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导入W20260201新题

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题库0.3/Problems.json
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@ -16297,7 +16297,9 @@
"20220624\t朱敏慧, 王伟叶"
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"024567": {
"id": "024567",
"content": "$-1005^{\\circ}$ 是第\\blank{20}象限角, $\\dfrac{145 \\pi}{7}$ 是第\\blank{20}象限角, $10$ 是第\\blank{20}象限角, $31415$ 是第\\blank{20}象限角.",
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"20240217\t王伟叶"
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"024568": {
"id": "024568",
"content": "(1) 终边在直线 $y=-\\sqrt{3}x$ 上的角的集合为 $\\{\\gamma | \\gamma=$\\blank{100}$\\}$;\\\\\n(2) 终边在第一象限或第二象限角平分线上的角的集合为 $\\{\\theta | \\theta=$\\blank{100}$\\}$.",
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"024569": {
"id": "024569",
"content": "(1) 在与 $2014^{\\circ}$ 角终边相同的角中, 最小的正角是\\blank{50}, 最大的负角是\\blank{50}.\\\\\n(2) 在与 $2014^{\\circ}$ 角终边垂直的角中, 最小的正角是\\blank{50}, 最大的负角是\\blank{50}.",
"objs": [],
"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
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"20240217\t王伟叶"
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"024570": {
"id": "024570",
"content": "若半径为 $r$ 的圆的弦 $AB$ 的长度等于 $\\sqrt{2}r$, 则弦 $AB$ 所对的劣弧长等于\\blank{50}.",
"objs": [],
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"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
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"024571": {
"id": "024571",
"content": "给出下列四个命题, 其中假命题有\\blank{50}.\\\\\n\\textcircled{1} 若 $\\alpha$ 是第二象限的角, 则 $2 \\alpha$ 是第四象限的角;\\\\\n\\textcircled{2} 若 $\\alpha$ 是第二象限的角, 则 $\\dfrac{\\alpha}{2}$ 是第一象限的角;\\\\\n\\textcircled{3} 若 $\\alpha, \\beta$ 都是第一象限的角, 则 $\\alpha+\\beta$ 是第二象限的角;\\\\\n\\textcircled{4} 若 $\\alpha, \\beta$ 都是第三象限的角, 则 $\\dfrac{\\alpha+\\beta}{2}$ 也是第三象限的角.",
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"024572": {
"id": "024572",
"content": "已知 $\\theta$ 是 $(0,2 \\pi)$ 中的角, 若 $5 \\theta$ 和 $\\theta$ 的终边重合, 则 $\\theta$ 的取值集合用列举法表示为\\blank{50}.",
"objs": [],
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"genre": "填空题",
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"024573": {
"id": "024573",
"content": "若 $\\alpha$ 与 $\\beta$ 的终边 (作为两条射线)关于 $\\gamma=\\dfrac{\\pi}{7}$ 的终边所在直线对称, 则 $\\alpha$ 与 $\\beta$ 之间的关系为\\blank{50}.",
"objs": [],
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"024574": {
"id": "024574",
"content": "已知角 $\\alpha$ 的终边在下图中的阴影部分内, 分别写出角 $\\alpha$ 的范围(图中所标出的不在坐标轴上的边界线均三等分所在象限).\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\fill [pattern = north east lines] (0,0) -- (90:2) arc (90:120:2) -- cycle;\n\\draw (0,0) -- (120:2);\n\\draw (0,-2.5) node {(1)};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {0,90,180,270}\n{\\fill [pattern = north east lines] (0,0) -- ({90+\\i}:2) arc ({90+\\i}:{120+\\i}:2) -- cycle;\n\\draw (0,0) -- ({120+\\i}:2);};\n\\draw (0,-2.5) node {(2)};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {0,90,180,270}\n{\\fill [pattern = north east lines] (0,0) -- ({60+\\i}:2) arc ({60+\\i}:{120+\\i}:2) -- cycle;\n\\draw (0,0) -- ({120+\\i}:2);\n\\draw (0,0) -- ({60+\\i}:2);};\n\\draw (0,-2.5) node {(3)};\n\\end{tikzpicture}\n\\end{center}\n(1) \\blank{100}; (2) \\blank{100}; (3) \\blank{100}.",
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"024575": {
"id": "024575",
"content": "若 $P(1, y)$ 是角 $\\alpha$ 终边上的一点, 且 $\\cos \\alpha=\\dfrac{\\sqrt{3}}{6}$, 则 $y$ 的值为\\blank{50}.",
"objs": [],
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"ans": "",
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"021470"
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"024576": {
"id": "024576",
"content": "已知 $\\sin \\alpha=-\\dfrac{1}{4}$, 且角 $\\alpha$ 是第三象限角, $\\tan \\alpha=$\\blank{50}.",
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"duration": -1,
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"024577": {
"id": "024577",
"content": "已知: $\\tan \\alpha=-\\dfrac{3}{4}$, 则 $\\sin ^2 \\alpha-\\sin \\alpha \\cos \\alpha-\\sqrt{3}=$\\blank{50}, $\\dfrac{2 \\sin \\alpha-3 \\cos \\alpha+4}{3 \\sin \\alpha+4 \\cos \\alpha}=$\\blank{50}.",
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"tags": [],
"genre": "填空题",
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"024578": {
"id": "024578",
"content": "设 $\\alpha$ 是第二象限角, $\\cos \\dfrac{\\alpha}{2}-\\sin \\dfrac{\\alpha}{2}=\\dfrac{1}{3}$, 则 $\\cos \\dfrac{\\alpha}{2}+\\sin \\dfrac{\\alpha}{2}=$\\blank{50}.",
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"genre": "填空题",
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"024579": {
"id": "024579",
"content": "已知在直角坐标系中, $O$ 是坐标原点, 角 $\\alpha$ 的终边上一点 $A$, 且 $|OA|=2$, 角 $\\beta$ 终边上一点 $B$, 且 $|OB|=4$.\\\\\n(1) 若 $\\alpha=\\dfrac{\\pi}{6}$, $\\beta=\\dfrac{3 \\pi}{4}$, 写出点 $A, B$ 的坐标, 并求出 $A, B$ 的距离 $|AB|$;\\\\\n(2) 若点 $P$ 为线段 $AB$ 的中点, 求以 $OP$ 为终边的角 $\\theta$ 的正切值(用 $\\alpha, \\beta$ 的三角比来表示).",
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"genre": "解答题",
"ans": "",
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"duration": -1,
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"edit": [
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"remark": "",
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"024580": {
"id": "024580",
"content": "已知 $\\sin \\theta=\\dfrac{m-3}{m+5}$, $\\cos \\theta=\\dfrac{4-2 m}{m+5}$, 其中 $\\dfrac{3 \\pi}{2}\\leq \\theta \\leq 2 \\pi$, 求 $\\cot \\theta$ 的值.",
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"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "自拟题目",
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"20240217\t王伟叶"
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"024581": {
"id": "024581",
"content": "已知 $\\sin \\theta+\\cos \\theta=\\dfrac{2}{3}$($0<\\theta<\\pi$), 求: \\\\\n(1) $\\sin \\theta-\\cos \\theta$ 的值;\\\\\n(2) $\\tan \\theta+\\cot \\theta$ 的值;\\\\\n(3) $\\sin ^3 \\theta-\\cos ^3 \\theta$ 的值;\\\\\n(4) $\\sin ^4 \\theta+\\cos ^4 \\theta$.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "自拟题目",
"edit": [
"20240217\t王伟叶"
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"024582": {
"id": "024582",
"content": "已知 $\\tan \\alpha, \\cot \\alpha$ 是方程 $2 x^2-2 k x+k^2-3=0$ 的两个实根, 且 $\\pi<\\alpha<\\dfrac{5 \\pi}{4}$, 求 $\\cos \\alpha-\\sin \\alpha$.",
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"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "自拟题目",
"edit": [
"20240217\t王伟叶"
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"003089"
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"024583": {
"id": "024583",
"content": "(1)已知扇形的周长为 $8 \\mathrm{cm}$ , 圆心角为 $2 \\mathrm{rad}$, 求扇形的面积 $S$.\\\\\n(2) 用总长为 $m$ 的绳子围成一个扇形, 如何围可以使得所围成的扇形面积 $S$ 最大?试确定扇形面积最大时的圆心角与半径.",
"objs": [],
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"ans": "",
"solution": "",
"duration": -1,
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"024584": {
"id": "024584",
"content": "新年的时候小明想要造一个扇形的围栏, 但是他爸爸只给了他 $300$ 元. 已知扇形围栏的弯的(圆弧)部分每米需要 $40$ 元, 直的(半径)部分每米只需要 $20$ 元. 问小明最多可以造一个多大的围栏, 应该如何建造?",
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"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
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"030001": {
"id": "030001",
"content": "若$x,y,z$都是实数, 则:(填写``\\textcircled{1} 充分非必要、\\textcircled{2} 必要非充分、\\textcircled{3} 充要、\\textcircled{4} 既非充分又非必要''之一)\\\\\n(1) ``$xy=0$''是``$x=0$''的\\blank{50}条件;\\\\\n(2) ``$x\\cdot y=y\\cdot z$''是``$x=z$''的\\blank{50}条件;\\\\\n(3) ``$\\dfrac xy=\\dfrac yz$''是``$xz=y^2$''的\\blank{50}条件;\\\\\n(4) ``$|x |>| y|$''是``$x>y>0$''的\\blank{50}条件;\\\\\n(5) ``$x^2>4$''是``$x>2$'' 的\\blank{50}条件;\\\\\n(6) ``$x=-3$''是``$x^2+x-6=0$'' 的\\blank{50}条件;\\\\\n(7) ``$|x+y|<2$''是``$|x|<1$且$|y|<1$'' 的\\blank{50}条件;\\\\\n(8) ``$|x|<3$''是``$x^2<9$'' 的\\blank{50}条件;\\\\\n(9) ``$x^2+y^2>0$''是``$x\\ne 0$'' 的\\blank{50}条件;\\\\\n(10) ``$\\dfrac{x^2+x+1}{3x+2}<0$''是``$3x+2<0$'' 的\\blank{50}条件;\\\\\n(11) ``$0<x<3$''是``$|x-1|<2$'' 的\\blank{50}条件.",
@ -736699,7 +737123,9 @@
"20230403\t王伟叶"
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