diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 213317c1..80232a81 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -472611,7 +472611,9 @@ "id": "018315", "content": "判断下列各角分别属于哪个象限:\\\\\n(1) $-240^{\\circ}$;\\\\\n(2) $2100^{\\circ}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -472631,7 +472633,9 @@ "id": "018316", "content": "写出与$-200^{\\circ}$的终边重合的所有角组成的集合$S$, 并列举$S$中满足不等式$-360^{\\circ} \\leq \\beta<720^{\\circ}$的所有元素$\\beta$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -472651,7 +472655,9 @@ "id": "018317", "content": "已知$\\alpha$为锐角, 且$\\alpha$的终边与$7 \\alpha$的终边关于$x$轴对称, 求$\\alpha$的大小.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -472671,7 +472677,9 @@ "id": "018318", "content": "已知集合$A=\\{\\alpha | 30^{\\circ}+k \\cdot 180^{\\circ}<\\alpha<90^{\\circ}+k \\cdot 180^{\\circ}, k \\in \\mathbf{Z}\\}$, 集合$B=\\{\\beta |-45^{\\circ}+k \\cdot 360^{\\circ}<\\beta<45^{\\circ}+k \\cdot 360^{\\circ}, k \\in \\mathbf{Z}\\}$, 求$A \\cap B$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -472691,7 +472699,9 @@ "id": "018319", "content": "已知$\\alpha$与$\\beta$都是锐角, $\\alpha+\\beta$的终边与$-280^{\\circ}$的终边重合、$\\alpha-\\beta$的终边与$-670^{\\circ}$的终边重合, 求$\\alpha$与$\\beta$的大小.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -472711,7 +472721,9 @@ "id": "018320", "content": "按下列要求, 将$75^{\\circ}$换算成弧度:\\\\\n(1) 精确值;\\\\\n(2) 近似值. (结果精确到 $0.001$ )", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -472731,7 +472743,9 @@ "id": "018321", "content": "将 $2.1$ 弧度换算成角度. (用度数表示, 结果保留两位小数)", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -472751,7 +472765,9 @@ "id": "018322", "content": "根据一些常用特殊角的角度与弧度的对应关系, 填写下表.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline 角度 &$0^{\\circ}$&$30^{\\circ}$&$45^{\\circ}$&$60^{\\circ}$& & &$135^{\\circ}$& &$180^{\\circ}$&$270^{\\circ}$&$360^{\\circ} $\\\\\n\\hline 弧度 & & & & &$\\dfrac{\\pi}{2}$&$\\dfrac{2 \\pi}{3}$& &$\\dfrac{5 \\pi}{6}$&$\\pi$& &$2 \\pi$\\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -472771,7 +472787,9 @@ "id": "018323", "content": "将$32^{\\circ} 18'$换算为弧度.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -472791,7 +472809,9 @@ "id": "018324", "content": "在弧度制的条件下, 证明下列关于扇形的公式:\\\\\n(1) $l=\\alpha r$;\\\\\n(2) $S=\\dfrac{1}{2} \\alpha r^2$;\\\\\n(3) $S=\\dfrac{1}{2} l r$.\\\\\n其中$r$是圆的半径, $\\alpha$($0<\\alpha<2 \\pi$)为圆心角, $l$是扇形的弧长, $S$是扇形的面积.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -472811,7 +472831,9 @@ "id": "018325", "content": "写出终边在$x$轴上的所有角组成的集合. (用弧度制表示)", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -472831,7 +472853,9 @@ "id": "018326", "content": "设$\\alpha$是第二象限的角, 判断$\\dfrac{\\alpha}{2}$是哪个象限的角.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -472851,7 +472875,9 @@ "id": "018327", "content": "已知扇形的圆心角为$2$弧度, 其圆心角所对弦长为$2$厘米, 则扇形面积是多少平方厘米? (结果精确到 $0.1$ 平方厘米)", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -472871,7 +472897,9 @@ "id": "018328", "content": "在直角$\\triangle POB$中 (如图所示), $\\angle PBO=90^{\\circ}$, 以$O$为圆心、$OB$为半径作圆弧交$OP$于点$A$. 若弧$AB$等分$\\triangle POB$的面积, 且$\\angle AOB=\\alpha$弧度, 则\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$B$} coordinate (B);\n\\draw (2,0) node [below right] {$O$} coordinate (O);\n\\draw (0,3) node [above] {$P$} coordinate (P);\n\\draw (B) arc (180:{180-atan(3/2)}:2) node [above right] {$A$} coordinate (A);\n\\draw (P)--(B)--(O)--cycle;\n\\draw (B) pic [draw, scale = 0.3] {right angle = O--B--P};\n\\draw (O) pic [draw, scale = 0.3, \"$\\alpha$\", angle eccentricity = 2.5] {angle = P--O--B};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\tan \\alpha=\\alpha$}{$\\tan \\alpha=2 \\alpha$}{$\\sin \\alpha=2 \\cos \\alpha$}{$2 \\sin \\alpha=\\cos \\alpha$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -472891,7 +472919,9 @@ "id": "018329", "content": "已知相互啮合的两个齿轮, 大轮有$48$齿, 小轮有$20$齿.\\\\\n(1) 当大轮转动一周时, 求小轮转动的角的大小;\\\\\n(2) 如果大轮的转速为$180$转/分, 小轮的半径为 $10.5$ 厘米, 那么小轮的圆周上一点每$1$秒转过的弧长是多少?", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -472911,7 +472941,9 @@ "id": "018330", "content": "已知角$\\alpha$的终边经过点$P(1,-2)$, 求角$\\alpha$的正弦、余弦、正切及余切值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -472931,7 +472963,9 @@ "id": "018331", "content": "已知角$\\alpha$的终边经过点$P(a,-2 a)$($a<0$), 求角$\\alpha$的正弦、余弦、正切及余切值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -472951,7 +472985,9 @@ "id": "018332", "content": "已知角$\\alpha$的终边经过点$P(-2,0)$, 求角$\\alpha$的正弦、余弦、正切及余切值. 填写下表.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline 角$\\alpha$& 0 &$\\dfrac{\\pi}{2}$&$\\pi$&$\\dfrac{3 \\pi}{2}$\\\\\n\\hline$\\sin \\alpha$& & & & \\\\\n\\hline$\\cos \\alpha$& & & & \\\\\n\\hline$\\tan \\alpha$& & & & \\\\\n\\hline$\\cot \\alpha$& & & & \\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -472971,7 +473007,9 @@ "id": "018333", "content": "若角$\\alpha$满足$\\sin \\alpha>0$, 且$\\tan \\alpha<0$, 则角$\\alpha$属于第几象限?", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -472991,7 +473029,9 @@ "id": "018334", "content": "已知角$\\alpha$的终边经过点$P(3 m,-4 m)$($m<0$), 求角$\\alpha$的正弦、正切值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473011,7 +473051,9 @@ "id": "018335", "content": "设$\\alpha$是三角形的一个内角, 在$\\sin \\alpha, \\cos \\alpha, \\tan \\alpha, \\cot \\dfrac{\\alpha}{2}$中, 有可能取负值的是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -473031,7 +473073,9 @@ "id": "018336", "content": "已知角$\\alpha$的终边在直线$y=-3 x$上, 求$\\sin \\alpha, \\cos \\alpha, \\tan \\alpha$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473051,7 +473095,9 @@ "id": "018337", "content": "已知$y=\\dfrac{\\sin \\alpha}{|\\sin \\alpha|}+\\dfrac{\\cos \\alpha}{|\\cos \\alpha|}+\\dfrac{\\tan \\alpha}{|\\tan \\alpha|}+\\dfrac{\\cot \\alpha}{|\\cot \\alpha|}$, 其中$\\alpha \\neq \\dfrac{k \\pi}{2}$($k \\in \\mathbf{Z}$), 请写出$y$所有可能的值.\n例1 求角$\\dfrac{5 \\pi}{4}$的正弦、余弦和正切值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473071,7 +473117,9 @@ "id": "018338", "content": "已知$\\sin \\alpha=\\dfrac{3}{5}$, 且$\\alpha$是第二象限的角.求$\\cos \\alpha, \\tan \\alpha$及$\\cot \\alpha$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473091,7 +473139,9 @@ "id": "018339", "content": "已知$\\tan \\alpha=-\\dfrac{5}{12}$, 求$\\sin \\alpha$、$\\cos \\alpha$及$\\cot \\alpha$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473111,7 +473161,9 @@ "id": "018340", "content": "已知$\\tan \\alpha=m$($m \\neq 0$), 且$\\alpha \\in(\\dfrac{\\pi}{2}, \\dfrac{3 \\pi}{2})$, 用$m$表示$\\sin \\alpha$、$\\cos \\alpha$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473131,7 +473183,9 @@ "id": "018341", "content": "已知$\\cos \\alpha=-\\dfrac{2 \\sqrt{2}}{3}$, 且$\\sin \\alpha>0$, 求$\\tan \\alpha+\\cot \\alpha$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473151,7 +473205,9 @@ "id": "018342", "content": "已知$\\tan \\alpha=2$, 且$\\alpha \\in(2 \\pi, \\dfrac{5 \\pi}{2})$, 求$\\sin \\alpha+\\cos \\alpha$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473171,7 +473227,9 @@ "id": "018343", "content": "已知$\\dfrac{4}{\\cos \\alpha}+\\tan \\alpha=8$, 求$\\sin \\alpha$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473191,7 +473249,9 @@ "id": "018344", "content": "已知$\\sin \\theta=\\dfrac{1-a}{1+a}$, $\\cos \\theta=\\dfrac{3 a-1}{1+a}$, 且$\\theta \\in(\\dfrac{3 \\pi}{2}, \\dfrac{5 \\pi}{2})$, 求$\\theta$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473211,7 +473271,9 @@ "id": "018345", "content": "已知$\\tan \\alpha=m$, 用$m$表示$\\sin \\alpha$、$\\cos \\alpha$及$\\cot \\alpha$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473231,7 +473293,9 @@ "id": "018346", "content": "已知$\\sin \\alpha+\\cos \\alpha=\\dfrac{1}{5}$, 求$\\sin \\alpha \\cos \\alpha$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473251,7 +473315,9 @@ "id": "018347", "content": "已知$\\sin \\alpha+\\cos \\alpha=\\dfrac{1}{5}$, 且$\\alpha \\in(0, \\pi)$, 求$\\sin \\alpha-\\cos \\alpha$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473271,7 +473337,9 @@ "id": "018348", "content": "已知$\\sin \\alpha+\\cos \\alpha=\\dfrac{1}{5}$, 求$\\tan \\alpha+\\cot \\alpha$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473291,7 +473359,9 @@ "id": "018349", "content": "已知$\\tan \\alpha=\\dfrac{1}{2}$, 求$\\sin ^2 \\alpha-\\sin \\alpha \\cos \\alpha-\\cos ^2 \\alpha$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473311,7 +473381,9 @@ "id": "018350", "content": "已知$\\tan \\alpha=\\dfrac{1}{2}$, 求$\\dfrac{\\sin ^2 \\alpha-\\sin \\alpha \\cos \\alpha}{\\sin ^2 \\alpha-2}$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473331,7 +473403,9 @@ "id": "018351", "content": "证明下列恒等式:\\\\\n(1) $1+\\tan ^2 \\alpha=\\dfrac{1}{\\cos ^2 \\alpha}$;\\\\\n(2) $1+\\cot ^2 \\alpha=\\dfrac{1}{\\sin ^2 \\alpha}$;\\\\\n(3) $\\dfrac{1+\\cos \\alpha}{\\sin \\alpha}=\\dfrac{\\sin \\alpha}{1-\\cos \\alpha}$;\\\\\n(4) $\\dfrac{\\sin ^2 \\alpha-\\sin ^2 \\beta}{\\tan ^2 \\alpha-\\tan ^2 \\beta}=\\cos ^2 \\alpha \\cos ^2 \\beta$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473351,7 +473425,9 @@ "id": "018352", "content": "已知$\\sin \\alpha+\\cos \\alpha=\\dfrac{7}{13}$, $\\alpha \\in(0, \\pi)$. 求$\\sin \\alpha$和$\\cos \\alpha$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473371,7 +473447,9 @@ "id": "018353", "content": "已知$\\sin \\alpha-\\cos \\alpha=\\dfrac{7}{5}$, $\\alpha \\in(0, \\pi)$, 求$\\sin ^3 \\alpha+\\cos ^3 \\alpha$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473391,7 +473469,9 @@ "id": "018354", "content": "证明: $\\dfrac{\\tan \\alpha \\cdot \\sin \\alpha}{\\tan \\alpha-\\sin \\alpha}=\\dfrac{\\tan \\alpha+\\sin \\alpha}{\\tan \\alpha \\cdot \\sin \\alpha}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473411,7 +473491,9 @@ "id": "018355", "content": "利用诱导公式求值:\\\\\n(1) $\\sin \\dfrac{20}{3} \\pi$;\\\\\n(2) $\\cos (-\\dfrac{7}{6} \\pi)$;\\\\\n(3) $\\tan (-\\dfrac{19}{4} \\pi)$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473431,7 +473513,9 @@ "id": "018356", "content": "化简: $\\dfrac{\\sin (2 \\pi-\\alpha) \\tan (\\pi+\\alpha) \\cot (-\\pi-\\alpha)}{\\cos (\\pi-\\alpha) \\tan (3 \\pi-\\alpha)}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473451,7 +473535,9 @@ "id": "018357", "content": "化简:\\\\\n(1) $\\dfrac{\\sin (-\\alpha)}{\\sin (\\pi+\\alpha)}-\\dfrac{\\cot (\\pi-\\alpha)}{\\cot (\\pi+\\alpha)}+\\dfrac{\\cos (2 \\pi-\\alpha)}{\\cos (3 \\pi-\\alpha)}$;\\\\\n(2) $\\dfrac{\\sin (\\pi-\\alpha)}{\\tan (4 \\pi+\\alpha)} \\cdot \\dfrac{\\tan (2 \\pi-\\alpha)}{\\cos (\\pi-\\alpha)} \\cdot \\dfrac{\\cos (-\\alpha)}{\\sin (\\pi+\\alpha)}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473471,7 +473557,9 @@ "id": "018358", "content": "求$\\cos 1^{\\circ}+\\cos 2^{\\circ}+\\cdots+\\cos 179^{\\circ}$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473491,7 +473579,9 @@ "id": "018359", "content": "已知$\\alpha \\neq \\dfrac{n \\pi}{2}$, $n \\in \\mathbf{Z}$, 化简: $\\dfrac{\\sin (k \\pi-\\alpha)}{\\sin (k \\pi+\\alpha)}+\\dfrac{\\cos (k \\pi-\\alpha)}{\\cos (k \\pi+\\alpha)}+\\dfrac{\\tan (k \\pi-\\alpha)}{\\tan (k \\pi+\\alpha)}$, 其中$k \\in \\mathbf{Z}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473511,7 +473601,9 @@ "id": "018360", "content": "证明:\\\\\n(1) $\\sin (\\dfrac{3 \\pi}{2}+\\alpha)=-\\cos \\alpha$;\\\\\n(2) $\\cos (\\dfrac{3 \\pi}{2}+\\alpha)=\\sin \\alpha$;\\\\\n(3) $\\tan (\\dfrac{3 \\pi}{2}+\\alpha)=-\\cot \\alpha$;\\\\\n(4) $\\cot (\\dfrac{3 \\pi}{2}+\\alpha)=-\\tan \\alpha$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473531,7 +473623,9 @@ "id": "018361", "content": "化简: $\\dfrac{\\sin (\\dfrac{\\pi}{2}+\\alpha) \\cos (\\dfrac{\\pi}{2}+\\alpha) \\sin (\\dfrac{\\pi}{2}-\\alpha)}{\\tan (\\dfrac{\\pi}{2}+\\alpha) \\cos (\\dfrac{3 \\pi}{2}+\\alpha) \\sin (-\\pi+\\alpha)}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473551,7 +473645,9 @@ "id": "018362", "content": "已知点$A$的坐标为$(-\\dfrac{3}{5}, \\dfrac{4}{5})$, 将$OA$绕坐标原点$O$逆时针旋转$\\dfrac{\\pi}{2}$至$OA'$. 求点$A'$的坐标.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473571,7 +473667,9 @@ "id": "018363", "content": "已知点$A$的坐标为$(1,1)$, 将$OA$绕坐标原点$O$逆时针旋转$\\dfrac{3 \\pi}{2}$至$OA'$. 求点$A'$的坐标.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473591,7 +473689,9 @@ "id": "018364", "content": "已知$\\alpha$为钝角, 且$\\sin (\\dfrac{\\pi}{4}+\\alpha)=\\dfrac{3}{4}$, 求$\\sin (\\dfrac{\\pi}{4}-\\alpha)$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473611,7 +473711,9 @@ "id": "018365", "content": "化简下列各式, 其中$k \\in \\mathbf{Z}$:\\\\\n(1) $\\sin (\\dfrac{k \\pi}{2}+\\alpha)$;\\\\\n(2) $\\cos (\\dfrac{k \\pi}{2}-\\alpha)$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473631,7 +473733,9 @@ "id": "018366", "content": "根据下列条件, 分别求角$x$:\\\\\n(1) 已知$\\sin x=\\dfrac{\\sqrt{3}}{2}$;\\\\\n(2) 已知$\\cos x=-\\dfrac{\\sqrt{2}}{2}$;\\\\\n(3) 已知$\\tan x=\\dfrac{\\sqrt{3}}{3}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473651,7 +473755,9 @@ "id": "018367", "content": "分别求满足下列条件的角$x$的集合:\n(1) $\\sin 2 x=\\dfrac{\\sqrt{3}}{2}$, $x \\in[0,2 \\pi]$;\\\\\n(2) $\\cos (x+\\dfrac{\\pi}{6})=\\dfrac{\\sqrt{2}}{2}$;\\\\\n(3) $\\tan (2 x+\\dfrac{\\pi}{3})=-\\dfrac{\\sqrt{3}}{3}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473671,7 +473777,9 @@ "id": "018368", "content": "已知角$x$满足$\\sin x=\\dfrac{\\sqrt{2}}{2}$.\\\\\n(1) 若$x \\in(0, \\dfrac{\\pi}{2})$, 则$x=$\\blank{50};\\\\\n(2) 若$x \\in(0, \\pi)$, 则$x=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -473691,7 +473799,9 @@ "id": "018369", "content": "根据下列条件, 分别求角$x$:\\\\\n(1) $\\sin ^2 x-2 \\sin x-3=0$;\\\\\n(2) $2 \\sqrt{3} \\sin ^2 x=\\cos x$;\\\\\n(3) $\\sin x=\\cos x$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473711,7 +473821,9 @@ "id": "018370", "content": "满足$\\cos (\\pi \\cos x)=0$, $x \\in[0, \\dfrac{3 \\pi}{2}]$的角$x$组成的集合为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -473731,7 +473843,9 @@ "id": "018371", "content": "满足$\\sin 5 x=\\cos x, x \\in(0, \\pi)$的角$x$的个数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -473751,7 +473865,9 @@ "id": "018372", "content": "利用两角和与差的余弦公式, 求$\\cos 75^{\\circ}$和$\\cos 15^{\\circ}$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473771,7 +473887,9 @@ "id": "018373", "content": "已知$\\sin \\alpha=\\dfrac{3}{5}$, $\\alpha \\in(\\dfrac{\\pi}{2}, \\pi)$, $\\cos \\beta=\\dfrac{5}{13}$, $\\beta \\in(\\dfrac{3 \\pi}{2}, 2 \\pi)$, 求$\\cos (\\alpha-\\beta)$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473791,7 +473909,9 @@ "id": "018374", "content": "化简:\\\\\n(1) $\\cos 72^{\\circ} \\cos 18^{\\circ}-\\sin 72^{\\circ} \\sin 18^{\\circ}$;\\\\\n(2) $\\cos (\\alpha-\\dfrac{\\pi}{4}) \\cos \\alpha+\\sin (\\alpha-\\dfrac{\\pi}{4}) \\sin \\alpha$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473811,7 +473931,9 @@ "id": "018375", "content": "若$\\alpha$、$\\beta$为锐角, $\\sin \\alpha=\\dfrac{4 \\sqrt{3}}{7}$, $\\cos (\\alpha+\\beta)=-\\dfrac{11}{14}$, 求角$\\beta$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473831,7 +473953,9 @@ "id": "018376", "content": "已知$\\cos (\\theta-\\dfrac{\\pi}{4})=-\\dfrac{\\sqrt{2}}{10}$, 且$\\theta$是第二象限的角. 求$\\cos \\theta$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473851,7 +473975,9 @@ "id": "018377", "content": "化简:\\\\\n(1) $\\cos ^215^{\\circ}-\\sin ^215^{\\circ}$;\\\\\n(2) $\\cos ^2 \\alpha-\\sin ^2 \\alpha$;\\\\\n(3) $\\sin ^2 \\dfrac{3}{2} x-\\cos ^2 \\dfrac{3}{2} x$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473871,7 +473997,9 @@ "id": "018378", "content": "利用两角差的正弦公式, 求$\\sin 15^{\\circ}$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473891,7 +474019,9 @@ "id": "018379", "content": "证明: $\\sin (\\alpha+\\beta) \\sin (\\alpha-\\beta)=\\sin ^2 \\alpha-\\sin ^2 \\beta$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473911,7 +474041,9 @@ "id": "018380", "content": "已知$\\tan \\alpha=\\dfrac{1}{3}$, $\\tan \\beta=-2$. 求:\\\\\n(1) $\\tan (\\alpha+\\beta)$;\\\\\n(2) $\\cot (\\alpha-\\beta)$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473931,7 +474063,9 @@ "id": "018381", "content": "利用两角和的正切公式, 求$\\dfrac{1+\\tan 75^{\\circ}}{1-\\tan 75^{\\circ}}$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473951,7 +474085,9 @@ "id": "018382", "content": "已知$\\tan \\alpha, \\tan \\beta$是方程$4 x^2-3 \\sqrt{3} x+1=0$的两根, 若$\\alpha, \\beta \\in(-\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2})$, 求$\\alpha+\\beta$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473971,7 +474107,9 @@ "id": "018383", "content": "求证: $\\tan 1^{\\circ}$不是有理数.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -473991,7 +474129,9 @@ "id": "018384", "content": "不用计算器, 求$\\tan 20^{\\circ}+\\tan 40^{\\circ}+\\sqrt{3} \\tan 20^{\\circ} \\tan 40^{\\circ}$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474011,7 +474151,9 @@ "id": "018385", "content": "若$\\triangle ABC$不是直角三角形, 求证: $\\tan A+\\tan B+\\tan C=\\tan A \\tan B \\tan C$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474031,7 +474173,9 @@ "id": "018386", "content": "如图, 已知点$A$的坐标为$(1,2)$, 将$OA$绕坐标原点$O$逆时针旋转$\\dfrac{\\pi}{4}$至$OA'$. 求点$A'$的坐标.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-3, 0) -- (3, 0) node [below] {$x$};\n\\draw [->] (0, -1) -- (0, 3) node [left] {$y$};\n\\draw (0, 0) node [below left] {$O$} coordinate (O);\n\\filldraw (1, 2) circle (0.05) node [right] {$A$} coordinate (A);\n\\filldraw ($(O)!1!45: (A)$) circle (0.05) node [left] {$A'$} coordinate (A');\n\\draw (A)--(O)--(A');\n\\draw (O) pic [draw, ->, scale = 0.6, \"$\\alpha$\", angle eccentricity = 1.8] {angle = A--O--A'};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474051,7 +474195,9 @@ "id": "018387", "content": "把下列各式化为$A \\sin (\\alpha+\\varphi)$($A>0$)的形式:\\\\\n(1) $\\dfrac{1}{2} \\sin \\alpha+\\dfrac{\\sqrt{3}}{2} \\cos \\alpha$;\\\\\n(2) $\\sin \\alpha-\\cos \\alpha$;\\\\\n(3) $3 \\sin \\alpha+4 \\cos \\alpha$;\\\\\n(4) $a \\sin \\alpha+b \\cos \\alpha$($a b \\neq 0$).", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474071,7 +474217,9 @@ "id": "018388", "content": "若存在角$\\alpha$, 使得$\\sqrt{3} \\sin \\alpha+\\cos \\alpha=4-m$, 求实数$m$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474091,7 +474239,9 @@ "id": "018389", "content": "$A, B, C$为$\\triangle ABC$的内角, $\\triangle ABC$不为直角三角形. 若$\\sqrt{3} \\tan C-1=\\dfrac{\\tan B+\\tan C}{\\tan A}$, 求角$B$的大小.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474111,7 +474261,9 @@ "id": "018390", "content": "已知角$\\alpha$的终边经过点$A(\\dfrac{\\sqrt{3}}{2}, \\dfrac{1}{2})$, 将$OA$绕坐标原点$O$顺时针旋转$\\dfrac{3 \\pi}{4}$至$OB$, 若角$\\beta$的终边经过点$B$, 求$\\cos (\\alpha+\\beta)$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474131,7 +474283,9 @@ "id": "018391", "content": "已知$\\alpha$、$\\beta$为任意角, ``$\\sin \\alpha \\sin \\beta+\\cos \\alpha \\cos \\beta=0$''是``$\\sin \\alpha \\cos \\beta-\\cos \\alpha \\sin \\beta=1$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -474151,7 +474305,9 @@ "id": "018392", "content": "已知角$\\alpha$的终边经过点$P(1-\\tan \\dfrac{\\pi}{12}, 1+\\tan \\dfrac{\\pi}{12})$, 且$0<\\alpha<2 \\pi$, 求角$\\alpha$的大小.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474171,7 +474327,9 @@ "id": "018393", "content": "已知$\\sin \\alpha=\\dfrac{4}{5}, \\alpha \\in(\\dfrac{\\pi}{2}, \\pi)$. 求$\\sin 2 \\alpha$、$\\cos 2 \\alpha$和$\\tan 2 \\alpha$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474191,7 +474349,9 @@ "id": "018394", "content": "已知$\\sin \\alpha=\\dfrac{4}{5}, \\alpha \\in(\\dfrac{\\pi}{2}, \\pi)$. 求$\\sin 4 \\alpha$、$\\cos 4 \\alpha$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474211,7 +474371,9 @@ "id": "018395", "content": "试用$\\cos \\theta$表示$\\cos 3 \\theta$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474231,7 +474393,9 @@ "id": "018396", "content": "证明:\\\\\n(1) $2 \\cos ^2 \\theta+2 \\sin \\theta \\cos \\theta-1=\\sqrt{2} \\sin (2 \\theta+\\dfrac{\\pi}{4})$;\\\\\n(2) $\\dfrac{1+\\sin 2 \\theta+\\cos 2 \\theta}{1+\\sin 2 \\theta-\\cos 2 \\theta}=\\cot \\theta$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474251,7 +474415,9 @@ "id": "018397", "content": "若$\\sin \\theta=\\dfrac{3}{5}, \\cos \\theta=-\\dfrac{4}{5}$, 则角$2 \\theta$的终边在第\\blank{50}象限.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -474271,7 +474437,9 @@ "id": "018398", "content": "已知$\\sin (\\dfrac{\\pi}{4}-x)=\\dfrac{3}{5}$, 求$\\sin 2 x$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474291,7 +474459,9 @@ "id": "018399", "content": "已知$\\alpha \\in(\\dfrac{3 \\pi}{2}, \\dfrac{5 \\pi}{2})$, 则$\\sqrt{\\dfrac{1}{2}+\\dfrac{1}{2} \\cos \\alpha}$等于\\bracket{20}.\n\\fourch{$\\sin \\dfrac{\\alpha}{2}$}{$\\cos \\dfrac{\\alpha}{2}$}{$-\\sin \\dfrac{\\alpha}{2}$}{$-\\cos \\dfrac{\\alpha}{2}$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -474311,7 +474481,9 @@ "id": "018400", "content": "如图是来自古希腊数学家希波克拉底所研究的几何图形, 此图由三个半圆构成, 三个半圆的直径分别为直角三角形$ABC$的斜边$BC$, 直角边$AB, AC$. 已知以直角边$AC, AB$为直径的半圆的面积之比为$\\dfrac{1}{4}$, 记$\\angle ABC=\\alpha$, 则$\\sin 2 \\alpha=$\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw (0, 0) node [below] {$B$} coordinate (B);\n\\draw (5, 0) node [below] {$C$} coordinate (C);\n\\draw (4, 2) node [above] {$A$} coordinate (A);\n\\draw (C) arc (0: 180: 2.5);\n\\draw (C) arc ({-atan(2)}: {180-atan(2)}: {sqrt(5)/2});\n\\draw (B) arc ({180+atan(0.5)}: {atan(0.5)}: {sqrt(5)});\n\\draw (A)--(B)--(C)--cycle;\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -474331,7 +474503,9 @@ "id": "018401", "content": "用$\\cos \\alpha$分别表示$\\cos ^2 \\dfrac{\\alpha}{2}, \\sin ^2 \\dfrac{\\alpha}{2}$及$\\tan ^2 \\dfrac{\\alpha}{2}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474351,7 +474525,9 @@ "id": "018402", "content": "证明: $\\tan \\dfrac{\\alpha}{2}=\\dfrac{\\sin \\alpha}{1+\\cos \\alpha}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474371,7 +474547,9 @@ "id": "018403", "content": "证明: $\\sin \\alpha \\cos \\beta=\\dfrac{1}{2}[\\sin (\\alpha+\\beta)+\\sin (\\alpha-\\beta)]$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474391,7 +474569,9 @@ "id": "018404", "content": "证明: $\\sin \\alpha+\\sin \\beta=2 \\sin \\dfrac{\\alpha+\\beta}{2} \\cos \\dfrac{\\alpha-\\beta}{2}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474411,7 +474591,9 @@ "id": "018405", "content": "证明下列恒等式:\\\\\n(1) $\\sin \\alpha=\\dfrac{2 \\tan \\dfrac{\\alpha}{2}}{1+\\tan ^2 \\dfrac{\\alpha}{2}}$;\\\\\n(2) $\\cos \\alpha=\\dfrac{1-\\tan ^2 \\dfrac{\\alpha}{2}}{1+\\tan ^2 \\dfrac{\\alpha}{2}}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474431,7 +474613,9 @@ "id": "018406", "content": "证明下列恒等式:\\\\\n(1) $\\dfrac{1-\\cos 4 \\alpha}{\\sin 4 \\alpha} \\cdot \\dfrac{\\cos 2 \\alpha}{1+\\cos 2 \\alpha}=\\tan \\alpha$;\\\\\n(2) $\\tan \\dfrac{3 \\alpha}{2}-\\tan \\dfrac{\\alpha}{2}=\\dfrac{2 \\sin \\alpha}{\\cos \\alpha+\\cos 2 \\alpha}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474451,7 +474635,9 @@ "id": "018407", "content": "已知$2 \\sin \\alpha=\\sin \\theta+\\cos \\theta$, $\\sin ^2 \\beta=\\sin \\theta \\cos \\theta$, 求证: $2 \\cos 2 \\alpha=\\cos 2 \\beta$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474471,7 +474657,9 @@ "id": "018408", "content": "某同学在一次研究性学习中发现, 以下五个式子的值都等于同一个常数:\\\\\n\\textcircled{1} $\\sin ^213^{\\circ}+\\cos ^217^{\\circ}-\\sin 13^{\\circ} \\cos 17^{\\circ}$;\n\\textcircled{2} $\\sin ^215^{\\circ}+\\cos ^215^{\\circ}-\\sin 15^{\\circ} \\cos 15^{\\circ}$;\n\\textcircled{3} $\\sin ^218^{\\circ}+\\cos ^212^{\\circ}-\\sin 18^{\\circ} \\cos 12^{\\circ}$;\n\\textcircled{4} $\\sin ^2(-18^{\\circ})+\\cos ^248^{\\circ}-\\sin (-18^{\\circ}) \\cos 48^{\\circ}$;\n\\textcircled{5} $\\sin ^2(-25^{\\circ})+\\cos ^255^{\\circ}-\\sin (-25^{\\circ}) \\cos 55^{\\circ}$.\\\\\n(1) 试从上述五个式子中选择一个, 求出这个常数;\\\\\n(2) 根据 (1) 的计算结果, 将该同学的发现推广为三角恒等式, 并证明你的结论.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474491,7 +474679,9 @@ "id": "018409", "content": "在$\\triangle ABC$中, 已知$\\angle CAB=130^{\\circ}$, $\\angle CBA=30^{\\circ}$, $AB=10 \\text{km}$. 求$AC$与$BC$的长. (结果精确到$0.1 \\text{km}$)", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474511,7 +474701,9 @@ "id": "018410", "content": "已知圆$O$是$\\triangle ABC$的外接圆, 其圆心为$O$, 直径为$2R$. 试用$R$与角$A$、$B$及$C$的正弦来表示三角形的边长$a$、$b$及$c$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474531,7 +474723,9 @@ "id": "018411", "content": "设$R$是$\\triangle ABC$的外接圆的半径, $S$为$\\triangle ABC$的面积. 求证:\\\\\n(1) $S=\\dfrac{a b c}{4R}$;\\\\\n(2) $S=2R^2 \\sin A \\sin B \\sin C$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474551,7 +474745,9 @@ "id": "018412", "content": "在$\\triangle ABC$中, 角$A$、$B$、$C$所对边的边长分别记作$a$、$b$、$c$. 若$b=5, B=\\dfrac{\\pi}{4}$, $\\sin A=\\dfrac{1}{3}$, 求$c$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474571,7 +474767,9 @@ "id": "018413", "content": "在$\\triangle ABC$中, 角$A$、$B$、$C$所对边的边长分别记作$a$、$b$、$c$. 已知$A=\\dfrac{\\pi}{4}$, $b \\sin (\\dfrac{\\pi}{4}+C)-c \\sin (\\dfrac{\\pi}{4}+B)=a$.\\\\\n(1) 求$B-C$;\\\\\n(2) 若$a=\\sqrt{2}$, 求$\\triangle ABC$的面积.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474591,7 +474789,9 @@ "id": "018414", "content": "在$\\triangle ABC$中, 已知$a=\\sqrt{6}$, $b=\\sqrt{3}+1$, $C=45^{\\circ}$. 求$c$、$A$及$B$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474611,7 +474811,9 @@ "id": "018415", "content": "在$\\triangle ABC$中, 已知$a=2$, $b=2 \\sqrt{3}$, $A=30^{\\circ}$, 求$B$、$C$及$c$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474631,7 +474833,9 @@ "id": "018416", "content": "在$\\triangle ABC$中, 已知$a=4$, $b=5$, $c=6$. 求角$A$的余弦值和$\\triangle ABC$的面积$S$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474651,7 +474855,9 @@ "id": "018417", "content": "已知三角形的三边长是三个连续的正整数.\\\\\n(1) 若此三角形是钝角三角形, 求三边的长;\\\\\n(2) 若此三角形最大角是最小角的两倍, 求三边的长.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474671,7 +474877,9 @@ "id": "018418", "content": "在$\\triangle ABC$中, 已知$b^2+c^2-b c=a^2$, 且$\\dfrac{b}{c}=\\dfrac{\\tan B}{\\tan C}$. 求证: $\\triangle ABC$为等边三角形.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474691,7 +474899,9 @@ "id": "018419", "content": "在$\\triangle ABC$中, 已知$b^2+c^2-b c=a^2$, 且$\\dfrac{b}{c}=\\dfrac{\\cos B}{\\cos C}$, 请判断$\\triangle ABC$的形状.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474711,7 +474921,9 @@ "id": "018420", "content": "在$\\triangle ABC$中, 已知$a=5$, $b=4$, 三角形的面积$S=8$. 求$c$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474731,7 +474943,9 @@ "id": "018421", "content": "根据下列条件, 分别求角$x$:\\\\\n(1) 已知$\\sin x=\\dfrac{1}{3}$;\\\\\n(2) 已知$\\cos x=-\\dfrac{3}{5}$, $x \\in[0, \\pi]$;\\\\\n(3) 已知$\\tan x=-3$, $x \\in(\\dfrac{\\pi}{2}, \\dfrac{3 \\pi}{2})$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474751,7 +474965,9 @@ "id": "018422", "content": "已知$\\cos 2 \\alpha=\\dfrac{7}{25}$, $\\alpha \\in(0, \\dfrac{\\pi}{2})$, $\\sin \\beta=-\\dfrac{5}{13}$, $\\beta \\in(\\pi, \\dfrac{3 \\pi}{2})$, 求$\\alpha+\\beta$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474771,7 +474987,9 @@ "id": "018423", "content": "在$\\triangle ABC$中, 已知$a \\sin A+b(\\sin A+\\sin B)-c \\sin C=0$.\\\\ \n(1) 求$C$;\\\\\n(2) 若$c=2$, 求$a+b$的最大值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474791,7 +475009,9 @@ "id": "018424", "content": "在$\\triangle ABC$中, 已知$\\dfrac{a}{\\cos A}=\\dfrac{b}{\\cos B}=\\dfrac{c}{\\cos C}$, 判断这个三角形的形状.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474811,7 +475031,9 @@ "id": "018425", "content": "金茂大厦是改革开放以来上海出现的超高层标志性建筑. 有一位测量爱好者在与金茂大厦底部同一水平线上的$B$处测得金茂大厦顶部$A$的仰角为$15.66^{\\circ}$, 再向金茂大厦前进$500 \\mathrm{m}$到达$C$处, 测得金茂大厦顶部$A$的仰角为$22.81^{\\circ}$. 请根据以上数据估算出金茂大厦的高度. (结果精确到$1 \\mathrm{m}$)", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474831,7 +475053,9 @@ "id": "018426", "content": "甲船在距离$A$港口$24$海里并在南偏西$20^{\\circ}$方向的$C$处驻留等候进港, 乙船在$A$港口南偏东$40^{\\circ}$方向的$B$处沿直线行驶入港, 甲、乙两船距离为$31$海里. 当乙船行驶$20$海里到达$D$处时, 接到港口指令, 前往救援忽然发生火灾的甲船. 求此时甲、乙两船之间的距离.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474851,7 +475075,9 @@ "id": "018427", "content": "甲船在距离$A$港口$24$海里并在南偏西$20^{\\circ}$方向的$C$处驻留等候进港, 乙船在$A$港口南偏东$40^{\\circ}$方向的$B$处沿直线行驶入港, 甲、乙两船距离为$31$海里. 当乙船行驶$20$海里到达$D$处时, 接到港口指令, 前往救援忽然发生火灾的甲船. 当乙船接到命令前往救援时, 应按何方向行驶可最快抵达甲船处?", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474871,7 +475097,9 @@ "id": "018428", "content": "如图, 有两条相交成$60^{\\circ}$角的直路$l_1, l_2$, 交点是$O$, 警务岗$A$、$B$分别在$l_1, l_2$上, 警务岗$A$离$O$点$1$千米, 警务岗$B$离$O$点$3$千米. 若警员甲、乙分别从$A$、$B$同时出发, 甲沿$AC$方向, 乙沿$BO$方向, 均以$4$千米/小时的速度沿途巡逻. \n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (-1, 0) -- (4, 0) node [below] {$l_2$};\n\\draw (60: -1)-- (60: 2.5) node [right] {$l_1$};\n\\filldraw (0, 0) circle (0.03) node [below] {$O$} coordinate (O);\n\\filldraw (3, 0) circle (0.03) node [below] {$B$} coordinate (B);\n\\filldraw (60: {8/(3+sqrt(3))}) circle (0.03) node [above left] {$C$} coordinate (C);\n\\filldraw (60: 1) circle (0.03) node [above left] {$A$} coordinate (A);\n\\draw (O) pic [draw, \"$60^\\circ$\", scale = 0.5, angle eccentricity = 2.5] {angle = B--O--A};\n\\end{tikzpicture}\n\\end{center}\n(1) 当警员甲行至点$C$处时, $\\angle OBC=45^{\\circ}$, 求$OC$之间的距离;\\\\\n(2) 设$t$小时后甲、乙两人之间的距离为$S$千米, 试将$S$表示为$t$的函数.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474891,7 +475119,9 @@ "id": "018429", "content": "化简$\\dfrac{\\sin (\\theta-5 \\pi)}{\\tan (3 \\pi-\\theta)} \\cdot \\dfrac{\\cot (\\dfrac{\\pi}{2}-\\theta)}{\\tan (\\theta-\\dfrac{3 \\pi}{2})} \\cdot \\dfrac{\\cos (8 \\pi-\\theta)}{\\cos (\\theta+\\dfrac{\\pi}{2})}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474911,7 +475141,9 @@ "id": "018430", "content": "若$\\cos (\\dfrac{\\pi}{4}+x)=\\dfrac{3}{5}, \\pi=latex, scale = 0.2]\n\\draw (0, 0) node [above left] {$A$} coordinate (A);\n\\draw (-11: 3) node [below] {$B$} coordinate (B);\n\\draw (43: {3/sin(11)*sin(115)}) node [above right] {$C$} coordinate (C);\n\\foreach \\i in {A, C}\n{\\draw [dashed] (\\i) ++ (-2, 0) --++ (4, 0) ++ (-2, -2) --++ (0, 4) node [above] {北};};\n\\draw (C) ++ (10, 0) node [above] {$P$} coordinate (P);\n\\draw (A)--(B)--(C)--cycle(C)--(P);\n\\end{tikzpicture}\n\\end{center}\n(1) 求$A$、$C$两点间的距离; (结果精确到 $0.01$ 海里)\\\\\n(2) 某时刻, 我国一渔船在点$A$处因故障抛针发出求救信号.一艘 R 国舰艇正从点$C$正东$10$海里的点$P$处以$18$海里/小时的速度接近渔船, 其航线为$P \\to C \\to A$(沿直线行进), 而我国的救援船位于点$A$南偏西$60^{\\circ}$方向$20$海里的点$Q$处, 收到信号后赶往救助, 其航线为先向正北航行$8$海里至点$M$处, 再折向点$A$沿直线航行, 航速为$22$海里/小时. 问救援船能否先于 R 国舰艇赶到进行救助? 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474951,7 +475185,9 @@ "id": "018432", "content": "至少用两种方法求$\\sin ^2 \\alpha \\cdot \\sin ^2 \\beta+\\cos ^2 \\alpha \\cdot \\cos ^2 \\beta-\\dfrac{1}{2} \\cos 2 \\alpha \\cos 2 \\beta$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474971,7 +475207,9 @@ "id": "018433", "content": "如图, 某段海岸线可近似看作一条曲线, 该曲线由线段$AB$和四分之一圆弧$\\overset\\frown{BC}$构成, $D$为一海岛, $B$在$D$的正北方向, 且$B$、$D$相距 $39.2$ 千米, $A$在$D$的北偏西$58^{\\circ}$方向, $C$在$D$的北偏东$22^{\\circ}$方向, $C$在$B$的南偏东$68^{\\circ}$方向.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.06]\n\\draw (0, 0) node [above] {$B$} coordinate (B);\n\\draw (B) ++ (0, -39.2) node [below] {$D$} coordinate (D);\n\\draw (B) ++ (-22: {39.2*sin(22)}) node [right] {$C$} coordinate (C);\n\\draw (B) ++ ({270-(180-58-asin(39.2/40*sin(58)))}: 40) node [left] {$A$} coordinate (A);\n\\draw (A)-- (B) arc (113: 23: {39.2/2*sin(22)*sqrt(2)});\n\\draw [dashed] (A)--(D)--(C)--(B)--(D);\n\\draw pic [draw, scale = 0.9, \"$22^\\circ$\", angle eccentricity = 2.5] {angle = C--D--B};\n\\draw pic [draw, scale = 0.6, \"$58^\\circ$\", angle eccentricity = 1.8] {angle = B--D--A};\n\\draw pic [draw, scale = 0.6, \"$68^\\circ$\", angle eccentricity = 1.8] {angle = D--B--C};\n\\end{tikzpicture}\n\\end{center}\n(1) 若沿$\\overset\\frown{BC}$修建观光道, 计算该观光道$\\overset\\frown{BC}$的长度(精确到 $0.001$ 千米);\\\\\n(2) 现规划在该海岸线上选取一处$E$, 修建从$E$直通$D$的公路桥. 已知$A$、$B$相距$40$千米, 求公路桥$DE$的最短长度 (精确到 $0.001$ 千米).", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -474991,7 +475229,9 @@ "id": "018434", "content": "用``五点法''作出函数$y=1-\\sin x$, $x \\in[0,2 \\pi]$的大致图像, 并写出使得$y<1$的$x$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -475011,7 +475251,9 @@ "id": "018435", "content": "作出函数$y=2-\\sin x$, $x \\in[-\\dfrac{\\pi}{2}, \\dfrac{3 \\pi}{2}]$的大致图像.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -475031,7 +475273,9 @@ "id": "018436", "content": "设$a$为常数, 若满足$a=\\sin x-\\dfrac{1}{2}$且$x \\in[-\\pi, \\pi]$的$x$的值只有一个, 则实数$a$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -475051,7 +475295,9 @@ "id": "018437", "content": "满足$2 \\sin |x|-\\sqrt{3}>0$且$x \\in[-2 \\pi, 2 \\pi]$的$x$的取值集合为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -475071,7 +475317,9 @@ "id": "018438", "content": "若函数$y=\\sin x+2|\\sin x|, x \\in[0,2 \\pi]$的图像与直线$y=k$有且只有两个公共点, 则实数$k$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -475091,7 +475339,9 @@ "id": "018439", "content": "求下列函数$y=f(x)$的最小正周期:\\\\\n(1) $f(x)=\\sin 3 x$;\\\\\n(2) $f(x)=2 \\sin (-\\dfrac{1}{2} x+\\dfrac{\\pi}{3})$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -475111,7 +475361,9 @@ "id": "018440", "content": "已知函数$y=\\sin (k x+\\dfrac{\\pi}{3}$) (其中常数$k \\neq 0$) 的最小正周期是 $2$ , 求$k$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -475131,7 +475383,9 @@ "id": "018441", "content": "函数$y=|\\sin x|$的最小正周期为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -475151,7 +475405,9 @@ "id": "018442", "content": "已知$f(n)=\\sin \\dfrac{\\pi n}{6}$, $n \\in \\mathbf{N}$, 求$f(1)+f(2)+f(3)+\\cdots+f(2021)$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -475171,7 +475427,9 @@ "id": "018443", "content": "求下列函数的最大值和最小值, 并求出取得最大值和最小值时所有$x$的值:\\\\\n(1) $y=-2 \\sin (3 x+\\dfrac{\\pi}{3})$;\\\\\n(2) $y=\\sin x+\\sqrt{3} \\cos x$;\\\\\n(3) $y=\\sin ^2 x-\\sin x$;\\\\\n(4) $y=\\sin ^2 x+2 \\sqrt{3} \\sin x \\cos x+3 \\cos ^2 x$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -475191,7 +475449,9 @@ "id": "018444", "content": "如图所示, 在一个半径为$r$的半圆形铁板中, 截取一块矩形$ABCD$, 使得矩形的顶点$A$、$B$在半圆的直径上, $C$、$D$在半圆弧上. 问: 如何截取矩形$ABCD$, 使其面积达到最大值? 并求出这个最大值.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (-2,0) coordinate (S) (2,0) coordinate (T);\n\\draw (S) arc (180:0:2);\n\\filldraw (0,0) circle (0.03) node [below] {$O$} coordinate (O);\n\\draw (40:2) node [above right] {$C$} coordinate (C);\n\\draw (140:2) node [above left] {$D$} coordinate (D);\n\\draw ($(S)!(D)!(T)$) node [below] {$A$} coordinate (A);\n\\draw ($(S)!(C)!(T)$) node [below] {$B$} coordinate (B);\n\\draw (S)--(T)(A)--(D)--(C)--(B);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -475211,7 +475471,9 @@ "id": "018445", "content": "求函数$y=|\\sin x|+|\\cos x|$的最大值和最小值, 并求使其取得最大值和最小值的$x$的集合.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -475231,7 +475493,9 @@ "id": "018446", "content": "如图, 某小区有一块半径为$10$米的扇形空地, 其圆心角$\\angle AOB=\\dfrac{\\pi}{3}$, 现规划在此空地上建造一块矩形的健身区域$CDEF$, 其中点$C$、$D$在半径$OA$上, 点$E$在$\\overset\\frown{AB}$上, 点$F$在半径$OB$上, 其余地方进行绿化. 设$\\angle EOA=\\theta$, 当$\\theta$为何值时, 健身区域$CDEF$的面积$S$最大? 并求出面积$S$的最大值.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (2,0) node [below] {$A$} coordinate (A);\n\\draw (60:2) node [above] {$B$} coordinate (B);\n\\draw (40:2) node [above right] {$E$} coordinate (E);\n\\draw ($(O)!(E)!(A)$) node [below] {$D$} coordinate (D);\n\\path [name path = OB] (O) -- (B);\n\\path [name path = EF] (E) --++ (-1.5,0);\n\\path [name intersections = {of = OB and EF, by = F}];\n\\draw (F) node [above left] {$F$};\n\\draw ($(O)!(F)!(A)$) node [below] {$C$} coordinate (C);\n\\draw (C)--(F)--(E)--(D)(A)--(O)--(B)(A) arc (0:60:2)(O)--(E);\n\\draw pic [draw, \"$\\theta$\", scale = 0.6, angle eccentricity = 1.8] {angle = A--O--E};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -475251,7 +475515,9 @@ "id": "018447", "content": "判断下列函数$y=f(x)$的奇偶性, 并说明理由:\\\\\n(1) $f(x)=\\sin |x|$;\\\\\n(2) $f(x)=\\sin (x+\\dfrac{\\pi}{2})$;\\\\\n(3) $f(x)=\\sin (x-\\dfrac{\\pi}{4})$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -475271,7 +475537,9 @@ "id": "018448", "content": "利用函数的单调性, 比较下列各组数的大小:\\\\\n(1) $\\sin \\dfrac{6 \\pi}{5}$与$\\sin \\dfrac{7 \\pi}{6}$;\\\\\n(2) $\\sin \\dfrac{43 \\pi}{7}$与$\\sin (-\\dfrac{47 \\pi}{8})$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -475291,7 +475559,9 @@ "id": "018449", "content": "求函数$y=\\sin (x+\\dfrac{\\pi}{2})$的单调减区间.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -475311,7 +475581,9 @@ "id": "018450", "content": "求函数$y=2 \\sin (-2 x+\\dfrac{\\pi}{6})$, $x \\in(-\\pi, 0]$的单调增区间.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -475331,7 +475603,9 @@ "id": "018451", "content": "设$\\varphi$为给定的实数. 函数$y=4 \\sin (2 x+\\varphi)$为偶函数的一个充要条件是$\\varphi=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -475351,7 +475625,9 @@ "id": "018452", "content": "已知函数$y=2 \\sqrt{3} \\sin x \\cos x+\\cos ^2 x-\\sin ^2 x-1, x \\in[0, \\dfrac{7 \\pi}{12}]$.\\\\\n(1) 求此函数的单调增区间;\\\\\n(2) 求此函数的值域.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -475371,7 +475647,9 @@ "id": "018453", "content": "求下列函数的最大值与最小值, 并求出取得最大值和最小值时所有$x$的值:\\\\\n(1) $y=\\cos ^2 x-4 \\cos x+1$, $x \\in \\mathbf{R}$;\\\\\n(2) $y=\\cos \\dfrac{x}{2}$, $x \\in[-\\dfrac{4 \\pi}{3}, \\dfrac{\\pi}{2}]$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -475391,7 +475669,9 @@ "id": "018454", "content": "求函数$y=2 \\cos (2 x-\\dfrac{\\pi}{3})$的最小正周期及单调增区间.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -475411,7 +475691,9 @@ "id": "018455", "content": "设常数$\\omega>0$, 求函数$y=\\cos ^2 \\omega x$的最小正周期和单调区间.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -475431,7 +475713,9 @@ "id": "018456", "content": "观察余弦曲线, 分别写出满足下列条件的所有$x$的集合:\\\\\n(1) $\\cos x<0$;\\\\\n(2) $\\cos x>-\\dfrac{\\sqrt{2}}{2}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -475451,7 +475735,9 @@ "id": "018457", "content": "当函数$y=A \\sin (\\omega x+\\varphi)$($A>0$, $\\omega>0$)中的常数$A$、$\\omega$、$\\varphi$分别取下列各组值时, 用计算器 (机) 在同一平面直角坐标系中作出它们的图像:\\\\\n(1) $A=2, \\omega=1, \\varphi=0$;\\\\\n(2) $A=1, \\omega=2, \\varphi=0$;\\\\\n(3) $A=1, \\omega=1, \\varphi=\\dfrac{\\pi}{2}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -475471,7 +475757,9 @@ "id": "018458", "content": "作出函数$y=3 \\sin (2 x+\\dfrac{\\pi}{4})$的大致图像, 并指出其振幅、频率和初始相位.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -475491,7 +475779,9 @@ "id": "018459", "content": "已知交流电的电流强度$I$关于时间$t$的函数为$I=I_0 \\sin (\\omega t+\\varphi)$, 其中$I_0>0$, $\\omega>0$, $0 \\leq \\varphi<2 \\pi$. 根据图像求出它的周期、频率和电流的最大值, 并写出$I_0$、$\\omega$和$\\varphi$的值.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 1.5, yscale = 0.08]\n\\draw [->] (0,0) -- (4,0) node [below] {$t$(s)};\n\\draw [->] (0,-15) -- (0,15) node [left] {$I$(A)};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0:3, samples = 100] plot (\\x,{10*sin(\\x*180)});\n\\foreach \\i/\\j in {0.01/below left,0.02/below right,0.03/below}\n{\\draw ({\\i*100},2) --++ (0,-2) node [\\j] {\\small$\\i$};};\n\\draw [dashed] (3.5,10) -- (0,10) node [left] {\\small$10$} (3.5,-10) -- (0,-10) node [left] {\\small$-10$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -475511,7 +475801,9 @@ "id": "018460", "content": "函数$y=2 \\sin 3 x$, $x \\in[\\dfrac{\\pi}{6}, \\dfrac{5 \\pi}{6}]$与函数$y=2$的图像所围成的封闭图形的面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -475531,7 +475823,9 @@ "id": "018461", "content": "已知函数$y=\\sin (\\omega x+\\varphi)$(其中常数$\\omega$、$\\varphi$满足$\\omega>0$且$0 \\leq \\varphi \\leq \\pi$) 是$\\mathbf{R}$上的偶函数, 其图像关于点$P(\\dfrac{2 \\pi}{3}, 0)$成中心对称, 且在区间$[0, \\dfrac{\\pi}{2}]$上是单调函数, 求$\\varphi$和$\\omega$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -475551,7 +475845,9 @@ "id": "018462", "content": "求函数$y=\\tan (\\dfrac{\\pi}{6} x+\\dfrac{\\pi}{3})$的定义域和单调区间.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -475571,7 +475867,9 @@ "id": "018463", "content": "求函数$y=\\tan (2 x+\\dfrac{\\pi}{6})$的最小正周期.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -475591,7 +475889,9 @@ "id": "018464", "content": "函数$y=2 \\tan x+\\tan (\\dfrac{\\pi}{2}-x)$, $x \\in(0, \\dfrac{\\pi}{2})$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -475611,7 +475911,9 @@ "id": "018465", "content": "已知函数$y=2 \\tan (n x-\\dfrac{\\pi}{3})$的最小正周期$T$满足$10$, $\\omega>0$, $0<\\varphi<\\pi$, 函数$y=A \\sin (\\omega x+\\varphi)$的部分图像如图所示.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [above right] {$O$};\n\\draw [domain = -1:{2*pi/3}, samples = 200] plot (\\x,{2*sin(2*\\x/pi*180+120)});\n\\filldraw ({pi/6},0) circle (0.03) node [below left] {\\small$\\dfrac{\\pi}{6}$};\n\\filldraw ({-pi/12},0) circle (0.03) node [below] {\\small$\\dfrac{\\pi}{12}$};\n\\filldraw (0,2) circle (0.03) node [right] {\\small$2$};\n\\draw [dashed] ({-pi/12},0) --++ (0,2) --(0,2);\n\\end{tikzpicture}\n\\end{center}\n(1) 求$A, \\omega, \\varphi$的值;\\\\\n(2) 求函数$y=A \\sin (\\omega x+\\varphi)$, $x \\in[0, \\dfrac{\\pi}{2}]$的最值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -475671,7 +475977,9 @@ "id": "018468", "content": "设$\\omega$为正数, 若函数$y=\\sin \\omega x$在区间$[-\\dfrac{\\pi}{4}, \\dfrac{2 \\pi}{3}]$上为严格增函数, 则$\\omega$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -475691,7 +475999,9 @@ "id": "018469", "content": "如图, 某公园有一块半径为$r$米的扇形绿地$AOB$, 其中$O$为扇形所在圆的圆心, $\\angle AOB=60^{\\circ}$. 欲在该绿地上修建小路$CD$、$CE$, 点$C$在$\\overset\\frown{AB}$上, $D$、$E$分别在半径$OA$、$OB$上, 且$CD\\parallel OB$, $CE\\parallel OA$, 小路$CD$与$CE$的总长为$l$米, 设$\\angle AOC=\\theta$. 试将$l$表示为$\\theta$的函数, 并求$l$的最大值及取得最大值时$\\theta$的值.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (2,0) node [below] {$A$} coordinate (A);\n\\draw (60:2) node [above] {$B$} coordinate (B);\n\\draw (40:2) node [above right] {$C$} coordinate (C);\n\\path [name path = OB] (O) -- (B);\n\\path [name path = CE] (C) --++ (-1.5,0);\n\\path [name intersections = {of = OB and CE, by = E}];\n\\draw (E) node [above left] {$E$};\n\\path [name path = OA] (O) -- (A);\n\\path [name path = CD] (C) --++ (-120:1.5);\n\\path [name intersections = {of = OA and CD, by = D}];\n\\draw (D) node [below] {$D$};\n\\draw (D)--(C)--(E)(O)--(C)(A)--(O)--(B)(A)arc(0:60:2);\n\\draw pic [draw, \"$\\theta$\", scale = 0.6, angle eccentricity = 1.8] {angle = A--O--C};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -475711,7 +476021,9 @@ "id": "018470", "content": "如图, 在等边三角形$ABC$中, $D$、$E$、$F$分别是边$BC$、$AB$、$AC$的中点. 写出图中与向量$\\overrightarrow{EF}$平行的非零向量和与$\\overrightarrow{EF}$相等的向量.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [above] {$A$} coordinate (A);\n\\draw (-60:2) node [right] {$C$} coordinate (C);\n\\draw (-120:2) node [left] {$B$} coordinate (B);\n\\draw ($(A)!0.5!(B)$) node [left] {$E$} coordinate (E);\n\\draw ($(A)!0.5!(C)$) node [right] {$F$} coordinate (F);\n\\draw ($(B)!0.5!(C)$) node [below] {$D$} coordinate (D);\n\\draw (A)--(B)--(C)--cycle(D)--(E)--(F)--cycle;\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -475731,7 +476043,9 @@ "id": "018471", "content": "如图, 在等边三角形$ABC$中, $D$、$E$、$F$分别是边$BC$、$AB$、$AC$的中点.写出向量$\\overrightarrow{AE}$的负向量.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [above] {$A$} coordinate (A);\n\\draw (-60:2) node [right] {$C$} coordinate (C);\n\\draw (-120:2) node [left] {$B$} coordinate (B);\n\\draw ($(A)!0.5!(B)$) node [left] {$E$} coordinate (E);\n\\draw ($(A)!0.5!(C)$) node [right] {$F$} coordinate (F);\n\\draw ($(B)!0.5!(C)$) node [below] {$D$} coordinate (D);\n\\draw (A)--(B)--(C)--cycle(D)--(E)--(F)--cycle;\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -475751,7 +476065,9 @@ "id": "018472", "content": "已知向量$\\overrightarrow {a}$与$\\overrightarrow {b}$不平行, 若$\\overrightarrow {c}\\parallel \\overrightarrow {a}$且$\\overrightarrow {c}\\parallel \\overrightarrow {b}$, 则下列说法正确的是\\bracket{20}.\n\\twoch{$\\overrightarrow {c}=\\overrightarrow{0}$}{$\\overrightarrow {c}=\\overrightarrow {a}$}{$\\overrightarrow {c}=\\overrightarrow {b}$}{不存在满足条件的向量$\\overrightarrow {c}$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -475771,7 +476087,9 @@ "id": "018473", "content": "判断下列命题的真假, 并说明理由:\\\\\n(1) 若$\\overrightarrow {a}$与$\\overrightarrow {b}$都是单位向量, 则$\\overrightarrow {a}=\\overrightarrow {b}$;\\\\\n(2) 方向为南偏西$60^{\\circ}$的向量与北偏东$60^{\\circ}$的向量是平行向量;\\\\\n(3) 若$\\overrightarrow {a}$与$\\overrightarrow {b}$是平行向量, 则$\\overrightarrow {a}=\\overrightarrow {b}$;\\\\\n(4) 若向量$\\overrightarrow{AM}$与$\\overrightarrow{AN}$不相等, 则点$M$与$N$不重合.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -475791,7 +476109,9 @@ "id": "018474", "content": "判断下列命题真假, 并说明理由:\\\\\n(1) 若$|\\overrightarrow {a}|=|\\overrightarrow {b}|$, $|\\overrightarrow {b}|=|\\overrightarrow {c}|$, 则$|\\overrightarrow {a}|=|\\overrightarrow {c}|$;\\\\\n(2) 若$\\overrightarrow {a}\\parallel \\overrightarrow {b}, \\overrightarrow {b}\\parallel \\overrightarrow {c}$, 则$\\overrightarrow {a}\\parallel \\overrightarrow {c}$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -475811,7 +476131,9 @@ "id": "018475", "content": "已知点$O$是矩形$ABCD$的对角线$AC$与$BD$的交点, 设点集$M=\\{A, B, C, D, O\\}$, 向量的集合$T=\\{\\overrightarrow{PQ} | P, Q \\in M$, 且$P$、$Q$不重合$\\}$. 试求集合$T$的元素个数.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -475831,7 +476153,9 @@ "id": "018476", "content": "物体受水平方向$6 \\mathrm{N}$和铅垂方向$8 \\mathrm{N}$的两个力的作用, 求合力的大小以及合力与铅垂方向偏离的角度. (结果精确到$0.01^{\\circ}$)", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -475851,7 +476175,9 @@ "id": "018477", "content": "已知$\\triangle ABC$是边长为$1$的等边三角形, 点$O$是$\\triangle ABC$所在平面上的任意一点.求向量$(\\overrightarrow{OA}-\\overrightarrow{OC})+(\\overrightarrow{OB}-\\overrightarrow{OC})$的模.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -475871,7 +476197,9 @@ "id": "018478", "content": "若点$O$是$\\triangle ABC$所在平面内的一点, 且满足$|\\overrightarrow{OB}-\\overrightarrow{OC}|=|(\\overrightarrow{OB}-\\overrightarrow{OA})+(\\overrightarrow{OC}-\\overrightarrow{OA})|$, 则$\\triangle ABC$的形状为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -475891,7 +476219,9 @@ "id": "018479", "content": "巳知$|\\overrightarrow {a}|=2$, $|\\overrightarrow {b}|=1$, $|\\overrightarrow {a}-\\overrightarrow {b}|=\\sqrt{3}$, 求$|\\overrightarrow {a}+\\overrightarrow {b}|$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -475911,7 +476241,9 @@ "id": "018480", "content": "化简下列向量线性运算:\\\\\n(1) $(-2) \\times(\\dfrac{1}{2} \\overrightarrow {a})$;\\\\\n(2) $2(\\overrightarrow {a}-\\overrightarrow {b})+3(\\overrightarrow {a}+\\overrightarrow {b})$;\\\\\n(3) $(\\lambda-\\mu)(\\overrightarrow {a}+\\overrightarrow {b})-(\\lambda+\\mu)(\\overrightarrow {a}-\\overrightarrow {b})$.($\\lambda, \\mu \\in \\mathbf{R}$)", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -475931,7 +476263,9 @@ "id": "018481", "content": "已知向量$\\overrightarrow {a}$、$\\overrightarrow {b}$、$\\overrightarrow {c}$满足$\\dfrac{1}{2}(\\overrightarrow {a}-3 \\overrightarrow {c})+2(2 \\overrightarrow {a}-3 \\overrightarrow {b})=\\overrightarrow{0}$, 试用$\\overrightarrow {a}$、$\\overrightarrow {b}$表示$\\overrightarrow {c}$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -475951,7 +476285,9 @@ "id": "018482", "content": "如图, 在$\\triangle ABC$中, $D$是$AB$的中点, $E$是$BC$延长线上一点, 且$BE=2BC$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$B$} coordinate (B);\n\\draw (1.2,0.2) node [below] {$C$} coordinate (C);\n\\draw (1,1.2) node [above] {$A$} coordinate (A);\n\\draw ($(A)!0.5!(B)$) node [above left] {$D$} coordinate (D);\n\\draw ($(B)!2!(C)$) node [right] {$E$} coordinate (E);\n\\draw (A)--(B)--(E)--(D)(A)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 用向量$\\overrightarrow{BA}$、$\\overrightarrow{BC}$表示$\\overrightarrow{DE}$;\\\\\n(2) 用向量$\\overrightarrow{CA}$、$\\overrightarrow{CB}$表示$\\overrightarrow{DE}$;", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -475971,7 +476307,9 @@ "id": "018483", "content": "已知任意两个非零向量$\\overrightarrow {a}$和$\\overrightarrow {b}$, 试作$\\overrightarrow{OA}=\\overrightarrow {a}+\\overrightarrow {b}$, $\\overrightarrow{OB}=\\overrightarrow {a}+2 \\overrightarrow {b}$, $\\overrightarrow{OC}=\\overrightarrow {a}+3 \\overrightarrow {b}$. 判断$A$、$B$、$C$三点之间的位置关系, 并证明你的猜想.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -475991,7 +476329,9 @@ "id": "018484", "content": "如图, 在$\\triangle ABC$中, $\\overrightarrow{BC}=3 \\overrightarrow{BD}$, $\\overrightarrow{AE}=\\dfrac{2}{3} \\overrightarrow{AD}$, $\\overrightarrow{CE}=\\lambda \\overrightarrow{AB}+\\mu \\overrightarrow{AC}$, 求实数$\\lambda$和$\\mu$的值.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$B$} coordinate (B);\n\\draw (0.9,0.1) node [below] {$D$} coordinate (D);\n\\draw (1,1.2) node [above] {$A$} coordinate (A);\n\\draw ($(B)!3!(D)$) node [right] {$C$} coordinate (C);\n\\draw ($(A)!{2/3}!(D)$) node [left] {$E$} coordinate (E);\n\\draw (A)--(B)--(C)--cycle(A)--(D)(E)--(C);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476011,7 +476351,9 @@ "id": "018485", "content": "已知$\\overrightarrow {a}$和$\\overrightarrow {b}$是两个不平行的向量, 向量$\\overrightarrow {b}-t \\overrightarrow {a}$与$\\dfrac{1}{2} \\overrightarrow {a}-\\dfrac{3}{2} \\overrightarrow {b}$平行, 求实数$t$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476031,7 +476373,9 @@ "id": "018486", "content": "已知向量$\\overrightarrow {a}$与$\\overrightarrow {b}$的夹角为$\\dfrac{2 \\pi}{3}$, 且$|\\overrightarrow {a}|=3$, $|\\overrightarrow {b}|=4$.\\\\\n(1) 求$\\overrightarrow {b}$在$\\overrightarrow {a}$方向上的投影与数量投影;\\\\\n(2) 求$\\overrightarrow {b} \\cdot \\overrightarrow {a}$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476051,7 +476395,9 @@ "id": "018487", "content": "已知$\\overrightarrow {a}$是非零向量, $\\overrightarrow {b}$与$\\overrightarrow {c}$是任意向量, 它们在$\\overrightarrow {a}$方向上的投影分别为$\\overrightarrow{b'}$与$\\overrightarrow{c'}$. 求证: $\\overrightarrow {b}+\\overrightarrow {c}$在$\\overrightarrow {a}$方向上的投影为$\\overrightarrow{b'}+\\overrightarrow{c'}$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476071,7 +476417,9 @@ "id": "018488", "content": "已知圆$O$中, 弦$AB$的长为$\\sqrt{3}$, 圆上的点$C$满足$\\overrightarrow{OA}+\\overrightarrow{OB}+\\overrightarrow{OC}=\\overrightarrow{0}$, 求$\\overrightarrow{AC}$在$\\overrightarrow{OA}$方向上的数量投影.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476091,7 +476439,9 @@ "id": "018489", "content": "如图, 已知正三角形$ABC, O$是三角形$ABC$外一点, 且$|OA|=1$, 若$\\angle OAB=\\theta$($\\theta \\in(0, \\dfrac{\\pi}{3})$), 试用向量的方法证明: $\\cos \\theta+\\cos (\\theta+\\dfrac{2 \\pi}{3})+\\cos (\\theta+\\dfrac{4 \\pi}{3})=0$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [above] {$A$} coordinate (A);\n\\draw (-60:1.5) node [right] {$C$} coordinate (C);\n\\draw (-120:1.5) node [left] {$B$} coordinate (B);\n\\draw (A) --++ (-160:1) node [left] {$O$} coordinate (O);\n\\draw pic [draw, \"$\\theta$\", scale = 0.6, angle eccentricity = 2.5] {angle = O--A--B};\n\\draw (A)--(B)--(C)--cycle;\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476111,7 +476461,9 @@ "id": "018490", "content": "证明:\\\\\n(1) $(\\overrightarrow {a}+\\overrightarrow {b})^2=\\overrightarrow {a}^2+2 \\overrightarrow {a} \\cdot \\overrightarrow {b}+\\overrightarrow {b}^2$;\\\\\n(2) $(\\overrightarrow {a}+\\overrightarrow {b}) \\cdot(\\overrightarrow {a}-\\overrightarrow {b})=\\overrightarrow {a}^2-\\overrightarrow {b}^2$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476131,7 +476483,9 @@ "id": "018491", "content": "设向量$\\overrightarrow {a}$、$\\overrightarrow {b}$满足$|\\overrightarrow {a}|=2$, $|\\overrightarrow {b}|=3$, $\\langle\\overrightarrow {a}, \\overrightarrow {b}\\rangle=\\dfrac{\\pi}{3}$. 求$|3 \\overrightarrow {a}-2 \\overrightarrow {b}|$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476151,7 +476505,9 @@ "id": "018492", "content": "已知向量$\\overrightarrow {a}$、$\\overrightarrow {b}$满足$|\\overrightarrow {a}|=4$, $\\overrightarrow {b}$在$\\overrightarrow {a}$方向上的数量投影为 $-2$ , 求$|\\overrightarrow {a}-3 \\overrightarrow {b}|$的最小值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476171,7 +476527,9 @@ "id": "018493", "content": "如图, $O$是线段$AB$外一点, $|OA|=3$, $|OB|=2$, $P$是线段$AB$的垂直平分线$l$上的动点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\filldraw (0,0) circle (0.03) node [left] {$B$} coordinate (B);\n\\filldraw (3,2) circle (0.03) node [right] {$A$} coordinate (A);\n\\draw ($(A)!0.5!(B)$) ++ (-1,1.5) node [below] {$l$} --++ (2,-3) ++ (-0.2,0.3) node [right] {$P$} coordinate (P);\n\\filldraw (P) circle (0.03);\n\\filldraw ({24/13},{-10/13}) node [below] {$O$} coordinate (O);\n\\draw (O)--(P)(A)--(B)--(O)--cycle;\n\\end{tikzpicture}\n\\end{center}\n(1) 求$\\overrightarrow{AB} \\cdot(\\overrightarrow{OA}+\\overrightarrow{OB})$;\\\\\n(2) 求$\\overrightarrow{OP} \\cdot \\overrightarrow{AB}$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476191,7 +476549,9 @@ "id": "018494", "content": "如图, 在平行四边形$ABCD$中, 两条对角线的交点是$M$, 设$\\overrightarrow{AB}=\\overrightarrow {a}$, $\\overrightarrow{AD}=\\overrightarrow {b}$. 试用$\\overrightarrow {a}$、$\\overrightarrow {b}$的线性组合分别表示$\\overrightarrow{MA}$、$\\overrightarrow{MB}$、$\\overrightarrow{MC}$与$\\overrightarrow{MD}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$A$} coordinate (A);\n\\draw (2.5,0) node [below] {$B$} coordinate (B);\n\\draw (1,2) node [above] {$D$} coordinate (D);\n\\draw (3.5,2) node [above] {$C$} coordinate (C);\n\\draw [->] (A)--(B) node [midway, below] {$\\overrightarrow{a}$};\n\\draw [->] (A)--(D) node [midway, left] {$\\overrightarrow{b}$};\n\\draw (A)--(C)(B)--(D)--(C)--cycle;\n\\draw ($(A)!0.5!(C)$) node [above] {$M$} coordinate (M);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476211,7 +476571,9 @@ "id": "018495", "content": "设$\\overrightarrow {a}$、$\\overrightarrow {b}$是平面内不平行的两个向量, 实数$\\lambda$、$\\mu$满足$3 \\lambda \\overrightarrow {a}+(10-\\mu) \\overrightarrow {b}=(2 \\mu+1) \\overrightarrow {a}+2 \\lambda \\overrightarrow {b}$, 试求$\\lambda$与$\\mu$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476231,7 +476593,9 @@ "id": "018496", "content": "已知$\\overrightarrow {a}$、$\\overrightarrow {b}$、$\\overrightarrow {c}$是平面上任意三个给定的向量, 其中$\\overrightarrow {a}$和$\\overrightarrow {b}$不平行, 将满足$x^2 \\overrightarrow {a}+x \\overrightarrow {b}+\\overrightarrow {c}=\\overrightarrow{0}$的实数$x$的个数记为$n$, 则$n$的值的集合为\\bracket{20}.\n\\fourch{$\\{0\\}$}{$\\{1\\}$}{$\\{2\\}$}{$\\{0,1\\}$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -476251,7 +476615,9 @@ "id": "018497", "content": "如图, 平面上三个向量$\\overrightarrow{OA}$、$\\overrightarrow{OB}$、$\\overrightarrow{OC}$满足: $|\\overrightarrow{OA}|=2,|\\overrightarrow{OB}|=\\sqrt{3}$, $|\\overrightarrow{OC}|=1$, 且$\\angle AOB=120^{\\circ}$, $\\angle AOC=150^{\\circ}$, 设$\\overrightarrow{OC}=m \\overrightarrow{OA}+n \\overrightarrow{OB}$, 求实数$m$、$n$的值.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$O$} coordinate (O);\n\\draw (O) ++ (2,0) node [right] {$A$} coordinate (A);\n\\draw (O) ++ (120:{sqrt(3)}) node [left] {$B$} coordinate (B);\n\\draw (O) ++ (-150:1) node [left] {$C$} coordinate (C);\n\\foreach \\i in {A,B,C}\n{\\draw [->] (O) -- (\\i);};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476271,7 +476637,9 @@ "id": "018498", "content": "如图, 写出向量$\\overrightarrow {a}$、$\\overrightarrow {b}$与$\\overrightarrow {c}$的坐标.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2.5,0) -- (2.5,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {-2,-1,1,2}\n{\\draw [gray, dashed] (-2,\\i) -- (2,\\i) (\\i,-2) -- (\\i,2);\n\\draw (\\i,0.2) -- (\\i,0) node [below] {$\\i$} (0.2,\\i) -- (0,\\i) node [left] {$\\i$};};\n\\draw [->] (0,0) -- (1,2) node [midway, below right] {$\\overrightarrow{a}$};\n\\draw [->] (1,-2) -- (2,0) node [midway, below right] {$\\overrightarrow{b}$};\n\\draw [->] (-1,0) -- (0,-2) node [midway, below left] {$\\overrightarrow{c}$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476291,7 +476659,9 @@ "id": "018499", "content": "给定向量$\\overrightarrow {a}=(4,-1)$与$\\overrightarrow {b}=(5,2)$, 求向量$2 \\overrightarrow {a}+3 \\overrightarrow {b}$的坐标.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476311,7 +476681,9 @@ "id": "018500", "content": "平面上$A$、$B$、$C$三点的坐标分别为$(2,1)$、$(-3,2)$、$(-1,3)$, 写出向量$\\overrightarrow{AC}$、$\\overrightarrow{BC}$的坐标.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476331,7 +476703,9 @@ "id": "018501", "content": "已知平面内两点$P$、$Q$的坐标分别为$(-2,4)$、$(2,1)$, 求$\\overrightarrow{PQ}$的单位向量$\\overrightarrow{a_0}$的坐标.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476351,7 +476725,9 @@ "id": "018502", "content": "已知平面上$A$、$B$、$C$三点的坐标分别为$(-1,-1)$、$(2,-7)$、$(-3,3)$, 请尝试判断$A$、$B$、$C$三点是否共线.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476371,7 +476747,9 @@ "id": "018503", "content": "已知$A$、$B$、$C$、$D$是一个平行四边形的四个顶点, 若$A$、$B$、$C$的坐标分别为$(-2,1)$、$(-1,3)$、$(3,4)$, 求顶点$D$的坐标.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476391,7 +476769,9 @@ "id": "018504", "content": "若$\\overrightarrow{a_1}, \\overrightarrow{a_2}, \\overrightarrow{a_3}$均为单位向量, 则$\\overrightarrow{a_1}=(\\dfrac{\\sqrt{3}}{3}$, $\\dfrac{\\sqrt{6}}{3})$是$\\overrightarrow{a_1}+\\overrightarrow{a_2}+\\overrightarrow{a_3}=(\\sqrt{3}, \\sqrt{6})$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": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -476411,7 +476791,9 @@ "id": "018505", "content": "已知向量$\\overrightarrow {a}=(1,2)$, $\\overrightarrow {b}=(2,-2)$. 求$|\\overrightarrow {a}|$、$|\\overrightarrow {b}|$与$\\langle\\overrightarrow {a}, \\overrightarrow {b}\\rangle$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476431,7 +476813,9 @@ "id": "018506", "content": "已知$\\triangle ABC$中$A$、$B$、$C$三点的坐标分别为$(2,-2)$、$(-2,3)$、$(3,7)$, 求证: $\\triangle ABC$为直角三角形.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476451,7 +476835,9 @@ "id": "018507", "content": "已知$x_1$、$x_2$、$y_1$、$y_2$都是实数, 求证: $(x_1 x_2+y_1 y_2)^2 \\leq(x_1^2+y_1^2)(x_2^2+y_2^2)$, 并且等号成立的一个充要条件是$x_1 y_2=x_2 y_1$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476471,7 +476857,9 @@ "id": "018508", "content": "设$x$、$y \\in \\mathbf{R}$, 向量$\\overrightarrow {a}=(x, 1)$, $\\overrightarrow {b}=(1, y)$, $\\overrightarrow {c}=(2,-4)$, 且$\\overrightarrow {a} \\perp \\overrightarrow {c}$, $\\overrightarrow {b}\\parallel \\overrightarrow {c}$, 则$|\\overrightarrow {a}+\\overrightarrow {b}|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -476491,7 +476879,9 @@ "id": "018509", "content": "已知两个向量$\\overrightarrow {a}$和$\\overrightarrow {b}$满足$\\overrightarrow {a}+\\overrightarrow {b}=(2,-8)$, $\\overrightarrow {a}-\\overrightarrow {b}=(-6,-4)$, 求$\\overrightarrow {a}$与$\\overrightarrow {b}$的夹角.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476511,7 +476901,9 @@ "id": "018510", "content": "证明对角线互相平分的四边形是平行四边形.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476531,7 +476923,9 @@ "id": "018511", "content": "已知$P$是直线$P_1P_2$上一点, 且$\\overrightarrow{P_1P}=\\lambda \\overrightarrow{PP_2}$($\\lambda$为实数, 且$\\lambda \\neq-1$), $P_1$、$P_2$的坐标分别为$(x_1, y_1)$、$(x_2, y_2)$, 求点$P$的坐标$(x, y)$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476551,7 +476945,9 @@ "id": "018512", "content": "已知$\\triangle ABC$的三个顶点$A$、$B$、$C$的坐标分别是$(x_1, y_1) 、(x_2, y_2)$、$(x_3, y_3)$, 求此三角形重心$G$的坐标.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476571,7 +476967,9 @@ "id": "018513", "content": "在$\\triangle ABC$中, 设$\\overrightarrow{CA}=\\overrightarrow {a}, \\overrightarrow{CB}=\\overrightarrow {b}$, 记$\\triangle ABC$的面积为$S$\n(1) 求证: $S=\\dfrac{1}{2} \\sqrt{|\\overrightarrow {a}|^2|\\overrightarrow {b}|^2-(\\overrightarrow {a} \\cdot \\overrightarrow {b})^2}$;\n(2)设$\\overrightarrow {a}=(x_1, y_1), \\overrightarrow {b}=(x_2, y_2)$, 求证: $S=\\dfrac{1}{2}|x_1 y_2-x_2 y_1|$.\n$\\overrightarrow{OC} \\perp$. 设$OA$、$B$、$C$是平面上四点, 已知$\\overrightarrow{OA} \\perp \\overrightarrow{BC}, \\overrightarrow{OB} \\perp \\overrightarrow{AC}$. 求证:", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476591,7 +476989,9 @@ "id": "018514", "content": "求证: 两组对边平方和相等的四边形的对角线互相垂直.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476611,7 +477011,9 @@ "id": "018515", "content": "用向量方法证明: $\\cos (\\alpha-\\beta)=\\cos \\alpha \\cos \\beta+\\sin \\alpha \\sin \\beta$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476631,7 +477033,9 @@ "id": "018516", "content": "如图, 平面上$A, B, C$三点的坐标分别是$(2,3)$、$(2,0)$、$(1,1)$, 已知小明在点$B$处休憩, 有只小狗沿着$AC$所在直线来回跑动. 问: 其在什么位置时, 离小明最近?\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (-0.5,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (2,0) node [below] {$B$} coordinate (B);\n\\draw (2,3) node [above] {$A$} coordinate (A);\n\\draw (1,1) node [left] {$C$} coordinate (C);\n\\draw (A) -- ($(A)!1.6!(C)$) (C)--(B)--(A);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476651,7 +477055,9 @@ "id": "018517", "content": "将质量为$20 \\mathrm{kg}$的物体用两根绳子悬挂起来, 如图, 两根绳子与铅锤线的夹角分别为$45^{\\circ}$与$30^{\\circ}$, 求它们分别提供的拉力的大小. (结果精确到$0.1 \\mathrm{N}$)\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) coordinate (O);\n\\draw (O) ++ ({-2/sqrt(3)},2) coordinate (S) (O) ++ (2,2) coordinate (T);\n\\draw ($(S)!-0.2!(T)$) -- ($(T)!-0.2!(S)$);\n\\draw (S)--(O)--(T);\n\\draw [->] (O) --++ (120:2) node [left] {$\\overrightarrow{f_1}$};\n\\draw [->] (O) --++ (1,1) node [right] {$\\overrightarrow{f_2}$};\n\\draw [dashed] (O) --++ (0,2) coordinate (P);\n\\draw pic [\"$30^\\circ$\", angle eccentricity = 2.5] {angle = P--O--S};\n\\draw pic [\"$45^\\circ$\", angle eccentricity = 2] {angle = T--O--P};\n\\draw (O) -- (0,-0.5) coordinate (A);\n\\draw (A) ++ (-0.3,-0.6) rectangle++ (0.6,0.6);\n\\draw (A) ++ (0.3,-0.3) node [right] {$20$kg};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476671,7 +477077,9 @@ "id": "018518", "content": "某人朝正南方向游去, 他在静水中游泳速度的大小是$\\sqrt{3} \\mathrm{m} / \\mathrm{s}$, 河水自西向东流速为$1 \\mathrm{m} / \\mathrm{s}$, 求他实际前进的速度的大小和方向.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476691,7 +477099,9 @@ "id": "018519", "content": "已知向量$\\overrightarrow {a}, \\overrightarrow {b}, \\overrightarrow {c}, \\overrightarrow {d}$, 且$\\overrightarrow {a}, \\overrightarrow {b}$中至少有一个不是零向量, 实数$t$的函数$y=|t \\overrightarrow {a}+\\overrightarrow {c}|^2+|t \\overrightarrow {b}+\\overrightarrow {d}|^2$, 当$t=$\\blank{50}时, $y$取得最小值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -476711,7 +477121,9 @@ "id": "018520", "content": "已知点$A(3,0)$, $B(0,3)$, $C(\\cos \\alpha, \\sin \\alpha)$, $\\alpha \\in(\\dfrac{\\pi}{2}, \\dfrac{3 \\pi}{2})$. 若$\\overrightarrow{AC} \\cdot \\overrightarrow{BC}=-1$, 求$\\dfrac{2 \\sin ^2 \\alpha+\\sin 2 \\alpha}{1+\\tan \\alpha}$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476731,7 +477143,9 @@ "id": "018521", "content": "在平行四边形$ABCD$中, $\\angle A=\\dfrac{\\pi}{3}$, 边$AB$、$AD$的长分别为 $2$、$1$. 若$M$、$N$分别是边$BC$、$CD$上的点, 且满足$\\dfrac{|\\overrightarrow{BM}|}{|\\overrightarrow{BC}|}=\\dfrac{|\\overrightarrow{CN}|}{|\\overrightarrow{CD}|}$, 则$\\overrightarrow{AM} \\cdot \\overrightarrow{AN}$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -476751,7 +477165,9 @@ "id": "018522", "content": "用向量的方法证明三角形的正弦定理.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476771,7 +477187,9 @@ "id": "018523", "content": "如图, $OM\\parallel AB$, 点$P$在由射线$OM$、 线段$OB$及$AB$的延长线围成的阴影区域内 (不含边界) 运动, 且$\\overrightarrow{OP}=x \\overrightarrow{OA}+y \\overrightarrow{OB}$($x, y \\in \\mathbf{R}$). 在阴影区域内作$BF\\parallel OA$, 使得$OABF$构成平行四边形, 当$x=-\\dfrac{1}{2}$, $y=1$时, 点$P$的位置在\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (2,0) node [below] {$A$} coordinate (A);\n\\draw (1.1,1.5) node [above right] {$B$} coordinate (B);\n\\draw ($(A)!1.7!(B)$) coordinate (T);\n\\draw ($(O) + 1.2*(B) - 1.2*(A)$) node [above] {$M$} coordinate (M);\n\\fill [pattern = north east lines] (M)--(T)--(B)--(O)--cycle;\n\\draw [dashed] (B)--(T) (O)--(M);\n\\draw [->] (O)--(A);\n\\draw [->] (O)--(B);\n\\draw (A)--(B);\n\\draw [->] (O) --++ (-0.1,0.6) node [above, fill = white] {$P$} coordinate (P); \n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\triangle OBF$的重心}{线段$OF$的中点}{线段$OB$的中点}{线段$BF$的中点}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -476791,7 +477209,9 @@ "id": "018524", "content": "如图, $OM\\parallel AB$, 点$P$在由射线$OM$、线段$OB$及$AB$的延长线围成的阴影区域内 (不含边界)运动, 且$\\overrightarrow{OP}=x \\overrightarrow{OA}+y \\overrightarrow{OB}$($x, y \\in \\mathbf{R}$). 当$x=-\\dfrac{1}{2}$时, $y$的取值范围是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (2,0) node [below] {$A$} coordinate (A);\n\\draw (1.1,1.5) node [above right] {$B$} coordinate (B);\n\\draw ($(A)!1.7!(B)$) coordinate (T);\n\\draw ($(O) + 1.2*(B) - 1.2*(A)$) node [above] {$M$} coordinate (M);\n\\fill [pattern = north east lines] (M)--(T)--(B)--(O)--cycle;\n\\draw [dashed] (B)--(T) (O)--(M);\n\\draw [->] (O)--(A);\n\\draw [->] (O)--(B);\n\\draw (A)--(B);\n\\draw [->] (O) --++ (-0.1,0.6) node [above, fill = white] {$P$} coordinate (P); \n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -476811,7 +477231,9 @@ "id": "018525", "content": "如图, 在$\\triangle ABC$中, 点$M$是边$BC$的中点, 点$N$是边$AC$的中点, $AM$与$BN$交于点$P$, 用向量的方法求$\\dfrac{AP}{AM}$的值.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$B$} coordinate (B);\n\\draw (2,0) node [right] {$C$} coordinate (C);\n\\draw (1.2,1.6) node [above] {$A$} coordinate (A);\n\\draw ($(A)!0.5!(C)$) node [above right] {$N$} coordinate (N);\n\\draw ($(B)!0.5!(C)$) node [below] {$M$} coordinate (M);\n\\draw ($(B)!{2/3}!(N)$) node [above left] {$P$} coordinate (P);\n\\draw (A)--(B)--(C)--cycle(A)--(M)(B)--(N);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476831,7 +477253,9 @@ "id": "018526", "content": "如图, 在一个平面内, $\\triangle ABC$为直角三角形, $A$为直角, $AB=3$, $BC=4$, 长为$10$的线段$PQ$以点$A$为中点, 当$\\overrightarrow{PQ}$与$\\overrightarrow{BC}$的夹角$\\theta$取何值时, $\\overrightarrow{BP} \\cdot \\overrightarrow{CQ}$的值最大? 并求出这个最大值.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.3]\n\\draw (0,0) node [below] {$A$} coordinate (A);\n\\draw (3,0) node [right] {$B$} coordinate (B);\n\\draw (0,4) node [above] {$C$} coordinate (C);\n\\draw (40:5) node [above] {$P$} coordinate (P);\n\\draw (220:5) node [below] {$Q$} coordinate (Q);\n\\draw (P)--(Q)(A)--(B)--(C)--cycle;\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476851,7 +477275,9 @@ "id": "018527", "content": "已知正三角形$ABC$边长为$2$, 圆$A$的半径是$1, PQ$为圆$A$的任意一条直径.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [above] {$A$} coordinate (A);\n\\draw (-60:2) node [right] {$C$} coordinate (C);\n\\draw (-120:2) node [left] {$B$} coordinate (B);\n\\draw (A) circle (1)(A)--(B)--(C)--cycle;\n\\draw (-10:1) node [right] {$Q$} coordinate (Q);\n\\draw (170:1) node [left] {$P$} coordinate (P);\n\\draw (P)--(Q);\n\\end{tikzpicture}\n\\end{center}\n(1) 判断$\\overrightarrow{BP} \\cdot \\overrightarrow{CQ}-\\overrightarrow{AP} \\cdot \\overrightarrow{CB}$是否为定值, 请说明理由;\\\\\n(2) 求$\\overrightarrow{BP} \\cdot \\overrightarrow{CQ}$的最大值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476871,7 +477297,9 @@ "id": "018528", "content": "已知向量$\\overrightarrow{OA}=(1,7)$, $\\overrightarrow{OB}=(5,1)$, $\\overrightarrow{OP}=(2,1)$, $K$为直线$OP$上的一个动点, 当$\\overrightarrow{KA} \\cdot \\overrightarrow{KB}$取最小值时, 求向量$\\overrightarrow{OK}$的坐标.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476891,7 +477319,9 @@ "id": "018529", "content": "计算:\\\\\n(1) $(1+3 \\mathrm{i})+(-4+2 \\mathrm{i})$;\\\\\n(2) $(3-2 \\mathrm{i})-(3+2 \\mathrm{i})$;\\\\\n(3) $(2-3 \\mathrm{i})(4+2 \\mathrm{i})$;\\\\\n(4) $(2+\\mathrm{i})(3+4 \\mathrm{i})(2-\\mathrm{i})$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476911,7 +477341,9 @@ "id": "018530", "content": "计算:\\\\\n(1) $\\dfrac{3+\\mathrm{i}}{2-\\mathrm{i}}$;\\\\\n(2) $\\dfrac{1+\\sqrt{2} \\mathrm{i}}{1-\\sqrt{2} \\mathrm{i}}$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476931,7 +477363,9 @@ "id": "018531", "content": "计算虚数单位$\\mathrm{i}$的整数次幂, 并找出规律.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476951,7 +477385,9 @@ "id": "018532", "content": "对任意整数$m$, 计算$\\mathrm{i}^m+\\mathrm{i}^{m+1}+\\mathrm{i}^{m+2}+\\mathrm{i}^{m+3}$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476971,7 +477407,9 @@ "id": "018533", "content": "计算: $(a+b \\mathrm{i})^2-(a-b \\mathrm{i})^2$($a$、$b \\in \\mathbf{R}$).", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -476991,7 +477429,9 @@ "id": "018534", "content": "对任意奇数$n$, 计算$(\\dfrac{1-\\mathrm{i}}{1+\\mathrm{i}})^{2 n}+(\\dfrac{1+\\mathrm{i}}{1-\\mathrm{i}})^{2 n}$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -477011,7 +477451,9 @@ "id": "018535", "content": "已知$m \\in \\mathbf{R}$, 设集合$M=\\{1,2,(m^2-3 m-1)+(m^2-5 m+6) \\mathrm{i}\\}$, $N=\\{-1,3\\}$. 若$M \\cap N \\neq \\varnothing$, 求$m$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -477031,7 +477473,9 @@ "id": "018536", "content": "填写下表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline$z$ & 是否复数 & 是否实数 & 是否虚数 & 是否纯虚数 & $\\mathrm{Re} z$ & $\\mathrm{Im} z$ \\\\\n\\hline$-0.5 \\mathrm{i}$ & & & & & & \\\\\n\\hline $\\frac{1}{2}-\\sqrt{2} \\mathrm{i}$ & & & & & &\\\\\n\\hline $\\pi$ & & & & & &\\\\\n\\hline $0$ & & & & & &\\\\\n\\hline $\\sqrt{3}$ & & & & & &\\\\\n\\hline $2 \\mathrm{i}-5$ & & & & & & \\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -477051,7 +477495,9 @@ "id": "018537", "content": "求实数$m$的值或取值范围, 使得复数$z=m^2+m-2+(m^2-1) \\mathrm{i}$分别是:\\\\\n(1) 实数;\\\\\n(2) 虚数;\\\\\n(3) 纯虚数;\\\\\n(4) $0$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -477071,7 +477517,9 @@ "id": "018538", "content": "设$z$是复数, 求证: $\\overline {z}=z$是$z \\in \\mathbf{R}$的充要条件.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -477091,7 +477539,9 @@ "id": "018539", "content": "设$z_1=1+3 \\mathrm{i}$, $z_2=1-\\mathrm{i}$. 求复数$z$, 使得$\\overline {z}=\\dfrac{z_1}{z_2}$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -477111,7 +477561,9 @@ "id": "018540", "content": "求角$\\theta$($\\theta \\in \\mathbf{R}$), 使得复数$z=(2 \\sin ^2 \\theta-\\sin \\theta)+(3 \\tan ^2 \\theta-1) \\mathrm{i}$分别是:\\\\\n(1) 实数;\\\\\n(2) 纯虚数;\\\\\n(3) 零.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -477131,7 +477583,9 @@ "id": "018541", "content": "设$z$是复数, 你能说出$z$是纯虚数的一个充要条件吗? (通过$z$与$\\overline {z}$之间的关系描述)", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -477151,7 +477605,9 @@ "id": "018542", "content": "在复平面上作出表示下列复数的向量: $z_1=2+2 \\mathrm{i}$, $z_2=-3-2 \\mathrm{i}$, $z_3=2 \\mathrm{i}$, $z_4=-4$, $z_5=2-2 \\mathrm{i}$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -477171,7 +477627,9 @@ "id": "018543", "content": "设复平面上的点$A$和点$B$所对应的复数分别为$z_A$和$z_B$, 试用$z_A$和$z_B$表示复平面上的向量$\\overrightarrow{AB}$所对应的复数$z$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -477191,7 +477649,9 @@ "id": "018544", "content": "设$z \\in \\mathbf{C}$, 复平面上的点$Z$与$Z'$分别表示$z$与$z \\mathrm{i}$. 求证: $\\overrightarrow{OZ} \\perp \\overrightarrow{OZ'}$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -477211,7 +477671,9 @@ "id": "018545", "content": "如图, 在复平面上给定平行四边形$OABC$, 其中点$A$与点$C$分别对应于复数$z_A=-1+\\mathrm{i}$与$z_C=3+2 \\mathrm{i}$, 求点$B$所对应的复数.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-2,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (-1,1) node [left] {$A$} coordinate (A);\n\\draw (3,2) node [right] {$C$} coordinate (C);\n\\draw ($(A)+(C)-(O)$) node [above] {$B$} coordinate (B);\n\\draw (O)--(A)--(B)--(C)--cycle;\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -477231,7 +477693,9 @@ "id": "018546", "content": "已知复数$z$、$2 \\overline {z}$在复平面上所对应的点分别为$Z$、$Z'$, 若向量$\\overrightarrow{ZZ'}=(1,3)$, 求$z+2 \\overline {z}$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -477251,7 +477715,9 @@ "id": "018547", "content": "如图, 已知复平面上点$A(-2,2)$, 点$B(1,3)$, 点$D$与点$B$关于$x$轴成轴对称. 若$ABCD$为平行四边形, 求点$C$所对应的复数.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.3]\n\\draw [->] (-3,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-2,2) node [left] {$A$} coordinate (A);\n\\draw (1,3) node [above] {$B$} coordinate (B);\n\\draw (1,-3) node [below] {$D$} coordinate (D);\n\\draw ($(B)+(D)-(A)$) node [right] {$C$} coordinate (C);\n\\draw (A)--(B)--(C)--(D)--cycle;\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -477271,7 +477737,9 @@ "id": "018548", "content": "已知复数$z$满足$|z|=1$, 求证: $z+\\dfrac{1}{z}$是实数.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -477291,7 +477759,9 @@ "id": "018549", "content": "求下列复数的模:\\\\\n(1) $\\dfrac{(1-\\mathrm{i})(1+2 \\mathrm{i})}{4+3 \\mathrm{i}}$;\\\\\n(2) $\\dfrac{(7-3 \\mathrm{i})(5+4 \\mathrm{i})}{(7+3 \\mathrm{i})(-4-5 \\mathrm{i})}$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -477311,7 +477781,9 @@ "id": "018550", "content": "设复数$-\\sqrt{5}+2 \\mathrm{i}$和复数$2+\\sqrt{5} \\mathrm{i}$在复平面上分别对应点$A$和点$B$, 求$A$、$B$两点间的距离.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -477331,7 +477803,9 @@ "id": "018551", "content": "若复数$z$满足$|z-3|+|z-4 \\mathrm{i}|=5$, 求$|z|$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -477351,7 +477825,9 @@ "id": "018552", "content": "设复平面上三点$A$、$B$、$C$所对应的复数分别是$z_A$、$z_B$、$z_C$, 若$\\dfrac{z_B-z_A}{z_C-z_A}=2 \\mathrm{i}$, 求:\\\\\n(1) $\\angle BAC$的大小;\\\\\n(2) $\\angle ABC$的正切值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -477371,7 +477847,9 @@ "id": "018553", "content": "在复数范围求$25$与$-25$的平方根.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -477391,7 +477869,9 @@ "id": "018554", "content": "在复数范围内解方程: $2 x^2-4 x+5=0$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -477411,7 +477891,9 @@ "id": "018555", "content": "如果$p$、$q$都是实数, 而关于$x$的方程$2 x^2+p x+q=0$有一个根$-2+3 \\mathrm{i}$, 求$p$、$q$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -477431,7 +477913,9 @@ "id": "018556", "content": "已知实系数一元二次方程$x^2-2 x+m=0$的两根为$x_1$、$x_2$, 且$|x_1|+|x_2|=8$, 求$m$的值. 某同学的解答如下:\\\\\n{\\it 解: 因为$x_1$、$x_2$是$x^2-2 x+m=0$的两个根, 所以$\\begin{cases}x_1+x_2=2, \\\\ x_1 x_2=m.\\end{cases}$\n由 $|x_1|+|x_2|=8$, 即$x_1^2+2|x_1 x_2|+x_2^2=64$, 即$(x_1+x_2)^2-2 x_1 x_2+2|x_1 x_2|=64$, 即$4-2 m+2|m|=64$, 解得$m=-15$.}\\\\\n上述解法是否正确? 若有误, 请指出错误之处, 并写出正确的解答过程.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -477451,7 +477935,9 @@ "id": "018557", "content": "分别写出下列复数的模$r$与辐角主值$\\theta$, 并把这些复数用三角形式表示:\\\\\n(1) $\\sqrt{3}+\\mathrm{i}$;\\\\\n(2) $-1+\\mathrm{i}$;\\\\\n(3) $-1$;\\\\\n(4) $-3-4 \\mathrm{i}$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -477471,7 +477957,9 @@ "id": "018558", "content": "把下列复数用三角形式表示:\\\\\n(1) $\\cos \\theta-\\mathrm{i} \\sin \\theta$;\\\\\n(2) $-2(\\cos \\alpha+\\mathrm{i} \\sin \\alpha)$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -477491,7 +477979,9 @@ "id": "018559", "content": "若复数$z$满足$|z|=2 \\sqrt{3}$, $\\arg z=\\dfrac{7 \\pi}{6}$, 则$z$的代数形式为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -477511,7 +478001,9 @@ "id": "018560", "content": "设$a \\in \\mathbf{R}$, 复数$z=(-2 a^2-1)+(a^2-a+1) \\mathrm{i}$的辐角主值是\\bracket{20}.\n\\fourch{第一象限的角}{第二象限的角}{第三象限的角}{第四象限的角}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -477531,7 +478023,9 @@ "id": "018561", "content": "将复数$\\sin \\alpha+\\mathrm{i} \\cos \\alpha$用三角形式表示.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -477551,7 +478045,9 @@ "id": "018562", "content": "设$\\theta \\in(0, \\dfrac{\\pi}{2}) \\cup(\\dfrac{\\pi}{2}, \\pi)$, 将复数$1+\\mathrm{i}\\tan \\theta$用三角形式表示.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -477571,7 +478067,9 @@ "id": "018563", "content": "已知$z_1=\\dfrac{3}{2}(\\cos \\dfrac{\\pi}{12}+\\mathrm{i} \\sin \\dfrac{\\pi}{12})$, $z_2=2(\\cos \\dfrac{\\pi}{4}+\\mathrm{i} \\sin \\dfrac{\\pi}{4})$, 计算$z_1 z_2$, 结果用复数的代数形式表示.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -477591,7 +478089,9 @@ "id": "018564", "content": "如图, 设复数$-2+2 \\mathrm{i}$在复平面上所对应的向量是$\\overrightarrow{OZ}$, 将$\\overrightarrow{OZ}$绕原点$O$逆时针旋转$120^{\\circ}$得到向量$\\overrightarrow{OZ'}$. 求向量$\\overrightarrow{OZ'}$所对应的复数. (结果用复数的代数形式表示)\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3.5) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {-2,-1,1}\n{\\draw [dashed, gray] (\\i,-3.5) -- (\\i,3);\n\\draw (\\i,0) node [below left] {$\\i$};};\n\\foreach \\i in {-3,-2,-1,1,2}\n{\\draw [dashed, gray] (-3,\\i) -- (3,\\i);\n\\draw (0,\\i) node [right] {$\\i$};};\n\\draw [->] (0,0) -- (-2,2) node [above] {$Z$} coordinate (Z);\n\\draw ($(O)!1!120:(Z)$) node [below] {$Z'$} coordinate (Z');\n\\draw [->] (O)--(Z');\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -477611,7 +478111,9 @@ "id": "018565", "content": "计算$\\dfrac{4(\\cos \\dfrac{4 \\pi}{3}+\\mathrm{i} \\sin \\dfrac{4 \\pi}{3})}{2(\\cos \\dfrac{5 \\pi}{6}+\\mathrm{i} \\sin \\dfrac{5 \\pi}{6})}$, 并用复数的代数形式表示.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -477631,7 +478133,9 @@ "id": "018566", "content": "计算$(1-\\mathrm{i})^{20}$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -477651,7 +478155,9 @@ "id": "018567", "content": "求$1$的三次方根.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -477671,7 +478177,9 @@ "id": "018568", "content": "在复数范围内解方程: $(\\sqrt{3}-\\mathrm{i}) x^4=32 \\mathrm{i}$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -477691,7 +478199,9 @@ "id": "018569", "content": "已知$n$为正整数, 若复数$z=(\\dfrac{3}{3+\\sqrt{3} \\mathrm{i}})^n$为实数, 求$n$的最小值及相应的$z$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -477711,7 +478221,9 @@ "id": "018570", "content": "求实数$m$的值或取值范围, 使得复数$z=(m+1)+(3 m-2) \\mathrm{i}$分别满足:\\\\\n(1) 复数$z$是虚数;\\\\\n(2) 复数$z$在复平面上所对应的点位于虚轴上;\\\\\n(3) 复数$z$在复平面上所对应的点位于第三象限;\\\\\n(4) 复数$z$在复平面上所对应的点到原点距离为$5$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -477731,7 +478243,9 @@ "id": "018571", "content": "已知复平面上如图所示的平行四边形$ABCD$的顶点$A$、$B$、$D$三点所对应的复数分别是$2+3 \\mathrm{i}$、$5-\\mathrm{i}$、$6+\\mathrm{i}$, 求向量$\\overrightarrow{AC}$所对应的复数.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.3]\n\\draw [->] (-1,0) -- (10,0) node [below] {$x$};\n\\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (2,3) node [above] {$A$} coordinate (A);\n\\draw (5,-1) node [below] {$B$} coordinate (B);\n\\draw (6,1) node [above] {$D$} coordinate (D);\n\\draw (9,-3) node [right] {$C$} coordinate (C);\n\\draw (A)--(B)--(C)--(D)--cycle;\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -477751,7 +478265,9 @@ "id": "018572", "content": "已知复数$z_1=\\sqrt{3}+\\mathrm{i},|z_2|=1, z_1 \\overline{z_2}$是虚部为负数的纯虚数, 求复数$z_2$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -477771,7 +478287,9 @@ "id": "018573", "content": "若虚数$z_1$、$z_2$满足$z_1^2=z_2$, 且$z_1$、$z_2$是一个实系数一元二次方程的两个根, 写出这样的一个实系数一元二次方程.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -477791,7 +478309,9 @@ "id": "018574", "content": "是否存在虚数$z$, 使$z+\\dfrac{5}{z} \\in \\mathbf{R}$, 且$z+3$的实部与虚部互为相反数? 若存在, 求出虚数$z$; 若不存在, 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -477811,7 +478331,9 @@ "id": "018575", "content": "已知复数$z$满足$z \\overline {z}=2$, $z^2$的虚部为$2$.\\\\\n(1) 求复数$z$;\\\\\n(2) 设$z, z^2, z-z^2$在复平面上所对应的点分别为$A, B, C$, 求$\\triangle ABC$的面积.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "",