参加$2$天, 其余每人各参加$1$天, 则所有不同的安排种数为\\blank{50}.(结果用数值表示)", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-上学期测验卷-测验11" ], "genre": "填空题", "ans": "$24$", @@ -322859,7 +325361,8 @@ "content": "如图, 正方形$OABC$的边长为$a$($a>1$), 函数$y=3 x^2$的图像交$AB$于点$Q$, 函数$y=x^{-\\frac 12}$与$BC$交于点$P$, 则当$|AQ|+|CP|$最小时, $a$的值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.5,0) -- (2.5,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (2,0) node [below] {$C$} coordinate (C) -- (2,2) node [above right] {$B$} coordinate (B) -- (0,2) node [left] {$A$} coordinate (A);\n\\draw [samples = 100, domain = 0.2:2.2] plot (\\x,{pow(\\x,-0.5)});\n\\draw [samples = 100, domain = 0:0.9] plot (\\x,{3*pow(\\x,2)});\n\\draw (2,{sqrt(2)/2}) node [below left] {$P$} coordinate (P);\n\\draw ({sqrt(2/3)},2) node [above right] {$Q$} coordinate (Q);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验11" ], "genre": "填空题", "ans": "$\\sqrt{3}$", @@ -322894,7 +325397,9 @@ "content": "已知$P$为椭圆$\\dfrac{x^2}4+\\dfrac{y^2}2=1$上的任意一点, $Q$与$P$关于$x$轴对称, $F_1$、$F_2$为椭圆的左、右焦点, 若有$\\overrightarrow{F_1 P} \\cdot \\overrightarrow{F_2P} \\leq 1$, 则向量$\\overrightarrow{F_1 P}$与$\\overrightarrow{F_2 Q}$的夹角范围为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-上学期测验卷-测验11", + "2023届高三-四月错题重做-04_解析几何" ], "genre": "填空题", "ans": "$[\\pi-\\arccos\\dfrac 13,\\pi]$", @@ -322931,7 +325436,8 @@ "content": "已知$t \\in \\mathbf{R}$, 集合$A=[t, t+1] \\cup[t+4, t+9]$, $0 \\not\\in A$, 若存在正数$\\lambda$, 对任意$a \\in A$, 都有$\\dfrac{\\lambda}a \\in A$, 则$t$的值为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期测验卷-测验11" ], "genre": "填空题", "ans": "$-3$或$1$", @@ -322967,7 +325473,8 @@ "objs": [], "tags": [ "第二单元", - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-测验11" ], "genre": "选择题", "ans": "B", @@ -323005,7 +325512,9 @@ "content": "已知$a$、$b \\in \\mathbf{R}$, 则``$a^2>b^2$''是``$|a|>|b|$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期测验卷-测验11", + "2023届高三-寒假作业-容易题" ], "genre": "选择题", "ans": "C", @@ -323040,7 +325549,9 @@ "content": "已知平面$\\alpha$、$\\beta$、$\\gamma$两两垂直, 直线$a$、$b$、$c$满足: $a \\subseteq \\alpha$, $b \\subseteq \\beta$, $c \\subseteq \\gamma$, 则直线$a$、$b$、$c$不可能是\\bracket{20}.\n\\fourch{两两垂直}{两两平行}{两两相交}{两两异面}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期测验卷-测验11", + "2023届高三-寒假作业-容易题" ], "genre": "选择题", "ans": "B", @@ -323077,7 +325588,8 @@ "content": "平面直角坐标系中, 两动圆$O_1$、$O_2$的圆心分别为$(a_1, 0)$、$(a_2, 0)$, 且两圆均过定点$(1,0)$, 两圆与$y$轴正半轴分别交于点$(0, y_1)$、$(0, y_2)$, 若$\\ln y_1+\\ln y_2=0$, 点$(\\dfrac 1{a_1}, \\dfrac 1{a_2})$的轨迹为$\\Gamma$, 则$\\Gamma$所在的曲线可能是\\bracket{20}.\n\\fourch{直线}{圆}{椭圆}{双曲线}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-上学期测验卷-测验11" ], "genre": "选择题", "ans": "A", @@ -323112,7 +325624,8 @@ "content": "如图, 正三棱锥$P-ABC$中, 侧棱长为$2$, 底面边长为$\\sqrt 3$, $M$、$N$分别是$PB$和$BC$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (1,0,{sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw (1,{sqrt(3)},{sqrt(3)/3}) node [above] {$P$} coordinate (P);\n\\draw ($(B)!0.5!(P)$) node [right] {$M$} coordinate (M);\n\\draw ($(B)!0.5!(C)$) node [below] {$N$} coordinate (N);\n\\draw (A) -- (B) -- (C) (A) -- (P) -- (C) (M) -- (N) (P) -- (B);\n\\draw [dashed] (A) -- (C); \n\\end{tikzpicture}\n\\end{center}\n(1) 求异面直线$MN$与$AC$所成角的大小;\\\\\n(2) 求三棱锥$P-ABC$的体积.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期测验卷-测验11" ], "genre": "解答题", "ans": "(1) $\\arccos \\dfrac{\\sqrt{3}}3$; (2) $\\dfrac 34$", @@ -323147,7 +325660,8 @@ "content": "已知数列$\\{a_n\\}$中, $a_1=3$, 前$n$项和为$S_n$.\\\\\n(1) 若$\\{a_n\\}$为等差数列, 且$a_4=15$, 求$S_n$;\\\\\n(2) 若$\\{a_n\\}$为等比数列, 且$\\displaystyle\\lim_{n\\to\\infty} S_n<12$, 求公比$q$的取值范围.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-上学期测验卷-测验11" ], "genre": "解答题", "ans": "(1) $S_n=2n^2+n$; (2) $(-1,0)\\cup (0,\\dfrac 34)$", @@ -323256,7 +325770,8 @@ "content": "集合$A=\\{1,3\\}, B=\\{1,2, a\\}$, 若$A \\subseteq B$, 则$a=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期周末卷-周末卷14" ], "genre": "填空题", "ans": "$3$", @@ -323284,7 +325799,8 @@ "content": "不等式$\\dfrac 1x>3$的解集为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期周末卷-周末卷14" ], "genre": "填空题", "ans": "$(0,\\dfrac 13)$", @@ -323312,7 +325828,8 @@ "content": "函数$y=\\tan 2 x$的最小正周期为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-周末卷14" ], "genre": "填空题", "ans": "$\\dfrac\\pi 2$", @@ -323340,7 +325857,8 @@ "content": "已知复数$z$满足$z+2 \\overline{z}=6+\\mathrm{i}$, 则$z$的实部为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期周末卷-周末卷14" ], "genre": "填空题", "ans": "$2$", @@ -323368,7 +325886,8 @@ "content": "已知$3 \\sin 2 x=2 \\sin x$, $x \\in(0, \\pi)$, 则$x=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-周末卷14" ], "genre": "填空题", "ans": "$\\arccos \\dfrac 13$", @@ -323396,7 +325915,8 @@ "content": "若函数$y=a \\cdot 3^x+\\dfrac 1{3^x}$为偶函数, 则$a=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷14" ], "genre": "填空题", "ans": "$1$", @@ -323424,7 +325944,8 @@ "content": "已知直线$l_1: x+a y=1$, $l_2: a x+y=1$, 若$l_1\\parallel l_2$, 则$l_1$与$l_2$的距离为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-上学期周末卷-周末卷14" ], "genre": "填空题", "ans": "$\\sqrt{2}$", @@ -323452,7 +325973,8 @@ "content": "已知二项式$(2 x+\\sqrt x)^5$, 则展开式中$x^3$的系数为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-上学期周末卷-周末卷14" ], "genre": "填空题", "ans": "$10$", @@ -323480,7 +326002,8 @@ "content": "三角形$ABC$中, $D$是$BC$中点, $AB=2$, $AC=3$, $BC=4$, 则$\\overrightarrow{AD} \\cdot \\overrightarrow{AB}=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-周末卷14" ], "genre": "填空题", "ans": "$\\dfrac 54$", @@ -323509,7 +326032,8 @@ "content": "已知$A=\\{-3,-2,-1,0,1,2,3\\}$, $a$、$b \\in A$, 则$|a|<|b|$的情况有\\blank{50}种.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-上学期周末卷-周末卷14" ], "genre": "填空题", "ans": "$18$", @@ -323537,7 +326061,8 @@ "content": "已知平面上有$A_1$、$A_2$、$A_3$、$A_4$、$A_5$五个点, 满足$\\overrightarrow{A_n A_{n+1}} \\cdot \\overrightarrow{A_{n+1} A_{n+2}}=0$($n=1,2,3$), $|\\overrightarrow{A_n A_{n+1}}|\\cdot|\\overrightarrow{A_{n+1} A_{n+2}}|=n+1$($n=1,2,3$), 则$|\\overrightarrow{A_1 A_5}|$的最小值为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期周末卷-周末卷14" ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{6}}3$", @@ -323565,7 +326090,8 @@ "content": "已知$f(x)=\\sqrt {x-1}$, 其反函数为$f^{-1}(x)$, 若$f^{-1}(x)-a=f(x+a)$有实数根, 则$a$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷14" ], "genre": "填空题", "ans": "$[\\dfrac 34,+\\infty)$", @@ -323615,7 +326141,8 @@ "content": "``$\\alpha=\\beta$''是``$\\sin ^2 \\alpha+\\cos ^2 \\beta=1$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-周末卷14" ], "genre": "选择题", "ans": "A", @@ -323643,7 +326170,8 @@ "content": "已知椭圆$\\dfrac{x^2}2+y^2=1$, 作垂直于$x$轴的垂线交椭圆于$A$、$B$两点, 作垂直于$y$轴的垂线交椭圆于$C$、$D$两点, 且$AB=CD$, 两垂线相交于点$P$, 则点$P$的轨迹是\\bracket{20}.\n\\fourch{椭圆的一部分}{双曲线的一部分}{圆的一部分}{抛物线的一部分}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-上学期周末卷-周末卷14" ], "genre": "选择题", "ans": "B", @@ -323672,7 +326200,10 @@ "content": "数列$\\{a_n\\}$各项均为实数, 对任意$n \\in \\mathbf{N}$, $n\\ge 1$满足$a_{n+3}=a_n$, 且$a_na_{n+3}-a_{n+1}a_{n+2}=c$为定值, 则下列选项中不可能的是\\bracket{20}.\n\\fourch{$a_1=1$, $c=1$}{$a_1=2$, $c=2$}{$a_1=-1$, $c=4$}{$a_1=2$, $c=0$}", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-上学期周末卷-周末卷14", + "2023届高三-四月错题重做-03_数列", + "2023届高三-四月错题重做-03_易错题-数列" ], "genre": "选择题", "ans": "B", @@ -323712,7 +326243,8 @@ "content": "已知四棱锥$P-ABCD$, 底面$ABCD$为正方形, 边长为$3$, $PD \\perp$平面$ABCD$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$D$} coordinate (D);\n\\draw (3,0,0) node [right] {$C$} coordinate (C);\n\\draw (3,0,3) node [right] {$B$} coordinate (B);\n\\draw (0,0,3) node [left] {$A$} coordinate (A);\n\\draw (D) ++ (0,2,0) node [above] {$P$} coordinate (P);\n\\draw (P) -- (A) (P) -- (B) (P) -- (C);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C) (D) -- (P);\n\\end{tikzpicture}\n\\end{center}\n(1) 若$PC=5$, 求四棱锥$P-ABCD$的体积;\\\\\n(2) 若直线$AD$与$BP$的夹角为$60^{\\circ}$, 求$PD$的长.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期周末卷-周末卷14" ], "genre": "解答题", "ans": "(1) $12$; (2) $3\\sqrt{2}$", @@ -323740,7 +326272,8 @@ "content": "已知各项均为正数的数列$\\{a_n\\}$, 其前$n$项和为$S_n$, $a_1=1$.\\\\\n(1) 若数列$\\{a_n\\}$为等差数列, $S_{10}=70$, 求数列$\\{a_n\\}$的通项公式;\\\\\n(2) 若数列$\\{a_n\\}$为等比数列, $a_4=\\dfrac 18$, 求满足$S_n>100 a_n$时$n$的最小值.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-上学期周末卷-周末卷14" ], "genre": "解答题", "ans": "(1) $a_n=\\dfrac 43n-\\dfrac 13$; (2) $n$的最小值为$7$", @@ -323768,7 +326301,8 @@ "content": "有一条长为$120$米的步行道$OA$, $A$是垃圾投放点$\\omega_1$, 若以$O$为原点, $OA$为$x$轴正半轴建立直角坐标系, 设点$B(x, 0)$, 现要建设另一座垃圾投放点$\\omega_2(t, 0)$, 函数$f_t(x)$表示与$B$点距离最近的垃圾投放点的距离.\\\\\n(1) 若$t=60$, 求$f_{60}(10)$、$f_{60}(80)$、$f_{60}(95)$的值, 并写出$f_{60}(x)$的函数解析式;\\\\\n(2) 若可以通过$f_t(x)$与坐标轴围成的面积来测算扔垃圾的便利程度, 面积越小越便利. 问: 垃圾投放点$\\omega_2$建在何处才能比建在中点时更加便利?", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷14" ], "genre": "解答题", "ans": "(1) $f_{60}(x)=\\begin{cases}|60-x|, & 0\\le x\\le 90,\\\\ |120-x|, & 900$, 若$f(x)$具有$A$性质, 求$a$的取值范围;\\\\\n(3) 当$A=\\{-2, m\\}$, $m \\in \\mathbf{Z}$, 若$D=\\mathbf{Z}$且具有$A$性质的函数均为常值函数, 求所有符合条件的$m$的值.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷14" ], "genre": "解答题", "ans": "(1) $f(x)=-x$具有$A$性质, $g(x)=2x$不具有$A$性质, 理由略; (2) $[1,+\\infty)$; (3) 全体正奇数, 理由略", @@ -323853,7 +326389,8 @@ "content": "已知等差数列$\\{a_n\\}$的首项为$3$, 公差为$2$, 则$a_{10}=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-上学期测验卷-测验12" ], "genre": "填空题", "ans": "$21$", @@ -323884,7 +326421,8 @@ "content": "已知$z=1-3 \\mathrm{i}$, 则$|\\overline{z}-\\mathrm{i}|=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期测验卷-测验12" ], "genre": "填空题", "ans": "$\\sqrt{5}$", @@ -323915,7 +326453,8 @@ "content": "已知圆柱的底面半径为$1$, 高为$2$, 则圆柱的侧面积为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期测验卷-测验12" ], "genre": "填空题", "ans": "$4\\pi$", @@ -323946,7 +326485,8 @@ "content": "不等式$\\dfrac{2 x+5}{x-2}<1$的解集为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期测验卷-测验12" ], "genre": "填空题", "ans": "$(-7,2)$", @@ -323977,7 +326517,8 @@ "content": "直线$x=-2$与直线$\\sqrt 3 x-y+1=0$的夹角为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-上学期测验卷-测验12" ], "genre": "填空题", "ans": "$\\dfrac \\pi 6$", @@ -324030,7 +326571,8 @@ "content": "已知$(1+x)^n$的展开式中, 唯有$x^3$的系数最大, 则$x^3$的系数为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-上学期测验卷-测验12" ], "genre": "填空题", "ans": "$64$", @@ -324061,7 +326603,8 @@ "content": "已知函数$f(x)=3^x+\\dfrac a{3^x+1}$($a>0$)的最小值为$5$, 则$a=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验12" ], "genre": "填空题", "ans": "$9$", @@ -324114,7 +326657,8 @@ "content": "某人某天需要运动总时长大于等于$60$分钟, 现有五项运动可以选择, 如下表所示, 共有\\blank{50}种运动方式组合.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline $A$运动 & $B$运动 & $C$运动 & $D$运动 & $E$运动 \\\\\n\\hline $7$点--$8$点 & $8$点--$9$点 & $9$点--$10$点 & $10$点--$11$点 & $11$点--$12$点 \\\\\n\\hline $30$分钟 & $20$分钟 & $40$分钟 & $30$分钟 & $30$分钟 \\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-上学期测验卷-测验12" ], "genre": "填空题", "ans": "$23$", @@ -324145,7 +326689,10 @@ "content": "已知椭圆$x^2+\\dfrac{y^2}{b^2}=1$($00$, 存在实数$\\varphi$, 使得对任意$n \\in \\mathbf{N}$, $n\\ge 1$, $\\cos (n \\theta+\\varphi)<\\dfrac{\\sqrt 3}2$, 则$\\theta$的最小值是\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-测验12" ], "genre": "填空题", "ans": "$\\dfrac{2\\pi}5$", @@ -324220,7 +326768,8 @@ "objs": [], "tags": [ "第二单元", - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-测验12" ], "genre": "选择题", "ans": "C", @@ -324251,7 +326800,8 @@ "content": "已知集合$A=\\{x|x>-1,\\ x \\in \\mathbf{R}\\}$, $B=\\{x|x^2-x-2 \\ge 0,\\ x \\in \\mathbf{R}\\}$, 则下列关系中, 正确的是\\bracket{20}\n\\fourch{$A \\subseteq B$}{$\\overline{A} \\subseteq \\overline{B}$}{$A \\cap B=\\varnothing$}{$A \\cup B=\\mathbf{R}$}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期测验卷-测验12" ], "genre": "选择题", "ans": "D", @@ -324282,7 +326832,8 @@ "content": "已知函数$y=f(x)$的定义域为$\\mathbf{R}$, 下列是$f(x)$无最大值的充分条件的是\\bracket{20}.\n\\twoch{$f(x)$为偶函数且关于点$(1,1)$对称}{$f(x)$为偶函数且关于直线$x=1$对称}{$f(x)$为奇函数且关于点$(1,1)$对称}{$f(x)$为奇函数且关于直线$x=1$对称}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验12" ], "genre": "选择题", "ans": "C", @@ -324313,7 +326864,8 @@ "content": "在$\\triangle ABC$中, $D$为$BC$中点, $E$为$AD$中点, 则以下结论: \\textcircled{1} 存在$\\triangle ABC$, 使得$\\overrightarrow{AB} \\cdot \\overrightarrow{CE}=0$; \\textcircled{2} 存在三角形$\\triangle ABC$, 使得$\\overrightarrow{CE} \\parallel (\\overrightarrow{CB}+\\overrightarrow{CA})$; 它们的成立情况是\\bracket{20}.\n\\fourch{\\textcircled{1}成立, \\textcircled{2}成立}{\\textcircled{1}成立, \\textcircled{2}不成立}{\\textcircled{1}不成立, \\textcircled{2}成立}{\\textcircled{1}不成立, \\textcircled{2}不成立}", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期测验卷-测验12" ], "genre": "选择题", "ans": "B", @@ -324344,7 +326896,8 @@ "content": "四棱锥$P-ABCD$, 底面为正方形$ABCD$, 边长为$4$, $E$为$AB$中点, $PE \\perp$平面$ABCD$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (4,0,0) node [right] {$D$} coordinate (D);\n\\draw (4,0,4) node [right] {$C$} coordinate (C);\n\\draw (0,0,4) node [left] {$B$} coordinate (B);\n\\draw ($(A)!0.5!(B)$) node [left] {$E$} coordinate (E) ++ (0,{2*sqrt(3)},0) node [above] {$P$} coordinate (P);\n\\draw (E) ++ (4,0,0) node [right] {$F$} coordinate (F) -- (P);\n\\draw (P) -- (B) (P) -- (C) (P) -- (D) (B) -- (C) -- (D);\n\\draw [dashed] (E) -- (F) (P) -- (A) (A) -- (B) (A) -- (D) (A) -- (C) (P) -- (E);\n\\end{tikzpicture}\n\\end{center}\n(1) 若$\\triangle PAB$为等边三角形, 求四棱锥$P-ABCD$的体积;\\\\ \n(2) 若$CD$的中点为$F$, $PF$与平面$ABCD$所成角为$45^{\\circ}$, 求$PC$与$AD$所成角的大小.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期测验卷-测验12" ], "genre": "解答题", "ans": "(1) $\\dfrac{32\\sqrt{3}}3$; (2) $\\arctan \\dfrac{\\sqrt{5}}2$", @@ -324375,7 +326928,8 @@ "content": "已知$A$、$B$、$C$为$\\triangle ABC$的三个内角, $a$、$b$、$c$是其三条边, $a=2$, $\\cos C=-\\dfrac 14$.\\\\\n(1) 若$\\sin A=2 \\sin B$, 求$b$、$c$;\\\\\n(2) 若$\\cos (A-\\dfrac{\\pi}4)=\\dfrac 45$, 求$c$.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-测验12" ], "genre": "解答题", "ans": "(1) $b=1$, $c=\\sqrt{6}$; (2) $\\dfrac{5\\sqrt{30}}2$", @@ -324406,7 +326960,8 @@ "content": "设$O$是地面上的一点, 团队在$O$点西侧、东侧$20$千米处设有$A$、$B$两站点.\\\\\n(1) 测量距离发现一点$P$满足$|PA|-|PB|=20$千米, 可知$P$在以$A$、$B$为焦点的双曲线上, 以$O$点为原点, 东侧为$x$轴正半轴, 北侧为$y$轴正半轴, 建立平面直角坐标系, 若$P$在$O$点北偏东$60^{\\circ}$处, 求双曲线标准方程和$P$点坐标;\\\\\n(2) 团队又在$O$点南侧、北侧$15$千米处设有$C$、$D$两站点, 测量距离发现一点$Q$满足$|QA|-|QB|=30$千米, $|QC|-|QD|=10$千米, 求$|OQ|$和$Q$点方位. (精确到$1$千米, $1^{\\circ}$)", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-上学期测验卷-测验12" ], "genre": "解答题", "ans": "(1) 双曲线的标准方程为$\\dfrac{x^2}{100}-\\dfrac{y^2}{300}=1$, $P$的坐标为$(\\dfrac{\\sqrt{15\\sqrt{2}}}2, \\dfrac{5\\sqrt{6}}2)$; (2) $|OQ|\\approx 19$千米, $Q$点的位置再$O$点北偏东约$66^\\circ$方向", @@ -324483,7 +327038,8 @@ "content": "已知$z=2+\\mathrm{i}$(其中$\\mathrm{i}$为虚数单位), 则$\\overline{z}=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期测验卷-测验13" ], "genre": "填空题", "ans": "$2-\\mathrm{i}$", @@ -324517,7 +327073,8 @@ "content": "已知集合$A=(-1,2)$, 集合$B=(1,3)$, 则$A \\cap B=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期测验卷-测验13" ], "genre": "填空题", "ans": "$(1,2)$", @@ -324551,7 +327108,8 @@ "content": "不等式$\\dfrac{x-1}x<0$的解集为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验13" ], "genre": "填空题", "ans": "$(0,1)$", @@ -324585,7 +327143,8 @@ "content": "若$\\tan \\alpha=3$, 则$\\tan (\\alpha+\\dfrac{\\pi}4)=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-测验13" ], "genre": "填空题", "ans": "$-2$", @@ -324619,7 +327178,8 @@ "content": "设函数$f(x)=x^3$的反函数为$f^{-1}(x)$, 则$f^{-1}(27)=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验13" ], "genre": "填空题", "ans": "$3$", @@ -324653,7 +327213,8 @@ "content": "在$(x^3+\\dfrac 1x)^{12}$的展开式中, 含$\\dfrac 1{x^4}$项的系数为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-上学期测验卷-测验13" ], "genre": "填空题", "ans": "$66$", @@ -324687,7 +327248,8 @@ "content": "若关于$x, y$的方程组$\\begin{cases}x+m y=2, \\\\m x+16 y=8\\end{cases}$有无穷多解, 则实数$m$的值为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期测验卷-测验13" ], "genre": "填空题", "ans": "$4$", @@ -324721,7 +327283,8 @@ "content": "已知在$\\triangle ABC$中, $\\angle A=\\dfrac{\\pi}3$, $AB=2$, $AC=3$, 则$\\triangle ABC$的外接圆半径为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-测验13" ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{21}}3$", @@ -324755,7 +327318,8 @@ "content": "用数字$1$、$2$、$3$、$4$组成没有重复数字的四位数, 则这些四位数中比$2134$大的数字个数为\\blank{50}.(用数字作答)", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-上学期测验卷-测验13" ], "genre": "填空题", "ans": "$17$", @@ -324789,7 +327353,8 @@ "content": "在$\\triangle ABC$中, $\\angle C=90^{\\circ}$, $BC=AC=2$, 点$M$为边$AB$的中点, 点$P$在边$BC$上, 则$\\overrightarrow{MP}\\cdot \\overrightarrow{CP}$的最小值为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期测验卷-测验13" ], "genre": "填空题", "ans": "$\\dfrac 78$", @@ -324823,7 +327388,8 @@ "content": "已知$P_1(x_1, y_1)$, $P_2(x_2, y_2)$两点均在双曲线$\\Gamma: \\dfrac{x^2}{a^2}-y^2=1(a>0)$的右支上, 若$x_1 x_2>y_1 y_2$恒成立, 则实数$a$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-上学期测验卷-测验13" ], "genre": "填空题", "ans": "$[1,+\\infty)$", @@ -324857,7 +327423,8 @@ "content": "已知函数$y=f(x)$为定义域为$\\mathbf{R}$的奇函数, 其图像关于$x=1$对称, 且当$x \\in (0, 1]$时, $f(x)=\\ln x$, 若将方程$f(x)=x+1$的正实数根从小到大依次记为$x_1,x_2,x_3,\\cdots,x_n$, 则$\\displaystyle\\lim_{n\\to\\infty}(x_{n+1}-x_n)=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验13" ], "genre": "填空题", "ans": "$2$", @@ -324891,7 +327458,8 @@ "content": "下列函数定义域为$\\mathbf{R}$的是\\bracket{20}.\n\\fourch{$y=x^{-\\frac 12}$}{$y=x^{-1}$}{$y=x^{\\frac 13}$}{$y=x^{\\frac 12}$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验13" ], "genre": "选择题", "ans": "C", @@ -324925,7 +327493,8 @@ "content": "若$a>b>c>d$, 则下列不等式恒成立的是\\bracket{20}.\n\\fourch{$a+d>b+c$}{$a+c>b+d$}{$a c>b d$}{$a d>b c$}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期测验卷-测验13" ], "genre": "选择题", "ans": "B", @@ -324959,7 +327528,8 @@ "content": "上海海关大楼的顶部为逐级收拢的四面钟楼, 如图, 四个大钟分布在正四棱柱的四个侧面, 则每天$0$点至$12$点(包含$0$点, 不含$12$点)相邻两钟面上的时针相互垂直的次数为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\begin{scope}[x = {(-10:0.9)}]\n\\draw (0,0) -- (-2,0) -- (-2,2) -- (0,2) -- cycle;\n\\draw (-1,1) circle (0.8);\n\\draw [->] (-1,1) --++ (-45:0.5);\n\\draw [->] (-1,1) --++ (-90:0.65);\n\\foreach \\i in {1,2,...,12} {\\draw (-1,1) ++ ({30*\\i}:0.7) --++ ({30*\\i}:0.05);};\n\\end{scope}\n\\begin{scope}[x = {(40:0.7)}]\n\\draw (0,0) -- (2,0) -- (2,2) -- (0,2) -- cycle;\n\\draw (1,1) circle (0.8);\n\\draw [->] (1,1) --++ (-45:0.5);\n\\draw [->] (1,1) --++ (-90:0.65);\n\\foreach \\i in {1,2,...,12} {\\draw (1,1) ++ ({30*\\i}:0.7) --++ ({30*\\i}:0.05);};\n\\end{scope}\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$0$}{$2$}{$4$}{$12$}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期测验卷-测验13" ], "genre": "选择题", "ans": "B", @@ -325039,7 +327609,8 @@ "content": "已知在数列$\\{a_n\\}$中, $a_2=1$, 其前$n$项和为$S_n$.\\\\\n(1) 若$\\{a_n\\}$是等比数列, $S_2=3$, 求$\\displaystyle\\lim_{n\\to\\infty} S_n$;\\\\\n(2) 若$\\{a_n\\}$是等差数列, $S_{2 n} \\geq n$, 求其公差$d$的取值范围.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-上学期测验卷-测验13" ], "genre": "解答题", "ans": "(1) $4$; (2) $(-\\infty,1]$", @@ -325073,7 +327644,8 @@ "content": "为有效塑造城市景观、提升城市环境品质, 上海市正在努力推进新一轮架空线入地工程的建设. 如图是一处要架空线入地的矩形地块$ABCD$, $AB=30\\text{m}$, $AD=15\\text{m}$. 为保护$D$处的一棵古树, 有关部门划定了以$D$为圆心、$DA$为半径的四分之一圆的地块为历史古迹封闭区. 若空线入线口为$AB$边上的点$E$, 出线口为$CD$边上的点$F$, 施工要求$EF$与封闭区边界相切, $EF$右侧的四边形地块$BCFE$将作为绿地保护生态区.(计算长度精确到$0.1\\text{m}$, 计算面积精确到$0.01 \\text{m}^2$)\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.5]\n\\draw (0,0) node [below] {$A$} coordinate (A) -- (2,0) node [below] {$B$} coordinate (B) -- (2,1) node [above] {$C$} coordinate (C) -- (0,1) node [above] {$D$} coordinate (D) -- cycle;\n\\draw (0,0) arc (-90:0:1);\n\\draw ({tan(20)},0) node [below] {$E$} coordinate (E) -- ({1/cos(50)},1) node [above] {$F$} coordinate (F);\n\\end{tikzpicture}\n\\end{center}\n(1) 若$\\angle ADE=20^{\\circ}$, 求$EF$的长;\\\\\n(2) 当入线口$E$在$AB$上的什么位置时, 生态区的面积最大? 最大面积是多少?", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-测验13" ], "genre": "解答题", "ans": "(1) 约$23.3\\text{m}$; (2) 当$AE$的长约为$8.7\\text{m}$时, 绿地保护生态区的面积最大, 最大面积为$255.14\\text{m}^2$", @@ -325107,7 +327679,8 @@ "content": "已知椭圆$\\Gamma: \\dfrac{x^2}{a^2}+y^2=1$($a>1$), $A$、$B$两点分别为$\\Gamma$的左顶点、下顶点, $C$、$D$两点均在直线$l: x=a$上, 且$C$在第一象限.\\\\\n(1) 设$F$是椭圆$\\Gamma$的右焦点, 且$\\angle AFB=\\dfrac{\\pi}6$, 求$\\Gamma$的标准方程;\\\\\n(2) 若$C$、$D$两点纵坐标分别为$2$、$1$, 请判断直线$AD$与直线$BC$的交点是否在椭圆$\\Gamma$上, 并说明理由;\\\\ \n(3) 设直线$AD$、$BC$分别交椭圆$\\Gamma$于点$P$、点$Q$, 若$P$、$Q$关于原点对称, 求$|CD|$的最小值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-上学期测验卷-测验13" ], "genre": "解答题", "ans": "(1) $\\dfrac{x^2}4+y^2=1$; (2) 交点在椭圆上, 理由略; (3) $|CD|$的最小值为$6$", @@ -325163,7 +327736,8 @@ "content": "已知集合$A=(-2,1]$, $B=\\mathbf{Z}$, 则$A \\cap B=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-简单题冲刺-简单题冲刺01" ], "genre": "填空题", "ans": "$\\{-1,0,1\\}$", @@ -325197,7 +327771,8 @@ "content": "函数$y=\\sin x \\cdot \\cos x$的最小正周期为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-简单题冲刺-简单题冲刺01" ], "genre": "填空题", "ans": "$\\pi$", @@ -325231,7 +327806,8 @@ "content": "已知$a$、$b \\in \\mathbf{R}$, $\\mathrm{i}$是虚数单位, 若$a-\\mathrm{i}$与$2+b \\mathrm{i}$互为共轭复数, 则$(a+b \\mathrm{i})^2=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-简单题冲刺-简单题冲刺01" ], "genre": "填空题", "ans": "$3+4\\mathrm{i}$", @@ -325265,7 +327841,8 @@ "content": "记$S_n$为等差数列$\\{a_n\\}$的前$n$项和, 若$2 S_3=3 S_2+6$, 则公差$d=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-简单题冲刺-简单题冲刺01" ], "genre": "填空题", "ans": "$2$", @@ -325299,7 +327876,8 @@ "content": "已知函数$y=a-\\dfrac 2{2^x+1}$为奇函数, 则实数$a=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-简单题冲刺-简单题冲刺01" ], "genre": "填空题", "ans": "$1$", @@ -325333,7 +327911,8 @@ "content": "已知圆锥的母线长为$5$, 侧面积为$20 \\pi$, 则此圆锥的体积为\\blank{50}. (结果保留$\\pi$)", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-简单题冲刺-简单题冲刺01" ], "genre": "填空题", "ans": "$16\\pi$", @@ -325367,7 +327946,8 @@ "content": "已知向量$\\overrightarrow a=(5,3)$, $\\overrightarrow b=(-1,2)$, 则$\\overrightarrow a$在$\\overrightarrow b$上的投影向量的坐标为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-简单题冲刺-简单题冲刺01" ], "genre": "填空题", "ans": "$(-\\dfrac 15,\\dfrac 25)$", @@ -325401,7 +327981,8 @@ "content": "对任意$x \\in \\mathbf{R}$, 不等式$|x-2|+|x-3|\\geq 2 a^2+a$恒成立, 则实数$a$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-简单题冲刺-简单题冲刺01" ], "genre": "填空题", "ans": "$[-1,\\dfrac 12]$", @@ -325436,7 +328017,8 @@ "objs": [], "tags": [ "第二单元", - "第一单元" + "第一单元", + "2023届高三-中档题冲刺-中档题冲刺01" ], "genre": "填空题", "ans": "$(4,5]$", @@ -325471,7 +328053,8 @@ "content": "已知$F_1$、$F_2$是双曲线$\\Gamma: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的左、右焦点, 点$M$是双曲线$\\Gamma$上的任意一点(不是顶点), 过$F_1$作$\\angle F_1MF_2$的角平分线的垂线, 垂足为$N$, 线段$F_1N$的延长线交$MF_2$于点$Q$, $O$是坐标原点, 若$|ON|=\\dfrac{|F_1F_2|}6$, 则双曲线$\\Gamma$的渐近线方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-中档题冲刺-中档题冲刺01" ], "genre": "填空题", "ans": "$y=\\pm 2\\sqrt{2} x$", @@ -325506,7 +328089,8 @@ "content": "动点$P$在棱长为$1$的正方体$ABCD-A_1B_1C_1D_1$表面上运动, 且与点$A$的距离是$\\dfrac{2 \\sqrt 3}3$, 点$P$的集合形成一条曲线, 这条曲线的长度为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-中档题冲刺-中档题冲刺01" ], "genre": "填空题", "ans": "$\\dfrac{5\\sqrt{3}}6\\pi$", @@ -325608,7 +328192,8 @@ "content": "在天文学中, 天体的明暗程度可以用星等或亮度来描述, 两颗星的星等与亮度满足$m_2-m_1=\\dfrac 52 \\lg \\dfrac{E_1}{E_2}$, 其中星等为$m_k$的星的亮度为$E_k$($k=1$、$2$), 已知太阳的星等是$-26.7$, 天狼星的星等是$-1.45$, 则太阳与天狼星的亮度的比值为\\bracket{20}.\n\\fourch{$10^{10.1}$}{$10.1$}{$\\lg 10.1$}{$10^{-10.1}$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-中档题冲刺-中档题冲刺01" ], "genre": "选择题", "ans": "A", @@ -325665,7 +328250,8 @@ "content": "如图, 已知$AB \\perp$平面$BCD$, $BC \\perp CD$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$B$} coordinate (B);\n\\draw ({2*sqrt(2)},0,0) node [right] {$D$} coordinate (D);\n\\draw ({sqrt(2)},0,{sqrt(2)}) node [below] {$C$} coordinate (C);\n\\draw (0,1,0) node [left] {$A$} coordinate (A);\n\\draw (A) -- (B) -- (C) -- (D) (A) -- (D) (A) -- (C);\n\\draw [dashed] (B) -- (D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: 平面$ACD \\perp$平面$ABC$;\\\\\n(2) 若$AB=1$, $CD=BC=2$, 求直线$AD$与平面$ABC$所成角的大小.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-简单题冲刺-简单题冲刺01" ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\arccos \\dfrac{\\sqrt{5}}3$", @@ -325699,7 +328285,8 @@ "content": "在$\\triangle ABC$中, 内角$A$、$B$、$C$所对边分别为$a$、$b$、$c$, 已知$b \\sin A=a \\cos (B-\\dfrac{\\pi}6)$.\\\\\n(1) 求角$B$的大小;\\\\\n(2) 若$c=2a$, $\\triangle ABC$的面积为$\\dfrac{2 \\sqrt 3}3$, 求$\\triangle ABC$的周长.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-简单题冲刺-简单题冲刺01" ], "genre": "解答题", "ans": "(1) $\\dfrac\\pi 3$; (2) $2\\sqrt{3}+2$", @@ -325733,7 +328320,8 @@ "content": "某地准备在山谷中建一座桥梁, 桥址位置的竖直截面图如图所示, 谷底$O$在水平线$MN$上、桥$AB$与$MN$平行, $OO'$为铅垂线($O'$在$AB$上). 经测量, 山谷左侧的轮廓曲线$AO$上任一点$D$到$MN$的距离$h_1$(米)与$D$到$OO'$的距离$a$(米) 之间满足关系式$h_1=\\dfrac 1{40} a^2$, 山谷右侧的轮廓曲线$BO$上任一点$F$到$MN$的距离$h_2$(米)与$F$到$OO'$的距离$b$(米)之间满足关系式$h_2=-\\dfrac 1{800} b^3+6 b$. 已知点$B$到$OO'$的距离为$40$米.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.2]\n\\draw (-8,16) node [left] {$A$} coordinate (A);\n\\draw (4,16) node [right] {$B$} coordinate (B);\n\\draw (-10,0) node [below] {$M$} coordinate (M);\n\\draw (6,0) node [below] {$N$} coordinate (N);\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (0,16) node [above] {$O'$} coordinate (O');\n\\draw (-6,16) node [above] {$C$} coordinate (C);\n\\draw (-6,9) node [left] {$D$} coordinate (D);\n\\draw (2,16) node [above] {$E$} coordinate (E);\n\\draw (2,12) node [right] {$F$} coordinate (F);\n\\draw [ultra thick] (A) -- (B) (C) -- (D) (E) -- (F);\n\\draw (M) -- (N);\n\\draw [dashed] (O) -- (O');\n\\draw [domain = -8:0] plot (\\x,{0.25*pow(\\x,2)});\n\\draw [domain = 0:4.2] plot (\\x,{16-pow(\\x-4,2)});\n\\draw [dashed] (D) --++ (0,-9) node [midway,left] {$h_1$} coordinate (h_1) (D) --++ (6,0) node [midway,above] {$a$} coordinate (a);\n\\draw [dashed] (F) --+ (0,-12) node [midway,right] {$h_2$} coordinate (h_2) (F) --++ (-2,0) node [midway,above] {$b$} coordinate (b);\n\\end{tikzpicture}\n\\end{center}\n(1) 求谷底$O$到桥面$AB$的距离和桥$AB$的长度;\\\\\n(2) 计划在谷底两侧建造平行于$OO'$的桥墩$CD$和$EF$, 且$CE$为$80$米, 其中$C$、$E$在$AB$上(不包括端点), 桥墩$EF$每米造价为$k$(万元)、桥墩$CD$每米造价为$\\dfrac 32 k$(万元)($k>0$). 问$O'E$为多少米时, 桥墩$CD$与$EF$的总造价最低?", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-中档题冲刺-中档题冲刺01" ], "genre": "解答题", "ans": "(1) $120$米; (2) $20$米时, 总造价最低", @@ -325815,7 +328403,8 @@ "content": "已知集合$A=\\{x | 00$, $a \\neq 1$)的图像经过点$(4,2)$, 则实数$a=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-简单题冲刺-简单题冲刺03" ], "genre": "填空题", "ans": "$2$", @@ -325955,7 +328547,8 @@ "content": "设等比数列$\\{a_n\\}$满足$a_1+a_2=-1$, $a_1-a_3=-3$, 则$a_4=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-简单题冲刺-简单题冲刺03" ], "genre": "填空题", "ans": "$-8$", @@ -325990,7 +328583,8 @@ "content": "已知方程组$\\begin{cases}x+m y=2, \\\\ m x+16 y=8\\end{cases}$无解, 则实数$m=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-简单题冲刺-简单题冲刺03" ], "genre": "填空题", "ans": "$-4$", @@ -326025,7 +328619,8 @@ "content": "已知角$\\alpha$的终边与单位圆$x^2+y^2=1$交于点$P(\\dfrac 12, y)$, 则$\\sin (\\dfrac{\\pi}2+\\alpha)=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-简单题冲刺-简单题冲刺03" ], "genre": "填空题", "ans": "$\\dfrac 12$", @@ -326060,7 +328655,8 @@ "content": "将半径为$2$的半圆形纸片卷成一个无盖的圆锥筒, 则该圆锥筒的高为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-简单题冲刺-简单题冲刺03" ], "genre": "填空题", "ans": "$\\sqrt{3}$", @@ -326095,7 +328691,8 @@ "content": "已知函数$f(x)=x^2$, 则曲线$y=f(x)$在点$P(1,1)$处的切线方程是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-中档题冲刺-中档题冲刺02" ], "genre": "填空题", "ans": "$y=2x-1$", @@ -326122,7 +328719,8 @@ "content": "设函数$f(x)=\\sin (\\omega x-\\dfrac{\\pi}6)+k$($\\omega>0$), 若$f(x) \\leq f(\\dfrac{\\pi}3)$对任意的实数$x$都成立, 则$\\omega$的最小值为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-中档题冲刺-中档题冲刺02" ], "genre": "填空题", "ans": "$2$", @@ -326149,7 +328747,8 @@ "content": "在边长为$2$的正六边形$ABCDEF$中, 点$P$为其内部或边界上一点, 则$\\overrightarrow{AD} \\cdot \\overrightarrow{BP}$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-中档题冲刺-中档题冲刺02" ], "genre": "填空题", "ans": "$[-4,12]$", @@ -326242,7 +328841,8 @@ "content": "设函数$f(x)=\\sin (x-\\dfrac{\\pi}6)$, 若对于任意$\\alpha \\in[-\\dfrac{5 \\pi}6,-\\dfrac{\\pi}2]$, 在区间$[0, m]$上总存在唯一确定的$\\beta$, 使得$f(\\alpha)+f(\\beta)=0$, 则$m$的最小值为\\bracket{20}.\n\\fourch{$\\dfrac{\\pi}6$}{$\\dfrac{\\pi}2$}{$\\dfrac{7 \\pi}6$}{$\\pi$}", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-中档题冲刺-中档题冲刺02" ], "genre": "选择题", "ans": "B", @@ -326291,7 +328891,8 @@ "content": "如图, 长方体$ABCD-A_1B_1C_1D_1$中, $AB=BC=\\sqrt 2$, $A_1C$与底面$ABCD$所成角为$45^{\\circ}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{{sqrt(2)}}\n\\def\\m{{sqrt(2)}}\n\\def\\n{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw (D1) -- (B1) (A1) -- (B);\n\\draw [dashed] (A) -- (D1) (A1) -- (D) (A1) -- (C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求四棱锥$A_1-ABCD$的体积;\\\\\n(2) 求异面直线$A_1B$与$B_1D_1$所成角的大小.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-简单题冲刺-简单题冲刺03" ], "genre": "解答题", "ans": "(1) $\\dfrac 43$; (2) $\\arccos \\dfrac{\\sqrt{6}}6$", @@ -326326,7 +328927,8 @@ "content": "已知函数$f(x)=\\sin x \\cos x-\\sin ^2 x+\\dfrac 12$.\\\\\n(1) 求$f(x)$的单调递增区间;\\\\\n(2) 在$\\triangle ABC$中, $a$、$b$、$c$为角$A$、$B$、$C$的对边, 且满足$b \\cos 2 A=b \\cos A-a \\sin B$, 且$0=latex,scale = 0.03]\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (100,0) node [below right] {$A$} coordinate (A);\n\\draw (0,100) node [above left] {$B$} coordinate (B);\n\\draw (50,50) node [above right] {$C$} coordinate (C);\n\\draw (C) -- (A) (A) -- (O) -- (B);\n\\draw [domain = 0:50, samples = 100] plot (\\x,{100-\\x*\\x/50});\n\\draw (40,68) node [above right] {$D$} coordinate (D);\n\\draw (D) -- ($(O)!(D)!(A)$) node [below] {$E$} coordinate (E);\n\\draw (D) -- ($(O)!(D)!(B)$) node [left] {$F$} coordinate (F);\n\\end{tikzpicture}\n\\end{center}\n(1) 试建立平面直角坐标系, 求曲线段$BC$的方程;\\\\\n(2) 求面积$S$关于$x$的函数解析式$S=f(x)$;\\\\\n(3) 试确定点$D$的位置, 使得游乐场的面积$S$最大.(结果精确到$0.1$米)", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-中档题冲刺-中档题冲刺02" ], "genre": "解答题", "ans": "(1) $y=-0.02x^2+100$($0\\le x\\le 50$); (2) $f(x)=\\begin{cases}x(-0.02x^2+100), & 30\\le x\\le 50, \\\\ x(-x+100), & 50b$, 则$a^3>b^3$''是\\blank{50}命题(填``真''、``假'').", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期测验卷-2023届杨浦区一模" ], "genre": "填空题", "ans": "真", @@ -330024,7 +332632,8 @@ "content": "设集合$A=\\{x | 0 \\leq x \\leq 2\\}$, 集合$B=\\{x | x-1 \\leq 0\\}$, 则$A \\cap B=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期测验卷-2023届杨浦区一模" ], "genre": "填空题", "ans": "$[0,1]$", @@ -330059,7 +332668,8 @@ "content": "方程$\\log_3(x^2-4 x-5)=\\log_3(x+1)$的解是$x=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-2023届杨浦区一模" ], "genre": "填空题", "ans": "$6$", @@ -330094,7 +332704,8 @@ "content": "已知$\\sin \\alpha=\\dfrac 12$, $\\alpha \\in(0, \\pi)$, 则$\\alpha=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-2023届杨浦区一模" ], "genre": "填空题", "ans": "$\\dfrac\\pi 6$或$\\dfrac{5\\pi}6$", @@ -330129,7 +332740,8 @@ "content": "设$\\mathrm{i}$是虚数单位, 则复数$z=2 \\mathrm{i}(1-\\mathrm{i})$的虚部是\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期测验卷-2023届杨浦区一模" ], "genre": "填空题", "ans": "$2$", @@ -330164,7 +332776,8 @@ "content": "向量$\\overrightarrow a=(3,4)$在向量$\\overrightarrow b=(1,0)$上的投影的坐标为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期测验卷-2023届杨浦区一模" ], "genre": "填空题", "ans": "$(3,0)$", @@ -330199,7 +332812,8 @@ "content": "一支田径队有男运动员$48$人, 女运动员$36$人, 若用分层抽样的方法从全体运动员中抽取一个容量为$21$的样本, 则抽取男运动员的人数为\\blank{50}.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-上学期测验卷-2023届杨浦区一模" ], "genre": "填空题", "ans": "$12$", @@ -330234,7 +332848,8 @@ "content": "若双曲线的渐近线方程为$y=\\pm \\dfrac 34 x$, 则双曲线的离心率为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-上学期测验卷-2023届杨浦区一模" ], "genre": "填空题", "ans": "$\\dfrac 54$或$\\dfrac 53$", @@ -330269,7 +332884,8 @@ "content": "若正数$x, y$满足$x+3 y=x y$, 则$x+y$的最小值为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期测验卷-2023届杨浦区一模" ], "genre": "填空题", "ans": "$4+2\\sqrt{3}$", @@ -330304,7 +332920,8 @@ "content": "已知$\\mathrm{C}_n^2=\\mathrm{C}_n^3$($n$是正整数), $(2 x-1)^n=a_0+a_1(x-1)+a_2(x-1)^2+\\cdots+a_n(x-1)^n$, \n则$a_0+a_1+a_2+\\cdots+a_n=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-上学期测验卷-2023届杨浦区一模" ], "genre": "填空题", "ans": "$243$", @@ -330339,7 +332956,8 @@ "content": "等差数列$\\{a_n\\}$的公差$d \\neq 0$, 其前$n$项和为$S_n$, 若$S_{10}=0$, 则$S_i$($i=1,2,3, \\cdots, 2022$)中不同的数值有\\blank{50}个.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-上学期测验卷-2023届杨浦区一模" ], "genre": "填空题", "ans": "$2018$", @@ -330374,7 +332992,8 @@ "content": "已知$f(x)=-x^2-2 a x-a^2+a+1$, 若方程$f(x)=0$与$f(f(x))=0$均恰有两个不同的实根, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期测验卷-2023届杨浦区一模" ], "genre": "填空题", "ans": "$(-\\dfrac 34,0)$", @@ -330409,7 +333028,8 @@ "content": "从学号为$1$--$10$的$10$名学生中, 用抽签法从中抽取$3$名学生进行问卷调查, 设$5$号同学被抽到的概率为$a$, $6$号同学被抽到的概率为$b$, 则\\bracket{20}.\n\\fourch{$a=\\dfrac 3{10}$, $b=\\dfrac 29$}{$a=\\dfrac 1{10}$, $b=\\dfrac 19$}{$a=\\dfrac 3{10}$, $b=\\dfrac 3{10}$}{$a=\\dfrac 1{10}$, $b=\\dfrac 1{10}$}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-上学期测验卷-2023届杨浦区一模" ], "genre": "选择题", "ans": "C", @@ -330444,7 +333064,8 @@ "content": "对于平面$\\alpha$和两条直线$m, n$, 下列说法正确的是\\bracket{20}.\n\\twoch{若$m \\perp \\alpha$, $m \\perp n$, 则$n\\parallel\\alpha$}{若$m, n$与$\\alpha$所成的角相等, 则$m\\parallel n$}{若$m\\parallel\\alpha$, $n\\parallel\\alpha$, 则$m\\parallel n$}{若$m \\subset \\alpha$, $m\\parallel n$, $n$在平面$\\alpha$外, 则$n\\parallel\\alpha$}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期测验卷-2023届杨浦区一模" ], "genre": "选择题", "ans": "D", @@ -330479,7 +333100,8 @@ "content": "若$\\triangle ABC$中, $A=\\dfrac{\\pi}3$, 则``$\\sin B<\\dfrac 12$''是``$\\triangle ABC$是钝角三角形''的\\bracket{20}.\n\\twoch{充分而不必要条件}{必要而不充分条件}{充分必要条件}{既不充分也不必要条件}", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-2023届杨浦区一模" ], "genre": "选择题", "ans": "A", @@ -330514,7 +333136,8 @@ "content": "已知定义在$\\mathbf{R}$上的函数$y=f(x)$对任意$x_1a$成立, 且满足$f(0)=-a^2$(其中$a$为常数), 关于$x$的方程:$f(a+x)=a x$的解的情况, 下面判断正确的是\\bracket{20}.\n\\twoch{存在常数$a$, 使得该方程无实数解}{对任意常数$a$, 方程均有且仅有 1 解}{存在常数$a$, 使得该方程有无数解}{对任意常数$a$, 方程解的个数大于 2}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-2023届杨浦区一模" ], "genre": "选择题", "ans": "B", @@ -330549,7 +333172,8 @@ "content": "在$\\triangle ABC$中, 内角$A, B, C$所对的边分别为$a, b, c$, 满足$a^2+c^2=b^2-a c$.\\\\\n(1) 求角$B$的大小;\\\\\n(2) 若$b=2 \\sqrt 3$, 求$\\triangle ABC$的面积的最大值.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-2023届杨浦区一模" ], "genre": "解答题", "ans": "(1) $\\dfrac{2\\pi}3$; (2) $\\sqrt{3}$", @@ -330584,7 +333208,8 @@ "content": "如图所示圆锥$P-O$中, $CD$为底面的直径, $A, B$分别为母线$PD$与$PC$的中点, 点$E$是底面圆周上一点, 若$\\angle DCE=30^{\\circ}$, $|AB|=\\sqrt 2$, 圆锥的高为$\\sqrt {14}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\r{1.6}\n\\def\\h{3}\n\\draw ({-\\r},0,0) node [left] {$D$} coordinate (D) -- (0,\\h,0) node [above] {$P$} coordinate (P) -- (\\r,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,0) node [above right] {$O$} coordinate (O);\n\\draw (D) arc (180:360:{\\r} and {\\r/4});\n\\draw [dashed] (D) arc (180:0:{\\r} and {\\r/4});\n\\draw [dashed] (D) -- (C) (O) -- (P);\n\\draw ({\\r*cos(-130)},{\\r/4*sin(-130)}) node [below] {$E$} coordinate (E);\n\\draw [dashed] (E) -- (C);\n\\draw ($(C)!0.5!(P)$) node [right] {$B$} coordinate (B);\n\\draw ($(D)!0.5!(P)$) node [left] {$A$} coordinate (A);\n\\draw [dashed] (E) -- (A) -- (B);\n\\end{tikzpicture}\n\\end{center}\n(1) 求圆锥的侧面积$S$;\\\\\n(2) 求证: $AE$与$PC$是异面直线, 并求其所成角的大小.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期测验卷-2023届杨浦区一模" ], "genre": "解答题", "ans": "(1) $4\\sqrt{2}\\pi$; (2) 证明略, 角的大小为$\\arccos\\dfrac{7\\sqrt{5}}{20}$", @@ -330619,7 +333244,8 @@ "content": "企业经营一款节能环保产品, 其成本由研发成本与生产成本两部分构成. 生产成本固定为每台$130$元. 根据市场调研, 若该产品产量为$x$万台时, 每万台产品的销售收入为$I(x)$万元, 两者满足关系:$I(x)=220-x$($00$)的两条直线$l_1$和$l_2$与曲线$E$都只有一个公共点, 且$l_1 \\perp l_2$, 求$h$的值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-上学期测验卷-2023届杨浦区一模" ], "genre": "解答题", "ans": "(1) $2\\sqrt{2}+2$; (2) $\\pm \\dfrac{\\sqrt{14}}2$; (3) $\\sqrt{2}$或$\\sqrt{3}$或$\\sqrt{\\dfrac{1+\\sqrt{17}}2}$", @@ -330690,7 +333317,8 @@ "objs": [], "tags": [ "第二单元", - "第四单元" + "第四单元", + "2023届高三-上学期测验卷-2023届杨浦区一模" ], "genre": "解答题", "ans": "(1) $[-\\dfrac{\\sqrt[3]{2}}2,+\\infty)$; (2) $(-2,-1)$; (3) $a=-2$", @@ -330725,7 +333353,8 @@ "content": "不等式$\\dfrac x{x+2} \\leq 0$的解集为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-简单题冲刺-简单题冲刺02" ], "genre": "填空题", "ans": "$(-2,0]$", @@ -330760,7 +333389,8 @@ "content": "对于正实数$x$, 代数式$x+\\dfrac 4x$的最小值为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-简单题冲刺-简单题冲刺02" ], "genre": "填空题", "ans": "$4$", @@ -330795,7 +333425,8 @@ "content": "已知一个球的半径为$3$, 则这个球的体积为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-简单题冲刺-简单题冲刺02" ], "genre": "填空题", "ans": "$36\\pi$", @@ -330830,7 +333461,8 @@ "content": "在$(x+\\dfrac 1{\\sqrt x})^7$的二项展开式中$x$项的系数为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-简单题冲刺-简单题冲刺02" ], "genre": "填空题", "ans": "$35$", @@ -330865,7 +333497,8 @@ "content": "设$m, n \\in \\mathbf{R}$, $\\mathrm{i}$为虚数单位, 若$1-\\sqrt 3 \\mathrm{i}$是关于$x$的二次方程$x^2+m x+n=0$的一个虚根, 则$m+n=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-简单题冲刺-简单题冲刺02" ], "genre": "填空题", "ans": "$2$", @@ -330900,7 +333533,8 @@ "content": "已知首项为$2$的等比数列$\\{b_n\\}$的公比为$\\dfrac 13$, 则这个数列所有项的和为\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-简单题冲刺-简单题冲刺02" ], "genre": "填空题", "ans": "$3$", @@ -330935,7 +333569,8 @@ "content": "设曲线$y=\\ln x+2 x$的斜率为$3$的切线为$l$, 则$l$的方程为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-简单题冲刺-简单题冲刺02" ], "genre": "填空题", "ans": "$y=3x-1$", @@ -330970,7 +333605,8 @@ "content": "第$5$届中国国际进口博览会在上海举行, 某高校派出了包括甲同学在内的$4$名同学参加了连续$5$天的志愿者活动. 已知甲同学参加了$2$天的活动, 其余同学各参加了$1$天的活动, 则甲同学参加连续两天活动的概率为\\blank{50}. (结果用分数表示)", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-简单题冲刺-简单题冲刺02" ], "genre": "填空题", "ans": "$\\dfrac 25$", @@ -331005,7 +333641,8 @@ "content": "设$a, b \\in \\mathbf{R}$, 若函数$f(x)=\\lg|a+\\dfrac 4{2-x}|+b$为奇函数, 则$a+b=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-中档题冲刺-中档题冲刺03" ], "genre": "填空题", "ans": "$-1$", @@ -331037,7 +333674,8 @@ "content": "设函数$f(x)=\\cos (\\omega x+\\varphi)$(其中$\\omega>0$, $|\\varphi|<\\dfrac{\\pi}2)$, 若函数$y=f(x)$图像的对称轴$x=\\dfrac{\\pi}6$与其对称中心的最小距离为$\\dfrac{\\pi}8$, 则$f(x)=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-中档题冲刺-中档题冲刺03" ], "genre": "填空题", "ans": "$\\cos(4x+\\dfrac \\pi 3)$", @@ -331069,7 +333707,8 @@ "content": "在$\\triangle ABC$中, $AB=5$, $AC=6$, $\\cos A=\\dfrac 15$, $O$是$\\triangle ABC$的外心, 若$\\overrightarrow{OP}=x \\overrightarrow{OB}+y \\overrightarrow{OC}$, 其中$x, y \\in[0,1]$, 则动点$P$的轨迹所覆盖图形的面积为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-中档题冲刺-中档题冲刺03" ], "genre": "填空题", "ans": "$\\dfrac{49}{24}\\sqrt{6}$", @@ -331167,7 +333806,8 @@ "content": "已知$F$是椭圆$C_1: \\dfrac{x^2}4+\\dfrac{y^2}3=1$与抛物线$C_2: y^2=2 p x(p>0)$的一个共同焦点, $C_1$与$C_2$相交于$A, B$两点, 则线段$AB$的长等于\\bracket{20}.\n\\fourch{$\\dfrac 23 \\sqrt 6$}{$\\dfrac 43 \\sqrt 6$}{$\\dfrac 53$}{$\\dfrac{10}3$}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-中档题冲刺-中档题冲刺03" ], "genre": "选择题", "ans": "B", @@ -331221,7 +333861,8 @@ "content": "设$\\triangle ABC$的内角$A, B, C$所对的边分别为$a, b, c$, 已知$2 \\cos (\\pi+A)+\\sin (\\dfrac{\\pi}2+2A)+\\dfrac 32=0$.\\\\\n(1) 求角$A$;\\\\\n(2) 若$c-b=\\dfrac{\\sqrt 3}3 a$, 求证: $\\triangle ABC$是直角三角形.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-简单题冲刺-简单题冲刺02" ], "genre": "解答题", "ans": "(1) $\\dfrac \\pi 3$; (2) 证明略", @@ -331256,7 +333897,8 @@ "content": "在等差数列$\\{a_n\\}$中, $a_1=2$, 且$a_2, a_3+2, a_8$构成等比数列.\\\\\n(1) 求数列$\\{a_n\\}$的通项公式;\\\\\n(2) 令$b_n=2^{a_n}+9$, 记$S_n$为数列$\\{b_n\\}$的前$n$项和, 若$S_n \\geq 2022$, 求正整数$n$的最小值.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-简单题冲刺-简单题冲刺02" ], "genre": "解答题", "ans": "(1) $a_n=2n$; (2) $6$", @@ -331291,7 +333933,8 @@ "content": "如图, 在三棱柱$ABC-A_1B_1C_1$中, 底面$ABC$是以$AC$为斜边的等腰直角三角形, 侧面$AA_1C_1C$为菱形, 点$A_1$在底面上的投影为$AC$的中点$D$, 且$AB=2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (0,0,0) node [below] {$D$} coordinate (D);\n\\draw (-1,0,0) node [left] {$A$} coordinate (A);\n\\draw (1,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,1) node [below] {$B$} coordinate (B);\n\\draw (0,{sqrt(3)},0) node [left] {$A_1$} coordinate (A_1);\n\\draw ($(B)+(A_1)-(A)$) node [below right] {$B_1$} coordinate (B_1);\n\\draw ($(C)+(A_1)-(A)$) node [right] {$C_1$} coordinate (C_1);\n\\draw ($(A_1)!0.5!(B_1)$) node [above right] {$E$} coordinate (E);\n\\draw (A) -- (B) -- (C) (A_1) -- (B_1) -- (C_1);\n\\draw (A) -- (A_1) -- (C_1) -- (C) (B) -- (B_1);\n\\draw [dashed] (B) -- (D) -- (A_1) (D) -- (E) (A) -- (C);\n\\end{tikzpicture}\n\\end{center}(1) 求证: $BD \\perp CC_1$;\\\\\n(2) 求点$C$到侧面$AA_1B_1B$的距离;\\\\\n(3) 在线段$A_1B_1$上是否存在点$E$, 使得直线$DE$与侧面$AA_1B_1B$所成角的正弦值为$\\dfrac{\\sqrt 6}7$? 若存在, 请求出$A_1E$的长; 若不存在, 请说明理由.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-中档题冲刺-中档题冲刺03" ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\dfrac{2\\sqrt{42}}7$; (3) 存在, $|A_1E|=1$", @@ -331368,7 +334011,9 @@ "content": "已知集合$A=\\{1,2,3,4\\}$, $B=\\{x |(x-1)(x-5)<0\\}$, 则$A \\cap B=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-下学期周末卷-高三下学期周末卷01", + "2023届高三-简单题冲刺-简单题冲刺04" ], "genre": "填空题", "ans": "$\\{2,3,4\\}$", @@ -331403,7 +334048,9 @@ "content": "若复数$z=\\dfrac{a+\\mathrm{i}}{\\mathrm{i}}$(其中$\\mathrm{i}$为虚数单位)的实部与虚部相等, 则实数$a=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-下学期周末卷-高三下学期周末卷01", + "2023届高三-简单题冲刺-简单题冲刺04" ], "genre": "填空题", "ans": "$-1$", @@ -331440,7 +334087,9 @@ "content": "从等差数列$84,80,76,72, \\cdots$的第\\blank{50}项起, 各项均为负值.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-下学期周末卷-高三下学期周末卷01", + "2023届高三-简单题冲刺-简单题冲刺04" ], "genre": "填空题", "ans": "$23$", @@ -331482,7 +334131,9 @@ "content": "不等式$2^{x^2-2 x-3}<(\\dfrac 12)^{3(x-1)}$的解集为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期周末卷-高三下学期周末卷01", + "2023届高三-简单题冲刺-简单题冲刺04" ], "genre": "填空题", "ans": "$(-3,2)$", @@ -331520,7 +334171,9 @@ "content": "在一次射击训练中, 某运动员$5$次射击的环数依次是$9,10,9,7,10$, 则该组数据的方差是\\blank{50}.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-下学期周末卷-高三下学期周末卷01", + "2023届高三-简单题冲刺-简单题冲刺04" ], "genre": "填空题", "ans": "$\\dfrac 65$", @@ -331562,7 +334215,9 @@ "content": "已知函数$f(x)=x^3-2 x$, 则$f(x)$在点$(1, f(1))$处的切线的倾斜角为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期周末卷-高三下学期周末卷01", + "2023届高三-简单题冲刺-简单题冲刺04" ], "genre": "填空题", "ans": "$\\dfrac\\pi 4$", @@ -331601,7 +334256,9 @@ "content": "若$(x+\\dfrac{\\sqrt a}{x^2})^6$的展开式的常数项是$45$, 则常数$a$的值为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-下学期周末卷-高三下学期周末卷01", + "2023届高三-简单题冲刺-简单题冲刺04" ], "genre": "填空题", "ans": "$3$", @@ -331643,7 +334300,9 @@ "content": "若函数$y=f(x)$的定义域和值域分别为$A=\\{1,2,3\\}$和$B=\\{1,2\\}$, 则$y=f(x)$是单调函数的概率是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期周末卷-高三下学期周末卷01", + "2023届高三-简单题冲刺-简单题冲刺04" ], "genre": "填空题", "ans": "$\\dfrac 23$", @@ -331685,7 +334344,9 @@ "content": "已知空间三点$A(-1,3,1)$, $B(2,4,0)$, $C(0,2,4)$, 则以$\\overrightarrow{AB}$、$\\overrightarrow{AC}$为一组邻边的平行四边形的面积大小为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期周末卷-高三下学期周末卷01", + "2023届高三-中档题冲刺-中档题冲刺04" ], "genre": "填空题", "ans": "$2\\sqrt{30}$", @@ -331728,7 +334389,9 @@ "content": "在平面直角坐标系中, $A(0,0)$, $B(1,2)$两点绕定点$P$按顺时针方向旋转$\\theta$角后, 分别到$A'(4,4)$, $B'(5,2)$两点位置, 则$\\cos \\theta$的值为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期周末卷-高三下学期周末卷01", + "2023届高三-中档题冲刺-中档题冲刺04" ], "genre": "填空题", "ans": "$-\\dfrac 35$", @@ -331771,7 +334434,9 @@ "content": "已知圆柱的轴截面是边长为$2$的正方形, $P$为上底面圆的圆心, $AB$为下底面圆的直径, $C$为下底面圆周上一点, 则三棱锥$P-ABC$外接球的体积为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期周末卷-高三下学期周末卷01", + "2023届高三-中档题冲刺-中档题冲刺04" ], "genre": "填空题", "ans": "$\\dfrac{125\\pi}{48}$", @@ -331814,7 +334479,8 @@ "content": "已知数列$\\{a_n\\}$中, $a_2=3 a_1$, 记$\\{a_n\\}$的前$n$项和为$S_n$, 且满足$S_{n+1}+S_n+S_{n-1}=3 n^2+2$($n \\geq 2$, $n \\in \\mathbf{N}$). 若对任意$n \\in \\mathbf{N}$, $n\\ge 1$, 都有$a_nb$''是``$\\dfrac 1a<\\dfrac 1b$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分也非必要条件}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-下学期周末卷-高三下学期周末卷01" ], "genre": "选择题", "ans": "D", @@ -331878,7 +334545,8 @@ "content": "已知$m, n$是两条不同直线,$\\alpha, \\beta$是两个不同平面, 则下列命题错误的是\\bracket{20}.\n\\onech{若$\\alpha, \\beta$不平行, 则在$\\alpha$内不存在与$\\beta$平行的直线}{若$m , n$平行于同一平面, 则$m$与$n$可能异面}{若$m, n$不平行, 则$m$与$n$不可能垂直于同一平面}{若$\\alpha, \\beta$垂直于同一平面, 则$\\alpha$与$\\beta$可能相交}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期周末卷-高三下学期周末卷01" ], "genre": "选择题", "ans": "A", @@ -331910,7 +334578,11 @@ "content": "已知函数$y=f(x)$定义域为$\\mathbf{R}$, 下列论断:\\\\ \n\\textcircled{1} 若对任意实数$a$, 存在实数$b$, 使得$f(a)=f(b)$, 且$b=-a$, 则$f(x)$是偶函数.\\\\\n\\textcircled{2} 若对任意实数$a$, 存在实数$b$, 使得$f(a)0$, 若对任意实数$a$, 存在实数$b$, 使得$f(a)=f(b)$, 且$|a-b|=T$, 则$f(x)$是周期函数.\n其中正确的论断的个数是\\bracket{20}.\n\\fourch{$0$个}{$1$个}{$2$个}{$3$个}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期周末卷-高三下学期周末卷01", + "2023届高三-中档题冲刺-中档题冲刺04", + "2023届高三-四月错题重做-01_函数一", + "2023届高三-四月错题重做-01_易错题-函数1" ], "genre": "选择题", "ans": "B", @@ -331965,7 +334637,8 @@ "content": "在直角坐标平面$xOy$中, 已知两定点$F_1(-2,0)$与$F_2(2,0)$, $F_1$, $F_2$到直线$l$的距离之差的绝对值等于$2 \\sqrt 2$, 则平面上不在任何一条直线$l$上的点组成的图形面积是\\bracket{20}.\n\\fourch{$4 \\pi$}{$8$}{$2 \\pi$}{$4+\\pi$}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期周末卷-高三下学期周末卷01" ], "genre": "选择题", "ans": "D", @@ -331997,7 +334670,9 @@ "content": "已知函数$f(x)=\\sqrt 3 \\sin x \\cos x-\\cos ^2 x$, $x \\in \\mathbf{R}$.\\\\\n(1) 求$f(x)$的单调递增区间;\\\\ \n(2) 求$f(x)$在区间$[-\\dfrac{\\pi}4, \\dfrac{\\pi}4]$上的最大值和最小值.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期周末卷-高三下学期周末卷01", + "2023届高三-简单题冲刺-简单题冲刺04" ], "genre": "解答题", "ans": "(1) $[k\\pi-\\dfrac\\pi 6,k\\pi+\\dfrac\\pi 3]$, $k\\in \\mathbf{Z}$; (2) 最大值为$\\dfrac{\\sqrt{3}-1}2$, 最小值为$-\\dfrac 32$", @@ -332039,7 +334714,9 @@ "content": "如图, 在正三棱柱$ABC-A_1B_1C_1$中, $E, F$分别为$BB_1$, $AC$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\h{2}\n\\draw ({-\\l/2},0,0) node [left] {$A$} coordinate (A);\n\\draw (0,0,{\\l/2*sqrt(3)}) node [below] {$C$} coordinate (C);\n\\draw ({\\l/2},0,0) node [right] {$B$} coordinate (B);\n\\draw (A) ++ (0,\\h) node [left] {$A_1$} coordinate (A_1);\n\\draw (C) ++ (0,\\h) node [below right] {$C_1$} coordinate (C_1);\n\\draw (B) ++ (0,\\h) node [right] {$B_1$} coordinate (B_1);\n\\draw (A) -- (C) -- (B) (A) -- (A_1) (C) -- (C_1) (B) -- (B_1) (A_1) -- (C_1) -- (B_1) (A_1) -- (B_1);\n\\draw ($(B)!0.5!(B_1)$) node [right] {$E$} coordinate (E);\n\\draw ($(A)!0.5!(C)$) node [below left] {$F$} coordinate (F);\n\\draw [dashed] (F) -- (B) (E) -- (A_1);\n\\draw (A_1) -- (C) (C) -- (E);\n\\draw [dashed] (A) -- (B);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $BF\\parallel$平面$A_1EC$;\\\\ \n(2) 求证: 平面$A_1EC \\perp$平面$ACC_1A_1$.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期周末卷-高三下学期周末卷01", + "2023届高三-简单题冲刺-简单题冲刺04" ], "genre": "解答题", "ans": "(1) 证明略; (2) 证明略", @@ -332081,7 +334758,9 @@ "content": "流行性感冒是由流感病毒引起的急性呼吸道传染病. 某市去年$11$月份曾发生流感, 据统计, $11$月$1$日该市的新感染者有$30$人, 以后每天的新感染者比前一天的新感染者增加$50$人. 由于该市医疗部门采取措施, 使该种病毒的传播得到控制, 从$11$月$k+1$($9 \\leq k \\leq 29$, $k \\in \\mathbf{N}$)日起每天的新感染者比前一天的新感染者减少$20$人.\\\\\n(1) 若$k=9$, 求$11$月$1$日至$11$月$10$日新感染者总人数;\\\\\n(2) 若到$11$月$30$日止, 该市在这$30$天内的新感染者总人数为$11940$人, 问$11$月几日, 该市新感染者人数最多? 并求这一天的新感染者人数.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-下学期周末卷-高三下学期周末卷01", + "2023届高三-中档题冲刺-中档题冲刺04" ], "genre": "解答题", "ans": "(1) $2480$人; (2) $11$月$13$日新感染者人数最多, 为$630$人", @@ -332124,7 +334803,10 @@ "content": "在平面直角坐标系$xOy$中, 已知椭圆$\\Gamma: \\dfrac{x^2}2+y^2=1$, 过右焦点$F$作两条互相垂直的弦$AB$, $CD$, 设$AB$, $CD$中点分别为$M$, $N$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,1.5) coordinate (S) (2,{2/3}) coordinate (T) (1,0) coordinate (F) node [below] {$F$};\n\\draw [name path = elli] (0,0) ellipse ({sqrt(2)} and 1);\n\\path [name path = l1] (S) -- ($(S)!1.5!(F)$);\n\\path [name path = l2] (T) -- ($(T)!2.5!(F)$);\n\\path [name intersections = {of = elli and l1, by = {A,B}}];\n\\draw (A) node [above] {$A$} coordinate (A)-- (B) node [below] {$B$} coordinate (B);\n\\path [name intersections = {of = elli and l2, by = {C,D}}];\n\\draw (C) node [right] {$C$} coordinate (C)-- (D) node [below] {$D$} coordinate (D);\n\\draw ($(A)!0.5!(B)$) node [above] {$M$} coordinate (M) -- ($(C)!0.5!(D)$) node [below] {$N$} coordinate (N);\n\\end{tikzpicture}\n\\end{center}\n(1) 写出椭圆右焦点$F$的坐标及该椭圆的离心率;\\\\ \n(2) 证明: 直线$MN$必过定点, 并求出此定点坐标; \\\\\n(3) 若弦$AB, CD$的斜率均存在, 求$\\triangle FMN$面积的最大值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期周末卷-高三下学期周末卷01", + "2023届高三-四月错题重做-04_易错题-解析几何", + "2023届高三-四月错题重做-04_解析几何" ], "genre": "解答题", "ans": "(1) $\\dfrac{\\sqrt{2}}2$; (2) 过定点$(\\dfrac 23,0)$; (3) $\\dfrac 19$", @@ -332165,7 +334847,8 @@ "content": "设函数$f_1(x)=x^2+a \\mathrm{e}^x$(其中$a$是非零常数, $\\mathrm{e}$是自然对数的底), 记$f_n(x)=f_{n-1}'(x)$($n \\geq 2$, $n \\in \\mathbf{N}$).\\\\\n(1) 求对任意实数$x$, 都有$f_n(x)=f_{n-1}(x)$成立的最小整数$n$的值($n \\geq 2$, $n \\in \\mathbf{N}$);\\\\ \n(2) 设函数$g_n(x)=f_2(x)+f_3(x)+\\cdots+f_n(x)$, 若对任意$n \\geq 3$, $n \\in \\mathbf{N}$, $y=g_n(x)$都存在极值点$x=t_n$, 求证: 点$A_n(t_n, g_n(t_n))$($n \\geq 3$, $n \\in \\mathbf{N}$)在一定直线上, 并求出该直线方程;\\\\\n(3) 是否存在正整数$k$($k \\geq 2$)和实数$x_0$, 使$f_k(x_0)=f_{k-1}(x_0)=0$且对于任意$n \\in \\mathbf{N}$, $n\\ge 1$, $f_n(x)$至多有一个极值点, 若存在, 求出所有满足条件的$k$和$x_0$, 若不存在, 说明理由.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期周末卷-高三下学期周末卷01" ], "genre": "解答题", "ans": "(1) $5$; (2) 在定直线$y=2x$上; (3) 当且仅当$a=-\\dfrac 2{\\mathrm{e}}$时存在, $k=3$, $x_0=2$", @@ -332197,7 +334880,8 @@ "content": "若集合$M=\\{0,1,2\\}$, $N=\\{x | 2 x-1>0\\}$, 则$M \\cap N=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-简单题冲刺-简单题冲刺05" ], "genre": "填空题", "ans": "$\\{1,2\\}$", @@ -332231,7 +334915,8 @@ "content": "若$x$满足$\\mathrm{i} x=1+\\mathrm{i}$(其中$\\mathrm{i}$为虚数单位), 则$x=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-简单题冲刺-简单题冲刺05" ], "genre": "填空题", "ans": "$1-\\mathrm{i}$", @@ -332265,7 +334950,8 @@ "content": "双曲线$x^2-\\dfrac{y^2}8=1$的离心率为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-简单题冲刺-简单题冲刺05" ], "genre": "填空题", "ans": "$3$", @@ -332299,7 +334985,8 @@ "content": "在$\\triangle ABC$中, 已知边$AB=4 \\sqrt 3$, 角$A=45^{\\circ}$, $C=60^{\\circ}$, 则边$BC=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-简单题冲刺-简单题冲刺05" ], "genre": "填空题", "ans": "$4\\sqrt{2}$", @@ -332333,7 +335020,8 @@ "content": "已知正实数$x$、$y$满足$\\lg x=m$, $y=10^{m-1}$, 则$\\dfrac xy=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-简单题冲刺-简单题冲刺05" ], "genre": "填空题", "ans": "$10$", @@ -332367,7 +335055,8 @@ "content": "将一颗骰子连掷两次, 每次结果相互独立, 则第一次点数小于$3$且第二次点数大于$3$的概率为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-简单题冲刺-简单题冲刺05" ], "genre": "填空题", "ans": "$\\dfrac 16$", @@ -332401,7 +335090,8 @@ "content": "如图, 对于直四棱柱$ABCD-A_1B_1C_1D_1$, 要使$A_1C \\perp B_1D_1$, 则在四边形$ABCD$中, 满足的条件可以是\\blank{50}.(只需写出一个正确的条件)\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{1.5}\n\\def\\m{1.5}\n\\def\\n{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw (B1) -- (D1);\n\\draw [dashed] (A1) -- (C);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-简单题冲刺-简单题冲刺05" ], "genre": "填空题", "ans": "如$AC\\perp BD$, 或以$ABCD$为底面的正四棱锥等", @@ -332435,7 +335125,8 @@ "content": "若曲线$\\Gamma: y=\\sqrt x$和直线$l: x-2 y-4=0$的某一条平行线相切, 则切点的横坐标是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-简单题冲刺-简单题冲刺05" ], "genre": "填空题", "ans": "$1$", @@ -332469,7 +335160,8 @@ "content": "已知二次函数$f(x)=a x^2+x+a$的值域为$(-\\infty, \\dfrac 34]$, 则函数$g(x)=2^x+a$的值域为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-中档题冲刺-中档题冲刺05" ], "genre": "填空题", "ans": "$(-\\dfrac 14,+\\infty)$", @@ -332502,7 +335194,8 @@ "content": "已知$A(x_1, y_1)$、$B(x_2, y_2)$是圆$x^2+y^2=1$上的两个不同的动点, 且$x_1 y_2=x_2 y_1$, 则$2 x_1+x_2+2 y_1+y_2$的最大值为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-中档题冲刺-中档题冲刺05" ], "genre": "填空题", "ans": "$\\sqrt{2}$", @@ -332535,7 +335228,8 @@ "content": "已知函数$f(x)=2 \\sin (\\omega x+\\dfrac{\\pi}4)$($\\omega>0$)在区间$[-1,1]$上的值域为$[m, n]$, 且$n-m=3$, 则$\\omega$的值为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-中档题冲刺-中档题冲刺05" ], "genre": "填空题", "ans": "$\\dfrac{5\\pi}{12}$", @@ -332634,7 +335328,8 @@ "content": "已知函数$y=f(x)$与它的导函数$y=f'(x)$的定义域均为$\\mathbf{R}$, 现有下述两个命题:\\\\\n\\textcircled{1} ``$y=f(x)$为奇函数''是``$y=f'(x)$为偶函数'' 的充分非必要条件;\\\\\n\\textcircled{2} ``$y=f(x)$为严格增函数''是``$y=f'(x)$为严格增函数'' 的必要非充分条件. 则说法正确的选项是\\bracket{20}.\n\\onech{命题\\textcircled{1}和\\textcircled{2}均为真命题}{命题\\textcircled{1}为真命题, 命题\\textcircled{2}为假命题}{命题\\textcircled{1}为假命题, 命题\\textcircled{2}为真命题}{命题\\textcircled{1}和\\textcircled{2}均为假命题}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-中档题冲刺-中档题冲刺05" ], "genre": "选择题", "ans": "B", @@ -332689,7 +335384,8 @@ "content": "在等差数列$\\{a_n\\}$中, $a_1=25$, $a_2 \\neq a_1$, $a_1$、$a_{11}$、$a_{13}$成等比数列, $\\{a_n\\}$的前$n$项和为$S_n$.\\\\\n(1) 求数列$\\{a_n\\}$的通项公式;\\\\\n(2) 求$S_n$的最大值.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-简单题冲刺-简单题冲刺05" ], "genre": "解答题", "ans": "(1) $a_n=27-2n$; (2) $169$", @@ -332723,7 +335419,8 @@ "content": "如图, 已知圆柱$OO_1$的底面半径为$1$, 正$\\triangle ABC$内接于圆柱的下底面圆$O$, 点$O_1$是圆柱的上底面的圆心, 线段$AA_1$是圆柱的母线.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\begin{scope}[x = {(-30:1cm)}, y = {(-150:1cm)}, z = {(90:1cm)}]\n\\draw (0,0,1) circle (1);\n\\draw (-45:1) node [right] {$A$} coordinate (A) arc (-45:135:1);\n\\draw [dashed] (-45:1) arc (-45:-225:1);\n\\draw (A) ++ (0,0,1) node [right] {$A_1$} coordinate (A_1);\n\\draw (75:1) node [below] {$B$} coordinate (B);\n\\draw (195:1) node [above] {$C$} coordinate (C);\n\\draw [dashed] (A) -- (B) -- (C) -- cycle;\n\\draw (A) -- (A_1);\n\\draw (135:1) --++ (0,0,1);\n\\draw [dashed] (B) -- (A_1);\n\\end{scope}\n\\filldraw (0,0) circle (0.01) node [left] {$O$} coordinate (O);\n\\filldraw (0,1) circle (0.01) node [left] {$O_1$} coordinate (O_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求点$C$到平面$A_1AB$的距离;\\\\\n(2) 在劣弧$\\overset\\frown{BC}$上是否存在一点$D$, 满足$O_1D\\parallel$平面$A_1AB$? 若存在, 求出$\\angle BOD$的大小; 若不存在, 请说明理由.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-简单题冲刺-简单题冲刺05" ], "genre": "解答题", "ans": "(1) $\\dfrac 32$; (2) 存在, $\\angle BOD=\\dfrac \\pi 6$", @@ -332757,7 +335454,8 @@ "content": "$2022$年, 第二十二届世界杯足球赛在卡塔尔举行, 某国家队$26$名球员的年龄分布茎叶图如图所示:\n\\begin{center}\n\\begin{tabular}{l|l}\n1 & 8 \\ 9 \\\\\n2 & 1 \\ 2 \\ 3 \\ 3 \\ 4 \\ 5 \\ 5 \\ 5 \\ 6 \\ 6 \\ 7 \\ 8 \\ 8 \\ 8 \\ 9 \\ 9 \\ 9 \\\\\n3 & 0 \\ 1 \\ 2 \\ 2 \\ 2 \\ 3 \\ 4 \n\\end{tabular}\n\\end{center}\n(1) 该国家队$25$岁的球员共有几位? 求该国家队球员年龄的第$75$百分位数;\\\\\n(2) 从这$26$名球员中随机选取$11$名球员参加某项活动, 求这$11$名球员中至少有一位年龄不小于$30$岁的概率.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-中档题冲刺-中档题冲刺05" ], "genre": "解答题", "ans": "(1) $3$位, 第$75$百分位数为$30$; (2) $\\dfrac{911}{920}$", @@ -332834,7 +335532,8 @@ "content": "已知集合$A=\\{x||x-1 |<1\\}$, $\\mathbf{Z}$是整数集, 则$A \\cap \\mathbf{Z}=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-简单题冲刺-简单题冲刺06" ], "genre": "填空题", "ans": "$\\{1\\}$", @@ -332868,7 +335567,8 @@ "content": "已知复数$z=\\dfrac 1{\\mathrm{i}}$, $\\mathrm{i}$是虚数单位, 则$z$的虚部为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-简单题冲刺-简单题冲刺06" ], "genre": "填空题", "ans": "$-1$", @@ -332902,7 +335602,8 @@ "content": "直线$x=1$与直线$\\sqrt 3 x-y+1=0$的夹角大小为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-简单题冲刺-简单题冲刺06" ], "genre": "填空题", "ans": "$\\dfrac\\pi 6$", @@ -332936,7 +335637,8 @@ "content": "已知$m \\in \\mathbf{R}$, 若关于$x$的方程$2 m x^2+3 x+m-1=m^2 \\cdot x^2+(m+1) x+1$解集为$\\mathbf{R}$, 则$m$的值为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-简单题冲刺-简单题冲刺06" ], "genre": "填空题", "ans": "$2$", @@ -332970,7 +335672,8 @@ "content": "已知某一个圆锥的侧面积为$20 \\pi$, 底面积为$16 \\pi$, 则这个圆锥的体积为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-简单题冲刺-简单题冲刺06" ], "genre": "填空题", "ans": "$16\\pi$", @@ -333004,7 +335707,8 @@ "content": "某果园种植了$100$棵苹果树, 随机抽取的$12$棵果树的产量(单位: 千克)分别为: $$24,25,36,27,28,32,20,26,29,30,26,33$$据此预计, 该果园的总产量为\\blank{50}千克以及第$75$百分位数为\\blank{50}千克.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-简单题冲刺-简单题冲刺06" ], "genre": "填空题", "ans": "$2800$, $31$", @@ -333039,7 +335743,8 @@ "content": "已知常数$m \\in \\mathbf{R}$, 在$(x+m y)^n$的二项展开式中, $x^3 y^3$项的系数等于$160$, 则$m=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-简单题冲刺-简单题冲刺06" ], "genre": "填空题", "ans": "$2$", @@ -333073,7 +335778,8 @@ "content": "若函数$y=\\dfrac 1{x-1}$的值域是$(-\\infty, 0) \\cup[\\dfrac 12,+\\infty)$, 则此函数的定义域为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-简单题冲刺-简单题冲刺06" ], "genre": "填空题", "ans": "$(-\\infty,1)\\cup (1,3]$", @@ -333107,7 +335813,8 @@ "content": "如图为正六棱柱$ABCDEF-A' B' C' D' E' F'$. 其 6 个侧面的 12 条面对角线所在直线中, 与直线$A' B$异面的共有\\blank{50}条.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, x = {(-135:0.5cm)}, y = {(0:1cm)}, z = {(90:1cm)}]\n\\draw (0,0) node [left] {$A$} coordinate (A) ++ (0,0,1) node [left] {$A'$} coordinate (A');\n\\draw (A) ++ (30:1) node [below] {$B$} coordinate (B) ++ (0,0,1) node [above] {$B'$} coordinate (B');\n\\draw (B) ++ (90:1) node [below] {$C$} coordinate (C) ++ (0,0,1) node [above] {$C'$} coordinate (C');\n\\draw (C) ++ (150:1) node [right] {$D$} coordinate (D) ++ (0,0,1) node [right] {$D'$} coordinate (D');\n\\draw (D) ++ (210:1) node [below] {$E$} coordinate (E) ++ (0,0,1) node [above] {$E'$} coordinate (E');\n\\draw (E) ++ (270:1) node [below] {$F$} coordinate (F) ++ (0,0,1) node [above] {$F'$} coordinate (F');\n\\draw (A) -- (A') (B) -- (B') (C) -- (C') (D) -- (D') (A') -- (B') -- (C') -- (D') -- (E') -- (F') -- cycle (A) -- (B) -- (C) -- (D);\n\\draw [dashed] (E) -- (E') (F) -- (F') (A) -- (F) -- (E) -- (D);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-中档题冲刺-中档题冲刺06" ], "genre": "填空题", "ans": "$5$", @@ -333139,7 +335846,8 @@ "content": "关于$x$的方程$|2 x-3|+|-x+2|=|x-1|$的解集为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-中档题冲刺-中档题冲刺06" ], "genre": "填空题", "ans": "$[\\dfrac 32,2]$", @@ -333171,7 +335879,8 @@ "content": "在空间直角坐标系中, 点$A(1,0,0)$, 点$B(5,-4,3)$, 点$C(2,0,1)$, 则$\\overrightarrow{AB}$在$\\overrightarrow{CA}$方向上的投影向量的坐标为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-中档题冲刺-中档题冲刺06" ], "genre": "填空题", "ans": "$(\\dfrac 72,0,\\dfrac 72)$", @@ -333281,7 +335990,8 @@ "content": "甲、乙两人弈棋, 根据以往总共$20$次的对弈记录, 甲取胜$10$次, 乙取胜$10$次. 两人进行一场五局三胜的比赛, 最终胜者赢得$200$元奖金. 第一局、第二局比赛都是甲胜, 现在比赛因意外中止. 鉴于公平, 奖金应该分给甲\\bracket{20}.\n\\fourch{$100$元}{$150$元}{$175$元}{$200$元}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-中档题冲刺-中档题冲刺06" ], "genre": "选择题", "ans": "C", @@ -333336,7 +336046,8 @@ "content": "如图, 已知正四棱柱$ABCD-A_1B_1C_1D_1$, 底面正方形$ABCD$的边长为$2$, $AA_1=3$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{3}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw (A1) -- (C1) (A1) -- (B);\n\\draw [dashed] (B) -- (D) -- (A1) (A) -- (C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: 平面$AA_1CC_1 \\perp$平面$A_1BD$;\\\\\n(2) 求点$A$到平面$A_1BD$的距离.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-简单题冲刺-简单题冲刺06" ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\dfrac{3\\sqrt{22}}{11}$", @@ -333370,7 +336081,8 @@ "content": "若数列$\\{\\dfrac 1{a_n}\\}$是等差数列, 则称数列$\\{a_n\\}$为调和数列. 若实数$a$、$b$、$c$依次成调和数列, 则称$b$是$a$和$c$的调和中项.\\\\\n(1) 求$\\dfrac 13$和$1$的调和中项;\\\\\n(2) 已知调和数列$\\{a_n\\}$, $a_1=6$, $a_4=2$, 求$\\{a_n\\}$的通项公式.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-简单题冲刺-简单题冲刺06" ], "genre": "解答题", "ans": "(1) $\\dfrac 12$; (2) $a_n=\\dfrac{18}{2n+1}$", @@ -333404,7 +336116,8 @@ "content": "李先生属于一年工作$250$天的上班族, 计划购置一辆新车用以通勤. 大致推断每天早八点从家出发, 晩上六点回家, 往返总距离为$40$公里. 考虑从$A$、$B$两款车型中选择其一, $A$款车是燃油车, $B$款车是电动车, 售价均为$30$万元. 现提供关于两种车型的相关信息:\\\\\n$A$款车的油耗为$6$升/百公里, 油价为每升$8$至$9$元. 车险费用$4000$元/年. 购置税为售价的$10\\%$. 购车后, 车价每年折旧率为$12\\%$. 保养费用平均$2000$元/万公里;\\\\\n$B$款车的电耗为$20$度/百公里, 电费为每度$0.6$至$0.7$元. 车险费用$6000$元/年. 国务院$2022$年出台文件, 宣布保持免除购置税政策. 电池使用寿命为$5$年, 更换费用为$10$万元. 购车后, 车价每年折旧率为$15\\%$. 保养费用平均$1000$元/万公里.\\\\\n(1) 除了上述了解到的情况, 还有哪些因素可能需要考虑? 写出这些因素(至少 3 个, 不超过 5 个);\\\\\n(2) 为了简化问题, 请对相关因素做出合情假设, 由此为李先生作出买车的决策, 并说明理由.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-中档题冲刺-中档题冲刺06" ], "genre": "解答题", "ans": "(1) 例如: 非通勤时段的车辆使用情况; 油价和电价的变化; 工作单位能否提供免费充电; 电动车的国家减免政策的变化; 车辆的外观、内饰与品牌效应; 车牌费用等; (2) 解答略", @@ -333480,7 +336193,8 @@ "content": "若$z=\\mathrm{i} \\cdot(1-\\mathrm{i})$(其中$\\mathrm{i}$表示虚数单位), 则$\\text{Im} z=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-简单题冲刺-简单题冲刺07" ], "genre": "填空题", "ans": "$1$", @@ -333511,7 +336225,8 @@ "content": "若正四棱柱的底面周长为$4$、高为$2$, 则该正四棱柱的体积为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-简单题冲刺-简单题冲刺07" ], "genre": "填空题", "ans": "$2$", @@ -333542,7 +336257,8 @@ "content": "设$y=x^{\\frac 12}-x^3$, 则满足$y<0$的$x$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-简单题冲刺-简单题冲刺07" ], "genre": "填空题", "ans": "$(1,+\\infty)$", @@ -333575,7 +336291,8 @@ "content": "函数$y=\\tan 2 x$在区间$(-\\dfrac{\\pi}4, \\dfrac{\\pi}4)$上的零点为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-简单题冲刺-简单题冲刺07" ], "genre": "填空题", "ans": "$0$", @@ -333606,7 +336323,8 @@ "content": "函数$y=1-2 \\sin ^2 x$的最小正周期为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-简单题冲刺-简单题冲刺07" ], "genre": "填空题", "ans": "$\\pi$", @@ -333637,7 +336355,8 @@ "content": "在$(x+1)^4+(x+1)^5$展开式中, 含有$x^2$项的系数为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-简单题冲刺-简单题冲刺07" ], "genre": "填空题", "ans": "$16$", @@ -333668,7 +336387,8 @@ "content": "双曲线$\\dfrac{x^2}3-y^2=1$的两条渐近线的夹角大小为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-简单题冲刺-简单题冲刺07" ], "genre": "填空题", "ans": "$60^\\circ$", @@ -333699,7 +336419,8 @@ "content": "``青山''饮料厂推出一款新产品——``绿水'', 该厂开展促销活动, 将$6$罐``绿水''装成一箱, 且每箱均有$2$罐可以中奖. 若从一箱中随机抽取$2$罐, 则能中奖的概率为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-简单题冲刺-简单题冲刺07" ], "genre": "填空题", "ans": "$\\dfrac 35$", @@ -333730,7 +336451,8 @@ "content": "设$m \\in \\mathbf{R}$. 若直线$l: x=-1$与曲线$C_m:(x-\\dfrac{m^2}4)^2+(y-m)^2=1$仅有一个公共点, 则$m=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-中档题冲刺-中档题冲刺07" ], "genre": "填空题", "ans": "$0$", @@ -333758,7 +336480,8 @@ "content": "某地``小康果''大丰收, 现抽取$5$个样本, 其质量分别为$125$、$a$、$121$、$b$、$127$(单位: 克). 若该样本的中位数和平均数均为$124$, 则此样本的标准差为\\blank{50}.(用数字作答).", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-中档题冲刺-中档题冲刺07" ], "genre": "填空题", "ans": "$2$", @@ -333786,7 +336509,8 @@ "content": "设$a$、$b \\in \\mathbf{R}$且$a \\leq b$. 若函数$y=f(x)$的表达式为$f(x)=|x-1|$($x \\in \\mathbf{R}$), 且$f(a)=f(b+1)$, 则$a \\cdot(b+1)$的最大值为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-中档题冲刺-中档题冲刺07" ], "genre": "填空题", "ans": "$\\dfrac 34$", @@ -333880,7 +336604,8 @@ "content": "设$k>0$, 若向量$\\overrightarrow a$、$\\overrightarrow b$、$\\overrightarrow c$满足$|\\overrightarrow a|:|\\overrightarrow b|:|\\overrightarrow c|=1: k: 3$, 且$\\overrightarrow b-\\overrightarrow a=2(\\overrightarrow c-\\overrightarrow b)$, 则满足条件的$k$的取值可以是\\bracket{20}.\n\\fourch{$1$}{$2$}{$3$}{$4$}", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-中档题冲刺-中档题冲刺07" ], "genre": "选择题", "ans": "B", @@ -333930,7 +336655,8 @@ "content": "如图所示, $BD$为四边形$ABCD$的对角线, 设$AB=AD=1$, $\\triangle BCD$为等边三角形. 记$\\angle BAD=\\theta$($0<\\theta<\\pi$).\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$A$} coordinate (A) node [right] {$\\theta$};\n\\draw (-70:1) node [below] {$B$} coordinate (B);\n\\draw (60:1) node [above] {$D$} coordinate (D);\n\\draw ($(B)!1!-60:(D)$) node [right] {$C$} coordinate (C);\n\\draw (A) -- (B) -- (C) -- (D) -- cycle;\n\\draw [dashed] (B) -- (D);\n\\end{tikzpicture}\n\\end{center}\n(1) 当$BD=\\sqrt 3$时, 求$\\theta$的值;\\\\\n(2) 设$S$为四边形$ABCD$的面积, 用含有$\\theta$的关系式表示$S$, 并求$S$的最大值.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-简单题冲刺-简单题冲刺07" ], "genre": "解答题", "ans": "(1) $\\dfrac {2\\pi}3$; (2) $S=\\sin(\\theta-\\dfrac\\pi 3)+\\dfrac{\\sqrt{3}}2$, $\\theta\\in (0,\\pi)$, $S$的最大值为$1+\\dfrac{\\sqrt{3}}2$", @@ -333961,7 +336687,8 @@ "content": "设$a$、$b$均为正整数, $\\{a_n\\}$为首项为$a$、公差为$b$的等差数列, $\\{b_n\\}$为首项为$b$、公比为$a$的等比数列.\\\\\n(1) 设$t$为正整数, 当$a=3$, $b=1$, $a_7=latex]\n\\draw [dashed] (-2,-1,0) -- (2,-1,0) (0,-1,-2) -- (0,-1,2);\n\\draw (-2,-1,-0.3) -- (-0.3,-1,-0.3) -- (-0.3,-1,-2);\n\\draw (-2,-1,0.3) -- (-0.3,-1,0.3) -- (-0.3,-1,2);\n\\draw (2,-1,-0.3) -- (0.3,-1,-0.3) -- (0.3,-1,-2);\n\\draw (2,-1,0.3) -- (0.3,-1,0.3) -- (0.3,-1,2);\n\\fill [domain = 0:360, white] plot ({2*cos(\\x)},0,{2*sin(\\x)});\n\\fill [domain = 0:360, pattern = north east lines] plot ({2*cos(\\x)},0,{2*sin(\\x)});\n\\fill [domain = 0:360, white] plot ({cos(\\x)},0,{sin(\\x)});\n\\draw [domain = 0:360,ultra thick,samples = 100] plot ({2*cos(\\x)},0,{2*sin(\\x)});\n\\draw [domain = 0:360,thick] plot ({cos(\\x)},0,{sin(\\x)});\n\\filldraw (0,0) circle (0.03) node [left] {$O$} coordinate (O);\n\\draw [domain = 0:180,thick] plot ({-sqrt(2)*cos(\\x)},{2*sin(\\x)},{sqrt(2)*cos(\\x)});\n\\draw [domain = 0:180,thick] plot ({sqrt(2)*cos(\\x)},{2*sin(\\x)},{sqrt(2)*cos(\\x)});\n\\draw (0,2,0) node [above] {$S$} coordinate (S);\n\\draw ({cos(45)},0,{sin(45)}) --++ (0,{sqrt(3)},0);\n\\draw ({cos(135)},0,{sin(135)}) node [right] {$D$} coordinate (D) --++ (0,{sqrt(3)},0) node [above left] {$C$} coordinate (C);\n\\draw ({cos(225)},0,{sin(225)}) --++ (0,{sqrt(3)},0);\n\\draw ({cos(315)},0,{sin(315)}) --++ (0,{sqrt(3)},0);\n\\draw [thick] ({2*cos(45)},0,{2*sin(45)}) node [below left] {$A$} coordinate (A) -- ({3*cos(45)},-1,{3*sin(45)}) node [below] {$B$} coordinate (B);\n\\draw [thick] ({2*cos(45)},0,{2*sin(45)}) node [below left] {$A$} coordinate (A) -- ({3*cos(45)},-1,{3*sin(45)}) node [below] {$B$} coordinate (B);\n\\draw [thick] ({2*cos(135)},0,{2*sin(135)}) -- ({3*cos(135)},-1,{3*sin(135)});\n\\draw [thick] ({2*cos(315)},0,{2*sin(315)}) -- ({3*cos(315)},-1,{3*sin(315)});\n\\draw [thick,dashed] ({2*cos(225)},0,{2*sin(225)}) -- ({3*cos(225)},-1,{3*sin(225)});\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $CD\\parallel$平面$SOA$;\\\\\n(2) 设$AB$为经过$A$的一条步道, 其长度为$12$米且与地面所成角的大小为$30^{\\circ}$. 桥面小圆与大圆的半径之比为$4: 5$, 当桥面大圆半径为$20$米时, 求点$C$到地面的距离.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-中档题冲刺-中档题冲刺07" ], "genre": "解答题", "ans": "(1) 证明略; (2) $18$米", @@ -334043,7 +336771,8 @@ "content": "若函数$y=f(x)$($x \\in D$)同时满足下列两个条件, 则称$y=f(x)$在$D$上具有性质$M$.\\\\\n\\textcircled{1} $y=f(x)$在$D$上的导数$f'(x)$存在;\\\\\n\\textcircled{2} $y=f'(x)$在$D$上的导数$f''(x)$存在, 且$f''(x)>0$(其中$f''(x)=[f'(x)]'$)恒成立.\\\\\n(1) 判断函数$y=\\lg \\dfrac 1x$在区间$(0,+\\infty)$上是否具有性质$M$? 并说明理由;\\\\\n(2) 设$a$、$b$均为实常数, 若奇函数$g(x)=2 x^3+a x^2+\\dfrac bx$在$x=1$处取得极值, 是否存在实数$c$, 使得$y=g(x)$在区间$[c,+\\infty)$上具有性质$M$? 若存在, 求出$c$的取值范围; 若不存在, 请说明理由;\\\\\n(3) 设$k \\in \\mathbf{Z}$且$k>0$, 对于任意的$x \\in(0,+\\infty)$, 不等式$\\dfrac{1+\\ln (x+1)}x>\\dfrac k{x+1}$成立, 求$k$的最大值.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-16_导数及其应用" ], "genre": "解答题", "ans": "(1) 具有性质$M$, 理由略; (2) 存在, $c$的范围为$(0,+\\infty)$; (3) $3$", @@ -334078,7 +336807,8 @@ "content": "设全集$U=\\{1,2,3,4\\}$, $A=\\{1,3\\}$, 则$\\overline A=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-简单题冲刺-简单题冲刺08" ], "genre": "填空题", "ans": "$\\{2,4\\}$", @@ -334108,7 +336838,8 @@ "content": "不等式$x^2-3 x+2<0$的解集为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-简单题冲刺-简单题冲刺08" ], "genre": "填空题", "ans": "$(1,2)$", @@ -334138,7 +336869,8 @@ "content": "复数$z$满足$\\overline z=\\dfrac 1{1+\\mathrm{i}}$(其中$\\mathrm{i}$为虚数单位), 则复数$z$在复平面上所对应的点$Z$到原点$O$的距离为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-简单题冲刺-简单题冲刺08" ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{2}}2$", @@ -334168,7 +336900,8 @@ "content": "设向量$\\overrightarrow a$、$\\overrightarrow b$满足$|\\overrightarrow a|=1 , \\overrightarrow a \\cdot \\overrightarrow b=2$, 则$\\overrightarrow a \\cdot(\\overrightarrow a+\\overrightarrow b)=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-简单题冲刺-简单题冲刺08" ], "genre": "填空题", "ans": "$3$", @@ -334198,7 +336931,8 @@ "content": "如图, 在三棱台$ABC-A_1B_1C_1$的$9$条棱所在直线中, 与直线$A_1B$是异面直线的共有\\blank{50}条.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$A$} coordinate (A);\n\\draw (2,-0.5) node [below] {$B$} coordinate (B);\n\\draw (3,0.3) node [right] {$C$} coordinate (C);\n\\path (1.4,2) coordinate (P);\n\\draw (A) -- ($(A)!0.5!(P)$) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) -- ($(B)!0.5!(P)$) node [below right] {$B_1$} coordinate (B_1);\n\\draw (C) -- ($(C)!0.5!(P)$) node [right] {$C_1$} coordinate (C_1);\n\\draw (A) -- (B) -- (C) (A_1) -- (B_1) -- (C_1) -- cycle;\n\\draw (A_1) -- (B);\n\\draw [dashed] (A) -- (C);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-简单题冲刺-简单题冲刺08" ], "genre": "填空题", "ans": "$3$", @@ -334228,7 +336962,8 @@ "content": "甲、乙两城市某月初连续$7$天的日均气温数据如图所示, 则在这$7$天中, \\\\\n\\textcircled{1} 甲城市日均气温的中位数与平均数相等;\\\\\n\\textcircled{2} 甲城市的日均气温比乙城市的日均气温稳定;\\\\\n\\textcircled{3} 乙城市日均气温的极差为$3^{\\circ} \\text{C}$;\\\\\n\\textcircled{4} 乙城市日均气温的众数为$5^{\\circ} \\text{C}$.\n以上判断正确的是\\blank{50}.(写出所有正确判断的序号)\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.4]\n\\foreach \\i in {1,2,...,7} {\n\\draw [gray] (\\i,0) -- (\\i,7) (0,\\i) -- (7,\\i); \n\\draw (\\i,0.2) -- (\\i,0) node [below] {$\\i$};\n\\draw (0.2,\\i) -- (0,\\i) node [left] {$\\i$};};\n\\draw [->] (0,0) -- (8,0) node [below] {日期};\n\\draw [->] (0,0) -- (0,8) node [left] {气温$^\\circ\\text{C}$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1,5) -- (2,3) -- (3,6) -- (4,3) -- (5,7) -- (6,5) -- (7,6);\n\\draw [dashed] (1,5) -- (2,4) -- (3,6) -- (4,5) -- (5,5) -- (6,4) -- (7,6);\n\\draw (7.5,5.5) -- (9.5,5.5) node [right] {甲};\n\\draw [dashed] (7.5,4) -- (9.5,4) node [right] {乙};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-简单题冲刺-简单题冲刺08" ], "genre": "填空题", "ans": "\\textcircled{1}\\textcircled{4}", @@ -334259,7 +336994,8 @@ "content": "有甲、乙、丙三项任务, 其中甲需$2$人承担, 乙、丙各需$1$人承担 . 现从$6$人中任选$4$人承担这三项任务, 则共有\\blank{50}种不同的选法.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-简单题冲刺-简单题冲刺08" ], "genre": "填空题", "ans": "$180$", @@ -334289,7 +337025,8 @@ "content": "研究发现, 某昆虫释放信息素$t$秒后, 在距释放处$x$米的地方测得的信息素浓度$y$满足$\\ln y=-\\dfrac 12 \\ln t-\\dfrac kt x^2+a$, 其中$k, a$为非零常数. 已知释放$1$秒后, 在距释放处$2$米的地方测得信息素浓度为$m$, 则释放信息素$4$秒后, 距释放处的\\blank{50}米的位置, 信息素浓度为$\\dfrac m2$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-简单题冲刺-简单题冲刺08" ], "genre": "填空题", "ans": "$4$", @@ -334319,7 +337056,8 @@ "content": "若$\\overrightarrow{OA}=(1,-2,0)$, $\\overrightarrow{OB}=(2,1,0)$, $\\overrightarrow{OC}=(1,1,3)$, 则三棱锥$O-ABC$的体积为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-中档题冲刺-中档题冲刺08" ], "genre": "填空题", "ans": "$\\dfrac 52$", @@ -334343,7 +337081,8 @@ "content": "已知函数$y=2 \\sin (\\omega x+\\dfrac{\\pi}6)(\\omega>0)$的图像向右平移$\\varphi(0<\\varphi<\\dfrac{\\pi}2)$个单位, 可得到函数$y=\\sin 2 x-a \\cos 2 x$的图像, 则$\\varphi=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-中档题冲刺-中档题冲刺08" ], "genre": "填空题", "ans": "$\\dfrac\\pi 4$", @@ -334367,7 +337106,8 @@ "content": "已知$AA_1$是圆柱的一条母线, $AB$是圆柱下底面的直径, $C$是圆柱下底面圆周上异于$A$、$B$的点. 若圆柱的侧面积为$4 \\pi$, 则三棱锥$A_1-ABC$外接球体积的最小值为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-中档题冲刺-中档题冲刺08" ], "genre": "填空题", "ans": "$\\dfrac{8\\sqrt{2}\\pi}3$", @@ -334458,7 +337198,8 @@ "content": "掷两颗骰子, 观察掷得的点数. 设事件$A$为: 至少一个点数是奇数; 事件$B$为: 点数之和是偶数, 事件$A$的概率为$P(A)$, 事件$B$的概率为$P(B)$. 则$1-P(A \\cap B)$是下列哪个事件的概率\\bracket{20}.\n\\twoch{两个点数都是偶数}{至多有一个点数是偶数}{两个点数都是奇数}{至多有一个点数是奇数}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-中档题冲刺-中档题冲刺08" ], "genre": "选择题", "ans": "D", @@ -334505,7 +337246,8 @@ "content": "已知数列$\\{a_n\\}$为等差数列, 数列$\\{b_n\\}$为等比数列, 数列$\\{a_n\\}$的公差为$2$.\\\\\n(1) 若$b_1=a_1$, $b_2=a_2$, $b_3=a_5$, 求数列$\\{b_n\\}$的通项公式;\\\\\n(2) 设数列$\\{a_n\\}$的前$n$项和为$S_n$, 若$S_{12}=3 a_k$, $a_1+a_{k+1}=6$, 求$a_1$.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-简单题冲刺-简单题冲刺08" ], "genre": "解答题", "ans": "(1) $b_n=3^{n-1}$; (2) $-8$", @@ -334535,7 +337277,8 @@ "content": "已知$\\triangle ABC$的三个内角$A$、$B$、$C$的对边分别为$a$、$b$、$c$.\\\\\n(1) 若$\\triangle ABC$的面积$S=\\dfrac{a^2+c^2-b^2}4$, 求$B$;\\\\\n(2) 若$a c=\\sqrt 3$, $\\sin A=\\sqrt 3 \\sin B$, $C=\\dfrac{\\pi}6$, 求$c$.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-简单题冲刺-简单题冲刺08" ], "genre": "解答题", "ans": "(1) $\\dfrac \\pi 4$; (2) $1$", @@ -334565,7 +337308,8 @@ "content": "如图, 在三棱锥$D-ABC$中, 平面$ACD \\perp$平面$ABC$, $AD \\perp AC$, $AB \\perp BC$, $E$、$F$分别为棱$BC$、$CD$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,1.6,0) node [above] {$D$} coordinate (D);\n\\draw (1,0,1) node [below] {$B$} coordinate (B);\n\\draw ($(B)!0.5!(C)$) node [below right] {$E$} coordinate (E);\n\\draw ($(C)!0.5!(D)$) node [above] {$F$} coordinate (F);\n\\draw (A) -- (B) -- (C) -- (D) -- cycle;\n\\draw (D) -- (B) (F) -- (E);\n\\draw [dashed] (A) -- (C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: 直线$EF\\parallel$平面$ABD$;\\\\\n(2) 求证: 直线$BC \\perp$平面$ABD$;\\\\\n(3) 若直线$CD$与平面$ABC$所成的角为$45^{\\circ}$, 直线$CD$与平面$ABD$所成角为$30^{\\circ}$, 求二面角$B-AD-C$的大小.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-中档题冲刺-中档题冲刺08" ], "genre": "解答题", "ans": "(1) 证明略; (2) 证明略; (3) $\\dfrac\\pi 4$", @@ -334635,7 +337379,8 @@ "content": "已知全集$U=\\mathbf{R}$, 集合$A=\\{x||x |>0\\}$, 则$\\overline A=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-简单题冲刺-简单题冲刺09" ], "genre": "填空题", "ans": "$\\{0\\}$", @@ -334665,7 +337410,8 @@ "content": "在复平面内, 复数$z$所对应的点的坐标为$(1,-1)$, 则$z \\cdot \\overline z=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-简单题冲刺-简单题冲刺09" ], "genre": "填空题", "ans": "$2$", @@ -334695,7 +337441,8 @@ "content": "不等式$\\dfrac{x+5}{x^2+2 x+3} \\geq 1$的解集为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-简单题冲刺-简单题冲刺09" ], "genre": "填空题", "ans": "$[-2,1]$", @@ -334725,7 +337472,8 @@ "content": "函数$y=\\tan x$在区间$(\\dfrac{\\pi}2, \\dfrac{3 \\pi}2)$上的零点是\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-简单题冲刺-简单题冲刺09" ], "genre": "填空题", "ans": "$\\pi$", @@ -334755,7 +337503,8 @@ "content": "已知$f(x)$是定义域为$\\mathbf{R}$的奇函数, 且$x \\leq 0$时, $f(x)=\\mathrm{e}^x-1$, 则$f(x)$的值域是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-简单题冲刺-简单题冲刺09" ], "genre": "填空题", "ans": "$(-1,1)$", @@ -334786,7 +337535,8 @@ "content": "在$(x-\\dfrac 2x)^9$的二项展开式中, $x^3$项的系数是\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-简单题冲刺-简单题冲刺09" ], "genre": "填空题", "ans": "$-672$", @@ -334816,7 +337566,8 @@ "content": "已知圆锥的侧面积为$2 \\pi$, 且侧面展开图为半圆, 则该圆锥底面半径为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-简单题冲刺-简单题冲刺09" ], "genre": "填空题", "ans": "$1$", @@ -334846,7 +337597,8 @@ "content": "在数列$\\{a_n\\}$中, $a_1=2$, 且$a_n=a_{n-1}+\\lg \\dfrac n{n-1}$($n \\geq 2$), 则$a_{100}=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-简单题冲刺-简单题冲刺09" ], "genre": "填空题", "ans": "$4$", @@ -334877,7 +337629,8 @@ "objs": [], "tags": [ "第八单元", - "第九单元" + "第九单元", + "2023届高三-中档题冲刺-中档题冲刺09" ], "genre": "填空题", "ans": "$\\dfrac 29$", @@ -334901,7 +337654,8 @@ "content": "在$\\triangle ABC$中, $AC=4$, 且$\\overrightarrow{AC}$在$\\overrightarrow{AB}$方向上的数量投影是$-2$, 则$|\\overrightarrow{BC}-\\lambda \\overrightarrow{BA}|$($\\lambda \\in \\mathbf{R}$)的最小值为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-中档题冲刺-中档题冲刺09" ], "genre": "填空题", "ans": "$2\\sqrt{3}$", @@ -334927,7 +337681,8 @@ "content": "设$k \\in \\mathbf{R}$, 函数$y=|x^2-4 x+3|$的图像与直线$y=k x+1$有四个交点, 且这些交点的横坐标分别为$x_1, x_2, x_3, x_4(x_1=latex,scale = 0.6]\n\\draw (0,0,0) node [left] {$B$} coordinate (B) ++ (0,4) node [left] {$B_1$} coordinate (B_1);\n\\draw ({sqrt(2)},0,{-sqrt(2)}) node [right] {$A$} coordinate (A) ++ (0,4)node [above] {$A_1$} coordinate (A_1);\n\\draw ({2*sqrt(2)},0,0) node [right] {$C$} coordinate (C) ++ (0,4) node [right] {$C_1$} coordinate (C_1);\n\\draw ($(C)!0.5!(C_1)$) node [right] {$D$} coordinate (D);\n\\draw ($(A)!0.25!(A_1)$) node [right] {$E$} coordinate (E);\n\\draw (B) -- (C) -- (C_1) -- (A_1) -- (B_1) -- cycle;\n\\draw (B_1) -- (C_1) (B_1) -- (D) (B_1) -- (C);\n\\draw [dashed] (B) -- (A) -- (C) (B) -- (E) (A) -- (A_1) (B_1) -- (A);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证:$BE \\perp$平面$AB_1C$;\\\\\n(2) 求直线$B_1D$与平面$AB_1C$所成角的大小.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-简单题冲刺-简单题冲刺09" ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\arcsin \\dfrac{\\sqrt{15}}{15}$", @@ -335093,7 +337850,8 @@ "content": "已知$f(x)=\\ln x-(a+1) x+\\dfrac 12 a x^2$($a \\in \\mathbf{R}$).\\\\\n(1) 当$a=0$时, 求函数$y=f(x)$在点$(1, f(1))$处的切线方程;\\\\\n(2) 当$a \\in(0,1]$时, 求函数$y=f(x)$的单调区间.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-简单题冲刺-简单题冲刺09" ], "genre": "解答题", "ans": "(1) $y+1=0$; (2) 当$0=latex]\n\\draw (0,0) node [below left] {$E$} coordinate (E) rectangle (3,3) node [above right] {$G$} coordinate (G);\n\\draw (3,0) node [below right] {$F$} coordinate (F) (0,3) node [above left] {$H$} coordinate (H);\n\\draw ($(E)!0.5!(F)$) node [below] {$A$} coordinate (A) --++ (60:3) node [right] {$B$} coordinate (B) arc (60:120:3) node [left] {$C$} coordinate (C) -- cycle;\n\\draw (1.5,0) ++ (80:3) node [above] {$P$} coordinate (P);\n\\draw (P) -- ($(A)!(P)!(B)$) node [below left] {$Q$} coordinate (Q);\n\\draw (P) -- ($(A)!(P)!(C)$) node [below left] {$R$} coordinate (R);\n\\draw (R) -- (Q);\n\\end{tikzpicture}\n\\end{center} \n(1) 如果点$P$位于弧$BC$的中点, 求三条步行道$PQ$、$PR$、$RQ$的总长度;\\\\\n(2) ``地摊经济''对于``拉动灵活就业、增加多源收入、便利居民生活''等都有积极作用. 为此街道允许在步行道$PQ$、$PR$、$RQ$开辟临时摊点, 积极推进``地摊经济''发展, 预计每年能产生的经济效益分别为每米$5$万元、$5$万元及$5.9$万元 . 则这三条步行道每年能产生的经济总效益最高为多少?(精确到 1 万元)", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-中档题冲刺-中档题冲刺09" ], "genre": "解答题", "ans": "(1) $200+100\\sqrt{3}$米; (2) $2022$万元", @@ -335191,7 +337950,8 @@ "content": "已知集合$A=\\{1,2\\}$, $B=\\{2,3\\}$, 则$A \\cap B=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-简单题冲刺-简单题冲刺10" ], "genre": "填空题", "ans": "$\\{2\\}$", @@ -335216,7 +337976,8 @@ "content": "函数$y=\\log_2 \\dfrac{1+x}{1-x}$的定义域是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-简单题冲刺-简单题冲刺10" ], "genre": "填空题", "ans": "$(-1,1)$", @@ -335241,7 +338002,8 @@ "content": "设复数$z=\\mathrm{i}(2-\\mathrm{i})$(其中$\\mathrm{i}$为虚数单位), 则$|z|=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-简单题冲刺-简单题冲刺10" ], "genre": "填空题", "ans": "$\\sqrt{5}$", @@ -335266,7 +338028,8 @@ "content": "设$x>1$, 则$x+\\dfrac 4{x-1}$的最小值是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-简单题冲刺-简单题冲刺10" ], "genre": "填空题", "ans": "$5$", @@ -335291,7 +338054,8 @@ "content": "若指数函数$y=a^x$($a>0$, $a \\neq 1$)在$[1, 2]$上的最大值与最小值之和等于$12$, 则实数$a=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-简单题冲刺-简单题冲刺10" ], "genre": "填空题", "ans": "$3$", @@ -335316,7 +338080,8 @@ "content": "两个篮球运动员罚球时的命中概率分别是$0.6$和$0.5$, 两人各投一次, 则他们同时命中的概率是\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-简单题冲刺-简单题冲刺10" ], "genre": "填空题", "ans": "$0.3$", @@ -335341,7 +338106,8 @@ "content": "将圆锥的侧面展开后得到一个半径为$2$的半圆, 则此圆锥的体积为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-简单题冲刺-简单题冲刺10" ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{3}}3\\pi$", @@ -335379,7 +338145,8 @@ "content": "已知平面向量$\\overrightarrow a$、$\\overrightarrow b$满足$|\\overrightarrow a|=3$, $|\\overrightarrow b|=4$, 则$2 \\overrightarrow a+\\overrightarrow b$在$\\overrightarrow a$方向上的数量投影的最小值是\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-简单题冲刺-简单题冲刺10" ], "genre": "填空题", "ans": "$2$", @@ -335404,7 +338171,8 @@ "content": "从$5$名志愿者中选出$4$名分别参加测温、扫码、做核酸和信息登记的工作 (每项$1$人), 其中甲不参加测温的分配方案有\\blank{50}种. (结果用数值表示)", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-中档题冲刺-中档题冲刺10" ], "genre": "填空题", "ans": "$96$", @@ -335428,7 +338196,8 @@ "content": "双曲线$C$的左、右焦点分别为$F_1$、$F_2$, 点$A$在$y$轴上. 双曲线$C$与线段$AF_1$交于点$P$, 与线段$AF_2$交于点$Q$, 直线$AF_1$平行于双曲线$C$的渐近线, 且$|AP|:|PQ|=5: 6$, 则双曲线$C$的离心率为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-中档题冲刺-中档题冲刺10" ], "genre": "填空题", "ans": "$\\dfrac 53$", @@ -335452,7 +338221,8 @@ "content": "某人去公园郊游, 在草地上搭建了如图所示的简易遮阳篷$ABC$, 遮阳篷是一个直角边长为$6$的等腰直角三角形, 斜边$AB$朝南北方向固定在地上, 正西方向射出的太阳光线与地面成$30^{\\circ}$角, 则当遮阳篷$ABC$与地面所成的角大小为\\blank{50}时, 所遮阴影面$ABC'$面积达到最大.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,2) node [below] {$A$} coordinate (A);\n\\draw (0,0,-2) node [right] {$B$} coordinate (B);\n\\draw ({2*sqrt(3)},0,0) node [right] {$C'$} coordinate (C');\n\\draw (0,2,0) node [above] {$C$} coordinate (C);\n\\fill [pattern = north east lines] (A) -- (B) -- (C) -- cycle;\n\\draw (A) -- (C) -- (C') -- cycle;\n\\draw [dashed] (A) -- (B) -- (C) (B) -- (C');\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-中档题冲刺-中档题冲刺10" ], "genre": "填空题", "ans": "$60^\\circ$", @@ -335542,7 +338312,8 @@ "content": "设对任意满足$\\sin \\alpha+\\cos \\alpha=x$的实数$\\alpha$与$x$, $\\sin ^3 \\alpha+\\cos ^3 \\alpha=a_3 x^3+a_2 x^2+a_1 x+a_0$总成立, 则$a_0+a_1+a_2+a_3=$\\bracket{20}.\n\\fourch{$-1$}{$\\dfrac 12$}{$1$}{$\\sqrt 2$}", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-中档题冲刺-中档题冲刺10" ], "genre": "选择题", "ans": "C", @@ -335589,7 +338360,8 @@ "content": "已知函数$f(x)=\\sin 2 x+\\sqrt 3 \\cos 2 x$, $x \\in \\mathbf{R}$.\\\\\n(1) 求函数$f(x)$的单调增区间;\\\\\n(2) 在$\\triangle ABC$中, 角$A$、$B$、$C$的对边分别为$a$、$b$、$c$, 当$f(A)=0$, $b=1$, 且三角形$ABC$的面积为$\\sqrt 3$时, 求$a$.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-简单题冲刺-简单题冲刺10" ], "genre": "解答题", "ans": "(1) $[k\\pi-\\dfrac{5\\pi}{12},k\\pi+\\dfrac\\pi{12}]$, $k\\in \\mathbf{Z}$; (2) $\\sqrt{13}$或$\\sqrt{61}$", @@ -335614,7 +338386,8 @@ "content": "已知数列$\\{a_n\\}$满足$a_1=1$, $a_n=3 a_{n-1}+4$($n \\geq 2$).\\\\\n(1) 求证: 数列$\\{a_n+2\\}$是等比数列;\\\\\n(2) 求数列$\\{a_n\\}$的通项公式;\\\\\n(3) 写出$\\displaystyle\\sum_{i=1}^5 a_{2 i-1}$的具体展开式, 并求其值.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-简单题冲刺-简单题冲刺10" ], "genre": "解答题", "ans": "(1) 证明略; (2) $a_n=3^n-2$; (3) $a_1+a_3+a_5+a_7+a_9=22133$", @@ -335639,7 +338412,8 @@ "content": "如图, 棱长为$2$的正方体$ABCD-A_1B_1C_1D_1$中, $M$、$N$、$P$分别是$C_1D_1$、$C_1C$、$A_1A$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\filldraw ($(C1)!0.5!(D1)$) circle (0.03) node [above] {$M$} coordinate (M);\n\\filldraw ($(A)!0.5!(A1)$) circle (0.03) node [left] {$P$} coordinate (P);\n\\filldraw ($(C)!0.5!(C1)$) circle (0.03) node [right] {$N$} coordinate (N);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $M$、$N$、$A_1$、$B$四点共面;\\\\\n(2) 求异面直线$PD_1$与$MN$所成角的大小; (结果用反三角函数值表示)\\\\\n(3) 求三棱锥$P-MNB$的体积.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-中档题冲刺-中档题冲刺10" ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\arccos \\dfrac{\\sqrt{10}}{10}$; (3) $\\dfrac 13$", @@ -335707,7 +338481,8 @@ "content": "设集合$A=(-2,2)$, $B=(-3,1)$, 则$A \\cap B=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-简单题冲刺-简单题冲刺11" ], "genre": "填空题", "ans": "$(-2,1)$", @@ -335729,7 +338504,8 @@ "content": "若幂函数$y=x^a$的图像经过点$(\\sqrt[4]3, 3)$, 则实数$a=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-简单题冲刺-简单题冲刺11" ], "genre": "填空题", "ans": "$4$", @@ -335751,7 +338527,8 @@ "content": "函数$y=\\log_2(2-x)$的定义域为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-简单题冲刺-简单题冲刺11" ], "genre": "填空题", "ans": "$(-\\infty,2)$", @@ -335773,7 +338550,8 @@ "content": "$(x+2)^5$的二项展开式中$x^2$的系数为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-简单题冲刺-简单题冲刺11" ], "genre": "填空题", "ans": "$80$", @@ -335795,7 +338573,8 @@ "content": "若圆锥的轴截面是边长为$1$的正三角形, 则圆锥的侧面积是\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-简单题冲刺-简单题冲刺11" ], "genre": "填空题", "ans": "$\\dfrac\\pi 2$", @@ -335817,7 +338596,8 @@ "content": "已知$\\alpha$为锐角, 若$\\sin (\\alpha+\\dfrac{\\pi}2)=\\dfrac 35$, 则$\\tan (\\alpha+\\dfrac{\\pi}4)=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-简单题冲刺-简单题冲刺11" ], "genre": "填空题", "ans": "$-7$", @@ -335839,7 +338619,8 @@ "content": "已知某射击爱好者的打靶成绩(单位: 环)的茎叶图如图所示, 其中整数部分为 ``茎'', 小数部分为``叶'', 则这组数据的方差为\\blank{50}(精确到0.01).\n\\begin{center}\n\\begin{tabular}{r|l}\n5 & 6 \\ 8\\\\\n6 & 2 \\ 3 \\ 6 \\ 6\\\\\n7 & 3 \\ 4\n\\end{tabular}\n\\end{center}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-简单题冲刺-简单题冲刺11" ], "genre": "填空题", "ans": "$0.36$", @@ -335861,7 +338642,8 @@ "content": "已知拋物线$C: y^2=16 x$的焦点为$F$, 在$C$上有一点$P$满足$|PF|=13$, 则点$P$到$x$轴的距离为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-简单题冲刺-简单题冲刺11" ], "genre": "填空题", "ans": "$12$", @@ -335883,7 +338665,8 @@ "content": "某医院需要从$4$名男医生和$3$名女医生中选出$3$名医生去担任``中国进博会''三个不同区域的核酸检测服务工作, 则选出的$3$名医生中, 恰有$1$名女医生的概率是\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-中档题冲刺-中档题冲刺11" ], "genre": "填空题", "ans": "$\\dfrac{18}{35}$", @@ -335906,7 +338689,8 @@ "objs": [], "tags": [ "第四单元", - "第一单元" + "第一单元", + "2023届高三-中档题冲刺-中档题冲刺11" ], "genre": "填空题", "ans": "$8$", @@ -335929,7 +338713,8 @@ "objs": [], "tags": [ "第三单元", - "第二单元" + "第二单元", + "2023届高三-中档题冲刺-中档题冲刺11" ], "genre": "填空题", "ans": "$(-\\pi,-\\dfrac\\pi 2]$和$[0,\\dfrac\\pi 2]$", @@ -336017,7 +338802,8 @@ "content": "已知直线$l$与平面$\\alpha$相交, 则下列命题中, 正确的个数为\\bracket{20}.\\\\\n\\textcircled{1} 平面$\\alpha$内的所有直线均与直线$l$异面;\\\\\n\\textcircled{2} 平面$\\alpha$内存在与直线$l$垂直的直线;\\\\\n\\textcircled{3} 平面$\\alpha$内不存在直线与直线$l$平行;\\\\\n\\textcircled{4} 平面$\\alpha$内所有直线均与直线$l$相交.\n\\fourch{1}{2}{3}{4}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-中档题冲刺-中档题冲刺11" ], "genre": "选择题", "ans": "B", @@ -336061,7 +338847,8 @@ "content": "已知数列$\\{a_n\\}$是公差不为$0$的等差数列, $a_1=4$, 且$a_1$、$a_3$、$a_4$成等比数列.\\\\\n(1) 求数列$\\{a_n\\}$的通项公式;\\\\\n(2) 求当$n$为何值时, 数列$\\{a_n\\}$的前$n$项和$S_n$取得最大值.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-简单题冲刺-简单题冲刺11" ], "genre": "解答题", "ans": "(1) $a_n=5-n$; (2) $n=4$或$5$", @@ -336083,7 +338870,8 @@ "content": "如图, 三棱锥$P-ABC$中, 侧面$PAB$垂直于底面$ABC$, $PA=PB$, 底面$ABC$是斜边为$AB$的直角三角形, 且$\\angle ABC=30^{\\circ}$, 记$O$为$AB$的中点, $E$为$OC$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [above] {$O$} coordinate (O);\n\\draw (-1,0,0) node [left] {$B$} coordinate (B);\n\\draw (1,0,0) node [right] {$A$} coordinate (A);\n\\draw (0.5,0,{sqrt(3)/2}) node [below] {$C$} coordinate (C);\n\\draw ($(O)!0.5!(C)$) node [left] {$E$} coordinate (E);\n\\draw (0,{sqrt(3)},0) node [above] {$P$} coordinate (P);\n\\draw (B) -- (C) -- (A) -- (P) -- cycle (P) -- (C);\n\\draw [dashed] (A) -- (B) (O) -- (C) (A) -- (E);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $PC \\perp AE$;\\\\\n(2) 若$AB=2$, 直线$PC$与底面$ABC$所成角的大小为$60^{\\circ}$, 求四面体$PAOC$的体积.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-简单题冲刺-简单题冲刺11" ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\dfrac 14$", @@ -336105,7 +338893,8 @@ "content": "在临港滴水湖畔拟建造一个四边形的露营基地, 如图$ABCD$所示. 为考虑露营客人娱乐休闲的需求, 在四边形$ABCD$区域中, 将$\\triangle ABD$区域设立成花卉观赏区, $\\triangle BCD$区域设立成烧烤区, 边$AB$、$BC$、$CD$、$DA$修建成观赏步道, 边$BD$修建隔离防护栏. 其中$CD=100$米, $BC=200$米, $\\angle A=\\dfrac{\\pi}3$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.7]\n\\draw (0,0) node [right] {$C$} coordinate (C);\n\\draw (-1,0) node [left] {$D$} coordinate (D);\n\\draw (-80:2) node [right] {$B$} coordinate (B);\n\\draw ($(B)!{1/sqrt(3)}!30:(D)$) coordinate (O);\n\\draw ($(O)!1!100:(D)$) node [left] {$A$} coordinate (A);\n\\draw (A) -- (B) -- (C) -- (D) -- cycle (D) -- (B);\n\\draw (barycentric cs:A=1,B=1,D=1) node {花卉观赏区};\n\\draw (barycentric cs:B=1,C=1,D=1) node {烧烤区};\n\\end{tikzpicture}\n\\end{center}\n(1) 如果烧烤区是一个占地面积为$9600$平方米的钝角三角形, 那么需要修建多长的隔离防护栏(精确到$0.1$米)?\\\\\n(2) 考虑到烧烤区的安全性, 在规划四边形$ABCD$区域时, 首先保证烧烤区的占地面积最大时, 再使得花卉观赏区的面积尽可能大, 则应如何设计观赏步道?", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-中档题冲刺-中档题冲刺11" ], "genre": "解答题", "ans": "(1) $247.4$米; (2) $AB=AD=100\\sqrt{5}$米, $\\angle C=\\dfrac\\pi 2$", @@ -336171,7 +338960,9 @@ "content": "设$A=\\{x |-1=latex]\n\\def\\l{1}\n\\def\\m{1}\n\\def\\n{1.5}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(A)!0.7!(A1)$) node [left] {$E$} coordinate (E);\n\\draw (B) -- (E);\n\\draw [dashed] (E) -- (C1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期周末卷-高三下学期周末卷03", + "2023届高三-中档题冲刺-中档题冲刺12" ], "genre": "填空题", "ans": "$2$", @@ -336522,7 +339331,9 @@ "content": "设$p>0, q>0$且满足$\\log_{16} p=\\log_{20} q=\\log_{25}(p+q)$, 则$\\dfrac pq=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期周末卷-高三下学期周末卷03", + "2023届高三-中档题冲刺-中档题冲刺12" ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{5}-1}2$", @@ -336557,7 +339368,8 @@ "content": "已知某商品的成本$C$和产量$q$满足关系$C=50000+200 q$, 该商品的销售单价$p$和产量$q$满足关系式$p=24200-\\dfrac 15 q^2$, 则当产量$q$等于\\blank{50}时, 利润最大.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期周末卷-高三下学期周末卷03" ], "genre": "填空题", "ans": "$200$", @@ -336592,7 +339404,8 @@ "content": "下列四组函数中, 同组的两个函数是相同函数的是\\bracket{20}.\n\\fourch{$y=x$与$y=(\\dfrac 1x)^{-1}$}{$y=|x|$与$y=(\\sqrt x)^2$}{$y=x$与$y=e^{\\ln x}$}{$y=x$与$y=\\sqrt [5]{x^5}$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期周末卷-高三下学期周末卷03" ], "genre": "选择题", "ans": "D", @@ -336627,7 +339440,8 @@ "content": "紫砂壶是中国特有的手工制造陶土工艺品, 其制作始于明朝正德年间. 紫砂壶的壸型众多, 经典的有西施壶、掇球壶、石瓢壸、潘壶等. 其中, 石瓢壸的壸体可以近似看成一个圆台. 若壶口的直径为$6\\text{cm}$, 壶底的直径为$10\\text{cm}$, 壶体高$4\\text{cm}$, 那么该壶的容积约接近于\\bracket{20}.\n\\fourch{$100 \\text{cm}^3$}{$200 \\text{cm}^3$}{$300 \\text{cm}^3$}{$400 \\text{cm}^3$}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期周末卷-高三下学期周末卷03" ], "genre": "选择题", "ans": "B", @@ -336662,7 +339476,9 @@ "content": "下列结论不正确的是\\bracket{20}.\n\\onech{若事件$A$与$B$互斥, 则$P(A \\cup B)=P(A) P(B)$}{若事件$A$与$B$相互独立, 则$P(A \\cap B)=P(A) P(B)$}{如果$X$、$Y$分别是两个独立的随机变量, 那么$D[X+Y]=D[X]+D[Y]$}{若随机变量$Y$的方差$D[Y]=3$, 则$D[2Y+1]=12$}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-下学期周末卷-高三下学期周末卷03", + "2023届高三-中档题冲刺-中档题冲刺12" ], "genre": "选择题", "ans": "A", @@ -336697,7 +339513,8 @@ "content": "已知$a, b, \\alpha, \\beta \\in \\mathbf{R}$, 满足$\\sin \\alpha+\\cos \\beta=a$, $\\cos \\alpha+\\sin \\beta=b$, $0=latex,scale = 0.5]\n\\draw (0,0,0) node [left] {$B$} coordinate (B);\n\\draw (5,0,0) node [right] {$D$} coordinate (D);\n\\draw ({16/5},0,{4*sqrt(34)/5}) node [below] {$C$} coordinate (C);\n\\draw ({7/5},{12*sqrt(2/17)},{36/5*sqrt(2/17)}) node [above] {$A$} coordinate (A);\n\\draw ($(A)!0.5!(D)$) node [above right] {$E$} coordinate (E);\n\\draw (A) -- (B) -- (C) -- (D) -- cycle (A) -- (C) -- (E);\n\\draw [dashed] (E) -- (B) -- (D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $AD \\perp$平面$BEC$;\\\\\n(2) 已知$AB=5$, $\\angle BDC=\\arccos \\dfrac 9{25}$, $AD=6$. 作出二面角$D-BC-E$的平面角, 并求它的正弦值.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期周末卷-高三下学期周末卷03", + "2023届高三-简单题冲刺-简单题冲刺12" ], "genre": "解答题", "ans": "(1) 证明略; (2) 图略, $\\dfrac{3\\sqrt{17}}{17}$", @@ -336802,7 +339623,9 @@ "content": "某地区$1997$年底沙漠面积为$9 \\times 10^5 \\text{hm}^2$(注:$\\text{hm}^2$是面积单位, 表示公顷). 地质工作者为了解这个地区沙漠面积的变化情况, 从$1998$年开始进行了连续$5$年的观测, 并在每年底将观测结果记录如下表:\n\\begin{center}\n\\begin{tabular}{|c|c|}\n\\hline 观测年份 & 该地区沙漠面积比原有($1997$年底)面积增加数: $\\text{hm}^2$\\\\\n\\hline $1998$ & $2000$ \\\\\n\\hline $1999$ & $4000$ \\\\\n\\hline $2000$ & $6001$ \\\\\n\\hline $2001$ & $7999$ \\\\\n\\hline $2002$ & $10001$ \\\\\n\\hline\n\\end{tabular} \n\\end{center} \n请根据上表所给的信息进行估计.\\\\\n(1) 如果不采取任何措施, 到$2020$年底, 这个地区的沙漠面积大约变成多少$\\text{hm}^2$?\\\\\n(2) 如果从$2003$年初开始, 采取植树造林等措施, 每年改造面积$8000 \\text{hm}^2$沙漠, 但沙漠面积仍按原有速度增加, 那么到哪一年年底, 这个地区的沙漠面积将首次小于$8 \\times 10^5 \\text{hm}^2$?", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-下学期周末卷-高三下学期周末卷03", + "2023届高三-中档题冲刺-中档题冲刺12" ], "genre": "解答题", "ans": "(1) 大约$9.46\\times 10^5\\text{hm}^2$; (2) 到$2021$年底这个地区的沙漠面积首次小于$8\\times 10^5\\text{hm}^2$", @@ -336837,7 +339660,8 @@ "content": "已知椭圆$C$的中心在原点$O$, 且它的一个焦点$F$为$(\\sqrt 3, 0)$. 点$A_1, A_2$分别是椭圆的左、右顶点, 点$B$为椭圆的上顶点, $\\triangle OFB$的面积为$\\dfrac{\\sqrt 3}2$. 点$M$是椭圆$C$上在第一象限内的一个动点.\\\\\n(1) 求椭圆$C$的标准方程;\\\\ \n(2) 若把直线$MA_1, MA_2$的斜率分别记作$k_1, k_2$, 若$k_1+k_2=-\\dfrac 34$, 求点$M$的坐标;\\\\\n(3) 设直线$MA_1$与$y$轴交于点$P$, 直线$MA_2$与$y$轴交于点$Q$. 令$\\overrightarrow{PB}=\\lambda \\overrightarrow{BQ}$, 求实数$\\lambda$的取值范围.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期周末卷-高三下学期周末卷03" ], "genre": "解答题", "ans": "(1) $\\dfrac{x^2}{4}+y^2=1$; (2) $M(\\dfrac 65,\\dfrac 45)$; (3) $(0,1)$", @@ -336872,7 +339696,8 @@ "content": "已知函数$y=f(x)$, $y=g(x)$, 其中$f(x)=\\dfrac 1{x^2}$, $g(x)=\\ln x$.\\\\\n(1) 求函数$y=g(x)$在点$(1, g(1))$的切线方程;\\\\ \n(2) 函数$y=m f(x)+2 g(x)$, $m \\in \\mathbf{R}$, $m \\neq 0$是否存在极值点, 若存在求出极值点, 若不存在, 请说明理由;\\\\\n(3) 若关于$x$的不等式$a f(x)+g(x) \\geq a$在区间$(0,1]$上恒成立, 求实数$a$的取值范围.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期周末卷-高三下学期周末卷03" ], "genre": "解答题", "ans": "(1) $y=x-1$; (2) 存在极小值点$x=\\sqrt{m}$, 无极大值点; (3) $[\\dfrac 12,+\\infty)$", @@ -336907,7 +339732,9 @@ "content": "函数$y=\\sin (2x-\\dfrac \\pi 4)$的最小正周期$T=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期周末卷-高三下学期周末卷02", + "2023届高三-简单题冲刺-简单题冲刺13" ], "genre": "填空题", "ans": "$\\pi$", @@ -336939,7 +339766,9 @@ "content": "已知集合$A=\\{-1,0,1,2\\}$, $B=\\{x | 00$, 则$x+\\dfrac 2x$的最小值为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-下学期周末卷-高三下学期周末卷02", + "2023届高三-简单题冲刺-简单题冲刺13" ], "genre": "填空题", "ans": "$2\\sqrt{2}$", @@ -337003,7 +339834,9 @@ "content": "已知抛物线$y^2=2px$($p>0$)的焦点坐标为$(2,0)$, 则$p$的值为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期周末卷-高三下学期周末卷02", + "2023届高三-简单题冲刺-简单题冲刺13" ], "genre": "填空题", "ans": "$4$", @@ -337035,7 +339868,9 @@ "content": "已知圆锥的底面半径为$3$, 高为$4$, 则该圆锥的侧面积为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期周末卷-高三下学期周末卷02", + "2023届高三-简单题冲刺-简单题冲刺13" ], "genre": "填空题", "ans": "$15\\pi$", @@ -337067,7 +339902,9 @@ "content": "已知$f(x)=x^2+x$, 则曲线$y=f(x)$在$x=0$处的切线方程是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期周末卷-高三下学期周末卷02", + "2023届高三-简单题冲刺-简单题冲刺13" ], "genre": "填空题", "ans": "$x-y=0$", @@ -337099,7 +339936,9 @@ "content": "若$x>0$时, 指数函数$y=(m^2-3)^x$的值总大于$1$, 则实数$m$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期周末卷-高三下学期周末卷02", + "2023届高三-简单题冲刺-简单题冲刺13" ], "genre": "填空题", "ans": "$m>2$或$m<-2$", @@ -337131,7 +339970,9 @@ "content": "已知$m$是实数, $\\mathrm{i}$是虚数单位, 若复数$z=\\dfrac{6+m \\mathrm{i}}{1+2 \\mathrm{i}}$的实部和虚部互为相反数, 则$|z|=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-下学期周末卷-高三下学期周末卷02", + "2023届高三-简单题冲刺-简单题冲刺13" ], "genre": "填空题", "ans": "$2\\sqrt{2}$", @@ -337163,7 +340004,9 @@ "content": "从$7$个人中选$4$人负责元旦三天假期的值班工作, 其中第一天安排$2$人, 第二天和第三天均安排$1$人, 且人员不重复, 则一共有\\blank{50}种安排方式(结果用数值表示).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-下学期周末卷-高三下学期周末卷02", + "2023届高三-中档题冲刺-中档题冲刺13" ], "genre": "填空题", "ans": "$420$", @@ -337195,7 +340038,9 @@ "content": "函数$y=3 \\sin ^2 x+2 \\sqrt 3 \\sin x \\cos x+\\cos ^2 x$, $x \\in[0, \\dfrac{\\pi}2]$的值域为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期周末卷-高三下学期周末卷02", + "2023届高三-中档题冲刺-中档题冲刺13" ], "genre": "填空题", "ans": "$[1,4]$", @@ -337227,7 +340072,9 @@ "content": "若集合$A=\\{(x, y) |(x+y)^2+x+y-2 \\leq 0\\}, B=\\{(x, y) |(x-a)^2+(y-2 a-1)^2 \\leq a^2-1\\}$, 且$A \\cap B \\neq \\varnothing$, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期周末卷-高三下学期周末卷02", + "2023届高三-中档题冲刺-中档题冲刺13" ], "genre": "填空题", "ans": "$[-\\dfrac{11}7,-1]$", @@ -337259,7 +340106,8 @@ "content": "设$\\{a_n\\}$是由正整数组成且项数为$m$的增数列, 已知$a_1=1$, $a_m=100$, 数列$\\{a_n\\}$任意相邻两项的差的绝对值不超过$1$, 若对于$\\{a_n\\}$中任意序数不同的两项$a_s$和$a_t$, 在剩下的项中总存在序数不同的两项$a_p$和$a_q$, 使得$a_s+a_t=a_p+a_q$, 则$\\displaystyle\\sum_{i=1}^{m} a_i$的最小值为\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-下学期周末卷-高三下学期周末卷02" ], "genre": "填空题", "ans": "$5454$", @@ -337291,7 +340139,8 @@ "content": "已知直线$l_1: 3 x-(a+2) y+6=0$, 直线$l_2: a x+(2 a-3) y+2=0$, 则``$a=-9$''是``$l_1\\parallel l_2$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期周末卷-高三下学期周末卷02" ], "genre": "选择题", "ans": "C", @@ -337324,7 +340173,8 @@ "objs": [], "tags": [ "第三单元", - "第四单元" + "第四单元", + "2023届高三-下学期周末卷-高三下学期周末卷02" ], "genre": "选择题", "ans": "D", @@ -337356,7 +340206,9 @@ "content": "已知正四面体$ABCD$的棱长为$6$, 设集合$\\Omega=\\{P||AP | \\leq 2 \\sqrt 7$, 点$P \\in$平面$BCD\\}$, 则$\\Omega$表示的区域的面积为 \\bracket{20}.\n\\fourch{$\\pi$}{$3 \\pi$}{$4 \\pi$}{$6 \\pi$}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期周末卷-高三下学期周末卷02", + "2023届高三-中档题冲刺-中档题冲刺13" ], "genre": "选择题", "ans": "C", @@ -337388,7 +340240,8 @@ "content": "对于函数$y=f(x)$, 若自变量$x$在区间$[a, b]$上变化时, 函数值$f(x)$的取值范围也恰为$[a, b]$, 则称区问$[a, b]$是函数$y=f(x)$的保值区间, 区间长度为$b-a$. 已知定义域为$\\mathbf{R}$的函数$y=g(x)$的表达式为$g(x)=|x^2-1|$, 给出下列命题: \\textcircled{1} 函数$y=g(x)$有且仅有$4$个保值区间; \\textcircled{2} 函数$y=g(x)$的所有保值区间长度之和为$\\dfrac{3+\\sqrt 5}2$. 下列说法正确的是\\bracket{20}.\n\\twoch{结论\\textcircled{1}成立, 结论\\textcircled{2}不成立}{结论\\textcircled{1}不成立, 结论\\textcircled{2}成立}{两个结论都成立}{两个结论都不成立}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期周末卷-高三下学期周末卷02" ], "genre": "选择题", "ans": "B", @@ -337420,7 +340273,9 @@ "content": "如图, 在四棱锥$P-ABCD$中, 已知$PA\\perp$底面$ABCD$, 底面$ABCD$是正方形, $PA=AB$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,0,2) node [left] {$B$} coordinate (B);\n\\draw (2,0,2) node [right] {$C$} coordinate (C);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw (P) -- (B) -- (C) -- (D) -- cycle (P) -- (C);\n\\draw [dashed] (P) -- (A) -- (B) (A) -- (D) (A) -- (C) (B) -- (D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: 直线$BD \\perp$平面$PAC$;\\\\\n(2) 求直线$PC$与平面$PBD$所成的角的大小.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期周末卷-高三下学期周末卷02", + "2023届高三-简单题冲刺-简单题冲刺13" ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\arcsin \\dfrac 13$", @@ -337452,7 +340307,9 @@ "content": "近两年, 直播带货逐渐成为一种新兴的营销模式, 带来电商行业的新增长点. 某直播平台第$1$年初的启动资金为$500$万元, 由于一些知名主播加入, 平台资金的年平均增长率可达$40\\%$, 每年年底扣除运营成本$a$万元, 再将剩余资金继续投入直播平台.\\\\\n(1) 若 $a=100$, 在第$3$年年底扣除运营成本后, 直播平合的资金有多少万元?\\\\\n(2) 每年的运营成本最多控制在多少万元, 才能使得直播平台在第$6$年年底扣除运营成本后资金达到$3000$万元?(结果精确到$0.1$万元)", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-下学期周末卷-高三下学期周末卷02", + "2023届高三-简单题冲刺-简单题冲刺13" ], "genre": "解答题", "ans": "(1) $936$万元; (2) $46.8$万元", @@ -337484,7 +340341,9 @@ "content": "在$\\triangle ABC$中, 设角$A$、$B$、$C$所对的边分别为$a$、$b$、$c$, 且$a \\cos B+(b-4 c) \\cos A=0$.\\\\\n(1) 求$\\cos A$;\\\\\n(2) 若$\\overrightarrow{BD}=2 \\overrightarrow{DC}$, $|\\overrightarrow{AD}|=1$, 求$c+2 b$的最大值.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期周末卷-高三下学期周末卷02", + "2023届高三-中档题冲刺-中档题冲刺13" ], "genre": "解答题", "ans": "(1) $\\dfrac 14$; (2) $\\dfrac{6\\sqrt{10}}5$", @@ -337516,7 +340375,8 @@ "content": "已知椭圆$\\Gamma:\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的左、右焦点分别为$F_1$、$F_2$.\\\\\n(1) 以$F_2$为圆心的圆经过椭圆的左焦点$F_1$和上顶点$B$, 求椭圆$\\Gamma$的离心率;\\\\\n(2) 已知$a=5$, $b=4$ 设点$P$是椭圆$\\Gamma$上一点, 且位于$x$轴的上方, 若$\\triangle PF_1F_2$是等腰三角形, 求点$P$的坐标;\\\\\n(3) 已知$a=2$, $b=\\sqrt{3}$, 过点$F_2$且倾斜角为$\\dfrac\\pi 2$的直线与椭圆$\\Gamma$在$x$轴上方的交点记作$A$, 若动直线$l$也过点$F_2$且与椭圆$\\Gamma$交于$M$、$N$两点(均不同于$A$), 是否存在定直线$l_0: x=x_0$, 使得动直线$l$与$l_0$的交点$C$满足直线$AM$、$AC$、$AN$的斜率总是成等差数列? 若存在, 求常数$x_0$的值; 若不存在, 请说明理由.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期周末卷-高三下学期周末卷02" ], "genre": "解答题", "ans": "(1) $\\dfrac 12$; (2) $(0,4)$, $(-\\dfrac 53, \\dfrac{8\\sqrt{2}}3)$或$(\\dfrac 53, \\dfrac{8\\sqrt{2}}3)$; (3) 存在, $x_0=4$", @@ -337548,7 +340408,8 @@ "content": "若函数$y=f(x)$是其定义域内的区间$I$上的严格增函数, 而$y=\\dfrac{f(x)}{x}$是$I$上的严格减函数, 则称$y=f(x)$是$I$上的``弱增函数''. 若数列$\\{a_n\\}$是严格增数列, 而$\\{\\dfrac{a_n}{n}\\}$是严格减数列, 则称$\\{a_n\\}$是``弱增数列''.\\\\\n(1) 判断函数$y=\\ln x$是否为$(\\mathrm{e},+\\infty)$上的``弱增函数'', 并说明理由(其中$\\mathrm{e}$是自然对数的底数);\\\\\n(2) 已知函数$y=f(x)$与函数$y=-2 x^2-4 x-8$的图像关于坐标原点对称, 若$y=f(x)$是$[m, n]$上的``弱增函数\", 求$n-m$的最大值;\\\\\n(3) 已知等差数列$\\{a_n\\}$是首项为$4$的``弱增数列'', 且公差$d$是偶数. 记$\\{a_n\\}$的前$n$项和为$S_n$, 设$T_n=\\dfrac{S_n+2 \\lambda}{2^n}$($n$是正整数, 常数$\\lambda \\geq -2$), 若存在正整数$k$和$m$, 使得$k>m>1$且$T_k=T_m$, 求$\\lambda$所有可能的值.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期周末卷-高三下学期周末卷02" ], "genre": "解答题", "ans": "(1) 是``弱增函数'', 理由略; (2) $1$; (3) $-1$和$-2$", @@ -338044,7 +340905,8 @@ "content": "函数$y=\\tan (3 x-\\dfrac{\\pi}{4})$的定义域是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-简单题冲刺-简单题冲刺14" ], "genre": "填空题", "ans": "$\\{x|x\\in \\mathbf{R}, \\ x\\ne \\dfrac{k\\pi}3+\\dfrac\\pi 4, \\ k \\in \\mathbf{Z}\\}$", @@ -338066,7 +340928,8 @@ "content": "已知复数$z=\\dfrac{-1+2 a \\mathrm{i}}{a-\\mathrm{i}}$($\\mathrm{i}$为虚数单位) 在复平面内对应的点位于第二象限, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-简单题冲刺-简单题冲刺14" ], "genre": "填空题", "ans": "$(\\dfrac{\\sqrt{2}}2,+\\infty)$", @@ -338088,7 +340951,8 @@ "content": "若直线$x+2 y+3=0$与直线$2 x+m y+10=0$平行, 则这两条直线间的距离是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-简单题冲刺-简单题冲刺14" ], "genre": "填空题", "ans": "$\\dfrac{2\\sqrt{5}}{5}$", @@ -338110,7 +340974,8 @@ "content": "16-17 岁未成年人的体重的主要百分位数表(单位: $\\text{kg}$)\n\\begin{center}\n\\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|}\n\\hline & P1 & P5 & P10 & P25 & P50 & P75 & P90 & P95 & P99 \\\\\n\\hline 男 &$40.1$&$45.1$&$47.9$&$51.5$&$56.7$&$63.7$&$72.4$&$80.4$&$95.5$\\\\\n\\hline 女 &$38.3$&$41.2$&$43.1$&$46.5$&$50.5$&$55.3$&$61.1$&$65.4$&$75.6$\\\\\n\\hline\n\\end{tabular}\\\\\n表中数据来源: 《中国未成年人人体尺寸》(标准号: GB/T 26158-2010)\n\\end{center}\n小王同学今年 17 岁, 她的体重$50 \\mathrm{~kg}$, 她所在城市女性同龄人约有$4.2$万人. 根据表中数据估计小王同学所在的城市有\\blank{50}万女性同龄人的体重高于她的体重. (单位: 万人, 结果保留一位小数)", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-简单题冲刺-简单题冲刺14" ], "genre": "填空题", "ans": "$2.1$", @@ -338132,7 +340997,9 @@ "content": "已知函数$f(x)=\\mathrm{e}^x \\cos 2 x-\\mathrm{e}^2$, 则$f(x)$的导数$f'(x)=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-16_导数及其应用", + "2023届高三-简单题冲刺-简单题冲刺14" ], "genre": "填空题", "ans": "$\\mathrm{e}^x\\cos 2x-2\\mathrm{e}^x\\sin 2x$", @@ -338167,7 +341034,8 @@ "content": "现有$5$根细木棍, 长度分别为$1$、$3$、$5$、$7$、$9$(单位: $\\text{cm}$), 从中任取$3$根, 能搭成一个三角形的概率是\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-简单题冲刺-简单题冲刺14" ], "genre": "填空题", "ans": "$0.3$", @@ -338189,7 +341057,8 @@ "content": "有一种空心钢球, 质量为$140.2 \\text{g}$, 测得球的外直径等于$5.0 \\text{cm}$, 若球壁厚度均匀, 则它的内直径为\\blank{50}$\\text{cm}$. (钢的密度是$7.9 \\text{g} / \\text{cm}^3$, 结果保留一位小数)", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-简单题冲刺-简单题冲刺14" ], "genre": "填空题", "ans": "$4.5$", @@ -338211,7 +341080,8 @@ "content": "$\\overline {A}$、$\\overline {B}$分别是事件$A$、$B$的对立事件, 如果$A$、$B$两个事件独立, 那么以下四个概率等式\n一定成立的是\\blank{50}.(填写所有成立的等式序号)\n\\textcircled{1} $P(\\overline {A} \\cup B)=P(A)+P(B)$; \\textcircled {2} $P(\\overline {A} \\cap B)=P(\\overline {A}) P(B)$;\n\\textcircled{3} $P(\\overline {A} \\cap \\overline {B})=[1-P(A)][1-P(B)]$;\n\\textcircled{4} $P(\\overline {A} \\cup \\overline {B})=P(\\overline {A})+P(\\overline {B})$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-简单题冲刺-简单题冲刺14" ], "genre": "填空题", "ans": "\\textcircled{2}\\textcircled{3}", @@ -338233,7 +341103,8 @@ "content": "2022 年 11 月 27 日上午 7 点, 时隔两年再度回归的上海马拉松赛在外滩金牛广场鸣枪开跑, 途径黄浦、静安和徐汇三区. 数千名志愿者为$1.8$万名跑者提供了良好的志愿服务. 现将$5$名志愿者分配到防疫组、检录组、起点管理组、路线垃圾回收组$4$个组, 每组至少分配$1$名志愿者, 则不同的分配方法共有\\blank{50}种.(结果用数值表示)", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-中档题冲刺-中档题冲刺14" ], "genre": "填空题", "ans": "$240$", @@ -338255,7 +341126,8 @@ "content": "已知全集为实数集$\\mathbf{R}$, 集合$M=\\{x | \\dfrac{1}{16} \\leq 2^{2 x} \\leq 256\\}, N=\\{x | \\log _5(x^2-4 x)>1\\}$, 则$\\overline {M} \\cap N=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-中档题冲刺-中档题冲刺14" ], "genre": "填空题", "ans": "$(-\\infty,-2)\\cup (5,+\\infty)$", @@ -338277,7 +341149,8 @@ "content": "在空间直角坐标系$O-x y z$中, 点$P(7,4,6)$关于坐标平面$x O y$的对称点$P'$在第\\blank{50}卦限; 若点$Q$的坐标为$(8,-1,5)$, 则向量$\\overrightarrow{P Q}$与向量$\\overrightarrow{P P'}$夹角的余弦值是\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-中档题冲刺-中档题冲刺14" ], "genre": "填空题", "ans": "五; $\\dfrac{\\sqrt{3}}9$", @@ -338365,7 +341238,8 @@ "content": "在$(3 x+x^{-\\frac{2}{3}})^n$的二项展开式中, $\\mathrm{C}_n^r 3^{n-r} x^{n-\\frac{5 r}{3}}$称为二项展开式的第$r+1$项, 其中$r=0,1,2,3,\\cdots,n$. 下列关于$(3x+x^{-\\frac 23})^n$的命题中, 不正确的一项是\\bracket{20}.\n\\onech{若$n=8$, 则二项展开式中系数最大的项是$\\mathrm{C}_8^2 3^6 x^{\\frac{14}{8}}$}{已知$x>0$, 若$n=9$, 则二项展开式中第$2$项不大于第$3$项的实数$x$的取值范围是$0=latex]\n\\draw (0,0) node [left] {$A$} coordinate (A) -- (4,0) node [right] {$B$} coordinate (B) -- (4,2) node [right] {$C$} coordinate (C) -- (0,2) node [left] {$D$} coordinate (D) -- cycle;\n\\draw ($(C)!0.5!(D)$) node [above] {$E$} coordinate (E) -- (A);\n\\draw ($(A)!0.5!(E)$) node [below] {$O$} coordinate (O) -- (D);\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$A$} coordinate (A) -- (4,0,0) node [right] {$B$} coordinate (B) -- (4,0,-2) node [right] {$C$} coordinate (C);\n\\draw (2,0,-2) node [below] {$E$} coordinate (E);\n\\draw ($(A)!0.5!(E)$) node [below] {$O$} coordinate (O);\n\\draw (O) ++ (0,{sqrt(2)},0) node [above] {$P$} coordinate (P);\n\\draw (P) -- (A) (P) -- (B) (P) -- (C);\n\\draw [dashed] (P) -- (O) (A) --(E) -- (C) (P) -- (E);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证:$PO \\perp$平面$ABCE$;\\\\\n(2) 求直线$AC$与平面$PAB$所成角$\\theta$的正弦值.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-中档题冲刺-中档题冲刺14" ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\dfrac{\\sqrt{30}}{15}$", @@ -338611,7 +341488,8 @@ "objs": [], "tags": [ "第五单元", - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-01_集合与逻辑" ], "genre": "填空题", "ans": "$-1$", @@ -338706,7 +341584,8 @@ "content": "已知集合$A=\\{(x, y) | y=x^2,\\ x \\in \\mathbf{R}\\}$, $B=\\{(x, y) | y-1=2^{2018} \\cdot(x-1),\\ x \\in \\mathbf{R}\\}$. 则$A \\cap B$的元素个数为\\bracket{20}.\n\\fourch{$0$}{$1$}{$2$}{无限}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-01_集合与逻辑" ], "genre": "选择题", "ans": "C", @@ -338734,7 +341613,8 @@ "content": "设$U$是全集, 集合$A$, $B$满足$A\\subset B$, 给出下列四个命题: \\textcircled{1} $A \\cap \\overline B=\\varnothing$; \\textcircled{2} $B \\cap \\overline A=\\overline A$; \\textcircled{3} $B \\cup \\overline A=U$; \\textcircled{4} $\\overline A \\cap \\overline B=\\overline B$. 四个命题中, 正确命题的序号是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-01_集合与逻辑" ], "genre": "填空题", "ans": "\\textcircled{1}\\textcircled{3}\\textcircled{4}", @@ -338762,7 +341642,8 @@ "content": "设$U$是全集, $A, B$是两个集合, 则``存在集合$C$使得$A \\subseteq C, B \\subseteq \\overline C$''是``$A \\cap B=\\varnothing$''的 \\bracket{20}\n\\twoch{充分不必要条件}{必要不充分条件}{充分必要条件}{既非充分又非必要条件}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-01_集合与逻辑" ], "genre": "选择题", "ans": "C", @@ -338796,7 +341677,8 @@ "objs": [], "tags": [ "第二单元", - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-01_集合与逻辑" ], "genre": "选择题", "ans": "A", @@ -338829,7 +341711,8 @@ "content": "已知集合$A=\\{x, 1\\}$, $B=\\{y, 1,2\\}$, 其中$x, y \\in\\{1,2,3,4,5\\}$, 且$A \\subseteq B$. 如果把满足上述条件的一对有序整数$(x, y)$作为一个点, 则这样的点的个数为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-01_集合与逻辑" ], "genre": "填空题", "ans": "$6$", @@ -338857,7 +341740,8 @@ "content": "设常数$a \\in \\mathbf{R}$, 集合$A=\\{x|| x-1 |<2,\\ x \\in \\mathbf{Z}\\}, B=\\{x | x \\geq a\\}$. 若$A \\cap B=\\{1,2\\}$, 则$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-01_集合与逻辑" ], "genre": "填空题", "ans": "$(0,1]$", @@ -338913,7 +341797,8 @@ "objs": [], "tags": [ "第一单元", - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-01_集合与逻辑" ], "genre": "解答题", "ans": "(1) 证明略; (2) 对任意非零实数$T$, 存在$x\\in \\mathbf{R}$, 使得$f(x+T)\\ne f(x)$", @@ -338964,7 +341849,8 @@ "content": "设常数$a \\in \\mathbf{R}$, 集合$A=\\{x | \\dfrac{6}{x+1} \\geq 1,\\ x \\in \\mathbf{R}\\}$, $B=\\{x | x^2-3 a x+2 a^2<0\\}$. 若$A \\cap B=B$, 求$a$的取值范围.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-01_集合与逻辑" ], "genre": "解答题", "ans": "$[-\\dfrac 12,\\dfrac 52]$", @@ -338992,7 +341878,8 @@ "content": "设常数$m \\in \\mathbf{R}$, 集合$A=\\{x | \\dfrac{6}{x+1} \\geq 1,\\ x \\in \\mathbf{R}\\}$, $B=\\{x | x^2-2 x+2 m<0\\}$. 若$A \\cup B=A$, 求$m$的取值范围.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-01_集合与逻辑" ], "genre": "解答题", "ans": "$[-\\dfrac 32,+\\infty)$", @@ -339047,7 +341934,8 @@ "content": "不等式$\\dfrac{5-x}{2 x-4} \\leq 1$的解集为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-02_等式与不等式" ], "genre": "填空题", "ans": "$(-\\infty,2)\\cup [3,+\\infty)$", @@ -339072,7 +341960,8 @@ "content": "不等式$|x-1|b c$}{$a d$与$b c$的大小不确定}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-02_等式与不等式" ], "genre": "选择题", "ans": "C", @@ -339422,7 +342314,8 @@ "content": "设常数$a, b \\in \\mathbf{R}$. 若$x=1$是不等式组$\\begin{cases}xb\\end{cases}$的唯一整数解, 则$a, b$满足的条件为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-02_等式与不等式" ], "genre": "填空题", "ans": "$0\\le b<1f(a)$, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-04_函数的概念与性质" ], "genre": "填空题", "ans": "$(-2,1)$", @@ -339741,7 +342639,8 @@ "content": "定义域和值域均为$[-a, a]$(常数$a>0$) 的函数$y=f(x)$和$y=g(x)$的图像如图所示, 给出下列四个命题: \\textcircled{1} 方程$f(g(x))=0$有且仅有三个解; \\textcircled{2} 方程$g(f(x))=0$有且仅有三个解; \\textcircled{3} 方程$f(f(x))=0$有且仅有九个解; \\textcircled{4} 方程$g(g(x))=0$有且仅有一个解. 那么, 其中正确命题的序号是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-2.5,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (-2,-2) rectangle (2,2);\n\\draw (-2,-2) .. controls +(75:1) and +(180:0.6) .. (-1,0.6) .. controls +(0:0.6) and +(225:1.5) .. (1,0) .. controls +(45:0.5) and +(255:1) .. (2,2);\n\\filldraw (1,0) circle (0.03);\n\\draw (1,0) node [below] {\\tiny $\\dfrac a2$};\n\\draw (-1,1) node [above] {\\small $y=f(x)$};\n\\draw (-2,0) node [below left] {$-a$};\n\\draw (2,0) node [below right] {$a$};\n\\draw (0,2) node [above right] {$a$};\n\\draw (0,-2) node [below right] {$-a$};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-2.5,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (-2,-2) rectangle (2,2);\n\\draw (-2,2) .. controls +(-45:1) and +(165:0.6) .. (0,0.5) .. controls +(-15:0.6) and +(135:0.5) .. (1,0) .. controls +(-45:0.5) and +(105:1) .. (2,-2);\n\\filldraw (1,0) circle (0.03);\n\\draw (1,0) node [below] {\\tiny $\\dfrac a2$};\n\\draw (1,1) node [above] {\\small $y=g(x)$};\n\\draw (-2,0) node [below left] {$-a$};\n\\draw (2,0) node [below right] {$a$};\n\\draw (0,2) node [above right] {$a$};\n\\draw (0,-2) node [below right] {$-a$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-04_函数的概念与性质" ], "genre": "填空题", "ans": "\\textcircled{1}\\textcircled{4}", @@ -339776,7 +342675,8 @@ "content": "设函数$f(x)=\\dfrac{a^x}{1+a^x}$($a>0$, $a \\neq 1$), $[m]$表示不超过实数$m$的最大整数, 则函数$g(x)=[f(x)-\\dfrac{1}{2}]+[f(-x)-\\dfrac{1}{2}]$的值域为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-04_函数的概念与性质" ], "genre": "填空题", "ans": "$\\{-1,0\\}$", @@ -339877,7 +342777,8 @@ "content": "函数$f(x)=\\sqrt{x+1}+\\dfrac{1}{2-x}$的定义域为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-04_函数的概念与性质" ], "genre": "填空题", "ans": "$[-1,2)\\cup (2,+\\infty)$", @@ -339913,7 +342814,8 @@ "objs": [], "tags": [ "第二单元", - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-04_函数的概念与性质" ], "genre": "填空题", "ans": "$[-1,10)$", @@ -339948,7 +342850,8 @@ "content": "``$a=1$''是``函数$f(x)=|x-a|$在区间$[1,+\\infty)$上为严格增函数''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充分必要条件}{既非充分又非必要条件}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-04_函数的概念与性质" ], "genre": "选择题", "ans": "A", @@ -339983,7 +342886,8 @@ "content": "若$x>1$, 则函数$y=\\dfrac{x^2-x+1}{x-1}$的最小值为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-04_函数的概念与性质" ], "genre": "填空题", "ans": "$3$", @@ -340040,7 +342944,8 @@ "content": "若函数$f(x)=\\log _a(2-a x)$在$[0,1]$上单调递减, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-04_函数的概念与性质" ], "genre": "填空题", "ans": "$(1,2)$", @@ -340097,7 +343002,8 @@ "content": "已知$f(x)=4^x-k \\cdot 2^x+1$, 当$x \\in \\mathbf{R}$时, $f(x)$恒为正值, 则$k$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-04_函数的概念与性质" ], "genre": "填空题", "ans": "$(-\\infty,2)$", @@ -340132,7 +343038,10 @@ "content": "已知定义在$\\mathbf{R}$上的奇函数$f(x)$, 满足$f(x-4)=-f(x)$, 且在区间$[0,2]$上是增函数, 若方程$f(x)=m$($m>0$)在区间$[-8,8]$上有四个不同的根$x_1, x_2, x_3, x_4$, 则$x_1+x_2+x_3+x_4=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-四月错题重做-02_函数二", + "2023届高三-四月错题重做-02_易错题-函数2", + "2023届高三-第二轮复习讲义-04_函数的概念与性质" ], "genre": "填空题", "ans": "$-8$", @@ -340179,7 +343088,8 @@ "content": "设函数$f(x)=\\dfrac{2 x}{1+|x|}$($x \\in \\mathbf{R}$), 区间$M=[a, b]$($a0$, $a \\neq 1$)的图像如图所示, 则$a, b$满足的关系是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-1.5,0) -- (2.5,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -0.95:2.5,samples = 100] plot (\\x,{ln(pow(2,\\x)-0.5)/ln(5)});\n\\draw (-0.2,-1) -- (0,-1) node [right] {$-1$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$00$且$a \\neq 1)$在$\\mathbf{R}$上既是奇函数, 又是减函数, 则$g(x)=\\log _a(x+k)$的大致图像是\\bracket{20}.\n\\fourch{\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-2.5,0) -- (2.5,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (-2,-2.5) -- (-2,2.5);\n\\draw (-2,0) node [fill = white, below] {$-2$};\n\\draw [domain = -2.5:2.5, samples = 100] plot ({pow(1.8,\\x)-2},-\\x);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-2.5,0) -- (2.5,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (-2,-2.5) -- (-2,2.5);\n\\draw (-2,0) node [fill = white, below] {$-2$};\n\\draw [domain = -2.5:2.5, samples = 100] plot ({pow(1.8,\\x)-2},\\x);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-0.5,0) -- (4.5,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (2,-2.5) -- (2,2.5);\n\\draw (2,0) node [fill = white, below] {$2$};\n\\draw [domain = -2.5:2.5, samples = 100] plot ({pow(1.4,\\x)+2},-\\x);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-0.5,0) -- (4.5,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (2,-2.5) -- (2,2.5);\n\\draw (2,0) node [fill = white, below] {$2$};\n\\draw [domain = -2.5:2.5, samples = 100] plot ({pow(1.4,\\x)+2},\\x);\n\\end{tikzpicture}}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-04_函数的概念与性质" ], "genre": "选择题", "ans": "A", @@ -340422,7 +343336,8 @@ "content": "已知$f(x)=\\begin{cases}(2-a) x+1, & x<1, \\\\ a^x, & x \\geq 1\\end{cases}$满足: 对任意$x_1 \\neq x_2$, 都有$\\dfrac{f(x_1)-f(x_2)}{x_1-x_2}>0$成立, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-04_函数的概念与性质" ], "genre": "填空题", "ans": "$[\\dfrac 32,2)$", @@ -340525,7 +343440,8 @@ "content": "设常数$a \\in \\mathbf{R}$, 已知函数$f(x)=x^2+\\dfrac{a}{x}$.\\\\\n(1) 判断函数$f(x)$的奇偶性;\\\\\n(2) 若$f(x)$在区间$[2,+\\infty)$上是严格增函数, 求实数$a$的取值范围.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-04_函数的概念与性质" ], "genre": "解答题", "ans": "(1) 当$a=0$时, $y=f(x)$是偶函数; 当$a\\ne 0$时, $y=f(x)$既不是奇函数, 又不是偶函数; (2) $(-\\infty,16]$", @@ -340561,7 +343477,10 @@ "objs": [], "tags": [ "第一单元", - "第二单元" + "第二单元", + "2023届高三-四月错题重做-02_函数二", + "2023届高三-四月错题重做-02_易错题-函数2", + "2023届高三-第二轮复习讲义-04_函数的概念与性质" ], "genre": "解答题", "ans": "$(-\\infty,-2]\\cup [2,+\\infty)$", @@ -340608,7 +343527,8 @@ "content": "函数$f(x)=x^2+\\dfrac{4}{x^2+3}$的最小值为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-04_函数的概念与性质" ], "genre": "填空题", "ans": "$\\dfrac 43$", @@ -340643,7 +343563,8 @@ "content": "若$f(x+2)=\\begin{cases}\\tan x, & x \\geq 0, \\\\ \\log _2(-x), & x<0,\\end{cases}$ 则$f(\\dfrac{\\pi}{4}+2) \\cdot f(-2)=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-04_函数的概念与性质" ], "genre": "填空题", "ans": "$2$", @@ -340678,7 +343599,10 @@ "content": "已知函数$y=f(x)$对任意实数$x$都有$f(-x)=f(x)$, $f(x)=-f(x+1)$, 且在$[0,1]$上是严格减函数, 则$f(\\dfrac{7}{2})$、$f(\\dfrac{7}{3})$、$f(\\dfrac{7}{5})$的大小顺序是\\blank{50}.(用``$>$''连接)", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-四月错题重做-01_函数一", + "2023届高三-四月错题重做-01_易错题-函数1", + "2023届高三-第二轮复习讲义-04_函数的概念与性质" ], "genre": "填空题", "ans": "$f(\\dfrac 73)>f(\\dfrac 72)>f(\\dfrac 75)$", @@ -340945,7 +343869,8 @@ "content": "函数$y=(x-1)^{\\frac{3}{5}}$的图像不经过第\\blank{50}象限.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-05_函数综合" ], "genre": "填空题", "ans": "二", @@ -340980,7 +343905,8 @@ "content": "若函数$y=x^2+(a+2) x+3$, $x \\in[a, b]$的图像关于直线$x=1$对称, 则$b=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-05_函数综合" ], "genre": "填空题", "ans": "$6$", @@ -341015,7 +343941,8 @@ "content": "若偶函数$f(x)$满足: 当$x>0$时, $f(x)$为严格减函数, 且$f(\\pi)=0$, 则$\\dfrac{f(x)}{x}<0$的解集是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-05_函数综合" ], "genre": "填空题", "ans": "$(-\\pi,0)\\cup (\\pi,+\\infty)$", @@ -341072,7 +343999,8 @@ "content": "已知关于$x$的方程$9^x+(a+4) \\cdot 3^x+4=0$有实数解, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-05_函数综合" ], "genre": "填空题", "ans": "$(-\\infty,-8]$", @@ -341107,7 +344035,8 @@ "content": "已知函数$f(x)$是定义在实数集$\\mathbf{R}$上的不恒为零的偶函数, 且对任意实数$x$都有$x f(x+1)=(1+x) f(x)$, 则$f(f(\\dfrac{5}{2}))$的值是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-05_函数综合" ], "genre": "填空题", "ans": "$0$", @@ -341252,7 +344181,8 @@ "content": "奇函数$y=f(x)$, 当$x<0$时, $f(x)=x+\\lg |x|$, 则$f(10)=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-05_函数综合" ], "genre": "填空题", "ans": "$9$", @@ -341287,7 +344217,8 @@ "content": "关于$x$的方程$\\lg ^2 x+\\lg x^2=0$的解是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-05_函数综合" ], "genre": "填空题", "ans": "$\\dfrac 1{100}$或$1$", @@ -341322,7 +344253,8 @@ "content": "函数$y=\\log _{\\frac{1}{3}}(1-x^2)$的单调增区间为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-05_函数综合" ], "genre": "填空题", "ans": "$[0,1)$", @@ -341357,7 +344289,8 @@ "content": "已知$x_0$是函数$f(x)=2^x+\\dfrac{1}{1-x}$的一个零点. 若$x_1 \\in(1, x_0)$, $x_2 \\in(x_0,+\\infty)$, 则\\bracket{20}.\n\\fourch{$f(x_1)<0$, $f(x_2)<0$}{$f(x_1)<0$, $f(x_2)>0$}{$f(x_1)>0$, $f(x_2)<0$}{$f(x_1)>0$, $f(x_2)>0$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-05_函数综合" ], "genre": "选择题", "ans": "B", @@ -341392,7 +344325,8 @@ "content": "函数$f(x)$的定义域关于原点对称, 对定义域中任一$x$值, 恒有$|f(x)|=|f(-x)|$成立, 则\\bracket{20}.\n\\twoch{$f(x)$是奇函数}{$f(x)$是偶函数}{$f(x)$不可能既非奇函数也非偶函数}{$f(x)$有可能既非奇函数也非偶函数}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-05_函数综合" ], "genre": "选择题", "ans": "D", @@ -341449,7 +344383,8 @@ "content": "设函数$f(x)$在$(-\\infty,+\\infty)$内有定义, 给出下列四个函数: \\textcircled{1} $y=-|f(x)|$; \\textcircled{2} $y=x f(x^2)$; \\textcircled{3} $y=-f(-x)$; \\textcircled{4} $y=f(x+1)-f(1-x)$. 其中必为奇函数的是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-05_函数综合" ], "genre": "填空题", "ans": "\\textcircled{2}\\textcircled{4}", @@ -341551,7 +344486,8 @@ "objs": [], "tags": [ "第二单元", - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-03_幂指对函数" ], "genre": "解答题", "ans": "(1) 当$a>0$时, $y=f(x)$为严格增函数; 当$a<0$时, $y=f(x)$为严格减函数; (2) 当$a>0$时, 取值范围为$(-\\infty,\\log_{\\frac 32}(-\\dfrac a{2b}))$; 当$a<0$时, 取值范围为$(\\log_{\\frac 32}(-\\dfrac a{2b}),+\\infty)$", @@ -341586,7 +344522,8 @@ "content": "已知函数$f(x)=\\log _4(4^x+1)$, $g(x)=(k-1) x$, 记$F(x)=f(x)-g(x)$, 并且$F(x)$为偶函数.\\\\\n(1) 求常数$k$的值;\\\\\n(2) 若对一切$a \\in \\mathbf{R}$, 不等式$F(a)>-\\dfrac{1}{2} m$恒成立, 求实数$m$的取值范围;\\\\\n(3) 设$M(x)=\\log _4(a \\cdot 2^x-\\dfrac{4}{3} a)$, 若函数$F(x)$与$M(x)$的图像有且只有一个公共点, 求实数$a$的取值范围.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-03_幂指对函数" ], "genre": "解答题", "ans": "(1) $\\dfrac 32$; (2) $(-1,+\\infty)$; (3) $\\{-3\\}\\cup (1,+\\infty)$", @@ -341621,7 +344558,8 @@ "content": "函数$y=\\log _{0.7}(x^2-3 x+2)$的单调减区间为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-05_函数综合" ], "genre": "填空题", "ans": "$(2,+\\infty)$", @@ -341659,7 +344597,8 @@ "content": "若$\\alpha \\in\\{-1,-3, \\dfrac{1}{3}, 2\\}$, 则使函数$y=x^\\alpha$的定义域为$\\mathbf{R}$且在$(-\\infty, 0)$上单调递增的$\\alpha$值为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-05_函数综合" ], "genre": "填空题", "ans": "$\\dfrac 13$", @@ -341694,7 +344633,8 @@ "content": "函数$y=\\dfrac{1}{x^2-4 x+5}$的图像关于\\bracket{20}.\n\\fourch{$y$轴对称}{原点对称}{直线$x=2$对称}{点$(2,1)$对称}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-05_函数综合" ], "genre": "选择题", "ans": "C", @@ -341732,7 +344672,8 @@ "content": "定义运算: $a \\otimes b=\\begin{cases}b, & a \\geq b, \\\\ a, & a0$.\\\\\n(1) 当$01$;\\\\\n(2) 是否存在实数$a$、$b$($a0$, 那么$f(x)$在$(1,+\\infty)$上是\\bracket{20}.\n\\fourch{增函数}{减函数}{非单调函数}{由$a$值决定单调性}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-05_函数综合" ], "genre": "选择题", "ans": "B", @@ -342128,7 +345080,8 @@ "content": "已知定义在$\\mathbf{R}$上的函数$f(x)$满足$f(-2-x)=f(-2+x)$, 且当$x \\geq -2$时, 函数的解析式为$f(x)=x^2-1$, 则当$x<-2$时, 函数的解析式$f(x)=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-05_函数综合" ], "genre": "填空题", "ans": "$x^2+8x+15$", @@ -342229,7 +345182,8 @@ "content": "已知函数$f(x)$满足: $f(1)=\\dfrac{1}{4}$, $4 f(x) f(y)=f(x+y)+f(x-y)$($x, y \\in \\mathbf{R}$), 则$f(2022)=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-05_函数综合" ], "genre": "填空题", "ans": "$\\dfrac 12$", @@ -342264,7 +345218,8 @@ "content": "设函数$f(x)=\\sqrt{a x^2+b x+c}$($a<0$)的定义域为$D$, 若所有点$(s, f(t))$($s, t \\in D$)构成一个正方形区域, 则$a$的值为\\bracket{20}.\n\\fourch{$-2$}{$-4$}{$-8$}{不能确定}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-05_函数综合" ], "genre": "选择题", "ans": "B", @@ -342322,7 +345277,8 @@ "objs": [], "tags": [ "第二单元", - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-03_幂指对函数" ], "genre": "解答题", "ans": "(1) $a\\in (0,9)\\cup (9,+\\infty)$, $b=3$; (2) $4\\sqrt{2}$", @@ -342445,7 +345401,8 @@ "content": "设$\\{a_n\\}$为等差数列, 公差$d=\\dfrac{1}{2}$, 前$100$项之和为$145$, 则$a_1+a_3+\\cdots+a_{99}=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-14_等差数列和等比数列" ], "genre": "填空题", "ans": "$60$", @@ -342524,7 +345481,8 @@ "content": "数列$\\{a_n\\}$的通项公式$a_n=n \\cos \\dfrac{n \\pi}{2}+1$, 前$n$项和为$S_n$, 则$S_{2012}=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-15_数列综合" ], "genre": "填空题", "ans": "$3018$", @@ -342621,7 +345579,8 @@ "content": "$2$与$6$的等差中项为\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-14_等差数列和等比数列" ], "genre": "填空题", "ans": "$4$", @@ -342656,7 +345615,8 @@ "content": "$2$与$6$的等比中项为\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-14_等差数列和等比数列" ], "genre": "填空题", "ans": "$\\pm 2\\sqrt{3}$", @@ -342759,7 +345719,8 @@ "content": "数列$\\{a_n\\}$中, $a_1=1$, $a_n$, $a_{n+1}$是方程$x^2-(2 n+1) x+\\dfrac{1}{b_n}=0$的两个根, 数列$\\{b_n\\}$的前$n$项和$S_n=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-15_数列综合" ], "genre": "填空题", "ans": "$\\dfrac{n}{n+1}$", @@ -342790,7 +345751,9 @@ "content": "已知数列$\\{a_n\\}$满足$a_{n+2}-a_n=2^n$, $a_1=1$, 则$a_{2 n-1}=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-四月错题重做-03_数列", + "2023届高三-第二轮复习讲义-14_等差数列和等比数列" ], "genre": "填空题", "ans": "$\\dfrac{2^{2n-1}+1}3$", @@ -342871,7 +345834,8 @@ "content": "数列$\\{a_n\\}$满足$a_{n+1}+(-1)^n a_n=2 n-1$, 则$\\{a_n\\}$的前$60$项和为\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-15_数列综合" ], "genre": "填空题", "ans": "$1830$", @@ -342926,7 +345890,8 @@ "content": "已知数列$\\{a_n\\}$的通项$a_n=2^n$, 删去该数列的第$2,5, \\cdots, 3 k-1, \\cdots$项, 得到一个新数列$\\{b_n\\}$.\\\\\n(1) 求$\\{b_n\\}$的通项;\\\\\n(2) 求$\\{b_n\\}$的前$n$项和$S_n$.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-15_数列综合" ], "genre": "解答题", "ans": "(1) $b_n=\\begin{cases} 2^{\\frac 32 n}, & n=2k, \\\\ 2^{\\frac{3n-1}2} & n=2k-1\\end{cases}$($k$为正整数); (2) $S_n=\\begin{cases} \\dfrac{10}{7}\\cdot 2^{\\frac 32 n}-\\dfrac{10}{7}, & n=2k, \\\\ \\dfrac{3}{7}\\cdot 2^{\\frac{3n+3}2}-\\dfrac{10}{7} & n=2k-1\\end{cases}$($k$为正整数)", @@ -343003,7 +345968,8 @@ "content": "已知数列$\\{a_n\\}$满足$\\dfrac{1}{a_{n+1}}-\\dfrac{1}{a_n}=1$, $a_1=2$, 则$a_{10}=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-15_数列综合" ], "genre": "填空题", "ans": "$\\dfrac{2}{19}$", @@ -343034,7 +346000,8 @@ "content": "已知数列$\\{a_n\\}$的前$n$项和$S_n=2^n$, 则通项$a_n=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-15_数列综合" ], "genre": "填空题", "ans": "$\\begin{cases} 2, & n=1, \\\\ 2^{n-1} & n\\ge 2\\end{cases}$", @@ -343157,7 +346124,8 @@ "content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n$, $a_1=2$且$3 a_{n+1}+2S_n=3$, 求数列$\\{a_n\\}$的通项公式.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-15_数列综合" ], "genre": "解答题", "ans": "$\\begin{cases} 2, & n=1, \\\\ -\\dfrac{1}{3^{n-1}} & n\\ge 2\\end{cases}$", @@ -343254,7 +346222,8 @@ "content": "已知$x$是实数, 数列$\\{a_n\\}$的通项为$a_n=\\log _x n$. 若$\\{a_n\\}$递增, 则$x$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-14_等差数列和等比数列" ], "genre": "填空题", "ans": "$(1,+\\infty)$", @@ -343311,7 +346280,8 @@ "content": "已知数列$\\{a_n\\}$满足$a_{n+1}=2 a_n+3$, $a_1=1$, 则$a_n=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-14_等差数列和等比数列" ], "genre": "填空题", "ans": "$2^{n+1}-3$", @@ -343368,7 +346338,8 @@ "content": "无穷等比数列$\\{a_n\\}$的各项和为 1 , 则首项$a_1$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-14_等差数列和等比数列" ], "genre": "填空题", "ans": "$(0,1)\\cup (1,2)$", @@ -343403,7 +346374,8 @@ "content": "若对任意的正整数$n, a_1 a_2 \\cdots a_n=2^{n+1}$, 则$a_n=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-15_数列综合" ], "genre": "填空题", "ans": "$\\begin{cases} 4, & n=1, \\\\ 2 & n\\ge 2\\end{cases}$", @@ -343480,7 +346452,8 @@ "content": "数列$\\{a_n\\}$的前$n-1$项之和$S_{n-1}=a_n$($n \\geq 2$, $n \\in \\mathbf{N}$), $a_1=1$, 则通项$a_n=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-15_数列综合" ], "genre": "填空题", "ans": "$\\begin{cases} 1, & n=1, \\\\ 2^{n-2} & n\\ge 2\\end{cases}$", @@ -343624,7 +346597,8 @@ "content": "已知等比数列$\\{a_n\\}$中$a_2=1$, 则其前 3 项的和$S_3$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-14_等差数列和等比数列" ], "genre": "填空题", "ans": "$(-\\infty,-1]\\cup [3,+\\infty)$", @@ -343659,7 +346633,8 @@ "content": "已知数列$\\{a_n\\}$的通项$a_n=\\dfrac{n-\\sqrt{60}}{n-\\sqrt{59}}$($1 \\leq n \\leq 100$), 则此数列中最大项为第\\blank{50}项, 最小项为第\\blank{50}项.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-15_数列综合" ], "genre": "填空题", "ans": "$8$", @@ -343690,7 +346665,8 @@ "content": "已知数列$\\{a_n\\}$是以$-2$为公差的等差数列, $S_n$是其前$n$项和, 若$S_7$是数列$\\{S_n\\}$中的唯一最大项, 则数列$\\{a_n\\}$的首项$a_1$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-14_等差数列和等比数列" ], "genre": "填空题", "ans": "$(12,14)$", @@ -343725,7 +346701,8 @@ "content": "等差数列$\\{a_n\\}$中, $S_n$为前$n$项和, 且$S_6S_8$, 给出下列命题: \\textcircled{1} 数列$\\{a_n\\}$中前$7$项是递增的, 从第$8$项开始递减; \\textcircled{2} $S_9$一定小于$S_6$; \\textcircled{3} $a_1$是各项中的最大的; \\textcircled{4} $S_7$不一定是$\\{S_n\\}$中最大项. 其中正确的序号是\\bracket{20}.\n\\fourch{\\textcircled{1}\\textcircled{3}}{\\textcircled{2}\\textcircled{3}}{\\textcircled{3}\\textcircled{4}}{\\textcircled{2}\\textcircled{4}}", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-14_等差数列和等比数列" ], "genre": "选择题", "ans": "B", @@ -343760,7 +346737,8 @@ "content": "设$\\{a_n\\}$是公比为实数$q$的等比数列, $|q|>1$, 令$b_n=a_n+1$($n=1,2, \\cdots$), 若数列$\\{b_n\\}$有连续四项在集合$\\{-53,-23,19,37,82\\}$中, 则$q=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-14_等差数列和等比数列" ], "genre": "填空题", "ans": "$-\\dfrac 32$", @@ -343817,7 +346795,8 @@ "content": "已知等差数列$\\{a_n\\}$通项$a_n=1000 n$, 等比数列$\\{b_n\\}$通项$b_n=2^n$.\\\\\n(1) 求数列$\\{a_n-b_n\\}$的最大项;\\\\\n(2) 解不等式: $a_n>b_n$.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-15_数列综合" ], "genre": "解答题", "ans": "(1) 最大项为$a_{10}-b_{10}=8976$; (2) $\\{1,2,3,\\cdots,13\\}$", @@ -343848,7 +346827,8 @@ "content": "已知等差数列$\\{a_n\\}$中$a_1=2$, 公差是正整数$d$, 等比数列$\\{b_n\\}$中, $b_1=a_1$, $b_2=a_2$.\\\\\n(1) 试给出一个$d$的值, 使得$n \\geq 3$时, $b_n$都不在$\\{a_n\\}$中, 并说明理由;\\\\\n(2) 判断$d=10$时, 是否数列$\\{b_n\\}$中的所有项都是$\\{a_n\\}$中的项, 并证明你的结论.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-14_等差数列和等比数列" ], "genre": "解答题", "ans": "(1) 如$d=1$(正奇数都可以); (2) 是的, 证明略", @@ -343949,7 +346929,8 @@ "content": "已知数列$\\{a_n\\}$满足$a_{n+1}=2^n a_n$, $a_1=1$, 则$a_n=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-15_数列综合" ], "genre": "填空题", "ans": "$2^{\\frac{n(n-1)}{2}}$", @@ -343980,7 +346961,8 @@ "content": "已知数列$\\{a_n\\}$的通项公式为$a_n=(\\dfrac{3}{4})^{n-1}[(\\dfrac{3}{4})^{n-1}-1]$, 则数列$\\{a_n\\}$的最大项的值为\\blank{50}, 最小项的值为\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-15_数列综合" ], "genre": "填空题", "ans": "$0$; $-\\dfrac{63}{256}$", @@ -344013,7 +346995,8 @@ "content": "数列$\\{a_n\\}$的通项公式为$a_n=6 n-3$, 数列$\\{b_n\\}$的通项公式为$b_n=5 n-4$, 若$a_n \\leq 1000$, $b_n \\leq 1000$, 由数列$\\{a_n\\}$与数列$\\{b_n\\}$中共有的项构成数列$\\{c_n\\}$, 则数列$\\{c_n\\}$中共有\\blank{50}项.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-15_数列综合" ], "genre": "填空题", "ans": "$33$", @@ -344044,7 +347027,9 @@ "content": "设等差数列$\\{a_n\\}$的前$n$项和为$S_n$, 若$S_4 \\geq 10$, $S_5 \\leq 15$, 则$a_4$的最大值为\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-四月错题重做-03_数列", + "2023届高三-第二轮复习讲义-14_等差数列和等比数列" ], "genre": "填空题", "ans": "$4$", @@ -344103,7 +347088,8 @@ "content": "在等差数列$\\{a_n\\}$中, $a_3+a_4+a_5=84$, $a_9=73$, 对任意$m \\in \\mathbf{N}$, $m\\ge 1$, 将数列$\\{a_n\\}$中落入区间$(9^m, 9^{2 m})$内的项的个数记为$b_m$, 则数列$b_m=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-14_等差数列和等比数列" ], "genre": "填空题", "ans": "$9^{2m-1}-9^{m-1}$", @@ -344138,7 +347124,8 @@ "content": "已知数列$\\{a_n\\}$满足$a_1=m$, $m \\in \\mathbf{N}$, $m\\ge 1$, $a_{n+1}=\\begin{cases}\\dfrac{a_n}{2}, & a_n \\text {是偶数}, \\\\ 3 a_n+1, & a_n \\text{是奇数}.\\end{cases}$ $a_6=1$, 则$m$的所有可能值为\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-15_数列综合" ], "genre": "填空题", "ans": "$4$或$5$或$32$", @@ -344171,7 +347158,8 @@ "content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n$, 且$a_2 a_n=S_2+S_n$对一切正整数$n$都成立.\\\\\n(1) 求$a_1$, $a_2$的值;\\\\\n(2) 设$a_1>0$, 数列$\\{\\lg \\dfrac{10 a_1}{a_n}\\}$的前$n$项和为$T_n$, 当$n$为何值时, $T_n$最大? 并求出$T_n$的最大值.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-15_数列综合" ], "genre": "解答题", "ans": "(1) $(a_1,a_2)=(0,0)$或$(1-\\sqrt{2},2-\\sqrt{2})$或$(1+\\sqrt{2},2+\\sqrt{2})$; (2) 当$n=7$时$T_n$最大, $T_7=7-\\dfrac{21}{2}\\lg 2$", @@ -344556,7 +347544,8 @@ "content": "若数列$\\{a_n\\}$满足$a_1=\\dfrac{1}{3}$, $a_n-a_{n-1}=\\dfrac{1}{n(n+2)}$, 则$a_n=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-15_数列综合" ], "genre": "填空题", "ans": "$\\dfrac 34-\\dfrac{2n+3}{2(n^2+3n+2)}$", @@ -344611,7 +347600,8 @@ "content": "若$f(\\dfrac{1}{2}+x)+f(\\dfrac{1}{2}-x)=2$对任意的实数$x$成立, 则$f(\\dfrac{1}{3000})+f(\\dfrac{2}{3000})+f(\\dfrac{3}{3000})+\\cdots+f(\\dfrac{2999}{3000})=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-15_数列综合" ], "genre": "填空题", "ans": "$2999$", @@ -344686,7 +347676,8 @@ "content": "设$a_1=2$, $a_{n+1}=\\dfrac{2}{a_n+1}$, $b_n=|\\dfrac{a_n+2}{a_n-1}|$, $n \\in \\mathbf{N}$, $n\\ge 1$, 则数列$\\{b_n\\}$的通项公式为\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-15_数列综合" ], "genre": "填空题", "ans": "$2^{n+1}$", @@ -344717,7 +347708,8 @@ "content": "设$S_n=1+\\dfrac{1}{2}+\\dfrac{1}{3}+\\cdots+\\dfrac{1}{n}$, $f(n)=S_{2 n+1}-S_{n+1}$.\\\\\n(1) 判断数列$f(n)$的单调性;\\\\\n(2) 试确定实数$t$的范围, 使得对于$n \\in \\mathbf{N}$, $n>1$, 不等式$f(n)>t^2-\\dfrac{11}{20} t^{-2}$恒成立.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-15_数列综合" ], "genre": "解答题", "ans": "(1) $\\{f(n)\\}$是严格增数列, 证明略; (2) $(-1,0)\\cup (0,1)$", @@ -344750,7 +347742,8 @@ "content": "已知数列$\\{a_n\\}$中, $a_1=\\dfrac{1}{2}$, $2 a_{n+1}-a_n=n$.\\\\\n(1) 令$b_n=a_{n+1}-a_n-1$, 求证:$\\{b_n\\}$是等比数列.\\\\\n(2) 求数列$\\{a_n\\}$的通项公式;\\\\\n(3) 是否存在实数$\\lambda$, 使得数列$\\{a_{n+1}-\\lambda a_n\\}$为等差数列? 若存在, 求出$\\lambda$的值; 若不存在, 说明理由.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-14_等差数列和等比数列" ], "genre": "解答题", "ans": "(1) 证明略; (2) $a_n=n-2+\\dfrac{3}{2^n}$; (3) $\\lambda$存在, $\\lambda=\\dfrac 12$", @@ -344829,7 +347822,8 @@ "content": "若$\\tan \\theta+\\dfrac{1}{\\tan \\theta}=4$, 则$\\sin 2 \\theta=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-06_三角与解三角形" ], "genre": "填空题", "ans": "$\\dfrac 12$", @@ -344864,7 +347858,8 @@ "content": "若扇形的圆心角为$120^{\\circ}$, 半径为$5$, 则扇形的面积为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-06_三角与解三角形" ], "genre": "填空题", "ans": "$\\dfrac{25\\pi}{3}$", @@ -344899,7 +347894,8 @@ "content": "若$0 \\leq \\alpha \\leq 2 \\pi$, $\\sin \\alpha>\\sqrt{3} \\cos \\alpha$, 则$\\alpha$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-06_三角与解三角形" ], "genre": "填空题", "ans": "$(\\dfrac\\pi 3,\\dfrac{4\\pi}3)$", @@ -344956,7 +347952,8 @@ "content": "$\\triangle ABC$中, $A=60^{\\circ}$, $b=1$, $S_{\\triangle ABC}=\\sqrt{3}$, 则$\\dfrac{a+b+c}{\\sin A+\\sin B+\\sin C}=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-06_三角与解三角形" ], "genre": "填空题", "ans": "$\\dfrac{2\\sqrt{39}}3$", @@ -344991,7 +347988,8 @@ "content": "若$\\alpha \\in(0, \\dfrac{\\pi}{2})$, $\\beta \\in(-\\dfrac{\\pi}{2}, 0)$, $\\cos (\\dfrac{\\pi}{4}+\\alpha)=\\dfrac{1}{3}$, $\\cos (\\dfrac{\\pi}{4}-\\dfrac{\\beta}{2})=\\dfrac{\\sqrt{3}}{3}$, 则$\\cos (\\alpha+\\dfrac{\\beta}{2})=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-06_三角与解三角形" ], "genre": "填空题", "ans": "$\\dfrac{5\\sqrt{3}}9$", @@ -345026,7 +348024,8 @@ "content": "满足方程$\\sin (2 x+\\dfrac{\\pi}{4})=\\cos (\\dfrac{\\pi}{6}-x)$的最小的正角是\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-06_三角与解三角形" ], "genre": "填空题", "ans": "$\\dfrac\\pi{12}$", @@ -345063,7 +348062,8 @@ "content": "设$\\triangle ABC$的内角$A, B, C$所对的边为$a, b, c$, 则下列命题正确的是\\blank{50}.\\\\\n\\textcircled{1} 若$a b>c^2$, 则$C<\\dfrac{\\pi}{3}$; \\textcircled{2} 若$a+b>2 c$, 则$C<\\dfrac{\\pi}{3}$; \\textcircled{3} 若$a^3+b^3=c^3$, 则$C<\\dfrac{\\pi}{2}$.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-06_三角与解三角形" ], "genre": "填空题", "ans": "\\textcircled{1}\\textcircled{2}\\textcircled{3}", @@ -345186,7 +348186,8 @@ "content": "已知$\\cos 2 \\alpha=\\dfrac{1}{5}$, 则$\\sin ^4 \\alpha-\\cos ^4 \\alpha$的值为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-06_三角与解三角形" ], "genre": "填空题", "ans": "$-\\dfrac 15$", @@ -345221,7 +348222,8 @@ "content": "若$\\tan \\theta=2$, 则$\\sin 2 \\theta=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-06_三角与解三角形" ], "genre": "填空题", "ans": "$\\dfrac 45$", @@ -345256,7 +348258,8 @@ "content": "若$\\alpha, \\beta$为锐角, 且$\\cos \\alpha=\\dfrac{4}{5}$, $\\cos (\\alpha+\\beta)=\\dfrac{3}{5}$, 则$\\sin \\beta=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-06_三角与解三角形" ], "genre": "填空题", "ans": "$\\dfrac 7{25}$", @@ -345291,7 +348294,8 @@ "content": "若$\\alpha \\in(-\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2})$, $\\beta \\in(-\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2})$, 且$\\tan \\alpha$、$\\tan \\beta$是方程$x^2+3 \\sqrt{3} x+4=0$的两个相异实根, 则$\\alpha+\\beta=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-06_三角与解三角形" ], "genre": "填空题", "ans": "$-\\dfrac{2\\pi}3$", @@ -345370,7 +348374,8 @@ "content": "$\\triangle ABC$的三内角$A, B, C$的对边边长分别为$a, b, c$, 若$a=\\dfrac{\\sqrt{5}}{2} b$, $A=2B$, 则$\\cos B=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-06_三角与解三角形" ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{5}}4$", @@ -345493,7 +348498,8 @@ "content": "某兴趣小组测量电视塔$AE$的高度$H$(单位: $\\text{m}$), 如示意图, 垂直放置的标杆$BC$的高度$h=4 \\text{m}$, 仰角$\\angle ABE=\\alpha$, $\\angle ADE=\\beta$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (0,0) node [left] {$D$} coordinate (D);\n\\draw (1,0) node [above left] {$B$} coordinate (B);\n\\draw (1,0.6) node [above] {$C$} coordinate (C);\n\\draw ($(D)!2.5!(B)$) node [below right] {$A$} coordinate (A);\n\\draw ($(D)!2.5!(C)$) node [right] {$E$} coordinate (E);\n\\draw (D) -- (A) (B) -- (C) (A) -- (E);\n\\draw [dashed] (D) -- (E) (B) -- (E);\n\\draw (B) ++ (0,-0.3) --++ (0,0.2) (A) ++ (0,-0.3) --++ (0,0.2);\n\\draw [<->] (B) ++ (0,-0.2) --++ (1.5,0) node [midway, fill = white] {$d$};\n\\draw (B) pic [draw, \"$\\alpha$\", angle eccentricity = 1.3] {angle = A--B--E};\n\\draw (D) pic [draw, \"$\\beta$\", angle eccentricity = 1.3] {angle = A--D--E};\n\\end{tikzpicture}\n\\end{center}\n(1) 该小组已经测得一组$\\alpha$、$\\beta$的值, $\\tan \\alpha=1.24$, $\\tan \\beta=1.20$, 请据此算出$H$的值;\\\\\n(2) 该小组分析若干测得的数据后, 认为适当调整标杆到电视塔的距离$d$(单位: $\\text{m}$), 使$\\alpha$与$\\beta$之差较大, 可以提高测量精确度. 若电视塔的实际高度为$125 \\text{m}$, 试问$d$为多少时, $\\alpha-\\beta$最大?", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-06_三角与解三角形" ], "genre": "解答题", "ans": "(1) $124$(塔高为$124$米); (2) $d=55\\sqrt{5}$时, $\\alpha-\\beta$最大", @@ -345572,7 +348578,8 @@ "content": "函数$y=2 \\sin (\\dfrac{\\pi}{3}-2 x)$的单调递增区间为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-07_三角函数" ], "genre": "填空题", "ans": "$[-\\dfrac {7\\pi}{12}+k\\pi,-\\dfrac{\\pi}{12}+k\\pi]$, $k\\in \\mathbf{Z}$", @@ -345685,7 +348692,8 @@ "content": "已知定义在$\\mathbf{R}$上的函数$y=f(x)$的周期为$2 \\pi$, 当$x \\in[0,2 \\pi)$时, $f(x)=\\sin \\dfrac{x}{2}$, 那么方程$f(x)=\\dfrac{1}{2}$的解集是\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-07_三角函数" ], "genre": "填空题", "ans": "$\\{x|x=\\dfrac\\pi 3+2k\\pi\\text{或}\\dfrac{5\\pi}3+2k\\pi, \\ k\\in \\mathbf{Z}\\}$", @@ -345732,7 +348740,8 @@ "content": "对函数$y=\\sin x+\\cos x$, 有下列命题: \\textcircled{1} 若$x \\in[0, \\dfrac{\\pi}{2}]$, 则$y \\in[0, \\sqrt{2}]$; \\textcircled{2} 与函数$y=\\sin x-\\cos x$的图像关于直线$x=k \\pi+\\dfrac{\\pi}{2}$($k \\in \\mathbf{Z}$)对称; \\textcircled{3} 在区间$[\\dfrac{\\pi}{4}, \\dfrac{5 \\pi}{4}]$上单调递减; \\textcircled{4} 其图像可由$y=\\sqrt{2} \\sin 2 x$的图像纵坐标不变横坐标变为原来的$\\dfrac{1}{2}$后再向左平移$\\dfrac{\\pi}{4}$个单位得到. 正确的命题是\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-07_三角函数" ], "genre": "填空题", "ans": "\\textcircled{1}\\textcircled{2}\\textcircled{3}", @@ -345933,7 +348942,8 @@ "content": "函数$f(x)=\\sqrt{3} \\sin x+\\sin (\\dfrac{\\pi}{2}+x)$的最大值是\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-07_三角函数" ], "genre": "填空题", "ans": "$2$", @@ -345980,7 +348990,8 @@ "content": "由函数$y=2 \\sin 3 x$, $x \\in[\\dfrac{\\pi}{6}, \\dfrac{5 \\pi}{6}]$与$y=2$($x \\in \\mathbf{R}$)的图像围成一个封闭图形, 这个封闭图形的面积为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-07_三角函数" ], "genre": "填空题", "ans": "$\\dfrac{4\\pi}{3}$", @@ -346014,7 +349025,8 @@ "content": "若函数$f(x)=a^2 \\sin 2 x+(a-2) \\cos 2 x$的图像关于直线$x=-\\dfrac{\\pi}{8}$对称, 则实数$a=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-07_三角函数" ], "genre": "填空题", "ans": "$1$或$-2$", @@ -346048,7 +349060,8 @@ "content": "以下四个命题中, 正确命题的序号是\\blank{50}.\\\\\n\\textcircled{1} 若$\\cos \\alpha=\\cos \\beta$, 则$\\alpha-\\beta=2 k \\pi$($k$是某个整数); \\textcircled{2} 函数$y=2 \\cos (2 x+\\dfrac{\\pi}{3})$的图像关于点$(\\dfrac{\\pi}{12}, 0)$对称; \\textcircled{3} 函数$y=\\sin |x|$是周期函数, 且$2 \\pi$是它的一个周期; \\textcircled{4} 函数$y=\\cos (\\sin x)$是偶函数.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-07_三角函数" ], "genre": "填空题", "ans": "\\textcircled{2}\\textcircled{4}", @@ -346082,7 +349095,8 @@ "content": "已知$-\\dfrac{\\pi}{2}0$.\\\\\n(1) 求函数$y=f(x)$的值域;\\\\\n(2) 若$f(x)$在区间$[-\\dfrac{3 \\pi}{2}, \\dfrac{\\pi}{2}]$上为增函数, 求$\\omega$的最大值.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-07_三角函数" ], "genre": "解答题", "ans": "(1) $[1-\\sqrt{3},1+\\sqrt{3}]$; (2) $\\dfrac 16$", @@ -346217,7 +349232,8 @@ "content": "已知直线$l_1: x+a y+2=0$和$l_2:(a-2) x+3 y+6 a=0$, 则$l_1\\parallel l_2$的充要条件是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-11_直线与圆" ], "genre": "填空题", "ans": "$a=3$", @@ -346252,7 +349268,8 @@ "content": "直线$a x+b y-a b=0$($a>0$, $b>0$)的倾斜角是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-11_直线与圆" ], "genre": "填空题", "ans": "$\\pi-\\arctan\\dfrac ab$", @@ -346287,7 +349304,8 @@ "content": "直线$l$上有两点$M(t-1,2)$, $N(t-3, t^2-4 t+4)$, 则直线$l$的倾斜角的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-11_直线与圆" ], "genre": "填空题", "ans": "$[0,\\dfrac\\pi 4]\\cup (\\dfrac \\pi 2,\\pi)$", @@ -346344,7 +349362,8 @@ "content": "将直线$l_1: n x+y-n=0$, $l_2: x+n y-n=0$($n \\in \\mathbf{N}$, $n \\geq 2$), $x$轴及$y$轴围成的封闭区域的面积记为$S_n$, 则$\\displaystyle\\lim_{n\\to\\infty} S_n=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-11_直线与圆" ], "genre": "填空题", "ans": "$1$", @@ -346379,7 +349398,10 @@ "content": "若实数$a, b, c$成等差数列, 点$P(-1,0)$在动直线$l: a x+b y+c=0$上的射影为$M$, 点$N(0,3)$, 则线段$MN$的长度的最小值是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-四月错题重做-04_易错题-解析几何", + "2023届高三-四月错题重做-04_解析几何", + "2023届高三-第二轮复习讲义-11_直线与圆" ], "genre": "填空题", "ans": "$4-\\sqrt{2}$", @@ -346467,7 +349489,8 @@ "content": "已知集合$M$是平面直角坐标系中方程为$x-2 k y+k^2=0$($k \\in \\mathbf{R}$)的直线的集合, 集合$S$是满足以下条件的点的集合: 对于集合$S$中的每一个点, 集合$M$中有且仅有一条直线经过该点.\\\\\n(1) 判断下列直线是否为集合$M$中的直线: $l_1: x-y+1=0$, $l_2: x-2 y+1=0$;\\\\\n(2) 判断下列各点是否为集合$S$中的点: $D(2,1)$, $E(1,1)$;\\\\\n(3) 求集合$S$中的点的轨迹方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-11_直线与圆" ], "genre": "解答题", "ans": "(1) $l_1$不是$M$中的直线, $l_2$是$M$中的直线; (2) $D$不是$S$中的点, $E$是$S$中的点; (3) $y^2=x$", @@ -346546,7 +349569,8 @@ "content": "已知三条直线$y=3 x+2$, $2 x+y+3=0$, $k x+y=0$. 若它们不能围成一个三角形, 则$k$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-11_直线与圆" ], "genre": "填空题", "ans": "$\\{-3,2,-1\\}$", @@ -346581,7 +349605,8 @@ "content": "点$P(-2,0)$关于直线$x-y-2=0$的对称点的坐标为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-11_直线与圆" ], "genre": "填空题", "ans": "$(2,-4)$", @@ -346616,7 +349641,8 @@ "content": "已知直线$l_1$的斜率是$\\dfrac{1}{2}$, 直线$l_2$的倾斜角是$l_1$的倾斜角的$2$倍, 则$l_2$的斜率是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-11_直线与圆" ], "genre": "填空题", "ans": "$\\dfrac 43$", @@ -346673,7 +349699,8 @@ "content": "直线$l$被两直线$l_1: x-3 y+10=0$和$l_2: 2 x+y+8=0$所截得的线段的中点为$P(0,1)$, 则直线$l$的方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-11_直线与圆" ], "genre": "填空题", "ans": "$23x-20y+20=0$", @@ -346708,7 +349735,8 @@ "content": "过直线$l: 3 x+4 y-5=0$上的一点$P$向圆$(x-3)^2+(y-4)^2=4$作两条切线$l_1, l_2$. 设$l_1$与$l_2$的夹角为$\\theta$, 则$\\theta$的最大值为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-11_直线与圆" ], "genre": "填空题", "ans": "$\\dfrac\\pi 3$", @@ -346765,7 +349793,8 @@ "content": "设$f(x, y)=A x+B y+C$, 这里$A, B, C$是常数, $A, B$不全为零. 若点$P(x_0, y_0)$不在直线$f(x, y)=0$上, 则曲线$f(x, y)-f(x_0, y_0)=0$表示\\bracket{20}.\n\\twoch{不过点$P$但平行于$l$的直线}{过点$P$且垂直于$l$的直线}{过点$P$且平行于$l$的直线}{不过点$P$但垂直于$l$的直线}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-11_直线与圆" ], "genre": "选择题", "ans": "C", @@ -346800,7 +349829,8 @@ "content": "已知直线$l: 5 x+2 y+3=0$.\\\\\n(1) 求直线$l_1: 3 x+7 y-13=0$与$l$所成的角的大小;\\\\\n(2) 若$l_2$经过点$P(2,1)$, 且与$l$的夹角等于$45^{\\circ}$, 求直线$l_2$的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-11_直线与圆" ], "genre": "解答题", "ans": "(1) $45^\\circ$; (2) $3x+7y-13=0$或$7x-3y-11=0$", @@ -346857,7 +349887,8 @@ "content": "设$m$是常数, 若点$F(0,5)$是双曲线$\\dfrac{y^2}{m^2}-\\dfrac{x^2}{9}=1$的一个焦点, 则$m=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-12_圆锥曲线" ], "genre": "填空题", "ans": "$\\pm 4$", @@ -346929,7 +349960,8 @@ "content": "已知双曲线$C$与椭圆$\\dfrac{x^2}{16}+\\dfrac{y^2}{8}=1$有相同的焦点, 直线$y=\\sqrt{3} x$为$C$的一条渐近线, 则双曲线$C$的方程是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-12_圆锥曲线" ], "genre": "填空题", "ans": "$\\dfrac{x^2}2-\\dfrac{y^2}6=1$", @@ -346964,7 +349996,8 @@ "content": "已知$F_1, F_2$是椭圆$\\dfrac{x^2}{16}+\\dfrac{y^2}{9}=1$的两个焦点, 过$F_2$的直线交椭圆于$A, B$两点, 若$|AB|=5$, 则$|AF_1|+|BF_1|=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-12_圆锥曲线" ], "genre": "填空题", "ans": "$11$", @@ -346998,7 +350031,8 @@ "content": "已知抛物线方程为$y^2=2 p x$($p>0$), 过焦点$F$的直线与抛物线交于$A, B$两点, 以$AB$为直径的圆$M$与抛物线的准线$l$的位置关系为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-12_圆锥曲线" ], "genre": "填空题", "ans": "相切", @@ -347032,7 +350066,8 @@ "content": "已知点$A$的坐标是$(1, \\dfrac{1}{2}), P$是椭圆$x^2+4 y^2=1$上的动点, 则线段$PA$中点的轨迹方程是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-12_圆锥曲线" ], "genre": "填空题", "ans": "$(2x-1)^2+(4y-1)^2=1$", @@ -347067,7 +350102,8 @@ "content": "某海域内有一孤岛, 岛四周的海平面(视为平面)上有一浅水区(含边界), 其边界是长轴长为$2 a$, 短轴长为$2 b$的椭圆. 已知岛上甲、乙导航灯的海拔高度分别为$h_1, h_2$, 且两个导航灯在海平面上的投影恰好落在椭圆的两个焦点上, 现有船只经过该海域(船只的大小忽略不计), 在船上测得甲、乙导航灯的仰角分别为$\\theta_1, \\theta_2$, 那么船只进入该浅水区的判断条件是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [domain = 0:360,samples = 100] plot ({2*cos(\\x)},0,{sqrt(3)*sin(\\x)});\n\\draw (-1,0,0,0) coordinate (A) -- (-1,1.2,0) coordinate (A1) node [midway, left] {$h_1$};\n\\draw (1,0,0) coordinate (B) -- (1,1,0) coordinate (B1) node [midway, right] {$h_2$};\n\\draw (0,0,1.2) coordinate (C);\n\\draw [dashed] (A) -- (C) (A1) -- (C) (B) -- (C) (B1) -- (C);\n\\draw (C) pic [draw, \"$\\theta_1$\", angle eccentricity = 1.3] {angle = A1--C--A};\n\\draw (C) pic [draw, \"$\\theta_2$\", angle eccentricity = 1.5] {angle = B--C--B1};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-12_圆锥曲线" ], "genre": "填空题", "ans": "$\\dfrac{h_1}{\\tan\\theta_1}+\\dfrac{h_2}{\\tan\\theta_2}\\le 2a$", @@ -347192,7 +350228,8 @@ "content": "抛物线$y=8 x^2$的准线方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-12_圆锥曲线" ], "genre": "填空题", "ans": "$y=-\\dfrac 1{32}$", @@ -347271,7 +350308,8 @@ "content": "若双曲线$8 m x^2-m y^2=8$的一个焦点是$(0,3)$, 则实数$m=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-12_圆锥曲线" ], "genre": "填空题", "ans": "$-1$", @@ -347327,7 +350365,8 @@ "content": "已知$F_1, F_2$为双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的焦点. 过$F_2$作垂直于$x$轴的直线交双曲线于点$P$, 且$\\angle PF_1F_2=30^{\\circ}$, 则双曲线的渐近线方程是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-12_圆锥曲线" ], "genre": "填空题", "ans": "$y=\\pm \\sqrt{2} x$", @@ -347361,7 +350400,8 @@ "content": "已知定圆$C_1:(x-7)^2+y^2=4$, $C_2:(x+7)^2+y^2=25$, 动圆$M$与两定圆外切, 则动圆圆心$M$的轨迹方程是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-12_圆锥曲线" ], "genre": "填空题", "ans": "$\\dfrac{x^2}{9/4}-\\dfrac{y^2}{187/4}=1$($x>0$)", @@ -347396,7 +350436,10 @@ "content": "若$F$是双曲线$x^2-y^2=1$的左焦点, 点$P$在第三象限的双曲线上, 则直线$FP$的倾斜角的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-四月错题重做-04_易错题-解析几何", + "2023届高三-四月错题重做-04_解析几何", + "2023届高三-第二轮复习讲义-12_圆锥曲线" ], "genre": "填空题", "ans": "$(\\dfrac\\pi 4,\\pi)$", @@ -347465,7 +350508,8 @@ "content": "记椭圆$E_n: \\dfrac{x^2}{4}+\\dfrac{n y^2}{4 n+1}=1$, 其中$n=1,2, \\cdots$. 当点$(x, y)$分别在$E_1, E_2, \\cdots$上时, $x+y$的最大值分别是$M_1, M_2, \\cdots$, 则$\\displaystyle\\lim_{n\\to\\infty} M_n=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-12_圆锥曲线" ], "genre": "填空题", "ans": "$2\\sqrt{2}$", @@ -347524,7 +350568,8 @@ "content": "已知圆$O: x^2+y^2=4$.\\\\\n(1) 直线$l_1: \\sqrt{3} x+y-2 \\sqrt{3}=0$与圆$O$相交于$A, B$两点, 求弦$AB$的长;\\\\\n(2) 设$M(x_1, y_1)$, $P(x_2, y_2)$是圆$O$上的两个动点, 点$M$关于原点的对称点为$M_1$, 点$M$关于$x$轴的对称点为$M_2$. 如果点$P$和$M_1, M_2$均不重合, 且直线$PM_1, PM_2$都和$y$轴相交, 且分别交于$S(0, m)$和$T(0, n)$, 问$m \\cdot n$是否为定值? 若是, 求出该定值; 若不是, 请说明理由.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-11_直线与圆" ], "genre": "解答题", "ans": "(1) $2$; (2) 恒等于$4$, 是定值", @@ -347559,7 +350604,8 @@ "content": "若动点$P$到点$F(2,0)$的距离与它到直线$x+2=0$的距离相等, 则点$P$的轨迹方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-12_圆锥曲线" ], "genre": "填空题", "ans": "$y^2=8x$", @@ -347615,7 +350661,8 @@ "content": "以原点为圆心, 且截直线$3 x+4 y+15=0$所得弦长为$8$的圆的方程是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-11_直线与圆" ], "genre": "填空题", "ans": "$x^2+y^2=25$", @@ -347650,7 +350697,8 @@ "content": "动直线$(2 k-1) x-(k+3) y-(k-11)=0$($k \\in \\mathbf{R}$)所过的定点是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-11_直线与圆" ], "genre": "填空题", "ans": "$(2,3)$", @@ -347729,7 +350777,8 @@ "content": "直线$y=x+3$与曲线$\\dfrac{y^2}{9}-\\dfrac{x|x|}{4}=1$的公共点的个数为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-13_解析几何综合" ], "genre": "填空题", "ans": "$3$", @@ -347829,7 +350878,8 @@ "content": "$P$是双曲线$\\dfrac{x^2}{4}-y^2=1$的右顶点, 过点$P$的两条互相垂直的直线分别与双曲线的右支交于点$A, B$, 问直线$AB$是否一定过$x$轴上一定点? 如果不存在这样的定点, 请说明理由; 如果存在这样的定点, 试求出这个定点的坐标.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-13_解析几何综合" ], "genre": "解答题", "ans": "存在, 坐标为$(-\\dfrac{10}3,0)$", @@ -347885,7 +350935,8 @@ "content": "双曲线$\\dfrac{x^2}{25}-\\dfrac{y^2}{39}=1$上一点$P$到双曲线一个焦点的距离为$12$, 则$P$到另一个焦点的距离为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-12_圆锥曲线" ], "genre": "填空题", "ans": "$22$", @@ -347941,7 +350992,8 @@ "content": "双曲线$x^2-y^2=1$, 点$F_1, F_2$为其两个焦点, 点$P$为双曲线上一点, 若$PF_1 \\perp PF_2$, 则$|PF_1|+|PF_2|$的值为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-13_解析几何综合" ], "genre": "填空题", "ans": "$2\\sqrt{3}$", @@ -347997,7 +351049,10 @@ "content": "若直线$y=x+k$与曲线$y=\\sqrt{2-x^2}$相交于两点, 则实数$k$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-四月错题重做-04_易错题-解析几何", + "2023届高三-四月错题重做-04_解析几何", + "2023届高三-第二轮复习讲义-13_解析几何综合" ], "genre": "填空题", "ans": "$[\\sqrt{2},2)$", @@ -348043,7 +351098,8 @@ "content": "过抛物线$y^2=4 x$的焦点作倾斜角为$\\dfrac{\\pi}{3}$的弦$AB$, 则$|AB|=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-13_解析几何综合" ], "genre": "填空题", "ans": "$\\dfrac{16}3$", @@ -348077,7 +351133,8 @@ "content": "设$P$是抛物线$y^2=4 x$上一动点, $F$是抛物线的焦点, 定点$B(3,2)$, 则$|PB|+|PF|$的最小值为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-12_圆锥曲线" ], "genre": "填空题", "ans": "$4$", @@ -348112,7 +351169,8 @@ "content": "已知两点$M(-5,0), N(5,0)$, 若直线上存在点$P$, 使$|PM|-|PN|=8$, 则称该直线为``$B$型直线'', 现给出下列直线: \\textcircled{1} $y=x+2$; \\textcircled{2} $y=2$; \\textcircled{3} $y=\\dfrac{2}{3} x$; \\textcircled{4} $y=2 x+3$. 其中是``$B$型直线''的是\\bracket{20}.\n\\fourch{\\textcircled{1}\\textcircled{2}}{\\textcircled{1}\\textcircled{3}}{\\textcircled{2}\\textcircled{3}}{\\textcircled{1}\\textcircled{4}}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-13_解析几何综合" ], "genre": "选择题", "ans": "C", @@ -348191,7 +351249,8 @@ "content": "已知直线$l: x-a y+a=0$与双曲线$x^2-y^2=1$的左支交于$A, B$两点, 过弦$AB$的中点$Q$与点$P(-2,1)$的直线交$y$轴于$(0, b)$点. 当$a$变化时, 求实数$b$的取值范围.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-13_解析几何综合" ], "genre": "解答题", "ans": "$(-\\infty,-3-2\\sqrt{2})\\cup (3,+\\infty)$", @@ -348292,7 +351351,8 @@ "content": "设$AB$是椭圆$\\Gamma$的长轴, 点$C$在$\\Gamma$上, 且$\\angle CBA=\\dfrac{\\pi}{4}$, 若$AB=4, BC=\\sqrt{2}$, 则$\\Gamma$的两个焦点之间的距离为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-12_圆锥曲线" ], "genre": "填空题", "ans": "$\\dfrac{4}{3}\\sqrt{6}$", @@ -348348,7 +351408,8 @@ "content": "中心在原点, 对称轴为坐标轴, 椭圆的短轴的一个顶点$B$与两个焦点$F_1, F_2$组成的三角形的周长为$4+2 \\sqrt{3}$, 且$\\angle F_1BF_2=\\dfrac{2 \\pi}{3}$, 则椭圆的方程是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-12_圆锥曲线" ], "genre": "填空题", "ans": "$\\dfrac{x^2}4+y^2=1$或$x^2+\\dfrac{y^2}{4}=1$", @@ -348383,7 +351444,8 @@ "content": "若动点$(x, y)$在曲线$\\dfrac{x^2}{4}+\\dfrac{y^2}{b^2}=1(01$), 点$P$是$C$上的动点, $M$是右顶点, 定点$A$的坐标为$(2,0)$.\\\\\n(1) 若$M$与$A$重合, 求$C$的焦点坐标;\\\\\n(2) 若$m=3$, 求$|PA|$的最大值与最小值;\\\\\n(3) 若$|PA|$的最小值为$|MA|$, 求$m$的取值范围.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-四月错题重做-04_易错题-解析几何", + "2023届高三-四月错题重做-04_解析几何", + "2023届高三-第二轮复习讲义-12_圆锥曲线" ], "genre": "解答题", "ans": "(1) $(\\pm \\sqrt{3},0)$; (2) 最大值为$5$, 最小值为$\\dfrac{\\sqrt{2}}2$; (3) $(1,1+\\sqrt{2}]$", @@ -348530,7 +351595,8 @@ "content": "直线$x-y-3=0$被双曲线$\\dfrac{x^2}{4}-y^2=1$所截得的弦长为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-13_解析几何综合" ], "genre": "填空题", "ans": "$\\dfrac{8\\sqrt{3}}3$", @@ -348565,7 +351631,8 @@ "content": "已知点$M(x, y)$到点$F_1(-5,0)$和$F_2(5,0)$的距离差是$8$, 则点$M$的轨迹方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-12_圆锥曲线" ], "genre": "填空题", "ans": "$\\dfrac{x^2}{16}-\\dfrac{y^2}{9}=1$($x>0$)", @@ -348622,7 +351689,8 @@ "content": "设$F_1, F_2$为双曲线$\\dfrac{x^2}{4}-y^2=1$的两焦点, 点$P$在双曲线上且满足$\\angle F_1PF_2=90^{\\circ}$, 则$\\triangle F_1PF_2$的面积为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-12_圆锥曲线" ], "genre": "填空题", "ans": "$1$", @@ -348722,7 +351790,8 @@ "content": "抛物线$x^2=2 p y$($p>0$)的焦点为$F$, 其准线与双曲线$\\dfrac{x^2}{3}-\\dfrac{y^2}{3}=1$相交于$A, B$两点, 若$\\triangle ABF$为等边三角形, 则$p=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-13_解析几何综合" ], "genre": "填空题", "ans": "$6$", @@ -348802,7 +351871,8 @@ "content": "已知双曲线$H$的中心在原点, 抛物线$y^2=8 x$的焦点是双曲线$H$的一个焦点, 且$H$经过点$(\\sqrt{2}, \\sqrt{3})$.\\\\\n(1) 求双曲线$H$的方程;\\\\\n(2) 设双曲线$H$的实轴左顶点为$A$, 右焦点为$F$, 在第一象限内任取双曲线$H$上一点$P$, 试问是否存在常数$\\lambda$, 使得$\\angle PFA=\\lambda \\angle PAF$恒成立? 证明你的结论.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-12_圆锥曲线" ], "genre": "解答题", "ans": "(1) $x^2-\\dfrac{y^2}3=1$; (2) $\\lambda$存在, $\\lambda=2$", @@ -348881,7 +351951,8 @@ "content": "若向量$\\overrightarrow {a}, \\overrightarrow {b}$满足$|\\overrightarrow {a}|=2$, $|\\overrightarrow {b}|=3$, 且$\\overrightarrow {a}$与$\\overrightarrow {b}$的夹角为$\\dfrac{\\pi}{3}$, 则$|\\overrightarrow {a}+\\overrightarrow {b}|=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第二轮复习讲义-08_平面向量与复数" ], "genre": "填空题", "ans": "$\\sqrt{19}$", @@ -348915,7 +351986,8 @@ "content": "已知$\\overrightarrow {m}, \\overrightarrow {n}$是夹角为$60^{\\circ}$的单位向量, 则$\\overrightarrow {a}=2 \\overrightarrow {m}+\\overrightarrow {n}$与$\\overrightarrow {b}=-3 \\overrightarrow {m}+2 \\overrightarrow {n}$的夹角是\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第二轮复习讲义-08_平面向量与复数" ], "genre": "填空题", "ans": "$\\dfrac{2\\pi}3$", @@ -348949,7 +352021,8 @@ "content": "在四面体$O-ABC$中, $\\overrightarrow{AB}=\\overrightarrow {a}$, $\\overrightarrow{OB}=\\overrightarrow {b}$, $\\overrightarrow{OC}=\\overrightarrow {c}$, $D$为$BC$的中点, $E$为$AD$的中点, 则用$\\overrightarrow {a}, \\overrightarrow {b}, \\overrightarrow {c}$表示$\\overrightarrow{OE}=$\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-08_平面向量与复数" ], "genre": "填空题", "ans": "$-\\dfrac 12\\overrightarrow{a}+\\dfrac 34\\overrightarrow{b}+\\dfrac 14\\overrightarrow{c}$", @@ -349160,7 +352233,8 @@ "content": "已知$\\overrightarrow{e_1}, \\overrightarrow{e_2}$是两个不共线的平面向量, 向量$\\overrightarrow {a}=2 \\overrightarrow{e_1}-\\overrightarrow{e_2}$, $\\overrightarrow {b}=\\overrightarrow{e_1}+\\lambda \\overrightarrow{e_2}$($\\lambda \\in \\mathbf{R}$), 若$\\overrightarrow {a}\\parallel \\overrightarrow {b}$, 则$\\lambda=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第二轮复习讲义-08_平面向量与复数" ], "genre": "填空题", "ans": "$-\\dfrac 12$", @@ -349195,7 +352269,8 @@ "content": "已知向量$\\overrightarrow {a}=(2,3)$, $\\overrightarrow {b}=(x, 6)$, 且$\\overrightarrow {a}\\parallel \\overrightarrow {b}$, 则$x=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第二轮复习讲义-08_平面向量与复数" ], "genre": "填空题", "ans": "$4$", @@ -349251,7 +352326,8 @@ "content": "已知向量$\\overrightarrow {a}=(2,3)$, $\\overrightarrow {b}=(-1,4)$, 则向量$\\overrightarrow {b}$在向量$\\overrightarrow {a}$方向上的数量投影为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第二轮复习讲义-08_平面向量与复数" ], "genre": "填空题", "ans": "$\\dfrac{10\\sqrt{13}}{13}$", @@ -349307,7 +352383,8 @@ "content": "在$\\triangle ABC$中, $M$是$BC$的中点, $AM=3$, $BC=10$, 则$\\overrightarrow{AB} \\cdot \\overrightarrow{AC}=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第二轮复习讲义-08_平面向量与复数" ], "genre": "填空题", "ans": "$-16$", @@ -349540,7 +352617,8 @@ "content": "已知某圆锥体的底面半径$r=3$, 沿圆锥体的母线把侧面展开后得到一个圆心角为$\\dfrac{2}{3} \\pi$的扇形, 则该圆锥体的表面积是\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-09_立体几何综合" ], "genre": "填空题", "ans": "$36\\pi$", @@ -349619,7 +352697,8 @@ "content": "给出下列命题, 其中正确命题的所有序号是\\blank{50}.\\\\\n\\textcircled{1} 直线上有两点到平面的距离相等, 则此直线与平面平行;\\\\\n\\textcircled{2} 夹在两个平行平面间的两条异面线段(端点均在相应平面上)的中点连线平行于这两个平面;\\\\\n\\textcircled{3} $\\alpha$内存在不共线的三点到$\\beta$的距离相等, 则平面$\\alpha$与$\\beta$平行;\\\\\n\\textcircled{4} 垂直于同一个平面的两条直线是平行直线;\\\\\n\\textcircled{5} $l$、$m$是两条异面直线, $\\alpha$、$\\beta$是两个平面, 且$l\\parallel \\alpha$, $m\\parallel \\alpha$, $l\\parallel \\beta$, $m\\parallel \\beta$, 则平面$\\alpha$与$\\beta$平行.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-09_立体几何综合" ], "genre": "填空题", "ans": "\\textcircled{2}\\textcircled{4}\\textcircled{5}", @@ -349678,7 +352757,8 @@ "content": "如图, $AD, BC$是四面体$ABCD$中互相垂直的棱, $BC=2$. 若$AD=2 c$, $AB=BD=AC=CD=a$, 其中$a, c$为常数, 则四面体$ABCD$的体积是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,1) node [below] {$B$} coordinate (B);\n\\draw (0,0,-1) node [right] {$C$} coordinate (C);\n\\draw (-1,-0.5,0) node [below] {$A$} coordinate (A);\n\\draw (-1,0.5,0) node [above] {$D$} coordinate (D);\n\\draw (D)--(A)--(B)--(C)--cycle (D)--(B);\n\\draw [dashed] (A)--(C);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-09_立体几何综合" ], "genre": "填空题", "ans": "$\\dfrac 23 c\\sqrt{a^2-c^2-1}$", @@ -349757,7 +352837,8 @@ "content": "如图, 圆锥顶点为$P$, 底面圆心为$O$, 其母线与底面所成的角为$22.5^{\\circ}$. $AB$和$CD$是底面圆$O$上的两条平行的弦, 轴$OP$与平面$PCD$所成的角为$60^{\\circ}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [above] {$O$} coordinate (O) ellipse (2 and 0.5);\n\\draw (0,-2) node [below] {$P$} coordinate (P);\n\\draw (P)--({sqrt(15)/2},-1/8) (P)--({-sqrt(15)/2},-1/8);\n\\draw ({2*cos(-50)},{sin(-50)/2}) node [above right] {$C$} coordinate (C);\n\\draw ({2*cos(50)},{sin(50)/2}) node [above right] {$D$} coordinate (D);\n\\draw ({2*cos(-140)},{sin(-140)/2}) node [above right] {$A$} coordinate (A);\n\\draw ({2*cos(140)},{sin(140)/2}) node [above right] {$B$} coordinate (B);\n\\draw (O)--(C)--(D)--cycle (P)--(C) (P)--(A)--(B);\n\\draw [dashed] (P)--(O) (P)--(D) (P)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: 平面$PAB$与平面$PCD$的交线平行于底面;\\\\ \n(2) 求$\\cos \\angle COD$.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-09_立体几何综合" ], "genre": "解答题", "ans": "(1) 证明略; (2) $17-12\\sqrt{2}$", @@ -349858,7 +352939,8 @@ "content": "下列四个命题中不正确的命题的序号是\\blank{50}.\\\\\n\\textcircled{1} 三个点确定一个平面; \\textcircled{2} 圆锥的侧面展开图可以是一个圆面; \\textcircled{3} 底面是等边三角形, 三个侧面都是等腰三角形的三棱锥是正三棱锥; \\textcircled{4} 过球面上任意两不同点的大圆有且只有一个.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-09_立体几何综合" ], "genre": "填空题", "ans": "\\textcircled{1}\\textcircled{2}\\textcircled{3}\\textcircled{4}", @@ -350115,7 +353197,8 @@ "content": "设关于$x$的实系数一元二次方程$x^2-2 a x+a^2-4 a+4=0$($a \\in \\mathbf{R}$)的两虚根为$x_1, x_2$且$|x_1|+|x_2|=3$, 则$a=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第二轮复习讲义-08_平面向量与复数" ], "genre": "填空题", "ans": "$\\dfrac 12$", @@ -350150,7 +353233,8 @@ "content": "若$z \\in \\mathbf{C}$且$|z+2-2 \\mathrm{i}|=1$, 则$|z-2-2 \\mathrm{i}|$的最小值是\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第二轮复习讲义-08_平面向量与复数" ], "genre": "填空题", "ans": "$3$", @@ -350296,7 +353380,8 @@ "content": "对于任意的复数$z=x+y \\mathrm{i}$($x, y \\in \\mathbf{R}$), 定义$z$ ``经运算$P$''为$P(z)=x^2[\\cos (y \\pi)+\\mathrm{i} \\sin (y \\pi)]$.\\\\\n(1) 集合$A=\\{\\omega|\\omega=P(z), \\ | z |\\leq 1, \\ \\text{Re} z, \\text{Im} z\\text{均为整数}\\}$, 试用列举法写出集合$A$;\\\\\n(2) 若$z=2+y\\mathrm{i}$($y \\in \\mathbf{R}$), $P(z)$为纯虚数, 求$|z|$的最小值;\\\\\n(3) 直线$l: y=x-9$上是否存在整点$(x, y)$(坐标$x, y$均为整数), 使复数$z=x+y \\mathrm{i}$ ``经运算$P$''后, $P(z)$对应的点也在直线$l$上? 若存在, 求出所有的点; 若不存在, 请说明理由.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第二轮复习讲义-08_平面向量与复数" ], "genre": "解答题", "ans": "(1) $A=\\{0,1\\}$; (2) $\\dfrac{\\sqrt{17}}{2}$; (3) $(3,-6)$与$(-3,-12)$", @@ -350397,7 +353482,8 @@ "content": "在复平面内, $O$是原点, $\\overrightarrow{OA}, \\overrightarrow{OC}, \\overrightarrow{AB}$表示的复数分别为$-2+\\mathrm{i}, 3+2\\mathrm{i}, 1+5\\mathrm{i}$, 那么$\\overrightarrow{BC}$表示的复数为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第二轮复习讲义-08_平面向量与复数" ], "genre": "填空题", "ans": "$4-4\\mathrm{i}$", @@ -350564,7 +353650,8 @@ "content": "已知复数$z_0=1-m \\mathrm{i}$($m>0$), $z=x+y \\mathrm{i}$, $w=x'+y' \\mathrm{i}$, $x, y, x', y'$均为实数, $\\mathrm{i}$为虚数单位, 且对于任意复数$z$, 当$w=\\overline{z_0} \\cdot \\overline {z}$时, 成立$|w|=2|z|$.\\\\\n(1) 求$m$的值, 并分别写出$x', y'$用$x, y$表示的关系式;\\\\\n(2) 将$(x, y)$看作点$P$的坐标, $(x', y')$看作点$Q$的坐标, 上述关系可以看作是坐标平面上点的一个变换: 它将平面上的点$P$变换到这一平面上的点$Q$; 已知点$P$经该点的变换后得到的点$Q$的坐标是$(\\sqrt{3}, 2)$, 试求点$P$的坐标;\\\\\n(3) 若直线$y=k x$上任一点经上述变换后得到的点仍在该直线上, 求$k$的值.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第二轮复习讲义-08_平面向量与复数" ], "genre": "解答题", "ans": "(1) $\\begin{cases}x'=x+\\sqrt{3}y,\\\\y'=\\sqrt{3}x-y;\\end{cases}$ (2) $P(\\dfrac{3\\sqrt{3}}4,\\dfrac 14)$; (3) $\\dfrac{\\sqrt{3}}3$或$-\\sqrt{3}$.", @@ -350598,7 +353685,8 @@ "content": "复数$z=a+b \\mathrm{i}$($a$、$b \\in \\mathbf{R})$, 将一颗骰子连续抛掷两次, 第一次点数记为$a$, 第二次点数记为$b$, 则使复数\n$z^2$为纯虚数的概率为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第二轮复习讲义-18_概率与统计" ], "genre": "填空题", "ans": "$\\dfrac 16$", @@ -350631,7 +353719,8 @@ "content": "某学校组织学生参加英语测试, 成绩的频率分布直方图如图, 数据的分组依次为$[20,40),[40,60)$, $[60,80),[80,100)$若低于$60$分的人数是$15$人, 则该班的学生人数是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (6,0) node [below] {成绩/分};\n\\draw [->] (0,0) -- (0,3) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j/\\k in {1/1/20,2/2/40,3/4/60,4/3/80}\n{\\draw (\\i,0) node [below] {$\\k$} --++ (0,\\j/2) --++ (1,0) --++ (0,-\\j/2);};\n\\draw (5,0) node [below] {$100$};\n\\foreach \\i/\\j/\\k in {1/1/0.005,2/2/0.01,3/4/0.015,4/3/0.02}\n{\\draw [dashed] (\\j,{\\i/2}) -- (0,{\\i/2}) node [left] {$\\k$};};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-第二轮复习讲义-19_统计" ], "genre": "填空题", "ans": "$50$", @@ -350688,7 +353777,8 @@ "content": "样本$(x_1, x_2, \\cdots, x_n)$的平均数为$\\overline {x}$, 样本$(y_1, y_2, \\cdots, y_m)$的平均数为$\\overline {y}$($\\overline {x} \\neq \\overline {y}$). 若样本$(x_1, x_2, \\cdots, x_n, y_1, y_2, \\cdots, y_m)$的平均数$\\overline {z}=\\alpha \\overline {x}+(1-\\alpha) \\overline {y}$, 其中$0<\\alpha<\\dfrac{1}{2}$, 则$n, m$的大小关系为\\blank{50}.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-第二轮复习讲义-19_统计" ], "genre": "填空题", "ans": "$m>n$", @@ -350721,7 +353811,8 @@ "content": "将$2$名教师, $4$名学生分成$2$个小组, 分别安排到甲、乙两地参加社会实践活动, 每个小组由$1$名教师和$2$名学生组成, 不同的安排方案共有\\blank{50}种.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-第二轮复习讲义-17_计数原理与二项式定理" ], "genre": "填空题", "ans": "$12$", @@ -350756,7 +353847,8 @@ "content": "若从$1,2, \\cdots \\cdots, 9$这$9$个整数中同时取$4$个不同的数, 其和为偶数, 则不同的取法共有\\blank{50}种.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-第二轮复习讲义-17_计数原理与二项式定理" ], "genre": "填空题", "ans": "$66$", @@ -350789,7 +353881,8 @@ "content": "若将函数$f(x)=x^5$表示为$f(x)=a_0+a_1(1+x)+a_2(1+x)^2+\\cdots+a_5(1+x)^5$其中$a_0, a_1, a_2, \\cdots, a_5$为实数, 则$a_3=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-第二轮复习讲义-17_计数原理与二项式定理" ], "genre": "填空题", "ans": "$10$", @@ -350822,7 +353915,8 @@ "content": "设$a \\in \\mathbf{Z}$, 且$0 \\leq a<13$, 若$51^{2012}+a$能被$13$整除, 则$a=$\\blank{50}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-第二轮复习讲义-17_计数原理与二项式定理" ], "genre": "填空题", "ans": "$12$", @@ -350924,7 +354018,8 @@ "content": "将$a, b, c, d, e, f$字母排成三行两列, 则不同的排列方法共有\\blank{50}种.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-第二轮复习讲义-17_计数原理与二项式定理" ], "genre": "填空题", "ans": "$720$", @@ -350957,7 +354052,8 @@ "content": "$(x-\\dfrac{1}{x})^8$的展开式中常数项为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-第二轮复习讲义-17_计数原理与二项式定理" ], "genre": "填空题", "ans": "$70$", @@ -351036,7 +354132,8 @@ "content": "$6$位同学互通电话, 任意两位同学之间最多通电话一次, 已知$6$位同学之间共通了$13$次电话, 则通了$4$次电话的同学人数为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-第二轮复习讲义-17_计数原理与二项式定理" ], "genre": "填空题", "ans": "$2$或$4$", @@ -351091,7 +354188,8 @@ "content": "某同学到银行取款时忘记了帐户密码, 但他记得: \\textcircled{1} 密码是四位数字, 如:$0235,1330,2351$等; \\textcircled{2} 四位数字中有$6,8,9$; \\textcircled{3} 四位数字各不相同. 于是他就用$6,8,9$这三个数字再随意加上一个与这三个数字不同的数字, 排成四位数输入取款机尝试, 那么他只试一次就成功的概率是\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-第二轮复习讲义-18_概率与统计" ], "genre": "填空题", "ans": "$\\dfrac{1}{168}$", @@ -351123,7 +354221,8 @@ "content": "甲、乙两人在一次射击比赛中各射靶$5$次, 两人成绩的条形统计图如图所示, 则下列命题正确的有\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.4]\n\\draw [->] (0,0) -- (0.1,0) -- (0.3,0.5) -- (0.7,-0.5) -- (0.9,0) -- (9,0) node [below right] {环数};\n\\draw [->] (0,0) -- (0,4) node [left] {频数};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {3,4,...,10}\n{\\draw ({\\i-2},0.3) -- ({\\i-2},0) node [below] {$\\i$};};\n\\foreach \\i in {1,2,3}\n{\\draw (0.3,\\i) -- (0,\\i) node [left] {$\\i$};};\n\\foreach \\i/\\j in {4/1,5/1,6/1,7/1,8/1}\n{\\filldraw [pattern = north east lines] ({\\i-2.3},0) --++ (0,\\j) --++ (0.6,0) --++ (0,-\\j);};\n\\draw (5,-2) node {(甲)};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,scale = 0.4]\n\\draw [->] (0,0) -- (0.1,0) -- (0.3,0.5) -- (0.7,-0.5) -- (0.9,0) -- (9,0) node [below right] {环数};\n\\draw [->] (0,0) -- (0,4) node [left] {频数};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {3,4,...,10}\n{\\draw ({\\i-2},0.3) -- ({\\i-2},0) node [below] {$\\i$};};\n\\foreach \\i in {1,2,3}\n{\\draw (0.3,\\i) -- (0,\\i) node [left] {$\\i$};};\n\\foreach \\i/\\j in {5/3,6/1,9/1}\n{\\filldraw [pattern = north east lines] ({\\i-2.3},0) --++ (0,\\j) --++ (0.6,0) --++ (0,-\\j);};\n\\draw (5,-2) node {(乙)};\n\\end{tikzpicture}\n\\end{center}\n\\textcircled{1} 甲的成绩的平均数小于乙的成绩的平均数; \\textcircled{2} 甲的成绩的中位数等于乙的成绩的中位数; \\textcircled{3} 甲的成绩的方差小于乙的成绩的方差.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-第二轮复习讲义-19_统计" ], "genre": "填空题", "ans": "\\textcircled{3}", @@ -351156,7 +354255,8 @@ "content": "将序号分别为$1,2,3,4,5$的$5$张参观券全部分给$4$人, 每人至少$1$张, 如果分给同一人的$2$张参观券连号, 那么不同的分法种数是\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-第二轮复习讲义-17_计数原理与二项式定理" ], "genre": "填空题", "ans": "$96$", @@ -351189,7 +354289,8 @@ "content": "为了考察某校各班参加课外书法小组的人数, 在全校随机抽取$5$个班级, 把每个班级参加该小组的人数作为样本数据. 已知样本平均数为$7$, 样本方差为$4$, 则该样本数据中最大值的所有可能值为\\blank{50}.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-第二轮复习讲义-19_统计" ], "genre": "填空题", "ans": "$8$或$10$或$11$", @@ -352364,7 +355465,8 @@ "content": "已知复数$(a+2 \\mathrm{i})(1+\\mathrm{i})$的实部为$0$, 其中$\\mathrm{i}$为虚数单位, 则实数的$a$值是\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-双基冲刺卷-双基训练01" ], "genre": "填空题", "ans": "$2$", @@ -352393,7 +355495,8 @@ "content": "设$a \\in \\mathbf{R}$, ``$x \\in\\{0,1,2\\}$''是``$x2$", @@ -352422,7 +355525,8 @@ "content": "已知一组数据为$85,87,88,90,92$, 则这组数据的第$60$百分位数为\\blank{50}.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-双基冲刺卷-双基训练01" ], "genre": "填空题", "ans": "$89$", @@ -352451,7 +355555,8 @@ "content": "方程$|x+1|+|x-2|=3$的解集为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-双基冲刺卷-双基训练01" ], "genre": "填空题", "ans": "$[-1,2]$", @@ -352480,7 +355585,8 @@ "content": "若$(x-\\dfrac{1}{\\sqrt{11}})^n$的展开式中第三项系数等于$6$, 则$n=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-双基冲刺卷-双基训练01" ], "genre": "填空题", "ans": "$12$", @@ -352509,7 +355615,8 @@ "content": "若$x, y \\in(0,+\\infty)$, 且$\\dfrac{1}{x}$与$2 y$的算术平均值为$\\dfrac{3}{2}$, 则$\\dfrac{1}{x}$与$y$的几何平均值的最大值为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-双基冲刺卷-双基训练01" ], "genre": "填空题", "ans": "$\\dfrac{3\\sqrt{2}}{4}$", @@ -352538,7 +355645,8 @@ "content": "已知数列$\\{a_n\\}$($n \\geq 1$, $n \\in \\mathbf{N}$)是等差数列, $S_n$是其前$n$项和. 若$a_2 a_5+a_8=0$, $S_9=27$, 则$S_8$的值是\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-双基冲刺卷-双基训练01" ], "genre": "填空题", "ans": "$16$", @@ -352567,7 +355675,8 @@ "content": "若正四棱锥的体积为$4 \\text{cm}^3$, 底面边长为$2 \\sqrt{3} \\text{cm}$, 则该正四棱锥的侧棱与底面所成角的大小为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-双基冲刺卷-双基训练01" ], "genre": "填空题", "ans": "$\\arctan \\dfrac{\\sqrt{6}}6$", @@ -352596,7 +355705,8 @@ "content": "投掷红、蓝两颗均匀的骰子, 设事件$A$: 蓝色骰子的点数为$5$或$6$; 事件$B$: 两骰子的点数之和大于$9$, 则在事件$B$发生的条件下事件$A$发生的概率$P(A | B)=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-双基冲刺卷-双基训练01" ], "genre": "填空题", "ans": "$\\dfrac 56$", @@ -352625,7 +355735,8 @@ "content": "已知椭圆$C: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的左、右焦点为$F_1, F_2$, 焦距$|F_1F_2|=2 c$($c>0$)过$F_2$的直线与圆$x^2+y^2=b^2$相切于点$A$, 并与椭圆$C$交于两点$P, Q$, 若$\\overrightarrow{PF_1} \\cdot \\overrightarrow{PF_2}=0$, 则椭圆$C$的离心率为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-双基冲刺卷-双基训练01" ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{5}}{3}$", @@ -352654,7 +355765,9 @@ "content": "已知函数$y=f(x)$, 其导函数$y=f'(x)$的图像如图所示, 则下列对函数$y=$$f(x)$表述不正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,1) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {1,2,3,4}\n{\\draw (\\i,0.1) -- (\\i,0);};\n\\draw [domain = -0.5:4.5,samples = 100] plot (\\x,{-\\x*(\\x-2)*(\\x-4)/8});\n\\draw (1,0) node [above] {$1$};\n\\draw (2,0) node [above] {$2$};\n\\draw (4,0) node [below] {$4$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{在$x=0$处取极小值}{在$x=2$处取极小值}{在$(0,2)$上为减函数}{在$(2,4)$上为增函数}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-双基冲刺卷-双基训练01", + "2023届高三-第二轮复习讲义-16_导数及其应用" ], "genre": "选择题", "ans": "A", @@ -352695,7 +355808,8 @@ "objs": [], "tags": [ "第八单元", - "第九单元" + "第九单元", + "2023届高三-双基冲刺卷-双基训练01" ], "genre": "选择题", "ans": "B", @@ -352724,7 +355838,8 @@ "content": "已知抛物线$y^2=2 p x(p>0)$的焦点为$F$, 点$P_1(x_1, y_1)$, $P_2(x_2, y_2)$, $P_3(x_3, y_3)$在抛物线上, 且$2 x_2=x_1+x_3$, 则有\\bracket{20}.\n\\twoch{$|FP_1|+|FP_2|=|FP_3|$}{$|FP_1|^2+|FP_2|^2=|FP_3|^2$}{$2|FP_2|=|FP_1|+|FP_3|$}{$|FP_2|^2=|FP_1| \\cdot|FP_3|$}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-双基冲刺卷-双基训练01" ], "genre": "选择题", "ans": "C", @@ -352753,7 +355868,8 @@ "content": "已知函数$y=f(x)$, 其中$f(x)=\\dfrac{a^x+1}{2^x}$($a>0$, $a \\neq 1$)是偶函数.\\\\\n(1) 求实数$a$的值;\\\\\n(2) 证明函数$y=f(x)$在$[0,+\\infty)$上严格递增.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-双基冲刺卷-双基训练01" ], "genre": "解答题", "ans": "(1) $a=4$; (2) 证明略", @@ -352782,7 +355898,8 @@ "content": "已知向量$\\overrightarrow {m}=(\\sin x,-1)$, 向量$\\overrightarrow {n}=(\\sqrt{3} \\cos x,-\\dfrac{1}{2})$, 函数$y=f(x)$, 其中$f(x)=(\\overrightarrow {m}+\\overrightarrow {n}) \\cdot \\overrightarrow {m}$.\\\\\n(1) 求单调递减区间;\\\\\n(2) 已知$a, b, c$分别为$\\triangle ABC$内角$A, B, C$的对边, $A$为锐角, $a=2 \\sqrt{3}$, $c=4$, 且$f(A)$恰是$y=f(x)$在$[0, \\dfrac{\\pi}{2}]$上的最大值, 求$A$, $b$和$\\triangle ABC$的面积$S$.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-双基冲刺卷-双基训练01" ], "genre": "解答题", "ans": "(1) $[k\\pi+\\dfrac\\pi 3,k\\pi+\\dfrac{5\\pi}6]$, $k\\in \\mathbf{Z}$; (2) $A=\\dfrac\\pi 3$, $b=2$, $S=2\\sqrt{3}$", @@ -352811,7 +355928,8 @@ "content": "$\\dfrac{1}{x}<1$的解集为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-双基冲刺卷-双基训练02" ], "genre": "填空题", "ans": "$(-\\infty,0)\\cup (1,+\\infty)$", @@ -352840,7 +355958,8 @@ "content": "在复平面内, $O$是坐标原点, 向量$\\overrightarrow{OZ_1}$, $\\overrightarrow{OZ_2}$对应的复数分别为$z_1=1-2 \\mathrm{i}$, $z_2=3+a \\mathrm{i}$($a \\in \\mathbf{R}$). 若$\\overrightarrow{OZ_1} \\perp \\overrightarrow{OZ_2}$, 则实数$a$的值为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-双基冲刺卷-双基训练02" ], "genre": "填空题", "ans": "$\\dfrac 32$", @@ -352870,7 +355989,8 @@ "content": "已知圆锥的母线长为$5$, 侧面积为$15 \\pi$, 则此圆锥的体积为\\blank{50}(结果保留$\\pi$).", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-双基冲刺卷-双基训练02" ], "genre": "填空题", "ans": "$12\\pi$", @@ -352902,7 +356022,8 @@ "content": "已知$\\theta$是第四象限角, 且$\\sin (\\theta+\\dfrac{\\pi}{4})=\\dfrac{3}{5}$, 则$\\tan (\\theta-\\dfrac{\\pi}{4})=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-双基冲刺卷-双基训练02" ], "genre": "填空题", "ans": "$-\\dfrac 43$", @@ -352931,7 +356052,8 @@ "content": "已知随机变量$X$的分布是$\\begin{pmatrix}0 & 1 \\\\ a & 3 a\\end{pmatrix}$, 则$D[X]=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-双基冲刺卷-双基训练02" ], "genre": "填空题", "ans": "$\\dfrac 3{16}$", @@ -352960,7 +356082,9 @@ "content": "过点$P(0,-\\mathrm{e})$作曲线$y=x \\ln x$的切线, 则切线方程是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-双基冲刺卷-双基训练02", + "2023届高三-赋能-赋能30" ], "genre": "填空题", "ans": "$y=2x-\\mathrm{e}$", @@ -352995,7 +356119,8 @@ "content": "若$(1+2 x)^n$展开式中含$x^3$项的系数等于含$x$项系数的$8$倍, 则正整数$n=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-双基冲刺卷-双基训练02" ], "genre": "填空题", "ans": "$5$", @@ -353024,7 +356149,8 @@ "content": "设直线$y=x+2 a$与圆$C: x^2+y^2-2 a y-2=0$相交于$A, B$两点, 若$|AB|=$$2 \\sqrt{3}$, 则圆$C$的面积为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-双基冲刺卷-双基训练02" ], "genre": "填空题", "ans": "$4\\pi$", @@ -353053,7 +356179,8 @@ "content": "已知函数$y=f(x)$, 其中$f(x)=\\begin{cases}|x^2-x|,& x \\leq 1, \\\\ -\\sqrt{x^2-1},& x>1,\\end{cases}$ 若不等式$f(x) \\geq a x-1$恒成立, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-双基冲刺卷-双基训练02" ], "genre": "填空题", "ans": "$-3\\le a\\le -1$", @@ -353082,7 +356209,8 @@ "content": "$36$的所有正约数之和可按如下方法得到 : 因为$36=2^2 \\times 3^2$, 所以$36$的所有正约数之和为$(1+3+3^2)+(2+2 \\times 3+2 \\times 3^2)+(2^2+2^2 \\times 3+2^2 \\times 3^2)=(1+2+2^2)(1+3+3^2)=91$, 参照上述方法, 可求得$4000$的所有正约数之和为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-双基冲刺卷-双基训练02" ], "genre": "填空题", "ans": "$9828$", @@ -353112,7 +356240,8 @@ "content": "有 $10$件产品, 其中$4$件是正品, 其余都是次品, 现不放回的从中依次抽$2$件, 则在第一次抽到次品的条件下, 第二次抽到次品的概率是\\bracket{20}.\n\\fourch{$\\dfrac 13$}{$\\dfrac 25$}{$\\dfrac 59$}{$\\dfrac 23$}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-双基冲刺卷-双基训练02" ], "genre": "选择题", "ans": "C", @@ -353141,7 +356270,8 @@ "content": "已知随机变量$X$服从二项分布, 记$X \\sim B(n, p)$, 若$4P(X=2)=3P(X=3)$, 则$p$的最大值为\\bracket{20}.\n\\fourch{$\\dfrac{5}{6}$}{$\\dfrac{4}{5}$}{$\\dfrac{3}{4}$}{$\\dfrac{2}{3}$}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-双基冲刺卷-双基训练02" ], "genre": "选择题", "ans": "B", @@ -353171,7 +356301,8 @@ "objs": [], "tags": [ "第二单元", - "第三单元" + "第三单元", + "2023届高三-双基冲刺卷-双基训练02" ], "genre": "选择题", "ans": "C", @@ -353200,7 +356331,8 @@ "content": "已知多面体$ABCDE$中, $DE \\perp$平面$ACD$, $AB\\parallel DE$, $AC=AD=CD=DE=2$, $AB=1$, $O$为$CD$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$C$} coordinate (C);\n\\draw ({2*sqrt(2)},0,0) node [right] {$E$} coordinate (E);\n\\draw ({sqrt(2)},0,{sqrt(2)}) node [below] {$D$} coordinate (D);\n\\draw ($(C)!0.5!(D)$) node [below] {$O$} coordinate (O);\n\\draw (O) ++ (0,{sqrt(3)},0) node [above] {$A$} coordinate (A);\n\\draw ($(A)+(E)-(D)$) coordinate (P);\n\\draw ($(A)!0.5!(P)$) node [above] {$B$} coordinate (B);\n\\draw (A)--(C)--(D)--(E)--(B)--cycle (O)--(A) (A)--(D) (B)--(D);\n\\draw [dashed] (B)--(C) (C)--(E);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证:$AO\\parallel$平面$BCE$;\\\\\n(2) 求直线$BD$与平面$BCE$所成角的正弦值.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-双基冲刺卷-双基训练02" ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\dfrac{\\sqrt{10}}5$", @@ -353229,7 +356361,8 @@ "content": "已知数列$\\{a_n\\}$是首项为$0$的严格增数列, 前$n$项和为$S_n$满足$S_n=\\dfrac{1}{2} a_n^2+\\dfrac{1}{2} a_n$($n \\geq 1$, $n \\in \\mathbf{N}$).\\\\\n(1) 求数列$\\{a_n\\}$的通项公式;\\\\\n(2) 设$b_n=\\dfrac{4}{15} \\cdot(-2)^{a_n}$($n \\geq 1$, $n \\in \\mathbf{N}$), 对任意的正整数$k$, 将集合$\\{b_{2 k-1}, b_{2 k}$, $b_{2 k+1}\\}$中的三个元素排成一个递增的等差数列, 其公差为$d_k$, 求证: 数列$\\{d_k\\}$为等比数列.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-双基冲刺卷-双基训练02" ], "genre": "解答题", "ans": "(1) $a_n=n-1$; (2) 证明略", @@ -353635,7 +356768,8 @@ "content": "已知函数$y=f(x)$, 其中函数$f(x)=a \\ln x+\\dfrac{b}{x}$, 当$x=1$时, 取得最大值$-2$, 则$f'(2)=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-16_导数及其应用" ], "genre": "填空题", "ans": "$-\\dfrac 12$", @@ -353934,7 +357068,8 @@ "content": "设$\\mathrm{i}$是虚数单位, 则$\\dfrac{1+\\mathrm{i}}{3+4\\mathrm{i}}$的虚部为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-双基冲刺卷-双基训练03" ], "genre": "填空题", "ans": "$-\\dfrac 1{25}$", @@ -353964,7 +357099,8 @@ "content": "已知角$\\theta$的终边过点$(1,-1)$, 则$\\sin \\theta=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-双基冲刺卷-双基训练03" ], "genre": "填空题", "ans": "$-\\dfrac{\\sqrt{2}}2$", @@ -353993,7 +357129,8 @@ "content": "数据$8$、$6$、$5$、$2$、$7$、$9$、$12$、$4$、$12$的第$40$百分位数是\\blank{50}.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-双基冲刺卷-双基训练03" ], "genre": "填空题", "ans": "$6$", @@ -354022,7 +357159,8 @@ "content": "已知向量$\\overrightarrow {a}=(3,6)$, $\\overrightarrow {b}=(3,-4)$, 则$\\overrightarrow {b}$在$\\overrightarrow {a}$方向上的投影向量的坐标为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-双基冲刺卷-双基训练03" ], "genre": "填空题", "ans": "$(-1,-2)$", @@ -354051,7 +357189,8 @@ "content": "设$x, y \\in(1,+\\infty)$, $\\log _2 x$, $\\log _2 y$的算术平均值为$1$, 则$2^x$, $2^y$的几何平均值的最小值为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-双基冲刺卷-双基训练03" ], "genre": "填空题", "ans": "$4$", @@ -354080,7 +357219,8 @@ "content": "已知函数$y=(x+a)^2 \\cdot \\sin x$是$\\mathbf{R}$上的奇函数, 则实数$a$的值为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-双基冲刺卷-双基训练03" ], "genre": "填空题", "ans": "$0$", @@ -354109,7 +357249,8 @@ "content": "某圆台的上、下底面圆的半径分别为$\\dfrac{3}{2}$、$5$, 且该圆台的体积为$139 \\pi$, 则该圆台的高为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-双基冲刺卷-双基训练03" ], "genre": "填空题", "ans": "$12$", @@ -354138,7 +357279,8 @@ "content": "幂函数$y=(m^2-3 m+3) x^{m^2-6 m+6}$在$(0,+\\infty)$上是严格增函数, 则$m$的值为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-双基冲刺卷-双基训练03" ], "genre": "填空题", "ans": "$1$", @@ -354167,7 +357309,8 @@ "content": "已知$F_1, B$分别是椭圆$C: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的左焦点和上顶点, 点$O$为坐标原点, 过$M(\\dfrac{a}{2}, 0)$作垂直于$x$轴的直线, 与椭圆$C$在第一象限的交点为$P$, 且$PO\\parallel F_1B$, 则椭圆$C$的离心率为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-双基冲刺卷-双基训练03" ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{3}}3$", @@ -354196,7 +357339,8 @@ "content": "设函数$y=f(x)$的定义域为$D$, 若函数$y=f(x)$满足条件: 存在$[a, b] \\subseteq D$, 使$y=f(x)$在$[a, b]$上的值域是$[\\dfrac{a}{2}, \\dfrac{b}{2}]$, 则称函数$y=f(x)$为``倍缩函数''. 若函数$y=\\log _3(3^x+t)$为``倍缩函数'', 则实数$t$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-双基冲刺卷-双基训练03" ], "genre": "填空题", "ans": "$(0,\\dfrac 14)$", @@ -354225,7 +357369,8 @@ "content": "已知数列$\\{a_n\\}$是等差数列, 其前$n$项和为$S_n$, 且$a_1=1, S_8=4S_4$, 若$a_k+a_3=18$, 则$k$的值为\\bracket{20}.\n\\fourch{$6$}{$7$}{$8$}{$9$}", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-双基冲刺卷-双基训练03" ], "genre": "选择题", "ans": "B", @@ -354254,7 +357399,8 @@ "content": "设函数$y=f(x)$, 其中$f(x)=2 k \\sin (x-1) \\cos (1-x), k$是非零实数, 则下列说法错误的是\\bracket{20}.\n\\onech{函数$y=f(x)$的最大值为$k$}{把函数$y=g(x)$, 其中$g(x)=k \\sin (x-2)$图像上的每个点的纵坐标不变, 横坐标变成原来的一半, 可得到函数$y=f(x)$的图像}{把函数$y=h(x)$, 其中$h(x)=k \\sin 2 x$的图像向右平移一个单位, 可得到函数$y=f(x)$的图像}{直线$x=1+\\dfrac{\\pi}{4}$是函数$y=f(x)$的一条对称轴}", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-双基冲刺卷-双基训练03" ], "genre": "选择题", "ans": "A", @@ -354283,7 +357429,8 @@ "content": "双纽线最早于 1694 年被瑞士数学家雅各布·伯努利用来描述他所发现的曲线. 经研究发现, 在平面直角坐标系$x O y$中, 到定点$A(-a, 0)$, $B(a, 0)$距离之积等于$a^2$($a>0$)的点的轨迹是双纽线$C$, 若点$P(x_0, y_0)$是轨迹$C$上一点, 则下列说法不正确的是\\bracket{20}.\n\\onech{曲线$C$关于原点$O$成中心对称}{$x_0$的取值范围是$[-\\sqrt{2} a, \\sqrt{2} a]$}{曲线$C$上有且仅有一个点$P$满足$|PA|=|PB|$}{$|PO|^2-a^2$的最大值为$2 a^2$}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-双基冲刺卷-双基训练03" ], "genre": "选择题", "ans": "D", @@ -354312,7 +357459,8 @@ "content": "已知圆$M: x^2+(y-2)^2=1$, 点$P$是直线$l: x+2 y=0$上的一动点, 过点$P$作圆$M$的切线$PA, PB$, 切点为$A, B$.\\\\\n(1) 当切线$PA$的长度为$\\sqrt{3}$时, 求点$P$的坐标;\\\\\n(2) 若$\\triangle PAM$的外接圆为圆$N$, 试问: 当$P$运动时, 圆$N$是否过定点 (不在坐标轴上)? 若存在, 求出所有的定点的坐标; 若不存在, 请说明理由.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-双基冲刺卷-双基训练03" ], "genre": "解答题", "ans": "(1) $P(0,0)$或$P(-\\dfrac 85,\\dfrac 45)$; (2) 过定点$(-\\dfrac 45,\\dfrac 25)$", @@ -354341,7 +357489,8 @@ "content": "甲、乙两人组成``星队''参加趣味知识竞赛. 比赛分两轮进行, 每轮比赛答一道趣味题. 在第一轮比赛中, 答对题者得$2$分, 答错题者得$0$分; 在第二轮比赛中, 答对题者得$3$分, 答错题者得$0$分. 已知甲、乙两人在第一轮比赛中答对题的概率都为$p$, 在第二轮比赛中答对题的概率都为$q$. 且在两轮比赛中答对与否互不影响. 设定甲、乙两人先进行第一轮比赛, 然后进行第二轮比赛, 甲、乙两人的得分之和为``星队''总得分. 已知在一次比赛中甲得$2$分的概率为$\\dfrac{1}{2}$, 乙得$5$分的概率为$\\dfrac{1}{6}$.\\\\\n(1) 求$p, q$的值;\\\\\n(2) 求``星队''在一次比赛中的总得分为$5$分的概率.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-双基冲刺卷-双基训练03" ], "genre": "解答题", "ans": "(1) $p=\\dfrac 23$, $q=\\dfrac 14$; (2) $\\dfrac 16$", @@ -354370,7 +357519,8 @@ "content": "设集合$M=\\{1,3,5,7,9\\}, N=\\{x | 2 x>7\\}$, 则$M \\cap N=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-双基冲刺卷-双基训练04" ], "genre": "填空题", "ans": "$\\{5,7,9\\}$", @@ -354399,7 +357549,8 @@ "content": "在用反证法证明``已知$a^3+b^3=2$, 求证: $a+b \\leq 2$''时应先假设\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-双基冲刺卷-双基训练04" ], "genre": "填空题", "ans": "$a+b>2$", @@ -354428,7 +357579,8 @@ "content": "函数$y=\\dfrac{1}{x+1}+\\ln x$的定义域是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-双基冲刺卷-双基训练04" ], "genre": "填空题", "ans": "$(0,+\\infty)$", @@ -354457,7 +357609,8 @@ "content": "若$\\cos x \\cos y+\\sin x \\sin y=\\dfrac{1}{2}$, $\\sin 2 x+\\sin 2 y=\\dfrac{2}{3}$, 则$\\sin (x+y)=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-双基冲刺卷-双基训练04" ], "genre": "填空题", "ans": "$\\dfrac 23$", @@ -354486,7 +357639,8 @@ "content": "记$S_n$为等差数列$\\{a_n\\}$的前$n$项和. 若$a_1=-2$, $a_2+a_6=2$, 则$S_{10}=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-双基冲刺卷-双基训练04" ], "genre": "填空题", "ans": "$25$", @@ -354515,7 +357669,8 @@ "content": "已知问题: ``$|x+3|+|x-a| \\geq 5$恒成立, 求实数$a$的取值范围''. 两位同学对此问题展开讨论: 小明说可以分类讨论, 将不等式左边的两个绝对值打开; 小新说可以利用三角不等式解决问题. 请你选择一个适合自己的方法求解此题, 并写出实数$a$的取值范围\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-双基冲刺卷-双基训练04" ], "genre": "填空题", "ans": "$(-\\infty,-8]\\cup [2,+\\infty)$", @@ -354544,7 +357699,8 @@ "content": "设函数$y=f(x)$的定义域为$\\mathbf{R}$, $f(x+1)$为奇函数, $f(x+2)$为偶函数, 当$x \\in$$[1,2]$时, $f(x)=a x^2+b$, 若$f(0)+f(3)=6$, 则$f(\\dfrac{9}{2})=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-双基冲刺卷-双基训练04" ], "genre": "填空题", "ans": "$\\dfrac 52$", @@ -354573,7 +357729,8 @@ "content": "某超市为庆祝开业举办酬宾抽奖活动, 凡在开业当天进店的顾客, 都能抽一次奖, 每位进店的顾客得到一个不透明的盒子, 盒子里装有红、黄、蓝三种颜色的小球共$6$个, 其中红球$2$个, 黄球$3$个, 蓝球$1$个, 除颜色外, 小球的其它方面, 诸如形状、大小、质地等完全相同, 每个小球上均写有获奖内容, 顾客先从自己得到的盒子里随机取出$2$个小球, 然后再依据取出的$2$个小球上的获奖内容去兑奖. 设$X$表示某顾客在一次抽奖时, 从自己得到的那个盒子取出的$2$个小球中红球的个数, 则$X$的数学期望$E[X]=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-双基冲刺卷-双基训练04" ], "genre": "填空题", "ans": "$\\dfrac 23$", @@ -354602,7 +357759,8 @@ "content": "给出下列命题:\\\\\n\\textcircled{1} 若两条不同的直线同时垂直于第三条直线, 则这两条直线互相平行;\\\\\n\\textcircled{2} 若两个不同的平面同时垂直于同一条直线, 则这两个平面互相平行;\\\\\n\\textcircled{3} 若两条不同的直线同时垂直于同一个平面, 则这两条直线互相平行;\\\\\n\\textcircled{4} 若两个不同的平面同时垂直于第三个平面, 则这两个平面互相垂直. 其中所有正确命题的序号为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-双基冲刺卷-双基训练04" ], "genre": "填空题", "ans": "\\textcircled{2}\\textcircled{3}", @@ -354631,7 +357789,8 @@ "content": "在$\\triangle ABC$中, 角$A, B, C$所对的边分别为$a, b, c$, $\\angle ABC=120^{\\circ}$, $\\angle ABC$的平分线交$AC$于点$D$, 且$BD=1$, 则$4 a+c$的最小值为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-双基冲刺卷-双基训练04" ], "genre": "填空题", "ans": "$9$", @@ -354660,7 +357819,8 @@ "content": "已知非零向量$\\overrightarrow {a}, \\overrightarrow {b}, \\overrightarrow {c}$, 则``$\\overrightarrow {a} \\cdot \\overrightarrow {c}=\\overrightarrow {b} \\cdot \\overrightarrow {c}$''是``$\\overrightarrow {a}=\\overrightarrow {b}$''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充分必要条件}{既不充分又不必要条件}", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-双基冲刺卷-双基训练04" ], "genre": "选择题", "ans": "B", @@ -354689,7 +357849,8 @@ "content": "从某网络平台推荐的影视作品中抽取$400$部, 统计其评分数据, 将所得$400$个评分数据分为$8$组: $[66,70)$, $[70,74)$, $\\cdots$, $[94,98]$, 并整理得到如下的频率分布直方图, 则评分在区间$[82,86)$内的影视作品数量是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) --++ (0.1,0) --++ (0.05,0.15) --++ (0.1,-0.3) --++ (0.05,0.15) -- (6.5,0) node [below] {评分};\n\\draw [->] (0,0) -- (0,4) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {0.020,0.025,0.030,0.035,0.040,0.045,0.050}\n{\\draw (0.1,{60*\\i}) -- (0,{60*\\i}) node [left] {$\\i$};};\n\\foreach \\i/\\j/\\k in {1.5/66/0.035,2/70/0.02,2.5/74/0.03,3/78/0.04,3.5/82/0.05,4/86/0.025,4.5/90/0.03,5/94/0.02}\n{\\draw [dashed] (0,{\\k*60}) --++ (\\i,0);\n\\draw [thick] (\\i,0) node [below] {$\\j$} --++ (0,{60*\\k}) --++ (0.5,0) --++ (0,{-60*\\k});\n};\n\\draw (5.5,0) node [below] {$98$};\n\\end{tikzpicture}\n\\end{center} \n\\fourch{$20$}{$40$}{$64$}{$80$}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-双基冲刺卷-双基训练04" ], "genre": "选择题", "ans": "D", @@ -354718,7 +357879,8 @@ "content": "若双曲线$C: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的一条渐近线与直线$y=2 x+1$垂直, 则$C$的离心率为\\bracket{20}.\n\\fourch{$5$}{$\\sqrt{5}$}{$\\dfrac{5}{4}$}{$\\dfrac{\\sqrt{5}}{2}$}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-双基冲刺卷-双基训练04" ], "genre": "选择题", "ans": "D", @@ -354747,7 +357909,8 @@ "content": "已知函数$y=x \\ln x$和$y=m(x^2-1)$($m \\in \\mathbf{R}$).\\\\\n(1) 当$m=1$时, 求方程$x \\ln x=m(x^2-1)$的实根;\\\\\n(2) 若对任意的$x \\in(1,+\\infty)$, 函数$y=m(x^2-1)$($m \\in \\mathbf{R}$)的图像总在函数$y=x \\ln x$的图像的上方, 求实数$m$的取值范围.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-双基冲刺卷-双基训练04" ], "genre": "解答题", "ans": "(1) $x=1$; (2) $[\\dfrac 12,+\\infty)$", @@ -354776,7 +357939,8 @@ "content": "已知椭圆$C: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的离心率为$\\dfrac{\\sqrt{2}}{2}$, 且经过点$(\\sqrt{2}, 1)$, 直线$l$经过$P(0,1)$, 且与椭圆$C$相交于$A$、$B$两点.\\\\\n(1) 求椭圆$C$的标准方程;\\\\\n(2) 当$|AB|=3$, 求此时直线$l$的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-双基冲刺卷-双基训练04" ], "genre": "解答题", "ans": "(1) $\\dfrac{x^2}{4}+\\dfrac{y^2}{2}=1$; (2) $y=\\pm \\dfrac{\\sqrt{2}}2x+1$", @@ -354805,7 +357969,8 @@ "content": "已知集合$A=\\{-1,3,2 m-1\\}$, 集合$B=\\{3, m^2\\}$, 若$B \\subseteq A$, 则实数$m=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-双基冲刺卷-双基训练05" ], "genre": "填空题", "ans": "$1$", @@ -354837,7 +358002,8 @@ "content": "直线$x+\\sqrt{3} y+1=0$的倾斜角的大小是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-双基冲刺卷-双基训练05" ], "genre": "填空题", "ans": "$\\dfrac{5\\pi}6$", @@ -354866,7 +358032,8 @@ "content": "函数$y=\\sin x \\cdot \\cos x$的最小正周期是\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-双基冲刺卷-双基训练05" ], "genre": "填空题", "ans": "$\\pi$", @@ -354895,7 +358062,8 @@ "content": "设椭圆$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的焦距为$2c$, 若$b^2=a c$, 则椭圆的离心率为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-双基冲刺卷-双基训练05" ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{5}-1}2$", @@ -354924,7 +358092,8 @@ "content": "已知两个球的表面积之比为$1: 2$, 则这两个球的体积之比为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-双基冲刺卷-双基训练05" ], "genre": "填空题", "ans": "$1:2\\sqrt{2}$", @@ -354953,7 +358122,8 @@ "content": "过点$M(-6,3)$且和双曲线$x^2-2 y^2=2$有相同的渐近线的双曲线方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-双基冲刺卷-双基训练05" ], "genre": "填空题", "ans": "$\\dfrac{x^2}{18}-\\dfrac{y^2}{9}=1$", @@ -354982,7 +358152,8 @@ "content": "已知甲、乙两位射手, 甲击中目标的概率为$0.7$, 乙击中目标的概率为$0.6$, 如果甲乙两位射手的射击相互独立, 那么甲乙两射手同时瞄准一个目标射击, 目标被射中的概率为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-双基冲刺卷-双基训练05" ], "genre": "填空题", "ans": "$0.88$", @@ -355012,7 +358183,8 @@ "content": "设随机变量$X$服从正态分布$N(0,1)$, 已知$P(X<-1.96)=0.025$, 则$P(|X|<1. 96)=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-双基冲刺卷-双基训练05" ], "genre": "填空题", "ans": "$0.95$", @@ -355041,7 +358213,8 @@ "content": "已知一组数据: $x_1 \\leq x_2 \\leq \\cdots \\leq x_{10}$, 且$x_i \\in\\{1,2,3,4,5,6,7,8,9,10\\}$($i=1,2,3, \\cdots,10$), 这组数据的中位数是$5$, 则这组数据的平均数的最大可能值是\\blank{50}.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-双基冲刺卷-双基训练05" ], "genre": "填空题", "ans": "$7$", @@ -355070,7 +358243,8 @@ "content": "如果$|x+1|+|x+9|>a$对任意实数$x$总成立, 那么$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-双基冲刺卷-双基训练05" ], "genre": "填空题", "ans": "$(-\\infty,8)$", @@ -355099,7 +358273,8 @@ "content": "已知平面$\\alpha$与平面$\\beta, \\gamma$都相交, 则这三个平面可能的交线有\\bracket{20}.\n\\fourch{$1$条或$2$条}{$2$条或$3$条}{$1$条或$3$条}{$1$条或$2$条或$3$条}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-双基冲刺卷-双基训练05" ], "genre": "选择题", "ans": "D", @@ -355128,7 +358303,8 @@ "content": "已知$y=\\begin{cases}(3 a-1) x+4 a,& x \\leq 1, \\\\ \\log _a x,& x>1\\end{cases}$在$(-\\infty,+\\infty)$上是严格减函数, 那么$a$的取值范围是\\bracket{20}.\n\\fourch{$[\\dfrac{1}{7}, \\dfrac{1}{3})$}{$[\\dfrac{1}{7}, 1)$}{$(0,1)$}{$(0, \\dfrac{1}{3})$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-双基冲刺卷-双基训练05" ], "genre": "选择题", "ans": "A", @@ -355157,7 +358333,8 @@ "content": "如图, 正方体$ABCD-A_1B_1C_1D_1$, 动点$M$从$B_1$点出发, 在正方体表面匀速运动一周后, 再回到$B_1$的运动过程中, 点$M$与平面$A_1DC_1$的距离保持不变, 运动的路程$x$与$l=MA_1+MC_1+$$MD$之间满足函数关系$l=f(x)$, 则此函数图像大致是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$B$} coordinate (B);\n\\draw (B) ++ (\\l,0,0) node [below right] {$C$} coordinate (C);\n\\draw (B) ++ (\\l,0,-\\l) node [right] {$D$} coordinate (D);\n\\draw (B) ++ (0,0,-\\l) node [left] {$A$} coordinate (A);\n\\draw (B) -- (C) -- (D);\n\\draw [dashed] (B) -- (A) -- (D);\n\\draw (B) ++ (0,\\l,0) node [left] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above right] {$D_1$} coordinate (D1);\n\\draw (A) ++ (0,\\l,0) node [above left] {$A_1$} coordinate (A1);\n\\draw (B1) -- (C1) -- (D1) -- (A1) -- cycle;\n\\draw (D) -- (D1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (A) -- (A1);\n\\draw (A1)--(C1)--(D);\n\\draw [dashed] (A1)--(D);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{\\begin{tikzpicture}[>=latex,scale = 0.8]\n\\draw [->] (0,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,2) node [left] {$l$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0:1] plot ({1.5*\\x},{(sqrt(1+2*\\x*\\x)+sqrt(\\x*\\x+(\\x-1)*(\\x-1))+sqrt(1+2*(\\x-1)*(\\x-1)))/3});\n\\draw [domain = 0:1] plot ({(\\x+1)*1.5},{(sqrt(1+2*\\x*\\x)+sqrt(\\x*\\x+(\\x-1)*(\\x-1))+sqrt(1+2*(\\x-1)*(\\x-1)))/3});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.8]\n\\draw [->] (0,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,2) node [left] {$l$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0:1] plot ({1.5*\\x},{2.5-(sqrt(1+2*\\x*\\x)+sqrt(\\x*\\x+(\\x-1)*(\\x-1))+sqrt(1+2*(\\x-1)*(\\x-1)))/3});\n\\draw [domain = 0:1] plot ({(\\x+1)*1.5},{2.5-(sqrt(1+2*\\x*\\x)+sqrt(\\x*\\x+(\\x-1)*(\\x-1))+sqrt(1+2*(\\x-1)*(\\x-1)))/3});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.8]\n\\draw [->] (0,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,2) node [left] {$l$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0:1] plot (\\x,{(sqrt(1+2*\\x*\\x)+sqrt(\\x*\\x+(\\x-1)*(\\x-1))+sqrt(1+2*(\\x-1)*(\\x-1)))/3});\n\\draw [domain = 0:1] plot (\\x+1,{(sqrt(1+2*\\x*\\x)+sqrt(\\x*\\x+(\\x-1)*(\\x-1))+sqrt(1+2*(\\x-1)*(\\x-1)))/3});\n\\draw [domain = 0:1] plot (\\x+2,{(sqrt(1+2*\\x*\\x)+sqrt(\\x*\\x+(\\x-1)*(\\x-1))+sqrt(1+2*(\\x-1)*(\\x-1)))/3});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.8]\n\\draw [->] (0,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,2) node [left] {$l$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0:1] plot (\\x,{2.5-(sqrt(1+2*\\x*\\x)+sqrt(\\x*\\x+(\\x-1)*(\\x-1))+sqrt(1+2*(\\x-1)*(\\x-1)))/3});\n\\draw [domain = 0:1] plot (\\x+1,{2.5-(sqrt(1+2*\\x*\\x)+sqrt(\\x*\\x+(\\x-1)*(\\x-1))+sqrt(1+2*(\\x-1)*(\\x-1)))/3});\n\\draw [domain = 0:1] plot (\\x+2,{2.5-(sqrt(1+2*\\x*\\x)+sqrt(\\x*\\x+(\\x-1)*(\\x-1))+sqrt(1+2*(\\x-1)*(\\x-1)))/3});\n\\end{tikzpicture}}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-双基冲刺卷-双基训练05" ], "genre": "选择题", "ans": "C", @@ -355186,7 +358363,8 @@ "content": "某蔬菜中转厂的每日进货的蔬菜量最多不超过 20 吨, 由于蔬菜采购, 运输, 管理等因素, 蔬菜每日浪费率$p$与日进货量$x$(吨) 之间近似地满足关系式$p=\\begin{cases}\\dfrac{2}{15-x}, & 1 \\leq x \\leq 9, \\\\ \\dfrac{x^2+60}{540},& 10 \\leq x \\leq 20, \\end{cases}x \\in \\mathbf{N}$(日浪费率$=\\dfrac{\\text {日浪费量}}{\\text {日进货量}} \\times 100 \\%$), 已知售出一吨蔬菜可赢利$2$千元, 而浪费一吨蔬菜则亏损$1$千元(蔬菜中转厂的日利润$y=$日售出赢利额$-$日浪费亏损额).\\\\\n(1) 将该蔬果中转厂的日利润$y$(千元) 表示成日进货量$x$(吨)的函数;\\\\\n(2) 当该蔬菜中转厂的日进货量为多少吨时, 日利润最大? 最大日利润是几千元?", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-双基冲刺卷-双基训练05" ], "genre": "解答题", "ans": "(1) $y=\\begin{cases}\\dfrac{24x-2x^2}{15-x}, & x\\in \\{1,2,\\cdots,9\\},\\\\ \\dfrac 53 x-\\dfrac{x^3}{180},& x\\in \\{10,11,\\cdots,20\\};\\end{cases}$ (2) 日进货量为$10$吨时, 日利润最大, 最大日利润是$\\dfrac{100}{9}$千元", @@ -355216,7 +358394,8 @@ "content": "已知函数$y=A \\sin (\\omega x+\\varphi)+B$($A>0$, $\\omega>0$, $|\\varphi|<\\dfrac{\\pi}{2}$)的部分图像如图所示.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.5,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = {-pi/12}:2.5,samples = 100] plot (\\x,{2*sin(2*\\x/pi*180+60)-1});\n\\draw [dashed] (0,1) -- ({pi/12},1) -- ({pi/12},0) (0,-3) -- ({7*pi/12},-3) -- ({7*pi/12},0);\n\\draw (0,1) node [left] {$1$} (0,-3) node [left] {$-3$};\n\\draw ({pi/12},0) node [below] {$\\dfrac{\\pi}{12}$} ({7*pi/12},0) node [above] {$\\dfrac{7\\pi}{12}$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求该函数的解析式;\\\\\n(2) 求该函数的单调递增区间; 当$x \\in[-\\dfrac{\\pi}{6}, \\dfrac{\\pi}{3}]$时, 求该函数的取值范围.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-双基冲刺卷-双基训练05" ], "genre": "解答题", "ans": "(1) $y=2\\sin(2x+\\dfrac\\pi 3)-1$; (2) 单调增区间为$[-\\dfrac{5\\pi}{12}+k\\pi,\\dfrac{\\pi}{12}+k\\pi]$, $k \\in \\mathbf{Z}$, 取值范围为$[-1,1]$", @@ -355575,7 +358754,8 @@ "content": "函数$y=\\dfrac{1}{\\lg x}+\\sqrt{2-x}$的定义域是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-双基冲刺卷-双基训练06" ], "genre": "填空题", "ans": "$(0,1)\\cup (1,2]$", @@ -355604,7 +358784,8 @@ "content": "已知向量$\\overrightarrow {a}=(-2,-3)$, $\\overrightarrow {b}=(-4,7)$, 则向量$\\overrightarrow {b}$在向量$\\overrightarrow {a}$的方向上的数量投影为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-双基冲刺卷-双基训练06" ], "genre": "填空题", "ans": "$-\\sqrt{13}$", @@ -355633,7 +358814,8 @@ "content": "已知双曲线的两条渐近线互相垂直, 则该双曲线的离心率为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-双基冲刺卷-双基训练06" ], "genre": "填空题", "ans": "$\\sqrt{2}$", @@ -355662,7 +358844,8 @@ "content": "从某小区抽取$100$名居民用户进行用电量调查, 发现他们的用电量都在$50 \\sim 350$(单位: $\\text{kW} \\cdot \\text{h}$)之间, 进行适当分组(每组为左闭右开的区间), 画出频率直方图如图所示, 则$x$的值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (8,0) node [below right] {月用电量/($\\text{kW}\\cdot\\text{h}$)};\n\\draw [->] (0,0) -- (0,3) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j in {0.0012,0.0021,0.0036,0.005/x,0.006}\n{\\draw (0.1,{300*\\i}) -- (0,{300*\\i}) node [left] {$\\j$};};\n\\foreach \\i/\\j/\\k in {1/50/0.0021,2/100/0.0036,3/150/0.006,4/200/0.005,5/250/0.0021,6/300/0.0012}\n{\\draw [dashed] (0,{\\k*300}) --++ (\\i,0);\n\\draw [thick] (\\i,0) node [below] {$\\j$} --++ (0,{300*\\k}) --++ (1,0) --++ (0,{-300*\\k});\n};\n\\draw (7,0) node [below] {$350$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-双基冲刺卷-双基训练06" ], "genre": "填空题", "ans": "$0.005$", @@ -355691,7 +358874,8 @@ "content": "若已知随机变量$X$服从二项分布$B(90, p)$, 且$E[2X+1]=61$, 则$D[X]=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-双基冲刺卷-双基训练06" ], "genre": "填空题", "ans": "$20$", @@ -355720,7 +358904,8 @@ "content": "函数$y=x^3-3 x$在$[-2,2]$上的最大值为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-双基冲刺卷-双基训练06" ], "genre": "填空题", "ans": "$2$", @@ -355749,7 +358934,8 @@ "content": "已知正项等比数列$\\{a_n\\}$, 若$a_1 \\cdot a_2 \\cdot \\cdots \\cdot a_7 \\cdot a_8=16$, 则$a_4+a_5$的最小值为\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-双基冲刺卷-双基训练06" ], "genre": "填空题", "ans": "$2\\sqrt{2}$", @@ -355778,7 +358964,8 @@ "content": "已知$F$是抛物线$y^2=2 p x$($p>0$)的焦点, 点$P$在抛物线上且横坐标为$8$, $O$为坐标原点, 若$\\triangle OFP$的面积为$2 \\sqrt{2}$, 则该抛物线的准线方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-双基冲刺卷-双基训练06" ], "genre": "填空题", "ans": "$x=-1$", @@ -355807,7 +358994,8 @@ "content": "已知函数$y=f(x)$, 其中$f(x)=2 \\sin (\\omega x+\\varphi)$($\\omega>0$), 若存在$x_0 \\in \\mathbf{R}$, 使得$f(x_0+2)-f(x_0)=4$, 则$\\omega$的最小值为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-双基冲刺卷-双基训练06" ], "genre": "填空题", "ans": "$\\dfrac\\pi 2$", @@ -355837,7 +359025,8 @@ "objs": [], "tags": [ "第一单元", - "第二单元" + "第二单元", + "2023届高三-双基冲刺卷-双基训练06" ], "genre": "填空题", "ans": "$-1$", @@ -355866,7 +359055,8 @@ "content": "在$2022$北京冬奥会单板滑雪U型场地技巧比赛中, $6$名评委给$4$选手打出了$6$个各不相同的原始分, 经过``去掉一个最高分和一个最低分''处理后, 得到$4$个有效分, 则仅处理后的$4$个有效分与$6$个原始分相比, 一定会变小的数字特征是\\bracket{20}.\n\\fourch{平均数}{中位数}{众数}{方差}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-双基冲刺卷-双基训练06" ], "genre": "选择题", "ans": "D", @@ -355895,7 +359085,8 @@ "content": "如图是函数$f(x)$的导函数$f'(x)$的图像, 则下列判断正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.4]\n\\draw [->] (-4,0) -- (6,0) node [below] {$x$};\n\\draw [->] (0,-6) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -3.5:5.2,samples = 100] plot (\\x,{(\\x+1)*(\\x-2)*(\\x-4)/20});\n\\foreach \\i/\\j in {-3/above,-2/above,1/below,3/above,5/below}\n{\\draw [dashed] (\\i,{(\\i+1)*(\\i-2)*(\\i-4)/20}) -- (\\i,0);\n\\draw (\\i,0) node [\\j] {\\tiny $\\i$};\n};\n\\filldraw (-1,0) circle (0.03) node [above] {\\tiny $-1$};\n\\filldraw (2,0) circle (0.03) node [above] {\\tiny $2$};\n\\filldraw (4,0) circle (0.03) node [above] {\\tiny $4$};\n\\end{tikzpicture}\n\\end{center}\n\\twoch{$f(x)$在$(-3,1)$上是严格增函数}{$f(x)$在$(1,+\\infty)$上是严格减函数}{$f(x)$在$[-3,4]$上的最大值是$f(1)$}{当$x=4$时, $f(x)$取得极小值}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-双基冲刺卷-双基训练06" ], "genre": "选择题", "ans": "D", @@ -355924,7 +359115,8 @@ "content": "甲乙两位同学同时解关于$x$的方程: $\\log _3 x-b \\log _x 3+c=0$, 甲写错了常数$b$, 得到两根为$3$和$\\dfrac{1}{9}$, 乙写错了常数$c$, 得到两根为$\\dfrac{1}{27}$和$81$, 则这个方程的两根应该是\\bracket{20}.\n\\fourch{$9$和$\\dfrac{1}{3}$}{$3$和$\\dfrac{1}{27}$}{$27$和$\\dfrac{1}{81}$}{$81$和$\\dfrac{1}{9}$}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-双基冲刺卷-双基训练06" ], "genre": "选择题", "ans": "C", @@ -355953,7 +359145,8 @@ "content": "如图, $ABCD$为圆柱$OO'$的轴截面, $EF$是圆柱上异于$AD, BC$的母线.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (-1,0) node [left] {$A$} coordinate (A);\n\\draw (1,0) node [right] {$B$} coordinate (B);\n\\draw (1,2) node [right] {$C$} coordinate (C);\n\\draw (-1,2) node [left] {$D$} coordinate (D);\n\\filldraw ($(A)!0.5!(B)$) circle (0.03) node [below] {$O$} coordinate (O);\n\\filldraw ($(C)!0.5!(D)$) circle (0.03) node [below right] {$O'$} coordinate (O');\n\\draw (O') ellipse (1 and 0.4);\n\\draw (A) arc (180:360:1 and 0.4);\n\\draw [dashed] (A) arc (180:0:1 and 0.4);\n\\draw (A)--(D)--(C)--(B);\n\\draw [dashed] (A)--(B);\n\\draw ({cos(115)},{0.4*sin(115)}) node [below] {$E$} coordinate (E);\n\\draw (E) ++ (0,2) node [above] {$F$} coordinate (F);\n\\draw [dashed] (E)--(F) (B)--(E)--(D) (B)--(D) (B)--(F);\n\\draw (D)--(F);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $BE \\perp$平面$DEF$;\\\\\n(2) 若$AB=BC=\\sqrt{6}$, $E, F$分别是$\\overset\\frown{AB}, \\overset\\frown{CD}$的中点, 求二面角$B-DF-E$的正弦值.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-双基冲刺卷-双基训练06" ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\dfrac{\\sqrt{3}}3$", @@ -355982,7 +359175,8 @@ "content": "如图, $A$、$B$、$C$、$D$都在同一个与水平面垂直的平面内, $B$、$D$为两岛上的两座灯塔的塔顶, 测量船于水面$A$处测得$B$点和$D$点的仰角分别为$75^{\\circ}$和$30^{\\circ}$, 于水面$C$处测得$B$点和$D$点的仰角均为$60^{\\circ}$, 且测得$AC=0.1 \\text{km}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.3]\n\\draw (0,0) node [below] {$A$} coordinate (A);\n\\draw (1,0) node [below] {$C$} coordinate (C);\n\\path [name path = AB] (A) --++ (105:3.5);\n\\path [name path = CB] (C) --++ (120:4);\n\\path [name path = CD] (C) --++ (60:1.1);\n\\path [name path = AD] (A) --++ (30:1.8);\n\\path [name intersections = {of = AB and CB, by = B}];\n\\path [name intersections = {of = AD and CD, by = D}];\n\\draw (A)--(D) node [above] {$D$} (A)--(B) node [above] {$B$} (C)--(D) (C)--(B) (B)--(D);\n\\draw ($(A)!-0.6!(C)$) coordinate (S) -- ($(A)!1.3!(C)$) coordinate (T);\n\\filldraw [pattern = north east lines] (B) ++ (0,-0.5) --++ (-1,-0.6) coordinate (P) -- ($(S)!(P)!(T)$) -- (S) --cycle;\n\\draw [ultra thick] (B)--++(0,-0.5);\n\\filldraw [pattern = north east lines] (D) ++ (0,-0.3) --++ (0.2,-0.3) coordinate (Q) -- ($(S)!(Q)!(T)$) -- (T) -- cycle;\n\\draw [ultra thick] (D)--++(0,-0.3);\n\\draw (A) pic [draw,\"\\tiny $30^\\circ$\",angle eccentricity = 1.9,scale = 0.5] {angle = C--A--D};\n\\draw (A) pic [draw,\"\\tiny $75^\\circ$\",angle eccentricity = 1.9,scale = 0.5] {angle = B--A--S};\n\\draw (A) pic [draw,\"\\tiny $60^\\circ$\",angle eccentricity = 1.9,scale = 0.5] {angle = B--C--A};\n\\draw (A) pic [draw,\"\\tiny $60^\\circ$\",angle eccentricity = 1.9,scale = 0.5] {angle = T--C--D};\n\\end{tikzpicture}\n\\end{center}\n(1) 试研究$B$、$D$间的距离与另外哪两点间的距离相等;\\\\\n(2) 计算$B$、$D$间的距离. (结果精确到$0.01 \\text{km}$)", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-双基冲刺卷-双基训练06" ], "genre": "解答题", "ans": "(1) $BD=BA$; (2) 约$0.33\\text{km}$", @@ -356671,7 +359865,8 @@ "content": "直线$x+y-4=0$的倾斜角$\\theta=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-双基冲刺卷-双基训练07" ], "genre": "填空题", "ans": "$\\dfrac 34\\pi$", @@ -356700,7 +359895,8 @@ "content": "已知$\\langle\\overrightarrow {a}, \\overrightarrow {b}\\rangle=\\dfrac{\\pi}{3}$, $|\\overrightarrow {a}|=2$, $|\\overrightarrow {b}|=1$, 则$|\\overrightarrow {a}+\\overrightarrow {b}|=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-双基冲刺卷-双基训练07" ], "genre": "填空题", "ans": "$\\sqrt{7}$", @@ -356729,7 +359925,8 @@ "content": "记$S_n$为等差数列$\\{a_n\\}$的前$n$项和. 若$a_4+a_5=24$, $S_6=48$, 则$\\{a_n\\}$的公差为\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-双基冲刺卷-双基训练07" ], "genre": "填空题", "ans": "$4$", @@ -356758,7 +359955,8 @@ "content": "$2002$年在北京召开的国际数学家大会, 会标是以我国古代数学家赵爽的弦图为基础设计的. 弦图是由四个全等直角三角形与一个小正方形拼成的一个大正方形(如图). 如果小正方形的面积为$1$, 大正方形的面积为$25$, 直角三角形中较小的锐角为$\\theta$, 那么$\\sin 2 \\theta$的值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.4]\n\\draw (0,0) rectangle (5,5);\n\\draw (0,0) --++ ({atan(3/4)}:4);\n\\draw (5,0) --++ ({90+atan(3/4)}:4);\n\\draw (5,5) --++ ({180+atan(3/4)}:4);\n\\draw (0,5) --++ ({270+atan(3/4)}:4);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-双基冲刺卷-双基训练07" ], "genre": "填空题", "ans": "$\\dfrac{24}{25}$", @@ -356787,7 +359985,8 @@ "content": "曲线$y=\\ln x$在点$(\\mathrm{e}, 1)$处的切线方程为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-双基冲刺卷-双基训练07" ], "genre": "填空题", "ans": "$y=\\dfrac {1}{\\mathrm{e}}x$", @@ -356816,7 +360015,8 @@ "content": "现有$7$张卡片, 分别写上数字$1,2,2,3,4,5,6$. 从这$7$张卡片中随机抽取$3$张, 记所抽取卡片上数字的最小值为$\\xi$, 则$P(\\xi=2)=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-双基冲刺卷-双基训练07" ], "genre": "填空题", "ans": "$\\dfrac{16}{35}$", @@ -356845,7 +360045,8 @@ "content": "函数$y=f(x)$在$(-\\infty,+\\infty)$上严格减, 且为奇函数. 若$f(1)=-1$, 则满足$-1 \\leq$$f(x-2) \\leq 1$的$x$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-双基冲刺卷-双基训练07" ], "genre": "填空题", "ans": "$[1,3]$", @@ -356874,7 +360075,8 @@ "content": "安排$3$名志愿者完成$4$项工作, 每人至少完成$1$项, 每项工作由$1$人完成, 则不同的安排方式共有\\blank{50}种.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-双基冲刺卷-双基训练07" ], "genre": "填空题", "ans": "$36$", @@ -356903,7 +360105,8 @@ "content": "$(1+\\dfrac{1}{x^2})(1+x)^6$的展开式中$x^2$的系数为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-双基冲刺卷-双基训练07" ], "genre": "填空题", "ans": "$30$", @@ -356932,7 +360135,8 @@ "content": "已知双曲线$C: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的右顶点为$A$, 以$A$为圆心, $b$为半径作圆$A$, 圆$A$与双曲线$C$的一条渐近线交于$M$、$N$两点. 若$\\angle MAN=60^{\\circ}$, 则离心率的值为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-双基冲刺卷-双基训练07" ], "genre": "填空题", "ans": "$\\dfrac{2\\sqrt{3}}3$", @@ -356961,7 +360165,8 @@ "content": "下列说法中不正确的是\\bracket{20}.\n\\onech{独立性检验是检验两个分类变量是否有关的一种统计方法}{独立性检验得到的结论一定是正确的}{独立性检验的样本不同, 其结论可能不同}{独立性检验的基本思想是带有概率性质的反证法}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-双基冲刺卷-双基训练07" ], "genre": "选择题", "ans": "B", @@ -356990,7 +360195,8 @@ "content": "设有下面四个命题\n$p_1$: 若复数$z$满足$\\dfrac{1}{z} \\in \\mathbf{R}$, 则$z \\in \\mathbf{R}$;\\\\\n$p_2$: 若复数$z$满足$z^2 \\in \\mathbf{R}$, 则$z \\in \\mathbf{R}$;\\\\\n$p_3$: 若复数$z_1, z_2$满足$z_1 z_2 \\in \\mathbf{R}$, 则$z_1=\\overline{z_2}$;\\\\\n$p_4$: 若复数$z \\in \\mathbf{R}$, 则$\\overline {z} \\in \\mathbf{R}$. 其中的真命题为\\bracket{20}.\n\\fourch{$p_1$, $p_3$}{$p_1$, $p_4$}{$p_2$, $p_3$}{$p_2$, $p_4$}", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-双基冲刺卷-双基训练07" ], "genre": "选择题", "ans": "B", @@ -357019,7 +360225,8 @@ "content": "甲、乙两个圆锥的母线长相等, 侧面展开图的圆心角之和为$2 \\pi$, 侧面积分别为$S_{\\text {甲}}$和$S_{\\text {乙}}$, 体积分别为$V_{\\text {甲}}$和$V_{\\text {乙}}$. 若$\\dfrac{S_{\\text {甲}}}{S_{\\text {乙}}}=2$, 则$\\dfrac{V_{\\text {甲}}}{V_{\\text {乙}}}=$\\bracket{20}.\n\\fourch{$\\sqrt{5}$}{$2 \\sqrt{2}$}{$\\sqrt{10}$}{$\\dfrac{5 \\sqrt{10}}{4}$}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-双基冲刺卷-双基训练07" ], "genre": "选择题", "ans": "C", @@ -357048,7 +360255,8 @@ "content": "在四棱锥$P-ABCD$中, $PD \\perp$底面$ABCD$, $CD\\parallel AB$, $AD=DC=CB=1$, $AB=2$, $DP=\\sqrt{3}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = {(210:1cm)}]\n\\draw (0,0,0) node [above right] {$D$} coordinate (D);\n\\draw (1,0,0) node [right] {$C$} coordinate (C);\n\\draw ({-1/2},0,{sqrt(3)/2}) node [left] {$A$} coordinate (A);\n\\draw ({3/2},0,{sqrt(3)/2}) node [right] {$B$} coordinate (B);\n\\draw (D)++(0,{sqrt(3)}) node [above] {$P$} coordinate (P);\n\\draw (A)--(B)--(P)--cycle;\n\\draw (B)--(C)--(P);\n\\draw [dashed] (P)--(D)--(A) (D)--(C) (D)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $BD \\perp PA$;\\\\\n(2) 求直线$PD$与平面$PAB$所成的角的正弦值.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-双基冲刺卷-双基训练07" ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\dfrac{\\sqrt{5}}5$", @@ -357077,7 +360285,8 @@ "content": "某农场有一块农田, 如图所示, 它的边界由圆$O$的一段圆弧$\\overset\\frown{MPN}$($P$为此圆弧的中点)和线段$MN$构成. 已知圆$O$的半径为$40$米, 点$P$到$MN$的距离为$50$米. 现规划在此农田上修建两个温室大棚, 大棚 I 内的地块形状为矩形$ABCD$, 大棚 II 内的地块形状为$\\triangle CDP$, 要求$A, B$均在线段$MN$上, $C, D$均在圆弧上. 设$OC$与$MN$所成的角为$\\theta$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (2, 0) coordinate (R) arc (0: 180: 2) coordinate (L);\n\\draw (R) arc (0: {-asin(0.25)}: 2) node [right] {$N$} coordinate (N);\n\\draw (L) arc (180: {180+asin(0.25)}: 2) node [left] {$M$} coordinate (M);\n\\draw (0, 0) node [below left] {$O$} coordinate (O);\n\\draw [dashed] (-2, 0) -- (2, 0) (0, 2) node [above] {$P$} coordinate (P) -- ($(M)!(P)!(N)$);\n\\draw (40: 2) node [above right] {$C$} coordinate (C);\n\\draw (140: 2) node [above left] {$D$} coordinate (D);\n\\draw ($(M)!(C)!(N)$) node [below] {$B$} coordinate (B);\n\\draw ($(M)!(D)!(N)$) node [below] {$A$} coordinate (A);\n\\draw (M) -- (N);\n\\draw (A) -- (D) -- (C) -- (B);\n\\draw (C) -- (P) -- (D) (O) -- (C);\n\\end{tikzpicture}\n\\end{center}\n(1) 用$\\theta$分别表示矩形$ABCD$和$\\triangle CDP$的面积, 并确定$\\sin \\theta$的取值范围;\\\\\n(2) 若大棚 I 内种植甲种蔬菜, 大棚 II 内种植乙种蔬菜, 且甲、乙两种蔬菜的单位面积年产值之比为$4: 3$. 求当$\\theta$为何值时, 能使甲、乙两种蔬菜的年总产值最大.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-双基冲刺卷-双基训练07" ], "genre": "解答题", "ans": "(1) 矩形面积为$800\\cos\\theta+3200\\cos\\theta\\sin\\theta$, 三角形面积为$1600\\cos\\theta-1600\\cos\\theta\\sin\\theta$, $\\sin\\theta$的范围为$[\\dfrac 14,1)$; (2) 当$\\theta=\\dfrac{\\pi}{6}$时", @@ -357106,7 +360315,8 @@ "content": "已知$a \\in \\mathbf{R}$, $\\mathrm{i}$为虚数单位, 若$\\dfrac{a-\\mathrm{i}}{2+\\mathrm{i}}$为实数, 则$a$的值为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-双基冲刺卷-双基训练08" ], "genre": "填空题", "ans": "$-2$", @@ -357135,7 +360345,8 @@ "content": "已知圆锥的底面半径为$\\sqrt{2}$, 其侧面展开图为一个半圆, 则该圆锥的母线长为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-双基冲刺卷-双基训练08" ], "genre": "填空题", "ans": "$2\\sqrt{2}$", @@ -357164,7 +360375,8 @@ "content": "某班有$48$名同学, 一次考试后的数学成绩服从正态分布, 平均分为$80$, 标准差为$10$, 理论上说在$80$分到$90$分的人数是\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-双基冲刺卷-双基训练08" ], "genre": "填空题", "ans": "$16$", @@ -357193,7 +360405,8 @@ "content": "已知$a, b \\in \\mathbf{R}$, 且$a-3 b+6=0$, 则$2^a+\\dfrac{1}{8^b}$的最小值为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-双基冲刺卷-双基训练08" ], "genre": "填空题", "ans": "$\\dfrac 14$", @@ -357222,7 +360435,8 @@ "content": "若$(3 x^2+\\dfrac{1}{\\sqrt{x}})^n$的展开式中含有常数项, 则最小的正整数$n=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-双基冲刺卷-双基训练08" ], "genre": "填空题", "ans": "$5$", @@ -357251,7 +360465,8 @@ "content": "若$y=\\cos x-\\sin x$在$[0, a]$是减函数, 则$a$的最大值是\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-双基冲刺卷-双基训练08" ], "genre": "填空题", "ans": "$\\dfrac{3\\pi}{4}$", @@ -357280,7 +360495,8 @@ "content": "已知点$O(0,0), A(-2,0), B(2,0)$. 设点$P$满足$|PA|-|PB|=2$, 且$P$为函数$y=3 \\sqrt{4-x^2}$图像上的点, 则$|OP|=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-双基冲刺卷-双基训练08" ], "genre": "填空题", "ans": "$\\sqrt{10}$", @@ -357309,7 +360525,8 @@ "content": "如图, 在平面四边形$ABCD$中, $AB \\perp BC$, $AD \\perp CD$, $\\angle BAD=120^{\\circ}$, $AB=AD=1$, 若点$E$为边$CD$上的动点, 则$\\overrightarrow{AE} \\cdot \\overrightarrow{BE}$的最小值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$A$} coordinate (A);\n\\draw (0,2) node [above] {$C$} coordinate (C);\n\\draw (30:1) node [right] {$B$} coordinate (B);\n\\draw (150:1) node [left] {$D$} coordinate (D);\n\\draw ($(C)!0.7!(D)$) node [above left] {$E$} coordinate (E);\n\\draw (A)--(E)--(B) (A)--(B)--(C)--(D)--cycle;\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-双基冲刺卷-双基训练08" ], "genre": "填空题", "ans": "$\\dfrac{21}{16}$", @@ -357338,7 +360555,8 @@ "content": "徳国数学家狄里克雷(Dirichlet, Johann Peter Gustav Lejeune, 1805-1859)在$1837$年时提出: ``如果对于$x$的每一个值, $y$总有一个完全确定的值与之对应, 那么$y$是$x$的函数.''\n这个定义较清楚地说明了函数的内涵. 只要有一个法则, 使得取值范围中的每一个$x$, 有一个确定的$y$和它对应就行了, 不管这个法则是用公式还是用图像、表格等形式表示, 例如狄里克雷函数$d(x)$, 即: 当自变量取有理数时, 函数值为$1$; 当自变量取无理数时, 函数值为$0$. 下列关于狄里克雷函数$d(x)$的性质表述正确的序号是\\blank{50}.\\\\\n\\textcircled{1} $d(\\pi)=0$; \\textcircled{2} $d(x)$的值域为$\\{0,1\\}$; \\textcircled{3} $d(x)$的图像关于直线$x=1$对称; \\textcircled{4} $d(x)$的图像关于直线$x=2$对称.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-双基冲刺卷-双基训练08" ], "genre": "填空题", "ans": "\\textcircled{1}\\textcircled{2}\\textcircled{3}\\textcircled{4}", @@ -357369,7 +360587,8 @@ "content": "数列$\\{a_n\\}$满足$a_{n+2}+(-1)^n a_n=3 n-1$, 前$16$项和为$540$, 则$a_1=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-双基冲刺卷-双基训练08" ], "genre": "填空题", "ans": "$7$", @@ -357398,7 +360617,8 @@ "content": "下列变量之间的关系是函数关系的是\\bracket{20}.\n\\onech{光照时间与大棚内蔬菜的产量}{已知二次函数$y=a x^2+b x+c$, 其中$a$、$c$是常数, $b$为自变量, 因变量是这个函数的判别式$\\Delta=b^2-4 a c$}{每亩施肥量与粮食亩产量之间的关系}{人的身高与所穿鞋子的号码之间的关系}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-双基冲刺卷-双基训练08" ], "genre": "选择题", "ans": "B", @@ -357427,7 +360647,8 @@ "content": "魏晋时刘徽撰写的《海岛算经》是有关测量的数学著作, 其中第一题是测海岛的高. 如图, 点$E, H, G$在水平线$AC$上, $DE$和$FG$是两个垂直于水平面且等高的测量标杆的高度, 称为``表高'', $EG$称为``表距'', $GC$和$EH$都称为``表目距'', $GC$与$EH$的差称为``表目距的差''则海岛的高$AB=$\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$A$} coordinate (A);\n\\draw (0,2) node [above] {$B$} coordinate (B);\n\\draw (3,0) node [below] {$H$} coordinate (H);\n\\draw (5,0) node [below] {$C$} coordinate (C);\n\\draw ($(A)!0.8!(H)$) node [below] {$E$} coordinate (E);\n\\draw ($(B)!0.8!(H)$) node [above] {$D$} coordinate (D);\n\\draw ($(A)!0.8!(C)$) node [below] {$G$} coordinate (G);\n\\draw ($(B)!0.8!(C)$) node [above] {$F$} coordinate (F);\n\\draw ($(A)!-0.3!(C)$) coordinate (S);\n\\filldraw [gray!20] (S)--(B)--($(S)!1.8!(A)$);\n\\draw [dashed] (B)--(A) (B)--(H) (B)--(C) (D)--(E) (F)--(G);\n\\draw (S)--(C);\n\\end{tikzpicture}\n\\end{center}\n\\twoch{$\\dfrac{\\text { 表高 } \\times \\text { 表距 }}{\\text { 表目距的差 }}+\\text{ 表高 }$}{$\\dfrac{\\text { 表高 } \\times \\text { 表距 }}{\\text { 表目距的差 }}-\\text{ 表高 }$}{$\\dfrac{\\text { 表高 } \\times \\text { 表距 }}{\\text { 表目距的差 }}+\\text{ 表距 }$}{$\\dfrac{\\text { 表高 } \\times \\text { 表距 }}{\\text { 表目距的差 }}-\\text{ 表距 }$}", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-双基冲刺卷-双基训练08" ], "genre": "选择题", "ans": "A", @@ -357456,7 +360677,8 @@ "content": "已知$f(x)$是定义域为$(-\\infty,+\\infty)$的奇函数, 满足$f(1-x)=f(1+x)$. 若$f(1)=2$, 则$f(1)+f(2)+f(3)+\\cdots+f(50)=$\\bracket{20}.\n\\fourch{$-50$}{$0$}{$2$}{$50$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-双基冲刺卷-双基训练08" ], "genre": "选择题", "ans": "C", @@ -357487,7 +360709,8 @@ "content": "某校为举办甲、乙两项不同活动, 分别设计了相应的活动方案: 为了解该校学生对活动方案是否支持, 对学生进行简单随机抽样, 获得数据如下表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline \\multicolumn{2}{|c|}{男生} & \\multicolumn{2}{|c|}{女生} \\\\\n\\hline 支持 & 不支持 & 支持 & 不支持 \\\\\n\\hline 200 人 & 400 人 & 300 人 & 100 人 \\\\\n\\hline\n\\end{tabular} \n\\end{center}\n假设所有学生对活动方案是否支持相互独立.\\\\\n(1) 分别估计该校男生支持方案的概率、该校女生支持方案的概率;\\\\\n(2) 从该校全体男生中随机抽取$2$人, 全体女生中随机抽取$1$人, 估计这$3$人中恰有$2$人支持方案的概率.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-双基冲刺卷-双基训练08" ], "genre": "解答题", "ans": "(1) 男生: $\\dfrac 13$, 女生: $\\dfrac 34$; (2) $\\dfrac{13}{36}$", @@ -357516,7 +360739,8 @@ "content": "已知函数$y=f(x)$, 其中$f(x)=a x-\\dfrac{1}{x}-(a+1) \\ln x$.\\\\\n(1) 当$a=0$时, 求$f(x)$的最大值;\\\\\n(2) 若$f(x)$恰有一个零点, 求$a$的取值范围.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-双基冲刺卷-双基训练08" ], "genre": "解答题", "ans": "(1) $-1$; (2) $(0,+\\infty)$", @@ -357545,7 +360769,8 @@ "content": "实系数一元二次方程$x^2+a x+b=0$的一根为$x_1=\\dfrac{3+\\mathrm{i}}{1+\\mathrm{i}}$(其中$\\mathrm{i}$为虚数单位), 则$a+b=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-双基冲刺卷-双基训练09" ], "genre": "填空题", "ans": "$1$", @@ -357574,7 +360799,8 @@ "content": "二项式$(\\sqrt{x}-\\dfrac{2}{x})^6$的展开式的常数项为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-双基冲刺卷-双基训练09" ], "genre": "填空题", "ans": "$60$", @@ -357603,7 +360829,8 @@ "content": "已知随机变量$X$服从二项分布$X \\sim B(6, \\dfrac{1}{3})$, 则$P(X=2)=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-双基冲刺卷-双基训练09" ], "genre": "填空题", "ans": "$\\dfrac{80}{243}$", @@ -357632,7 +360859,8 @@ "content": "从$1$、$2$、$3$、$4$这四个数中一次随机地抽取两个数, 则其中一个数是另一个数的两倍的概率是\\blank{50}(结果用数值表示).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-双基冲刺卷-双基训练09" ], "genre": "填空题", "ans": "$\\dfrac 13$", @@ -357661,7 +360889,8 @@ "content": "如图, 三棱锥$P-ABC$中, $PA \\perp$底面$ABC$, 底面$ABC$是边长为$2$的正三角形, 且$PA=2 \\sqrt{3}$, 若$M$是$BC$的中点, 则异面直线$PM$与$AC$所成角的大小是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (1,0,{sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw ($(B)!0.5!(C)$) node [below right] {$M$} coordinate (M);\n\\draw (0,{2*sqrt(3)},0) node [above] {$P$} coordinate (P);\n\\draw (P)--(A)--(B)--(C)--cycle (P)--(M) (P)--(B);\n\\draw [dashed] (A)--(C);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-双基冲刺卷-双基训练09" ], "genre": "填空题", "ans": "$\\arccos\\dfrac{\\sqrt{15}}{10}$", @@ -357690,7 +360919,8 @@ "content": "已知数列$\\{a_n\\}$满足$a_1=1, a_{n+1}=a_n+\\dfrac{1}{n(n+1)}(n \\geq 1, n \\in \\mathbf{N})$, 则$a_n=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-双基冲刺卷-双基训练09" ], "genre": "填空题", "ans": "$2-\\dfrac 1n$", @@ -357719,7 +360949,8 @@ "content": "某市举行了首届阅读大会, 为调查市民对阅读大会的满意度, 相关部门随机抽取男女市民各$50$名, 每位市民对大会给出满意或不满意的评价, 得到下面列联表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline & 满意 & 不满意 \\\\\n\\hline 男市民 &$60-m$&$m-10$\\\\\n\\hline 女市民 &$m+10$&$40-m$\\\\\n\\hline\n\\end{tabular} \n\\end{center}\n当$1\\le m \\leq 25, m \\in \\mathbf{N}$时, 若没有$95 \\%$的把握认为男、女市民对大会的评价有差异, 则$m$的最小值为\\blank{50}.\\\\\n附:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline$\\alpha=P(\\chi^2 \\geq k)$&$0.10$&$0.05$&$0.005$\\\\\n\\hline$k$&$2.706$&$3.841$&$7.879$\\\\\n\\hline\n\\end{tabular} \n\\end{center}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-双基冲刺卷-双基训练09" ], "genre": "填空题", "ans": "$21$", @@ -357748,7 +360979,8 @@ "content": "已知$y=f(x)$是定义在$\\mathbf{R}$上的奇函数, 且当$x>0$时, $f(x)=x^2+\\dfrac{1}{x}$, 则函数$y=$$f(x)$的解析式为$y=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-双基冲刺卷-双基训练09" ], "genre": "填空题", "ans": "$y=\\begin{cases}x^2+\\dfrac 1x, & x>0, \\\\ 0, & x=0, \\\\ -x^2+\\dfrac 1x, & x<0\\end{cases}$", @@ -357777,7 +361009,8 @@ "content": "张老师整理旧资料时发现一题部分字迹模糊不清, 只能看到 : 在$\\triangle ABC$中, $a, b, c$分别是角$A, B, C$的对边, 已知$b=2 \\sqrt{2}$, $\\angle A=45^{\\circ}$, 求边$c$. 显然缺少条件, 若他打算补充$a$的大小, 并使得$c$只有一解. 那么, $a$的可能取值是\\blank{50}. (只需填写一个适合的答案)", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-双基冲刺卷-双基训练09" ], "genre": "填空题", "ans": "$a=2$或$a\\ge 2\\sqrt{2}$其中的一个即可", @@ -357806,7 +361039,8 @@ "content": "如图, 直径$AB=4$的半圆, $D$为圆心, 点$C$在半圆弧上, $\\angle ADC=\\dfrac{\\pi}{3}$, 线段$AC$上有动点$P$, 则$\\overrightarrow{DP} \\cdot \\overrightarrow{BA}$的最小值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (-1,0) node [below] {$A$} coordinate (A) -- (1,0) node [below] {$B$} coordinate (B) arc (0:180:1);\n\\draw (120:1) node [above] {$C$} coordinate (C);\n\\draw (0,0) node [below] {$D$} coordinate (D);\n\\draw (A)--(C)--(D);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-双基冲刺卷-双基训练09" ], "genre": "填空题", "ans": "$4$", @@ -357835,7 +361069,8 @@ "content": "已知集合$A=\\{x | x^2-3 x+2 \\leq 0\\}$, $B=\\{x | \\dfrac{x-a}{x+2}>0\\}$, 其中常数$a>0$. 若``$x \\in A$''是``$x \\in$$B$''的充分非必要条件, 则$a$的取值范围是\\bracket{20}.\n\\fourch{$0=latex]\n\\def\\l{2}\n\\draw (0,0,0) coordinate (A);\n\\draw (A) ++ (\\l,0,0) coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) coordinate (C);\n\\draw (A) ++ (0,0,-\\l) coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) coordinate (A1);\n\\draw (B) ++ (0,\\l,0) coordinate (B1);\n\\draw (C) ++ (0,\\l,0) coordinate (C1);\n\\draw (D) ++ (0,\\l,0) coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(A1)!0.5!(B1)$) -- ($(C1)!0.5!(D1)$) ($(A1)!0.5!(D1)$) -- ($(B1)!0.5!(C1)$);\n\\draw [dashed] ($(A)!0.5!(D)$) -- ($(B)!0.5!(C)$);\n\\draw [domain = 0:360] plot ({1+cos(\\x)},{1+sin(\\x)},0);\n\\draw [domain = 0:360] plot (2,{1+cos(\\x)},{-1+sin(\\x)});\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\dfrac{1}{2}$}{$\\dfrac{\\sqrt{2}}{2}$}{$\\sqrt{2}$}{$\\sqrt{3}$}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-双基冲刺卷-双基训练09" ], "genre": "选择题", "ans": "C", @@ -357895,7 +361131,8 @@ "content": "$2022$年第二十四届北京冬奧会开幕式上由$96$片小雪花组成的大雪花惊艳了全世界, 数学中也有一朵美丽的雪花--``科赫雪花''. 它可以这样画, 任意画一个正三角形$P_1$, 并把每一边三等分: 取三等分后的一边中间一段为边向外作正三角形, 并把这``中间一段''擦掉, 形成雪花曲线$P_2$; 重复上述两步, 画出更小的三角形.一直重复, 直到无穷, 形成雪花曲线, $P_3, P_4, \\cdots, P_n, \\cdots$. 设雪花曲线$P_n$的边长为$a_n$, 边数为$b_n$, 周长为$l_n$, 面积为$S_n$, 若$a_1=3$, 则下列说法正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[scale = 2,thick]\n\\draw (0,0) ++ (90:{1/sqrt(3)}) coordinate (A1) (0,0) ++ (210:{1/sqrt(3)}) coordinate (B1) (0,0) ++ (-30:{1/sqrt(3)}) coordinate (C1);\n\\draw (A1) -- (B1) -- (C1) -- cycle;\n\\draw (0,-1) node {$P_1$};\n\\draw (B1) ++ (1.5,0) coordinate (B2) --++ (0:{1/3}) --++ (-60:{1/3}) --++ (60:{1/3}) --++ (0:{1/3}) --++ (120:{1/3}) --++ (60:{1/3}) --++ (180:{1/3}) --++ (120:{1/3}) --++ (240:{1/3}) --++ (180:{1/3}) --++ (-60:{1/3}) --++ (-120:{1/3});\n\\draw (1.5,-1) node {$P_2$};\n\\draw (B2) ++ (1.5,0) coordinate (B3) coordinate (P);\n\\foreach \\i in {0,120,240}\n{\\foreach \\j in {0,-60,60,0}\n{\\foreach \\k in {0,-60,60,0}\n{\\draw (P) --++ ({\\i+\\j+\\k}:{1/9}) coordinate (P);};};};\n\\draw (3,-1) node {$P_3$};\n\\draw (B3) ++ (1.5,0) coordinate (P);\n\\foreach \\i in {0,120,240}\n{\\foreach \\j in {0,-60,60,0}\n{\\foreach \\k in {0,-60,60,0}\n{\\foreach \\m in {0,-60,60,0}\n{\\draw (P) --++ ({\\i+\\j+\\k+\\m}:{1/27}) coordinate (P);};};};};\n\\draw (4.5,-1) node {$P_4$};\n\\draw (6,-1) node {$\\cdots$} (6,0) node {$\\cdots$};\n\\end{tikzpicture}\n\\end{center}\n\\twoch{$a_5=\\dfrac{1}{27}$, $l_5=9 \\times(\\dfrac{3}{2})^3$}{$S_1 \\leq S_3<\\dfrac{8}{5} S_1$}{$\\{a_n\\},\\{b_n\\},\\{l_n\\},\\{S_n\\}$均构成等比数列}{$S_n=S_{n-1}+\\dfrac{\\sqrt{3}}{4} b_{n-1} a_{n-1}^2$}", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-双基冲刺卷-双基训练09" ], "genre": "选择题", "ans": "B", @@ -357924,7 +361161,8 @@ "content": "已知函数$f(x)=\\dfrac{1}{3} x^3+\\dfrac{m}{2} x^2-x+\\dfrac{1}{6}$.\\\\\n(1) 当$m=1$时, 求$f(x)$在点$(1, f(1))$的切线方程;\\\\\n(2) 若$f(x)$在$(\\dfrac{1}{2}, 2)$上存在单调减区间, 求实数$m$的取值范围.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-双基冲刺卷-双基训练09" ], "genre": "解答题", "ans": "(1) $y=x-1$; (2) $(-\\infty,\\dfrac 32)$", @@ -357953,7 +361191,8 @@ "content": "如图, 已知点$P$是$y$轴左侧 (不含$y$轴)一点, 抛物线$C: y^2=4 x$上存在不同的两点$A$、$B$, 满足$PA$、$PB$的中点均在抛物线$C$上.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-2, 0) -- (6, 0) node [below] {$x$};\n\\draw [->] (0, -5) -- (0, 5) node [left] {$y$};\n\\draw (0, 0) node [below left] {$O$};\n\\path [domain = -5:5, samples = 100, name path = para, draw] plot ({\\x*\\x/4}, \\x);\n\\draw (-1, 1) node [above] {$P$} coordinate (P);\n\\filldraw ({(7-2*sqrt(10))/8}, {(2-sqrt(10))/2}) circle (0.06) coordinate (D);\n\\filldraw ({(7+2*sqrt(10))/8}, {(2+sqrt(10))/2}) circle (0.06) coordinate (C);\n\\draw ($(P)!2!(C)$) node [above] {$A$} coordinate (A)-- ($(P)!2!(D)$) node [below] {$B$} coordinate (B);\n\\draw (A)--(P)--(B);\n\\draw ($(A)!0.5!(B)$) node [above] {$M$} coordinate (M)--(P);\n\\end{tikzpicture}\n\\end{center}\n(1) 设$AB$中点为$M$, 且$P(x_P, y_P)$, $M(x_M, y_M)$, 证明: $y_P=y_M$;\\\\\n(2) 若$P$是曲线$x^2+\\dfrac{y^2}{4}=1(x<0)$上的动点, 求$\\triangle PAB$面积的最小值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-双基冲刺卷-双基训练09" ], "genre": "解答题", "ans": "(1) 证明略; (2) $6\\sqrt{2}$", @@ -357982,7 +361221,8 @@ "content": "函数$y=x^2$在区间$[2,4]$上的平均变化率等于\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-双基冲刺卷-双基训练10" ], "genre": "填空题", "ans": "$6$", @@ -358011,7 +361251,8 @@ "content": "点$(2,1)$到直线$3 x+4 y=0$的距离为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-双基冲刺卷-双基训练10" ], "genre": "填空题", "ans": "$2$", @@ -358040,7 +361281,8 @@ "content": "已知随机变量$X$服从正态分布$N(-2, \\sigma^2)$, 且$P(X \\leq-1)=k$, 则$P(X \\leq-3)=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-双基冲刺卷-双基训练10" ], "genre": "填空题", "ans": "$1-k$", @@ -358069,7 +361311,8 @@ "content": "抛物线$y^2=4 x$上一点$M$到焦点的距离为 5 , 则点$M$的横坐标是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-双基冲刺卷-双基训练10" ], "genre": "填空题", "ans": "$4$", @@ -358098,7 +361341,8 @@ "content": "已知数据$x_1, x_2, \\cdots, x_9$的标准差为 5 , 则数据$3 x_1+1,3 x_2+1, \\cdots, 3 x_9+1$的标准差为\\blank{50}.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-双基冲刺卷-双基训练10" ], "genre": "填空题", "ans": "$15$", @@ -358127,7 +361371,8 @@ "content": "已知函数$y=\\ln x-a x-2$在区间$(1,2)$上不单调, 则实数$a$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-双基冲刺卷-双基训练10" ], "genre": "填空题", "ans": "$(\\dfrac 12,1)$", @@ -358156,7 +361401,8 @@ "content": "一名工人维护$3$台独立的游戏机, 一天内这$3$台需要维护的概率分别为$0.9$、$0.8$和$0.6$, 则一天内至少有一台游戏机不需要维护的概率为\\blank{50}(结果用小数表示).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-双基冲刺卷-双基训练10" ], "genre": "填空题", "ans": "$0.568$", @@ -358185,7 +361431,8 @@ "content": "已知$\\triangle ABC$的内角$A$、$B$、$C$的对边分别为$a$、$b$、$c$, 若$\\triangle ABC$的面积为$\\dfrac{a^2+b^2-c^2}{4}$, $c=\\sqrt{2}$, 则该三角形外接圆的半径等于\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-双基冲刺卷-双基训练10" ], "genre": "填空题", "ans": "$1$", @@ -358214,7 +361461,8 @@ "content": "若关于$x$的不等式$\\log _{\\frac{1}{2}}(4^{x+1}+\\lambda \\cdot 2^x)<0$在$x>0$时恒成立, 则实数$\\lambda$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-双基冲刺卷-双基训练10" ], "genre": "填空题", "ans": "$\\lambda\\ge -3$", @@ -358243,7 +361491,8 @@ "content": "《九章算术》中将正四棱台体 (棱台的上下底面均为正方形) 称为方亭. 如图, 现有一方亭$ABCD-EFGH$, 其中上底面与下底面的面积之比为$1:4$, $BF=\\dfrac{\\sqrt{6}}{2}EF$, 方亭的四个侧面均为全等的等腰梯形, 已知方亭四个侧面的面积之和为$12\\sqrt{5}$, 则方亭的体积为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (-1,0,1) node [left] {$A$} coordinate (A);\n\\draw (1,0,1) node [right] {$B$} coordinate (B);\n\\draw (1,0,-1) node [right] {$C$} coordinate (C);\n\\draw (-1,0,-1) node [left] {$D$} coordinate (D);\n\\draw ($(A)!0.5!(0,2,0)$) node [left] {$E$} coordinate (E);\n\\draw ($(B)!0.5!(0,2,0)$) node [right] {$F$} coordinate (F);\n\\draw ($(C)!0.5!(0,2,0)$) node [right] {$G$} coordinate (G);\n\\draw ($(D)!0.5!(0,2,0)$) node [left] {$H$} coordinate (H);\n\\draw (A)--(B)--(C) (E)--(F)--(G)--(H)--cycle (A)--(E) (B)--(F) (C)--(G);\n\\draw [dashed] (A)--(D)--(C) (D)--(H);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-双基冲刺卷-双基训练10" ], "genre": "填空题", "ans": "$\\dfrac{56}3$", @@ -358274,7 +361523,8 @@ "content": "将函数$y=\\sin (x-\\dfrac{\\pi}{6})$的图像上所有的点向右平移$\\dfrac{\\pi}{4}$个单位长度, 再把图形上各点的横坐标扩大到原来的$2$倍(纵坐标不变), 则所得图像的解析式为\\bracket{20}.\n\\fourch{$y=\\sin (\\dfrac{x}{2}-\\dfrac{5 \\pi}{12})$}{$y=\\sin (\\dfrac{x}{2}+\\dfrac{5 \\pi}{12})$}{$y=\\sin (2 x-\\dfrac{5 \\pi}{12})$}{$y=\\sin (\\dfrac{x}{2}-\\dfrac{5 \\pi}{24})$}", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-双基冲刺卷-双基训练10" ], "genre": "选择题", "ans": "A", @@ -358303,7 +361553,9 @@ "content": "已知函数$y=f(x)(a=latex]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,0) .. controls ++ (0.3,0.6) and ++ (-0.2,0) .. (0.8,1) .. controls ++ (0.2,0) and ++ (-0.1,0.6) .. (1.5,-1);\n\\draw (0,0) .. controls ++(-0.2,0.4) and ++ (0.2,0) .. (-0.5,0.7) .. controls ++ (-0.2,0) and ++ (0.2,0) .. (-1,-0.8) .. controls ++ (-0.2,0) and ++(0.1,-0.3).. (-1.6,0.9);\n\\draw [dashed] (-1.6,0.9) -- (-1.6,0) node [below] {$a$};\n\\draw [dashed] (1.5,-1) -- (1.5,0) node [above] {$b$};\n\\draw (0.8,1) node [above] {$y=f'(x)$};\n\\filldraw [fill = white] (1.5,-1) circle (0.03);\n\\filldraw [fill = white] (-1.6,0.9) circle (0.03);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$3$个驻点}{$4$个极值点}{$1$个极小值点}{$1$个极大值点}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-双基冲刺卷-双基训练10", + "2023届高三-第二轮复习讲义-16_导数及其应用" ], "genre": "选择题", "ans": "C", @@ -358344,7 +361596,8 @@ "content": "已知正态分布的密度函数$\\varphi_{\\mu, \\sigma}(x)=\\dfrac{1}{\\sqrt{2 \\pi \\sigma^2}} \\mathrm{e}^{-\\frac{(x-\\mu)^2}{2 \\sigma^2}}, x \\in(-\\infty,+\\infty)$, 以下关于正态曲线的说法错误的是\\bracket{20}.\n\\onech{曲线与$x$轴之间的面积为$1$}{曲线在$x=\\mu$处达到峰值$\\dfrac{1}{\\sqrt{2 \\pi} \\sigma}$}{当$\\sigma$一定时, 曲线的位置由$\\mu$确定, 曲线随着$\\mu$的变化而沿$x$轴平移}{当$\\mu$一定时, 曲线的形状由$\\sigma$确定, $\\sigma$越小, 曲线越``矮胖''}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-双基冲刺卷-双基训练10" ], "genre": "选择题", "ans": "D", @@ -358374,7 +361627,8 @@ "content": "已知$p: x^2-7 x+10<0$, $q: x^2-3 m x+2 m^2<0$, 其中$m>0$.\\\\\n(1) 若$m=3$, 且$p$、$q$同时为真命题, 求$x$的取值范围;\\\\\n(2) 若$p$是$q$的必要非充分条件, 求实数$m$的取值范围.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-双基冲刺卷-双基训练10" ], "genre": "解答题", "ans": "(1) $(3,5)$; (2) $[2,\\dfrac 52]$", @@ -358403,7 +361657,8 @@ "content": "某工厂去年$12$月试生产新工艺消毒剂$1050$升, 产品合格率为$90 \\%$. 从今年$1$月开始, 工厂在接下来的两年中将生产这款消毒剂. $1$月按去年$12$月的产量和产品合格率生产, 以后每月的产量都在前一个月产量的基础上提高$5 \\%$, 产品合格率比前一个月增加$0.4 \\%$.\\\\\n(1) 求今年该消毒剂的年产量 (精确到$1$升);\\\\\n(2) 从第几个月起, 月产消毒剂中不合格的量能一直控制在$100$升以内?", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-双基冲刺卷-双基训练10" ], "genre": "解答题", "ans": "(1) 约$16713$升; (2) 从第$13$个月起", @@ -358762,7 +362017,8 @@ "content": "已知集合$A=\\{(x, y) | y=x^2+1\\}$, $B=\\{(x, y) | y=2 x+1\\}$, 则$A \\cap B=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-双基冲刺卷-双基训练11" ], "genre": "填空题", "ans": "$\\{(0,1),(2,5)\\}$", @@ -358790,7 +362046,8 @@ "content": "用反证法证明: ``若$x+y \\leq 2$, 则$x \\leq 1$或$y \\leq 1$''时, 需假设\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-双基冲刺卷-双基训练11" ], "genre": "填空题", "ans": "$x>1$且$y>1$", @@ -358818,7 +362075,8 @@ "content": "若$x>0$, 则$\\dfrac{2}{x^3}$与$x^3$的算术平均值的最小值为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-双基冲刺卷-双基训练11" ], "genre": "填空题", "ans": "$\\sqrt{2}$", @@ -358846,7 +362104,8 @@ "content": "已知问题: ``$|x+3|+|x-a| \\geq 5$恒成立, 求实数$a$的取值范围''. 两位同学对此问题展开讨论: 小明说可以分类讨论, 将不等式左边的两个绝对值去掉; 小新说可以利用三角不等式解决问题. 请你选择一个适合自己的方法求解此题, 并写出实数$a$的取值范围: \\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-双基冲刺卷-双基训练11" ], "genre": "填空题", "ans": "$(-\\infty,-8]\\cup [2,+\\infty)$", @@ -358874,7 +362133,8 @@ "content": "函数$y=\\dfrac{x}{x^2+4}$的严格增区间是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-双基冲刺卷-双基训练11" ], "genre": "填空题", "ans": "$[-2,2]$", @@ -358902,7 +362162,8 @@ "content": "已知直线$l_1$的斜率为$2$, $l_2$的方程为$y=x+2$, 那么直线$l_1$与直线$l_2$的夹角的正切值为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-双基冲刺卷-双基训练11" ], "genre": "填空题", "ans": "$\\dfrac 13$", @@ -358930,7 +362191,8 @@ "content": "已知$O(0,0)$, $A(-\\sin \\theta, 1)$, $B(1, \\sqrt{3} \\cos \\theta)$, $\\theta \\in(\\dfrac{\\pi}{2}, \\dfrac{3 \\pi}{2})$, 若$|\\overrightarrow{OA}+\\overrightarrow{OB}|=|\\overrightarrow{AB}|$, 则$\\theta=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-双基冲刺卷-双基训练11" ], "genre": "填空题", "ans": "$\\dfrac{4\\pi}{3}$", @@ -358958,7 +362220,8 @@ "content": "首钢滑雪大跳台是东奥历史上第一座与工业遗产再利用直接结合的竞赛场馆, 大跳台的设计中融人了世界文化遗产敦煌壁画中``飞天''的元素. 如图, 研究性学习小组为了估算赛道造型最高点$A$距离地面的高度$AB$($AB$与底面垂直), 在赛道一侧找到一座建筑物$CD$, 测得$CD$的高度为$h$, 并从$C$点测得$A$点的仰角为$30^{\\circ}$; 在赛道与建筑物$CD$之间的地面上的点$E$处测得$A$点, $C$点的仰角分\n别为$60^{\\circ}$和$30^{\\circ}$(其中$B, E, D$三点共线), 该学习小组利用这些数据估算出$AB$约为$60$米, 则$CD$的高$h$约为\\blank{50}米.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw (0,0) node [below] {$E$} coordinate (E);\n\\draw (2,0) node [below] {$D$} coordinate (D);\n\\draw (2,{2/sqrt(3)}) node [right] {$C$} coordinate (C);\n\\path [name path = CA] (C) --++ (150:4.7);\n\\path [name path = EA] (E) --++ (120:4.2);\n\\path [name intersections = {of = CA and EA, by = A}];\n\\draw ($(E)!(A)!(D)$) node [below] {$B$} coordinate (B);\n\\draw (A) node [left] {$A$} -- (B)--(D)--(C)--(E)--(A);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-双基冲刺卷-双基训练11" ], "genre": "填空题", "ans": "$20$", @@ -358987,7 +362250,8 @@ "content": "半正多面体(semiregular polyhedron) 亦称``阿基米德多面体'', 是由边数不全相同的正多边形围成的多面体, 体现了数学的对称美. 二十四等边体就是一种半正多面体, 是由正方体切截而成的, 它由八个正三角形和六个正方形构成(如图所示), 若它的所有棱长都为$\\sqrt{2}$, 则正确的序号是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = {(220:0.5cm)}]\n\\draw (-1,0,0) node [left] {$A$} coordinate (A);\n\\draw (0,0,1) node [below] {$B$} coordinate (B);\n\\draw (1,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,-1) node [below] {$D$} coordinate (D);\n\\draw (-1,1,1) node [left] {$E$} coordinate (E);\n\\draw (1,1,1) node [left] {$F$} coordinate (F);\n\\draw (1,1,-1) node [left] {$G$} coordinate (G);\n\\draw (-1,1,-1) node [right] {$H$} coordinate (H);\n\\draw (-1,2,0) node [left] {$M$} coordinate (M);\n\\draw (0,2,1) node [above] {$N$} coordinate (N);\n\\draw (1,2,0) node [right] {$P$} coordinate (P);\n\\draw (0,2,-1) node [above] {$Q$} coordinate (Q);\n\\draw (A)--(B)--(E)--cycle (B)--(C)--(F)--cycle (C)--(G) (G)--(P) (F)--(P)--(N) --cycle (M)--(N)--(E)--cycle (M)--(Q)--(P);\n\\draw [dashed] (A)--(D)--(H)--cycle (C)--(D)--(G) (M)--(H)--(Q)--(G);\n\\end{tikzpicture}\n\\end{center}\n\\textcircled{1} $BF \\perp$平面$EAB$;\\\\\n\\textcircled{2} $AB$与$PF$所成角为$45^{\\circ}$;\\\\\n\\textcircled{3} 该二十四等边体的体积为$\\dfrac{20}{3}$;\\\\\n\\textcircled{4} 该二十四等边体外接球的表面积为$8 \\pi$.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-双基冲刺卷-双基训练11" ], "genre": "填空题", "ans": "\\textcircled{3}\\textcircled{4}", @@ -359015,7 +362279,8 @@ "content": "某学校为了加强学生数学核心素养的培养, 锻炼学生自主探究学习的能力, 他们以函数$y=f(x)$, 其中$f(x)=\\lg \\dfrac{1-x}{1+x}$为基本素材, 研究该函数的相关性质, 取得部分研究成果如下:\\\\\n\\textcircled{1} 同学甲发现: 函数$y=f(x)$的定义域为$(-1,1)$;\\\\\n\\textcircled{2} 同学乙发现: 函数$y=f(x)$是偶函数;\\\\\n\\textcircled{3} 同学丙发现: 对于任意的$x \\in(-1,1)$都有$f(\\dfrac{2 x}{x^2+1})=2 f(x)$;\\\\\n\\textcircled{4} 同学丁发现: 对于任意的$a, b \\in(-1,1)$, 都有$f(a)+f(b)=f(\\dfrac{a+b}{1+a b})$;\\\\\n\\textcircled{5} 同学戊发现: 对于函数$y=f(x)$定义域中任意的两个不同实数$x_1, x_2$, 总满足$\\dfrac{f(x_1)-f(x_2)}{x_1-x_2}>0$.\\\\\n其中所有正确研究成果的序号是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-双基冲刺卷-双基训练11" ], "genre": "填空题", "ans": "\\textcircled{1}\\textcircled{3}\\textcircled{4}", @@ -359043,7 +362308,8 @@ "content": "数学赋予建筑美惑与活力, 许多建筑融人数学元索, 更具神韵. 某单叶双曲面(由双曲线绕虚轴旋转形成的立体图形)型建筑过轴的部分截面图像如下图, 上、下底面与地面平行. 现测得下底直径$AB=20 \\sqrt{10}$米, 上底直径$CD=$$20 \\sqrt{2}$米, $AB$与$CD$间的距离为$80$米, 与上下底面等距离的$E$点处的直径等于$CD$, 则该建筑最细处的直径为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.3]\n\\draw [domain = -6:2] plot ({sqrt(1+\\x*\\x/4)},\\x);\n\\draw [domain = -6:2] plot ({-sqrt(1+\\x*\\x/4)},\\x);\n\\draw ({sqrt(10)},-6) node [below] {$B$} coordinate (B);\n\\draw ({-sqrt(10)},-6) node [below] {$A$} coordinate (A);\n\\draw ({sqrt(2)},2) node [above] {$D$} coordinate (D);\n\\draw ({-sqrt(2)},2) node [above] {$C$} coordinate (C);\n\\draw [dashed] (0,-6) -- (0,2);\n\\draw (A)--(B) (C)--(D);\n\\filldraw (0,-2) circle (0.1) node [right] {$E$} coordinate (E);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$10$}{$20$}{$10 \\sqrt{3}$}{$10 \\sqrt{5}$}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-双基冲刺卷-双基训练11" ], "genre": "选择题", "ans": "B", @@ -359071,7 +362337,8 @@ "content": "明代朱载堉创造了音乐上极为重要的``等程律''. 在创造律制的过程中, 他不仅给出了求三项等比数列的等比中项的方法, 还给出了求解四项等比数列的中间两项的方法, 比如, 若已知黄钟, 大吕, 太簇, 夹钟四个音律成等比数列, 则有$\\text{大吕}=\\sqrt{\\text {黄钟}\\times\\text{太簇}}$, $\\text{大吕}=\\sqrt[3]{(\\text {黄钟})^2 \\times \\text {夹钟}}$, $\\text{太簇}=\\sqrt[3]{\\text {黄钟} \\times(\\text {夹钟})^2}$. 据此, 可得正项等比数列$\\{a_n\\}$中, $a_k=$\\bracket{20}.\n\\fourch{$\\sqrt[n-k+1]{a_1^{n-k} \\cdot a_n}$}{$\\sqrt[n-k+1]{a_1 \\cdot a_n^{n-k}}$}{$\\sqrt[n-1]{a_1^{n-k} \\cdot a_n^{k-1}}$}{$\\sqrt[n-1]{a_1^{k-1} \\cdot a_n^{n-k}}$}", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-双基冲刺卷-双基训练11" ], "genre": "选择题", "ans": "C", @@ -359099,7 +362366,8 @@ "content": "已知函数$y=\\mathrm{e}^x$, $y=\\ln \\dfrac{x}{2}+\\dfrac{1}{2}$的图像分别与直线$y-m=0$交于$A, B$两点, 则使得$|AB|$取得最小值时的$m$的值为\\bracket{20}.\n\\fourch{$1$}{$-\\dfrac{1}{2}$}{$\\dfrac{1}{3}$}{$\\dfrac{1}{2}$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-双基冲刺卷-双基训练11" ], "genre": "选择题", "ans": "D", @@ -359127,7 +362395,8 @@ "content": "甲、乙两人在相同条件下各射击$10$次, 每次命中的环数如表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline 甲 & 8 & 6 & 7 & 8 & 6 & 5 & 9 & 10 & 4 & 7 \\\\\n\\hline 乙 & 6 & 7 & 7 & 8 & 6 & 7 & 8 & 7 & 9 & 5 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 分别计算以上两组数据的平均数;\\\\\n(2) 分别计算以上两组数据的方差;\\\\\n(3) 根据计算的结果, 对甲乙两人的射击成绩作出评价.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-双基冲刺卷-双基训练11" ], "genre": "解答题", "ans": "(1) 甲与乙的平均数分别为$7$和$7$; (2) 甲与乙的方差分别为$3$和$1.2$; (3) 两人射击水平相当, 乙的发挥更稳定", @@ -359155,7 +362424,8 @@ "content": "已知: 椭圆$C: \\dfrac{x^2}{16}+\\dfrac{y^2}{12}=1$, 直线$l: x-2 y-12=0$.\\\\\n(1) 求椭圆$C$的离心率;\\\\\n(2) 求椭圆$C$上一点$P$到直线$l$的距离的最小值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-双基冲刺卷-双基训练11" ], "genre": "解答题", "ans": "(1) $\\dfrac 12$; (2) $\\dfrac{4\\sqrt{5}}5$", @@ -359513,7 +362783,8 @@ "content": "函数$y=\\log _2(x^2-1)$的定义域是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-双基冲刺卷-双基训练12" ], "genre": "填空题", "ans": "$(-\\infty,-1)\\cup (1,+\\infty)$", @@ -359541,7 +362812,8 @@ "content": "$(x^2+\\dfrac{2}{x})^6$的展开式中常数项是\\blank{50}. (用数字作答)", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-双基冲刺卷-双基训练12" ], "genre": "填空题", "ans": "$240$", @@ -359569,7 +362841,8 @@ "content": "已知$a, b \\in \\mathbf{R}$, 且$a+b=1$, 则$(a+1)^2+(b+1)^2$的最小值为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-双基冲刺卷-双基训练12" ], "genre": "填空题", "ans": "$\\dfrac 92$", @@ -359597,7 +362870,8 @@ "content": "已知$|x-a|+|x+3|>-a$对所有实数$x$均成立, 则$a$的取值范围\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-双基冲刺卷-双基训练12" ], "genre": "填空题", "ans": "$(-\\dfrac 32,+\\infty)$", @@ -359625,7 +362899,8 @@ "content": "经过点$P(6,-2)$, 且在两坐标轴上的截距相等的直线的方程是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-双基冲刺卷-双基训练12" ], "genre": "填空题", "ans": "$x+3y=0$或$x+y-4=0$", @@ -359653,7 +362928,8 @@ "content": "关于$x$的不等式$x^2-2 a x-8 a^2<0$($a>0$)的解集为$(x_1, x_2)$, 且$x_2-x_1=15$, 则$a=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-双基冲刺卷-双基训练12" ], "genre": "填空题", "ans": "$\\dfrac 52$", @@ -359681,7 +362957,8 @@ "content": "埃及胡夫金字塔是古代世界建筑奇迹之一, 它的形状可视为一个正四棱锥, 以该四棱锥的高为边长的正方形面积等于该四棱锥一个侧面三角形的面积, 则其侧面三角形底边上的高与底面正方形的边长的比值为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-双基冲刺卷-双基训练12" ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{5}+1}4$", @@ -359709,7 +362986,8 @@ "content": "无穷等比数列$\\{a_n\\}$中, 首项$a_1=\\dfrac{1}{2}$, 公比$q=\\dfrac{1}{2}$, $T_n=a_2^2+a_4^2+\\cdots+a_{2 n}^2$, 则$\\displaystyle\\lim_{n\\to\\infty} T_n=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-双基冲刺卷-双基训练12" ], "genre": "填空题", "ans": "$\\dfrac{1}{15}$", @@ -359737,7 +363015,8 @@ "content": "设$\\{a_n\\}$是公差为$d$的等差数列, $\\{b_n\\}$是公比为$q$的等比数列. 已知数列$\\{a_n+b_n\\}$的前$n$项和$S_n=n^2-n+2^n-1$, 则$d+q$的值是\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-双基冲刺卷-双基训练12" ], "genre": "填空题", "ans": "$4$", @@ -359765,7 +363044,8 @@ "content": "设$F_1$、$F_2$分别为椭圆$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的左右焦点, 与直线$y=b$相切的圆$F_2$交椭圆于点$E$, 且$E$是直线$EF_1$与圆$F_2$相切的切点, 则椭圆的离心率为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (-4,2) -- (4,2);\n\\draw ({-sqrt(5)},0) node [below] {$F_1$} coordinate (F_1);\n\\draw ({sqrt(5)},0) node [below] {$F_2$} coordinate (F_2);\n\\draw (F_2) ++ (0,2) node [above] {$y=b$};\n\\path [draw,name path = elli] (0,0) circle (3 and 2);\n\\path [draw,name path = circ] (F_2) circle (2);\n\\path [name intersections = {of = elli and circ, by = {E,E1}}];\n\\draw (F_1)--(E) node [below] {$E$} --(F_2);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-双基冲刺卷-双基训练12" ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{5}}{3}$", @@ -359793,7 +363073,8 @@ "content": "设有下面四个命题:\\\\\n$p_1$: 若复数$z$满足$\\dfrac{1}{z} \\in \\mathbf{R}$, 则$z \\in \\mathbf{R}$;\\\\\n$p_2: $: 若复数$z$满足$z^2 \\in \\mathbf{R}$, 则$z \\in \\mathbf{R}$;\\\\\n$p_3: $若复数$z_1$, $z_2$满足$z_1 z_2 \\in \\mathbf{R}$, 则$z_1=\\overline{z_2}$;\\\\\n$p_4$: 若复数$z \\in \\mathbf{R}$, 则$\\overline {z} \\in \\mathbf{R}$.\n其中的真命题为\\bracket{20}.\n\\fourch{$p_1$, $p_3$}{$p_1$, $p_4$}{$p_2$, $p_3$}{$p_2$, $p_4$}", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-双基冲刺卷-双基训练12" ], "genre": "选择题", "ans": "B", @@ -359821,7 +363102,8 @@ "content": "已知向量$\\overrightarrow {a}$, $\\overrightarrow {b}$满足$|\\overrightarrow {a}|=5$, $|\\overrightarrow {b}|=6$, $\\overrightarrow {a} \\cdot \\overrightarrow {b}=-6$, 则$\\cos \\langle\\overrightarrow {a}, \\overrightarrow {a}+\\overrightarrow {b}\\rangle=$\\bracket{20}.\n\\fourch{$-\\dfrac{31}{35}$}{$-\\dfrac{19}{35}$}{$\\dfrac{17}{35}$}{$\\dfrac{19}{35}$}", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-双基冲刺卷-双基训练12" ], "genre": "选择题", "ans": "D", @@ -359849,7 +363131,8 @@ "content": "《九章算术》中, 称底面为矩形而有一侧棱垂直于底面的四棱锥为阳马, 设$AA_1$是正六棱柱的一条侧棱, 如图, 若阳马以该正六棱柱的顶点为顶点、以$AA_1$为底面矩形的一边, 则这样的阳马的个数是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (1,0,0) node [above] {$A_1$} coordinate (A1)--++ ({1/2},0,{-sqrt(3)/2}) coordinate (B1) --++ ({-1/2},0,{-sqrt(3)/2}) coordinate (C1) --++ (-1,0,0) coordinate (D1) --++ ({-1/2},0,{sqrt(3)/2}) coordinate (E1) --++ ({1/2},0,{sqrt(3)/2}) coordinate (F1) -- cycle;\n\\draw (A1) --++ (0,-1.5,0) coordinate (A) node [below] {$A$};\n\\draw (B1) --++ (0,-1.5,0) coordinate (B);\n\\draw (F1) --++ (0,-1.5,0) coordinate (F);\n\\draw (E1) --++ (0,-1.5,0) coordinate (E);\n\\draw [dashed] (C1) --++ (0,-1.5,0) coordinate (C);\n\\draw [dashed] (D1) --++ (0,-1.5,0) coordinate (D);\n\\draw (E)--(F)--(A)--(B);\n\\draw [dashed] (E)--(D)--(C)--(B);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$4$}{$8$}{$12$}{$16$}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-双基冲刺卷-双基训练12" ], "genre": "选择题", "ans": "D", @@ -359886,7 +363169,8 @@ "objs": [], "tags": [ "第八单元", - "第九单元" + "第九单元", + "2023届高三-双基冲刺卷-双基训练12" ], "genre": "解答题", "ans": "(1) $\\chi^2=24>6.635$, 有$99.9\\%$的把握认为患该疾病群体与为患该疾病群体的卫生习惯有差异; (2) 证明略, $P(A|B)$的估计值为$0.4$, $P(A|\\overline{B})$的估计值为$0.1$, $R$的估计值为$6$", @@ -359915,7 +363199,8 @@ "content": "已知函数$y=f(x)$, 其中$f(x)=\\ln (1+x)+a x \\mathrm{e}^{-x}$.\\\\\n(1) 当$a=1$时, 求曲线$y=f(x)$在点$(0, f(0))$处的切线方程;\\\\\n(2) 若$y=f(x)$在区间$(-1,0),(0,+\\infty)$各恰有一个零点, 求$a$的取值范围.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-双基冲刺卷-双基训练12" ], "genre": "解答题", "ans": "(1) $y=2x$; (2) $(-\\infty,-1)$", @@ -361295,7 +364580,8 @@ "content": "设函数$y=f(x)$, 其中$f(x)=\\dfrac{\\mathrm{e}^x}{x+a}$, 若$f'(1)=\\dfrac{\\mathrm{e}}{4}$, 则$a=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-16_导数及其应用" ], "genre": "填空题", "ans": "$1$", @@ -362628,7 +365914,9 @@ "content": "已知集合$A=\\{1,2\\}, B=\\{a, a^2, 3\\}, A \\cap B=\\{1\\}$, 则$a$的值为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-01_集合与逻辑", + "2023届高三-赋能-赋能20" ], "genre": "填空题", "ans": "$-1$", @@ -362665,7 +365953,8 @@ "content": "设集合$A=\\{x | x=\\sqrt{5 k+1},\\ k \\in \\mathbf{N}, \\ k\\ge 1\\}, B=\\{x | x \\leq 6,\\ x \\in \\mathbf{Q}\\}$, 则$A \\cap B=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-01_集合与逻辑" ], "genre": "填空题", "ans": "$\\{4,6\\}$", @@ -362808,7 +366097,8 @@ "content": "已知$f(x)=(a^2-1) x^2+(a-1) x+2$, 写出``$f(x)>0$($x \\in \\mathbf{R}$)恒成立''的充要条件是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-01_集合与逻辑" ], "genre": "填空题", "ans": "$(-\\infty,-\\dfrac 97)\\cup [1,+\\infty)$", @@ -362858,7 +366148,8 @@ "content": "设$A, B$两点坐标分别为$(-1,0),(1,0)$. 条件甲: $A$、$B$、$C$三点是以$\\angle C$为钝角的三角形的三个顶点; 条件乙: 点$C$的坐标是方程$x^2+2 y^2=1$($y \\neq 0$)的解, 则甲是乙的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充要条件}{既不充分又不必要条件}", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-01_集合与逻辑" ], "genre": "选择题", "ans": "B", @@ -362930,7 +366221,8 @@ "content": "定义集合运算: $A \\odot B=\\{z | z=x y(x+y),\\ x \\in A,\\ y \\in B\\}$, 设集合$A=\\{0,1\\}$, $B=\\{2,3\\}$, 则集合$A \\odot B$的所有元素之和为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-01_集合与逻辑" ], "genre": "填空题", "ans": "$18$", @@ -363026,7 +366318,8 @@ "content": "已知集合$P=\\{x \\| x-a |<4\\}$, $Q=\\{x | x^2-4 x+3<0\\}$, 且``$x \\in P$''是``$x \\in Q$''的必要非充分条件, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-01_集合与逻辑" ], "genre": "填空题", "ans": "$[-1,5]$", @@ -363078,7 +366371,8 @@ "content": "已知集合$M=\\{1,2,3, \\cdots, 10\\}$, 集合$A \\subseteq M$, 定义$M(A)$为$A$中元素的最小值, 当$A$取遍$M$的所有非空子集时, 对应的$M(A)$的和记为$S_{10}$, 则$S_{10}=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-第二轮复习讲义-01_集合与逻辑" ], "genre": "填空题", "ans": "$2036$", @@ -363289,7 +366583,8 @@ "content": "设$a$、$b$、$c\\in (0,+\\infty)$且$\\dfrac{1}{a}+\\dfrac{9}{b}=1$, 则不等式$a+b-c \\geq 0$对任意满足条件的$a$、$b$恒成立的$c$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-02_等式与不等式" ], "genre": "填空题", "ans": "$(0,16]$", @@ -363323,7 +366618,10 @@ "objs": [], "tags": [ "第一单元", - "第二单元" + "第二单元", + "2023届高三-四月错题重做-01_函数一", + "2023届高三-四月错题重做-01_易错题-函数1", + "2023届高三-第二轮复习讲义-02_等式与不等式" ], "genre": "填空题", "ans": "$(3,+\\infty)$", @@ -363406,7 +366704,9 @@ "objs": [], "tags": [ "第一单元", - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-02_等式与不等式", + "2023届高三-第二轮复习讲义-02_等式与不等式" ], "genre": "填空题", "ans": "\\textcircled{4}", @@ -363483,7 +366783,8 @@ "content": "正方形$ABCD$的边长为$1, K$为对角线$BD$上一动点, 联结$CK$并延长, 交$BA$于点$M$, 则$\\triangle CKD$与$\\triangle BKM$的面积之和最小时, $DK=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-02_等式与不等式" ], "genre": "填空题", "ans": "$1$", @@ -363692,7 +366993,8 @@ "content": "已知关于$x$的方程$a x+2 a+1=0$. 若该方程在区间$[-1,1]$上有解, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-02_等式与不等式" ], "genre": "填空题", "ans": "$[-1,-\\dfrac 13]$", @@ -363749,7 +367051,8 @@ "objs": [], "tags": [ "第一单元", - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-02_等式与不等式" ], "genre": "解答题", "ans": "$[\\dfrac 14,\\dfrac 34)$", @@ -363782,7 +367085,8 @@ "content": "已知$a_n=(n-1) n$, $b_n=n^2$, $n \\in \\mathbf{N}$, $n \\ge 1$.\\\\\n(1) 若$ma_n^2+b_m^2-2 a_m b_n$;\\\\\n(2) 求最小的自然数$k$, 使得当$n \\geq k$时, 对任意实数$\\lambda \\in[0,1]$, 不等式$(2 \\lambda-3) b_n \\geq(2 \\lambda-4) a_n+(\\lambda-3)$恒成立.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-02_等式与不等式" ], "genre": "解答题", "ans": "(1) 证明略; (2) $k$的最小值为$3$", @@ -363807,7 +367111,8 @@ "content": "不等式$\\dfrac{x+5}{(x-1)^2} \\geq 2$的解集为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-02_等式与不等式" ], "genre": "填空题", "ans": "$[-\\dfrac 12,1)\\cup (1,3]$", @@ -363855,7 +367160,8 @@ "objs": [], "tags": [ "第一单元", - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-02_等式与不等式" ], "genre": "填空题", "ans": "$(1,2)\\cup (\\sqrt{10},+\\infty)$", @@ -363955,7 +367261,8 @@ "content": "关于$x$的不等式$a x-b<0$的解集是$(1,+\\infty)$, 则关于$x$的不等式$(a x+b)(x-3)>0$的解集为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-02_等式与不等式" ], "genre": "填空题", "ans": "$(-1,3)$", @@ -363989,7 +367296,10 @@ "objs": [], "tags": [ "第一单元", - "第二单元" + "第二单元", + "2023届高三-四月错题重做-02_函数二", + "2023届高三-四月错题重做-02_易错题-函数2", + "2023届高三-第二轮复习讲义-02_等式与不等式" ], "genre": "填空题", "ans": "$(0,\\dfrac 12]\\cup [8,+\\infty)$", @@ -364079,7 +367389,8 @@ "objs": [], "tags": [ "第一单元", - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-02_等式与不等式" ], "genre": "填空题", "ans": "$(-\\dfrac 14,+\\infty)$", @@ -364126,7 +367437,8 @@ "content": "已知$a, b\\in (0,+\\infty)$, 且$a \\neq b$, $n$是正整数. 求证: $(a+b)(a^n+b^n)<2(a^{n+1}+b^{n+1})$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-02_等式与不等式" ], "genre": "解答题", "ans": "证明略", @@ -364181,7 +367493,9 @@ "content": "集合$A=\\{(x, y) |(y-x)(y-\\dfrac{1}{x}) \\geq 0\\}$, $B=\\{(x, y) |(x-1)^2+(y-1)^2 \\leq 1\\}$, 则$A \\cap B$所表示的平面图形的面积为\\bracket{20}.\n\\fourch{$\\pi$}{$\\dfrac{\\pi}{2}$}{$\\dfrac{\\pi}{3}$}{$\\dfrac{1}{2}$}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-01_集合与逻辑", + "2023届高三-第二轮复习讲义-02_等式与不等式" ], "genre": "选择题", "ans": "B", @@ -364285,7 +367599,8 @@ "content": "设$a, b$是两个实数, $A=\\{(x, y) | x=n, y=n a+b, n \\in \\mathbf{Z}\\}$, $B=\\{(x, y) | x=m, \\ y=3(m^2+5),\\ m \\in \\mathbf{Z}\\}$, $C=\\{x \\cdot y | x^2+y^2 \\leq 144\\}$, 讨论是否存在$a, b$使得$A \\cap B \\neq \\varnothing$且$(a, b) \\in C$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-01_集合与逻辑" ], "genre": "解答题", "ans": "不存在, 理由略", @@ -365225,7 +368540,8 @@ "content": "已知函数$f(x)=4^x-2^x$, 实数$s, t$满足$f(s)+f(t)=0$, $a=2^s+2^t$, $b=2^{s+t}$.\\\\\n(1) 当函数$f(x)$的定义域为$[-1,1]$时, 求函数$f(x)$的值域;\\\\\n(2) 求函数关系式$b=g(a)$, 并求函数$g(a)$的定义域$D$;\\\\\n(3) 在 (2) 的结论中, 对任意$x_1 \\in D$, 都存在$x_2 \\in[-1,1]$, 使得$g(x_1)=f(x_2)+m$成立, 求实数$m$的取值范围.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-03_幂指对函数" ], "genre": "解答题", "ans": "(1) $[-\\dfrac 14,2]$; (2) $g(a)=\\dfrac 12 a^2-\\dfrac 12 a$, $D=(1,2]$; (3) $[-1,\\dfrac 14]$", @@ -366126,7 +369442,10 @@ "content": "已知函数$f(x)=\\begin{cases}|x|+2, & x<1, \\\\ x+\\dfrac{2}{x}, & x \\geq 1,\\end{cases}$ 设$a \\in \\mathbf{R}$, 若关于$x$的不等式$f(x) \\geq|\\dfrac{x}{2}+a|$在$\\mathbf{R}$上恒成立, 则$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-四月错题重做-02_函数二", + "2023届高三-四月错题重做-02_易错题-函数2", + "2023届高三-第二轮复习讲义-05_函数综合" ], "genre": "填空题", "ans": "$[-2,2]$", @@ -366174,7 +369493,8 @@ "objs": [], "tags": [ "第二单元", - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-05_函数综合" ], "genre": "选择题", "ans": "B", @@ -366209,7 +369529,8 @@ "content": "已知函数$f(x)=\\log _4(4^x+1)+k x$是偶函数.\\\\\n(1) 求$k$的值;\\\\\n(2) 若方程$f(x)-m=0$有解, 求实数$m$的取值范围.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-05_函数综合" ], "genre": "解答题", "ans": "(1) $2$; (2) $[\\dfrac 12,+\\infty)$", @@ -366244,7 +369565,8 @@ "content": "已知函数$f(x)=|2 x+a|+|2 x-1|$, $g(x)=\\dfrac{6 x-5}{2 x-1}$.\\\\\n(1) 当$a=3$时, 解不等式$f(x) \\leq 6$;\\\\\n(2) 若对任意$x_1 \\in[1, \\dfrac{5}{2}]$都存在$x_2 \\in \\mathbf{R}$, 使得$g(x_1)=f(x_2)$成立, 求实数$a$的取值范围.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-05_函数综合" ], "genre": "解答题", "ans": "(1) $[-2,-1]$; (2) $[-2,0]$", @@ -366279,7 +369601,8 @@ "content": "已知函数$f(x)=x^2+(x-1)|x-a|$, 是否存在实数$a$, 使不等式$f(x) \\geq 2 x-3$对一切实数$x$恒成立? 若存在, 求出$a$的取值范围, 若不存在, 请说明理由.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-05_函数综合" ], "genre": "解答题", "ans": "存在, 范围为$[-3,1]$", @@ -366380,7 +369703,8 @@ "content": "在$\\triangle ABC$中, 角$A, B, C$所对的边分别为$a, b, c$.\\\\\n(1) 若$\\sin C+\\sin (B-A)=\\sin 2A$, 试判断$\\triangle ABC$的形状;\\\\\n(2) 若$c=2$, $C=\\dfrac{\\pi}{3}$, 且$S_{\\triangle ABC}=\\sqrt{3}$, 求$a, b$的值.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-06_三角与解三角形" ], "genre": "解答题", "ans": "(1) 等腰三角形或直角三角形; (2) $a=b=2$", @@ -366437,7 +369761,8 @@ "content": "在$\\triangle ABC$中, 角$A, B, C$的对边分别为$a, b, c$, 若$a^2+b^2+4 \\sqrt{2}=c^2$, $a b=4$, 则$\\dfrac{\\sin C}{\\tan ^2A \\sin 2B}$的最小值是\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-06_三角与解三角形" ], "genre": "填空题", "ans": "$2+\\dfrac{3}{2}\\sqrt{2}$", @@ -366494,7 +369819,8 @@ "content": "已知$f(x)=\\sqrt{\\dfrac{1-x}{1+x}}$, 若$\\alpha \\in(\\dfrac{\\pi}{2}, \\pi)$, 则$f(\\cos \\alpha)+f(-\\cos \\alpha)=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-06_三角与解三角形" ], "genre": "填空题", "ans": "$\\dfrac{2}{\\sin\\alpha}$", @@ -366705,7 +370031,8 @@ "content": "函数$y=2 \\cos ^2 x+\\sin 2 x$的最小值是\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-07_三角函数" ], "genre": "填空题", "ans": "$1-\\sqrt{2}$", @@ -366761,7 +370088,8 @@ "content": "已知函数$f(x)=\\sin x$.\\\\\n(1) 设$a \\in \\mathbf{R}$, 判断函数$g(x)=a \\cdot f(x)+f(x+\\dfrac{\\pi}{2})$的奇偶性, 并说明理由;\\\\\n(2) 设函数$F(x)=2 f(x)-\\sqrt{3}$. 对任意$b \\in \\mathbf{R}$, 求$y=F(x)$在区间$[b, b+10 \\pi]$上零点个数的所有可能值.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-07_三角函数" ], "genre": "解答题", "ans": "(1) 当$a=0$时, $g(x)$是偶函数; 当$a\\ne 0$时, $g(x)$既不是奇函数也不是偶函数; (2) $10$或$11$", @@ -366795,7 +370123,8 @@ "content": "若函数$F(x)=2 \\sin x-\\sqrt{3}$在$[a, b]$上至少含有$10$个零点, 则$b-a$的最小值为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-07_三角函数" ], "genre": "填空题", "ans": "$\\dfrac{25\\pi}3$", @@ -366895,7 +370224,8 @@ "content": "设函数$f(x)=\\sin 2 x$, 若$f(x+t)$是偶函数, 则$t$的一个可能值是\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-07_三角函数" ], "genre": "填空题", "ans": "(如)$\\dfrac\\pi 4$(所有可能的解为$\\dfrac\\pi 4+\\dfrac{k\\pi}2$, $k\\in \\mathbf{Z}$)", @@ -366929,7 +370259,8 @@ "content": "已知$-\\dfrac{\\pi}{6} \\leq x<\\dfrac{\\pi}{3}$, $\\cos x=\\dfrac{m-1}{m+1}$, 则$m$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-07_三角函数" ], "genre": "填空题", "ans": "$(3,+\\infty)$", @@ -366998,7 +370329,8 @@ "content": "设$\\omega>0$, 函数$y=\\sin (\\omega x+\\dfrac{\\pi}{3})+2$的图像向右平移$\\dfrac{4 \\pi}{3}$个单位后与原图像重合, 则$\\omega$的最小值是\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-07_三角函数" ], "genre": "填空题", "ans": "$\\dfrac 32$", @@ -367113,7 +370445,8 @@ "content": "$\\triangle ABC$中, $B=60^{\\circ}, AC=\\sqrt{3}$, 则$AB+2BC$的最大值为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-06_三角与解三角形" ], "genre": "填空题", "ans": "$2\\sqrt{7}$", @@ -367215,7 +370548,8 @@ "content": "某动物园要为刚入园的小动物建造一间两面靠墙的三角形露天活动室, 地面形状如图所示, 已知已有两面墙的夹角为$\\dfrac{\\pi}{3}$(即$\\angle ACB=\\dfrac{\\pi}{3}$), 墙$AB$的长度为$6$米 (已有两面墙的可利用长度足够大), 记$\\angle ABC=\\theta$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$C$} coordinate (C);\n\\draw (2,0) node [above right] {$B$} coordinate (B);\n\\draw (60:3) node [right] {$A$} coordinate (A);\n\\draw (A) ++ (60:0.5) coordinate (A1);\n\\draw (B) ++ (0.5,0) coordinate (B1);\n\\draw (A1) ++ (150:0.1) coordinate (A2);\n\\draw (B1) ++ (0,-0.1) coordinate (B2);\n\\draw (C) ++ (-150:0.2) coordinate (C2);\n\\fill [gray] (B2) -- (C2) -- (A2) -- (A1) -- (C) -- (B1) -- cycle;\n\\draw (A1) -- (C) -- (B1);\n\\draw (A)--(B);\n\\draw pic (B) [draw, \"$\\theta$\", angle eccentricity = 1.5] {angle = A--B--C};\n\\end{tikzpicture}\n\\end{center}\n(1) 若$\\theta=\\dfrac{\\pi}{4}$, 求$\\triangle ABC$的周长(结果精确到$0.01$米);\\\\\n(2) 为了使小动物能健康成长, 要求所建造的三角形露天活动室面积即$\\triangle ABC$的面积尽可能大. 问当$\\theta$为何值时, 该活动室面积最大? 并求出最大面积.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-06_三角与解三角形" ], "genre": "解答题", "ans": "(1) 约为$17.59$米($6+3\\sqrt{6}+3\\sqrt{2}$); (2) 当$\\theta=\\dfrac\\pi 3$时, 活动室面积最大, 最大面积为$9\\sqrt{3}$平方米.", @@ -367250,7 +370584,8 @@ "content": "设$\\alpha_1$、$\\alpha_2 \\in \\mathbf{R}$, 且$\\dfrac{1}{2+\\sin \\alpha_1}+\\dfrac{1}{2+\\sin (2 \\alpha_2)}=2$, 则$|10 \\pi-\\alpha_1-\\alpha_2|$的最小值等于\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-06_三角与解三角形" ], "genre": "填空题", "ans": "$\\dfrac\\pi 4$", @@ -367462,7 +370797,8 @@ "content": "在直角三角形$ABC$中, $\\angle A=\\dfrac{\\pi}{2}$, $AB=1$, $AC=2$, $M$是$\\triangle ABC$内一点, 且$AM=\\dfrac{1}{2}$, 若$\\overrightarrow{AM}=\\lambda \\overrightarrow{AB}+\\mu \\overrightarrow{AC}$, 则$\\lambda+2 \\mu$的最大值为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第二轮复习讲义-08_平面向量与复数" ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{2}}2$", @@ -367497,7 +370833,8 @@ "content": "如图所示, 在同一个平面内, 向量$\\overrightarrow{OA}, \\overrightarrow{OB}, \\overrightarrow{OC}$的模分别为$1,1, \\sqrt{2}$, $\\overrightarrow{OA}$与$\\overrightarrow{OC}$的夹角为$\\alpha$, 且$\\tan \\alpha=7$, $\\overrightarrow{OB}$与$\\overrightarrow{OC}$的夹角为$45^{\\circ}$. 若$\\overrightarrow{OC}=m \\overrightarrow{OA}+n \\overrightarrow{OB}$($m, n \\in \\mathbf{R}$), 则$m+n=$\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (1,0) node [below] {$A$} coordinate (A);\n\\draw ($(O)!{sqrt(2)}!{atan(7)}:(A)$) node [above] {$C$} coordinate (C);\n\\draw ($(O)!{sqrt(2)/2}!45:(C)$) node [left] {$B$} coordinate (B);\n\\draw [->] (O)--(A);\n\\draw [->] (O)--(B);\n\\draw [->] (O)--(C);\n\\draw (O) pic [draw, \"$\\alpha$\", angle eccentricity = 1.5] {angle = A--O--C};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第二轮复习讲义-08_平面向量与复数" ], "genre": "填空题", "ans": "$3$", @@ -367553,7 +370890,8 @@ "content": "在正方形$ABCD$中, $AB=2$, 点$E$为$BC$的中点, 点$F$在边$CD$上. 若$\\overrightarrow{AE} \\cdot \\overrightarrow{BF}=0$, 则$\\overrightarrow{AE} \\cdot \\overrightarrow{AF}=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第二轮复习讲义-08_平面向量与复数" ], "genre": "填空题", "ans": "$4$", @@ -367720,7 +371058,8 @@ "content": "在$\\triangle ABC$中, 已知$\\overrightarrow{AB} \\cdot \\overrightarrow{AC}=\\overrightarrow{BA} \\cdot \\overrightarrow{BC}$.\\\\\n(1) 求证: $|\\overrightarrow{AC}|=|\\overrightarrow{BC}|$;\\\\\n(2) 若$|\\overrightarrow{AC}+\\overrightarrow{BC}|=|\\overrightarrow{AC}-\\overrightarrow{BC}|=\\sqrt{6}$, 求$|\\overrightarrow{BA}-t \\overrightarrow{BC}|$的最小值及相应的$t$值.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第二轮复习讲义-08_平面向量与复数" ], "genre": "解答题", "ans": "(1) 证明略; (2) 最小值为$\\sqrt{3}$, 取到最小值时$t=1$", @@ -367821,7 +371160,8 @@ "content": "已知实数$x_1$、$x_2$、$y_1$、$y_2$满足: $x_1^2+y_1^2=1$, $x_2^2+y_2^2=1$, $x_1 x_2+y_1 y_2=\\dfrac{1}{2}$, 则$\\dfrac{|x_1+y_1-1|}{\\sqrt{2}}+\\dfrac{|x_2+y_2-1|}{\\sqrt{2}}$的最大值为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-08_平面向量与复数" ], "genre": "填空题", "ans": "$\\sqrt{3}+\\sqrt{2}$", @@ -368124,7 +371464,8 @@ "content": "如图, 点$P$位于两条平行直线$l_1, l_2$的下方, 它到直线$l_1, l_2$距离分别为$1$, $3$, 动点$N, M$分别在$l_1, l_2$上, 满足$|\\overrightarrow{PM}+\\overrightarrow{PN}|=8$, 则$\\overrightarrow{PM} \\cdot \\overrightarrow{PN}$的最大值为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (-2,0) -- (3,0) node [right] {$l_1$} (-2,2) -- (3,2) node [right] {$l_2$};\n\\draw [->] ({2-sqrt(14)},-1) node [below] {$P$} coordinate (P) -- (1,0) node [below right] {$N$};\n\\draw [->] (P) -- (2,2) node [below right] {$M$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$15$}{$12$}{$10$}{$9$}", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第二轮复习讲义-08_平面向量与复数" ], "genre": "选择题", "ans": "A", @@ -368203,7 +371544,8 @@ "content": "设$S_n$为等比数列$\\{a_n\\}$的前$n$项和. 若$a_1=1$, 且$3S_1, 2S_2, S_3$成等差数列, 则$a_n=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-14_等差数列和等比数列" ], "genre": "填空题", "ans": "$3^{n-1}$", @@ -368260,7 +371602,10 @@ "content": "数列$\\{a_n\\}$的前$n$项和为$S_n$, $2S_n-n a_n=n$($n \\in \\mathbf{N}$, $n\\ge 1$), 若$S_{20}=-360$, 则$a_2=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-四月错题重做-03_数列", + "2023届高三-四月错题重做-03_易错题-数列", + "2023届高三-第二轮复习讲义-14_等差数列和等比数列" ], "genre": "填空题", "ans": "$-1$", @@ -368329,7 +371674,8 @@ "content": "设数列$\\{a_n\\}$的前$n$项和为$S_n$, 且$a_1=a_2=1$, $\\{n S_n+(n+2) a_n\\}$为等差数列, 则$\\{a_n\\}$的通项公式为$a_n=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-14_等差数列和等比数列" ], "genre": "填空题", "ans": "$n\\cdot (\\dfrac 12)^{n-1}$", @@ -368386,7 +371732,10 @@ "content": "设$S_n$是等比数列$\\{a_n\\}$的前$n$项和, 若$\\dfrac{S_4}{S_2}=3$, 则$\\dfrac{S_6}{S_4}=$\\bracket{20}.\n\\fourch{$2$}{$\\dfrac{7}{3}$}{$\\dfrac{3}{10}$}{$1$或$2$}", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-四月错题重做-03_数列", + "2023届高三-四月错题重做-03_易错题-数列", + "2023届高三-第二轮复习讲义-14_等差数列和等比数列" ], "genre": "选择题", "ans": "B", @@ -368432,7 +371781,8 @@ "content": "等差数列$\\{a_n\\}$中, $a_1=\\dfrac{1}{2015}$, $a_m=\\dfrac{1}{n}$, $a_n=\\dfrac{1}{m}$($m \\neq n$), 则数列$\\{a_n\\}$的公差为\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-14_等差数列和等比数列" ], "genre": "填空题", "ans": "$\\dfrac{1}{2015}$", @@ -368645,7 +371995,8 @@ "content": "已知数列$\\{a_n\\}$满足$a_n\\dfrac{m}{32}$成立? 若存在, 求出$m$的值; 若不存在, 请说明理由.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-14_等差数列和等比数列" ], "genre": "解答题", "ans": "(1) $a_n=10-2n$; (2) $S_n=\\begin{cases}-n^2+9n, & n=1,2,3,4,5, \\\\ n^2-9n+40, & n=6,7,8,\\cdots.\\end{cases}$ (3) 存在, 整数$m$的最大值为$7$", @@ -369119,7 +372472,8 @@ "content": "若实数$a, b$是函数$f(x)=x^2-p x+q$($p>0$, $q>0$)的两个不同的零点, 且$a, b,-2$这三个数可适当排序后成等差数列, 也可适当排序后成等比数列, 则$p+q$的值等于\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-15_数列综合" ], "genre": "填空题", "ans": "$9$", @@ -369241,7 +372595,8 @@ "content": "已知数列$\\{a_n\\}$中, $a_1=3$, $a_2=5$, $\\{a_n\\}$的前$n$项和为$S_n$, 且满足$S_n+S_{n-2}=2S_{n-1}+2^{n-1}$($n \\geq 3$).\\\\\n(1) 试求数列$\\{a_n\\}$的通项公式;\\\\\n(2) 令$b_n=\\dfrac{2^{n-1}}{a_n \\cdot a_{n+1}}$, $T_n$是数列$\\{b_n\\}$的前$n$项和, 证明: $T_n<\\dfrac{1}{6}$;\\\\ \n(3) 证明: 对任意给定的$m \\in(0, \\dfrac{1}{6})$, 均存在$n_0 \\in \\mathbf{N}$, 使得当$n>n_0$时, (2)中的$T_n>m$恒成立.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-15_数列综合" ], "genre": "解答题", "ans": "(1) $a_n=2^n+1$; (2) 证明略; (3) 证明略", @@ -369453,7 +372808,8 @@ "objs": [], "tags": [ "第四单元", - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-11_直线与圆" ], "genre": "填空题", "ans": "$(1,-2)$", @@ -369488,7 +372844,8 @@ "content": "已知实数$x, y$满足$2 x+y+5=0$, 那么$x^2+y^2$的最小值是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-11_直线与圆" ], "genre": "填空题", "ans": "$5$", @@ -369545,7 +372902,8 @@ "content": "平面直角坐标系$xOy$中, 以点$(1,0)$为圆心且与直线$m x-y-2 m-1=0$($m \\in \\mathbf{R}$)相切的所有圆中, 半径最大的圆的标准方程是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-11_直线与圆" ], "genre": "填空题", "ans": "$(x-1)^2+y^2=2$", @@ -369602,7 +372960,8 @@ "content": "如图, 正方形$ABCD$的边长为$20$米, 圆$O$的半径为$1$米, 圆心是正方形的中心, 点$P$、$Q$分别在线段$AD$、$CB$上, 若线段$PQ$与圆$O$有公共点, 则称点$Q$在点$P$的``盲区''中, 已知点$P$以$1.5$米/秒的速度从$A$出发向$D$移动, 同时, 点$Q$以$1$米/秒的速度从$C$出发向$B$移动, 则在点$P$$A$移动到$D$的过程中, 点$Q$在点$P$的盲区中的时长约为\\blank{50}秒. (精确到$0.1$)\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0) node [right] {$B$} coordinate (B);\n\\draw (2,2) node [right] {$C$} coordinate (C);\n\\draw (0,2) node [left] {$D$} coordinate (D);\n\\draw (A) rectangle (C);\n\\draw (1,1) circle (0.1) node [below] {$O$};\n\\draw (0,0.45) node [left] {$P$} coordinate (P);\n\\draw (2,1.7) node [right] {$Q$} coordinate (Q);\n\\draw (P)--(Q);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-11_直线与圆" ], "genre": "填空题", "ans": "$4.4$", @@ -369661,7 +373020,8 @@ "content": "如图, 为了保护河上古桥$OA$, 规划建一座新桥$BC$, 同时设立一个圆形保护区. 规划要求: 新桥$BC$与河岸$AB$垂直; 保护区的边界为圆心$M$在线段$OA$上并与$BC$相切的圆. 且古桥两端$O$和$A$到该圆上任意一点的距离均不少于$80 \\text{m}$. 经测量, 点$A$位于点$O$正北方向$60 \\text{m}$处, 点$C$位于点$O$正东方向$170 \\text{m}$处($OC$为河岸), $\\tan \\angle BCO=\\dfrac{4}{3}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (3.4,0) node [below right] {$C$} coordinate (C);\n\\draw (0,1.2) node [above left] {$A$} coordinate (A);\n\\draw (C) ++ ({90+atan(3/4)}:3) coordinate (T);\n\\draw ($(C)!(A)!(T)$) node [above] {$B$} coordinate (B);\n\\draw (A)--(B)--(C) (A)--(O)--(C);\\\n\\draw (O) --++ (0,-0.5) coordinate (O1) (C) --++ (0,-0.5) coordinate (C1);\n\\draw [<->] ($(O)!0.5!(O1)$) -- ($(C)!0.5!(C1)$) node [midway, fill = white] {$170\\text{m}$};\n\\draw (O) --++ (-0.5,0) coordinate (O2) (A) --++ (-0.5,0) coordinate (A2);\n\\draw [<->] ($(O)!0.5!(O2)$) -- ($(A)!0.5!(A2)$) node [midway, fill = white, rotate = 90] {$60\\text{m}$};\n\\draw [->] (C) -- ($(O)!1.3!(C)$) node [below] {东};\n\\draw [->] (A) -- ($(O)!1.6!(A)$) node [left] {北};\n\\end{tikzpicture}\n\\end{center}\n(1) 求新桥$BC$的长;\\\\\n(2) 当$OM$为多长时, 圆形保护区的面积最大?", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-11_直线与圆" ], "genre": "解答题", "ans": "(1) $150$米; (2) ($OM\\in [10,35]$)当$OM$的长为$10$米时, 圆形保护区的面积最大", @@ -369740,7 +373100,8 @@ "content": "若点$A(4,3)$在椭圆$\\Gamma: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)上, 则在点$B(4,-3)$、$C(3,-4)$、$D(-4,-3)$、$F(-4,3)$中, 不在椭圆$\\Gamma$上的点有\\blank{50}.(请写出所有满足要求的点)", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-13_解析几何综合" ], "genre": "填空题", "ans": "点$C$", @@ -369797,7 +373158,8 @@ "content": "已知一个酒杯的轴截面是抛物线的一部分, 酒杯杯口直径和酒杯深都是$8 \\text{cm}$. 现准备在酒杯内放入一个玻璃球, 使得玻璃球触及酒杯底部, 求玻璃球半径的取值范围.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-12_圆锥曲线" ], "genre": "解答题", "ans": "$(0,1]$(单位: $\\text{cm}$)", @@ -369832,7 +373194,10 @@ "content": "如图, 从双曲线$C: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的左焦点$F_1$引圆$x^2+y^2=a^2$的切线, 切点为$T$, 延长$F_1T$交双曲线右支于点$P$, 若$M$是线段$F_1P$的中点, 且$M$在线段$PT$上, $O$为坐标原点, 则$|MT|-|MO|$的值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,0) circle (1);\n\\path [domain = -4:4, samples = 100, name path = cr, draw] plot ({sqrt(1+\\x*\\x/3)},\\x);\n\\draw [domain = -4:4, samples = 100] plot ({-sqrt(1+\\x*\\x/3)},\\x);\n\\draw (-2,0) node [below] {$F_1$} coordinate (F_1);\n\\draw (2,0) node [below] {$F_2$} coordinate (F_2);\n\\path [name path = F1P] (F_1) --++ (30:5);\n\\path [name intersections = {of = F1P and cr, by = P}];\n\\draw (P) node [right] {$P$} -- (F_1) (P) -- (F_2);\n\\draw ($(F_1)!0.5!(P)$) node [above] {$M$} coordinate (M) -- (0,0);\n\\draw ($(F_1)!(0,0)!(P)$) node [above left] {$T$} coordinate (T) -- (0,0);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-四月错题重做-04_易错题-解析几何", + "2023届高三-四月错题重做-04_解析几何", + "2023届高三-第二轮复习讲义-13_解析几何综合" ], "genre": "填空题", "ans": "$a-b$", @@ -369985,7 +373350,8 @@ "content": "已知点$A(3,0)$和圆$B: (x+3)^2+y^2=16$, 动圆$C$与圆$B$外切, 且过点$A$, 则动圆圆心$C$的轨迹方程是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-12_圆锥曲线" ], "genre": "填空题", "ans": "$\\dfrac{x^2}{4}-\\dfrac{y^2}{5}=1$($x>0$)", @@ -370019,7 +373385,8 @@ "content": "已知椭圆$C$的焦点为$F_1(-1,0)$, $F_2(1,0)$, 过$F_2$的直线与椭圆$C$交于$A$、$B$两点. 若$|AF_2|=2|F_2B|$, $|AB|=|BF_1|$, 则椭圆$C$的方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-13_解析几何综合" ], "genre": "填空题", "ans": "$\\dfrac{x^2}3+\\dfrac{y^2}2=1$", @@ -370053,7 +373420,8 @@ "content": "设$F_1$、$F_2$分别是椭圆$C: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的左、右焦点, 过点$F_1$的直线交椭圆$C$于$A$、$B$两点, 且$|AF_1|=3|BF_1|$.\\\\\n(1) 若$|AB|=4$, $\\triangle ABF_2$的周长为$16$, 求$|AF_2|$;\\\\\n(2) 若$\\cos \\angle AF_2B=\\dfrac{3}{5}$, $a=\\lambda b$, 求实数$\\lambda$的值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-12_圆锥曲线" ], "genre": "解答题", "ans": "(1) $5$; (2) $\\lambda = \\sqrt{2}$", @@ -370109,7 +373477,8 @@ "content": "若经过点$F_2(2,0)$的直线$l$与双曲线$x^2-\\dfrac{y^2}{3}=1$相交于$A$、$B$两点, 且$|AB|=6$, 则满足条件的直线$l$共有\\blank{50}条.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-13_解析几何综合" ], "genre": "填空题", "ans": "$3$", @@ -370231,7 +373600,8 @@ "content": "过抛物线$y^2=4 x$的焦点作直线交抛物线于$A(x_1, y_1)$、$B(x_2, y_2)$两点, 且$x_1+x_2=6$, 则$|AB|$的值为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-13_解析几何综合" ], "genre": "填空题", "ans": "$8$", @@ -370287,7 +373657,8 @@ "content": "若经过点$M(1, m)$且与双曲线$x^2-\\dfrac{y^2}{4}=1$恰有一个公共点的直线有且仅有$2$条, 则实数$m$的值是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-13_解析几何综合" ], "genre": "填空题", "ans": "$2$或$-2$", @@ -370589,7 +373960,8 @@ "content": "直线$l$与抛物线$y^2=2 x$相交于$A$、$B$两点, 与$x$轴正半轴不相交. 若$\\overrightarrow{OA} \\cdot \\overrightarrow{OB}=3$, 则直线$l$过定点\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-13_解析几何综合" ], "genre": "填空题", "ans": "$(-1,0)$", @@ -370646,7 +374018,8 @@ "content": "(1) 已知直线$l: 4 x-y-1=0$与抛物线$x^2=2 y$交于$A(x_A, y_A)$、$B(x_B, y_B)$两点, 直线$l$与$x$轴相交于点$C(x_C, 0)$, 求证: $\\dfrac{1}{x_A}+\\dfrac{1}{x_B}=\\dfrac{1}{x_C}$;\\\\\n(2) 试将第(1)题中的命题加以推广, 使得第(1)题中的命题是推广后得到的命题的特例, 并证明推广后得到的命题正确.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-13_解析几何综合" ], "genre": "解答题", "ans": "(1) 证明略; (2) (如)不经过原点的直线$l$与抛物线$x^2=2py$交于$A(x_A,y_A)$与$B(x_B,y_B)$两点, 直线$l$与$x$轴相交于点$C(x_C,y_C)$, 则$\\dfrac{1}{x_A}+\\dfrac{1}{x_B}=\\dfrac{1}{x_C}$, 证明略", @@ -370770,7 +374143,8 @@ "content": "设椭圆$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的左焦点为$F$, 上顶点为$B$, 焦距为$2 \\sqrt{5}$. 已知点$A$的坐标为$(b, 0)$, 且$|FB| \\cdot|AB|=6 \\sqrt{2}$.\\\\\n(1) 求椭圆的方程;\\\\\n(2) 设直线$l: y=k x$($k>0$)与椭圆在第一象限的交点为$P$, 且$l$与直线$AB$交于点$Q$. 若$\\dfrac{|AQ|}{|PQ|}=\\dfrac{5 \\sqrt{2}}{4} \\sin \\angle AOQ$($O$为原点), 求$k$的值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-13_解析几何综合" ], "genre": "解答题", "ans": "(1) $\\dfrac{x^2}9+\\dfrac{y^2}4=1$; (2) $k=\\dfrac 12$或$\\dfrac{11}{28}$", @@ -370805,7 +374179,8 @@ "content": "已知点$P(1,2)$在抛物线$C: y^2=2 p x$上. 过点$Q(0,1)$的直线$l$与抛物线$C$有两个不同的交点$A, B$, 且直线$PA$交$y$轴于$M$, 直线$PB$交$y$轴于$N$.\\\\\n(1) 求抛物线$C$的方程;\\\\\n(2) 求直线$l$的斜率的取值范围;\\\\\n(3) 设$O$为原点, $\\overrightarrow{QM}=\\lambda \\overrightarrow{QO}$, \n$\\overrightarrow{QN}=\\mu \\overrightarrow{QO}$, 求证: $\\dfrac{1}{\\lambda}+\\dfrac{1}{\\mu}$为定值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-13_解析几何综合" ], "genre": "解答题", "ans": "(1) $y^2=4x$; (2) $(-\\infty,-3)\\cup (-3,0)\\cup (0,1)$; (3) 定值为$2$, 证明略", @@ -370993,7 +374368,8 @@ "content": "如图, 正方体$ABCD-A_1B_1C_1D_1$的棱长为$2$, 动点$E, F$在棱$A_1B_1$上, 动点$P, Q$分别在棱$AD, CD$上, 若$EF=1$, $A_1E=x$, $DQ=y$, $DP=z$($x, y, z$大于零), 则四面体$PEFQ$的体积 \\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\filldraw ($(A1)!0.3!(B1)$) node [below] {$E$} coordinate (E) circle (0.03);\n\\filldraw ($(A1)!0.8!(B1)$) node [below] {$F$} coordinate (F) circle (0.03);\n\\filldraw ($(C)!0.6!(D)$) node [below] {$Q$} coordinate (Q) circle (0.03);\n\\filldraw ($(A)!0.3!(D)$) node [right] {$P$} coordinate (P) circle (0.03);\n\\end{tikzpicture}\n\\end{center}\n\\twoch{与$x, y, z$都有关}{与$x$有关, 与$y, z$无关}{与$y$有关, 与$x, z$无关}{与$z$有关, 与$x, y$无关}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-09_立体几何综合" ], "genre": "选择题", "ans": "D", @@ -371204,7 +374580,8 @@ "content": "如图, $AD$与$BC$是四面体$ABCD$中互相垂直的棱, $BC=2$, 若$AD=2 c$, 且$AB+BD=AC+CD=2 a$, 其中$a$、$c$为常数, 则四面体$ABCD$的体积的最大值是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,-0.8,0) node [below] {$A$} coordinate (A);\n\\draw (0,1.2,0) node [above] {$D$} coordinate (D);\n\\draw (1,0,0.8) node [below] {$B$} coordinate (B);\n\\draw (1,0,-0.8) node [right] {$C$} coordinate (C);\n\\draw (A)--(D) (B)--(D) (C)--(D) (A)--(B)--(C);\n\\draw [dashed] (A)--(C);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-09_立体几何综合" ], "genre": "填空题", "ans": "$\\dfrac 23 c\\sqrt{a^2-c^2-1}$", @@ -371261,7 +374638,8 @@ "content": "已知平面$ABC$中, $\\overrightarrow{AC}=(0,1,0)$, $\\overrightarrow{AB}=(1,0,1)$, 则平面$ABC$的一个法向量为$\\overrightarrow {n}=$\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-10_空间向量与应用" ], "genre": "填空题", "ans": "$(1,0,-1)$", @@ -371295,7 +374673,8 @@ "content": "已知直线$l_1$与直线$l_2$异面, 若$l_1$的一个方向向量为$\\overrightarrow {d_1}=(2,0,-2)$, $l_2$的一个方向向量为$\\overrightarrow{d_2}=(1,-1,-1)$, 则异面直线$l_1$与$l_2$所成角的大小为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-10_空间向量与应用" ], "genre": "填空题", "ans": "$\\arccos\\dfrac{\\sqrt{6}}3$", @@ -371329,7 +374708,8 @@ "content": "平面$\\alpha$的一个法向量为$\\overrightarrow {m}=(a,-3,2)$, 直线$l$的一个方向向量为$\\overrightarrow {d}=(-2,-1,3)$, 若直线$l$与平面$\\alpha$所成的角为$\\dfrac{\\pi}{6}$, 则$a=$\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-10_空间向量与应用" ], "genre": "填空题", "ans": "$1$或$71$", @@ -371364,7 +374744,8 @@ "content": "如图的几何体是由一个棱长为$2$的正方体$ABCD-A_1B_1C_1D_1$与一个侧棱长为$2$的正四棱锥$P-A_1B_1C_1D_1$组合而成. 若点$E$是棱$BC$的中点, $F$是直线$A_1B_1$上的一个动点, 当异面直线$PE$与$AF$所成角最大时, 求线段$A_1F$的长.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1);\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1) (C1) -- (D1) -- (A1);\n\\draw ($(A1)!0.5!(C1)$) ++ (0,{sqrt(2)}) node [above] {$P$} coordinate (P);\n\\draw (A1) -- (P) -- (B1) (P) -- (C1);\n\\draw [dashed] (D1) -- (P);\n\\filldraw ($(B)!0.5!(C)$) node [right] {$E$} coordinate (E) circle (0.03);\n\\filldraw ($(A1)!0.45!(B1)$) node [below] {$F$} coordinate (F) circle (0.03);\n\\draw (A)--(F);\n\\draw [dashed] (P)--(E);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-09_立体几何综合" ], "genre": "解答题", "ans": "$4+2\\sqrt{2}$", @@ -371421,7 +374802,8 @@ "content": "如图, 在四棱锥$P-ABCD$中, 已知$PA \\perp$平面$ABCD$, 且四边形$ABCD$为直角梯形, $\\angle ABC=\\angle BAD=\\dfrac{\\pi}{2}$, $PA=AD=2$, $AB=BC=1$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [above right] {$A$} coordinate (A);\n\\draw (0,0,1) node [left] {$B$} coordinate (B);\n\\draw (B) ++ (1,0,0) node [below] {$C$} coordinate (C);\n\\draw (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw ($(B)!0.5!(P)$) node [left] {$Q$} coordinate (Q);\n\\draw (P)--(B) (P)--(C) (P)--(D) (B)--(C)--(D) (Q)--(C);\n\\draw [dashed] (B)--(A)--(D) (A)--(P); \n\\end{tikzpicture}\n\\end{center}\n(1) 求四棱锥$P-ABCD$的表面积;\\\\\n(2) 若$P, A, C, D$四点在同一球面上, 求该球的体积;\\\\\n(3) 点$Q$是线段$BP$上的动点, 当直线$CQ$与$DP$所成的角最小时, 求线段$BQ$的长.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-09_立体几何综合" ], "genre": "解答题", "ans": "(1) $\\dfrac 92+\\dfrac{\\sqrt{5}}2+\\sqrt{3}$; (2) $\\dfrac{8\\sqrt{2}}3\\pi$; (3) $\\dfrac{2\\sqrt{5}}5$", @@ -371456,7 +374838,8 @@ "content": "已知$M(2,1,8)$是平面$\\alpha$外的一点, $A(-1,-2,5)$是平面$\\alpha$内的一点, $\\overrightarrow {n}=(-4,0,3)$是平面$\\alpha$的一个法向量, 则点$M$到平面$\\alpha$的距离$d=$\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-10_空间向量与应用" ], "genre": "填空题", "ans": "$\\dfrac 35$", @@ -371513,7 +374896,8 @@ "content": "如图, 在棱长为$1$的正方体$ABCD-A_1B_1C_1D_1$中, 点$E$是棱$AB$上的动点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(A)!0.7!(B)$) node [below] {$E$} coordinate (E);\n\\draw [dashed] (D1)--(E) (A1)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $DA_1 \\perp ED_1$;\\\\\n(2) 若直线$DA_1$与平面$CED_1$所成的角是$45^{\\circ}$, 请你确定点$E$的位置, 并证明你的结论.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-10_空间向量与应用" ], "genre": "解答题", "ans": "(1) 证明略; (2) $E$是棱$AB$的中点", @@ -371569,7 +374953,8 @@ "content": "如图所示, 正四棱锥$V-ABCD$的表面积为$12$, $AB=2$, 将正四棱锥$V-ABCD$绕棱$AB$旋转, 若$AB \\subset$平面$\\alpha$, $M$、$N$分别是$AB$、$CD$的中点, 点$V$在平面$\\alpha$上的射影为点$O$, 则$ON$的最大值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,1) node [below] {$B$} coordinate (B);\n\\draw (0,0,-1) node [right] {$A$} coordinate (A);\n\\draw (-1,{sqrt(3)},1) node [left] {$C$} coordinate (C);\n\\draw (-1,{sqrt(3)},-1) node [above] {$D$} coordinate (D);\n\\draw ($(A)!0.5!(C)$) ++ ({3/2},{sqrt(3)/2},0) node [above] {$V$} coordinate (V);\n\\draw (1,0,0) node [below] {$O$} coordinate (O);\n\\draw (0,0,0) node [below] {$M$} coordinate (M);\n\\draw ($(C)!0.5!(D)$) node [above left] {$N$} coordinate (N);\n\\draw (V)--(D)(V)--(C)(V)--(B)(V)--(A)(D)--(C)--(B)--(A)(V)--(O)--(M);\n\\draw [dashed] (M)--(N)(A)--(D);\n\\path [name path = BC] (B)--(C);\n\\path [name path = AV] (A)--(V);\n\\path [name path = line] (-3,0,-2) -- (2,0,-2);\n\\path [name intersections = {of = line and AV, by = S}];\n\\path [name intersections = {of = line and BC, by = T}];\n\\draw (-3,0,-2) -- (-3,0,2) -- (2,0,2) -- (2,0,-2);\n\\draw (-3,0,-2) -- (T) (S) -- (2,0,-2);\n\\draw (-2.5,0,1.5) node {$\\alpha$};\n\\draw [dashed] (S)--(T);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-09_立体几何综合" ], "genre": "填空题", "ans": "$1+\\sqrt{3}$", @@ -371604,7 +374989,8 @@ "content": "已知棱长为$1$的正方体$ABCD-A_1B_1C_1D_1$中, $E$为侧面$BB_1C_1C$的中心, $F$在棱$AD$上运动, 正方体表面上有一点$P$满足$\\overrightarrow{D_1P}=\\lambda \\overrightarrow{D_1F}+\\mu \\overrightarrow{D_1E}$($\\lambda \\geq 0$, $\\mu \\geq 0$), 则所有满足条件的$P$点构成图形的面积为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(B)!0.5!(C1)$) node [right] {$E$} coordinate (E);\n\\draw ($(A)!0.4!(D)$) node [right] {$F$} coordinate (F);\n\\draw [dashed] (F)--(D1);\n\\draw (B)--(C1);\n\\filldraw (E) circle (0.03);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-10_空间向量与应用" ], "genre": "填空题", "ans": "$\\dfrac{11}8$", @@ -371639,7 +375025,8 @@ "content": "三棱柱$ABC-A_1B_1C_1$的棱长均为$2$, $\\angle AA_1C_1=\\angle AA_1B_1=60^{\\circ}$, 则该棱柱的体积等于\\bracket{20}.\n\\fourch{$\\sqrt{2}$}{$2 \\sqrt{2}$}{$3 \\sqrt{2}$}{$4 \\sqrt{2}$}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-09_立体几何综合" ], "genre": "选择题", "ans": "B", @@ -371696,7 +375083,8 @@ "content": "若三棱锥$P-ABC$的四个顶点在球$O$的球面上, $PA=PB=PC$, $\\triangle ABC$是边长为$2$的正三角形, $E, F$分别是$PA, AB$的中点, $\\angle CEF=90^{\\circ}$, 求球$O$的体积.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 2]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$B$} coordinate (B);\n\\draw (1,0,{-sqrt(3)}) node [above] {$C$} coordinate (C);\n\\draw (1,0,{-sqrt(3)/3}) ++ (0,{sqrt(2/3)},0) node [above] {$P$} coordinate (P);\n\\draw ($(P)!0.5!(A)$) node [above left] {$E$} coordinate (E);\n\\draw ($(A)!0.5!(B)$) node [below] {$F$} coordinate (F);\n\\draw (E)--(F);\n\\draw (A)--(B)(P)--(A)(P)--(B)(P)--(C)--(B);\n\\draw [dashed] (A)--(C)(C)--(E)(C)--(F);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-09_立体几何综合" ], "genre": "解答题", "ans": "$\\sqrt{6}\\pi$", @@ -371753,7 +375141,8 @@ "content": "已知$a, b$为空间中两条互相垂直的直线, 等腰直角三角形$ABC$的直角边$AC$所在直线与$a, b$都垂直, 斜边$AB$以直线$AC$为旋转轴旋转, 有下列结论:\\\\\n\\textcircled{1} 当直线$AB$与$a$成$60^{\\circ}$角时, $AB$与$b$成$30^{\\circ}$角;\\\\\n\\textcircled{2} 当直线$AB$与$a$成$60^{\\circ}$角时, $AB$与$b$成$60^{\\circ}$角;\\\\\n\\textcircled{3} 直线$AB$与$a$所成角的最小值为$45^{\\circ}$;\\\\\n\\textcircled{4} 直线$AB$与$a$所成角的最大值为$60^{\\circ}$.\n其中正确的是\\blank{50}(填写所有正确结论的编号).", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-09_立体几何综合" ], "genre": "填空题", "ans": "\\textcircled{2}\\textcircled{3}", @@ -371788,7 +375177,8 @@ "content": "如图, 正四棱锥$P-ABCD$中, $B_1$为$PB$的中点, $D_1$为$PD$的中点, 则棱锥$A-B_1CD_1$与$P-ABCD$的体积之比为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (-1,0,1) node [left] {$B$} coordinate (B);\n\\draw (1,0,1) node [right] {$C$} coordinate (C);\n\\draw (1,0,-1) node [right] {$D$} coordinate (D);\n\\draw (-1,0,-1) node [below] {$A$} coordinate (A);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw ($(B)!0.5!(P)$) node [above left] {$B_1$} coordinate (B_1);\n\\draw ($(P)!0.5!(D)$) node [above right] {$D_1$} coordinate (D_1);\n\\draw (P)--(B)(P)--(C)(P)--(D)(B)--(C)--(D);\n\\draw [dashed] (P)--(A)(B)--(A)--(D);\n\\draw (C)--(B_1)(C)--(D_1);\n\\draw [dashed] (C)--(A)--(B_1)(D_1)--(A)(D_1)--(B_1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-09_立体几何综合" ], "genre": "填空题", "ans": "$1:4$", @@ -372109,7 +375499,8 @@ "content": "在$-3,-2,-1,0,1,2,3,4$这$8$个数中, 任取$3$个不重复的数作为二次函数$f(x)=a x^2+b x+c$的系数$a, b, c$, 问:\\\\\n(1) 能组成多少个不同的二次函数;\\\\\n(2) 能组成多少条对称轴为$y$轴的抛物线?\\\\\n(3) 能组成多少经过原点且顶点在第一或第三象限的抛物线?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-第二轮复习讲义-17_计数原理与二项式定理" ], "genre": "解答题", "ans": "(1) $294$; (2) $42$; (3) $24$", @@ -372144,7 +375535,8 @@ "content": "把由$1,2,3,4,5$这五个数字组成的无重复数字的五位数, 把它们按从小到大的顺序排成一列, 构成一个数列.\\\\\n(1) $43251$是这个数列的第几项?\\\\\n(2) 这个数列的第$96$项是多少?\\\\\n(3) 求这个数列的各项和.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-第二轮复习讲义-17_计数原理与二项式定理" ], "genre": "解答题", "ans": "(1) $88$; (2) $45321$; (3) $3999960$", @@ -372223,7 +375615,8 @@ "content": "某餐厅供应客饭, 每位顾客可在餐厅提供的菜肴中任选$2$荤$2$素共$4$种不同品种, 现餐厅准备了$5$种不同的荤菜, 若保证每位顾客有$200$种以上的不同选择, 则餐厅至少还需准备\\blank{50}种不同的素菜品种.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-第二轮复习讲义-17_计数原理与二项式定理" ], "genre": "填空题", "ans": "$7$", @@ -372256,7 +375649,8 @@ "content": "$A, B, C, D, E$五个人并排站成一排, 若$B$必须站在$A$的右边(可以相邻, 也可以不相邻), 那么不同的排法共有\\bracket{20}种.\n\\fourch{$24$}{$60$}{$90$}{$120$}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-第二轮复习讲义-17_计数原理与二项式定理" ], "genre": "选择题", "ans": "B", @@ -372289,7 +375683,8 @@ "content": "在$5$双不同的手套中, 任取$4$只, 四只手套中至少有两只配成一双的可能取法种数是\\bracket{20}.\n\\fourch{$20$}{$30$}{$130$}{$140$}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-第二轮复习讲义-17_计数原理与二项式定理" ], "genre": "选择题", "ans": "C", @@ -372368,7 +375763,8 @@ "content": "已知$a_n$是函数$f_n(x)=(1+2 x)(1+2^2 x)(1+2^3 x) \\cdots(1+2^n x)$($n \\in \\mathbf{N}$, $n\\ge 1$)的展开式中的$x^2$的系数.\\\\\n(1) 计算$a_1, a_2, a_3$;\\\\\n(2) 求证: $a_{n+1}=a_n+2^{n+1}(2+2^2+\\cdots+2^n)$;\\\\\n(3) 是否存在常数$a$、$b$, 使得对不小于$2$的自然数$n$, 下列关系式$a_n=\\dfrac{8}{3}(2^{n-1}-1)(a \\cdot 2^n+b)$恒成立? 证明你的结论.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-第二轮复习讲义-17_计数原理与二项式定理" ], "genre": "解答题", "ans": "(1) $a_1=0$, $a_2=8$, $a_3=56$; (2) 证明略; (3) 存在, $a=1$, $b=-1$", @@ -372401,7 +375797,8 @@ "content": "规定$\\mathrm{C}_x^m=\\dfrac{x(x-1) \\cdots(x-m+1)}{m !}$, 其中$x \\in \\mathbf{R}$, $m$是正整数, 且$\\mathrm{C}_x^0=1$, 这是组合数$\\mathrm{C}_n^m$($n$、$m$是正整数, 且$m \\leq n)$的一种推广.\\\\\n(1) 求$\\mathrm{C}_{-15}^5$的值;\\\\\n(2) 组合数的性质$\\mathrm{C}_n^m=\\mathrm{C}_n^{n-m}$; $\\mathrm{C}_n^m+\\mathrm{C}_n^{m-1}=\\mathrm{C}_{n+1}^m$, 是否都能推广到$\\mathrm{C}_x^m$($x \\in \\mathbf{R}$, $m$是正整数)的情形? 若能, 写出推广的形式, 并给出证明; 若不能, 说明理由;\\\\\n(3) 已知组合数$\\mathrm{C}_n^m$是正整数, 证明: 当$x \\in \\mathbf{Z}$, $m$是正整数时, $\\mathrm{C}_x^m \\in \\mathbf{Z}$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-第二轮复习讲义-17_计数原理与二项式定理" ], "genre": "解答题", "ans": "(1) $-11628$; (2) 前一个性质不能推广, 后一个性质能推广, 证明略; (3) 证明略", @@ -372436,7 +375833,8 @@ "content": "某同学从物理、化学、生物、政治、历史、地理六科中随机选择三科参加考试, 则物理和化学不同时被选中的概率为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-第二轮复习讲义-18_概率与统计" ], "genre": "填空题", "ans": "$\\dfrac 45$", @@ -372490,7 +375888,8 @@ "content": "某三位数密码, 每位数字可在$0-9$这$10$个数字中任选一个, 则该三位数密码中, 恰有两位数字相同的概率是\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-第二轮复习讲义-18_概率与统计" ], "genre": "填空题", "ans": "$\\dfrac{27}{100}$", @@ -372546,7 +375945,8 @@ "objs": [], "tags": [ "第九单元", - "第八单元" + "第八单元", + "2023届高三-第二轮复习讲义-19_统计" ], "genre": "解答题", "ans": "(1) $400$; (2) $4$; (3) $0.7$", @@ -372579,7 +375979,8 @@ "content": "甲、乙两队进行篮球决赛, 采取七场四胜制(当一队赢得四场胜利时, 该队获胜, 决赛结束). 根据前期比赛成绩, 甲队的主客场安排依次为``主主客客主客主''. 设甲队主场取胜的概率为$0.6$, 客场取胜的概率为$0.5$, 且各场比赛结果相互独立, 则甲队以$4: 1$获胜的概率是\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-第二轮复习讲义-18_概率与统计" ], "genre": "填空题", "ans": "$\\dfrac{9}{50}$", @@ -372611,7 +376012,9 @@ "content": "气象意义上从春季进入夏季的标志为: ``连续$5$天的日平均温度均不低于$22$($^\\circ\\text{C}$). 现有甲、乙、丙三地连续$5$天的日平均温度的记录数据(记录数据都是正整数): \\textcircled{1} 甲地: $5$个数据的中位数为$24$, 众数为$22$; \\textcircled{2} 乙地: $5$个数据的中位数为$27$, 总体均值为$24$; \\textcircled{3} 丙地: $5$个数据中有一个数据是$32$, 总体均值为$26$, 总体方差为$10.8$; 则肯定进入夏季的地区有\\bracket{20}.\n\\fourch{$0$个}{$1$个}{$2$个}{$3$个}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-下学期测验卷-高三下学期测验02", + "2023届高三-第二轮复习讲义-19_统计" ], "genre": "选择题", "ans": "C", @@ -372655,7 +376058,8 @@ "content": "一个容量为$n$的样本, 分成若干组, 已知某数的频数和频率分别为$40$、$0.125$, 则$n$的值为\\bracket{20}.\n\\fourch{$640$}{$320$}{$240$}{$160$}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-第二轮复习讲义-19_统计" ], "genre": "选择题", "ans": "B", @@ -372710,7 +376114,8 @@ "content": "从$1$、$2$、$3$、$4$这四个数中一次随机地抽取两个数, 其中一个数是另一个数的两倍的概率是\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-第二轮复习讲义-18_概率与统计" ], "genre": "填空题", "ans": "$\\dfrac 13$", @@ -372743,7 +376148,10 @@ "content": "袋中有$3$个五分硬币, $3$个二分硬币和$4$个一分硬帀, 从中任取三个, 则总金额超过$8$分的概率是\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-四月错题重做-05_易错题-概率与统计", + "2023届高三-四月错题重做-05_概率与统计", + "2023届高三-第二轮复习讲义-18_概率与统计" ], "genre": "填空题", "ans": "$\\dfrac{31}{120}$", @@ -372782,7 +376190,8 @@ "content": "一个总体中有$100$个个体, 随机编号为$0,1,2, \\cdots, 99$, 依编号顺序平均分成$10$个小组, 组号依次为$1,2,3, \\cdots, 10$. 现用某种抽样方法抽取一个容量为$10$的样本, 规定如果在第$1$组随机抽取的号码为$m$, 那么在第$k$小组中抽取的号码个位数字与$m+k$的个位数字相同($k\\ge 2$). 若$m=6$, 则在第$7$组中抽取的号码是\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第二轮复习讲义-19_统计" ], "genre": "填空题", "ans": "$63$", @@ -372837,7 +376246,8 @@ "content": "某人有$5$把钥匙, 其中有$k$把是房门钥匙, 但忘记了开房门的是哪一把. 于是, 他逐把不重复地试开, 问:\\\\\n(1) 若$k=1$, 则恰好第三次打开房门锁的概率是多少?\\\\\n(2) 若$k=1$, 则三次内打开的概率是多少?\\\\\n(3) 若$k=2$, 则三次内打开的概率是多少?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-第二轮复习讲义-18_概率与统计" ], "genre": "解答题", "ans": "(1) $\\dfrac 15$; (2) $\\dfrac 35$; (3) $\\dfrac 9{10}$", @@ -372892,7 +376302,8 @@ "content": "已知$03$或$x<1\\}$, 则$A \\cup \\overline {B}=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-01_集合与逻辑" ], "genre": "填空题", "ans": "$(0,3]$", @@ -373157,7 +376569,8 @@ "content": "用反证法证明命题``设$x_1, x_2, \\cdots, x_n \\in \\mathbf{R}$($n$是正整数). 求证: 若$x_1+x_2+\\cdots+x_n>n$, 则$x_1, x_2, \\cdots, x_n$中至少有一个大于$1$.''时, 第一步需假设\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-01_集合与逻辑" ], "genre": "填空题", "ans": "$x_1,x_2,\\cdots,x_n$均小于等于$1$", @@ -373209,7 +376622,8 @@ "content": "已知$m, a \\in \\mathbf{R}$, $f(x)=x^2+(a-1) x+1$, $g(x)=m x^2+2 a x+\\dfrac{m}{4}$, 若``对于一切实数$x, f(x)>0$''是``对一切实数$x, g(x)>0$''的充分条件, 求实数$m$的取值范围.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-01_集合与逻辑" ], "genre": "解答题", "ans": "$[6,+\\infty)$", @@ -373259,7 +376673,8 @@ "content": "集合$A=\\{x | a x^2+2 x+1=0, x \\in \\mathbf{R}\\}$的元素个数组成的集合为\\blank{50}.(用列举法表示)", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-01_集合与逻辑" ], "genre": "填空题", "ans": "$\\{0,1,2\\}$", @@ -373287,7 +376702,8 @@ "content": "设$a, b$是实数, \\textcircled{1} $a+b>1$; \\textcircled{2} $a+b>2$; \\textcircled{3} $a^2+b^2>2$; \\textcircled{4} $a b>1$, 其中能作为``$a, b$中至少有一个大于$1$''的充分条件的是\\blank{50}.(写出所有正确结论的序号)", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-01_集合与逻辑" ], "genre": "填空题", "ans": "\\textcircled{2}", @@ -373408,7 +376824,8 @@ "content": "若集合$M=\\{y | y=x^2,\\ x \\in \\mathbf{R}\\}$, $N=\\{y | y=-3 x^2+4, \\ x \\in \\mathbf{R}\\}$, 则$M \\cap N=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-01_集合与逻辑" ], "genre": "填空题", "ans": "$[0,4]$", @@ -373436,7 +376853,8 @@ "content": "判断下列各题中甲是乙的什么条件(充分非必要条件、必要非充分条件、充要条件、既非充分又非必要条件), 并说明理由.\\\\\n(1) 已知$\\triangle ABC$. 甲: $A>B$; 乙: $\\sin A>\\sin B$;\\\\\n(2) 已知$\\overrightarrow {a}$, $\\overrightarrow {b}$, $\\overrightarrow {c}$是非零的共面向量. 甲: $\\overrightarrow {a} \\cdot \\overrightarrow {b}=\\overrightarrow {b} \\cdot \\overrightarrow {c}$; 乙: $\\overrightarrow {a}=\\overrightarrow {c}$;\\\\\n(3) 已知函数$y=f(x)$的定义域为$\\mathbf{R}$, 且存在导函数$f'(x)$. 甲: $f'(x)>0$恒成立; 乙: $y=f(x)$是$\\mathbf{R}$上的严格增函数.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-01_集合与逻辑" ], "genre": "解答题", "ans": "(1) 充要条件, 理由略; (2) 必要非充分条件, 理由略; (3) 充分非必要条件, 理由略", @@ -373491,7 +376909,8 @@ "content": "给定实数$a$, $a \\neq 0$且$a \\neq 1$, 设函数$y=\\dfrac{x-1}{a x-1}$(其中$x \\in \\mathbf{R}$且$x \\neq \\dfrac{1}{a}$). 求证: 经过这个函数图像上任意两个不同点的直线不平行于$x$轴.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-01_集合与逻辑" ], "genre": "解答题", "ans": "证明略", @@ -373546,7 +376965,8 @@ "content": "设集合$P_1=\\{x | x^2+a x+1>0\\}$, $P_2=\\{x | x^2+a x+2>0\\}$, $Q_1=\\{x | x^2+x+b>0\\}$, $Q_2=\\{x | x^2+2 x+b>0\\}$, 其中$a, b \\in \\mathbf{R}$. 下列说法正确的是\\bracket{20}.\n\\onech{对任意$a$, $P_1$是$P_2$的子集; 对任意$b$, $Q_1$不是$Q_2$的子集}{对任意$a$, $P_1$是$P_2$的子集; 存在$b$, 使得$Q_1$是$Q_2$的子集}{存在$a$, 使得$P_1$不是$P_2$的子集; 对任意$b$, $Q_1$不是$Q_2$的子集}{存在$a$, 使得$P_1$不是$P_2$的子集; 存在$b$, 使得$Q_1$是$Q_2$的子集}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-01_集合与逻辑" ], "genre": "选择题", "ans": "B", @@ -373801,7 +377221,8 @@ "content": "不等式$\\dfrac{x+5}{x^2-2 x+1} \\geq 1$的解集为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-02_等式与不等式" ], "genre": "填空题", "ans": "$[-1,1)\\cup (1,4]$", @@ -373834,7 +377255,8 @@ "content": "不等式$|x-2|+2|x+1|<5$的解集为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-02_等式与不等式" ], "genre": "填空题", "ans": "$(-\\dfrac 53,1)$", @@ -373881,7 +377303,8 @@ "content": "若关于$x$的不等式$0 \\leq x^2-m x+2 \\leq 1$有且仅有一个实数解, 则实数$m=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-02_等式与不等式" ], "genre": "填空题", "ans": "$2$或$-2$", @@ -374258,7 +377681,8 @@ "content": "已知$f(x)=2 \\lg x-1, g(x)=2 \\lg x-3$.\\\\\n(1) 若$|f(x)+g(x)|=|f(x)|+|g(x)|$, 求满足条件的$x$的取值范围;\\\\\n(2) 若$|f(x)|+|g(x)|$的最小值为$M, 00$, $a \\neq 1$)在区间$[1,2]$上的最大值与最小值之和等于$12$, 则实数$a$的值为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-03_幂指对函数" ], "genre": "填空题", "ans": "$3$", @@ -374677,7 +378107,8 @@ "content": "设幂函数$y=x^a$, $a \\in\\{-2,-1, \\dfrac{1}{2}, 1,2,3\\}$, 则``函数$y=x^a$的图像经过点$(-1,-1)$''是``函数$y=x^a$为奇函数''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-03_幂指对函数" ], "genre": "选择题", "ans": "C", @@ -374712,7 +378143,8 @@ "content": "证明: 函数$y=\\log _2 \\dfrac{x-1}{x+1}$在区间$(1,+\\infty)$上是严格增函数.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-03_幂指对函数" ], "genre": "解答题", "ans": "证明略", @@ -374747,7 +378179,8 @@ "content": "设非零实数$x, y, z$满足$3^x=4^y=6^z$, 求$\\dfrac{2 z}{x}+\\dfrac{z}{y}$的值.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-03_幂指对函数" ], "genre": "解答题", "ans": "$2$", @@ -374782,7 +378215,8 @@ "content": "设正数$p$、$q$满足$\\log _{16} p=\\log _{20} q=\\log _{25}(p+q)$, 求$\\dfrac{p}{q}$的值.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-03_幂指对函数" ], "genre": "解答题", "ans": "$\\dfrac{\\sqrt{5}-1}2$", @@ -374839,7 +378273,8 @@ "content": "若正实数$x$、$y$满足$\\lg x=m$, $y=10^{m-1}$, 则$\\dfrac{x}{y}=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-03_幂指对函数" ], "genre": "填空题", "ans": "$10$", @@ -374874,7 +378309,8 @@ "content": "若幂函数$y=x^a$的图像经过点$(\\sqrt[4]{3}, 3)$, 则实数$a=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-03_幂指对函数" ], "genre": "填空题", "ans": "$4$", @@ -374909,7 +378345,8 @@ "content": "研究发现, 某昆虫释放信息素$t$秒后, 在距释放处$x$米的地方测得的信息素浓度$y$满足$\\ln y=-\\dfrac{1}{2} \\ln t-\\dfrac{k}{t} x^2+a$, 其中$k, a$为非零常数. 已知该昆虫释放信息素$1$秒后, 在距释放处$2$米的地方测得信息素浓度为$m$, 则该昆虫释放信息素$4$秒后, 距释放处的\\blank{50}米的位置, 信息素浓度为$\\dfrac{m}{2}$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-03_幂指对函数" ], "genre": "填空题", "ans": "$4$", @@ -374944,7 +378381,8 @@ "content": "若对任意负数$x$, 代数式$|x|+2 \\cdot \\sqrt[2022]{x^{2022}}+a \\cdot \\sqrt[2023]{x^{2023}}$恒为定值, 则实数$a$的取值是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-03_幂指对函数" ], "genre": "填空题", "ans": "$3$", @@ -374979,7 +378417,8 @@ "content": "若$g(x)=\\begin{cases}\\log _2(x+1), & x \\geq 0, \\\\ 2^x+1, & x<0,\\end{cases}$则满足方程$g(x)=2$的$x$的值为\\blank{50}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-03_幂指对函数" ], "genre": "填空题", "ans": "$3$", @@ -375014,7 +378453,8 @@ "content": "设$y=x^{\\frac{1}{2}}-x^3$, 则满足不等式$y<0$的$x$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-03_幂指对函数" ], "genre": "填空题", "ans": "$(1,+\\infty)$", @@ -375049,7 +378489,8 @@ "content": "服用某种感冒药, 每次服用的药物含量为$a$, 随着时间$t$的变化, 体内的药物含量为$y=0.57^t\\cdot a$(其中$t$以小时为单位), 则服药$2$小时后体内药物的含量为服药 $4$小时后体内药物含量的\\blank{50}倍.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-03_幂指对函数" ], "genre": "填空题", "ans": "$\\dfrac{1}{0.57^2}$", @@ -375084,7 +378525,8 @@ "content": "下列命题中正确的是\\bracket{20}.\n\\twoch{若$a>0$, $x_2>x_1>0$, 则$(\\dfrac{x_2}{x_1})^a<1$}{若$a>0$, $x_1>x_2>0$, 则$(\\dfrac{x_2}{x_1})^a>1$}{若$a<0$, $x_2>x_1>0$, 则$(\\dfrac{x_2}{x_1})^a>1$}{若$a<0$, $x_1>x_2>0$, 则$(\\dfrac{x_2}{x_1})^a>1$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-03_幂指对函数" ], "genre": "选择题", "ans": "D", @@ -375119,7 +378561,8 @@ "content": "已知常数$a>0$且$a \\neq 1$, $b \\in \\mathbf{R}$, 函数$y=a^x+b$的定义域和值域都是$[-1,0]$, 求$a+b$的值.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-03_幂指对函数" ], "genre": "解答题", "ans": "$-\\dfrac 32$", @@ -375154,7 +378597,8 @@ "content": "定义: 若一个对数可以用$\\lg 2$、$\\lg 3$和整数的线性组合表示(系数也为整数), 则称$\\lg 2$和$\\lg 3$为该对数的``基本对数''. 如: $\\lg 12=2 \\lg 2+\\lg 3$, 故$\\lg 2$和$\\lg 3$为$\\lg 12$的``基本对数''. 下列对数中, 所有以$\\lg 2$和$\\lg 3$为``基本对数''的对数的序号为\\blank{50}.\n\\textcircled{1} $\\lg 4$; \\textcircled{2} $\\lg 56$; \\textcircled{3} $\\lg 15$; \\textcircled{4} $\\lg 225$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-03_幂指对函数" ], "genre": "填空题", "ans": "\\textcircled{1}\\textcircled{3}\\textcircled{4}", @@ -375321,7 +378765,8 @@ "content": "(1) 是否存在实数$a$, 使函数$y=\\lg |2 x-a|$为偶函数?\\\\\n(2) 是否存在实数$a$、$b$, 使函数$y=\\lg |a+\\dfrac{4}{2-x}|+b$为奇函数?", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-04_函数的概念与性质" ], "genre": "解答题", "ans": "(1) 存在, $a=0$; (2) 存在, $a=-1$, $b=0$", @@ -375356,7 +378801,10 @@ "content": "设$a$、$b$为常数, 函数$y=x^2-2 b x+1, x \\in[a, a+2]$.\\\\\n(1) 当$b=1$时, 若该函数在定义域上为单调函数, 求实数$a$的取值范围;\\\\\n(2) 当$a=0$时, 将该函数的最小值记为$f(b)$, 求$f(b)$的表达式.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-四月错题重做-02_函数二", + "2023届高三-四月错题重做-02_易错题-函数2", + "2023届高三-第二轮复习讲义-04_函数的概念与性质" ], "genre": "解答题", "ans": "(1) $(-\\infty,-1]\\cup [1,+\\infty)$; (2) $f(b)=\\begin{cases}1, & b\\le 0,\\\\1-b^2, & 0\\le b\\le 2,\\\\ 5-4b, & b\\ge 2.\\end{cases}$", @@ -375403,7 +378851,8 @@ "content": "请利用函数$y=x^3+x$解决下列问题:\\\\\n(1) 方程$x^3+x=2023$是否有整数解? 请说明理由;\\\\\n(2) 解不等式:$(x+2023)^3+x^3+2 x+2023>0$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-04_函数的概念与性质" ], "genre": "解答题", "ans": "(1) 无整数解, 理由略; (2) $(-\\dfrac{2023}2,+\\infty)$", @@ -375438,7 +378887,8 @@ "content": "若函数$y=a-\\dfrac{2}{2^x+1}$为奇函数, 则实数$a=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-03_幂指对函数" ], "genre": "填空题", "ans": "$1$", @@ -375473,7 +378923,8 @@ "content": "满足不等式$\\ln x+x<1$的实数$x$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-03_幂指对函数" ], "genre": "填空题", "ans": "$(0,1)$", @@ -375596,7 +379047,8 @@ "content": "已知集合$A=\\{x | \\dfrac{2}{x-2} \\geq 1, \\ x \\in \\mathbf{R}\\}$, 设函数$y=\\log _{\\frac{1}{2}} x+a,\\ x \\in A$的值域为$B$, 若$B \\subseteq A$, 则实数$a$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-03_幂指对函数" ], "genre": "填空题", "ans": "$(4,5]$", @@ -375697,7 +379149,8 @@ "content": "求满足方程$12^x+5^x=13^x$的实数$x$的值.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-03_幂指对函数" ], "genre": "解答题", "ans": "$2$", @@ -375732,7 +379185,8 @@ "content": "若扇形$AOB$的周长是$32$, 则该扇形面积的最大值是\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-06_三角与解三角形" ], "genre": "填空题", "ans": "$64$", @@ -375767,7 +379221,8 @@ "content": "在直角坐标系$xOy$中, 角$\\alpha$的顶点与坐标原点重合, 始边与$x$轴的正半轴重合. 若角$\\alpha$的终边经过点$(-3,4)$, 则$\\sin (\\alpha+\\pi)=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-06_三角与解三角形" ], "genre": "填空题", "ans": "$-\\dfrac 45$", @@ -375802,7 +379257,8 @@ "content": "若$\\sin (\\alpha+\\beta)=\\dfrac{1}{3}$, $\\sin (\\alpha-\\beta)=\\dfrac{1}{2}$, 则$\\sin \\alpha \\cos \\beta=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-06_三角与解三角形" ], "genre": "填空题", "ans": "$\\dfrac{5}{12}$", @@ -375837,7 +379293,8 @@ "content": "若$\\cos \\alpha=-\\dfrac{\\sqrt{5}}{5}$, $\\alpha \\in(0, \\pi)$, 则$\\tan 2 \\alpha=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-06_三角与解三角形" ], "genre": "填空题", "ans": "$\\dfrac 43$", @@ -375872,7 +379329,8 @@ "content": "若角$x$满足$3 \\sin 2 x=2 \\sin x$, $x \\in(0, \\pi)$, 则$x=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-06_三角与解三角形" ], "genre": "填空题", "ans": "$\\arccos \\dfrac 13$", @@ -376083,7 +379541,8 @@ "content": "若角$\\alpha$的终边经过点$P(3,4)$, 将角$\\alpha$的终边绕原点$O$逆时针旋转$\\dfrac{\\pi}{2}$得到角$\\beta$的终边, 则$\\tan \\beta=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-06_三角与解三角形" ], "genre": "填空题", "ans": "$-\\dfrac 34$", @@ -376184,7 +379643,8 @@ "content": "已知$\\cos (2 \\alpha-\\beta)=-\\dfrac{2\\sqrt{7}}{7}$, $\\sin (\\alpha-2 \\beta)=\\dfrac{\\sqrt{21}}{14}$, $0<\\beta<\\dfrac{\\pi}{4}<\\alpha<\\dfrac{\\pi}{2}$, 求$\\alpha+\\beta$的值.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-06_三角与解三角形" ], "genre": "解答题", "ans": "$\\dfrac{2\\pi}3$", @@ -376308,7 +379768,8 @@ "content": "在$\\triangle ABC$中, 若$AB=2$, $AC=2$, $A=60^{\\circ}$, 则$\\triangle ABC$外接圆的半径为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-06_三角与解三角形" ], "genre": "填空题", "ans": "$\\dfrac{2\\sqrt{3}}3$", @@ -376343,7 +379804,8 @@ "content": "已知角$A$、$B$、$C$是$\\triangle ABC$的三个内角, 若$\\sin A: \\sin B: \\sin C=4: 5: 7$, 则该三角形的最大内角等于\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-06_三角与解三角形" ], "genre": "填空题", "ans": "$\\pi-\\arccos\\dfrac 15$", @@ -376444,7 +379906,8 @@ "content": "在$\\triangle ABC$中, 角$A$、$B$及$C$所对边的边长分别为$a$、$b$及$c$.\\\\\n(1) 若$a=3$, $b=2c$, $2 \\sin B-\\sin C=1$, 求$\\triangle ABC$的周长;\\\\\n(2) 若$B=\\dfrac{2 \\pi}{3}$, $b=2 \\sqrt{3}$, 求$\\triangle ABC$面积的最大值.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-06_三角与解三角形" ], "genre": "解答题", "ans": "(1) $3+4\\sqrt{2}\\pm \\sqrt{5}$; (2) $\\sqrt{3}$", @@ -376766,7 +380229,8 @@ "content": "若函数$y=\\sin (\\omega x)$(其中常数$\\omega \\neq 0$) 的最小正周期为$\\dfrac{\\pi}{3}$, 则$\\omega=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-07_三角函数" ], "genre": "填空题", "ans": "$6$或$-6$", @@ -376800,7 +380264,8 @@ "content": "函数$y=2 \\sin (x-\\dfrac{\\pi}{3})$, $x \\in[0, \\dfrac{\\pi}{2}]$的值域是\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-07_三角函数" ], "genre": "填空题", "ans": "$[-\\sqrt{3},1]$", @@ -376878,7 +380343,8 @@ "content": "已知$f(x)=2 \\sin x \\cos x+\\sqrt{3} \\cos 2 x$.\\\\\n(1) 求函数$y=f(x)$的单调减区间;\\\\\n(2) 求函数$y=f(x)$在区间$[0, \\dfrac{\\pi}{4}]$上的最大值和最小值.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-07_三角函数" ], "genre": "解答题", "ans": "(1) $[\\dfrac\\pi{12}+k\\pi,\\dfrac{7\\pi}{12}+k\\pi]$, $k\\in \\mathbf{Z}$; (2) 最大值为$2$, 最小值为$1$", @@ -377035,7 +380501,8 @@ "content": "若函数$y=3 \\cos (2 x+\\varphi)$的图像关于点$(\\dfrac{\\pi}{3}, 0)$对称, 则$|\\varphi|$的最小值为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-07_三角函数" ], "genre": "填空题", "ans": "$\\dfrac\\pi 6$", @@ -377148,7 +380615,8 @@ "content": "函数$y=\\tan 2 x$的最小正周期为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-07_三角函数" ], "genre": "填空题", "ans": "$\\dfrac \\pi 2$", @@ -377195,7 +380663,8 @@ "content": "函数$y=\\tan 2 x$在区间$(-\\dfrac{\\pi}{4}, \\dfrac{\\pi}{4})$上的零点为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-07_三角函数" ], "genre": "填空题", "ans": "$0$", @@ -377252,7 +380721,8 @@ "objs": [], "tags": [ "第二单元", - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-07_三角函数" ], "genre": "选择题", "ans": "C", @@ -377299,7 +380769,8 @@ "content": "已知$f(x)=\\sin (\\omega x-\\dfrac{\\pi}{6})+k$($\\omega>0$), 若$f(x) \\leq f(\\dfrac{\\pi}{3})$对任意的实数$x$成立, 则$\\omega$的最小值为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-07_三角函数" ], "genre": "填空题", "ans": "$2$", @@ -377355,7 +380826,8 @@ "content": "已知$f(x)=\\sin x$, 若存在$x_1, x_2, \\cdots, x_m$满足$0 \\leq x_10$, $0<\\varphi<\\pi$), 函数$y=f(x)$的最小正周期为$\\pi$, 且直线$x=-\\dfrac{\\pi}{2}$是其图像的一条对称轴.\\\\\n(1) 求函数$y=f(x)$的表达式;\\\\\n(2) 将函数$y=f(x)$的图像向右平移$\\dfrac{\\pi}{4}$个单位, 再将所得的图像上每一点的纵坐标不变, 横坐标伸长为原来的$2$倍后得到函数$y=g(x)$的图像, 设常数$\\lambda \\in \\mathbf{R}$, $n$为正整数, 且函数$y=f(x)+\\lambda g(x)$在区间$(0, n \\pi)$内恰有$2023$个零点, 求常数$\\lambda$与$n$的值.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-07_三角函数" ], "genre": "解答题", "ans": "(1) $f(x)=\\cos 2x$; (2) $\\lambda=1$, $n=1349$", @@ -377467,7 +380940,8 @@ "content": "若函数$y=\\sin (x+\\theta)$(其中常数$\\theta \\in[0, \\pi)$) 是$\\mathbf{R}$上的偶函数, 则满足$\\sin (x+\\theta)=\\dfrac{1}{2}$的角$x$的集合为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-07_三角函数" ], "genre": "填空题", "ans": "$\\{x|x=\\pm \\dfrac\\pi 3+2k\\pi, \\ k\\in \\mathbf{Z}\\}$", @@ -377514,7 +380988,8 @@ "content": "已知函数$y=A \\sin (\\omega x+\\varphi)$($A>0$, $\\omega>0$, $|\\varphi| \\leq \\dfrac{\\pi}{2}$)图像的一部分如图所示, 则$y=$\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -2:3, samples = 100] plot (\\x,{2*sin(2*\\x/pi*180-30)});\n\\draw ({-5*pi/12},0) node [below left] {$-\\dfrac{5\\pi}{12}$};\n\\draw ({pi/3},0) node [below] {$\\dfrac\\pi 3$};\n\\draw [dashed] ({pi/3},0) --++ (0,2) -- (0,2) node [left] {$2$};\n\\draw [dashed] ({-pi/6},-2) -- (0,-2) node [right] {$-2$};\n\\draw (0,-1) node [left] {$-1$};\n\\filldraw (0,-1) circle (0.03);\n\\filldraw ({-5*pi/12},0) circle (0.03);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$2 \\sin (2 x-\\dfrac{\\pi}{6})$}{$2 \\sin (2 x+\\dfrac{\\pi}{6})$}{$2 \\sin (2 x-\\dfrac{\\pi}{3})$}{$2 \\sin (2 x+\\dfrac{\\pi}{3})$}", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-07_三角函数" ], "genre": "选择题", "ans": "A", @@ -377606,7 +381081,8 @@ "content": "已知$f(x)=2 \\sin (x+\\dfrac{\\pi}{3})$.\\\\\n(1) 若对任意的$x \\in[-\\dfrac{\\pi}{6}, \\dfrac{\\pi}{3}]$, 不等式$|f(x)-m| \\leq 3$恒成立, 求实数$m$的取值范围;\\\\\n(2) 画出函数$y=f(x-\\dfrac{\\pi}{3})+f(x)$, $x \\in[0, \\dfrac{\\pi}{2}]$的大致图像, \n并写出满足$f(x-\\dfrac{\\pi}{3})+f(x)=\\sqrt{10}$的锐角$x$的集合.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-07_三角函数" ], "genre": "解答题", "ans": "(1) $[-1,4]$; (2) 图像如下:\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw [->] (-0.5,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0:{pi/2}] plot (\\x,{2*(sin(\\x/pi*180)+sin(\\x/pi*180+60))});\n\\draw [dashed] (0,{2*sqrt(3)}) node [left] {$2\\sqrt{3}$} -- ({pi/3},{2*sqrt(3)}) --({pi/3},0) node [below] {$\\dfrac\\pi 3$};\n\\draw [dashed] (0,3) node [left] {$3$} -- ({pi/2},3) --({pi/2},0) node [below] {$\\dfrac\\pi 2$};\n\\draw (0,{sqrt(3)}) node [left] {$\\sqrt{3}$};\n\\end{tikzpicture}\n\\end{center}\n满足条件的锐角的集合为$\\{\\arcsin\\dfrac{\\sqrt{30}}6-\\dfrac\\pi 6,\\dfrac{5\\pi}6-\\arcsin\\dfrac{\\sqrt{30}}6\\}$", @@ -377852,7 +381328,8 @@ "content": "设$\\omega>0$, 若函数$y=\\sin \\omega x$在区间$[0, \\pi]$上恰有两个零点, 则实数$\\omega$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-07_三角函数" ], "genre": "填空题", "ans": "$[1,2)$", @@ -377921,7 +381398,8 @@ "content": "如图, 某地有三家工厂分别位于矩形$ABCD$的两个顶点$A$、$B$及$CD$的中点$P$处. $AB=20 \\text{km}, BC=10 \\text{km}$. 为了处理这三家工厂的污水, 现要在该矩形区域内 (含边界) 且与$A$、$B$等距离的一点$O$处, 建造一个污水处理厂, 并铺设三条排污管道$OA$、$OB$、$OP$. 记排污管道的总长度为$y \\text{km}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$A$} coordinate (A);\n\\draw (3,0) node [right] {$B$} coordinate (B);\n\\draw (3,1.5) node [right] {$C$} coordinate (C);\n\\draw (0,1.5) node [left] {$D$} coordinate (D);\n\\draw (1.5,0.75) node [below] {$O$} coordinate (O);\n\\draw ($(C)!0.5!(D)$) node [above] {$P$} coordinate (P);\n\\draw (A) -- (B) -- (C) -- (D) -- cycle (P) -- (O) (A) -- (O) -- (B);\n\\end{tikzpicture}\n\\end{center}\n(1) 设$\\angle BAO=\\theta$, 将$y$表示成$\\theta$的函数并求其定义域;\\\\\n(2) 确定污水处理厂的位置, 使排污管道的总长度$y$最短, 并求出其值.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-07_三角函数" ], "genre": "解答题", "ans": "(1) $y=10+\\dfrac{20}{\\cos\\theta}-10\\tan\\theta$, $\\theta\\in [0,\\dfrac\\pi 4)$; (2) 应使$|OP|=10-\\dfrac{10\\sqrt{3}}3\\text{km}$, 此时总长度的最小值为$10\\sqrt{3}+10\\text{km}$.", @@ -377968,7 +381446,8 @@ "content": "已知函数$y=\\sin (\\omega x-\\dfrac{\\pi}{3})$($\\omega>0$), 记$y=f(x)$. 对任意$x_1, x_2 \\in \\mathbf{R}$, 当$|f(x_1)-f(x_2)|=2$时, $|x_1-x_2|$的最小值是$\\dfrac{\\pi}{3}$, 则函数$y=f(x)$, $x \\in[0, \\dfrac{\\pi}{2}]$的单调减区间是\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第二轮复习讲义-07_三角函数" ], "genre": "填空题", "ans": "$[\\dfrac{5\\pi}{18},\\dfrac{\\pi}2]$", @@ -378024,7 +381503,8 @@ "content": "函数$y=\\begin{cases}x^2-2, & x \\leq 0, \\\\ 2 x-6+\\ln x, & x>0\\end{cases}$的零点的个数为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-03_幂指对函数" ], "genre": "填空题", "ans": "$2$", @@ -378081,7 +381561,8 @@ "content": "不等式$\\log _{\\frac{1}{2}}(1-2 x)>\\log _{\\frac{1}{2}}(3 x)$的解集为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-03_幂指对函数" ], "genre": "填空题", "ans": "$(\\dfrac 15,\\dfrac 12)$", @@ -378977,7 +382458,8 @@ "content": "已知平面向量$\\overrightarrow {a}$、$\\overrightarrow {b}$满足$|\\overrightarrow {a}|=|\\overrightarrow {b}|=1$, 且$\\overrightarrow {a}$、$\\overrightarrow {b}$的夹角是$120^{\\circ}$, 问$t$为何实数值时, $|\\overrightarrow {a}-t \\overrightarrow {b}|$的值最小? 并求此时$\\overrightarrow {b}$与$\\overrightarrow {a}-t \\overrightarrow {b}$夹角的大小.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第二轮复习讲义-08_平面向量与复数" ], "genre": "解答题", "ans": "当$t=-\\dfrac 12$时$|\\overrightarrow{a}-t\\overrightarrow{b}|$的值最小, 此时夹角的大小为$\\dfrac \\pi 2$", @@ -379012,7 +382494,8 @@ "content": "如图, 在直角三角形$ABC$中, $|CA|=|CB|=2, M$、$N$是斜边$AB$上的两个动点(点$N$在线段$BM$上), 且$|MN|=\\sqrt{2}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$C$} coordinate (C);\n\\draw (2,0) node [right] {$A$} coordinate (A);\n\\draw (0,2) node [above] {$B$} coordinate (B);\n\\draw ($(A)!0.2!(B)$) node [above right] {$M$} coordinate (M);\n\\draw ($(A)!0.7!(B)$) node [above right] {$N$} coordinate (N);\n\\draw (A)--(B)--(C)--cycle;\n\\draw (C)--(M)(C)--(N);\n\\end{tikzpicture}\n\\end{center}\n(1) 求向量$\\overrightarrow{MN}$在$\\overrightarrow{CB}$方向上的投影与数量投影;\\\\\n(2) 求$\\overrightarrow{CM} \\cdot \\overrightarrow{CN}$的取值范围.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第二轮复习讲义-08_平面向量与复数" ], "genre": "解答题", "ans": "(1) 投影为$\\dfrac 12\\overrightarrow{CB}$, 数量投影为$1$; (2) $[\\dfrac 32,2]$", @@ -379201,7 +382684,8 @@ "content": "已知向量$\\overrightarrow {a}=(1,1)$, $\\overrightarrow {b}=(-2,1)$, $\\lambda \\in \\mathbf{R}$. 若$\\overrightarrow {a}+\\overrightarrow {b}$与$2 \\overrightarrow {a}+\\lambda \\overrightarrow {b}$的夹角为锐角, 则$\\lambda$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第二轮复习讲义-08_平面向量与复数" ], "genre": "填空题", "ans": "$(-\\dfrac 12,2)\\cup (2,+\\infty)$", @@ -379367,7 +382851,8 @@ "content": "已知正方体$ABCD-A_1B_1C_1D_1$的棱长为$2$, 正方体上底面$A_1B_1C_1D_1$内(含边界)一动点$P$到点$A$的距离为$2 \\sqrt{2}$, 点$P$的轨迹形成一条曲线, 这条曲线的长度为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-09_立体几何综合" ], "genre": "填空题", "ans": "$\\pi$", @@ -379424,7 +382909,8 @@ "content": "如图, 在棱长为$2$的正方体$ABCD-A_1B_1C_1D_1$中, $M$、$N$、$P$分别是$C_1D_1$、$C_1C$、$A_1A$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(A)!0.5!(A1)$) node [left] {$P$} coordinate (P) circle (0.03);\n\\draw ($(C)!0.5!(C1)$) node [right] {$N$} coordinate (N) circle (0.03);\n\\draw ($(C1)!0.5!(D1)$) node [above] {$M$} coordinate (M) circle (0.03);\n\\draw (M)--(A1)--(B)--(N) (P)--(B);\n\\draw [dashed] (M)--(N) (C)--(D1) (P)--(D1) (P)--(M)(P)--(N)(M)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $M$、$N$、$A_1$、$B$四点共面;\\\\\n(2) 求异面直线$PD_1$与$MN$所成角的余弦值;\\\\\n(3) 求三棱锥$P-MNB$的体积.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-10_空间向量与应用" ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\arccos \\dfrac{\\sqrt{10}}{10}$; (3) $\\dfrac 13$", @@ -379459,7 +382945,8 @@ "content": "如图甲所示, 在平面五边形$PABCD$中, $PD=PA$, $AC=CD=BD=\\sqrt{5}$, $AB=1$, $AD=2$, $PD \\perp PA$, 现将图甲所示中的$\\triangle PAD$沿$AD$边折起, 使平面$PAD \\perp$平面$ABCD$得到四棱锥$P-ABCD$, 如图乙所示.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$D$} coordinate (D);\n\\draw (2,0) node [right] {$A$} coordinate (A);\n\\draw (1,1) node [above] {$P$} coordinate (P);\n\\draw (2,-1) node [right] {$B$} coordinate (B);\n\\draw (1,{-sqrt(3)}) node [below] {$C$} coordinate (C);\n\\draw (A)--(B)--(C)--(D)--(P)--cycle(B)--(D)--(A)--(C);\n\\draw (1,-2.5) node [below] {图甲};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$D$} coordinate (D);\n\\draw (2,0,0) node [right] {$A$} coordinate (A);\n\\draw (1,1,0) node [above] {$P$} coordinate (P);\n\\draw (2,0,1) node [right] {$B$} coordinate (B);\n\\draw (1,0,{sqrt(3)}) node [below] {$C$} coordinate (C);\n\\draw (A)--(B)--(C)--(D)--(P)--cycle (B)--(P)--(C);\n\\draw [dashed] (B)--(D)--(A)--(C);\n\\draw (1,-2.5) node [below] {图乙};\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $PD \\perp$平面$PAB$;\\\\\n(2) 求二面角$A-PB-C$的大小;\\\\\n(3) 在棱$PA$上是否存在点$M$使得$BM$与平面$PCB$所成的角的正弦值为$\\dfrac{1}{3}$? 说明理由.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-10_空间向量与应用" ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\dfrac{5\\pi}6$; (3) 存在($M$满足$\\overrightarrow{OM}=(4-\\sqrt{10})\\overrightarrow{OA}+(\\sqrt{10}-3)\\overrightarrow{OP}$)", @@ -379537,7 +383024,8 @@ "content": "在正四棱锥$S-ABCD$中, $E$是线段$AB$上的点(不含端点). 设$SE$与$BC$所成的角为$\\alpha$, $SE$与平面$ABCD$所成的角为$\\beta$, 二面角$S-AB-C$的平面角为$\\gamma$, 则\\bracket{20}.\n\\fourch{$\\alpha \\leq \\beta \\leq \\gamma$}{$\\beta \\leq \\alpha \\leq \\gamma$}{$\\beta \\leq \\gamma \\leq \\alpha$}{$\\gamma \\leq \\beta \\leq \\alpha$}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-09_立体几何综合" ], "genre": "选择题", "ans": "C", @@ -379594,7 +383082,8 @@ "content": "正三棱锥$S-ABC$中, $\\angle BSC=40^{\\circ}$, $SB=2$, \n一质点自点$B$出发, 沿着三棱锥的侧面绕行一周回到点$B$的最短路线的长为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-09_立体几何综合" ], "genre": "填空题", "ans": "$2\\sqrt{3}$", @@ -379717,7 +383206,8 @@ "content": "在直三棱柱$ABC-A_1B_1C_1$中, 底面为直角三角形, $\\angle ACB=90^{\\circ}$, $AC=6$, $BC=CC_1=\\sqrt{2}$, $P$是$BC_1$上一动点, 则$CP+PA_1$的最小值为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-09_立体几何综合" ], "genre": "填空题", "ans": "$5\\sqrt{2}$", @@ -379752,7 +383242,8 @@ "content": "如图$1$是由矩形$ADEB$, Rt$\\triangle ABC$和菱形$BFGC$组成的一个平面图形, 其中$AB=1$, $BE=BF=2$, $\\angle FBC=60^{\\circ}$, 将其沿$AB, BC$折起使得$BE$与$BF$重合, 连结$DG$, 如图$2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$B$} coordinate (B);\n\\draw (-2,0) node [below] {$E$} coordinate (E);\n\\draw (2,0) node [right] {$C$} coordinate (C);\n\\draw (0,1) node [above] {$A$} coordinate (A);\n\\draw (-2,1) node [above] {$D$} coordinate (D);\n\\draw (-60:2) node [below] {$F$} coordinate (F);\n\\draw (F) ++ (2,0) node [below] {$G$} coordinate (G);\n\\draw (E)--(C)(D)--(A)--(C)(D)--(E)(A)--(B)--(F)--(G)(C)--(G);\n\\draw (0.5,-2.5) node {图$1$};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex, z = {(120:0.5cm)}]\n\\draw (0,0,0) node [below] {$B$} coordinate (B);\n\\draw (2,0,0) node [below] {$C$} coordinate (C);\n\\draw (0,0,1) node [left] {$A$} coordinate (A);\n\\draw (B) ++ (1,{sqrt(3)},0) node [below right] {$E$($F$)} coordinate (E);\n\\draw (C) ++ (1,{sqrt(3)},0) node [above] {$G$} coordinate (G);\n\\draw (A) ++ (1,{sqrt(3)},0) node [above] {$D$} coordinate (D);\n\\draw (A)--(B)--(C)--(G)--(D)--cycle (D)--(E)--(G) (E)--(B);\n\\draw [dashed] (A)--(C);\n\\draw (0.875,-0.75) node {图$2$};\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: 图$2$中的$A, C, G, D$四点共面, 且平面$ABC \\perp$平面$BCGE$;\\\\\n(2) 求图$2$中的二面角$B-CG-A$的大小.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-10_空间向量与应用" ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\dfrac\\pi 6$", @@ -379853,7 +383344,8 @@ "content": "已知点$A(1,0)$, $B(-1,2)$.\\\\\n(1) 若直线$AB$与直线$x-m y+1=0$垂直, 求实数$m$的值;\\\\\n(2) 若直线$AB$与直线$x-m y+1=0$平行, 求实数$m$的值, 以及这两条平行直线之间的距离;\\\\\n(3) 求过点$B$与直线$2 x-y+1=0$夹角的余弦值为$\\dfrac{2 \\sqrt{5}}{5}$的直线方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-11_直线与圆" ], "genre": "解答题", "ans": "(1) $1$; (2) $m=-1$, 距离为$\\sqrt{2}$; (3) $x+1=0$或$3x-4y+11=0$", @@ -379912,7 +383404,8 @@ "content": "已知圆$N: (x-2)^2+(y+1)^2=4$.\\\\\n(1) 直线$l$经过点$A(3,2)$, 且被圆$N$截得长为$2 \\sqrt{2}$的弦, 求直线$l$的方程;\\\\\n(2) 过点$B(3,0)$的直线$l$与圆$N$交于$P$、$Q$两点, 分别求弦$PQ$最短和最长时其所在直线的方程;\\\\\n(3) 讨论圆$C: (x-a)^2+y^2=1$($a \\in \\mathbf{R}$)与圆$N$的位置关系.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-11_直线与圆" ], "genre": "解答题", "ans": "(1) $x-y-1=0$或$7x+2y-23=0$; (2) 弦$PQ$最短时, 所在直线的方程为$x+y-3=0$; 弦$PQ$最长时, 所在直线的方程为$x-y+3=0$; (3) 当$a=2$时, 两圆内切; 当$a\\in (2-2\\sqrt{2},2)\\cup (2,2+2\\sqrt{2})$时, 两圆相交; 当$a=2\\pm 2\\sqrt{2}$时, 两圆外切; 当$a\\in (-\\infty,2-2\\sqrt{2})\\cup (2+2\\sqrt{2},+\\infty)$时, 两圆外离", @@ -379947,7 +383440,8 @@ "content": "若方程$x^2+y^2-x+y+k=0$表示圆, 则实数$k$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-11_直线与圆" ], "genre": "填空题", "ans": "$(-\\infty,\\dfrac 12)$", @@ -380004,7 +383498,8 @@ "content": "设$\\theta \\in \\mathbf{R}$, 直线$x \\cos \\theta+y-1=0$的倾斜角的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-11_直线与圆" ], "genre": "填空题", "ans": "$[0,\\dfrac\\pi 4]\\cup (\\dfrac {3\\pi}4,\\pi)$", @@ -380105,7 +383600,8 @@ "content": "若直线$l_1: 2 x+y-3=0$与直线$l_2: 4 x+2 y+a=0$的距离为$\\dfrac{\\sqrt{5}}{2}$, 则实数$a$的值为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-11_直线与圆" ], "genre": "填空题", "ans": "$-1$或$-11$", @@ -380206,7 +383702,8 @@ "content": "如图, 动点$C$在以$AB$为直径的半圆$O$上(异于$A, B$), $\\angle DCB=\\dfrac{\\pi}{2}$, 且$DC=CB$, 若$|AB|=2$, 则$\\overrightarrow{OC} \\cdot \\overrightarrow{OD}$的取值范围为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (-1,0) node [left] {$A$} coordinate (A);\n\\draw (1,0) node [right] {$B$} coordinate (B);\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw ($(O)!1!110:(B)$) node [above left] {$C$} coordinate (C);\n\\draw ($(C)!1!90:(B)$) node [above] {$D$} coordinate (D);\n\\draw (O)--(C)--(D)--cycle(A)--(B)--(C) (A)arc(180:0:1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第二轮复习讲义-11_直线与圆" ], "genre": "填空题", "ans": "$(1,2]$", @@ -380285,7 +383782,8 @@ "content": "已知$m$为实数, 复数$z=m^2+m-2+(m^2-4) \\mathrm{i}$.\\\\\n(1) 当$z$为实数时, $m=$\\blank{50};\\\\\n(2) 当$z$为纯虚数时, $m=$\\blank{50};\\\\\n(3) 当$z=0$时, $m=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第二轮复习讲义-08_平面向量与复数" ], "genre": "填空题", "ans": "(1) $2$或$-2$; (2) $1$; (3) $-2$", @@ -380319,7 +383817,8 @@ "content": "设复数$3-4 \\mathrm{i}$与$5-6 \\mathrm{i}$在复平面上所对应的向量分别为$\\overrightarrow{OA}$与$\\overrightarrow{OB}$, 则向量$\\overrightarrow{AB}$所对应的复数为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第二轮复习讲义-08_平面向量与复数" ], "genre": "填空题", "ans": "$2-2\\mathrm{i}$", @@ -380353,7 +383852,8 @@ "content": "如果复数$z$满足$(1+2 \\mathrm{i}) \\overline {z}=4+3 \\mathrm{i}$, 则$|z|=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第二轮复习讲义-08_平面向量与复数" ], "genre": "填空题", "ans": "$\\sqrt{5}$", @@ -380387,7 +383887,8 @@ "content": "若关于$x$的实系数一元二次方程$x^2-b x+c=0$的一个根为$1-3 \\mathrm{i}$, 则$3 b+c=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第二轮复习讲义-08_平面向量与复数" ], "genre": "填空题", "ans": "$16$", @@ -380465,7 +383966,8 @@ "content": "已知复数$z$满足下列条件, 根据条件分别求复数$z$在复平面上对应点的轨迹方程, 并分别求出$|z|$的最大值.\\\\\n(1) $|z-1-\\mathrm{i}|=1$;\\\\\n(2) $|z-3 \\mathrm{i}|+|z+3 \\mathrm{i}|=10$.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第二轮复习讲义-08_平面向量与复数" ], "genre": "解答题", "ans": "(1) 轨迹方程为$(x-1)^2+(y-1)^2=1$, $|z|$的最大值为$1+\\sqrt{2}$; (2) 轨迹方程为$\\dfrac{y^2}{25}+\\dfrac{x^2}{16}=1$, $|z|$的最大值为$5$", @@ -380521,7 +384023,8 @@ "content": "已知复数$z=\\dfrac{(1+3 \\mathrm{i})^2(3-\\mathrm{i})}{(1-2 \\mathrm{i})^2}$, 则$|z|=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第二轮复习讲义-08_平面向量与复数" ], "genre": "填空题", "ans": "$2\\sqrt{10}$", @@ -380622,7 +384125,8 @@ "content": "若复数$z=(m^2-5 m+6)+(2 m^2-5 m+2) \\mathrm{i}$为纯虚数, 则实数$m=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第二轮复习讲义-08_平面向量与复数" ], "genre": "填空题", "ans": "$3$", @@ -380657,7 +384161,8 @@ "content": "已知$|z_1|=3$, $|z_2|=4$, $|z_1+z_2|=5$, , 则$|z_1-z_2|=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第二轮复习讲义-08_平面向量与复数" ], "genre": "填空题", "ans": "$5$", @@ -380692,7 +384197,8 @@ "content": "已知复数$z=\\dfrac{\\mathrm{i}+\\mathrm{i}^2+\\mathrm{i}^3+\\cdots+\\mathrm{i}^{2003}}{1+\\mathrm{i}}$, 则复数$z=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第二轮复习讲义-08_平面向量与复数" ], "genre": "填空题", "ans": "$-\\dfrac 12+\\dfrac 12\\mathrm{i}$", @@ -381057,7 +384563,8 @@ "content": "如图, 在一个$60^{\\circ}$的二面角$\\alpha-l-\\beta$的棱上有两个点$A$、$B$, 其中$AC$、$BD$分别是在这个二面角的两个半平面内垂直于$AB$的线段, 且$AB=4$, $AC=6$, $BD=8$, 则$CD$的长为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = {(-120:0.5cm)}]\n\\draw (0,0,0) -- (3,0,0) --++ (0,0,3) --++ (-3,0,0) coordinate (S) -- cycle;\n\\draw (0,0,0) --++ (0,{sqrt(3)},1) coordinate (T) --++ (3,0,0) --++ (0,{-sqrt(3)},-1);\n\\draw (1,0,0) node [below] {$A$} coordinate (A);\n\\draw (A) --++ (0,{0.6*sqrt(3)},0.6) node [above] {$C$} coordinate (C);\n\\draw (1.8,0,0) node [above] {$B$} coordinate (B);\n\\draw (B) --++ (0,0,1.6) node [below] {$D$} coordinate (D);\n\\draw (C)--(D);\n\\draw (0.5,0,0) node [above] {$l$} coordinate (l);\n\\draw (S) ++ (0.3,0,-0.3) node {$\\beta$};\n\\draw (T) ++ (0.3,{-0.15*sqrt(3)},-0.15) node {$\\alpha$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-09_立体几何综合" ], "genre": "填空题", "ans": "$2\\sqrt{17}$", @@ -381114,7 +384621,8 @@ "content": "在正方体$ABCD-A_1B_1C_1D_1$中, 给出下列四个命题:\\\\\n\\textcircled{1} 点$P$在直线$BC_1$上运动时, 三棱锥$A-D_1PC$的体积不变;\\\\\n\\textcircled{2} 点$P$在直线$BC_1$上运动时, 直线$AP$与平面$ACD_1$所成的角的大小不变;\\\\\n\\textcircled{3} 点$P$在直线$BC_1$上运动时, 二面角$P-AD_1-C$的大小不变;\\\\\n\\textcircled{4} 点$P$是平面$ABCD$上到点$D$和$C_1$距离相等的动点, 则$P$的轨迹是过点$B$的直线.\n其中的真命题是\\bracket{20}.\n\\fourch{\\textcircled{1}\\textcircled{3}}{\\textcircled{1}\\textcircled{3}\\textcircled{4}}{\\textcircled{1}\\textcircled{2}\\textcircled{4}}{\\textcircled{3}\\textcircled{4}}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-09_立体几何综合" ], "genre": "选择题", "ans": "B", @@ -381193,7 +384701,8 @@ "content": "如图, 已知圆柱$OO_1$的底面半径为$1$, 正三角形$ABC$内接于圆柱的下底面圆$O$, 点$O_1$是圆柱的上底面的圆心, 线段$AA_1$是圆柱的母线.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\filldraw (0,0) node [left] {$O$} coordinate (O) circle (0.03);\n\\filldraw (0,2) node [left] {$O_1$} coordinate (O_1) circle (0.03);\n\\draw (1,0) node [right] {$A$} coordinate (A) --++ (0,2) node [right] {$A_1$} coordinate (A_1);\n\\draw (-1,0) -- (-1,2);\n\\draw (O_1) ellipse (1 and 0.3);\n\\draw (A) arc (0:-180:1 and 0.3);\n\\draw [dashed] (A) arc (0:180:1 and 0.3);\n\\draw (135:1 and 0.3) node [above] {$C$} coordinate (C);\n\\draw (-105:1 and 0.3) node [below] {$B$} coordinate (B);\n\\draw [dashed] (A)--(B)--(C)--cycle;\n\\draw [dashed] (B)--(A_1)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求点$C$到平面$A_1AB$的距离;\\\\\n(2) 在劣弧$\\overset\\frown{BC}$上是否存在一点$D$, 满足$O_1D\\parallel$平面$A_1AB$? 若存在, 求出$\\angle BOD$的大小; 若不存在, 请说明理由.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-09_立体几何综合" ], "genre": "解答题", "ans": "(1) $\\dfrac 32$; (2) 存在, $\\angle BOD=\\dfrac\\pi 6$", @@ -381294,7 +384803,8 @@ "content": "如图, 已知正四面体$ABCD$的棱长为$2$, 用平行于底面$BCD$的平面截这个棱锥, 得到一个小棱锥和一个棱台. 若截面与底面之间的距离为$\\dfrac{\\sqrt{6}}{2}$, 则棱台的体积为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw ({-2/sqrt(3)},0,0) node [left] {$B$} coordinate (B);\n\\draw ({1/sqrt(3)},0,1) node [below] {$C$} coordinate (C);\n\\draw (C)++(0,0,-2) node [right] {$D$} coordinate (D);\n\\draw (0,{2*sqrt(6)/3},0) node [above] {$A$} coordinate (A);\n\\draw (A)--(B)(A)--(C)(A)--(D)(B)--(C)--(D);\n\\draw [dashed] (B)--(D);\n\\draw ($(A)!{1/4}!(B)$) -- ($(A)!{1/4}!(C)$) -- ($(A)!{1/4}!(D)$);\n\\draw [dashed] ($(A)!{1/4}!(B)$) -- ($(A)!{1/4}!(D)$);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-09_立体几何综合" ], "genre": "填空题", "ans": "$\\dfrac{21\\sqrt{2}}{32}$", @@ -381395,7 +384905,8 @@ "content": "已知球$O$的表面积为$900 \\pi$, $ABCD-A_1B_1C_1D_1$是该球的内接长方体(即该长方体的八个顶点均在球面上).\\\\\n(1) 若$AB=12$, $BC=9$, 求球心$O$到平面$ABCD$的距离;\\\\\n(2) 若$ABCD-A_1B_1C_1D_1$是正四棱柱, 当该正四棱柱的侧面积最大时, 求其体积.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-09_立体几何综合" ], "genre": "解答题", "ans": "(1) $\\dfrac{15}2\\sqrt{3}$; (2) $3375\\sqrt{2}$", @@ -381474,7 +384985,8 @@ "content": "如图, 在直径$AB=4$的半圆$O$内作一个内接直角三角形$ABC$, 使$\\angle BAC=30^{\\circ}$, 将图中阴影部分以直线$AB$为旋转轴旋转$180^{\\circ}$形成一个几何体, 则该几何体的体积为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\filldraw (0,0) node [left] {$O$} coordinate (O) circle (0.03);\n\\draw (0,1.5) node [above] {$A$} coordinate (A);\n\\draw (0,-1.5) node [below] {$B$} coordinate (B);\n\\draw (-30:1.5) node [right] {$C$} coordinate (C);\n\\fill [pattern = north east lines] (A)--(C) arc (-30:90:1.5);\n\\fill [pattern = north east lines] (B)--(C) arc (-30:-90:1.5);\n\\draw (A)--(B)--(C)--cycle;\n\\draw (A) arc (90:-90:1.5);\n\\draw pic [draw, \"$30^\\circ$\", angle eccentricity = 2] {angle = B--A--C};\n\\draw pic [draw, \"$60^\\circ$\", angle eccentricity = 1.5] {angle = C--B--A};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-09_立体几何综合" ], "genre": "填空题", "ans": "$\\dfrac{10}3\\pi$", @@ -381509,7 +385021,8 @@ "content": "如图, 已知正四棱柱$ABCD-A_1B_1C_1D_1$的底面边长为$2$, 体积为$27$, $E$、$F$都是棱$BC$上的任意点且$EF=1$, $P$、$Q$分别在棱$A_1D_1$、$D_1C_1$上运动, 则四面体$P-EFQ$的体积\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{2.5}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(A1)!0.4!(D1)$) node [left] {$P$} coordinate (P);\n\\draw ($(C1)!0.3!(D1)$) node [above] {$Q$} coordinate (Q);\n\\draw ($(B)!0.2!(C)$) node [right] {$E$} coordinate (E);\n\\draw ($(B)!0.7!(C)$) node [right] {$F$} coordinate (F);\n\\draw (P)--(Q);\n\\draw [dashed] (P)--(E)(P)--(F)(Q)--(E)(Q)--(F);\n\\end{tikzpicture}\n\\end{center}\n\\twoch{与点$E$、$F$、$P$、$Q$位置都有关}{与点$P$位置有关, 与点$E$、$F$、$Q$位置无关}{与点$Q$位置有关, 与点$E$、$F$、$P$位置无关}{与点$E$、$F$、$P$、$Q$位置都无关}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-09_立体几何综合" ], "genre": "选择题", "ans": "C", @@ -381632,7 +385145,8 @@ "content": "《九章算术》是我国古代的数学巨著, 其卷第五``商功''有如下的问题: ``今有刍甍, 下广三丈, 袤四丈, 上袤二丈, 无广, 高一丈. 问积几何? ''意思为: 今有底面为矩形的屋脊形状的多面体(如图), 下底面宽$AD=3$丈, 长$AB=4$丈, 上棱$EF=2$丈, $EF\\parallel AB$, $EF$与平面$ABCD$的距离为$1$丈, 则它的体积是\\blank{50}(立方丈).\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = {(-120:0.5cm)}]\n\\draw (-2,0,1.5) node [below] {$A$} coordinate (A);\n\\draw (A)++(4,0,0) node [below] {$B$} coordinate (B);\n\\draw (A)++(0,0,-3) node [left] {$D$} coordinate (D);\n\\draw (D)++(4,0,0) node [right] {$C$} coordinate (C);\n\\draw (-1,1,0) node [above] {$E$} coordinate (E);\n\\draw (1,1,0) node [above] {$F$} coordinate (F);\n\\draw (E)--(F)(E)--(A)(F)--(B)(F)--(C)(A)--(B)--(C)(A)--(D)--(E);\n\\draw [dashed] (D)--(C);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-09_立体几何综合" ], "genre": "填空题", "ans": "$5$", @@ -381711,7 +385225,8 @@ "content": "已知正四棱锥的侧棱长为$l$, 其各顶点都在同一球面上. 若该球的体积为$36 \\pi$, 且$3 \\leq l \\leq 3 \\sqrt{3}$, 求该正四棱锥体积的取值范围.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-09_立体几何综合" ], "genre": "解答题", "ans": "$[\\dfrac{27}4,\\dfrac{64}3]$", @@ -381746,7 +385261,8 @@ "content": "如图, 在平行六面体$ABCD-A_1B_1C_1D_1$中, $AC$与$BD$的交点为$M$, 设$\\overrightarrow{A_1B_1}=\\overrightarrow {a}$, $\\overrightarrow{A_1D_1}=\\overrightarrow {b}$, $\\overrightarrow{A_1A}=\\overrightarrow {c}$, 则下列向量中与$\\overrightarrow{B_1M}$相等的向量是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$A$} coordinate (A);\n\\draw (2,0,0) node [below] {$B$} coordinate (B);\n\\draw (B) ++ (1,0,{-sqrt(3)}) node [right] {$C$} coordinate (C);\n\\draw ($(A)+(C)-(B)$) node [above left] {$D$} coordinate (D);\n\\draw (A) ++ (0,{sqrt(3)},-1) node [left] {$A_1$} coordinate (A_1);\n\\draw ($(B)-(A)+(A_1)$) node [below right] {$B_1$} coordinate (B_1);\n\\draw ($(C)-(A)+(A_1)$) node [right] {$C_1$} coordinate (C_1);\n\\draw ($(D)-(A)+(A_1)$) node [above] {$D_1$} coordinate (D_1);\n\\draw (A)--(A_1)--(B_1)--(B)--cycle (A_1)--(D_1)--(C_1)--(C)--(B)(B_1)--(C_1);\n\\draw [dashed] (A)--(D)--(C)(D)--(D_1);\n\\draw ($(A)!0.5!(C)$) node [below] {$M$} coordinate (M);\n\\draw [dashed] (A)--(C)(B)--(D)(M)--(B_1);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$-\\dfrac{1}{2} \\overrightarrow {a}-\\dfrac{1}{2} \\overrightarrow {b}+\\overrightarrow {c}$}{$-\\dfrac{1}{2} \\overrightarrow {a}+\\dfrac{1}{2} \\overrightarrow {b}+\\overrightarrow {c}$}{$\\dfrac{1}{2} \\overrightarrow {a}-\\dfrac{1}{2} \\overrightarrow {b}+\\overrightarrow {c}$}{$\\dfrac{1}{2} \\overrightarrow {a}+\\dfrac{1}{2} \\overrightarrow {b}+\\overrightarrow {c}$}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-10_空间向量与应用" ], "genre": "选择题", "ans": "B", @@ -381780,7 +385296,8 @@ "content": "在如图所示的空间直角坐标系中, $ABCD-A_1B_1C_1D_1$为长方体, $AB=BC=1$, $AA_1=2$, 点$A$关于$x$轴对称的点$A'$的坐标为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{1}\n\\def\\m{1}\n\\def\\n{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw [->] (B1) -- ($(C1)!2!(B1)$) node [below] {$x$};\n\\draw [->] (C1) -- ($(D1)!1.5!(C1)$) node [below] {$y$};\n\\draw [->] (C1) -- ($(C)!1.25!(C1)$) node [left] {$z$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-10_空间向量与应用" ], "genre": "填空题", "ans": "$(1,1,2)$", @@ -381814,7 +385331,8 @@ "content": "已知向量$\\overrightarrow {a}=(1,-1,3)$, $\\overrightarrow {b}=(-1,4,-2)$, $\\overrightarrow {c}=(1,5,x)$, 若$\\overrightarrow {a}$、$\\overrightarrow {b}$、$\\overrightarrow {c}$共面, 则实数$x=$\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-10_空间向量与应用" ], "genre": "填空题", "ans": "$5$", @@ -381849,7 +385367,8 @@ "content": "已知空间三点$A(-2,0,2)$, $B(-1,1,2)$, $C(-3,0,4)$. 设$\\overrightarrow {a}=\\overrightarrow{AB}$, $\\overrightarrow {b}=\\overrightarrow{AC}$. 若向量$k \\overrightarrow {a}+\\overrightarrow {b}$与$k \\overrightarrow {a}-2 \\overrightarrow {b}$互相垂直, 则实数$k=$\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-10_空间向量与应用" ], "genre": "填空题", "ans": "$2$或$-\\dfrac 52$", @@ -381884,7 +385403,8 @@ "content": "已知平面$ABC$中, $\\overrightarrow{AC}=(-1,-1,0)$, $\\overrightarrow{AB}=(0,1,2)$, 则平面$ABC$的一个法向量为$\\overrightarrow {n}=$\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-10_空间向量与应用" ], "genre": "填空题", "ans": "$(2,-2,1)$", @@ -381919,7 +385439,8 @@ "content": "如图, 在正四面体$ABCD$中, $E$、$F$分别为棱$DA$、$BC$的中点, 又设$\\overrightarrow{DA}=\\overrightarrow {a}$, $\\overrightarrow{DB}=\\overrightarrow {b}$, $\\overrightarrow{DC}=\\overrightarrow {c}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (1,0,{sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw (${1/3}*(A)+{1/3}*(B)+{1/3}*(C)+(0,{2*sqrt(6)/3},0)$) node [above] {$D$} coordinate (D);\n\\draw (A)--(B)--(C)--(D)--cycle(B)--(D);\n\\draw [dashed] (A)--(C);\n\\draw ($(A)!0.5!(D)$) node [left] {$E$} coordinate (E);\n\\draw ($(B)!0.5!(C)$) node [below right] {$F$} coordinate (F);\n\\draw (E)--(B)(F)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 用向量$\\overrightarrow {a},\\overrightarrow {b},\\overrightarrow {c}$的线性组合表示向量$\\overrightarrow{BE}$, $\\overrightarrow{DF}$;\\\\\n(2) 求$\\langle\\overrightarrow{BE},\\overrightarrow{DF}\\rangle$的大小.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-10_空间向量与应用" ], "genre": "解答题", "ans": "(1) $\\overrightarrow{BE}=\\dfrac 12\\overrightarrow{a}-\\overrightarrow{b}$, $\\overrightarrow{DF}=\\dfrac 12\\overrightarrow{b}+\\dfrac 12\\overrightarrow{c}$; (2) $\\pi-\\arccos \\dfrac 23$", @@ -381976,7 +385497,8 @@ "content": "如图, 在棱长为$2$的正方体$ABCD-A_1B_1C_1D_1$中, $E$、$F$、$M$、$N$分别是棱$AB$、$AD$、$A_1B_1$、$A_1D_1$的中点, 点$P$、$Q$分别在棱$DD_1$、$BB_1$上移动, 且$DP=BQ=\\lambda$($0<\\lambda<2$).\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [below] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(A)!0.5!(B)$) node [below] {$E$} coordinate (E);\n\\draw ($(A)!0.5!(D)$) node [left] {$F$} coordinate (F);\n\\draw ($(A1)!0.5!(B1)$) node [above] {$M$} coordinate (M);\n\\draw ($(A1)!0.5!(D1)$) node [left] {$N$} coordinate (N);\n\\draw ($(B)!0.6!(B1)$) node [right] {$Q$} coordinate (Q);\n\\draw ($(D)!0.5!(D1)$) node [left] {$P$} coordinate (P);\n\\draw (B)--(C1)(E)--(Q)--(M)--(N);\n\\draw [dashed] (E)--(F)--(P)--(N)(P)--(Q);\n\\end{tikzpicture}\n\\end{center}\n(1) 当$\\lambda=1$时, 证明: 直线$BC_1\\parallel$平面$EFPQ$;\\\\\n(2) 是否存在$\\lambda$, 使平面$EFPQ$与平面$PQMN$所成的二面角为直二面角? 若存在, 求出$\\lambda$的值; 若不存在, 说明理由.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-10_空间向量与应用" ], "genre": "解答题", "ans": "(1) 证明略; (2) 存在, $\\lambda=1\\pm \\dfrac{\\sqrt{2}}2$", @@ -382099,7 +385621,8 @@ "content": "如图, 以长方体$ABCD-A_1B_1C_1D_1$的顶点$D$为坐标原点, 过$D$的三条棱所在的直线为坐标轴, 建立空间直角坐标系, 若$\\overrightarrow{DB_1}$的坐标为$(4,3,2)$, 则$\\overrightarrow{AC}_1$的坐标为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\def\\l{4}\n\\def\\m{3}\n\\def\\n{2}\n\\draw (0,0,0) node [below right] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [below right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw [->] (A)-- ($(D)!1.5!(A)$) node [below left] {$x$};\n\\draw [->] (C)-- ($(D)!1.4!(C)$) node [below] {$y$};\n\\draw [->] (D1)-- ($(D)!1.5!(D1)$) node [right] {$z$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-10_空间向量与应用" ], "genre": "填空题", "ans": "$(-4,3,2)$", @@ -382177,7 +385700,8 @@ "content": "在空间直角坐标系中, 点$A(-1,3,1)$、点$B(2,4,0)$、点$C(0,2,4)$, 则以$\\overrightarrow{AB}$、$\\overrightarrow{AC}$为一组邻边的平行四边形的面积为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-10_空间向量与应用" ], "genre": "填空题", "ans": "$2\\sqrt{30}$", @@ -382211,7 +385735,8 @@ "content": "在空间直角坐标系中, 点$A(1,0,0)$, 点$B(5,-4,3)$, 点$C(2,0,1)$, 则$\\overrightarrow{AB}$在$\\overrightarrow{CA}$方向上的投影向量的坐标为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-10_空间向量与应用" ], "genre": "填空题", "ans": "$(\\dfrac 72,0,\\dfrac 72)$", @@ -382277,7 +385802,8 @@ "content": "已知半径为$1$的球$O$内切于正四面体$ABCD$, 线段$MN$是球$O$的一条动直径 ($M$、$N$是直径的两端点), 点$P$是正四面体$ABCD$的表面上的一个动点, 则$\\overrightarrow{PM} \\cdot \\overrightarrow{PN}$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-09_立体几何综合" ], "genre": "填空题", "ans": "$[0,8]$", @@ -382312,7 +385838,8 @@ "content": "如图, 在三棱柱$ABC-A_1B_1C_1$中, 底面$ABC$是以$AC$为斜边的等腰直角三角形, 侧面$AA_1C_1C$为菱形, 点$A_1$在底面上的投影为$AC$的中点$D$, 且$AB=2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (1,0,1) node [below] {$B$} coordinate (B);\n\\draw (1,{sqrt(3)},0) node [above] {$A_1$} coordinate (A_1);\n\\draw (A_1) ++ (2,0,0) node [above] {$C_1$} coordinate (C_1);\n\\draw (1,0,0) node [above right] {$D$} coordinate (D);\n\\draw ($(A_1)+(B)-(A)$) node [below right] {$B_1$} coordinate (B_1);\n\\draw ($(A_1)!0.4!(B_1)$) node [above right] {$E$} coordinate (E);\n\\draw (A)--(B)--(C)--(C_1)--(A_1)--cycle(A_1)--(B_1)--(C_1)(B_1)--(B);\n\\draw [dashed] (A_1)--(D)--(E)(A)--(C)(B)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $BD \\perp CC_1$;\\\\\n(2) 求点$C$到侧面$AA_1B_1B$的距离;\\\\\n(3) 在线段$A_1B_1$上是否存在点$E$, 使得直线$DE$与侧面$AA_1B_1B$所成角的正弦值为$\\dfrac{\\sqrt{6}}{7}$? 若存在, 请求出$A_1E$的长; 若不存在, 请说明理由.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-10_空间向量与应用" ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\dfrac{2\\sqrt{42}}7$; (3) 存在, $A_1E=1$", @@ -382412,7 +385939,8 @@ "content": "双曲线$\\dfrac{x^2}{3}-y^2=1$的两条渐近线的夹角的大小为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-13_解析几何综合" ], "genre": "填空题", "ans": "$\\dfrac{\\pi}3$", @@ -382535,7 +386063,8 @@ "content": "已知抛物线$C: y^2=4 x$的焦点为$F$.\\\\\n(1) 若抛物线$C$的焦点$F$为双曲线$\\Gamma: \\dfrac{x^2}{a^2}-2 y^2=1$($a>0$)的一个焦点, 求双曲线$\\Gamma$的离心率$e$;\\\\\n(2) 设抛物线$C$的准线$l$与$x$轴的交点为$E$, 点$P$在抛物线$C$上, 且在第一象限, 若$\\dfrac{|PF|}{|PE|}=\\dfrac{\\sqrt{2}}{2}$, 求直线$PE$的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-12_圆锥曲线" ], "genre": "解答题", "ans": "(1) $\\sqrt{2}$; (2) $y=x+1$", @@ -382855,7 +386384,8 @@ "content": "若过点$P(0,-1)$的直线$l$与抛物线$y^2=2 x$恰有一个公共点, 则直线$l$的方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-13_解析几何综合" ], "genre": "填空题", "ans": "$x=0$或$y=-1$或$y=-\\dfrac 12 x-1$", @@ -382889,7 +386419,8 @@ "content": "若$P(x, y)$是抛物线$y^2=x$上一动点, 则点$P$到直线$y=x+3$的距离的最小值为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-13_解析几何综合" ], "genre": "填空题", "ans": "$\\dfrac{11\\sqrt{2}}8$", @@ -382990,7 +386521,8 @@ "content": "设椭圆$\\Gamma: \\dfrac{x^2}{a^2}+y^2=1$($a>0$), $F_1$、$F_2$分别是椭圆$\\Gamma$的左、右焦点, 椭圆$\\Gamma$的离心率为$\\dfrac{\\sqrt{2}}{2}$, 直线$l$与椭圆$\\Gamma$交于不同的两点$A$、$B$.\\\\\n(1) 求椭圆$\\Gamma$的方程;\\\\\n(2) 已知直线$l$经过椭圆$\\Gamma$的右焦点$F_2$, $P$、$Q$是椭圆$\\Gamma$上两点, 四边形$ABQP$是菱形, 求直线$l$的方程;\\\\\n(3) 已知直线$l$与$x$轴的正半轴和$y$轴分别交于点$M$、$N$, 且$\\overrightarrow{AN}=\\lambda \\overrightarrow{AM}$, $\\overrightarrow{BN}=\\mu \\overrightarrow{BM}$, 若$\\lambda+\\mu=3$, 证明: 直线$l$过定点, 并求此定点的坐标.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-13_解析几何综合" ], "genre": "解答题", "ans": "(1) $\\dfrac{x^2}2+y^2=1$; (2) $y=\\pm\\sqrt{2}(x-1)$; (3) $(\\dfrac{\\sqrt{6}}3,0)$", @@ -383046,7 +386578,8 @@ "content": "若经过点$F_2(2,0)$的直线$l$与双曲线$x^2-\\dfrac{y^2}{3}=1$相交于$A$、$B$两点, 且$|AB|=6$, 则满足条件的直线$l$共有\\blank{50}条.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-13_解析几何综合" ], "genre": "填空题", "ans": "$3$", @@ -383191,7 +386724,8 @@ "content": "设$k \\in \\mathbf{R}$, 已知双曲线$\\Gamma: \\dfrac{x^2}{4}-y^2=1$, 若对任意的$m \\in \\mathbf{R}$, 直线$y=k x+m$与双曲线$\\Gamma$总有公共点, 则$k$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-13_解析几何综合" ], "genre": "填空题", "ans": "$(-\\dfrac 12,\\dfrac 12)$", @@ -383468,7 +387002,8 @@ "content": "如图, 汽车前灯反射镜与轴截面的交线是抛物线的一部分, 灯口所在的圆面与反射镜的轴垂直, 灯泡位于抛物线的焦点处. 经灯口直径是$24$厘米, 灯深$10$厘米, 则灯泡与反射镜顶点的距离是\\blank{50}厘米.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.15]\n\\filldraw (3.6, 0) circle (0.1) node [right] {$F$} coordinate (F);\n\\filldraw (10, 0) circle (0.1);\n\\draw [domain = -12:12] plot ({pow(\\x, 2)/14.4}, \\x);\n\\draw (10, 0) ellipse (3 and 12);\n\\draw (0, 0) node [left] {$O$} coordinate (O) --++ (0, -14);\n\\draw (10, -12) --++ (0, -2);\n\\draw [<->] (0, -13) -- (10, -13) node [midway, below] {$10\\text{cm}$};\n\\draw (10, -12) --++ (5, 0) (10, 12) --++ (5, 0);\n\\draw [<->] (14, -12) -- (14, 12) node [midway, right] {\\rotatebox{90}{$24\\text{cm}$}};\n\\draw [dashed] (10, 0) --++ (0, -12);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-13_解析几何综合" ], "genre": "填空题", "ans": "$3.6$", @@ -383524,7 +387059,10 @@ "content": "直线$l$与抛物线$y^2=2 x$相交于$A$、$B$两点, 与$x$轴正半轴不相交. 若$\\overrightarrow{OA} \\cdot \\overrightarrow{OB}=3$, 其中$O$为坐标原点, 则直线$l$过定点\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-四月错题重做-04_易错题-解析几何", + "2023届高三-四月错题重做-04_解析几何", + "2023届高三-第二轮复习讲义-13_解析几何综合" ], "genre": "填空题", "ans": "$(-1,0)$", @@ -383567,7 +387105,8 @@ "content": "已知椭圆$\\dfrac{x^2}{2}+y^2=1$, 作垂直于$x$轴的垂线交椭圆于$A$、$B$两点, 作垂直于$y$轴的垂线交椭圆于$C$、$D$两点, 且$AB=CD$, 两垂线相交于点$P$, 则点$P$的轨迹是\\bracket{20}的一部分.\n\\fourch{椭圆}{双曲线}{圆}{抛物线}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-13_解析几何综合" ], "genre": "选择题", "ans": "B", @@ -383646,7 +387185,8 @@ "content": "已知$F_1, F_2$是双曲线$C: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a, b>0$)的左、右焦点, 过$F_2$的直线交双曲线的右支于$A, B$两点, 且$|AF_1|=2|AF_2|$, $\\angle AF_1F_2=\\angle F_1BF_2$, 则在下列结论中, 正确结论的序号为\\blank{50}.\\\\\n\\textcircled{1} 双曲线$C$的离心率为$2$;\\\\\n\\textcircled{2} 双曲线$C$的一条渐近线的斜率为$\\sqrt{2}$;\\\\\n\\textcircled{3} 线段$AB$的长为$6 a$;\\\\\n\\textcircled{4} $\\triangle AF_1F_2$的面积为$\\sqrt{15} a^2$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-13_解析几何综合" ], "genre": "填空题", "ans": "\\textcircled{1}\\textcircled{4}", @@ -383703,7 +387243,9 @@ "content": "函数$y=\\lg (2-x)$的定义域为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期周末卷-高三下学期周末卷04", + "2023届高三-简单题冲刺-简单题冲刺15" ], "genre": "填空题", "ans": "$(-\\infty,2)$", @@ -383735,7 +387277,9 @@ "content": "已知集合$A=(-2,2)$, $B=(-3,-1) \\cup(1,5)$, 则$A \\cup B=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-下学期周末卷-高三下学期周末卷04", + "2023届高三-简单题冲刺-简单题冲刺15" ], "genre": "填空题", "ans": "$(-3,5)$", @@ -383767,7 +387311,9 @@ "content": "$(2 x+1)^5$的二项展开式中$x^3$项的系数是\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-下学期周末卷-高三下学期周末卷04", + "2023届高三-简单题冲刺-简单题冲刺15" ], "genre": "填空题", "ans": "$80$", @@ -383799,7 +387345,9 @@ "content": "已知向量$\\overrightarrow {a}=(-m, 1,3)$, $\\overrightarrow {b}=(2, n, 1)$, 若$\\overrightarrow {a}\\parallel \\overrightarrow {b}$, 则$m n$的值为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期周末卷-高三下学期周末卷04", + "2023届高三-简单题冲刺-简单题冲刺15" ], "genre": "填空题", "ans": "$-2$", @@ -383831,7 +387379,9 @@ "content": "已知复数$z$满足$(1+\\mathrm{i}) z=4-2 \\mathrm{i}$($\\mathrm{i}$为虚数单位), 则复数$z$的模等于\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-下学期周末卷-高三下学期周末卷04", + "2023届高三-简单题冲刺-简单题冲刺15" ], "genre": "填空题", "ans": "$\\sqrt{10}$", @@ -383863,7 +387413,9 @@ "content": "某个品种的小麦麦穗长度(单位: $\\text{cm}$)的样本数据如下: $10.2$、$9.7$、$10.8$、$9.1$、$8.9$、$8.6$、$9.8$、$9.6$、$9.9$、$11.2$、$10.6$、$11.7$, 则这组数据的第$80$百分数为\\blank{50}.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-下学期周末卷-高三下学期周末卷04", + "2023届高三-简单题冲刺-简单题冲刺15" ], "genre": "填空题", "ans": "$10.8$", @@ -383895,7 +387447,9 @@ "content": "在平面直角坐标系$xOy$中, 若角$\\theta$的顶点为坐标原点, 始边与$x$轴的非负半轴重合, 终边与以点$O$为圆心的单位圆交于点$P(-\\dfrac{3}{5}, \\dfrac{4}{5})$, 则$\\sin (2 \\theta-\\dfrac{\\pi}{2})$的值为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期周末卷-高三下学期周末卷04", + "2023届高三-简单题冲刺-简单题冲刺15" ], "genre": "填空题", "ans": "$\\dfrac{7}{25}$", @@ -383927,7 +387481,9 @@ "content": "已知一个圆锥的侧面展开图是一个面积为$2 \\pi$的半圆, 则该圆锥的体积为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期周末卷-高三下学期周末卷04", + "2023届高三-简单题冲刺-简单题冲刺15" ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{3}}3\\pi$", @@ -383959,7 +387515,9 @@ "content": "已知$\\triangle ABC$的三边长分别为$4$、$5$、$7$, 记$\\triangle ABC$的三个内角的正切值所组成的集合为$M$, 则集合$M$中的最大元素为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期周末卷-高三下学期周末卷04", + "2023届高三-中档题冲刺-中档题冲刺15" ], "genre": "填空题", "ans": "$\\dfrac{2\\sqrt{6}}5$", @@ -383991,7 +387549,9 @@ "content": "现有$5$人参加抽奖活动, 每人依次从装有$5$张奖票(其中$3$张为中奖票)的箱子中不放回地随机抽取一张, 直到$3$张中奖票都被抽出时活动结束, 则活动恰好在第$4$人抽完后结束的概率为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-下学期周末卷-高三下学期周末卷04", + "2023届高三-中档题冲刺-中档题冲刺15" ], "genre": "填空题", "ans": "$\\dfrac{3}{10}$", @@ -384023,7 +387583,9 @@ "content": "已知四边形$ABCD$是平行四边形, 若$\\overrightarrow{AD}=2\\overrightarrow{DE}$, $\\overrightarrow{BF}\\parallel \\overrightarrow{BE}$, $\\overrightarrow{AF} \\cdot \\overrightarrow{BE}=0$, \n且$\\overrightarrow{AF} \\cdot \\overrightarrow{AC}=60$, 则$\\overrightarrow{AC}$在$\\overrightarrow{AF}$上的数量投影为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-下学期周末卷-高三下学期周末卷04", + "2023届高三-中档题冲刺-中档题冲刺15" ], "genre": "填空题", "ans": "$10$", @@ -384055,7 +387617,8 @@ "content": "已知曲线$C_1: y=\\sqrt{1-x^2}$与曲线$C_2: y=\\sqrt{2-x^2}$, 长度为$1$的线段$AB$的两端点$A$、$B$分别在曲线$C_1$、$C_2$上沿顺时针方向运动, 若点$A$从点$(-1,0)$开始运动, 点$B$到达点$(\\sqrt{2}, 0)$时停止运动, 则线段$AB$所扫过的区域的面积为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期周末卷-高三下学期周末卷04" ], "genre": "填空题", "ans": "$\\dfrac{3\\pi}{8}$", @@ -384087,7 +387650,8 @@ "content": "在平面直角坐标系$xOy$中, ``$m<0$''是``方程$x^2+m y^2=1$表示的曲线是双曲线''的\\bracket{20}条件.\n\\fourch{充分不必要}{必要不充分}{充要}{既不充分也不必要}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期周末卷-高三下学期周末卷04" ], "genre": "选择题", "ans": "C", @@ -384119,7 +387683,8 @@ "content": "如图, 四边形$ABCD$是边长为$1$的正方形, $MD \\perp$平面$ABCD$, $NB \\perp$平面$ABCD$, 且$MD=NB=1$, 点$G$为$MC$的中点. 则下列结论中不正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 2,z = {(240:0.5cm)}]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (1,0,0) node [right] {$B$} coordinate (B);\n\\draw (1,0,-1) node [right] {$C$} coordinate (C);\n\\draw (0,0,-1) node [above right] {$D$} coordinate (D);\n\\draw (B)++(0,1,0) node [above right] {$N$} coordinate (N);\n\\draw (D)++(0,1,0) node [above] {$M$} coordinate (M);\n\\draw ($(C)!0.5!(M)$) node [below left] {$G$} coordinate (G);\n\\draw (A)--(B)--(C)--(N)--(M)--cycle(A)--(N)--(B);\n\\draw [dashed] (A)--(D)--(C)(D)--(M)--(C)(B)--(G);\n\\end{tikzpicture}\n\\end{center}\n\\twoch{$MC \\perp AN$}{平面$DCM\\parallel$平面$ABN$}{直线$GB$与$AM$是异面直线}{直线$GB$与平面$AMD$无公共点}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期周末卷-高三下学期周末卷04" ], "genre": "选择题", "ans": "D", @@ -384151,7 +387716,9 @@ "content": "已知$f(x)=\\sin (\\omega x+\\dfrac{\\pi}{6})(\\omega>0)$, 且函数$y=f(x)$恰有两个极大值点在$[0, \\dfrac{\\pi}{3}]$内, 则$\\omega$的取值范围是\\bracket{20}.\n\\fourch{$(7,13]$}{$[7,13)$}{$(7,10]$}{$[7,10)$}", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期周末卷-高三下学期周末卷04", + "2023届高三-中档题冲刺-中档题冲刺15" ], "genre": "选择题", "ans": "B", @@ -384183,7 +387750,8 @@ "content": "设$a$、$b$、$c$、$p$为实数, 若同时满足不等式$a x^2+b x+c>0$、$b x^2+c x+a>0$与$c x^2+a x+b>0$的全体实数$x$所组成的集合等于$(p,+\\infty)$. 则关于结论: \\textcircled{1} $a$、$b$、$c$至少有一个为$0$; \\textcircled{2} $p=0$. 下列判断中正确的是\\bracket{20}.\n\\fourch{\\textcircled{1}和\\textcircled{2}都正确}{\\textcircled{1}和\\textcircled{2}都错误}{\\textcircled{1}正确, \\textcircled{2}错误}{\\textcircled{1}错误, \\textcircled{2}正确}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-下学期周末卷-高三下学期周末卷04" ], "genre": "选择题", "ans": "A", @@ -384215,7 +387783,9 @@ "content": "已知$\\{a_n\\}$是等差数列, $\\{b_n\\}$是等比数列, 且$b_2=3$, $b_3=9$, $a_1=b_1$, $a_{14}=b_4$.\\\\\n(1) 求$\\{a_n\\}$的通项公式;\\\\\n(2) 设$c_n=a_n+(-1)^n b_n$($n \\in \\mathbf{N}$, $n\\ge 1$), 求数列$\\{c_n\\}$的前$2 n$项和.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-下学期周末卷-高三下学期周末卷04", + "2023届高三-简单题冲刺-简单题冲刺15" ], "genre": "解答题", "ans": "(1) $2n-1$; (2) $4n^2+\\dfrac{9^n}{4}-\\dfrac 14$", @@ -384247,7 +387817,9 @@ "content": "如图所示, 四棱锥$P-ABCD$中, 底面$ABCD$为菱形, 且$PA \\perp$平面$ABCD$, 又棱$PA=AB=2, E$为棱$CD$的中点, $\\angle ABC=60^{\\circ}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [above left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (1,0,{sqrt(3)}) node [below] {$C$} coordinate (C);\n\\draw (C)++(-2,0,0) node [below] {$B$} coordinate (B);\n\\draw ($(C)!0.5!(D)$) node [below right] {$E$} coordinate (E);\n\\draw (A)++(0,2,0) node [above] {$P$} coordinate (P);\n\\draw (P)--(B)--(C)--(D)--cycle(P)--(C);\n\\draw [dashed] (B)--(A)--(D)(A)--(E)(A)--(P);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: 直线$AE \\perp$平面$PAB$;\\\\\n(2) 求直线$AE$与平面$PCD$所成角的正切值.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期周末卷-高三下学期周末卷04", + "2023届高三-简单题冲刺-简单题冲刺15" ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\dfrac{2\\sqrt{3}}3$", @@ -384280,7 +387852,9 @@ "objs": [], "tags": [ "第三单元", - "第二单元" + "第二单元", + "2023届高三-下学期周末卷-高三下学期周末卷04", + "2023届高三-中档题冲刺-中档题冲刺15" ], "genre": "解答题", "ans": "(1) 如设$AE=x$百米, 则$y=4x+8-2\\sqrt{x^2-9}$($3b>0$)的离心率为$\\dfrac{\\sqrt{2}}{2}$, 以其四个顶点为顶点的四边形的面积等于$8 \\sqrt{2}$. 动直线$l_1$、$l_2$都过点$M(0, m)$($0=latex, scale = 0.7]\n\\draw [->] (-3.5,0) -- (3.5,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\path [draw, name path = elli] (0,0) ellipse ({2*sqrt(2)} and 2);\n\\path [name path = PQ] (1.5,-2) -- (1.5,2);\n\\path [name intersections = {of = elli and PQ, by = {P,Q}}];\n\\draw (P) node [above] {$P$} --(Q) node [below] {$Q$};\n\\draw ($(P)!0.25!(Q)$) ++ (-1.5,0) node [left] {$M$} coordinate (M);\n\\path [name path = AP] (M)--($(P)!3!(M)$);\n\\path [name path = BQ] (M)--($(Q)!1.5!(M)$);\n\\path [name intersections = {of = AP and elli, by = A}];\n\\path [name intersections = {of = BQ and elli, by = B}];\n\\draw (A) node [left] {$A$} --(B) node [above] {$B$}(A)--(P)(B)--(Q);\n\\end{tikzpicture}\n\\end{center}\n(1) 求椭圆$C$的标准方程;\\\\\n(2) 若直线$l_1$与$x$轴交于点$N$, 求证: $|NP|=2|NM|$;\\\\\n(3) 求直线$AB$的斜率的最小值, 并求直线$AB$的斜率取最小值时的直线$l_1$的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期周末卷-高三下学期周末卷04" ], "genre": "解答题", "ans": "(1) $\\dfrac{x^2}8+\\dfrac{y^2}4=1$; (2) 证明略; (3) 斜率的最小值为$\\dfrac{\\sqrt{6}}2$, 此时直线$l_1$的方程为$y=\\dfrac{\\sqrt{6}}6x+\\dfrac{2\\sqrt{7}}7$", @@ -384344,7 +387919,8 @@ "content": "已知集合$A$和定义域为$\\mathbf{R}$的函数$y=f(x)$, 若对任意$t \\in A, x \\in \\mathbf{R}$, 都有$f(x+t)-f(x) \\in A$, 则称$f(x)$是关于$A$的同变函数.\\\\\n(1) 当$A=(0,+\\infty)$与$(0,1)$时, 分别判断$f(x)=2^x$是否为关于$A$的同变函数, 并说明理由;\\\\\n(2) 若$f(x)$是关于$\\{2\\}$的同变函数, 且当$x \\in[0,2)$时, $f(x)=\\sqrt{2 x}$, 试求$f(x)$在$[2 k, 2 k+2)$($k \\in \\mathbf{Z}$)上的表达式, 并比较$f(x)$与$x+\\dfrac{1}{2}$的大小;\\\\\n(3) 若$n$为正整数, 且$f(x)$是关于$[2^{-n}, 2^{1-n}]$的同变函数, 求证: $f(x)$既是关于$\\{m \\cdot 2^{-n}\\}$($m \\in \\mathbf{Z}$)的同变函数, 也是关于$[0,+\\infty)$的同变函数.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期周末卷-高三下学期周末卷04" ], "genre": "解答题", "ans": "(1) $f(x)=2^x$是关于$(0,+\\infty)$的同变函数, 不是关于$(0,1)$的同变函数; (2) $f(x)=\\sqrt{2(x-2k)}+2k$, 当$x=2k+\\dfrac 12$($k\\in\\mathbf{Z}$)时, $f(x)=x+\\dfrac 12$, 当$x\\ne 2k+\\dfrac 12$($k\\in\\mathbf{Z}$)时, $f(x)0$, $0.60$, $0.72.5)=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-第二轮复习讲义-18_概率与统计" ], "genre": "填空题", "ans": "$0.14$", @@ -386130,7 +389737,8 @@ "content": "实力相当的甲、乙两人参加乒乓球比赛, 规定$5$局$3$胜制(即$5$局内谁先赢$3$局就算胜出并停止比赛).\\\\\n(1) 试求甲打完$5$局才获胜的概率;\\\\\n(2) 按比赛规则甲获胜的概率.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-第二轮复习讲义-18_概率与统计" ], "genre": "解答题", "ans": "(1) $\\dfrac 3{16}$; (2) $\\dfrac 12$", @@ -386162,7 +389770,8 @@ "content": "某工厂有三个车间生产同一产品, 第一车间的次品率为$0.05$, 第一车间的次品率为$0.03$, 第一车间的次品率为$0.01$, 各车间的产品数量分别为$1500$件、$2000$件、$1500$件, 出厂时, 三个车间的产品完全混合, 现从中任取$1$件产品, 求该产品是次品的概率.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-第二轮复习讲义-18_概率与统计" ], "genre": "解答题", "ans": "$0.03$", @@ -386194,7 +389803,8 @@ "content": "甲、乙两个学校进行体育比赛, 比赛共设三个项目, 每个项目胜方得$10$分, 负方得$0$分, 没有平局. 三个项目比赛结束后, 总得分高的学校获得冠军. 已知甲学校在三个项目中获胜的概率分别为$0.5$、$0.4$、$0.8$, 各项目的比赛结果相互独立.\\\\\n(1) 求甲学校获得冠军的概率;\\\\\n(2) 用$X$表示乙学校的总得分, 求$X$的分布与期望.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-第二轮复习讲义-18_概率与统计" ], "genre": "解答题", "ans": "(1) $0.6$; (2) $X\\sim \\begin{pmatrix}0 & 10 & 20 & 30 \\\\ 0.16 & 0.44 & 0.34 & 0.06\\end{pmatrix}$, $E[X]=13$", @@ -386227,7 +389837,8 @@ "content": "已知随机变量$X$服从二项分布$B(12,0.25)$, 且$E[a X-3]=3$($a \\in \\mathbf{R}$), 则$D[a X-3]=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-第二轮复习讲义-18_概率与统计" ], "genre": "填空题", "ans": "$9$", @@ -386260,7 +389871,8 @@ "content": "有若干个大小与质地均相同的红球和白球. 已知甲袋中有$6$个红球, $4$个白球, 乙袋中有$8$个红球, $6$个白球, 若随机取一个袋子, 再从该袋中随机取一个球, 则该球是红球的概率是\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-第二轮复习讲义-18_概率与统计" ], "genre": "填空题", "ans": "$\\dfrac{41}{70}$", @@ -386293,7 +389905,8 @@ "content": "袋中装有大小与质地均相同的$2$个白球和$3$个黑球.\\\\\n(1) 从中有放回地摸两次, 每次摸$1$个球, 求两球颜色不同的概率;\\\\\n(2) 从中不放回地摸两次, 每次摸$1$个球, 记$X$为摸出两球中白球的个数, 求$X$的期望和方差.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-第二轮复习讲义-18_概率与统计" ], "genre": "解答题", "ans": "(1) $\\dfrac{12}{25}$; (2) $\\dfrac 9{25}$", @@ -386325,7 +389938,8 @@ "content": "哥德巴赫猜想是指``每个大于$2$的偶数都可以表示为两个素数的和'', 例如$10=7+3$, $16=13+3$. 在不超过$32$的所有素数中, 随机选取两个不同的数, 其和等于$32$的概率为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-第二轮复习讲义-18_概率与统计" ], "genre": "填空题", "ans": "$\\dfrac 2{55}$", @@ -386358,7 +389972,8 @@ "content": "袋中有大小与质地均相同的$5$个红球, $4$个白球, 现随机地从中取出一个球, 记录颜色后, 将其放回袋中, 并随之放入$2$个与之颜色相同的球, 再从袋中第二次取出一球, 则第二次取出的是白球的概率为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-第二轮复习讲义-18_概率与统计" ], "genre": "填空题", "ans": "$\\dfrac 49$", @@ -386390,7 +390005,8 @@ "content": "某实验测试的规则如下: 每位学生最多可做$3$次实验, 一旦实验成功, 则停止实验, 否则做完$3$次为止. 设某学生每次实验成功的概率为$p$($01.39$, 则$p$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-第二轮复习讲义-18_概率与统计" ], "genre": "填空题", "ans": "$(0,0.7)$", @@ -386423,7 +390039,8 @@ "content": "已知两个随机变量$X, Y$, 其中$X \\sim B(4, \\dfrac{1}{4})$, $Y \\sim N(\\mu, \\sigma^2)$($\\sigma>0$), 若$E[X]=E[Y]$, 且$P(|Y|<1)=0.4$, 则$P(Y>3)=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-第二轮复习讲义-18_概率与统计" ], "genre": "填空题", "ans": "$0.1$", @@ -386455,7 +390072,8 @@ "content": "某射击小组共有$25$名射手, 其中一级射手$5$人, 二级射手$10$人, 三级射手$10$人, 若一、二、三级射手能通过选拔进入比赛的概率分别是$0.9$, $0.8$, $0.4$, 则从中任选一名射手能通过选拔进入比赛的概率为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-第二轮复习讲义-18_概率与统计" ], "genre": "填空题", "ans": "$0.66$", @@ -386487,7 +390105,8 @@ "content": "在四次独立重复试验中, 事件$A$在每次试验中发生的概率相同, 若事件$A$至少发生一次的概率为$\\dfrac{65}{81}$, 则事件$A$发生次数$X$的期望是\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-第二轮复习讲义-18_概率与统计" ], "genre": "填空题", "ans": "$\\dfrac 43$", @@ -386519,7 +390138,8 @@ "content": "某批零件(个数非常多)的尺寸$X$服从正态分布$N(10, \\sigma^2)$, 且满足$p(X<9)=\\dfrac{1}{6}$, 零件的尺寸与$10$的误差的绝对值不超过$1$即合格, 从这批产品中随机抽取$n$件, 若要保证抽取的合格零件不少于$2$件的概率不低于$0.9$, 则$n$的最小值为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-第二轮复习讲义-18_概率与统计" ], "genre": "填空题", "ans": "$5$", @@ -386551,7 +390171,8 @@ "content": "甲、乙两人各拿两颗骰子做抛掷游戏, 规则如下: 若掷出的点数之和为$3$的倍数, 原掷骰子的人再继续掷; 若掷出的点数之和不是$3$的倍数, 就由对方接着掷. 第一次由甲开始掷, 求第$n$次由甲掷的概率$P_n$(用含$n$的式子表示).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-第二轮复习讲义-18_概率与统计" ], "genre": "解答题", "ans": "$P_n=\\dfrac 12(1+(-\\dfrac 13)^{n-1})$", @@ -386672,7 +390293,8 @@ "content": "已知$\\{a_n\\}$是首项为$a$, 公差为$1$的等差数列, $b_n=\\dfrac{1+a_n}{a_n}$, 若对于任意正整数$n$, 都有$b_n \\geq b_8$成立, 则$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第三轮复习讲义-01_函数方程与不等式" ], "genre": "填空题", "ans": "$(-8,-7)$", @@ -386723,7 +390345,8 @@ "objs": [], "tags": [ "第一单元", - "第二单元" + "第二单元", + "2023届高三-第三轮复习讲义-01_函数方程与不等式" ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{2}}2$", @@ -386929,7 +390552,8 @@ "content": "设数列$\\{a_n\\}$的首项$a_1$为常数, 且$a_1 \\neq \\dfrac{3}{5}$, 又$a_{n+1}=3^n-2 a_n$. 若$\\{a_n\\}$是严格递增数列, 求$a_1$的取值范围.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第三轮复习讲义-01_函数方程与不等式" ], "genre": "解答题", "ans": "$(0,\\dfrac 35)\\cup (\\dfrac 35,1)$", @@ -387046,7 +390670,8 @@ "content": "某校小卖部为研究学生对饮料的喜好情况, 从该校$450$名同学中用随机数法抽取$30$人参加这一项调查. 将这$450$名同学编号为$001,002, \\ldots, 449,450$, 在以下随机数表中从任意一个随机数开始读出三位数组, 假设从第$2$行第$7$列的数字开始, 则第$5$个被抽到的同学的编号为\\blank{50}.\n\\begin{center}\n\\begin{tabular}{llllll}\n16227794 & 39495443 & 54821737 & 93237887 & 35209643 & 84263491 \\\\ 64844217 & 55721754 & 55068331 & 04744767 & 21763350 & 25839212 \\\\ 06766301 & 63785916 & 95556719 & 98105071 & 75128673 & 58074439\n\\end{tabular}\n\\end{center}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-第二轮复习讲义-19_统计" ], "genre": "填空题", "ans": "$447$", @@ -387101,7 +390726,8 @@ "content": "某校课题小组为了研究高一年级学生甲、乙两门学科成绩的线性相关关系, 在高一第二学期期末考试后随机抽取了$5$名同学 (记为$A$、$B$、$C$、$D$、$E$) 的甲、乙两门学科成绩(满分均为$100$分), 如图. 后来发现$D$同学数据记录有误, 那么去掉数据$D(82,88)$后, 下列说法错误的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.25]\n\\draw [->] (80,75) -- (95,75) node [below] {学科甲成绩$x$};\n\\draw [->] (80,75) -- (80,90) node [left] {学科乙成绩$y$};\n\\filldraw (81,78) circle (0.1) node [right] {$A(81,78)$};\n\\filldraw (83,81) circle (0.1) node [right] {$E(83,81)$};\n\\filldraw (87,84) circle (0.1) node [right] {$B(87,84)$};\n\\filldraw (89,87) circle (0.1) node [right] {$C(89,87)$};\n\\filldraw (82,88) circle (0.1) node [right] {$D(82,88)$};\n\\end{tikzpicture}\n\\end{center}\n\\twoch{样本线性相关系数$r$变大}{最小二乘拟合误差变大}{变量$x$、$y$的相关程度变高}{线性相关系数$r$越接近于$1$}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-第二轮复习讲义-19_统计" ], "genre": "选择题", "ans": "B", @@ -387134,7 +390760,8 @@ "content": "甲、乙两城之间的长途客车均由$A$和$B$两家公司运营, 为了解这两家公司长途客车的运行情况, 随机调查了甲、乙两城之间的$500$个班次, 得到下面的$2 \\times 2$列联表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline & 准点班次数 & 未准点班次数 \\\\\n\\hline$A$& 240 & 20 \\\\\n\\hline$B$& 210 & 30 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n根据上表数据, 判断甲、乙两城之间的长途客车准点情况与客车所属公司\\blank{50}. (填写``有关''或``无关'')\\\\\n($\\chi^2=\\dfrac{n(a d-b c)^2}{(a+b)(c+d)(a+c)(b+d)}$, 显著水平取$0.05$, $P(\\chi^2 \\geq 3.841) \\approx 0.05$).", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-第二轮复习讲义-19_统计" ], "genre": "填空题", "ans": "无关", @@ -387167,7 +390794,8 @@ "content": "深入实施科教兴国战略是中华人民伟大复兴的必由之路. $2020$年第七次全国人口普查中(未包括中国香港、澳门特别行政区和台湾省的人口数据), 我国$31$个省级行政区$15$岁及以上男性和女性的文盲人口比重($\\%$)情况, 经统计得到如下的茎叶图(其中$a$是$0$至$9$中的一个数字).\n\\begin{center}\n\\begin{tabular}{cccccccc|c|cccccccccccc}\n&&&&&&&女&&男\\\\\n&&&&&&&&0&4&6&7&7&7&8&9&9&9\\\\\n&&&&&&5&4&1&0&0&0&1&2&3&3&5&5&8&8&9\\\\\n&&9&9&2&1&1&1&2&1&6&7&7&7\\\\\n&&&&&2&0&0&3&3\\\\\n9&9&9&8&4&3&3&0&4&1&7\\\\\n&&&&&&7&0&5\\\\\n&&&&&8&5&2&6&2\\\\\n&&&&&&&6&7\\\\\n&&&&&&5&5&8\\\\\n&&&&&&&1&12\\\\\n&&&&&&&5&13\\\\\n&&&&&&&1&14\\\\\n&&&&&&&&20&4\\\\\n&&&&&&&$a$&36\n\\end{tabular}\n\\end{center}\n(1) 根据茎叶图判断男性样本数据和女性样本数据的离散程度, 并求离散程度较小的样本数据的第$80$百分位数;\\\\\n(2) 若女性样本数据的极差为$35.3(\\%)$, 求该样本数据的的平均数与方差(结果精确到$0.1$);\\\\\n(3) 为了调查今年某地区$15$岁及以上男性和女性文盲人口情况, 研究小组准备采用分层随机抽样方法抽取$5000$人进行调查. 已知该地区$15$岁及以上的男性约有$4.2$百万人, 女性约有$3.8$百万人. 分别求出抽取的男性人数和女性人数.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-第二轮复习讲义-19_统计" ], "genre": "解答题", "ans": "(1) 女性样本离散程度较高, 男性样本数据的第$80$百分位数为$2.7$($\\%$); (2) $a=8$, 平均数约为$6.3$, 方差约为$41.3$; (3) 男性$2625$人, 女性$2375$人", @@ -387200,7 +390828,8 @@ "content": "某研究小组通过部分直辖市的统计年鉴, 整理了各区$15$岁及以上人口中大学专科、 本科和研究生学历的人口比重($\\%$)如下表所示:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline 地区 & \\makecell{专科\\\\学历\\\\比重} & \\makecell{本科\\\\学历\\\\比重} & \\makecell{研究生\\\\学历\\\\比重} & 地区 & \\makecell{专科\\\\学历\\\\比重} & \\makecell{本科\\\\学历\\\\比重} & \\makecell{研究生\\\\学历\\\\比重} & 地区 & \\makecell{专科\\\\学历\\\\比重} & \\makecell{本科\\\\学历\\\\比重} & \\makecell{研究生\\\\学历\\\\比重} \\\\\n\\hline 1 区 & 12.37 & 18.14 & 4.84 & 12 区 & 10.74 & 10.22 & 1.01 & 23 区 & 16.37 & 17.21 & 3.15 \\\\\n\\hline 2 区 & 14.69 & 26.5 & 9.74 & 13 区 & 13.27 & 17.92 & 3.12 & 24 区 & 15.07 & 17.1 & 3.21 \\\\\n\\hline 3 区 & 14.63 & 26.74 & 8.28 & 14 区 & 11.35 & 12.1 & 1.56 & 25 区 & 16.65 & 20.04 & 3.29 \\\\\n\\hline 4 区 & 15.67 & 23.65 & 5.38 & 15 区 & 10.46 & 11.86 & 1.13 & 26 区 & 13.62 & 17.3 & 2.72 \\\\\n\\hline 5 区 & 15.89 & 23.47 & 5.77 & 16 区 & 7.03 & 6.41 & 0.42 & 27 区 & 16.7 & 26.34 & 6.46 \\\\\n\\hline 6 区 & 15.33 & 23.92 & 5.43 & 17 区 & 15.82 & 28.71 & 8.44 & 28 区 & 16.05 & 19.51 & 4.23 \\\\\n\\hline 7 区 & 14.52 & 23.55 & 8.01 & 18 区 & 15.01 & 29.25 & 12.28 & 29 区 & 12.74 & 12.62 & 3.84 \\\\\n\\hline 8 区 & 14.68 & 22.21 & 5.89 & 19 区 & 15.28 & 30.76 & 9.56 & 30 区 & 13.7 & 11.41 & 0.98 \\\\\n\\hline 9 区 & 15.45 & 17.82 & 2.83 & 20 区 & 16.29 & 26.37 & 7.27 & 31 区 & 13 & 11.73 & 1.14 \\\\\n\\hline 10 区 & 13.01 & 14.93 & 3.03 & 21 区 & 16.97 & 28.17 & 8.21 & 32 区 & 12.79 & 12.51 & 1.26 \\\\\n\\hline 11 区 & 13.91 & 19.98 & 6.14 & 22 区 & 13.08 & 33.14 & 17.81 &&&&\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n请从以下几个方面对上表中的数据进行统计分析:\\\\\n(1) 各类学历的频率分布直方图;\\\\\n(2) 各类学历的集中趋势和离散程度;\\\\\n(3) 相关指标的相关性分析.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-第二轮复习讲义-19_统计" ], "genre": "解答题", "ans": "(1) 图略; (2) \\begin{tabular}{|c||c|c||c|c|c|}\n\\hline\n学历 & 均值 & 中位数 & 极差 & 方差 & 标准差 \\\\ \\hline\n专科 & $14.13$ & $14.66$ & $9.94$ & $4.52$ & $2.13$ \\\\ \\hline\n本科 & $20.05$ & $19.75$ & $26.73$ & $45.84$ & $6.77$ \\\\ \\hline\n研究生 & $5.20$ & $4.54$ & $17.39$ & $14.03$ & $3.75$ \\\\ \\hline\n\\end{tabular} (3) 专科与本科的相关系数约为$0.68$, 有较强的正相关性; 专科与研究生的相关系数约为$0.41$, 有较弱的正相关性; 本科与研究生的相关性约为$0.90$, 有很强的正相关性", @@ -387233,7 +390862,8 @@ "content": "某同学要调查一个地区博物馆的展品数量, 有以下说法:\n\\textcircled{1} 可以通过调查获取相关数据;\n\\textcircled{2} 可以通过统计报表获取相关数据;\n\\textcircled{3} 可以通过实验获取相关数据;\n\\textcircled{4} 可以通过互联网获取相关数据. 其中所有正确说法的序号是\\blank{50}.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-第二轮复习讲义-19_统计" ], "genre": "填空题", "ans": "\\textcircled{1}\\textcircled{2}\\textcircled{4}", @@ -387266,7 +390896,8 @@ "content": "分别统计了甲、乙两位同学$16$周的各周课外体育运动时长(单位: $\\text{h}$), 得如下茎叶图: 则下列结论中错误的是\\bracket{20}.\n\\begin{center}\n\\begin{tabular}{cccc|c|cccccccc}\n&&&甲&&乙\\\\\n&&6&1&5&\\\\\n8&5&3&0&6&3\\\\\n7&5&3&2&7&4&6\\\\\n6&4&2&1&8&1&2&2&5&6&6&6&6\\\\\n&&4&2&9&0&2&3&8\\\\\n&&&&10&1\n\\end{tabular}\n\\end{center}\n\\onech{甲同学周课外体育运动时长的样本中位数为$7.4$}{乙同学周课外体育运动时长的样本平均数大于$8$}{甲同学周课外体育运动时长大于$8$的概率的估计值大于$0.4$}{乙同学周课外体育运动时长大于$8$的概率的估计值大于$0.6$}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-第二轮复习讲义-19_统计" ], "genre": "选择题", "ans": "C", @@ -387299,7 +390930,8 @@ "content": "为了解甲、乙两种离子在小鼠体内的残留程度, 进行如下试验: 将$200$只小鼠随机分成$A$、$B$两组, 每组$100$只, 其中给$A$组小鼠服甲离子溶液, 给$B$组小鼠服乙离子溶液, 给每只小鼠服的溶液体积相同、摩尔浓度相同. 经过一段时间后, 用某种科学方法测算出残留在小鼠体内离子的百分比. 根据试验数据分别得到如下直方图:\n记$C$为事件: ``乙离子残留在体内的百分比不低于$5.5$'', 根据直方图得到概率$P(C)$的估计值为$0.70$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,xscale = 0.6, yscale = 8]\n\\draw [->] (0,0) -- (0,0.45) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw [->] (0,0) -- (0.1,0) -- (0.2,-0.02) -- (0.4,0.02) -- (0.6,-0.02) -- (0.8,0) -- (9.5,0) node [below] {百分比};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j/\\k in {6.5/0.05/0.05,5.5/0.1/0.1,1.5/0.15/0.15,4.5/0.2/0.2,3.5/0.3/0.3}\n{\\draw [dashed] (\\i,\\j) -- (0,\\j) node [left] {$\\k$};};\n\\foreach \\i/\\j/\\k in {1.5/0.15/0.15,2.5/0.2/0.2,3.5/0.3/0.3,4.5/0.2/0.2,5.5/0.1/0.1,6.5/0.05/0.05}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (1,0) --++ (0,-\\j);\n};\n\\draw (7.5,0) node [below] {$7.5$};\n\\draw (4.5,-0.1) node {甲离子残留百分比直方图};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,xscale = 0.6, yscale = 8]\n\\draw [->] (0,0) -- (0,0.45) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw [->] (0,0) -- (0.1,0) -- (0.2,-0.02) -- (0.4,0.02) -- (0.6,-0.02) -- (0.8,0) -- (9.5,0) node [below] {百分比};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j/\\k in {2.5/0.05/0.05,3.5/0.1/b,7.5/0.15/0.15,6.5/0.2/0.2,5.5/0.35/a}\n{\\draw [dashed] ({\\i-1},\\j) -- (0,\\j) node [left] {$\\k$};};\n\\foreach \\i/\\j/\\k in {2.5/0.05/0.05,3.5/0.1/0.1,4.5/0.15/0.15,5.5/0.35/0.35,6.5/0.2/0.2,7.5/0.15/0.15}\n{\\draw ({\\i-1},0) node [below] {$\\i$} --++ (0,\\j) --++ (1,0) --++ (0,-\\j);\n};\n\\draw (7.5,0) node [below] {$8.5$};\n\\draw (4.5,-0.1) node {乙离子残留百分比直方图};\n\\end{tikzpicture}\n\\end{center}\n(1) 求乙离子残留百分比直方图中$a$、$b$的值;\\\\\n(2) 分别估计甲、乙离子残留百分比的平均值(同一组中的数据用该组区间的中点值为代表).", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-第二轮复习讲义-19_统计" ], "genre": "解答题", "ans": "(1) $a=0.3$, $b=0.1$; (2) 甲离子残留百分比的平均值约为$4.05$($\\%$), 乙离子残留百分比的平均值约为$6$($\\%$)", @@ -387332,7 +390964,8 @@ "content": "通过随机抽样, 我们获得某种产品加工前含水率与加工后含水率的一组测试数据, 如下表所示.\n\\begin{center}\n\\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|}\n\\hline 加工前含水率 ($\\%$) & 16.7 & 18.2 & 18.2 & 17.9 & 17.4 & 16.6 & 17.2 & 17.7 & 15.7 & 17.1 \\\\\n\\hline 加工后含水率($\\%$)& 17.1 & 18.4 & 18.6 & 18.5 & 18.2 & 17.1 & 18.0 & 18.2 & 16.0 & 17.5 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 请绘制上述数据的散点图, 并依据散点图观察两组数据的相关性;\\\\\n(2) 计算产品加工前含水率与加工后含水率之间的相关系数, 并判断两个变量的相关程度;\\\\\n(3) 依据表中给出的某种产品加工前含水率与加工后含水率的一组测试数据, 试预测当产品加工前含水率为$19 \\%$时, 加工后含水率的数值.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-第二轮复习讲义-19_统计" ], "genre": "解答题", "ans": "(1) 图略, 有明显的正相关关系; (2) $r\\approx 0.97$, 相关性明显, 为正相关; (3) 拟合直线为$y=1.0173x+0.1909$, 当产品加工前含水率为$19\\%$时, 加工后含水率估计为$19.5\\%$", @@ -387387,7 +391020,8 @@ "content": "甲、乙两城市某月初连续$7$天的日均气温数据如图所示, 则在这$7$天中, 有以下说法:\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.4]\n\\foreach \\i in {1,2,...,7} {\n\\draw [gray] (\\i,0) -- (\\i,7) (0,\\i) -- (7,\\i); \n\\draw (\\i,0.2) -- (\\i,0) node [below] {$\\i$};\n\\draw (0.2,\\i) -- (0,\\i) node [left] {$\\i$};};\n\\draw [->] (0,0) -- (8,0) node [below] {日期};\n\\draw [->] (0,0) -- (0,8) node [left] {气温$^\\circ\\text{C}$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1,5) -- (2,3) -- (3,6) -- (4,3) -- (5,7) -- (6,5) -- (7,6);\n\\draw [dashed] (1,5) -- (2,4) -- (3,6) -- (4,5) -- (5,5) -- (6,4) -- (7,6);\n\\draw (7.5,5.5) -- (9.5,5.5) node [right] {甲};\n\\draw [dashed] (7.5,4) -- (9.5,4) node [right] {乙};\n\\end{tikzpicture}\n\\end{center}\n\\textcircled{1} 甲城市日均气温的平均数与中位数相等;\\\\\n\\textcircled{2} 甲城市的日均气温比乙城市的日均气温稳定;\\\\\n\\textcircled{3} 乙城市日均气温的极差为$3^{\\circ} \\text{C}$;\\\\\n\\textcircled{4} 乙城市日均气温的众数为$5^{\\circ} \\text{C}$.\n其中所有正确说法的序号是\\blank{50}.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-第二轮复习讲义-19_统计" ], "genre": "填空题", "ans": "\\textcircled{1}\\textcircled{4}", @@ -387420,7 +391054,8 @@ "content": "某校抽取$100$名学生测身高, 其中身高最大值为$186 \\text{cm}$, 最小值为$154 \\text{cm}$, 根据身高数据绘制频率组距分布直方图, 组距为$5$, 且第一组下限为$153.5$, 则组数为\\blank{50}.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-第二轮复习讲义-19_统计" ], "genre": "填空题", "ans": "$7$", @@ -387453,7 +391088,8 @@ "content": "翠冠梨是盛产于江南夏季的一种丰水蜜梨, 是盛夏解渴消暑的时令佳品. 某产区标准化的果园每亩梨树$42$株. 为调查该产区$8000$亩翠冠梨的产量, 在收获期从绿港村$600$亩梨园中随机抽取了$10$棵梨树, 测得其产量 (单位: $\\text{kg}$) 分别为$72$、$75$、$84$、$94$、$60$、$78$、$99$、$78$、$91$、$90$. 预估该产区翠冠梨的总产量为\\blank{50}吨.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-第二轮复习讲义-19_统计" ], "genre": "填空题", "ans": "$27.6\\times 10^3$", @@ -387486,7 +391122,8 @@ "content": "某国家足球队$26$名球员的年龄分布茎叶图如图所示:\n\\begin{center}\n\\begin{tabular}{c|ccccccccccccccccc}\n1 & 8 & 9 \\\\\n2 & 1 & 2 & 3 & 3 & 4 & 5 & 5 & 5 & 6 & 6 & 7 & 8 & 8 & 8 & 9 & 9 & 9 \\\\ \n3 & 0 & 1 & 2 & 2 & 2 & 3 & 4 \n\\end{tabular}\n\\end{center}\n该国家足球队$25$岁的球员共有\\blank{50}位, 该国家足球队球员年龄的第$75$百分位数为\\blank{50}.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-第二轮复习讲义-19_统计" ], "genre": "填空题", "ans": "$3$; $30$", @@ -387519,7 +391156,8 @@ "content": "某地经过多年的环境治理, 已将荒山改造成了绿水青山. 为估计一林区某种树木的总材积量, 随机选取了$10$棵这种树木, 测量每棵树的根部横截面积 (单位: $\\text{m}^2$) 和材积量 (单位: $\\text{m}^3$), 得到如下数据:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline 样本号$i$& 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 总和 \\\\\\hline\n\\makecell{根部横截\\\\面积$x_i$} & 0.04 & 0.06 & 0.04 & 0.08 & 0.08 & 0.05 & 0.05 & 0.07 & 0.07 & 0.06 & 0.6 \\\\\\hline\n材积量$y_i$ & 0.25 & 0.40 & 0.22 & 0.54 & 0.51 & 0.34 & 0.36 & 0.46 & 0.42 & 0.40 & 3.9 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 估计该林区这种树木平均一棵的根部横截面积与平均一棵的材积量;\\\\\n(2) 求该林区这种树木的根部横截面积与材积量的样本相关系数(精确到$0.01$);\\\\\n(3) 现测量了该林区所有这种树木的根部横截面积, 并得到所有这种树木的根部横截面积总和为$186 \\text{m}^2$. 已知树木的材积量与其根部横截面积近似成正比. 利用以上数据给出该林区这种树木的总材积量的估计值.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-第二轮复习讲义-19_统计" ], "genre": "解答题", "ans": "(1) 横截面积的平均值约为$0.06\\text{m}^2$; 材积量的平均值约为$0.39\\text{m}^3$; (2) $r\\approx 0.97$; (3) 比例系数的估计值$\\hat{k}=\\dfrac{\\displaystyle\\sum_{i=1}^{10}x_iy_i}{\\displaystyle\\sum_{i=1}^{10}x_i^2}$, 约为$1210.9\\text{m}^3$", @@ -387553,7 +391191,8 @@ "objs": [], "tags": [ "第八单元", - "第九单元" + "第九单元", + "2023届高三-第二轮复习讲义-19_统计" ], "genre": "解答题", "ans": "(1) $\\chi^2=24$, 认为有关; (2) 证明略; (3) $R$的估计值为$6$", @@ -387587,7 +391226,8 @@ "objs": [], "tags": [ "第八单元", - "第九单元" + "第九单元", + "2023届高三-第二轮复习讲义-19_统计" ], "genre": "解答题", "ans": "(1) $X\\sim \\begin{pmatrix} 155 & 165 & 175 & 185 & 195 & 205 \\\\ 0.22 & 0.27 & 0.25 & 0.15 & 0.1 & 0.01\\end{pmatrix}$, $E[X]=171.7$($\\text{cm}$); (2) $0.0312$; (3) $27.25$", @@ -389354,7 +392994,8 @@ "content": "已知$\\triangle ABC$, 角$A$、$B$、$C$所对的边长分别记作$a$、$b$、$c$. 若$a=1$, $b=2$, 且三角形面积$S=\\dfrac{\\sqrt{15}}{4}$, 则$c=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第三轮复习讲义-03_分类讨论" ], "genre": "填空题", "ans": "$2$或$\\sqrt{6}$", @@ -389537,7 +393178,8 @@ "content": "设$a \\in \\mathbf{R}$, 求函数$y=x^3-a x$的单调增区间.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第三轮复习讲义-03_分类讨论" ], "genre": "解答题", "ans": "当$a>0$时, 单调增区间为$(-\\infty,-\\dfrac{\\sqrt{3a}}3]$和$[\\dfrac{\\sqrt{3a}}3,+\\infty)$; 当$a\\le 0$时, 单调增区间为$(-\\infty,+\\infty)$", @@ -389612,7 +393254,8 @@ "objs": [], "tags": [ "第一单元", - "第二单元" + "第二单元", + "2023届高三-第三轮复习讲义-03_分类讨论" ], "genre": "填空题", "ans": "$(-\\infty,-2]\\cup [\\dfrac 12,1)\\cup (1,+\\infty)$", @@ -389774,7 +393417,8 @@ "content": "函数$y=\\sin (2 x+\\dfrac{\\pi}{3})$的最小正周期$T=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期测验卷-高三下学期测验05" ], "genre": "填空题", "ans": "$\\pi$", @@ -389809,7 +393453,8 @@ "content": "设$\\mathrm{i}$为虚数单位, 若复数$z=\\dfrac{1+2 \\mathrm{i}}{\\mathrm{i}}$, 则$z$的实部与虚部的和为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-下学期测验卷-高三下学期测验05" ], "genre": "填空题", "ans": "$1$", @@ -389844,7 +393489,8 @@ "content": "设向量$\\overrightarrow {a}$、$\\overrightarrow {b}$满足$|\\overrightarrow {a}|=2$, $\\overrightarrow {a} \\cdot \\overrightarrow {b}=1$, 则$\\overrightarrow {a} \\cdot(2 \\overrightarrow {a}+\\overrightarrow {b})=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-下学期测验卷-高三下学期测验05" ], "genre": "填空题", "ans": "$9$", @@ -389879,7 +393525,8 @@ "content": "在$(1+2 x)^5$的二项展开式中, $x^3$项的系数是\\blank{50}(结果用数值表示)", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-下学期测验卷-高三下学期测验05" ], "genre": "填空题", "ans": "$80$", @@ -389914,7 +393561,8 @@ "content": "若双曲线$x^2-\\dfrac{y^2}{m}=1$的离心率$e=2$, 则实数$m=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期测验卷-高三下学期测验05" ], "genre": "填空题", "ans": "$3$", @@ -389949,7 +393597,8 @@ "content": "已知事件$A$与事件$B$相互独立, 如果$P(A)=0.5$, $P(B)=0.4$, 那么$P(A \\cap \\overline {B})=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-下学期测验卷-高三下学期测验05" ], "genre": "填空题", "ans": "$0.3$", @@ -389984,7 +393633,8 @@ "content": "已知一个圆锥的底面半径为$1 \\text{cm}$, 侧面积为$2 \\pi \\text{cm}^2$, 则该圆锥的体积为\\blank{50}$\\text{cm}^3$.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期测验卷-高三下学期测验05" ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{3}}3\\pi$", @@ -390019,7 +393669,8 @@ "content": "已知$x$、$y>0$且$x+2 y=1$, 则$\\dfrac{1}{x}+\\dfrac{2}{y}$的最小值为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-下学期测验卷-高三下学期测验05" ], "genre": "填空题", "ans": "$9$", @@ -390054,7 +393705,8 @@ "content": "如图所示的茎叶图记录了甲、乙两组各$5$名工人某日的产量数据. 若这两组数据的中位数相等, 且平均值也相等, 则$x+y=$\\blank{50}.\n\\begin{center}\n\\begin{tabular}{cc|c|ccc}\n\\multicolumn{2}{c|}{甲组} & & \\multicolumn{3}{c}{乙组}\\\\ \\hline\n& 6 & 5 & 9 \\\\ \n2 & 5 & 6 & 1 & 7 & $y$ \\\\\n$x$ & 4 & 7 & 8\n\\end{tabular}\n\\end{center}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-下学期测验卷-高三下学期测验05" ], "genre": "填空题", "ans": "$8$", @@ -390089,7 +393741,8 @@ "content": "对于两个均不等于$1$的正数$m$、$n$, 定义: $m * n=\\begin{cases}\\log _m n,& m \\geq n, \\\\ \\log _n m, & m=latex]\n\\draw (0,0) node [above] {$A$} coordinate (A) arc (120:180:2) node [below left] {$B$} coordinate (B) arc (240:300:2) node [below right] {$C$} coordinate (C) arc (0:60:2);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-下学期测验卷-高三下学期测验05" ], "genre": "填空题", "ans": "$10-4\\sqrt{7}$", @@ -390194,7 +393849,8 @@ "content": "在下列条件下, 能确定一个平面的是\\bracket{20}.\n\\twoch{空间的任意三点}{空间的任意一条直线和任意一点}{空间的任意两条直线}{梯形的两条腰所在的直线}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期测验卷-高三下学期测验05" ], "genre": "选择题", "ans": "D", @@ -390229,7 +393885,8 @@ "content": "已知集合$A=\\{x|| x-1 |>2\\}$,$B=\\{x | x^2+p x+q \\leq 0\\}$, 若$A \\cup B=\\mathbf{R}$, 且$A \\cap B=[-2,-1)$, 则$p$、$q$的值分别为\\bracket{20}.\n\\fourch{$-1$、$-6$}{$1$、$-6$}{$3$、$2$}{$-3$、$2$}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-下学期测验卷-高三下学期测验05" ], "genre": "选择题", "ans": "A", @@ -390264,7 +393921,8 @@ "content": "已知函数$f(x)=3^x-(\\dfrac{1}{3})^x+2$, 若$f(a^2)+f(a-2)>4$, 则实数$a$的取值范围是\\bracket{20}.\n\\fourch{$(-\\infty, 1)$}{$(-\\infty,-2) \\cup(1,+\\infty)$}{$(-2,1)$}{$(-1,2)$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期测验卷-高三下学期测验05" ], "genre": "选择题", "ans": "B", @@ -390299,7 +393957,8 @@ "content": "数学家们在探寻自然对数底$\\mathrm{e} \\approx 2.71828$与圆周率$\\pi$之间的联系时, 发现了以下公式($\\mathrm{i}$为虚数单位):\\\\\n(I) $\\mathrm{e}^x=1+\\dfrac{x}{1 !}+\\dfrac{x^2}{2 !}+\\dfrac{x^3}{3 !}+\\cdots+\\dfrac{x^n}{n !}+\\cdots$;\\\\\n(II) $\\sin x=\\dfrac{x}{1 !}-\\dfrac{x^3}{3 !}+\\dfrac{x^5}{5 !}-\\dfrac{x^7}{7 !}+\\cdots+(-1)^{n-1} \\dfrac{x^{2 n-1}}{(2 n-1) !}+\\cdots$;\\\\\n(III) $\\cos x=1-\\dfrac{x^2}{2 !}+\\dfrac{x^4}{4 !}-\\dfrac{x^6}{6 !}+\\cdots+(-1)^{n-1} \\dfrac{x^{2 n-2}}{(2 n-2) !}+\\cdots$.\\\\\n上述公式中, $x \\in \\mathbf{C}$, $n \\in \\mathbf{N}$, $n\\ge 1$, 据此判断, 当$x\\in \\mathbf{C}$时, 以下命题\n\\textcircled{1} $\\mathrm{e}^{\\mathrm{i}x}=\\cos x+\\mathrm{i} \\sin x$; \n\\textcircled{2} $\\mathrm{e}^{\\mathrm{i}x}=\\sin x+\\mathrm{i} \\cos x$; \n\\textcircled{3} $\\mathrm{e}^{\\mathrm{i} \\pi}+1=0$; \n\\textcircled{4} $\\mathrm{e}^{\\mathrm{i} \\pi}+\\mathrm{i}=0$; \n\\textcircled{5} $|\\mathrm{e}^{\\mathrm{i}x}+\\mathrm{e}^{-\\mathrm{i}x}| \\leq 2$中, 正确的个数是\\bracket{20}.\n\\fourch{$1$个}{$2$个}{$3$个}{$4$个}", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-下学期测验卷-高三下学期测验05" ], "genre": "选择题", "ans": "B", @@ -390335,7 +393994,8 @@ "content": "锐角$\\triangle ABC$中, 角$A$、$B$、$C$的对边分别为$a$、$b$、$c$. 已知$\\sin ^2B+\\sin ^2C=\\sin ^2A+\\sin B \\sin C$.\\\\\n(1) 求$A$;\\\\\n(2) 若$a=3$, 求$b+c$的最大值.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期测验卷-高三下学期测验05" ], "genre": "解答题", "ans": "(1) $\\dfrac\\pi 3$; (2) 最大值为$6$", @@ -390370,7 +394030,8 @@ "content": "如图, 在多面体$EFG-ABCD$中, 四边形$ABCD$、$CFGD$、$ADGE$均是边长为$1$的正方形, 点$H$在棱$EF$上.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$D$} coordinate (D);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,2) node [below] {$A$} coordinate (A);\n\\draw (2,0,2) node [below] {$B$} coordinate (B);\n\\draw (A) ++ (0,2,0) node [left] {$E$} coordinate (E);\n\\draw (D) ++ (0,2,0) node [above] {$G$} coordinate (G);\n\\draw (C) ++ (0,2,0) node [right] {$F$} coordinate (F);\n\\draw ($(E)!0.6!(F)$) node [below left] {$H$} coordinate (H);\n\\draw (A)--(B)--(C)--(F)--(G)--(E)--cycle;\n\\draw (B)--(F)--(E)--cycle(B)--(H)--(G);\n\\draw [dashed] (G)--(D)--(C)(D)--(B)(D)--(A);\n\\end{tikzpicture}\n\\end{center}\n(1) 求该几何体的体积;\\\\\n(2) 证明: 存在点$H$, 使得$DH \\perp BF$;\\\\\n(3) 求$BD$与平面$BEF$所成角的大小.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期测验卷-高三下学期测验05" ], "genre": "解答题", "ans": "(1) $\\dfrac 56$; (2) 证明略($H$和点$E$重合); (3) $\\arcsin\\dfrac{\\sqrt{6}}3$", @@ -390405,7 +394066,8 @@ "content": "高斯是德国著名的数学家, 近代数学奠基者之一, 享有``数学王子''的称号. 以他的名字定义的函数称为高斯函数$f(x)=[x]$, 其中$[x]$表示不超过$x$的最大整数. 已知数列$\\{a_n\\}$满足$a_1=2$, $a_2=6$, $a_{n+2}+5 a_n=6 a_{n+1}$, 若$b_n=[\\log _5 a_{n+1}]$, $S_n$为数列$\\{\\dfrac{1000}{b_n b_{n+1}}\\}$的前$n$项和.\\\\\n(1) 证明: 数列$\\{a_{n+1}-a_n\\}$是等比数列, 并求数列$\\{a_n\\}$的通项公式;\\\\\n(2) 求$[S_{2023}]$的值.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-下学期测验卷-高三下学期测验05" ], "genre": "解答题", "ans": "(1) 证明略; $a_n=5^{n-1}+1$; (2) $999$", @@ -390440,7 +394102,8 @@ "content": "已知椭圆$E: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$), 依次连接椭圆$E$的四个顶点构成的四边形面积为$4 \\sqrt{3}$.\\\\\n(1) 若$a=2$, 求椭圆$E$的标准方程;\\\\\n(2) 以椭圆$E$的右顶点为焦点、以原点为顶点的抛物线$G$, 若$G$上动点$M$到点$H(10,0)$的最短距离为$4 \\sqrt{6}$, 求$a$的值;\\\\\n(3) 当$a=2$时, 设点$F$为椭圆$E$的右焦点, $A(-2,0)$, 直线$l$交$E$于$P$、$Q$(均不与点$A$重合) 两点, 直线$l$、$AP$、$AQ$的斜率分别为$k$、$k_1$、$k_2$, 若$k k_1+k k_2+3=0$, 求$\\triangle FPQ$的周长.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期测验卷-高三下学期测验05" ], "genre": "解答题", "ans": "(1) $\\dfrac{x^2}4+\\dfrac{y^2}3=1$; (2) $4$; (3) $8$", @@ -390475,7 +394138,8 @@ "content": "已知函数$f(x)=a x^3-b x^2+c$, 其中实数$a>0$, $b \\in \\mathbf{R}$, $c \\in \\mathbf{R}$.\\\\\n(1) $b=3 a$时, 求函数$y=f(x)$的极值点;\\\\\n(2) $a=1$时, $x^2 \\ln x \\geq f(x)-2 x-c$在$[3,4]$上恒成立, 求$b$的取值范围;\\\\\n(3) 证明: 当$b=3 a$, 且$5 a=latex, xscale = 0.08, yscale = 50]\n\\draw [->] (80,0) -- (155,0) node [below] {成绩/分};\n\\draw [->] (80,0) -- (80,0.055) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\foreach \\i/\\j in {90/0.03,100/0.04,110/0.015,120/0.01,130/0.005}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (10,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {90/0.03,100/0.04,110/0.015,120/0.01,130/0.005}\n{\\draw [dashed] (\\i,\\j) -- (80,\\j) node [left] {$\\k$};};\n\\draw (140,0) node [below] {$140$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-下学期测验卷-2023届杨浦区二模" ], "genre": "填空题", "ans": "$107$", @@ -391233,7 +394905,8 @@ "content": "$\\triangle ABC$内角$A, B, C$的对边是$a, b, c$, 若$a=3$, $b=\\sqrt{6}$, $\\angle A=\\dfrac{\\pi}{3}$, 则$\\angle B=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期测验卷-2023届杨浦区二模" ], "genre": "填空题", "ans": "$\\dfrac{\\pi}4$", @@ -391268,7 +394941,8 @@ "content": "$F_1, F_2$分别是双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$的左右焦点, 过$F_1$的直线$l$与双曲线的左右两支分别交于$A, B$两点. 若$\\triangle ABF_2$为等边三角形, 则双曲线的离心率为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期测验卷-2023届杨浦区二模" ], "genre": "填空题", "ans": "$\\sqrt{7}$", @@ -391303,7 +394977,8 @@ "content": "若存在实数$\\varphi$, 使函数$f(x)=\\cos (\\omega x+\\varphi)-\\dfrac{1}{2}$($\\omega>0$)在$x \\in[\\pi, 3 \\pi]$上有且仅有$2$个零点, 则$\\omega$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期测验卷-2023届杨浦区二模" ], "genre": "填空题", "ans": "$[\\dfrac 13,\\dfrac 53)$", @@ -391338,7 +395013,8 @@ "content": "已知非零平面向量$\\overrightarrow {a}, \\overrightarrow {b}, \\overrightarrow {c}$, 满足$|\\overrightarrow {a}|=5$, $2|\\overrightarrow {b}|=|\\overrightarrow {c}|$, 且$(\\overrightarrow {b}-\\overrightarrow {a}) \\cdot(\\overrightarrow {c}-\\overrightarrow {a})=0$, 则$|\\overrightarrow {b}|$的最小值是\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-下学期测验卷-2023届杨浦区二模" ], "genre": "填空题", "ans": "$\\sqrt{5}$", @@ -391373,7 +395049,8 @@ "content": "已知$a, b \\in \\mathbf{R}$, 则``$a>b$''是``$a^3>b^3$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-下学期测验卷-2023届杨浦区二模" ], "genre": "选择题", "ans": "C", @@ -391408,7 +395085,8 @@ "content": "对成对数据$(x_1, y_1)$、$(x_2, y_2)$、$\\cdots \\cdots$、$(x_n, y_n)$用最小二乘法求回归方程是为了使\\bracket{20}.\n\\fourch{$\\displaystyle\\sum_{i=1}^n(y_i-\\overline {y})=0$}{$\\displaystyle\\sum_{i=1}^n(y_i-\\hat{y}_i)=0$}{$\\displaystyle\\sum_{i=1}^n(y_i-\\hat{y}_i)$最小}{$\\displaystyle\\sum_{i=1}^n(y_i-\\hat{y}_i)^2$最小}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-下学期测验卷-2023届杨浦区二模" ], "genre": "选择题", "ans": "D", @@ -391443,7 +395121,8 @@ "content": "下列函数中, 既是偶函数, 又在区间$(-\\infty, 0)$上严格递减的是\\bracket{20}.\n\\fourch{$y=2^{|x|}$}{$y=\\ln (-x)$}{$y=x^{-\\frac{2}{3}}$}{$y=-\\sqrt{x^2}$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期测验卷-2023届杨浦区二模" ], "genre": "选择题", "ans": "A", @@ -391478,7 +395157,8 @@ "content": "如图, 一个由四根细铁杆$PA$、$PB$、$PC$、$PD$组成的支架($PA$、$PB$、$PC$、$PD$按照逆时针排布), 若$\\angle APB=\\angle BPC=\\angle CPD=\\angle DPA=\\dfrac{\\pi}{3}$, 一个半径为$1$的球恰好放在支架上与四根细铁杆均有接触, 则球心$O$到点$P$的距离是\n\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.5]\n\\path [name path = circ, draw] (0,0) circle (1);\n\\filldraw (0,0) node [right] {$O$} circle (0.03);\n\\draw [dashed] (0,0) ellipse (1 and 0.25);\n\\draw [dashed] (0,{-sqrt(2)/2}) ellipse ({sqrt(2)/2} and {sqrt(2)/8});\n\\draw (0,{-sqrt(2)}) node [below] {$P$} coordinate (P);\n\\draw (P) --++ (-1.5,1.5) node [above] {$A$};\n\\draw (P) --++ (-0.6,2.5) node [above] {$B$};\n\\draw (P) --++ (1.5,1.5) node [above] {$C$};\n\\path [name path = PD] (P) --++ (0.6,2.5) node [above] {$D$} coordinate (D);\n\\path [name intersections = {of = PD and circ, by = {M,N}}];\n\\draw (M)--(D)(P)--(N);\n\\draw [dashed] (M)--(N);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\sqrt{3}$}{$\\sqrt{2}$}{2}{$\\dfrac{3}{2}$}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期测验卷-2023届杨浦区二模" ], "genre": "选择题", "ans": "B", @@ -391513,7 +395193,8 @@ "content": "已知一个随机变量$X$的分布为: $\\begin{pmatrix}6 & 7 & 8 & 9 & 10 \\\\ 0.1 & a & 0.2 & 0.3 & b\\end{pmatrix}$.\\\\\n(1) 已知$E[X]=\\dfrac{43}{5}$, 求$a, b$的值;\\\\\n(2) 记事件$A: X$为偶数; 事件$B: X \\leq 8$. 已知$P(A)=\\dfrac{1}{2}$, 求$P(B)$, $P(A \\cap B)$, 并判断$A, B$是否相互独立?", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-下学期测验卷-2023届杨浦区二模" ], "genre": "解答题", "ans": "(1) $a=0.1$, $b=0.3$; (2) $P(B)=0.5$, $P(A\\cap B)=0.3$, $P(A)P(B)=0.25\\ne 0.3$, 不相互独立", @@ -391548,7 +395229,8 @@ "content": "四边形$ABCD$是边长为$1$的正方形, $AC$与$BD$交于$O$点, $PA \\perp$平面$ABCD$, 且二面角$P-BC-A$的大小为$45^{\\circ}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 2.5]\n\\draw (0,0,0) node [above right] {$A$} coordinate (A);\n\\draw (1,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,0,1) node [left] {$B$} coordinate (B);\n\\draw (1,0,1) node [right] {$C$} coordinate (C);\n\\draw (0.5,0,0.5) node [below] {$O$} coordinate (O);\n\\draw (0,1,0) node [above] {$P$} coordinate (P);\n\\draw (P)--(B)--(C)--(D)--cycle(P)--(C);\n\\draw [dashed] (P)--(A)--(C)(B)--(D)(B)--(A)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求点$A$到平面$PBD$的距离;\\\\\n(2) 求直线$AC$与平面$PCD$所成的角.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期测验卷-2023届杨浦区二模" ], "genre": "解答题", "ans": "(1) $\\dfrac{\\sqrt{3}}3$; (2) $\\dfrac\\pi 6$", @@ -391583,7 +395265,8 @@ "content": "如图, 某国家森林公园的一区域$OAB$为人工湖, 其中射线$OA, OB$为公园边界. 已知$OA \\perp OB$, 以点$O$为坐标原点, 以$OB$为$x$轴正方向, 建立平面直角坐标系(单位: 千米), 曲线$AB$的轨迹方程为: $y=-x^2+4$($0 \\leq x \\leq 2$). 计划修一条与湖边$AB$相切于点$P$的直路\n$l$(宽度不计), 直路$l$与公园边界交于点$C, D$两点, 把人工湖围成一片景区$\\triangle OCD$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw [->] (0,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,5.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {0.5,1,1.5,2,2.5,3,3.5}{\\draw (\\i,0.1)--(\\i,0) node [below] {\\tiny$\\i$};};\n\\foreach \\i in {0.5,1,1.5,2,2.5,3,3.5,4,4.5,5}{\\draw (0.1,\\i)--(0,\\i) node [left] {\\tiny$\\i$};};\n\\draw [domain = 0:2, samples = 100] plot (\\x,{4-\\x*\\x});\n\\filldraw (0,4) circle (0.05) node [below right] {$A$};\n\\filldraw (2,0) circle (0.05) node [above right] {$B$};\n\\filldraw (0.8,3.36) circle (0.05) node [above right] {$P$};\n\\filldraw (0,4.64) circle (0.05) node [above right] {$C$} coordinate (C);\n\\filldraw (2.9,0) circle (0.05) node [above right] {$D$} coordinate (D);\n\\draw ($(C)!-0.1!(D)$) node [left] {$l$}-- ($(C)!1.1!(D)$);\n\\end{tikzpicture}\n\\end{center}\n(1) 若$P$点坐标为$(1,3)$, 计算直路$CD$的长度; (精确到$0.1$千米)\\\\\n(2) 若$P$为曲线$AB$(不含端点)上的任意一点, 求景区$\\triangle OCD$面积的最小值. (精确到$0.1$平方千米)", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期测验卷-2023届杨浦区二模" ], "genre": "解答题", "ans": "(1) 约$5.6$千米($\\sqrt{31.25}$); (2) 约$6.2$平方千米($\\dfrac{32\\sqrt{3}}{9}$)", @@ -391618,7 +395301,8 @@ "content": "已知椭圆$C: \\dfrac{x^2}{4 a^2}+\\dfrac{y^2}{3 a^2}=1$($a>0$)的右焦点为$F$, 直线$l: x+y-4=0$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2.5,0) -- (2.5,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\path [name path = elli, draw] (0,0) ellipse (2 and {sqrt(3)});\n\\filldraw (1,0) circle (0.03) node [below] {$F$} coordinate (F);\n\\path [name path = l,draw] (-1.5,2.1) -- (2.1,-1.5);\n\\path [name intersections = {of = l and elli, by = {A,B}}];\n\\draw (A) node [above] {$A$};\n\\draw (B) node [below] {$B$};\n\\end{tikzpicture}\n\\end{center}\n(1) 若$F$到直线$l$的距离为$2 \\sqrt{2}$, 求$a$;\\\\\n(2) 若直线$l$与椭圆$C$交于$A, B$两点, 且$\\triangle ABO$的面积为$\\dfrac{48}{7}$, 求$a$;\\\\\n(3) 若椭圆$C$上存在点$P$, 过$P$作直线$l$的垂线$l_1$, 垂足为$H$, 满足直线$l_1$和直线$FH$的夹角为$\\dfrac{\\pi}{4}$, 求$a$的取值范围.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期测验卷-2023届杨浦区二模" ], "genre": "解答题", "ans": "(1) $a=8$; (2) $a=2$; (3) $[\\dfrac{4\\sqrt{7}-8}3,4)\\cup (4,+\\infty)$", @@ -391653,7 +395337,8 @@ "content": "已知数列$\\{a_n\\}$是由正实数组成的无穷数列, 满足$a_1=3$, $a_2=7$, $a_n=|a_{n+1}-a_{n+2}|$($n$为正整数).\\\\\n(1) 写出数列$\\{a_n\\}$前$4$项的所有可能取法;\\\\\n(2) 判断: 在满足条件的所有数列$\\{a_n\\}$中, 是否可能存在正整数$k$, 满足$a_k=1$, 并说明理由;\\\\\n(3) $c_n$为数列$\\{a_n\\}$的前$n$项中不同取值的个数, 求$c_{100}$的最小值.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-下学期测验卷-2023届杨浦区二模" ], "genre": "解答题", "ans": "(1) $3,7,10,17$或$3,7,10,3$或$3,7,4,11$; (2) 不存在, 证明略; (3) $51$, 证明略", @@ -391688,7 +395373,8 @@ "content": "若实数$x$满足$|x-2|<1$, 则$x$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-下学期测验卷-高三下学期测验07" ], "genre": "填空题", "ans": "$(1,3)$", @@ -391723,7 +395409,8 @@ "content": "设复数$z$满足$(1+\\mathrm{i}) z=2 \\mathrm{i}$($\\mathrm{i}$为虚数单位), 则$z=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-下学期测验卷-高三下学期测验07" ], "genre": "填空题", "ans": "$1+\\mathrm{i}$", @@ -391758,7 +395445,8 @@ "content": "已知集合$A=\\{1,2\\}$, $B=\\{a, a^2+1\\}$, 若$A \\cap B=\\{1\\}$, 则实数$a=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-下学期测验卷-高三下学期测验07" ], "genre": "填空题", "ans": "$0$", @@ -391793,7 +395481,8 @@ "content": "已知函数$y=\\sin (2 \\omega x+\\varphi)$($\\omega>0$)的最小正周期为$1$, 则$\\omega=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期测验卷-高三下学期测验07" ], "genre": "填空题", "ans": "$\\pi$", @@ -391828,7 +395517,8 @@ "content": "已知正实数$a$、$b$满足$a b=1$, 则$a+4 b$的最小值等于\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-下学期测验卷-高三下学期测验07" ], "genre": "填空题", "ans": "$4$", @@ -391863,7 +395553,8 @@ "content": "在$(x^4+\\dfrac{1}{x})^{10}$的展开式中常数项是\\blank{50}.(用数字作答)", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-下学期测验卷-高三下学期测验07" ], "genre": "填空题", "ans": "$45$", @@ -391898,7 +395589,8 @@ "content": "以下数据为参加某次数学竞赛的$15$人的成绩 (单位: 分), 分数从低到高依次是: 56、70、72、78、79、80、81、83、84、86、88、90、91、94、98, 则这$15$人成绩的第$80$百分位数是\\blank{50}.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-下学期测验卷-高三下学期测验07" ], "genre": "填空题", "ans": "$90.5$", @@ -391933,7 +395625,8 @@ "content": "某单位为了解用电量$y$度与气温$x^{\\circ} \\text{C}$之间的关系, 随机统计了某$4$天的用电量与当天气温.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline 气温$({ }^{\\circ} \\text{C})$& 14 & 12 & 8 & 6 \\\\\n\\hline 用电量 (度) & 22 & 26 & 34 & 38 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n由表中数据所得回归直线方程为$y=-2 x+\\hat{b}$, 据此预测当气温为$5^{\\circ} \\text{C}$时, 用电量的度数约为\\blank{50}度.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-下学期测验卷-高三下学期测验07" ], "genre": "填空题", "ans": "$40$", @@ -391969,7 +395662,8 @@ "content": "已知抛物线$x^2=2 y$上的两个不同的点$A$、$B$的横坐标恰好是方程$x^2+6 x+4=0$的根, 则直线$AB$的方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期测验卷-高三下学期测验07" ], "genre": "填空题", "ans": "$y=-3x-2$", @@ -392004,7 +395698,8 @@ "content": "在一个十字路口, 每次亮绿灯的时长为$30$秒, 那么, 每次绿灯亮时, 在一条直行道路上能有多少汽车通过? 这个问题涉及车长、车距、车速、堵塞的干扰等多种因素, 不同型号车的车长是不同的, 驾驶员的习惯不同也会使车距、车速不同, 行人和非机动车的干扰因素则复杂且不确定. 面对这些不同和不确定, 需要作出假设, 例如小明发现虽然通过路口的车辆各种各样, 但多数是小轿车, 因此小明给出如下假设: 通过路口的车辆长度都相等, 请写出一个你认为合理的假设\\blank{100}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期测验卷-高三下学期测验07" ], "genre": "填空题", "ans": "如: 等待时, 前后相邻两辆车的车距都相等; 绿灯亮后, 汽车都是在静止状态匀加速启动; 前一辆车启动后, 下一辆车启动的间隔时间相等; 车辆行驶秩序良好, 不会发生堵塞; 等等", @@ -392039,7 +395734,8 @@ "content": "设平面向量$\\overrightarrow {a}$、$\\overrightarrow {b}$、$\\overrightarrow {c}$满足: $|\\overrightarrow {a}|=2$, $|\\overrightarrow {b}|=|\\overrightarrow {c}|$, $|\\overrightarrow {a}-\\overrightarrow {b}|=1$, $\\overrightarrow {b} \\perp \\overrightarrow {c}$, 则$|\\overrightarrow {b}-\\overrightarrow {c}|$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-下学期测验卷-高三下学期测验07" ], "genre": "填空题", "ans": "$[\\sqrt{2},3\\sqrt{2}]$", @@ -392074,7 +395770,8 @@ "content": "若函数$y=\\begin{cases}\\dfrac{x^3}{\\mathrm{e}^x}, & x \\geq 0, \\\\ a x^2, & x<0\\end{cases}$的图像上点$A$与点$B$, 点$C$与点$D$分别关于原点对称且这四点两两均不重合, 除此之外, 不存在函数图像上的其它不重合的两点关于原点对称, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期测验卷-高三下学期测验07" ], "genre": "填空题", "ans": "$(-\\dfrac{1}{\\mathrm{e}},0)$", @@ -392109,7 +395806,8 @@ "content": "下列函数在其定义域上既是严格增函数, 又是奇函数的是\\bracket{20}.\n\\fourch{$f(x)=\\tan x$}{$f(x)=-\\dfrac{1}{x}$}{$f(x)=x-\\cos x$}{$f(x)=\\mathrm{e}^x-\\mathrm{e}^{-x}$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期测验卷-高三下学期测验07" ], "genre": "选择题", "ans": "D", @@ -392144,7 +395842,8 @@ "content": "设两个正态分布$N(\\mu_1, \\sigma_1^2)$($\\sigma_1>0$)和$N(\\mu_2, \\sigma_2^2)$($\\sigma_2>0$)的正态密度函数图像如图所示, 则\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\def\\s{0.3}\n\\def\\m{0.5}\n\\draw [domain = -0.3:1.3, samples = 100] plot (\\x,{1/sqrt(2*pi)/\\s*exp(-(\\x-\\m)*(\\x-\\m)/\\s/\\s/2)});\n\\def\\s{0.2}\n\\def\\m{-0.1}\n\\draw [domain = -1:0.8, samples = 100] plot (\\x,{1/sqrt(2*pi)/\\s*exp(-(\\x-\\m)*(\\x-\\m)/\\s/\\s/2)});\n\\draw (-0.1,2) node [left] {$N(\\mu_1,\\sigma_1^2)$};\n\\draw (0.5,1.5) node [right] {$N(\\mu_2,\\sigma_2^2)$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\mu_1<\\mu_2$, $\\sigma_1<\\sigma_2$}{$\\mu_1<\\mu_2$, $\\sigma_1>\\sigma_2$}{$\\mu_1>\\mu_2$, $\\sigma_1<\\sigma_2$}{$\\mu_1>\\mu_2$, $\\sigma_1>\\sigma_2$}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-下学期测验卷-高三下学期测验07" ], "genre": "选择题", "ans": "A", @@ -392179,7 +395878,8 @@ "content": "《九章算术》中将底面为直角三角形且侧棱垂直于底面的三棱柱称为``堑堵''; 底面为矩形, 一条侧棱垂直于底面的四棱锥称之为``阳马''; 四个面均为直角三角形的四面体称为``鳖臑''. 如图, 在堑堵$ABC-A_1B_1C_1$中, $AC \\perp BC$, 且$AA_1=AB=2$. 下列说法错误的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$B$} coordinate (B);\n\\draw (1.5,0,{sqrt(3)/2}) node [below] {$C$} coordinate (C);\n\\draw (A) ++ (0,2,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,2,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,2,0) node [right] {$C_1$} coordinate (C_1);\n\\draw ($(A_1)!0.5!(B)$) node [above] {$E$} coordinate (E);\n\\draw ($(C)!{2.5/7}!(A_1)$) node [right] {$F$} coordinate (F);\n\\draw (A_1)--(C)(A_1)--(A)--(C)--(B)--(B_1)--cycle(A_1)--(C_1)--(B_1)(C)--(C_1)(A)--(F);\n\\draw [dashed] (A_1)--(B)(A)--(B)(A)--(E)(E)--(F);\n\\end{tikzpicture}\n\\end{center}\n\\onech{四棱锥$B-A_1ACC_1$为``阳马''}{四面体$A_1C_1CB$为``鳖臑''}{四棱锥$B-A_1ACC_1$体积的最大值为$\\dfrac{2}{3}$}{过$A$点作$AE \\perp A_1B$于点$E$, 过$E$点作$EF \\perp A_1B$并交$A_1C$于点$F$, 则$A_1B \\perp$平面$AEF$}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期测验卷-高三下学期测验07" ], "genre": "选择题", "ans": "C", @@ -392214,7 +395914,8 @@ "content": "已知数列$\\{a_n\\}$是各项为正数的等比数列, 公比为$q$, 在$a_1$、$a_2$之间插入$1$个数, 使这$3$个数成等差数列, 记公差为$d_1$, 在$a_2$、$a_3$之间插入$2$个数, 使这$4$个数成等差数列, 公差为$d_2$, $\\cdots$, 在$a_n$、$a_{n+1}$之间插入$n$个数, 使这$n+2$个数成等差数列, 公差为$d_n$, 则\\bracket{20}.\n\\twoch{当$01$时, 数列$\\{d_n\\}$严格增}{当$d_1>d_2$时, 数列$\\{d_n\\}$严格减}{当$d_1=latex]\n\\filldraw (0,0) node [above left] {$O$} coordinate (O) circle (0.03);\n\\filldraw (0,2.3) node [above] {$O_1$} coordinate (O_1) circle (0.03);\n\\draw (O_1) ellipse (1.5 and 0.5);\n\\draw (O) ++ (1.5,0) node [right] {$B$} coordinate (B) --++ (0,2.3) node [right] {$B_1$} coordinate (B_1);\n\\draw (O) ++ (-1.5,0) node [left] {$A$} coordinate (A) --++ (0,2.3) node [left] {$A_1$} coordinate (A_1) -- (B_1);\n\\draw (A) arc (180:360:1.5 and 0.5);\n\\draw [dashed] (A) arc (180:0:1.5 and 0.5);\n\\draw (-60:1.5 and 0.5) node [below] {$P$} coordinate (P);\n\\draw [dashed] (A_1)--(P)(A_1)--(B)--(P)--(A)--(B)(O)--(P);\n\\end{tikzpicture}\n\\end{center}\n(1) 求直线$A_1P$与平面$ABP$所成角的大小;\\\\\n(2) 求点$A$到平面$A_1BP$的距离.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期测验卷-高三下学期测验07" ], "genre": "解答题", "ans": "(1) $\\arctan\\dfrac{\\sqrt{3}}2$; (2) $\\dfrac{6\\sqrt{7}}7$", @@ -392284,7 +395986,8 @@ "content": "在$\\triangle ABC$中, $a$、$b$、$c$分别是内角$A$、$B$、$C$的对边, $\\overrightarrow {m}=(2 a+c, b)$, $\\overrightarrow {n}=(\\cos B, \\cos C)$, $\\overrightarrow {m} \\cdot \\overrightarrow {n}=0$.\\\\\n(1) 求角$B$的大小;\\\\\n(2) 设$f(x)=2 \\cos x \\sin (x+\\dfrac{\\pi}{3})-2 \\sin ^2 x \\sin B+2 \\sin x \\cos x \\cos (A+C)$, 当$x \\in[\\dfrac{\\pi}{6}, \\dfrac{2 \\pi}{3}]$时, 求$f(x)$的最小值及相应的$x$的值.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期测验卷-高三下学期测验07" ], "genre": "解答题", "ans": "(1) $B=\\dfrac{2\\pi}{3}$; (2) 最小值为$-2$, 取到最小值当且仅当$x=\\dfrac{7\\pi}{12}$", @@ -392320,7 +396023,8 @@ "objs": [], "tags": [ "第八单元", - "第九单元" + "第九单元", + "2023届高三-下学期测验卷-高三下学期测验07" ], "genre": "解答题", "ans": "(1) $\\dfrac 12$; (2) 分布列为$\\begin{pmatrix} 0 & 1 & 2 \\\\ \\dfrac 17 & \\dfrac 47 & \\dfrac 27\\end{pmatrix}$, $E[X]=\\dfrac 87$; (3) $3$月$3$日", @@ -392356,7 +396060,8 @@ "content": "已知椭圆$\\Gamma: \\dfrac{x^2}{m^2}+\\dfrac{y^2}{2}=1$($m>0$, $m \\neq \\sqrt{2}$), 点$A$、$B$分别是椭圆$\\Gamma$与$y$轴的交点(点$A$在点$B$的上方), 过点$D(0,1)$且斜率为$k$的直线$l$交椭圆$\\Gamma$于$E$、$G$两点.\\\\\n(1) 若椭圆$\\Gamma$焦点在$x$轴上, 且其离心率是$\\dfrac{\\sqrt{2}}{2}$, 求实数$m$的值;\\\\\n(2) 若$m=k=1$, 求$\\triangle BEG$的面积;\\\\\n(3) 设直线$AE$与直线$y=2$交于点$H$, 证明: $B$、$G$、$H$三点共线.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期测验卷-高三下学期测验07" ], "genre": "解答题", "ans": "(1) $m=2$; (2) $\\dfrac{2\\sqrt{2}+2}3$; (3) 证明略", @@ -392391,7 +396096,8 @@ "content": "已知定义域为区间$D$的函数$y=f(x)$, 其导函数为$y'=f'(x)$, 满足对任意的$x \\in D$都有$|f'(x)|<1$.\\\\\n(1) 若$f(x)=a x+\\ln x$, $x \\in[1,2]$, 求实数$a$的取值范围;\\\\\n(2) 证明: 方程$f(x)-x=0$至多只有一个实根;\\\\\n(3) 若$y=f(x)$, $x \\in \\mathbf{R}$是周期为$2$的周期函数, 证明: 对任意的实数$x_1$、$x_2$, 都有$|f(x_1)-f(x_2)|<1$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期测验卷-高三下学期测验07" ], "genre": "解答题", "ans": "(1) $(-\\dfrac 32,0)$; (2) 证明略; (3) 证明略", @@ -393664,7 +397370,8 @@ "content": "正数列$\\{a_n\\}$的前$n$项和$S_n$满足: $r S_n=a_n a_{n+1}-1$, $a_1=a>0$, 常数$r \\in \\mathbf{N}$.\\\\\n(1) 求证: $a_{n+2}-a_n$是一个定值;\\\\\n(2) 若数列$\\{a_n\\}$是一个周期数列, 求该数列的周期;\\\\\n(3) 若数列$\\{a_n\\}$是一个有理数等差数列, 求$S_n$.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第三轮复习讲义-03_分类讨论" ], "genre": "解答题", "ans": "(1) 定值为$r$, 证明略; (2) 当$a=1$时, 周期为$1$; 当$a\\in (0,1)\\cup (1,+\\infty)$时, 周期为$2$; (3) $S_n=n$($r=0$时)或$S_n=\\dfrac 34n^2+\\dfrac 54n$($r=3$时)", @@ -393715,7 +397422,8 @@ "content": "函数$y=\\cos x+2|\\cos x|$($x \\in[0,2 \\pi]$)的图像与直线$y=k$有且仅有两个不同的公共点, 则实数$k$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第三轮复习讲义-02_数形结合" ], "genre": "填空题", "ans": "$\\{0\\}\\cup (1,3]$", @@ -393789,7 +397497,8 @@ "content": "设有函数$f(x)=a+\\sqrt{-x^2-4 x}$和$g(x)=\\dfrac{4}{3} x+1$, 已知$x \\in[-4,0]$时恒有$f(x) \\leq g(x)$成立, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第三轮复习讲义-02_数形结合" ], "genre": "填空题", "ans": "$(-\\infty,-5]$", @@ -393884,7 +397593,8 @@ "content": "已知函数$f(x)$的定义域是$\\mathbf{R}$, 满足对任意$x\\in \\mathbf{R}$, 都成立$f(x+1)=\\dfrac{1-f(x)}{1+f(x)}$.\\\\\n(1) 证明: $2$是函数$f(x)$的一个周期;\\\\\n(2) 当$x \\in[0,1)$时, $f(x)=x$, 求$f(x)$在$[-1,0)$上的解析式;\\\\\n(3) 设$a>0$, 对于 (2) 中的函数$f(x)$, 关于$x$的方程$f(x)=a x$恰有$100$个根, 求正实数$a$的取值范围.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第三轮复习讲义-02_数形结合" ], "genre": "解答题", "ans": "(1) 证明略; (2) $f(x)=-\\dfrac x{2+x}$; (3) $(\\dfrac{1}{101},\\dfrac{1}{99})$", @@ -393959,7 +397669,8 @@ "objs": [], "tags": [ "第七单元", - "第二单元" + "第二单元", + "2023届高三-第三轮复习讲义-02_数形结合" ], "genre": "选择题", "ans": "B", @@ -394032,7 +397743,8 @@ "content": "若关于$x$的方程$\\dfrac{|x|}{x-3}=k x^2$有四个不同的实数根, 求实数$k$的取值范围.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第三轮复习讲义-02_数形结合" ], "genre": "解答题", "ans": "$(-\\infty,-\\dfrac 49)$", @@ -394127,7 +397839,8 @@ "content": "关于$x$的方程$x^2-1-|x^2-1|+k=0$, 给出下列四个命题, 其中假命题的序号是\\blank{50}.\\\\\n\\textcircled{1} 存在实数$k$, 使得方程恰有$1$个不同的实根;\\\\\n\\textcircled{2} 存在实数$k$, 使得方程恰有$2$个不同的实根;\\\\\n\\textcircled{3} 存在实数$k$, 使得方程恰有$4$个不同的实根;\\\\\n\\textcircled{4} 存在实数$k$, 使得方程有无数个实根.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第三轮复习讲义-06_表达与论证" ], "genre": "填空题", "ans": "\\textcircled{3}", @@ -394222,7 +397935,8 @@ "content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n$, 点$(n, \\dfrac{S_n}{n})$在直线$y=\\dfrac{1}{2} x+\\dfrac{11}{2}$上. 数列$\\{b_n\\}$满足$b_{n+2}-2 b_{n+1}+b_n=0$($n \\in \\mathbf{N}$, $n\\ge 1$)且$b_3=11$, 前$9$项和为$153$.\\\\\n(1) 求数列$\\{a_n\\}$、$\\{b_n\\}$的通项公式;\\\\\n(2) 设$f(n)=\\begin{cases}a_n,& n=2 l-1, \\\\ b_n, &n=2 l\\end{cases}$($l \\in \\mathbf{N}$, $l\\ge 1$), 问是否存在正整数$m$, 使得$f(m+15)=5 f(m)$成立? 若不存在, 请说明理由.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第三轮复习讲义-06_表达与论证" ], "genre": "解答题", "ans": "(1) $a_n=n+5$, $b_n=3n+2$; (2) 存在, $m=11$", @@ -394273,7 +397987,8 @@ "content": "定义函数$y=f(x)$, $x \\in D$($D$为定义域)图像上的点到坐标原点的距离为函数的$y=f(x)$, $x \\in D$的模. 若模存在最大值, 则称之为函数$y=f(x)$, $x \\in D$的长距; 若模存在最小值, 则称之为函数$y=f(x)$, $x \\in D$的短距.\\\\\n(1) 分别判断函数$f_1(x)=\\dfrac{1}{x}$与$f_2(x)=\\sqrt{-x^2-4 x+5}$是否存在长距与短距;\\\\\n(2) 对于任意$x \\in[1,2]$, 是否存在实数$a$, 使得函数$f(x)=\\sqrt{2 x|x-a|}$的短距不小于$2$且长距不大于$4$? 若存在, 求出$a$的取值范围; 若不存在, 说明理由.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第三轮复习讲义-06_表达与论证" ], "genre": "解答题", "ans": "(1) $f_1(x)$存在短距, 不存在长距, $f_2(x)$存在短距, 也存在长距; (2) 存在满足条件的实数$a$, $a$的范围为$[-1,-\\dfrac 12]\\cup [\\dfrac 52,5]$", @@ -394324,7 +398039,8 @@ "content": "设$g(x)$是定义在$\\mathbf{R}$上, 以$1$为周期的函数. 若函数$f(x)=x+g(x)$在区间$[3,4]$上的值域为$[-2,5]$. 则\\\\\n(1) $f(x)$在区间$[4,5]$上的值域为\\blank{50};\\\\\n(2) $f(x)$在区间$[-8,8]$上的值域为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第三轮复习讲义-04_转化与化归" ], "genre": "填空题", "ans": "(1) $[-1,6]$; (2) $[-13,9]$", @@ -394352,7 +398068,8 @@ "content": "在平面直角坐标系$xOy$中, 对于直线$l: a x+b y+c=0$和点$P_1(x_1, y_1)$, $P_2(x_2, y_2)$, 记$\\eta=(a x_1+b y_1+c)(a x_2+b y_2+c)$. 若$\\eta<0$, 则称点$P_1, P_2$被直线$l$分割. 若曲线$C$与直线$l$没有公共点, 且曲线$C$上存在点$P_1, P_2$被直线$l$分割, 则称直线$l$为曲线$C$的一条分割线.\\\\\n(1) 设点$A$的坐标为$(2,2)$, 直线$l: x+y-1=0$.\\\\\n(i) 求证: 点$A$、$B(-1,0)$被直线$l$分割;\\\\\n(ii) 求证: 存在一点$C$, 点$C$、$A$不被直线$l$分割;\\\\\n(2) 设曲线$C_1: x y=1$是否存在分割线? 若存在, 写出一条分割线, 并证明其为曲线$C_1$的分割线; 若不存在, 说明理由;\\\\\n(3) 求证: 曲线$C_2: |x y|=1$恰存在两条分割线.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第三轮复习讲义-04_转化与化归" ], "genre": "解答题", "ans": "(1) (i) $A$与$B$被直线$l$分割; (ii) $A$与$C$不被直线$l$分割; (2) 如$x+y=0$, 理由略; (3) 证明略", @@ -394379,7 +398096,8 @@ "content": "若实数$x, y, m$满足$\\lg (x-m)>\\lg (y-m)$, 则称$x$比$y$``更真''于$m$. 若$4 x-x^2-1$比$x-1$``更真''于$1$, 则$x$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第三轮复习讲义-05_新情境中的问题" ], "genre": "填空题", "ans": "$(2,3)$", @@ -394431,7 +398149,8 @@ "content": "已知平面上的线段$l$及点$P$. 任取$l$上一点$Q$, 线段$PQ$长度的最小值称为点$P$到线段$l$的距离, 记作$d(P, l)$. 设线段$l_1: y=2$($-2 \\leq x \\leq 2)$.\\\\\n(1) 分别求点$P_1(0,3)$、$P_2(4,3)$到线段$l_1$的距离$d(P_1, l_1)$、$d(P_2, l_1)$;\\\\\n(2) 求点的集合$D=\\{P | d(P, l_1)=1\\}$;\\\\\n(3) 设$A(0,2)$, 写出$\\Omega=\\{P|d(P, l_1)=| PA |\\}$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第三轮复习讲义-05_新情境中的问题" ], "genre": "解答题", "ans": "(1) $d(P_1,l_1)=1$, $d(P_2,l_1)=\\sqrt{5}$;\\\\\n(2) $D=\\left\\{(x,y)|\\begin{cases}x\\ge 2, \\\\ (x-2)^2+(y-2)^2=1,\\end{cases}\\text{ 或 } \\begin{cases} x\\le =2, \\\\ (x+2)^2+(y-2)^2=1, \\end{cases}\\text{ 或 }\\begin{cases} -2b_n$, $n=1,2, \\cdots$, 其中$\\overrightarrow {j}$为方向与$y$轴正方向相同的单位向量, 则称$\\{A_n\\}$为$T$点列.\\\\\n(1) 判断$A_1(1,1)$, $A_2(2, \\dfrac{1}{2})$, $A_3(3, \\dfrac{1}{3})$, $\\cdots$, $A_n(n, \\dfrac{1}{n})$, $\\cdots$是否为$T$点列, 并说明理由;\\\\\n(2) 若$\\{A_n\\}$为$T$点列, 且点$A_2$在点$A_1$的右上方. 任取其中连续三点$A_k, A_{k+1}, A_{k+2}$, 判断$\\triangle A_k A_{k+1} A_{k+2}$的形状(锐角三角形、直角三角形、钝角三角形), 并予以证明;\\\\\n(3) 若$\\{A_n\\}$为$T$点列, 从小到大排列的四个不同的正整数$m,n,p,q$满足$m+q=n+p$, 求证: $\\overrightarrow{A_n A_q} \\cdot \\overrightarrow {j}>\\overrightarrow{A_m A_p} \\cdot \\overrightarrow {j}$.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第三轮复习讲义-05_新情境中的问题" ], "genre": "解答题", "ans": "(1) 是$T$点列, 理由略; (2) 是钝角三角形, 证明略; (3) 证明略", @@ -394488,7 +398208,8 @@ "content": "已知$a, b, c$均为正实数, 求证: $\\dfrac{1}{a}+\\dfrac{1}{b}+\\dfrac{1}{c} \\geq \\dfrac{1}{\\sqrt{a b}}+\\dfrac{1}{\\sqrt{b c}}+\\dfrac{1}{\\sqrt{c a}}$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第三轮复习讲义-06_表达与论证" ], "genre": "解答题", "ans": "证明略", @@ -394517,7 +398238,8 @@ "content": "已知函数$f(x)$、$g(x)$的定义域均为$\\mathbf{R}$, $x_1$、$x_2$是在$\\mathbf{R}$上任意选取的两个实数. 若$f(x)$是奇函数且不等式$|f(x_1)+f(x_2)| \\geq|g(x_1)+g(x_2)|$恒成立, 问$g(x)$是否也为奇函数? 证明你的结论.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第三轮复习讲义-06_表达与论证" ], "genre": "解答题", "ans": "是奇函数, 证明略", @@ -394545,7 +398267,8 @@ "content": "已知函数$f(x)$、$g(x)$的定义域均为$\\mathbf{R}$, $x_1$、$x_2$是在$\\mathbf{R}$上任意选取的两个实数. 若$f(x)$是周期函数且不等式$|f(x_1)-f(x_2)| \\geq|g(x_1)-g(x_2)|$恒成立, 问$g(x)$是否为周期函数? 证明你的结论.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第三轮复习讲义-06_表达与论证" ], "genre": "解答题", "ans": "是周期函数, 证明略", @@ -394574,7 +398297,8 @@ "content": "若$a, b\\in \\mathbf{R}$, 满足$2 a+b+2 \\leq 0$.\\\\\n(1) 求证: 关于$t$的方程$t^2+a t+b-2=0$有实数解;\\\\\n(2) 求证: 关于$x$的方程$x^2+\\dfrac{1}{x^2}+a(x+\\dfrac{1}{x})+b=0$有正实数解.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第三轮复习讲义-06_表达与论证" ], "genre": "解答题", "ans": "(1) 证明略; (2) 证明略", @@ -394603,7 +398327,8 @@ "content": "对于无穷数列$\\{a_n\\}$, 若存在正常数$M$, 使得对任意正整数$n$, 总成立$|a_n|0$且$a \\neq b$, 由$a, b, \\dfrac{a+b}{2}, \\sqrt{a b}$按一定顺序构成的数列\\bracket{20}.\n\\twoch{可能是等差数列, 也可能是等比数列}{可能是等差数列, 但不可能是等比数列}{不可能是等差数列, 但可能是等比数列}{不可能是等差数列, 也不可能是等比数列}", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第三轮复习讲义-03_分类讨论" ], "genre": "选择题", "ans": "B", @@ -394879,7 +398608,8 @@ "content": "我们称点$P$到图形$C$上任意一点距离的最小值为点$P$到图形$C$的距离, 那么平面内到定圆$C$的距离与到定点$A$的距离相等的点的轨迹不可能是\\bracket{20}.\n\\fourch{圆}{椭圆}{双曲线的一支}{直线}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第三轮复习讲义-03_分类讨论" ], "genre": "选择题", "ans": "D", @@ -394908,7 +398638,8 @@ "content": "在集合$U=\\{a, b, c, d\\}$的子集中选出$4$个不同的子集, 需同时满足以下两个条件: \\textcircled{1} $\\varnothing, U$都要选出; \\textcircled{2} 对选出的任意两个子集$A$和$B$, 必有$A \\subseteq B$或$B \\subseteq A$. 那么共有\\blank{50}种不同的选法.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-第三轮复习讲义-03_分类讨论" ], "genre": "填空题", "ans": "$36$", @@ -395196,7 +398927,8 @@ "content": "若关于$x$的不等式$|2^x-m|-\\dfrac{1}{2^x}<0$在区间$[0,1]$内恒成立, 则实数$m$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第三轮复习讲义-04_转化与化归" ], "genre": "填空题", "ans": "$(\\dfrac 32,2)$", @@ -395224,7 +398956,8 @@ "content": "在$xOy$平面上, 将曲线$\\dfrac{x^2}{9}-\\dfrac{y^2}{16}=1$($x>0$)、直线$y=\\dfrac{4}{3} x$、直线$y=0$和直线$y=4$围成的封闭图形记为$D$, 记$D$绕$y$轴旋转一周所得的几何体为$\\Omega$, 利用祖暅原理得出$\\Omega$的体积为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第三轮复习讲义-04_转化与化归" ], "genre": "填空题", "ans": "$36\\pi$", @@ -395273,7 +399006,8 @@ "content": "已知曲线$C_1: \\dfrac{x^2}{4}-\\dfrac{y^2}{3}=1$和曲线$C_2: |y|=|x|+2$, 求证: 不存在过点$(1,0)$的直线, 同时与$C_1, C_2$都有公共点.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第三轮复习讲义-05_新情境中的问题" ], "genre": "解答题", "ans": "证明略", @@ -395301,7 +399035,8 @@ "content": "若关于$x$的方程$|x^2-2 a x-3|=4 a$有且仅有两个实数解, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第三轮复习讲义-01_函数方程与不等式" ], "genre": "填空题", "ans": "$\\{0\\}\\cup (1,3)$", @@ -395374,7 +399109,8 @@ "content": "若关于$x$的方程$\\cos 2 x-2 \\cos x+m=0$在$[0, \\pi]$内有且仅有两个解, 则实数$m$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第三轮复习讲义-01_函数方程与不等式" ], "genre": "填空题", "ans": "$[1,\\dfrac 32)$", @@ -395447,7 +399183,8 @@ "content": "已知函数$f(x)=\\dfrac{4 x+a}{x^2+1}$的值域为$[-1, b]$, 求实数$a, b$的值.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第三轮复习讲义-01_函数方程与不等式" ], "genre": "解答题", "ans": "$a=3$, $b=4$", @@ -395519,7 +399256,8 @@ "content": "已知点$P(0,1)$, 椭圆$\\dfrac{x^2}{4}+y^2=m$($m>1$)上两点$A, B$满足$\\overrightarrow{AP}=2 \\overrightarrow{PB}$, 则当$m=$\\blank{50}时, 点$B$横坐标的绝对值最大.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第三轮复习讲义-01_函数方程与不等式" ], "genre": "填空题", "ans": "$5$", @@ -395548,7 +399286,8 @@ "content": "已知实数$a, b$满足$\\dfrac{8}{(a+1)^3}+\\dfrac{10}{a+1}-b^3-5 b>0$, 对于命题: \\textcircled{1} 若$a, b$两数中有一个大于$1$, 则另一个必小于$1$; \\textcircled{2} 若$a \\in(-2,-1)$, 则$a>b$, 下列判断正确的是\\bracket{20}.\n\\twoch{\\textcircled{1}和\\textcircled{2}均为真命题}{\\textcircled{1}和\\textcircled{2}均为假命题}{\\textcircled{1}为真命题, \n\\textcircled{2}为假命题}{\\textcircled{1}为假命题, \\textcircled{2}为真命题}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第三轮复习讲义-01_函数方程与不等式" ], "genre": "选择题", "ans": "A", @@ -395643,7 +399382,8 @@ "content": "已知函数$f(x)=\\dfrac{(1-t) x-t^2}{x}$($t \\in \\mathbf{R}$)的定义域为$D$, 若存在区间$[a, b] \\subseteq D$, 使得当$x \\in[a, b]$时, $f(x)$的取值范围也是$[a, b]$, 则当$t$变化时, $b-a$的最大值为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第三轮复习讲义-01_函数方程与不等式" ], "genre": "填空题", "ans": "$\\dfrac 23\\sqrt{3}$", @@ -395672,7 +399412,8 @@ "objs": [], "tags": [ "第二单元", - "第四单元" + "第四单元", + "2023届高三-第三轮复习讲义-01_函数方程与不等式" ], "genre": "解答题", "ans": "(1) 证明略; (2) 证明略", @@ -395745,7 +399486,8 @@ "objs": [], "tags": [ "第一单元", - "第七单元" + "第七单元", + "2023届高三-第三轮复习讲义-02_数形结合" ], "genre": "填空题", "ans": "$(-\\infty,8-4\\sqrt{2}]$", @@ -395775,7 +399517,8 @@ "objs": [], "tags": [ "第六单元", - "第二单元" + "第二单元", + "2023届高三-第三轮复习讲义-02_数形结合" ], "genre": "选择题", "ans": "D", @@ -395826,7 +399569,8 @@ "content": "如图, 在平面直角坐标系$xOy$中, 菱形$ABCD$的边长为$4$, 且$|OB|=|OD|=6$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-1,0) -- (7,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (5,2) node [right] {$C$} coordinate (C);\n\\path [draw, name path = arc] (2,-2) arc (-90:90:2);\n\\path [name path = OC] (O) -- (C);\n\\path [name intersections = {of = OC and arc, by = A}];\n\\draw (A) node [below left] {$A$} coordinate (A);\n\\path [name path = circle1] (A) circle (4);\n\\path [name path = circle2] (C) circle (4);\n\\path [name intersections = {of = circle1 and circle2, by = {B,D}}];\n\\draw (A)--(B)--(C)--(D)--cycle(B) node [above] {$B$} --(O)--(D) node [below] {$D$};\n\\draw [dashed] (2,-2)--(2,2);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $|OA| \\cdot|OC|$为定值;\\\\\n(2) 当点$A$在半圆$(x-2)^2+y^2=4$($2 \\leq x \\leq 4$)上运动时, 求点$C$的轨迹方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第三轮复习讲义-02_数形结合" ], "genre": "解答题", "ans": "(1) 定值为$20$, 证明略; (2) $x=5$且$-5\\le y\\le 5$", @@ -395855,7 +399599,8 @@ "content": "已知圆$O$的半径为$1$, $PA$、$PB$为该圆的两条切线, $A$、$B$为两个切点, 则$\\overrightarrow{PA} \\cdot \\overrightarrow{PB}$的最小值为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第三轮复习讲义-02_数形结合" ], "genre": "填空题", "ans": "$2\\sqrt{2}-3$", @@ -395884,7 +399629,8 @@ "content": "已知$f(x)$是定义在$\\mathbf{R}$上的偶函数, 且$f(x)$在$[0,+\\infty)$上是严格增函数, 如果对于任意$x \\in[1,2]$, 不等式$f(a x+1) \\leq f(x-3)$恒成立, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第三轮复习讲义-02_数形结合" ], "genre": "填空题", "ans": "$[-1,0]$", @@ -395914,7 +399660,8 @@ "content": "已知$f(x)$是定义在$[1,+\\infty)$上的函数, 且$f(x)=\\begin{cases}1-|2 x-3|,& 1 \\leq x<2, \\\\ \\dfrac{1}{2} f(\\dfrac{1}{2} x), & x \\geq 2,\\end{cases}$则函数$y=2 x f(x)-3$在区间$(1,2022)$上的零点个数为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第三轮复习讲义-02_数形结合" ], "genre": "填空题", "ans": "$11$", @@ -395943,7 +399690,8 @@ "content": "设$\\theta \\in(0, \\dfrac{\\pi}{2}]$, 函数$f(x)=5 \\sin (2 x-\\theta)$, $x \\in[0,5 \\pi]$, 若函数$F(x)=f(x)-3$的所有零点依次记为$x_1, x_2, x_3, \\cdots, x_n$, 且$x_1=latex]\n\\draw [->] (-0.5,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\filldraw [domain = -2:2, pattern = horizontal lines] plot ({sqrt(\\x*\\x+1)},\\x) -- (2,2) -- (0,0) -- (2,-2) -- cycle;\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第三轮复习讲义-05_新情境中的问题" ], "genre": "填空题", "ans": "$4\\pi$", @@ -396339,7 +400089,8 @@ "content": "数学中有许多形状优美的曲线, 如星形线. 设定大圆的半径是小圆半径的$4$倍, 让小圆在大圆内部沿着大圆的圆周滚动, 小圆圆周上的任一点形成的轨迹即为星形线. 已知小圆圆周上某一点$P$形成的星形线$C$的方程为$x^{\\frac{2}{3}}+y^{\\frac{2}{3}}=a^{\\frac{2}{3}}$($a>0$), 有如下结论:\\\\\n\\textcircled{1} 曲线$D: |x|+|y|=a$的周长大于星形线$C$的周长;\\\\\n\\textcircled{2} 星形线$C$上任意两点间距离的最大值为$2 a$;\\\\\n\\textcircled{3} 星形线$C$与圆$E: x^2+y^2=\\dfrac{a^2}{4}$有且仅有$4$个公共点.\\\\\n其中所有正确结论的序号是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第三轮复习讲义-04_转化与化归" ], "genre": "填空题", "ans": "\\textcircled{2}\\textcircled{3}", @@ -396366,7 +400117,8 @@ "content": "对于函数$f(x)$和实数$t$, 若在其定义域内存在实数$x_0$, 使得$f(x_0+t)=f(x_0)+f(t)$成立, 则称$f(x)$是``$t$跃点''函数, 并称$x_0$是函数$f(x)$的``$t$跃点''.\\\\\n(1) 求证: 函数$f(x)=2^x+3 x^2$, $x \\in[0,3]$是``$2$跃点''函数;\\\\\n(2) 若函数$g(x)=x^3-\\dfrac{a}{2} x+3 a$, $x \\in(-3,+\\infty)$是``$1$跃点''函数, 求实数$a$的取值范围.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第三轮复习讲义-05_新情境中的问题" ], "genre": "解答题", "ans": "(1) 证明略; (2) $[-\\dfrac 14,+\\infty)$", @@ -396395,7 +400147,8 @@ "objs": [], "tags": [ "第一单元", - "第五单元" + "第五单元", + "2023届高三-第三轮复习讲义-05_新情境中的问题" ], "genre": "选择题", "ans": "A", @@ -396423,7 +400176,8 @@ "content": "``阿基米德多面体''也称为半正多面体, 是由边数不全相同的正多边形为面围成的多面体, 它体现了数学的对称美. 如图, 将正方体沿交于一顶点的三条棱的中点截去一个三棱锥, 共可截去八个三棱锥, 得到八个面为正三角形、六个面为正方形的``阿基米德多面体'', 则异面直线$AB$与$CD$所成角的大小是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = {(220:0.5cm)}]\n\\draw (-1,0,0) coordinate (A);\n\\draw (0,0,1) node [below] {$C$} coordinate (B);\n\\draw (1,0,0) coordinate (C);\n\\draw (0,0,-1) coordinate (D);\n\\draw (-1,1,1) coordinate (E);\n\\draw (1,1,1) node [left] {$D$} coordinate (F);\n\\draw (1,1,-1) coordinate (G);\n\\draw (-1,1,-1) coordinate (H);\n\\draw (-1,2,0) node [left] {$A$} coordinate (M);\n\\draw (0,2,1) coordinate (N);\n\\draw (1,2,0) coordinate (P);\n\\draw (0,2,-1) node [above] {$B$} coordinate (Q);\n\\draw (A)--(B)--(E)--cycle (B)--(C)--(F)--cycle (C)--(G) (G)--(P) (F)--(P)--(N) --cycle (M)--(N)--(E)--cycle (M)--(Q)--(P);\n\\draw [dashed] (A)--(D)--(H)--cycle (C)--(D)--(G) (M)--(H)--(Q)--(G);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$30^{\\circ}$}{$45^{\\circ}$}{$60^{\\circ}$}{$120^{\\circ}$}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第三轮复习讲义-05_新情境中的问题" ], "genre": "选择题", "ans": "C", @@ -396475,7 +400229,8 @@ "content": "在平面直角坐标系$xOy$中, 已知任意角$\\theta$的顶点与原点$O$重合, 始边与$x$轴的正半轴重合. 若角$\\theta$的终边经过点$P(x_0, y_0)$且$|OP|=r$($r>0$), 定义$\\mathrm{sicos} \\theta=\\dfrac{x_0+y_0}{r}$, 称``函数$y=\\mathrm{sicos} x$, $x \\in \\mathbf{R}$''为``正余弦函数''. 对于正余弦函数, 有如下结论:\\\\\n\\textcircled{1} 该函数是偶函数;\\\\\n\\textcircled{2} 该函数图像的一个对称中心是点$(\\dfrac{3 \\pi}{4}, 0)$;\\\\\n\\textcircled{3} 该函数的单调递减区间是$[2 k \\pi-\\dfrac{3 \\pi}{4}, 2 k \\pi+\\dfrac{\\pi}{4}]$($k \\in \\mathbf{Z}$);\\\\\n\\textcircled{4} 该函数的图像与直线$y=\\dfrac{3}{2}$没有公共点.\\\\\n其中所有正确结论的序号是\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第三轮复习讲义-05_新情境中的问题" ], "genre": "填空题", "ans": "\\textcircled{2}\\textcircled{4}", @@ -396503,7 +400258,8 @@ "content": "如果有穷数列$a_1, a_2, a_3, \\cdots, a_n$($n$为正整数) 满足条件$a_1=a_n$, $a_2=a_{n-1}$, $\\cdots$, $a_n=a_1$, 即$a_i=a_{n-i+1}$($i=1,2, \\cdots, n$), 我们称其为``对称数列''. 例如, 由组合数组成的数列$\\mathrm{C}_m^0, \\mathrm{C}_m^1, \\cdots, \\mathrm{C}_m^m$就是``对称数列''.\\\\\n(1) 设$\\{b_n\\}$是项数为$7$的``对称数列'', 其中$b_1, b_2, b_3, b_4$是等差数列, 且$b_1=2$, $b_4=11$. 依次写出$\\{b_n\\}$的每一项;\\\\\n(2) 设$\\{c_n\\}$是项数为$2 k-1$的``对称数列''($k$是大于$1$的正整数), 其中$c_k, c_{k+1}, \\cdots, c_{2 k-1}$是首项为$50$, 公差为$-4$的等差数列. 记$\\{c_n\\}$各项的和为$S_{2 k-1}$. 当$k$为何值时, $S_{2 k-1}$取得最大值? 并求出$S_{2 k-1}$的最大值;\\\\\n(3) 对于确定的$m$($m$是大于$1$的正整数), 写出所有项数不超过$2 m$的``对称数列'', 使得$1,2,2^2, \\cdots, 2^{m-1}$依次是该数列中连续的项; 当$m>1500$时, 求其中一个``对称数列''前$2022$项的和$S_{2022}$.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第三轮复习讲义-05_新情境中的问题" ], "genre": "解答题", "ans": "(1) $2,5,8,11,8,5,2$;\\\\\n(2) $k=13$时$S_{2k-1}$取到最大, 最大值为$626$;\\\\\n(3) 共有四种满足要求的数列:\\\\\n第一种: $1,2,\\cdots,2^{m-2},2^{m-1},2^{m-1},2^{m-2},\\cdots,2,1$($2m$项), $S_{2022}=\\begin{cases}2^{m+1}-2^{2m-2022}-1, & 1500-2)=0.9$, 则$P(X>2)=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-下学期周末卷-高三下学期周末卷10" ], "genre": "填空题", "ans": "$0.1$", @@ -396686,7 +400447,8 @@ "content": "双曲线$C: \\dfrac{x^2}{2}-\\dfrac{y^2}{4}=1$的右焦点$F$到其一条渐近线的距离为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期周末卷-高三下学期周末卷10" ], "genre": "填空题", "ans": "$2$", @@ -396717,7 +400479,8 @@ "content": "投掷一颗骰子, 记事件$A=\\{2,4,5\\}$, $B=\\{1,2,4,6\\}$, 则$P(A | B)=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-下学期周末卷-高三下学期周末卷10" ], "genre": "填空题", "ans": "$\\dfrac 12$", @@ -396748,7 +400511,8 @@ "content": "在$\\triangle ABC$中, 角$A$、$B$、$C$的对边分别记为$a$、$b$、$c$, 若$5 a \\cos A=b \\cos C+c \\cos B$, 则$\\sin 2A=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期周末卷-高三下学期周末卷10" ], "genre": "填空题", "ans": "$\\dfrac{4\\sqrt{6}}{25}$", @@ -396779,7 +400543,8 @@ "content": "函数$y=\\log _2 x+\\dfrac{1}{\\log _4(2 x)}$在区间$(\\dfrac{1}{2},+\\infty)$上的最小值为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期周末卷-高三下学期周末卷10" ], "genre": "填空题", "ans": "$2\\sqrt{2}-1$", @@ -396810,7 +400575,8 @@ "content": "已知$\\omega \\in \\mathbf{R}$, $\\omega>0$, 函数$y=\\sqrt{3} \\sin \\omega x-\\cos \\omega x$在区间$[0,2]$上有唯一的最小值$-2$, 则$\\omega$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期周末卷-高三下学期周末卷10" ], "genre": "填空题", "ans": "$[\\dfrac{5\\pi}6,\\dfrac{11\\pi}6)$", @@ -396841,7 +400607,8 @@ "content": "已知边长为$2$的菱形$ABCD$中, $\\angle A=120^{\\circ}$, $P$、$Q$是菱形内切圆上的两个动点, 且$PQ \\perp BD$, 则$\\overrightarrow{AP} \\cdot \\overrightarrow{CQ}$的最大值是\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-下学期周末卷-高三下学期周末卷10" ], "genre": "填空题", "ans": "$\\dfrac 14$", @@ -396872,7 +400639,8 @@ "content": "已知$01$''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充要条件}{既不充分也不必要条件}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-下学期周末卷-高三下学期周末卷10" ], "genre": "选择题", "ans": "B", @@ -396934,7 +400703,8 @@ "content": "某种产品的广告支出$x$与销售额$y$(单位: 万元) 之间有下表关系, $y$与$x$的线性回归方程为$y=10.5 x+5.4$, 当广告支出$6$万元时, 随机误差的效应即离差(真实值减去预报值)为\\bracket{20}.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline$x$& 2 & 4 & 5 & 6 & 8 \\\\\n\\hline$y$& 30 & 40 & 60 & 70 & 80 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\fourch{$1.6$}{$8.4$}{$11.6$}{$7.4$}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-下学期周末卷-高三下学期周末卷10" ], "genre": "选择题", "ans": "A", @@ -396965,7 +400735,8 @@ "content": "在空间中, 下列命题为真命题的是\\bracket{20}.\n\\onech{若两条直线垂直于第三条直线, 则这两条直线互相平行}{若两个平面分别平行于两条互相垂直的直线, 则这两个平面互相垂直}{若两个平面垂直, 则过一个平面内一点垂直于交线的直线与另外一个平面垂直}{若一条直线平行于一个平面, 另一条直线与这个平面垂直, 则这两条直线互相垂直}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期周末卷-高三下学期周末卷10" ], "genre": "选择题", "ans": "D", @@ -396996,7 +400767,8 @@ "content": "已知函数$y=f(x)$($x \\in \\mathbf{R}$), 其导函数为$y=f'(x)$, 有以下两个命题: \\textcircled{1} 若$y=f'(x)$为偶函数, 则$y=f(x)$为奇函数; \\textcircled{2} 若$y=f'(x)$为周期函数, 则$y=f(x)$也为周期函数. 那么\\bracket{20}.\n\\twoch{\\textcircled{1}是真命题, \\textcircled{2}是假命题}{\\textcircled{1}是假命题, \\textcircled{2}是真命题}{\\textcircled{1}、\\textcircled{2}都是真命题}{\\textcircled{1}、\\textcircled{2}都是假命题}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期周末卷-高三下学期周末卷10" ], "genre": "选择题", "ans": "D", @@ -397027,7 +400799,8 @@ "content": "已知数列$\\{a_n\\}$是首项为$9$, 公比为$\\dfrac{1}{3}$的等比数列.\\\\\n(1) 求$\\dfrac{1}{a_1}+\\dfrac{1}{a_2}+\\dfrac{1}{a_3}+\\dfrac{1}{a_4}+\\dfrac{1}{a_5}$的值;\\\\\n(2) 设数列$\\{\\log _3 a_n\\}$的前$n$项和为$S_n$, 求$S_n$的最大值, 并指出$S_n$取最大值时$n$的取值.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-下学期周末卷-高三下学期周末卷10" ], "genre": "解答题", "ans": "(1) $\\dfrac{121}9$; (2) $S_n$的最大值为$3$, 当且仅当$n=2$或$3$时取得", @@ -397058,7 +400831,8 @@ "content": "如图, 三角形$EAD$与梯形$ABCD$所在的平面互相垂直, $AE \\perp AD$, $AB \\perp AD$, $BC\\parallel AD$, $AB=AE=BC=2$, $AD=4$, $F$、$H$分别为$ED$、$EA$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$A$} coordinate (A);\n\\draw (4,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,2,0) node [above] {$E$} coordinate (E);\n\\draw (0,0,2) node [below] {$B$} coordinate (B);\n\\draw (2,0,2) node [below] {$C$} coordinate (C);\n\\draw ($(E)!0.5!(D)$) node [above] {$F$} coordinate (F);\n\\draw ($(A)!0.5!(E)$) node [right] {$H$} coordinate (H);\n\\draw (B)--(C)--(D)--(E)--cycle(C)--(F)(E)--(C);\n\\draw [dashed] (B)--(H)(E)--(A)--(B)(A)--(C)(A)--(D)(A)--(F);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $BH\\parallel$平面$AFC$;\\\\\n(2) 求平面$ACF$与平面$EAB$所成锐二面角的余弦值.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期周末卷-高三下学期周末卷10" ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\dfrac{\\sqrt{6}}6$", @@ -397089,7 +400863,8 @@ "content": "为了庆祝党的二十大顺利召开, 某学校特举办主题为``重温光辉历史展现坚定信心''的百科知识小测试比赛. 比赛分抢答和必答两个环节, 两个环节均设置$10$道题, 其中$5$道人文历史题和$5$道地理环境题.\\\\\n(1) 在抢答环节, 某代表队非常积极, 抢到$4$次答题机会, 求该代表队至少抢到$1$道地理环境题的概率;\\\\\n(2) 在必答环节, 每个班级从$5$道人文历史题和$5$道地理环境题各选$2$题, 各题答对与否相互独立, 每个代表队可以先选择人文历史题, 也可以先选择地理环境题开始答题. 若中间有一题答错就退出必答环节, 仅当第一类问题中$2$题均答对, 才有资格开始第二类问题答题. 已知答对$1$道人文历史题得$2$分, 答对$1$道地理环境题得$3$分. 假设某代表队答对人文历史题的概率都是$\\dfrac{3}{5}$, 答对地理环境题的概率都是$\\dfrac{1}{3}$. 请你为该代表队作出答题顺序的选择, 使其得分期望值更大, 并说明理由.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-下学期周末卷-高三下学期周末卷10" ], "genre": "解答题", "ans": "(1) $\\dfrac{41}{42}$; (2) $\\dfrac{12}{5}>\\dfrac{116}{75}$, 故应该先答人文历史题, 再答地理环境题", @@ -397120,7 +400895,8 @@ "content": "椭圆$C$的方程为$x^2+3 y^2=4$, $A$、$B$为椭圆的左右顶点, $F_1$、$F_2$为左右焦点, $P$为椭圆上的动点.\\\\\n(1) 求椭圆的离心率;\\\\\n(2) 若$\\triangle PF_1F_2$为直角三角形, 求$\\triangle PF_1F_2$的面积;\\\\\n(3) 若$Q$、$R$为椭圆上异于$P$的点, 直线$PQ$、$PR$均与圆$x^2+y^2=r^2$($00$)在区间$(0, \\pi)$没有最值, 则$\\omega$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期测验卷-高三下学期测验11" ], "genre": "填空题", "ans": "$(0,\\dfrac 56]$", @@ -397969,7 +401756,8 @@ "content": "已知函数$y=f(x)$和$y=g(x)$的表达式分别为$f(x)=\\sqrt{-x^2-4 x}$, $g(x)=x|x^2-a|$, 若对任意$x_1 \\in[1, \\sqrt{2}]$, 总存在$x_2 \\in[-3,0]$, 使得$g(x_1)b^2>0$, 则下列不等式中成立的是\\bracket{20}.\n\\fourch{$a>b$}{$2^a>2^b$}{$a>|b|$}{$\\log _2 a^2>\\log _2 b^2$}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-下学期测验卷-高三下学期测验11" ], "genre": "选择题", "ans": "D", @@ -398065,7 +401855,8 @@ "content": "某社区通过公益讲座宣传交通法规. 为了解讲座效果, 随机抽取$10$位居民, 分别在讲座前、\n后各回答一份交通法规知识问卷, 满分为$100$分. 他们得分的茎叶图如图所示 (``叶''是个位数字), 则下列选项叙述错误的是\\bracket{20}.\n\\begin{center}\n\\begin{tabular}{ccc|c|cccc} \n\\multicolumn{3}{r|}{讲座前} & & \\multicolumn{4}{l}{讲座后} \\\\\n& 5 & 0 & 5 & & & & \\\\\n5 & 0 & 0 & 6 & & & & \\\\\n5 & 0 & 0 & 7 & & & & \\\\\n& & 0 & 8 & 0 & 5 & 5 & 5 \\\\\n& & 0 & 9 & 0 & 0 & 5 & 5 \\\\\n& & & 10 & 0 & 0 & &\n\\end{tabular}\n\\end{center}\n\\twoch{讲座后的答卷得分整体上高于讲座前的得分}{讲座前的答卷得分分布较讲座后分散}{讲座后答卷得分的第$80$百分位数为$95$}{讲座前答卷得分的极差大于讲座后得分的极差}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-下学期测验卷-高三下学期测验11" ], "genre": "选择题", "ans": "C", @@ -398097,7 +401888,8 @@ "content": "如图, 在矩形$ABCD$中, $E$、$F$分别为边$AD$、$BC$上的点, 且$AD=3AE$, $BC=3BF$, 设$P$、$Q$分别为线段$AF$、$CE$的中点, 将四边形$ABFE$沿着直线$EF$进行翻折, 使得点$A$不在平面$CDEF$上, 在这一过程中, 下列关系不能恒成立的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$B$} coordinate (B);\n\\draw (1,0) node [below] {$F$} coordinate (F);\n\\draw (3,0) node [right] {$C$} coordinate (C);\n\\draw (3,2) node [right] {$D$} coordinate (D);\n\\draw (1,2) node [above] {$E$} coordinate (E);\n\\draw (0,2) node [left] {$A$} coordinate (A);\n\\draw (B) rectangle (D) (E)--(F)(A)--(F)(E)--(C);\n\\filldraw ($(A)!0.5!(F)$) node [left] {$P$} coordinate (P) circle (0.03);\n\\filldraw ($(C)!0.5!(E)$) node [left] {$Q$} coordinate (Q) circle (0.03);\n\\end{tikzpicture}\n\\end{center}\n\\twoch{直线$AB\\parallel$直线$CD$}{直线$PQ\\parallel$直线$ED$}{直线$AB \\perp$直线$PQ$}{直线$PQ\\parallel$平面$ADE$}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期测验卷-高三下学期测验11" ], "genre": "选择题", "ans": "B", @@ -398129,7 +401921,8 @@ "content": "设$\\{a_n\\}$是项数为$n_0$的有穷数列, 其中$n_0 \\geq 2$. 当$n \\leq \\dfrac{n_0}{2}$时, $a_n=\\dfrac{1}{2^n}$, 且对任意正整数$n \\leq n_0$都有$a_n+a_{n_0+1-n}=0$. 给出下列两个命题: \\textcircled{1} 若对任意正整数$n \\leq n_0$都有$\\displaystyle\\sum_{i=1}^n a_i \\leq \\dfrac{511}{512}$, 则$n_0$的最大值为$18$; \\textcircled{2} 对于任意满足$1 \\leq s=latex]\n\\def\\l{2}\n\\def\\h{2}\n\\draw ({-\\l/2},0,0) node [left] {$C$} coordinate (C);\n\\draw (0,0,{\\l/2*sqrt(3)}) node [below] {$A$} coordinate (A);\n\\draw ({\\l/2},0,0) node [right] {$B$} coordinate (B);\n\\draw (C) ++ (0,\\h) node [left] {$C_1$} coordinate (C_1);\n\\draw (A) ++ (0,\\h) node [below right] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\h) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) -- (A) -- (B) (C) -- (C_1) (A) -- (A_1) (B) -- (B_1) (C_1) -- (A_1) -- (B_1) (C_1) -- (B_1);\n\\draw [dashed] (C) -- (B);\n\\draw ($(A)!0.5!(B)$) node [below] {$D$} coordinate (D);\n\\draw (D)--(B_1);\n\\draw [dashed] (D)--(C)--(B_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求直线$CC_1$与$DB_1$所成的角的大小;\\\\\n(2) 求证: 平面$CDB_1 \\perp$平面$ABB_1A_1$, 并求点$B$到平面$CDB_1$的距离.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期测验卷-高三下学期测验11" ], "genre": "解答题", "ans": "(1) $\\arctan \\dfrac 12$; (2) 证明略, 距离为$\\dfrac{2\\sqrt{5}}5$", @@ -398225,7 +402020,8 @@ "content": "某网站计划$4$月份订购草莓在网络销售, 每天的进货量相同, 成本价为每盒$15$元. 决定每盒售价为$20$元, 未售出的草莓降价处理, 每盒$10$元. 假设当天进货能全部售完. 根据销售经验, 每天的购买量与网站每天的浏览量(单位: 万次) 有关. 为确定草莓的进货量, 相关人员统计了前两年$4$月份(共$60$天)网站每天的浏览量(单位: 万次)、购买草莓的数量(单位: 盒) 以及达到该流量的天数, 如下表所示:\n\\begin{center} \n\\begin{tabular}{|c|c|c|}\n\\hline 每天的浏览量 &$(0,1)$& {$[1,+\\infty)$} \\\\\n\\hline 每天的购买量 & 600 & 900 \\\\\n\\hline 天数 & 36 & 24 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n以每天的浏览量位于各区间的频率代替浏览量位于该区间的概率.\\\\\n(1) 求$4$月份草莓一天的购买量$X$(单位: 盒)的分布;\\\\\n(2) 设$4$月份销售草莓一天的利润为$Y$(单位: 元), 一天的进货量为$n$(单位: 盒), $n$为正整数且$n \\in[600,900]$, 当$n$为多少时, $Y$的期望达到最大值, 并求此最大值.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-下学期测验卷-高三下学期测验11" ], "genre": "解答题", "ans": "(1) 分布为$\\begin{pmatrix}600 & 900 \\\\ \\dfrac 35 & \\dfrac 25\\end{pmatrix}$; (2) 当$n$为$600$时, $Y$的期望达到最大值, 最大值为$3000$元", @@ -398259,7 +402055,8 @@ "content": "已知椭圆$\\Gamma: \\dfrac{x^2}{4}+\\dfrac{y^2}{b^2}=1$($0=latex]\n\\fill [pattern = north east lines] (-1,1) arc (180:360:1 and 0.25) -- (0,0) -- cycle;\n\\fill [pattern = north west lines] (0,1) ellipse (1 and 0.25);\n\\draw (0,1) ellipse (1 and 0.25);\n\\draw (-1,0) arc (180:360:1 and 0.25);\n\\draw [dashed] (-1,0) arc (180:0:1 and 0.25);\n\\draw (-1,0) --++ (0,1) (1,0) --++ (0,1);\n\\draw [dashed] (-1,1) -- (0,0) -- (1,1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期测验卷-高三下学期测验10" ], "genre": "填空题", "ans": "$(300+100\\sqrt{2})\\pi$", @@ -398587,7 +402393,8 @@ "content": "若函数$y=f(x)$的图像可由函数$y=3 \\sin 2 x-\\sqrt{3} \\cos 2 x$的图像向右平移$\\varphi$($0<\\varphi<\\pi$)个单位所得到, 且函数$y=f(x)$在区间$[0, \\dfrac{\\pi}{2}]$上是严格减函数, 则$\\varphi=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期测验卷-高三下学期测验10" ], "genre": "填空题", "ans": "$\\dfrac{2\\pi}{3}$", @@ -398620,7 +402427,8 @@ "content": "若每经过一天某种物品的价格变为原来的$1.02$倍的概率为$0.5$, 变为原来的$0.98$倍的概率也为$0.5$, 则经过$6$天该物品的价格较原来价格增加的概率为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-下学期测验卷-高三下学期测验10" ], "genre": "填空题", "ans": "$\\dfrac{11}{32}$", @@ -398653,7 +402461,8 @@ "content": "如图, 在直角梯形$ABCD$中, $AD\\parallel BC$, $\\angle ABC=90^{\\circ}$, $AD=2$, $BC=1$, 点$P$是腰$AB$上的动点, 则$|2 \\overrightarrow{PC}+\\overrightarrow{PD}|$的最小值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$A$} coordinate (A);\n\\draw (2.5,0) node [below] {$B$} coordinate (B);\n\\draw (0,2) node [left] {$D$} coordinate (D);\n\\draw (2.5,1) node [right] {$C$} coordinate (C);\n\\draw ($(A)!0.6!(B)$) node [below] {$P$} coordinate (P);\n\\draw (D)--(A)--(B)--(C)--cycle;\n\\draw [->] (P) -- (C);\n\\draw [->] (P) -- (D);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-下学期测验卷-高三下学期测验10" ], "genre": "填空题", "ans": "$4$", @@ -398686,7 +402495,8 @@ "content": "已知实数$a, b, c$满足: $a+b+c=0$与$a^2-b c=3$, 则$a b c$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期测验卷-高三下学期测验10" ], "genre": "填空题", "ans": "$[-2,2]$", @@ -398719,7 +402529,8 @@ "content": "若直线$(a-1) x+y-1=0$与直线$3 x-a y+2=0$垂直, 则实数$a$的值为\\bracket{20}.\n\\fourch{$\\dfrac{1}{2}$}{$\\dfrac{3}{2}$}{$\\dfrac{1}{4}$}{$\\dfrac{3}{4}$}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期测验卷-高三下学期测验10" ], "genre": "选择题", "ans": "B", @@ -398752,7 +402563,8 @@ "content": "从装有两个红球和两个白球的口袋内任取两个球, 那么互斥而不对立的事件是\\bracket{20}.\n\\twoch{``恰好有一个白球''与``都是红球''}{``至多有一个白球''与``都是红球''}{``至多有一个白球''与``都是白球''}{``至多有一个白球''与``至多有一个红球''}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-下学期测验卷-高三下学期测验10" ], "genre": "选择题", "ans": "A", @@ -398785,7 +402597,8 @@ "content": "如图, $\\triangle ABD$与$\\triangle BCD$都是等腰直角三角形, 其底边分别为$BD$与$BC$, 点$E$、$F$分别为线段$BD$、$AC$的中点, 设二面角$A-BD-C$的大小为$\\alpha$, 当$\\alpha$在区间$(0, \\pi)$内变化时, 下列结论正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.5]\n\\draw (-1,0,0) node [left] {$B$} coordinate (B);\n\\draw (1,0,0) node [right] {$D$} coordinate (D);\n\\draw (1,0,2) node [below] {$C$} coordinate (C);\n\\draw (0,1,0) node [above] {$A$} coordinate (A);\n\\draw ($(B)!0.5!(D)$) node [below] {$E$} coordinate (E);\n\\draw ($(A)!0.5!(C)$) node [right] {$F$} coordinate (F);\n\\draw (A)--(B)--(C)--(D)--cycle (A)--(C);\n\\draw [dashed] (B)--(D)(E)--(F);\n\\end{tikzpicture}\n\\end{center}\n\\twoch{存在某一$\\alpha$值, 使得$AC \\perp BD$}{存在某一$\\alpha$值, 使得$EF \\perp BD$}{存在某一$\\alpha$值, 使得$EF \\perp CD$}{存在某一$\\alpha$值, 使得$AB \\perp CD$}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期测验卷-高三下学期测验10" ], "genre": "选择题", "ans": "D", @@ -398819,7 +402632,8 @@ "content": "设数列$\\{a_n\\}$的前$n$项的和为$S_n$, 若对任意的$n \\in \\mathbf{N}$, $n\\ge 1$, 都有$S_n=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1)--(C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (A1)--(B)--(C1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(A1)!{1/3}!(C1)$) node [below] {$E$} coordinate (E);\n\\draw ($(B)!{1/3}!(C1)$) node [left] {$F$} coordinate (F);\n\\draw (D1)--(E)--(F)--(C);\n\\draw [dashed] (C)--(D1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $EF\\parallel A_1B$;\\\\\n(2) 若$C_1E=2EA_1$, 求点$E$到平面$A_1D_1CB$的距离以及$ED_1$与平面$A_1D_1CB$所成角的大小.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期测验卷-高三下学期测验10" ], "genre": "解答题", "ans": "(1) 证明略; (2) 距离为$\\dfrac{\\sqrt{2}}2$, 所成角为$\\arcsin\\dfrac{\\sqrt{10}}{10}$", @@ -398918,7 +402734,8 @@ "content": "将某工厂的工人按年龄分成两组: ``$35$周岁及以上''、``$35$周岁以下'', 从每组中随机抽取$80$人, 将他们的绩效分数分成$5$组: $[50,60),[60,70),[70,80),[80,90),[90,100]$, 分别加以统计, 得到下列频率分布直方图. 该工厂规定绩效分数不少于$80$者为生产标兵.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.05, yscale = 60]\n\\draw [->] (40,0) -- (42,0) -- (44,-0.003) -- (46,0.003) -- (48,0)-- (120,0) node [below] {绩效分数};\n\\draw [->] (40,0) -- (40,0.05) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (40,0) node [below left] {$O$};\n\\foreach \\i/\\j in {50/0.005,60/0.035,70/0.035,80/0.020,90/0.005}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (10,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {70/0.035,80/0.020,90/0.005}\n{\\draw [dashed] (\\i,\\j) -- (40,\\j) node [left] {$\\k$};};\n\\draw (100,0) node [below] {$100$};\n\\draw (75,-0.01) node {$35$周岁及以上组};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex, xscale = 0.05, yscale = 60]\n\\draw [->] (40,0) -- (42,0) -- (44,-0.003) -- (46,0.003) -- (48,0)-- (120,0) node [below] {绩效分数};\n\\draw [->] (40,0) -- (40,0.05) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (40,0) node [below left] {$O$};\n\\foreach \\i/\\j in {50/0.005,60/0.025,70/0.0325,80/0.0325,90/0.005}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (10,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {60/0.025,80/0.0325,90/0.005}\n{\\draw [dashed] (\\i,\\j) -- (40,\\j) node [left] {$\\k$};};\n\\draw (100,0) node [below] {$100$};\n\\draw (75,-0.01) node {$35$周岁以下组};\n\\end{tikzpicture}\n\\end{center}\n(1) 请列出$2 \\times 2$列联表, 并判断能否有$95 \\%$的把握认为是否为生产标兵与工人所在的年龄组有关;\\\\\n(2) 若已知该工厂工人中生产标兵的占比为$30 \\%$, 试估计该厂$35$周岁以下的工人所占的百分比以及生产标兵中$35$周岁以下的工人所占的百分比.\\\\\n附: $\\chi^2=\\dfrac{n(a d-b c)^2}{(a+b)(c+d)(a+c)(b+d)}$.\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline$P(x^2 \\geq k)$& 0.100 & 0.050 & 0.010 & 0.001 \\\\\n\\hline$k$& 2.706 & 3.841 & 6.635 & 10.828 \\\\\n\\hline\n\\end{tabular}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-下学期测验卷-高三下学期测验10" ], "genre": "解答题", "ans": "(1) \\begin{tabular}{|c|c|c|c|} \\hline & 生产标兵 & 非生产标兵 & 总计\\\\\\hline\n$35$周岁及以上组 & $20$ & $60$ & $80$\\\\\\hline\n$35$周岁以下组 & $30$ & $50$ & $80$ \\\\\\hline\n总计& $50$ & $110$ & $160$ \\\\ \\hline\n\\end{tabular}, $\\chi^2\\approx 2.91$, 因此没有$95\\%$的把握认为是否为生产标兵与工人所在的年龄有关; (2) 估计该厂工人中$35$周岁以下占$40\\%$, 该厂生产标兵中$35$周岁以下占$50\\%$", @@ -398951,7 +402768,8 @@ "content": "已知双曲线$C$的中心在坐标原点, 左焦点$F_1$与右焦点$F_2$都在$x$轴上, 离心率为$3$, 过点$F_2$的动直线$l$与双曲线$C$交于点$A$、$B$, 设$\\dfrac{|AF_2| \\cdot|BF_2|}{|AB|^2}=\\lambda$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.2]\n\\draw [->] (-6,0) -- (6,0) node [below] {$x$};\n\\draw [->] (0,-12) -- (0,12) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = {-2*sqrt(30)}:{2*sqrt(30)}, samples = 100] plot ({sqrt(1+\\x*\\x/8)},\\x);\n\\draw [domain = {-2*sqrt(30)}:{2*sqrt(30)}, samples = 100] plot ({-sqrt(1+\\x*\\x/8)},\\x);\n\\filldraw (3,0) circle (0.15) node [below] {$F_2$} coordinate (F_2);\n\\filldraw (-3,0) circle (0.15) node [below] {$F_1$} coordinate (F_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求双曲线$C$的渐近线方程;\\\\\n(2) 若点$A$、$B$都在双曲线$C$的右支上, 求$\\lambda$的最大值以及$\\lambda$取最大值时$\\angle AF_1B$的正切值; (关于求$\\lambda$的最值, 某学习小组提出了如下的思路可供参考: \\textcircled{1} 利用基本不等式求最值; \\textcircled{2} 设$\\dfrac{|AF_2|}{|AB|}$为$\\mu$, 建立相应数量关系并利用它求最值; \\textcircled{3} 设直线$l$的斜率为$k$, 建立相应数量关系并利用它求最值)\\\\\n(3) 若点$A$在双曲线$C$的左支上(点$A$不是该双曲线的顶点), 且$\\lambda=1$, 求证: $\\triangle AF_1B$是等腰三角形, 且$AB$边的长等于双曲线$C$的实轴长的$2$倍.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期测验卷-高三下学期测验10" ], "genre": "解答题", "ans": "(1) $y=\\pm 2\\sqrt{2}x$; (2) 最大值为$\\dfrac 14$, 此时$\\angle AF_1B$的正切值为$-\\dfrac{24}{7}$; (3) 证明略", @@ -398985,7 +402803,8 @@ "content": "三个互不相同的函数$y=f(x), y=g(x)$与$y=h(x)$在区间$D$上恒有$f(x) \\geq h(x) \\geq g(x)$或恒有$f(x) \\leq h(x) \\leq g(x)$, 则称$y=h(x)$为$y=f(x)$与$y=g(x)$在区间$D$上的``分割函数''.\\\\\n(1) 设$h_1(x)=4 x$, $h_2(x)=x+1$, 试分别判断$y=h_1(x)$、$y=h_2(x)$是否是$y=2 x^2+2$与$y=-x^2+4 x$在区间$(-\\infty,+\\infty)$上的``分割函数'', 请说明理由;\\\\\n(2) 求所有的二次函数, 使得该函数是$y=2 x^2+2$与$y=4 x$在区间$(-\\infty,+\\infty)$上的``分割函数'';\\\\\n(3) 若$[m, n] \\subseteq[-2,2]$, 且存在实数$k, b$, 使得$y=k x+b$为$y=x^4-4 x^2$与$y=4 x^2-16$在区间$[m, n]$\n上的``分割函数'', 求$n-m$的最大值.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期测验卷-高三下学期测验10" ], "genre": "解答题", "ans": "(1) $h_1(x)$是, $h_2(x)$不是; (2) $y=ax^2+(4-2a)x+a$($0a\\}$, 若$A \\cap B=\\varnothing$, 则实数$a$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-下学期测验卷-高三下学期测验09" ], "genre": "填空题", "ans": "$[3,+\\infty)$", @@ -399193,7 +403017,8 @@ "content": "已知圆柱的底面直径和高都等于球的直径, 圆柱的体积为$16 \\pi$, 则球的表面积为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期测验卷-高三下学期测验09" ], "genre": "填空题", "ans": "$16\\pi$", @@ -399228,7 +403053,8 @@ "content": "已知函数$y=a x^2+b x+c$的图像如图所示, 则不等式$(a x+b)(b x+c)(c x+a)<0$的解集是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-1,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -0.5:3.5] plot (\\x,{(\\x-1)*(\\x-2)});\n\\draw (1,0) node [below] {$1$} (2,0) node [below] {$2$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期测验卷-高三下学期测验09" ], "genre": "填空题", "ans": "$(-\\dfrac 12,\\dfrac 23)\\cup (3,+\\infty)$", @@ -399263,7 +403089,8 @@ "content": "已知函数$y=f(x)$是定义在$\\mathbf{R}$上的奇函数, 且满足$f(2+x)=-f(2-x)$, $f(1)=1$, 则$f(1)+f(2)+\\cdots+f(2023)=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期测验卷-高三下学期测验09" ], "genre": "填空题", "ans": "$0$", @@ -399298,7 +403125,8 @@ "content": "如图所示, 要在两山顶$M$、$N$间建一索道, 需测量两山顶$M$、$N$间的距离. 已知两山的海拔高度分别是$MC=100 \\sqrt{3}$米和$NB=50 \\sqrt{2}$米, 现选择海平面上一点$A$为观测点, 从$A$点测得$M$点的仰角$\\angle MAC=60^{\\circ}$, $N$点的仰角$\\angle NAB=30^{\\circ}$以及$\\angle MAN=45^{\\circ}$, 则$MN$等于\\blank{50}米.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw (0,0) node [left] {$C$} coordinate (C);\n\\draw (3,0) node [right] {$B$} coordinate (B);\n\\draw (1.7,-0.5) node [below] {$A$} coordinate (A);\n\\draw (C) ++ (0,3) node [above] {$M$} coordinate (M);\n\\draw (B) ++ (0,1.5) node [above] {$N$} coordinate (N);\n\\draw (C)--(A)--(B)--(N)--(M)--cycle(M)--(A)--(N);\n\\draw [dashed] (C)--(B);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期测验卷-高三下学期测验09" ], "genre": "填空题", "ans": "$100\\sqrt{2}$", @@ -399333,7 +403161,8 @@ "content": "已知数列$\\{a_n\\}$满足$a_n=a n^2+n$, 若满足$a_1a_{n+1}$, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-下学期测验卷-高三下学期测验09" ], "genre": "填空题", "ans": "$(-\\dfrac 1{11},-\\dfrac 1{19})$", @@ -399368,7 +403197,8 @@ "content": "如图, 已知$F_1, F_2$分别是椭圆$C: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的左、右焦点, $M, N$为椭圆上两点, 满足$F_1M\\parallel F_2N$, 且$|F_2N|: |F_2M|: |F_1M|=1: 2: 3$, 则椭圆$C$的离心率为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (-2.5,0) -- (2.5,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,0) ellipse ({sqrt(5)} and {sqrt(3)});\n\\draw ({-sqrt(2)},0) node [below] {$F_1$} coordinate (F_1);\n\\draw ({sqrt(2)},0) node [below] {$F_2$} coordinate (F_2);\n\\draw ({sqrt(2)/2},{3*sqrt(3)/sqrt(10)}) node [above] {$M$} coordinate (M);\n\\draw ($(F_2)+1/3*(M)-1/3*(F_1)$) node [above] {$N$} coordinate (N);\n\\draw (F_1)--(M)--(F_2)--(N);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期测验卷-高三下学期测验09" ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{10}}5$", @@ -399403,7 +403233,8 @@ "content": "已知函数$y=\\sqrt{1-x^2}$, $-\\dfrac{1}{2} \\leq x \\leq \\dfrac{1}{2}$的图像绕着原点按逆时针方向旋转$\\theta$($0 \\leq \\theta \\leq \\pi$)弧度, 若得到的图像仍是函数图像, 则$\\theta$可取值的集合为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期测验卷-高三下学期测验09" ], "genre": "填空题", "ans": "$[0,\\dfrac\\pi 3]\\cup [\\dfrac{2\\pi}3,\\pi]$", @@ -399438,7 +403269,8 @@ "content": "设$\\overrightarrow{e_1}$、$\\overrightarrow{e_2}$是两个不平行的向量, 则下列四组向量中, 不能组成平面向量的一个基的是\\bracket{20}.\n\\fourch{$\\overrightarrow{e_1}+\\overrightarrow{e_2}$和$\\overrightarrow{e_1}-\\overrightarrow{e_2}$}{$\\overrightarrow{e_1}+2 \\overrightarrow{e_2}$和$\\overrightarrow{e_2}+2 \\overrightarrow{e_1}$}{$3 \\overrightarrow{e_1}-2 \\overrightarrow{e_2}$和$4 \\overrightarrow{e_2}-6 \\overrightarrow{e_1}$}{$\\overrightarrow{e_2}$和$\\overrightarrow{e_2}+\\overrightarrow{e_1}$}", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-下学期测验卷-高三下学期测验09" ], "genre": "选择题", "ans": "C", @@ -399473,7 +403305,8 @@ "content": "已知$n$为正整数, 则``$n$是$3$的倍数''是``$(x^4-\\dfrac{2}{x^2})^n$的二项展开式中存在常数项''的 \\bracket{20}条件.\n\\fourch{充分非必要}{必要非充分}{充要}{既不充分也不必要}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-下学期测验卷-高三下学期测验09" ], "genre": "选择题", "ans": "C", @@ -399508,7 +403341,8 @@ "content": "某产品的广告费$x$(单位: 万元) 与销售额$y$(单位: 万元) 的统计数据如下表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline 广告费$x$(万元) & 2 & 3 & 4 & 5 \\\\\n\\hline 销售额$y$(万元) & 26 & 39 & 49 & 54 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n根据上表可得回归方程$y=\\hat{a} x+\\hat{b}$中$\\hat{a}=9.4$, 据此模型可预测当广告费为$6$万元时, 销售额约为\\bracket{20}.\n\\fourch{$63.6$万元}{$65.5$万元}{$67.7$万元}{$72.0$万元}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-下学期测验卷-高三下学期测验09" ], "genre": "选择题", "ans": "B", @@ -399543,7 +403377,8 @@ "content": "已知数列$\\{a_n\\}$满足$a_1=1$, $a_{n+1}-a_n=(-\\dfrac{1}{2})^n$, 存在正偶数$n$使得$(a_n-\\lambda)(a_{n+1}+\\lambda)>0$, 且对任意正奇数$n$有$(a_n-\\lambda)(a_{n+1}+\\lambda)<0$, 则实数$\\lambda$的取值范围是\\bracket{20}.\n\\fourch{$(-\\dfrac{2}{3}, 1]$}{$(-\\infty,-\\dfrac{2}{3}] \\cup(1,+\\infty)$}{$(-\\dfrac{3}{4}, \\dfrac{2}{3})$}{$(-\\dfrac{3}{4},-\\dfrac{2}{3}]$}", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-下学期测验卷-高三下学期测验09" ], "genre": "选择题", "ans": "D", @@ -399578,7 +403413,8 @@ "content": "已知函数$y=f(x)$的表达式为$f(x)=\\sqrt{3} \\sin (x+\\dfrac{\\pi}{6}) \\cos (x+\\dfrac{\\pi}{6})+\\cos ^2(x-\\dfrac{\\pi}{3})$.\\\\\n(1) 求函数$y=f(x)$的最小正周期及图像的对称轴的方程;\\\\\n(2) 求函数$y=f(x)$在$(0, \\dfrac{\\pi}{2})$上的值域.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期测验卷-高三下学期测验09" ], "genre": "解答题", "ans": "(1) 最小正周期为$\\pi$, 对称轴的方程为$x=\\dfrac\\pi 6+\\dfrac{k\\pi}2$, $k\\in \\mathbf{Z}$; (2) $(0,\\dfrac 32]$", @@ -399613,7 +403449,8 @@ "content": "如图, 在直三棱柱$ABC-A_1B_1C_1$中, 底面$\\triangle ABC$是等腰直角三角形, $AC=BC=AA_1=2$, $D$为侧棱$AA_1$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$C$} coordinate (C);\n\\draw (0,0,2) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$B$} coordinate (B);\n\\draw (0,2,0) node [above] {$C_1$} coordinate (C_1);\n\\draw (2,2,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (0,2,2) node [left] {$A_1$} coordinate (A_1);\n\\draw ($(A)!0.5!(A_1)$) node [left] {$D$} coordinate (D);\n\\draw (A)--(B)--(B_1)--(C_1)--(A_1)--cycle(A_1)--(B_1)(D)--(B_1);\n\\draw [dashed] (A)--(C)--(B)(C)--(C_1)(C)--(B_1)(C)--(D)--(C_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $BC \\perp$平面$ACC_1A_1$;\\\\\n(2) 求二面角$B_1-CD-C_1$的正弦值.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期测验卷-高三下学期测验09" ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\dfrac{\\sqrt{5}}3$", @@ -399649,7 +403486,8 @@ "objs": [], "tags": [ "第八单元", - "第九单元" + "第九单元", + "2023届高三-下学期测验卷-高三下学期测验09" ], "genre": "解答题", "ans": "(1) $a=0.26$, $b=0.38$; (2) $\\dfrac 23$; (3) $X$的分布为$\\begin{pmatrix} 0 & 1 & 2 \\\\ \\dfrac 5{14} & \\dfrac{15}{28} & \\dfrac 3{28} \\end{pmatrix}$, 期望为$\\dfrac 34$", @@ -399684,7 +403522,8 @@ "content": "如图, 已知$A$、$B$、$C$是抛物线$\\Gamma_1: x^2=y$上的三个点, 且直线$CB$、$CA$分别与抛物线$\\Gamma_2: y^2=4 x$相切, $F$为抛物线$\\Gamma_1$的焦点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain = -2:2, samples = 100] plot (\\x,{\\x*\\x});\n\\draw [domain = {-sqrt(12)}:{sqrt(12)}, samples = 100] plot ({\\x*\\x/4},\\x);\n\\filldraw (0,0.25) node [right] {$F$} coordinate (F) circle (0.03);\n\\draw (-1,1) node [below] {$C$} coordinate (C);\n\\draw ({(1+sqrt(5))/2},{(3+sqrt(5))/2}) node [above] {$B$} coordinate (B);\n\\draw ({(1-sqrt(5))/2},{(3-sqrt(5))/2}) node [below] {$A$} coordinate (A);\n\\draw [thick] ($(B)!-0.5!(C)$) -- ($(B)!1.2!(C)$);\n\\draw [thick] ($(A)!-5!(C)$) -- ($(A)!1.5!(C)$);\n\\draw [thick] (A)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 若点$C$的横坐标为$x_3$, 用$x_3$表示线段$CF$的长;\\\\\n(2) 若$CA \\perp CB$, 求点$C$的坐标;\\\\\n(3) 证明: 直线$AB$与抛物线$\\Gamma_2$相切.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期测验卷-高三下学期测验09" ], "genre": "解答题", "ans": "(1) $x_3^2+\\dfrac 14$; (2) $(-1,1)$; (3) 证明略", @@ -399719,7 +403558,8 @@ "content": "设$y=f(x)$、$y=g(x)$是定义域为$\\mathbf{R}$的函数, 当$g(x_1) \\neq g(x_2)$时, 记$\\delta(x_1, x_2)=\\dfrac{f(x_1)-f(x_2)}{g(x_1)-g(x_2)}$.\\\\\n(1) 已知$y=g(x)$在区间$I$上严格增, 且对任意$x_1, x_2 \\in I$, $x_1 \\neq x_2$, 有$\\delta(x_1, x_2)>0$, 证明: 函数$y=f(x)$在区间$I$上严格增;\\\\\n(2) 已知$g(x)=\\dfrac{1}{3} x^3+a x^2-3 x$, 且对任意$x_1, x_2 \\in \\mathbf{R}$, 当$g(x_1) \\neq g(x_2)$时, 有$\\delta(x_1, x_2)>0$, 若当$x=1$时, 函数$y=f(x)$取得极值, 求实数$a$的值;\\\\\n(3) 已知$g(x)=\\sin x$, $f(\\dfrac{\\pi}{2})=1$, $f(-\\dfrac{\\pi}{2})=-1$, 且对任意$x_1, x_2 \\in \\mathbf{R}$, 当$g(x_1) \\neq g(x_2)$时, 有$|\\delta(x_1, x_2)| \\leq 1$, 证明: $f(x)=\\sin x$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期测验卷-高三下学期测验09" ], "genre": "解答题", "ans": "(1) 证明略; (2) $1$; (3) 证明略", @@ -399754,7 +403594,8 @@ "content": "已知集合$A=\\{x |-20$时, $f(x)=2^x+\\dfrac{9}{2^x+1}$, 则该函数的值域为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期测验卷-高三下学期测验08" ], "genre": "填空题", "ans": "$(-\\infty,-5]\\cup \\{0\\}\\cup [5,+\\infty)$", @@ -400034,7 +403882,8 @@ "content": "端午节吃粽子是我国的传统习俗. 一盘中放有$10$个外观完全相同的粽子, 其中豆沙粽$3$个, 肉粽$3$个, 白米粽$4$个, 现从盘子任意取出$3$个, 则取到白米粽的个数的数学期望为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-下学期测验卷-高三下学期测验08" ], "genre": "填空题", "ans": "$\\dfrac 65$", @@ -400069,7 +403918,8 @@ "content": "已知$A, B$是球$O$的球面上两点, $\\angle AOB=60^{\\circ}$, $P$为该球面上的动点, 若三棱锥$P-OAB$体积的最大值为$6$, 则球$O$的表面积为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期测验卷-高三下学期测验08" ], "genre": "填空题", "ans": "$48\\pi$", @@ -400104,7 +403954,8 @@ "content": "过原点的直线$l$与双曲线$C: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a, b>0$)的左、右两支分别交于$M, N$两点, $F(2,0)$为$C$的右焦点, 若$\\overrightarrow{FM} \\cdot \\overrightarrow{FN}=0$, 且$|\\overrightarrow{FM}|+|\\overrightarrow{FN}|=2 \\sqrt{5}$, 则双曲线$C$的方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期测验卷-高三下学期测验08" ], "genre": "填空题", "ans": "$\\dfrac{x^2}3-y^2=1$", @@ -400139,7 +403990,8 @@ "content": "已知平面向量$\\overrightarrow {a}, \\overrightarrow {b}, \\overrightarrow {c}, \\overrightarrow {e}$满足$|\\overrightarrow {a}|=3$, $|\\overrightarrow {e}|=1$, $|\\overrightarrow {b}-\\overrightarrow {a}|=1$, $\\langle\\overrightarrow {a}, \\overrightarrow {e}\\rangle=\\dfrac{2 \\pi}{3}$, 且对任意的实数$t$, 均有$|\\overrightarrow {c}-t \\overrightarrow {e}| \\geq|\\overrightarrow {c}-2 \\overrightarrow {e}|$, 则$|\\overrightarrow {c}-\\overrightarrow {b}|$的最小值为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-下学期测验卷-高三下学期测验08" ], "genre": "填空题", "ans": "$\\dfrac 52$", @@ -400174,7 +404026,8 @@ "content": "已知复数$z=\\dfrac{1}{1-\\mathrm{i}}-\\mathrm{i}$($\\mathrm{i}$为虚数单位$)$, 则$z \\cdot \\overline {z}=$\\bracket{20}.\n\\fourch{$\\dfrac{1}{2}$}{$\\dfrac{\\sqrt{2}}{2}$}{$\\dfrac{\\sqrt{3}}{2}$}{2}", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-下学期测验卷-高三下学期测验08" ], "genre": "选择题", "ans": "A", @@ -400209,7 +404062,8 @@ "content": "某同学上学路上有$4$个红绿灯的路口, 假设他走到每个路口遇到绿灯的概率为$\\dfrac{2}{3}$, 且在各个路口遇到红灯或绿灯互不影响, 则该同学上学路上至少遇到$2$次绿灯的概率为\\bracket{20}.\n\\fourch{$\\dfrac{1}{8}$}{$\\dfrac{3}{8}$}{$\\dfrac{7}{8}$}{$\\dfrac{8}{9}$}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-下学期测验卷-高三下学期测验08" ], "genre": "选择题", "ans": "D", @@ -400244,7 +404098,8 @@ "content": "对于函数$f(x)=\\sqrt{3} \\sin x \\cos x+\\sin ^2 x-\\dfrac{1}{2}$, 给出下列结论:\\\\\n\\textcircled{1} 函数$y=f(x)$的图像关于点$(\\dfrac{5 \\pi}{12}, 0)$对称;\\\\\n\\textcircled{2} 函数$y=f(x)$在区间$[\\dfrac{\\pi}{6}, \\dfrac{2 \\pi}{3}]$上的值域为$[-\\dfrac{1}{2}, 1]$;\\\\\n\\textcircled{3} 将函数$y=f(x)$的图像向左平移$\\dfrac{\\pi}{3}$个单位长度得到函数$y=-\\cos 2 x$的图像;\\\\\n\\textcircled{4} 曲线$y=f(x)$在$x=\\dfrac{\\pi}{4}$处的切线的斜率为$1$.\n则所有正确的结论是\\bracket{20}.\n\\fourch{\\textcircled{1}\\textcircled{2}}{\\textcircled{2}\\textcircled{3}}{\\textcircled{2}\\textcircled{4}}{\\textcircled{1}\\textcircled{3}}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期测验卷-高三下学期测验08" ], "genre": "选择题", "ans": "C", @@ -400279,7 +404134,8 @@ "content": "在数列$\\{b_n\\}$中, 若有$b_m=b_n$($m, n$均为正整数, 且$m \\neq n$), 就有$b_{m+1}=b_{n+1}$, 则称数列$\\{b_n\\}$为``递等数列''. 已知数列$\\{a_n\\}$满足$a_5=5$, 且$a_n=n(a_{n+1}-a_n)$, 将``递等数列''$\\{b_n\\}$的前$n$项和记为$S_n$, 若$b_1=a_1=b_4$, $b_2=a_2$, $S_5=a_{10}$, 则$S_{2023}=$\\bracket{20}.\n\\fourch{$4720$}{$4719$}{$4718$}{$4716$}", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-下学期测验卷-高三下学期测验08" ], "genre": "选择题", "ans": "B", @@ -400314,7 +404170,8 @@ "content": "记$S_n$为数列$\\{a_n\\}$的前$n$项和, 已知$a_1=2$, $a_{n+1}=S_n$($n$为正整数).\\\\\n(1) 求数列$\\{a_n\\}$的通项公式;\\\\\n(2) 设$b_n=\\log _2 a_n$, 若$b_m+b_{m+1}+b_{m+2}+\\cdots+b_{m+9}=145$, 求正整数$m$的值.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-下学期测验卷-高三下学期测验08" ], "genre": "解答题", "ans": "(1) $\\begin{cases}2, & n=1, \\\\ 2^{n-1}, & n\\ge 2;\\end{cases}$ (2) $m=11$", @@ -400349,7 +404206,8 @@ "content": "如图, 在圆锥$PO$中, $AB$是底面的直径, $C$是底面圆周上的一点, 且$PO=3$, $AB=4$, $\\angle BAC=30^{\\circ}$, $M$是$BC$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (0,3) node [above] {$P$} coordinate (P);\n\\draw (-2,0) node [left] {$A$} coordinate (A);\n\\draw (2,0) node [right] {$B$} coordinate (B);\n\\draw (A) arc (180:360:2 and 0.5) -- (P)--cycle;\n\\draw [dashed] (A) arc (180:0:2 and 0.5) -- cycle(O)--(P);\n\\draw (-60:2 and 0.5) node [below] {$C$} coordinate (C);\n\\draw (P)--(C);\n\\draw ($(B)!0.5!(C)$) node [below] {$M$} coordinate (M);\n\\draw [dashed] (B)--(C)(O)--(C)(P)--(M)--(O);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: 平面$PBC \\perp$平面$POM$;\\\\\n(2) 求二面角$O-PB-C$的余弦值.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期测验卷-高三下学期测验08" ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\dfrac{\\sqrt{3}}4$", @@ -400384,7 +404242,8 @@ "content": "电解电容是常见的电子元件之一. 检测组在$85^{\\circ} \\text{C}$的温度条件下对电解电容进行质量检测, 按检测结果将其分为次品、正品, 其中正品分合格品、优等品两类.\\\\\n(1) 铝箔是组成电解电容必不可少的材料. 现检测组在$85^{\\circ} \\text{C}$的温度条件下, 对铝箔质量与电解电容质量进行测试, 得到如下$2 \\times 2$列联表, 那么他们是否有$99.9 \\%$的把握认为电解电容质量与铝箔质量有关? 请说明理由;\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline & 电解电容为次品 & 电解电容为正品 \\\\\n\\hline 铝箔为次品 & 174 & 76 \\\\\n\\hline 铝箔为正品 & 108 & 142 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(2) 电解电容经检验为正品后才能装箱, 已知两箱电解电容 (每箱$50$个), 第一箱和第二箱中分别有优等品$8$件与$9$件. 现用户从两箱中随机挑选出一箱, 并从该箱中先后随机抽取两个元件, 求在第一次取出的是优等品的情况下, 第二次取出的是合格品的概率.\\\\\n附录: $\\chi^2=\\dfrac{n(a d-b c)^2}{(a+b)(c+d)(a+c)(b+d)}$, 其中 $n=a+b+c+d$.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline $P\\left(\\chi^2 \\geq k\\right)$ & 0.100 & 0.050 & 0.025 & 0.010 & 0.001 \\\\\n\\hline $k$ & 2.706 & 3.841 & 5.024 & 6.635 & 10.828 \\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-下学期测验卷-高三下学期测验08" ], "genre": "解答题", "ans": "(1) $\\chi^2\\approx 35.4$, 有$99.9\\%$的把握认为电解电容质量与铝箔质量有关; (2) 约为$0.846$", @@ -400419,7 +404278,8 @@ "content": "已知动点$R(x, y)$到点$F(1,0)$的距离和它到直线$x=2$的距离之比等于$\\dfrac{\\sqrt{2}}{2}$, 动点$R$的轨迹记为曲线$C$, 过点$F$的直线$l$与曲线$C$相交于$P, Q$两点.\\\\\n(1) 求曲线$C$的方程;\\\\\n(2) 若$\\overrightarrow{FP}=-2 \\overrightarrow{FQ}$, 求直线$l$的方程;\\\\\n(3) 已知$A(-\\sqrt{2}, 0)$, 直线$AP, AQ$分别与直线$x=2$相交于$M, N$两点, 求证: 以$MN$为直径的圆经过点$F$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期测验卷-高三下学期测验08" ], "genre": "解答题", "ans": "(1)$\\dfrac{x^2}2+y^2=1$; (2) $x\\pm \\dfrac{\\sqrt{14}}7y-1=0$; (3) 证明略", @@ -400455,7 +404315,8 @@ "content": "设$f(x)=\\mathrm{e}^x$, $g(x)=\\ln x$, $h(x)=\\sin x+\\cos x$.\\\\\n(1) 求函数$y=\\dfrac{h(x)}{f(x)}$, $x \\in(-\\pi, 3 \\pi)$的单调区间和极值;\\\\\n(2) 若关于$x$不等式$f(x)+h(x) \\geq a x+2$在区间$[0,+\\infty)$上恒成立, 求实数$a$的取值范围;\\\\\n(3) 若存在直线$y=t$, 其与曲线$y=\\dfrac{x}{f(x)}$和$y=\\dfrac{g(x)}{x}$共有$3$个不同交点$A(x_1, t)$, $B(x_2, t)$, $C(x_3, t)$($x_10$, $(x+\\dfrac{m}{x})^6$的二项展开式中$x^2$项的系数是$60$, 则$m$的值为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-下学期周末卷-高三下学期周末卷08" ], "genre": "填空题", "ans": "$2$", @@ -401589,7 +405455,8 @@ "content": "已知事件$A$与事件$B$互斥, 如果$P(A)=0.3$, $P(B)=0.5$, 那么$P(\\overline{A \\cup B})=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-下学期周末卷-高三下学期周末卷08" ], "genre": "填空题", "ans": "$0.2$", @@ -401624,7 +405491,8 @@ "content": "今年春季流感爆发期间, 某医院准备将$2$名医生和$4$名护士分配到两所学校, 给学校老师和学生接种流感疫苗. 若每所学校分配$1$名医生和$2$名护士, 则不同的分配方法数为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-下学期周末卷-高三下学期周末卷08" ], "genre": "填空题", "ans": "$12$", @@ -401659,7 +405527,8 @@ "content": "$\\displaystyle\\lim _{h \\to 0} \\dfrac{\\ln (h+4)-2 \\ln 2}{h}=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期周末卷-高三下学期周末卷08" ], "genre": "填空题", "ans": "$\\dfrac 14$", @@ -401694,7 +405563,8 @@ "content": "若关于$x$的方程$(\\dfrac{1}{2})^x+m=\\sqrt{x+1}$在实数范围内有解, 则实数$m$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期周末卷-高三下学期周末卷08" ], "genre": "填空题", "ans": "$[-2,+\\infty)$", @@ -401729,7 +405599,8 @@ "content": "已知在等比数列$\\{a_n\\}$中, $a_3$、$a_7$分别是函数$y=x^3-6 x^2+6 x-1$的两个驻点, 则$a_5=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期周末卷-高三下学期周末卷08" ], "genre": "填空题", "ans": "$\\sqrt{2}$", @@ -401764,7 +405635,8 @@ "content": "已知抛物线$C_1: y^2=8 x$, 圆$C_2: (x-2)^2+y^2=1$, 点$M$的坐标为$(4,0)$, $P$、$Q$分别为$C_1$、$C_2$上的动点, 且满足$|PM|=|PQ|$, 则点$P$的横坐标的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期周末卷-高三下学期周末卷08" ], "genre": "填空题", "ans": "$[\\dfrac 76,\\dfrac{15}2]$", @@ -401799,7 +405671,8 @@ "content": "平面上有一组互不相等的单位向量$\\overrightarrow{OA_1}, \\overrightarrow{OA_2}, \\cdots, \\overrightarrow{OA_n}$, 若存在单位向量$\\overrightarrow{OP}$满足$\\overrightarrow{OP} \\cdot \\overrightarrow{OA_1}+\\overrightarrow{OP} \\cdot \\overrightarrow{OA_2}+\\cdots+\\overrightarrow{OP} \\cdot \\overrightarrow{OA_n}=0$, 则称$\\overrightarrow{OP}$是向量组$\\overrightarrow{OA_1}, \\overrightarrow{OA_2}, \\cdots, \\overrightarrow{OA_n}$的平衡向量. 已知$\\langle\\overrightarrow{OA_1}, \\overrightarrow{OA_2}\\rangle=\\dfrac{\\pi}{3}$, 向量$\\overrightarrow{OP}$是向量组$\\overrightarrow{OA_1}, \\overrightarrow{OA_2}, \\overrightarrow{OA_3}$的平衡向量, 当$\\overrightarrow{OP} \\cdot \\overrightarrow{OA_3}$取得最大值时, $\\overrightarrow{OA_1} \\cdot \\overrightarrow{OA_3}$的值为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-下学期周末卷-高三下学期周末卷08" ], "genre": "填空题", "ans": "$\\dfrac{-3\\pm \\sqrt{6}}6$", @@ -401834,7 +405707,8 @@ "content": "下列函数中, 既不是奇函数, 也不是偶函数的为\\bracket{20}.\n\\fourch{$y=0$}{$y=\\dfrac{1}{x}$}{$y=x^2$}{$y=2^x$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期周末卷-高三下学期周末卷08" ], "genre": "选择题", "ans": "D", @@ -401869,7 +405743,8 @@ "content": "在某区高三年级举行的一次质量检测中, 某学科共有$3000$人参加考试. 为了解本次考试学生的成绩情况, 从中抽取了部分学生的成绩 (成绩均为正整数, 满分为$100$分) 作为样本进行统计, 样本容量为$n$. 按照$[50,60),[60,70),[70,80),[80,90),[90,100]$的分组作出频率分布直方图 (如图所示). 已知成绩落在$[50,60)$内的人数为$16$, 则下列结论正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.05, yscale = 60]\n\\draw [->] (40,0) -- (42,0) -- (44,-0.003) -- (46,0.003) -- (48,0)-- (120,0) node [below] {成绩(分)};\n\\draw [->] (40,0) -- (40,0.05) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (40,0) node [below left] {$O$};\n\\foreach \\i/\\j in {50/0.016,60/0.03,70/0.04,80/0.01,90/0.004}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (10,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {50/0.016,60/0.03/x,70/0.04,80/0.01,90/0.004}\n{\\draw [dashed] (\\i,\\j) -- (40,\\j) node [left] {$\\k$};};\n\\draw (100,0) node [below] {$100$};\n\\end{tikzpicture}\n\\end{center}\n\\onech{样本容量$n=1000$}{图中$x=0.025$}{估计全体学生该学科成绩的平均分为$70.6$分}{若将该学科成绩由高到低排序, 前$15 \\%$的学生该学科成绩为A等, 则成绩为$78$分的学生该学科成绩肯定不是A等}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-下学期周末卷-高三下学期周末卷08" ], "genre": "选择题", "ans": "C", @@ -401904,7 +405779,8 @@ "content": "已知$f(x)=\\cos 2 x-a \\sin x$, 若存在正整数$n$, 使函数$y=f(x)$在区间$(0, n \\pi)$内有$2023$个零点, 则实数$a$所有可能的值为\\bracket{20}.\n\\fourch{$1$}{$-1$}{$0$}{$1$或$-1$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期周末卷-高三下学期周末卷08" ], "genre": "选择题", "ans": "B", @@ -401939,7 +405815,8 @@ "content": "若数列$\\{b_n\\}$、$\\{c_n\\}$均为严格增数列, 且对任意正整数$n$, 都存在正整数$m$, 使得$b_m \\in[c_n, c_{n+1}]$, 则称数列$\\{b_n\\}$为数列$\\{c_n\\}$的``M数列''. 已知数列$\\{a_n\\}$的前$n$项和为$S_n$, 则下列选项中为假命题的是\\bracket{20}.\n\\onech{存在等差数列$\\{a_n\\}$, 使得$\\{a_n\\}$是$\\{S_n\\}$的``M数列''}{存在等比数列$\\{a_n\\}$, 使得$\\{a_n\\}$是$\\{S_n\\}$的``M数列''}{存在等差数列$\\{a_n\\}$, 使得$\\{S_n\\}$是$\\{a_n\\}$的``M数列''}{存在等比数列$\\{a_n\\}$, 使得$\\{S_n\\}$是$\\{a_n\\}$的``M数列''}", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-下学期周末卷-高三下学期周末卷08" ], "genre": "选择题", "ans": "C", @@ -401974,7 +405851,8 @@ "content": "在$\\triangle ABC$中, 角$A$、$B$、$C$所对的边分别为$a$、$b$、$c$, 已知$\\sin A=\\sin 2B, a=4$, $b=6$.\\\\\n(1) 求$\\cos B$的值;\\\\\n(2) 求$\\triangle ABC$的面积.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期周末卷-高三下学期周末卷08" ], "genre": "解答题", "ans": "(1) $\\dfrac 13$; (2) $8\\sqrt{2}$", @@ -402009,7 +405887,8 @@ "content": "如图, 在四棱锥$P-ABCD$中, 底面$ABCD$为矩形, $PD \\perp$平面$ABCD$, $PD=AD=2$, $AB=4$, 点$E$在线段$AB$上, 且$BE=\\dfrac{1}{4} AB$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$D$} coordinate (D);\n\\draw (0,0,2) node [left] {$A$} coordinate (A);\n\\draw (4,0,0) node [right] {$C$} coordinate (C);\n\\draw (4,0,2) node [right] {$B$} coordinate (B);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw ($(A)!0.75!(B)$) node [below] {$E$} coordinate (E);\n\\draw (P)--(A)--(B)--(C)--cycle(P)--(E)(P)--(B);\n\\draw [dashed] (P)--(D)--(A)(D)--(C)(D)--(B)(A)--(C)(C)--(E);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $CE \\perp$平面$PBD$;\\\\\n(2) 求二面角$P-CE-A$的余弦值.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期周末卷-高三下学期周末卷08" ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\dfrac{4\\sqrt{21}}{21}$", @@ -402044,7 +405923,8 @@ "content": "在临床检测试验中, 某地用某种抗原来诊断试验者是否患有某种疾病. 设事件$A$表示试验者的检测结果为阳性, 事件$B$表示试验者患有此疾病. 据临床统计显示, $P(A | B)$$=0.99, P(\\overline {A} | \\overline {B})=0.98$. 已知该地人群中患有此种疾病的概率为$0.001$. (下列两小题计算结果中的概率值精确到$0.00001$)\\\\\n(1) 对该地某人进行抗原检测, 求事件$A$与$\\overline {B}$同时发生的概率;\\\\\n(2) 对该地$3$个患有此疾病的患者进行抗原检测, 用随机变量$X$表示检测结果为阳性的人数, 求$X$的分布和期望.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-下学期周末卷-高三下学期周末卷08" ], "genre": "解答题", "ans": "(1) $0.01998$; (2) 分布为$\\begin{pmatrix}0 & 1 & 2 & 3 \\\\ 0.00000 & 0.00030 & 0.02940 & 0.97030 \\end{pmatrix}$, 期望为$2.97$", @@ -402079,7 +405959,8 @@ "content": "已知$O$为坐标原点, 曲线$C_1: \\dfrac{x^2}{a^2}-y^2=1$($a>0$)和曲线$C_2: \\dfrac{x^2}{4}+\\dfrac{y^2}{2}=1$有公共点, 直线$l_1: y=k_1 x+b_1$与曲线$C_1$的左支相交于$A$、$B$两点, 线段$AB$的中点为$M$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.75]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\path [draw, name path = elli] (0,0) ellipse ({sqrt(2)} and 1);\n\\path [draw, name path = hypbo, samples = 100, domain = -2:3] plot ({-sqrt(\\x*\\x+1)},\\x); \n\\path [draw, name path = hypbo2, samples = 100, domain = -2:2] plot ({sqrt(\\x*\\x+1)},\\x); \n\\path [name path = AB] (-3,3) -- (-1.75,-2);\n\\path [name path = CD] (0,-2) -- (1,2);\n\\path [draw, name path = MN] (-3,0.75) -- (3,-0.75);\n\\path [name intersections = {of = AB and hypbo, by = {B,A}}];\n\\path [name intersections = {of = CD and elli, by = {C,D}}];\n\\path [name intersections = {of = AB and MN, by = M}];\n\\path [name intersections = {of = CD and MN, by = N}];\n\\draw (A) node [above] {$A$} -- (B) node [below] {$B$};\n\\draw (C) node [above] {$C$} -- (D) node [below] {$D$};\n\\draw (M) node [above] {$M$} (N) node [below] {$N$};\n\\end{tikzpicture}\n\\end{center}\n(1) 若曲线$C_1$和$C_2$有且仅有两个公共点, 求曲线$C_1$的离心率和渐近线方程;\\\\\n(2) 若直线$OM$经过曲线$C_2$上的点$T(\\sqrt{2},-1)$, 且$a^2$为正整数, 求$a$的值;\\\\\n(3) 若直线$l_2: y=k_2 x+b_2$与曲线$C_2$相交于$C$、$D$两点, 且直线$OM$经过线段$CD$中点$N$, 求证: $k_1^2+k_2^2>1$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期周末卷-高三下学期周末卷08" ], "genre": "解答题", "ans": "(1) 离心率为$\\dfrac{\\sqrt{5}}2$, 渐近线方程为$y=\\pm \\dfrac 12 x$; (2) $1$; (3) 证明略", @@ -402114,7 +405995,8 @@ "content": "如果曲线$y=f(x)$存在相互垂直的两条切线, 称函数$y=f(x)$是``正交函数''. 已知$f(x)=x^2+a x+2 \\ln x$, 设曲线$y=f(x)$在点$M(x_0, f(x_0))$处的切线为$l_1$.\\\\\n(1) 当$f'(1)=0$时, 求实数$a$的值;\\\\\n(2) 当$a=-8$, $x_0=8$时, 是否存在直线$l_2$满足$l_1 \\perp l_2$, 且$l_2$与曲线$y=f(x)$相切? 请说明理由;\\\\\n(3) 当$a \\geq-5$时, 如果函数$y=f(x)$是``正交函数'', 求满足要求的实数$a$的集合$D$; 若对任意$a \\in D$, 曲线$y=f(x)$都不存在与$l_1$垂直的切线$l_2$, 求$x_0$的取值范围.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期周末卷-高三下学期周末卷08" ], "genre": "解答题", "ans": "(1) $-4$; (2) 存在切线$l_2$与$l_1$垂直, 理由略; (3) $a$的取值范围为$[-5,-4)$, $x_0$的取值范围为$(\\dfrac{3-\\sqrt{5}}{2},\\dfrac 12]\\cup [2,\\dfrac{3+\\sqrt{5}}{2})$", @@ -402612,7 +406494,8 @@ "content": "已知集合$A=\\{1,2,3,4\\}$, $B=\\{x | \\dfrac{2}{x}>1\\}$, 则$A \\cap B=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-下学期周末卷-高三下学期周末卷07" ], "genre": "填空题", "ans": "$\\{1\\}$", @@ -402645,7 +406528,8 @@ "content": "若复数$z$满足$\\mathrm{i} \\cdot z=3-4 \\mathrm{i}$, 则$|\\overline {z}|=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-下学期周末卷-高三下学期周末卷07" ], "genre": "填空题", "ans": "$5$", @@ -402678,7 +406562,8 @@ "content": "已知空间向量$\\overrightarrow {a}=(1,2,3)$, $\\overrightarrow {b}=(2,-2,0)$, $\\overrightarrow {c}=(1,1, \\lambda)$, 若$\\overrightarrow {c} \\perp(2 \\overrightarrow {a}+\\overrightarrow {b})$, 则$\\lambda=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-下学期周末卷-高三下学期周末卷07" ], "genre": "填空题", "ans": "$-1$", @@ -402711,7 +406596,8 @@ "content": "已知随机变量$X$服从正态分布$N(0,1)$, 若$P(X<-1.96)=0.03$, 则$P(|X|<1.96)=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-下学期周末卷-高三下学期周末卷07" ], "genre": "填空题", "ans": "$0.94$", @@ -402744,7 +406630,8 @@ "content": "已知$\\dfrac{\\pi}{2}<\\theta<\\pi$, 且$\\cos \\theta=-\\dfrac{4}{5}$, 则$\\tan 2 \\theta=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期周末卷-高三下学期周末卷07" ], "genre": "填空题", "ans": "$-\\dfrac{24}{7}$", @@ -402777,7 +406664,8 @@ "content": "在二项式$(x-\\dfrac{1}{x})^8$的展开式中, 含$x^4$的项的系数是\\blank{50}.(用数字作答)", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-下学期周末卷-高三下学期周末卷07" ], "genre": "填空题", "ans": "$28$", @@ -402810,7 +406698,8 @@ "content": "将右图所示的圆锥形容器内的液体全部倒入底面半径为$50 \\text{mm}$的直立的圆柱形容器内, 则液面高度为\\blank{50}$\\text{mm}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\fill [pattern = north east lines] (-0.5,1.5) arc (180:0:0.5 and 0.125) -- (0,0)--cycle;\n\\draw (0,0) -- (1,3) (0,0) -- (-1,3) -- (1,3);\n\\draw (0,3) ellipse (1 and 0.25);\n\\draw (-0.5,1.5) arc (180:360:0.5 and 0.125);\n\\draw [dashed] (-0.5,1.5) arc (180:0:0.5 and 0.125);\n\\draw (-1.1,0) -- (-1.5,0) (-1.1,3) -- (-1.5,3);\n\\draw [<->] (-1.3,0) -- (-1.3,3) node [midway, fill = white] {\\tiny$300\\text{mm}$};\n\\draw (-1,3.1) -- (-1,3.7) (1,3.1) -- (1,3.7);\n\\draw [<->] (-1,3.5) -- (1,3.5) node [midway, fill=white] {\\tiny$200\\text{mm}$};\n\\draw (0.8,1.5) -- (1.2,1.5) (0.8,0) -- (1.2,0);\n\\draw [<->] (1,1.5) -- (1,0) node [midway, fill=white] {\\tiny$150\\text{mm}$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期周末卷-高三下学期周末卷07" ], "genre": "填空题", "ans": "$50$", @@ -402843,7 +406732,8 @@ "content": "从$4$名男生和$3$名女生中抽取两人加入志愿者服务队. 用$A$表示事件``抽到的两名学生性别相同'', 用$B$表示事件``抽到的两名学生都是女生'', 则$P(B | A)=$\\blank{50}.(结果用最简分数表示)", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-下学期周末卷-高三下学期周末卷07" ], "genre": "填空题", "ans": "$\\dfrac 13$", @@ -402876,7 +406766,8 @@ "content": "参考《九章算术》中``竹九节''问题, 提出: 一根$9$节的竹子, 自上而下各节的容积成等差数列, 上面$4$节的容积共$2$升, 下面$3$节的容积共$3$升, 则第$5$节的容积为\\blank{50}升.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-下学期周末卷-高三下学期周末卷07" ], "genre": "填空题", "ans": "$\\dfrac 8{11}$", @@ -402909,7 +406800,8 @@ "content": "已知$x \\in(0, \\dfrac{\\pi}{2})$, 则$\\dfrac{1}{\\sin ^2 x}+\\dfrac{4}{\\cos ^2 x}$的最小值为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期周末卷-高三下学期周末卷07" ], "genre": "填空题", "ans": "$9$", @@ -402942,7 +406834,8 @@ "content": "已知函数$y=f(x)$为$\\mathbf{R}$上的奇函数, 且$f(x)+f(2-x)=0$, 当$-10$). 若存在$m, n \\in \\mathbf{R}$, 使得$m \\overrightarrow{AB}+\\overrightarrow{OA}$与$n \\overrightarrow{AB}+\\overrightarrow{OB}$垂直, 且$|(m \\overrightarrow{AB}+\\overrightarrow{OA})-(n \\overrightarrow{AB}+\\overrightarrow{OB})|=a$, 则$|\\overrightarrow{AB}|$的最小值为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-下学期周末卷-高三下学期周末卷07" ], "genre": "填空题", "ans": "$\\sqrt{15}a$", @@ -403008,7 +406902,8 @@ "content": "已知直线$l_1: a x+y+1=0$与直线$l_2: x+a y-2=0$, 则``$l_1\\parallel l_2$''是``$a=1$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期周末卷-高三下学期周末卷07" ], "genre": "选择题", "ans": "B", @@ -403041,7 +406936,8 @@ "content": "为了解某社区居民的家庭年收入与年支出的关系, 随机调查了该社区$5$户家庭, 得到如下统计数据表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline 收入$x$(万元) & 8.2 & 8.6 & 10.0 & 11.3 & 11.9 \\\\\n\\hline 支出$y$(万元) & 6.2 & 7.5 & 8.0 & 8.5 & 9.8 \\\\\n\\hline\n\\end{tabular} \n\\end{center}\n根据上表可得回归直线方程$y=\\hat{a} x+\\hat{b}$, 其中$\\hat{a}=0.76$, $\\hat{b}=\\overline {y}-\\hat{a} \\overline {x}$, 据此估计, 该社区一户收入为$15$万元家庭年支出为\\bracket{20}.\n\\fourch{$11.4$万元}{$11.8$万元}{$12.0$万元}{$12.2$万元}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-下学期周末卷-高三下学期周末卷07" ], "genre": "选择题", "ans": "B", @@ -403074,7 +406970,8 @@ "content": "若方程$f(x) \\cdot g(x)=0$的解集为$M$, 则以下结论一定正确的是\\bracket{20}.\\\\\n\\textcircled{1} $M=\\{x | f(x)=0\\} \\cup\\{x | g(x)=0\\}$;\\\\\n\\textcircled{2} $M=\\{x | f(x)=0\\} \\cap\\{x | g(x)=0\\}$;\\\\\n\\textcircled{3} $M \\subseteq\\{x | f(x)=0\\} \\cup\\{x | g(x)=0\\}$;\\\\\n\\textcircled{4} $M \\supseteq\\{x | f(x)=0\\} \\cap\\{x | g(x)=0\\}$.\n\\fourch{\\textcircled{1}\\textcircled{4}}{\\textcircled{2}\\textcircled{4}}{\\textcircled{3}\\textcircled{4}}{\\textcircled{1}\\textcircled{3}\\textcircled{4}}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-下学期周末卷-高三下学期周末卷07" ], "genre": "选择题", "ans": "C", @@ -403107,7 +407004,8 @@ "content": "已知函数$y=\\dfrac{1}{3} x^3-x^2-3 x+a$, $a \\in \\mathbf{R}$, 在区间$(t-3, t+5)$上有最大值, 则实数$t$的取值范围是\\bracket{20}.\n\\fourch{$-6=latex]\n\\draw (0,0,0) node [below] {$O$} coordinate (O);\n\\draw (0,0,{2*sin(22.5)}) node [below] {$A$} coordinate (A);\n\\draw (0,0,{-2*sin(22.5)}) node [below] {$C$} coordinate (C);\n\\draw ({-2*cos(22.5)},0,0) node [left] {$D$} coordinate (D);\n\\draw ({2*cos(22.5)},0,0) node [right] {$B$} coordinate (B);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw ($(P)!0.5!(D)$) node [left] {$M$} coordinate (M);\n\\draw (P)--(D)--(A)--(B)--cycle(P)--(A)(M)--(A);\n\\draw [dashed] (D)--(C)--(B)(D)--(B)(A)--(C)(P)--(O)(P)--(C)(C)--(M);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $PB\\parallel$平面$ACM$;\\\\\n(2) 求直线$AM$与平面$ABCD$所成角的大小.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期周末卷-高三下学期周末卷07" ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\arctan\\dfrac{2\\sqrt{5}}5$", @@ -403206,7 +407106,8 @@ "content": "某城市响应国家号召, 积极调整能源结构, 推出多种价位的新能源电动汽车. 根据前期市场调研, 有购买新能源车需求的约有$2$万人, 他们的选择意向统计如下:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline 车型 &A&B&C&D&E&F\\\\\n\\hline 价格 & 9 万元 & 12 万元 & 18 万元 & 24 万元 & 30 万元 & 40 万元 \\\\\n\\hline 占比 &$5 \\%$&$15 \\%$&$25 \\%$&$35 \\%$&$15 \\%$&$5 \\%$\\\\\n\\hline\n\\end{tabular} \n\\end{center}\n(1) 如果有购车需求的这些人今年都购买了新能源车, 今年新能源车的销售额预计约为多少亿元?\\\\\n(2) 车企推出两种付款方式: 全款购车: 购车时一次性付款可优惠车价的$3 \\%$; 分期付款: 无价格优惠, 购车时先付车价的一半, 余下的每半年付一次, 分$4$次付完, 每次付车价的$\\dfrac{1}{8}$.\\\\\n(i) 某位顾客现有$a$万元现金, 欲购买价值$a$万元的某款车, 付款后剩余的资金全部用于购买半年期的理财产品(该理财产品半年期到期收益率为$1.8 \\%$), 到期后, 可用资金(含理财收益)继续购买半年期的理财产品. 问: 顾客选择哪一种付款方式收益更多? (计算结果精确到$0.0001$)\\\\\n(ii) 为了激励购买理财产品, 银行对采用分期付款方式的顾客, 赠送价值$1888$元的大礼包, 试问: 这一措施对哪些车型有效? (计算结果精确到$0.0001$)", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期周末卷-高三下学期周末卷07" ], "genre": "解答题", "ans": "(1) $43.3$亿元; (2) (i) 全款购车两年后资产总额为$0.0322a$万元, 分期付款购车两年后资产总额为$0.0233a$万元, 应选择全款购车; (ii) $a<21.2134$, 这一措施对购买A, B, C车型有效", @@ -403239,7 +407140,8 @@ "content": "已知椭圆$C_1: \\dfrac{x^2}{2}+\\dfrac{y^2}{b^2}=1$的左、右焦点分别为$F_1$、$F_2$, 离心率为$e_1$; 双曲线\n$C_2: \\dfrac{x^2}{2}-\\dfrac{y^2}{b^2}=1$的左、右焦点分别为$F_3$、$F_4$, 离心率为$e_2$, $e_1 \\cdot e_2=\\dfrac{\\sqrt{3}}{2}$. 过点$F_1$作不垂直于$y$轴的直线$l$交曲线$C_1$于点$A$、$B$, 点$M$为线段$AB$的中点, 直线$OM$交曲线$C_2$于$P$、$Q$两点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,0) ellipse ({sqrt(2)} and 1);\n\\draw [domain = -1.6:1.6, samples = 100] plot ({sqrt(2*\\x*\\x+2)},\\x);\n\\draw [domain = -1.6:1.6, samples = 100] plot ({-sqrt(2*\\x*\\x+2)},\\x);\n\\filldraw (-1,0) circle (0.03) node [below] {$F_1$} coordinate (F_1);\n\\filldraw (1,0) circle (0.03) node [below] {$F_2$} coordinate (F_2);\n\\filldraw ({-sqrt(3)},0) circle (0.03) node [below] {$F_3$} coordinate (F_3);\n\\filldraw ({sqrt(3)},0) circle (0.03) node [below] {$F_4$} coordinate (F_4);\n\\end{tikzpicture}\n\\end{center}\n(1) 求$C_1$、$C_2$的方程;\\\\\n(2) 若$\\overrightarrow{AF_1}=3 \\overrightarrow{F_1B}$, 求直线$PQ$的方程;\\\\\n(3) 求四边形$APBQ$面积的最小值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期周末卷-高三下学期周末卷07" ], "genre": "解答题", "ans": "(1) $C_1:\\dfrac{x^2}2+y^2=1$, $C_2:\\dfrac{x^2}2-y^2=1$; (2) $y=\\dfrac 12 x$或$y=-\\dfrac 12 x$; (3) $2$", @@ -403272,7 +407174,8 @@ "content": "已知$x>0$, 记$f(x)=\\mathrm{e}^x$, $g(x)=x^x$, $h(x)=\\ln g(x)$.\\\\\n(1) 试将$y=f(x)$、$y=g(x)$、$y=h(x)$中的一个函数表示为另外两个函数复合而成的复合函数;\\\\\n(2) 借助 (1) 的结果, 求函数$y=g(2 x)$的导函数和最小值;\\\\\n(3) 记$H(x)=\\dfrac{f(x)-h(x)}{x}+x+a$, $a$是实常数, 函数$y=H(x)$的导函数是$y'=H'(x)$. 已知函数$y=H(x) \\cdot H'(x)$有三个不相同的零点$x_1$、$x_2$、$x_3$. 求证: $x_1 \\cdot x_2 \\cdot x_3<1$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期周末卷-高三下学期周末卷07" ], "genre": "解答题", "ans": "(1) $g(x)=f(h(x))$; (2) 导函数为$2(2x)^{2x}(1+\\ln (2x))$, 最小值为$(\\dfrac 1{\\mathrm{e}})^{\\frac 1{\\mathrm{e}}}$; (3) 证明略", @@ -405239,7 +409142,8 @@ "content": "已知集合$A=\\{x \\| x |<3\\}$, $B=\\{x | y=\\sqrt{2-x}\\}$, 则$A \\cup B=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-下学期周末卷-高三下学期周末卷09" ], "genre": "填空题", "ans": "$(-\\infty,3)$", @@ -405271,7 +409175,8 @@ "content": "若角$\\alpha$的终边过点$P(4,-3)$, 则$\\sin (\\dfrac{3 \\pi}{2}+\\alpha)=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期周末卷-高三下学期周末卷09" ], "genre": "填空题", "ans": "$-\\dfrac 45$", @@ -405310,7 +409215,8 @@ "content": "抽取某校高一年级$10$名女生, 测得她们的身高 (单位: $\\text{cm}$) 数据如下: $163$, $165$, $161$, $157$, $162$, $165$, $158$, $155$, $164$, $162$, 据此估计该校高一年级女生身高的第$25$百分位数是\\blank{50}.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-下学期周末卷-高三下学期周末卷09" ], "genre": "填空题", "ans": "$158$", @@ -405342,7 +409248,8 @@ "content": "命题``若$x>a$, 则$\\dfrac{x-1}{x}>0$''是真命题, 实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-下学期周末卷-高三下学期周末卷09" ], "genre": "填空题", "ans": "$[1,+\\infty)$", @@ -405374,7 +409281,8 @@ "content": "在正项等比数列$\\{a_n\\}$中, $a_5^2+2 a_6 a_8+a_9^2=100$, 则$a_5+a_9=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-下学期周末卷-高三下学期周末卷09" ], "genre": "填空题", "ans": "$10$", @@ -405406,7 +409314,8 @@ "content": "设一组样本数据$x_1, x_2, \\cdots, x_n$的方差为$0.01$, 则数据$10 x_1, 10 x_2, \\cdots, 10 x_n$的方差为\\blank{50}.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-下学期周末卷-高三下学期周末卷09" ], "genre": "填空题", "ans": "$1$", @@ -405438,7 +409347,8 @@ "content": "如图所示, 圆锥$SO$的底面圆半径$OA=1$, 侧面的平面展开图的面积为$3 \\pi$, 则此圆锥的体积为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw (0,0) node [left] {$O$} coordinate (O);\n\\draw (1,0) node [below] {$A$} coordinate (A);\n\\draw (0,{2*sqrt(2)}) node [above left] {$S$} coordinate (S);\n\\draw (A) arc ({-acos(1/3)}:{-acos(1/3)+120}:3) node [above] {$B$} coordinate (B);\n\\draw (A)--(S)--(-1,0) arc (180:360:1 and 0.25) (B)--(S);\n\\draw [dashed] (S)--(O)--(A) arc (0:180:1 and 0.25);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期周末卷-高三下学期周末卷09" ], "genre": "填空题", "ans": "$\\dfrac{2\\sqrt{2}}3\\pi$", @@ -405470,7 +409380,8 @@ "content": "若$(1+x)(1-2 x)^{2023}=a_0+a_1 x+a_2 x^2+\\cdots+a_{2024} x^{2024}$, $a_i \\in \\mathbf{R}$($i=0,1,2, \\cdots, 2024$), 则$a_1+a_2+\\cdots+a_{2024}=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-下学期周末卷-高三下学期周末卷09" ], "genre": "填空题", "ans": "$-3$", @@ -405504,7 +409415,8 @@ "content": "己知双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的左焦点为$F(-1,0)$, 过$F$且与$x$轴垂直的直线与\n双曲线交于$A$、$B$两点, $O$为坐标原点, $\\triangle AOB$的面积为$\\dfrac{3}{2}$, 则$F$到双曲线的浙近线距离为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期周末卷-高三下学期周末卷09" ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{3}}2$", @@ -405536,7 +409448,8 @@ "content": "甲、乙、丙、丁四名同学互不影响地报名参加高中社会实践活动, 高中社会实践活动共有博物馆讲解、养老院慰问、交通宣传、超市导购四个项目, 每人限报其中一项, 记事件$A$为``$4$名同学所报项目各不相同'', 事件$B$为``只有甲同学一人报交通宣传项目, 则$P(A | B)=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-下学期周末卷-高三下学期周末卷09" ], "genre": "填空题", "ans": "$\\dfrac 29$", @@ -405568,7 +409481,8 @@ "content": "已知函数$f(x)=x+\\dfrac{a}{x}+b$, $x \\in[b,+\\infty)$, $a \\in \\mathbf{R}$, 若存在$b\\in (0,+\\infty)$, 使得$f(x)$的最小值为$2$, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期周末卷-高三下学期周末卷09" ], "genre": "填空题", "ans": "$(-\\infty,1)$", @@ -405603,7 +409517,8 @@ "tags": [ "第二单元", "第三单元", - "第四单元" + "第四单元", + "2023届高三-下学期周末卷-高三下学期周末卷09" ], "genre": "填空题", "ans": "$10$", @@ -405636,7 +409551,8 @@ "content": "已知$z \\in \\mathbf{C}$, 则``$z+\\overline {z}=0$''是``$z$为纯虚数''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-下学期周末卷-高三下学期周末卷09" ], "genre": "选择题", "ans": "B", @@ -405669,7 +409585,8 @@ "content": "某社区通过公益讲座宣传中国非物质文化遗产保护知识. 为了解讲座效果, 随机抽取$10$位社区居民, 让他们在讲座前和讲座后各回答一份相关知识问卷, 这$10$位社区居民在讲座前和讲座后问卷答题的正确率如下图. 则下列选项正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.7, yscale = 0.08]\n\\draw [->] (0,55) -- (11,55);\n\\draw [->] (0,55) -- (0,56) -- (0.2,57) -- (-0.2,58) -- (0,59) -- (0,105);\n\\draw (0,55) node [below left] {$O$};\n\\foreach \\i in {1,2,...,10}\n{\\draw (\\i,55.5) -- (\\i,55) node [below] {$\\i$};};\n\\foreach \\i in {60,65,...,100}\n{\\draw [dotted] (10.5,\\i) -- (0,\\i) node [left] {$\\i\\%$};};\n\\draw (5.5,45) node {居民编号};\n\\draw (-2,80) node [rotate = 90] {正确率};\n\\filldraw (12,70) circle (0.05 and {7/16}) ++ (1,0) node {讲座后};\n\\filldraw (12,80) node {\\tiny$\\times$} node {\\tiny$+$} ++ (1,0) node {讲座前};\n\\foreach \\i/\\j/\\k in {1/65/90,2/60/85,3/70/80,4/60/90,5/65/85,6/75/85,7/90/95,8/85/100,9/80/85,10/95/100}\n{\\filldraw (\\i,\\j) node {\\tiny$\\times$} node {\\tiny$+$} (\\i,\\k) circle (0.05 and {7/16});};\n\\end{tikzpicture}\n\\end{center}\n\\onech{讲座前问卷答题的正确率的中位数小于$70 \\%$}{讲座后问卷答题的正确率的平均数大于$85 \\%$}{讲座前问卷答题的正确率的标准差小于讲座后正确率的标准差}{讲座后问卷答题的正确率的极差大于讲座前正确率的极差}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-下学期周末卷-高三下学期周末卷09" ], "genre": "选择题", "ans": "B", @@ -405701,7 +409618,8 @@ "content": "设函数$f(x)=\\begin{cases}-x^2-2 x,& x \\leq 0, \\\\ |\\ln x|,& x>0,\\end{cases}$ 现有如下命题: \\textcircled{1} 若方程$f(x)=a$有四个不同的实根$x_1$、$x_2$、$x_3$、$x_4$, 则$x_1 \\cdot x_2 \\cdot x_3 \\cdot x_4$的取值范围是$(0,1)$; \\textcircled{2} 方程$f^2(x)-(a+\\dfrac{1}{a}) f(x)+1=0$的不同实根的个数只能是$1 , 2 , 3 , 8$, 下列判断正确的是\\bracket{20}.\n\\twoch{\\textcircled{1}和\\textcircled{2}均为真命题}{\\textcircled{1}和\\textcircled{2}均为假命题}{\\textcircled{1}为真命题, \\textcircled{2}为假命题}{\\textcircled{1}为假命题, \\textcircled{2}为真命题}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期周末卷-高三下学期周末卷09" ], "genre": "选择题", "ans": "C", @@ -405734,7 +409652,8 @@ "content": "如图: 棱长为$2$的正方体$ABCD-A_1B_1C_1D_1$的内切球为球$O$, $E$、$F$分别是棱$AB$和棱$CC_1$的中点, $G$在棱$BC$上移动, 则下列命题正确的个数是\\bracket{20}.\\\\\n\\textcircled{1} 存在点$G$, 使$OD$垂直于平面$EFG$;\\\\\n\\textcircled{2} 对于任意点$G, OA$平行于平面$EFG$;\\\\\n\\textcircled{3} 直线$EF$被球$O$截得的弦长为$\\sqrt{2}$;\\\\\n\\textcircled{4} 过直线$EF$的平面截球$O$所得的所有截面圆中, 半径最小的圆的面积为$\\dfrac{\\pi}{2}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$B$} coordinate (B);\n\\draw (B) ++ (\\l,0,0) node [below right] {$C$} coordinate (C);\n\\draw (B) ++ (\\l,0,-\\l) node [right] {$D$} coordinate (D);\n\\draw (B) ++ (0,0,-\\l) node [above right] {$A$} coordinate (A);\n\\draw (B) -- (C) -- (D);\n\\draw [dashed] (B) -- (A) -- (D);\n\\draw (A) ++ (0,\\l,0) node [above] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [left] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [right] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (D) -- (D1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (A) -- (A1);\n\\filldraw ($(A1)!0.5!(C)$) node [right] {$O$} coordinate (O) circle (0.03);\n\\draw [dashed] (O) circle (1) ellipse (1 and 0.25);\n\\draw ($(A)!0.5!(B)$) node [left] {$E$} coordinate (E);\n\\draw ($(C)!0.5!(C1)$) node [right] {$F$} coordinate (F);\n\\draw ($(B)!0.8!(C)$) node [below] {$G$} coordinate (G);\n\\draw (F)--(G);\n\\draw [dashed] (F)--(E)--(G);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{0}{1}{2}{3}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期周末卷-高三下学期周末卷09" ], "genre": "选择题", "ans": "D", @@ -405767,7 +409686,8 @@ "objs": [], "tags": [ "第八单元", - "第九单元" + "第九单元", + "2023届高三-下学期周末卷-高三下学期周末卷09" ], "genre": "解答题", "ans": "(1) $2\\times 2$列联表为\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n& 年龄在$50$周岁以上(含$50$周岁) & 年龄在$50$周岁以下 & 总计\\\\\\hline\n支持态度人数 & $60$ & $180$ & $240$ \\\\\\hline\n不支持态度人数 & $30$ & $30$ & $60$ \\\\\\hline\n总计 & $90$ & $210$ & $300$ \\\\ \\hline\n\\end{tabular}. $\\chi^2\\approx 14.3>3.841$, 有$95\\%$的把握认为年龄与所持态度具有相关性; (2) $X\\sim \\begin{pmatrix}0 & 1 & 2 & 3 & 4 \\\\ \\dfrac 1{81} & \\dfrac 8{81} & \\dfrac {24}{81} & \\dfrac {32}{81} & \\dfrac {16}{81}\\end{pmatrix}$, $E[X]=\\dfrac 83$", @@ -405799,7 +409719,8 @@ "content": "已知向量$\\overrightarrow {m}=(2 \\sqrt{3} \\cos \\dfrac{x}{2},-2 \\sin \\dfrac{x}{2})$, $\\overrightarrow {n}=(\\cos \\dfrac{x}{2}, \\cos \\dfrac{x}{2})$, 函数$y=f(x)=\\overrightarrow {m} \\cdot \\overrightarrow {n}$.\\\\\n(1) 设$\\theta \\in[-\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2}]$, 且$f(\\theta)=\\sqrt{3}+1$, 求$\\theta$的值;\\\\\n(2) 在$\\triangle ABC$中, $AB=1$, $f(C)=\\sqrt{3}+1$, 且$\\triangle ABC$的面积为$\\dfrac{\\sqrt{3}}{2}$, 求$\\sin A+\\sin B$的值.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期周末卷-高三下学期周末卷09" ], "genre": "解答题", "ans": "(1) $-\\dfrac\\pi 2$或$\\dfrac \\pi 6$; (2) $1+\\dfrac{\\sqrt{3}}2$", @@ -405831,7 +409752,8 @@ "content": "如图, 在直三棱柱$ABC-A_1B_1C_1$中, $\\angle BAC=90^{\\circ}$, $AB=AC=a$, $AA_1=b$, 点$E$、$F$分別在棱$BB_1$、$CC_1$上, 且$BE=\\dfrac{1}{3} BB_1$, $C_1F=\\dfrac{1}{3} CC_1$. 设$\\lambda=\\dfrac{b}{a}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,-1) node [below] {$A$} coordinate (A);\n\\draw (1,0,0) node [right] {$C$} coordinate (C);\n\\draw (-1,0,0) node [left] {$B$} coordinate (B);\n\\draw (A) ++ (0,3,0) node [above] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,3,0) node [left] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,3,0) node [right] {$C_1$} coordinate (C_1);\n\\draw ($(B)!{1/3}!(B_1)$) node [left] {$E$} coordinate (E);\n\\draw ($(C)!{2/3}!(C_1)$) node [right] {$F$} coordinate (F);\n\\draw (B)--(C)--(C_1)--(A_1)--(B_1)--cycle(B_1)--(C_1) (E)--(F);\n\\draw [dashed] (A)--(E)(A)--(F)(A_1)--(E)(A_1)--(F)(A)--(A_1)(B)--(A)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 当$\\lambda=3$时, 求异面直线$AE$与$A_1F$所成的角的大小;\\\\\n(2) 当平面$AEF \\perp$平面$A_1EF$时, 求$\\lambda$的值.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期周末卷-高三下学期周末卷09" ], "genre": "解答题", "ans": "(1) $60^\\circ$; (2) $\\lambda = \\dfrac 32$", @@ -405863,7 +409785,8 @@ "content": "已知椭圆$C: \\dfrac{x^2}{t}+y^2=1$($t>1$)的左、右焦点分别为$F_1, F_2$, 直线$l: y=k x+m$($m \\neq 0$)与椭圆$C$交于$M$、$N$两点 ($M$点在$N$点的上方), 与$y$轴交于点$E$.\\\\\n(1) 当$t=2$时, 点$A$为椭圆$C$上除顶点外任一点, 求$\\triangle AF_1F_2$的周长;\\\\\n(2) 当$t=3$且直线$l$过点$D(-1,0)$时, 设$\\overrightarrow{EM}=\\lambda \\overrightarrow{DM}$, $\\overrightarrow{EN}=\\mu \\overrightarrow{DN}$, 求证: $\\lambda+\\mu$为定值, 并求出该值;\\\\\n(3) 若椭圆$C$的离心率为$\\dfrac{\\sqrt{3}}{2}$, 当$k$为何值时, $|OM|^2+|ON|^2$恒为定值? 并求此时$\\triangle MON$面积的最大值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期周末卷-高三下学期周末卷09" ], "genre": "解答题", "ans": "(1) $2+2\\sqrt{2}$; (2) $\\lambda+\\mu=3$, 证明略; (3) $k=\\pm \\dfrac 12$, 面积的最大值为$1$", @@ -405895,7 +409818,8 @@ "content": "已知常数$k$为非零整数, 若函数$y=f(x)$, $x \\in[0,1]$满足: 对任意$x_1, x_2 \\in[0,1]$, $|f(x_1)-f(x_2)| \\leq|(x_1+1)^k-(x_2+1)^k|$, 则称函数$y=f(x)$为$L(k)$函数.\\\\\n(1) 函数$y=2 x$, $x \\in[0,1]$是否为$L(2)$函数? 请说明理由;\\\\\n(2) 若$y=f(x)$为$L(1)$函数, 图像在$x \\in [0,1]$是一条连续的曲线, $f(0)=0$, $f(1)=\\dfrac{1}{2}$, 且$f(x)$在区间$(0,1)$上仅存在一个极值点, 分别记$f(x)_{\\max}$、$f(x)_{\\min}$为函数$y=f(x)$的最大、最小值, 求$f(x)_{\\max}-f(x)_{\\min}$的取值范围;\\\\\n(3) 若$a>0$, $f(x)=0.05 x^2+0.1 x+a \\ln (x+1)$, 且$y=f(x)$为$L(-1)$函数, $g(x)=f'(x)$, 对任意$x, y \\in[0,1]$, 恒有$|g(x)-g(y)| \\leq M$, 记$M$的最小值为$M(a)$, 求$a$的取值范围及$M(a)$关于$a$的表达式.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期周末卷-高三下学期周末卷09" ], "genre": "解答题", "ans": "(1) 是$L(2)$函数, 理由略; (2) $(\\dfrac 12,\\dfrac 34]$; (3) $a$的取值范围为$(0,\\dfrac 1{10}]$, $M(a)=0.1-\\dfrac 12a$", @@ -448002,7 +451926,8 @@ "content": "集合$A=\\{-1,0,1,2\\}$, $B=\\{x | 00$), 若$E[X]=E[Y]$, 且$P(|Y|<1)=0.4$, 则$P(Y>3)=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-下学期测验卷-高三下学期月考02" ], "genre": "填空题", "ans": "$0.1$", @@ -448317,7 +452250,8 @@ "content": "已知甲袋中有$3$个白球和$2$个红球, 乙袋中有$2$个白球和$4$个红球. 若先随机取一只袋, 再从该袋中先后随机取$2$个球, 则在第一次取出的球是红球的前提下, 第二次取出的球是白球的概率为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-下学期测验卷-高三下学期月考02" ], "genre": "填空题", "ans": "$\\dfrac{17}{32}$", @@ -448353,7 +452287,8 @@ "objs": [], "tags": [ "第八单元", - "第三单元" + "第三单元", + "2023届高三-下学期测验卷-高三下学期月考02" ], "genre": "填空题", "ans": "$[2,2^{2023}]$", @@ -448388,7 +452323,8 @@ "content": "已知平面向量$\\overrightarrow {a}, \\overrightarrow {b}, \\overrightarrow {c}$满足$|\\overrightarrow {a}|=1$, $\\langle\\overrightarrow {a}, \\overrightarrow {b}\\rangle=\\langle 7 \\overrightarrow {a}-\\overrightarrow {c}, 9 \\overrightarrow {a}-\\overrightarrow {c}\\rangle=\\dfrac{\\pi}{6}$, 则$|\\overrightarrow {b}-\\overrightarrow {c}|$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-下学期测验卷-高三下学期月考02" ], "genre": "填空题", "ans": "$[\\dfrac{1}{2},+\\infty)$", @@ -448423,7 +452359,8 @@ "content": "若$a<0\\dfrac{1}{b}$}{$-a>b$}{$a^2>b^2$}{$a^3a_m$''是``$\\{a_n\\}$是严格递增数列''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充要条件}{既不充分也不必要条件}", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-下学期测验卷-高三下学期月考02" ], "genre": "选择题", "ans": "C", @@ -448528,7 +452467,8 @@ "content": "已知函数$f(x)$是定义在$\\mathbf{R}$上的连续可导函数, 其导函数为$f'(x)$, 且对任意$x \\in \\mathbf{R}$均有$f(x)-f(-x)=2 x$. 若当$x<0$时, $f'(x)>1$恒成立, 且$f(a-2)-f(1-2 a)>3 a-3$, 则实数$a$的取值范围是\\bracket{20}.\n\\fourch{$(-1,1)$}{$(1,+\\infty)$}{$(-\\infty,-1)$}{$(-\\infty,-1) \\cup(1,+\\infty)$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期测验卷-高三下学期月考02" ], "genre": "选择题", "ans": "D", @@ -448563,7 +452503,8 @@ "content": "直三棱柱$ABC-A_1B_1C_1$中, 底面$ABC$为等腰直角三角形, $AB \\perp AC$, $AB=AC=2$, $AA_1=4, M$是侧棱$CC_1$上一点, 设$MC=h$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,2) node [left] {$B$} coordinate (B);\n\\draw (A) ++ (0,3,0) node [above] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,3,0) node [left] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,3,0) node [right] {$C_1$} coordinate (C_1);\n\\draw ($(C)!0.3!(C_1)$) node [right] {$M$} coordinate (M);\n\\draw (B)--(C)--(C_1)--(B_1)--cycle (B_1)--(A_1)--(C_1)(B)--(M);\n\\draw [dashed] (B)--(A)--(C)(B)--(A_1)--(C)(A)--(M)(A)--(A_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 若$BM \\perp A_1C$, 求$h$的值;\\\\\n(2) 若$h=2$, 求直线$BA_1$与平面$ABM$所成的角.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期测验卷-高三下学期月考02" ], "genre": "解答题", "ans": "(1) $h=1$; (2) $\\arcsin\\dfrac{\\sqrt{10}}{5}$", @@ -448598,7 +452539,8 @@ "content": "已知$\\triangle ABC$的内角$A, B, C$的对边分别为$a, b, c$.\\\\\n(1) 若$B=\\dfrac{\\pi}{3}$, $b=\\sqrt{5}$, $\\triangle ABC$的面积$S=\\sqrt{3}$, 求$a-c$的值;\\\\\n(2) 若$2 \\cos C(\\overrightarrow{BA} \\cdot \\overrightarrow{BC}+\\overrightarrow{AB} \\cdot \\overrightarrow{AC})=c^2$, 求角$C$的大小.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期测验卷-高三下学期月考02" ], "genre": "解答题", "ans": "(1) $a-c=\\pm 1$; (2) $C=\\dfrac{\\pi}{3}$", @@ -448633,7 +452575,8 @@ "content": "疫苗在上市前必须经过严格的检测, 并通过临床实验获得相关数据, 以保证疫苗使用的安全和有效.某生物制品研究所将某一型号疫苗用在动物小白鼠身上进行科研和临床实验, 得到统计数据如下:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline & 未感染病毒 & 感染病毒 & 总计 \\\\\n\\hline 未注射疫苗 & 40 &$p$&$x$\\\\\n\\hline 注射疫苗 & 60 &$q$&$y$\\\\\n\\hline 总计 & 100 & 100 & 200 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n现从未注射疫苗的小白鼠中任取$1$只, 取到``感染病毒''的小白鼠的概率为$\\dfrac{3}{5}$.\\\\\n(1) 求$2 \\times 2$列联表中的数据$p, q, x, y$的值;\\\\\n(2) 是否有$95 \\%$的把握认为注射此种疫苗有效? 说明理由;\\\\\n(3) 在感染病毒的小白鼠中, 按未注射疫苗和注射疫苗的比例抽取$10$只进行病例分析, 然后从这$10$只小白鼠中随机抽取$4$只对注射疫苗情况进行核实, 记$X$为$4$只中未注射疫苗的小白鼠的只数, 求$X$的分布与期望$E[X]$.\\\\\n附: $\\chi^2=\\dfrac{n(a d-b c)^2}{(a+b)(c+d)(a+c)(b+d)}$, 其中$n=a+b+c+d$.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline$P(\\chi^2 \\geq k)$& 0.10 & 0.05 & 0.01 & 0.005 & 0.001 \\\\\n\\hline$k$& 2.706 & 3.841 & 6.635 & 7.879 & 10.828 \\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-下学期测验卷-高三下学期月考02" ], "genre": "解答题", "ans": "(1) $p=60$, $q=40$, $x=100$, $y=100$; (2) $\\chi^2=8$, 有$95 \\%$的把握认为注射此种疫苗有效; (3) 分布为$\\begin{pmatrix}0&1&2&3&4\\\\\\dfrac{1}{210}&\\dfrac{4}{35}&\\dfrac{3}{7}&\\dfrac{8}{21}&\\dfrac{1}{14}\\end{pmatrix}$, $E[X]=\\dfrac{12}{5}$", @@ -448668,7 +452611,8 @@ "content": "已知椭圆$C: \\dfrac{x^2}{4}+\\dfrac{y^2}{3}=1$.\\\\\n(1) 求该椭圆的离心率;\\\\\n(2) 设点$P(x_0, y_0)$是椭圆$C$上一点, 求证: 过点$P$的椭圆$C$的切线方程为$\\dfrac{x_0 x}{4}+\\dfrac{y_0 y}{3}=1$;\\\\\n(3) 若点$M$为直线$l: x=4$上的动点, 过点$M$作该椭圆的切线$MA, MB$, 切点分别为$A, B$, 求$\\triangle MAB$的面积的最小值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期测验卷-高三下学期月考02" ], "genre": "解答题", "ans": "(1) $\\dfrac 12$; (2) 证明略; (3) $\\dfrac 92$", @@ -448703,7 +452647,8 @@ "content": "设函数$f(x)=\\ln (x+1)$, $g(x)=\\dfrac{x}{x+1}$.\\\\\n(1) 记$x_1=g(1)$, $x_{n+1}=g(x_n)$, $n \\in \\mathbf{N}$, $n \\geq 1$. 证明: 数列$\\{\\dfrac{1}{x_n}\\}$为等差数列;\\\\\n(2) 设$m \\in \\mathbf{Z}$. 若对任意$x>0$均有$f(x)>m g(x)-1$成立, 求$m$的最大值;\\\\\n(3) 是否存在正整数$t$使得对任意$n \\in \\mathbf{N}$, $n \\geq t$, 都有$\\displaystyle f(n-t)=latex, z = {(-150:0.5cm)}]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [below] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [above] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw (B) -- (C1) (A) -- (B1);\n\\draw [dashed] (A) -- (C1) (A) -- (D1);\n\\end{tikzpicture}\n\\end{center}\n(1) 点$B_1$到平面$ABC_1$的距离;\\\\\n(2) 二面角$C_1-AB-B_1$的大小.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期测验卷-高三下学期测验02" ], "genre": "解答题", "ans": "(1) $\\dfrac{2\\sqrt{5}}5$; (2) $\\arctan 2$", @@ -539657,7 +543616,8 @@ "objs": [], "tags": [ "第二单元", - "导数" + "导数", + "2023届高三-第二轮复习讲义-16_导数及其应用" ], "genre": "填空题", "ans": "$-\\dfrac 12$", @@ -540006,7 +543966,8 @@ "objs": [], "tags": [ "第二单元", - "导数" + "导数", + "2023届高三-第二轮复习讲义-16_导数及其应用" ], "genre": "填空题", "ans": "$1$；$4$", @@ -540690,7 +544651,8 @@ "objs": [], "tags": [ "第二单元", - "导数" + "导数", + "2023届高三-第二轮复习讲义-16_导数及其应用" ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{2}}2$", @@ -540737,7 +544699,8 @@ "objs": [], "tags": [ "第二单元", - "导数" + "导数", + "2023届高三-第二轮复习讲义-16_导数及其应用" ], "genre": "填空题", "ans": "$1$", @@ -541442,7 +545405,8 @@ "objs": [], "tags": [ "第二单元", - "导数" + "导数", + "2023届高三-第二轮复习讲义-16_导数及其应用" ], "genre": "填空题", "ans": "$18$", @@ -541744,7 +545708,8 @@ "objs": [], "tags": [ "第二单元", - "导数" + "导数", + "2023届高三-第二轮复习讲义-16_导数及其应用" ], "genre": "解答题", "ans": "(1) 当$a\\le 2$时, $f(x)$是定义域$(0,+\\infty)$上的严格减函数; 当$a>2$时, $f(x)$在$(0,\\dfrac{a-\\sqrt{a^2-4}}2]$上是严格减函数, 在$[\\dfrac{a-\\sqrt{a^2-4}}2,\\dfrac{a+\\sqrt{a^2-4}}2]$上是严格增函数; (2) 证明略", @@ -541784,7 +545749,8 @@ "objs": [], "tags": [ "第二单元", - "导数" + "导数", + "2023届高三-第二轮复习讲义-16_导数及其应用" ], "genre": "解答题", "ans": "(1) 最大值为$\\dfrac 1{\\mathrm{e}}$, 最小值为$-\\mathrm{e}$; (2) 最大值为$10$, 最小值为$-71$; (3) 最大值为$\\dfrac{\\pi}6+\\sqrt{3}$, 最小值为$\\dfrac{\\pi}2$; (4) 最大值为$\\sqrt{\\mathrm{e}}$, 最小值为$-\\dfrac 2{\\mathrm{e}}$", @@ -541853,7 +545819,8 @@ "objs": [], "tags": [ "第二单元", - "导数" + "导数", + "2023届高三-第二轮复习讲义-16_导数及其应用" ], "genre": "填空题", "ans": "\\textcircled{3}", @@ -563695,7 +567662,8 @@ "K0222002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第一轮复习讲义-10_有关函数的应用问题" ], "genre": "解答题", "ans": "(1) $M=10+mx-x-10\\sqrt{x}, \\ x\\in \\{1,2,3,\\cdots,16\\}$; (2) $[\\dfrac 72,\\dfrac{19}4]$.", @@ -563719,7 +567687,8 @@ "K0222002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第一轮复习讲义-10_有关函数的应用问题" ], "genre": "解答题", "ans": "(1) $2197.2\\text{m}/\\text{s}$; (2) $53.6$倍.", @@ -563743,7 +567712,8 @@ "K0222002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第一轮复习讲义-10_有关函数的应用问题" ], "genre": "解答题", "ans": "(1) $19$万元; (2) 当促销费为$7$万元时, 该网店售出商品的总利润最大, 此时商品的剩余量为$0.25$万件.", @@ -563793,7 +567763,8 @@ "K0222002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第一轮复习讲义-10_有关函数的应用问题" ], "genre": "解答题", "ans": "(1) $(0,80]$; (2) 车流量的最大值约为$3250\\text{辆}/\\text{小时}$, 此时车流密度约为$87\\text{辆}/\\text{千米}$.", @@ -563837,7 +567808,9 @@ "content": "若函数$f(x)=\\log_2(x+1)+a$的图像经过点$(1,4)$, 则实数$a=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷01", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$3$", @@ -563873,7 +567846,8 @@ "content": "将函数$y=\\sqrt{3}\\sin 2x-\\cos 2x$的图像向左平移$m$($m>0$)个单位, 所得图像对应的函数为偶函数, 则$m$的最小值为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-周末卷01" ], "genre": "填空题", "ans": "$\\dfrac{\\pi}3$", @@ -563912,7 +567886,8 @@ "K0302002B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-11_三角比的定义及直接性质" ], "genre": "填空题", "ans": "$\\dfrac{5\\pi}{6}$, $10$, 钝, $\\dfrac{\\pi}{24}$", @@ -563954,7 +567929,8 @@ "K0305001B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第一轮复习讲义-11_三角比的定义及直接性质" ], "genre": "解答题", "ans": "证明略", @@ -563989,7 +567965,8 @@ "content": "函数$y=\\dfrac{1}{2^x}$的反函数为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷02" ], "genre": "填空题", "ans": "$y=-\\log_2 x$", @@ -564062,7 +568039,8 @@ "K0311002B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-月考01" ], "genre": "填空题", "ans": "$\\arccos \\dfrac 5{13}$", @@ -564101,7 +568079,8 @@ "K0305001B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-月考01" ], "genre": "填空题", "ans": "$-\\dfrac{\\sqrt{3}}2$", @@ -564140,7 +568119,8 @@ "K0216002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-赋能-赋能01" ], "genre": "填空题", "ans": "$2$", @@ -564183,7 +568163,8 @@ "tags": [ "第四单元", "第八单元", - "二项式定理" + "二项式定理", + "2023届高三-赋能-赋能01" ], "genre": "填空题", "ans": "$\\dfrac 12 (1 - (\\dfrac 13)^{100})$", @@ -564223,7 +568204,8 @@ "K0402004X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-上学期周末卷-周末卷03_暂未使用" ], "genre": "填空题", "ans": "$\\dfrac{3n^2-5n}2$", @@ -564250,7 +568232,8 @@ "K0407002X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-上学期周末卷-周末卷03_暂未使用" ], "genre": "填空题", "ans": "$\\dfrac 23(\\dfrac 1{4^n}-1)$", @@ -564277,7 +568260,8 @@ "K0215003B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷03_暂未使用" ], "genre": "填空题", "ans": "$[0,\\dfrac 34]$", @@ -564306,7 +568290,8 @@ ], "tags": [ "第七单元", - "直线" + "直线", + "2023届高三-赋能-赋能01" ], "genre": "填空题", "ans": "$1$", @@ -564344,7 +568329,8 @@ "content": "已知$\\sin \\alpha +\\sin \\beta =\\dfrac 35$, $\\cos \\alpha +\\cos \\beta =\\dfrac 45$, 则$\\cos \\alpha \\cdot \\cos \\beta$的值为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-周末卷03" ], "genre": "填空题", "ans": "$-\\dfrac{11}{100}$", @@ -564383,7 +568369,8 @@ "content": "已知$\\tan (\\alpha -\\beta)=\\dfrac 12$, $\\tan \\beta =-\\dfrac 17$, 且$\\alpha,\\beta \\in (0,\\pi)$, 则$2\\alpha -\\beta$的值为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-周末卷03" ], "genre": "填空题", "ans": "$-\\dfrac{3\\pi}{4}$", @@ -564421,7 +568408,8 @@ "content": "证明:集合$A=\\{x|x=14n-1, \\ n\\in\\mathbf{Z}\\}$是$B=\\{x|x=7n+6, \\ n\\in\\mathbf{Z}\\}$的真子集.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期周末卷-国庆卷" ], "genre": "解答题", "ans": "证明略", @@ -564459,7 +568447,9 @@ "content": "已知集合$A=\\{1,2\\}$, $B=\\{x|mx^2+2mx-1>0, x \\in\\mathbf{R}\\}$. 已知$A \\cap B=\\{2\\}$, 则实数$m$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期周末卷-国庆卷", + "2023届高三-寒假作业-中档题" ], "genre": "填空题", "ans": "$(\\dfrac 18,\\dfrac 13]$", @@ -564497,7 +568487,8 @@ "content": "已知$x,y,z\\in \\mathbf{R}$, ``$x^2+y^2+z^2>0$''是``$x\\ne 0$或$y\\ne 0$或$z\\ne 0$''的\\bracket{20}.\n\\twoch{充分而不必要条件}{必要而不充分条件}{充要条件}{既不充分又不必要条件}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期周末卷-国庆卷" ], "genre": "选择题", "ans": "C", @@ -564535,7 +568526,8 @@ "content": "已知$m$是实常数. 命题甲: 关于$x$的方程$x^2+x+2m=0$有两个相异的负根; 命题乙: 关于$x$的方程$4x^2+x+2m=0$无实根, 若这两个命题有且只有一个是真命题, 则实数$m$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期周末卷-国庆卷" ], "genre": "填空题", "ans": "$(0,\\dfrac 1{32}]\\cup [\\dfrac 18,+\\infty)$", @@ -564573,7 +568565,8 @@ "content": "判断题: (如果正确请在题目前面的横线上写``\\checkmark'', 错误请在题目前面的横线上写``$\\times$'')\\\\ \n\\blank{30}(1) 若$a>b$, $c\\geq d$, 则$a+c>b+d$;\\\\ \n\\blank{30}(2) 若$a>b$, $c\\geq d$, 则$a+c\\geq b+d$;\\\\ \n\\blank{30}(3) 若$a\\ge b$, $c\\ge d$, 则$a+c\\ge b+d$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期周末卷-国庆卷" ], "genre": "填空题", "ans": "(1) \\checkmark; (2) \\checkmark; (3) \\checkmark", @@ -564611,7 +568604,8 @@ "content": "利用绝对值的三角不等式$|a+b|\\le |a|+|b|$, 证明: 对任意$x,y\\in\\mathbf{R}$, $|x-y|\\ge ||x|-|y||$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期周末卷-国庆卷" ], "genre": "解答题", "ans": "证明略", @@ -564675,7 +568669,8 @@ "content": "已知关于$x$的不等式$|x-4|+|x+3|0$的解集为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期周末卷-国庆卷" ], "genre": "填空题", "ans": "$(-\\infty,-2)$", @@ -564789,7 +568786,8 @@ "content": "不等式$\\dfrac1{|x-2|}\\ge 3 $的解集为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期周末卷-国庆卷" ], "genre": "填空题", "ans": "$[\\dfrac 53,2)\\cup (2,\\dfrac 73]$", @@ -564827,7 +568825,8 @@ "content": "已知关于$x$的不等式$ax^2-bx+c>0$的解集是$(-\\dfrac 32,\\dfrac 12)$, 对于$a,b,c$有以下结论: \\textcircled{1} $a>0$; \\textcircled{2} $b>0$; \\textcircled{3} $c>0$; \\textcircled{4} $a+b+c>0$; \\textcircled{5} $a-b+c>0$. 其中正确的序号有\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期周末卷-国庆卷" ], "genre": "填空题", "ans": "\\textcircled{2}\\textcircled{3}\\textcircled{4}", @@ -564868,7 +568867,8 @@ "K0206002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-国庆卷" ], "genre": "填空题", "ans": "$a=b$或$x=1$($a,b,x\\in (0,+\\infty)$)", @@ -564909,7 +568909,8 @@ "K0219003B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-国庆卷" ], "genre": "填空题", "ans": "$(1,+\\infty)$", @@ -564950,7 +568951,8 @@ "K0214002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-国庆卷" ], "genre": "填空题", "ans": "$(0,1)\\cup (1,+\\infty)$", @@ -564990,7 +568992,8 @@ "K0214002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-国庆卷" ], "genre": "填空题", "ans": "$(\\dfrac 25,1)\\cup (3,+\\infty)$", @@ -565030,7 +569033,8 @@ "K0215003B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-国庆卷" ], "genre": "填空题", "ans": "$[\\dfrac 23,\\dfrac 83]$", @@ -565072,7 +569076,8 @@ "K0220001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-国庆卷" ], "genre": "解答题", "ans": "图像略, 在$(-\\infty,1]$上是严格增函数; 在$[1,+\\infty)$上是严格减函数, 证明略", @@ -565112,7 +569117,8 @@ "K0219003B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-国庆卷" ], "genre": "解答题", "ans": "在$[0,+\\infty)$上是严格增函数, 证明略", @@ -565152,7 +569158,8 @@ "K0219001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-国庆卷" ], "genre": "填空题", "ans": "$(-4,-2]$", @@ -565192,7 +569199,8 @@ "K0218001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-国庆卷" ], "genre": "解答题", "ans": "奇函数, 证明略", @@ -565233,7 +569241,8 @@ "K0223004B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-国庆卷" ], "genre": "填空题", "ans": "$[-\\dfrac 1{12},-\\dfrac 1{20}]$", @@ -565274,7 +569283,10 @@ "K0234003X" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-国庆卷", + "2023届高三-四月错题重做-01_函数一", + "2023届高三-四月错题重做-01_易错题-函数1" ], "genre": "填空题", "ans": "$\\dfrac{283}4$, $12\\sqrt{3}$", @@ -565326,7 +569338,8 @@ "K0221002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-国庆卷" ], "genre": "解答题", "ans": "$-1$或$2$", @@ -565366,7 +569379,8 @@ "K0221002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-国庆卷" ], "genre": "填空题", "ans": "$(1,\\sqrt{3})$", @@ -565404,7 +569418,8 @@ "content": "直角坐标平面内, 终边过点$(1,\\sqrt{2})$的所有角组成的集合可表示成\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-国庆卷" ], "genre": "填空题", "ans": "$\\{x|x=2k\\pi+\\arctan \\sqrt{2}, \\ k\\in \\mathbf{Z}\\}$", @@ -565442,7 +569457,8 @@ "content": "已知角$\\alpha$的终边上一点的坐标为$\\left(\\sin\\dfrac{3\\pi}{7},\\cos\\dfrac{3\\pi}{7}\\right)$, 则角$\\alpha$的最小正值为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-国庆卷" ], "genre": "填空题", "ans": "$\\dfrac{\\pi}{14}$", @@ -565480,7 +569496,8 @@ "content": "四点三十分时, 时针和分针的夹角的弧度数为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-国庆卷" ], "genre": "填空题", "ans": "$\\dfrac\\pi 4$", @@ -565519,7 +569536,8 @@ "content": "已知$\\theta$是象限角, 化简下列各式.\\\\ \n(1) $\\dfrac{\\cos(-\\theta)\\sin(\\pi-\\theta)}{\\cos(\\theta-3\\pi)}+\\dfrac{\\sin(-2\\pi-\\theta)\\sin(\\theta+\\pi)}{\\sin(4\\pi-\\theta)}=$\\blank{50}.\\\\ \n(2) $\\sin\\left(\\dfrac{\\pi}{2}-\\theta\\right)+\\sin\\left(\\dfrac{5\\pi}{2}+\\theta\\right)\n-\\sin\\left(\\dfrac{3\\pi}{2}-\\theta\\right)+\\sin\\left(\\dfrac{3\\pi}{2}+\\theta\\right)=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-国庆卷" ], "genre": "填空题", "ans": "(1) $-2\\sin \\theta$; (2) $2\\cos\\theta$", @@ -565557,7 +569575,8 @@ "content": "已知点$A$的坐标为$(3, 4)$, 将$OA$绕坐标原点$O$逆时针旋转$\\dfrac\\pi 4$至$OA'$. 求点$A'$的坐标.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-国庆卷" ], "genre": "解答题", "ans": "$(-\\dfrac{\\sqrt{2}}2,\\dfrac{7\\sqrt{2}}2)$", @@ -565595,7 +569614,8 @@ "content": "方程$\\sin x+4\\cos x=0$的解集为\\blank{150}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-国庆卷" ], "genre": "填空题", "ans": "$\\{x|x=-\\arctan 4+k\\pi, \\ k\\in \\mathbf{Z}\\}$", @@ -565636,7 +569656,10 @@ "K0221002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-国庆卷", + "2023届高三-四月错题重做-01_函数一", + "2023届高三-四月错题重做-01_易错题-函数1" ], "genre": "解答题", "ans": "$y=-\\pi x^2+lx, \\ x\\in (0,\\dfrac{l}{2\\pi})$", @@ -565687,7 +569710,8 @@ "content": "下列各式中, 不正确的是\\bracket{20}.\n\\twoch{$\\sin \\alpha +\\sin \\beta =2\\sin \\dfrac{\\beta +\\alpha}2\\cos \\dfrac{\\beta -\\alpha}2$}{$\\sin \\alpha -\\sin \\beta =2\\cos \\dfrac{\\beta +\\alpha}2\\sin \\dfrac{\\alpha -\\beta}2$}{$\\cos \\alpha +\\cos \\beta =2\\cos \\dfrac{\\beta +\\alpha}2\\cos \\dfrac{\\beta -\\alpha}2$}{$\\cos \\alpha -\\cos \\beta =2\\sin \\dfrac{\\beta +\\alpha}2\\sin \\dfrac{\\alpha -\\beta}2$}", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-国庆卷" ], "genre": "选择题", "ans": "D", @@ -565725,7 +569749,8 @@ "content": "在$\\triangle ABC$中, $b=3,c=2$, $B-C=\\dfrac{\\pi}2$, 则$\\tan A=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-国庆卷" ], "genre": "填空题", "ans": "$\\dfrac 5{12}$", @@ -565763,7 +569788,8 @@ "content": "在$\\triangle ABC$中, 已知$A=60^\\circ$, $b=6$. 若$a=2\\sqrt{7}$, 则$B=$\\blank{50}; 若$a=6$, 则$B=$\\blank{50}; 若$a=9$, 则$B=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-国庆卷" ], "genre": "填空题", "ans": "$\\arcsin\\dfrac{3\\sqrt{21}}{14}$或$\\pi-\\arcsin\\dfrac{3\\sqrt{21}}{14}$, $\\dfrac\\pi 3$, $\\arcsin\\dfrac{\\sqrt{3}}3$", @@ -565801,7 +569827,8 @@ "content": "函数$y=\\sqrt{\\cos x}+\\dfrac 1{\\sqrt{36-x^2}}$的定义域为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-国庆卷" ], "genre": "填空题", "ans": "$(-6,-\\dfrac{3\\pi}2]\\cup [-\\dfrac\\pi 2,\\dfrac\\pi 2]\\cup [\\dfrac{3\\pi}2,6)$", @@ -565839,7 +569866,8 @@ "content": "求函数$y=\\sin x(\\sin x+\\sqrt{3}\\cos x)$的最大值与最小值.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-国庆卷" ], "genre": "解答题", "ans": "最小值为$-\\dfrac 12$, 最大值为$\\dfrac 32$", @@ -565877,7 +569905,8 @@ "content": "求函数$y=\\dfrac{\\cos^2 x-3}{1-\\sin x}$的最大值.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-国庆卷" ], "genre": "解答题", "ans": "$2-2\\sqrt{3}$", @@ -565915,7 +569944,8 @@ "content": "已知函数$f(x)=(2\\sin(x+\\dfrac{\\pi}3)+\\sin x)\\cos x-\\sqrt 3\\sin^2 x$, 求$f(x)$的单调递减区间", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-国庆卷" ], "genre": "解答题", "ans": "$[\\dfrac\\pi{12}+k\\pi,\\dfrac{7\\pi}{12}+k\\pi], \\ k\\in \\mathbf{Z}$", @@ -565953,7 +569983,8 @@ "content": "设常数$a\\in \\mathbf{R}$. 若函数$y=\\sin 2x+a\\cos 2x$的图像关于直线$x=\\dfrac{\\pi}3$对称, 则$a=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期周末卷-国庆卷" ], "genre": "填空题", "ans": "$-\\dfrac{\\sqrt{3}}3$", @@ -566044,7 +570075,8 @@ ], "tags": [ "第八单元", - "二项式定理" + "二项式定理", + "2023届高三-赋能-赋能03" ], "genre": "填空题", "ans": "$6$", @@ -566085,7 +570117,10 @@ "K0402004X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-四月错题重做-03_数列", + "2023届高三-四月错题重做-03_易错题-数列", + "2023届高三-赋能-赋能03" ], "genre": "填空题", "ans": "$2n^2+6n$", @@ -566137,7 +570172,8 @@ "K0219001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-赋能-赋能03" ], "genre": "填空题", "ans": "\\textcircled{1}\\textcircled{2}", @@ -566175,7 +570211,8 @@ "content": "函数$f(x)=1+\\log_2 x$的值域为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-赋能-赋能04" ], "genre": "填空题", "ans": "$\\mathbf{R}$", @@ -566215,7 +570252,9 @@ "content": "设地球半径为$R$, 圆心为$O$. 若$A$、$B$两地均位于北纬$45^\\circ$, 且两地所在纬度圈上的弧长为$\\dfrac{\\sqrt{2}}{4}\\pi R$, 则$\\triangle OAB$的面积为\\blank{50}(结果用含有$R$的代数式表示).", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-寒假作业-较难题", + "2023届高三-赋能-赋能04" ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{3}}4R^2$", @@ -566253,7 +570292,8 @@ "content": "不等式$|2x+5|+|2x-3|\\le 8$的解集是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-国庆卷" ], "genre": "填空题", "ans": "$[-\\dfrac 52,\\dfrac 32]$", @@ -566290,7 +570330,8 @@ "K0601002B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-20_描述空间位置关系的公理" ], "genre": "填空题", "ans": "(1) $A\\in \\alpha$, $AB\\subset \\alpha$; (2) $C\\not \\in \\beta$, $CD\\cap \\beta = \\varnothing$", @@ -566318,7 +570359,8 @@ "content": "画出下列符号语言表示的点、直线、平面的位置关系的示意图.\\\\ \n(1) $a\\subset \\alpha$, $b\\subset \\alpha$, $a\\cap b=A$;\\\\\n(2) $a\\subset \\alpha$, $b\\cap \\alpha =A$, $A\\notin a$.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-20_描述空间位置关系的公理" ], "genre": "解答题", "ans": "(1) 图略; (2) 图略", @@ -566368,7 +570410,8 @@ "content": "已知三条直线$l_1$, $l_2$和$l_3$两两相交, 且不交于同一个点. 求证: 直线$l_1$, $l_2$和$l_3$在同一个平面上.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-20_描述空间位置关系的公理" ], "genre": "解答题", "ans": "证明略", @@ -566399,7 +570442,8 @@ "K0603005B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-20_描述空间位置关系的公理" ], "genre": "解答题", "ans": "图略", @@ -566427,7 +570471,8 @@ "content": "如图, 已知$E$、$F$分别是正方体$ABCD-A_1B_1C_1D_1$的棱$A_1A$、$C_1C$的中点. 求证: 四边形$BED_1F$是平行四边形.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{1.5}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(A)!0.5!(A1)$) node [left] {$E$} coordinate (E);\n\\draw ($(C)!0.5!(C1)$) node [right] {$F$} coordinate (F);\n\\draw (E) -- (B) -- (F);\n\\draw [dashed] (E) -- (D1) -- (F);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-20_描述空间位置关系的公理" ], "genre": "解答题", "ans": "证明略", @@ -566481,7 +570526,8 @@ "content": "如图, 在四面体$A-BCD$中, $E,F,G$分别为$AB,AC,AD$上的点. 若$EF\\parallel BC$, $FG\\parallel CD$, 则$\\triangle EFG$和$\\triangle BCD$有什么关系? 为什么?\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw (0,0) node [above] {$A$} coordinate (A);\n\\draw (-0.5,-1) node [left] {$E$} coordinate (E);\n\\draw (0.5,-1) node [right] {$G$} coordinate (G);\n\\draw (0.2,-1.4) node [below left] {$F$} coordinate (F);\n\\draw ($(A)!2!(E)$) node [left] {$B$} coordinate (B);\n\\draw ($(A)!2!(F)$) node [below] {$C$} coordinate (C);\n\\draw ($(A)!2!(G)$) node [right] {$D$} coordinate (D);\n\\draw (A) -- (B) (A) -- (C) (A) -- (D) (B) -- (C) -- (D) (E) -- (F) -- (G);\n\\draw [dashed] (E) -- (G) (B) -- (D);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-20_描述空间位置关系的公理" ], "genre": "解答题", "ans": "两三角形相似, 证明略", @@ -566511,7 +570557,8 @@ "K0606005B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-20_描述空间位置关系的公理" ], "genre": "解答题", "ans": "若$a\\subset\\alpha$, $B\\cap \\alpha = P$, 且$P\\not\\in a$, 则$a$与$b$是异面直线. 证明略", @@ -566541,7 +570588,8 @@ "K0607002B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-20_描述空间位置关系的公理" ], "genre": "填空题", "ans": "$(0,\\dfrac\\pi 2]$", @@ -566571,7 +570619,9 @@ "K0601005B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-20_描述空间位置关系的公理" ], "genre": "解答题", "ans": "若直线$l$上有两点在平面$\\alpha$上, 则直线$l$上的每一点都在平面$\\alpha$上", @@ -566599,7 +570649,8 @@ "content": "复述``空间直线与平面''一章中的公理2.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-20_描述空间位置关系的公理" ], "genre": "解答题", "ans": "过不共线的三点有且仅有一个平面", @@ -566629,7 +570680,8 @@ "K0603001B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-20_描述空间位置关系的公理" ], "genre": "解答题", "ans": "若两平面有公共点, 则它们的公共部分是(通过该点)的一条直线", @@ -566659,7 +570711,9 @@ "K0605001B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-20_描述空间位置关系的公理" ], "genre": "解答题", "ans": "若两直线均平行于第三条直线, 那么它们平行", @@ -566689,7 +570743,8 @@ "K0608001B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-21_空间直线与平面的位置关系" ], "genre": "解答题", "ans": "若平面$\\alpha$外的一条直线$l$与平面$\\alpha$上的一条直线$m$平行, 则$l$与平面$\\alpha$平行", @@ -566719,7 +570774,8 @@ "K0608003B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-21_空间直线与平面的位置关系" ], "genre": "解答题", "ans": "若直线$l$平行于平面$\\alpha$, $l$在平面$\\beta$上, 且$\\alpha\\cap \\beta=m$, 则$l$与$m$平行. 证明略", @@ -566749,7 +570805,9 @@ "K0609002B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-21_空间直线与平面的位置关系" ], "genre": "解答题", "ans": "若直线$l$垂直于平面$\\alpha$上的两条相交直线, 则$l\\perp \\alpha$", @@ -566779,7 +570837,8 @@ "K0609004B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-21_空间直线与平面的位置关系" ], "genre": "解答题", "ans": "垂直于同一平面的两条直线平行. 证明略", @@ -566809,7 +570868,9 @@ "K0612001B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-22_空间平面与平面的位置关系" ], "genre": "解答题", "ans": "若一平面上的两条相交直线都平行与另一平面, 则两平面平行", @@ -566839,7 +570900,9 @@ "K0613006B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-22_空间平面与平面的位置关系" ], "genre": "解答题", "ans": "一平面与两平行平面的交线平行, 证明略", @@ -566868,7 +570931,9 @@ "K0613008B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-22_空间平面与平面的位置关系" ], "genre": "解答题", "ans": "若平面$\\alpha$经过平面$\\beta$的垂线, 则$\\alpha\\perp \\beta$", @@ -566895,7 +570960,8 @@ "content": "叙述并证明平面与平面垂直的性质定理.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-22_空间平面与平面的位置关系" ], "genre": "解答题", "ans": "若$\\alpha\\perp \\beta$, $\\alpha\\cap \\beta=l$, $m\\subset \\alpha$, $m\\perp l$, 则$m\\perp \\beta$. 证明略", @@ -566953,7 +571019,9 @@ "K0612004B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-寒假作业-较难题", + "2023届高三-第一轮复习讲义-22_空间平面与平面的位置关系" ], "genre": "解答题", "ans": "证明略", @@ -566980,7 +571048,8 @@ "content": "已知锐角$\\alpha ,\\beta$满足$\\cos \\alpha =\\dfrac 45$, $\\tan (\\alpha -\\beta)=-\\dfrac 13$, 则$\\cos \\beta=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-测验03" ], "genre": "填空题", "ans": "$\\dfrac{9\\sqrt{10}}{50}$", @@ -567018,7 +571087,9 @@ "content": "如图, 已知函数$y=A\\cos (\\omega x+\\varphi)$($A>0$, $\\omega >0$, $0<\\varphi <2\\pi$)的图像与$y$轴的交点为$(0, 1)$, 并已知其在$y$轴右侧的第一个最高点和第一个最低点的坐标分别为$(x_0, 2)$和$(x_0+2\\pi , -2)$. 则此函数的表达式为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-1.5,0) -- (12,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1.5:12, samples = 100] plot (\\x,{2*sin(\\x/2/pi*180+30)});\n\\foreach \\i in {-2,1,2} {\\draw (0,\\i) node [left] {$\\i$};};\n\\draw [dashed] (0,2) -- ({2*pi/3},2) -- ({2*pi/3},0) node [below] {$x_0$};\n\\draw [dashed] (0,-2) -- ({2*pi/3+2*pi},-2) -- ({2*pi/3+2*pi},0) node [above] {$x_0+2\\pi$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-测验03", + "2023届高三-寒假作业-较难题" ], "genre": "填空题", "ans": "$y=2\\cos(\\dfrac x2+\\dfrac{5\\pi}3)$", @@ -567057,7 +571128,8 @@ "objs": [], "tags": [ "第二单元", - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-测验03" ], "genre": "解答题", "ans": "(1) $h(x)=\\begin{cases}\\dfrac{x^2}{x-1}, & x\\ne 1,\\\\ 1, & x=1;\\end{cases}$ (2) $(-\\infty,0]\\cup \\{1\\}\\cup [4,+\\infty)$; (3) 答案不唯一, 如$f(x)=\\cos 2x+\\sin 2x$, $\\alpha=\\dfrac\\pi 4$等", @@ -567095,7 +571167,8 @@ "K0307003B" ], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-测验03" ], "genre": "填空题", "ans": "$\\cos\\alpha$", @@ -567158,7 +571231,8 @@ "content": "已知$|\\overrightarrow a|=2$, $|\\overrightarrow b|=4$, $\\overrightarrow a$与$\\overrightarrow b$的夹角为$120^{\\circ }$, 且向量$\\overrightarrow a+k\\overrightarrow b$与$k\\overrightarrow a+\\overrightarrow b$的夹角是锐角, 则实数$k$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期周末卷-周末卷04" ], "genre": "填空题", "ans": "$(\\dfrac{5-\\sqrt{21}}2,1)\\cup (1,\\dfrac{5+\\sqrt{21}}2)$", @@ -567192,7 +571266,8 @@ "content": "已知向量$\\overrightarrow a=(3,4)$与$\\overrightarrow b=(1,0)$. 则$\\overrightarrow a$在$\\overrightarrow b$的方向上的投影为\\blank{50}; $\\overrightarrow b$在$\\overrightarrow a$的方向上的投影为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期周末卷-周末卷04" ], "genre": "填空题", "ans": "$(\\dfrac 9{25},\\dfrac{12}{25})$", @@ -567226,7 +571301,8 @@ "content": "已知$4\\overrightarrow{m}=\\overrightarrow{a}+2\\overrightarrow{n}$, $6\\overrightarrow{b}=\\overrightarrow{a}-2\\overrightarrow{n}$. 其中$|\\overrightarrow{a}|=3$, $|\\overrightarrow{m}|=\\dfrac{\\sqrt{13}}{2}$, $|\\overrightarrow{n}|=\\dfrac{5}{2}$, 则$\\overrightarrow{a},\\overrightarrow{b}$的夹角的大小为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期周末卷-周末卷04" ], "genre": "填空题", "ans": "$\\dfrac\\pi 2$", @@ -567260,7 +571336,8 @@ "content": "$|\\dfrac{(2+2\\mathrm{i})^5}{(-1+\\sqrt{3}\\mathrm{i})^4}|=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第一轮复习讲义-18_复数的代数运算与性质" ], "genre": "填空题", "ans": "$8\\sqrt{2}$", @@ -567300,7 +571377,8 @@ "K0515004B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第一轮复习讲义-19_复数的几何意义与实系数二次方程" ], "genre": "填空题", "ans": "(1) $-1\\pm \\dfrac{\\sqrt{2}}2\\mathrm{i}$; (2) $2(x+1+\\dfrac{\\sqrt{2}}2\\mathrm{i})(x+1-\\dfrac{\\sqrt{2}}2\\mathrm{i})$", @@ -567339,7 +571417,8 @@ "objs": [], "tags": [ "第一单元", - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-20_描述空间位置关系的公理" ], "genre": "填空题", "ans": "(1) F; (2) F; (3) T; (4) T; (5) F; (6) F; (7) F; (8) T; (9) F; (10) F", @@ -567371,7 +571450,8 @@ "objs": [], "tags": [ "第一单元", - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-20_描述空间位置关系的公理" ], "genre": "填空题", "ans": "(1) F; (2) T; (3) F; (4) F; (5) F; (6) T", @@ -567480,7 +571560,9 @@ "K0621002B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-23_多面体及旋转体的概念与性质" ], "genre": "填空题", "ans": "$4$, 三棱锥", @@ -567563,7 +571645,9 @@ "K0615002B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-23_多面体及旋转体的概念与性质" ], "genre": "填空题", "ans": "矩形", @@ -567596,7 +571680,8 @@ "K0615004B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-23_多面体及旋转体的概念与性质" ], "genre": "填空题", "ans": "(1) \\checkmark; (2) $\\times$; (3) \\checkmark; (4) $\\times$", @@ -567652,7 +571737,8 @@ "K0618002B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-23_多面体及旋转体的概念与性质" ], "genre": "填空题", "ans": "$2\\sqrt{10}$, $6$", @@ -567708,7 +571794,8 @@ "K0618007B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-23_多面体及旋转体的概念与性质" ], "genre": "填空题", "ans": "$\\sqrt{22}\\text{cm}$", @@ -567737,7 +571824,8 @@ "K0615007B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-23_多面体及旋转体的概念与性质" ], "genre": "填空题", "ans": "平行, 平行, 相交, 平行", @@ -567770,7 +571858,8 @@ "K0622006B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-23_多面体及旋转体的概念与性质" ], "genre": "填空题", "ans": "$30660$", @@ -567826,7 +571915,9 @@ "K0618005B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-23_多面体及旋转体的概念与性质" ], "genre": "填空题", "ans": "$10\\sqrt{3}$", @@ -568052,7 +572143,9 @@ "K0622006B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-23_多面体及旋转体的概念与性质" ], "genre": "填空题", "ans": "$\\dfrac 12$", @@ -568107,7 +572200,8 @@ "K0618007B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-23_多面体及旋转体的概念与性质" ], "genre": "填空题", "ans": "(1) $\\times$; (2) $\\times$; (3) $\\times$", @@ -568185,7 +572279,8 @@ "K0621002B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-23_多面体及旋转体的概念与性质" ], "genre": "填空题", "ans": "\\textcircled{1}\\textcircled{3}\\textcircled{4}\\textcircled{5}", @@ -568212,7 +572307,8 @@ "content": "如图, 平面$\\alpha$上的斜线$l$与平面$\\alpha$所成的角为$\\theta$, $l'$是$l$在平面$\\alpha$上的投影, $O$是$l$与平面$\\alpha$的交点, 点$B$是$l$上一点$A$在$\\alpha$上的投影, $OC$是$\\alpha$上的任意一条直线.\\\\\n(1) 如果$\\theta =45^\\circ$, $\\angle BOC=45^\\circ$, 求$\\angle AOC$;\\\\\n(2) 试证明: $\\angle AOC>\\theta$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (-1,0,0) -- (3,0,0) node [right] {$l'$} (2,2,0) node [above] {$A$} coordinate (A) -- (2,0,0) coordinate (B) node [below] {$B$} (2.5,2.5,0) node [right] {$l$} -- (0,0,0) coordinate (O) node [below] {$O$};\n\\draw (1,0,1) node [below] {$C$} coordinate (C);\n\\draw ($(O)!-0.5!(C)$) -- ($(O)!1.8!(C)$);\n\\draw [name path = edge] (-1.5,0,-2.5) coordinate (L) -- (-1.5,0,2.5) --++ (5,0,0) --++ (0,0,-5) coordinate (R);\n\\path [name path = LR] (L) -- (R);\n\\path [name path = OA] (O) -- (A);\n\\path [name path = AB] (A) -- (B);\n\\path [name intersections = {of = OA and LR, by = A1}];\n\\path [name intersections = {of = AB and LR, by = B1}];\n\\draw (L) -- (A1) (B1) -- (R);\n\\draw [dashed] (A1) -- (B1);\n\\path [name path = down] ($(O)!-0.6!(A)$) -- (O);\n\\path [name intersections = {of = down and edge, by = T}];\n\\draw (T) -- ($(O)!-0.6!(A)$);\n\\draw [dashed] (T) -- (O);\n\\draw (O) pic [\"$\\theta$\",draw,angle eccentricity = 1.5] {angle = B--O--A};\n\\draw (O) pic [scale = 1.1,draw,angle eccentricity = 1.7]{angle = C--O--B};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-21_空间直线与平面的位置关系" ], "genre": "解答题", "ans": "(1) $\\dfrac{\\pi}3$; (2) 证明略", @@ -568242,7 +572338,9 @@ "content": "已知不共面的三条射线$a,b,c$均以点$P$为端点, 平面$\\alpha,\\beta$与直线$a,b,c$分别相交于$A,B,C$和$A_1,B_1,C_1$, 且$\\dfrac{PA}{PA_1}=\\dfrac{PB}{PB_1}=\\dfrac{PC}{PC_1}$, 求证: $\\alpha \\parallel\\beta$.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-寒假作业-较难题", + "2023届高三-第一轮复习讲义-22_空间平面与平面的位置关系" ], "genre": "解答题", "ans": "证明略", @@ -568272,7 +572370,8 @@ "content": "如图, 已知$AB\\perp$平面$BCD$, $BC\\perp CD$, 有哪些平面互相垂直? 选择其中一对互相垂直的平面给出证明. \n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$B$} coordinate (B) -- (2.4,0,1.6) node [below] {$C$} coordinate (C) -- (3.6,0,0) node [right] {$D$} coordinate (D) -- (0,2,0) node [above] {$A$} coordinate (A);\n\\draw (A) -- (B) (A) -- (C);\n\\draw [dashed] (B) -- (D); \n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-22_空间平面与平面的位置关系" ], "genre": "解答题", "ans": "平面$ABC\\perp$平面$BCD$, 平面$ABD\\perp$平面$BCD$, 平面$ACD\\perp$平面$ABC$. 证明略", @@ -568302,7 +572401,9 @@ "content": "过$60^\\circ$的二面角$\\alpha-l-\\beta$的棱上一点$A$, 分别在$\\alpha,\\beta$内引两条射线, 使得它们与$l$都成$45^\\circ$角, 则这两条射线夹角的余弦值为\\blank{60}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-寒假作业-较难题", + "2023届高三-第一轮复习讲义-22_空间平面与平面的位置关系" ], "genre": "填空题", "ans": "$\\dfrac 34$或$-\\dfrac 14$", @@ -568332,7 +572433,8 @@ "content": "下列命题是否为真命题? 如果是, 请回答``真''; 如果不是, 请说明理由:\\\\\n(1) 垂直于同一直线的两个平面平行;\\\\\n(2) 平行于同一平面的两条直线平行;\\\\\n(3) 垂直于同一平面的两条直线平行.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-21_空间直线与平面的位置关系" ], "genre": "解答题", "ans": "(1) 真; (2) 如正方体$ABCD-A_1B_1C_1D_1$中, $AB$与$BC$均平行与平面$A_1B_1C_1D_1$, 但它们不平行; (3) 真", @@ -568365,7 +572467,8 @@ "content": "下列命题是否为真命题? 如果是, 请回答``真''; 如果不是, 请说明理由:\\\\\n(1) 若直线$l$与平面$M$斜交, 则$M$内不存在与$l$垂直的直线;\\\\\n(2) 若直线$l\\perp\\text{平面}M$, 则$M$内不存在与$l$不垂直的直线;\\\\\n(3) 若直线$l$与平面$M$斜交, 则$M$内不存在与$l$平行的直线;\\\\\n(4) 若直线$l\\parallel\\text{平面}M$, 则$M$内不存在与$l$不平行的直线.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-21_空间直线与平面的位置关系" ], "genre": "解答题", "ans": "(1) 如正方体$ABCD-A_1B_1C_1D_1$中的直线$AB_1$与平面$ABCD$斜交, 但$AB_1\\perp BC$; (2) 真; (3) 真; (4) 如正方体$ABCD-A_1B_1C_1D_1$中, $AB\\parallel$平面$A_1B_1C_1D_1$, 但$AB\\not\\parallel B_1C_1$", @@ -568396,7 +572499,8 @@ "content": "下列命题是否为真命题? 如果是, 请回答``真''; 如果不是, 请说明理由:\\\\\n(1) 若两直线$a$、$b$互相平行, 则$a$平行于经过$b$的任何平面;\\\\\n(2) 若直线$a$与平面$\\alpha$平行, 则$a$平行于$\\alpha$内的任何直线;\\\\\n(3) 若两直线$a$、$b$都与平面$\\alpha$平行, 则$a\\parallel b$;\\\\\n(4) 若直线$a$平行于平面$\\alpha$, 直线$b$在平面$\\alpha$上, 则$a\\parallel b$或者$a$与$b$为异面直线.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-21_空间直线与平面的位置关系" ], "genre": "解答题", "ans": "(1) $a$不平行与由$a,b$所确定的平面; (2) 如正方体$ABCD-A_1B_1C_1D_1$中, $A_1B_1\\parallel$平面$ABCD$, 但是$AB\\not\\parallel BC$; (3) 如正方体$A_1B_1C_1D_1$中, $A_1B_1$与$B_1C_1$均平行与平面$ABCD$, 但$A_1B_1\\not\\parallel B_1C_1$; (4) 真", @@ -568426,7 +572530,8 @@ "content": "已知平面$\\alpha\\perp$平面$\\beta$, 下列命题是否为真命题? 如果是, 请回答``真''; 如果不是, 请说明理由:\\\\ \n(1) 平面$\\alpha$上的任意一条直线都垂直于平面$\\beta$上的任意一条直线;\\\\\n(2) 平面$\\alpha$上的任意一条直线都垂直于平面$\\beta$上的无数条直线;\\\\\n(3) 平面$\\alpha$上的任意一条直线都垂直于平面$\\beta$;\\\\\n(4) 过平面$\\alpha$上任意一点作平面$\\alpha$与$\\beta$交线的垂线$l$, 则$l\\perp \\beta$.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-22_空间平面与平面的位置关系" ], "genre": "解答题", "ans": "(1) 和棱平行的直线就和$\\beta$不垂直; (2) 真; (3) 和棱平行的直线就和$\\beta$不垂直; (4) 直线可以与$\\alpha$相交", @@ -568456,7 +572561,8 @@ "content": "下列命题是否为真命题? 如果是, 请回答``真''; 如果不是, 请说明理由:\\\\\n(1) 若一个平面内的两条直线均平行于另一个平面, 则这两个平面平行;\\\\\n(2) 若一个平面内两条不平行的直线都平行于另一个平面, 则这两个平面平行;\\\\\n(3) 若两个平面平行, 则其中一个平面中的任何直线都平行于另一个平面;\\\\\n(4) 平行于同一个平面的两个平面平行;\\\\\n(5) 若一个平面内的任何一条直线都平行于另一个平面, 则这两个平面平行.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-22_空间平面与平面的位置关系" ], "genre": "解答题", "ans": "(1) 两直线到平面距离不等时, 所在的平面与原平面不平行; (2) 真; (3) 真; (4) 真; (5) 真", @@ -568485,7 +572591,9 @@ "content": "在复数范围内, 方程$2x^2-2x+1=0$的解为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期测验卷-测验04", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$\\dfrac 12\\pm \\dfrac 12\\mathrm{i}$", @@ -568522,7 +572630,8 @@ "content": "已知平面上四点$O,A,B,C$满足$OA=OB=1, OC=2\\sqrt3, \\angle AOC=25^\\circ, \\angle BOC=95^\\circ,\\angle AOB=120^\\circ$, 则用向量$\\overrightarrow{OA},\\overrightarrow{OB}$表示向量$\\overrightarrow{OC}$为$\\overrightarrow{OC}=$\\blank{80}(系数精确到$0.001$).", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期测验卷-测验04" ], "genre": "填空题", "ans": "$3.985\\overrightarrow{OA}+1.690\\overrightarrow{OB}$", @@ -568560,7 +572669,9 @@ "content": "已知$|\\overrightarrow{a}|$=$\\sqrt 2$, $|\\overrightarrow{b}|=3$, $\\overrightarrow{a}$和$\\overrightarrow{b}$的夹角为$45^\\circ$, 求当向量$\\overrightarrow{a}+\\lambda \\overrightarrow{b}$与$\\lambda \\overrightarrow{a}+\\overrightarrow{b}$夹角为锐角时, $\\lambda$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期测验卷-线上测验01", + "2023届高三-赋能-赋能31" ], "genre": "填空题", "ans": "$(-\\infty,\\dfrac{-11-\\sqrt{85}}6)\\cup (\\dfrac{-11+\\sqrt{85}}6,1)\\cup (1,+\\infty)$", @@ -568600,7 +572711,8 @@ "content": "若关于$x$的实系数方程$2x^2+3ax+a^2-a=0$至少有一个模为$1$的根, 则实数$a$的值为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期测验卷-线上测验01" ], "genre": "填空题", "ans": "$2\\pm \\sqrt{2}$或$-1$", @@ -568636,7 +572748,8 @@ "K0505005B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期测验卷-线上测验01" ], "genre": "解答题", "ans": "\\textcircled{1}与\\textcircled{3}可能成立, \\textcircled{2}不可能成立. 证明略", @@ -568669,7 +572782,8 @@ "content": "已知$z-2|z|=-7+4\\mathrm{i}$, 则复数$z=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期测验卷-线上测验01" ], "genre": "填空题", "ans": "$3+4\\mathrm{i}$或$\\dfrac 53+4\\mathrm{i}$", @@ -568702,7 +572816,8 @@ "content": "在底面边长为$1$的正三棱锥$P-ABC$中, 二面角$P-AB-C$为$\\dfrac{\\pi}{3}$, $G$是侧面$PAB$的重心, $H$是侧面$PAC$的重心, 则$GH$的长为\\blank{80}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-23_多面体及旋转体的概念与性质" ], "genre": "填空题", "ans": "$\\dfrac 13$", @@ -568735,7 +572850,8 @@ "content": "判断下列说法是否正确. 如果正确, 请说明理由; 如果不正确, 请举一个反例.\\\\\n(1) 有两个相邻的侧面是矩形的棱柱是直棱柱;\\\\\n(2) 正四棱柱是正方体.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-23_多面体及旋转体的概念与性质" ], "genre": "解答题", "ans": "(1) 正确, 侧棱垂直于底面上的两相交直线, 故侧棱垂直于底面; (2) 错误, 如长宽高分别为$1,1,3$的长方体", @@ -568767,7 +572883,9 @@ "K0620004B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-24_体积及表面积的计算" ], "genre": "填空题", "ans": "$8\\sqrt{3}$", @@ -568800,7 +572918,8 @@ "K0616003B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-24_体积及表面积的计算" ], "genre": "填空题", "ans": "$300\\sqrt{3}\\text{cm}$", @@ -568833,7 +572952,9 @@ "K0623002B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-24_体积及表面积的计算", + "2023届高三-赋能-赋能32" ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt[3]{3}}3R$", @@ -568873,7 +572994,8 @@ "K0619003B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-24_体积及表面积的计算" ], "genre": "填空题", "ans": "$9\\sqrt{2}\\pi$", @@ -568909,7 +573031,8 @@ "K0623004B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-24_体积及表面积的计算" ], "genre": "填空题", "ans": "$2500\\pi$, $\\dfrac{62500}3\\pi$", @@ -568943,7 +573066,8 @@ "K0620005B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-24_体积及表面积的计算" ], "genre": "填空题", "ans": "$60\\sqrt{3}$", @@ -568976,7 +573100,8 @@ "K0619003B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-24_体积及表面积的计算" ], "genre": "填空题", "ans": "$\\sqrt{3}$, $\\dfrac 16$", @@ -569009,7 +573134,8 @@ "K0617007B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-24_体积及表面积的计算" ], "genre": "填空题", "ans": "$\\dfrac{3+\\sqrt{3}}2\\pi$, $\\dfrac \\pi 2$", @@ -569041,7 +573167,8 @@ "K0617006B" ], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第一轮复习讲义-24_体积及表面积的计算" ], "genre": "填空题", "ans": "$(8+6\\sqrt{3})\\pi$", @@ -569074,7 +573201,8 @@ "K0514001B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-上学期周末卷-周末卷05" ], "genre": "填空题", "ans": "$2$", @@ -569186,7 +573314,8 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-赋能-赋能33" ], "genre": "填空题", "ans": "$\\dfrac{1}{12}$", @@ -569511,7 +573640,8 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-第一轮复习讲义-26_大数定律及独立性" ], "genre": "填空题", "ans": "$\\dfrac 13$", @@ -569547,7 +573677,8 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-第一轮复习讲义-26_大数定律及独立性" ], "genre": "填空题", "ans": "$\\dfrac 8{81}$", @@ -569583,7 +573714,8 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-第一轮复习讲义-26_大数定律及独立性" ], "genre": "填空题", "ans": "\\textcircled{1}\\textcircled{2}\\textcircled{4}", @@ -569614,7 +573746,8 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-第一轮复习讲义-26_大数定律及独立性" ], "genre": "选择题", "ans": "B", @@ -569646,7 +573779,8 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-第一轮复习讲义-26_大数定律及独立性" ], "genre": "选择题", "ans": "C", @@ -569680,7 +573814,8 @@ "objs": [], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-第一轮复习讲义-26_大数定律及独立性" ], "genre": "解答题", "ans": "(1) $\\dfrac 1{16}$; (2) $\\dfrac 34$; (3) $\\dfrac 7{16}$", @@ -569734,7 +573869,8 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-第一轮复习讲义-26_大数定律及独立性" ], "genre": "填空题", "ans": "$\\dfrac 12$", @@ -569766,7 +573902,8 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-第一轮复习讲义-26_大数定律及独立性" ], "genre": "解答题", "ans": "(1) $\\dfrac 16$; (2) $\\dfrac 5{72}$", @@ -569823,7 +573960,8 @@ "K0903006B" ], "tags": [ - "第九单元" + "第九单元", + "2023届高三-第一轮复习讲义-27_统计初步中的术语" ], "genre": "解答题", "ans": "(1) 分层抽样, 因不同区域的体验不同; (2) 简单随机抽样, 因高峰时无论在地铁何区域体验都基本相同", @@ -569923,7 +574061,9 @@ "K0903005B" ], "tags": [ - "第九单元" + "第九单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-27_统计初步中的术语" ], "genre": "填空题", "ans": "$24$与$30$", @@ -569975,7 +574115,9 @@ "K0903005B" ], "tags": [ - "第九单元" + "第九单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-27_统计初步中的术语" ], "genre": "选择题", "ans": "C", @@ -570100,7 +574242,8 @@ "K0904005B" ], "tags": [ - "第九单元" + "第九单元", + "2023届高三-第一轮复习讲义-27_统计初步中的术语" ], "genre": "填空题", "ans": "$32$", @@ -570129,7 +574272,8 @@ "K0905002B" ], "tags": [ - "第九单元" + "第九单元", + "2023届高三-第一轮复习讲义-27_统计初步中的术语" ], "genre": "填空题", "ans": "$0.32$, $91$, $60$, $75$, $74.84$, $8.54$", @@ -570159,7 +574303,8 @@ "K0906009B" ], "tags": [ - "第九单元" + "第九单元", + "2023届高三-第一轮复习讲义-27_统计初步中的术语" ], "genre": "解答题", "ans": "(1) $\\overline{x}=10.0$, $\\overline{y}=10.3$, $S_1^2=0.036$, $S_2^2=0.04$; (2) 不认为新设备的各项指标均值有显著提高($0.3<0.39$)", @@ -570212,7 +574357,8 @@ "K0906004B" ], "tags": [ - "第九单元" + "第九单元", + "2023届高三-第一轮复习讲义-27_统计初步中的术语" ], "genre": "填空题", "ans": "$3$与$5$", @@ -570242,7 +574388,9 @@ "K0905003B" ], "tags": [ - "第九单元" + "第九单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-27_统计初步中的术语" ], "genre": "填空题", "ans": "存在", @@ -570294,7 +574442,8 @@ "K0906004B" ], "tags": [ - "第九单元" + "第九单元", + "2023届高三-第一轮复习讲义-27_统计初步中的术语" ], "genre": "选择题", "ans": "A", @@ -570322,7 +574471,9 @@ "K0907001B" ], "tags": [ - "第九单元" + "第九单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-27_统计初步中的术语" ], "genre": "选择题", "ans": "C", @@ -570494,7 +574645,8 @@ "K0901004B" ], "tags": [ - "第九单元" + "第九单元", + "2023届高三-第一轮复习讲义-27_统计初步中的术语" ], "genre": "选择题", "ans": "D", @@ -570618,7 +574770,10 @@ "K0906010B" ], "tags": [ - "第九单元" + "第九单元", + "2023届高三-四月错题重做-05_易错题-概率与统计", + "2023届高三-四月错题重做-05_概率与统计", + "2023届高三-第一轮复习讲义-27_统计初步中的术语" ], "genre": "解答题", "ans": "(1) $49$人与$51$人, $165.406\\text{cm}$; (2) $165.406\\text{cm}$", @@ -570924,7 +575079,8 @@ "K0904005B" ], "tags": [ - "第九单元" + "第九单元", + "2023届高三-第一轮复习讲义-27_统计初步中的术语" ], "genre": "解答题", "ans": "(1) $9$, $12$; (2) 说明时长在$[20,25)$与$[25,30]$分的通话次数(频数, 频率)都小于$[15,20)$中的.", @@ -571031,7 +575187,8 @@ "K0907003B" ], "tags": [ - "第九单元" + "第九单元", + "2023届高三-第一轮复习讲义-27_统计初步中的术语" ], "genre": "解答题", "ans": "$\\text{P}25=155.5$, $\\text{P}50=161.0$, $\\text{P}75=164.0$", @@ -571110,7 +575267,9 @@ "K0906005B" ], "tags": [ - "第九单元" + "第九单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-27_统计初步中的术语" ], "genre": "解答题", "ans": "$S_1^20$''是``$\\dfrac ab+\\dfrac ba>2$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分也非必要条件}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-上学期测验卷-测验06", + "2023届高三-寒假作业-中档题" ], "genre": "选择题", "ans": "B", @@ -576864,7 +581141,8 @@ "content": "符号$[x]$表示不超过$x$的最大整数, 如$[\\pi]=3$, $[-1.08]=-2$, 定义函数$\\{x\\}=x-[x]$, 那么下列命题中正确的序号是\\bracket{20}.\\\\\n\\textcircled{1} 函数$\\{x\\}$的定义域为$\\mathbf{R}$, 值域为$[0,1]$; \\textcircled{2} 方程$\\{x\\}=\\dfrac 12$有无数解; \\textcircled{3} 函数$\\{x\\}$是周期函数; \\textcircled{4} 函数$\\{x\\}$是增函数.\n\\fourch{\\textcircled{1}\\textcircled{2}}{\\textcircled{2}\\textcircled{3}}{\\textcircled{3}\\textcircled{4}}{\\textcircled{4}\\textcircled{1}}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验06" ], "genre": "选择题", "ans": "B", @@ -576899,7 +581177,8 @@ "content": "如图所示, 已知$PA\\perp$平面$ABC$, $AD\\perp BC$于$D$, $BC=CD=AD=1$. 令$PD=x$, $\\angle BPC=\\theta$, 则\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [right] {$A$} coordinate (A);\n\\draw (-1,0,2) node [below] {$D$} coordinate (D);\n\\draw (-5,0,0) node [below] {$B$} coordinate (B);\n\\draw ($(D)!0.5!(B)$) node [below] {$C$} coordinate (C);\n\\draw (0,2,0) node [right] {$P$} coordinate (P);\n\\draw (B) -- (D) -- (A) -- (P) (B) -- (P) (C) -- (P) (D) -- (P);\n\\draw [dashed] (B) -- (A) (C) -- (A);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\tan\\theta = \\dfrac{x}{x^2+2}$}{$\\tan\\theta = \\dfrac{x}{x^2+1}$}{$\\tan\\theta = \\dfrac{1}{x^2+2}$}{$\\tan\\theta = \\dfrac{1}{x^2+1}$}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期测验卷-测验06" ], "genre": "选择题", "ans": "A", @@ -576934,7 +581213,8 @@ "content": "在$\\triangle ABC$中, 角$A,B,C$所对的边分别为$a,b,c$. $b=\\sqrt{5}$, $B=\\dfrac\\pi 4$.\\\\\n(1) 若$a=3$, 求$\\sin A$的值;\\\\\n(2) 若$\\triangle ABC$的面积等于$1$, 求$a$的值.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-上学期测验卷-测验06" ], "genre": "解答题", "ans": "(1) $\\dfrac{3\\sqrt{10}}{10}$; (2) $1$或$2\\sqrt{2}$", @@ -576969,7 +581249,8 @@ "content": "如图, 圆锥的顶点是$P$, 底面中心是$O$, 已知$OP=\\sqrt{2}$, 圆$O$的直径是$AB=2$, 点$C$在弧$AB$上, 且$\\angle CAB=30^\\circ$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (-1,0) node [left] {$A$} coordinate (A) (1,0) node [right] {$B$} coordinate (B) (0,0) node [above right] {$O$} coordinate (O) (0,{sqrt(2)}) node [above] {$P$} coordinate (P);\n\\draw ({cos(-60)},{0.3*sin(-60)}) node [below] {$C$} coordinate (C);\n\\draw (A) arc (180:360:1 and 0.3);\n\\draw [dashed] (A) arc (180:0:1 and 0.3) (A) -- (B) (O) -- (P) (A) -- (C);\n\\draw (A) -- (P) (B) -- (P) (C) -- (P);\n\\end{tikzpicture}\n\\end{center}\n(1) 求圆锥的侧面积;\\\\\n(2) 求$O$到平面$APC$的距离.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-上学期测验卷-测验06" ], "genre": "解答题", "ans": "(1) $\\sqrt{3}\\pi$; (2) $\\dfrac{\\sqrt{2}}3$", @@ -577004,7 +581285,9 @@ "content": "科学家发现某种特别物质的温度$y$(单位: 摄氏度)随时间$x$(单位: 分钟)的变化规律满足关系式: $y=m\\cdot 2^x+2^{1-x}$($0\\le x\\le 4$, $m>0$).\\\\\n(1) 若$m=2$, 求经过多少分钟, 该物质的温度为$5$摄氏度;\\\\\n(2) 如果该物质温度总不低于$2$摄氏度, 求$m$的取值范围.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验06", + "2023届高三-下学期测验卷-高三下学期测验01" ], "genre": "解答题", "ans": "(1) $1$分钟; (2) $[\\dfrac 12,+\\infty)$", @@ -577096,7 +581379,8 @@ "content": "从小到大排列的$9$个数据$1.23,1.35,2.14,2.55,3.67,3.89,4.21,4.43,5.51$的第$60$百分位数$\\mathrm{P}60$为\\blank{50}.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-上学期测验卷-测验06" ], "genre": "填空题", "ans": "$3.89$", @@ -577131,7 +581415,9 @@ "content": "函数$y=\\sqrt{\\dfrac 1{x^2-1}}$的定义域为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期测验卷-测验06", + "2023届高三-寒假作业-容易题" ], "genre": "填空题", "ans": "$(-\\infty,-1)\\cup (1,+\\infty)$", @@ -577169,7 +581455,8 @@ ], "tags": [ "第二单元", - "导数" + "导数", + "2023届高三-第一轮复习讲义-29_导数的应用" ], "genre": "解答题", "ans": "证明略", @@ -577262,7 +581549,8 @@ ], "tags": [ "第六单元", - "空间向量" + "空间向量", + "2023届高三-第一轮复习讲义-32_空间向量的概念与性质及立体几何中的证明问题" ], "genre": "填空题", "ans": "\\textcircled{1}\\textcircled{3}", @@ -577354,7 +581642,8 @@ ], "tags": [ "第六单元", - "空间向量" + "空间向量", + "2023届高三-第一轮复习讲义-32_空间向量的概念与性质及立体几何中的证明问题" ], "genre": "解答题", "ans": "(1) $\\dfrac{\\sqrt{3}}2$; (2) 投影为$(-\\dfrac 12,0,0)$, 数量投影为$-\\dfrac 12$", @@ -577494,7 +581783,9 @@ ], "tags": [ "第六单元", - "空间向量" + "空间向量", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-33_立体几何中的定量计算" ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\dfrac{\\sqrt{2}}3$", @@ -577556,7 +581847,9 @@ ], "tags": [ "第六单元", - "空间向量" + "空间向量", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-32_空间向量的概念与性质及立体几何中的证明问题" ], "genre": "填空题", "ans": "$\\dfrac 18$", @@ -577665,7 +581958,9 @@ ], "tags": [ "第六单元", - "空间向量" + "空间向量", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-33_立体几何中的定量计算" ], "genre": "选择题", "ans": "B", @@ -577727,7 +582022,8 @@ ], "tags": [ "第六单元", - "空间向量" + "空间向量", + "2023届高三-第一轮复习讲义-32_空间向量的概念与性质及立体几何中的证明问题" ], "genre": "填空题", "ans": "$\\dfrac{11}8$", @@ -577792,7 +582088,8 @@ ], "tags": [ "第六单元", - "空间向量" + "空间向量", + "2023届高三-第一轮复习讲义-33_立体几何中的定量计算" ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\dfrac{\\sqrt{15}}{5}$; (3) 存在, $\\dfrac{PM}{PC}=\\dfrac 13$", @@ -577828,7 +582125,11 @@ "K0403004X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-四月错题重做-03_数列", + "2023届高三-四月错题重做-03_易错题-数列", + "2023届高三-寒假作业-较难题", + "2023届高三-第一轮复习讲义-30_等差数列与等比数列" ], "genre": "填空题", "ans": "$a_n=-2^{n-8}$或$a_n=(-2)^{n-8}$", @@ -577876,7 +582177,8 @@ "K0406002X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第一轮复习讲义-31_数列的递推与通项及数学归纳法" ], "genre": "解答题", "ans": "是第$15$项", @@ -577915,7 +582217,8 @@ "K0407001X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第一轮复习讲义-31_数列的递推与通项及数学归纳法" ], "genre": "填空题", "ans": "$0,1,3,6,10$", @@ -577954,7 +582257,8 @@ "K0409001X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-第一轮复习讲义-31_数列的递推与通项及数学归纳法" ], "genre": "解答题", "ans": "(1) $a_2=\\dfrac 65$, $a_3=\\dfrac{10}9$, $a_4=\\dfrac{18}{17}$, 猜想$a_n=\\dfrac{2^n+1}{2^n+1}$; (2) 证明略", @@ -577993,7 +582297,9 @@ "K0406003X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-寒假作业-容易题", + "2023届高三-第一轮复习讲义-31_数列的递推与通项及数学归纳法" ], "genre": "解答题", "ans": "(1) $a_n=4n$; (2) $a_n=\\dfrac{2n-1}{2n^2}$; (3) $a_n=(-1)^n\\dfrac 1{2^n}$; (4) $a_n=(-1)^{n+1}\\sqrt[3]{n}$", @@ -578031,7 +582337,9 @@ "K0407003X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-寒假作业-中档题", + "2023届高三-第一轮复习讲义-31_数列的递推与通项及数学归纳法" ], "genre": "解答题", "ans": "$a_n=2^{n+1}-3n\\cdot 2^{n-1}$", @@ -578070,7 +582378,8 @@ ], "tags": [ "第六单元", - "空间向量" + "空间向量", + "2023届高三-第一轮复习讲义-32_空间向量的概念与性质及立体几何中的证明问题" ], "genre": "解答题", "ans": "$(1,0,\\sqrt{2})$或$(1,0,-\\sqrt{2})$", @@ -578108,7 +582417,9 @@ "content": "已知$\\omega,t>0$, 函数$f(x)=\\sqrt 3 \\cos \\omega x -\\sin \\omega x$的最小正周期为$2\\pi$, 将$f(x)$的图像向左平移$t$个单位, 所得图像对应的函数为偶函数, 则$t$的最小值为\\blank{50}.", "objs": [], "tags": [ - "暂无对应" + "暂无对应", + "2023届高三-上学期测验卷-测验07", + "2023届高三-第一轮复习讲义-35_圆及曲线方程" ], "genre": "填空题", "ans": "$\\dfrac{5\\pi}{6}$", @@ -578158,7 +582469,8 @@ "objs": [], "tags": [ "第七单元", - "圆" + "圆", + "2023届高三-第一轮复习讲义-35_圆及曲线方程" ], "genre": "填空题", "ans": "$[0,\\sqrt{3}]$", @@ -578226,7 +582538,8 @@ "tags": [ "第七单元", "椭圆", - "双曲线" + "双曲线", + "2023届高三-第一轮复习讲义-36_椭圆与双曲线的概念及性质" ], "genre": "填空题", "ans": "$\\pm 1$", @@ -578268,7 +582581,8 @@ ], "tags": [ "第七单元", - "双曲线" + "双曲线", + "2023届高三-第一轮复习讲义-36_椭圆与双曲线的概念及性质" ], "genre": "解答题", "ans": "$90^{\\circ}$", @@ -578537,7 +582851,10 @@ ], "tags": [ "第八单元", - "二项式定理" + "二项式定理", + "2023届高三-四月错题重做-05_易错题-概率与统计", + "2023届高三-四月错题重做-05_概率与统计", + "2023届高三-第一轮复习讲义-39_二项式定理" ], "genre": "解答题", "ans": "$15360$", @@ -578584,7 +582901,8 @@ "KNONE" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-赋能-赋能10" ], "genre": "填空题", "ans": "$1$", @@ -578624,7 +582942,8 @@ "KNONE" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-赋能-赋能11" ], "genre": "填空题", "ans": "$(0,\\dfrac 13]$", @@ -578662,7 +582981,8 @@ "content": "如果实数$x$、$y$满足$(x-1)^2+(y-2)^2=4$, 则$2x+y$的最大值是\\blank{50}.", "objs": [], "tags": [ - "暂无对应" + "暂无对应", + "2023届高三-赋能-赋能12" ], "genre": "填空题", "ans": "$4+2\\sqrt{5}$", @@ -578704,7 +583024,10 @@ "tags": [ "第四单元", "第八单元", - "二项式定理" + "二项式定理", + "2023届高三-四月错题重做-03_数列", + "2023届高三-四月错题重做-03_易错题-数列", + "2023届高三-赋能-赋能12" ], "genre": "填空题", "ans": "$39$", @@ -578755,7 +583078,8 @@ "content": "函数$f(x)=\\sin^2 x-\\cos^2 x$的最小正周期是\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-赋能-赋能12" ], "genre": "填空题", "ans": "$\\pi$", @@ -578794,7 +583118,8 @@ "content": "已知抛物线$y^2=4 x$, $F$为焦点,$P$为抛物线准线$l$上一动点, 线段$PF$与抛物线交于点$Q$, 定义$d(P)=\\dfrac{|FP|}{|FQ|}$.\\\\\n(1) 若点$P$坐标为$(-1,-\\dfrac 83)$, 求$d(P)$;\\\\\n(2) 求证: 存在常数$a$, 使得$2 d(P)=|FP|+a$关于点$P$恒成立.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-上学期测验卷-测验11" ], "genre": "解答题", "ans": "(1) $\\dfrac 83$; (2) $a=2$, 证明略", @@ -578833,7 +583158,8 @@ "content": "已知数列$\\{a_n\\}$满足$a_n \\ge 0$, 对任意$n \\ge 2$, $a_n$和$a_{n+1}$中存在一项使其为另一项与$a_{n-1}$的等差中项.\\\\\n(1) 已知$a_1=5$, $a_2=3$, $a_4=2$, 求$a_3$的所有可能取值;\\\\\n(2) 已知$a_1=a_4=a_7=0$, $a_2$、$a_5$、$a_8$为正数, 求证:$a_2$、$a_5$、$a_8$成等比数列, 并求出公比$q$.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-上学期测验卷-测验12" ], "genre": "解答题", "ans": "(1) $a_3=1$; (2) 证明略, 公比为$\\dfrac 14$", @@ -578867,7 +583193,8 @@ "content": "设$S_n$为数列$\\{a_n\\}$的前$n$项和, ``$\\{a_n\\}$是严格递增数列''是``$\\{S_n\\}$是严格递增数列''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-上学期周末卷-周末卷13" ], "genre": "选择题", "ans": "D", @@ -578898,7 +583225,8 @@ "content": "设$a>0$, 函数$f(x)=\\dfrac 1{1+a \\cdot 2^x}$.\\\\\n(1) 求函数$y=f(x) \\cdot f(-x)$的最大值(用$a$表示);\\\\\n(2) 设$g(x)=f(x)-f(x-1)$, 若对任意$x \\in(-\\infty, 0]$, $g(x) \\ge g(0)$恒成立, 求$a$的取值范围.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-上学期周末卷-周末卷13" ], "genre": "解答题", "ans": "(1) $\\dfrac 1{(1+a)^2}$; (2) $(0,\\sqrt{2}]$", @@ -578929,7 +583257,8 @@ "content": "已知等比数列$\\{a_n\\}$的前$n$项和为$S_n$, 前$n$项积为$T_n$, 则下列选项判断正确的是\\bracket{20}.\n\\onech{若$S_{2022}>S_{2021}$, 则数列$\\{a_n\\}$是严格递增数列}{若$T_{2022}>T_{2021}$, 则数列$\\{a_n\\}$是严格递增数列}{若数列$\\{S_n\\}$是严格递增数列, 则$a_{2022} \\ge a_{2021}$}{若数列$\\{T_n\\}$是严格递增数列, 则$a_{2022} \\ge a_{2021}$}", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-上学期测验卷-测验13" ], "genre": "选择题", "ans": "D", @@ -579175,7 +583504,8 @@ "objs": [], "tags": [ "第八单元", - "正态分布" + "正态分布", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "选择题", "ans": "D", @@ -579206,7 +583536,8 @@ "objs": [], "tags": [ "第八单元", - "正态分布" + "正态分布", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "选择题", "ans": "A", @@ -579237,7 +583568,8 @@ "objs": [], "tags": [ "第八单元", - "正态分布" + "正态分布", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "解答题", "ans": "$34.1\\%$", @@ -579268,7 +583600,8 @@ "objs": [], "tags": [ "第八单元", - "正态分布" + "正态分布", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "解答题", "ans": "$13.6\\%$", @@ -579299,7 +583632,8 @@ "objs": [], "tags": [ "第八单元", - "正态分布" + "正态分布", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "解答题", "ans": "$1-a$", @@ -579330,7 +583664,10 @@ "objs": [], "tags": [ "第八单元", - "正态分布" + "正态分布", + "2023届高三-四月错题重做-05_易错题-概率与统计", + "2023届高三-四月错题重做-05_概率与统计", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "解答题", "ans": "(1) $0.1\\%$; (2) $95.4\\%$", @@ -579367,7 +583704,8 @@ "content": "下表给出了一些地区鸟的种类数与该地区的海拔高度的数据, 试绘制散点图, 并通过观察散点图大致判断鸟的种类数与海拔高度之间的相关性.\n\\begin{center}\n\\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline 地区 & A & B & C & D & E & F & G & H & I & J & K \\\\\n\\hline 海拔高度 /m & 1250 & 1158 & 1067 & 457 & 701 & 731 & 610 & 670 & 1493 & 762 & 549 \\\\\n\\hline 鸟的种类 / 种 & 36 & 30 & 37 & 11 & 11 & 13 & 17 & 13 & 29 & 4 & 15 \\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "解答题", "ans": "散点图略; 总体来说鸟类的种类随海拔高度增加而变多, 但在几个区域内有相反的趋势, 有一定相关性", @@ -579395,7 +583733,8 @@ "content": "随机抽取某地区的 7 家超市, 得到其广告支出与销售额数据如下:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline 超市 & A & B & C & D & E & F & G \\\\\n\\hline 广告支出/万 & 1 & 2 & 4 & 6 & 10 & 14 & 20 \\\\\n\\hline 销售额 /万元 & 19 & 32 & 44 & 40 & 52 & 53 & 54 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n请推断该地区超市的销售额与广告支出之间是否存在线性相关关系.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "解答题", "ans": "存在线性相关关系(相关系数$r\\approx 0.83$)", @@ -579423,7 +583762,8 @@ "content": "随机抽取 10 家航空公司, 对其最近一年的航班正点率顾客投诉次数进行调查, 所得数据如下:\n\\begin{center}\n\\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|}\n\\hline 航空公司编 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\\\n\\hline 航班正点率/\\% &$81.8$&$76.8$&$76.6$&$75.7$&$73.8$&$72.2$&$71.2$&$70.8$&$91.4$&$68.5$\\\\\n\\hline 顾客投诉/次 & 21 & 58 & 85 & 68 & 74 & 93 & 72 & 122 & 18 & 125 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n试判断航班正点率与顾客投诉次数之间是否存在线性相关关系.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "解答题", "ans": "存在线性相关关系(相关系数$r\\approx -0.87$)", @@ -579451,7 +583791,8 @@ "content": "某地区的环境条件适合天鹅栖息繁衍. 有人发现了一个有趣的现象, 该地区有 5 个村庄, 其中 3 个村庄附近栖息的天鹅较多, 婴儿出生率也较高; 2 个村庄附近栖息的天鹅较少, 婴儿出生率也较低. 有人认为婴儿出生率和天鹅数之间存在相关关系, 并得出一个结论: 天鹅能够带来孩子. 你同意这个结论吗?", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "解答题", "ans": "不同意, 相关性不是因果关系. 可能是巧合(样本容量小), 可能是因为其它更基础的因素导致的", @@ -579479,7 +583820,8 @@ "content": "有人收集了某城市居民年收入(所有居民在一年内收入的总和)与$A$商品销售额的$10$年数据, 如下表所示:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline\n第$n$年 &$1$&$2$&$3$&$4$&$5$&$6$&$7$&$8$&$9$&$10$\\\\ \\hline\n居民收入/亿元 &$32.2$&$31.1$&$32.9$&$35.8$&$37.1$&$38.0$&$39.0$&$43.0$&$44.6$&$46.0$\\\\ \\hline\n$A$商品销售额/万元 &$25.0$&$30.0$&$34.0$&$37.0$&$39.0$&$41.0$&$42.0$&$44.0$&$48.0$&$51.0$\\\\ \\hline\n\\end{tabular}\n\\end{center}\n请画出散点图, 观察成对数据是否线性相关, 并通过计算相关系数判断``居民年收入'' 和``$A$商品销售额''的相关程度.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "解答题", "ans": "散点图略, 数据是线性相关的, 相关系数$r\\approx 0.952$, 线性相关程度较强", @@ -579507,7 +583849,8 @@ "content": "对下面两组数据\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline $x$ & 1 & 2 & 3 & 4 & 10 & 10 \\\\\n\\hline $y$ & 1 & 3 & 3 & 5 & 1 & 11 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n计算相关系数, 大概在$0.5$左右. 通过观察散点图, 发现对这两组大部分数据来说, 变量$x$与$y$有很强的线性相关关系, 是什么因素导致相关系数只有$0.5$左右?", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "解答题", "ans": "$(10,1)$这一数据偏离得很厉害, 异常值对相关系数有很大影响", @@ -579535,7 +583878,8 @@ "content": "已知根据某样本数据可得到回归方程为$y=4 x+\\hat{b}$, 且$\\overline x=3$, $\\overline y=6$, 则$\\hat{b}=$\\blank{50}.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "填空题", "ans": "$-6$", @@ -579563,7 +583907,8 @@ "content": "某小吃店的日盈利$y$(单位: 百元)与当天平均气温$x$(单位:$^{\\circ} \\text{C}$) 之间有如下数据:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline $x/^{\\circ} \\text{C}$&$-2$&$-1$& 0 & 1 & 2 \\\\\n\\hline $y$/百元 & 5 & 4 & 2 & 2 & 1 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n甲、乙、丙$3$位同学对上述数据进行了分析, 发现$y$与$x$之间具有线性相关关系, 他们通过计算分别得到$3$个线性回归方程:\\\\\n\\textcircled{1} $y=-x+2.8$; \\textcircled{2} $y=-x+3$; \\textcircled{3} $y=-1.2 x+2.6$. 使得拟合误差$Q$取到最小值的方程是\\blank{50}(填序号).", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "填空题", "ans": "\\textcircled{1}", @@ -579613,7 +583958,8 @@ "content": "为了解小麦种子是否灭菌与小麦发生黑穗病的关系, 经试验观察, 得到如下数据. 根据这组数据, 能否认为发生黑穗病与种子是否灭菌有关?\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline & 种子灭菌 & 种子未灭菌 & 总计 \\\\\n\\hline 有黑穗病 & 26 & 184 & 210 \\\\\n\\hline 无黑穗病 & 50 & 200 & 250 \\\\\n\\hline 总计 & 76 & 384 & 460 \\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "解答题", "ans": "与种子是否灭菌有关($\\chi^2=4.80$)", @@ -579640,7 +583986,8 @@ "content": "一医疗团队为研究某地的一种地方性疾病与当地居民的卫生习惯 (卫生习惯分为良好与不够良好两类) 的关系, 在已患该疾病的病例中随机调查了 100 例 (称为病例组), 同时在未患该疾病的人群中随机调查了 100 人 (称为对照组), 得到如下数据:\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline\n& 不够良好 & 良好 \\\\ \\hline\n 病例组 & $40$&$60$\\\\ \\hline\n 对照组 & $10$&$90$ \\\\ \\hline\n\\end{tabular}\n\\end{center}\n试问: 能否认为患该疾病群体与未患该疾病群体的卫生习惯有差异?", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "解答题", "ans": "有差异($\\chi^2=24$)", @@ -579667,7 +584014,8 @@ "content": "一个随机抽样的样本包括 110 位女士和 90 位男士, 女士中约有$9 \\%$是左利手 (又称左撇子), 男士中约有$11 \\%$是左利手. 基于这些数据, 你认为左利手与性别有关吗? 为什么?", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "解答题", "ans": "与性别无关($\\chi^2=0.224$)", @@ -579716,7 +584064,8 @@ "content": "设随机变量$X \\sim N(0,1)$, 则$X$的密度函数为\\blank{50}, $P(X \\leq 0)=$\\blank{50}, $P(|X|\\leq 1)\\approx$\\blank{50}, $P(X \\leq 1)\\approx$\\blank{50}, $P(X>1)\\approx$\\blank{50}.(精确到$0.0001$)", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "填空题", "ans": "$\\varphi(x)=\\dfrac{1}{\\sqrt{2\\pi}}\\mathrm{e}^{-\\frac{x^2}2}$; $0.5$; $0.6827$; $0.8413$; $0.1587$", @@ -579746,7 +584095,8 @@ "content": "设随机变量$X \\sim N(0,2^2)$, 随机变量$Y \\sim N(0,3^2)$, 画出分布密度曲线草图, 并指出$P(X \\leq -2)$与$P(X \\leq 2)$的关系, 以及$P(|X|\\leq 1)$与$P(|Y|\\leq 1)$之间的大小关系.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "解答题", "ans": "图略, $P(X\\le -2)+P(X\\le 2)=1$, $P(|X|\\le 1)>P(|Y|\\le 1)$", @@ -579820,7 +584170,8 @@ "content": "某市高二年级男生的身高$X$(单位: $\\text{cm}$) 近似服从正态分布$N(170,5^2)$, 随机选择一名该市高二年级的男生, 求下列事件的概率:(精确到$0.01$)\\\\ \n(1) $165175$.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "解答题", "ans": "(1) $0.68$; (2) $0.16$; (3) $0.16$", @@ -579872,7 +584223,10 @@ "content": "袋装食盐标准质量为$400 \\text{g}$, 规定误差的绝对值不超过$4 \\text{g}$就认为合格. 假设误差服从正态分布, 随机抽取$100$袋食盐, 误差的样本均值为$0$, 样本方差为$4$. 请你估计这批袋装食盐的合格率.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-四月错题重做-05_易错题-概率与统计", + "2023届高三-四月错题重做-05_概率与统计", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "解答题", "ans": "$95.45\\%$", @@ -580156,7 +584510,8 @@ "content": "某城市高中数学统考, 假设考试成绩服从正态分布$N(75,8^2)$. 如果按照$16 \\%, 34 \\%, 34 \\%, 16 \\%$的比例将考试成绩分为A, B, C, D四个等级, 试确定各等级的分数线(精确到$1$分).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "解答题", "ans": "$\\ge 83$分为A, $\\ge 75$分而小于$83$分为B, $\\ge 67$分而$<75$分为C, 小于$67$分为D", @@ -580208,7 +584563,8 @@ "content": "根据下面的散点图, 推断图中的两个变量是否存在相关关系.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.7, xscale = 0.3, yscale = 0.3]\n\\draw [->] (0, 0) -- (11, 0) node [below] {$x$};\n\\draw [->] (0, 0) -- (0, 11) node [left] {$y$};\n\\draw (0, 0) node [below left] {$O$};\n\\foreach \\i in {1, 2, ..., 10}\n{\\draw (\\i, 0.2) -- (\\i, 0) node [below] {\\tiny $\\i$};\n\\draw (0.2, \\i) -- (0, \\i) node [left] {\\tiny $\\i$};};\n\\foreach \\i/\\j in {3.194/7.070, 4.188/6.777, 4.382/5.639, 3.005/7.971, 3.840/7.151, 0.780/9.321, 6.540/4.264, 0.603/9.743, 1.839/9.140, 9.091/1.618, 9.234/1.275, 9.281/1.361, 8.411/1.979, 8.380/1.909, 0.638/9.390, 8.357/1.919, 1.087/9.054, 2.021/8.972, 2.992/7.973, 2.861/7.222, 1.578/9.048, 2.142/8.763, 7.934/2.865, 5.403/5.335, 4.604/5.715, 5.260/4.999, 3.708/6.483, 1.626/9.189, 0.387/9.917, 7.994/2.412, 7.912/2.171, 8.185/2.307, 8.526/1.651, 8.612/1.861, 1.069/9.810, 8.300/1.821, 2.551/8.104, 7.184/3.441, 5.890/4.590, 0.416/10.263}\n{\\filldraw (\\i, \\j) circle (0.05);}\n\\draw (5.5,-3) node {(1)};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex, scale = 0.7, xscale = 0.3, yscale = 0.3]\n\\draw [->] (0, 0) -- (11, 0) node [below] {$x$};\n\\draw [->] (0, 0) -- (0, 11) node [left] {$y$};\n\\draw (0, 0) node [below left] {$O$};\n\\foreach \\i in {1, 2, ..., 10}\n{\\draw (\\i, 0.2) -- (\\i, 0) node [below] {\\tiny $\\i$};\n\\draw (0.2, \\i) -- (0, \\i) node [left] {\\tiny $\\i$};};\n\\foreach \\i/\\j in {8.756/3.876, 0.250/6.803, 3.767/10.185, 3.255/10.784, 1.464/8.429, 6.011/9.004, 5.890/9.864, 1.519/8.964, 4.443/10.778, 8.263/5.680, 5.434/10.160, 0.580/7.541, 8.382/4.983, 0.251/6.512, 4.669/10.289, 0.908/7.976, 5.407/9.626, 7.372/7.046, 0.400/6.566, 7.772/6.091, 8.620/4.368, 8.738/4.059, 3.481/10.037, 8.276/5.701, 1.606/9.025, 7.562/7.091, 6.841/7.972, 1.940/9.046, 5.549/9.596, 8.203/5.539, 0.390/6.556, 4.180/9.997, 8.152/5.724, 0.354/7.083, 5.676/9.782, 2.571/10.091, 7.333/7.182, 6.613/8.861, 8.470/4.686, 5.516/10.077}\n{\\filldraw (\\i, \\j) circle (0.05);}\n\\draw (5.5,-3) node {(2)};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex, scale = 0.7, xscale = 0.3, yscale = 0.3]\n\\draw [->] (0, 0) -- (11, 0) node [below] {$x$};\n\\draw [->] (0, 0) -- (0, 11) node [left] {$y$};\n\\draw (0, 0) node [below left] {$O$};\n\\foreach \\i in {1, 2, ..., 10}\n{\\draw (\\i, 0.2) -- (\\i, 0) node [below] {\\tiny $\\i$};\n\\draw (0.2, \\i) -- (0, \\i) node [left] {\\tiny $\\i$};};\n\\foreach \\i/\\j in {4.722/0.150, 6.323/0.162, 5.318/5.970, 4.774/8.597, 0.486/0.178, 2.309/3.744, 7.638/1.408, 7.708/5.855, 3.677/0.007, 1.170/4.449, 1.580/0.716, 1.865/9.864, 7.739/0.171, 2.098/4.754, 9.045/0.978, 5.171/2.419, 8.343/6.755, 9.007/9.753, 4.743/0.395, 9.272/5.289, 9.031/7.690, 6.157/9.380, 9.372/6.672, 3.103/8.398, 8.863/6.656, 8.010/8.238, 1.871/9.275, 1.807/0.524, 1.216/4.293, 8.080/2.223, 8.373/9.335, 4.525/7.231, 1.374/2.144, 1.671/4.441, 3.436/0.622, 9.401/6.552, 0.932/7.255, 5.049/0.744, 3.128/3.868, 2.686/5.655}\n{\\filldraw (\\i, \\j) circle (0.05);}\n\\draw (5.5,-3) node {(3)};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex, scale = 0.7, xscale = 0.3, yscale = 0.3]\n\\draw [->] (0, 0) -- (11, 0) node [below] {$x$};\n\\draw [->] (0, 0) -- (0, 11) node [left] {$y$};\n\\draw (0, 0) node [below left] {$O$};\n\\foreach \\i in {1, 2, ..., 10}\n{\\draw (\\i, 0.2) -- (\\i, 0) node [below] {\\tiny $\\i$};\n\\draw (0.2, \\i) -- (0, \\i) node [left] {\\tiny $\\i$};};\n\\foreach \\i/\\j in {1.108/3.652, 6.471/8.427, 7.997/9.438, 9.573/10.167, 6.033/8.445, 3.193/5.893, 9.248/9.521, 5.931/7.792, 8.461/9.309, 1.709/4.774, 1.619/4.191, 7.233/8.444, 6.255/8.520, 2.723/5.802, 0.624/2.270, 1.226/3.435, 8.010/9.071, 6.083/7.958, 9.750/9.841, 5.552/7.568, 9.944/10.066, 1.066/3.591, 7.446/8.399, 3.939/6.989, 0.575/2.607, 2.701/5.971, 8.189/9.468, 6.667/8.479, 6.922/8.100, 3.722/6.171, 3.902/6.886, 0.446/1.580, 2.471/5.804, 3.774/6.897, 9.831/9.563, 0.233/1.208, 7.847/9.393, 5.374/7.433, 7.215/9.197, 3.568/6.796}\n{\\filldraw (\\i, \\j) circle (0.05);}\n\\draw (5.5,-3) node {(4)};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "解答题", "ans": "(1) 存在线性相关关系; (2) 存在非线性相关关系; (3) 不存在相关关系; (4) 存在非线性相关关系", @@ -580500,7 +584856,8 @@ "content": "假如女儿身高$y$(单位: $\\text{cm}$) 关于父亲身高$x$(单位: $\\text{cm}$)的经验回归方程为$\\hat{y}=0.81 x+25.82$. 已知父亲身高为$175 \\text{cm}$, 请估计女儿的身高.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "解答题", "ans": "约$168\\text{cm}$", @@ -580638,7 +584995,8 @@ "content": "根据某城市居民年收人与$A$商品销售额的数据:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline 第$n$年 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\\\n\\hline 居民年收人/亿元 & 32.2 & 31.1 & 32.9 & 35.8 & 37.1 & 38.0 & 39.0 & 43.0 & 44.6 & 46.0 \\\\\n\\hline A 商品销售额/万元 & 25.0 & 30.0 & 34.0 & 37.0 & 39.0 & 41.0 & 42.0 & 44.0 & 48.0 & 51.0 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 建立 A 商品销售额关于居民年收人的一元线性回归模型;\\\\\n(2) 如果这座城市居民的年收人达到$40$亿元, 估计 A 商品的销售额是多少.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "解答题", "ans": "(1) 设$A$商品的销售额为$y$(万元), 居民年收入为$x$(亿元). 则$y=1.447x-15.8427$; (2) $42.04$万元", @@ -580754,7 +585112,8 @@ "content": "根据有关规定, 香烟盒上必须印上``吸烟有害健康''的警示语. 那么\\\\\n(1) 吸烟是否对每位烟民一定会引发健康问题?\\\\\n(2) 有人说吸烟不一定引起健康问题, 因此可以吸烟. 这种说法对吗?", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "解答题", "ans": "(1) 不一定, 标示语的含义是有较显著的联系, 并不是``一定''; (2) 不正确, 一来有健康风险, 二来影响他人", @@ -580781,7 +585140,8 @@ "content": "假设随机抽取了$44$名学生, 按照性别和体育锻炼情况整理为如下的列联表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\backslashbox{性别}{锻炼} & 不经常 & 经常 & 合计 \\\\\n\\hline 女生 & 5 & 15 & 20 \\\\\n\\hline 男生 & 6 & 18 & 24 \\\\\n\\hline 合计 & 11 & 33 & 44 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 据此推断性别因素是否影响学生锻炼的经常性;\\\\\n(2) 说明你的推断结论是否可能犯错, 并解释原因.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "解答题", "ans": "(1) 不影响($\\chi^2=0$); (2) 可能犯错, 样本量较小", @@ -580808,7 +585168,8 @@ "content": "某儿童医院用甲、乙两种疗法治疗小儿消化不良. 采用有放回简单随机抽样的 方法对治疗情况进行检查, 得到了如下数据: 抽到接受甲种疗法的患儿$67$名, 其中未治愈$15$名, 治愈$52$名; 抽到接受乙种疗法的患儿$69$名, 其中未治愈$6$名, 治愈 $63$名. 试根据小概率值 $\\alpha=0.05$的独立性检验, 分析乙种疗法的效果是否比甲种疗法好.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "解答题", "ans": "认为乙种疗法的效果比甲种疗法更好($\\chi^2=4.881$, $P(\\text{治愈}|\\text{使用乙疗法})>P(\\text{治愈}|\\text{使用甲疗法})$)", @@ -580857,7 +585218,8 @@ "content": "为考察某种药物$A$对预防疾病$B$的效果, 进行了动物试验, 根据$105$个有放回简单随机样本的数据, 得到如下列联表: \n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\backslashbox{药物A}{疾病B} & 未患病 & 患病 & 合计 \\\\\n\\hline 未服用 & 29 & 15 & 44 \\\\\n\\hline 服用 & 47 & 14 & 61 \\\\\n\\hline 合计 & 76 & 29 & 105 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n依据$\\alpha=0.05$的独立性检验, 分析药物$A$对预防疾病$B$的有效性.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "解答题", "ans": "认为对预防无效($\\chi^2=1.587$)", @@ -580994,7 +585356,8 @@ "content": "为了研究高三年级学生的性别和身高是否大于$170 \\text{cm}$的关联性, 调查了某中学所有高三年级的学生, 整理得到如下列联表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\backslashbox{性别}{身高} & 低于$170 \\text{cm}$& 不低于$170 \\text{cm}$& 合计 \\\\\n\\hline 女 & 81 & 16 & 97 \\\\\n\\hline 男 & 28 & 75 & 103 \\\\\n\\hline 合计 & 109 & 91 & 200 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 请画出列联表的等高堆积条形图, 判断该中学高三年级学生的性别和身高是否有关联. 如果结论是性别与身高有关联, 请解释它们之间如何相互影响;\\\\\n(2) 从这$200$名高三学生中获取容量为$40$的有放回简单随机样本, 由样本数据整理得到如下列联表: \n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\backslashbox{性别}{身高} & 低于$170 \\text{cm}$& 不低于$170 \\text{cm}$& 合计 \\\\\n\\hline 女 & 14 & 7 & 21 \\\\\n\\hline 男 & 8 & 11 & 19 \\\\\n\\hline 合计 & 22 & 18 & 40 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n依据$\\alpha=0.05$的独立性检验, 能否认为该中学高三年级学生的性别与身高有关联? 解释所得结论的实际含义;\\\\\n(3) (2)得到的结论与(1)得到的结论一致吗? 如果不一致, 你认为原因是什么.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "解答题", "ans": "(1) 图略; (2) 不能认为性别与身高有关($\\chi^2=2.43$); (3) 不一致, (1)是准确的, 是普查的结果, 不一致是由于样本选取的偏差导致", @@ -581065,7 +585428,8 @@ "content": "变量$x$与$y$的成对样本数据的散点图如下图所示, 据此可以推断变量$x$与$y$之间\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0, 0) -- (3.5, 0) node [below] {$x$};\n\\draw [->] (0, 0) -- (0, 2.5) node [left] {$y$};\n\\draw (0, 0) node [below left] {$O$};\n\\foreach \\i in {0.5, 1.0, 1.5, 2.0, 2.5, 3.0} {\\draw (\\i, 0.1) -- (\\i, 0.05) node [below] {$\\i$};};\n\\foreach \\i in {2, 4, 6, 8} {\\draw (0.1, {\\i/4}) -- (0, {\\i/4}) node [left] {$\\i$};};\n\\foreach \\i/\\j in {0.2/0.2, 0.7/1.0, 1.3/2.1, 1.9/4.9, 2.6/5.2, 3.2/6.9}\n{\\filldraw (\\i, \\j/4) circle (0.03);};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{很可能存在负相关}{一定存在正相关}{很可能存在正相关}{一定不存在负相关}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "选择题", "ans": "C", @@ -581115,7 +585479,8 @@ "content": "根据分类变量$x$与$y$的成对样本数据, 计算得到$\\chi^2=2.974$. 依据$\\alpha=0.05$的独立性检验, 结论为 \\bracket{20}.\n\\onech{变量$x$与$y$不独立}{变量$x$与$y$不独立, 这个结论犯错误的概率不超过$0.05$}{变量$x$与$y$独立}{变量$x$与$y$独立, 这个结论犯错误的概率不超过$0.05$}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-正态分布及成对数据新课-一月新课作业" ], "genre": "选择题", "ans": "C", @@ -581274,7 +585639,8 @@ "content": "设全集$U=\\{x | x^3-x=0\\}$, 集合$A=\\{0,1\\}$, 则$\\overline A=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-赋能-赋能22" ], "genre": "填空题", "ans": "$-1$", @@ -583287,7 +587653,8 @@ "content": "已知非空集合$A$、$B$满足$A \\cup B=\\mathbf{R}$, $A \\cap B=\\varnothing$, 函数$f(x)=\\begin{cases}x^2,& x \\in A, \\\\2 x-1, & x \\in B,\\end{cases}$ 对于下列两个命题: \\textcircled{1} 存在唯一的非空集合对$(A, B)$, 使得$f(x)$为偶函数; \\textcircled{2} 存在无穷多非空集合对$(A, B)$, 使得方程$f(x)=2$无解. 下面判断正确的是\\bracket{20}.\n\\fourch{\\textcircled{1}正确, \\textcircled{2}错误}{\\textcircled{1}错误, \\textcircled{2}正确}{\\textcircled{1}\\textcircled{2}都正确}{\\textcircled{1}\\textcircled{2}都错误}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期测验卷-高三下学期月考01" ], "genre": "选择题", "ans": "B", @@ -593620,7 +597987,10 @@ "content": "在一个袋子里有$4$个红球和$2$个白球, 球除了颜色之外, 其它物理属性均相同. 某人从袋中有放回地摸$8$次球, 设所摸出的球中红球的数量为$X$, 则$E[X]=$\\blank{50}, $D[X]=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-四月错题重做-05_易错题-概率与统计", + "2023届高三-四月错题重做-05_概率与统计", + "2023届高三-正态分布及成对数据新课-正态分布及成对数据测试" ], "genre": "填空题", "ans": "$\\dfrac{16}{3}$, $\\dfrac{16}{9}$", @@ -593659,7 +598029,8 @@ "content": "某小组有$5$名男生和$4$名女生, 现要从中随机选出$3$名参加活动. 选中的男生人数恰为$1$的概率为\\blank{50}; 在已知有男生入选的条件下, 选中的男生人数恰为$1$的概率为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-正态分布及成对数据新课-正态分布及成对数据测试" ], "genre": "填空题", "ans": "$\\dfrac{3}{8}$", @@ -593691,7 +598062,8 @@ "content": "已知随机变量$X$的分布为$\\begin{pmatrix}\n 0 & 1 & 2 & 3 \\\\ \\dfrac 18 & \\dfrac 38 & \\dfrac 38 & \\dfrac 18\n\\end{pmatrix}$, 随机变量$Y$的分布为$\\begin{pmatrix}\n 1 & 2 & 3 \\\\ \\dfrac 15 & \\dfrac 35 & \\dfrac 15\n\\end{pmatrix}$, 则\\bracket{20}.\n\\twoch{$X$是二项分布, $Y$是二项分布}{$X$是二项分布, $Y$是超几何分布}{$X$是超几何分布, $Y$是二项分布}{$X$是超几何分布, $Y$是超几何分布}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-正态分布及成对数据新课-正态分布及成对数据测试" ], "genre": "选择题", "ans": "B", @@ -593723,7 +598095,8 @@ "content": "设$X\\sim N(3,2^2)$.\\\\\n(1) 求$P(00$)与$N(\\mu_2,\\sigma_2^2)$($\\sigma_2>0$)的随机变量的正态分布密度函数的大致图像如下, 则\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-5,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,-0) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\def\\s{1}\n\\draw [domain = -5:5, samples = 100, dashed] plot (\\x,{5*1/sqrt(2*pi)/pow(\\s,2)*exp(-1/2*pow(\\x-0.5,2)/pow(\\s,2))});\n\\def\\s{1.5}\n\\draw [domain = -5:5, samples = 100] plot (\\x,{5*1/sqrt(2*pi)/pow(\\s,2)*exp(-1/2*pow(\\x+1,2)/pow(\\s,2))});\n\\draw (5.5,1.5) -- (6,1.5) node [right] {$N(\\mu_1,\\sigma_1^2)$};\n\\draw [dashed] (5.5,1) -- (6,1) node [right] {$N(\\mu_2,\\sigma_2^2)$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\mu_1<\\mu_2$, $\\sigma_1<\\sigma_2$}{$\\mu_1<\\mu_2$, $\\sigma_1>\\sigma_2$}{$\\mu_1>\\mu_2$, $\\sigma_1<\\sigma_2$}{$\\mu_1>\\mu_2$, $\\sigma_1>\\sigma_2$}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-正态分布及成对数据新课-正态分布及成对数据测试" ], "genre": "选择题", "ans": "B", @@ -593787,7 +598161,8 @@ "content": "设某总体服从正态分布$N(\\mu,\\sigma^2)$($\\sigma>0$), 其$100$个随机选取的样本的数据如下表, 则该正态分布的参数最有可能是\\bracket{20}.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline\n四舍五入之后的值 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18\\\\ \\hline\n频数 & 1 & 0 & 1 & 2 & 7 & 8 & 9 & 13 & 14 & 8 & 16 & 5 & 7 & 6 & 0 & 1 & 2 \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\fourch{$\\mu = 10$, $\\sigma = 2$}{$\\mu = 10$, $\\sigma = 3$}{$\\mu = 12$, $\\sigma = 2$}{$\\mu = 12$, $\\sigma = 3$}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-正态分布及成对数据新课-正态分布及成对数据测试" ], "genre": "选择题", "ans": "B", @@ -593819,7 +598194,8 @@ "content": "设$\\Phi(x)$是标准正态分布函数(标准正态密度函数$y=\\varphi(x)$从$-\\infty$到$x$的累计面积), 且$\\Phi(x_1)=3\\Phi(x_2)$, 则\\bracket{20}.\n\\twoch{$x_1$一定大于零, $x_2$一定小于零}{$x_1$一定大于零, $x_2$可能大于零}{$x_1$可能小于零, $x_2$一定小于零}{$x_1$可能小于零, $x_2$可能大于零}", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-正态分布及成对数据新课-正态分布及成对数据测试" ], "genre": "选择题", "ans": "C", @@ -594159,7 +598535,8 @@ "content": "为了解发动机的动力$x$(单位: $\\text{PH}$) 与排气温度$y$(单位:${ }^{\\circ} \\text{C}$) 之间的关系, 某部门进行相关试验, 得到如下数据:\n\\begin{center}\n\\begin{tabular}{|c|c||c|c|}\n\\hline$x / \\text{PH}$&$y /{ }^{\\circ} \\text{C}$&$x / \\text{PH}$&$y /{ }^{\\circ} \\text{C}$\\\\\n\\hline 4300 & 960 & 4010 & 907 \\\\\n\\hline 4650 & 900 & 3810 & 843 \\\\\n\\hline 3200 & 807 & 4500 & 927 \\\\\n\\hline 3150 & 755 & 3008 & 688 \\\\\n\\hline 4950 & 993 & & \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 求相关系数;\\\\\n(2) 求线性回归方程;\\\\\n(3) 估计当$x=3100$时对应$y$的值.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-正态分布及成对数据新课-正态分布及成对数据测试" ], "genre": "解答题", "ans": "(1) $r\\approx 0.925$; (2) $y=0.13 x+350.5$; (3) 约$754$", @@ -594367,7 +598744,8 @@ "content": "下表所示的是关于$11$岁儿童患花粉热与湿疹情况的调查数据. 若按$95 \\%$的可靠性的要求, 则对$11$岁儿童能否做出花粉热与湿疹有关的结论?\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline & 患花粉热 & 未患花粉热 & 合计 \\\\\n\\hline 患湿疹 & 141 & 420 & 561 \\\\\n\\hline 未患湿疹 & 928 & 13525 & 14453 \\\\\n\\hline 合计 & 1069 & 13945 & 15014 \\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-正态分布及成对数据新课-正态分布及成对数据测试" ], "genre": "解答题", "ans": "$\\chi^2\\approx 285.96$, 按$95\\%$的可靠性要求, 对$11$岁儿童可以作出花粉热与湿疹有关的结论", @@ -594553,7 +598931,10 @@ "content": "已知$x$, $y$的取值如下表所示, 从散点图分析可知$y$与$x$线性相关, 如果线性回归方程为$y=0.95 x+2.6$, 那么表格中的数据$m$的值为\\blank{50}.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline$x$& 0 & 1 & 3 & 4 \\\\\n\\hline$y$&$2.2$&$4.3$&$4.8$&$m$\\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-四月错题重做-05_易错题-概率与统计", + "2023届高三-四月错题重做-05_概率与统计", + "2023届高三-正态分布及成对数据新课-正态分布及成对数据测试" ], "genre": "填空题", "ans": "$6.7$", @@ -594685,7 +599066,8 @@ "content": "设$x_i$($i=1,2,3,\\cdots,100$)和$y_i$($i=1,2,\\cdots,100$)是两组两两不同的数据, 以$x_i$为解释变量, $y_i$为反应变量计算可得相关系数$r_1$, 拟合直线的斜率为$\\hat{a}_1$; 以$y_i$为解释变量, $x_i$为反应变量计算可得相关系数$r_2$, 拟合直线的斜率为$\\hat{a}_2$. 则关于$r_1,r_2,\\hat{a}_1,\\hat{a}_2$的以下两个结论: \\textcircled{1} $r_1$一定与$r_2$相等; \\textcircled {2} $\\hat{a}_1\\cdot \\hat{a}_2$一定等于$1$, 它们的真假情况为\\bracket{20}.\n\\fourch{\\textcircled{1}和\\textcircled{2}都为真}{\\textcircled{1}和\\textcircled{2}都为假}{\\textcircled{1}为真, \\textcircled{2}为假}{\\textcircled{1}为假, \\textcircled{2}为真}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-正态分布及成对数据新课-正态分布及成对数据测试" ], "genre": "选择题", "ans": "C", @@ -594717,7 +599099,8 @@ "content": "若一组成对数据$x_i$($i=1,2,\\cdots,n$)与$y_i$($i=1,2,\\cdots,n$)的相关系数为$0.4$, 则$x_i$($i=1,2,\\cdots,n$)与$-y_i+0.1$($i=1,2,\\cdots,n$)的相关系数为\\bracket{20}.\n\\fourch{$0.5$}{$0.4$}{$-0.3$}{$-0.4$}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-正态分布及成对数据新课-正态分布及成对数据测试" ], "genre": "选择题", "ans": "D", @@ -594749,7 +599132,8 @@ "content": "为了判断甲、乙两种药物对某疾病的疗效是否有显著差异, 在两个地区分别进行了采样, 所得的列联表如下:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\nA地区 & 治愈 & 未治愈 & 合计 \\\\ \\hline\n药物甲 & $a$ & $b$ & $a+b$ \\\\ \\hline\n药物乙 & $c$ & $d$ & $c+d$ \\\\ \\hline\n合计 & $a+c$ & $b+d$ & $a+b+c+d$\\\\ \\hline\n\\end{tabular}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\nB地区 & 治愈 & 未治愈 & 合计 \\\\ \\hline\n药物甲 & $3a$ & $3b$ & $3a+3b$ \\\\ \\hline\n药物乙 & $3c$ & $3d$ & $3c+3d$ \\\\ \\hline\n合计 & $3a+3c$ & $3b+3d$ & $3a+3b+3c+3d$\\\\ \\hline\n\\end{tabular}\n\\end{center}\n巧合的是, B地区每一类型的人数恰好是A地区的$3$倍. 规定显著性水平为$p=0.05$, 则以下两个论断:\\\\\n\\textcircled{1} 如果A地区的数据支持两种药物的疗效有显著差异, 那么B地区的数据一定支持两种药物的疗效有显著差异;\\\\\n\\textcircled{2} 如果A地区的数据支持两种药物的疗效无显著差异, 那么B地区的数据一定支持两种药物的疗效无显著差异. 正确与否的情况为\\bracket{20}.\n\\fourch{\\textcircled{1}和\\textcircled{2}都正确}{\\textcircled{1}和\\textcircled{2}都错误}{\\textcircled{1}正确, \\textcircled{2}错误}{\\textcircled{1}错误, \\textcircled{2}正确}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-正态分布及成对数据新课-正态分布及成对数据测试" ], "genre": "选择题", "ans": "C", @@ -594781,7 +599165,8 @@ "content": "$\\displaystyle\\sum_{n=1}^{\\infty} (\\dfrac 14)^n=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-赋能-赋能13" ], "genre": "填空题", "ans": "$\\dfrac 13$", @@ -594816,7 +599201,8 @@ "content": "等差数列$\\{a_n\\}$中, $a_{10}=7$, $a_1=4$, 则$\\{a_n\\}$的前$20$项之和$S_{20}=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-赋能-赋能17" ], "genre": "填空题", "ans": "$\\dfrac{430}{3}$", @@ -594851,7 +599237,8 @@ "content": "已知$z\\in \\mathbf{C}$. 若$\\dfrac{1}{2z-3}=\\mathrm{i}$($\\mathrm{i}$为虚数单位), 则$z=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-寒假作业-寒假作业反馈练习" ], "genre": "填空题", "ans": "$\\dfrac 32-\\dfrac 12\\mathrm{i}$", @@ -594892,7 +599279,8 @@ "K0512002B" ], "tags": [ - "第五单元" + "第五单元", + "2023届高三-寒假作业-寒假作业反馈练习" ], "genre": "填空题", "ans": "$7$", @@ -594930,7 +599318,8 @@ "content": "已知集合$A=(-\\infty ,a]$, $B=[2,5]$且$A\\cap B$非空, 则实数$a$的取值范围\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-寒假作业反馈练习" ], "genre": "填空题", "ans": "$[2,+\\infty)$", @@ -594971,7 +599360,8 @@ "K0217004B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-寒假作业反馈练习" ], "genre": "填空题", "ans": "$-2$", @@ -595012,7 +599402,8 @@ ], "tags": [ "第七单元", - "双曲线" + "双曲线", + "2023届高三-寒假作业-寒假作业反馈练习" ], "genre": "填空题", "ans": "$2\\sqrt{10}$", @@ -595051,7 +599442,8 @@ "content": "在$\\triangle ABC$中, $AC=6$, $3 \\sin A=2 \\sin B$, 且$\\cos C=\\dfrac 14$, 则$AB=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-寒假作业-寒假作业反馈练习" ], "genre": "填空题", "ans": "$2\\sqrt{10}$", @@ -595091,7 +599483,8 @@ "K0203005B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-寒假作业反馈练习" ], "genre": "填空题", "ans": "$\\sqrt{5}$", @@ -595129,7 +599522,8 @@ "content": "已知$f(x)$是定义域为$\\mathbf{R}$的奇函数, 满足$f(1+x)=f(1-x)$. 若$f(1)=2$, 则$f(1)+f(2)+f(3)+\\cdots+f(2023)=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-寒假作业反馈练习" ], "genre": "填空题", "ans": "$0$", @@ -595168,7 +599562,8 @@ "objs": [], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-寒假作业-寒假作业反馈练习" ], "genre": "填空题", "ans": "$\\dfrac{14}{15}$", @@ -595207,7 +599602,8 @@ "objs": [], "tags": [ "第一单元", - "第二单元" + "第二单元", + "2023届高三-寒假作业-寒假作业反馈练习" ], "genre": "填空题", "ans": "$\\dfrac{5}{14}$", @@ -595245,7 +599641,8 @@ "content": "已知平面$\\alpha$、$\\beta$、$\\gamma$两两垂直, 直线$a$、$b$、$c$满足: $a \\parallel \\alpha$, $b \\parallel \\beta$, $c \\parallel \\gamma$, 则直线$a$、$b$、$c$不可能是\\bracket{20}.\n\\fourch{两两垂直}{两两平行}{两两相交}{两两异面}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-寒假作业-寒假作业反馈练习" ], "genre": "选择题", "ans": "B", @@ -595283,7 +599680,8 @@ "content": "已知$a,b\\in \\mathbf{R}$, 则``$ab>0$''是``$\\dfrac ab+\\dfrac ba>1$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分也非必要条件}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-寒假作业-寒假作业反馈练习" ], "genre": "选择题", "ans": "C", @@ -595321,7 +599719,8 @@ "content": "为了得到函数$y=\\sin(2x+\\dfrac{5\\pi}6)$的图像, 可将函数$y=\\sin 2x$的图像\\bracket{20}.\n\\twoch{左移$\\dfrac{5\\pi}6$个长度}{右移$\\dfrac{5\\pi}6$个长度}{左移$\\dfrac{5\\pi}{12}$个长度}{右移$\\dfrac{5\\pi}{12}$个长度}", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-寒假作业-寒假作业反馈练习" ], "genre": "选择题", "ans": "C", @@ -595363,7 +599762,8 @@ ], "tags": [ "第二单元", - "导数" + "导数", + "2023届高三-寒假作业-寒假作业反馈练习" ], "genre": "解答题", "ans": "$f'(x)=x+3-\\dfrac 4x$, $f'(x)>0$的解集为$(1,+\\infty)$", @@ -595406,7 +599806,8 @@ ], "tags": [ "第一单元", - "第二单元" + "第二单元", + "2023届高三-寒假作业-寒假作业反馈练习" ], "genre": "解答题", "ans": "(1) 当$a<-2$时, 解集为$(-\\infty,-2)\\cup (0,+\\infty)$; 当$a=-2$时, 解集为$\\varnothing$; 当$a>-2$时, 解集为$(-2,0)$; (2) $(-\\infty,-\\dfrac{26}{15})$", @@ -595448,7 +599849,8 @@ "K0223004B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-寒假作业-寒假作业反馈练习" ], "genre": "解答题", "ans": "(1) $(-3,-1)$; (2) $[-\\dfrac 12,-\\dfrac{1}{30}]$", @@ -595487,7 +599889,8 @@ "content": "设常数$a \\in \\mathbf{R}$, 集合$A=\\{x|| 2-x |<5, x \\in \\mathbf{R}\\}$, $B=\\{x|| x+a | \\geq 4, \\ x \\in \\mathbf{R}\\}$. 若$A \\cup B=\\mathbf{R}$, 则$a$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-01_集合与逻辑" ], "genre": "填空题", "ans": "$[-3,-1]$", @@ -595523,7 +599926,8 @@ "content": "集合$A=\\{x | x^2-5 x-6=0\\}$, $B=\\{x | a x^2-x+6=0,\\ x \\in \\mathbf{R}\\}$, 且$A \\cup B=A$. 则实数$a$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-01_集合与逻辑" ], "genre": "填空题题", "ans": "$\\{0\\}\\cup (\\dfrac 1{24},+\\infty)$", @@ -595559,7 +599963,8 @@ "content": "若$x \\geq 0$, $y \\geq 0$, 且$x^2+\\dfrac{y^2}{2}=1$, 求$x \\sqrt{1+y^2}$的最大值.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第二轮复习讲义-02_等式与不等式" ], "genre": "解答题", "ans": "$\\dfrac{3\\sqrt{2}}4$", @@ -595595,7 +600000,8 @@ "content": "设常数$a, b \\in \\mathbf{R}$. 若不等式组$\\begin{cases}xb\\end{cases}$的解集与$\\mathbf{Z}$的交集为$\\{1,2,3\\}$, 则$a, b$满足的条件为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-下学期测验卷-高三下学期测验01" ], "genre": "填空题", "ans": "$0\\le b<1$, $34$)", @@ -595869,7 +600283,8 @@ "content": "已知平面上的线段$l$及点$P$, 在$l$上任取一点$Q$, 线段$PQ$长度的最小值称为点$P$到线段$l$的距离, 记作$d(P, l)$, 则点$P(1,1)$到线段$l: x-y-3=0$($3 \\leq x \\leq 5$)的距离$d(P, l)=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第二轮复习讲义-12_圆锥曲线" ], "genre": "填空题", "ans": "$\\sqrt{5}$", @@ -595906,7 +600321,9 @@ "content": "设抛物线$C: y^2=4 x$的焦点为$F$, 过$F$且斜率为$k(k>0)$的直线$l$与$C$交于$A$、$B$两点, $|AB|=8$. 则直线$l$的方程为\\blank{50}; 过点$A, B$且与抛物线$C$的准线相切的圆的方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-四月错题重做-04_解析几何", + "2023届高三-第二轮复习讲义-13_解析几何综合" ], "genre": "填空题", "ans": "$y=x-1$; $(x-3)^2+(y-2)^2=16$或$(x-11)^2+(y+6)^2=144$", @@ -595945,7 +600362,8 @@ "content": "半径为$1$的球面上的四点$A$、$B$、$C$、$D$是正四面体的顶点, $|AB|=$\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-09_立体几何综合" ], "genre": "填空题", "ans": "$\\dfrac 23\\sqrt{6}$", @@ -595983,7 +600401,8 @@ "content": "如图, 半径为$R$的半球$O$的底面圆$O$在平面$\\alpha$内, 过点$O$作平面$\\alpha$的垂线交半球面于点$A$, 过圆$O$的直径$CD$作平面$\\alpha$成$45^{\\circ}$角的平面与半球面相交, 所得交线上到平面$\\alpha$的距离最大的点为$B$, 该交线上的一点$P$满足$\\angle BOP=60^{\\circ}$, 则$|AP|=$\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [domain = 0:180,dashed,samples=100] plot ({2*cos(\\x)},0,{-2*sin(\\x)});\n\\draw [domain = 0:180,samples=100] plot ({2* cos(\\x)},0,{2* sin(\\x)});\n\\draw [dashed] (0,0,2) node [below left] {$C$} coordinate (C) -- (0,0,-2) node [above right] {$D$} coordinate (D);\n\\draw [dashed] (0,0,0) node [below right] {$O$} coordinate (O) -- (0,2,0) node [above] {$A$} coordinate (A);\n\\draw [dashed] (O) -- ({-sqrt(2)},{sqrt(2)},0) node [above left] {$B$} coordinate (B);\n\\path [draw,domain = 0:180,samples=100,name path = semi] plot ({2*cos (\\x)},{2*sin(\\x)},0);\n\\draw [domain = 0:90,samples=100] plot ({-sqrt(2)*sin(\\x)},{sqrt(2)*sin(\\x)},{2*cos(\\x)});\n\\draw [domain = 90:180,dashed,samples=100] plot ({-sqrt(2)*sin(\\x)},{sqrt(2)*sin(\\x)},{2*cos(\\x)});\n\\draw ({-sqrt(2)*sin(30)},{sqrt(2)*sin(30)},{2*cos(30)}) node [left] {$P$} coordinate (P);\n\\draw [dashed] (O)--(P);\n\\draw (O) pic [draw,\"$60^\\circ$\",angle eccentricity=1.5] {angle = B--O--P};\n\\path [name path = outline] (-3,0,3) -- (3,0,3) -- (3,0,-3) -- (-3,0,-3) -- cycle;\n\\path [name intersections = {of = semi and outline, by = {S,T}}];\n\\draw (T) -- (-3,0,-3) -- (-3,0,3) -- (3,0,3) -- (3,0,-3) -- (S);\n\\draw [dashed] (S) -- (T);\n\\draw (-2.6,0,2.6) node {$\\alpha$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第二轮复习讲义-09_立体几何综合" ], "genre": "填空题", "ans": "$\\sqrt{2-\\dfrac{\\sqrt{2}}2}R$", @@ -596021,7 +600440,8 @@ "content": "已知函数 $f(x)=\\mathrm{e}^x \\ln (1+x)$.\\\\\n(1) 求曲线$y=f(x)$在点$(0, f(0))$处的切线方程;\\\\\n(2) 设$g(x)=f'(x)$, 讨论函数$g(x)$在$[0,+\\infty)$上的单调性;\\\\\n(3) 证明: 对任意$s,t\\in (0,+\\infty)$, 均成立$f(s+t)>f(s)+f(t)$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期测验卷-高三下学期月考01" ], "genre": "解答题", "ans": "(1) $y=x$; (2) $g'(x)=\\mathrm{e}^x(\\ln (x+1)+\\dfrac 1{x+1}+\\dfrac{x}{(x+1)^2})$, 故$g(x)$在$[1,+\\infty)$上是严格增函数; (3) 证明略.", @@ -596060,7 +600480,8 @@ ], "tags": [ "第八单元", - "概率" + "概率", + "2023届高三-下学期测验卷-高三下学期月考01" ], "genre": "填空题", "ans": "$\\dfrac 35$", @@ -596101,7 +600522,8 @@ ], "tags": [ "第八单元", - "二项式定理" + "二项式定理", + "2023届高三-下学期测验卷-高三下学期月考01" ], "genre": "填空题", "ans": "$17$", @@ -596139,7 +600561,8 @@ "content": "已知$\\tan \\alpha =3$, 则$\\dfrac 1{\\sin ^2\\alpha +2\\sin \\alpha \\cos \\alpha}$的值为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期测验卷-高三下学期月考01" ], "genre": "填空题", "ans": "$\\dfrac 23$", @@ -596180,7 +600603,8 @@ "content": "设向量$\\overrightarrow a$、$\\overrightarrow b$满足$|\\overrightarrow a|=5$, $|\\overrightarrow b|=6$, $(\\overrightarrow a+\\overrightarrow b)\\cdot \\overrightarrow b=21$, 则$\\langle \\overrightarrow a, \\overrightarrow b\\rangle =$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-下学期测验卷-高三下学期月考01" ], "genre": "填空题", "ans": "$\\dfrac{2\\pi}3$", @@ -596218,7 +600642,8 @@ "content": "函数$y=\\dfrac{x^2+7}{\\sqrt{x^2+4}}$的值域是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期测验卷-高三下学期测验02" ], "genre": "填空题", "ans": "$[\\dfrac 72,+\\infty)$", @@ -596255,7 +600680,8 @@ "content": "已知$f(x)=\\begin{cases}(2-a) x+1, & x<1, \\\\ 2ax, & x \\geq 1\\end{cases}$满足: 对任意$x_1 \\neq x_2$, 都有$\\dfrac{f(x_1)-f(x_2)}{x_1-x_2}>0$成立, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期测验卷-高三下学期测验02" ], "genre": "填空题", "ans": "$[1,2)$", @@ -596292,7 +600718,8 @@ "content": "若数列$\\{a_n\\}$的前$n$项和$S_n=-3n^2+2n+1$($n\\in \\mathbf{N}$, $n\\ge 1$), 则$\\{a_n\\}$的通项公式为$a_n=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-赋能-赋能21" ], "genre": "填空题", "ans": "$\\begin{cases} 0, & n=1,\\\\ 5-6n, & n \\ge 2\\end{cases}$", @@ -596812,7 +601239,8 @@ "content": "已知$18$个整数的中位数为$5$, 第$75$百分位数也为$5$, 那么这$18$个数中, $5$的个数的最小可能值为\\blank{50}.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-下学期测验卷-高三下学期月考01" ], "genre": "填空题", "ans": "$6$", @@ -597373,7 +601801,8 @@ "content": "设$a, b$均为非零实数, 则直线$y=a x+b$和曲线$a y^2-b x^2=a b$在同一坐标系下的图形可能是\\bracket{20}.\n\\fourch{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,0) ellipse ({sqrt(1.3)} and {sqrt(0.6)});\n\\draw (-0.6,{-1.3*(-0.6)+0.6}) -- (1.2,{-1.3*1.2+0.6});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -0.9:0.9] plot ({sqrt(0.5*(1+\\x*\\x/0.3))},\\x) plot ({-sqrt(0.5*(1+\\x*\\x/0.3))},\\x);\n\\draw (-1.5,{0.3*(-1.5)-0.3}) -- (1.5,{0.3*1.5-0.3});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,0) ellipse ({sqrt(0.6)} and {sqrt(1.3)});\n\\draw (-1,{-1.5*(-1)-0.3}) -- (0.7,{-1.5*0.7-0.3});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1.5:1.5] plot (\\x,{sqrt(0.4*(1+\\x*\\x/0.8))}) plot (\\x,{-sqrt(0.4*(1+\\x*\\x/0.8))});\n\\draw (-0.2,{2*(-0.2)-1.1}) -- (1.3,{2*1.3-1.1});\n\\end{tikzpicture}}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期测验卷-高三下学期测验03" ], "genre": "选择题", "ans": "A", @@ -598362,7 +602791,8 @@ "K0219001B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期测验卷-高三下学期测验03" ], "genre": "选择题", "ans": "B", @@ -598400,7 +602830,8 @@ "content": "已知集合$A=\\{x | x^2-4 x<0\\}$, $B=\\{1,2,3,4,5\\}$, 则$\\overline {A} \\cap B=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-下学期测验卷-高三下学期测验04" ], "genre": "填空题", "ans": "$\\{4,5\\}$", @@ -598435,7 +602866,8 @@ "content": "不等式$\\lg (x-1)<1$的解集为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期测验卷-高三下学期测验04" ], "genre": "填空题", "ans": "$(1,11)$", @@ -598470,7 +602902,8 @@ "content": "已知复数$z=1+2 \\mathrm{i}$, 则$z\\cdot (\\overline{z})^{-1}=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-下学期测验卷-高三下学期测验04" ], "genre": "填空题", "ans": "$-\\dfrac 35+\\dfrac 45\\mathrm{i}$", @@ -598505,7 +602938,8 @@ "content": "函数$y=\\sin ^2(\\pi x)$的最小正周期为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期测验卷-高三下学期测验04" ], "genre": "填空题", "ans": "$1$", @@ -598540,7 +602974,8 @@ "content": "平行直线$x+\\sqrt{3} y+\\sqrt{3}=0$与$\\sqrt{3} x+3 y-9=0$之间的距离为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期测验卷-高三下学期测验04" ], "genre": "填空题", "ans": "$2\\sqrt{3}$", @@ -598575,7 +603010,8 @@ "content": "若$12^a=3^b=m$, 且$\\dfrac{1}{a}-\\dfrac{1}{b}=2$, 则$m=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期测验卷-高三下学期测验04" ], "genre": "填空题", "ans": "$2$", @@ -598611,7 +603047,8 @@ "objs": [], "tags": [ "第七单元", - "第五单元" + "第五单元", + "2023届高三-下学期测验卷-高三下学期测验04" ], "genre": "填空题", "ans": "$(-\\dfrac 15,\\dfrac 25)$", @@ -598646,7 +603083,8 @@ "content": "长方体$ABCD-A_1B_1C_1D_1$为不计容器壁厚度的密封容器, 里面盛有体积为$V$的水, 已知$AB=3$, $AA_1=2$, $AD=1$, 如果将该密封容器任意摆放均不能使水面呈三角形, 则$V$的取值范围为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale= 0.6]\n\\def\\l{3}\n\\def\\m{2}\n\\def\\n{2}\n\\begin{scope}[x = {(10:1cm)}, y = {(100:1cm)}]\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\end{scope}\n\\draw ($(A)!0.6!(A1)$) coordinate (P1);\n\\draw (P1) ++ (0,0,-\\m) coordinate (P4);\n\\draw (P1) ++ ({\\l/cos(10)},0) coordinate (P2);\n\\draw (P2) ++ (0,0,-\\m) coordinate (P3);\n\\fill [gray!30] (P1)--(P4)--(P3)--(C)--(B)--(A)--cycle;\n\\draw [thick] (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw [thick] (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [thick, dashed] (D) -- (D1);\n\\draw [thick] (A) -- (B) -- (C);\n\\draw [thick, dashed] (A) -- (D) -- (C);\n\\draw (P1)--(P2)--(P3);\n\\draw [dashed] (P1)--(P4)--(P3);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期测验卷-高三下学期测验04" ], "genre": "填空题", "ans": "$(1,5)$", @@ -598681,7 +603119,8 @@ "content": "在$\\triangle ABC$中, $\\angle A=150^{\\circ}, D_1, D_2, \\cdots, D_{2022}$依次为边$BC$上的点, \n且$BD_1=D_1D_2=D_2D_3=\\cdots=D_{2021} D_{2022}=D_{2022} C$, 设$\\angle BAD_1=\\alpha_1$, $\\angle D_1AD_2=\\alpha_2$, $\\cdots$, \n$\\angle D_{2021} AD_{2022}=\\alpha_{2022}$, $\\angle D_{2022} AC=\\alpha_{2023}$, 则$\\dfrac{\\sin \\alpha_1 \\sin \\alpha_3 \\cdots \\sin \\alpha_{2023}}{\\sin \\alpha_2 \\sin \\alpha_4 \\cdots \\sin \\alpha_{2022}}$的值为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期测验卷-高三下学期测验04" ], "genre": "填空题", "ans": "$\\dfrac 1{4046}$", @@ -598716,7 +603155,10 @@ "content": "上海电视台五星体育频道有一档四人扑克牌竞技节目``上海三打一'', 在打法中有一种``三带二''的牌型, 即点数相同的三张牌外加一对牌, (三张牌的点数必须和对牌的点数不同). 在一副不含大小王的$52$张扑克牌中不放回的抽取五次, 已知前三次抽到两张A, 一张K, 则接下来两次抽取能抽到``三带二''的牌型(AAAKK或KKKAA)的概率为\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-下学期测验卷-高三下学期测验04", + "2023届高三-四月错题重做-05_易错题-概率与统计", + "2023届高三-四月错题重做-05_概率与统计" ], "genre": "填空题", "ans": "$\\dfrac 3{392}$", @@ -598764,7 +603206,10 @@ "objs": [], "tags": [ "第一单元", - "第二单元" + "第二单元", + "2023届高三-下学期测验卷-高三下学期测验04", + "2023届高三-四月错题重做-01_函数一", + "2023届高三-四月错题重做-01_易错题-函数1" ], "genre": "填空题", "ans": "$(-6,\\dfrac{19}{54})$", @@ -598811,7 +603256,8 @@ "content": "已知抛物线$y^2=2 p x$($p>0$), $P(2,1)$为抛物线内一点, 不经过点$P$的直线$l: y=2 x+m$与抛物线相交于$A, B$两点, 直线$AP, BP$分别交抛物线于$C, D$两点, 若对任意直线$l$, 总存在$\\lambda$, 使得$\\overrightarrow{AP}=\\lambda \\overrightarrow{PC}$, $\\overrightarrow{BP}=\\lambda \\overrightarrow{PD}$($\\lambda>0$, $\\lambda \\neq 1$)成立, 则$p=$\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.3]\n\\draw [->] (-2,0) -- (10,0) node [below] {$x$};\n\\draw [->] (0,-6.5) -- (0,6.5) node [left] {$y$};\n\\draw (0,0) node [above left] {$O$};\n\\draw [domain = -6.3:6.3] plot ({\\x*\\x/4},\\x);\n\\draw (2,1) node [right] {$P$} coordinate (P);\n\\draw (4,-4) node [below] {$C$} coordinate (C);\n\\draw (9,6) node [above] {$D$} coordinate (D);\n\\draw (1.44,2.4) node [above left] {$A$} coordinate (A);\n\\draw (0.04,-0.4) node [below left] {$B$} coordinate (B);\n\\draw ($(A)!-0.5!(B)$)--($(A)!1.5!(B)$) (C)--(D) (B)--(D)(A)--(C);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期测验卷-高三下学期测验04" ], "genre": "填空题", "ans": "$2$", @@ -598846,7 +603292,10 @@ "content": "已知直线$l_1: x+a y-2=0$, $l_2: (a+1) x-a y+1=0$, 则$a=-2$是$l_1\\parallel l_2$的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充要条件}{既不充分也不必要条件}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期测验卷-高三下学期测验04", + "2023届高三-四月错题重做-04_易错题-解析几何", + "2023届高三-四月错题重做-04_解析几何" ], "genre": "选择题", "ans": "A", @@ -598893,7 +603342,8 @@ "content": "已知函数$y=\\dfrac{\\mathrm{e}^x}{\\mathrm{e}^x-1}$, 则其图像大致是\\bracket{20}.\n\\fourch{\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\begin{scope}\n\\clip (-3,-3) rectangle (3,3);\n\\draw [domain = -3.5:-0.1, samples = 100] plot (\\x+0.5,{exp(\\x)/(exp(\\x)-1)+0.3});\n\\draw [domain = 0.1:3, samples = 100] plot (\\x+0.5,{exp(\\x)/(exp(\\x)-1)+0.3});\n\\end{scope}\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\end{tikzpicture}\n}{\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\begin{scope}\n\\clip (-3,-3) rectangle (3,3);\n\\draw [domain = -3:-0.1, samples = 100] plot (\\x,{exp(\\x)/(exp(\\x)-1)});\n\\draw [domain = 0.1:3, samples = 100] plot (\\x,{exp(\\x)/(exp(\\x)-1)});\n\\end{scope}\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\begin{scope}\n\\clip (-3,-3) rectangle (3,3);\n\\draw [domain = -3.5:-0.1, samples = 100] plot (\\x+0.6,{exp(\\x)/(exp(\\x)-1)+1.3});\n\\draw [domain = 0.1:3, samples = 100] plot (\\x+0.5,{exp(\\x)/(exp(\\x)-1)+0.3});\n\\end{scope}\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\begin{scope}\n\\clip (-3,-3) rectangle (3,3);\n\\draw [domain = -3:-0.1, samples = 100] plot (\\x,{exp(\\x)/(exp(\\x)-1)-1});\n\\draw [domain = 0.1:3, samples = 100] plot (\\x,{exp(\\x)/(exp(\\x)-1)-1});\n\\end{scope}\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\end{tikzpicture}}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期测验卷-高三下学期测验04" ], "genre": "选择题", "ans": "B", @@ -598928,7 +603378,8 @@ "content": "如图所示, 正三棱柱$ABC-A_1B_1C_1$的所有棱长均为$1$, 点$P$、$M$、$N$分别为棱$AA_1$、$AB$、$A_1B_1$的中点, 点$Q$为线段$MN$上的动点. 当点$Q$由点$N$出发向点$M$运动的过程中, 以下结论中正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$A$} coordinate (A);\n\\draw (2,0,0) node [below] {$B$} coordinate (B);\n\\draw (1,0,{-sqrt(3)}) node [below] {$C$} coordinate (C);\n\\draw (A) ++ (0,2,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,2,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,2,0) node [above] {$C_1$} coordinate (C_1);\n\\draw ($(A)!0.5!(B)$) node [below] {$M$} coordinate (M);\n\\draw ($(A_1)!0.5!(B_1)$) node [above] {$N$} coordinate (N);\n\\draw ($(A)!0.5!(A_1)$) node [left] {$P$} coordinate (P);\n\\draw ($(N)!0.3!(M)$) node [left] {$Q$} coordinate (Q);\n\\draw (A)--(B)--(B_1)--(A_1)--cycle(A_1)--(C_1)--(B_1)(M)--(N)(P)--(B);\n\\draw [dashed] (A)--(C)--(B)(C)--(P)(C)--(C_1)--(Q);\n\\end{tikzpicture}\n\\end{center}\n\\twoch{直线$C_1Q$与直线$CP$可能相交}{直线$C_1Q$与直线$CP$始终异面}{直线$C_1Q$与直线$CP$可能垂直}{直线$C_1Q$与直线$BP$不可能垂直}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期测验卷-高三下学期测验04" ], "genre": "选择题", "ans": "B", @@ -598963,7 +603414,8 @@ "content": "下列用递推公式表示的数列中, 使得$\\displaystyle\\lim _{n \\to+\\infty} a_n=\\sqrt{2}$成立的是\\bracket{20}.\n\\twoch{$\\begin{cases}a_n=\\dfrac{1}{2}(a_{n-1}+\\dfrac{2}{a_{n-1}})(n \\geq 2), \\\\ a_1=-1\\end{cases}$}{$\\begin{cases}a_n=\\dfrac{2-3 a_{n-1}}{a_{n-1}-3}(n \\geq 2), \\\\ a_1=1\\end{cases}$}{$\\begin{cases}a_n=\\dfrac{a_{n-1}+99}{49 a_{n-1}+1}(n \\geq 2), \\\\ a_1=1\\end{cases}$}{$\\begin{cases}a_n=\\dfrac{2+a_{n-1} \\ln a_{n-1}}{a_{n-1}+\\ln a_{n-1}}(n \\geq 2), \\\\ a_1=1\\end{cases}$}", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-下学期测验卷-高三下学期测验04" ], "genre": "选择题", "ans": "D", @@ -598998,7 +603450,8 @@ "content": "如图, 四棱锥$P-ABCD$中, 等腰$\\triangle PAB$的边长分别为$PA=PB=5$, $AB=6$, 矩形$ABCD$所在的平面与平面$PAB$垂直.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (6,0,0) node [right] {$B$} coordinate (B);\n\\draw (A) ++ (0,3,0) node [left] {$D$} coordinate (D);\n\\draw (B) ++ (0,3,0) node [right] {$C$} coordinate (C);\n\\draw (3,0,5) node [below] {$P$} coordinate (P);\n\\draw (A)--(P)--(B)--(C)--(D)--cycle(D)--(P)--(C);\n\\draw [dashed] (A)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 如果$BC=3$, 求直线$PC$与平面$PAB$所成的角的大小;\\\\\n(2) 如果$PC \\perp BD$, 求$BC$的长.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期测验卷-高三下学期测验04" ], "genre": "解答题", "ans": "(1) $\\arctan 35$; (2) $3\\sqrt{2}$", @@ -599033,7 +603486,8 @@ "content": "自$2015$年上海启动 《上海绿道专项规划(2035)》至今上海已建成绿道总长度近$1600$公里. 根据 《上海市生态空间专项规划(2021-2035)》, 到$2035$年, 上海绿道总长度将超过$2000$公里. 届时, 绿道会像城市的毛细血管一样, 延伸到市民生活的各个角落. 绿荫下的绿道 (步道、骑行道) 给市民提供了散步休憩、跑步骑行运动的生态空间. 某一线品牌自行车制造商在布局线下自行车体验与销售店时随机调研了$1000$位市民, 调研数据如左下表所示. $166$位有意愿购买万元级运动自行车的受访者的年龄(单位: 岁), 在各区间内的频数记录如右下表所示.\\\\\n(1) 试估计有意愿购买万元级运动自行车人群的平均年龄 (结果精确到$0.1$岁).\\\\\n(2) 将表$1$的$2 \\times 2$列联表中的数据补充完整, 并判断是否有$95\\%$的把握认为``离家附近($2$千米内)有骑行绿道与万元级运动自行车消费有关''? \\\\\n附: $\\chi^2=\\dfrac{n(a d-b c)^2}{(a+b)(c+d)(a+c)(b+d)}$, 其中$n=a+b+c+d$.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline$P(\\chi^2 \\geq k)$& 0.10 & 0.05 & 0.01 & 0.005 \\\\\n\\hline$k$& 2.706 & 3.841 & 6.635 & 7.879 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline &\\makecell{有意愿购买 \\\\ 万元级 \\\\ 运动自行车 }& \\makecell{没有意愿购买 \\\\ 万元级 \\\\ 运动自行车 }& 总计 \\\\ \\hline\n\\makecell{距家$2$千米内\\\\有骑行绿道}& 118 & 270 & \\\\ \\hline\n\\makecell{距家$2$千米内\\\\无骑行绿道}& & & \\\\\\hline \n总计 & 166 & & 1000 \\\\\\hline\n\\end{tabular}\n\\begin{tabular}{|c|c|}\n\\hline\n\\makecell{ 年龄分 \\\\ 组区间 } & 频数 \\\\\n\\hline$[12,18)$& 16 \\\\\n\\hline$[18,24)$& 24 \\\\\n\\hline$[24,30)$& 35 \\\\\n\\hline$[30,36)$& 30 \\\\\n\\hline$[36,42)$& 21 \\\\\n\\hline$[42,48)$& 15 \\\\\n\\hline$[48,54)$& 11 \\\\\n\\hline$[54,60)$& 6 \\\\\n\\hline$[60,66)$& 5 \\\\\n\\hline$[66,72)$& 3 \\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-下学期测验卷-高三下学期测验04" ], "genre": "解答题", "ans": "(1) $33.7$岁; (2) $\\chi^2\\approx 87.366>3.841$, 所以有骑行绿道与万元级运动自行车购买意愿有关", @@ -599068,7 +603522,8 @@ "content": "雨天外出虽然有撑雨伞, 时常却总免不了淋湿衣袖、裤脚、背包等, 小明想通过数学建模的方法研究如何撑伞可以让淋湿的面积尽量小. 为了简化问题小明做出下列假设:\\\\\n假设 1: 在网上查阅了人均身高和肩宽的数据后, 小明把人假设为身高、肩宽分别为$170 \\text{cm}$、$40 \\text{cm}$的矩形``纸片人'';\\\\\n假设 2: 受风的影响, 雨滴下落轨迹视为与水平地面所成角为$60^{\\circ}$的直线;\\\\\n假设 3: 伞柄$OT$长为$60 \\text{cm}$, 可绕矩形``纸片人''上点$O$旋转;\\\\\n假设 4: 伞面为被伞柄$OT$垂直平分的线段$AB$, $AB=120 \\text{cm}$.\\\\\n以如图$1$方式撑伞矩形``纸片人''将淋湿``裤脚''; 以如图$2$方式撑伞矩形``纸片人''将淋湿``头和肩膀''(``裤脚''也会有一小部分被淋湿).\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\begin{scope}\n\\clip (-3,0) rectangle (3,6);\n\\draw (0,0) -- (0,3.4) -- (0.8,3.4) node [below right] {$O$} coordinate (O) -- (0.8,0);\n\\draw (O) --++ (60:1.2) node [above right] {$T$} coordinate (T);\n\\draw (T) --++ (150:1.2) node [left] {$A$} coordinate (A);\n\\draw (T) --++ (-30:1.2) node [right] {$B$} coordinate (B);\n\\draw (A) ++ (60:1) --++ (60:-15);\n\\draw (B) ++ (60:1) --++ (60:-15);\n\\end{scope}\n\\draw (-3,0) -- (3,0);\n\\draw (0,0) node [below] {图$1$};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\begin{scope}\n\\clip (-3,0) rectangle (3,6);\n\\draw (0,0) -- (0,3.4) -- (0.8,3.4) node [below right] {$O$} coordinate (O) -- (0.8,0);\n\\draw (O) --++ (20:1.2) node [above right] {$T$} coordinate (T);\n\\draw (T) --++ (110:1.2) node [left] {$A$} coordinate (A);\n\\draw (T) --++ (-70:1.2) node [right] {$B$} coordinate (B);\n\\draw (A) ++ (60:1) --++ (60:-15);\n\\draw (B) ++ (60:1) --++ (60:-15);\n\\end{scope}\n\\draw (-3,0) -- (3,0);\n\\draw (0,0) node [below] {图$2$};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\begin{scope}\n\\clip (-3,0) rectangle (3.5,6);\n\\filldraw [gray!50] (0.62586,0) -- (0.8,0) -- (0.8,0.3016) -- cycle;\n\\draw (0,0) -- (0,3.4) -- (0.8,3.4) node [below right] {$O$} coordinate (O) -- (0.8,0);\n\\draw (O) --++ (39.09484:1.2) node [above right] {$T$} coordinate (T);\n\\draw (T) --++ (129.09484:1.2) node [left] {$A$} coordinate (A);\n\\draw (T) --++ ({39.09484-90}:1.2) node [right] {$B$} coordinate (B);\n\\draw (A) ++ (60:1) --++ (60:-15);\n\\draw (B) ++ (60:1) --++ (60:-15);\n\\end{scope}\n\\draw (-3,0) -- (3,0);\n\\draw (0,0) node [below] {图$3$};\n\\end{tikzpicture}\n\\end{center}\n(1) 如图$3$在矩形``纸片人''上身恰好不被淋湿时, 求其``裤脚''被淋湿 (阴影) 部分的面积(结果精确到$0.1 \\text{cm}^2)$;\\\\\n(2) 请根据你的生活经验对小明建立的数学模型提两条改进建议. (无需求解改进后的模型, 如果建议超过两条仅对前两条评分)", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期测验卷-高三下学期测验04" ], "genre": "解答题", "ans": "(1) 约为$65.7\\text{cm}^2$; (2) 参考改进建议: \\textcircled{1} 雨伞不遮挡视线; \\textcircled{2} 伞面为弧形，改进模型将伞设为一段圆弧; \\textcircled{3} 考虑伞柄可以伸缩; \\textcircled{4} 人体改进为立体模型; \\textcircled{5} 考虑风速、风向; \\textcircled{6} 考虑撑伞的省力、稳定等.", @@ -599103,7 +603558,8 @@ "content": "已知椭圆$c: \\dfrac{x^2}{4}+\\dfrac{y^2}{b^2}=1$($00$, 记直线$QF_1$与椭圆$C$在$x$轴上方的交点为$A(x_1, y_1)$, 直线$QF_2$与椭圆$c$在$x$轴上方的交点为$B(x_2, y_2)$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [name path = elli] (0,0) ellipse (2 and {sqrt(3)});\n\\filldraw (-1,0) circle (0.03) node [below] {$F_1$} coordinate (F_1);\n\\filldraw (1,0) circle (0.03) node [below] {$F_2$} coordinate (F_2);\n\\path [name path = line1] (F_1) --++ (50:2.5);\n\\path [name path = line2] (F_2) --++ (50:1.5);\n\\path [name intersections = {of = line1 and elli, by = B}];\n\\path [name intersections = {of = line2 and elli, by = A}];\n\\path [draw, name path = line3] (F_1) -- (A) node [above] {$A$};\n\\path [draw, name path = line4] (F_2) -- (B) node [above] {$B$};\n\\path [name intersections = {of = line3 and line4, by = Q}];\n\\draw (Q) node [below] {$Q$};\n\\draw (F_1)--(B) (F_2)--(A);\n\\end{tikzpicture}\n\\end{center}\n(1) 求椭圆$c$的的离心率;\\\\\n(2) 若$AF_2\\parallel BF_1$, 证明: $\\dfrac{1}{y_1}+\\dfrac{1}{y_2}=\\dfrac{1}{y_0}$;\\\\\n(3) 若$\\dfrac{1}{y_1}+\\dfrac{1}{y_2}=\\dfrac{4}{3 y_0}$, 求点$Q$的轨迹方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期测验卷-高三下学期测验04" ], "genre": "解答题", "ans": "(1) $\\dfrac 12$; (2) 证明略; (3) $\\dfrac 49x^2+\\dfrac 45 y^2=1$($y>0$)", @@ -599138,7 +603594,8 @@ "content": "已知函数$y=f(x)$的表达式为$f(x)=\\dfrac{1}{2} a x^2+(a+1) x+\\ln x$($a \\in \\mathbf{R}$).\\\\\n(1) 若$1$是$f(x)$的极值点, 求$a$的值;\\\\\n(2) 求$f(x)$的单调区间;\\\\\n(3) 若$f(x)=\\dfrac{1}{2} a x^2+x$有两个实数解$x_1, x_2$($x_11$)的解集为$\\mathbf{R}$, 则$\\dfrac{1+2 b+4 c}{b-1}$的最小值为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-下学期测验卷-高三下学期测验06" ], "genre": "填空题", "ans": "$8$", @@ -599490,7 +603956,8 @@ "objs": [], "tags": [ "第八单元", - "第六单元" + "第六单元", + "2023届高三-下学期测验卷-高三下学期测验06" ], "genre": "填空题", "ans": "$\\dfrac 6{11}$", @@ -599525,7 +603992,8 @@ "content": "设$a>b>0$, 椭圆$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$的离心率为$e_1$, 双曲线$\\dfrac{x^2}{b^2}-\\dfrac{y^2}{a^2-2 b^2}=1$的离心率为$e_2$, 若$e_1 e_2<1$, 则$\\dfrac{e_2}{e_1}$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期测验卷-高三下学期测验06" ], "genre": "填空题", "ans": "$(\\sqrt{2},\\dfrac{\\sqrt{5}+1}2)$", @@ -599560,7 +604028,8 @@ "content": "在$\\triangle ABC$中, $AB=3$, $BC=4$, $AC=5, P$为$\\triangle ABC$内部一动点(含边界), 在空间中, 若到点$P$的距离不超过$1$的点的轨迹为$L$, 则几何体$L$的体积等于\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期测验卷-高三下学期测验06" ], "genre": "填空题", "ans": "$\\dfrac{22\\pi}3+12$", @@ -599595,7 +604064,8 @@ "content": "设$m \\in \\mathbf{R}$, 若幂函数$y=x^{m^2-2 m+1}$定义域为$\\mathbf{R}$, 且其图像关于$y$轴成轴对称, 则$m$的值可以为\\bracket{20}.\n\\fourch{$1$}{$4$}{$7$}{$10$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期测验卷-高三下学期测验06" ], "genre": "选择题", "ans": "C", @@ -599630,7 +604100,8 @@ "content": "已知$x \\in \\mathbf{R}$, 下列不等式中正确的是\\bracket{20}.\n\\twoch{$\\dfrac{1}{2^x}>\\dfrac{1}{3^x}$}{$\\dfrac{1}{x^2-x+1}>\\dfrac{1}{x^2+x+1}$}{$\\dfrac{1}{2|x|}>\\dfrac{1}{x^2+1}$}{$\\dfrac{1}{x^2+1}>\\dfrac{1}{x^2+2}$}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-下学期测验卷-高三下学期测验06" ], "genre": "选择题", "ans": "D", @@ -599665,7 +604136,8 @@ "content": "设$a$、$b$、$c$、$d \\in \\mathbf{R}$, 若函数$y=a x^3+b x^2+c x+d$的部分图像如图所示, 则下列结论正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1.1:1.5] plot (\\x,{(\\x+0.5)*(\\x-0.4)*(\\x-1.1)+1.5});\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$b>0$, $c>0$}{$b>0$, $c<0$}{$b<0$, $c>0$}{$b<0$, $c<0$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期测验卷-高三下学期测验06" ], "genre": "选择题", "ans": "D", @@ -599700,7 +604172,8 @@ "content": "设数列$\\{a_n\\}$的前$n$项和为$S_n$, 若对任意正整数$n$, 总存在正整数$m$, 使得$S_n=a_m$, 有结论: \\textcircled{1} $\\{a_n\\}$可能为等差数列; \\textcircled{2} $\\{a_n\\}$可能为等比数列. 关于以上两个结论, 正确的判断是\\bracket{20}.\n\\fourch{\\textcircled{1}成立, \\textcircled{2}成立}{\\textcircled{1}成立, \\textcircled{2}不成立}{\\textcircled{1}不成立, \\textcircled{2}成立}{\\textcircled{1}不成立, \\textcircled{2}不成立}", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-下学期测验卷-高三下学期测验06" ], "genre": "选择题", "ans": "B", @@ -599735,7 +604208,8 @@ "content": "深入实施科教兴国战略是中华人民伟大复兴的必由之路, 2020 年第七次全国人口普查对$6$岁及以上人口的受教育程度进行统计 (未包括中国香港、澳门特别行政区和台湾省的人口数据), 我国$31$个省级行政区具有初中及以上文化程度人口比例情况经统计得到如下的频率分布直方图.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 8, yscale = 0.7]\n\\draw [->] (0,0) -- (1.1,0) node [below] {比例};\n\\draw [->] (0,0) -- (0,7) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j in {0.35/0.323,0.45/0.323,0.55/1.613,0.65/5.161,0.75/1.935,0.85/0.645}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (0.1,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {0.35/0.323,0.55/1.613,0.65/5.161,0.75/1.935/a,0.85/0.645}\n{\\draw [dashed] (\\i,\\j) -- (0,\\j) node [left] {$\\k$};};\n\\draw (0.95,0) node [below] {$0.95$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求具有初中及以上文化程度人口比例在区间$[0.75,0.85)$内的省级行政区有几个?\\\\\n(2) 已知上海具有初中及以上文化程度人口比例是这组数据的第$41$百分位数, 求该比例落在哪个区间内?", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-下学期测验卷-高三下学期测验06" ], "genre": "解答题", "ans": "(1) $6$个; (2) 落在区间$[0.65,0.75)$内", @@ -599770,7 +604244,8 @@ "content": "如图, $AB$为圆$O$的直径, 点$E$、$F$在圆$O$上, $AB\\parallel EF$, 矩形$ABCD$所在平面和圆$O$所在的平面互相垂直. 已知$AB=2$, $EF=1$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = {(225:0.5cm)}]\n\\draw [domain = -30:{180+atan(sqrt(2)*7/8)}, samples = 100] plot ({2*cos(\\x)},0,{2*sin(\\x)});\n\\draw [domain = -30:{-180+atan(sqrt(2)*7/8)}, samples = 100, dashed] plot ({2*cos(\\x)},0,{2*sin(\\x)});\n\\draw (0,0,2) node [below] {$A$} coordinate (A);\n\\draw (0,0,-2) node [above left] {$B$} coordinate (B);\n\\draw ({sqrt(3)},0,1) node [below] {$F$} coordinate (F);\n\\draw ({sqrt(3)},0,-1) node [above] {$E$} coordinate (E);\n\\draw (A) --++ (0,2) node [above] {$D$} coordinate (D);\n\\draw [dashed] (B) --++ (0,2) node [above] {$C$} coordinate (C);\n\\filldraw (0,0) node [right] {$O$} coordinate (O) circle (0.03);\n\\draw (A)--(F)--(C) (D)--(C);\n\\draw [dashed] (A)--(B)--(E)(B)--(F);\n\\draw (D)--(F)(C)--(E);\n\\draw (E)--(F);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: 平面$DAF \\perp$平面$CBF$;\\\\\n(2) 当$AD$的长为何值时, 二面角$C-EF-B$的大小为$60^{\\circ}$?", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期测验卷-高三下学期测验06" ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\dfrac 32$", @@ -599805,7 +604280,8 @@ "content": "设$a \\in \\mathbf{R}$, $f(x)=\\sin 2 x+a \\cos x$.\\\\\n(1) 是否存在$a$使得$y=f(x)$为奇函数? 说明理由;\\\\\n(2) 当$a<-4$时, 求证: 函数$y=f(x)$在区间$(0, \\dfrac{\\pi}{2})$上是严格增函数.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期测验卷-高三下学期测验06" ], "genre": "解答题", "ans": "(1) 存在, $a=0$; (2) 证明略", @@ -599840,7 +604316,8 @@ "content": "已知圆$C: x^2+y^2=4$, 点$P(2,2)$.\\\\\n(1) 直线$l$过点$P$且与圆$C$相交于$A$、$B$两点, 若$\\overrightarrow{CA} \\cdot \\overrightarrow{CB}=0$, 求直线$l$的方程;\\\\\n(2) 若动圆$D$经过点$P$且与圆$C$外切, 求动圆的圆心$D$的轨迹方程;\\\\\n(3) 是否存在异于点$P$的点$Q$, 使得对于圆$C$上任意一点$M$, 均有$\\dfrac{|MP|}{|MQ|}=\\lambda$? 若存在, 求出$Q$点坐标和常数$\\lambda$的值; 若不存在, 也请说明理由.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期测验卷-高三下学期测验06" ], "genre": "解答题", "ans": "(1) $y-2=(2\\pm \\sqrt{3})(x-2)$; (2) $2xy-2x-2y+1=0$($x>1$); (3) 存在, $Q(1,1)$, $\\lambda=2$", @@ -599875,7 +604352,8 @@ "content": "定义在$\\mathbf{R}$上的函数$y=f(x)$、$y=g(x)$, 若$|f(x_1)-f(x_2)| \\geq|g(x_1)-g(x_2)|$对任意的$x_1$、$x_2 \\in \\mathbf{R}$成立, 则称函数$y=g(x)$是函数$y=f(x)$的``从属函数''.\\\\\n(1) 若函数$y=g(x)$是函数$y=f(x)$的``从属函数''且$y=f(x)$是偶函数, 求证: $y=g(x)$是偶函数;\\\\\n(2) 若$f(x)=a x+\\mathrm{e}^x$, $g(x)=\\sqrt{x^2+1}$, 求证: 当$a \\geq 1$时, 函数$y=g(x)$是函数$y=f(x)$的``从属函数'';\\\\\n(3) 设定义在$\\mathbf{R}$上的函数$y=f(x)$与$y=g(x)$, 它们的图像各为一条连续的曲线, 且函数$y=g(x)$是函数$y=f(x)$的``从属函数''. 设$\\alpha: $``函数$y=f(x)$在$\\mathbf{R}$上是严格增函数或严格减函数''; $\\beta$: ``函数$y=g(x)$在$\\mathbf{R}$上为严格增函数或严格减函数''. 试判断$\\alpha$是$\\beta$的什么条件? 请说明理由.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期测验卷-高三下学期测验06" ], "genre": "解答题", "ans": "(1) 证明略; (2) 证明略; (3) $\\alpha$是$\\beta$的必要非充分条件", @@ -599911,7 +604389,8 @@ "objs": [], "tags": [ "第二单元", - "导数" + "导数", + "2023届高三-第二轮复习讲义-16_导数及其应用" ], "genre": "填空题", "ans": "(1) $[0,+\\infty)$; (2) $(-\\infty,-6]$; (3) $-6$; (4) $(-6,0)$", @@ -599949,7 +604428,8 @@ "objs": [], "tags": [ "第二单元", - "导数" + "导数", + "2023届高三-第二轮复习讲义-16_导数及其应用" ], "genre": "填空题", "ans": "$\\mathrm{e}^2$", @@ -599987,7 +604467,8 @@ "objs": [], "tags": [ "第二单元", - "导数" + "导数", + "2023届高三-第二轮复习讲义-16_导数及其应用" ], "genre": "填空题", "ans": "$(2,3)$或$(-2,-29)$", @@ -600029,7 +604510,8 @@ ], "tags": [ "第二单元", - "导数" + "导数", + "2023届高三-第二轮复习讲义-16_导数及其应用" ], "genre": "填空题", "ans": "$-2$; $[-3,-1]$", @@ -600070,7 +604552,8 @@ ], "tags": [ "第八单元", - "二项式定理" + "二项式定理", + "2023届高三-赋能-赋能23" ], "genre": "填空题", "ans": "$12$", @@ -600106,7 +604589,8 @@ "content": "若直线$x+y-c=0$, $x-y+2=0$, $y+2=0$无法围成三角形, 则实数$c$的值为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-赋能-赋能25" ], "genre": "填空题", "ans": "$-6$", @@ -600136,7 +604620,8 @@ "content": "已知$2^x<3^x$, 则$x$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-赋能-赋能25" ], "genre": "填空题", "ans": "$(0,+\\infty)$", @@ -600166,7 +604651,8 @@ "content": "点集$\\{(x,y)||x|+2|y|\\le 2\\}$在平面直角坐标系中所对应图形的面积为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-赋能-赋能27" ], "genre": "填空题", "ans": "$4$", @@ -600665,7 +605151,8 @@ "content": "已知数列$\\{a_n\\}$满足$a_{n+2}-a_n=2^n$, $a_1=1$, $a_2=2$, 则$a_n=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-四月错题重做-03_易错题-数列" ], "genre": "填空题", "ans": "$\\begin{cases}\\dfrac{2^n+1}{3}, & n = 2k+1, \\\\ \\dfrac{2^n+2}{3}, & n = 2k+2\\end{cases}$($k\\in \\mathbf{N}$)", @@ -600702,7 +605189,8 @@ "content": "设等差数列$\\{a_n\\}$的前$n$项和为$S_n$, 若$S_4 \\geq 10$, $S_5 \\leq 15$, 则$a_5$的最大值为\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-四月错题重做-03_易错题-数列" ], "genre": "填空题", "ans": "$5$", @@ -600743,7 +605231,8 @@ "K0406005X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-四月错题重做-03_易错题-数列" ], "genre": "解答题", "ans": "$a_9$最大, $a_{10}$最小", @@ -600783,7 +605272,8 @@ "K0406005X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-四月错题重做-03_易错题-数列" ], "genre": "解答题", "ans": "$a_1=-\\dfrac{10}{11}$最小, $a_{12}=a_{13}=\\dfrac{10^{13}}{11^{12}}$最大", @@ -600821,7 +605311,8 @@ "content": "已知$P$为椭圆$\\dfrac{x^2}4+\\dfrac{y^2}2=1$上的任意一点, $Q$与$P$关于$x$轴对称, $F_1$、$F_2$为椭圆的左、右焦点, 若有$\\overrightarrow{F_1 P} \\cdot \\overrightarrow{F_2P} \\leq 1$, 求向量$\\overrightarrow{F_1 P}$与$\\overrightarrow{F_2 Q}$夹角的取值范围.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-四月错题重做-04_易错题-解析几何" ], "genre": "解答题", "ans": "$[\\pi-\\arccos\\dfrac 13,\\pi]$", @@ -600863,7 +605354,8 @@ ], "tags": [ "第七单元", - "直线" + "直线", + "2023届高三-四月错题重做-04_易错题-解析几何" ], "genre": "填空题", "ans": "$y=-\\dfrac 23 x$或$y=-x+1$", @@ -600901,7 +605393,8 @@ "content": "设抛物线$C: y^2=4 x$的焦点为$F$, 过$F$且斜率为$k(k>0)$的直线$l$与$C$交于$A$、$B$两点, $|AB|=8$. 求直线$l$的方程为及过点$A, B$且与抛物线$C$的准线相切的圆的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-四月错题重做-04_易错题-解析几何" ], "genre": "解答题", "ans": "$y=x-1$; $(x-3)^2+(y-2)^2=16$或$(x-11)^2+(y+6)^2=144$", @@ -600940,7 +605433,8 @@ "K0720003X" ], "tags": [ - "第七单元" + "第七单元", + "2023届高三-四月错题重做-04_易错题-解析几何" ], "genre": "填空题", "ans": "$a=0$或$-\\dfrac 12$", @@ -600981,7 +605475,8 @@ ], "tags": [ "第七单元", - "双曲线" + "双曲线", + "2023届高三-四月错题重做-04_易错题-解析几何" ], "genre": "填空题", "ans": "$x^2-\\dfrac{y^2}9=1$或$\\dfrac{y^2}{81}-\\dfrac{x^2}9=1$", @@ -601021,7 +605516,8 @@ ], "tags": [ "第七单元", - "直线" + "直线", + "2023届高三-四月错题重做-04_易错题-解析几何" ], "genre": "填空题", "ans": "$y=-\\dfrac{\\sqrt 3}{3}(x+1)$", @@ -601056,7 +605552,8 @@ "content": "已知定义在$(-3,3)$上的偶函数$y=f(x)$的导函数是$f'(x)$, 当$x\\ge 0$时, $y=f(x)$的图像如图所示, 则关于$x$的不等式$\\dfrac{f'(x)}{x}>0$的解集为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=stealth, line cap = round, line join = round,scale = 0.6]\n\\draw [->] (-1,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0:3] plot (\\x,{1-pow(\\x-1,2)});\n\\draw [dashed] (1,0) node [below] {$1$} -- (1,1) (3,-3) -- (3,0) node [below right] {$3$};\n\\draw (2,0) node [below left] {$2$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-四月错题重做-01_易错题-函数1" ], "genre": "填空题", "ans": "$(-1,0)\\cup (0,1)$", @@ -601096,7 +605593,8 @@ "K0405001X" ], "tags": [ - "第四单元" + "第四单元", + "2023届高三-赋能-赋能31" ], "genre": "填空题", "ans": "$45$", @@ -601132,7 +605630,8 @@ ], "tags": [ "第八单元", - "二项式定理" + "二项式定理", + "2023届高三-赋能-赋能36" ], "genre": "填空题", "ans": "$\\dfrac{n(n+1)}{2}$", @@ -601163,7 +605662,8 @@ "K0803002B" ], "tags": [ - "第二单元" + "第二单元", + "2023届高三-赋能-赋能40" ], "genre": "填空题", "ans": "$\\frac 37$", @@ -601190,7 +605690,8 @@ "content": "如图, 长方体$ABCD-A_1B_1C_1D_1$的边长$AB=AA_1=1$ ,$AD=\\sqrt2$ , 它的外接球是球$O$, 则$\\angle AOA_1=$\\blank{50}.\n\\begin{center}\n \\begin{tikzpicture}\n \\draw (0,0) node [below left] {$A$} coordinate (A) --++ (2,0) node [below right] {$B$} coordinate (B) --++ (45:{2*sqrt(2)/2}) node [right] {$C$} coordinate (C)\n --++ (0,2) node [above right] {$C_1$} coordinate (C1)\n --++ (-2,0) node [above left] {$D_1$} coordinate (D1) --++ (225:{2*sqrt(2)/2}) node [left] {$A_1$} coordinate (A1) -- cycle;\n \\draw (A) ++ (2,2) node [right] {$B_1$} coordinate (B1) -- (B) (B1) --++ (45:{2*sqrt(2)/2}) (B1) --++ (-2,0);\n \\draw [dashed] (A) --++ (45:{2*sqrt(2)/2}) node [left] {$D$} coordinate (D) --++ (2,0) (D) --++ (0,2);\n \\draw [dashed] (A1) -- (C) (A) --(C1);\n \\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-赋能-赋能42" ], "genre": "填空题", "ans": "$\\frac{\\pi}3$", @@ -601215,7 +605716,8 @@ "content": "对于非零向量$\\overrightarrow {a}$、$\\overrightarrow {b}$, 定义一种向量的运算: $\\overrightarrow {a} \\otimes \\overrightarrow {b}=\\dfrac{\\overrightarrow {a} \\cdot \\overrightarrow {b}}{\\overrightarrow {b} \\cdot \\overrightarrow {b}}$. 设集合$P=\\{\\dfrac{n}{2} | n \\in \\mathbf{N}\\}$, 若非零向量$\\overrightarrow {a}$、$\\overrightarrow {b}$满足$\\overrightarrow {a} \\otimes \\overrightarrow {b} \\in P$, $\\overrightarrow {b} \\otimes \\overrightarrow {a} \\in P$, 且其夹角$\\theta \\in(\\dfrac{\\pi}{4}, \\dfrac{\\pi}{2})$, 求$\\overrightarrow {a} \\otimes \\overrightarrow {b}$的所有可能的值组成的集合.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-第三轮复习讲义-04_转化与化归" ], "genre": "解答题", "ans": "$\\{\\dfrac 12\\}$", @@ -601246,7 +605748,8 @@ "content": "已知函数$f(x)=2^x+x$, 若$a,b,c\\in \\mathbf{R}$, 满足$2f(b)=f(a)+f(c)$, 求证: $2b\\ge a+c$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第三轮复习讲义-04_转化与化归" ], "genre": "解答题", "ans": "证明略", @@ -601277,7 +605780,8 @@ "content": "(1) 是否存在第一象限的角$\\alpha$和第三象限的角$\\beta$, 使得$\\tan \\alpha \\tan \\beta=\\tan (\\alpha-\\beta)$? 请说明理由;\\\\\n(2) 是否存在第二象限的角$\\alpha$和第四象限的角$\\beta$, 使得$\\tan \\alpha \\tan \\beta=\\tan (\\alpha-\\beta)$? 请说明理由;\\\\\n(3) 是否存在第一象限的角$\\alpha$和第三象限的角$\\beta$, 使得$\\sin \\alpha \\sin \\beta=\\sin (\\alpha-\\beta)$? 请说明理由.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-第三轮复习讲义-04_转化与化归" ], "genre": "解答题", "ans": "(1) 存在, 理由略; (2) 存在, 理由略; (3) 存在, 理由略", @@ -601310,7 +605814,8 @@ "content": "若方程$x^4+a x-4=0$的各个实根$x_1, x_2, \\cdots, x_k$($k \\leq 4$)所对应的点$(x_i, \\dfrac{4}{x_i})$($i=1,2, \\cdots, k$)均在直线$y=x$的同侧, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第三轮复习讲义-04_转化与化归" ], "genre": "填空题", "ans": "$(-\\infty,-6)\\cup (6,+\\infty)$", @@ -601340,7 +605845,8 @@ "content": "如图, 棱长为$2$的正方体$ABCD-A_1B_1C_1D_1$中, $E$为棱$CC_1$的中点, 点$P$、$Q$分别为面$A_1B_1C_1D_1$和线段$B_1C$上的动点, 则$\\triangle PEQ$周长的最小值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(C)!0.5!(C1)$) node [right] {$E$} coordinate (E);\n\\draw ($(B1)!0.3!(C)$) node [below] {$Q$} coordinate (Q);\n\\draw ($1/3*(A1)+1/3*(B1)+1/3*(C1)$) node [left] {$P$} coordinate (P);\n\\draw (B1)--(C)(E)--(Q);;\n\\draw [dashed] (Q)--(P)--(E);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-第三轮复习讲义-04_转化与化归" ], "genre": "填空题", "ans": "$\\sqrt{10}$", @@ -619379,7 +623885,8 @@ "content": "设函数$y=f_i(x)$($i=1,2,3$)的导数分别为$y'=f_i'(x)$($i=1,2,3$).\\\\\n(1) 若$f_1(x)=\\sqrt{x}$, $f_2(x)=\\sqrt[3]{x}$, $f_3(x)=\\dfrac{1}{\\sqrt{x}}$, 则\\\\$f_1'(x)=$\\blank{50}; $f_2'(x)=$\\blank{50}; $f_3'(x)=$\\blank{50}.\\\\\n(2) 若$f_1(x)=x^2-\\mathrm{e}^x$, $f_2(x)=\\ln x+\\cos x$, $f_3(x)=\\sqrt{x}+\\sin x$, 则\\\\$f_1'(x)=$\\blank{50}; $f_2'(x)=$\\blank{50}; $f_3'(x)=$\\blank{50}.\\\\\n(3) 若$f_1(x)=x^3 \\cdot \\mathrm{e}^x$, $f_2(x)=x \\ln x$, $f_3(x)=\\dfrac{x}{\\sin x}$, 则\\\\$f_1'(x)=$\\blank{50}; $f_2'(x)=$\\blank{50}; $f_3'(x)=$\\blank{50}.\\\\\n(4) 若$f_1(x)=\\mathrm{e}^{2 x-1}$, $f_2(x)=2^x$, $f_3(x)=(\\dfrac{1}{3})^{x+1}$, 则\\\\$f_1'(x)=$\\blank{50}; $f_2'(x)=$\\blank{50}; $f_3'(x)=$\\blank{50}.\\\\\n(5) 若$f_1(x)=\\lg x$, $f_2(x)=\\ln (2 x-1)$, $f_3(x)=\\log _3(3-2 x)$, 则\\\\$f_1'(x)=$\\blank{50}; $f_2'(x)=$\\blank{50}; $f_3'(x)=$\\blank{50}.\\\\\n(6) 若$f_1(x)=\\sin (x-\\dfrac{\\pi}{3})$, $f_2(x)=\\cos (\\dfrac{x}{3}+\\dfrac{\\pi}{4})$, $f_3(x)=\\tan (2 x)$, 则\\\\$f_1'(x)=$\\blank{50}; $f_2'(x)=$\\blank{50}; $f_3'(x)=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-16_导数及其应用" ], "genre": "填空题", "ans": "(1) $\\dfrac 12 x^{-\\frac 12}$; (2) $\\dfrac 13 x^{-\\frac 23}$; (3) $-\\dfrac 12 x^{-\\frac 32}$; (4) $2x-\\mathrm{e}^x$; (5) $\\dfrac 1x-\\sin x$; (6) $\\dfrac 12 x^{-\\frac 12}+\\cos x$; (7) $\\mathrm{e}^x(x^3+3x^2)$; (8) $1+\\ln x$; (9) $\\dfrac{\\sin x-x\\cos x}{\\sin^2 x}$; (10) $2\\mathrm{e}^{2x-1}$; (11) $2^x\\ln 2$; (12) $-(\\dfrac 13)^{x+1}\\ln 3$; (13) $\\dfrac{1}{x\\ln 10}$; (14) $\\dfrac{2}{2x-1}$; (15) $\\dfrac{2}{(2x-3)\\ln 3}$; (16) $\\cos (x-\\dfrac\\pi 3)$; (17) $-\\dfrac 13 \\sin(\\dfrac x 3+\\dfrac \\pi 4)$; (18) $\\dfrac{2}{\\cos^2(2x)}$", @@ -619476,7 +623983,8 @@ "content": "设函数$y=f(x)$的导数为$y'=f'(x)$. 若$f'(1)=-3$.\\\\\n(1) $\\displaystyle\\lim _{h \\to 0} \\dfrac{f(1+3 h)-f(1)}{h}=$\\blank{50};\\\\\n(2) $\\displaystyle\\lim _{h \\to 0} \\dfrac{f(1+h)-f(1-h)}{h}=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-16_导数及其应用" ], "genre": "填空题", "ans": "(1) $-9$; (2) $-6$", @@ -619964,7 +624472,8 @@ "content": "已知函数$y=f(x)$的导数$y=f'(x)$的图像如下图所示, 则$y=f(x)$的图像可能是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-1,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1:3] plot (\\x,{\\x*(\\x-2)});\n\\draw (2,0.2) -- (2,0) node [below] {$2$};\n\\draw (1,0.2) -- (1,0) node [below] {$1$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{\\begin{tikzpicture}[>=latex, scale = 0.8]\n\\draw [->] (-1,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -0.4:2.2] plot (\\x,{-\\x*(\\x-1)*(\\x-2)});\n\\draw [dashed] ({(3-sqrt(3))/3},{-2/3/sqrt(3)}) -- ({(3-sqrt(3))/3},0) node [above] {$1$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.8]\n\\draw [->] (-1,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -0.2:2.4] plot (\\x,{1.1*\\x*(\\x-1)*(\\x-2)});\n\\draw (2,0.2) -- (2,0) node [below] {$2$};\n\\draw (1,0.2) -- (1,0) node [below] {$1$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.8]\n\\draw [->] (-1,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [above left] {$O$};\n\\draw [domain = -1:2.85] plot (\\x,{\\x*\\x-\\x*\\x*\\x/3-0.5});\n\\draw [dashed] (2,{4-8/3-0.5}) -- (2,0) node [below] {$2$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.8]\n\\draw [->] (-1,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1:2.85] plot (\\x,{-\\x*\\x+\\x*\\x*\\x/3+1});\n\\draw [dashed] (2,{-4+8/3+1}) -- (2,0) node [above] {$2$};\n\\end{tikzpicture}}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第二轮复习讲义-16_导数及其应用" ], "genre": "选择题", "ans": "D", @@ -625800,7 +630309,8 @@ "content": "设集合$A=\\{x \\| x |<2, x \\in \\mathbf{R}\\}$, $B=\\{x | x^2-4 x+3 \\geq 0,\\ x \\in \\mathbf{R}\\}$, 则$A \\cap B=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-下学期周末卷-高三下学期周末卷05" ], "genre": "填空题", "ans": "$(-2,1]$", @@ -625835,7 +630345,8 @@ "content": "已知$\\mathrm{i}$为虚数单位, 复数$z$满足$\\dfrac{1-z}{1+z}=\\mathrm{i}$, 则$|z|=$\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-下学期周末卷-高三下学期周末卷05" ], "genre": "填空题", "ans": "$1$", @@ -625884,7 +630395,8 @@ "content": "在平面直角坐标系内, 直线$l: 2 x+y-2=0$, 将$l$与两条坐标轴围成的封闭图形绕$x$轴旋转一周, 所得几何体的体积为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期周末卷-高三下学期周末卷05" ], "genre": "填空题", "ans": "$\\dfrac{4 \\pi}{3}$", @@ -625919,7 +630431,8 @@ "content": "已知$\\sin 2 \\theta+\\sin \\theta=0$, $\\theta \\in(\\dfrac{\\pi}{2}, \\pi)$, 则$\\tan 2 \\theta=$\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期周末卷-高三下学期周末卷05" ], "genre": "填空题", "ans": "$\\sqrt{3}$", @@ -625957,7 +630470,8 @@ "content": "设定义在$\\mathbf{R}$上的奇函数$y=f(x)$, 当$x>0$时, $f(x)=2^x-4$, 则不等式$f(x) \\leq 0$的解集是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期周末卷-高三下学期周末卷05" ], "genre": "填空题", "ans": "$(-\\infty,-2] \\cup[0,2]$", @@ -625996,7 +630510,8 @@ "content": "在平面直角坐标系$xOy$中, 有一定点$A(1,1)$, 若线段$OA$的垂直平分线过抛物线$C: y^2=2 p x(p>0)$的焦点, 则抛物线$C$的方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期周末卷-高三下学期周末卷05" ], "genre": "填空题", "ans": "$y^2=4 x$", @@ -626034,7 +630549,8 @@ "content": "设某产品的一个部件来自三个供应商, 这三个供应商的良品率分别是$0.92$、$0.95$、$0.94$, 若这三个供应商的供货比例为$3: 2: 1$, 那么这个部件的总体良品率是\\blank{50}(用最简分数作答).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-下学期周末卷-高三下学期周末卷05" ], "genre": "填空题", "ans": "$\\dfrac{14}{15}$", @@ -626069,7 +630585,8 @@ "content": "记$(2 x+\\dfrac{1}{x})^n$($n \\in \\mathbf{N}$, $n\\ge 1$)的展开式中第$m$项的系数为$b_m$, 若$b_3=2 b_4$, 则$n=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-下学期周末卷-高三下学期周末卷05" ], "genre": "填空题", "ans": "$5$", @@ -626104,7 +630621,8 @@ "content": "已知一个正四棱锥的每条棱长均为$2$, 从该正四棱锥的$5$个顶点中任取$3$个点, 设随机变量$X$表示这三个点所构成的三角形的面积, 则其数学期望$E[X]=$\\blank{50}.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-下学期周末卷-高三下学期周末卷05" ], "genre": "填空题", "ans": "$\\dfrac{6+2 \\sqrt{3}}{5}$", @@ -626139,7 +630657,8 @@ "content": "已知函数$f(x)=x^2+p x+q$有两个零点$1$、$2$, 数列$\\{x_n\\}$满足$x_{n+1}=x_n-\\dfrac{f(x_n)}{f'(x_n)}$, 若$a_n=\\ln \\dfrac{x_n-2}{x_n-1}$, 且$a_1=-2$, 则数列$\\{a_n\\}$的前$2023$项的和为\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-下学期周末卷-高三下学期周末卷05" ], "genre": "填空题", "ans": "$2-2^{2024}$", @@ -626174,7 +630693,8 @@ "content": "平面直角坐标系$xOy$中, 抛物线$y^2=2 x$的焦点为$F$, 设$M$是抛物线上的动点, 则$\\dfrac{|MO|}{|MF|}$的最大值是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期周末卷-高三下学期周末卷05" ], "genre": "填空题", "ans": "$\\dfrac{2 \\sqrt{3}}{3}$", @@ -626209,7 +630729,8 @@ "content": "已知$a>0$, 函数$f(x)=x-\\dfrac{a}{x}$($x \\in[1,2]$)的图像的两个端点分别为$A$、$B$, 设$M$是函数$f(x)$图像上任意一点, 过$M$作垂直于$x$轴的直线$l$, 且$l$与线段$AB$交于点$N$, 若$|MN| \\leq 1$恒成立, 则$a$的最大值是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期周末卷-高三下学期周末卷05" ], "genre": "填空题", "ans": "$6+4 \\sqrt{2}$", @@ -626245,7 +630766,8 @@ "objs": [], "tags": [ "第三单元", - "第一单元" + "第一单元", + "2023届高三-下学期周末卷-高三下学期周末卷05" ], "genre": "选择题", "ans": "B", @@ -626280,7 +630802,8 @@ "content": "设$l_1$、$l_2$为两条不同的直线, $\\alpha$为一个平面, 则下列命题正确的是\\bracket{20}.\n\\onech{若直线$l_1\\parallel$平面$\\alpha$, 直线$l_2\\parallel$平面$\\alpha$, 则$l_1\\parallel l_2$}{若直线$l_1$上有两个点到平面$\\alpha$的距离相等, 则$l_1\\parallel \\alpha$}{直线$l_2$与平面$\\alpha$所成角的取值范围是$(0, \\dfrac{\\pi}{2})$}{若直线$l_1 \\perp$平面$\\alpha$, 直线$l_2 \\perp$平面$\\alpha$, 则$l_1\\parallel l_2$}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期周末卷-高三下学期周末卷05" ], "genre": "选择题", "ans": "D", @@ -626315,7 +630838,8 @@ "content": "已知$\\overrightarrow {a}$、$\\overrightarrow {b}$是平面内两个互相垂直的单位向量, 若向量$\\overrightarrow {c}$满足$(\\overrightarrow {c}-\\overrightarrow {a}) \\cdot(\\overrightarrow {c}-\\overrightarrow {b})=0$, 则$|\\overrightarrow {c}|$的最大值是\\bracket{20}\n\\fourch{$1$}{$2$}{$\\sqrt{2}$}{$\\dfrac{\\sqrt{2}}{2}$}", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期周末卷-高三下学期周末卷05" ], "genre": "选择题", "ans": "C", @@ -626353,7 +630877,8 @@ "content": "已知$f(x)=\\begin{cases}|\\log _3 x|, & 0=latex]\n\\draw (0,0,0) node [below] {$C$} coordinate (C);\n\\draw (2,0,0) node [right] {$B$} coordinate (B);\n\\draw (0,0,2) node [left] {$A$} coordinate (A);\n\\draw (0,2,0) node [above] {$C_1$} coordinate (C_1);\n\\draw (2,2,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (0,2,2) node [left] {$A_1$} coordinate (A_1);\n\\draw ($(A)!0.5!(A_1)$) node [left] {$D$} coordinate (D);\n\\draw (A)--(B)--(B_1)--(C_1)--(A_1)--cycle(A_1)--(B_1);\n\\draw [dashed] (A)--(C)--(B)(C)--(C_1)(C_1)--(D)--(C)(D)--(B_1)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $BC \\perp$平面$ACC_1A_1$;\\\\\n(2) 求二面角$B_1-CD-C_1$的大小.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期周末卷-高三下学期周末卷05" ], "genre": "解答题", "ans": "(1) 略; (2) $\\arccos \\dfrac{2}{3}$", @@ -626423,7 +630949,8 @@ "content": "已知$f(x)=2 \\sin x \\cos x+2 \\cos ^2 x$.\\\\\n(1) 求函数$f(x)$的单调增区间;\\\\\n(2) 将函数$y=f(x)$图像向右平移$\\dfrac{\\pi}{4}$个单位后, 得到函数$y=g(x)$的图像, 求方程$g(x)=1$的解集.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期周末卷-高三下学期周末卷05" ], "genre": "解答题", "ans": "(1) $[k \\pi-\\dfrac{3 \\pi}{8}, k \\pi+\\dfrac{\\pi}{8}]$, $k \\in \\mathbf{Z}$; (2) $\\{x | x=\\dfrac{k \\pi}{2}+\\dfrac{\\pi}{8},\\ k \\in \\mathbf{Z}\\}$", @@ -626458,7 +630985,8 @@ "content": "如图, 一智能扫地机器人在$A$处发现位于它正西方向的$B$处和北偏东$30^{\\circ}$方向上的$C$处分别有需要清扫的垃圾, 红外线感应测量发现机器人到$B$的距离比到$C$的距离少$0.4 \\text{m}$, 于是选择沿$A \\to B \\to C$路线清扫. 已知智能扫地机器人的直线行走速度为$0.2 \\text{m} / \\text{s}$, 忽略机器人吸入垃圾及在$B$处旋转所用时间, $10$秒钟完成了清扫任务.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 2]\n\\draw (0,0) node [right] {$A$} coordinate (A);\n\\draw (-0.6,0) node [left] {$B$} coordinate (B);\n\\draw (60:1) node [right] {$C$} coordinate (C);\n\\draw (A)--(B)--(C)--cycle;\n\\draw (-0.6,0.6) coordinate (T);\n\\draw [->] (T) --++ (0.3,0) node [right] {东};\n\\draw [->] (T) --++ (0,0.3) node [above] {北};\n\\end{tikzpicture}\n\\end{center}\n(1) $B$、$C$两处垃圾的距离是多少?\\\\\n(2) 智能扫地机器人此次清扫行走路线的夹角$\\angle ABC$是多少?", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期周末卷-高三下学期周末卷05" ], "genre": "解答题", "ans": "(1) $1.4$米; (2) $\\arcsin \\dfrac{5 \\sqrt{3}}{14}$", @@ -626493,7 +631021,8 @@ "content": "如图, 设$F$是椭圆$\\dfrac{x^2}{3}+\\dfrac{y^2}{4}=1$的下焦点, 直线$y=k x-4$($k>0$)与椭圆相交于$A$、$B$两点, 与$y$轴交于$P$点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-5) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (O) ellipse ({sqrt(3)} and 2);\n\\draw (0,-1) node [left] {$F$} coordinate (F);\n\\draw (0,-4) node [left] {$P$} coordinate (P);\n\\draw ({3*sqrt(5)/8},{-7/4}) node [below right] {$A$} coordinate (A);\n\\draw ($(P)!2!(A)$) node [right] {$B$} coordinate (B);\n\\draw ($(P)!-0.2!(B)$) -- ($(B)!-0.2!(P)$);\n\\draw (A)--(F)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 若$\\overrightarrow{PA}=\\overrightarrow{AB}$, 求$k$的值;\\\\\n(2) 求证: $\\angle AFP=\\angle BFO$;\\\\\n(3) 求$\\triangle ABF$面积的最大值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期周末卷-高三下学期周末卷05" ], "genre": "解答题", "ans": "(1) $\\dfrac{6 \\sqrt{5}}{5}$; (2) 略; (3) $\\dfrac{3 \\sqrt{3}}{4}$", @@ -626528,7 +631057,8 @@ "content": "已知正项数列$\\{a_n\\}$、$\\{b_n\\}$满足: 对任意$n \\in \\mathbf{N}$, $n\\ge 1$都有$a_n$、$b_n$、$a_{n+1}$成等差数列, $b_n$、$a_{n+1}$、$b_{n+1}$成等比数列, 且$a_1=10$, $a_2=15$.\\\\\n(1) 求证: 数列$\\{\\sqrt{b_n}\\}$是等差数列;\\\\\n(2) 求数列$\\{a_n\\}$、$\\{b_n\\}$的通项公式;\\\\\n(3) 设$S_n=\\dfrac{1}{a_1}+\\dfrac{1}{a_2}+\\cdots+\\dfrac{1}{a_n}$, 如果对任意正整数$n$, 不等式$2 a S_n<2-\\dfrac{b_n}{a_n}$恒成立, 求实数$a$的取值范围.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-下学期周末卷-高三下学期周末卷05" ], "genre": "解答题", "ans": "(1) 略; (2) $a_n=\\dfrac{(n+3)(n+4)}{2}$, $b_n=\\dfrac{(n+4)^2}{2}$; (3) $(-\\infty,1]$", @@ -626563,7 +631093,8 @@ "content": "设$a$、$b$、$c$是互不相等的实数, 则满足条件$\\{a, b\\} \\cup A=\\{a, b, c\\}$的所有集合$A$有\\blank{50}个.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-下学期周末卷-高三下学期周末卷06" ], "genre": "填空题", "ans": "$4$", @@ -626598,7 +631129,8 @@ "content": "已知复数$z=(1+2 \\mathrm{i})(3-\\mathrm{i})$, 则$\\dfrac{1}{z}$对应的点在第\\blank{50}象限.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-下学期周末卷-高三下学期周末卷06" ], "genre": "填空题", "ans": "四", @@ -626633,7 +631165,8 @@ "content": "若扇形的弧长和面积都是$4$, 那么这个扇形的圆心角的弧度数是\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期周末卷-高三下学期周末卷06" ], "genre": "填空题", "ans": "$2$", @@ -626668,7 +631201,8 @@ "content": "首项为$1$, 公比为$-\\dfrac{1}{2}$的无穷等比数列$\\{a_n\\}$的各项和为\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "2023届高三-下学期周末卷-高三下学期周末卷06" ], "genre": "填空题", "ans": "$\\dfrac 23$", @@ -626705,7 +631239,8 @@ "content": "已知数据$x_1, x_2, x_3, \\cdots, x_8$的方差为$16$, 则数据$3 x_1+1$、$3 x_2+1$、$\\cdots$、$3 x_8+1$的标准差为\\blank{50}.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-下学期周末卷-高三下学期周末卷06" ], "genre": "填空题", "ans": "$12$", @@ -626740,7 +631275,8 @@ "content": "若$(x^2-\\dfrac{2}{x^3})^5$展开式中的常数项为\\blank{50}(用数字作答).", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-下学期周末卷-高三下学期周末卷06" ], "genre": "填空题", "ans": "$40$", @@ -626776,7 +631312,8 @@ "objs": [], "tags": [ "第二单元", - "第一单元" + "第一单元", + "2023届高三-下学期周末卷-高三下学期周末卷06" ], "genre": "填空题", "ans": "$2$", @@ -626811,7 +631348,8 @@ "content": "某校数学兴趣小组给一个底面边长互不相等的直四棱柱容器的侧面和下底面染色, 提出如下的``四色问题'': 要求相邻两个面不得使用同一种颜色, 现有$4$种颜色可以选择, 则不同的染色方案有\\blank{50}种.", "objs": [], "tags": [ - "第八单元" + "第八单元", + "2023届高三-下学期周末卷-高三下学期周末卷06" ], "genre": "填空题", "ans": "$72$", @@ -626846,7 +631384,8 @@ "content": "有一种卫星接收天线, 其曲面与轴截面的交线为抛物线的一部分, 已知该卫星接收天线的口径$AB=6$, 深度$MO=2$, 信号处理中心$F$位于焦点处, 以顶点$O$为坐标原点, 建立如图所示的平面直角坐标系$xOy$, 若$P$是该抛物线上一点, 点$Q(\\dfrac{15}{8}, 2)$, 则$|PF|+|PQ|$的最小值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.3]\n\\draw [->] (-1,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -3:3, samples = 100] plot ({\\x*\\x/4.5},\\x);\n\\draw [dashed, domain = -4:-3, samples = 100] plot ({\\x*\\x/4.5},\\x);\n\\draw [dashed, domain = 3:4, samples = 100] plot ({\\x*\\x/4.5},\\x);\n\\draw (2,3) node [above] {$A$} coordinate (A);\n\\draw (2,-3) node [below] {$B$} coordinate (B);\n\\draw (2,0) node [below right] {$M$} coordinate (M);\n\\draw (1.25,0) node [below] {$F$} coordinate (F);\n\\filldraw (F) circle (0.1);\n\\draw [dashed] (A)--(B);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期周末卷-高三下学期周末卷06" ], "genre": "填空题", "ans": "$3$", @@ -626881,7 +631420,8 @@ "content": "$y=\\sin x+\\sin 2 x$在$(-a, a)$上恰有$5$个零点, 则实数$a$的最大值为\\blank{50}.", "objs": [], "tags": [ - "第三单元" + "第三单元", + "2023届高三-下学期周末卷-高三下学期周末卷06" ], "genre": "填空题", "ans": "$\\dfrac{4\\pi}3$", @@ -626916,7 +631456,8 @@ "content": "设向量$\\overrightarrow{OA}, \\overrightarrow{OB}$满足$|\\overrightarrow{OA}|=|\\overrightarrow{OB}|=2$, $\\overrightarrow{OA} \\cdot \\overrightarrow{OB}=2$, 若$m, n \\in \\mathbf{R}$, $m+n=1$, 则$|m \\overrightarrow{AB}|+|\\dfrac{1}{2} \\overrightarrow{BO}-n \\overrightarrow{BA}|$的最小值为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "2023届高三-下学期周末卷-高三下学期周末卷06" ], "genre": "填空题", "ans": "$\\sqrt{3}$", @@ -626951,7 +631492,8 @@ "content": "函数$f(x)=\\begin{cases}-3 x,& x<0, \\\\ x^2-1,& x \\geq 0.\\end{cases}$ 若方程$f(x)+3 \\sqrt{1-x^2}+|f(x)-3 \\sqrt{1-x^2}|-2 a x-6=0$有三个根, 且$x_1=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw ({sqrt(6)},0,0) node [right] {$B$} coordinate (B);\n\\draw (B) ++ (0,{sqrt(2)},0) node [above] {$P$} coordinate (P);\n\\draw (A) ++ ({sqrt(6)/2},0,{sqrt(6)/2}) node [below] {$C$} coordinate (C);\n\\draw ($(C)!0.5!(P)$) node [above] {$D$} coordinate (D);\n\\draw (A)--(C)--(B)--(P)--cycle(P)--(C);\n\\draw (A)--(D)--(B);\n\\draw [dashed] (A)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $BC \\perp AC$;\\\\\n(2) 若$AC=\\sqrt{3}$, 求直线$BC$与平面$ADB$所成角的正弦值.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "2023届高三-下学期周末卷-高三下学期周末卷06" ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\dfrac{\\sqrt{14}}7$", @@ -627149,7 +631695,8 @@ "content": "为了促进地方经济的快速发展, 国家鼓励地方政府实行积极灵活的人才引进政策, 被引进的人才, 可享受地方的福利待遇, 发放高标准的安家补贴费和生活津贴. 某市政府从本年度的$1$月份开始进行人才招聘工作, 参加报名的人员通过笔试和面试两个环节的审查后, 符合一定标准的人员才能被录用. 现对该市$1\\sim 4$月份的报名人员数和录用人才数(单位: 千人)进行统计, 得到如下表格.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline 月份 & 1 月份 & 2 月份 & 3 月份 & 4 月份 \\\\\n\\hline 报名人员数$x /$千人 & 3.5 & 5 & 6.5 & 7 \\\\\n\\hline 录用人才数$y /$千人 & 0.2 & 0.33 & 0.4 & 0.47 \\\\\n\\hline\n\\end{tabular} \n\\end{center}\n(1) 求出$y$关于$x$的经验回归方程;\\\\\n(2) 假设该市对被录用的人才每人发放$2$万元的生活津贴\\\\\n(i) 若该市$5$月份报名人员数为$8000$人, 试估计该市对$5$月份招聘的人才需要发放的生活津贴的总金额;\\\\\n(ii) 假设在参加报名的人员中, 小王和小李两人被录用的概率分别为$p, 3p-1$. 若两人的生活津贴之和的均值不超过$3$万元, 求$p$的取值范围.\\\\\n附: 经验回归方程$\\hat{y}=\\hat{a}+\\hat{b} x$中, 斜率和截距的最小二乘法估计公式分别为$\\hat{b}=\\dfrac{\\displaystyle\\sum_{i=1}^n x_i y_i-n \\overline{x}\\cdot \\overline{y}}{\\displaystyle\\sum_{i=1}^n x_i^2-n \\overline {x}^2}$, $\\hat{a}=\\overline {y}-\\hat{b} \\overline {x}$; $\\displaystyle\\sum_{i=1}^4 x_i^2=128.5$, $\\displaystyle\\sum_{i=1}^4 x_i y_i=8.24$.", "objs": [], "tags": [ - "第九单元" + "第九单元", + "2023届高三-下学期周末卷-高三下学期周末卷06" ], "genre": "解答题", "ans": "(1) $y=0.072x-0.046$; (2) (i) $1060$万; (ii) $(\\dfrac 13,\\dfrac 58]$", @@ -627185,7 +631732,8 @@ "content": "已知函数$f(x)=\\cos ^2 x+\\sin x \\cos x-\\dfrac{1}{2}$, 其中$x \\in \\mathbf{R}$.\\\\\n(1) 求不等式$f(x) \\geq \\dfrac{1}{2}$的解集;\\\\\n(2) 若函数$g(x)=\\dfrac{\\sqrt{2}}{2} \\sin (2 x+\\dfrac{3 \\pi}{4})$, 且对任意的$0 \\leq x_10$, $b>0$)的一条渐近线, 且双曲线$C$经过点$(2 \\sqrt{2}, 1)$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.3]\n\\draw [->] (-10,0) -- (10,0) node [below] {$x$};\n\\draw [->] (0,-5) -- (0,5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (10,5) -- (-10,-5) (10,-5) -- (-10,5);\n\\draw [domain = -10:{-2*sqrt(2)}, samples = 100] plot (\\x,{sqrt(\\x*\\x/4-2)});\n\\draw [domain = -10:{-2*sqrt(2)}, samples = 100] plot (\\x,{-sqrt(\\x*\\x/4-2)});\n\\draw [domain = {2*sqrt(2)}:10, samples = 100] plot (\\x,{sqrt(\\x*\\x/4-2)});\n\\draw [domain = {2*sqrt(2)}:10, samples = 100] plot (\\x,{-sqrt(\\x*\\x/4-2)}); \n\\end{tikzpicture}\n\\end{center}\n(1) 求双曲线$C$的方程;\\\\\n(2) 设直线$l': x=t y+4$与$C$交于$M, N$, 三角形$OMN$面积为$S$, 判断: 是否存在$t$使得$S=8 \\sqrt{15}$成立? 若存在, 求出$t$的值, 否则说明理由;\\\\\n(3) 设$A, B$是双曲线右支上两点, 若直线$l$上存在点$P$, 使得$\\triangle ABP$为正三角形, 求直线$AB$的斜率的取值范围.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-下学期周末卷-高三下学期周末卷06" ], "genre": "解答题", "ans": "(1) $\\dfrac{x^2}{4}-y^2=1$; (2) 存在, $t=\\pm \\sqrt{3}$或$t=\\pm \\sqrt{\\dfrac{76}{15}}$; (3) $(-8-5\\sqrt{3},-\\dfrac 12)\\cup (\\dfrac 12,-8+5\\sqrt{3})$", @@ -627256,7 +631805,8 @@ "content": "已知函数$f(x)=a x$($a \\in \\mathbf{R}$), $g(x)=\\cos x$.\\\\\n(1) 分别写出函数$y=f(g(x))$与$y=g(f(x))$的导函数;\\\\\n(2) 当$x \\in[\\pi,+\\infty)$时, 若不等式$f(x-\\pi) \\leq g(\\dfrac{\\pi}{2}-x)$恒成立, 求实数$a$的取值范围;\\\\\n(3) 令函数$h(x)=f(x)+g(x)$, $x \\in[0, \\pi]$, 若$y=h(x)$恰有两个极值点, 记极大值和极小值分别为$m, n$, 求$2 m-n$的取值范围.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-下学期周末卷-高三下学期周末卷06" ], "genre": "解答题", "ans": "(1) $f(g(x))$的导函数为$y=-a\\sin x$, $g(f(x))$的导函数为$y=-a\\sin (ax)$; (2) $(-\\infty,-1]$; (3) $[\\dfrac 32,3)$", @@ -628394,7 +632944,8 @@ "content": "已知实数$x$、$y$满足$y=\\sqrt{9-x^2}$, 请分别解决以下问题:\\\\\n(1) $\\dfrac{y}{x-5}$的取值范围是\\blank{50};\\\\\n(2) $(x-5)^2+(y-4)^2$的最小值是\\blank{50};\\\\\n(3) $x+y+5$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第三轮复习讲义-02_数形结合" ], "genre": "填空题", "ans": "(1) $[-\\dfrac 34,0]$; (2) $50-6\\sqrt{41}$; (3) $[2,5+3\\sqrt{2}]$", @@ -628447,7 +632998,8 @@ "content": "设$k \\in \\mathbf{R}$, 已知$f(x)=\\begin{cases}-x,& x \\leq 1, \\\\ \\ln (2x-2),& x>1,\\end{cases}$若不等式$f(x) \\leq|x-k|$对任意的$x \\in \\mathbf{R}$成立, 则$k$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第三轮复习讲义-02_数形结合" ], "genre": "填空题", "ans": "$[0,2-\\ln 2]$", @@ -628499,7 +633051,8 @@ "content": "设$a \\in \\mathbf{R}$, 若不等式$|x-2 a| \\geq \\dfrac{1}{2} x+a-1$对任意的$x \\in \\mathbf{R}$成立, 则$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "2023届高三-第三轮复习讲义-02_数形结合" ], "genre": "填空题", "ans": "$(-\\infty,\\dfrac 12]$", @@ -628572,7 +633125,8 @@ "content": "设$m \\in \\mathbf{R}$, 若方程$x^2-4|x|+5=m$有$4$个不相等的实根, 则$m$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "2023届高三-第三轮复习讲义-02_数形结合" ], "genre": "填空题", "ans": "$(1,5)$", @@ -628602,7 +633156,8 @@ "objs": [], "tags": [ "第二单元", - "第三单元" + "第三单元", + "2023届高三-第三轮复习讲义-02_数形结合" ], "genre": "填空题", "ans": "$8$", @@ -628701,7 +633256,8 @@ "content": "若正实数$a$、$b$满足$3 a+2 b=6$, 则$b+\\sqrt{a^2+b^2-2 b+1}$的最小值为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "2023届高三-第三轮复习讲义-02_数形结合" ], "genre": "填空题", "ans": "$\\dfrac{29}{13}$",