diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 619e4d2a..0bda03d7 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -6,7 +6,8 @@ "K0102001B" ], "tags": [ - "第一单元" + "第一单元", + "G20260155-必修第一章集合与逻辑复习" ], "genre": "解答题", "ans": "(1) $\\{$鼠, 牛, 虎, 兔, 龙, 蛇, 马, 羊, 猴, 鸡, 狗, 猪$\\}$; (2) $\\{$红色, 黄色$\\}$.", @@ -43,7 +44,8 @@ "K0102002B" ], "tags": [ - "第一单元" + "第一单元", + "G20260155-必修第一章集合与逻辑复习" ], "genre": "解答题", "ans": "(1) $\\{(x,y)|y=x, \\ x\\ge 0\\}$; (2) $\\{y|y=3n, \\ n\\in \\mathbf{Z}\\}$", @@ -81,7 +83,8 @@ ], "tags": [ "第一单元", - "2023届高三-第一轮复习讲义-02-常用逻辑用语" + "2023届高三-第一轮复习讲义-02-常用逻辑用语", + "G20260155-必修第一章集合与逻辑复习" ], "genre": "填空题", "ans": "(1) 必要非充分; (2) 充分非必要", @@ -127,7 +130,8 @@ "K0103001B" ], "tags": [ - "第一单元" + "第一单元", + "G20260155-必修第一章集合与逻辑复习" ], "genre": "解答题", "ans": "$\\varnothing,\\{1\\},\\{4\\},\\{-5\\},\\{1,4\\},\\{1,-5\\},\\{4,-5\\},\\{1,4,-5\\}$", @@ -166,7 +170,8 @@ "K0104003B" ], "tags": [ - "第一单元" + "第一单元", + "G20260155-必修第一章集合与逻辑复习" ], "genre": "解答题", "ans": "$A\\cup B=(-\\infty,-2)\\cup (-2,+\\infty)$; (2) $A\\cap B=\\varnothing$", @@ -205,7 +210,8 @@ "K0104006B" ], "tags": [ - "第一单元" + "第一单元", + "G20260155-必修第一章集合与逻辑复习" ], "genre": "解答题", "ans": "$(-\\infty,-1]\\cup [2,3)$", @@ -243,7 +249,8 @@ "K0104004B" ], "tags": [ - "第一单元" + "第一单元", + "G20260155-必修第一章集合与逻辑复习" ], "genre": "解答题", "ans": "$p=2$, $q=1$, $r=-2$", @@ -292,7 +299,8 @@ "K0106001B" ], "tags": [ - "第一单元" + "第一单元", + "G20260155-必修第一章集合与逻辑复习" ], "genre": "填空题", "ans": "$(-\\infty,1)$", @@ -331,7 +339,8 @@ ], "tags": [ "第一单元", - "2023届高三-第一轮复习讲义-02-常用逻辑用语" + "2023届高三-第一轮复习讲义-02-常用逻辑用语", + "G20260155-必修第一章集合与逻辑复习" ], "genre": "选择题", "ans": "A", @@ -375,7 +384,8 @@ "K0107003B" ], "tags": [ - "第一单元" + "第一单元", + "G20260155-必修第一章集合与逻辑复习" ], "genre": "解答题", "ans": "证明略", @@ -412,7 +422,8 @@ "K0103001B" ], "tags": [ - "第一单元" + "第一单元", + "G20260155-必修第一章集合与逻辑复习" ], "genre": "选择题", "ans": "C", @@ -477,7 +488,8 @@ "K0104004B" ], "tags": [ - "第一单元" + "第一单元", + "G20260155-必修第一章集合与逻辑复习" ], "genre": "填空题", "ans": "$\\{3,5,11,13\\}$, $\\{7,11,13,19\\}$", @@ -514,7 +526,8 @@ "K0103001B" ], "tags": [ - "第一单元" + "第一单元", + "G20260155-必修第一章集合与逻辑复习" ], "genre": "解答题", "ans": "$(-\\infty,3]$", @@ -552,7 +565,8 @@ "K0103001B" ], "tags": [ - "第一单元" + "第一单元", + "G20260155-必修第一章集合与逻辑复习" ], "genre": "解答题", "ans": "$(-\\infty,-2)\\cup [5,+\\infty)$", @@ -592,7 +606,8 @@ "K0109001B" ], "tags": [ - "第一单元" + "第一单元", + "G20260155-必修第一章集合与逻辑复习" ], "genre": "解答题", "ans": "$a=1$, 两个子集为$\\varnothing,\\dfrac{2}{3}$, 或者$a=-\\dfrac{1}{8}$, 两个子集为$\\varnothing,\\dfrac{4}{3}$", @@ -631,7 +646,8 @@ "K0107003B" ], "tags": [ - "第一单元" + "第一单元", + "G20260155-必修第一章集合与逻辑复习" ], "genre": "解答题", "ans": "证明略", @@ -671,7 +687,8 @@ ], "tags": [ "第一单元", - "2023届高三-第一轮复习讲义-02-常用逻辑用语" + "2023届高三-第一轮复习讲义-02-常用逻辑用语", + "G20260155-必修第一章集合与逻辑复习" ], "genre": "解答题", "ans": "证明略", @@ -716,7 +733,8 @@ "K0107003B" ], "tags": [ - "第一单元" + "第一单元", + "G20260155-必修第一章集合与逻辑复习" ], "genre": "解答题", "ans": "(1) $0$; (2) $\\{0\\}$; (3) $S=\\{n|n=5k, \\ k\\in \\mathbf{Z}\\}$", @@ -754,7 +772,8 @@ ], "tags": [ "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-第一轮复习讲义-04-方程与不等式的求解", + "G20260156-必修第二章等式与不等式复习" ], "genre": "解答题", "ans": "(1) $\\dfrac{5}{2}$; (2) $-\\dfrac{35}{4}$.", @@ -795,7 +814,8 @@ "K0111003B" ], "tags": [ - "第一单元" + "第一单元", + "G20260156-必修第二章等式与不等式复习" ], "genre": "解答题", "ans": "$\\dfrac{b+2a}{a+2b}<\\dfrac ab$", @@ -871,7 +891,8 @@ "tags": [ "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" + "2023届高三-第一轮复习讲义-04-方程与不等式的求解", + "G20260156-必修第二章等式与不等式复习" ], "genre": "解答题", "ans": "$(-\\infty,\\dfrac 53)$", @@ -917,7 +938,8 @@ "K0114001B" ], "tags": [ - "第一单元" + "第一单元", + "G20260156-必修第二章等式与不等式复习" ], "genre": "解答题", "ans": "(1) $(-\\infty,-3)\\cup (5,+\\infty)$; (2) $(-\\dfrac{2}{3},5)$; (3) $[\\dfrac{3-\\sqrt{19}}{5},\\dfrac{3+\\sqrt{19}}{5}]$; (4) $[-4-2\\sqrt{6},-4+2\\sqrt{6}]$; (5) $\\{9\\}$; (6) $\\mathbf{R}$", @@ -979,7 +1001,8 @@ "K0104001B" ], "tags": [ - "第一单元" + "第一单元", + "G20260156-必修第二章等式与不等式复习" ], "genre": "解答题", "ans": "$3,4,5$", @@ -1019,7 +1042,8 @@ "K0116002B" ], "tags": [ - "第一单元" + "第一单元", + "G20260156-必修第二章等式与不等式复习" ], "genre": "解答题", "ans": "(1) $(-\\infty,-7)\\cup (-\\dfrac{22}{5},+\\infty)$; (2) $[1,2]$", @@ -1081,7 +1105,8 @@ "K0117002B" ], "tags": [ - "第一单元" + "第一单元", + "G20260156-必修第二章等式与不等式复习" ], "genre": "解答题", "ans": "(1) $[\\dfrac{1}{3},1]$; (2) $(-\\dfrac{7}{3},3)$", @@ -1167,7 +1192,8 @@ "K0120001B" ], "tags": [ - "第一单元" + "第一单元", + "G20260156-必修第二章等式与不等式复习" ], "genre": "解答题", "ans": "(1) $M(\\dfrac{a+b}{2},0)$, $|MQ|=|\\dfrac{a+b}{2}|$, $|MN|=\\dfrac{1}{2}(|a|+|b|)$; (2) $|\\dfrac{a+b}{2}|\\le \\dfrac{1}{2}(|a|+|b|)$", @@ -1206,7 +1232,8 @@ "tags": [ "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" + "2023届高三-第一轮复习讲义-04-方程与不等式的求解", + "G20260156-必修第二章等式与不等式复习" ], "genre": "解答题", "ans": "$-1$", @@ -1252,7 +1279,8 @@ "K0109004B" ], "tags": [ - "第一单元" + "第一单元", + "G20260156-必修第二章等式与不等式复习" ], "genre": "解答题", "ans": "$(-3,-1]$", @@ -1291,7 +1319,8 @@ "tags": [ "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" + "2023届高三-第一轮复习讲义-04-方程与不等式的求解", + "G20260156-必修第二章等式与不等式复习" ], "genre": "解答题", "ans": "$(-\\infty,-1)$", @@ -1335,7 +1364,8 @@ "K0117001B" ], "tags": [ - "第一单元" + "第一单元", + "G20260156-必修第二章等式与不等式复习" ], "genre": "解答题", "ans": "(1) $(-\\infty,-\\dfrac{3}{4})\\cup [-\\dfrac{1}{3},+\\infty)$; (2) $[-4,-3)\\cup (1,2]$", @@ -1374,7 +1404,8 @@ "K0103001B" ], "tags": [ - "第一单元" + "第一单元", + "G20260156-必修第二章等式与不等式复习" ], "genre": "解答题", "ans": "$[0,1]$", @@ -1413,7 +1444,8 @@ "K0118003B" ], "tags": [ - "第一单元" + "第一单元", + "G20260156-必修第二章等式与不等式复习" ], "genre": "解答题", "ans": "证明略, 等号成立的条件为$x=0$", @@ -1450,7 +1482,8 @@ "K0119001B" ], "tags": [ - "第一单元" + "第一单元", + "G20260156-必修第二章等式与不等式复习" ], "genre": "解答题", "ans": "$2$", @@ -1511,7 +1544,8 @@ "K0119001B" ], "tags": [ - "第一单元" + "第一单元", + "G20260156-必修第二章等式与不等式复习" ], "genre": "解答题", "ans": "$\\dfrac{1}{4}$", @@ -1548,7 +1582,8 @@ "K0120002B" ], "tags": [ - "第一单元" + "第一单元", + "G20260119-三角不等式" ], "genre": "解答题", "ans": "证明略, 等号成立的条件为$a=\\pm b$", @@ -1586,7 +1621,8 @@ "K0111003B" ], "tags": [ - "第一单元" + "第一单元", + "G20260156-必修第二章等式与不等式复习" ], "genre": "解答题", "ans": "(1) 证明略, 等号成立的条件为$a=b=0$; (2) 证明略", @@ -1624,7 +1660,8 @@ "K0117001B" ], "tags": [ - "第一单元" + "第一单元", + "G20260156-必修第二章等式与不等式复习" ], "genre": "解答题", "ans": "(1) $(-\\infty,3)\\cup (3,4]\\cup [5,+\\infty)$; (2) $(-\\infty,\\dfrac{2}{3}]\\cup [4,+\\infty)$", @@ -1663,7 +1700,8 @@ "K0115002B" ], "tags": [ - "第一单元" + "第一单元", + "G20260156-必修第二章等式与不等式复习" ], "genre": "解答题", "ans": "$p=-1$, $q=-6$", @@ -1701,7 +1739,8 @@ ], "tags": [ "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-第一轮复习讲义-03-等式与不等式的性质及证明", + "G20260156-必修第二章等式与不等式复习" ], "genre": "解答题", "ans": "证明略", @@ -1774,7 +1813,8 @@ "tags": [ "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" + "2023届高三-第一轮复习讲义-05-幂指数与对数", + "G20260157-必修第三章幂、指数与对数复习" ], "genre": "填空题", "ans": "(1) $\\sqrt[3]{5}$, $\\log_3 5$; (2) $a^{\\frac 13}$; (3) $\\dfrac 14$; (4) $125$", @@ -1820,7 +1860,8 @@ "tags": [ "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" + "2023届高三-第一轮复习讲义-05-幂指数与对数", + "G20260157-必修第三章幂、指数与对数复习" ], "genre": "选择题", "ans": "(1) B (2) D", @@ -1865,7 +1906,9 @@ "K0202002B" ], "tags": [ - "第二单元" + "第二单元", + "G20260121-指数幂的拓展(2)", + "G20260157-必修第三章幂、指数与对数复习" ], "genre": "解答题", "ans": "$\\dfrac{3}{2}$", @@ -1909,7 +1952,8 @@ "K0203005B" ], "tags": [ - "第二单元" + "第二单元", + "G20260157-必修第三章幂、指数与对数复习" ], "genre": "解答题", "ans": "(1) $1$; (2) $8$", @@ -1946,7 +1990,9 @@ "K0205002B" ], "tags": [ - "第二单元" + "第二单元", + "G20260123-对数的运算性质", + "G20260157-必修第三章幂、指数与对数复习" ], "genre": "解答题", "ans": "$1-\\lg a$", @@ -1998,7 +2044,8 @@ ], "tags": [ "第二单元", - "2023届高三-第一轮复习讲义-05-幂指数与对数" + "2023届高三-第一轮复习讲义-05-幂指数与对数", + "G20260157-必修第三章幂、指数与对数复习" ], "genre": "解答题", "ans": "$5$", @@ -2043,7 +2090,8 @@ ], "tags": [ "第二单元", - "2023届高三-第一轮复习讲义-05-幂指数与对数" + "2023届高三-第一轮复习讲义-05-幂指数与对数", + "G20260157-必修第三章幂、指数与对数复习" ], "genre": "填空题", "ans": "(1) $-3$; (2) $64$; (3) $1$", @@ -2087,7 +2135,8 @@ "K0206002B" ], "tags": [ - "第二单元" + "第二单元", + "G20260157-必修第三章幂、指数与对数复习" ], "genre": "选择题", "ans": "B", @@ -2127,7 +2176,9 @@ "K0119001B" ], "tags": [ - "第二单元" + "第二单元", + "G20260123-对数的运算性质", + "G20260157-必修第三章幂、指数与对数复习" ], "genre": "解答题", "ans": "$2$", @@ -2175,7 +2226,8 @@ "K0203005B" ], "tags": [ - "第二单元" + "第二单元", + "G20260121-指数幂的拓展(2)" ], "genre": "解答题", "ans": "$\\dfrac{1}{1-2^x}$", @@ -2213,7 +2265,9 @@ ], "tags": [ "第二单元", - "2023届高三-第一轮复习讲义-05-幂指数与对数" + "2023届高三-第一轮复习讲义-05-幂指数与对数", + "G20260121-指数幂的拓展(2)", + "G20260157-必修第三章幂、指数与对数复习" ], "genre": "解答题", "ans": "证明略", @@ -2268,7 +2322,8 @@ "K0206002B" ], "tags": [ - "第二单元" + "第二单元", + "G20260157-必修第三章幂、指数与对数复习" ], "genre": "解答题", "ans": "$4$与$8$", @@ -2307,7 +2362,8 @@ "tags": [ "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" + "2023届高三-第一轮复习讲义-05-幂指数与对数", + "G20260157-必修第三章幂、指数与对数复习" ], "genre": "解答题", "ans": "证明略", @@ -2356,7 +2412,8 @@ "tags": [ "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-第一轮复习讲义-06-幂指对函数", + "G20260158-必修第四章幂函数、指数函数与对数函数复习" ], "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) $(2,2)$", @@ -2398,7 +2455,8 @@ "K0213007B" ], "tags": [ - "第二单元" + "第二单元", + "G20260158-必修第四章幂函数、指数函数与对数函数复习" ], "genre": "选择题", "ans": "(1) C; (2) C", @@ -2440,7 +2498,8 @@ "K0213007B" ], "tags": [ - "第二单元" + "第二单元", + "G20260158-必修第四章幂函数、指数函数与对数函数复习" ], "genre": "解答题", "ans": "(1) $[1,+\\infty)$; (2) $[1,+\\infty)$; (3) $(-1,1)$", @@ -2503,7 +2562,8 @@ "K0207001B" ], "tags": [ - "第二单元" + "第二单元", + "G20260125-幂函数的定义与图像" ], "genre": "解答题", "ans": "$1$或$-1$", @@ -2542,7 +2602,8 @@ "K0213008B" ], "tags": [ - "第二单元" + "第二单元", + "G20260158-必修第四章幂函数、指数函数与对数函数复习" ], "genre": "解答题", "ans": "$c=latex, scale = 0.5]\n\\draw [->] (-2,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0:3, samples = 100] plot (\\x,{sqrt(\\x)});\n\\draw [domain = {-ln(3)/ln(2)}:3, samples = 100] plot (\\x,{exp(\\x*ln(1/2))});\n\\end{tikzpicture}, 1个", @@ -2819,7 +2887,8 @@ "K0213007B" ], "tags": [ - "第二单元" + "第二单元", + "G20260158-必修第四章幂函数、指数函数与对数函数复习" ], "genre": "解答题", "ans": "$\\{1\\}$", @@ -2909,7 +2978,8 @@ "K0215003B" ], "tags": [ - "第二单元" + "第二单元", + "G20260159-必修第五章函数的概念、性质及应用复习" ], "genre": "解答题", "ans": "$(-\\infty,-1]\\cup [1,2)\\cup (2,+\\infty)$", @@ -2949,7 +3019,8 @@ "K0217004B" ], "tags": [ - "第二单元" + "第二单元", + "G20260159-必修第五章函数的概念、性质及应用复习" ], "genre": "解答题", "ans": "(1) 偶函数, 理由略; (2) 奇函数, 理由略; (3) 当$k=-2$时, 是偶函数; 当$k\\ne -2$($k<2$)时, 既不是奇函数, 又不是偶函数", @@ -2987,7 +3058,8 @@ "K0218001B" ], "tags": [ - "第二单元" + "第二单元", + "G20260159-必修第五章函数的概念、性质及应用复习" ], "genre": "解答题", "ans": "$m=1$, $n=2$", @@ -3049,7 +3121,8 @@ "K0220001B" ], "tags": [ - "第二单元" + "第二单元", + "G20260159-必修第五章函数的概念、性质及应用复习" ], "genre": "解答题", "ans": "(1) \\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-1,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1:5, samples = 200] plot (\\x,{abs(\\x*\\x-4*\\x)});\n\\end{tikzpicture}, 在$(-\\infty,0]$和$[2,4]$上是严格减函数, 在$[0,2]$和$[4,+\\infty)$上是严格增函数;\\\\\n(2) \\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-3.5,0) -- (3.5,0) node [below] {$x$};\n\\draw [->] (0,-3.5) -- (0,3.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-3,3) -- (0,-3) -- (3,3);\n\\end{tikzpicture}, 在$(-\\infty,0]$上是严格减函数, 在$[0,+\\infty)$上是严格增函数", @@ -3085,7 +3158,8 @@ "K0219002B" ], "tags": [ - "第二单元" + "第二单元", + "G20260159-必修第五章函数的概念、性质及应用复习" ], "genre": "解答题", "ans": "$f(2)2$时, 解集为$\\varnothing$; 当$a< 2$时, 解集为$\\{\\dfrac{2+\\sqrt{4-2a}}{a},\\dfrac{2-\\sqrt{4-2a}}{a}\\}$.", @@ -264159,7 +264318,8 @@ "K0109004B" ], "tags": [ - "第一单元" + "第一单元", + "G20260109-一元二次方程和韦达定理" ], "genre": "解答题", "ans": "(1) $3$; (2) $\\dfrac{4}{3}$; (3) $7$; (4) $-17$", @@ -264197,7 +264357,8 @@ "K0105002B" ], "tags": [ - "第一单元" + "第一单元", + "G20260110-不等式的性质(1)" ], "genre": "解答题", "ans": "(1) 假, 如$a=1$, $b=0$, $c=100$, $d=0$; (2) 假, 如$a=-1$, $b=-1$, $c=0$; (3) 真, 根据符号的定义; (4) 假, 如$a=2$, $b=1$, $c=\\dfrac{1}{2}$, $d=1$; (5) 假, 如$a=-1$, $b=-1$, $c=1$, $d=-1$", @@ -264222,7 +264383,8 @@ "K0111003B" ], "tags": [ - "第一单元" + "第一单元", + "G20260110-不等式的性质(1)" ], "genre": "解答题", "ans": "证明略", @@ -264294,7 +264456,8 @@ "K0112001B" ], "tags": [ - "第一单元" + "第一单元", + "G20260112-一元一次不等式, 一元二次不等式的求解(1)" ], "genre": "解答题", "ans": "当$a=1$时, 解集为$\\varnothing$; 当$a>1$时, 解集为$(-\\infty,a+1)$; 当$a<1$时, 解集为$(a+1,+\\infty)$", @@ -264331,7 +264494,8 @@ "K0113001B" ], "tags": [ - "第一单元" + "第一单元", + "G20260112-一元一次不等式, 一元二次不等式的求解(1)" ], "genre": "填空题", "ans": "(1) $(-3,2)$; (2) $(-\\infty,-3)\\cup (2,+\\infty)$; (3) $(-\\infty,-3]\\cup [2,+\\infty)$", @@ -264379,7 +264543,8 @@ "K0114001B" ], "tags": [ - "第一单元" + "第一单元", + "G20260113-一元二次不等式的求解(2)" ], "genre": "解答题", "ans": "(1) 解集为$\\mathbf{R}$; (2) 解集为$\\varnothing$; (3) 解集为$(-\\infty,\\dfrac{1}{3})\\cup (\\dfrac{1}{3},+\\infty)$; (4) 解集为$\\varnothing$; (5) 解集为$\\mathbf{R}$; (6) 解集为$\\mathbf{R}$", @@ -264403,7 +264568,8 @@ "K0115002B" ], "tags": [ - "第一单元" + "第一单元", + "G20260113-一元二次不等式的求解(2)" ], "genre": "解答题", "ans": "(1) 如$x^2-6x+7<0$; (2) 如$x^2-6x+7>0$; (3) 如$x^2+1>0$; (4) 如$x^2+1<0$", @@ -264428,7 +264594,8 @@ "K0104001B" ], "tags": [ - "第一单元" + "第一单元", + "G20260112-一元一次不等式, 一元二次不等式的求解(1)" ], "genre": "解答题", "ans": "(1) 解集为$(3,+\\infty)$; (2) 解集为$[5,6)$.", @@ -264465,7 +264632,8 @@ "K0115002B" ], "tags": [ - "第一单元" + "第一单元", + "G20260114-一元二次不等式的求解(3)" ], "genre": "解答题", "ans": "$[\\dfrac{1}{4},+\\infty)$", @@ -264490,7 +264658,8 @@ "K0114001B" ], "tags": [ - "第一单元" + "第一单元", + "G20260114-一元二次不等式的求解(3)" ], "genre": "解答题", "ans": "$(-\\dfrac{1}{2},-\\dfrac{1}{3})$", @@ -264527,7 +264696,8 @@ "K0116002B" ], "tags": [ - "第一单元" + "第一单元", + "G20260115-分式不等式的求解" ], "genre": "解答题", "ans": "(1) 解集为$(-\\infty,-1)\\cup (\\dfrac{3}{2},+\\infty)$; (2) 解集为$(-2,\\dfrac{1}{2}]$; (3) 解集为$(1,+\\infty)$; (4) 解集为$(-2,0]$; (5) 解集为$(2,3)$; (6) 解集为$\\varnothing$", @@ -264565,7 +264735,8 @@ "K0117002B" ], "tags": [ - "第一单元" + "第一单元", + "G20260116-含绝对值不等式的求解" ], "genre": "解答题", "ans": "(1) 解集为$(-7,1)$; (2) 解集为$(-\\infty,-1)\\cup (2,+\\infty)$; (3) 解集为$(1,+\\infty)$; (4) 解集为$(-\\infty,-2)\\cup (5,+\\infty)$", @@ -264589,7 +264760,8 @@ "K0114001B" ], "tags": [ - "第一单元" + "第一单元", + "G20260117-平均值不等式及其应用(1)" ], "genre": "解答题", "ans": "证明略", @@ -264614,7 +264786,8 @@ "K0118003B" ], "tags": [ - "第一单元" + "第一单元", + "G20260117-平均值不等式及其应用(1)" ], "genre": "解答题", "ans": "证明略, 等号成立当且仅当$x=-1$", @@ -264640,7 +264813,8 @@ "K0119002B" ], "tags": [ - "第一单元" + "第一单元", + "G20260118-平均值不等式及其应用(2)" ], "genre": "解答题", "ans": "长与宽均为$\\dfrac{l}{4}$时", @@ -264667,7 +264841,8 @@ "K0119002B" ], "tags": [ - "第一单元" + "第一单元", + "G20260118-平均值不等式及其应用(2)" ], "genre": "解答题", "ans": "面积的最大值为$2$, 此时矩形的长与宽均为$\\sqrt{2}$", @@ -264691,7 +264866,8 @@ "K0120002B" ], "tags": [ - "第一单元" + "第一单元", + "G20260119-三角不等式" ], "genre": "解答题", "ans": "证明略", @@ -264717,7 +264893,8 @@ "tags": [ "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" + "2023届高三-第一轮复习讲义-03-等式与不等式的性质及证明", + "G20260119-三角不等式" ], "genre": "解答题", "ans": "证明略", @@ -264749,7 +264926,8 @@ "K0201003B" ], "tags": [ - "第二单元" + "第二单元", + "G20260120-指数幂的拓展(1)" ], "genre": "解答题", "ans": "$-\\dfrac{2}{3}$", @@ -264773,7 +264951,8 @@ "K0201003B" ], "tags": [ - "第二单元" + "第二单元", + "G20260120-指数幂的拓展(1)" ], "genre": "解答题", "ans": "$\\pm \\sqrt{3}$", @@ -264797,7 +264976,8 @@ "K0201002B" ], "tags": [ - "第二单元" + "第二单元", + "G20260120-指数幂的拓展(1)" ], "genre": "解答题", "ans": "(1) $-4$; (2) $b-a$", @@ -264835,7 +265015,8 @@ "K0202002B" ], "tags": [ - "第二单元" + "第二单元", + "G20260121-指数幂的拓展(2)" ], "genre": "解答题", "ans": "(1) $10$; (2) $\\dfrac{1}{4}$", @@ -264859,7 +265040,8 @@ "K0202002B" ], "tags": [ - "第二单元" + "第二单元", + "G20260121-指数幂的拓展(2)" ], "genre": "解答题", "ans": "(1) $a^{\\frac{59}{15}}$; (2) $a^{\\frac{4}{9}}$", @@ -264883,7 +265065,8 @@ "K0202002B" ], "tags": [ - "第二单元" + "第二单元", + "G20260121-指数幂的拓展(2)" ], "genre": "解答题", "ans": "(1) $a^6$; (2) $-6b^{\\frac{1}{6}}$", @@ -264907,7 +265090,8 @@ "K0211001B" ], "tags": [ - "第二单元" + "第二单元", + "G20260121-指数幂的拓展(2)" ], "genre": "解答题", "ans": "证明略", @@ -264931,7 +265115,8 @@ "K0204004B" ], "tags": [ - "第二单元" + "第二单元", + "G20260122-对数的定义" ], "genre": "选择题", "ans": "A", @@ -264979,7 +265164,8 @@ "K0204004B" ], "tags": [ - "第二单元" + "第二单元", + "G20260122-对数的定义" ], "genre": "解答题", "ans": "(1) $16$; (2) $2$", @@ -265003,7 +265189,8 @@ "K0205002B" ], "tags": [ - "第二单元" + "第二单元", + "G20260123-对数的运算性质" ], "genre": "解答题", "ans": "(1) $A+B$; (2) $2A+\\dfrac{1}{2}B$", @@ -265027,7 +265214,8 @@ "K0205002B" ], "tags": [ - "第二单元" + "第二单元", + "G20260123-对数的运算性质" ], "genre": "解答题", "ans": "(1) $1$; (2) $\\dfrac{2}{3}$; (3) $-1$", @@ -265052,7 +265240,8 @@ "K0205002B" ], "tags": [ - "第二单元" + "第二单元", + "G20260123-对数的运算性质" ], "genre": "解答题", "ans": "$3b+2a$", @@ -265076,7 +265265,8 @@ "K0205002B" ], "tags": [ - "第二单元" + "第二单元", + "G20260124-对数的换底" ], "genre": "解答题", "ans": "(1) $-\\dfrac{2}{3}$; (2) $1$; (3) $18$; (4) $-2$", @@ -265101,7 +265291,8 @@ "K0206002B" ], "tags": [ - "第二单元" + "第二单元", + "G20260124-对数的换底" ], "genre": "解答题", "ans": "$5+\\dfrac{1}{a}$", @@ -265127,7 +265318,8 @@ "K0206002B" ], "tags": [ - "第二单元" + "第二单元", + "G20260124-对数的换底" ], "genre": "解答题", "ans": "证明略", @@ -265230,7 +265422,8 @@ "tags": [ "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" + "2023届高三-第一轮复习讲义-06-幂指对函数", + "G20260126-幂函数的性质" ], "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}", @@ -265259,7 +265452,8 @@ "K0208004B" ], "tags": [ - "第二单元" + "第二单元", + "G20260126-幂函数的性质" ], "genre": "解答题", "ans": "(1) $2.5^{-3}>3.1^{-3}$; (2) $1.7^{\\frac 32}>1.6^{\\frac 32}$", @@ -265285,7 +265479,8 @@ "tags": [ "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" + "2023届高三-第一轮复习讲义-06-幂指对函数", + "G20260126-幂函数的性质" ], "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}", @@ -265321,7 +265516,8 @@ "K0209001B" ], "tags": [ - "第二单元" + "第二单元", + "G20260127-指数函数的定义与图像" ], "genre": "解答题", "ans": "(1) 幂函数; (2) 幂函数; (3) 指数函数; (4) 幂函数; (5) 指数函数; (6) 指数函数", @@ -265345,7 +265541,8 @@ "K0209002B" ], "tags": [ - "第二单元" + "第二单元", + "G20260127-指数函数的定义与图像" ], "genre": "解答题", "ans": "(1) $\\mathbf{R}$; (2) $(-\\infty,2)\\cup (2,+\\infty)$", @@ -265370,7 +265567,8 @@ "K0210005B" ], "tags": [ - "第二单元" + "第二单元", + "G20260127-指数函数的定义与图像" ], "genre": "解答题", "ans": "\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,5.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -3:{ln(5.5)/ln(4)}, samples = 100] plot (\\x,{exp(\\x*ln(4))}) node [right] {$y=4^x$};\n\\draw [domain = {ln(5.5)/ln(0.25)}:3, samples = 100] plot (\\x,{exp(\\x*ln(0.25))}) node [right] {$y=(\\frac{1}{4})^x$};\n\\end{tikzpicture}", @@ -265394,7 +265592,8 @@ "K0210006B" ], "tags": [ - "第二单元" + "第二单元", + "G20260128-指数函数的性质(1)" ], "genre": "解答题", "ans": "(1) $1.4^{0.3}<1.4^{0.4}$; (2) $0.3^{1.4}>0.3^{1.5}$; (3) 当$a>1$时, $a^{-3. 14}>(\\dfrac 1a)^\\pi$; 当$0=latex,scale = 0.5]\n\\draw [->] (-0.5,0) -- (5.5,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0.001:5, samples = 100] plot (\\x,{ln(\\x)/ln(10)}) node [right] {$y=\\lg x$};\n\\draw [domain = 0.001:5, samples = 100] plot (\\x,{ln(\\x)/ln(0.1)}) node [below right] {$y=\\log_{\\frac{1}{10}} x$};\n\\end{tikzpicture}", @@ -265592,7 +265796,8 @@ "K0213002B" ], "tags": [ - "第二单元" + "第二单元", + "G20260131-对数函数的性质(1)" ], "genre": "解答题", "ans": "$(2,0)$", @@ -265618,7 +265823,8 @@ "K0213008B" ], "tags": [ - "第二单元" + "第二单元", + "G20260131-对数函数的性质(1)" ], "genre": "解答题", "ans": "(1) $\\log_{0.2}3>\\log_{0.2}6$; (2) $\\log_{0.2}3>\\log_{0.3}3$", @@ -265642,7 +265848,8 @@ "K0213004B" ], "tags": [ - "第二单元" + "第二单元", + "G20260131-对数函数的性质(1)" ], "genre": "解答题", "ans": "证明略", @@ -265666,7 +265873,8 @@ "K0214002B" ], "tags": [ - "第二单元" + "第二单元", + "G20260132-对数函数的性质(2)" ], "genre": "解答题", "ans": "(1) $y=0.99988^t$; (2) 至少经过$87$年", @@ -265690,7 +265898,8 @@ "K0214001B" ], "tags": [ - "第二单元" + "第二单元", + "G20260132-对数函数的性质(2)" ], "genre": "解答题", "ans": "$3$", @@ -265716,7 +265925,8 @@ "K0215003B" ], "tags": [ - "第二单元" + "第二单元", + "G20260133-函数" ], "genre": "解答题", "ans": "(1) $(-\\infty,-3]\\cup [2,+\\infty)$; (2) $[1,2)\\cup (2,+\\infty)$", @@ -265741,7 +265951,8 @@ ], "tags": [ "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" + "2023届高三-第一轮复习讲义-07-函数的概念与奇偶性", + "G20260133-函数" ], "genre": "选择题", "ans": "C", @@ -265775,7 +265986,8 @@ "K0215005B" ], "tags": [ - "第二单元" + "第二单元", + "G20260133-函数" ], "genre": "解答题", "ans": "(1) $[1,+\\infty)$; (2) $[-\\dfrac{1}{3},1]$", @@ -265827,7 +266039,8 @@ "tags": [ "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-第一轮复习讲义-07-函数的概念与奇偶性", + "G20260134-函数的表示方法" ], "genre": "解答题", "ans": "$y=\\begin{cases} 3x+10, & x\\in [-3,-2],\\\\ 4, & x\\in [-2,2],\\\\ 10-3x, & x\\in [2,3]. \\end{cases}$", @@ -265861,7 +266074,8 @@ "K0218001B" ], "tags": [ - "第二单元" + "第二单元", + "G20260135-函数的奇偶性(1)" ], "genre": "解答题", "ans": "都不一定, 如$y=0$, $x\\in \\{-1,1\\}$", @@ -265885,7 +266099,8 @@ "K0218001B" ], "tags": [ - "第二单元" + "第二单元", + "G20260135-函数的奇偶性(1)" ], "genre": "解答题", "ans": "(1) \\begin{tikzpicture}[>=latex, samples = 200]\n\\draw [->] (-3,0) -- (0,0) node [above left] {$O$} -- (3,0) node [below right] {$x$};\n\\draw [->] (0,-1) -- (0,3) node [left] {$y$};\n\\draw [domain = -3:0, dashed] plot (\\x , {-\\x * \\x * \\x / 8+ \\x * \\x /8 +\\x});\n\\draw [domain = -3:0] plot (-\\x , {-\\x * \\x * \\x / 8+ \\x * \\x /8 +\\x});\n\\end{tikzpicture}\\\\\n(2) \\begin{tikzpicture}[>=latex, samples = 200]\n\\draw [->] (-3,0) -- (0,0) node [below left] {$O$} -- (3,0) node [below right] {$x$};\n\\draw [->] (0,-1) -- (0,3) node [left] {$y$};\n\\draw [domain = 0:3, dashed] plot (\\x, {sin(\\x * 60) * \\x - 0.5});\n\\draw [domain = 0:3] plot (-\\x, {sin(\\x * 60) * \\x - 0.5});\n\\end{tikzpicture}", @@ -265909,7 +266124,8 @@ "K0217004B" ], "tags": [ - "第二单元" + "第二单元", + "G20260135-函数的奇偶性(1)" ], "genre": "解答题", "ans": "(1) 证明略; (2) 证明略", @@ -265933,7 +266149,8 @@ "K0217004B" ], "tags": [ - "第二单元" + "第二单元", + "G20260136-函数的奇偶性(2)" ], "genre": "解答题", "ans": "(1) 偶函数, 理由略; (2) 奇函数, 理由略; (3) 既不是奇函数, 又不是偶函数, 理由略; (4) 既是奇函数, 又是偶函数, 理由略", @@ -265957,7 +266174,8 @@ "K0218001B" ], "tags": [ - "第二单元" + "第二单元", + "G20260136-函数的奇偶性(2)" ], "genre": "解答题", "ans": "(1) 不存在, 理由略; (2) 存在($a=0$), 理由略", @@ -265983,7 +266201,8 @@ "tags": [ "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" + "2023届高三-第一轮复习讲义-08-函数的单调性", + "G20260137-函数的单调性(1)" ], "genre": "解答题", "ans": "不正确, 如$y=\\begin{cases} 0, & x=0, \\\\ \\dfrac 1x, & x>0.\\end{cases}$", @@ -266018,7 +266237,8 @@ "tags": [ "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" + "2023届高三-第一轮复习讲义-08-函数的单调性", + "G20260137-函数的单调性(1)" ], "genre": "解答题", "ans": "证明略", @@ -266053,7 +266273,8 @@ "K0219003B" ], "tags": [ - "第二单元" + "第二单元", + "G20260137-函数的单调性(1)" ], "genre": "解答题", "ans": "如$y=-x^2-2x$", @@ -266078,7 +266299,8 @@ "K0220002B" ], "tags": [ - "第二单元" + "第二单元", + "G20260138-函数的单调性(2)" ], "genre": "解答题", "ans": "(1) 在$[-2,1]$和$[2,3]$上是严格增函数; 在$[-3,-2]$和$[1,2]$上是严格减函数; (2) 在$[-\\pi,-\\dfrac{\\pi}{2}]$和$[\\dfrac{\\pi}{2},\\pi]$上是严格增函数, 在$[-\\dfrac{\\pi}{2},\\dfrac{\\pi}{2}]$上是严格减函数", @@ -266102,7 +266324,8 @@ "K0219003B" ], "tags": [ - "第二单元" + "第二单元", + "G20260138-函数的单调性(2)" ], "genre": "解答题", "ans": "在$[-2,-1]$上是严格减函数, 在$[-1,2]$上是严格增函数, 理由略", @@ -266128,7 +266351,8 @@ "tags": [ "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" + "2023届高三-第一轮复习讲义-08-函数的单调性", + "G20260138-函数的单调性(2)" ], "genre": "解答题", "ans": "(1) 证明略; (2) 是的, 证明略", @@ -266198,7 +266422,8 @@ "K0221002B" ], "tags": [ - "第二单元" + "第二单元", + "G20260139-函数的最值" ], "genre": "解答题", "ans": "(1) 最大值为$1$, 最小值不存在; (2) 最大值为$1$, 最小值为$-3$; (3) 最大值不存在, 最小值为$-8$; (4) 最大值为$0$, 最小值为$-6$", @@ -266248,7 +266473,8 @@ ], "tags": [ "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" + "2023届高三-第一轮复习讲义-10-有关函数的应用问题", + "G20260140-函数关系的建立" ], "genre": "解答题", "ans": "$y=6-\\dfrac x2, \\ x\\in (0,6)$", @@ -266272,7 +266498,8 @@ "K0222002B" ], "tags": [ - "第二单元" + "第二单元", + "G20260140-函数关系的建立" ], "genre": "解答题", "ans": "$S=\\begin{cases}\\dfrac{\\sqrt{3}}{2}t^2, & 01000\\end{cases}$", @@ -266346,7 +266574,8 @@ "tags": [ "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-第一轮复习讲义-09-函数的零点与最值", + "G20260141-用函数观点解方程与不等式" ], "genre": "解答题", "ans": "$(1,+\\infty)$", @@ -266382,7 +266611,8 @@ "tags": [ "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-第一轮复习讲义-09-函数的零点与最值", + "G20260142-用二分法求函数的零点" ], "genre": "解答题", "ans": "不是, 例如$f(x)=x^2-1$, $a=-10$, $b=10$等", @@ -266418,7 +266648,8 @@ "tags": [ "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-第一轮复习讲义-09-函数的零点与最值", + "G20260142-用二分法求函数的零点" ], "genre": "解答题", "ans": "$1.6$", @@ -266452,7 +266683,8 @@ "K0225003B" ], "tags": [ - "第二单元" + "第二单元", + "G20260143-反函数的概念" ], "genre": "解答题", "ans": "$[0,+\\infty)$", @@ -266476,7 +266708,8 @@ "K0225005B" ], "tags": [ - "第二单元" + "第二单元", + "G20260143-反函数的概念" ], "genre": "解答题", "ans": "(1) $y=\\dfrac{x-2}{3}$, $x\\in \\mathbf{R}$; (2) $y=-\\dfrac{3}{x}$, $x\\in \\{x|x\\ne 0\\}$; (3) $y=-\\sqrt{x}$, $x\\in [0,+\\infty)$; (4) $y=(x-1)^2$, $x\\in [1,+\\infty)$", @@ -266501,7 +266734,8 @@ "K0225002B" ], "tags": [ - "第二单元" + "第二单元", + "G20260143-反函数的概念" ], "genre": "解答题", "ans": "存在反函数, 反函数为$y=\\begin{cases}x, & -1\\le x\\le 0, \\\\ 1-x, & 0(x-1)(x^2+x+1)$", @@ -279941,7 +280221,8 @@ "K0111003B" ], "tags": [ - "第一单元" + "第一单元", + "G20260111-不等式的性质(2)" ], "genre": "解答题", "ans": "当$a=1$且$b=-1$时, $a^2+b^2=2a-2b-2$; 当$a\\ne 1$或$b\\ne -1$时, $a^2+b^2>2a-2b-2$.", @@ -279977,7 +280258,8 @@ "K0111003B" ], "tags": [ - "第一单元" + "第一单元", + "G20260111-不等式的性质(2)" ], "genre": "解答题", "ans": "证明略", @@ -280004,7 +280286,8 @@ "K0111003B" ], "tags": [ - "第一单元" + "第一单元", + "G20260111-不等式的性质(2)" ], "genre": "解答题", "ans": "证明略", @@ -280040,7 +280323,8 @@ "K0111003B" ], "tags": [ - "第一单元" + "第一单元", + "G20260111-不等式的性质(2)" ], "genre": "解答题", "ans": "证明略, 等号成立当且仅当$a=b(>0)$", @@ -280112,7 +280396,8 @@ "K0109002B" ], "tags": [ - "第一单元" + "第一单元", + "G20260109-一元二次方程和韦达定理" ], "genre": "解答题", "ans": "$a=2$, $b=1$, $c=-2$", @@ -280151,7 +280436,8 @@ "tags": [ "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" + "2023届高三-第一轮复习讲义-04-方程与不等式的求解", + "G20260109-一元二次方程和韦达定理" ], "genre": "解答题", "ans": "证明略", @@ -280247,7 +280533,8 @@ "K0111003B" ], "tags": [ - "第一单元" + "第一单元", + "G20260110-不等式的性质(1)" ], "genre": "解答题", "ans": "证明略", @@ -280308,7 +280595,8 @@ "K0111003B" ], "tags": [ - "第一单元" + "第一单元", + "G20260111-不等式的性质(2)" ], "genre": "解答题", "ans": "证明略", @@ -280502,7 +280790,8 @@ "K0114001B" ], "tags": [ - "第一单元" + "第一单元", + "G20260113-一元二次不等式的求解(2)" ], "genre": "解答题", "ans": "(1) 解集为$\\{3\\}$; (2) 解集为$\\mathbf{R}$; (3) 解集为$\\varnothing$; (4) 解集为$\\mathbf{R}$", @@ -280615,7 +280904,8 @@ "K0104001B" ], "tags": [ - "第一单元" + "第一单元", + "G20260113-一元二次不等式的求解(2)" ], "genre": "解答题", "ans": "(1) 解集为$(-4,-3]$; (2) 解集为$(3-\\sqrt{5},\\dfrac{3}{4})$; (3) 解集为$(-\\infty,-1]\\cup (3,+\\infty)$", @@ -280653,7 +280943,8 @@ "K0116002B" ], "tags": [ - "第一单元" + "第一单元", + "G20260115-分式不等式的求解" ], "genre": "解答题", "ans": "(1) 解集为$(-\\infty,-1)\\cup (2,+\\infty)$; (2) 解集为$(-\\infty,0)\\cup (1,+\\infty)$; (3) 解集为$[\\dfrac{1}{4},\\dfrac{3}{4})$; (4) 解集为$(-\\infty,-2)\\cup [\\dfrac{1}{2},+\\infty)$; (5) 解集为$(1,8)$", @@ -280703,7 +280994,8 @@ "K0117002B" ], "tags": [ - "第一单元" + "第一单元", + "G20260116-含绝对值不等式的求解" ], "genre": "解答题", "ans": "(1) 解集为$(-1,\\dfrac{3}{2})$; (2) 解集为$(\\dfrac{4}{3},+\\infty)$; (3) 解集为$(-\\infty,\\dfrac{1}{2}]\\cup [3,+\\infty)$; (4) 解集为$(-1,4)$", @@ -280740,7 +281032,8 @@ "K0112002B" ], "tags": [ - "第一单元" + "第一单元", + "G20260112-一元一次不等式, 一元二次不等式的求解(1)" ], "genre": "解答题", "ans": "约$46\\text{km/h}$($\\dfrac{67+\\sqrt{561}}{3}\\text{km/h}$)", @@ -280777,7 +281070,8 @@ "K0112001B" ], "tags": [ - "第一单元" + "第一单元", + "G20260112-一元一次不等式, 一元二次不等式的求解(1)" ], "genre": "解答题", "ans": "当$a=0$且$b<0$时, 解集为$\\mathbf{R}$; 当$a=0$且$b\\ge 0$时, 解集为$\\varnothing$; 当$a>0$时, 解集为$(\\dfrac{b}{a},+\\infty)$; 当$a<0$时, 解集为$(-\\infty,\\dfrac{b}{a})$", @@ -280801,7 +281095,8 @@ "K0113001B" ], "tags": [ - "第一单元" + "第一单元", + "G20260113-一元二次不等式的求解(2)" ], "genre": "解答题", "ans": "(1) 当$a=-3$时, 解集为$\\mathbf{R}$; 当$a>-3$时, 解集为$(-\\infty,-3]\\cup [a,+\\infty)$; 当$a<-3$时, 解集为$(-\\infty,a]\\cup [-3,+\\infty)$;\\\\\n(2) 当$a=0$时, 解集为$(-\\infty,0)\\cup (0,+\\infty)$; 当$a>0$时, 解集为$(-\\infty,a)\\cup (2a,+\\infty)$; 当$a<0$时, 解集为$(-\\infty,2a)\\cup (a,+\\infty)$\n(3) 解集为$(-\\infty,a]\\cup [a+1,+\\infty)$", @@ -280874,7 +281169,8 @@ "K0116002B" ], "tags": [ - "第一单元" + "第一单元", + "G20260115-分式不等式的求解" ], "genre": "解答题", "ans": "(1) 解集为$[-1,\\dfrac{1}{2}]$; (2) 解集为$[1,2)\\cup (2,+\\infty)$", @@ -280965,7 +281261,8 @@ "K0115001B" ], "tags": [ - "第一单元" + "第一单元", + "G20260114-一元二次不等式的求解(3)" ], "genre": "解答题", "ans": "$[0,8)$", @@ -281001,7 +281298,8 @@ "K0111002B" ], "tags": [ - "第一单元" + "第一单元", + "G20260117-平均值不等式及其应用(1)" ], "genre": "选择题", "ans": "D", @@ -281125,7 +281423,8 @@ "K0119002B" ], "tags": [ - "第一单元" + "第一单元", + "G20260118-平均值不等式及其应用(2)" ], "genre": "解答题", "ans": "$25\\text{cm}^2$", @@ -281199,7 +281498,8 @@ "K0118003B" ], "tags": [ - "第一单元" + "第一单元", + "G20260117-平均值不等式及其应用(1)" ], "genre": "解答题", "ans": "$2ab=latex,scale = 0.3]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0:{exp(ln(3)/5)}, samples = 100] plot ({\\x*\\x*\\x*\\x*\\x},\\x) plot ({-\\x*\\x*\\x*\\x*\\x},-\\x);\n\\end{tikzpicture}; (2) $\\{x|x\\in\\mathbf{R}, \\ x\\in 0\\}$, 大致图像为\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-2,0) -- (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 = 0.5:2, samples = 100] plot (\\x,{exp(-2*ln(\\x))}) plot (-\\x,{exp(-2*ln(\\x))});\n\\end{tikzpicture}; (3) $(0,+\\infty)$, 大致图像为\\begin{tikzpicture}[>=latex,scale = 0.3]\n\\draw [->] (-0.5,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0.6:1.45, samples = 100] plot ({\\x*\\x*\\x*\\x},{1/\\x/\\x/\\x});\n\\end{tikzpicture}", @@ -282306,7 +282626,8 @@ "K0207001B" ], "tags": [ - "第二单元" + "第二单元", + "G20260125-幂函数的定义与图像" ], "genre": "解答题", "ans": "约$3086\\text{cm}^3/\\text{s}$", @@ -282343,7 +282664,8 @@ "K0208004B" ], "tags": [ - "第二单元" + "第二单元", + "G20260126-幂函数的性质" ], "genre": "解答题", "ans": "(1) $3.1^{-\\frac 12}>3.2^{-\\frac 12}$; (2) $(a+2)^{\\frac 13}>a^{\\frac 13}$", @@ -282381,7 +282703,8 @@ ], "tags": [ "第二单元", - "2023届高三-上学期测验卷-测验02" + "2023届高三-上学期测验卷-测验02", + "G20260126-幂函数的性质" ], "genre": "填空题", "ans": "\\textcircled{2}", @@ -282453,7 +282776,8 @@ "K0207003B" ], "tags": [ - "第二单元" + "第二单元", + "G20260125-幂函数的定义与图像" ], "genre": "填空题", "ans": "一、二", @@ -282533,7 +282857,8 @@ "tags": [ "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-第一轮复习讲义-06-幂指对函数", + "G20260126-幂函数的性质" ], "genre": "选择题", "ans": "D", @@ -282599,7 +282924,8 @@ "K0208002B" ], "tags": [ - "第二单元" + "第二单元", + "E20260103-高一上学期测验卷03" ], "genre": "解答题", "ans": "存在, $a$的值为$-1$或$0$", @@ -282660,7 +282986,8 @@ "K0209002B" ], "tags": [ - "第二单元" + "第二单元", + "G20260127-指数函数的定义与图像" ], "genre": "解答题", "ans": "(1) $(-\\infty,3]$; (2) $(-\\infty,0)\\cup (0,+\\infty)$", @@ -282698,7 +283025,8 @@ "K0208005B" ], "tags": [ - "第二单元" + "第二单元", + "G20260127-指数函数的定义与图像" ], "genre": "解答题", "ans": "\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,4.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -3:3, samples = 100] plot (\\x,{exp(\\x*ln(1.5))}) node [right] {\\textcircled{1}};\n\\draw [domain = -3:3, samples = 100] plot (\\x,{exp(\\x*ln(2/3))}) node [right] {\\textcircled{2}} plot (\\x,{exp(\\x*ln(2/3))-1}) node [right] {\\textcircled{3}};\n\\draw [dashed] (-3,-1) -- (3,-1);\n\\end{tikzpicture}, \\textcircled{1}与\\textcircled{2}关于$y$轴对称, \\textcircled{2}向下平移$1$个单位得到\\textcircled{3}", @@ -282736,7 +283064,8 @@ "K0210005B" ], "tags": [ - "第二单元" + "第二单元", + "G20260128-指数函数的性质(1)" ], "genre": "解答题", "ans": "$(2,3)$", @@ -282760,7 +283089,8 @@ "K0210002B" ], "tags": [ - "第二单元" + "第二单元", + "G20260128-指数函数的性质(1)" ], "genre": "解答题", "ans": "$(2,1)$", @@ -282784,7 +283114,8 @@ "K0210006B" ], "tags": [ - "第二单元" + "第二单元", + "G20260128-指数函数的性质(1)" ], "genre": "解答题", "ans": "(1) $1.2^{2.6}<1.2^{2.61}$; (2) $(\\sqrt 3)^{-\\frac 13}>(\\dfrac{\\sqrt 3}3)^\\frac 12$", @@ -282821,7 +283152,8 @@ "K0210006B" ], "tags": [ - "第二单元" + "第二单元", + "G20260129-指数函数的性质(2)" ], "genre": "解答题", "ans": "(1) $(1,3)$; (2) $[16,+\\infty)$", @@ -282858,7 +283190,8 @@ "K0211001B" ], "tags": [ - "第二单元" + "第二单元", + "G20260129-指数函数的性质(2)" ], "genre": "解答题", "ans": "$2$", @@ -282895,7 +283228,8 @@ "K0211002B" ], "tags": [ - "第二单元" + "第二单元", + "G20260129-指数函数的性质(2)" ], "genre": "解答题", "ans": "约$81$台", @@ -282932,7 +283266,8 @@ "K0210005B" ], "tags": [ - "第二单元" + "第二单元", + "G20260129-指数函数的性质(2)" ], "genre": "选择题", "ans": "C", @@ -282956,7 +283291,8 @@ "K0210005B" ], "tags": [ - "第二单元" + "第二单元", + "G20260129-指数函数的性质(2)" ], "genre": "选择题", "ans": "D", @@ -282993,7 +283329,8 @@ "K0211001B" ], "tags": [ - "第二单元" + "第二单元", + "G20260129-指数函数的性质(2)" ], "genre": "解答题", "ans": "$(-\\infty,-\\sqrt{2})\\cup (\\sqrt{2},+\\infty)$", @@ -283030,7 +283367,8 @@ "K0210006B" ], "tags": [ - "第二单元" + "第二单元", + "G20260128-指数函数的性质(1)" ], "genre": "解答题", "ans": "$3^{2x}<3^x<3^{-x}$", @@ -283054,7 +283392,8 @@ "K0211001B" ], "tags": [ - "第二单元" + "第二单元", + "G20260129-指数函数的性质(2)" ], "genre": "解答题", "ans": "$(-\\infty,0)\\cup (5,+\\infty)$", @@ -283116,7 +283455,8 @@ "K0212002B" ], "tags": [ - "第二单元" + "第二单元", + "G20260130-对数函数的定义和图像" ], "genre": "解答题", "ans": "(1) $(-12,+\\infty)$; (2) $\\mathbf{R}$", @@ -283152,7 +283492,8 @@ "K0204004B" ], "tags": [ - "第二单元" + "第二单元", + "G20260130-对数函数的定义和图像" ], "genre": "解答题", "ans": "$9$", @@ -283188,7 +283529,8 @@ "K0213001B" ], "tags": [ - "第二单元" + "第二单元", + "G20260130-对数函数的定义和图像" ], "genre": "解答题", "ans": "\\begin{tikzpicture}[>=latex]\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 = {1/9}:2, samples = 100] plot (\\x,{ln(\\x)/ln(3)}) node [right] {\\textcircled{1}};\n\\draw [domain = {1/9}:2, samples = 100] plot (\\x,{ln(\\x)/ln(1/3)}) node [right] {\\textcircled{2}};\n\\draw [domain = {1/9}:2, samples = 100] plot ({ln(\\x)/ln(1/3)},\\x) node [above] {\\textcircled{3}};\n\\end{tikzpicture}; \\textcircled{1}的图像与\\textcircled{2}的图像关于$x$轴对称, \\textcircled{2}的图像与\\textcircled{3}的图像关于直线$y=x$对称, \\textcircled{1}的图像绕原点逆时针旋转$90^\\circ$后得到\\textcircled{3}的图像", @@ -283325,7 +283667,8 @@ "K0214002B" ], "tags": [ - "第二单元" + "第二单元", + "G20260131-对数函数的性质(1)" ], "genre": "填空题", "ans": "\\textcircled{2}\\textcircled{3}", @@ -283363,7 +283706,8 @@ "K0211001B" ], "tags": [ - "第二单元" + "第二单元", + "G20260130-对数函数的定义和图像" ], "genre": "解答题", "ans": "当$a>1$时, 定义域为$(-\\infty,1)$; 当$01$时, $mn$; (3) $m0$时, 定义域为$(-\\infty,-a)\\cup (-a,a)\\cup (a,+\\infty)$; (2) 当$a<0$时, 定义域为$(-\\infty,a]\\cup [0,+\\infty)$; 当$a=0$时, 定义域为$\\mathbf{R}$; 当$a>0$时, 定义域为$(-\\infty,0]\\cup [a,+\\infty)$", @@ -283696,7 +284046,8 @@ "KNONE" ], "tags": [ - "第二单元" + "第二单元", + "G20260135-函数的奇偶性(1)" ], "genre": "选择题", "ans": "D", @@ -283738,7 +284089,8 @@ "K0218001B" ], "tags": [ - "第二单元" + "第二单元", + "G20260135-函数的奇偶性(1)" ], "genre": "解答题", "ans": "(1) 证明略; (2) 证明略", @@ -283775,7 +284127,8 @@ "K0218001B" ], "tags": [ - "第二单元" + "第二单元", + "G20260135-函数的奇偶性(1)" ], "genre": "解答题", "ans": "(1) 证明略; (2) 证明略", @@ -283812,7 +284165,8 @@ "K0218001B" ], "tags": [ - "第二单元" + "第二单元", + "G20260136-函数的奇偶性(2)" ], "genre": "解答题", "ans": "(1) 奇函数, 理由略; (2) 偶函数, 理由略; (3) 既不是奇函数, 又不是偶函数, 理由略; (4) 既不是奇函数, 又不是偶函数, 理由略; (5) 奇函数, 理由略", @@ -283839,7 +284193,8 @@ "K0219003B" ], "tags": [ - "第二单元" + "第二单元", + "G20260137-函数的单调性(1)" ], "genre": "解答题", "ans": "证明略", @@ -283924,7 +284279,8 @@ "K0221002B" ], "tags": [ - "第二单元" + "第二单元", + "G20260139-函数的最值" ], "genre": "解答题", "ans": "最大值为$-2$, 最小值为$-3$", @@ -283963,7 +284319,8 @@ "K0221002B" ], "tags": [ - "第二单元" + "第二单元", + "G20260139-函数的最值" ], "genre": "解答题", "ans": "$8$", @@ -284087,7 +284444,8 @@ "KNONE" ], "tags": [ - "第二单元" + "第二单元", + "G20260139-函数的最值" ], "genre": "解答题", "ans": "\\begin{tikzpicture}[>=latex,scale = 0.5]\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\\draw [domain = -3:3, samples = 100] plot (\\x,{\\x*\\x-2*abs(\\x)});\n\\end{tikzpicture}\\\\\n定义域为$\\mathbf{R}$, 是偶函数, 在$(-\\infty,-1]$和$[0,1]$上是严格减函数, 在$[-1,0]$和$[1,+\\infty)$上是严格增函数, 最小值为$-1$", @@ -284209,7 +284567,8 @@ "K0221002B" ], "tags": [ - "第二单元" + "第二单元", + "G20260139-函数的最值" ], "genre": "解答题", "ans": "$y_{\\min}=\\begin{cases}0, & t\\le 2, \\\\ 2^{t+1}-8, & 2\\le t<4 \\end{cases}$", @@ -284252,7 +284611,8 @@ "K0222002B" ], "tags": [ - "第二单元" + "第二单元", + "G20260140-函数关系的建立" ], "genre": "解答题", "ans": "定义域为$\\{1,2,3,4\\}$, 图像: \\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (0,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,0.2) -- (0.3,0.3) -- (0,0.4) -- (-0.3,0.5) -- (0,0.6) -- (0,5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (0.1,0.1) -- (4.5,4.5);\n\\foreach \\i/\\j/\\k in {1/1/10,2/2/12,3/3/14,4/4/16}\n{\\draw [dashed] (\\i,0) node [below] {$\\i$} --++ (0,\\j) -- (0,\\j) node [left] {$\\k$};\n\\filldraw (\\i,\\j) circle (0.06);};\n\\end{tikzpicture}", @@ -284346,7 +284706,8 @@ "tags": [ "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" + "2023届高三-第一轮复习讲义-09-函数的零点与最值", + "G20260142-用二分法求函数的零点" ], "genre": "解答题", "ans": "证明略, 近似值为$0.5$", @@ -284397,7 +284758,8 @@ "K0222002B" ], "tags": [ - "第二单元" + "第二单元", + "G20260140-函数关系的建立" ], "genre": "解答题", "ans": "(1) $P=\\dfrac{2}{3}v^2$, $v\\in (0,48]$; (2) $y=\\dfrac{200}{3}v+\\dfrac{86400}{v}$, $v\\in (0,48]$; (3) 船速为$36$海里/小时时所需总费用最少", @@ -284459,7 +284821,8 @@ "K0222002B" ], "tags": [ - "第二单元" + "第二单元", + "G20260140-函数关系的建立" ], "genre": "解答题", "ans": "$y=10600-200x$, $x\\in \\{4,5,6,7,8,9,10\\}$", @@ -284497,7 +284860,8 @@ "tags": [ "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" + "2023届高三-第一轮复习讲义-09-函数的零点与最值", + "G20260141-用函数观点解方程与不等式" ], "genre": "解答题", "ans": "证明略", @@ -284591,7 +284955,8 @@ "K0225002B" ], "tags": [ - "第二单元" + "第二单元", + "G20260143-反函数的概念" ], "genre": "解答题", "ans": "不存在, 理由略", @@ -284630,7 +284995,9 @@ "K0225005B" ], "tags": [ - "第二单元" + "第二单元", + "G20260143-反函数的概念", + "W20260111-高一上学期周末卷11" ], "genre": "解答题", "ans": "(1) $y=-\\sqrt[3]{x}$, $x\\in \\mathbf{R}$; (2) $y=\\dfrac{2x}{1-x}$, $x\\in \\{x|x\\ne 1\\}$; (3) $y=-\\sqrt{x-1}$, $x\\in (1,+\\infty)$", @@ -284679,7 +285046,8 @@ "K0225005B" ], "tags": [ - "第二单元" + "第二单元", + "G20260143-反函数的概念" ], "genre": "解答题", "ans": "(1) $y=\\lg (x-1)$, $x\\in (1,+\\infty)$; (2) $y=2^x-1$, $x\\in \\mathbf{R}$; (3) $y=2^{x-1}$, $x\\in \\mathbf{R}$", @@ -284716,7 +285084,8 @@ "K0225004B" ], "tags": [ - "第二单元" + "第二单元", + "G20260143-反函数的概念" ], "genre": "解答题", "ans": "$16$", @@ -298636,7 +299005,8 @@ "KNONE" ], "tags": [ - "第四单元" + "第四单元", + "G20260145-等差数列及其通项公式" ], "genre": "解答题", "ans": "(1) $29$; (2) $10$; (3) $3$; (4) $\\dfrac{23}{2}$", @@ -298660,7 +299030,8 @@ "K0401003X" ], "tags": [ - "第四单元" + "第四单元", + "G20260145-等差数列及其通项公式" ], "genre": "解答题", "ans": "证明略", @@ -298858,7 +299229,8 @@ "K0402004X" ], "tags": [ - "第四单元" + "第四单元", + "G20260146-等差数列的前$n$项和" ], "genre": "解答题", "ans": "(1) $80$; (2) $\\displaystyle_{i=1}^{10} a_{2i}=320$", @@ -298882,7 +299254,8 @@ "K0402004X" ], "tags": [ - "第四单元" + "第四单元", + "G20260146-等差数列的前$n$项和" ], "genre": "解答题", "ans": "$-4$", @@ -299050,7 +299423,8 @@ "K0403003X" ], "tags": [ - "第四单元" + "第四单元", + "G20260147-等比数列及其通项公式" ], "genre": "解答题", "ans": "证明略", @@ -299101,7 +299475,8 @@ "K0403002X" ], "tags": [ - "第四单元" + "第四单元", + "G20260147-等比数列及其通项公式" ], "genre": "解答题", "ans": "约$13.1\\%$", @@ -299125,7 +299500,8 @@ "K0404003X" ], "tags": [ - "第四单元" + "第四单元", + "G20260148-等比数列的前$n$项和(1)" ], "genre": "解答题", "ans": "(1) $605$; (2) $\\dfrac{255}{16}$; (3) $-\\dfrac{4095}{3072}$; (4) $24$", @@ -299186,7 +299562,8 @@ "K0404003X" ], "tags": [ - "第四单元" + "第四单元", + "G20260149-等比数列的前$n$项和(2)" ], "genre": "解答题", "ans": "(1) $299.609375\\text{m}$; (2) 不会超过, 理由略", @@ -299375,7 +299752,8 @@ "K0406002X" ], "tags": [ - "第四单元" + "第四单元", + "G20260150-数列的概念与性质" ], "genre": "解答题", "ans": "是的, 第$15$项", @@ -299429,7 +299807,8 @@ ], "tags": [ "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" + "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", + "G20260150-数列的概念与性质" ], "genre": "解答题", "ans": "有最大项$a_3$, 无最小项", @@ -299475,7 +299854,8 @@ "K0407002X" ], "tags": [ - "第四单元" + "第四单元", + "G20260151-利用递推公式表示数列" ], "genre": "解答题", "ans": "$a_n=2^{\\frac{n(n-1)}{2}}$", @@ -299527,7 +299907,8 @@ "K0407004X" ], "tags": [ - "第四单元" + "第四单元", + "G20260151-利用递推公式表示数列" ], "genre": "解答题", "ans": "(1) $73$个; (2) $a_n=8a_{n-1}$", @@ -299554,7 +299935,8 @@ "tags": [ "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" + "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", + "G20260150-数列的概念与性质" ], "genre": "解答题", "ans": "第$10$项最大, 第$9$项最小", @@ -299704,7 +300086,8 @@ "K0408003X" ], "tags": [ - "第四单元" + "第四单元", + "G20260152-数学归纳法" ], "genre": "选择题", "ans": "C", @@ -299810,7 +300193,8 @@ "K0408003X" ], "tags": [ - "第四单元" + "第四单元", + "G20260152-数学归纳法" ], "genre": "解答题", "ans": "证明略", @@ -325512,7 +325896,8 @@ "K0101002B" ], "tags": [ - "第一单元" + "第一单元", + "G20260101-集合的概念" ], "genre": "解答题", "ans": "(1) 是集合, 是空集; (2) 不是集合, 对象是否在其中不确定; (3) 不是集合, 未确定``哪一部分''; (4) 是集合, 对象是否在其中是确定的", @@ -325536,7 +325921,8 @@ "K0101002B" ], "tags": [ - "第一单元" + "第一单元", + "G20260101-集合的概念" ], "genre": "解答题", "ans": "(1) 是无限集, 正整数有无穷多个; (2) 是有限集, 因为是有限集$\\{1,2,3,\\cdots,600\\}$的一部分组成的; (3) 是有限集, 地球上出现过的人的数量是有限的; (4) 是无限集, 到$A$的距离为$\\dfrac{1}{2},\\dfrac{1}{3},\\dfrac{1}{4},\\cdots$的点就已经是无穷多个了", @@ -325561,7 +325947,8 @@ "K0101003B" ], "tags": [ - "第一单元" + "第一单元", + "G20260101-集合的概念" ], "genre": "填空题", "ans": "(1) $\\in$; (2) $\\in$; (3) $\\not\\in$; (4) $\\in$", @@ -325586,7 +325973,8 @@ "K0101003B" ], "tags": [ - "第一单元" + "第一单元", + "G20260101-集合的概念" ], "genre": "填空题", "ans": "(1) $\\in$; (2) $\\not\\in$; (3) $\\in$; (4) $\\in$; (5) $\\not\\in$; (6) $\\not\\in$", @@ -325610,7 +325998,8 @@ "K0101002B" ], "tags": [ - "第一单元" + "第一单元", + "G20260101-集合的概念" ], "genre": "解答题", "ans": "(1) 是集合, 是有限集; (2) 是集合, 是无限集; (3) 不是集合, ``影响力比较大''定义模糊, 无法明确界定; (4) 是集合, 是有限集; (5) 是集合, 是无限集; (6) 是集合, 是无限集", @@ -325661,7 +326050,8 @@ "K0101002B" ], "tags": [ - "第一单元" + "第一单元", + "G20260101-集合的概念" ], "genre": "填空题", "ans": "$a=4$", @@ -325700,7 +326090,8 @@ "K0102001B" ], "tags": [ - "第一单元" + "第一单元", + "G20260102-集合的表示方法" ], "genre": "解答题", "ans": "(1) $\\{1,2,3,\\cdots,10\\}$; (2) $\\{1,2,3,4\\}$; (3) $\\{3,4,5,6,7\\}$", @@ -325726,7 +326117,8 @@ "K0102003B" ], "tags": [ - "第一单元" + "第一单元", + "G20260102-集合的表示方法" ], "genre": "解答题", "ans": "(1) $A=\\{2,4,6,8\\}$; (2) $B=\\{x|x=2m, \\ m\\in \\mathbf{Z}\\}$; (3) $C=\\{x|\\dfrac{x-2}{3}\\in \\mathbf{Z}, \\ x\\in \\mathbf{N}\\}$; (4) $D=\\{x|x=3m+2, \\ m\\in \\mathbf{Z}\\}$; (5) $E=\\{(x,y)|xy<0\\}$", @@ -325751,7 +326143,8 @@ "K0102004B" ], "tags": [ - "第一单元" + "第一单元", + "G20260102-集合的表示方法" ], "genre": "解答题", "ans": "(1) $[1,2)$; (2) $(-\\infty,3]$", @@ -325778,7 +326171,8 @@ "K0102002B" ], "tags": [ - "第一单元" + "第一单元", + "G20260102-集合的表示方法" ], "genre": "解答题", "ans": "(1) 描述法, $A=\\{0,3,4,5,7,8,9,12\\}$; (2) 列举法, $B=\\{(x,y)|x^2=y^2=1\\}$", @@ -325851,7 +326245,8 @@ "K0101004B" ], "tags": [ - "第一单元" + "第一单元", + "G20260103-集合之间的关系" ], "genre": "解答题", "ans": "$\\begin{cases}x=2, \\\\ y=6\\end{cases}$或$\\begin{cases}x=4, \\\\ y=0.\\end{cases}$", @@ -325877,7 +326272,8 @@ "K0103005B" ], "tags": [ - "第一单元" + "第一单元", + "G20260103-集合之间的关系" ], "genre": "解答题", "ans": "(1) $B\\subset A$; (2) $C\\subset D$; (3) $E\\subset F$", @@ -325902,7 +326298,8 @@ "K0103005B" ], "tags": [ - "第一单元" + "第一单元", + "G20260103-集合之间的关系" ], "genre": "解答题", "ans": "所有子集为$\\varnothing,\\{a\\},\\{b\\},\\{c\\},\\{a,b\\},\\{a,c\\},\\{b,c\\},\\{a,b,c\\}$, 除$\\{a,b,c\\}$外其余七个均为真子集", @@ -325926,7 +326323,8 @@ "K0103001B" ], "tags": [ - "第一单元" + "第一单元", + "G20260103-集合之间的关系" ], "genre": "解答题", "ans": "$[2,+\\infty)$", @@ -326005,7 +326403,8 @@ "content": "已知集合$A=(-2,2)$, $B=(-3,-1)\\cup (1,+\\infty)$. 求$A\\cap B$及$A\\cup B$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260104-集合之间的运算" ], "genre": "解答题", "ans": "$A\\cap B=(-2,-1)\\cup (1,2)$; (2) $A\\cup B=(-3,+\\infty)$", @@ -326029,7 +326428,8 @@ "content": "已知集合$A=\\{(x,y)|2x+y=5\\}$, $B=\\{(x,y)|3x+2y=8\\}$. 求$A\\cap B$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260104-集合之间的运算" ], "genre": "解答题", "ans": "$\\{(2,1)\\}$", @@ -326051,7 +326451,8 @@ "content": "已知集合$A=\\{1,2\\}$, 求所有满足$A\\cup B= \\{1,2,3\\}$的集合$B$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260104-集合之间的运算" ], "genre": "解答题", "ans": "$\\{3\\}$, $\\{1,3\\}$, $\\{2,3\\}$, $\\{1,2,3\\}$", @@ -326073,7 +326474,8 @@ "content": "设全集$U=\\{a, b, c, d, e\\}$, 集合$A=\\{a, b, c\\}$, 集合$B=\\{c, d\\}$. 分别求: $\\overline{A\\cup B}$, $\\overline A\\cap \\overline B$, $\\overline{A\\cap B}$及$\\overline A\\cup \\overline B$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260104-集合之间的运算" ], "genre": "解答题", "ans": "$\\overline{A\\cup B}=\\{e\\}$, $\\overline A\\cap \\overline B=\\{e\\}$, $\\overline{A\\cap B}=\\{a,b,d,e\\}$, $\\overline A\\cup \\overline B=\\{a,b,d,e\\}$", @@ -326187,7 +326589,8 @@ "content": "下列语句哪些是命题? 如果是命题, 那么它们是真命题还是假命题? 为什么?\\\\\n(1) 个位数字是$5$的自然数能被$5$整除;\\\\\n(2) 凡直角三角形都相似;\\\\\n(3) 请起立;\\\\\n(4) 若两个角互为补角, 则这两个角不相等;\\\\\n(5) 若两个三角形的三组对应边分别相等, 则这两个三角形全等;\\\\\n(6) 你是高一学生吗?\\\\\n(7) $x>3$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260105-命题" ], "genre": "解答题", "ans": "(1) 是真命题, $n=10k+5=5(2k+1)$, $k\\in \\mathbf{N}$; (2) 是假命题, 如一副三角板的两块; (3) 不是命题, 无法判断真假; (4) 是假命题, 两个角都是$90^\\circ$时它们既互补, 又相等; (5) 是真命题, 是全等三角形的判定定理; (6) 不是命题, 无法判断真假; (7) 不是命题, 无法判断真假(是``开语句'', 当$x$的值确定后, 就是命题了)", @@ -326209,7 +326612,8 @@ "content": "判断下列命题的真假, 并说明理由:\\\\\n(1) 若一个数是偶数, 则这个数不是素数;\\\\\n(2) 若菱形的一组邻角相等, 则这个菱形是正方形;\\\\\n(3) 如果集合$A$是集合$B$的子集, 那么$B$不是$A$的子集.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260105-命题" ], "genre": "解答题", "ans": "(1) 假命题, $2$; (2) 真命题, $A+B=180^\\circ$, $A=B$, 故$A=90^\\circ$; (3) 假命题, 如$A=B=\\{1\\}$", @@ -326231,7 +326635,8 @@ "content": "将下列命题改写成``若$\\alpha$, 则$\\beta$''的形式, 并判断``$\\alpha \\Rightarrow \\beta$''是否成立.\\\\\n(1) 等腰三角形的两底角相等;\\\\\n(2) 凡是素数都是奇数;\\\\\n(3) 对顶角相等.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260105-命题" ], "genre": "解答题", "ans": "(1) 若一个三角形是等腰三角形, 则它的两底角相等. 成立; (2) 若$n$是素数, 则$n$是奇数. 不成立; (3) 若两个角是对顶角, 则它们相等. 成立", @@ -326253,7 +326658,8 @@ "content": "已知下列三组陈述句:\\\\\n\\textcircled{1} $\\alpha$: $a+b$是偶数, $\\beta$: $a,b$都是偶数;\\\\\n\\textcircled{2} $\\alpha$: $q<0$, $\\beta$: 关于$x$的方程$x^2+2x+q=0$($q\\in \\mathbf{R}$)有两个不相等的实数根;\\\\\n\\textcircled{3} $\\alpha$: $ab=0$, $\\beta$: $a^2+b^2=0$.\n其中满足关系``$\\alpha \\Rightarrow \\beta$''的题号是\\blank{50}. 满足关系``$\\alpha \\Leftarrow \\beta$''的题号是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260105-命题" ], "genre": "填空题", "ans": "\\textcircled{2}, \\textcircled{1}\\textcircled{3}", @@ -326322,7 +326728,8 @@ "content": "判断下列各组中的$\\alpha$分别是$\\beta$的什么条件, 并说明理由.\\\\\n(1) $\\alpha$: 四边形$ABCD$是正方形, $\\beta$: 四边形$ABCD$的四个内角都是直角;\\\\\n(2) $\\alpha$: $x^2$是有理数, $\\beta$: $x$是有理数.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260106-充分条件与必要条件" ], "genre": "解答题", "ans": "(1) 充分非必要, 正方形的四个内角都是直角, 但是长为$2$, 宽为$1$的矩形不是正方形, 它的四个内角也都是直角; (2) 必要非充分, 当$x=\\sqrt{2}$时, $x^2$是有理数, $x$不是有理数, 而当$x=\\dfrac{p}{q}$($p,q\\in \\mathbf{Z}$)时, $x^2=\\dfrac{p^2}{q^2}$也是有理数", @@ -326344,7 +326751,8 @@ "content": "已知$m$是实数, 集合$M=\\{2,3,m+6\\}$, $N=\\{0,7\\}$. 求证: ``$m=1$''是``$M\\cap N=\\{7\\}$''的充要条件.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260106-充分条件与必要条件" ], "genre": "解答题", "ans": "证明略", @@ -326410,7 +326818,8 @@ "content": "设$n\\in \\mathbf{Z}$. 证明: 若$n^2$是偶数, 则$n$也是偶数.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260107-反证法" ], "genre": "解答题", "ans": "反证法, 证明略", @@ -326435,7 +326844,8 @@ "content": "设$x,y\\in \\mathbf{R}$. 证明: 若$x+y>2$, 则$x>1$或$y>1$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260107-反证法" ], "genre": "解答题", "ans": "反证法, 证明略", @@ -326457,7 +326867,8 @@ "content": "证明: $\\sqrt 2$是无理数.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260107-反证法" ], "genre": "解答题", "ans": "证明略", @@ -326482,7 +326893,8 @@ "content": "已知$a_1,a_2,\\cdots,a_{10}$是实数, 且满足$a_1+a_2+\\cdots +a_{10}\\le 100$. 求证: $a_1,a_2,\\cdots,a_{10}$中至少有一个不大于$10$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260107-反证法" ], "genre": "解答题", "ans": "证明略", @@ -326504,7 +326916,8 @@ "content": "已知$x,y$是实数, 若$x$是有理数, $y$是无理数, 求证: $x+y$是无理数.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260107-反证法" ], "genre": "解答题", "ans": "证明略", @@ -326578,7 +326991,8 @@ "content": "设$a$、$b$、$c$、$d$是实数, 判断下列命题的真假, 并说明理由:\\\\\n(1) 如果$a=b$, 且$c=d$, 那么$a+c=b+d$;\\\\\n(2) 如果$a=b$, 且$c=d$, 那么$ac=bd$;\\\\\n(3) 如果$a=b\\ne 0$, 那么$\\dfrac 1a=\\dfrac 1b$;\\\\\n(4) 如果$a=b$, 那么$a^n=b^n$, 其中$n$是正整数;\\\\\n(5) 如果$ac=bc$, 那么$a=b$;\\\\\n(6) 如果$(a-b)^2+(b-c)^2=0$, 那么$a=b=c$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260108-等式的性质与方程的解集" ], "genre": "解答题", "ans": "(1) 正确, 根据加法性质; (2) 正确, 根据乘法性质; (3) 正确, $\\dfrac{1}{a}-\\dfrac{1}{b}=\\dfrac{b-a}{ab}=0$; (4) 正确, 反复使用乘法性质; (5) 错误, 如$a=2$, $b=-1$, $c=0$; (6) 正确, 因为实数的平方非负, 因此$a-b=0$, $b-c=0$, 从而$a=b=c$.", @@ -326600,7 +327014,8 @@ "content": "设$a$、$b\\in \\mathbf{R}$, 分别求下列关于$x$的方程的解集:\\\\\n(1) $ax=1$;\\\\\n(2) $1-ax=a-x$;\\\\\n(3) $ax=b$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260108-等式的性质与方程的解集" ], "genre": "解答题", "ans": "(1) 当$a\\ne 0$时, 解集为$\\{\\dfrac{1}{a}\\}$; 当$a=0$时, 解集为$\\varnothing$;\\\\\n(2) 当$a\\ne 1$时, 解集为$\\{-1\\}$; 当$a=1$时, 解集为$\\mathbf{R}$;\\\\\n(3) 当$a=0$且$b=0$时, 解集为$\\mathbf{R}$; 当$a=0$且$b\\ne 0$时, 解集为$\\varnothing$; 当$a\\ne 0$时, 解集为$\\{\\dfrac{b}{a}\\}$.", @@ -326784,7 +327199,8 @@ "content": "求证: $a_1=a_2$, $b_1=b_2$, $c_1=c_2$是等式$a_1x^2+b_1x+c_1=a_2x^2+b_2x+c_2$恒成立的充要条件.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260109-一元二次方程和韦达定理" ], "genre": "解答题", "ans": "证明略", @@ -326806,7 +327222,8 @@ "content": "已知方程$x^2+x-3=0$的两个根为$x_1$、$x_2$, 求下列各式的值:\\\\\n(1) $x_1^2x_2+x_2^2x_1$;\\\\\n(2) $|x_1-x_2|$;\\\\\n(3)$x_1^3+x_2^3$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260109-一元二次方程和韦达定理" ], "genre": "解答题", "ans": "(1) $3$; (2) $\\sqrt{13}$; (3) $-10$", @@ -326828,7 +327245,8 @@ "content": "已知一元二次方程$x^2+3x-3=0$的两个实根分别为$x_1$、$x_2$, 求作二次项系数是$1$, 且分别以下列数值为根的一元二次方程:\\\\\n(1)$-x_1$, $-x_2$;\\\\\n(2)$\\dfrac 1{x_1}$, $\\dfrac 1{x_2}$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260109-一元二次方程和韦达定理" ], "genre": "解答题", "ans": "(1) $x^2-3x-3=0$; (2) $x^2-x-\\dfrac{1}{3}=0$", @@ -326852,7 +327270,8 @@ "content": "$a$、$b$、$c$均为非零实数, 若关于$x$的一元二次方程$ax^2+bx+c=0$的解集为$\\{1,-3\\}$, 求关于$x$的一元二次方程$cx^2-bx+a=0$的解集.(要求: 用两种不同方法解答.)", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260109-一元二次方程和韦达定理" ], "genre": "解答题", "ans": "$x=\\dfrac{1}{3}$或$-1$", @@ -326896,7 +327315,8 @@ "content": "证明: 如果$a+b>c$, 那么$a>c-b$; 反之亦然.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260110-不等式的性质(1)" ], "genre": "解答题", "ans": "证明略", @@ -326918,7 +327338,8 @@ "content": "已知$a>b$, $c>d$. 求证: $a+c>b+d$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260110-不等式的性质(1)" ], "genre": "解答题", "ans": "证明略", @@ -326944,7 +327365,8 @@ "content": "已知$a>b$, $c>d$. 求证: $a-d>b-c$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260110-不等式的性质(1)" ], "genre": "解答题", "ans": "证明略", @@ -327058,7 +327480,8 @@ "content": "(1) 已知$a>b>0$, $c>d>0$. 求证: $ac>bd$;\\\\\n(2) 已知$a>b>0$, 求证: $a^n>b^n$, 其中$n$是正整数.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260111-不等式的性质(2)" ], "genre": "解答题", "ans": "证明略", @@ -327080,7 +327503,8 @@ "content": "已知$a,b$为正数, $n$为正整数, 求证: 如果$a^n>b^n$, 那么$a>b$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260111-不等式的性质(2)" ], "genre": "解答题", "ans": "证明略", @@ -327103,7 +327527,8 @@ "content": "设$a$是实数, 比较$(a+1)^2$与$a^2-a+1$的值的大小.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260111-不等式的性质(2)" ], "genre": "解答题", "ans": "当$a>0$时, $(a+1)^2>a^2-a+1$; 当$a=0$时, $(a+1)^2=a^2-a+1$; 当$a<0$时, $(a+1)^20$时, 解集为$(-\\infty,\\dfrac{1}{a})$; 当$a=0$时, 解集为$\\mathbf{R}$; 当$a<0$时, 解集为$(\\dfrac{1}{a},+\\infty)$", @@ -327213,7 +327640,8 @@ "content": "设$a$为实数, 解关于$x$的一元一次不等式组$\\begin{cases} 2x+a>0, \\\\ 3x-6a<0. \\end{cases}$", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260112-一元一次不等式, 一元二次不等式的求解(1)" ], "genre": "解答题", "ans": "当$a\\le 0$时, 解集为$\\varnothing$; 当$a>0$时, 解集为$(-\\dfrac{a}{2},2a)$.", @@ -327237,7 +327665,8 @@ "content": "解不等式$(x-3)(x+1)>0$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260112-一元一次不等式, 一元二次不等式的求解(1)" ], "genre": "解答题", "ans": "解集为$(-1,3)$", @@ -327287,7 +327716,8 @@ "content": "解不等式$-2x^2+3x-\\dfrac 12\\ge 0$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260112-一元一次不等式, 一元二次不等式的求解(1)" ], "genre": "解答题", "ans": "解集为$[\\dfrac{3-\\sqrt{5}}{4},\\dfrac{3+\\sqrt{5}}{4}]$", @@ -327331,7 +327761,8 @@ "content": "某厂计划全年完成产值$6000$万元, 前三个季度已完成$4300$万元. 如果$10$月份的产值是$500$万元, 设在最后两个月里, 月增长率是$x$($x\\ge 0$). 若要完成全年任务, 求$x$的最小值(精确到$1\\%$).", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260113-一元二次不等式的求解(2)" ], "genre": "解答题", "ans": "$13\\%$($x^2+3x-0.4\\ge 0$)", @@ -327353,7 +327784,8 @@ "content": "解下列不等式:\\\\\n(1) $x^2\\le 4x-4$;\\\\\n(2) $x(x+1)\\ge 7x-9$;\\\\\n(3) $4x^2-4x+3>0$;\\\\\n(4) $x^2\\le x-2$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260113-一元二次不等式的求解(2)" ], "genre": "解答题", "ans": "(1) $\\{2\\}$; (2) $\\mathbf{R}$; (3) $\\mathbf{R}$; (4) $\\varnothing$", @@ -327397,7 +327829,8 @@ "content": "设$a\\in \\mathbf{R}$, 解关于$x$的不等式: $x^2-(a+1)x+a\\le 0$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260113-一元二次不等式的求解(2)" ], "genre": "解答题", "ans": "当$a>1$时, 解集为$[1,a]$; 当$a=1$时, 解集为$\\{1\\}$; 当$a<1$时, 解集为$[a,1]$", @@ -327441,7 +327874,8 @@ "content": "设$a_1,a_2,b_1,b_2,c_1,c_2$均为非零实数, 关于$x$的不等式$a_1x^2+b_1x+c_1>0$与$a_2x^2+b_2x+c_2>0$的解集分别为$M$和$N$, 那么``$\\dfrac{a_1}{a_2}=\\dfrac{b_1}{b_2}=\\dfrac{c_1}{c_2}$''是``$M=N$''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", + "W20260104-高一上学期周末卷04" ], "genre": "选择题", "ans": "D", @@ -327500,7 +327934,8 @@ "content": "若关于$x$的不等式$x^2+(k-1)x+4>0$的解集为$\\mathbf{R}$, 求实数$k$的取值范围.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260114-一元二次不等式的求解(3)" ], "genre": "解答题", "ans": "$(-3,5)$", @@ -327522,7 +327957,8 @@ "content": "若关于$x$的不等式$(k-5)x^2-(5-k)x-k+10>0$的解集为$\\mathbf{R}$, 求实数$k$的取值范围.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260114-一元二次不等式的求解(3)" ], "genre": "解答题", "ans": "$(5,9)$", @@ -327544,7 +327980,8 @@ "content": "已知一元二次不等式$x^2+bx+c<0$的解集为$(1,2)$, 求实数$b$、$c$的值以及不等式$bx^2-5x+c\\le 0$的解集.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260114-一元二次不等式的求解(3)" ], "genre": "解答题", "ans": "$b=-3$, $c=2$, 解集为$(-\\infty,-2]\\cup [\\dfrac{1}{3},+\\infty)$", @@ -327566,7 +328003,8 @@ "content": "已知关于$x$的一元二次方程$4x^2+(m-2)x+m-5=0$有两个负实根, 求实数$m$的取值范围.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260114-一元二次不等式的求解(3)" ], "genre": "解答题", "ans": "$(5,+\\infty)$", @@ -327600,7 +328038,8 @@ "content": "设关于$x$的一元二次不等式$ax^2+bx+c>0$的解集为$(\\alpha ,\\beta)$, 其中$0<\\alpha <\\beta$, 求关于$x$的不等式$cx^2+bx+a<0$的解集.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260114-一元二次不等式的求解(3)" ], "genre": "解答题", "ans": "$(-\\infty,\\dfrac{1}{\\beta})\\cup (\\dfrac{1}{\\alpha},+\\infty)$", @@ -327644,7 +328083,8 @@ "content": "解分式不等式$\\dfrac{x+3}{4-x}>0$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260115-分式不等式的求解" ], "genre": "解答题", "ans": "$(-3,4)$", @@ -327666,7 +328106,8 @@ "content": "解不等式$\\dfrac{5x+3}{x-1}\\le 3$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260115-分式不等式的求解" ], "genre": "解答题", "ans": "$[-3,1)$", @@ -327688,7 +328129,8 @@ "content": "解不等式$\\dfrac{x+5}{x^2+2x+3}\\le 1$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260115-分式不等式的求解" ], "genre": "解答题", "ans": "$(-\\infty,-2]\\cup [1,+\\infty)$", @@ -327710,7 +328152,8 @@ "content": "某服装公司生产的衬衫每件定价$80$元, 在某城市年销售$8$万件.现该公司计划在该市招收代理商来销售衬衫, 以降低管理和营销成本.已知代理商要收取的代理费为总销售金额的$r\\%$(即每$100$元销售额收取$r$元), 为确保单件衬衫的利润保持不变, 服装公司将每件衬衫的价格提高到$\\dfrac{80}{1-r\\%}$元, 但提价后每年的销量会减少$0.62r$万件.求$r$的取值范围, 以确保代理商每年收取的代理费不少于$16$万元.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260115-分式不等式的求解" ], "genre": "解答题", "ans": "$[3.1,100]$", @@ -327754,7 +328197,8 @@ "content": "设$a\\in \\mathbf{R}$, 且$a\\ne 1$, 比较$\\dfrac{a+2}{a-1}$与$-1$的大小.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260115-分式不等式的求解" ], "genre": "解答题", "ans": "当$a<-\\dfrac{1}{2}$或$a>1$时, $\\dfrac{a+2}{a-1}>-1$; 当$a=-\\dfrac{1}{2}$时, $\\dfrac{a+2}{a-1}=-1$; 当$-\\dfrac{1}{2}x$;\\\\\n(2) 解不等式$|x-2|\\le \\dfrac{x+1}2$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260116-含绝对值不等式的求解" ], "genre": "解答题", "ans": "(1) $(-\\infty,\\dfrac{1}{3})\\cup (1,+\\infty)$; (2) $[1,5]$", @@ -327844,7 +328291,8 @@ "content": "解不等式$|x-3|+|x-5|<4$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260116-含绝对值不等式的求解" ], "genre": "解答题", "ans": "$(2,6)$", @@ -327912,7 +328360,8 @@ "content": "已知$x>0$, 求证: $x+\\dfrac 1x\\ge 2$, 并指出等号成立的条件.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260117-平均值不等式及其应用(1)" ], "genre": "解答题", "ans": "证明略, 等号成立当且仅当$a=1$", @@ -327936,7 +328385,8 @@ "content": "已知$ab>0$, 求证: $\\dfrac ba+\\dfrac ab\\ge 2$, 并指出等号成立的条件.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260117-平均值不等式及其应用(1)" ], "genre": "解答题", "ans": "证明略, 等号当且仅当$a=b(\\ne 0)$", @@ -327982,7 +328432,8 @@ "content": "判断下列结论是否正确:\\\\\n(1) 对任意的正整数$n$, 不等式$x^n+\\dfrac 1{x^n}\\ge 2$当$x\\ne 0$时总是成立的;\\\\\n(2) 对任意的实数$x$, 不等式$\\sqrt {x^2+2}+\\dfrac 1{\\sqrt {x^2+2}}\\ge 2$恒成立.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260117-平均值不等式及其应用(1)" ], "genre": "解答题", "ans": "(1) 如$x=-1$, $n=3$, 结论不正确;\\\\\n(2) 结论正确.", @@ -328004,7 +328455,8 @@ "content": "若$a>0$, $b>0$, $a+b=2$, 则下列不等式对一切满足条件的$a$、$b$恒成立的是\\blank{50}(写出所有恒成立的不等式的编号).\\\\\n\\textcircled{1} $ab\\le 1$; \\textcircled{2} $\\sqrt a+\\sqrt b\\le \\sqrt 2$; \\textcircled{3} $a^2+b^2\\ge 2$; \\textcircled{4} $\\dfrac 1a+\\dfrac 1b\\ge 2$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260117-平均值不等式及其应用(1)" ], "genre": "填空题", "ans": "\\textcircled{1}\\textcircled{3}\\textcircled{4}", @@ -328026,7 +328478,8 @@ "content": "设$a$、$b$为正数, 且$a+2b=1$, 比较$ab$的值与$\\dfrac 18$的大小.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260118-平均值不等式及其应用(2)" ], "genre": "解答题", "ans": "当$a=\\dfrac{1}{2}$时, $ab=\\dfrac{1}{8}$; 当$a\\ne \\dfrac{1}{2}$时, $ab<\\dfrac{1}{8}$", @@ -328048,7 +328501,8 @@ "content": "证明: (1) 在周长为常数的所有矩形中, 正方形的面积最大;\\\\\n(2) 在面积相同的所有矩形中, 正方形的周长最小.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260118-平均值不等式及其应用(2)" ], "genre": "解答题", "ans": "(1) 证明略; (2) 证明略", @@ -328070,7 +328524,8 @@ "content": "某新建居民小区欲建一面积为$700\\text{m}^2$的矩形绿地, 并在绿地四周铺设人行道, 设计要求绿地外南北两侧人行道宽$3\\text{m}$, 东西两侧人行道宽$4\\text{m}$, 如图所示(图中单位: $\\text{m}$), 问如何设计绿地的边长, 才能使人行道的占地面积最小. (结果精确到$0.1\\text{m}$)\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (-2,-1.5) rectangle (2,1.5) (-1.2,-0.9) rectangle (1.2,0.9);\n\\draw (0,0) node {绿地};\n\\draw (0,-1.5) node [below] {南} (0,1.5) node [above] {北};\n\\draw [<->] (0,0.9) -- (0,1.5) node [midway,right] {$3$};\n\\draw [<->] (0,-0.9) -- (0,-1.5) node [midway,right] {$3$};\n\\draw [<->] (1.2,0) -- (2,0) node [midway,above] {$4$};\n\\draw [<->] (-1.2,0) -- (-2,0) node [midway,above] {$4$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260118-平均值不等式及其应用(2)" ], "genre": "解答题", "ans": "东西向边长设计为约$30.6$米($20\\sqrt{\\dfrac{7}{3}}$米)", @@ -328092,7 +328547,8 @@ "content": "如果正数$a,b,c,d$满足$a+b=cd=4$, 那么\\bracket{20}.\n\\onech{$ab\\le c+d$, 且等号成立时$a,b,c,d$的取值唯一}{$ab\\ge c+d$, 且等号成立时$a,b,c,d$的取值唯一}{$ab\\le c+d$, 且等号成立时$a,b,c,d$的取值不唯一}{$ab\\ge c+d$, 且等号成立时$a,b,c,d$的取值不唯一}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260118-平均值不等式及其应用(2)" ], "genre": "选择题", "ans": "A", @@ -328152,7 +328608,8 @@ "content": "已知$a$、$b$为实数, 求证: $|a+b|+|a-b|\\ge 2|a|$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260119-三角不等式" ], "genre": "解答题", "ans": "证明略", @@ -328174,7 +328631,8 @@ "content": "已知为$a$、$b$为实数, 求证: $|a|-|b|\\le|a-b|$, 并指出等号成立的条件.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260119-三角不等式" ], "genre": "解答题", "ans": "证明略, 等号成立当且仅当$b(a-b)\\ge 0$", @@ -328198,7 +328656,8 @@ "content": "证明: $|x-3|+|x-5|\\ge 2$对所有实数$x$恒成立, 并求等号成立时的$x$取值范围.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260119-三角不等式" ], "genre": "解答题", "ans": "证明略, 等号成立时$x$的范围为$[3,5]$", @@ -328267,7 +328726,8 @@ "content": "(1) 求$-\\dfrac 1{32}$的$5$次方根;\\\\\n(2) 求$81$的$4$次方根.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260120-指数幂的拓展(1)" ], "genre": "解答题", "ans": "(1) $-\\dfrac{1}{2}$; (2) $\\pm 3$", @@ -328289,7 +328749,8 @@ "content": "求下列各根式的值:\\\\\n(1) $\\sqrt [5]{(-2)^5}$;\\\\\n(2) $\\sqrt [6]{(-8)^2}$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260120-指数幂的拓展(1)" ], "genre": "解答题", "ans": "(1) $-2$; (2) $2$", @@ -328357,7 +328818,8 @@ "content": "求下列各式的值:\\\\\n(1) $8^{\\frac 23}$;\\\\\n(2) $(\\dfrac{81}{625})^{-\\frac 34}$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260121-指数幂的拓展(2)" ], "genre": "解答题", "ans": "(1) $4$, (2) $\\dfrac{125}{27}$", @@ -328379,7 +328841,8 @@ "content": "用有理数指数幂的形式表示下列各式(其中$a>0)$:\\\\\n(1) $\\sqrt [3]{a^2}$;\\\\\n(2) $a^3\\cdot \\sqrt [4]{a^3}$;\\\\ \n(3) $\\sqrt {a\\sqrt a}$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260121-指数幂的拓展(2)" ], "genre": "解答题", "ans": "(1) $a^{\\frac{2}{3}}$; (2) $a^{\\frac{15}{4}}$; (3) $a^{\\frac{3}{4}}$", @@ -328401,7 +328864,8 @@ "content": "化简下列各式:\\\\\n(1) $(x^{\\frac{\\sqrt 3}2})^{\\sqrt 3}\\cdot \\sqrt x$(其中$x>0$);\\\\ \n(2) $\\dfrac{(a^{\\frac 23}b^{\\frac 12})\\cdot (-3a^{\\frac 12}b^{\\frac 13})}{\\dfrac 13a^{\\frac 16}b^{\\frac 56}}$(其中$a>0,b>0$).", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260121-指数幂的拓展(2)" ], "genre": "解答题", "ans": "(1) $x^2$; (2) $-9a$", @@ -328445,7 +328909,8 @@ "content": "已知$a-a^{-1}=1$, 则$a^{12}+a^{-12}$的值为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260120-指数幂的拓展(1)" ], "genre": "填空题", "ans": "$322$", @@ -328467,7 +328932,8 @@ "content": "求下列各式的值:\\\\\n(1) $\\log_28$;\\\\\n(2) $\\log_2\\sqrt 2$;\\\\\n(3) $\\log_{10}0.00001$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260122-对数的定义" ], "genre": "解答题", "ans": "(1) $3$; (2) $\\dfrac{1}{2}$; (3) $-5$", @@ -328489,7 +328955,8 @@ "content": "求下列各式中$x$的值:\\\\\n(1) $\\log_2x=-1$;\\\\\n(2) $\\log_{\\frac 12}x=3$;\\\\ \n(3) $\\ln x=-1$;\\\\\n(4) $\\log_x8=6$;\\\\ \n(5) $-\\ln \\mathrm{e}^2=x$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260122-对数的定义" ], "genre": "解答题", "ans": "(1) $\\dfrac{1}{2}$; (2) $\\dfrac{1}{8}$; (3) $\\dfrac{1}{\\mathrm{e}}$; (4) $\\sqrt{2}$; (5) $-2$", @@ -328511,7 +328978,8 @@ "content": "求下列各式中$x$的取值范围:\\\\\n(1) $\\log_a(1-x^2)$($a>0$且$a\\ne 1$);\\\\\n(2) $\\log_{(1+x)}(1-x)$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260122-对数的定义" ], "genre": "解答题", "ans": "(1) $(-1,1)$; (2) $(-1,0)\\cup (0,1)$", @@ -328535,7 +329003,8 @@ "content": "求下列各式的值:\\\\\n(1) $3^{\\log_312}$;\\\\\n(2) $4^{\\log_23}$;\\\\ \n(3) $2^{\\lg 7}\\cdot 5^{\\lg 7}$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260122-对数的定义" ], "genre": "解答题", "ans": "(1) $12$; (2) $9$; (3) $7$", @@ -328601,7 +329070,8 @@ "content": "求下列各式的值:\\\\\n(1) $\\log_3(9^4\\times 3^2)$;\\\\ \n(2) $\\log_3\\sqrt[5]9$;\\\\ \n(3) $2\\log_510+\\log_53-\\log_512$;\\\\ \n(4) $\\lg 2+\\lg 2\\times \\lg 5+(\\lg 5)^2$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260123-对数的运算性质" ], "genre": "解答题", "ans": "(1) $10$; (2) $\\dfrac{2}{5}$; (3) $2$; (4) $1$", @@ -328623,7 +329093,8 @@ "content": "已知$\\log_52=a$, $5^b=3$, 用$a$及$b$表示$\\log_512$和$\\log_5\\dfrac{81}{100}$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260123-对数的运算性质" ], "genre": "解答题", "ans": "$4b-2a-2$", @@ -328645,7 +329116,8 @@ "content": "在有声世界, 声强级是表示声强度相对大小的指标. 其值$y$[单位: $\\text{dB}$(分贝)]定义为$y=10\\lg \\dfrac I{I_0}$. 其中, $I$为声场中某点的声强度, 其单位为$\\text{W}/\\text{m}^2$(瓦$/$平方米), $I_0=10^{-12}\\text{W}/\\text{m}^2$为基准值.\\\\\n(1)如果$I=10\\text{W}/\\text{m}^2$, 求相应的声强级;\\\\\n(2)声强级为$60\\text{dB}$时的声强度$I_{60}$是声强级为$50\\text{dB}$时的声强度$I_{50}$的多少倍?", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260123-对数的运算性质" ], "genre": "解答题", "ans": "(1) $130\\text{dB}$; (2) $10$倍", @@ -328711,7 +329183,8 @@ "content": "求下列各式的值:\\\\\n(1) $\\log_225\\times \\log_34\\times \\log_59$;\\\\\n(2) $\\dfrac 1{\\log_23}+\\dfrac{\\lg 13.5}{\\lg 3}$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260124-对数的换底" ], "genre": "解答题", "ans": "(1) $8$; (2) $3$", @@ -328733,7 +329206,8 @@ "content": "设$a>0$, $a\\ne 1$, 且$N>0$. 求证: 若$m\\ne 0$, 则$\\log_{a^m}N^n=\\dfrac nm\\log_aN$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260124-对数的换底" ], "genre": "解答题", "ans": "证明略", @@ -328755,7 +329229,8 @@ "content": "(1) 已知$\\lg 2=a$, $\\lg 6=b$, 试用$a$及$b$表示$\\log_26$与$\\log_{12}15$;\\\\\n(2) 已知$\\log_{14}2=a$, 试用$a$表示$\\log_{49}16$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260124-对数的换底" ], "genre": "解答题", "ans": "(1) $\\log_2 6=\\dfrac{b}{a}$, $log_{12}15=\\dfrac{b+1-2a}{a+b}$; (2) $\\log_{49}16=\\dfrac{2a}{1-a}$", @@ -328821,7 +329296,8 @@ "content": "已知非零实数$a,b,c$满足$3^a=4^b=6^c$, 求证: $\\dfrac 2a+\\dfrac 1b=\\dfrac 2c$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260124-对数的换底" ], "genre": "解答题", "ans": "证明略", @@ -328854,7 +329330,8 @@ "content": "写出幂函数$y=x^{\\frac 12}$的定义域, 并作出它的大致图像.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260125-幂函数的定义与图像" ], "genre": "解答题", "ans": "定义域为$[0,+\\infty)$, 大致图像为\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0:2, samples = 100] plot ({\\x*\\x},\\x);\n\\end{tikzpicture}", @@ -328935,7 +329412,8 @@ "content": "下列幂函数中, 其图像不经过原点的有\\blank{50}.(请填入全部正确的序号)\\\\\n\\textcircled{1} $y=x^4$; \\textcircled{2} $y=x^{-\\frac 14}$; \\textcircled{3} $y=x^{\\frac 52}$; \\textcircled{4} $y=x^{-3}$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260125-幂函数的定义与图像" ], "genre": "填空题", "ans": "\\textcircled{2}\\textcircled{4}", @@ -328970,7 +329448,8 @@ "content": "已知幂函数$y=x^{n^2-2n-3}$($n$为正整数)的图像关于原点成中心对称, 且与两坐标轴都无公共点, 求$n$的值.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260125-幂函数的定义与图像" ], "genre": "解答题", "ans": "$2$", @@ -329005,7 +329484,8 @@ "content": "比较下列各题中两个数的大小:\\\\\n(1) $2.5^{-2}$与$1.8^{-2}$;\\\\\n(2) $1.32^{\\frac 45}$与$(-\\sqrt 2)^{\\frac 45}$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260126-幂函数的性质" ], "genre": "解答题", "ans": "(1) $2.5^{-2}<1.8^{-2}$; (2) $1.32^{\\frac{4}{5}}<(-\\sqrt{2})^{\\frac{4}{5}}$", @@ -329027,7 +329507,8 @@ "content": "已知函数$y=\\dfrac 1x$和$y=\\dfrac 1{x-2}$, 说明这两个函数图像之间的关系, 并在同一平面直角坐标系中作出它们的大致图像.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260126-幂函数的性质" ], "genre": "解答题", "ans": "$y=\\dfrac{1}{x}$的图像向右平移$2$个单位后得到$y=\\dfrac{1}{x-2}$的图像, 理由略, 大致图像为\\\\\n\\begin{tikzpicture}[>=latex, scale = 0.25]\n\\draw [->] (-8,0) -- (8,0) node [below] {$x$};\n\\draw [->] (0,-8) -- (0,8) node [left] {$y$};\n\\draw (0,0) node [above left] {$O$};\n\\draw [domain = 0.125:8, samples = 100, thin] plot (\\x,1/\\x) plot (-\\x,-1/\\x);\n\\draw [domain = 0.125:6, samples = 100, thick] plot (\\x+2,1/\\x);\n\\draw [domain = 0.125:10, samples = 100, thick] plot (-\\x+2,-1/\\x);\n\\end{tikzpicture}", @@ -329051,7 +329532,8 @@ "content": "已知函数$y=\\dfrac 1{x-2}$和$y=\\dfrac{x-1}{x-2}$, 说明这两个函数图像之间的关系, 并在同一平面直角坐标系中作出它们的大致图像.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260126-幂函数的性质" ], "genre": "解答题", "ans": "$y=\\dfrac{1}{x}$的图像向上平移$1$个单位后得到$y=\\dfrac{x-1}{x-2}$的图像, 理由略, 大致图像为\\\\\n\\begin{tikzpicture}[>=latex, scale = 0.25]\n\\draw [->] (-8,0) -- (8,0) node [below] {$x$};\n\\draw [->] (0,-8) -- (0,8) node [left] {$y$};\n\\draw (0,0) node [above left] {$O$};\n\\draw [domain = 0.125:6, samples = 100, thin] plot (\\x+2,1/\\x);\n\\draw [domain = 0.125:10, samples = 100, thin] plot (-\\x+2,-1/\\x);\n\\draw [domain = {15/7}:8, samples = 100, thick] plot (\\x,{(\\x-1)/(\\x-2)});\n\\draw [domain = -8:{17/9}, samples = 100, thick] plot (\\x,{(\\x-1)/(\\x-2)});\n\\end{tikzpicture}", @@ -329075,7 +329557,8 @@ "content": "若函数$y=\\dfrac x{x-m}$($m$为实常数)的图像先向下平移$\\text1$个单位, 再左平移$\\text1$个单位后, 能得到函数$y=\\dfrac 1x$的图像, 则$m=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260126-幂函数的性质" ], "genre": "填空题", "ans": "$1$", @@ -329119,7 +329602,8 @@ "content": "若指数函数$y=a^x$($a>0$且$a\\ne 1$)的图像经过点$(2,9)$, 求该指数函数的表达式.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260127-指数函数的定义与图像" ], "genre": "解答题", "ans": "$y=3^x$", @@ -329143,7 +329627,8 @@ "content": "分别作出指数函数$y=2^x$和$y=3^x$的大致图像.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260127-指数函数的定义与图像" ], "genre": "解答题", "ans": "\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,5.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -3:{ln(5.5)/ln(2)}, samples = 100] plot (\\x,{exp(\\x*ln(2))}) node [right] {$y=2^x$};\n\\draw [domain = -3:{ln(5.5)/ln(3)}, samples = 100] plot (\\x,{exp(\\x*ln(3))}) node [above] {$y=3^x$};\n\\end{tikzpicture}", @@ -329165,7 +329650,8 @@ "content": "作出指数函数$y=(\\dfrac 12)^x$的大致图像.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260127-指数函数的定义与图像" ], "genre": "解答题", "ans": "\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,5.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = {-ln(5.5)/ln(2)}:3, samples = 100] plot (\\x,{exp(\\x*ln(0.5))});\n\\end{tikzpicture}", @@ -329187,7 +329673,8 @@ "content": "已知指数函数图像的一部分如下图所示, 求该指数函数的表达式. \n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.4]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,8) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain = -1.8:3, samples = 100] plot (\\x,{pow(1/3,\\x)});\n\\draw [dashed] (0,3) -- (-1,3) -- (-1,0);\n\\draw (-1,0) node [below] {$-1$};\n\\draw (0,3) node [right] {$3$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260127-指数函数的定义与图像" ], "genre": "解答题", "ans": "$y=(\\dfrac{1}{3})^x$", @@ -329209,7 +329696,8 @@ "content": "在平面直角坐标系中作出函数$y=-4^x$的大致图像.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260127-指数函数的定义与图像" ], "genre": "解答题", "ans": "\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-5.5) -- (0,0.5) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain = -3:{ln(5.5)/ln(4)}, samples = 100] plot (\\x,{-exp(\\x*ln(4))});\n\\end{tikzpicture}", @@ -329231,7 +329719,8 @@ "content": "利用指数函数的性质, 比较下列各题中两个数的大小:\\\\\n(1) $1.7^{2.5}$与$1.7^{3}$;\\\\\n(2) $(\\frac{3}{4})^{\\frac{1}{6}}$与$(\\frac{4}{3})^{-\\frac{1}{5}}$;\\\\ \n(3) $a^{\\frac 12}$与$a^{\\frac 13}$($a>0$且$a\\ne 1$).", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260128-指数函数的性质(1)" ], "genre": "解答题", "ans": "(1) $1.7^{2.5}<1.7^3$; (2) $(\\frac{3}{4})^{\\frac{1}{6}}>(\\frac{4}{3})^{-\\frac{1}{5}}$; (3) 当$a>1$时, $a^{\\frac 12}>a^{\\frac 13}$; 当$0\\dfrac 1{27}$;\\\\\n(2) $a^{x^2-2x+3}>a^6$($01$)在区间$[1,2]$上的最大值比最小值大$\\dfrac a3$, 求实数$a$的值.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260128-指数函数的性质(1)" ], "genre": "解答题", "ans": "$\\dfrac{4}{3}$", @@ -329345,7 +329837,8 @@ "content": "统计资料显示: 某外来入侵物种现有种群数量为$k$, 若有理想的外部环境条件, 该物种的年平均增长率约为$20\\%$.试建立该物种的种群数量增长模型, 并预测$30$年后该物种的种群数量约为现有种群数量的多少倍. (结果精确到个位)", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260129-指数函数的性质(2)" ], "genre": "解答题", "ans": "$30$年后该物种的种群数量约为现有种群数量的$237$倍", @@ -329367,7 +329860,8 @@ "content": "当生物死亡后, 它机体内原有的碳$14$含量会按确定的比率衰减(称为衰减率), 大约每经过$5730$年衰减为原来的一半, 这个时间称为``半衰期''.假设死亡生物体内碳$14$的年衰减率为$p$, 将刚死亡的生物体内碳$14$含量看成$1$个单位.求$p$的值, 并按照上述变化规律, 写出生物体内碳$14$含量与死亡年数之间的关系式.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260129-指数函数的性质(2)" ], "genre": "解答题", "ans": "$y=(\\dfrac{1}{2})^{\\frac{x}{5730}}$", @@ -329411,7 +329905,8 @@ "content": "设$a>0$且$a\\ne 1$, 若函数$y=a^{2x}+2a^x-1$在区间$[-1,1]$上的最大值为14, 则$a=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260129-指数函数的性质(2)" ], "genre": "填空题", "ans": "$3$或$\\dfrac{1}{3}$", @@ -329455,7 +329950,8 @@ "content": "求下列函数的定义域:\\\\\n(1) $y=\\log_2(x-1)$;\\\\\n(2) $y=\\log_a(x^2-4x-5)$, 其中常数$a>0,a\\ne 1$;\\\\\n(3) $y=\\ln \\dfrac{1-2x}{x+1}$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260130-对数函数的定义和图像" ], "genre": "解答题", "ans": "(1) $(1,+\\infty)$; (2) $(-\\infty,-1)\\cup (5,+\\infty)$; (3) $(-1,\\dfrac{1}{2})$", @@ -329477,7 +329973,8 @@ "content": "在同一坐标系中, 作出函数$y=\\log_{\\frac 12}x$与$y=\\log_{\\frac 13}x$的大致图像.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260130-对数函数的定义和图像" ], "genre": "解答题", "ans": "\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-0.5,0) -- (5.5,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = {1/8}:5, samples = 100] plot (\\x,{ln(\\x)/ln(1/2)}) node [right] {$y=\\log_{\\frac{1}{2}} x$};\n\\draw [domain = {1/27}:5, samples = 100] plot (\\x,{ln(\\x)/ln(1/3)}) node [right] {$y=\\log_{\\frac{1}{3}} x$};\n\\end{tikzpicture}", @@ -329499,7 +329996,8 @@ "content": "观察函数$y=\\log_2x$和$y=\\log_{\\frac 12}x$的图像, 判断这两个函数的图像是否关于$x$轴对称?", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260130-对数函数的定义和图像" ], "genre": "解答题", "ans": "是的, 理由略", @@ -329543,7 +330041,8 @@ "content": "$a,b,c$是图中三个对数函数的底数, 它们之间的大小关系是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.5,0) -- (3,0) 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 = 0.5:2.5] plot (\\x,{ln(\\x)/ln(2)}) node [right] {$y=\\log_c x$};\n\\draw [domain = 0.5:2.5] plot (\\x,{ln(\\x)/ln(0.53)}) node [right] {$y=\\log_a x$};\n\\draw [domain = 0.5:2.5] plot (\\x,{ln(\\x)/ln(0.3)}) node [right] {$y=\\log_b x$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$c>a>b$}{$c>b>a$}{$a>b>c$}{$b>a>c$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260130-对数函数的定义和图像" ], "genre": "选择题", "ans": "A", @@ -329587,7 +330086,8 @@ "content": "利用对数函数的单调性, 比较下列各题中两个对数的大小:\\\\\n(1) $\\log_25$与$\\log_26$;\\\\\n(2) $\\log_a0.1$与$\\log_a0.2$(其中常数$a>0$, $a\\ne 1$);\\\\\n(3) $\\log_57$与$\\log_67$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260131-对数函数的性质(1)" ], "genre": "解答题", "ans": "(1) $\\log_25<\\log_26$; (2) 当$a>1$时, $\\log_a0.1<\\log_a0.2$; 当$0\\log_67$", @@ -329609,7 +330109,8 @@ "content": "比较$99^{89}$与$89^{99}$的大小关系.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260131-对数函数的性质(1)" ], "genre": "解答题", "ans": "$99^{89}<89^{99}$", @@ -329653,7 +330154,8 @@ "content": "已知$\\log_a\\dfrac 34<1$, 那么$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260131-对数函数的性质(1)" ], "genre": "填空题", "ans": "$(0,\\dfrac{3}{4})\\cup (1,+\\infty)$", @@ -329697,7 +330199,8 @@ "content": "试利用对数函数的单调性估算对数$\\log_23$的第一位小数的值.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260132-对数函数的性质(2)" ], "genre": "解答题", "ans": "$5$", @@ -329721,7 +330224,8 @@ "content": "如果不考虑空气阻力, 火箭的最大速度$v$(单位: $\\text{km}/\\text{s}$)和燃料质量$M$(单位: $\\text{kg}$)、火箭(除燃料外)的质量$m_0$(单位: $\\text{kg}$)之间的关系是$v=2\\ln (1+\\dfrac M{m_0})$. 问当燃料质量至少是火箭质量的多少倍时, 火箭的最大速度才能超过$8\\text{km}/\\text{s}$? (结果精确到$0.1$倍)", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260132-对数函数的性质(2)" ], "genre": "解答题", "ans": "$53.6$倍", @@ -329743,7 +330247,8 @@ "content": "某人在银行存入$1$万元, 若年利率为$5\\%$, 且按年计复利的条件下, 经过多少年存款才能连本带利超过$5$万元? (结果精确到$1$年)", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260132-对数函数的性质(2)" ], "genre": "解答题", "ans": "经过$33$年", @@ -329765,7 +330270,8 @@ "content": "求函数$y=(\\log_2x)\\cdot (\\log_22x)$的最小值.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260132-对数函数的性质(2)" ], "genre": "解答题", "ans": "$-\\dfrac{1}{4}$", @@ -329787,7 +330293,8 @@ "content": "已知$x>0$, $x\\ne 1$, 比较$1+\\log_x3$与$2\\log_x2$的大小.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260132-对数函数的性质(2)" ], "genre": "解答题", "ans": "当$02\\log_x 2$; 当$1\\dfrac{4}{3}$时, $1+\\log_x 3>2\\log_x 2$", @@ -329831,7 +330338,8 @@ "content": "求下列函数的定义域:\\\\\n(1) $y=\\log_2(x+1)$;\\\\\n(2) $y=\\dfrac{\\sqrt {x+3}}{x-1}$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260133-函数" ], "genre": "解答题", "ans": "(1) $(-1,+\\infty)$; (2) $[-3,1)\\cup (1,+\\infty)$", @@ -329853,7 +330361,8 @@ "content": "判断下列函数与函数$y=x$是否相同, 并说明理由:\\\\\n(1) $y=(\\sqrt x)^2$;\\\\\n(2) $y=\\ln \\mathrm{e}^x$;\\\\ \n(3) $y=\\sqrt [4]{x^4}$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260133-函数" ], "genre": "解答题", "ans": "(1) 不同, $-1$不在定义域中; (2) 是的, 定义域与对应关系均相同; (3) 不是, $x=-1$时, $y=1$", @@ -329875,7 +330384,9 @@ "content": "求函数$y=\\dfrac 1{2^x+1}$的值域.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260133-函数", + "V20260106-2026届高一寒假作业06" ], "genre": "解答题", "ans": "$(0,1)$", @@ -329950,7 +330461,8 @@ "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\\onech{$y=\\sqrt {x+1}\\cdot \\sqrt {x-1}$与$y=\\sqrt {(x+1)(x-1)}$}{$y=\\dfrac{x^2}{|x|}$与$y=|x|$}{$y=x^0$与$y=\\dfrac{x^2+1}{x^2+1}$}{$y=1$, $x\\in \\{1,2\\}$与$y=(x^2-5x+5)^2$, $x\\in \\{1,2\\}$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260133-函数" ], "genre": "选择题", "ans": "D", @@ -330036,7 +330548,8 @@ "content": "某车辆装配车间每$2\\text{h}$装配完成一辆车.按照计划, 该车间今天生产$8\\text{h}$.用解析法和图像法分别表示从当天开始生产的时刻起所经过的时间$x$(单位: $\\text{h}$)与装配完成的车辆数$y$(单位: 辆)之间的函数$y=f(x)$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260134-函数的表示方法" ], "genre": "解答题", "ans": "$y=\\begin{cases}0, & 0\\le x < 2,\\\\ 1, & 2\\le x<4, \\\\ 2, & 4\\le x<6, \\\\ 3, & 6\\le x<8,\\\\ 4, & x=8\\end{cases}$, 图像: \\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-1,0) -- (8.5,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,4.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j in {1/2,2/4,3/6,4/8}\n{\\draw [dashed] ({2*\\i},0) node [below] {$\\j$} --++ (0,\\i) -- (0,\\i) node [left] {$\\i$};};\n\\foreach \\i in {0,2,4,6}\n{\\draw [thick] (\\i,{\\i/2}) --++ (2,0);\n\\filldraw (\\i,{\\i/2}) circle (0.09);\n\\filldraw [fill = white] ({\\i+2},{\\i/2}) circle (0.09);};\n\\filldraw (8,4) circle (0.09);\n\\end{tikzpicture}", @@ -330102,7 +330615,8 @@ "content": "设函数$y=f(x)$的表达式为$f(x)=\\begin{cases} (x+1)^2, & x<1, \\\\ 4-x, & x\\ge 1, \\end{cases}$则不等式$f(x)\\ge 1$的解集为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260134-函数的表示方法" ], "genre": "填空题", "ans": "$(-\\infty,-2]\\cup [0,3]$", @@ -330146,7 +330660,8 @@ "content": "设函数$y=f(x)$的表达式为$f(x)=\\begin{cases} x^2+bx+c, & x\\le 0, \\\\ 2,& x>0 \\end{cases}$(其中$b,c\\in \\mathbf{R}$). 若$f(-4)=f(0)$, $f(-2)=-2$, 则关于$x$的方程$f(x)=x$解的个数为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260134-函数的表示方法" ], "genre": "填空题", "ans": "$3$", @@ -330181,7 +330696,8 @@ "content": "证明: 函数$y=2x^4-3x^2$是一个偶函数.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260135-函数的奇偶性(1)" ], "genre": "解答题", "ans": "证明略", @@ -330203,7 +330719,8 @@ "content": "证明: 函数$y=x^3-\\dfrac 1x$是一个奇函数.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260135-函数的奇偶性(1)" ], "genre": "解答题", "ans": "证明略", @@ -330225,7 +330742,8 @@ "content": "是否存在定义在$\\mathbf{R}$上的, 且既是奇函数又是偶函数的函数? 若存在, 求出所有满足此条件的函数; 若不存在, 说明理由.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260135-函数的奇偶性(1)" ], "genre": "解答题", "ans": "存在, $y=0$($x\\in \\mathbf{R}$)", @@ -330247,7 +330765,8 @@ "content": "已知$y=f(x)$是奇函数, $y=g(x)$是偶函数, 且$f(-1)+g(1)=3$, $f(1)+g(-1)=5$, 则$f(-1)=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260135-函数的奇偶性(1)" ], "genre": "填空题", "ans": "$-1$", @@ -330315,7 +330834,8 @@ "content": "判断函数$y=\\begin{cases} x(x+1), & x>0, \\\\ x(1-x), & x<0 \\end{cases}$的奇偶性, 并说明理由.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260136-函数的奇偶性(2)" ], "genre": "解答题", "ans": "奇函数, 理由略", @@ -330404,7 +330924,8 @@ "content": "证明: 函数$y=x^2-2x$在区间$(-\\infty ,1]$上是严格减函数.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260137-函数的单调性(1)" ], "genre": "解答题", "ans": "证明略", @@ -330426,7 +330947,8 @@ "content": "判断函数$y=\\log_2(3x+2)$在其定义域上的单调性, 并说明理由.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260137-函数的单调性(1)" ], "genre": "解答题", "ans": "是严格增函数, 证明略", @@ -330470,7 +330992,8 @@ "content": "函数$y=x^3-ax$在区间$[1,+\\infty)$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", + "G20260138-函数的单调性(2)" ], "genre": "解答题", "ans": "$(-\\infty,3]$", @@ -330492,7 +331015,8 @@ "content": "判断函数$y=x^2-2x$, $x\\in [-2,2]$的单调性, 并求出它的单调区间.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260138-函数的单调性(2)" ], "genre": "解答题", "ans": "在$[-2,1]$上是严格减函数, 在$[1,2]$上是严格增函数, 理由略", @@ -330514,7 +331038,8 @@ "content": "设$y=f(x)$是偶函数, 且它在区间$[-2,-1]$上是严格减函数, 判断它在区间$[1,2]$上的单调性, 并说明理由.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260138-函数的单调性(2)" ], "genre": "解答题", "ans": "在$[1,2]$上是严格增函数, 理由略", @@ -330646,7 +331171,8 @@ "content": "求函数$y=2x^2-3x+1$, $x\\in \\mathbf{R}$的最大值与最小值.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260139-函数的最值" ], "genre": "解答题", "ans": "最大值不存在, 最小值为$-\\dfrac{1}{8}$", @@ -330668,7 +331194,8 @@ "content": "求函数$y=\\dfrac 2x$, $x\\in [1,2]$的最大值与最小值.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260139-函数的最值" ], "genre": "解答题", "ans": "最大值为$2$, 最小值为$1$", @@ -330695,7 +331222,8 @@ "content": "已知$a<2$, 求函数$y=|x-1|$, $x\\in [a,2]$的最大值.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260139-函数的最值" ], "genre": "解答题", "ans": "$y_{\\max}=\\begin{cases}1-a, & a\\le 0, \\\\ 1, & 0\\le a<2\\end{cases}$", @@ -330754,7 +331282,8 @@ "content": "已知$t$为实常数, 求函数$y=x^2-2x+2$, $x\\in [t,t+1]$的最小值$g(t)$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260139-函数的最值" ], "genre": "解答题", "ans": "$g(t)=\\begin{cases}t^2+1, & t\\le 0, \\\\ 1, & 0\\le t\\le 1, \\\\ t^2-2t+2, & t\\ge 1\\end{cases}$", @@ -330776,7 +331305,8 @@ "content": "如图所示, 一个边长为$a,b$($a=latex]\n\\filldraw [pattern = north east lines] (0.8,0) rectangle (3,1.2);\n\\filldraw [pattern = north east lines] (0,1.2) rectangle (0.8,2);\n\\draw (0,0) rectangle (3,2);\n\\draw (-0.1,0) -- (-0.7,0) (-0.1,1.2) -- (-0.4,1.2) (-0.1,2) -- (-0.7,2);\n\\draw [<->] (-0.25,1.2) -- (-0.25,2) node [midway, fill = white] {\\rotatebox{90}{$x$}};\n\\draw [<->] (-0.55,0) -- (-0.55,2) node [midway, fill = white] {\\rotatebox{90}{$a$}};\n\\draw (0,2.1) -- (0,2.7) (0.8,2.1) -- (0.8,2.4) (3,2.1) -- (3,2.7);\n\\draw [<->] (0,2.25) -- (0.8,2.25) node [midway, fill = white] {$x$};\n\\draw [<->] (0,2.55) -- (3,2.55) node [midway, fill = white] {$b$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260140-函数关系的建立" ], "genre": "解答题", "ans": "$S=2x^2-(a+b)x+ab$, $x\\in (0,a)$", @@ -330798,7 +331328,8 @@ "content": "如图所示, 四边形$OABC$是平面直角坐标系中边长为1的正方形. 一直线$y=-x+t$($t\\in (0,2)$)与正方形$OABC$相交, 将正方形分为两个部分, 其中包含原点$O$的部分的面积记为$S$. 试将$S$表示为$t$的函数.\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] {$A$} coordinate (A) -- (2,2) node [above right] {$B$} coordinate (B) -- (0,2) node [left] {$C$} coordinate (C);\n\\draw (-0.5,1.4) -- (1.4,-0.5);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260140-函数关系的建立" ], "genre": "解答题", "ans": "$S=\\begin{cases}\\dfrac{1}{2}t^2, & 0=latex]\n\\filldraw [pattern = north east lines] (0,0) rectangle (5,0.2);\n\\draw (0.5,0) rectangle (4.5,-1.3);\n\\draw (2.5,0) -- (2.5,-1.3);\n\\draw (4.6,-1.3) -- (5,-1.3);\n\\draw [<->] (4.8,-1.3) -- (4.8,0) node [midway, fill = white] {\\rotatebox{90}{$x$}};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260140-函数关系的建立" ], "genre": "解答题", "ans": "当宽为$5\\text{m}$时, 才能使所建造的居室面积最大, 最大面积为$75\\text{m}^2$", @@ -330842,7 +331374,8 @@ "content": "如图, 某小区要建造一个直径为$16\\text{m}$的圆形喷水池, 并在池的周边靠近水面的位置安装一圈喷水头, 使喷出的水柱在离池中心水平距离$3\\text{m}$的地方达到最高高度$4\\text{m}$. 各方向喷来的水柱在池中心上方某一点汇合, 求该点离水面的高度.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.3]\n\\filldraw [pattern = north east lines] (-9,0) rectangle (9,-1);\n\\draw [domain = 0:8] plot (\\x,{(\\x-8)*(\\x+2)/25*(-4)});\n\\draw [domain = 0:8] plot (-\\x,{(\\x-8)*(\\x+2)/25*(-4)});\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260140-函数关系的建立" ], "genre": "解答题", "ans": "$2.56\\text{m}$", @@ -330864,7 +331397,8 @@ "content": "提高黄浦江上各大桥的通行能力, 可改善浦江两岸的交通状况. 在一般情况下, 大桥上的车流速度$v$(单位: 千米/小时)是车流密度$x$(单位: 辆/千米)的函数, 车流量$f(x)$(单位: 辆/小时)(单位时间内通过桥上某观测点的车辆数)满足$f(x)=xv$. 当车流密度$x\\in[20, 200]$时, 车流速度随着车流密度的增大而减小. 当桥上的车流密度达到或超过$200$辆/千米时, 车辆通行缓慢, 车流速度很小, 可将其视作为$0$.\\\\\n(1) 已知当车流密度不超过$20$辆/千米时, 车流速度为$60$千米/小时; 当$x\\in[20,200]$时, 车流速度$v$是车流密度$x$的一次函数.\\\\\n\\textcircled{1} 当$x\\in [0,200]$时, 试用解析法将$v$表示为$x$的函数;\\\\\n\\textcircled{2} 当车流密度$x$为多大时, 车流量可以达到最大, 并求出最大值(精确到$1$辆/小时);\\\\\n(2) 为减轻周边道路通行压力, 现规定当车流密度$x\\in [60,200]$时, 车流量始终不能超过$3000$辆/小时. 当车流密度$x\\in [60, 200]$时, 请参考教材例题或习题中出现过的函数, 写出一个符合要求的车流速度函数并说明其合理性.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260140-函数关系的建立" ], "genre": "解答题", "ans": "(1) \\textcircled{1} $f(x)=\\begin{cases}60x, & 02$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260141-用函数观点解方程与不等式" ], "genre": "解答题", "ans": "$(1,+\\infty)$", @@ -330952,7 +331488,8 @@ "content": "已知$a$是实常数, 设关于$x$的不等式$\\dfrac 1x\\ge x^2+a$的解集为$A$, 若$A$与区间$[1,+\\infty)$的交集非空, 求$a$的取值范围.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260141-用函数观点解方程与不等式" ], "genre": "解答题", "ans": "$(-\\infty,0]$", @@ -330974,7 +331511,8 @@ "content": "如图所示, 在一块边长为$13\\text{cm}$的正方形金属薄片的四个角上都剪去一个边长为$x\\text{cm}$的小正方形, 做成一个容积是$140\\text{cm}^3$的无盖长方体盒子. 问: $x$是多少? (结果精确到$0.1\\text{cm}$)\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\filldraw [pattern = north east lines] (0,0) rectangle (1,1) (5,0) rectangle (4,1) (0,5) rectangle (1,4) (5,5) rectangle (4,4);\n\\draw (0,0) rectangle (5,5);\n\\draw [dashed] (1,1) rectangle (4,4);\n\\draw (-0.1,0) -- (-1.1,0) (-0.1,5) -- (-1.1,5) (-0.1,1) -- (-0.6,1) (-0.1,4) -- (-0.6,4);\n\\draw [<->] (-0.35,0) -- (-0.35,1) node [midway,fill = white] {\\rotatebox{90}{$x$}};\n\\draw [<->] (-0.35,4) -- (-0.35,5) node [midway,fill = white] {\\rotatebox{90}{$x$}};\n\\draw [<->] (-0.85,0) -- (-0.85,5) node [midway,fill = white] {\\rotatebox{90}{$13$}};\n\\draw (0,-0.1) -- (0,-1.1) (5,-0.1) -- (5,-1.1) (1,-0.1) -- (1,-0.6) (4,-0.1) -- (4,-0.6);\n\\draw [<->] (0,-0.35) -- (1,-0.35) node [midway,fill = white] {$x$};\n\\draw [<->] (4,-0.35) -- (5,-0.35) node [midway,fill = white] {$x$};\n\\draw [<->] (0,-0.85) -- (5,-0.85) node [midway,fill = white] {$13$};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\path (-2,-2,0);\n\\draw (0,0,0) coordinate (A);\n\\draw (3,0,0) coordinate (B);\n\\draw (3,0,-3) coordinate (C);\n\\draw (0,0,-3) coordinate (D);\n\\draw (A) ++ (0,1,0) coordinate (A1);\n\\draw (B) ++ (0,1,0) coordinate (B1);\n\\draw (C) ++ (0,1,0) coordinate (C1);\n\\draw (D) ++ (0,1,0) coordinate (D1);\n\\path [name path = up] (A1) -- (B1) -- (C1);\n\\path [name path = line1] (A) -- (D);\n\\path [name intersections = {of = up and line1, by = P}];\n\\path [name path = line2] (D) -- (C);\n\\path [name intersections = {of = up and line2, by = Q}];\n\\draw (P) -- (D) -- (Q);\n\\draw (A) -- (B) -- (C) (A) -- (A1) (B) -- (B1) (C) -- (C1) (A1) -- (B1) -- (C1) -- (D1) -- cycle (D) -- (D1);\n\\draw [dashed] (A) -- (P) (C) -- (Q);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260142-用二分法求函数的零点" ], "genre": "解答题", "ans": "$1.3$或$3.2$", @@ -330996,7 +331534,8 @@ "content": "已知函数$y=x^3+2x-99$在区间$(4, 5)$上有且仅有一个零点, 求该零点的近似值. (结果精确到$0.1$)", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260142-用二分法求函数的零点" ], "genre": "解答题", "ans": "$4.5$", @@ -331040,7 +331579,8 @@ "content": "求方程$0.8^x-1=\\ln x$的近似解. (结果精确到$0.1$)", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260142-用二分法求函数的零点" ], "genre": "解答题", "ans": "$0.8$", @@ -331084,7 +331624,8 @@ "content": "若$f(x)=1+\\log_3x$, 并设$y=f^{-1}(x)$是$y=f(x)$的反函数, 求$f^{-1}(2)$, $f^{-1}(a)$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260143-反函数的概念" ], "genre": "解答题", "ans": "$3$, $3^{a-1}$", @@ -331106,7 +331647,8 @@ "content": "求下列函数的反函数:\\\\\n(1) $y=4x+2$;\\\\ \n(2) $y=x^2+1$, $x\\in [1,3]$;\\\\ \n(3) $y=\\dfrac{3x+1}{4x+2}$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260143-反函数的概念" ], "genre": "解答题", "ans": "(1) $y=\\dfrac{x-2}{4}$, $x\\in \\mathbf{R}$; (2) $y=\\sqrt{x-1}$, $x\\in [2,10]$; (3) $y=\\dfrac{1-2x}{4x-3}$, $x\\in (-\\infty,\\dfrac{3}{4})\\cup (\\dfrac{3}{4},+\\infty)$", @@ -331172,7 +331714,8 @@ "content": "求函数$y=x^3$的反函数, 并在同一坐标系中作出函数$y=x^3$和它的反函数的图像.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260144-反函数的图像" ], "genre": "解答题", "ans": "$y=x^{\\frac{1}{3}}$, $x\\in \\mathbf{R}$, 图像: \\begin{tikzpicture}[>=latex, scale = 0.3]\n\\draw [->] (-8,0) -- (8,0) node [below] {$x$};\n\\draw [->] (0,-8) -- (0,8) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain = -2:2, thin] plot (\\x,{\\x*\\x*\\x}) node [above] {$y=x^3$};\n\\draw [domain = -2:2, ultra thick] plot ({\\x*\\x*\\x},\\x) node [right] {$y=x^{\\frac{1}{3}}$};\n\\end{tikzpicture}", @@ -331194,7 +331737,8 @@ "content": "已知函数$y=a^x+b$的图像经过点$(1,7)$, 而其反函数的图像经过点$(4,0)$, 求实数$a$、$b$的值.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260144-反函数的图像" ], "genre": "解答题", "ans": "$a=4$, $b=3$", @@ -331216,7 +331760,8 @@ "content": "已知常数$a>0$且$a\\ne 1$, 若函数$y=a^x$的反函数的图像经过点$(2,-1)$, 则$a=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260144-反函数的图像" ], "genre": "填空题", "ans": "$2$", @@ -331251,7 +331796,8 @@ "content": "已知函数$y=f(x)$的图像经过点$(1,2)$, 设$y=f(x+4)$的反函数为$y=g(x)$, 那么函数$y=g(x)$的图像必经过点\\bracket{20}. \n\\fourch{$(2,-3)$}{$(-3,2)$}{$(2,5)$}{$(5,2)$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260144-反函数的图像" ], "genre": "选择题", "ans": "A", @@ -380569,7 +381115,8 @@ "content": "若$02 x-1$;\\\\\n(2) 讨论函数$f(x)$的奇偶性, 并说明理由.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260105-2026届高一寒假作业05" ], "genre": "解答题", "ans": "", @@ -434478,7 +435028,8 @@ "content": "函数$y=\\dfrac{2^x}{2^x+1}$的值域为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260106-2026届高一寒假作业06" ], "genre": "填空题", "ans": "", @@ -447847,7 +448398,8 @@ "content": "数列$\\{a_n\\}$满足$S_n=2 n-a_n$.\\\\\n(1) 计算$a_1, a_2, a_3, a_4$并由此猜想通项$a_n$的表达式;\\\\\n(2) 用数学归纳法证明 (1) 中的猜想.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "E20260105-高一上学期测验卷05" ], "genre": "解答题", "ans": "(1) $a_1=1$, $a_2=\\dfrac{3}{2}$, $a_3=\\dfrac{7}{4}$, $a_4=\\dfrac{15}{7}$, 猜$a_n=2-2^{1-n}$; (2) 证明略", @@ -489715,7 +490267,8 @@ "content": "已知等差数列$-5,-9,-13, \\cdots$.\\\\\n(1) 求该等差数列的第$20$项;\\\\\n(2) $-401$ 是不是该等差数列的项? 如果是, 指明是第几项; 如果不是, 请说明理由.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260145-等差数列及其通项公式" ], "genre": "解答题", "ans": "(1) $-85$; (2) 是的, 第$100$项", @@ -489737,7 +490290,8 @@ "content": "假设体育场一角看台的座位从第$2$排起每一排都比前一排多相等数目的座位. 若第$3$排有$10$个座位, 第$9$排有$28$个座位, 则第$12$排有多少个座位?", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260145-等差数列及其通项公式" ], "genre": "解答题", "ans": "$37$个座位", @@ -489759,7 +490313,8 @@ "content": "已知$a_n=p n+q$是数列$\\{a_n\\}$的通项公式, 其中$p$和$q$均为常数. 试判断数列$\\{a_n\\}$是否为等差数列, 并证明你的结论.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260145-等差数列及其通项公式" ], "genre": "解答题", "ans": "是的, 证明略", @@ -489825,7 +490380,8 @@ "content": "设数列$\\{a_n\\}$为等差数列, 其前$n$项和为$S_n$.\\\\\n(1) 已知$a_1=50$, $a_8=15$, 求$S_8$;\\\\\n(2) 已知$a_1=0.7$, $a_2=1.5$, 求$S_7$;\\\\\n(3) 已知$a_4=7$, 求$S_7$.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260146-等差数列的前$n$项和" ], "genre": "解答题", "ans": "(1) $260$; (2) $21.7$; (3) $4.9$", @@ -489847,7 +490403,8 @@ "content": "已知等差数列$\\{a_n\\}$的前$10$项和$S_{10}=310$, 前$20$项和$S_{20}=1220$, 由此可以确定数列$\\{a_n\\}$前$30$项和$S_{30}$吗?", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260146-等差数列的前$n$项和" ], "genre": "解答题", "ans": "可以确定, $S_{30}=2730$", @@ -489869,7 +490426,8 @@ "content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n=n^2+2 n$.\\\\\n(1) 求数列$\\{a_n\\}$的通项公式;\\\\\n(2) 求证: 数列$\\{a_n\\}$是等差数列.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260146-等差数列的前$n$项和" ], "genre": "解答题", "ans": "(1) $a_n=2n+1$; (2) 证明略", @@ -489913,7 +490471,8 @@ "content": "设数列$\\{a_n\\}$为等比数列.\\\\\n(1) 已知$a_1=3$, 公比$q=-2$, 求$a_6$;\\\\\n(2) 已知$a_3=20$, $a_6=160$, 求$a_n$.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260147-等比数列及其通项公式" ], "genre": "解答题", "ans": "(1) $-96$; (2) $5\\cdot 2^{n-1}$", @@ -489935,7 +490494,8 @@ "content": "某种放射性物质不断衰变为其他物质, 设每经过一年剩余的这种放射性物质是年初的$84 \\%$. 这种放射性物质的半衰期约为多少? (结果精确到$1$年)", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260147-等比数列及其通项公式" ], "genre": "解答题", "ans": "约$4$年", @@ -489957,7 +490517,8 @@ "content": "已知$a, b, c$成等差数列, 其公差为$d$. 试证明$3^a, 3^b, 3^c$成等比数列, 并写出其公比.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260147-等比数列及其通项公式" ], "genre": "解答题", "ans": "证明略, 公比为$3^d$", @@ -489979,7 +490540,8 @@ "content": "已知正实数$a, b, c$成等比数列, 其公比为$q$. 试证明$\\lg a, \\lg b, \\lg c$成等差数列, 并写出其公差.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260147-等比数列及其通项公式" ], "genre": "解答题", "ans": "证明略, 公差为$\\lg q$", @@ -490045,7 +490607,8 @@ "content": "设数列$\\{a_n\\}$为等比数列, 其前$n$项和为$S_n$.\\\\\n(1) 已知$a_1=-4$, 公比$q=-\\dfrac{1}{2}$, 求$S_{10}$;\\\\\n(2)已知$a_1=27$, $a_n=\\dfrac{1}{243}$, 公比$q=-\\dfrac{1}{3}$, 求$S_n$.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260148-等比数列的前$n$项和(1)" ], "genre": "解答题", "ans": "(1) $\\dfrac{341}{128}$; (2) $\\dfrac{4921}{243}$", @@ -490067,7 +490630,8 @@ "content": "在等比数列$\\{a_n\\}$中, 其前$n$项和为$S_n$. 已知$S_3=\\dfrac{7}{2}$, $S_6=\\dfrac{63}{2}$, 求$a_n$.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260148-等比数列的前$n$项和(1)" ], "genre": "解答题", "ans": "$2^{n-2}$", @@ -490089,7 +490653,8 @@ "content": "已知数列$\\{a_n\\}$的前$n$项和$S_n=3^n+a$($a$是实数). 当常数$a$满足什么条件时, 数列$\\{a_n\\}$是等比数列?", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260148-等比数列的前$n$项和(1)" ], "genre": "解答题", "ans": "当且仅当$a=-1$时, 数列$\\{a_n\\}$是等比数列", @@ -490178,7 +490743,8 @@ "content": "化下列循环小数为分数:\\\\\n(1) $0 . \\dot{2} \\dot{9}$;\\\\\n(2) $0.4 \\dot{3} \\dot{1}$.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260149-等比数列的前$n$项和(2)" ], "genre": "解答题", "ans": "(1) $\\dfrac{29}{99}$; (2) $\\dfrac{427}{990}$", @@ -490200,7 +490766,8 @@ "content": "如图, 正方形$ABCD$的边长等于 1, 连接这个正方形各边的中点得到一个小正方形$A_1B_1C_1D_1$; 又连接正方形$A_1B_1C_1D_1$各边的中点得到一个更小的正方形$A_2B_2C_2D_2$; 如此无限继续下去. 求所有这些正方形的周长的和与面积的和.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, line cap = round, line join = round, scale = 0.6]\n\\draw (0, 0) rectangle (4, 4);\n\\draw (0, 2) -- (2, 4) -- (4, 2) -- (2, 0) -- cycle;\n\\draw (1, 3) -- (3, 3) -- (3, 1) -- (1, 1) -- cycle;\n\\draw (2, 3) -- (3, 2) -- (2, 1) -- (1, 2) -- cycle;\n\\draw (0, 4) node [above left] {$A$} (4, 4) node [above right] {$B$} (4, 0) node [below right] {$C$} (0, 0) node [below left] {$D$};\n\\draw (2, 4) node [above] {$A_1$} (4, 2) node [right] {$B_1$} (2, 0) node [below] {$C_1$} (0, 2) node [left] {$D_1$};\n\\draw (3, 3) node [above right] {$A_2$} (3, 1) node [below right] {$B_2$} (1, 1) node [below left] {$C_2$} (1, 3) node [above left] {$D_2$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260149-等比数列的前$n$项和(2)" ], "genre": "解答题", "ans": "周长的和为$8+4\\sqrt{2}$, 面积的和为$2$", @@ -490223,7 +490790,8 @@ "content": "计算$\\displaystyle\\sum_{i=1}^{+\\infty}(\\dfrac{1}{4})^{i-1}$.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260149-等比数列的前$n$项和(2)" ], "genre": "解答题", "ans": "$\\dfrac{4}{3}$", @@ -490333,7 +490901,8 @@ "content": "已知数列$\\{a_n\\}$的通项公式, 写出这些数列的前$5$项:\\\\\n(1) $a_n=\\dfrac{n-2}{n+1}$;\\\\\n(2) $a_n=1+(-\\dfrac{1}{2})^n$.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260150-数列的概念与性质" ], "genre": "解答题", "ans": "(1) $-\\dfrac{1}{2},0,\\dfrac{1}{4},\\dfrac{2}{5},\\dfrac{1}{2}$; (2) $\\dfrac{1}{2},\\dfrac{5}{4},\\dfrac{7}{8},\\dfrac{17}{16},\\dfrac{31}{32}$", @@ -490355,7 +490924,8 @@ "content": "给出数列$\\{a_n\\}$的下述通项公式, 判断这些数列是否为单调数列, 请说明理由.\\\\\n(1) $a_n=1+(\\dfrac{1}{2})^n$;\\\\\n(2) $a_n=n-\\dfrac{1}{n}$.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260150-数列的概念与性质" ], "genre": "解答题", "ans": "(1) 是的, 理由略; (2) 是的, 理由略", @@ -490377,7 +490947,8 @@ "content": "已知数列$\\{a_n\\}$的通项公式是$a_n=(n+1)(\\dfrac{9}{10})^{n-1}$. 试问: 该数列是否有最大项? 若有, 指出第几项最大; 若没有, 试说明理由.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260150-数列的概念与性质" ], "genre": "解答题", "ans": "有最大项, 第$8$和第$9$项最大", @@ -490443,7 +491014,8 @@ "content": "在平面上画$n$条直线, 假设其中任意$2$条直线都相交, 且任意$3$条直线都不共点. 设这$n$条直线将平面分成了$a_n$个部分.\\\\\n(1) 写出数列$\\{a_n\\}$的一个递推公式;\\\\\n(2) 写出数列$\\{a_n\\}$的一个通项公式.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260151-利用递推公式表示数列" ], "genre": "解答题", "ans": "(1) $a_n=a_{n-1}+n$; (2) $a_n=\\dfrac{n^2+n+2}{2}$", @@ -490465,7 +491037,8 @@ "content": "已知数列$\\{a_n\\}$满足$\\begin{cases}a_n=2 a_{n-1}+1(n \\geq 2), \\\\ a_1=1 .\\end{cases}$\\\\\n(1) 求证: 数列$\\{a_n+1\\}$为等比数列;\\\\\n(2) 求数列$\\{a_n\\}$的通项公式.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260151-利用递推公式表示数列" ], "genre": "解答题", "ans": "(1) 证明略; (2) $a_n=2^n-1$", @@ -490531,7 +491104,8 @@ "content": "用数学归纳法证明: $1+3+5+\\cdots+(2 n-1)=n^2$($n$为正整数).", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260152-数学归纳法" ], "genre": "解答题", "ans": "证明略", @@ -490553,7 +491127,8 @@ "content": "用数学归纳法证明: $1^3+2^3+3^3+\\cdots+n^3=[\\dfrac{n(n+1)}{2}]^2$($n$为正整数).", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260152-数学归纳法" ], "genre": "解答题", "ans": "证明略", @@ -490641,7 +491216,8 @@ "content": "已知数列$\\{a_n\\}$满足$\\begin{cases}a_{n+1}=a_n+\\dfrac{n}{a_n}, \\\\ a_1=1 .\\end{cases}$尝试通过计算数列$\\{a_n\\}$的前四项, 猜想数列$\\{a_n\\}$的通项公式, 并用数学归纳法加以证明.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260153-数学归纳法的应用" ], "genre": "解答题", "ans": "前四项依次为$1,2,3,4$, 猜通项公式为$a_n=n$, 证明略", @@ -490663,7 +491239,8 @@ "content": "是否存在常数$a$、$b$、$c$, 使等式$1^2+2^2+3^2+\\cdots+n^2=a n^3+b n^2+c n$对任意正整数$n$都成立?", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260153-数学归纳法的应用" ], "genre": "解答题", "ans": "$a=\\dfrac{1}{3}$, $b=\\dfrac{1}{2}$, $c=\\dfrac{1}{6}$, 证明略", @@ -490685,7 +491262,8 @@ "content": "设数列$\\{a_n\\}$, 其前$n$项和为$S_n$, 且$S_n+a_n=n$. 计算$a_1, a_2, a_3$; 根据计算的结果, 猜想数列$\\{a_n\\}$的通项公式, 并用数学归纳法加以证明.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260153-数学归纳法的应用" ], "genre": "解答题", "ans": "$a_1=\\dfrac{1}{2}$, $a_2=\\dfrac{3}{4}$, $a_3=\\dfrac{7}{8}$, 猜$a_n=1-\\dfrac{1}{2^n}$, 证明略", @@ -490839,7 +491417,8 @@ "content": "仿照计算$\\sqrt{2}$的巴比伦算法, 构造计算$\\sqrt{5}$的迭代算法的递推公式, 并选取初值$x_1=1$. 通过计算器操作, 写出该迭代序列$\\{x_n\\}$的前$5$项.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260154-用迭代序列求$\\sqrt{2}$的近似值" ], "genre": "解答题", "ans": "递推公式可以是$x_{n+1}=\\dfrac{1}{2}(x_n+\\dfrac{5}{x_n})$, 前$5$项依次为$1, 3, 2.3333, 2.2381, 2.2361$", @@ -512116,7 +512695,8 @@ "content": "设$k$为实数, 求以下关于$x$与$y$的二元一次方程组的解集.\\\\\n(1) $\\begin{cases} y=(3-2k)x-2, \\\\ y=k^2x-1; \\end{cases}$\\\\\n(2) $\\begin{cases} y=x+k^2+2k, \\\\ y=k^2x+3. \\end{cases}$", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260108-等式的性质与方程的解集" ], "genre": "4em", "ans": "(1) 当$k=-3$或$k=1$时, 解集为$\\varnothing$; 当$k\\ne -3$且$k\\ne 1$时, 解集为$\\{\\dfrac{1}{k^2+2k-3}\\}$;\\\\\n(2) 当$k=1$时, 解集为$\\mathbf{R}$; 当$k=-1$时, 解集为$\\varnothing$; 当$k\\ne \\pm 1$时, 解集为$\\{\\dfrac{k+3}{k+1}\\}$.", @@ -512168,7 +512748,8 @@ "content": "写出幂函数$y=x^3$的定义域, 说明其图像关于原点对称的理由, 并作出它的大致图像.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260125-幂函数的定义与图像" ], "genre": "4em", "ans": "定义域为$\\mathbf{R}$, 对称的理由略, 大致图像为\\begin{tikzpicture}[>=latex,scale = 0.2]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-8) -- (0,8) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -2:2, samples = 100] plot (\\x,{\\x*\\x*\\x});\n\\end{tikzpicture}", @@ -512196,7 +512777,8 @@ "content": "写出幂函数$y=x^{-\\frac 23}$的定义域, 说明其图像关于$y$轴对称的理由, 并作出它的大致图像.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260125-幂函数的定义与图像" ], "genre": "4em", "ans": "定义域为$\\{x|x\\in\\mathbf{R}, \\ x\\ne 0\\}$, 对称的理由略, 大致图像为\\begin{tikzpicture}[>=latex,scale = 0.5]\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 = 0.2:2, samples = 100] plot (\\x,{exp(-2/3*ln(\\x))}) plot (-\\x,{exp(-2/3*ln(\\x))});\n\\end{tikzpicture}", @@ -512271,7 +512853,8 @@ "objs": [], "tags": [ "第一单元", - "不等式" + "不等式", + "G20260110-不等式的性质(1)" ], "genre": "选择题", "ans": "B", @@ -512307,7 +512890,8 @@ "objs": [], "tags": [ "第一单元", - "不等式" + "不等式", + "G20260110-不等式的性质(1)" ], "genre": "填空题", "ans": "(1) $\\alpha-\\beta$的最大值为$0$, 最小值为$-\\pi$;\\\\\n(2) $2\\alpha-\\beta$的最大值为$2\\pi$, 最小值为$-\\pi$.", @@ -512343,7 +512927,8 @@ "objs": [], "tags": [ "第一单元", - "不等式" + "不等式", + "G20260110-不等式的性质(1)" ], "genre": "解答题", "ans": "$b-a>b>-a>a-b$", @@ -512379,7 +512964,8 @@ "objs": [], "tags": [ "第一单元", - "不等式" + "不等式", + "W20260104-高一上学期周末卷04" ], "genre": "", "ans": "A", @@ -512418,7 +513004,8 @@ "objs": [], "tags": [ "第一单元", - "不等式" + "不等式", + "G20260113-一元二次不等式的求解(2)" ], "genre": "解答题", "ans": "当$a<-\\dfrac{1}{4}$时, 解集为$\\mathbf{R}$; 当$a=-\\dfrac{1}{4}$时, 解集为$\\mathbf{R}$; 当$a>-\\dfrac{1}{4}$时, 解集为$(-\\infty,\\dfrac{-1-\\sqrt{1+4a}}{2}]\\cup [\\dfrac{-1+\\sqrt{1+4a}}{2},+\\infty)$", @@ -512464,7 +513051,8 @@ "objs": [], "tags": [ "第一单元", - "不等式" + "不等式", + "G20260114-一元二次不等式的求解(3)" ], "genre": "解答题", "ans": "(1) $(-\\infty,2)$; (2) $(6,+\\infty)$", @@ -512557,7 +513145,8 @@ "objs": [], "tags": [ "第一单元", - "不等式" + "不等式", + "G20260115-分式不等式的求解" ], "genre": "解答题", "ans": "$[-\\dfrac{1}{2},1)\\cup (1,\\dfrac{4}{3}]$", @@ -512616,7 +513205,8 @@ "objs": [], "tags": [ "第一单元", - "不等式" + "不等式", + "G20260115-分式不等式的求解" ], "genre": "解答题", "ans": "$a=4$, $b=2$", @@ -512652,7 +513242,8 @@ "objs": [], "tags": [ "第一单元", - "不等式" + "不等式", + "G20260115-分式不等式的求解" ], "genre": "解答题", "ans": "$(-\\infty,-7)\\cup (\\dfrac{4}{3},+\\infty)$; $(-\\infty,\\dfrac{1}{2}]\\cup (3,+\\infty)$", @@ -512688,7 +513279,8 @@ "objs": [], "tags": [ "第一单元", - "不等式" + "不等式", + "G20260116-含绝对值不等式的求解" ], "genre": "4em", "ans": "(1) $(-\\sqrt{5},-1)\\cup (1,\\sqrt{5})$; (2) $(\\dfrac{3}{2},2)\\cup (2,\\dfrac{5}{2})$; (3) $\\{1,2\\}$", @@ -512772,7 +513364,8 @@ "objs": [], "tags": [ "第一单元", - "不等式" + "不等式", + "G20260116-含绝对值不等式的求解" ], "genre": "解答题", "ans": "$(1,+\\infty)$", @@ -512807,7 +513400,8 @@ "objs": [], "tags": [ "第一单元", - "不等式" + "不等式", + "G20260117-平均值不等式及其应用(1)" ], "genre": "解答题", "ans": "证明略", @@ -512875,7 +513469,8 @@ "content": "数列$\\{a_n\\}$中, $a_1=1$, $n \\geq 2$时, $a_n, S_n, S_n-\\dfrac 12$成等比数列.\\\\\n(1) 证明: $\\{\\dfrac{1}{S_n}\\}$是等差数列;\\\\\n(2) 求数列$\\{a_n\\}$通项公式.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260151-利用递推公式表示数列" ], "genre": "4em", "ans": "(1) 证明略; (2) $a_n=\\begin{cases}1, & n=1,\\\\ -\\dfrac{2}{4n^2-8n+3}, & n\\ge 2\\end{cases}$", @@ -512915,7 +513510,8 @@ "K0410002X" ], "tags": [ - "第四单元" + "第四单元", + "G20260154-用迭代序列求$\\sqrt{2}$的近似值" ], "genre": "4em", "ans": "递推公式可以是$x_{n+1}=\\dfrac{3}{4}x_n+\\dfrac{1}{2x_n^3}$, 前$5$项依次为$1, 1.25, 1.1935, 1.1892, 1.1892$", @@ -523885,7 +524481,8 @@ "K0213007B" ], "tags": [ - "第二单元" + "第二单元", + "G20260130-对数函数的定义和图像" ], "genre": "", "ans": "C", @@ -523934,7 +524531,8 @@ "content": "证明指数函数 $y=a^x$ 与 $y=(\\dfrac{1}{a})^x$($a>0$ 且 $a \\neq 1$) 的图像关于 $y$ 轴对称.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260128-指数函数的性质(1)" ], "genre": "解答题", "ans": "证明略", @@ -523956,7 +524554,8 @@ "content": "已知 $f(x)=3^x$, $u, v \\in \\mathbf{R}$.\\\\\n (1) 求证: 对于任意 $u, v$ 都有 $f(u) \\cdot f(v)=f(u+v)$;\\\\\n (2) 写出一个关于 $\\dfrac{f(u)}{f(v)}$ 的类似等式.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260128-指数函数的性质(1)" ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\dfrac{f(u)}{f(v)}=f(u-v)$", @@ -523991,7 +524590,8 @@ "content": "证明: 当 $a>1$, $N>1$ 时, $\\log _a N>0$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260131-对数函数的性质(1)" ], "genre": "解答题", "ans": "证明略", @@ -524013,7 +524613,8 @@ "content": "证明: 当 $a>1$ 时, 对数函数 $y=\\log _a x$ 在区间 ($0,+\\infty$) 上是严格增函数.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260131-对数函数的性质(1)" ], "genre": "解答题", "ans": "证明略", @@ -524057,7 +524658,8 @@ "content": "证明: $\\sqrt 3$是无理数.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260102-高一上学期周末卷02" ], "genre": "4em", "ans": "证明略", @@ -524096,7 +524698,8 @@ "content": "(1) 已知$a>b>0$, 求证: $\\dfrac{1}{b}>\\dfrac{1}{a}>0$;\\\\\n (2) 已知$a>b>0$, $c>d>0$, 求证: $\\dfrac{a}{d}>\\dfrac{b}{c}$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260110-不等式的性质(1)" ], "genre": "解答题", "ans": "证明略", @@ -524118,7 +524721,8 @@ "content": "已知$a,b$是实数, 且$a>b$, 求证: $a^3>b^3$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260111-不等式的性质(2)" ], "genre": "解答题", "ans": "证明略", @@ -524140,7 +524744,8 @@ "content": "指出陈述句 $P$ 是陈述句 $Q$ 的什么条件. (用选项回答: A. 充分不必要 B. 必要不充分 C. 充分必要 D. 既不充分又不必要)\\\\\n(1) $P$: $x=0$, $Q$: $x^2=0$: \\bracket{20};\\\\\n(2) $P$: $x+y+z>0$, $Q$: $x>0$ 或 $y>0$ 或 $z>0$: \\bracket{20};\\\\\n(3) $P$: $x+y+z>0$, $Q$: $x>0$ 且 $y>0$ 且 $z>0$: \\bracket{20};\\\\\n(4) $P$: $x / y>0$, $Q$: $x y>0$: \\bracket{20};\\\\\n(5) $P$: $x / y \\geq 0$, $Q$: $x y \\geq 0$: \\bracket{20};\\\\\n(6) $P$: $(x-3)^2+(y+4)^2=0$, $Q$: $(x-3)(y-4)=0$: \\bracket{20};\\\\\n(7) $P$: $x+y>0$ 且 $x y>0$, $Q$: $x>0$ 且 $y>0$: \\bracket{20};\\\\\n(8) $P$: $x+y>4$ 且 $x y>4$, $Q$: $x>2$ 且 $y>2$: \\bracket{20};\\\\\n(9) $P$: 函数 $y=k x+b$ 的图像通过原点 (其中 $k, b$ 为常数), $Q$: $b=0$: \\bracket{20};\\\\\n(10) $P$: $x z>y z$ 且 $z \\neq 0$, $Q$: $x>y$ 且 $z \\neq 0$: \\bracket{20};\\\\\n(11) $P$: $|x|<2$, $Q$: $x<2$: \\bracket{20};\\\\\n(12) $P$: $x^2+1=y$ 且 $x+1=y$, $Q$: $x^2=x$ 且 $x^2+1=y$: \\bracket{20}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260102-高一上学期周末卷02" ], "genre": "", "ans": "(1) C; (2) A; (3) B; (4) C; (5) A; (6) A; (7) C; (8) B; (9) C; (10) D; (11) A; (12) C", @@ -524941,7 +525546,8 @@ "content": "已知$\\alpha,\\beta$是方程$2x^2-\\sqrt{15}x+1=0$的两根, 则$|\\alpha-\\beta|$的值为\\blank{50}.", "objs": [], "tags": [ - "第五单元" + "第五单元", + "I20260102-高一上学期随堂练习02" ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{7}}{2}$", @@ -524966,7 +525572,8 @@ "content": "已知$x,y$是正实数, 则下列条件中, ``$x\\le y$''的必要条件为\\bracket{20}.\n\\fourch{$x+\\dfrac{2}{y}\\le y+\\dfrac{1}{x}$}{$x+\\dfrac{1}{2y}\\le y+\\dfrac{1}{x}$}{$x-\\dfrac{2}{y}\\le y-\\dfrac{1}{x}$}{$x-\\dfrac{1}{2y}\\le y-\\dfrac{1}{x}$}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "I20260102-高一上学期随堂练习02" ], "genre": "选择题", "ans": "B", @@ -524991,7 +525598,8 @@ "content": "已知三角形的三边长分别为$a,b,c$. 设$M=\\dfrac{a}{1+a}+\\dfrac{b}{1+b}$, $N=\\dfrac{c}{1+c}$, $Q=\\dfrac{a+b}{1+a+b}$, 则$M,N,Q$的大小关系为\\bracket{20}.\n\\fourch{$M1$时, 解集为$(m^2+m+1,+\\infty)$; 当$m=1$时, 解集为$\\varnothing$; 当$m<1$时, 解集为$(-\\infty,m^2+m+1)$", @@ -525046,7 +525655,8 @@ "K0114001B" ], "tags": [ - "第一单元" + "第一单元", + "G20260113-一元二次不等式的求解(2)" ], "genre": "4em", "ans": "(1) $(-\\infty,-\\dfrac{1}{6})\\cup (1,+\\infty)$; (2) $(-\\dfrac{1}{6},1)$", @@ -525071,7 +525681,8 @@ "content": "已知$a\\in \\mathbf{R}$, 求关于$x$的不等式 $x^2-(a+1)^2x+2a(a^2+1)<0$的解集.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260113-一元二次不等式的求解(2)" ], "genre": "4em", "ans": "当$a=1$时, 解集为$\\varnothing$; 当$a\\ne 1$时, 解集为$(2a,a^2+1)$", @@ -525135,7 +525746,8 @@ "content": "已知$a\\in \\mathbf{R}$, 当$|x^2-4|\\ge 1$成立时, $|x-2|\\ge a$也成立, 求$a$的取值范围.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260116-含绝对值不等式的求解" ], "genre": "4em", "ans": "$(-\\infty,-2+\\sqrt{5}]$", @@ -525174,7 +525786,8 @@ "K0119001B" ], "tags": [ - "第一单元" + "第一单元", + "G20260118-平均值不等式及其应用(2)" ], "genre": "4em", "ans": "$4$", @@ -525213,7 +525826,8 @@ "content": "已知$a,b>0$, 满足$a+b=4$.\\\\\n(1) 求$a^2+b^2$的最小值;\\\\\n(2) 求$\\dfrac 1a+\\dfrac 4b$的最小值;\\\\\n(3) 求$\\dfrac 1{a+1}+\\dfrac 8{2b+5}$的最小值.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260118-平均值不等式及其应用(2)" ], "genre": "4em", "ans": "(1) $8$; (2) $\\dfrac{9}{4}$; (3) $\\dfrac{6}{5}$", @@ -525255,7 +525869,8 @@ ], "tags": [ "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-第一轮复习讲义-03-等式与不等式的性质及证明", + "G20260119-三角不等式" ], "genre": "4em", "ans": "证明略, 等号成立时$x$的取值范围为$(-\\infty,-2]$", @@ -525293,7 +525908,8 @@ "content": "用三角不等式证明: $|x-3|-|x-5|\\le 2$对一切实数$x$恒成立, 并指出等号成立的条件.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260119-三角不等式" ], "genre": "4em", "ans": "证明略, 等号成立当且仅当$x\\ge 5$", @@ -526051,7 +526667,8 @@ "tags": [ "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" + "2023届高三-第一轮复习讲义-05-幂指数与对数", + "G20260122-对数的定义" ], "genre": "", "ans": "B", @@ -526089,7 +526706,8 @@ "objs": [], "tags": [ "第二单元", - "第三章" + "第三章", + "G20260124-对数的换底" ], "genre": "4em", "ans": "(1) $\\dfrac{1}{3}$; (2) $\\dfrac{5}{6}$; (3) $\\dfrac{5}{3}$; (4) $2$", @@ -526125,7 +526743,8 @@ "content": "解下列关于$x$的不等式:\\\\\n(1) $|2x-4|>5-x$;\\\\\n(2) $\\dfrac{x-a}{x-2}>1$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "E20260102-高一上学期测验卷02" ], "genre": "4em", "ans": "(1) $(-\\infty,-1)\\cup (3,+\\infty)$; (2) 当$a<2$时, 解集为$(2,+\\infty)$; 当$a=2$时, 解集为$\\varnothing$; 当$a>2$时, 解集为$(-\\infty,2)$", @@ -526163,7 +526782,8 @@ "content": "设$a,b,c$均为正数, 且$ab+bc+ca=1$, 求证: $a+b+c\\ge \\sqrt{3}$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "E20260102-高一上学期测验卷02" ], "genre": "4em", "ans": "证明略", @@ -526201,7 +526821,8 @@ "content": "已知 $a>0$, $b>0$, 且 $a+b=1$.\\\\\n(1) 求: $\\dfrac{1}{a}+\\dfrac{1}{b}$ 的最小值;\\\\\n(2) 求 $\\sqrt{a+1}+\\sqrt{b+1}$ 的最大值.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "E20260102-高一上学期测验卷02" ], "genre": "4em", "ans": "(1) $4$; (2) $\\sqrt{6}$", @@ -526396,7 +527017,8 @@ "content": "已知幂函数 $f(x)=x^m$.\\\\\n(1) 若 $m>0$, 求证: 在区间$[0,+\\infty)$上, 函数 $y=f(x)$ 中$y$随$x$的(严格)增大而(严格)增大;\\\\\n(2) 若 $m<0$, 求证: 在区间$(0,+\\infty)$上, 函数 $y=f(x)$ 中$y$随$x$的(严格)增大而(严格)减小.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260126-幂函数的性质" ], "genre": "4em", "ans": "(1) 证明略; (2) 证明略", @@ -526421,7 +527043,8 @@ "content": "作出下列函数的大致图像:\\\\\n(1) $y=(x-1)^{-\\frac 23}$;\\\\\n(2) $y=\\dfrac{2 x-3}{x-3}$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260126-幂函数的性质" ], "genre": "4em", "ans": "(1) \\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-3,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0.13:4, samples = 100] plot ({\\x+1},{exp(-2/3*ln(\\x))}) plot ({-\\x+1},{exp(-2/3*ln(\\x))});\n\\end{tikzpicture}; (2) \\begin{tikzpicture}[>=latex, scale = 0.25]\n\\draw [->] (-2,0) -- (8,0) node [below] {$x$};\n\\draw [->] (0,-4) -- (0,6) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -2:2.5, samples = 100] plot (\\x,{(2*\\x-3)/(\\x-3)});\n\\draw [domain = {15/4}:8, samples = 100] plot (\\x,{(2*\\x-3)/(\\x-3)});\n\\end{tikzpicture}", @@ -526458,7 +527081,8 @@ "content": "写出下列函数的定义域:\\\\\n(1) $y=3^{1-2 x}$:\\blank{50};\\\\\n(2) $y=0.5^{\\frac 1{2-3 x}}$:\\blank{50};\\\\\n(3) $y=\\dfrac 1{5^{\\frac x{x-1}}-1}$:\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260127-指数函数的定义与图像" ], "genre": "", "ans": "(1) $\\mathbf{R}$; (2) $(-\\infty,\\dfrac{2}{3})\\cup (\\dfrac{2}{3},+\\infty)$; (3) $(-\\infty,0)\\cup (0,1)\\cup (1,+\\infty)$", @@ -526496,7 +527120,8 @@ "content": "证明: 函数$y=2^x$的图像与函数$y=2^{2-x}$的图像关于直线$x=1$对称.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260128-指数函数的性质(1)" ], "genre": "解答题", "ans": "证明略", @@ -526531,7 +527156,8 @@ "content": "写出下列函数的定义域:\\\\\n(1) $y=\\lg (x-2)+\\lg (10 x-3-3 x^2)$:\\blank{50};\\\\\n(2) $y=\\dfrac{\\sqrt {2 x-1}}{\\lg x}$:\\blank{50};\\\\\n(3) $y=\\log_{x-1}(3-x)$:\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260130-对数函数的定义和图像" ], "genre": "", "ans": "(1) $(2,3)$; (2) $[\\dfrac{1}{2},1)\\cup (1,+\\infty)$; (3) $(1,2)\\cup (2,3)$", @@ -527006,7 +527632,8 @@ "content": "若设 $m \\in \\mathbf{Z}$, 若幂函数 $y=x^{(m-1)(m-4)}$ 的定义域为 $\\{x | x \\neq 0\\}$, 则 $m$ 的值为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "I20260104-高一上学期随堂练习04" ], "genre": "", "ans": "$1,2,3$或$4$", @@ -527069,7 +527696,8 @@ "content": "由下列不等式, 分别求出实数$a$的取值范围.\\\\\n(1) $(a+2)^{\\frac 23}>(1-2 a)^{\\frac 23}$, $a$的范围为\\blank{50};\\\\\n(2) $(a+2)^{\\frac 13}>(3-2 a)^{\\frac 13}$, $a$的范围为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "I20260104-高一上学期随堂练习04" ], "genre": "", "ans": "(1) $(-\\dfrac{1}{3},3)$; (2) $(\\dfrac{1}{3},+\\infty)$", @@ -527107,7 +527735,8 @@ "content": "关于 $x$ 的方程 $3^{2 x+1}+(2m+6)3^x-m-1=0$ 有两个不同的实数解, 求实数 $m$ 的取值范围.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260107-高一上学期周末卷07" ], "genre": "4em", "ans": "$(-\\infty,-\\dfrac{9+\\sqrt{33}}{2})$", @@ -527147,7 +527776,8 @@ "content": "利用计算器判断: 函数$y=(\\dfrac{4}{5})^{|x-1|}$在区间\\blank{50}上为严格减函数.(写出最大的区间)", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260107-高一上学期周末卷07" ], "genre": "填空题", "ans": "$[1,+\\infty)$", @@ -527182,7 +527812,8 @@ "content": "已知函数$y=\\log_4(4^x+1)-\\dfrac{1}{2} x$, 当$x$取何值时函数值最小? 并求出该函数的最小值.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260132-对数函数的性质(2)" ], "genre": "4em", "ans": "当且仅当$x=0$时函数值最小, 最小值为$\\dfrac{1}{2}$", @@ -527220,7 +527851,8 @@ "content": "利用计算器猜测: 函数$y=\\log_{0.5}(3-2 x-x^2)$在哪个区间上是严格增函数(写出最大的区间)? 并证明此函数在你所给出的区间上确实是严格增函数.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260132-对数函数的性质(2)" ], "genre": "4em", "ans": "$[-1,1)$, 证明略", @@ -527258,7 +527890,8 @@ "content": "函数$y=\\log_{\\frac 12}(x^2-6 x+ 11)$的最\\blank{20}值是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260132-对数函数的性质(2)" ], "genre": "", "ans": "大, $-1$", @@ -527472,7 +528105,8 @@ "content": "函数 $y=a^{x-1}+2$($a>0$, $a \\neq 1$) 的图像恒过一定点, 此定点的坐标为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "E20260103-高一上学期测验卷03" ], "genre": "填空题", "ans": "$(1,3)$", @@ -527507,7 +528141,8 @@ "content": "函数 $y=(m^2-m-1) x^{m^2-2 m-1}$ 是幂函数, 则 $m$ 的值为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "E20260103-高一上学期测验卷03" ], "genre": "填空题", "ans": "$2$或$-1$", @@ -527542,7 +528177,8 @@ "content": "设 $\\alpha \\in\\{-1,1, \\dfrac{1}{2}, 3\\}$, 则使函数 $y=x^\\alpha$ 的定义域为 $\\mathbf{R}$, 且图像关于原点对称的所有 $\\alpha$ 的取值集合为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "E20260103-高一上学期测验卷03" ], "genre": "填空题", "ans": "$\\{1,3\\}$", @@ -527577,7 +528213,8 @@ "content": "已知 $\\log _{14}7=a$, $14^b=5$, 则 $\\log _{35}28=$\\blank{50}.(用 $a, b$ 表示)", "objs": [], "tags": [ - "第二单元" + "第二单元", + "E20260103-高一上学期测验卷03" ], "genre": "填空题", "ans": "$\\dfrac{2-a}{a+b}$", @@ -527612,7 +528249,8 @@ "content": "已知 $a^{\\frac{2}{3}}=\\dfrac{4}{9}$($a>0$), 则 $\\log _{\\frac{2}{3}}a=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "E20260103-高一上学期测验卷03" ], "genre": "填空题", "ans": "$3$", @@ -527647,7 +528285,8 @@ "content": "方程 $\\lg (x-1)+\\lg (3-x)=\\lg (2-x)$ 的解为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "E20260103-高一上学期测验卷03" ], "genre": "填空题", "ans": "$\\dfrac{5-\\sqrt{5}}{2}$", @@ -527682,7 +528321,8 @@ "content": "若函数 $y=\\lg [(a^2-1) x^2+(a+1) x+1]$ 的定义域为 $\\mathbf{R}$, 则实数 $a$ 的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "E20260103-高一上学期测验卷03" ], "genre": "填空题", "ans": "$(-\\infty,-1]\\cup (\\dfrac{5}{3},+\\infty)$", @@ -527717,7 +528357,8 @@ "content": "已知函数 $y=(x-a)(x-b)$ (其中 $a>b$) 的图像如图所示, 则函数 $y=a^x+b$ 的图像大致为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1,0.1) -- (1,0) node [below] {$1$};\n\\draw (-1,0.1) -- (-1,0) node [above] {$-1$};\n\\draw [thick, domain = -2.4:1.4] plot (\\x,{(\\x-0.4)*(\\x+1.4)});\n\\end{tikzpicture}\n\\end{center}\n\\fourch{\\begin{tikzpicture}[>=latex, scale = 0.7]\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 (0.1,1) -- (0,1) node [right] {$1$};\n\\draw [domain = -1.3:2,thick] plot (\\x,{exp(\\x*ln(0.4))-1.4});\n\\draw [dashed] (-2,-1.4) -- (2,-1.4);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.7]\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 (0.1,1) -- (0,1) node [right] {$1$};\n\\draw [domain = -1.1:2,thick] plot (\\x,{exp(\\x*ln(0.4))-0.8});\n\\draw [dashed] (-2,-0.8) -- (2,-0.8);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.7]\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 (0.1,1) -- (0,1) node [right] {$1$};\n\\draw [domain = -1.1:2,thick] plot (-\\x,{exp(\\x*ln(0.4))-0.8});\n\\draw [dashed] (-2,-0.8) -- (2,-0.8);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.7]\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 (0.1,1) -- (0,1) node [right] {$1$};\n\\draw [domain = -1.3:2,thick] plot (-\\x,{exp(\\x*ln(0.4))-1.4});\n\\draw [dashed] (-2,-1.4) -- (2,-1.4);\n\\end{tikzpicture}}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "E20260103-高一上学期测验卷03" ], "genre": "选择题", "ans": "A", @@ -527752,7 +528393,8 @@ "content": "对于 $a>0$, $a \\neq 1$, 给出下列四个命题, 其中真命题是\\bracket{20}.\n\\onech{$\\log _a(1+a)$总是小于$\\log _a(1+\\dfrac{1}{a})$}{$\\log _a(1+a)$总是大于$\\log _a(1+\\dfrac{1}{a})$}{$a^{1+a}$总是小于$a^{1+\\frac{1}{a}}$}{$a^{1+a}, a^{1+\\frac{1}{a}}$ 的大小不能确定}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "E20260103-高一上学期测验卷03" ], "genre": "选择题", "ans": "B", @@ -527787,7 +528429,8 @@ "content": "已知函数 $y=x^{-4}$ 和函数 $y=(x-1)^{-4}+2$, 指出这两个函数图像之间的关系,并在平面直角坐标系中作出函数 $y=(x-1)^{-4}+2$ 的大致图像.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "E20260103-高一上学期测验卷03" ], "genre": "解答题", "ans": "函数$y=x^{-4}$的图像向右平移$1$个单位, 再向上平移$2$个单位后得到$y=(x-1)^{-4}+2$的图像, 大致图像为: \\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-3,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,6) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (-3,2) -- (5,2) (1,-0.5) -- (1,6);\n\\draw [domain = -3:{1-sqrt(2)/2}, samples = 100] plot (\\x,{1/(\\x-1)/(\\x-1)/(\\x-1)/(\\x-1)+2});\n\\draw [domain = {1+sqrt(2)/2}:5, samples = 100] plot (\\x,{1/(\\x-1)/(\\x-1)/(\\x-1)/(\\x-1)+2});\n\\end{tikzpicture}", @@ -527822,7 +528465,8 @@ "content": "已知 $2^x \\leq 256$ 且 $\\log _2 x \\geq \\dfrac{1}{2}$, 求函数 $f(x)=(\\log _2 x-1) \\cdot(\\log _{\\sqrt{2}}\\sqrt{x}-2)$的最小值.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "E20260103-高一上学期测验卷03" ], "genre": "解答题", "ans": "$-\\dfrac{1}{4}$", @@ -530116,7 +530760,8 @@ "objs": [], "tags": [ "第一单元", - "第二单元" + "第二单元", + "W20260108-高一上学期周末卷08" ], "genre": "", "ans": "$(-\\infty,3)$", @@ -530154,7 +530799,8 @@ "content": "判断下列函数的奇偶性, 并说明理由:\\\\\n(1) $f(x)=x^2+x$;\\\\\n(2) $f(x)=\\begin{cases}x(1-x), & x<0,\\\\x(1+x),& x>0;\\end{cases}$\\\\\n(3) $f(x)=|x-a|$(其中$a$是常数);\\\\\n(4) $f(x)=x^n-x^{-n}$($n \\in \\mathbf{Z}$);\\\\\n(5) $f(x)=\\dfrac{\\sqrt{1-x^2}}{2-|x+2|}$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260108-高一上学期周末卷08" ], "genre": "4em", "ans": "(1) 既不是奇函数, 又不是偶函数, 理由略; (2) 奇函数, 理由略; (3) 当$a\\ne 0$时, 既不是奇函数, 又不是偶函数, 理由略; 当$a=0$时, 是偶函数, 理由略; (4) 当$n=0$时, 既是奇函数, 又是偶函数, 理由略; 当$n$为非零偶数时, 是偶函数, 理由略; 当$n$为奇数时, 是奇函数, 理由略; (5) 奇函数, 理由略", @@ -530192,7 +530838,8 @@ "content": "函数$y=\\dfrac 1{\\sqrt {x^2+2 x+3}}$的值域为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260133-函数" ], "genre": "", "ans": "$(0,\\dfrac{\\sqrt{2}}{2}]$", @@ -530232,7 +530879,8 @@ "K0216003B" ], "tags": [ - "第二单元" + "第二单元", + "G20260134-函数的表示方法" ], "genre": "4em", "ans": "(1) \\begin{tikzpicture}[>=latex]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain = -3:3, samples = 100] plot (\\x,{1/(1+\\x*\\x)});\n\\end{tikzpicture}; (2) \\begin{tikzpicture}[>=latex]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,1) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain = -3:3, samples = 100] plot (\\x,{\\x/(1+\\x*\\x)});\n\\end{tikzpicture}", @@ -530257,7 +530905,8 @@ "content": "下列各图像中, \\blank{50}是函数的图像(填入序号).\\\\\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.9]\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 = {-pi/2}:{pi/2}] plot ({\\x},{-sin(2*\\x/pi*180)});\n\\draw (0,-2) node [below] {\\textcircled{1}};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,scale = 0.9]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain = {-pi/2}:{pi/2}] plot ({sin(2*\\x/pi*180)},{\\x});\n\\draw (0,-2) node [below] {\\textcircled{2}}; \n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-1,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {0,1,2,3} \n{ \n\\draw (\\i,{\\i+1}) -- ({\\i+1},{\\i+1});\n\\filldraw [fill = white, draw = black] (\\i,{\\i+1}) circle (0.05);\n\\filldraw ({\\i+1},{\\i+1}) circle (0.05);\n};\n\\filldraw [fill = white, draw = black] (0,0) circle (0.05);\n\\foreach \\i in {1,2,3,4}\n{\n\\draw (\\i,0.1) -- (\\i,0) node [below] {$\\i$};\n\\draw (0.1,\\i) -- (0,\\i) node [left] {$\\i$};\n};\n\\draw (2,-1) node [below] {\\textcircled{3}};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-1,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {0,1,2,3} \n{ \n\\draw ({\\i+1},\\i) -- ({\\i+1},{\\i+1});\n\\filldraw [fill = white, draw = black] ({\\i+1},{\\i+1}) circle (0.05);\n\\filldraw ({\\i+1},\\i) circle (0.05);\n};\n\\filldraw [fill = white, draw = black] (0,0) circle (0.05);\n\\foreach \\i in {1,2,3,4}\n{\n\\draw (\\i,0.1) -- (\\i,0) node [below] {$\\i$};\n\\draw (0.1,\\i) -- (0,\\i) node [left] {$\\i$};\n};\n\\draw (2,-1) node [below] {\\textcircled{4}};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260134-函数的表示方法" ], "genre": "", "ans": "\\textcircled{1}\\textcircled{3}", @@ -530413,7 +531062,8 @@ "K0102001B" ], "tags": [ - "第一单元" + "第一单元", + "G20260102-集合的表示方法" ], "genre": "解答题", "ans": "(1) $\\{2,3,5,7,11\\}$; (2) $\\{-2,-1,0,1,2\\}$; (3) $\\{2,1,0,-3\\}$; (4) $\\{-1,0,3\\}$; (5) $\\{(-2,3),(-1,0),(0,-1),(1,0),(2,3)\\}$; (6) $\\{(0,5),(1,4),(2,3),(3,2),(4,1),(5,0)\\}$", @@ -530450,7 +531100,8 @@ "K0102002B" ], "tags": [ - "第一单元" + "第一单元", + "G20260102-集合的表示方法" ], "genre": "解答题", "ans": "(1) $\\{x|x=2k+1, \\ k\\in \\mathbf{Z}\\}$; (2) $\\{x|x=3k+2, \\ k \\in \\mathbf{N}\\}$; (3) $\\{x|x=10\\}$; (4) $\\{x||x|>4, \\ x\\in \\mathbf{Z}\\}$; (5) $\\{(x,y)|x=0, \\ y\\in \\mathbf{R}\\}$; (6) $\\{(x,y)|y=2x+1, \\ x\\in \\mathbf{R}\\}$", @@ -530487,7 +531138,8 @@ "K0102004B" ], "tags": [ - "第一单元" + "第一单元", + "G20260102-集合的表示方法" ], "genre": "解答题", "ans": "(1) $(-2,7)$; (2) $[-2,7]$; (3) $[-2,7)$; (4) $(-\\infty,\\dfrac{5}{2})$; (5) $(-5,+\\infty)$; (6) $[0,+\\infty)$", @@ -530526,7 +531178,8 @@ "K0102002B" ], "tags": [ - "第一单元" + "第一单元", + "G20260102-集合的表示方法" ], "genre": "解答题", "ans": "(1) $\\{1,2,5,10\\}$; (2) $\\{n|n=10k, \\ k\\in \\mathbf{N}, k\\ge 1$; (3) $\\varnothing$; (4) $\\{(-1,2)\\}$; (5) $\\{(-3,5)\\}$", @@ -530615,7 +531268,8 @@ "K0102001B" ], "tags": [ - "第一单元" + "第一单元", + "G20260102-集合的表示方法" ], "genre": "解答题", "ans": "$P=\\{0,6,14,21\\}$", @@ -530652,7 +531306,8 @@ "K0102001B" ], "tags": [ - "第一单元" + "第一单元", + "G20260102-集合的表示方法" ], "genre": "解答题", "ans": "$-1$", @@ -530742,7 +531397,8 @@ "K0101001B" ], "tags": [ - "第一单元" + "第一单元", + "G20260102-集合的表示方法" ], "genre": "解答题", "ans": "$-1$或$-\\dfrac{1}{2}$", @@ -530853,7 +531509,8 @@ "K0103005B" ], "tags": [ - "第一单元" + "第一单元", + "G20260103-集合之间的关系" ], "genre": "填空题", "ans": "6", @@ -530892,7 +531549,8 @@ "K0103005B" ], "tags": [ - "第一单元" + "第一单元", + "G20260103-集合之间的关系" ], "genre": "填空题", "ans": "\\textcircled{1}\\textcircled{2}", @@ -530931,7 +531589,9 @@ "K0102002B" ], "tags": [ - "第一单元" + "第一单元", + "G20260102-集合的表示方法", + "G20260103-集合之间的关系" ], "genre": "填空题", "ans": "\\textcircled{2}\\textcircled{5}\\textcircled{6}", @@ -530982,7 +531642,8 @@ "K0103001B" ], "tags": [ - "第一单元" + "第一单元", + "G20260103-集合之间的关系" ], "genre": "填空题", "ans": "\\textcircled{1}\\textcircled{3}", @@ -531143,7 +531804,8 @@ ], "tags": [ "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-01-集合", + "G20260103-集合之间的关系" ], "genre": "解答题", "ans": "$B=\\{\\varnothing, \\{1\\}\\}$, $A\\in B$.", @@ -531184,7 +531846,8 @@ "K0103005B" ], "tags": [ - "第一单元" + "第一单元", + "G20260103-集合之间的关系" ], "genre": "填空题", "ans": "$5$", @@ -531223,7 +531886,8 @@ ], "tags": [ "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-01-集合", + "G20260103-集合之间的关系" ], "genre": "填空题", "ans": "$[5,+\\infty)$; $\\ge 5$; $x<5\\le a$; $<5$, $\\dfrac{5+a}2$; $a10$, (如) $x>0$", @@ -532974,7 +533670,8 @@ ], "tags": [ "第一单元", - "2023届高三-第一轮复习讲义-02-常用逻辑用语" + "2023届高三-第一轮复习讲义-02-常用逻辑用语", + "G20260106-充分条件与必要条件" ], "genre": "填空题", "ans": "$k<0$且$b<0$", @@ -533018,7 +533715,8 @@ "K0106001B" ], "tags": [ - "第一单元" + "第一单元", + "G20260106-充分条件与必要条件" ], "genre": "填空题", "ans": "$a>0$", @@ -533104,7 +533802,8 @@ "K0107001B" ], "tags": [ - "第一单元" + "第一单元", + "G20260106-充分条件与必要条件" ], "genre": "选择题", "ans": "D", @@ -533142,7 +533841,8 @@ "K0111003B" ], "tags": [ - "第一单元" + "第一单元", + "G20260106-充分条件与必要条件" ], "genre": "解答题", "ans": "证明略", @@ -533284,7 +533984,8 @@ "K0107002B" ], "tags": [ - "第一单元" + "第一单元", + "G20260107-反证法" ], "genre": "填空题", "ans": "(1) 真, $\\pi$不是无理数, 假; (2) 假, $2+1\\ne 4$, 真; (3) 假, 存在实数既不是正数、也不是负数, 真; (4) 假, 存在实数不是正数, 且存在实数不是负数, 真; (5) 假, 存在实数$x$, 使得$x^2+1\\ne 0$, 真; (6) 真, 对任意实数$x$, $x^2+1\\ne 0$, 真; (7) 假, 存在实数$k$, 使关于$x$的方程$x^2+x+k=0$无实数根, 真; (8) 真, 存在三角形, 其中至少有两个钝角, 假; (9) 真, 存在$a,b$满足$a>1$, $b>1$, 但$ab\\le 1$, 假; (10) 假, 存在能整除$2$的正整数不是质数, 真", @@ -533350,7 +534051,8 @@ "K0107003B" ], "tags": [ - "第一单元" + "第一单元", + "G20260105-命题" ], "genre": "选择题", "ans": "D", @@ -533390,7 +534092,8 @@ "K0107002B" ], "tags": [ - "第一单元" + "第一单元", + "G20260107-反证法" ], "genre": "选择题", "ans": "D", @@ -533433,7 +534136,8 @@ "K0107001B" ], "tags": [ - "第一单元" + "第一单元", + "G20260107-反证法" ], "genre": "解答题", "ans": "证明略", @@ -533495,7 +534199,8 @@ "K0107002B" ], "tags": [ - "第一单元" + "第一单元", + "W20260102-高一上学期周末卷02" ], "genre": "选择题", "ans": "B", @@ -533580,7 +534285,8 @@ "content": "设$a,b,x,y\\in \\mathbf{R}$, 判断下列命题的真假.\\\\\n\\blank{20}(1) 若$a=b$, 则$\\sqrt a=\\sqrt b$;\\\\\n\\blank{20}(2) 若$a^2=b^2$, 则$|a|=|b|$;\\\\\n\\blank{20}(3) 若$\\dfrac ab=\\dfrac xy$, 且$b-y\\ne 0$, 则$\\dfrac{a-x}{b-y}=\\dfrac ab$;\\\\\n\\blank{20}(4) 若$ax=b$, 则$x=\\dfrac ba$;\\\\\n\\blank{20}(5) 若$a^2x=a^2y$, 则$x=y$;\\\\\n\\blank{20}(6) 若$(\\sqrt a+1)x=(\\sqrt a+1)y$, 则$x=y$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260108-等式的性质与方程的解集" ], "genre": "填空题", "ans": "(1) 假; (2) 真; (3) 真; (4) 假; (5) 假; (6) 真.", @@ -533615,7 +534321,8 @@ "content": "若$(m^2-1)x^2+(m-1)x+3=0$是关于$x$的一元一次方程, 则这个方程的解集是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260108-等式的性质与方程的解集" ], "genre": "填空题", "ans": "$\\{\\dfrac{3}{2}\\}$", @@ -533650,7 +534357,8 @@ "content": "已知$a,b\\in \\mathbf{R}$, 解关于$x$的方程$(a-2)x-1=x+b$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260108-等式的性质与方程的解集" ], "genre": "解答题", "ans": "当$a=3$且$b=-1$时, 解集为$\\mathbf{R}$; 当$a=3$且$b\\ne -1$时, 解集为$\\varnothing$; 当$a\\ne 3$时, 解集为$\\{\\dfrac{b+1}{a-2}\\}$", @@ -533685,7 +534393,8 @@ "content": "方程组 $\\begin{cases}\\dfrac{x+y}2+\\dfrac{x-y}3=6, \\\\2(x+y)-x+y=-4 \\end{cases}$的解集是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260108-等式的性质与方程的解集" ], "genre": "填空题", "ans": "$\\{(8,-4)\\}$", @@ -533720,7 +534429,8 @@ "content": "已知关于$x$的方程$2a(x-1)=(5-a)x+3b$的解集为$\\mathbf{R}$, 则$a=$\\blank{50}, $b=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260108-等式的性质与方程的解集" ], "genre": "填空题", "ans": "$a=\\dfrac{5}{3}$, $b=-\\dfrac{10}{9}$", @@ -533755,7 +534465,8 @@ "content": "如果$a,b$为定值, 关于$x$的方程$\\dfrac{2kx+a}3=2+\\dfrac{x-bk}6,$无论$k$为何值时, 它总是至少有一个根为$1$, 则$a=$\\blank{50}, $b=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260108-等式的性质与方程的解集" ], "genre": "填空题", "ans": "$a=\\dfrac{13}{2}$, $b=-4$", @@ -533790,7 +534501,8 @@ "content": "已知$k\\in \\mathbf{R}$, 求关于$x,y$的一元二次方程组$\\begin{cases}\n 2x+ky=1, \\\\2kx+y=3 \\end{cases}$的解集.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260108-等式的性质与方程的解集" ], "genre": "解答题", "ans": "当$k=\\pm 1$时, 解集为$\\varnothing$; 当$k\\ne \\pm 1$时, 解集为$\\{(\\dfrac{3k-1}{2k^2-2},\\dfrac{k-3}{k^2-1})\\}$.", @@ -533828,7 +534540,8 @@ "content": "已知$a\\in \\mathbf{Z}$, 要使方程组$\\begin{cases} 2x+ay=16, \\\\x-2y=0 \\end{cases}$有正整数解, 求整数$a$组成的集合.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260108-等式的性质与方程的解集" ], "genre": "解答题", "ans": "$a=-3,-2,0,4,12$", @@ -533864,7 +534577,8 @@ "content": "设$a\\in \\mathbf{R}$, 求关于$x$的方程$ax^2+2x-3=0$的解集.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260109-一元二次方程和韦达定理" ], "genre": "解答题", "ans": "当$a>-\\dfrac{1}{3}$且$a\\ne 0$时, 解集为$\\{\\dfrac{-1-\\sqrt{1+3a}}{a},\\dfrac{-1+\\sqrt{1+3a}}{a}\\}$; 当$a=0$时, 解集为$\\{\\dfrac{3}{2}\\}$; 当$a=-\\dfrac{1}{3}$时, 解集为$\\{3\\}$; 当$a<-\\dfrac{1}{3}$时, 解集为$\\varnothing$.", @@ -533899,7 +534613,8 @@ "content": "证明: 对于任意实数$a$, 关于$x$的方程$x^2-3(a+1)x+6a+1=0$总有实数根.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260109-一元二次方程和韦达定理" ], "genre": "解答题", "ans": "证明略", @@ -533934,7 +534649,8 @@ "content": "若关于$x$的方程$2x^2-3x+1-m=0$有一个正根和一个负根, 则$m$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260109-一元二次方程和韦达定理" ], "genre": "填空题", "ans": "$(1,+\\infty)$", @@ -533983,7 +534699,8 @@ "content": "若关于$x$的方程$2x^2-3x+1-m=0$有两个不同正根, 则$m$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260109-一元二次方程和韦达定理" ], "genre": "填空题", "ans": "$(-\\dfrac{1}{8},1)$", @@ -534032,7 +534749,8 @@ "content": "已知关于$x$的方程$2x^2+2x+c=0$的两个根为$x_1$, $x_2$, 并且$|x_1-x_2|=\\sqrt 3$, 则实数$c$的值是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260109-一元二次方程和韦达定理" ], "genre": "填空题", "ans": "$c=-1$", @@ -534068,7 +534786,8 @@ "content": "已知$x_1$、$x_2$是一元二次方程$ax^2-2x-a=0$的两根, 且$\\dfrac{x_2}{x_1-1}+\\dfrac{x_1}{x_2-1}=6$, 求实数$a$的值.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260109-一元二次方程和韦达定理" ], "genre": "解答题", "ans": "$a=\\dfrac{5\\pm \\sqrt{17}}{2}$", @@ -534103,7 +534822,8 @@ "content": "已知实数$p$、$q$, 满足$p^2-3p-5=0$, $5q^2+3q-1=0$, 且$pq\\ne 1$, 则$p^2+\\dfrac 1{q^2}=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260109-一元二次方程和韦达定理" ], "genre": "填空题", "ans": "$19$", @@ -534205,7 +534925,8 @@ "content": "判断下列语句是否正确, 并在相应横线上填入``\\checkmark''或``$\\times$''.\\\\\n\\blank{20}(1) 若$ax>b,a\\ne 0$则$x>\\dfrac ba$;\\\\\n\\blank{20}(2) 若$a^2x>a^2y$, 则$x>y$;\\\\\n\\blank{20}(3) 若$a>b$, 则$a^2>ab$;\\\\\n\\blank{20}(4) 若$a\\dfrac 1a>\\dfrac 1b$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260110-不等式的性质(1)" ], "genre": "填空题", "ans": "(1) $\\times$; (2) \\checkmark; (3) $\\times$; (4) \\checkmark", @@ -534240,7 +534961,8 @@ "content": "设$x$为实数, 下列各式中, 值恒大于$x$的是\\bracket{20}.\n\\fourch{$x^2+4$}{$2x+4$}{$x^3+4$}{$\\dfrac 1{x^2+4}$}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260111-不等式的性质(2)" ], "genre": "选择题", "ans": "A", @@ -534274,7 +534996,8 @@ "content": "``$a(a-b)<0$''是``$\\dfrac ba>1$''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": [ - "第一单元" + "第一单元", + "W20260115-2026届高一上学期国庆复习卷" ], "genre": "选择题", "ans": "", @@ -534330,7 +535053,8 @@ "content": "比较下列各题中两式值的大小:\\\\\n(1) $x(x-y)$与$y(x-y),(x\\ne y)$;\\\\\n(2) $(3a+1)(a+1)$与$2(a+1)^2-3$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260111-不等式的性质(2)" ], "genre": "解答题", "ans": "(1) $x(x-y)>y(x-y)$;\\\\\n(2) $(3a+1)(a+1)<2(a+1)^2-3$.", @@ -534368,7 +535092,8 @@ "content": "已知实数$ab^2$, 那么下列不等式中正确的是\\bracket{20}.\n\\fourch{$a>0>b$}{$a>b>0$}{$|a|>|b|$}{$a>|b|$}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260111-不等式的性质(2)" ], "genre": "选择题", "ans": "C", @@ -534605,7 +535332,8 @@ "content": "若$x\\dfrac 1x$}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260111-不等式的性质(2)" ], "genre": "选择题", "ans": "B", @@ -534773,7 +535501,8 @@ "content": "已知$a,b,m,n$都是正实数, 且$m+n=1$, 比较$\\sqrt {ma+nb}$和$m\\sqrt a+n\\sqrt b$的大小.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260111-不等式的性质(2)" ], "genre": "解答题", "ans": "当$a=b$时, $\\sqrt {ma+nb}=m\\sqrt a+n\\sqrt b$; 当$a\\ne b$时, $\\sqrt {ma+nb}>m\\sqrt a+n\\sqrt b$.", @@ -534838,7 +535567,8 @@ "content": "已知$m$为常数. 若对于任意的$c$, 关于$x$的一元二次方程$mx^2+2x+c+1=0$总有实根, 求$m$的取值范围.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260115-2026届高一上学期国庆复习卷" ], "genre": "解答题", "ans": "", @@ -534905,7 +535635,8 @@ "content": "已知关于$x$的不等式$(4a-3b)x>2b-a$的解集为$(-\\infty ,\\dfrac 49)$, 则$ax>b$的解集是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260112-一元一次不等式, 一元二次不等式的求解(1)" ], "genre": "填空题", "ans": "$(-\\infty,\\dfrac{5}{6})$", @@ -534975,7 +535706,8 @@ "content": "设$a$为实数, 求关于$x$的一元一次不等式组$\\begin{cases}\n3x<5a+6, \\\\-2x<4a \\end{cases}$的解集.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260112-一元一次不等式, 一元二次不等式的求解(1)" ], "genre": "解答题", "ans": "当$a>-\\dfrac{6}{11}$时, 解集为$(-\\infty,\\dfrac{5a+6}{3})$; 当$a\\le -\\dfrac{6}{11}$时, 解集为$\\varnothing$", @@ -535018,7 +535750,8 @@ "content": "解关于$x$的不等式: $a^2(x-1)>b^2(1+x)+2ab$, 其中$a,b>0$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260112-一元一次不等式, 一元二次不等式的求解(1)" ], "genre": "解答题", "ans": "当$a=b$时, 解集为$\\varnothing$; 当$a>b$时, 解集为$(\\dfrac{a+b}{a-b},+\\infty)$; 当$a0, \\\\x-3b\\le 0 \\end{cases}$的整数解有且仅有$4,5$, 则$a$的取值范围是\\blank{50}, $b$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260112-一元一次不等式, 一元二次不等式的求解(1)" ], "genre": "填空题", "ans": "$[\\dfrac{3}{2},2)$; $[\\dfrac{5}{3},2)$", @@ -535100,7 +535834,8 @@ "content": "设$a$为实数, 求关于$x$的一元一次不等式$\\begin{cases} ax<1, \\\\ -2x<4a \\end{cases}$的解集.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260112-一元一次不等式, 一元二次不等式的求解(1)" ], "genre": "解答题", "ans": "当$a=0$时, 解集为$[0,+\\infty)$; 当$a>0$时, 解集为$(-2a,\\dfrac{1}{a})$; 当$a<0$时, 解集为$(-2a,+\\infty)$", @@ -535198,7 +535933,8 @@ "content": "已知$a\\in \\mathbf{R}$, 求下列关于$x$的不等式的解集:\\\\ \n(1) $(x-a)(x-1)<0(a>1)$;\\\\\n(2) $(x-a)(x-2a)<0(a>0)$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260113-一元二次不等式的求解(2)" ], "genre": "解答题", "ans": "(1) 解集为$(1,a)$; (2) 解集为$(a,2a)$", @@ -535347,7 +536083,8 @@ "content": "若函数$y=ax^2+2ax+1$的图像与$x$轴无交点, 求实数$a$的取值范围.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260113-一元二次不等式的求解(2)" ], "genre": "解答题", "ans": "$[0,1)$", @@ -535382,7 +536119,8 @@ "content": "已知集合$A=\\{x|x^2+(a-3)x+a<0\\}$非空, 且$A\\subset (0,+\\infty)$, 求实数$a$的取值范围.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260114-一元二次不等式的求解(3)" ], "genre": "解答题", "ans": "$[0,1)$", @@ -535440,7 +536178,8 @@ "content": "已知不等式$x^2+ax+b<0$的解集为$(-3,-1)$, 求实数$a$、$b$的值.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260114-一元二次不等式的求解(3)" ], "genre": "解答题", "ans": "$a=4$, $b=3$", @@ -535502,7 +536241,8 @@ "content": "已知关于$x$的不等式$ax^2+bx+c>0$的解集为$(-1,2)$, 求不等式$cx^2+ax-b\\le 0$的解集.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260114-一元二次不等式的求解(3)" ], "genre": "解答题", "ans": "$[-\\dfrac{1}{2},1]$", @@ -535536,7 +536276,8 @@ "content": "已知关于$x$的不等式组$\\begin{cases} (2x-3)(3x+2)\\le 0, \\\\ x-a>0 \\end{cases}$无实数解, 求实数$a$的取值范围.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260113-一元二次不等式的求解(2)" ], "genre": "解答题", "ans": "$[\\dfrac{3}{2},+\\infty)$", @@ -535596,7 +536337,8 @@ "content": "若不等式组$\\begin{cases} ax^2-x-2\\le 0, \\\\ x^2-x\\ge a(1-x) \\end{cases}$的解集为$\\mathbf{R}$, 求$a$的取值范围.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260113-一元二次不等式的求解(2)" ], "genre": "解答题", "ans": "$a=-1$", @@ -535631,7 +536373,8 @@ "content": "若关于$x$的不等式$ax^2-(a+1)x+1<0$的解集为$\\varnothing$, 求实数$a$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260114-一元二次不等式的求解(3)" ], "genre": "解答题", "ans": "$a=1$", @@ -535665,7 +536408,8 @@ "content": "若关于$x$的不等式$(a^2-1)x^2+2ax+1>0$有实数解, 求$a$的取值范围.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260114-一元二次不等式的求解(3)" ], "genre": "解答题", "ans": "$\\mathbf{R}$", @@ -535767,7 +536511,8 @@ "content": "定义运算``$\\ast$''满足如下法则: $a\\ast b=\\dfrac{a^2-1}b$, 则不等式$x\\ast (x+1)<0$的解集是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260115-分式不等式的求解" ], "genre": "填空题", "ans": "$(-\\infty,-1)\\cup (-1,1)$", @@ -535802,7 +536547,8 @@ "content": "若$a0$的解集是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260115-分式不等式的求解" ], "genre": "填空题", "ans": "$(-\\infty,-b)\\cup (-a,+\\infty)$", @@ -535839,7 +536585,8 @@ "content": "已知$f(x),g(x)$是定义在$\\mathbf{R}$上的函数, 若不等式$f(x)\\ge 0$的解集为$[1,2]$, 不等式$g(x)\\ge 0$的解集为$\\varnothing$, 则不等式$\\dfrac{f(x)}{g(x)}>0$的解集是\\bracket{20}.\n\\fourch{$\\varnothing$}{$(-\\infty ,1)\\cup (2,+\\infty)$}{$[1,2)$}{$\\mathbf{R}$}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260115-分式不等式的求解" ], "genre": "选择题", "ans": "B", @@ -535921,7 +536668,8 @@ "content": "若$x=5$是不等式$|x-a|\\le 4$的解中的最大值, 则$a$的值等于\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260116-含绝对值不等式的求解" ], "genre": "填空题", "ans": "$1$", @@ -535955,7 +536703,8 @@ "content": "已知集合$A=\\{x|$存在实数$y$, 使得$y=\\sqrt {x^2-2x-8}$成立$\\}$, \n集合$B=\\{x|1-|x-a|>0,\\ x\\in \\mathbf{R}\\}$, 若$A\\cap B=\\varnothing$, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260116-含绝对值不等式的求解" ], "genre": "填空题", "ans": "$(-\\infty,-3]\\cup [5,+\\infty)$", @@ -535989,7 +536738,8 @@ "content": "求下列不等式的解集.\\\\\n(1) $|x+3|-|x-1|<1$;\\\\\n(2) $|x+1|<\\dfrac 1{x-1}$;\\\\\n(3)\t$x^2-2|x|-15\\le 0$;\\\\\n(4) $x^2-x-5>|2x-1|$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260116-含绝对值不等式的求解" ], "genre": "解答题", "ans": "(1) $(-\\infty,-\\dfrac{1}{2})$; (2) $(1,\\sqrt{2})$; (3) $[-5,5]$; (4) $(-\\infty,-3)\\cup (4,+\\infty)$", @@ -536023,7 +536773,8 @@ "content": "已知不等式$|ax+1|\\le b$的解集是$[-1,3]$, 求$a$、$b$的值.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260116-含绝对值不等式的求解" ], "genre": "解答题", "ans": "$a=-1$, $b=2$", @@ -536059,7 +536810,8 @@ "content": "设$x\\in \\mathbf{R}$, 则$(1-|x|)(1+x)>0$成立的充要条件是\\bracket{20}.\n\\fourch{$|x|<1$}{$x<1$}{$|x|>1$}{$x<1$且$x\\ne -1$}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260116-含绝对值不等式的求解" ], "genre": "选择题", "ans": "D", @@ -536095,7 +536847,8 @@ "content": "若实数$a,b$满足$ab>0$, 则在``\\textcircled{1} $|a+b|>|a|$, \\textcircled{2} $|a+b|<|b|$, \\textcircled{3} $|a+b|<|a-b|$, \\textcircled{4} $|a+b|>|a-b|$''这四个式子中, 正确的是\\bracket{20}\n\\fourch{\\textcircled{1}\\textcircled{2}}{\\textcircled{1}\\textcircled{3}}{\\textcircled{1}\\textcircled{4}}{\\textcircled{2}\\textcircled{4}}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260119-三角不等式" ], "genre": "选择题", "ans": "C", @@ -536137,7 +536890,8 @@ "content": "若$|a+b|<-c$, 则在``\\textcircled{1} $a<-b-c$, \\textcircled{2} $a+b>c$, \\textcircled{3} $a+c0$的解集是$\\{x|x\\ne 2\\}$, 求实数$a$与$b$的值.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260114-一元二次不等式的求解(3)" ], "genre": "解答题", "ans": "$a=\\dfrac{1}{4}$, $b=-1$", @@ -536320,7 +537075,8 @@ "content": "已知集合$A=\\{x|x^2-6x+5<0,\\ x\\in \\mathbf{R}\\}$, $B=\\{x|x^2-3ax+2a^2<0, \\ x\\in \\mathbf{R}\\}$.\\\\\n(1) 若$A\\cap B=\\varnothing$, 求实数$a$的取值范围;\\\\\n(2) 若$B\\subseteq A$, 求实数$a$的取值范围.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260113-一元二次不等式的求解(2)" ], "genre": "解答题", "ans": "(1) $(-\\infty,\\dfrac{1}{2}]\\cup [5,+\\infty)$; (2) $\\{0\\}\\cup [1,\\dfrac{5}{2}]$", @@ -536377,7 +537133,8 @@ "content": "若$q<0\\dfrac 1p\\}$}{$\\{x|-\\dfrac 1p0$, $b>0$, 且$a\\ne b$, 则下列各式恒成立的是\\bracket{20}.\n\\twoch{$\\dfrac{2ab}{a+b}<\\dfrac{a+b}2<\\sqrt {ab}$}{$\\sqrt {ab}<\\dfrac{2ab}{a+b}<\\dfrac{a+b}2$}{$\\dfrac{2ab}{a+b}<\\sqrt {ab}<\\dfrac{a+b}2$}{$\\sqrt {ab}<\\dfrac{a+b}2<\\dfrac{2ab}{a+b}$}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260117-平均值不等式及其应用(1)" ], "genre": "选择题", "ans": "C", @@ -536702,7 +537465,8 @@ "content": "已知$a,b\\in \\mathbf{R}$. 若$M=a^2+b^2+1$, $N=a+b+ab$, 则$M$、$N$的大小关系是\\bracket{20}. \n\\fourch{$M\\ge N$}{$M\\le N$}{$M=N$}{无法确定}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260117-平均值不等式及其应用(1)" ], "genre": "选择题", "ans": "A", @@ -536737,7 +537501,8 @@ "content": "设$ab\\ne 0$, 利用基本不等式有如下证明: $\\dfrac ba+\\dfrac ab=\\dfrac{b^2+a^2}{ab}\\ge \\dfrac{2ab}{ab}=2$.试判断这个证明过程是否正确. 若正确, 请说明每一步的依据; 若不正确, 请说明理由.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260117-平均值不等式及其应用(1)" ], "genre": "解答题", "ans": "不正确, 在基本不等式两边同除$ab$时没有考虑到$ab$的符号.", @@ -536775,7 +537540,8 @@ "content": "若$a>0$, $b>0$, $c>0$, $d>0$, \n则$\\dfrac ba+\\dfrac ab\\ge$\\blank{50}, $\\dfrac{b+c}a+\\dfrac{c+a}b+\\dfrac{a+b}c\\ge$\\blank{50}, $(a+b)(\\dfrac 1a+\\dfrac 1b)\\ge$\\blank{50}, $(\\dfrac ba+\\dfrac dc)(\\dfrac cb+\\dfrac ad)\\ge$\\blank{50}.\n(填能使不等式成立的最大值)", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260117-平均值不等式及其应用(1)" ], "genre": "填空题", "ans": "$2$, $6$, $4$, $4$", @@ -536832,7 +537598,8 @@ "content": "求证:\\\\\n(1) 若$x>0$, $y>0$, 则$\\sqrt {(1+x)(1+y)}\\ge 1+\\sqrt {xy}$;\\\\\n(2) 若$a>0$, $b>0$, 则$a+b+\\dfrac 1{\\sqrt {ab}}\\ge 2\\sqrt 2$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260117-平均值不等式及其应用(1)" ], "genre": "解答题", "ans": "(1) 证明略; (2) 证明略", @@ -536867,7 +537634,8 @@ "content": "若实数$x+y=4$, 则$x^2+y^2$有最\\blank{50}值, 且此最值为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260118-平均值不等式及其应用(2)" ], "genre": "填空题", "ans": "小, $8$", @@ -536902,7 +537670,8 @@ "content": "已知$00$, 则$2x+\\dfrac 8{x+1}$的最小值是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260118-平均值不等式及其应用(2)" ], "genre": "填空题", "ans": "$6$", @@ -536986,7 +537757,8 @@ "content": "已知$x>-1$, 则当$x$取\\blank{50}时, $x+\\dfrac 4{x+1}$的值最小.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260118-平均值不等式及其应用(2)" ], "genre": "填空题", "ans": "$1$", @@ -537043,7 +537815,8 @@ "content": "若$x<1$, 则$\\dfrac{x^2-2x+3}{x-1}$有最\\blank{50}值, 且此最值为\\blank{50}, 此时$x=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260118-平均值不等式及其应用(2)" ], "genre": "填空题", "ans": "大, $-2\\sqrt{2}$, $1-\\sqrt{2}$", @@ -537125,7 +537898,8 @@ "content": "已知$x,y>0$, 且$x+y=1$, 求当$x,y$分别取何值时, $\\dfrac 1x+\\dfrac 2y$的值最小.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260118-平均值不等式及其应用(2)" ], "genre": "解答题", "ans": "$x=\\sqrt{2}-1$, $y=2-\\sqrt{2}$", @@ -537277,7 +538051,8 @@ "content": "已知实数$a,b,c$满足$|a-c|<1$, $|b-c|<1$, 证明: $|a-b|<2$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260119-三角不等式" ], "genre": "解答题", "ans": "证明略", @@ -537310,7 +538085,8 @@ "content": "若$|x-3|+|x-4|+|x-5|\\ge a$对一切实数$x$恒成立, 则$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260119-三角不等式" ], "genre": "填空题", "ans": "$(-\\infty,2]$", @@ -537343,7 +538119,8 @@ "content": "如果实数$a,b,c,d$满足$|a-2b|\\le 6$, $|b-d|\\le 7$, $|a-3b+d|\\ge 13$, 则$|a-2b|-|b-d|=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260119-三角不等式" ], "genre": "填空题", "ans": "$-1$", @@ -537376,7 +538153,8 @@ "content": "已知$|x|\\le 3$, $|y|\\le 1$, $|z|\\le 4$, 且$|x-2y+z| =9$, 则$2x-3y+z=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260119-三角不等式" ], "genre": "填空题", "ans": "$13$或$-13$", @@ -537526,7 +538304,8 @@ "content": "已知$a>0$, $b>0$, 求证: $\\dfrac a{\\sqrt b}+\\dfrac b{\\sqrt a}\\ge \\sqrt a+\\sqrt b$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260111-不等式的性质(2)" ], "genre": "解答题", "ans": "证明略", @@ -540833,7 +541612,8 @@ "objs": [], "tags": [ "第二单元", - "第三章" + "第三章", + "G20260120-指数幂的拓展(1)" ], "genre": "填空题", "ans": "(1) $2$; (2) $-2$; (3) $4$; (4) $-3$; (5) $\\sqrt{3}-\\sqrt{2}$; (6) $|a-b|^3$", @@ -540868,7 +541648,8 @@ "objs": [], "tags": [ "第二单元", - "第三章" + "第三章", + "G20260121-指数幂的拓展(2)" ], "genre": "填空题", "ans": "(1) $a^{\\frac{3}{4}}$; (2) $a^{\\frac{23}{5}}$; (3) $a^{\\frac{1}{2}}\\cdot b^{-\\frac{4}{3}}$; (4) $a^{\\frac{3}{4}}\\cdot b^{\\frac{11}{4}}$; (5) $2\\cdot 3^{-\\frac{1}{2}}\\cdot a^{-\\frac{1}{2}}\\cdot b^{-\\frac{3}{2}}$; (6) $|a-b|^{-\\frac{1}{3}}$", @@ -540995,7 +541776,8 @@ "objs": [], "tags": [ "第二单元", - "第三章" + "第三章", + "G20260120-指数幂的拓展(1)" ], "genre": "选择题", "ans": "C", @@ -541030,7 +541812,8 @@ "objs": [], "tags": [ "第二单元", - "第三章" + "第三章", + "G20260120-指数幂的拓展(1)" ], "genre": "选择题", "ans": "B", @@ -541065,7 +541848,8 @@ "objs": [], "tags": [ "第二单元", - "第三章" + "第三章", + "G20260120-指数幂的拓展(1)" ], "genre": "填空题", "ans": "$|a|+2a$", @@ -541146,7 +541930,8 @@ "objs": [], "tags": [ "第二单元", - "第三章" + "第三章", + "G20260121-指数幂的拓展(2)" ], "genre": "填空题", "ans": "$1$", @@ -541181,7 +541966,8 @@ "objs": [], "tags": [ "第二单元", - "第三章" + "第三章", + "G20260121-指数幂的拓展(2)" ], "genre": "填空题", "ans": "$2^{-\\frac{9}{4}}$", @@ -541239,7 +542025,8 @@ "objs": [], "tags": [ "第二单元", - "第三章" + "第三章", + "G20260120-指数幂的拓展(1)" ], "genre": "选择题", "ans": "B", @@ -541343,7 +542130,8 @@ "objs": [], "tags": [ "第二单元", - "第三章" + "第三章", + "G20260120-指数幂的拓展(1)" ], "genre": "解答题", "ans": "$-2$", @@ -541424,7 +542212,8 @@ "objs": [], "tags": [ "第二单元", - "第三章" + "第三章", + "G20260122-对数的定义" ], "genre": "解答题", "ans": "(1) $16^x=32$; (2) $\\sqrt{5}^4=x$; (3) $x^=9$($x>0$); (4) $16^{-\\frac{1}{4}}=\\dfrac{x}{2}$", @@ -541459,7 +542248,8 @@ "objs": [], "tags": [ "第二单元", - "第三章" + "第三章", + "G20260122-对数的定义" ], "genre": "解答题", "ans": "(1) $2$; (2) $-3$; (3) $-\\dfrac{1}{4}$; (4) $3$", @@ -541495,7 +542285,8 @@ "objs": [], "tags": [ "第二单元", - "第三章" + "第三章", + "G20260122-对数的定义" ], "genre": "解答题", "ans": "(1) $\\pm \\dfrac{1}{2}$; (2) $-3$; (3) $\\dfrac{3}{2}$; (4) $2$", @@ -541530,7 +542321,8 @@ "objs": [], "tags": [ "第二单元", - "第三章" + "第三章", + "G20260122-对数的定义" ], "genre": "填空题", "ans": "(1) $5$; (2) $1$; (3) $3$; (4) $7$", @@ -541611,7 +542403,8 @@ "objs": [], "tags": [ "第二单元", - "第三章" + "第三章", + "G20260123-对数的运算性质" ], "genre": "选择题", "ans": "B", @@ -541646,7 +542439,8 @@ "objs": [], "tags": [ "第二单元", - "第三章" + "第三章", + "G20260122-对数的定义" ], "genre": "解答题", "ans": "$64$", @@ -541681,7 +542475,8 @@ "objs": [], "tags": [ "第二单元", - "第三章" + "第三章", + "G20260122-对数的定义" ], "genre": "解答题", "ans": "(1) $\\mathbf{R}$; (2) $(-\\infty,-3)\\cup (1,+\\infty)$; (3) $(-2,+\\infty)$; (4) $(1,2)\\cup (2,3)$", @@ -541716,7 +542511,8 @@ "objs": [], "tags": [ "第二单元", - "第三章" + "第三章", + "G20260122-对数的定义" ], "genre": "解答题", "ans": "$32$", @@ -541751,7 +542547,8 @@ "objs": [], "tags": [ "第二单元", - "第三章" + "第三章", + "G20260123-对数的运算性质" ], "genre": "解答题", "ans": "(1) $\\dfrac{5}{2}$; (2) $\\dfrac{5}{2}$; (3) $1$; (4) $1$", @@ -541832,7 +542629,8 @@ "objs": [], "tags": [ "第二单元", - "第三章" + "第三章", + "G20260123-对数的运算性质" ], "genre": "选择题", "ans": "A", @@ -541868,7 +542666,8 @@ "objs": [], "tags": [ "第二单元", - "第三章" + "第三章", + "G20260123-对数的运算性质" ], "genre": "选择题", "ans": "C", @@ -541903,7 +542702,8 @@ "objs": [], "tags": [ "第二单元", - "第三章" + "第三章", + "G20260123-对数的运算性质" ], "genre": "填空题", "ans": "$a+b+1$", @@ -541938,7 +542738,8 @@ "objs": [], "tags": [ "第二单元", - "第三章" + "第三章", + "G20260123-对数的运算性质" ], "genre": "填空题", "ans": "$1-\\dfrac{1}{2}m$", @@ -541973,7 +542774,8 @@ "objs": [], "tags": [ "第二单元", - "第三章" + "第三章", + "G20260122-对数的定义" ], "genre": "填空题", "ans": "$\\{\\dfrac{1}{3},2,3\\}$", @@ -542008,7 +542810,8 @@ "objs": [], "tags": [ "第二单元", - "第三章" + "第三章", + "G20260123-对数的运算性质" ], "genre": "填空题", "ans": "$\\dfrac{1}{6}$", @@ -542043,7 +542846,8 @@ "objs": [], "tags": [ "第二单元", - "第三章" + "第三章", + "G20260124-对数的换底" ], "genre": "解答题", "ans": "$1$", @@ -542136,7 +542940,8 @@ "objs": [], "tags": [ "第二单元", - "第三章" + "第三章", + "G20260124-对数的换底" ], "genre": "填空题", "ans": "(1) $1-a$; (2) $\\dfrac{b}{a}$; (3) $\\dfrac{2-2a}{2a_b}$", @@ -542218,7 +543023,8 @@ "objs": [], "tags": [ "第二单元", - "第三章" + "第三章", + "G20260124-对数的换底" ], "genre": "填空题", "ans": "$\\dfrac{a+b-2}{2}$", @@ -542253,7 +543059,8 @@ "objs": [], "tags": [ "第二单元", - "第三章" + "第三章", + "G20260157-必修第三章幂、指数与对数复习" ], "genre": "填空题", "ans": "如$bc+2ac=3ab$", @@ -542289,7 +543096,8 @@ "objs": [], "tags": [ "第二单元", - "第三章" + "第三章", + "G20260124-对数的换底" ], "genre": "解答题", "ans": "$\\dfrac{3+a}{a+ab}$", @@ -542346,7 +543154,8 @@ "objs": [], "tags": [ "第二单元", - "第三章" + "第三章", + "G20260124-对数的换底" ], "genre": "解答题", "ans": "$2$", @@ -542380,7 +543189,8 @@ "objs": [], "tags": [ "第二单元", - "第三章" + "第三章", + "G20260124-对数的换底" ], "genre": "解答题", "ans": "$-4$", @@ -542435,7 +543245,8 @@ "content": "若函数$y=(m^2-2 m-2) x^{m+1}$是幂函数, 则实数$m=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260125-幂函数的定义与图像" ], "genre": "填空题", "ans": "$-1$或$3$", @@ -542470,7 +543281,8 @@ "content": "已知幂函数$y=x^a$的图像经过点$(2, \\dfrac{\\sqrt 2}2)$, 则当$x=4$时的函数值为\\bracket{20}.\n\\fourch{$16$}{$\\dfrac 1{16}$}{$\\dfrac 12$}{$2$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260125-幂函数的定义与图像" ], "genre": "选择题", "ans": "C", @@ -542492,7 +543304,8 @@ "content": "下列幂函数中, 定义域为$\\{x | x>0\\}$的是\\bracket{20}.\n\\fourch{$y=x^{\\frac 23}$}{$y=x^{\\frac 32}$}{$y=x^{-\\frac 23}$}{$y=x^{-\\frac 32}$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260125-幂函数的定义与图像" ], "genre": "选择题", "ans": "D", @@ -542514,7 +543327,8 @@ "content": "幂函数$y=x^n(n \\in \\mathbf{Z})$的图像一定不经过\\bracket{20}.\n\\fourch{第一象限}{第二象限}{第三象限}{第四象限}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260125-幂函数的定义与图像" ], "genre": "选择题", "ans": "D", @@ -542536,7 +543350,8 @@ "content": "求下列函数的定义域, 并作出它们的大致图像:\\\\\n(1) $y=x^{\\frac 15}$;\\\\\n(2) $y=x^{\\frac 43}$;\\\\\n(3) $y=x^{-\\frac 34}$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260125-幂函数的定义与图像" ], "genre": "解答题", "ans": "(1) $\\mathbf{R}$, 大致图像为\\begin{tikzpicture}[>=latex,scale = 0.3]\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 [domain = 0:{exp(ln(3)/5)}, samples = 100] plot ({\\x*\\x*\\x*\\x*\\x},\\x) plot ({-\\x*\\x*\\x*\\x*\\x},-\\x);\n\\end{tikzpicture}; (2) $\\mathbf{R}$, 大致图像为\\begin{tikzpicture}[>=latex,scale = 0.3]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0:1.5, samples = 100] plot ({\\x*\\x*\\x},{\\x*\\x*\\x*\\x}) plot ({-\\x*\\x*\\x},{\\x*\\x*\\x*\\x});\n\\end{tikzpicture}; (3) $(0,+\\infty)$, 大致图像为\\begin{tikzpicture}[>=latex,scale = 0.3]\n\\draw [->] (-0.5,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0.6:1.45, samples = 100] plot ({\\x*\\x*\\x*\\x},{1/\\x/\\x/\\x});\n\\end{tikzpicture}", @@ -542558,7 +543373,8 @@ "content": "若幂函数$y=x^{-m^2+2 m+3}$($m$为整数)的定义域是$\\mathbf{R}$, 求$m$的值.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260125-幂函数的定义与图像" ], "genre": "解答题", "ans": "$0,1$或$2$", @@ -542580,7 +543396,8 @@ "content": "若函数$y=(m x^2+m x+2)^{\\frac 34}$的定义域是$\\mathbf{R}$, 求实数$m$的取值范围.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260125-幂函数的定义与图像" ], "genre": "解答题", "ans": "$[0,8]$", @@ -542681,7 +543498,8 @@ "content": "已知函数$y=x^a$的图像关于$y$轴对称, 且在区间$(0,+\\infty)$上为严格减函数, 其中$a \\in\\{-1,1,-2,2, \\dfrac 12, 3, \\dfrac 13\\}$, 则$a=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "I20260104-高一上学期随堂练习04" ], "genre": "填空题", "ans": "$-2$", @@ -542782,7 +543600,8 @@ "content": "已知函数$y=\\dfrac x{x-1}$, 则下列四个命题错误的是\\bracket{20}.\n\\onech{该函数的图像关于点$(1,1)$对称}{该函数的图像关于直线$y=2-x$对称}{该函数在定义域内是严格减函数}{将该函数图像向左平移一个单位, 再向下平移一个单位后与函数$y=\\dfrac 1x$的图像重合}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260126-幂函数的性质" ], "genre": "选择题", "ans": "C", @@ -542816,7 +543635,8 @@ "content": "利用幂函数的性质比较下列两个值的大小:\\\\\n(1) $(-0.2)^{\\frac 25}$\\blank{50}$(-0.3)^{\\frac 25}$;\\\\\n(2) $(-2)^{-\\frac 25}$\\blank{50}$3^{-\\frac 25}$;\\\\\n(3) $1.68^{0.8}$\\blank{50}$(-1.71)^{0.8}$;\\\\\n(4) $0.3^{-\\frac 12}$\\blank{50}$0.2^{-\\frac 12}$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260126-幂函数的性质" ], "genre": "填空题", "ans": "(1) $(-0.2)^{\\frac 25}<(-0.3)^{\\frac 25}$; (2) $(-2)^{-\\frac 25}>3^{-\\frac 25}$; (3) $1.68^{0.8}<(-1.71)^{0.8}$; (4) $0.3^{-\\frac 12}<0.2^{-\\frac 12}$", @@ -542942,7 +543762,8 @@ "content": "由下列不等式, 分别求出实数$a$的取值范围.\\\\\n(1) $(a+2)^{\\frac 23}>(1-2 a)^{\\frac 23}$;\\\\\n(2) $(a+2)^{\\frac 13}>(3-2 a)^{\\frac 13}$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260126-幂函数的性质" ], "genre": "解答题", "ans": "(1) $(-\\dfrac{1}{3},3)$; (2) $(\\dfrac{1}{3},+\\infty)$", @@ -543000,7 +543821,8 @@ "content": "下列函数是指数函数的序号为\\blank{50}, 是幂函数的序号为\\blank{50}.\\\\\n\\textcircled{1} $y=x$; \\textcircled{2} $y=x^3$; \\textcircled{3} $y=\\mathrm{e}^x$; \\textcircled{4} $y=\\sqrt[3]x$; \\textcircled{5} $y=2^{-x}$; \\textcircled{6} $y=2^x$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260127-指数函数的定义与图像" ], "genre": "填空题", "ans": "\\textcircled{3}\\textcircled{5}\\textcircled{6}, \\textcircled{1}\\textcircled{2}\\textcircled{4}", @@ -543035,7 +543857,8 @@ "content": "下列函数是指数函数的序号为\\blank{50}.(请填入全部正确序号)\\\\\n\\textcircled{1} $y=(-2)^x$; \\textcircled{2} $y=3 \\cdot 2^x$; \\textcircled{3} $y=\\pi^x$; \\textcircled{4} $y=x^3$; \\textcircled{5} $y=2^{-x}$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260127-指数函数的定义与图像" ], "genre": "填空题", "ans": "\\textcircled{3}\\textcircled{5}", @@ -543094,7 +543917,8 @@ "content": "已知常数$a>0$且$a \\neq 1$. 若无论$a$取何值, 函数$y=a^{x+2}+2$的图像恒经过一个定点, 则此定点的坐标是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260128-指数函数的性质(1)" ], "genre": "填空题", "ans": "$(-2,3)$", @@ -543129,7 +543953,8 @@ "content": "函数$y=2^x$的图像与函数$y=2^{-x}$的图像关于\\blank{50}对称, 它们的交点坐标是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260127-指数函数的定义与图像" ], "genre": "填空题", "ans": "$y$轴, $(0,1)$", @@ -543164,7 +543989,8 @@ "content": "在同一坐标系中分别作出下列函数的大致图像:\\\\\n(1) $y=3^x$;\n(2) $y=(\\dfrac 13)^x$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260127-指数函数的定义与图像" ], "genre": "解答题", "ans": "\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,5.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -3:{ln(5.5)/ln(3)}, samples = 100] plot (\\x,{exp(\\x*ln(3))}) node [right] {$y=3^x$};\n\\draw [domain = {ln(5.5)/ln(1/3)}:3, samples = 100] plot (\\x,{exp(\\x*ln(1/3))}) node [right] {$y=(\\frac{1}{3})^x$};\n\\end{tikzpicture}", @@ -543265,7 +544091,9 @@ "content": "若指数函数$y=(m-1)^x$在$\\mathbf{R}$上是严格减函数, 则实数$m$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260128-指数函数的性质(1)", + "I20260104-高一上学期随堂练习04" ], "genre": "填空题", "ans": "$(1,2)$", @@ -543344,7 +544172,8 @@ "content": "下列条件中, 能够使得指数函数$y=a^x$为减函数的是\\bracket{20}.\n\\fourch{$a>1$}{$a<1$}{$a(a-1)<0$}{$a(a-1)>0$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260128-指数函数的性质(1)" ], "genre": "选择题", "ans": "C", @@ -543379,7 +544208,8 @@ "content": "若函数$y=2^x-m$的图像不经过第二象限, 则实数$m$的取值范围是\\bracket{20}.\n\\fourch{$m \\geq 1$}{$m<1$}{$m>-1$}{$m \\le -1$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260127-指数函数的定义与图像" ], "genre": "选择题", "ans": "A", @@ -543414,7 +544244,8 @@ "content": "若指数函数\\textcircled{1} $y=a^x$, \\textcircled{2} $y=b^x$, \\textcircled{3} $y=c^x$, \\textcircled{4} $y=d^x$在同一坐标系内的图像如图所示, 则$a$、$b$、$c$、$d$的大小顺序是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-0) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1:2] plot (\\x,{pow(3,-\\x)}) (-1,3) node [above] {\\textcircled{1}};\n\\draw [domain = -0.5:2] plot (\\x,{pow(9,-\\x)}) (-0.5,3) node [above] {\\textcircled{2}};\n\\draw [domain = -2:0.5] plot (\\x,{pow(9,\\x)}) (0.5,3) node [above] {\\textcircled{3}};\n\\draw [domain = -2:1] plot (\\x,{pow(3,\\x)}) (1,3) node [above] {\\textcircled{4}};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$b=latex,scale = 0.7]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1.8:2] plot (\\x,{0.8*\\x+0.45});\n\\draw [domain = -1.6:2] plot (\\x,{pow(0.45,0.8*\\x)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1.3:0.8] plot (\\x,{1.8*\\x+1.45});\n\\draw [domain = -1.6:2] plot (\\x,{pow(0.45,0.8*\\x)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1:1.3] plot (\\x,{-1.6*\\x+1.3});\n\\draw [domain = -1.6:2] plot (-\\x,{pow(0.45,0.8*\\x)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -2:2] plot (\\x,{-0.5*\\x+0.3});\n\\draw [domain = -1.6:2] plot (\\x,{pow(0.45,0.8*\\x)});\n\\end{tikzpicture}}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260127-指数函数的定义与图像" ], "genre": "选择题", "ans": "A", @@ -543506,7 +544338,8 @@ "content": "已知$m, n$分别满足下列条件, 比较$m, n$的大小:\\\\\n(1) $(\\dfrac 54)^{-m}<(\\dfrac 54)^{-n}$;\\\\\n(2) $(0.7)^{\\frac 1m}>(0.7)^{\\frac 1n}$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260128-指数函数的性质(1)" ], "genre": "解答题", "ans": "(1) $m>n$; (2) $m,n$同号时, $m>n$; $m,n$异号时, $m0$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260129-指数函数的性质(2)" ], "genre": "解答题", "ans": "(1) $(-\\infty,1)\\cup (3,+\\infty)$; (2) $(1,+\\infty)$", @@ -543576,7 +544410,9 @@ "content": "若不等式$(2 a^2-1)^x<1$的解集为$(-\\infty, 0)$, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260129-指数函数的性质(2)", + "I20260104-高一上学期随堂练习04" ], "genre": "填空题", "ans": "$(-\\infty,-1)\\cup (1,+\\infty)$", @@ -543611,7 +544447,8 @@ "content": "若函数$y=a^x$(其中$a>0$且$a \\neq 1$)在区间$[-1,2]$上的最大值为$4$, 最小值为$m$, 且函数$y=(1-4 m) \\sqrt x$在$[0,+\\infty)$上是严格增函数, 则$a=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260129-指数函数的性质(2)" ], "genre": "填空题", "ans": "$\\dfrac{1}{4}$", @@ -543646,7 +544483,8 @@ "content": "已知$m, n$为常数, 且$a=0.9^m \\cdot 0.8^n$, $b=0.9^n \\cdot 0.8^m$, 试比较$a, b$的大小.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260128-指数函数的性质(1)" ], "genre": "解答题", "ans": "若$m>n$, 则$a>b$; 若$m=n$, 则$a=b$; 若$my>1$, 且$0a^{\\frac 1y}$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260128-指数函数的性质(1)" ], "genre": "填空题", "ans": "\\textcircled{1}\\textcircled{4}", @@ -543741,7 +544580,8 @@ "content": "函数值域为$(0,+\\infty)$的是\\bracket{20}.\n\\fourch{$y=\\dfrac 1{x-2}$}{$y=3^{x-1}$}{$y=\\sqrt{2^x-1}$}{$y=\\sqrt{5-3x}$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260133-函数" ], "genre": "选择题", "ans": "B", @@ -543776,7 +544616,8 @@ "content": "函数$y=\\sqrt {(\\dfrac 1{16})^x-64}$的定义域为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260129-指数函数的性质(2)" ], "genre": "填空题", "ans": "$(-\\infty,-\\dfrac{3}{2}]$", @@ -543813,7 +544654,8 @@ "content": "已知集合$M=\\{y | y=2^x,\\ x \\in \\mathbf{R}\\}$, 集合$N=\\{y | y=x^2, \\ x \\in \\mathbf{R}\\}$, 求$M \\cap N$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260128-指数函数的性质(1)" ], "genre": "解答题", "ans": "$(0,+\\infty)$", @@ -543870,7 +544712,8 @@ "content": "已知$0 \\leq x \\leq 2$, 求函数$y=4^{x-\\frac 12}-3 \\cdot 2^x+5$的最大值, 并求此时$x$的值.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260129-指数函数的性质(2)" ], "genre": "解答题", "ans": "最大值为$\\dfrac{5}{2}$, 此时$x=0$", @@ -543998,7 +544841,8 @@ "content": "已知常数$a>0$且$a \\neq 1$. 若无论$a$取何值, 函数$y=\\log_a(2 x+1)$恒经过一个定点, 则此定点的坐标是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260131-对数函数的性质(1)" ], "genre": "填空题", "ans": "$(0,0)$", @@ -544159,7 +545003,8 @@ "content": "判断函数$y=\\ln|x|$的图像是否关于某条直线对称? 请说明理由.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260130-对数函数的定义和图像" ], "genre": "解答题", "ans": "关于$y$轴对称, 理由略", @@ -544248,7 +545093,8 @@ "content": "若函数$y=\\log_a(x+1)$是严格减函数, 则实数$a$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260131-对数函数的性质(1)" ], "genre": "填空题", "ans": "$(0,1)$", @@ -544314,7 +545160,8 @@ "content": "根据下列不等式, 确定实数$a$的取值范围:\\\\ \n(1) $\\log_a 0.2<\\log_a 0.1$;\\\\\n(2) $\\log_a \\pi>\\log_a \\mathrm{e}$;\\\\\n(3) $\\log_a 3<0$;\\\\\n(4) $\\log_a \\dfrac 45<1$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260131-对数函数的性质(1)" ], "genre": "解答题", "ans": "(1) $(0,1)$; (2) $(1,+\\infty)$; (3) $(0,1)$; (4) $(0,\\dfrac{4}{5})\\cup (1,+\\infty)$", @@ -544353,7 +545200,8 @@ "content": "已知$10$且$a \\neq 1$. 若无论$a$取何值, 函数$y=4+\\log_a(x^2-x-1)$的图像恒经过定点$P$, 求点$P$的坐标.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260131-对数函数的性质(1)" ], "genre": "解答题", "ans": "$(2,4)$与$(-1,4)$", @@ -544431,7 +545280,8 @@ "content": "已知集合$P$是函数$y=\\dfrac 1{\\sqrt {1-x^2}}$的定义域, 集合$Q$是函数$y=\\log_{\\frac 12}(2+x-6 x^2)$的定义域, 求$P \\cap Q$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260130-对数函数的定义和图像" ], "genre": "解答题", "ans": "$(-\\dfrac{1}{2},\\dfrac{2}{3})$", @@ -544468,7 +545318,8 @@ "content": "已知常数$a \\in \\mathbf{R}$, 集合$A=\\{x | 2^{x^2-x-6}>1\\}$, 集合\n$B=\\{x | \\log_4(x+1)=latex,scale = 0.7]\n\\draw [->] (0,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0.3:5] plot (\\x,{ln(\\x)/ln(1.8)}) (5,{ln(5)/ln(1.8)}) node [right] {$y=\\log_a x$};\n\\draw [domain = 0.3:5] plot (\\x,{ln(\\x)/ln(2.5)}) (5,{ln(5)/ln(2.5)}) node [right] {$y=\\log_b x$};\n\\draw [domain = 0.3:5] plot (\\x,{ln(\\x)/ln(0.4)}) (5,{ln(5)/ln(0.4)}) node [right] {$y=\\log_c x$};\n\\draw [domain = 0.3:5] plot (\\x,{ln(\\x)/ln(0.6)}) (5,{ln(5)/ln(0.6)}) node [right] {$y=\\log_d x$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$a>b>c>d$}{$a>b>d>c$}{$b>a>c>d$}{$b>a>d>c$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260130-对数函数的定义和图像" ], "genre": "选择题", "ans": "D", @@ -544609,7 +545463,8 @@ "content": "函数$y=\\log_a x$在区间$[a, 2 a]$上的最大值与最小值的差为$\\dfrac 12$, 则$a=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260132-对数函数的性质(2)" ], "genre": "填空题", "ans": "$4$或$\\dfrac{1}{4}$", @@ -544648,7 +545503,8 @@ "content": "设$a>0$且$a \\neq 1$, 比较$\\log_a 2 a$与$\\log_a 3 a$的大小.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260131-对数函数的性质(1)" ], "genre": "解答题", "ans": "当$a>1$时, $\\log_a 2a<\\log_a 3a$; 当$0\\log_a 3a$", @@ -544801,7 +545657,8 @@ "content": "不等式$\\log_{\\frac 12}(4+3 x-x^2)>\\log_{\\frac 12}(4 x-2)$的解是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260132-对数函数的性质(2)" ], "genre": "填空题", "ans": "$(2,4)$", @@ -544921,7 +545778,8 @@ "content": "若函数$f(x)=\\lg (k x^2-4x+k+3)$的定义域是$\\mathbf{R}$, 求实数$k$的取值范围.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260132-对数函数的性质(2)" ], "genre": "解答题", "ans": "$(1,+\\infty)$", @@ -544989,7 +545847,8 @@ "content": "已知$x$满足不等式$\\log_2^2 x-5 \\log_2 x+6 \\leq 0$, 求函数$f(x)=\\log_2 \\dfrac x2 \\log_2 \\dfrac x4$的最大值和最小值.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260132-对数函数的性质(2)" ], "genre": "解答题", "ans": "最大值为$2$, 最小值为$0$", @@ -545055,7 +545914,8 @@ "content": "若$0>\\log_m 2>\\log_n 2$, 则实数$m, n$的大小关系为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260131-对数函数的性质(1)" ], "genre": "填空题", "ans": "$m=latex,xscale = 0.1, yscale = 0.3]\n\\draw [->] (0,0) -- (50,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,9) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j in {6/4,16/5,22/6,28/7,34/8}\n{\\draw [dashed] (\\i,0) node [below] {$\\i$} --++ (0,\\j) -- (0,\\j) node [left] {$\\j$};};\n\\draw (0,3) node [left] {$3$};\n\\foreach \\i/\\j/\\k in {0/3/6,6/4/16,16/5/22,22/6/28,28/7/34}\n{\\draw (\\i,\\j) -- (\\k,\\j);\n\\filldraw [fill = white] (\\i,\\j) circle (0.45 and 0.15);\n\\filldraw (\\k,\\j) circle (0.45 and 0.15);\n};\n\\draw (34,8) -- (50,8);\n\\filldraw [fill = white] (34,8) circle (0.45 and 0.15);\n\\end{tikzpicture}; (2) 不能超过$22$千米", @@ -545557,7 +546419,8 @@ "content": "已知$f(x)=\\begin{cases}x-2, & x>8 \\\\f(x+3), & x \\leq 8,\\end{cases}$ 则$f(2)=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260134-函数的表示方法" ], "genre": "填空题", "ans": "$9$", @@ -545597,7 +546460,8 @@ "content": "如果函数$f(x)=\\dfrac{x-1}{x-2}$, 那么$f(x+1)=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260133-函数" ], "genre": "填空题", "ans": "$\\dfrac{x}{x-1}$", @@ -545637,7 +546501,8 @@ "content": "如果函数$f(x)=2 x^2-1$, 那么$f(f(x))=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260133-函数" ], "genre": "填空题", "ans": "$8x^4-8x^2+1$", @@ -545677,7 +546542,8 @@ "content": "已知函数$f(x)$的定义域为$[1,4]$, 则$f(x+1)+f(x-1)$的定义域是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260133-函数" ], "genre": "填空题", "ans": "$[2,3]$", @@ -545745,7 +546611,8 @@ "content": "函数$f(x)=\\begin{cases}-x, & -10$, 均成立$f(x^6)=\\log_2 x$, 则$f(8)=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260133-函数" ], "genre": "填空题", "ans": "(1) $\\lg 5$; (2) $\\dfrac{1}{2}$", @@ -545937,7 +546806,8 @@ "content": "若函数$f(x)$是定义在$\\mathbf{R}$上的奇函数, 则$f(1+\\sqrt 2)+f(\\dfrac 1{1-\\sqrt 2})=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260135-函数的奇偶性(1)" ], "genre": "填空题", "ans": "$0$", @@ -545978,7 +546848,8 @@ "content": "已知函数$f(x)=a x^5+b x^{\\frac 13}+2$. 若$f(2)=3$, 则$f(-2)=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260135-函数的奇偶性(1)" ], "genre": "填空题", "ans": "$1$", @@ -546018,7 +546889,8 @@ "content": "函数$y=\\sqrt {x^2-4}+\\sqrt {4-x^2}$的奇偶性是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260135-函数的奇偶性(1)" ], "genre": "填空题", "ans": "既是奇函数又是偶函数", @@ -546080,7 +546952,8 @@ "content": "``函数$f(x)$的定义域关于原点对称''是``函数$f(x)$为奇函数''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": [ - "第二单元" + "第二单元", + "G20260135-函数的奇偶性(1)" ], "genre": "选择题", "ans": "B", @@ -546115,7 +546988,8 @@ "content": "下列命题中, 真命题是\\bracket{20}.\n\\onech{偶函数的图像一定与$y$轴相交}{奇函数的图像一定通过原点}{偶函数的图像关于$y$轴对称}{即是奇函数又是偶函数的函数只能是$f(x)=0$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260135-函数的奇偶性(1)" ], "genre": "选择题", "ans": "C", @@ -546180,7 +547054,8 @@ "content": "已知$f(x)$为偶函数, 当$x \\geq 0$时, $f(x)=(x-1)^3+1$, 求$f(x)$的解析式.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260136-函数的奇偶性(2)" ], "genre": "解答题", "ans": "$f(x)=\\begin{cases}(x-1)^3+1, & x \\ge 0,\\\\ 1-(x+1)^3, & x<0\\end{cases}$", @@ -546296,7 +547171,8 @@ "content": "是否存在一个实数$a$, 使得函数$y=\\log_2 \\dfrac{a-x}{2+x}$是奇函数? 若存在, 求$a$的值; 若不存在, 说明理由.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260136-函数的奇偶性(2)" ], "genre": "解答题", "ans": "存在, $a=2$", @@ -546409,7 +547285,9 @@ "content": "已知$f(x)=a x^2+b x+3 a+b$是偶函数, 且其定义域为$[a-1,2 a]$, 则$a=$\\blank{50}, $b=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260135-函数的奇偶性(1)", + "V20260105-2026届高一寒假作业05" ], "genre": "填空题", "ans": "$\\dfrac{1}{3}$, $0$", @@ -546471,7 +547349,8 @@ "content": "若$f(x)=(k-1) x^2+2 k x+3$是偶函数, 则$f(-1)$、$f(-\\sqrt 2)$、$f(\\sqrt 3)$由小到大的顺序是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260135-函数的奇偶性(1)" ], "genre": "填空题", "ans": "$f(\\sqrt{3})0$时, 在$(-\\infty,+\\infty)$上是严格增函数; 当$a<0$时, 在$(-\\infty,+\\infty)$上是严格减函数; (3) 在$[-3,2]$上是严格减函数, 在$[2,3]$上是严格增函数; (4) 当$a>0$时, 在$(-\\infty,-1]$上是严格减函数; 当$a<0$时, 在$(-\\infty,-1]$上是严格增函数", @@ -547009,7 +547895,8 @@ "content": "已知函数$f(x)=\\begin{cases}(3 a-1) x+4 a, & x<1, \\\\\\log_a x, & x \\geq 1\\end{cases}$是$\\mathbf{R}$上的严格减函数, 那么$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260137-函数的单调性(1)" ], "genre": "填空题", "ans": "$[\\dfrac{1}{7},\\dfrac{1}{3})$", @@ -547096,7 +547983,8 @@ "content": "已知函数$f(x)=\\lg \\dfrac{1-x}{1+x}$.\\\\\n(1) 求$f(x)$的定义域;\\\\\n(2) 判断$f(x)$的奇偶性;\\\\ \n(3) 判断$f(x)$的单调性.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260137-函数的单调性(1)" ], "genre": "解答题", "ans": "(1) $(-1,1)$; (2) 奇函数, 理由略; (3) 是定义域上的严格减函数, 理由略", @@ -547158,7 +548046,8 @@ "content": "若函数$y=f(x)$的定义域为$\\mathbf{R}$, 则``对于任意$x \\in \\mathbf{R}$, 恒有$f(x)f(a^2-a+1)$}{$f(-\\dfrac 34) \\geq f(a^2-a+1)$}{$f(-\\dfrac 34)=latex, scale = 0.5]\n\\draw [->] (-4,0) -- (6,0) node [below] {$x$};\n\\draw [->] (0,-4) -- (0,6) node [left] {$y$};\n\\draw (0,0) node [above left] {$O$};\n\\draw [dashed] (-4,-4) -- (6,6) (1,-4) -- (1,6);\n\\draw (1,0) node [fill = white, below] {$1$};\n\\draw [domain = {1/2* (-3 - sqrt(21))}:{1/2* (-3 + sqrt(21))}, samples = 100] plot (\\x,{\\x+1/(\\x-1)});\n\\draw [domain = {1/2* (7 - sqrt(21))}:{1/2* (7 + sqrt(21))}, samples = 100] plot (\\x,{\\x+1/(\\x-1)});\n\\end{tikzpicture}, 图像关于点$(1,1)$对称", @@ -548558,7 +549456,8 @@ "content": "若函数$f(x)=x+\\dfrac 4x$($1 \\leq x \\leq 5$), 则函数$y=f(x)$的递减区间是\\blank{50}, 最小值是\\blank{50}, 最大值是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260139-函数的最值" ], "genre": "填空题", "ans": "$[1,2]$, $4$, $\\dfrac{29}{5}$", @@ -549042,7 +549941,8 @@ "content": "试用解析式将圆的面积$S$表示成圆的周长$C$的函数.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260140-函数关系的建立" ], "genre": "解答题", "ans": "$S=\\dfrac{C^2}{4\\pi}$, $C\\in (0,+\\infty)$", @@ -549080,7 +549980,8 @@ "content": "一个矩形的对角线长为$12$厘米, 试用解析式将它的一条边长$y$(厘米)表示成与这条边相邻的另一条边长$x$(厘米)的函数.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260140-函数关系的建立" ], "genre": "解答题", "ans": "$y=\\sqrt{144-x^2}$, $x\\in (0,12)$", @@ -549145,7 +550046,8 @@ "content": "某中学的高一学生进行野外生存训练, 从甲地步行到乙地. 已知甲乙两地相距 $32$千米, 在前$3$小时内学生们每小时走$4$千米, 随后以每小时$5$千米的速度一直走到乙地. 设他们离开甲地的距离为$s$(千米)时, 所用的时间为$t$(时), 试用解析式将$s$(千米)表示成$t$(时)的函数.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260140-函数关系的建立" ], "genre": "解答题", "ans": "$s=\\begin{cases}4t, & 0\\le t\\le 3, \\\\ 5t-3, & 3\\le t\\le 7\\end{cases}$", @@ -549237,7 +550139,8 @@ "content": "$\\triangle ABC$是边长为$1$的正三角形, $AD$为$BC$边上的高, 动点$P$由顶点$A$出发, 按逆时针方向在$\\triangle ABC$边界上移动一周, 设点$P$所移动的路程为$x$, 点$P$到$AD$的距离$PQ=y$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 2]\n\\draw (0,0) node [below left] {$A$} coordinate (A);\n\\draw (1,0) node [below right] {$B$} coordinate (B);\n\\draw (60:1) node [above] {$C$} coordinate (C);\n\\draw ($(B)!0.5!(C)$) node [above right] {$D$} coordinate (D);\n\\draw ($(A)!0.7!(B)$) node [below] {$P$} coordinate (P);\n\\draw ($(A)!(P)!(D)$) node [above left] {$Q$} coordinate (Q);\n\\draw (A) -- (B) -- (C) -- cycle (A) -- (D) (P) -- (Q);\n\\end{tikzpicture}\n\\end{center}\n(1) 求$y$关于$x$的函数, 并画出函数的图像;\\\\\n(2) 当点$P$所移动路程为$\\dfrac 43$时, 点$P$到$AD$的距离为多少? 当点$P$到$AD$的距离为$\\dfrac 14$时, 点$P$所移动的路程为多少?", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260140-函数关系的建立" ], "genre": "解答题", "ans": "(1) $y=\\begin{cases}\\dfrac{1}{2} x, & x\\in [0,1], \\\\ \\dfrac{3}{2}-x, & x\\in [1,\\dfrac{3}{2}], \\\\ x-\\dfrac{3}{2}, & x\\in [\\dfrac{3}{2},3],\\\\ \\dfrac{3}{2}-\\dfrac{1}{2} x, & x\\in [2,3],\\end{cases}$ 图像: \\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.5,0) -- (3.5,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,1) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (1,0) -- (1,0.5) (2,0) -- (2,0.5) (0,0.5) -- (2,0.5);\n\\draw (0,0.5) node [left] {$\\frac{1}{2}$} (1.5,0) node [below] {$\\frac{3}{2}$};\n\\foreach \\i in {1,2,3}\n{\\draw (\\i,0) node [below] {$\\i$};};\n\\draw (0,0) -- (1,0.5) -- (1.5,0) -- (2,0.5) -- (3,0);\n\\end{tikzpicture}\\\\\n(2) $\\dfrac{1}{2}$或$\\dfrac{5}{4}$或$\\dfrac{7}{4}$或$\\dfrac{5}{2}$", @@ -549407,7 +550310,8 @@ "content": "函数$f(x)=x+3-\\dfrac 1x$的零点是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260141-用函数观点解方程与不等式" ], "genre": "填空题", "ans": "$\\dfrac{-3\\pm \\sqrt{13}}{2}$", @@ -549442,7 +550346,8 @@ "content": "若函数$y=x^2-k|x|+1$恰有 $2$ 个零点, 则$k$的取值范围为\\blank{50}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260141-用函数观点解方程与不等式" ], "genre": "填空题", "ans": "$\\{2\\}$", @@ -549466,7 +550371,8 @@ "content": "已知方程$x^2-4|x|+5=m$有四个互不相等的实数根, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260141-用函数观点解方程与不等式" ], "genre": "填空题", "ans": "$(1,5)$", @@ -549523,7 +550429,8 @@ "content": "已知常数$a \\in \\mathbf{R}$, 关于$x$的不等式$x^2-a x+2>0$, 在区间$[1,2]$中有解, 则$a$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260141-用函数观点解方程与不等式" ], "genre": "填空题", "ans": "$(-\\infty,3)$", @@ -549603,7 +550510,8 @@ "content": "方程$\\dfrac{x^4+x^2+1}{x^4+x^2}=\\dfrac{201}{200}$是否有整数解? 说明理由.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260141-用函数观点解方程与不等式" ], "genre": "解答题", "ans": "无整数解, 理由略", @@ -549726,7 +550634,8 @@ "content": "已知关于$x$的方程$x^2-2 x=\\dfrac{3 a-2}{5-a}$在$(\\dfrac 12, 2)$上有实数根, 求实数$a$的取值范围.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260141-用函数观点解方程与不等式" ], "genre": "解答题", "ans": "$[-\\dfrac{3}{2},\\dfrac{2}{3})$", @@ -549806,7 +550715,8 @@ "content": "方程$|a^x-1|=2 a$($a>0$, $a \\neq 1$)有两个不相等的实数解, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260141-用函数观点解方程与不等式" ], "genre": "填空题", "ans": "$(0,\\dfrac{1}{2})$", @@ -549863,7 +550773,8 @@ "content": "已知函数$f(x)=\\begin{cases}\\log_2 x, & x>0, \\\\3^x, & x \\leq 0,\\end{cases}$ 且函数$F(x)=f(x)+x-a$有且仅有两个零点, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260141-用函数观点解方程与不等式" ], "genre": "填空题", "ans": "$(-\\infty,1]$", @@ -549898,7 +550809,8 @@ "content": "用函数观点解方程: $3^x+x^3+1=5$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260141-用函数观点解方程与不等式" ], "genre": "解答题", "ans": "解集为$\\{1\\}$", @@ -550089,7 +551001,8 @@ "content": "函数$y=\\sqrt [3]{x^3+2 x-3}$在区间$(0,+\\infty)$有\\blank{50}个零点.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260141-用函数观点解方程与不等式" ], "genre": "填空题", "ans": "$1$", @@ -550148,7 +551061,8 @@ "content": "方程$\\dfrac 35 x+\\log_3 x=3$实数解可在的区间是\\bracket{20}.\n\\fourch{$(2,3)$}{$(3,4)$}{$(4,5)$}{$(5,6)$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260142-用二分法求函数的零点" ], "genre": "选择题", "ans": "B", @@ -550183,7 +551097,8 @@ "content": "函数$y=-x^2+(m-1) x+m^2-m+2$在下列哪个区间内一定有零点\\bracket{20}.\n\\fourch{$(-\\infty, 0]$}{$(0,1]$}{$(1,+\\infty)$}{$[3,+\\infty)$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260142-用二分法求函数的零点" ], "genre": "选择题", "ans": "A", @@ -550218,7 +551133,8 @@ "content": "若连续函数$y=f(x)$($x \\in \\mathbf{R}$)满足: $f(a) f(b)<0$, 则$f(x)$在区间$(a, b)$上\\bracket{20}.\n\\fourch{有且只有一个零点}{至少有一个零点}{至多有一个零点}{可能没有零点}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260142-用二分法求函数的零点" ], "genre": "选择题", "ans": "B", @@ -550253,7 +551169,8 @@ "content": "若$an$, $m$、$n$均为正整数, 则$a_m=$\\blank{50}. (用$A$、$B$表示)", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260145-等差数列及其通项公式" ], "genre": "填空题", "ans": "(1) $90$, $120$; (2) $3$; (3) $24$; (4) $\\dfrac{A+B}{2}$", @@ -550744,7 +551669,8 @@ "content": "已知等差数列$\\{a_n\\}$的公差为$3$. 若$a_3+a_5+a_7=39$, 则$a_{2022}=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260145-等差数列及其通项公式" ], "genre": "填空题", "ans": "$6064$", @@ -550779,7 +551705,8 @@ "content": "在等差数列$\\{a_n\\}$中, $a_1=\\dfrac 18$, 若恰好从第$8$项开始有$a_n>1$($n \\in \\mathbf{N}$, $n \\geq 8$), 则公差$d$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260145-等差数列及其通项公式" ], "genre": "填空题", "ans": "$(\\dfrac{1}{8},\\dfrac{7}{48}]$", @@ -550858,7 +551785,8 @@ "content": "已知$a, b, c, d$成等差数列, 求证:$2 a-3 b, 2 b-3 c, 2 c-3 d$成等差数列.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260145-等差数列及其通项公式" ], "genre": "解答题", "ans": "证明略", @@ -550981,7 +551909,8 @@ "content": "已知等差数列$\\{a_n\\}$分别满足下列条件, 求解相应问题.\\\\\n(1) $d=\\dfrac 13$, $n=37$, $S_n=629$, 则$a_1=$\\blank{50};\\\\\n(2) $d=2$, $n=15$, $a_n=-10$, 则$S_n=$\\blank{50};\\\\\n(3) $a_1=20$, $a_n=54$, $S_n=999$, 则$d=$\\blank{50};\\\\\n(4) $a_1=\\dfrac 56$, $d=-\\dfrac 16$, $S_n=-5$, 则$a_n=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260146-等差数列的前$n$项和" ], "genre": "填空题", "ans": "(1) $11$; (2) $-360$; (3) $\\dfrac{17}{13}$; (4) $-\\dfrac{3}{2}$", @@ -551060,7 +551989,8 @@ "content": "等差数列$\\{a_n\\}$中,\\\\ \n(1) 若$a_4+a_6=6$, 若其前$5$项和$S_5=10$, 则其公差$d=$\\blank{50};\\\\\n(2) 若其前$15$项和为$135$, 则其第$8$项$a_8=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260146-等差数列的前$n$项和" ], "genre": "填空题", "ans": "(1) $\\dfrac{1}{2}$; (2) $9$", @@ -551095,7 +552025,9 @@ "content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n=n^2+n+1$, 则其通项公式$a_n=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260146-等差数列的前$n$项和", + "I20260107-高一上学期随堂练习07" ], "genre": "填空题", "ans": "$\\begin{cases}3, & n=1,\\\\ 2n, & n\\ge 2\\end{cases}$", @@ -551189,7 +552121,8 @@ "content": "已知数列$\\{a_n\\}$的通项公式为$a_n=47-2 n$, 当其前$n$项和$S_n$取最大值时, $n=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260146-等差数列的前$n$项和" ], "genre": "填空题", "ans": "$23$", @@ -551291,7 +552224,8 @@ "content": "已知$\\{a_n\\}$是等差数列, 记$b_n=\\dfrac{a_1+a_2+\\cdots+a_n}n$, 求证: 数列$\\{b_n\\}$是等差数列.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260146-等差数列的前$n$项和" ], "genre": "解答题", "ans": "证明略", @@ -551436,7 +552370,8 @@ "content": "已知两等差数列$\\{a_n\\}$、$\\{b_n\\}$前$n$项和分别为$A_n$、$B_n$, 若$\\dfrac{A_n}{B_n}=\\dfrac{4 n+3}{2 n+5}$, 求$a_8: b_8$.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260146-等差数列的前$n$项和" ], "genre": "解答题", "ans": "$\\dfrac{9}{5}$", @@ -551515,7 +552450,8 @@ "content": "一个有限项的等差数列, 它的前$6$项和为$48$, 后$6$项和为$132$, 所有项和为$255$, 则该数列共有\\blank{50}项.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260146-等差数列的前$n$项和" ], "genre": "填空题", "ans": "$17$", @@ -551616,7 +552552,8 @@ "content": "已知$\\{a_n\\}$是等比数列, 根据所给的条件填写下表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline$a_1$&$q$&$n$&$a_n$\\\\\n\\hline 9 & \\blank{20} & 4 & 243 \\\\\n\\hline \\blank{20} &$-2$& 7 & 32 \\\\\n\\hline 4 &$\\dfrac 12$&\\blank{20} &$\\dfrac 1{32}$\\\\\n\\hline 3 &$\\dfrac 23$& 3 & \\blank{20}\\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260147-等比数列及其通项公式" ], "genre": "填空题", "ans": "\\begin{tabular}{|c|c|c|c|}\n\\hline$a_1$&$q$&$n$&$a_n$\\\\\n\\hline 9 & \\underline{$3$} & 4 & 243 \\\\\n\\hline \\underline{$\\dfrac{1}{2}$} &$-2$& 7 & 32 \\\\\n\\hline 4 &$\\dfrac 12$& \\underline{$8$} &$\\dfrac 1{32}$\\\\\n\\hline 3 &$\\dfrac 23$& 3 & \\underline{$\\dfrac{4}{3}$}\\\\\n\\hline\n\\end{tabular}", @@ -551672,7 +552609,8 @@ "content": "在等比数列$\\{a_n\\}$中, 如果公比$q<0$, $a_1+a_2=30$, $a_3+a_4=120$, 那么这个数列的通项公式为\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260147-等比数列及其通项公式" ], "genre": "填空题", "ans": "$15\\cdot (-2)^n$", @@ -551706,7 +552644,8 @@ "content": "在等比数列$\\{a_n\\}$中, 若$a_1=1$, $a_3=4$, 则$a_2=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260147-等比数列及其通项公式" ], "genre": "填空题", "ans": "$\\pm 2$", @@ -551740,7 +552679,8 @@ "content": "在等比数列$\\{a_n\\}$中, 若$a_1=1$, $a_5=4$, 则$a_3=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260147-等比数列及其通项公式" ], "genre": "填空题", "ans": "$2$", @@ -551774,7 +552714,8 @@ "content": "在等比数列$\\{b_n\\}$中, $b_4=3$, 则该数列前七项之积是\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260147-等比数列及其通项公式" ], "genre": "填空题", "ans": "$2187$", @@ -551808,7 +552749,8 @@ "content": "已知直角三角形的斜边长为$c$, 两条直角边长分别为$a$、$b$, 且$a, b, c$成等比数列, 则$a: c$的值是\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260147-等比数列及其通项公式" ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{5}-1}{2}$", @@ -551842,7 +552784,8 @@ "content": "在$2, x, 8, y$四个数中, 前三个数成等比数列, 后三个数成等差数列, 则$x y=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260147-等比数列及其通项公式" ], "genre": "填空题", "ans": "$48$或$-80$", @@ -551898,7 +552841,8 @@ "content": "已知$\\{a_n\\}$是等比数列. 下列命题中, 不正确的是\\bracket{20}.\n\\onech{若$a_n$恒正, 则$\\{\\lg a_n\\}$是等差数列}{若$a_n$恒正, 则$\\dfrac{a_1+a_{n+2}}2 \\geq \\sqrt {a_2 a_{n+1}}$}{$a_{n+1}$一定是$a_n$与$a_{n+2}$的等比中项}{$a_{n-r}$与$a_{n+r}$($r0$, $q\\in (0,1)$均可以)", @@ -552766,7 +553723,8 @@ "content": "如右图, 正方形上连接等腰直角三角形, 直角三角形边上再连接正方形, $\\cdots \\cdots$, 无限重复, 若第一个正方形边长为$2$, 则所有这些正方形和三角形的面积总和为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw (1,0) node [below] {$2$};\n\\draw (0,0) rectangle (2,2);\n\\draw (0,2) --++ (1,1) --++ (1,-1);\n\\draw (0,2) --++ (-1,1) coordinate (A) --++ (1,1) --++ (1,-1);\n\\draw (A) --++ (0,1) --++ (1,0);\n\\draw (A) --++ (-1,0) --++ (0,1) --++ (1,0);\n\\draw (A) ++ (-1,0) node [below left] {$\\cdots$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260149-等比数列的前$n$项和(2)" ], "genre": "填空题", "ans": "$10$", @@ -552845,7 +553803,8 @@ "content": "设$\\{a_n\\}$为无穷等比数列, $\\{a_n\\}$中每一项都是它后面所有项和的$4$ 倍, 且$a_5=\\dfrac{16}{625}$, 求它的所有偶数项之和.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260149-等比数列的前$n$项和(2)" ], "genre": "解答题", "ans": "$\\dfrac{10}{3}$", @@ -552990,7 +553949,8 @@ "content": "已知数列$\\{a_n\\}$的通项公式为$a_n=2(1+3 n)$, 填写下表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}\n\\hline$n$& 1 & 2 & 3 &$\\cdots$& 11 &$\\cdots$&\\blank{20}&$\\cdots$&\\blank{20} \\\\\n\\hline$a_n$&\\blank{20} &\\blank{20}&\\blank{20} &$\\cdots$& \\blank{20}&$\\cdots$& 128 &$\\cdots$& 602 \\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260150-数列的概念与性质" ], "genre": "填空题", "ans": "\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}\n\\hline$n$& 1 & 2 & 3 &$\\cdots$& 11 &$\\cdots$&\\underline{$21$}&$\\cdots$&\\underline{$100$} \\\\\n\\hline$a_n$&\\underline{$8$} &\\underline{$14$}&\\underline{$20$} &$\\cdots$& \\underline{$68$}&$\\cdots$& 128 &$\\cdots$& 602 \\\\\n\\hline\n\\end{tabular}", @@ -553021,7 +553981,8 @@ "content": "$a_n=\\dfrac 1n+\\dfrac 1{n+1}+\\dfrac 1{n+2}+\\cdots+\\dfrac 1{n^2}$中每个$a_n$有\\blank{50}个加数项.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260150-数列的概念与性质" ], "genre": "填空题", "ans": "$n^2-n+1$", @@ -553074,7 +554035,8 @@ "content": "在数列$\\{a_n\\}$中, 若$a_n=3 n+1$, 则$2017$是这数列的第\\blank{50}项.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260150-数列的概念与性质" ], "genre": "填空题", "ans": "$672$", @@ -553331,7 +554293,8 @@ "content": "已知数列$\\{a_n\\}$, $a_1=1$, $a_{n+1}=a_n+\\dfrac 1{2^n}$, 其通项公式$a_n=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260151-利用递推公式表示数列" ], "genre": "填空题", "ans": "$2-\\dfrac{1}{2^{n-1}}$", @@ -553366,7 +554329,8 @@ "content": "已知数列$\\{a_n\\}$, $a_1=1$, $a_{n+1}=\\dfrac{2(n+1)}n a_n$, 其通项公式$a_n=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260151-利用递推公式表示数列" ], "genre": "填空题", "ans": "$2^{n-1}\\cdot n$", @@ -553401,7 +554365,8 @@ "content": "已知数列$\\{a_n\\}$, $a_1=1$, $a_{n+1}=\\dfrac 23 a_n+1$, 其通项公式$a_n=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260151-利用递推公式表示数列" ], "genre": "填空题", "ans": "$3-2(\\dfrac{2}{3})^{n-1}$", @@ -553436,7 +554401,8 @@ "content": "已知数列$\\{a_n\\}$, $a_1=1$, $a_{n+1}=\\dfrac{a_n}{2 a_n+1}$, 其通项公式$a_n=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260151-利用递推公式表示数列" ], "genre": "填空题", "ans": "$\\dfrac{1}{2n-1}$", @@ -553471,7 +554437,8 @@ "content": "已知数列$\\{a_n\\}$, $a_1=1$, $a_{n+1}=a_n+2 n+5$, 其通项$a_n=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260151-利用递推公式表示数列" ], "genre": "填空题", "ans": "$n^2+4n-4$", @@ -553493,7 +554460,8 @@ "content": "已知数列$\\{a_n\\}$, $a_1=1$, $a_{n+1}=5^n a_n$, 其通项公式$a_n=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260151-利用递推公式表示数列" ], "genre": "填空题", "ans": "$5^{\\frac{n(n-1)}{2}}$", @@ -553515,7 +554483,8 @@ "content": "已知数列$\\{a_n\\}$, 其前$n$项和记为$S_n$, $a_1=1$, $S_{n+1}=4 a_n+2$, 其通项公式$a_n=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260151-利用递推公式表示数列" ], "genre": "填空题", "ans": "$a_n=(3n-1)\\cdot 2^{n-2}$", @@ -553538,7 +554507,8 @@ "content": "已知数列$\\{a_n\\}$, $a_1=2$, $a_{n+1}=1-\\dfrac 1{a_n}$, 其通项公式$a_n=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260151-利用递推公式表示数列" ], "genre": "填空题", "ans": "$\\begin{cases}2, & n=3k+1, \\\\ \\dfrac{1}{2}, & n=3k+2, \\\\ -1, & n = 3k+3\\end{cases}$($k\\in \\mathbf{N}$)", @@ -553573,7 +554543,8 @@ "content": "已知数列$\\{a_n\\}$, 前$n$项和$S_n=n^2+n+1$, 求其通项公式$a_n$.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260151-利用递推公式表示数列" ], "genre": "解答题", "ans": "$a_n=\\begin{cases}3, & n=1, \\\\ 2n, & n\\ge 2\\end{cases}$", @@ -553729,7 +554700,8 @@ "content": "用数学归纳法证明: ``$2+3+4+\\cdots+n=\\dfrac{(n-1)(n+2)}2$($n$为正整数)''时, 第一步是取$n=$\\blank{50}验证.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260152-数学归纳法" ], "genre": "填空题", "ans": "$2$", @@ -553764,7 +554736,8 @@ "content": "设$n$为正整数, 用数学归纳法证明: $1+2+3+\\cdots+(n+4)=\\dfrac 12(n+4)(n+5)$时, 当$n=1$时, 左边应为\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260152-数学归纳法" ], "genre": "填空题", "ans": "$1+2+3+4+5$", @@ -553843,7 +554816,8 @@ "content": "用数学归纳法证明: $1 \\times 2+2 \\times 5+\\cdots+n(3 n-1)=n^2(n+1)$($n$为正整数).", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260152-数学归纳法" ], "genre": "解答题", "ans": "证明略", @@ -553878,7 +554852,8 @@ "content": "若$f(n)=\\dfrac 1{n+1}+\\dfrac 1{n+2}+\\dfrac 1{n+3}+\\cdots+\\dfrac 1{2 n}$($n$为正整数), 则$f(k+1)=f(k)+$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260152-数学归纳法" ], "genre": "填空题", "ans": "$\\dfrac{1}{(2k+1)(2k+2)}$", @@ -553913,7 +554888,8 @@ "content": "如果$f(n)=1+\\dfrac 12+\\dfrac 13+\\cdots+\\dfrac 1n$($n$为正整数), 则$f(2^{k+1})-f(2^k)=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260152-数学归纳法" ], "genre": "填空题", "ans": "$\\dfrac{1}{2^k+1}+\\dfrac{1}{2^k+2}+\\cdots+\\dfrac{1}{2^{k+1}}$", @@ -553948,7 +554924,8 @@ "content": "用数学归纳法证明:\n$1-2^2+3^2-4^2+\\cdots+(-1)^{n-1} n^2=(-1)^{n-1} \\dfrac{n(n+1)}2$($n$为正整数).", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260152-数学归纳法" ], "genre": "解答题", "ans": "证明略", @@ -553983,7 +554960,8 @@ "content": "用数学归纳法证明:\n``$1+\\dfrac 1{2^2}+\\dfrac 1{3^2}+\\cdots+\\dfrac 1{n^2}<\\dfrac{2 n-1}n$($n>1$, $n \\in \\mathbf{N}$)''的过程中, 第一步是验证不等式\\blank{50}成立.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260152-数学归纳法" ], "genre": "填空题", "ans": "$1+\\dfrac{1}{4}<\\dfrac{3}{2}$", @@ -554040,7 +555018,8 @@ "content": "在数列$\\{a_n\\}$中, 已知$a_1=1$, $a_2=2$. 若$a_n=2 a_{n-1}-a_{n-2}$($n \\geq 3$且$n \\in \\mathbf{N}$), 则$a_3=$\\blank{50}, $a_4=$\\blank{50}, $a_5=$\\blank{50}, 进而猜想$a_n=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260153-数学归纳法的应用" ], "genre": "填空题", "ans": "$3$, $4$, $5$, $n$", @@ -554099,7 +555078,8 @@ "content": "观察下列数字:\n\\begin{center}\n\\begin{tabular}{ccccccc}\n1 \\\\\n2 & 3 & 4\\\\\n3 & 4 & 5 & 6 & 7 \\\\\n4 & 5 & 6 & 7 & 8 & 9 & 10\\\\\n\\end{tabular}\\\\\n$\\cdots\\cdots\\cdots$\n\\end{center}\n猜想第$n$行的各数之和$S_n=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260153-数学归纳法的应用" ], "genre": "填空题", "ans": "$(2n-1)^2$", @@ -554178,7 +555158,8 @@ "content": "对于数列$1,3,5, \\cdots,(2 n-1), \\cdots$, 按如下规律分组: $1,(3,5),(7,9,11),(13,15,17,19), \\cdots$即依次取$1$项, $2$项, $3$项, $\\cdots$各为一组, 则第$n$组的各数之和为\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260153-数学归纳法的应用" ], "genre": "填空题", "ans": "$n^3$", @@ -554213,7 +555194,8 @@ "content": "已知$f(x)=2 x+1$, $f^{(1)}(x)=f(x)$, $f^{(n)}(x)=f[f^{(n-1)}(x)]$($n \\geq 2$, $n \\in \\mathbf{N}$), 猜想$f^{(n)}(x)=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260153-数学归纳法的应用" ], "genre": "填空题", "ans": "$2^nx+(2^n-1)$", @@ -554248,7 +555230,8 @@ "content": "是否存在常数$a$、$b$、$c$, 使等式$1 \\cdot(n^2-1^2)+2 \\cdot(n^2-2^2)+\\cdots+n \\cdot(n^2-n^2)=$$a n^4+b n^2+c$对一切正整数$n$都成立? 证明你的结论.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260153-数学归纳法的应用" ], "genre": "解答题", "ans": "存在, $a=\\dfrac{1}{4}$, $b=-\\dfrac{1}{4}$, 证明略", @@ -554393,7 +555376,8 @@ "content": "求证: $1+\\dfrac{1}{\\sqrt{2}}+\\dfrac{1}{\\sqrt{3}}+\\cdots+\\dfrac{1}{\\sqrt{n}}>\\sqrt{n}$($n \\in \\mathbf{N}$, $n \\geq 2$).", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260153-数学归纳法的应用" ], "genre": "解答题", "ans": "证明略", @@ -554516,7 +555500,8 @@ "content": "设数列$\\{a_n\\}$满足$a_1=1$且$a_{n+1}=a_n+\\dfrac 1{n(n+1)}$, 求数列$\\{a_n\\}$的通项公式.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260151-利用递推公式表示数列" ], "genre": "解答题", "ans": "$a_n=2-\\dfrac{1}{n}$", @@ -554551,7 +555536,8 @@ "content": "设数列$\\{a_n\\}$满足$a_1=1$且$a_{n+1}=2 a_n+2$, 求数列$\\{a_n\\}$的通项公式.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260151-利用递推公式表示数列" ], "genre": "解答题", "ans": "$a_n=3\\cdot 2^{n-1}-2$", @@ -554674,7 +555660,8 @@ "content": "设数列$\\{a_n\\}$满足$a_1=2$, $a_{n+1}=a_n^2$, 求通项公式$a_n$.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260151-利用递推公式表示数列" ], "genre": "解答题", "ans": "$a_n=2^{2^{n-1}}$", @@ -595509,7 +596496,8 @@ "content": "已知 $a$、$b\\in \\mathbf{R}$, 元素 $(1,2) \\in A \\cap B$, 且 $A=\\{(x, y) | a x-y^2+b=0\\}$, $B=\\{(x, y) | x^2-a y-b=0\\}$, 求 $a$、$b$ 的值.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260115-2026届高一上学期国庆复习卷" ], "genre": "解答题", "ans": "", @@ -595631,7 +596619,8 @@ "content": "用列举法表示:\\\\\n(1) $\\{y | y=-x^2+\\dfrac{9}{2}, x \\in \\mathbf{R}, y \\in \\mathbf{N}\\}=$\\blank{100};\\\\\n(2) $\\{(x, y) | x+(y+2)^2=5, x \\in \\mathbf{N}, y \\in \\mathbf{Q}, y<0\\}=$\\blank{100};\\\\\n(3) 若集合 $M=\\{0,3,4\\}$, $P=\\{(x, y) | x=a+b, y=a b, a,b \\in M\\}=$\\blank{100};\\\\\n(4) 平面上的两条曲线 $y=x^2+1$ 和 $y=x+6$ 的交点组成的集合\\blank{100}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260115-2026届高一上学期国庆复习卷" ], "genre": "填空题", "ans": "", @@ -595687,7 +596676,8 @@ "content": "给出四个命题: \\textcircled{1} 若 $x^2=y^2$, 则 $x=y$; \\textcircled{2}若 $x \\neq y$, 则 $x^2 \\neq y^2$; \\textcircled{3}若 $x^2 \\neq y^2$, 则 $x \\neq y$;\\\\\n\\textcircled{4} 若 $x \\neq y$ 且 $x \\neq-y$, 则 $x^2 \\neq y^2$, 其中真命题的序号是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260115-2026届高一上学期国庆复习卷" ], "genre": "填空题", "ans": "", @@ -596009,7 +596999,8 @@ "content": "集合 $\\{y | y=x^2-6, x \\in \\mathbf{R}\\}\\cap\\{y | y=-x^2+2 x+6, x \\in \\mathbf{R}\\}=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260102-高一上学期周末卷02" ], "genre": "填空题", "ans": "$[-6,7]$", @@ -596066,7 +597057,8 @@ "content": "已知 $m$ 是实常数, 若集合 $A=\\{0,1\\}$, $B=\\{x | x^2-m x-m>0\\}$ 且 $A \\cap B=\\{1\\}$, 则实数 $m$ 的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260102-高一上学期周末卷02" ], "genre": "填空题", "ans": "$[0,\\dfrac{1}{2})$", @@ -596101,7 +597093,8 @@ "content": "已知 $m$ 是实常数, 若 $\\alpha: 2 \\leq x \\leq 4$, $\\beta: 3 m-1a$ 且 $x<-3 a\\}$ 且满足 $\\overline{A}\\cap \\overline{B}\\subset C$, 则 $\\overline{A}\\cap \\overline{B}=$\\blank{50}, 实数 $a$ 的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260102-高一上学期周末卷02" ], "genre": "填空题", "ans": "$(-2,1)$, $(-\\infty,0]$", @@ -596252,7 +597247,8 @@ "content": "设 $m$ 为实常数, 命题甲: 关于 $x$ 的方程 $x^2+x+m=0$ 有两个不同的负根; 命题乙: 关于 $x$ 的方程 $4 x^2+x+m=0$ 无实根. 若这两个命题有且只有一个成立, 求实数 $m$ 的取值范围.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260102-高一上学期周末卷02" ], "genre": "解答题", "ans": "$(0,\\dfrac{1}{16}]\\cup [\\dfrac{1}{4},+\\infty)$", @@ -596287,7 +597283,8 @@ "content": "已知 $a$ 为实常数, 集合 $A=\\{x | x^2-2 a x+a=0, x \\in \\mathbf{R}\\}$, 集合 $B=\\{x | x^2-4 x+a+5=0, x \\in \\mathbf{R}\\}$.\\\\\n(1) 若 $A=B=\\varnothing$, 求实数 $a$ 的取值范围;\\\\\n(2) 若 $A$、$B$ 中有且只有一个是 $\\varnothing$, 求实数 $a$ 的取值范围;\\\\\n(3) 若 $A$、$B$ 中至少有一个是 $\\varnothing$, 求实数 $a$ 的取值范围.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260102-高一上学期周末卷02" ], "genre": "解答题", "ans": "(1) $(0,1)$; (2) $(-1,0]\\cup [1,+\\infty)$; (3) $(-1,+\\infty)$", @@ -596324,7 +597321,8 @@ "content": "``$a \\in \\mathbf{R}$''是`` $a^2>0$''的\\blank{50}条件.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260102-高一上学期周末卷02" ], "genre": "填空题", "ans": "必要非充分", @@ -596361,7 +597359,8 @@ "content": "``$a<0$''是``$a^2>a$''的\\blank{50}条件.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260102-高一上学期周末卷02" ], "genre": "填空题", "ans": "充分非必要", @@ -596528,7 +597527,8 @@ "content": "已知 $A=\\{y | y=x^2-6 x+10, x, y \\in \\mathbf{N}\\}$, $B=\\{y | y=-x^2-6 x+18, x \\in \\mathbf{Z}, y \\in \\mathbf{N}\\}$, 求 $A \\cap B$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260102-高一上学期周末卷02" ], "genre": "解答题", "ans": "$\\{2,26\\}$", @@ -596763,7 +597763,8 @@ "content": "已知 $a$、$b$、$c$ 是实数, 下列三个命题:\\\\\n\\textcircled{1} 若 $(a-2) c=(b-2) c$, 则 $a=b$;\\\\\n\\textcircled{2} $\\dfrac{a}{c}=\\dfrac{b}{c}$, 则 $a=b$;\\\\\n\\textcircled{3} 若 $ab^2$; \\textcircled{2} $a^3>b^3$; \\textcircled{3} $\\dfrac{1}{a}<\\dfrac{1}{b}$; \\textcircled{4} $\\dfrac{a}{b}>1$; \\textcircled{5} $\\dfrac{1}{a-b}>\\dfrac{1}{a}$; \\textcircled{6} $|a|>-b$. 其中正确项的序号是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260103-高一上学期周末卷03" ], "genre": "填空题", "ans": "\\textcircled{1}\\textcircled{4}\\textcircled{6}", @@ -597052,7 +598059,8 @@ "content": "已知互不相等的正数 $a$、$b$、$c$, 满足 $a^2+c^2=2 b c$, 则下列不等式中可能成立的是\\bracket{20}.\n\\fourch{$a>b>c$}{$b>a>c$}{$b>c>a$}{$c>a>b$}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260115-2026届高一上学期国庆复习卷" ], "genre": "选择题", "ans": "", @@ -597086,7 +598094,8 @@ "content": "设常数 $a \\in \\mathbf{R}$, 求关于 $x$ 的方程 $|x-1|=a-1$ 的解集.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260103-高一上学期周末卷03" ], "genre": "解答题", "ans": "当$a<1$时, 解集为$\\varnothing$; 当$a=1$时, 解集为$\\{1\\}$; 当$a>1$时, 解集为$\\{a,2-a\\}$", @@ -597121,7 +598130,8 @@ "content": "已知关于 $x$ 的方程 $x^2-2 k x+k^2-k-1=0$ 有两个不相等的实数根 $x_1$、$x_2$.\\\\\n(1) 求实数 $k$ 的取值范围;\\\\\n(2) 若 $x_1-3 x_2=2$, 求实数 $k$ 的值.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260103-高一上学期周末卷03" ], "genre": "解答题", "ans": "(1) $(-1,+\\infty)$; (2) $3$", @@ -597200,7 +598210,8 @@ "content": "设常数 $m \\in \\mathbf{R}$, $A=\\{(x, y) | y=-x^2+m x-1, x \\in \\mathbf{R}\\}$, $B=\\{(x, y) | x+y=3, x \\in \\mathbf{R}\\}$, 且 $A \\cap B$ 的子集有两个, 则实数 $m$ 的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260103-高一上学期周末卷03" ], "genre": "填空题", "ans": "$\\{-5,3\\}$", @@ -597257,7 +598268,8 @@ "content": "设常数 $a \\in \\mathbf{R}$, 求关于 $x$ 的方程 $\\dfrac{x}{x+1}=\\dfrac{1}{x+a}$ 的解集.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260115-2026届高一上学期国庆复习卷" ], "genre": "解答题", "ans": "", @@ -597291,7 +598303,8 @@ "content": "设常数 $a \\in \\mathbf{R}$, 求关于 $x, y$ 的方程组 $\\begin{cases}a x+2 y=1+a,\\\\2 x+2(a-1) y=3\\end{cases}$ 的解集.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260103-高一上学期周末卷03" ], "genre": "解答题", "ans": "当$a=2$时, 解集为$\\{(x,y)|y=-x+\\dfrac{3}{2}, \\ x\\in \\mathbf{R}\\}$; 当$a=-1$时, 解集为$\\varnothing$; 当$a\\ne 2$且$a\\ne -1$时, 解集为$\\{(\\dfrac{a+2}{a+1},\\dfrac{1}{2a+2})\\}$", @@ -597348,7 +598361,8 @@ "content": "设实数 $a$、$b$、$c$ 满足 $\\begin{cases}b+c=3 a^2-4 a+6,\\\\c-b=a^2-4 a+4,\\end{cases}$ 试确定 $a$、$b$、$c$ 的大小关系, 并说明理由.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260103-高一上学期周末卷03" ], "genre": "解答题", "ans": "当$a=2$时, $a0, x \\in \\mathbf{R}\\}$, $B=\\{x | b x<1, x \\in \\mathbf{R}\\}$, 其中 $a, b$ 均为实常数, 且 $b \\neq 0$.\\\\\n(1) 若 $A \\cap B=\\{x | 20$且$ab+1=0$", @@ -597418,7 +598433,8 @@ "content": "判断下列命题是否为真 (T or F):\\\\\n(1) 若 $\\dfrac{a}{c^2}<\\dfrac{b}{c^2}$, 则 $ab$, $cb d$: \\blank{20};\\\\\n(5) 若 $a>b>0$, $c>d>0$, 则 $\\dfrac{a}{c}>\\dfrac{b}{d}$: \\blank{20};\\\\\n(6) 若 $\\sqrt[3]{a}>\\sqrt[3]{b}$, 则 $a \\geq b$: \\blank{20};\\\\\n(7) 若 $\\dfrac{c}{a}=\\dfrac{c}{b}$, 则 $a=b \\neq 0$: \\blank{20};\\\\ \n(8) 若 $a=b$, 则 $\\sqrt[n]{a}=\\sqrt[n]{b}$, 其中 $n$ 是正整数: \\blank{20}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "I20260102-高一上学期随堂练习02" ], "genre": "填空题", "ans": "(1) T; (2) F; (3) F; (4) F; (5) F; (6) T; (7) F; (8) F", @@ -597444,7 +598460,8 @@ "content": "集合 $A=\\{x | x^2+p x+q=0\\}$, $B=\\{x | x^2-x+r=0\\}$, 且 $A \\cap B=\\{-1\\}$, $A \\cup B=\\{-1,2\\}$, 则 $p=$\\blank{50}, $q=$\\blank{50}, $r=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "I20260102-高一上学期随堂练习02" ], "genre": "填空题", "ans": "$2$, $1$, $-2$", @@ -597469,7 +598486,8 @@ "content": "已知集合 $E=\\{x | x^2-3 x+2=0\\}$, 集合 $F=\\{x | x^2-a x+(a-1)=0\\}$, 若 $F$ 不是 $E$ 的真子集, 则实数 $a$ 所有值的集合是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "I20260102-高一上学期随堂练习02" ], "genre": "填空题", "ans": "$(-\\infty,2)\\cup (2,+\\infty)$", @@ -597494,7 +598512,8 @@ "content": "设 $k \\in \\mathbf{R}$, 若关于 $x$ 与 $y$ 的二元一次方程组 $\\begin{cases}3 x-y=1,\\\\k x-y=2\\end{cases}$ 的解集为空集, 则 $k=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "I20260102-高一上学期随堂练习02" ], "genre": "填空题", "ans": "$3$", @@ -597519,7 +598538,8 @@ "content": "解关于 $x$ 的不等式: $m(x+2) \\leq x+m$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "I20260102-高一上学期随堂练习02" ], "genre": "解答题", "ans": "当$m>1$时, 解集为$(-\\infty,-\\dfrac{m}{m-1}]$; 当$m=1$时, 解集为$\\varnothing$; 当$m<1$时, 解集为$[-\\dfrac{m}{m-1},+\\infty)$", @@ -597808,7 +598828,8 @@ "content": "写出下列一元二次不等式的解集:\\\\\n(1) $x^2+2 x-15>0$: \\blank{100};\\\\\n(2) $x^2+4 x-45 \\geq 0$: \\blank{100};\\\\\n(3) $-3 x^2+2 x-4 \\geq 0$: \\blank{100};\\\\\n(4) $x^2+12 x+36 \\leq 0$: \\blank{100}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260104-高一上学期周末卷04" ], "genre": "填空题", "ans": "(1) $(-\\infty,-5)\\cup (3,+\\infty)$; (2) $(-\\infty,-9]\\cup [5,+\\infty)$; (3) $\\varnothing$; (4) $\\{-6\\}$", @@ -597887,7 +598908,8 @@ "content": "设关于 $x$ 的二次不等式 $a x^2+b x+c>0$ 的解集为 $(-1,3)$. 求二次不等式 $c x^2-b x+a \\leq 0$ 的解集.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260104-高一上学期周末卷04" ], "genre": "解答题", "ans": "$[-\\dfrac{1}{3},1]$", @@ -597922,7 +598944,8 @@ "content": "设 $a$ 是实常数, 已知 $M=(0,+\\infty)$, 二次不等式 $x^2+x-a \\leq 0$ 的解集为 $S$.\\\\\n(1) 若 $S \\subseteq M$, 求 $a$ 的取值范围;\\\\\n(2) 若 $S \\cap M=\\varnothing$, 求 $a$ 的取值范围.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260104-高一上学期周末卷04" ], "genre": "解答题", "ans": "(1) $(-\\infty,-\\dfrac{1}{4})$; (2) $(-\\infty,0]$", @@ -597958,7 +598981,8 @@ "content": "设 $a$ 是实常数, 求下列关于 $x$ 的二次不等式的解集:\\\\\n(1) $x^2-a x+1 \\leq 0$;\\\\\n(2) $a x^2-2(a+1) x+4>0$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260104-高一上学期周末卷04" ], "genre": "解答题", "ans": "(1) $[\\dfrac{a-\\sqrt{a^2-4}}{2},\\dfrac{a+\\sqrt{a^2-4}}{2}]$; (2) 当$a<0$时, 解集为$(\\dfrac{2}{a},2)$; 当$02 x+\\dfrac{1}{x-4}+3$: \\blank{100};\\\\\n(6) $\\dfrac{4}{x-1}\\leq x-1$: \\blank{100};\\\\\n(7) $\\dfrac{2}{x+1}\\geq \\dfrac{x+1}{2}$: \\blank{100}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260104-高一上学期周末卷04" ], "genre": "填空题", "ans": "(1) $(-\\dfrac{3}{2},2)$; (2) $[-\\dfrac{3}{2},-\\dfrac{5}{4})$; (3) $\\{0\\}\\cup (1,+\\infty)$; (4) $(-\\infty,0)\\cup (1,+\\infty)$; (5) $(-2,4)\\cup (4,+\\infty)$; (6) $[-1,1)\\cup [3,+\\infty)$; (7) $(-\\infty,-3]\\cup (-1,1]$", @@ -598051,7 +599076,8 @@ "content": "若对一切实数 $x$, 均有 $\\dfrac{2 x^2+x-2}{x^2+2 x+2}>k$, 则实数 $k$ 的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260104-高一上学期周末卷04" ], "genre": "填空题", "ans": "$(-\\infty,\\dfrac{1-\\sqrt{3}}{2})$", @@ -598086,7 +599112,8 @@ "content": "若关于 $x$ 的不等式 $-80$, $b>0$, 且 $2 a+3 b=1$, 则 $a b$ 的最大值为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260105-高一上学期周末卷05" ], "genre": "填空题", "ans": "$\\dfrac{1}{24}$", @@ -598947,7 +599982,8 @@ "content": "若 $0k$, 则实数 $k$ 的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260105-高一上学期周末卷05" ], "genre": "填空题", "ans": "$(-\\infty,\\dfrac{1-3\\sqrt{2}}{2})$", @@ -599071,7 +600109,8 @@ "content": "使不等式 $|x-4|+|x-3|>a$ 的解集为 $\\mathbf{R}$ 的实数 $a$ 的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260105-高一上学期周末卷05" ], "genre": "填空题", "ans": "$(-\\infty,1)$", @@ -599105,7 +600144,8 @@ "content": "$x$ 为实数, 且 $|x-5|+|x-3|-1$ 时, $\\dfrac{(x+5)(x+2)}{x+1}$ 的最小值是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260105-高一上学期周末卷05" ], "genre": "填空题", "ans": "$9$", @@ -599195,7 +600236,8 @@ "content": "下列命题中正确的是\\bracket{20}.\n\\twoch{$x+\\dfrac{1}{x}$ 的最小值是 $2$}{$\\dfrac{x^2+2}{\\sqrt{x^2+1}}$ 的最小值是 $2$}{$\\dfrac{x^2+5}{\\sqrt{x^2+4}}$ 的最小值是 $2$}{$2-3 x-\\dfrac{4}{x}$ 的最小值是 $2$}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260105-高一上学期周末卷05" ], "genre": "选择题", "ans": "B", @@ -599427,7 +600469,8 @@ "content": "甲乙两位同学分别解``$x, y$ 为正实数, 求 $(x+y)(\\dfrac{1}{x}+\\dfrac{2}{y})$ 的最小值''的过程如下:\\\\\n甲: $(x+y)(\\dfrac{1}{x}+\\dfrac{2}{y}) \\geq 2 \\sqrt{x y}\\cdot 2 \\sqrt{\\dfrac{2}{x y}}=4 \\sqrt{2}$, 所以$(x+y)(\\dfrac{1}{x}+\\dfrac{2}{y})_{\\min}=4 \\sqrt{2}$.\\\\\n乙: $(x+y)(\\dfrac{1}{x}+\\dfrac{2}{y})=3+\\dfrac{y}{x}+\\dfrac{2 x}{y}\\geq 3+2 \\sqrt{\\dfrac{y}{x}\\cdot \\dfrac{2 x}{y}}=3+2 \\sqrt{2}$,\n当 $y=\\sqrt{2}x$ 时, $(x+y)(\\dfrac{1}{x}+\\dfrac{2}{y})_{\\text{min}}=3+2 \\sqrt{2}$.\\\\\n试判断谁的结论有错? 为什么错?", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260105-高一上学期周末卷05" ], "genre": "解答题", "ans": "甲错误, 等号取不到", @@ -599461,7 +600504,8 @@ "content": "已知 $a, b, c$ 为正数, $a+b+c=1$, 求证:\\\\\n(1) $\\dfrac{1}{a}+\\dfrac{1}{b}+\\dfrac{1}{c}\\geq 9$;\\\\\n(2) $a b+b c+c a \\leq \\dfrac{1}{3}$;\\\\\n(3) $a^2+b^2+c^2 \\geq \\dfrac{1}{3}$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260105-高一上学期周末卷05" ], "genre": "解答题", "ans": "(1) 证明略; (2) 证明略; (3) 证明略", @@ -599739,7 +600783,8 @@ "content": "$\\dfrac{\\sqrt[3]{x \\sqrt{x}}}{x}=$\\blank{50}.(用分数指数幂表示)", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260106-高一上学期周末卷06" ], "genre": "填空题", "ans": "$x^{-\\frac{1}{2}}$", @@ -599773,7 +600818,8 @@ "content": "已知$(5 x-\\dfrac{1}{2})^{-\\frac{3}{4}}+(x-10)^0$ 有意义, 则 $x$ 的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260106-高一上学期周末卷06" ], "genre": "填空题", "ans": "$(\\dfrac{1}{10},10)\\cup (10,+\\infty)$", @@ -599807,7 +600853,8 @@ "content": "若$\\log _{(a-5)}(9-a)$ 有意义, 则 $a$ 的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260106-高一上学期周末卷06" ], "genre": "填空题", "ans": "$(5,6)\\cup (6,9)$", @@ -599841,7 +600888,8 @@ "content": "化简: $(a^{\\frac{2}{3}}b^{\\frac{1}{2}})(-3 a^{\\frac{1}{2}}b^{\\frac{1}{3}}) \\div(\\dfrac{1}{3}a^{\\frac{1}{6}}b^{\\frac{5}{6}})=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260106-高一上学期周末卷06" ], "genre": "填空题", "ans": "$-9a$", @@ -599875,7 +600923,8 @@ "content": "已知 $a>0$ 且 $a \\neq 1, M$, $N \\in \\mathbf{R}$. ``$M=N$''是``$\\log _a M=\\log _a N$''的\\blank{50}条件.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260106-高一上学期周末卷06" ], "genre": "填空题", "ans": "必要非充分", @@ -599909,7 +600958,8 @@ "content": "已知自然常数 $\\mathrm{e} \\approx 2.71828 \\cdots$, 若 $\\mathrm{e}^\\alpha=4$, $\\beta=\\ln 2$, 则 $\\mathrm{e}^{2 \\alpha+\\beta}=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260106-高一上学期周末卷06" ], "genre": "填空题", "ans": "$32$", @@ -599943,7 +600993,8 @@ "content": "方程 $\\log _2 x^2=-2$ 的解集是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260106-高一上学期周末卷06" ], "genre": "填空题", "ans": "$\\{\\pm\\dfrac{1}{2}\\}$", @@ -599977,7 +601028,8 @@ "content": "计算下列各式的值(不能使用计算器).\\\\\n(1) $\\log _535+2 \\log _{\\frac{1}{2}}\\sqrt{2}-\\log _5 \\dfrac{1}{50}-\\log _514=$\\blank{50}.\\\\\n(2) $\\dfrac{\\log _5 \\sqrt{2}\\times \\log _79}{\\log _5 \\dfrac{1}{3}\\times \\log _7 \\sqrt[3]{4}}=$\\blank{50}.\\\\\n(3) $(\\log _43+\\log _83) \\times \\dfrac{\\log _72}{\\log _73}=$\\blank{50}.\\\\\n(4) $2^{\\log _4(\\sqrt{3}-2)^2}=$\\blank{50}.\\\\\n(5) $\\log _6^23+\\dfrac{\\log _618}{\\log _26}=$\\blank{50}.\\\\\n(6) $\\log _2 \\dfrac{\\sqrt{2}}{2}-\\log _3 \\dfrac{\\sqrt{3}}{3}+(\\sqrt{7})^{\\frac{1}{2}\\log _714-\\frac{1}{2}}=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260106-高一上学期周末卷06" ], "genre": "填空题", "ans": "(1) $2$; (2) $-\\dfrac{3}{2}$; (3) $\\dfrac{5}{6}$; (4) $2-\\sqrt{3}$; (5) $1$; (6) $2^{\\frac{1}{4}}$", @@ -600011,7 +601063,8 @@ "content": "用 $a, b$ 的不含对数的式子表示下列各式的值.\\\\\n(1) 若 $\\log _53=a$, 则 $\\log _{45}9=$\\blank{50}.\\\\\n(2) 若 $\\log _{18}6=b$, 则 $\\log _{12}27=$\\blank{50}.\\\\\n(3) 若 $\\log _35=a$, $\\log _25=b$, 则 $\\log _65=$\\blank{50}.\\\\\n(4) 若 $\\log _754=a$, $\\log _772=b$, 则 $\\log _712=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260106-高一上学期周末卷06" ], "genre": "填空题", "ans": "(1) $\\dfrac{2a}{1+2a}$; (2) $\\dfrac{3-3b}{3b-1}$; (3) $\\dfrac{ab}{a+b}$; (4) $-\\dfrac{1}{7}a+\\dfrac{5}{7}b$", @@ -600045,7 +601098,8 @@ "content": "若方程 $(\\lg x)^2+(\\lg 7+\\lg 5) \\lg x+\\lg 7 \\cdot \\lg 5=0$ 的两根为 $\\alpha, \\beta$, 则 $\\alpha \\beta=$\\bracket{20}.\n\\fourch{$\\lg 7 \\cdot \\lg 5$}{$\\lg 35$}{$35$}{$\\dfrac{1}{35}$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260106-高一上学期周末卷06" ], "genre": "选择题", "ans": "D", @@ -600080,7 +601134,8 @@ "content": "若 $a$、$b$、$c$ 都是正数, 且 $3^a=4^b=6^c$, 则\\bracket{20}.\n\\fourch{$\\dfrac{1}{c}=\\dfrac{1}{a}+\\dfrac{1}{b}$}{$\\dfrac{2}{c}=\\dfrac{2}{a}+\\dfrac{1}{b}$}{$\\dfrac{1}{c}=\\dfrac{2}{a}+\\dfrac{2}{b}$}{$\\dfrac{2}{c}=\\dfrac{1}{a}+\\dfrac{2}{b}$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260106-高一上学期周末卷06" ], "genre": "选择题", "ans": "B", @@ -600159,7 +601214,8 @@ "content": "已知 $x_1$、$x_2$ 是方程 $x^2-m x+3=0$ 的两个根, 且 $\\dfrac{\\ln (x_1+x_2)}{\\ln x_1+\\ln x_2}=2$, 求 $m$ 的值.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260106-高一上学期周末卷06" ], "genre": "解答题", "ans": "$9$", @@ -600193,7 +601249,8 @@ "content": "已知 $\\log _a(x^2+1)+\\log _a(y^2+4)=\\log _a 8+\\log _a x+\\log _a y$, 求 $\\log _8(x y)$ 的值.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260106-高一上学期周末卷06" ], "genre": "解答题", "ans": "$\\dfrac{1}{3}$", @@ -600227,7 +601284,8 @@ "content": "已知 $a>0$ 且 $a \\neq 1$, $x>y>0$, 下列四个等式中, 恒成立的是\\blank{50}.\\\\\n\\textcircled{1} $\\log _a x \\log _a y=\\log _a(x+y)$;\\\\\n\\textcircled{2} $\\log _a x+\\log _a y=\\log _a(x+y)$;\\\\\n\\textcircled{3} $\\log _a \\dfrac{x}{y}=\\log _a x-\\log _a y$;\\\\\n\\textcircled{4} $\\log _a(x-y)=\\dfrac{\\log _a x}{\\log _a y}$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260106-高一上学期周末卷06" ], "genre": "填空题", "ans": "\\textcircled{3}", @@ -600261,7 +601319,8 @@ "content": "将 $a^{2 k}=N$($a>0$, $a \\neq 1$, $k\\ne 0$) 转化成对数式, 其中错误的是\\blank{50}.\\\\\n\\textcircled{1} $k=\\log _{a^2}N$\\\\\n\\textcircled{2} $2=\\log _{a^k}N$\\\\\n\\textcircled{3} $k=\\dfrac{1}{2}\\log _a N$\\\\\n\\textcircled{4} $k=\\log _a \\dfrac{N}{2}$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260106-高一上学期周末卷06" ], "genre": "填空题", "ans": "\\textcircled{4}", @@ -600296,7 +601355,8 @@ "content": "设 $x$、$y>1$, 且 $2 \\log _x y-2 \\log _y x+3=0$, 求 $T=x^2-4 y^2$ 的最小值.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260106-高一上学期周末卷06" ], "genre": "解答题", "ans": "$-4$", @@ -600660,7 +601720,8 @@ "content": "已知 $a$、$b$、$c$ 为正实数, $a^x=b^y=c^z$, $\\dfrac{1}{x}+\\dfrac{1}{y}+\\dfrac{1}{z}=0$, 则 $a b c$ 的值为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "I20260104-高一上学期随堂练习04" ], "genre": "填空题", "ans": "$1$", @@ -600718,7 +601779,8 @@ "content": "写出下列函数的定义域:\\\\\n(1) $y=x^{-\\frac{4}{5}}$: \\blank{100};\\\\\n(2) $y=x^{-\\frac{3}{2}}$: \\blank{100};\\\\\n(3) $y=x^{\\frac{7}{4}}$: \\blank{100};\\\\\n(4) $y=x^{\\frac{2}{3}}$: \\blank{100}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260107-高一上学期周末卷07" ], "genre": "填空题", "ans": "(1) $(-\\infty,0)\\cup (0,+\\infty)$; (2) $(0,+\\infty)$; (3) $[0,+\\infty)$; (4) $\\mathbf{R}$", @@ -600753,7 +601815,8 @@ "content": "设 $m \\in \\mathbf{Z}$, 若幂函数 $y=x^{-m^2+2 m}$ 的定义域为 $\\mathbf{R}$, 则 $m$ 的值为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260107-高一上学期周末卷07" ], "genre": "填空题", "ans": "$1$", @@ -600788,7 +601851,8 @@ "content": "若幂函数 $f(x)=x^{m^2+2 m-3}$($m \\in \\mathbf{Z}$) 的图像关于 $y$ 轴对称, 且在区间 $(0,+\\infty)$ 上是严格减函数, 则 $m=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260107-高一上学期周末卷07" ], "genre": "填空题", "ans": "$-1$", @@ -600847,7 +601911,8 @@ "content": "已知 $x>y>1$, 且 $0a^{\\frac{1}{y}}$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260107-高一上学期周末卷07" ], "genre": "填空题", "ans": "\\textcircled{1}\\textcircled{4}", @@ -600904,7 +601969,8 @@ "content": "设常数 $b \\in \\mathbf{R}$, 则``$b=2$''是``函数 $y=(2 b^2-3 b-1) x^{b-2}$ 为幂函数''的\\blank{50}条件.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260107-高一上学期周末卷07" ], "genre": "填空题", "ans": "充分非必要", @@ -600940,7 +602006,8 @@ "content": "若点 $A(a, b)$ 在幂函数 $y=x^n$($n \\in \\mathbf{R}$) 的图像上, 那么下列结论中不能成立的是 \\bracket{20}.\n\\fourch{$\\begin{cases}a>0,\\\\b>0\\end{cases}$}{$\\begin{cases}a>0,\\\\b<0\\end{cases}$}{$\\begin{cases}a<0,\\\\b<0\\end{cases}$}{$\\begin{cases}a<0,\\\\b>0\\end{cases}$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260107-高一上学期周末卷07" ], "genre": "选择题", "ans": "B", @@ -600999,7 +602066,8 @@ "content": "如果幂函数 $y=x^k$ 的图像, 当 $0=latex,scale = 0.5]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain = {-3^(1/4)}:{3^(1/4)}] plot (\\x*\\x*\\x,\\x*\\x*\\x*\\x);\n\\draw (0,-3) node [below] {(A)};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain = {-3^(1/5)}:{3^(1/5)}] plot (\\x*\\x*\\x,\\x*\\x*\\x*\\x*\\x);\n\\draw (0,-3) node [below] {(B)};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,scale =0.5]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain = {0}:{3^(1/3)}] plot (\\x*\\x,\\x*\\x*\\x);\n\\draw (0,-3) node [below] {(C)};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain = {3^(-3/2)}:3] plot (\\x,{\\x^(-2/3)}) plot (-\\x,{\\x^(-2/3)});\n\\draw (0,-3) node [below] {(D)};\n\\end{tikzpicture}\n\\end{center}\n(1) $y=x^\\frac 32$\\blank{50}; (2) $y=x^\\frac 43$\\blank{50}; (3) $y=x^\\frac 53$\\blank{50}; (4) $y=x^{-\\frac 23}$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260107-高一上学期周末卷07" ], "genre": "填空题", "ans": "(1) C; (2) A; (3) B; (4) D", @@ -601135,7 +602204,8 @@ "content": "若 $(a+1)^{-\\frac{2}{3}}>(3-2 a)^{-\\frac{2}{3}}$, 则 $a$ 取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260107-高一上学期周末卷07" ], "genre": "填空题", "ans": "$(-\\infty,-1)\\cup (-1,\\dfrac{2}{3})\\cup (4,+\\infty)$", @@ -601238,7 +602308,8 @@ "content": "作出函数 $y=\\dfrac{x^2+4 x+6}{x^2+4 x+4}$ 的大致图像, 并根据图像判断 $x$ 分别取 $-\\pi$ 和 $-\\dfrac{\\sqrt{2}}{2}$ 时, 函数值 $y_1$ 和 $y_2$ 的大小.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260107-高一上学期周末卷07" ], "genre": "解答题", "ans": "\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-6,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (-6,1) -- (2,1) (-2,-0.5) -- (-2,5);\n\\draw [domain = -6:{1/2 *(-4 - sqrt(2))}, samples = 100] plot (\\x,{1+2/(\\x+2)/(\\x+2)});\n\\draw [domain = {1/2 *(-4 + sqrt(2))}:2, samples = 100] plot (\\x,{1+2/(\\x+2)/(\\x+2)});\n\\end{tikzpicture}, $y_1>y_2$", @@ -601449,7 +602520,8 @@ "content": "幂函数 $y=x^{\\frac{1}{2}}$ 的图像可由函数 $y=1+\\sqrt{x-1}$ 的图像向\\blank{20}(填``左''或``右'') 平移一个单位, 再向\\blank{20}(填``上''或``下'') 平移一个单位得到(仅需写出一种可行的方式).", "objs": [], "tags": [ - "第二单元" + "第二单元", + "I20260104-高一上学期随堂练习04" ], "genre": "填空题", "ans": "右, 上(或上, 右)", @@ -601508,7 +602580,8 @@ "content": "$(\\log _43+\\log _83)(\\log _32+\\log _92)+\\log _{\\frac{1}{2}}\\sqrt[4]{32}=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "I20260104-高一上学期随堂练习04" ], "genre": "填空题", "ans": "$0$", @@ -601543,7 +602616,8 @@ "content": "设 $\\lg x+\\lg y=\\lg (x-2 y)^2$, 则 $\\log _{\\sqrt{2}}\\dfrac{x}{y}=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "I20260104-高一上学期随堂练习04" ], "genre": "填空题", "ans": "$0$或$4$", @@ -601734,7 +602808,8 @@ "content": "设常数 $a>0$ 且 $a \\neq 1$. 若无论 $a$ 取何值, 函数 $y=a^{x-3}+3$ 图像恒经过一个定点, 则此定点的坐标是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260107-高一上学期周末卷07" ], "genre": "填空题", "ans": "$(3,4)$", @@ -601813,7 +602888,8 @@ "content": "不等式$(\\dfrac{1}{4})^x<2^{-x^2+2 x-3}$ 的解集是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260107-高一上学期周末卷07" ], "genre": "填空题", "ans": "$(1,3)$", @@ -601870,7 +602946,8 @@ "content": "若指数函数 $y=(a^2-1)^x$ 的值在区间 ($-\\infty, 0$) 上恒大于 $1$, 则实数 $a$ 的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260107-高一上学期周末卷07" ], "genre": "填空题", "ans": "$(-\\sqrt{2},-1)\\cup (1,\\sqrt{2})$", @@ -601951,7 +603028,8 @@ "content": "已知函数\\textcircled{1} $y=3^x-1$, \\textcircled{2} $y=3^{x-1}$, \\textcircled{3} $y=-3^x$, \\textcircled{4} $y=-3^{-x}$, \\textcircled{5} $y=(\\dfrac{1}{3})^x$, \\textcircled{6} $y=(\\dfrac{1}{3})^{-x}$. 将 $y=3^x$ 的图像向右平移 $1$ 个单位得到\\blank{50}的图像; 将 $y=3^x$ 的图像向下平移 $1$ 个单位得到\\blank{50}的图像; $y=3^x$ 的图像与\\blank{50}的图像关于 $x$ 轴对称, 与\\blank{50}的图像关于 $y$ 轴对称, 与\\blank{50}的图像关于原点对称, 与\\blank{50}的图像完全相同. (用序号作答)", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260107-高一上学期周末卷07" ], "genre": "填空题", "ans": "\\textcircled{2}, \\textcircled{1}, \\textcircled{3}, \\textcircled{5}, \\textcircled{4}, \\textcircled{6}", @@ -602052,7 +603130,8 @@ "content": "一个小朋友买了一个体积为 $a$ 的彩色大气球, 放在自己的房间内, 由于气球的密封性不好, 经过 $t$ 天后气球体积变为 $V=a \\cdot \\mathrm{e}^{-k t}$. 若经过 $25$ 天后, 气球体积变为原来的 $\\dfrac{2}{3}$, 则至少经过多少天后, 气球的体积小于原来的 $\\dfrac{1}{3}$ ? (结果精确到 1 天)", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260107-高一上学期周末卷07" ], "genre": "解答题", "ans": "$68$天", @@ -602310,7 +603389,8 @@ "content": "若 $f(x)$ 的定义域为 $[0,10]$, 则函数 $F(x)=f(x+2)+f(x-2)$ 的定义域是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260108-高一上学期周末卷08" ], "genre": "填空题", "ans": "$[2,8]$", @@ -602345,7 +603425,8 @@ "content": "设函数 $f(x)$ 的定义域为 $(0,6)$, $g(x)$ 的定义域为 $[2,7]$, 若 $f(x)>g(x)$ 的解集是 $(3,5)$, 则 $f(x) \\leq g(x)$ 的解集是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260108-高一上学期周末卷08" ], "genre": "填空题", "ans": "$[2,3]\\cup [5,6)$", @@ -602381,7 +603462,8 @@ "objs": [], "tags": [ "第二单元", - "第一单元" + "第一单元", + "W20260108-高一上学期周末卷08" ], "genre": "填空题", "ans": "$(2,+\\infty)$", @@ -602416,7 +603498,8 @@ "content": "函数 $f(x)=\\begin{cases}x^2,& x \\geq 4,\\\\f(x+3), & x<4,\\end{cases}$ 则 $f(2)=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260108-高一上学期周末卷08" ], "genre": "填空题", "ans": "$25$", @@ -602474,7 +603557,8 @@ "content": "若函数 $f(x)=\\sqrt{2 x^2+3}$, 函数 $g(x)=f(\\sqrt{x})$, 则函数 $y=g(x)$ 的解析式为\\blank{50}, 定义域为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260108-高一上学期周末卷08" ], "genre": "填空题", "ans": "$y=\\sqrt{2x+3}$, $[0,+\\infty$", @@ -602509,7 +603593,8 @@ "content": "函数 $f(x)=\\dfrac{-5 x+3}{x-3}$, $x \\in(-\\infty, 2) \\cup[4,+\\infty)$ 的值域为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260108-高一上学期周末卷08" ], "genre": "填空题", "ans": "$[-17,-5)\\cup (-5,7)$", @@ -602566,7 +603651,8 @@ "content": "下列各组函数中, 表示同一函数的是\\bracket{20}.\n\\twoch{$y=1$, $y=\\dfrac{x}{x}$}{$y=\\sqrt{x-1}\\cdot \\sqrt{x+1}$, $y=\\sqrt{x^2-1}$}{$y=x$, $y=\\sqrt[3]{x^3}$}{$y=|x|$, $y=(\\sqrt{x})^2$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260108-高一上学期周末卷08" ], "genre": "选择题", "ans": "C", @@ -602601,7 +603687,8 @@ "content": "若 $a, b \\in \\mathbf{R}$, 则``$b^2<4 a c$''是``函数 $y=\\dfrac{1}{a x^2+b x+c}$ 的定义域为 $\\mathbf{R}$''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": [ - "第二单元" + "第二单元", + "W20260108-高一上学期周末卷08" ], "genre": "选择题", "ans": "A", @@ -602636,7 +603723,8 @@ "content": "若定义在 $\\mathbf{R}$ 上的函数 $f(x)$、$g(x)$ 均为奇函数, 设 $F(x)=a f(x)+b g(x)+1$, 若 $F(-2)=10$, 则 $F(2)$ 的值是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260108-高一上学期周末卷08" ], "genre": "填空题", "ans": "$-8$", @@ -602671,7 +603759,8 @@ "content": "已知函数 $f(x)$ 为定义在 $\\mathbf{R}$ 上的函数, 则命题:``存在 $x_0 \\in \\mathbf{R}$, 使 $f(-x_0) \\neq f(x_0)$ 且 $f(-x_0) \\neq-f(x_0)$''是命题``$f(x)$ 为非奇非偶函数''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": [ - "第二单元" + "第二单元", + "W20260108-高一上学期周末卷08" ], "genre": "选择题", "ans": "A", @@ -602728,7 +603817,8 @@ "content": "满足解析式为 $y=2 x^2+1$, 且值域为 $\\{5,19\\}$ 的函数有\\bracket{20}.\n\\fourch{4 个}{6 个}{8 个}{9 个}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260108-高一上学期周末卷08" ], "genre": "选择题", "ans": "D", @@ -602855,7 +603945,8 @@ "content": "已知定义域为 $[0,1]$ 的函数 $f(x)=1-|1-2 x|$ 和 $g(x)=(x-1)^2$, 且 $F(x)=\\begin{cases}g(x), & f(x) \\geq g(x),\\\\f(x),& f(x)f(2 a-1)$, 则实数 $a$ 的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260109-高一上学期周末卷09" ], "genre": "填空题", "ans": "$(-\\dfrac{1}{2},\\dfrac{1}{3})\\cup (1,\\dfrac{3}{2})$", @@ -603499,7 +604601,8 @@ "content": "下列函数中, 在区间 $(-\\infty, 0)$ 上是减函数的是\\bracket{20}.\n\\fourch{$y=1-x^2$}{$y=x^2+2 x$}{$y=\\dfrac{x+2}{x+1}$}{$y=\\dfrac{x}{x-1}$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260109-高一上学期周末卷09" ], "genre": "选择题", "ans": "D", @@ -603534,7 +604637,8 @@ "content": "下列函数中, 是奇函数的是\\bracket{20}.\n\\twoch{$f(x)=x^2+x$}{$g(x)=1-x^2$}{$h(x)=2 x+1$}{$m(x)=\\lg (\\sqrt{1+x^2}-x)$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260109-高一上学期周末卷09" ], "genre": "选择题", "ans": "D", @@ -603570,7 +604674,8 @@ "content": "下列命题中, 正确的是\\bracket{20} .\n\\onech{若函数 $f(x)$ 在 $(-\\infty, 0]$ 上严格递增, 在 $(0,+\\infty)$ 上也严格递增, 则它在 $(-\\infty, +\\infty)$ 上严格递增}{若函数 $f(x)$ 在 $(-\\infty, 2]$ 上严格递增, 在 $[2,+\\infty)$ 上也严格递增, 则它在 $(-\\infty, +\\infty)$ 上严格递增}{若 $f(x)$ 在 $D$ 上严格递增, $g(x)$ 在 $D$ 上严格递减, 则 $F(x)=f(x)+g(x)$ 在 $D$ 上严格递减}{若 $f(x)$ 在 $D$ 上严格递增, $g(x)$ 在 $D$ 上严格递增, 则 $G(x)=f(x) \\cdot g(x)$ 在 $D$ 上严格递增}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260109-高一上学期周末卷09" ], "genre": "选择题", "ans": "B", @@ -603698,7 +604803,8 @@ "content": "已知函数 $f(x)$ 的定义域是 $x \\neq 0$ 的一切实数, 对定义域内的任意 $x_1, x_2$ 都有 $f(x_1 \\cdot x_2)=f(x_1)+f(x_2)$, 且当 $x>1$ 时 $f(x)>0$, $f(2)=1$.\\\\\n(1) 求证: $f(x)$ 是偶函数;\\\\\n(2) 求证: $f(x)$ 在 ($0,+\\infty$) 上是增函数;\\\\\n(3) 解不等式 $f(2 x^2-1)<2$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260109-高一上学期周末卷09" ], "genre": "解答题", "ans": "(1) 证明略; (2) 证明略; (3) $(-\\dfrac{\\sqrt{10}}{2},-\\dfrac{\\sqrt{2}}{2})\\cup (-\\dfrac{\\sqrt{2}}{2},\\dfrac{\\sqrt{2}}{2})\\cup (\\dfrac{\\sqrt{2}}{2},\\dfrac{\\sqrt{10}}{2})$", @@ -603755,7 +604861,8 @@ "content": "函数 $y=\\sqrt{1-2^x}$ 的值域是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "I20260105-高一上学期随堂练习05" ], "genre": "填空题", "ans": "$[0,1)$", @@ -603925,7 +605032,8 @@ "content": "设 $m>0$, 若二次函数 $f(x)=-x^2+m x+1+n$, $x \\in[-1,1]$ 的最大值是 $9$ , 最小值是 $6$, 求实数 $m$、$n$ 的值.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260109-高一上学期周末卷09" ], "genre": "解答题", "ans": "$m=-2+2\\sqrt{3}$, $n=4+2\\sqrt{3}$", @@ -604646,7 +605754,8 @@ "content": "已知函数 $y=x^3-2 x^2+8 x^{\\frac{5}{3}}-1$ 在区间 $(0,1)$ 上有一个零点, 由二分法得这个零点约是\\blank{50}. (结果精确到$0.1$)", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260110-高一上学期周末卷10" ], "genre": "填空题", "ans": "$0.3$", @@ -604682,7 +605791,8 @@ "objs": [], "tags": [ "第一单元", - "第二单元" + "第二单元", + "W20260110-高一上学期周末卷10" ], "genre": "填空题", "ans": "$(\\dfrac{1}{2},+\\infty)$", @@ -604717,7 +605827,8 @@ "content": "(1) 已知 $f(x)$ 是定义域为 $[1,5]$ 上的连续函数, 则``$f(1) f(5)>0$''是``$f(x)$ 不存在零点''的\\blank{50}条件;\\\\\n(2) 已知 $f(x)$ 是定义域为 $[1,5]$ 上的函数, 则``$f(1) f(5)<0$''是``$f(x)$ 存在零点''的\\blank{50}条件.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260110-高一上学期周末卷10" ], "genre": "填空题", "ans": "(1) 必要非充分; (2) 既非充分又非必要", @@ -604774,7 +605885,8 @@ "content": "若函数 $f(x)=4^x+(m-3) \\cdot 2^x+m$ 有两个不相等的零点, 则实数 $m$ 的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260110-高一上学期周末卷10" ], "genre": "填空题", "ans": "$(0,1)$", @@ -604809,7 +605921,8 @@ "content": "若对任意 $x \\in \\mathbf{R}$, 都有 $4^x a+(3 a-1) 2^x+a>0$ 成立, 则实数 $a$ 的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260110-高一上学期周末卷10" ], "genre": "填空题", "ans": "$(\\dfrac{1}{5},+\\infty)$", @@ -604844,7 +605957,8 @@ "content": "若存在 $x \\in \\mathbf{R}$, 使得 $4^x a+(3 a-1) 2^x+a>0$ 成立, 则实数 $a$ 的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260110-高一上学期周末卷10" ], "genre": "填空题", "ans": "$(0,+\\infty)$", @@ -604879,7 +605993,8 @@ "content": "已知不等式 $3 x^2<\\log _a x$($a>0$, $a \\neq 1$) 在 $x \\in(0, \\dfrac{1}{3})$ 上恒成立, 则实数 $a$ 的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260110-高一上学期周末卷10" ], "genre": "填空题", "ans": "$[\\dfrac{1}{27},1)$", @@ -604914,7 +606029,8 @@ "content": "设 $n \\in \\mathbf{Z}$, 且函数 $y=10 x^3-10 x+1$ 在区间 $[n, n+2]$ 上有零点, 则 $n$ 的所有可能值为\\blank{50}. (注: 三次函数至多有三个零点)", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260110-高一上学期周末卷10" ], "genre": "填空题", "ans": "$-3,-2,-1,0$", @@ -605083,7 +606199,8 @@ "content": "判断方程 $2 x^3+3 x-2=50$ 是否有整数解, 并证明你的结论.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260110-高一上学期周末卷10" ], "genre": "解答题", "ans": "无整数解, 理由略", @@ -605338,7 +606455,8 @@ "content": "已知函数 $f(x)=\\dfrac{2 x+1}{x+a}$($x \\neq-a$, $a \\neq \\dfrac{1}{2}$).\\\\\n(1) 求 $f(x)$ 的反函数 $f^{-1}(x)$;\\\\\n(2) 设 $f(x)=f^{-1}(x)$, 求实数 $a$ 的值.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260111-高一上学期周末卷11" ], "genre": "解答题", "ans": "(1) $f^{-1}(x)=\\dfrac{1-ax}{x-2}$, $x\\in (-\\infty,2)\\cup (2,+\\infty)$; (2) $a=2$", @@ -605373,7 +606491,8 @@ "content": "在等差数列 $\\{a_n\\}$ 中, 已知 $a_1=\\dfrac{5}{6}$, $d=-\\dfrac{1}{6}$, 若 $a_n=-\\dfrac{3}{2}$, 则 $n=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "I20260107-高一上学期随堂练习07" ], "genre": "填空题", "ans": "$15$", @@ -605408,7 +606527,8 @@ "content": "等差数列 $\\{a_n\\}$ 中, 若 $a_6+a_9+a_{12}+a_{15}=20$, 则该数列的前$20$项之和 $S_{20}=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "I20260107-高一上学期随堂练习07" ], "genre": "填空题", "ans": "$100$", @@ -605444,7 +606564,8 @@ "content": "若 3 与 $x$ 的等比中项为 $3 x$, 则 $x$ 的值为\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "W20260112-高一上学期周末卷12" ], "genre": "填空题", "ans": "$\\dfrac{1}{3}$", @@ -605479,7 +606600,8 @@ "content": "$S_n$ 是等差数列 $\\{a_n\\}$ 的前 $n$ 项和, 满足 $S_{10}=100$, $S_{20}=110$, 则 $S_{40}=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "W20260112-高一上学期周末卷12" ], "genre": "填空题", "ans": "$-140$", @@ -605514,7 +606636,8 @@ "content": "在等比数列 $\\{a_n\\}$ 中, 若 $a_3 \\cdot a_7 \\cdot a_{17}\\cdot a_{21}=25$, 则 $a_5 \\cdot a_{19}=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "I20260107-高一上学期随堂练习07" ], "genre": "填空题", "ans": "$5$", @@ -605549,7 +606672,8 @@ "content": "在等差数列 $\\{a_n\\}$ 中, 满足 $3 a_4=7 a_7$, 且 $a_1>0, S_n$ 是数列 $\\{a_n\\}$ 前 $n$ 项的和. 若 $S_n$ 取得最大值, 则 $n=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "W20260112-高一上学期周末卷12" ], "genre": "填空题", "ans": "$9$", @@ -605584,7 +606708,8 @@ "content": "在等比数列 $\\{a_n\\}$ 中, 首项 $a_1<0$, 则 $\\{a_n\\}$ 是严格递增数列的一个充要条件是公比 $q$ 满足\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "W20260112-高一上学期周末卷12" ], "genre": "填空题", "ans": "$0S_8$, 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 $\\{a_n\\}$ 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\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "W20260112-高一上学期周末卷12" ], "genre": "填空题", "ans": "\\textcircled{2}\\textcircled{3}", @@ -605938,7 +607072,8 @@ "content": "在等差数列 $\\{a_n\\}$ 中, $a_4=10$, 且 $a_3, a_6, a_{10}$ 成等比数列, 求数列 $\\{a_n\\}$ 的前 $n$ 项和 $S_n$.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "W20260112-高一上学期周末卷12" ], "genre": "解答题", "ans": "$S_n=10n$或$S_n=\\dfrac{n^2+13n}{2}$", @@ -605973,7 +607108,8 @@ "content": "等差数列 $\\{a_n\\}$ 中, 设$S_n$表示$\\{a_n\\}$的前$n$项之和. 若 $S_{12}>0$, $S_{13}<0$, 则 $S_1, S_2, S_3, \\cdots, S_n$ 中最大项为\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "I20260107-高一上学期随堂练习07" ], "genre": "填空题", "ans": "$S_6$", @@ -606009,7 +607145,8 @@ "content": "有固定项的数列 $\\{a_n\\}$ 的前 $n$ 项和 $S_n=2 n^2+n$, 现从中抽取某一项(不包括首项、末项)后, 余下的项的平均值是 79 .\\\\\n(1) 求数列 $\\{a_n\\}$ 的通项 $a_n$;\\\\\n(2) 求这个数列的项数, 并回答抽取的是第几项?", "objs": [], "tags": [ - "第四单元" + "第四单元", + "W20260112-高一上学期周末卷12" ], "genre": "解答题", "ans": "(1) $a_n=4n-1$; (2) 共有$39$项, 抽取的是第$20$项", @@ -606158,7 +607295,8 @@ "content": "设 $n$ 是正整数, 求和: $2+2^4+\\cdots+2^{3 n+4}=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "W20260113-高一上学期周末卷13" ], "genre": "填空题", "ans": "$\\dfrac{2^{3n+7}-2}{7}$", @@ -606214,7 +607352,8 @@ "content": "数列 $\\{a_n\\}$ 的前 $n$ 项和 $S_n$ 满足: $S_n=\\begin{cases}3 n+1, &n \\leq 5,\\\\ n^2, & n \\geq 6,\\end{cases}$ 则数列 $\\{a_n\\}$ 的通项公式为\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "W20260114-高一上学期周末卷14" ], "genre": "填空题", "ans": "$a_n=\\begin{cases}4, & n=1, \\\\ 3, & 2\\le n\\le 5, \\\\ 20, & n=6, \\\\ 2n-1, & n\\ge 7\\end{cases}$", @@ -606245,7 +607384,8 @@ "content": "已知公比为 $q$($0=latex]\n\\draw (0,0) circle (1);\n\\draw (1,0) --++ (120:1) --++ (180:1) --++ (240:1) --++ (300:1) --++ (0:1) --++ (60:1);\n\\draw (0,0) circle ({sqrt(3)/2});\n\\draw ({sqrt(3)/2},0) --++ (120:{sqrt(3)/2}) --++ (180:{sqrt(3)/2}) --++ (240:{sqrt(3)/2}) --++ (300:{sqrt(3)/2}) --++ (0:{sqrt(3)/2}) --++ (60:{sqrt(3)/2});\n\\draw (0,0) circle (0.75);\n\\draw (0,0) node {$\\cdots$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第四单元" + "第四单元", + "W20260113-高一上学期周末卷13" ], "genre": "解答题", "ans": "$4\\pi r^2$", @@ -606672,7 +607815,8 @@ "content": "在各项均为正数的等比数列 $\\{a_n\\}$ 中, $a_5=3$, 则 $\\log _3 a_1+\\log _3 a_2+\\cdots+\\log _3 a_9$ 的值为\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "W20260113-高一上学期周末卷13" ], "genre": "填空题", "ans": "$9$", @@ -606729,7 +607873,8 @@ "content": "已知等比数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 且 $S_n=3 \\cdot 2^n+a$, 则 $a_1+a_9=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "W20260113-高一上学期周末卷13" ], "genre": "填空题", "ans": "$771$", @@ -606851,7 +607996,8 @@ "content": "``数列 $\\{a_n\\}$ 为等比数列''是``数列 $\\{a_n^2\\}$ 为等比数列''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", + "W20260113-高一上学期周末卷13" ], "genre": "选择题", "ans": "A", @@ -606885,7 +608031,8 @@ "content": "已知 $\\{a_n\\}$ 是等比数列, $a_2=2$, $a_3=1$, 则 $a_1 a_2+a_2 a_3+\\cdots+a_n a_{n+1}$ 等于\\bracket{20}.\n\\fourch{$16(1-4^{-n})$}{$16(1-2^{-n})$}{$\\dfrac{32}{3}(1-4^{-n})$}{$\\dfrac{32}{3}(1-2^{-n})$}", "objs": [], "tags": [ - "第四单元" + "第四单元", + "W20260113-高一上学期周末卷13" ], "genre": "选择题", "ans": "C", @@ -606919,7 +608066,8 @@ "content": "关于问题``函数 $f(x)=\\dfrac{x-1}{2 x-11}$ 的最值, 以及数列 $a_n=\\dfrac{n-1}{2 n-11}$ 的最值'', 下列说法正确的是\\bracket{20}.\n\\onech{函数 $f(x)$ 既有最大值又有最小值, 数列 $\\{a_n\\}$ 既有最大值又有最小值}{函数 $f(x)$ 既无最大值又无最小值, 数列 $\\{a_n\\}$ 既有最大值又有最小值}{函数 $f(x)$ 既有最大值又有最小值, 数列 $\\{a_n\\}$ 有最大值无最小值}{函数 $f(x)$ 既无最大值又无最小值, 数列 $\\{a_n\\}$ 无最大值有最小值}", "objs": [], "tags": [ - "第四单元" + "第四单元", + "W20260113-高一上学期周末卷13" ], "genre": "选择题", "ans": "B", @@ -607109,7 +608257,8 @@ "content": "陈述句``存在 $x \\in \\mathbf{R}$, 使得 $2 x+c>0$''的否定形式为\\blank{150}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260115-2026届高一上学期国庆复习卷" ], "genre": "填空题", "ans": "", @@ -607407,7 +608556,8 @@ "content": "不等式 $-3 x^2+x+2>0$ 的解集是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "E20260102-高一上学期测验卷02" ], "genre": "填空题", "ans": "$(-\\dfrac{2}{3},1)$", @@ -607442,7 +608592,8 @@ "content": "不等式 $\\dfrac{x-1}{x-3}\\leq 0$ 的解集为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "E20260102-高一上学期测验卷02" ], "genre": "填空题", "ans": "$[1,3)$", @@ -607477,7 +608628,8 @@ "content": "设常数 $a, b \\in \\mathbf{R}$, 写出等式 $|a|-|b|=|a-b|$ 成立的一个充分非必要条件: \\blank{100}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "E20260102-高一上学期测验卷02" ], "genre": "填空题", "ans": "(如)$a=b=100$(充要条件为$b(a-b)\\ge 0$", @@ -607534,7 +608686,8 @@ "content": "已知常数 $m \\in \\mathbf{R}$, 若关于 $x$ 的方程 $x^2-2 m x-m+1=0$ 的两实根分别为 $x_1$、$x_2$ 且满足 $x_1^2+x_2^2=4$, 则实数 $m$ 的值是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "E20260102-高一上学期测验卷02" ], "genre": "填空题", "ans": "$1$", @@ -607569,7 +608722,8 @@ "content": "若 $x>1$, 则 $2 x+\\dfrac{1}{x-1}$ 的最小值是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "E20260102-高一上学期测验卷02" ], "genre": "填空题", "ans": "$2\\sqrt{2}+2$", @@ -607626,7 +608780,8 @@ "content": "已知关于 $x$ 的不等式 $(1-a^2) x^2+(a-1) x+1>0$ 的解集为 $\\mathbf{R}$, 则实数 $a$ 的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "E20260102-高一上学期测验卷02" ], "genre": "填空题", "ans": "$(-\\dfrac{3}{5},1]$", @@ -607705,7 +608860,8 @@ "content": "下列不等式中, 与不等式 $x(x-2)>0$ 解集相同的是\\bracket{20}.\n\\twoch{$x^2+\\dfrac{1}{x-3}>2 x+\\dfrac{1}{x-3}$}{$x^2+\\dfrac{1}{x-1}>2 x+\\dfrac{1}{x-1}$}{$x^2 \\cdot \\sqrt{x-1}>2 x \\cdot \\sqrt{x-1}$}{$\\dfrac{x^2}{\\sqrt{x-1}}>\\dfrac{2 x}{\\sqrt{x-1}}$}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "E20260102-高一上学期测验卷02" ], "genre": "选择题", "ans": "B", @@ -607740,7 +608896,8 @@ "content": "对任意实数 $x, y$, 在下列命题中, 是假命题的是\\bracket{20}.\n\\twoch{$x^2>y^2$ 是 $x>|y|$ 的必要条件}{$x^2>y^2$ 是 $|x|>y$ 的充分条件}{$x>2$ 的一个充分条件是 $x^2>4$}{$\\dfrac{1}{x}>\\dfrac{1}{y}$ 的一个充分条件是 $0=latex, scale = 0.6]\n\\foreach \\i in {-3,-2,...,3}\n{\\draw [dashed, gray] (\\i,-4) -- (\\i,4) (-4,\\i) -- (4,\\i);};\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1,0) node [below] {$1$} (0,1) node [left] {$1$};\n\\end{tikzpicture}\\hspace*{3em}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\foreach \\i in {-3,-2,...,3}\n{\\draw [dashed, gray] (\\i,-4) -- (\\i,4) (-4,\\i) -- (4,\\i);};\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1,0) node [below] {$1$} (0,1) node [left] {$1$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260101-高一上学期周末卷01" ], "genre": "解答题", "ans": "(1) \\begin{tikzpicture}[>=latex, scale = 0.6]\n\\foreach \\i in {-3,-2,...,3}\n{\\draw [dashed, gray] (\\i,-4) -- (\\i,4) (-4,\\i) -- (4,\\i);};\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1,0) node [below] {$1$} (0,1) node [left] {$1$};\n\\filldraw (2,2) circle (0.05) (-2,2) circle (0.05);\n\\end{tikzpicture}; (2) \\begin{tikzpicture}[>=latex, scale = 0.6]\n\\foreach \\i in {-3,-2,...,3}\n{\\draw [dashed, gray] (\\i,-4) -- (\\i,4) (-4,\\i) -- (4,\\i);};\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1,0) node [below] {$1$} (0,1) node [left] {$1$};\n\\fill [pattern = north east lines] (1,-4) -- (1,1) -- (-4,1) -- (-4,2) -- (1,2) -- (1,4) -- (3,4) -- (3,2) -- (4,2) -- (4,1) -- (3,1) -- (3,-4) -- cycle;\n\\draw [dashed] (1,-4) -- (1,1) -- (-4,1) (-4,2) -- (1,2) -- (1,4) (3,4) -- (3,2) -- (4,2) (4,1) -- (3,1) -- (3,-4);\n\\end{tikzpicture}", @@ -608516,7 +609677,8 @@ "content": "已知 $A=\\{y | y=\\sqrt{x}-2, x \\geq 0\\}$, $B=\\{x | y=\\sqrt{-2 x+8}\\}$, 则集合 $A \\cap B=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260101-高一上学期周末卷01" ], "genre": "填空题", "ans": "$[-2,4]$", @@ -608551,7 +609713,8 @@ "content": "已知 $A=[-3,2]$, $B=(-\\infty, a]$, 若 $A \\subset B$, 则实数 $a$ 的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260101-高一上学期周末卷01" ], "genre": "填空题", "ans": "$[2,+\\infty)$", @@ -608586,7 +609749,8 @@ "content": "满足关系式 $\\{a_1, a_2\\}\\subseteq A \\subset\\{a_1, a_2, a_3, a_4, a_5\\}$ 的集合 $A$ 的个数是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260101-高一上学期周末卷01" ], "genre": "填空题", "ans": "$7$", @@ -608621,7 +609785,8 @@ "content": "若集合 $P=\\{x, y, 1\\}$, $Q=\\{x^2, x y, x\\}$, 满足 $P=Q$, 实数 $x=$\\blank{50}, $y=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260101-高一上学期周末卷01" ], "genre": "填空题", "ans": "$-1$, $0$", @@ -608656,7 +609821,8 @@ "content": "已知 $m$是常数. $\\alpha:-1 \\leq x \\leq 3$, $\\beta: x \\leq m-1$, 若 $\\alpha \\Rightarrow \\beta$, 则实数 $m$ 的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "W20260101-高一上学期周末卷01" ], "genre": "填空题", "ans": "$[4,+\\infty)$", @@ -608691,7 +609857,8 @@ "content": "设全集 $U=\\mathbf{R}$, 对集合 $A$、$B$ 定义: $A-B=A \\cap \\overline{B}$, $A \\triangle B=(A-B) \\cup(B-A)$, 若集合 $A=\\{x | 10, x \\in \\mathbf{R}\\}$, 且 $A \\cap B=\\{1\\}$, 则实数 $m$ 的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "E20260101-高一上学期测验卷01" ], "genre": "填空题", "ans": "$[0,\\dfrac{1}{2})$", @@ -609114,7 +610292,8 @@ "content": "在下列陈述句: \\textcircled{1} $a<0$ 或 $b<0$; \\textcircled{2} $a<0$ 且 $b<1$; \\textcircled{3} $a<-1$ 且 $b<0$; \\textcircled{4} $a<0$ 或 $b<-1$ 中, 能成为``$a<0$ 且 $b<0$'' 的必要非充分条件是\\blank{50}(填正确的序号).", "objs": [], "tags": [ - "第一单元" + "第一单元", + "E20260101-高一上学期测验卷01" ], "genre": "填空题", "ans": "\\textcircled{1}\\textcircled{2}\\textcircled{4}", @@ -609149,7 +610328,8 @@ "content": "已知 $a$ 为实常数, 集合 $A=[1,4]$, $B=\\{x | a \\leq x \\leq 2 a+1\\}$, 且 $A \\cap B \\neq \\varnothing$, 则实数 $a$ 的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "E20260101-高一上学期测验卷01" ], "genre": "填空题", "ans": "$[0,4]$", @@ -609184,7 +610364,8 @@ "content": "以下几个关系中正确的是\\bracket{20}.\n\\fourch{$0 \\in \\varnothing$}{$\\varnothing \\in\\{x | x^2+1=0\\}$}{$\\varnothing \\subset\\{0\\}$}{$A\\cap B\\subset A\\cup B$}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "E20260101-高一上学期测验卷01" ], "genre": "选择题", "ans": "C", @@ -609219,7 +610400,8 @@ "content": "``任意 $c \\in \\mathbf{R}$, 都有 $a c=b c$''是``$a=b$''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": [ - "第一单元" + "第一单元", + "E20260101-高一上学期测验卷01" ], "genre": "选择题", "ans": "C", @@ -609254,7 +610436,8 @@ "content": "命题``对任意的 $x \\in \\mathbf{R}$, 恒有 $x^3-x^2+1 \\leq 0$''的否定是\\bracket{20}.\n\\twoch{不存在 $x \\in \\mathbf{R}$, 使得 $x^3-x^2+1 \\leq 0$}{存在 $x \\in \\mathbf{R}$, 使得 $x^3-x^2+1 \\leq 0$}{存在 $x \\in \\mathbf{R}$, 使得 $x^3-x^2+1>0$}{对任意 $x \\in \\mathbf{R}$, 恒有 $x^3-x^2+1>0$}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "E20260101-高一上学期测验卷01" ], "genre": "选择题", "ans": "C", @@ -609289,7 +610472,8 @@ "content": "设 $a \\in \\mathbf{R}$, $b \\in \\mathbf{R}$, 集合 $A=\\{x | x^2-x=0\\}$, $B=\\{x | x^2+a x+b=0\\}$. 若 $B \\neq \\varnothing, B \\subseteq A$, 求 $a, b$ 的值.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "E20260101-高一上学期测验卷01" ], "genre": "解答题", "ans": "$(a,b)=(0,0)$或$(-2,1)$或$(-1,0)$", @@ -609324,7 +610508,8 @@ "content": "已知实数 $x_1, x_2, x_3, x_4, x_5$ 满足 $x_1+x_2+x_3+x_4+x_5=5$.\\\\\n(1) 求证: $x_1, x_2, x_3, x_4, x_5$ 中至少有一个不小于 $1$;\\\\\n(2) 若 $x_1, x_2, x_3, x_4, x_5$ 均为非零整数, 求 $\\dfrac{1}{x_1}+\\dfrac{1}{x_2}+\\dfrac{1}{x_3}+\\dfrac{1}{x_4}+\\dfrac{1}{x_5}$ 的最大值;\\\\\n(3) 设 $x_1, x_2, x_3, x_4, x_5$ 这五个实数两两不等, 集合 $A=\\{x_1, x_2, x_3, x_4, x_5\\}$. 若 $B \\neq \\varnothing$, 且 $B \\subseteq A$, 记 $G(B)$ 是 $B$ 中所有元素之和, 对所有的 $B$, 求 $G(B)$ 的平均值.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "E20260101-高一上学期测验卷01" ], "genre": "解答题", "ans": "(1) 证明略; (2) $5$; (3) $\\dfrac{80}{31}$", @@ -609359,7 +610544,8 @@ "content": "数集 $\\{a, a^2-a\\}$ 中 $a$ 的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "I20260101-高一上学期随堂练习01" ], "genre": "填空题", "ans": "$(-\\infty,0)\\cup (0,2)\\cup (2,+\\infty)$", @@ -609381,7 +610567,8 @@ "content": "设集合 $A=\\{x | x>2\\}$, $B=\\{x | x0$ 的解集是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "I20260103-高一上学期随堂练习03" ], "genre": "填空题", "ans": "$(-\\infty,8)\\cup (8,+\\infty)$", @@ -609613,7 +610809,8 @@ "content": "不等式 $\\dfrac{2 x-1}{x-1}<0$ 的解集是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "I20260103-高一上学期随堂练习03" ], "genre": "填空题", "ans": "$(\\dfrac{1}{2},1)$", @@ -609647,7 +610844,8 @@ "content": "不等式组 $\\begin{cases}x^2-16<0\\\\x^2-4 x+3 \\geq 0\\end{cases}$ 的解集是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "I20260103-高一上学期随堂练习03" ], "genre": "填空题", "ans": "$(-4,1]\\cup [3,4)$", @@ -609681,7 +610879,8 @@ "content": "不等式 $\\dfrac{2 x+3}{2-x}\\geq 1$ 的解集是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "I20260103-高一上学期随堂练习03" ], "genre": "填空题", "ans": "$[-\\dfrac{1}{3},2)$", @@ -609715,7 +610914,8 @@ "content": "关于 $x$ 的方程 $2 x+k(x+3)=4$ 的解为正数, 则实数 $k$ 的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "I20260103-高一上学期随堂练习03" ], "genre": "填空题", "ans": "$(-2,\\dfrac{4}{3})$", @@ -609749,7 +610949,8 @@ "content": "已知关于 $x$ 的不等式 $2 x^2-2(a-1) x+(a+3)>0$ 的解集是 $\\mathbf{R}$, 则实数 $a$ 的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "I20260103-高一上学期随堂练习03" ], "genre": "填空题", "ans": "$(-1,5)$", @@ -609783,7 +610984,8 @@ "content": "不等式 $\\dfrac{3 x^2+2 x+2}{x^2+x+1}0$ 的解集是 $(-\\dfrac{1}{2}, 2)$, 对于 $a, b, c$ 有以下几个结论:\\\\\n\\textcircled{1} $a>0$; \\textcircled{2} $b>0$; \\textcircled{3} $c>0$; \\textcircled{4} $a+b+c>0$; \\textcircled{5} $a-b+c>0$.\\\\\n其中正确结论的序号是\\blank{50}(填上所有你认为正确的命题的序号).", "objs": [], "tags": [ - "第一单元" + "第一单元", + "I20260103-高一上学期随堂练习03" ], "genre": "填空题", "ans": "\\textcircled{2}\\textcircled{3}\\textcircled{4}", @@ -609852,7 +611055,8 @@ "content": "解关于 $x$ 的不等式 $\\dfrac{a x+a}{x+a+1}>0$, $a \\in \\mathbf{R}$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "I20260103-高一上学期随堂练习03" ], "genre": "解答题", "ans": "当$a<0$时, 解集为$(-1,-a-1)$; 当$a=0$时, 解集为$\\varnothing$; 当$a>0$时, 解集为$(-\\infty,-a-1)\\cup (-1,+\\infty)$", @@ -609886,7 +611090,8 @@ "content": "是否存在正数$a$, 使函数$y=\\dfrac{a^x+1}{2^x}$是偶函数?", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260136-函数的奇偶性(2)" ], "genre": "解答题", "ans": "存在(当$a=4$时)", @@ -609908,7 +611113,8 @@ "content": "已知函数$y=f(x)$, $x\\in \\mathbf{R}$, 且当$x\\ge 0$时, $f(x)=2x^3+2^x-1$.\\\\\n(1) 若函数$y=f(x)$是偶函数, 求$f(-2)$;\\\\\n(2) $y=f(x)$是否可能是奇函数? 若可能, 求$f(x)$的表达式; 若不可能, 说明理由.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260136-函数的奇偶性(2)" ], "genre": "解答题", "ans": "(1) $19$; (2) 可能, $f(x)=\\begin{cases}2x^3-2^{-x}+1, & x<0,\\\\ 0, & x=0, \\\\ 2x^3+2^x-1, & x>0 \\end{cases}$", @@ -609930,7 +611136,8 @@ "content": "判断下列函数$y=f(x)$的奇偶性, 并给出证明:\\\\\n(1) $f(x)=2 x-x^2$;\\\\\n(2) $f(x)=\\dfrac{2-\\log_2 x}{2+\\log_2 x}$;\\\\\n(3) $f(x)=\\begin{cases}-x^2+x, & x \\geq 0, \\\\x^2+x, & x<0.\\end{cases}$", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260136-函数的奇偶性(2)" ], "genre": "4em", "ans": "(1) 既不是奇函数, 又不是偶函数, 证明略; (2) 既不是奇函数, 又不是偶函数, 证明略; (3) 奇函数, 证明略", @@ -609967,7 +611174,8 @@ "content": "已知集合 $A=\\{-1,0\\}$, 集合 $B=\\{2, a\\}$, 若 $A \\cap B=\\{0\\}$, 则 $a=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "E20260106-2026届高一上学期期中考试" ], "genre": "填空题", "ans": "$0$.", @@ -610002,7 +611210,8 @@ "content": "``$x>1$ 或 $y>1$''的否定形式为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "E20260106-2026届高一上学期期中考试" ], "genre": "填空题", "ans": "$x \\leq 1$ 且 $y \\leq 1$.", @@ -610039,7 +611248,8 @@ "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", + "E20260106-2026届高一上学期期中考试" ], "genre": "填空题", "ans": "$(1,2)$.", @@ -610074,7 +611284,8 @@ "content": "已知幂函数 $y=x^a$ 的图像经过点 $(\\sqrt[4]{2}, 2)$, 则实数 $a$ 的值为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "E20260106-2026届高一上学期期中考试" ], "genre": "填空题", "ans": "$4$.", @@ -610109,7 +611320,8 @@ "content": "关于 $x$ 的方程 $x^2-4 x+1=0$ 的两根为 $x_1, x_2$, 则 $x_1^2+x_2^2$ 的值为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "E20260106-2026届高一上学期期中考试" ], "genre": "填空题", "ans": "$14$.", @@ -610144,7 +611356,8 @@ "content": "已知 $\\log _23=a$, 用 $a$ 表示 $\\log _46=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "E20260106-2026届高一上学期期中考试" ], "genre": "填空题", "ans": "$\\dfrac{a+1}{2}$.", @@ -610179,7 +611392,8 @@ "content": "已知正实数 $a, b$ 满足 $a+2 b=1$, 则 $a b$ 的最大值为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "E20260106-2026届高一上学期期中考试" ], "genre": "填空题", "ans": "$\\dfrac{1}{8}$.", @@ -610214,7 +611428,8 @@ "content": "对于任意实数 $x$, 不等式 $a x^2+a x+1>0$ 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", + "E20260106-2026届高一上学期期中考试" ], "genre": "填空题", "ans": "$[0,4)$.", @@ -610249,7 +611464,8 @@ "content": "对任意 $x \\leq 1$, 指数函数 $y=a^x$ 的值总大于 $\\dfrac{1}{2}$, 则实数 $a$ 的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "E20260106-2026届高一上学期期中考试" ], "genre": "填空题", "ans": "$(\\dfrac{1}{2}, 1)$.", @@ -610285,7 +611501,8 @@ "objs": [], "tags": [ "第一单元", - "第二单元" + "第二单元", + "E20260106-2026届高一上学期期中考试" ], "genre": "填空题", "ans": "$(-\\infty, \\dfrac{1}{2}]$.", @@ -610320,7 +611537,8 @@ "content": "已知正实数 $a, b$ 满足 $a^{\\lg b}=2$, $a^{\\lg a}b^{\\lg b}=5$, 则 $(a b)^{\\lg (a b)}$ 的值为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "E20260106-2026届高一上学期期中考试" ], "genre": "填空题", "ans": "$20$.", @@ -610355,7 +611573,8 @@ "content": "若``对于任意的实数 $a$, 关于 $x$ 的不等式 $|2^x+a| \\geq m$ 在区间 $[0,1]$ 上总有解''是真命题, 则实数 $m$ 的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "E20260106-2026届高一上学期期中考试" ], "genre": "填空题", "ans": "$(-\\infty, \\dfrac{1}{2}]$.", @@ -610390,7 +611609,8 @@ "content": "函数 $y=x^{\\frac{2}{3}}$ 的图像是\\bracket{20}.\n\\fourch{\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1.8) -- (0,1.8) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0.0001:2] plot (\\x,{pow(\\x,0.6)}) plot ({-\\x},{-pow(\\x,0.6)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-0.6) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0.0001:2] plot (\\x,{pow(\\x,1.5)}) plot ({-\\x},{pow(\\x,1.5)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-0.6) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0.55:2] plot (\\x,{pow(\\x,-1.5)}) plot ({-\\x},{pow(\\x,-1.5)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-0.6) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0.0001:2] plot (\\x,{pow(\\x,2/3)}) plot ({-\\x},{pow(\\x,2/3)});\n\\end{tikzpicture}}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "E20260106-2026届高一上学期期中考试" ], "genre": "选择题", "ans": "D", @@ -610425,7 +611645,8 @@ "content": "已知实数 $a$ 满足 $0b$, 则 $a^3>b^3$; \\textcircled{2} 若 $a>b>1$, 则 $\\log _a 2>\\log _b 2$; \\textcircled{3} 若 $ab d$; \\textcircled{4} 若 $10\\}$ (其中常数 $a>0$, $a \\neq 1$), $B=\\{y | y=x^k, x \\in A\\}$ (其中 $k$ 是常数), 则``$k<0$''是``$A \\cap B=\\varnothing$''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", + "E20260106-2026届高一上学期期中考试" ], "genre": "选择题", "ans": "A", @@ -610531,7 +611754,8 @@ "objs": [], "tags": [ "第二单元", - "第一单元" + "第一单元", + "E20260106-2026届高一上学期期中考试" ], "genre": "解答题", "ans": "(1) $(-3,4]$; (2) $(-1,1)$", @@ -610566,7 +611790,8 @@ "content": "已知 $a>0$, $b>0$.\\\\\n(1) 比较 $a^3+b^3$ 与 $a^2 b+b^2 a$ 的大小;\\\\\n(2) 若 $a+b=1$, 求 $\\dfrac{a^2}{b}+\\dfrac{b^2}{a}$ 的最小值.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "E20260106-2026届高一上学期期中考试" ], "genre": "解答题", "ans": "(1) 当 $a=b$ 时, $a^3+b^3=a^2 b+a b^2$; 当 $a \\neq b$ 时, $a^3+b^3>a^2 b+a b^2$; (2) 最小值为$1$", @@ -610601,7 +611826,8 @@ "content": "《上海市生活垃圾管理条例》于 2019 年 7 月 1 日正式实施. 某小区全面实施垃圾分类处理. 已知该小区每月垃圾分类处理量不超过 $300$ 吨, 每月垃圾分类处理成本 $y$ (元) 与每月分类处理量 $x$ (吨) 之间的函数关系式可近似表示为 $y=x^2-200 x+40000$, 而分类处理一吨垃圾小区也可以获得 $300$ 元的收益.\\\\\n(1) 该小区每月分类处理多少吨垃圾, 才能使得每吨垃圾分类处理的平均成本最低?\\\\\n(2) 要保证该小区每月的垃圾分类处理不亏损, 每月的垃圾分类处理量应控制在什么范围?", "objs": [], "tags": [ - "第二单元" + "第二单元", + "E20260106-2026届高一上学期期中考试" ], "genre": "解答题", "ans": "(1) $200$吨; (2) $100$至$300$吨(含$100$吨与$300$吨)", @@ -610636,7 +611862,8 @@ "content": "已知 $a$ 为实常数, 函数 $y=2^x+\\dfrac{a}{2^x}$.\\\\\n(1) 当 $a=-3$ 时, 求所有满足 $y=2$ 的 $x$ 的值;\\\\\n(2) 若对任意的 $x \\in \\mathbf{R}$, 都有 $y \\geq 3$ 成立, 求实数 $a$ 的取值范围;\\\\\n(3) 若方程 $2^x+\\dfrac{a}{2^x}=6$ 有两个不相等的实数根 $x_1, x_2$, 且 $|x_1-x_2| \\leq 1$, 求实数 $a$ 的取值范围.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "E20260106-2026届高一上学期期中考试" ], "genre": "解答题", "ans": "(1) $\\log_2 3$; (2) $[\\dfrac{9}{4},+\\infty)$; (3) $[8,9)$", @@ -610671,7 +611898,8 @@ "content": "集合 $A=\\{a_1, a_2, \\cdots, a_n\\}$ 是由 $n$($n \\geq 3$) 个正整数组成的集合, 如果任意去掉其中一个元素 $a_i$($i=1,2, \\cdots, n$)之后, 剩余的所有元素组成的集合都能分为两个交集为空的集合, 且这两个集合的所有元素之和相等, 就称集合 $A$ 为``可分集合''.\\\\\n(1) 判断集合 $\\{1,2,3,4\\}$, $\\{1,3,5,7,9,11,13\\}$ 是否为``可分集合''(不用说明理由);\\\\\n(2) 求证: 五个元素的集合 $A=\\{a_1, a_2, a_3, a_4, a_5\\}$ 一定不是``可分集合'';\\\\\n(3) 若集合 $A=\\{a_1, a_2, \\cdots, a_n\\}$ 是``可分集合'', 证明 $n$ 是奇数.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "E20260106-2026届高一上学期期中考试" ], "genre": "解答题", "ans": "(1) $\\{1,2,3,4\\}$ 不是``可分集合'', $\\{1,3,5,7,9,11,13\\}$ 是``可分集合''; (2) 证明略; (3) 证明略", @@ -610990,7 +612218,8 @@ "content": "已知等比数列 $\\{a_n\\}$ 中每一项均为正数, 且 $a_1$、$\\dfrac{1}{2}a_3$、$2 a_2$ 成等差数列, 则 $\\dfrac{a_{2025}+a_{2022}}{a_{2023}+a_{2020}}=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "V20260108-2026届高一寒假作业08" ], "genre": "填空题", "ans": "$3+2\\sqrt{2}$", @@ -611200,7 +612429,8 @@ "content": "若无穷等比数列 $\\{a_n\\}$ 的首项为 $\\dfrac{1}{2}$, 公比为 $\\dfrac{5}{2}-a$, 且 $\\{a_n\\}$ 的各项的和为 $a-1$, 则实数 $a$ 的值为\\bracket{20}.\n\\fourch{$2$ 或 $\\dfrac{1}{2}$}{$2$}{$\\dfrac{1}{2}$}{$3$}", "objs": [], "tags": [ - "第四单元" + "第四单元", + "V20260108-2026届高一寒假作业08" ], "genre": "选择题", "ans": "B", @@ -611340,7 +612570,8 @@ "content": "由市场调查得知: 某公司生产的一种食品, 如果不做广告宣传且每千克盈利 $a$ 元, 那么销售量为 $a_0$ 千克; 如果做广告宣传且每件售价不变, 那么广告费用为 $n \\times 1000$ 元时的销售量比广告费用为 $(n-1) \\times 1000$ 元时的销售量多 $a_0 \\times \\dfrac{1}{2^n}$ 千克 ($n>0$ 且 $n \\in \\mathbf{Z}$).\\\\\n(1) 设广告费用为 $n \\times 1000$ 元时的销售量为 $a_n$, 求销售量 $a_n$ 关于 $n$ 的代数表达式;\\\\\n(2) 当 $a=10$, $a_0=4000$ 时, 公司应做几千元广告, 才能使得去掉广告费用后的获利最大? 此时销售量为多少千克?", "objs": [], "tags": [ - "第四单元" + "第四单元", + "V20260108-2026届高一寒假作业08" ], "genre": "解答题", "ans": "(1) $a_n=a_0(2-\\dfrac{1}{2^n})$; (2) $5000$元广告费时, 获利最大, 此时销售量为$7875$千克.", @@ -612183,7 +613414,8 @@ "content": "已知$a,b,c$为常数, $a>0$. 证明: 函数$y=ax^2+bx+c$在区间$(-\\infty,-\\dfrac{b}{2a}]$上是严格减函数, 在$[-\\dfrac{b}{2a},+\\infty)$上是严格增函数.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260137-函数的单调性(1)" ], "genre": "解答题", "ans": "证明略", @@ -612205,7 +613437,8 @@ "content": "证明: 函数$y=x+\\dfrac{1}{x}$在区间$(0,1]$上是严格减函数.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260137-函数的单调性(1)" ], "genre": "解答题", "ans": "证明略", @@ -612240,7 +613473,8 @@ "content": "证明: 函数$y=2^{3-2x}$在定义域上是严格减函数.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260137-函数的单调性(1)" ], "genre": "解答题", "ans": "证明略", @@ -612275,7 +613509,8 @@ "content": "写出下列函数的最大值与最小值(若不存在, 就写``不存在''), 并写出取最值时相应自变量的值:\\\\\n(1) $y=x^2-4x-2$: \\blank{150};\\\\\n(2) $y=6x-3x^2$: \\blank{150};\\\\\n(3) $y=-x^2-4x-3$, $x\\in [-3,1]$: \\blank{150};\\\\\n(4) $y=x^2-2x-3$, $x\\in [-2,0]$: \\blank{150}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260139-函数的最值" ], "genre": "填空题", "ans": "(1) 最大值不存在; 最小值为$-6$(当且仅当$x=2$时取); (2) 最大值为$3$(当且仅当$x=1$时取; 最小值不存在; (3) 最大值为$1$(当且仅当$x=-2$时取); 最小值为$-8$(当且仅当$x=1$时取); (4) 最大值为$5$(当且仅当$x=-2$时取), 最小值为$-3$(当且仅当$x=0$时取)", @@ -612310,7 +613545,8 @@ "content": "函数$f(x)=x+8-\\dfrac ax$在$[1,+\\infty)$上是增函数, 求$a$的取值范围.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260138-函数的单调性(2)" ], "genre": "4em", "ans": "$[-1,+\\infty)$", @@ -612935,7 +614171,8 @@ "content": "函数 $y=\\log_{\\frac{1}{2}}(x^2+4 x-5)$ 的单调减区间是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "I20260105-高一上学期随堂练习05" ], "genre": "填空题", "ans": "$(1,+\\infty)$", @@ -612972,7 +614209,8 @@ "content": "已知定义在区间 $(-1,1)$ 上的函数 $f(x)=\\dfrac{x+b}{1+x^2}$ 为奇函数.\\\\\n(1) 证明: 函数 $y=f(x)$ 在区间 $(-1,1)$ 上是严格增函数;\\\\\n(2) 解关于 $t$ 的不等式 $f(t-1)+f(t)<0$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "I20260105-高一上学期随堂练习05" ], "genre": "4em", "ans": "(1) 证明略; (2) $(0,\\dfrac{1}{2})$", @@ -613010,7 +614248,8 @@ "content": "函数$y=|x^2-1|$的单调增区间为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "I20260105-高一上学期随堂练习05" ], "genre": "填空题", "ans": "$[-1,0]$和$[1,+\\infty)$", @@ -613045,7 +614284,8 @@ "content": "已知函数 $f(x)$ 是奇函数且定义域为 $\\mathbf{R}$, 若 $x>0$ 时, $f(x)=x^2-2 x+1$, 则 $f(x)=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "I20260105-高一上学期随堂练习05" ], "genre": "", "ans": "$\\begin{cases}-x^2-2x-1, & x<0, \\\\ 0, & x=0, \\\\ x^2-2x+1, & x>0\\end{cases}$", @@ -613083,7 +614323,8 @@ "content": "下列函数既是奇函数, 又在区间 $[-1,1]$ 上是增函数的是\\bracket{20}.\n\\twoch{$f(x)=|x+1|$}{$f(x)=\\dfrac{x}{|x|-2}$}{$f(x)=\\begin{cases}x^2,& x \\geq 0,\\\\-x^2, & x<0\\end{cases}$}{$f(x)=-2x^3$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "I20260105-高一上学期随堂练习05" ], "genre": "", "ans": "C", @@ -613121,7 +614362,8 @@ "content": "已知函数 $f(x)=|x|+\\dfrac{1}{x}-3$. 判断函数 $y=f(x)$ 的奇偶性, 并说明理由.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "I20260105-高一上学期随堂练习05" ], "genre": "4em", "ans": "既不是奇函数, 又不是偶函数, 理由略", @@ -613159,7 +614401,8 @@ "content": "函数$y=0.5^{x^2+4x-12}$的单调减区间为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260109-高一上学期周末卷09" ], "genre": "填空题", "ans": "$[-2,+\\infty)$", @@ -613194,7 +614437,8 @@ "content": "函数$y=0.5^{x^2+4x-12}$的值域为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260109-高一上学期周末卷09" ], "genre": "填空题", "ans": "$(0,65536]$", @@ -613229,7 +614473,8 @@ "content": "已知 $m \\in \\mathbf{R}$, $m<0$. 分别求函数 $f(x)=x^2+2 x+3$ 在区间 $[m, 0]$ 上的最大值与最小值.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260109-高一上学期周末卷09" ], "genre": "4em", "ans": "$y_{\\max}=\\begin{cases}m^2+2m+3, & m\\le -2,\\\\ 3, & -2\\le m<0;\\end{cases}$ $y_{\\min}=\\begin{cases}2, & m\\le -1, \\\\ m^2+2m+3, & -1\\le m<0.\\end{cases}$", @@ -613267,7 +614512,8 @@ "content": "已知 $a$ 为实常数, 设 $f(x)=\\begin{cases}(x-a)^2,& x \\leq 0,\\\\x+\\dfrac{4}{x}+3 a,& x>0 .\\end{cases}$ 若 $f(0)$ 是 $f(x)$ 的最小值, 则 $a$ 的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260109-高一上学期周末卷09" ], "genre": "", "ans": "$[0,4]$", @@ -613305,7 +614551,8 @@ "content": "用二分法求出函数$y=x^3-2 x^2-x+1$在区间$(2,3)$内的一个零点为\\blank{50}.(精确到$0.1$)", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260142-用二分法求函数的零点" ], "genre": "", "ans": "$2.2$", @@ -613343,7 +614590,8 @@ "content": "已知函数$y=2 x^3-3 x^2-18 x+28$在区间$(1,2)$上有且仅有一个零点. 用二分法求出该零点的近似值为\\blank{50}.(结果精确到$0.1$)", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260142-用二分法求函数的零点" ], "genre": "", "ans": "$1.6$", @@ -613382,7 +614630,8 @@ "objs": [], "tags": [ "第七单元", - "第二单元" + "第二单元", + "G20260142-用二分法求函数的零点" ], "genre": "", "ans": "$\\sqrt{3}$", @@ -616147,7 +617396,8 @@ "content": "函数$y=\\dfrac{x-5}{3x+2}$, $x\\in (-\\dfrac{2}{3},3]$的值域为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "I20260106-高一上学期随堂练习06" ], "genre": "填空题", "ans": "$(-\\infty,-\\dfrac{2}{11}]$", @@ -616182,7 +617432,8 @@ "content": "用函数的观点解关于 $x$ 的不等式: $\\dfrac{x+2}{x-1}-a<0$($a \\in \\mathbf{R}$), 请直接根据$a$的不同取值写出解集.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "W20260110-高一上学期周末卷10" ], "genre": "4em", "ans": "当$a<1$时, 解集为$(\\dfrac{a+2}{a-1},1)$; 当$a=1$时, 解集为$(-\\infty,1)$; 当$a>1$时, 解集为$(-\\infty,1)\\cup (\\dfrac{a+2}{a-1},+\\infty)$", @@ -616757,7 +618008,8 @@ ], "tags": [ "第二单元", - "2023届高三-赋能-赋能03" + "2023届高三-赋能-赋能03", + "E20260104-高一上学期测验卷04" ], "genre": "", "ans": "A", @@ -616798,7 +618050,8 @@ "objs": [], "tags": [ "第二单元", - "2023届高三-第三轮复习讲义-02-数形结合" + "2023届高三-第三轮复习讲义-02-数形结合", + "E20260104-高一上学期测验卷04" ], "genre": "4em", "ans": "$[-1,0]$", @@ -616837,7 +618090,8 @@ "content": "已知函数$y=\\ln x-(\\dfrac{1}{2})^x+x^3-3$唯一的零点在区间$(d,d+0.1)$中, 且$10d\\in \\mathbf{Z}$, 则$d=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "E20260104-高一上学期测验卷04" ], "genre": "填空题", "ans": "$1.4$", @@ -616873,7 +618127,8 @@ "content": "证明: 函数$y=x+\\dfrac{4}{x}$在$[2,+\\infty)$上是严格增函数.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "E20260104-高一上学期测验卷04" ], "genre": "解答题", "ans": "证明略", @@ -616912,7 +618167,8 @@ "tags": [ "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" + "2023届高三-第一轮复习讲义-08-函数的单调性", + "W20260111-高一上学期周末卷11" ], "genre": "4em", "ans": "(1); $a=1$; (2) 证明略; (3) 证明略", @@ -616956,7 +618212,8 @@ "tags": [ "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" + "2023届高三-第一轮复习讲义-08-函数的单调性", + "W20260111-高一上学期周末卷11" ], "genre": "4em", "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}", @@ -616995,7 +618252,8 @@ "content": "设数列$\\{a_n\\}$为等差数列, 其前$n$项和为$S_n$.\\\\\n(1) 已知$a_4+a_{14}=1$, 求$S_{17}$;\\\\\n(2) 已知$S_{21}=420$, 求$a_{11}$.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260146-等差数列的前$n$项和" ], "genre": "解答题", "ans": "(1) $\\dfrac{17}{2}$; (2) $20$", @@ -617017,7 +618275,8 @@ "content": "设数列$\\{a_n\\}$为等差数列, 其前$n$项和为$S_n$.\\\\\n(1) 已知$a_1+a_2+a_3=-3$, $a_{18}+a_{19}+a_{20}=6$, 求$S_{20}$;\\\\\n(2) 已知$S_4=2$, $S_8=6$, 求$S_{16}$.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260146-等差数列的前$n$项和" ], "genre": "解答题", "ans": "(1) $10$; (2) $20$", @@ -617462,7 +618721,8 @@ "content": "已知数列 $\\{b_n\\}$ 的各项均为正数的等比数列, 且数列 $\\{a_n\\}$ 和 $\\{b_n\\}$ 满足 $a_n=\\dfrac{1}{n}\\displaystyle\\sum_{i=1}^n \\lg b_i $($n \\in \\mathbf{N}$, $n \\geq 1$), 求证: $\\{a_n\\}$ 为等差数列.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "I20260107-高一上学期随堂练习07" ], "genre": "解答题", "ans": "证明略", @@ -617499,7 +618759,8 @@ "content": "数列 $\\{a_n\\}$ 中, $a_1=8$, $a_4=2$, 且满足 $a_{n+2}-2 a_{n+1}+a_n=0$, 则数列 $\\{a_n\\}$ 的通项公式为\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "I20260107-高一上学期随堂练习07" ], "genre": "", "ans": "$a_n=-2n+10$", @@ -617537,7 +618798,8 @@ "content": "已知等比数列$\\{a_n\\}$的前$5$项和为$10$, 前$10$项和为$60$, 则这个数列的前$15$项和是\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260148-等比数列的前$n$项和(1)" ], "genre": "", "ans": "$310$", @@ -617575,7 +618837,8 @@ "content": "已知$a$为常数, 求$a+a^3+a^5+\\cdots+a^{2 n+1}$.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260148-等比数列的前$n$项和(1)" ], "genre": "4em", "ans": "$\\begin{cases}\\dfrac{a-a^{2n+3}}{1-a^2}, & a\\ne \\pm 1, \\\\ (n+1)a, & a = \\pm 1\\end{cases}$", @@ -617614,7 +618877,8 @@ "content": "写出无穷数列$\\{a_n\\}$的一个通项公式, 使它的前$4$项分别是下列各数:\\\\\n(1) $4,8,12,16$: \\blank{50};\\\\\n(2) $\\dfrac 12, \\dfrac 23, \\dfrac 34, \\dfrac 45$: \\blank{50};\\\\\n(3) $-\\dfrac 1{2 \\times 1}, \\dfrac 1{2 \\times 2},-\\dfrac 1{2 \\times 3}, \\dfrac 1{2 \\times 4}$: \\blank{50};\\\\\n(4) $1,-\\sqrt[3]2, \\sqrt[3]3,-\\sqrt[3]4$: \\blank{50};\\\\\n(5) $\\dfrac{2}{3}, -\\dfrac{3}{8}, \\dfrac{4}{15}, -\\dfrac{5}{24}$: \\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260150-数列的概念与性质" ], "genre": "", "ans": "(1) $a_n=4n$; (2) $a_n=\\dfrac{n}{n+1}$; (3) $a_n=(-1)^n\\cdot \\dfrac{1}{2n}$; (4) $a_n=(-1)^{n+1}\\cdot \\sqrt[3]{n}$; (5) $a_n=(-1)^{n+1}\\cdot \\dfrac{n+1}{n(n+2)}$", @@ -617648,7 +618912,8 @@ "content": "求下列各式的值:\\\\\n(1) $\\displaystyle\\lim_{n \\to +\\infty}(\\dfrac{1}{2}+\\dfrac{1}{4}+\\cdots+\\dfrac{1}{2^n})=$\\blank{50};\\\\\n(2) $\\displaystyle\\lim_{n \\to +\\infty}(\\dfrac{1}{3}-\\dfrac{1}{9}+\\cdots+(-1)^{(n+1)}\\dfrac{1}{3^n})=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260149-等比数列的前$n$项和(2)" ], "genre": "填空题", "ans": "(1) $1$; (2) $\\dfrac{1}{4}$", @@ -617683,7 +618948,8 @@ "content": "将下列循环小数化为分数:\\\\\n(1) $0.\\dot{3}2\\dot{5}$;\\\\\n(2) $2.6\\dot{4}\\dot{5}$.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260149-等比数列的前$n$项和(2)" ], "genre": "解答题", "ans": "(1) $\\dfrac{325}{999}$; (2) $\\dfrac{291}{110}$", @@ -617718,7 +618984,8 @@ "content": "将以下四则运算的结果用最简分数表示:\\\\\n(1) $0.\\dot{6}+1.\\dot{7}$;\\\\\n(2) $0.3\\times 0.\\dot{3}\\times 0.\\dot{3}\\dot{6}$.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260149-等比数列的前$n$项和(2)" ], "genre": "解答题", "ans": "(1) $\\dfrac{22}{9}$; (2) $\\dfrac{2}{55}$", @@ -617755,7 +619022,8 @@ "content": "已知数列$\\{a_n\\}$的通项公式为$a_n=n^2-5n+1$, 分析该数列的单调性, 并求其最小项的序数.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260150-数列的概念与性质" ], "genre": "解答题", "ans": "$a_1>a_2$, $a_2=a_3$, $a_3=latex]\n\\draw (1,0) arc (0:180:1) -- cycle;\n\\draw (80:0.8) node {$P_1$};\n\\draw (4,0) arc (0:180:1) arc (180:0:0.5) -- cycle;\n\\draw (3,0) ++ (80:0.8) node {$P_2$};\n\\draw (7,0) arc (0:180:1) arc (180:0:0.5) arc (180:0:0.25) -- cycle;\n\\draw (6,0) ++ (80:0.8) node {$P_3$};\n\\draw (10,0) arc (0:180:1) arc (180:0:0.5) arc (180:0:0.25) arc (180:0:0.125) -- cycle;\n\\draw (9,0) ++ (80:0.8) node {$P_4$};\n\\draw (12,0.5) node {$\\cdots$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第四单元" + "第四单元", + "I20260108-高一上学期随堂练习08" ], "genre": "填空题", "ans": "$\\dfrac{\\pi}{3}$", @@ -618218,7 +619500,8 @@ "content": "已知数列 $\\{a_n\\}$ 的通项公式 $a_n=(n^2-5 n+4) \\cdot 0.9^n$,\\\\\n(1) 数列中有多少项是负数?\\\\\n(2) 是否存在正整数 $n_0$ , 使得对于任意正整数 $n$, 都有 $a_n \\leq a_{n_0}$ 成立? 若存在, 求$n_0$的值; 若不存在, 说明理由.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "I20260108-高一上学期随堂练习08" ], "genre": "解答题", "ans": "(1) $2$项; (2) 存在, $n_0=22$", @@ -618245,7 +619528,8 @@ "content": "已知数列$\\{a_n\\}$满足$a_1=1$, $a_{n+1}=\\dfrac{n(n+1)}{a_n}$. 证明: 数列$\\{a_n\\}$的通项公式为$a_n=n$($n\\in \\mathbf{N}$, $n\\ge 1$).", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260152-数学归纳法" ], "genre": "解答题", "ans": "证明略", @@ -618280,7 +619564,8 @@ "content": "已知数列$\\{a_n\\}$满足$(a_{n+1}-a_n)^2-2(a_{n+1}+a_n)+1=0$, 且$a_1=1$, $a_{n+1} \\geq a_n$($n$为正整数).\\\\\n(1) 求$a_2,a_3,a_4$的值;\\\\\n(2) 猜想$\\{a_n\\}$的通项公式, 并用数学归纳法证明.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "G20260153-数学归纳法的应用" ], "genre": "4em", "ans": "(1) $a_2=4$, $a_3=9$, $a_4=16$; (2) 猜$a_n=n^2$, 证明略", @@ -618318,7 +619603,8 @@ "content": "已知 $\\log _{18}9=a$, $18^b=5$, 则 $\\log _{36}45=$\\blank{50}.(用 $a, b$ 表示)", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260161-上半学期内容补充复习" ], "genre": "填空题", "ans": "$\\dfrac{a+b}{2-a}$", @@ -618352,7 +619638,8 @@ "content": "若``$|x-1|>2 x-3$ \" 是``$x \\leq a$''成立的必要非充分条件, 则实数 $a$ 的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "G20260161-上半学期内容补充复习" ], "genre": "填空题", "ans": "$(-\\infty,2)$", @@ -618386,7 +619673,8 @@ "content": "化简: $\\log _23 \\cdot \\log _34 \\cdot \\log _45 \\cdot \\log _56 \\cdot \\log _67 \\cdot \\log _7 m=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260161-上半学期内容补充复习" ], "genre": "填空题", "ans": "$\\log_2 m$", @@ -618420,7 +619708,8 @@ "content": "已知函数 $y=x^{n^2-2 n-3}$($n \\in \\mathbf{Z}$) 的图像与两坐标轴都无公共点, 且图像关于 $y$ 轴对称, 则 $n=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260161-上半学期内容补充复习" ], "genre": "填空题", "ans": "$-1$或$1$或$3$", @@ -618456,7 +619745,8 @@ "content": "若 $00$, $a \\neq 1$. 若 $\\log _a 2<\\log _2 a$, 则实数 $a$ 的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260161-上半学期内容补充复习" ], "genre": "填空题", "ans": "$(\\dfrac{1}{2},1)\\cup (2,+\\infty)$", @@ -618593,7 +619886,8 @@ "content": "函数 $f(x)=\\log _{\\frac{1}{3}}(x^2-6 x+5)$ 的单调减区间是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260161-上半学期内容补充复习" ], "genre": "填空题", "ans": "$(5,+\\infty)$", @@ -618629,7 +619923,8 @@ "content": "已知 $a+\\lg a=10$, $b+10^b=10$, 则 $a+b$ 等于\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260161-上半学期内容补充复习" ], "genre": "填空题", "ans": "$10$", @@ -618663,7 +619958,8 @@ "content": "函数 $f(x)=x^2-b x+c$ 满足 $f(1+x)=f(1-x)$, $f(0)=3$, 则 $f(b^2)$\\blank{20}$f(c^2)$. (填写你认为正确的大小关系)", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260161-上半学期内容补充复习" ], "genre": "填空题", "ans": "$<$", @@ -618697,7 +619993,8 @@ "content": "若 $2(\\log _{0.5}x)^2+7(\\log _{0.5}x)+3 \\leq 0$, 求 $f(x)=(\\log _2 \\dfrac{x}{2})(\\log _2 \\dfrac{x}{4})$ 的最大值和最小值.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "G20260161-上半学期内容补充复习" ], "genre": "解答题", "ans": "最大值为$2$, 最小值为$-\\dfrac{1}{4}$", @@ -618731,7 +620028,8 @@ "content": "已知数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n=5-2 n$, 则通项 $a_n=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "E20260105-高一上学期测验卷05" ], "genre": "填空题", "ans": "$\\begin{cases}3, & n=1,\\\\ -2, & n\\ge 2\\end{cases}$", @@ -618768,7 +620066,8 @@ "content": "已知各项均为正数的数列 $\\{a_n\\}$ 满足 $\\sqrt{a_1}+\\sqrt{a_2}+\\cdots+\\sqrt{a_n}=n^2+3 n$($n \\in \\mathbf{N}$ 且 $n \\geq 1$), 则 $\\dfrac{a_1}{2}+\\dfrac{a_2}{3}+\\cdots+ \\dfrac{a_n}{n+1}=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "E20260105-高一上学期测验卷05" ], "genre": "填空题", "ans": "$2n^2+6n$", @@ -618803,7 +620102,8 @@ "content": "已知数列 $\\{x_n\\}$ 的通项公式为 $x_n=n a^n$, $n \\in \\mathbf{N}$ 且 $n \\geq 1$, 其中实数 $a>0$. 若数列 $\\{x_n\\}$ 是严格减数列, 则 $a$ 的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "E20260105-高一上学期测验卷05" ], "genre": "填空题", "ans": "$(0,\\dfrac{1}{2})$", @@ -618838,7 +620138,8 @@ "content": "设 $y=f(x)$ 是定义在正整数集上的函数, 且满足:``当 $f(k) \\geq k^2$ 成立时, 总可以推出 $f(k+1) \\geq(k+1)^2$ 成立'', 则当 $k \\in \\mathbf{N}$ 且 $k \\geq 1$ 时, 下列说法正确的是\\bracket{20}.\n\\onech{若 $f(2) \\geq 4$ 成立, 则当 $k \\geq 1$ 时, 均有 $f(k) \\geq k^2$ 成立}{若 $f(3) \\geq 9$ 成立, 则当 $k \\leq 3$ 时, 均有 $f(k) \\geq k^2$ 成立}{若 $f(4)=16$ 成立, 则当 $k \\geq 5$ 时, 均有 $f(k) \\geq k^2$ 成立}{若 $f(5)<25$ 成立, 则当 $k \\geq 6$ 时, 均有 $f(k)=latex,scale = 0.3]\n\\fill [gray!50] (1,0) --++ (120:1) --++ (180:1) --++ (240:1) --++ (300:1) --++ (0:1) -- cycle;\n\\foreach \\i in {0,60,...,300}\n{\\draw (\\i:1) --++ (\\i:1);\n\\draw (\\i:1) -- ({\\i+60}:1);\n\\draw (\\i:2) --++ ({\\i+60}:1) --++ ({\\i+120}:1) --++ ({\\i+180}:1);};\n\\draw (0,-4) node {第1个};\n\\end{tikzpicture}\n\\hspace*{3em}\n\\begin{tikzpicture}[>=latex,scale = 0.3]\n\\fill [gray!50] (1,0) --++ (120:1) --++ (180:1) --++ (240:1) --++ (300:1) --++ (0:1) -- cycle;\n\\fill [gray!50] (4,0) --++ (120:1) --++ (180:1) --++ (240:1) --++ (300:1) --++ (0:1) -- cycle;\n\\foreach \\i in {0,60,...,300}\n{\\draw (\\i:1) --++ (\\i:1);\n\\draw (\\i:1) -- ({\\i+60}:1);\n\\draw (\\i:2) --++ ({\\i+60}:1) --++ ({\\i+120}:1) --++ ({\\i+180}:1);};\n\\foreach \\i in {0,60,...,300}\n{\\draw (3,0) ++ (\\i:1) --++ (\\i:1);\n\\draw (3,0) ++ (\\i:1) --++ ({\\i+120}:1);\n\\draw (3,0) ++ (\\i:2) --++ ({\\i+60}:1) --++ ({\\i+120}:1) --++ ({\\i+180}:1);};\n\\draw (1.5,-4) node {第2个};\n\\end{tikzpicture}\n\\hspace*{3em}\n\\begin{tikzpicture}[>=latex,scale = 0.3]\n\\fill [gray!50] (1,0) --++ (120:1) --++ (180:1) --++ (240:1) --++ (300:1) --++ (0:1) -- cycle;\n\\fill [gray!50] (4,0) --++ (120:1) --++ (180:1) --++ (240:1) --++ (300:1) --++ (0:1) -- cycle;\n\\fill [gray!50] (7,0) --++ (120:1) --++ (180:1) --++ (240:1) --++ (300:1) --++ (0:1) -- cycle;\n\\foreach \\i in {0,60,...,300}\n{\\draw (\\i:1) --++ (\\i:1);\n\\draw (\\i:1) -- ({\\i+60}:1);\n\\draw (\\i:2) --++ ({\\i+60}:1) --++ ({\\i+120}:1) --++ ({\\i+180}:1);};\n\\foreach \\i in {0,60,...,300}\n{\\draw (3,0) ++ (\\i:1) --++ (\\i:1);\n\\draw (3,0) ++ (\\i:1) --++ ({\\i+120}:1);\n\\draw (3,0) ++ (\\i:2) --++ ({\\i+60}:1) --++ ({\\i+120}:1) --++ ({\\i+180}:1);};\n\\foreach \\i in {0,60,...,300}\n{\\draw (6,0) ++ (\\i:1) --++ (\\i:1);\n\\draw (6,0) ++ (\\i:1) --++ ({\\i+120}:1);\n\\draw (6,0) ++ (\\i:2) --++ ({\\i+60}:1) --++ ({\\i+120}:1) --++ ({\\i+180}:1);};\n\\draw (1.5,-4) node {第3个};\n\\end{tikzpicture}\n\\end{center}\n按此规律继续, 第 $n$ 个图案中有白色地面砖块\\blank{50}块.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "W20260114-高一上学期周末卷14" ], "genre": "填空题", "ans": "$4n+2$", @@ -619166,7 +620476,8 @@ "content": "已知数列 $\\{a_n\\}$ 满足 $a_{n+1}=-a_n^2+2 a_n$, 且 $a_1=0.9$, 令 $b_n=\\lg (1-a_n)$.\\\\\n(1) 求证: 数列 $\\{b_n\\}$ 是等比数列;\\\\\n(2) 求数列 $\\{\\dfrac{1}{b_n}\\}$ 所有项的和.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "W20260114-高一上学期周末卷14" ], "genre": "解答题", "ans": "(1) 证明略; (2) $-2$", @@ -619197,7 +620508,8 @@ "content": "用数学归纳法证明: 当 $n \\geq 2$ 时, $\\dfrac{1}{n+1}+\\dfrac{1}{n+2}+\\cdots+\\dfrac{1}{2 n}>\\dfrac{13}{24}$.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "W20260114-高一上学期周末卷14" ], "genre": "解答题", "ans": "证明略", @@ -619228,7 +620540,8 @@ "content": "已知数列 $\\{a_n\\}$, 满足 $a_1=a$($a>0$), $a_n=\\dfrac{2 a_{n-1}}{1+a_{n-1}}$($n \\geq 2$).\\\\\n(1) 求出 $a_2, a_3, a_4$;\\\\\n(2) 由(1)猜想 $a_n$ 的表达式, 并用数学归纳法证明.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "W20260114-高一上学期周末卷14" ], "genre": "解答题", "ans": "(1) $a_2=\\dfrac{2a}{1+a}$, $a_3=\\dfrac{4a}{1+3a}$, $a_4=\\dfrac{8a}{1+7a}$; (2) 猜想$a_n=\\dfrac{2^{n-1}a}{1+(2^{n-1}-1)a}$, 证明略", @@ -619259,7 +620572,8 @@ "content": "已知数列 $\\{a_n\\}$ 中, 前 $n$ 项和 $S_n$ 满足 $S_n=\\dfrac{3}{2}(a_n-1)$, 探索并猜想通项公式 $a_n$ 的表达式, 并用数学归纳法证明.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "W20260114-高一上学期周末卷14" ], "genre": "解答题", "ans": "猜想$a_n=3^n$, 证明略", @@ -619290,7 +620604,8 @@ "content": "在数列 $\\{a_n\\}$ 中, 若 $a_1=3$, $a_2=7$, $a_{n+2}=3 a_{n+1}-2 a_n$, 令$b_n=a_{n+1}-a_n$.\\\\\n(1) 求数列$\\{b_n\\}$的通项公式;\\\\\n(2) 求数列$\\{a_n\\}$的通项公式.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "W20260114-高一上学期周末卷14" ], "genre": "解答题", "ans": "(1) $b_n=2^{n+1}$; (2) $a_n=2^{n+1}-1$", @@ -619321,7 +620636,8 @@ "content": "已知数列 $\\{a_n\\}$ 的通项公式为 $a_n=\\begin{cases}2 \\times 3^n,& n \\text{为正奇数},\\\\5-4 n,& n \\text{为正偶数},\\end{cases}$ 求数列 $\\{a_n\\}$ 的前 $n$ 项和 $S_n$.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "W20260114-高一上学期周末卷14" ], "genre": "解答题", "ans": "$S_n=\\begin{cases}-n^2+\\dfrac{5}{2}n+\\dfrac{3^{n+2}}{4}-\\dfrac{9}{4}, & n=2k+1, \\\\ -n^2+\\dfrac{n}{2}+\\dfrac{3^{n+1}}{4}-\\dfrac{3}{4}, & n=2k+2,\\end{cases}$($k\\in \\mathbf{N}$)", @@ -628080,7 +629396,8 @@ "content": "已知 $A=\\{x |-30$ 或 $b \\leq 0$''的否定形式为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "V20260101-2026届高一寒假作业01" ], "genre": "填空题", "ans": "", @@ -628198,7 +629519,8 @@ "content": "语句``对任意 $x \\in A$, 都有 $x \\in B$''的否定形式为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "V20260101-2026届高一寒假作业01" ], "genre": "填空题", "ans": "", @@ -628222,7 +629544,8 @@ "content": "(1)已知 $x$、$y \\in \\mathbf{R}$, $``x y>0$, $x+y>0$''是``$x>0$, $y>0''$ 的\\blank{50}条件;\\\\\n(2) 已知 $x$、$y \\in \\mathbf{R}$, ``$x y>1$, $x+y>2$''是``$x>1$, $y>1$''的\\blank{50}条件;\\\\\n(3) 已知 $x$、$y \\in \\mathbf{R}$, ``$(x-1)^2+(y-2)^2=0$''是``$(x-1)(y-2)=0$''的\\blank{50}条件.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "V20260101-2026届高一寒假作业01" ], "genre": "填空题", "ans": "", @@ -628245,7 +629568,8 @@ "content": "知 $a$ 是实常数, 若 $P=\\{x | x^2+x-6=0\\}$, $S=\\{x | a x+1=0\\}$, 写出使 $S \\subseteq P$ 成立的一个充分非必要条件: \\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "V20260101-2026届高一寒假作业01" ], "genre": "填空题", "ans": "", @@ -628269,7 +629593,8 @@ "content": "若函数 $y=f(x)$、$y=g(x)$ 的定义域都是 $\\mathbf{R}$, 则``$f(x)>g(x)$ 对一切 $x \\in \\mathbf{R}$ 成立''的充要条件是\\bracket{20}.\n\\twoch{存在 $x \\in \\mathbf{R}$, 使得 $f(x)>g(x)$}{有无穷多个 $x \\in \\mathbf{R}$, 使得 $f(x)>g(x)$}{不存在 $x \\in \\mathbf{R}$, 使得 $f(x) \\leq g(x)$}{任取 $x \\in \\mathbf{R}$, 都有 $f(x)>g(x)+1$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260101-2026届高一寒假作业01" ], "genre": "选择题", "ans": "", @@ -628291,7 +629616,8 @@ "content": "设全集 $U=\\mathbf{R}$, $A, B$ 是两个集合, 则``存在集合 $C$, 使得 $A \\subseteq C$ 且 $B \\subseteq \\overline{C}$''是``$A \\cap B=\\varnothing$''的\\blank{50}条件.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "V20260101-2026届高一寒假作业01" ], "genre": "填空题", "ans": "", @@ -628315,7 +629641,8 @@ "content": "已知陈述句 $\\alpha:-10$ 且 $b>0$''是``对于任意 $x \\in[-1,1]$, $a x+b>0$ 成立''的\\blank{50}条件.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "V20260101-2026届高一寒假作业01" ], "genre": "填空题", "ans": "", @@ -628409,7 +629739,8 @@ "content": "设 $A$、$B$ 是非空集合, 定义 $A \\triangle B=\\{x | x \\in A \\cup B, x \\notin A \\cap B\\}$, 已知 $A=\\{x |$ 存在 $y \\in \\mathbf{R}$, 使得 $y=\\sqrt{2 x-x^2}\\}$, $B=\\{y | y=\\dfrac{2^x}{2^x-1}, x>0\\}$, 则 $A \\triangle B$ 等于\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "V20260101-2026届高一寒假作业01" ], "genre": "填空题", "ans": "", @@ -628434,7 +629765,8 @@ "objs": [], "tags": [ "第一单元", - "第二单元" + "第二单元", + "V20260101-2026届高一寒假作业01" ], "genre": "解答题", "ans": "", @@ -628458,7 +629790,8 @@ "content": "设 $A=\\{(x, y) | y=x^2+3\\}$, $B=\\{(x, y) | y=2 x+m, 0 \\leq x \\leq 2\\}$. 若 $A \\cap B \\neq \\varnothing$, 则实数 $m$ 的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "V20260101-2026届高一寒假作业01" ], "genre": "填空题", "ans": "", @@ -628480,7 +629813,8 @@ "content": "设 $a$ 是非零的实常数, $f(x)=x^2+a|x-m+1|+1$ 是偶函数.\\\\\n(1) 求实数 $m$ 的值.\\\\\n(2) 问:``$a<-2$''是``函数 $f(x)$ 在区间 $(-2,0)$ 上存在两个零点''的什么条件(充分非必要条件、必要非充分条件、充要条件、既非充分也非必要条件) ? 并证明你的结论.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260101-2026届高一寒假作业01" ], "genre": "解答题", "ans": "", @@ -628502,7 +629836,8 @@ "content": "不等式 $|x-2|+|x+1|>4$ 的解集是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "V20260102-2026届高一寒假作业02" ], "genre": "填空题", "ans": "", @@ -628526,7 +629861,8 @@ "content": "关于 $x$ 的不等式 $\\begin{cases}x \\leq 2 a,\\\\3 x+4>5 a\\end{cases}$ 有解, 则实数 $a$ 的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "V20260102-2026届高一寒假作业02" ], "genre": "填空题", "ans": "", @@ -628548,7 +629884,8 @@ "content": "已知 $-3b$, 则 $a c^2>b c^2$;\\\\\n\\textcircled{2} 若 $a c^2>b c^2$, 则 $a>b$;\\\\\n\\textcircled{3} 若 $a\\dfrac{1}{b}$;\\\\\n\\textcircled{4} 若 $ab d$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "V20260102-2026届高一寒假作业02" ], "genre": "填空题", "ans": "", @@ -628592,7 +629930,8 @@ "content": "已知不等式 $x^2-2 x-3<0$ 的解集为 $A$, $x^2+x-6<0$ 的解集为 $B$, 不等式 $x^2+a x+b<0$ 的解集为 $A \\cap B$, 则 $a+b=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "V20260102-2026届高一寒假作业02" ], "genre": "填空题", "ans": "", @@ -628614,7 +629953,8 @@ "content": "已知 $a \\in \\mathbf{R}$, 若不等式 $(a-2) x^2+2(a-2) x-4<0$ 对一切 $x \\in \\mathbf{R}$ 成立, 则 $a$ 的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "V20260102-2026届高一寒假作业02" ], "genre": "填空题", "ans": "", @@ -628636,7 +629976,8 @@ "content": "设 $x>\\dfrac{1}{2}$, 则函数 $y=x+\\dfrac{8}{2 x-1}$ 的最小值是\\blank{50}, 此时 $x=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "V20260102-2026届高一寒假作业02" ], "genre": "填空题", "ans": "", @@ -628660,7 +630001,8 @@ "content": "已知关于 $x$ 的一元二次方程 $x^2-m x+m+1=0$ 的两个实数根为 $x_1, x_2$, 且 $x_1^2+x_2^2=1$, 则实数 $m$ 的值为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "V20260102-2026届高一寒假作业02" ], "genre": "填空题", "ans": "", @@ -628682,7 +630024,8 @@ "content": "已知 $a>b>c$, 求证: $\\dfrac{1}{a-b}+\\dfrac{1}{b-c}+\\dfrac{1}{c-a}>0$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "V20260102-2026届高一寒假作业02" ], "genre": "解答题", "ans": "", @@ -628704,7 +630047,8 @@ "content": "已知 $a, b \\in \\mathbf{R}$, 求证: $|a-b| \\geq|a|-|b|$.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "V20260102-2026届高一寒假作业02" ], "genre": "解答题", "ans": "", @@ -628726,7 +630070,8 @@ "content": "已知关于 $x$ 的方程 $x^2+(m-3) x+m=0$.\\\\\n(1) 若该方程有两个不同的正数根, 求实数 $m$ 的集合\\\\\n(2) 若该方程有一个正数根和一个负数根, 求实数 $m$ 的集合.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "V20260102-2026届高一寒假作业02" ], "genre": "解答题", "ans": "", @@ -628748,7 +630093,8 @@ "content": "如果不等式 $a x^2+b x+c>0$ 的解集是 $(\\alpha, \\beta)$($0<\\alpha<\\beta$), 那么不等式 $c x^2+b x+a<0$的解集为\\bracket{20}.\n\\twoch{$(\\dfrac{1}{\\beta}, \\dfrac{1}{\\alpha})$}{$(-\\dfrac{1}{\\alpha},-\\dfrac{1}{\\beta})$}{$(-\\infty, \\dfrac{1}{\\beta}) \\cup(\\dfrac{1}{\\alpha},+\\infty)$}{$(-\\infty,-\\dfrac{1}{\\alpha}) \\cup(-\\dfrac{1}{\\beta},+\\infty$)}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "V20260102-2026届高一寒假作业02" ], "genre": "选择题", "ans": "", @@ -628772,7 +630118,8 @@ "content": "若关于 $x$ 的不等式 $|x-1|+|x-2| \\leq a^2+a+1$ 的解集是空集, 则实数 $a$ 的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "V20260102-2026届高一寒假作业02" ], "genre": "填空题", "ans": "", @@ -628794,7 +630141,8 @@ "content": "已知 $x, y>0$, $x y=x+y+1$, 则 $x+y$ 的最小值是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "V20260102-2026届高一寒假作业02" ], "genre": "填空题", "ans": "", @@ -628816,7 +630164,8 @@ "content": "设关于 $x$ 的不等式 $a x^2-(a+2) x+2>0$ 的解集为 $A$.\\\\\n(1) 求集合 $A$;\\\\\n(2) 是否存在实数 $a$, 使 $A \\cap\\{x | \\dfrac{1}{x}<1\\}\\cap \\mathbf{Z}$ 是单元素集合? 如果存在, 求实数 $a$ 的取值范围; 如果不存在, 说明理由.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "V20260102-2026届高一寒假作业02" ], "genre": "解答题", "ans": "", @@ -628838,7 +630187,8 @@ "content": "设 $a>b>0$, 则 $a^2+\\dfrac{1}{a b}+\\dfrac{1}{a(a-b)}$ 的最小值是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "V20260102-2026届高一寒假作业02" ], "genre": "填空题", "ans": "", @@ -628860,7 +630210,8 @@ "content": "已知 $P=\\{x | x^2-x-2>0\\}$, $Q=\\{x | x^2+4 x+m<0\\}$, 若 $Q \\subseteq P$, 则实数 $m$ 的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "V20260102-2026届高一寒假作业02" ], "genre": "填空题", "ans": "", @@ -628882,7 +630233,8 @@ "content": "用 $\\log _ax, \\log _ay, \\log _a(x+y), \\log _a(x-y)$ 表示下列各式:\\\\\n(1) $\\log _a(x^2-y^2)=$\\blank{50}.\\\\\n(2) $\\log _a\\dfrac{x^3y}{(x+y)^4}=$\\blank{50}.\\\\\n(3) $\\log _a(\\dfrac{\\sqrt{x}}{\\sqrt{y}}-\\dfrac{\\sqrt{y}}{\\sqrt{x}})=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260103-2026届高一寒假作业03" ], "genre": "填空题", "ans": "", @@ -628906,7 +630258,8 @@ "content": "改正下列式子中的错误:\\\\\n(1) $\\log _2(8-2)=\\log _28-\\log _22$: \\blank{50};\\\\\n(2) $\\dfrac{\\log _24}{\\log _28}=\\log _24-\\log _28$: \\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260103-2026届高一寒假作业03" ], "genre": "填空题", "ans": "", @@ -628928,7 +630281,8 @@ "content": "化简: $\\sqrt{(\\log _35)^2-4 \\log _35+4}=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260103-2026届高一寒假作业03" ], "genre": "填空题", "ans": "", @@ -628950,7 +630304,8 @@ "content": "$2^{-\\log _4x^2}$ 的化简结果是 \\bracket{20}.\n\\fourch{$x^2$}{$\\dfrac{1}{x}$}{$\\dfrac{1}{|x|}$}{$|x|$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260103-2026届高一寒假作业03" ], "genre": "选择题", "ans": "", @@ -628972,7 +630327,8 @@ "content": "已知 $\\log _32=m$, 则 $\\log _{32}18=$\\blank{50} (用 $m$ 表示)", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260103-2026届高一寒假作业03" ], "genre": "填空题", "ans": "", @@ -628997,7 +630353,8 @@ "content": "已知 $5.4^x=3$, $0.6^y=3$, 则 $\\dfrac{1}{x}-\\dfrac{1}{y}$ 的值为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260103-2026届高一寒假作业03" ], "genre": "填空题", "ans": "", @@ -629022,7 +630379,8 @@ "content": "不等式 $\\dfrac{1}{128}<(\\dfrac{1}{2})^x\\leq 16$ 的整数解的个数为\\bracket{20}.\n\\fourch{10}{11}{12}{13}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260103-2026届高一寒假作业03" ], "genre": "选择题", "ans": "", @@ -629044,7 +630402,8 @@ "content": "不等式 $\\log _3|x-2|<2$ 的整数解的个数为\\bracket{20}.\n\\fourch{15}{16}{17}{18}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260103-2026届高一寒假作业03" ], "genre": "选择题", "ans": "", @@ -629066,7 +630425,8 @@ "content": "若 $\\log _a\\dfrac{2}{3}<1$, 则 $a$ 的取值范围是\\bracket{20}.\n\\fourch{$(0, \\dfrac{2}{3})$}{($\\dfrac{2}{3},+\\infty$)}{$(\\dfrac{2}{3}, 1)$}{$(0, \\dfrac{2}{3}) \\cup$($1,+\\infty$)}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260103-2026届高一寒假作业03" ], "genre": "选择题", "ans": "", @@ -629089,7 +630449,8 @@ "objs": [], "tags": [ "第一单元", - "第二单元" + "第二单元", + "V20260103-2026届高一寒假作业03" ], "genre": "填空题", "ans": "", @@ -629111,7 +630472,8 @@ "content": "指数方程 $(\\dfrac{1}{2})^x+(\\dfrac{2}{3})^x+(\\dfrac{5}{6})^x=3$ 的解为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260103-2026届高一寒假作业03" ], "genre": "填空题", "ans": "", @@ -629133,7 +630495,8 @@ "content": "解方程: $\\dfrac{1+3^{-x}}{1+3^x}=3$;", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260103-2026届高一寒假作业03" ], "genre": "解答题", "ans": "", @@ -629155,7 +630518,8 @@ "content": "解方程: $4^x+4^{-x}-6(2^x+2^{-x})+10=0$ (提示: $t=2^x+2^{-x}$ )", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260103-2026届高一寒假作业03" ], "genre": "解答题", "ans": "", @@ -629179,7 +630543,8 @@ "content": "解不等式组: $\\begin{cases}\\log _{\\frac{1}{2}}(3 x-2)>\\log _{\\frac{1}{2}}(x+1),\\\\\\log _{\\frac{1}{2}}(4-2^x)+1 \\geq \\log _{\\frac{1}{4}}(2^x-1).\\end{cases}$", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260103-2026届高一寒假作业03" ], "genre": "解答题", "ans": "", @@ -629202,7 +630567,8 @@ "objs": [], "tags": [ "第一单元", - "第二单元" + "第二单元", + "V20260103-2026届高一寒假作业03" ], "genre": "选择题", "ans": "", @@ -629224,7 +630590,8 @@ "content": "如果不考虑空气阻力, 火箭的最大速度 $v(\\mathrm{km}/ \\mathrm{s})$ 和燃料的质量 $M(\\mathrm{kg})$ 、火箭 (除燃料外) 的质量 $m(\\mathrm{kg})$ 之间的关系是 $v=2 \\ln (1+\\dfrac{M}{m})$. 当燃料质量是火箭质量的\\blank{50}倍时, 火箭的最大速度能达到 $8 \\mathrm{km}/ \\mathrm{s}$ (精确到 $0.1$倍);", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260103-2026届高一寒假作业03" ], "genre": "填空题", "ans": "", @@ -629248,7 +630615,8 @@ "content": "利用函数、方程与不等式的关系, 不等式 $-x^2+3>2^x$ 的解集为\\blank{50}.(如有必要, 请精确到 $0.01$)", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260103-2026届高一寒假作业03" ], "genre": "填空题", "ans": "", @@ -629270,7 +630638,8 @@ "content": "已知对于任意正整数 $n$, 不等式 $n \\lg a<(n+1) \\lg a^a$ 都成立, 求实数 $a$ 的取值范围.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260103-2026届高一寒假作业03" ], "genre": "解答题", "ans": "", @@ -629292,7 +630661,8 @@ "content": "函数 $y=x^{-2}$ 在区间 $[\\dfrac{1}{2}, 2]$ 上的最大值是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260104-2026届高一寒假作业04" ], "genre": "填空题", "ans": "", @@ -629314,7 +630684,8 @@ "content": "若幂函数 $y=f(x)$ 的图像过点 $(3, \\sqrt[4]{27})$, 则 $y=f^{-1}(x)$ 的解析式是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260104-2026届高一寒假作业04" ], "genre": "填空题", "ans": "", @@ -629336,7 +630707,8 @@ "content": "若函数 $y=f(x)$ 的图像与函数 $g(x)=(\\dfrac{1}{2})^x$ 的图像关于直线 $y=x$ 对称, 则 $f(2 x-x^2)$ 的严格减区间为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260104-2026届高一寒假作业04" ], "genre": "填空题", "ans": "", @@ -629360,7 +630732,8 @@ "content": "若 $y=x^{a^2-4 a-9}$ 是偶函数, 且在 ($0,+\\infty$) 是减函数, 则整数 $a$ 的值是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260104-2026届高一寒假作业04" ], "genre": "填空题", "ans": "", @@ -629382,7 +630755,8 @@ "content": "若幂函数 $y=x^{(-1)^k\\frac{n}{m}}$($m, n, k \\in \\mathbf{N}$, $m$、$n$、$k>0$, $m, n$ 互质)的图像在一、二象限, 不过原点, 则 $k, m, n$ 奇偶性为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260104-2026届高一寒假作业04" ], "genre": "填空题", "ans": "", @@ -629405,7 +630779,8 @@ "content": "已知函数 $y=f(x)$ 的定义域是 $[1,2)$, 则函数 $f(2^x)$ 的定义域是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260104-2026届高一寒假作业04" ], "genre": "填空题", "ans": "", @@ -629429,7 +630804,8 @@ "content": "当 $a>0$ 且 $a \\neq 1$ 时, 函数 $f(x)=a^{x-2}-3$ 必过定点\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260104-2026届高一寒假作业04" ], "genre": "填空题", "ans": "", @@ -629453,7 +630829,8 @@ "content": "函数 $y=(\\dfrac{1}{2})^{\\sqrt{-x^2+x+2}}$ 的值域是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260104-2026届高一寒假作业04" ], "genre": "填空题", "ans": "", @@ -629475,7 +630852,8 @@ "content": "函数 $y=\\dfrac{1-3 x}{2+x}$ 的图像的对称中心是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260104-2026届高一寒假作业04" ], "genre": "填空题", "ans": "", @@ -629499,7 +630877,8 @@ "content": "设函数 $f(x)=a+\\dfrac{1}{3^x-1}$ 为奇函数, 则实数 $a$ 的值是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260104-2026届高一寒假作业04" ], "genre": "填空题", "ans": "", @@ -629521,7 +630900,8 @@ "content": "已知 $f(x)=\\dfrac{\\mathrm{e}^x-\\mathrm{e}^{-x}}{2}$, 则下列正确的是\\bracket{20}.\n\\twoch{奇函数, 在 $\\mathbf{R}$ 上为增函数}{偶函数, 在 $\\mathbf{R}$ 上为增函数}{奇函数, 在 $\\mathbf{R}$ 上为减函数}{偶函数, 在 $\\mathbf{R}$ 上为减函数}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260104-2026届高一寒假作业04" ], "genre": "选择题", "ans": "", @@ -629543,7 +630923,8 @@ "content": "如图所示, 幂函数 $y=x^{\\alpha}$ 在第一象限的图像, 比较 $0, \\alpha_1, \\alpha_2, \\alpha_3, \\alpha_4, 1$ 的大小关系为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0:{sqrt(3)}, samples = 100] plot (\\x,{\\x*\\x}) node [above] {$y=x^{\\alpha_1}$};\n\\draw [domain = {1/3}:3, samples = 100] plot (\\x,{1/\\x}) node [above right] {$y=x^{\\alpha_2}$};\n\\draw [domain = {1/sqrt(3)}:3, samples = 100] plot (\\x,{1/\\x/\\x}) node [right] {$y=x^{\\alpha_3}$};\n\\draw [domain = 0:3, samples = 100] plot (\\x,{sqrt(\\x)}) node [above] {$y=x^{\\alpha_4}$};\n\\end{tikzpicture}\n\\end{center}\n\\twoch{$\\alpha_1<\\alpha_3<0<\\alpha_4<\\alpha_2<1$}{$0<\\alpha_1<\\alpha_2<\\alpha_3<\\alpha_4<1$}{$\\alpha_2<\\alpha_4<0<\\alpha_3<1<\\alpha_1$}{$\\alpha_3<\\alpha_2<0<\\alpha_4<1<\\alpha_1$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260104-2026届高一寒假作业04" ], "genre": "选择题", "ans": "", @@ -629565,7 +630946,8 @@ "content": "函数 $f(x)=\\begin{cases}2^{-x}-1,& x \\leq 0,\\\\x^{\\frac{1}{2}},& x>0,\\end{cases}$ 求满足 $f(x)>1$ 的 $x$ 的取值范围.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260104-2026届高一寒假作业04" ], "genre": "解答题", "ans": "", @@ -629590,7 +630972,8 @@ "content": "若函数 $y=\\log _2(k x^2+4 k x+3)$ 的定义域为 $\\mathbf{R}$, 求实数 $k$ 的取值范围.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260104-2026届高一寒假作业04" ], "genre": "解答题", "ans": "", @@ -629615,7 +630998,8 @@ "content": "画出下列函数的图像, 并指出该函数图像可经由哪个幂函数的图像经过一系列的图像变换得到(写清楚图像变换的过程). \\\\\n(1) $y=(x-1)^{-\\frac{7}{5}}-2$;\\\\\n(2) $y=\\dfrac{1}{|x|-1}$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260104-2026届高一寒假作业04" ], "genre": "解答题", "ans": "", @@ -629637,7 +631021,8 @@ "content": "对于幂函数 $f(x)=x^{\\frac{4}{5}}$, 若 $00$, $a \\neq 1$, 则函数 $f(x)=a^x+x^2-2 x-2 a+1$ 的零点个数为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260104-2026届高一寒假作业04" ], "genre": "填空题", "ans": "", @@ -629707,7 +631094,8 @@ "content": "已知函数 $f(x)=\\log _2\\dfrac{x+1}{x-1}+\\log _2(x-1)+\\log _2(p-x), p$ 为常数, 且 $p \\in \\mathbf{R}$.\\\\\n(1) 求函数 $f(x)$ 的定义域; \\\\\n(2) 求函数 $f(x)$ 的值域.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260104-2026届高一寒假作业04" ], "genre": "解答题", "ans": "", @@ -629731,7 +631119,8 @@ "content": "设函数 $f(x)=\\lg (x+\\sqrt{x^2+1})$.\\\\\n(1) 确定函数 $f(x)$ 的定义域;\\\\\n(2) 判断函数 $f(x)$ 的奇偶性;\\\\\n(3) 证明函数 $f(x)$ 在其定义域上是严格增函数;\\\\\n(4) 求函数 $f(x)$ 的反函数.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260104-2026届高一寒假作业04" ], "genre": "解答题", "ans": "", @@ -629753,7 +631142,8 @@ "content": "函数 $y=\\dfrac{(x+1)^0}{\\sqrt{|x|-x}}$ 的定义域是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260105-2026届高一寒假作业05" ], "genre": "填空题", "ans": "", @@ -629775,7 +631165,8 @@ "content": "函数 $y=\\sqrt{\\dfrac{2+x}{1-x}}+\\sqrt{x^2-x-2}$ 的定义域是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260105-2026届高一寒假作业05" ], "genre": "填空题", "ans": "", @@ -629797,7 +631188,8 @@ "content": "设函数 $f(x)=\\begin{cases}\\sqrt{x-1},& x \\geq 1,\\\\1,& x<1,\\end{cases}$ 则 $f(f(f(2)))=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260105-2026届高一寒假作业05" ], "genre": "填空题", "ans": "", @@ -629819,7 +631211,8 @@ "content": "已知 $f(x)$ 是奇函数, 当 $x<0$ 时, $f(x)=x(x-1)$, 则 $f(x)=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260105-2026届高一寒假作业05" ], "genre": "填空题", "ans": "", @@ -629843,7 +631236,8 @@ "content": "函数 $f(x)=\\dfrac{1}{x}$ 的严格减区间是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260105-2026届高一寒假作业05" ], "genre": "填空题", "ans": "", @@ -629865,7 +631259,8 @@ "content": "已知 $a$ 是常数, 函数 $y=|x-a|$ 在 $[2,+\\infty)$ 上是严格增函数, 则 $a$ 的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260105-2026届高一寒假作业05" ], "genre": "填空题", "ans": "", @@ -629887,7 +631282,8 @@ "content": "已知 $a$ 是常数, $f(x)$ 是定义在 $\\mathbf{R}$ 上的偶函数, 它在 $[0,+\\infty)$ 上是严格减函数, 则 $f(-0.74)$ 与 $f(a^2-a+1)$ 的大小关系是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260105-2026届高一寒假作业05" ], "genre": "填空题", "ans": "", @@ -629909,7 +631305,8 @@ "content": "若 $00$}{$f(x)-f(-x) \\leq 0$}{$f(x) \\cdot f(-x) \\leq 0$}{$f(x) \\cdot f(-x)>0$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260106-2026届高一寒假作业06" ], "genre": "选择题", "ans": "", @@ -630383,7 +631800,8 @@ "content": "已知函数 $f(x), g(x)$ 定义在同一区间 $D$ 上, $f(x)$ 是增函数, $g(x)$ 是减函数, 且 $g(x) \\neq 0$, 则在 $D$ 上\\bracket{20}.\n\\twoch{$f(x)+g(x)$ 一定是减函数}{$f(x)-g(x)$ 一定是增函数}{$f(x) \\cdot g(x)$ 一定是增函数}{$\\dfrac{f(x)}{g(x)}$ 一定是减函数}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260106-2026届高一寒假作业06" ], "genre": "选择题", "ans": "", @@ -630405,7 +631823,8 @@ "content": "某种奥运纪念品进货价 50 元/件, 据市场调查, 当销售价格 $x$ (元/件) 在 $x \\in[50,80]$ 时,每天售出件数 $p=\\dfrac{100000}{x-40}$, 若想每天获得的利润最大, 销售价格应定为多少元?", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260106-2026届高一寒假作业06" ], "genre": "解答题", "ans": "", @@ -630427,7 +631846,8 @@ "content": "设 $x_1, x_2$ 是关于 $x$ 的一元二次方程 $x^2-2(m-1) x+m+1=0$ 的两个实根, 又 $y=x_1^2+x_2^2$.\\\\\n(1) 求 $y=f(m)$ 的解析式及此时函数的定义域;\\\\\n(2) 指出 $f(m)$ 的单调区间.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260106-2026届高一寒假作业06" ], "genre": "解答题", "ans": "", @@ -630449,7 +631869,8 @@ "content": "设函数 $f(x)$ 是定义在 $\\mathrm{R}$ 上的偶函数, 并在区间 $(-\\infty, 0)$ 上是严格增函数, $f(2 a^2+a+1)0$ 时, $f(x)<0$, $f(1)=-2$.\\\\\n(1) 判断函数 $f(x)$ 的奇偶性;\\\\\n(2) 当 $x \\in[-3,3]$ 时, 函数 $f(x)$ 是否有最值?如果有, 求出最值; 如果没有, 请说明理由.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260106-2026届高一寒假作业06" ], "genre": "解答题", "ans": "", @@ -630567,7 +631992,8 @@ "objs": [], "tags": [ "第二单元", - "第一单元" + "第一单元", + "V20260106-2026届高一寒假作业06" ], "genre": "解答题", "ans": "", @@ -630589,7 +632015,8 @@ "content": "对于 $y=f(x)$ 是定义在 $(0,+\\infty)$ 上的函数, 若其满足: 对任意 $x>0$, 都成立 $f(x)=f(\\dfrac{1}{x})$, 就称 $y=f(x)$ 是``倒数对称''的.\\\\\n(1) 判断函数 $y=x-\\dfrac{1}{x}$ 与函数 $y=x^2+\\dfrac{1}{x^2}$ 是否是``倒数对称''的;\\\\\n(2) 若``倒数对称''的函数 $y=f(x)$ 在区间 $[1,2]$ 上是严格减函数, 判断它在区间 $[\\dfrac{1}{2}, 1]$ 上的单调性, 并说明理由;\\\\\n(3) 证明: 若 $y=f(x)$ 是``倒数对称''的函数, 则存在定义在 $[2,+\\infty)$ 上的函数 $y=g(x)$,使得对任意 $x \\in (0,+\\infty)$, 总成立 $f(x)=g(x+\\dfrac{1}{x})$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260106-2026届高一寒假作业06" ], "genre": "解答题", "ans": "", @@ -630611,7 +632038,8 @@ "content": "已知数列 $\\{a_n\\}$ 中, $a_1=1$, $a_na_{n-1}=a_{n-1}+(-1)^n(n \\geq 2, n$ 为正整数), 则 $\\dfrac{a_3}{a_5}$ 的值是\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "V20260107-2026届高一寒假作业07" ], "genre": "填空题", "ans": "", @@ -630633,7 +632061,8 @@ "content": "已知数列 $\\{\\dfrac{n+2}{n}\\}$, 欲使它的前 $n$ 项的乘积大于 36 , 则 $n$ 的最小值为\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "V20260107-2026届高一寒假作业07" ], "genre": "填空题", "ans": "", @@ -630655,7 +632084,8 @@ "content": "等差数列 $7,4,1,-2,-5, \\cdots$ 的第 20 项是\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "V20260107-2026届高一寒假作业07" ], "genre": "填空题", "ans": "", @@ -630677,7 +632107,8 @@ "content": "对于无穷数列$\\{a_n\\}$, ``数列 $\\{a_n\\}$ 是等差数列''是``数列 $\\{a_n+a_{n+1}\\}$ 是等差数列''成立的\\blank{50}条件.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "V20260107-2026届高一寒假作业07" ], "genre": "填空题", "ans": "", @@ -630702,7 +632133,8 @@ "content": "有穷数列 $5,8,11, \\cdots, 3 n+11$ ($n$ 为正整数) 的项数是\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "V20260107-2026届高一寒假作业07" ], "genre": "填空题", "ans": "", @@ -630724,7 +632156,8 @@ "content": "已知等差数列中, $a_3=13$, $a_{15}=41$, 则 $a_9=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "V20260107-2026届高一寒假作业07" ], "genre": "填空题", "ans": "", @@ -630748,7 +632181,8 @@ "content": "在等比数列 $\\{a_n\\}$ 中, 已知 $a_1+a_2+a_3=6$, $a_4+a_5+a_6=48$, 则 $a_5+a_6+a_7=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "V20260107-2026届高一寒假作业07" ], "genre": "填空题", "ans": "", @@ -630770,7 +632204,8 @@ "content": "已知数列 $\\{a_n\\}$ 满足 $a_1a_2\\cdots a_n=2^{n^2}$, 则数列 $\\{a_n\\}$ \\blank{50}为等比数列; (填``是''``不是'')", "objs": [], "tags": [ - "第四单元" + "第四单元", + "V20260107-2026届高一寒假作业07" ], "genre": "填空题", "ans": "", @@ -630792,7 +632227,8 @@ "content": "已知数列 $\\{a_n\\}$ 的通项公式为 $a_n=4 n+\\dfrac{9}{n}$, 则 $n=$\\blank{50}时, 数列 $\\{a_n\\}$ 有最小项.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "V20260107-2026届高一寒假作业07" ], "genre": "填空题", "ans": "", @@ -630816,7 +632252,8 @@ "content": "设 $S_n$ 是等差数列 $\\{a_n\\}$ 的前 $n$ 项和, 若 $\\dfrac{S_3}{S_6}=\\dfrac{1}{3}$, 则 $\\dfrac{S_6}{S_{12}}=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "V20260107-2026届高一寒假作业07" ], "genre": "填空题", "ans": "", @@ -630840,7 +632277,8 @@ "content": "在两个不等正数 $a, b$ 之间插入 $n$ 个数, 使它们与 $a, b$ 组成等差数列 $\\{a_n\\}$, 公差为 $d_1$, 再插入 $m$ 个数, 使它们与 $a, b$ 组成等差数列 $\\{b_n\\}$, 公差为 $d_2$, 则 $\\dfrac{d_1}{d_2}=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "V20260107-2026届高一寒假作业07" ], "genre": "填空题", "ans": "", @@ -630864,7 +632302,8 @@ "content": "有穷等差数列 $\\{a_n\\}$ 的公差 $d=-3$, 共有偶数项, 其中奇数项之和与偶数项之和的比是 $11: 8$, 所有项之和为 95 , 求其项数 $n$.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "V20260107-2026届高一寒假作业07" ], "genre": "解答题", "ans": "", @@ -630888,7 +632327,8 @@ "content": "已知数列 $\\{a_n\\}$ 的前 $n$ 项和 $S_n=12 n-n^2$, 求数列 $\\{|a_n|\\}$ 的前 $n$ 项和 $T_n$.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "V20260107-2026届高一寒假作业07" ], "genre": "解答题", "ans": "", @@ -630912,7 +632352,8 @@ "content": "数列 $\\{b_n\\}$ 中, $b_1=1$, $b_2=2$, $b_{n+2}=b_{n+1}-b_n$ ($n$ 为正整数), 则 $b_6=$\\blank{50}, $b_{2014}=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "V20260107-2026届高一寒假作业07" ], "genre": "填空题", "ans": "", @@ -630936,7 +632377,8 @@ "content": "已知数列 $a_n=3 n^2-\\lambda n$ ($n$ 为正整数) 是一个递增数列, 则实数 $\\lambda$ 的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "V20260107-2026届高一寒假作业07" ], "genre": "填空题", "ans": "", @@ -630962,7 +632404,8 @@ "content": "等差数列 $\\{a_n\\}$ 中, $a_3=10$, 且 $a_3, a_7, a_{16}$ 组成等比数列中相邻的项, 则公比 $q=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "V20260107-2026届高一寒假作业07" ], "genre": "填空题", "ans": "", @@ -630984,7 +632427,8 @@ "content": "已知数列 $\\{a_n\\}$ 中, $a_n=n^5-2 n+1$ ($n$ 为正整数), 试判断 2015 是不是这个数列中的项? 若是, 是第几项? 若不是, 为什么?", "objs": [], "tags": [ - "第四单元" + "第四单元", + "V20260107-2026届高一寒假作业07" ], "genre": "解答题", "ans": "", @@ -631006,7 +632450,8 @@ "content": "在等差数列 $\\{a_n\\}$ 中, 公差 $d \\neq 0$, $a_kx^2+2 a_{k+1}x+a_{k+2}=0$ ($k$ 为正整数).\\\\\n(1) 求证: 对不同的 $\\mathrm{k}$ 值, 方程都有公共根.\\\\\n(2) 若方程除公共根外的根依次为 $b_1, b_2, b_3, \\cdots, b_k, \\cdots$, 求证: 数列 $\\{\\dfrac{1}{b_k+1}\\}$ 是等差数列.\\\\\n(3) 设$\\{b_n\\}$是(2)中定义的数列. 若 $b_1=2$, 求 $b_{10}$.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "V20260107-2026届高一寒假作业07" ], "genre": "解答题", "ans": "", @@ -631028,7 +632473,8 @@ "content": "如图, $64$ 个正数排成 $8$ 行 $8$ 列方阵. 符号 $a_{i j}$($1 \\leq i \\leq 8$, $1 \\leq j \\leq 8$, $i$、$j$ 均为正整数) 表示位于第 $i$ 行第 $j$ 列的正数. 已知每一行的数成等差数列, 每一列的数成等比数列, 且各列数的公比都等于 $q$. 若 $a_{11}=\\dfrac{1}{2}$, $a_{24}=1$, $a_{32}=\\dfrac{1}{4}$. \n\\begin{center}\n\\begin{tabular}{ccccc}\n$a_{11}$& $a_{12}$& $a_{13}$& $\\cdots$ & $a_{18}$\\\\\n$a_{21}$& $a_{22}$& $a_{23}$& $\\cdots$ & $a_{28}$\\\\\n$\\cdots$ & $\\cdots$ & $\\cdots$ & $\\cdots$ & $\\cdots$\\\\\n$a_{81}$& $a_{82}$& $a_{83}$& $\\cdots$ & $a_{88}$\n\\end{tabular}\n\\end{center}\n(1) 求 $\\{a_{i j}\\}$ 的通项公式;\\\\\n(2) 已知 $k$ 为正整数, 记第 $k$ 行各项和为 $A_k$ , 求 $A_1$ 的值及数列 $\\{A_k\\}$ 的通项公式;\\\\\n(3) 接着上面的第(2)题, 若 $A_k<1$, 求 $k$ 的值.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "V20260107-2026届高一寒假作业07" ], "genre": "解答题", "ans": "", @@ -631050,7 +632496,8 @@ "content": "设函数 $f(x)=(\\sqrt{x+1}+\\sqrt{x})(\\sqrt{x+1}-\\sqrt{x})$, 则函数 $y=f(x)$ 的定义域为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260108-2026届高一寒假作业08" ], "genre": "填空题", "ans": "", @@ -631072,7 +632519,8 @@ "content": "已知函数 $f(x)$ 满足 $f(\\sqrt{x})=x$, 则 $f(4)=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260108-2026届高一寒假作业08" ], "genre": "填空题", "ans": "", @@ -631094,7 +632542,8 @@ "content": "将函数 $f(x)=x^3$ 的图像向右平移 2 个单位后, 得到函数 $g(x)$ 的图像, 则 $g(2)=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260108-2026届高一寒假作业08" ], "genre": "填空题", "ans": "", @@ -631116,7 +632565,8 @@ "content": "已知数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n=n^2+2 n+3$, 则数列 $\\{a_n\\}$ 的通项公式 $a_n=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "V20260108-2026届高一寒假作业08" ], "genre": "填空题", "ans": "", @@ -631141,7 +632591,8 @@ "content": "设函数 $f(x)=\\log _2(3 x-1)$ 的反函数为 $f^{-1}(x)$. 若 $f^{-1}(a)=3$, 则 $a=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260108-2026届高一寒假作业08" ], "genre": "填空题", "ans": "", @@ -631165,7 +632616,8 @@ "content": "已知实常数 $a>0$, 函数 $f(x)=-\\dfrac{2^x-1}{2^x+a}$ 为奇函数, 则 $a=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260108-2026届高一寒假作业08" ], "genre": "填空题", "ans": "", @@ -631189,7 +632641,8 @@ "content": "已知常数 $a \\in \\mathbf{R}$, 函数 $f(x)=x^2-4 x+a$ 在 [1,4] 上有两个不同的零点, 则 $a$ 的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260108-2026届高一寒假作业08" ], "genre": "填空题", "ans": "", @@ -631211,7 +632664,8 @@ "content": "设 $x, y, z>0$, 满足 $2^x=3^y=6^z$, 则 $2 x+\\dfrac{1}{z}-\\dfrac{1}{y}$ 的最小值为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260108-2026届高一寒假作业08" ], "genre": "填空题", "ans": "", @@ -631233,7 +632687,8 @@ "content": "已知实常数 $a>0$, 函数 $f(x)=\\log _2(x^2+a)$, $g(x)=f[f(x)]$. 若 $f(x)$ 与 $g(x)$ 有相同的值域, 则 $a$ 的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260108-2026届高一寒假作业08" ], "genre": "填空题", "ans": "", @@ -631255,7 +632710,8 @@ "content": "已知常数 $a \\in \\mathbf{R}$. 设函数 $f(x)=3 x^3+(2 a-1) x+a \\sqrt{2-2 x^2}$, 定义域为 $(0, \\dfrac{\\sqrt{3}}{3})$. 若 $f(x)$的最小值为 0, 则 $a=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260108-2026届高一寒假作业08" ], "genre": "填空题", "ans": "", @@ -631277,7 +632733,8 @@ "content": "如图为幂函数 $y=x^{\\alpha}$ 的图像, 则实数 $\\alpha$ 的值可以为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = {1/sqrt(27)}:3, samples = 100] plot (\\x,{exp(-2/3*ln(\\x))}) plot (-\\x,{exp(-2/3*ln(\\x))});\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\dfrac{2}{3}$}{$\\dfrac{3}{2}$}{$-\\dfrac{2}{3}$}{$-\\dfrac{3}{2}$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260108-2026届高一寒假作业08" ], "genre": "选择题", "ans": "", @@ -631299,7 +632756,8 @@ "content": "设 $\\{a_n\\}$ 是等比数列, 则``$a_10$, 使得 $g(x)$具有性质 $P$; \\textcircled{2} 存在 $b>0$, 使得 $h(x)$ 具有性质 $P$, 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", + "V20260108-2026届高一寒假作业08" ], "genre": "选择题", "ans": "", @@ -631343,7 +632802,8 @@ "content": "已知常数 $a \\in \\mathbf{R}$, 函数 $f(x)=|2 x-1|+a$.\\\\\n(1) 若 $a=-3$, 解不等式 $f(x) \\leq 0$;\\\\\n(2) 若关于 $x$ 的不等式 $f(x) \\geq 1$ 对任意 $x \\in \\mathbf{R}$ 恒成立, 求 $a$ 的取值范围.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260108-2026届高一寒假作业08" ], "genre": "解答题", "ans": "", @@ -631365,7 +632825,8 @@ "content": "已知函数 $f(x)$ 的定义域为 $\\mathbf{R}$, 当 $x \\geq 0$ 时, $f(x)=2 x-\\dfrac{2}{x+1}$.\\\\\n(1) 求函数 $g(x)=f(x)-x$($x \\geq 0$) 的零点;\\\\\n(2) 若 $f(x)$ 为偶函数. 当 $x<0$ 时, 解不等式 $f(x)<-4 x-3$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260108-2026届高一寒假作业08" ], "genre": "解答题", "ans": "", @@ -631387,7 +632848,8 @@ "content": "已知数列 $\\{a_n\\}$ 是等差数列, 满足 $a_1=-3$, $a_3=1$, 数列 $\\{b_n\\}$ 满足 $a_n(b_n-1)=1$\\\\\n(1) 求数列 $\\{a_n\\}$ 的通项公式;\\\\\n(2) 求数列 $\\{b_n\\}$ 的通项公式, 并求出其最大项与最小项;\\\\\n(3) 记 $c_n=\\dfrac{(-1)^{n+1}}{a_na_{n+2}}$, 求 $c_1+c_2+\\cdots+c_n$.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "V20260108-2026届高一寒假作业08" ], "genre": "解答题", "ans": "", @@ -631409,7 +632871,8 @@ "content": "已知函数 $f(x), g(x)$ 的定义域分别为 $D_1, D_2$. 若存在常数 $C>0$, 满足:\\\\\n\\textcircled{1} 对任意 $x_0\\in D_1$, 恒有 $x_0+C \\in D_1$, 且 $f(x_0) \\leq f(x_0+C)$;\\\\\n\\textcircled{2} 对任意 $x_0\\in D_1$, 关于 $x$ 的不等式组 $f(x_0) \\leq g(x) \\leq g(x+C) \\leq f(x_0+C)$ 恒有解,则称 $g(x)$ 为 $f(x)$ 的一个``$C$ 型函数''.\\\\\n(1) 设函数 $f(x)=\\begin{cases}-1,&0 \\leq x \\leq \\dfrac{1}{3},\\\\1,& x>\\dfrac{1}{3}\\end{cases}$ 和 $g(x)=\\begin{cases}1,&0 \\leq x \\leq \\dfrac{1}{2},\\\\0,& x>\\dfrac{1}{2}.\\end{cases}$ 求证: $g(x)$ 为 $f(x)$ 的一个``$\\dfrac{1}{2}$型函数'';\\\\\n(2) 设常数 $a \\in \\mathbf{R}$, 函数 $f(x)=x^3+a x(x \\geq-1)$, $g(x)=2 x$($x \\geq-1$). 若 $g(x)$ 为 $f(x)$ 的一个``1 型函数'', 求 $a$ 的取值范围;\\\\\n(3) 设函数 $f(x)=x^2-4 x$($x \\geq 0$). 问: 是否存在常数 $t>0$, 使得函数 $g(x)=x+\\dfrac{2 t^2}{x}$($x>0$) 为 $f(x)$ 的一个``$t$ 型函数''? 若存在, 求 $t$ 的取值范围; 若不存在, 说明理由.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260108-2026届高一寒假作业08" ], "genre": "解答题", "ans": "", @@ -631431,7 +632894,8 @@ "content": "设实数 $a$ 满足 $\\log _2a=4$, 则 $a=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260109-2026届高一寒假作业09" ], "genre": "填空题", "ans": "", @@ -631453,7 +632917,8 @@ "content": "已知幂函数 $f(x)=(m-1) x^{m^2-3 m-5}$ 的图像不经过原点, 则实数 $m=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260109-2026届高一寒假作业09" ], "genre": "填空题", "ans": "", @@ -631477,7 +632942,8 @@ "content": "函数 $f(x)=\\log _2(1-x^2)$ 的定义域为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260109-2026届高一寒假作业09" ], "genre": "填空题", "ans": "", @@ -631499,7 +632965,8 @@ "content": "若函数 $f(x)=a^x$($a>1$) 在 $[-1,2]$ 上的最大值为 $4$, 则其最小值为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260109-2026届高一寒假作业09" ], "genre": "填空题", "ans": "", @@ -631521,7 +632988,8 @@ "content": "在同一平面直角坐标系中, 函数 $y=g(x)$ 的图像与 $y=3^x$ 的图像关于直线 $y=x$ 对称, 而函数 $y=f(x)$ 的图像与 $y=g(x)$ 的图像关于 $y$ 轴对称, 若 $f(a)=-1$, 则 $a$ 的值是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260109-2026届高一寒假作业09" ], "genre": "填空题", "ans": "", @@ -631545,7 +633013,8 @@ "content": "在数列 $\\{a_n\\}$ 中, 已知 $a_1=1$, $a_{n+1}=\\dfrac{n+1}{n+2}a_n$($n \\geq 1$), 则数列的通项公式为\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "V20260109-2026届高一寒假作业09" ], "genre": "填空题", "ans": "", @@ -631569,7 +633038,8 @@ "content": "若定义在 $\\mathbf{R}$ 上的奇函数 $f(x)$ 在 $(0,+\\infty)$ 上是严格增函数, 且 $f(-4)=0$, 则使得 $x f(x)>0$成立的 $x$ 的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260109-2026届高一寒假作业09" ], "genre": "填空题", "ans": "", @@ -631591,7 +633061,8 @@ "content": "函数 $f(x)=\\lg (2^x+2^{-x}+a-1)$ 的值域是 $\\mathbf{R}$, 则实数 $a$ 的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260109-2026届高一寒假作业09" ], "genre": "填空题", "ans": "", @@ -631613,7 +633084,8 @@ "content": "若直角坐标平面内两点 $P$、$Q$ 满足条件: \\textcircled{1} $P$、$Q$ 都在函数 $f(x)$ 的图像上; \\textcircled{2} $P$、$Q$ 关于原点对称, 则对称点 $(P, Q)$ 是函数 $f(x)$ 的一个``匹配点对''(点对 $(P, Q)$ 与 $(Q, P)$ 看作同一个``匹配点对''). 已知函数 $f(x)=\\begin{cases}2 x^2+4 x+1,& x<0,\\\\\\dfrac{2}{\\mathrm{e}^2},& x \\geq 0,\\end{cases}$ 则 $f(x)$ 的``匹配点对''有\\blank{50}个.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260109-2026届高一寒假作业09" ], "genre": "填空题", "ans": "", @@ -631635,7 +633107,8 @@ "content": "函数 $y=1-\\dfrac{1}{x + 1}$ 的值域是\\bracket{20}.\n\\fourch{$(-\\infty, 1)$}{$(1,+\\infty)$}{$(-\\infty, 1) \\cup(1,+\\infty)$}{$(-\\infty, +\\infty)$}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260109-2026届高一寒假作业09" ], "genre": "选择题", "ans": "", @@ -631657,7 +633130,8 @@ "content": "已知函数 $f(x)=\\begin{cases}1, & x<0,\\\\0, & x=0, \\\\ -1, & x<0\\dots\\end{cases}$ 设 $F(x)=x^2 \\cdot f(x)$, 则 $F(x)$ 是\\bracket{20}.\n\\onech{奇函数, 在 $(-\\infty, +\\infty)$ 上为严格减函数}{奇函数, 在 $(-\\infty, +\\infty)$ 上为严格增函数}{偶函数, 在 $(-\\infty, 0)$ 上严格减, 在$(0, +\\infty)$ 上严格增}{偶函数, 在 $(-\\infty, 0)$ 上严格增, 在 $(0,+\\infty)$ 上严格减}", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260109-2026届高一寒假作业09" ], "genre": "选择题", "ans": "", @@ -631679,7 +633153,8 @@ "content": "设 $a>b>c>0$, 则 $2 a^2+\\dfrac{1}{a b}+\\dfrac{1}{a(a-b)}-10 a c+25 c^2$ 取得最小值时, $a$ 的值为\\bracket{20}.\n\\fourch{$\\sqrt{2}$}{$2$}{$4$}{$2 \\sqrt{5}$}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "V20260109-2026届高一寒假作业09" ], "genre": "选择题", "ans": "", @@ -631701,7 +633176,8 @@ "content": "已知函数 $f(x)=a x^2+2 a x+1$.\\\\\n(1) 若实数 $a=1$ , 请写出函数 $y=3^{f(x)}$ 的单调区间 (不需要过程); \\\\\n(2) 已知函数 $y=f(x)$ 在区间 $[-3,2]$ 上的最大值为 $2$, 求实数 $a$ 的值.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260109-2026届高一寒假作业09" ], "genre": "解答题", "ans": "", @@ -631723,7 +633199,8 @@ "content": "设函数 $f(x)=|2 x-a|$, $g(x)=x+2$.\\\\\n(1) 当 $a=1$ 时, 求不等式 $f(x) \\leq g(x)$ 的解集;\\\\\n(2) 求证: $f(\\dfrac{b}{2}), f(-\\dfrac{b}{2}), f(\\dfrac{1}{2})$ 中至少有一个不小于 $\\dfrac{1}{2}$.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260109-2026届高一寒假作业09" ], "genre": "解答题", "ans": "", @@ -631746,7 +633223,8 @@ "objs": [], "tags": [ "第二单元", - "第一单元" + "第一单元", + "V20260109-2026届高一寒假作业09" ], "genre": "解答题", "ans": "", @@ -631768,7 +633246,8 @@ "content": "已知数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 满足 $a_1=\\dfrac{1}{2}$, $S_{n+1}=S_n+\\dfrac{2 a_n}{2 a_n+1}$.\\\\\n(1) 证明数列 $\\{\\dfrac{1}{a_n}\\}$ 是等差数列, 并求出数列 $\\{a_n\\}$ 的通项公式;\\\\\n(2) 若数列 $\\{b_n\\}$ 满足 $b_n=(2 n + 1)^2a_na_{n+1}$, 求数列 $\\{b_n\\}$ 的前 $n$ 项和 $T_n$;\\\\\n(3) 若数列 $\\{C_n\\}$ 满足 $C_n=(2^n-8) a_n$, 则数列 $C_n$ 最大项和最小项是否存在? 若存在请指出, 若不存在请说明理由.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "V20260109-2026届高一寒假作业09" ], "genre": "解答题", "ans": "", @@ -631790,7 +633269,8 @@ "content": "若函数 $f(x)$ 的定义域为 $D$, 集合 $M \\subseteq D$, 若存在非零实数 $t$ 使得任意 $x \\in M$ 都有 $x+t \\in D$, 且 $f(x+t)>f(x)$, 则称 $f(x)$ 为 $M$ 上的 $t-$增长函数.\\\\\n(1) 已知函数 $g(x)=x$ , 判断 $g(x)$ 是否为区间 $[-1,0]$ 上的 $\\dfrac{3}{2}-$ 增长函数, 并说明理由;\\\\\n(2) 已知函数 $f(x)=|x|$, 且 $f(x)$ 是区间 $[-4,-2]$ 上的 $n-$ 增长函数, 求正整数 $n$ 的最小值;\\\\\n(3) 如果 $f(x)$ 是定义域为 $\\mathbf{R}$ 的奇函数, 当 $x \\geq 0$ 时, $f(x)=|x-a^2|-a^2$, 且 $f(x)$ 为 $\\mathbf{R}$ 上的 $4-$增长函数, 求实数 $a$ 的取值范围.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "V20260109-2026届高一寒假作业09" ], "genre": "解答题", "ans": "", @@ -632274,7 +633754,8 @@ "content": "$3$和$7$的等差中项是\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "E20260107-2026届高一上学期期末考试" ], "genre": "填空题", "ans": "$5$", @@ -632309,7 +633790,8 @@ "content": "陈述句``$a=0$ 且 $b=0$''的否定形式为\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "E20260107-2026届高一上学期期末考试" ], "genre": "填空题", "ans": "$a\\neq 0$或$b\\neq 0$", @@ -632346,7 +633828,8 @@ "content": "数列$\\{a_n\\}$是等差数列, $a_1=1$,公差$d=2$, 该数列的前$10$项和$S_{10}=$\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "E20260107-2026届高一上学期期末考试" ], "genre": "填空题", "ans": "$100$", @@ -632381,7 +633864,8 @@ "content": "已知$\\log_2 5=a$, 则$\\log_2 25=$\\blank{50}(请用$a$表示).", "objs": [], "tags": [ - "第二单元" + "第二单元", + "E20260107-2026届高一上学期期末考试" ], "genre": "填空题", "ans": "$2a$", @@ -632416,7 +633900,8 @@ "content": "函数$f(x)=2^x+m$的反函数为$y=f^{-1}(x)$, 且$y=f^{-1}(x)$的图像过点$Q(5,2)$, 那么实数$m=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "E20260107-2026届高一上学期期末考试" ], "genre": "填空题", "ans": "$1$", @@ -632453,7 +633938,8 @@ "content": "函数$y=\\sqrt{-2x^2+3x-1}$的定义域是\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "E20260107-2026届高一上学期期末考试" ], "genre": "填空题", "ans": "$[\\dfrac{1}{2},1]$", @@ -632492,7 +633978,8 @@ "content": "无穷等比数列首项为$2$,公比为$q \\ (00$. 设$S_n$是数列$\\{a_n\\}$的前$n$项和, 若$S_k>0$, 则正整数$k$的最小值为\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "E20260107-2026届高一上学期期末考试" ], "genre": "填空题", "ans": "$40$", @@ -632602,7 +634091,8 @@ "objs": [], "tags": [ "第二单元", - "第一单元" + "第一单元", + "E20260107-2026届高一上学期期末考试" ], "genre": "填空题", "ans": "$[0,\\dfrac{1}{2})$", @@ -632639,7 +634129,8 @@ "content": "已知函数$f(x)$是定义在$\\mathbf{R}$上的严格增函数, 且$y=f(x)$是奇函数. 若关于$x$的不等式$f(m x)+f(-x^2-2)<0$在区间$[1,5]$上恒成立, 则实数$m$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "E20260107-2026届高一上学期期末考试" ], "genre": "填空题", "ans": "$(-\\infty,2\\sqrt{2})$", @@ -632674,7 +634165,8 @@ "content": "已知数列 $\\{a_n\\}$ 的各项均为正数, 其前 $n$ 项和 $S_n$ 满足 $a_n \\cdot S_n=9$($n=1,2, \\cdots$). 给出下列四个结论:\n\\textcircled{1} $\\{a_n\\}$ 的第 2 项小于 3 ;\n\\textcircled{2} $\\{a_n\\}$ 为等比数列;\n\\textcircled{3} $\\{a_n\\}$ 为严格减数列;\n\\textcircled{4} $\\{a_n\\}$ 中存在小于 $\\dfrac{1}{100}$ 的项.其中所有正确结论的序号是\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "E20260107-2026届高一上学期期末考试" ], "genre": "填空题", "ans": "\\textcircled{1}\\textcircled{3}\\textcircled{4}", @@ -632709,7 +634201,8 @@ "content": "已知实数$a,b$满足$a>b$, 则下列不等式中恒成立的是\\bracket{20}.\n\\fourch{$a^2>b^2$}{$\\dfrac 1a<\\dfrac 1b$}{$|a|>|b|$}{$2^a>2^b$}", "objs": [], "tags": [ - "第一单元" + "第一单元", + "E20260107-2026届高一上学期期末考试" ], "genre": "选择题", "ans": "D", @@ -632746,7 +634239,8 @@ "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", + "E20260107-2026届高一上学期期末考试" ], "genre": "选择题", "ans": "A", @@ -632783,7 +634277,8 @@ "content": "斐波那契数列$\\{a_n\\}$满足$a_1=a_2=1$, $a_{n+2}=a_{n+1}+a_n(n\\geq 1, n\\in \\mathbf{N})$, 设$a_1+a_3+a_5+a_7+a_9+\\cdots+a_{2023}=a_k$, 则$k=$\\bracket{20}.\n\\twoch{2022}{2023}{2024}{2025}", "objs": [], "tags": [ - "第四单元" + "第四单元", + "E20260107-2026届高一上学期期末考试" ], "genre": "选择题", "ans": "C", @@ -632818,7 +634313,8 @@ "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为\\bracket{20}.\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 (0.1,1) -- (0,1) node [right] {\\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 (0.1,1) -- (0,1) node [left] {\\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}\n\\twoch{ \\textcircled{1} \\textcircled{3} }{ \\textcircled{1} \\textcircled{4} }{ \\textcircled{2} \\textcircled{3} }{ \\textcircled{2} \\textcircled{4} }", "objs": [], "tags": [ - "第二单元" + "第二单元", + "E20260107-2026届高一上学期期末考试" ], "genre": "选择题", "ans": "B", @@ -632855,7 +634351,8 @@ "content": "已知函数$f(x)=x^2-\\dfrac{1}{x}$.\\\\\n(1) 判断函数$f(x)$是否是偶函数，并说明理由;\\\\\n(2) 判断$f(x)$在$(0, +\\infty)$上的单调性，并说明理由.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "E20260107-2026届高一上学期期末考试" ], "genre": "解答题", "ans": "(1) 不是偶函数, 理由略; (2) 是严格增函数, 理由略", @@ -632890,7 +634387,8 @@ "content": "已知数列$\\{a_n\\}$的各项均不为零, 且$a_{n+1}=\\dfrac{3a_n}{a_n+3}$, $b_n=\\dfrac{1}{a_n}$. \\\\ \n(1) 求证: 数列$\\{b_n\\}$是等差数列;\\\\ \n(2) 若$a_1=1$, 求数列$\\{a_n\\}$的通项公式.", "objs": [], "tags": [ - "第四单元" + "第四单元", + "E20260107-2026届高一上学期期末考试" ], "genre": "解答题", "ans": "(1) 证明略; (2) $a_n=\\dfrac{3}{n+2}$", @@ -632927,7 +634425,8 @@ "content": "某企业是用电大户, 去年的用电量达到$20$万度, 经预测, 在去年的基础上, 今年该企业若减少用电$x$万度, 今年的受损效益$S(x)$(万元)满足 $S(x)=\\begin{cases} 50x^2, &1\\le x\\le 4, \\\\ 100x-\\dfrac{400}{x}+500, & 40$)是函数$f(x)=-x^2+2x$的``$\\Omega$区间'', 求$m$的取值范围;\\\\\n(3) 已知定义在$\\mathbf{R}$上且图像是一段连续曲线的函数$f(x)$满足: 对任意$x_1,x_2\\in \\mathbf{R}$, 且$x_1\\neq x_2$, 有$\\dfrac{f(x_2)-f(x_1)}{x_2-x_1}<-1$. 求证：$f(x)$存在``$\\Omega$区间'', 且存在$x_0\\in \\mathbf{R}$, 使得$x_0$不属于$f(x)$的所有``$\\Omega$区间''.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "E20260107-2026届高一上学期期末考试" ], "genre": "解答题", "ans": "(1) 是, 不是; (2) $[1,2]$; (3) 证明略", @@ -633789,7 +635290,8 @@ "K0106001B" ], "tags": [ - "第一单元" + "第一单元", + "G20260155-必修第一章集合与逻辑复习" ], "genre": "填空题", "ans": "(1) \\textcircled{2}; (2) \\textcircled{2}; (3) \\textcircled{1}; (4) \\textcircled{2}; (5) \\textcircled{2}; (6) \\textcircled{1}; (7) \\textcircled{2}; (8) \\textcircled{3}; (9) \\textcircled{2}; (10) \\textcircled{3}; (11) \\textcircled{1}", @@ -646212,7 +647714,8 @@ "tags": [ "第二单元", "导数", - "2023届高三-第一轮复习讲义-29-导数的应用" + "2023届高三-第一轮复习讲义-29-导数的应用", + "E20260104-高一上学期测验卷04" ], "genre": "选择题", "ans": "D", @@ -656310,7 +657813,8 @@ "content": "函数$f(x)=\\sqrt x+1$的反函数为$f^{-1}(x)$, 则$f^{-1}(3)=$\\blank{50}.", "objs": [], "tags": [ - "第二单元" + "第二单元", + "E20260104-高一上学期测验卷04" ], "genre": "填空题", "ans": "$4$", @@ -669655,7 +671159,8 @@ "content": "已知全集$U=\\{x | 20\\end{cases}$ 若$f(a)=9$, 则$a=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "V20260109-2026届高一寒假作业09" ], "genre": "填空题", "ans": "$-9$或$3$", @@ -673564,7 +675070,8 @@ ], "tags": [ "第一单元", - "集合" + "集合", + "G20260101-集合的概念" ], "genre": "解答题", "ans": "(1) 是集合, 是有限集; (2) 是集合, 是无限集; (3) 不是集合", @@ -673603,7 +675110,8 @@ ], "tags": [ "第一单元", - "集合" + "集合", + "G20260101-集合的概念" ], "genre": "填空题", "ans": "(1) $\\not\\in$; (2) $\\in$; (3) $\\in$; (4) $\\in$; (5) $\\not\\in$", @@ -699188,7 +700696,8 @@ "content": "已知集合$A=\\{1,2\\}$, $B=\\{2,3\\}$, 则$A \\cup B=$\\blank{50}.", "objs": [], "tags": [ - "第一单元" + "第一单元", + "V20260108-2026届高一寒假作业08" ], "genre": "填空题", "ans": "$\\{1,2,3\\}$",