From c7acbd74e716f8d787268577dafece0c14328e25 Mon Sep 17 00:00:00 2001 From: "weiye.wang" Date: Sat, 17 Feb 2024 16:53:37 +0800 Subject: [PATCH] =?UTF-8?q?=E5=AF=BC=E5=85=A5W20260201=E6=96=B0=E9=A2=98?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 题库0.3/Problems.json | 466 ++++++++++++++++++++++++++++++++++++++++-- 1 file changed, 446 insertions(+), 20 deletions(-) diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 21f4377b..4d47402d 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -16297,7 +16297,9 @@ "20220624\t朱敏慧, 王伟叶" ], "same": [], - "related": [], + "related": [ + "024576" + ], "remark": "", "space": "", "unrelated": [] @@ -96922,7 +96924,9 @@ "20220701\t王伟叶" ], "same": [], - "related": [], + "related": [ + "024581" + ], "remark": "", "space": "4em", "unrelated": [] @@ -97076,7 +97080,9 @@ "20220701\t王伟叶" ], "same": [], - "related": [], + "related": [ + "024577" + ], "remark": "", "space": "4em", "unrelated": [] @@ -97120,7 +97126,9 @@ "20220701\t王伟叶" ], "same": [], - "related": [], + "related": [ + "024582" + ], "remark": "", "space": "4em", "unrelated": [] @@ -176665,7 +176673,8 @@ ], "same": [], "related": [ - "008110" + "008110", + "024568" ], "remark": "", "space": "", @@ -178620,7 +178629,8 @@ ], "same": [], "related": [ - "005959" + "005959", + "024581" ], "remark": "", "space": "4em", @@ -232207,7 +232217,8 @@ ], "same": [], "related": [ - "005879" + "005879", + "024568" ], "remark": "", "space": "", @@ -232934,7 +232945,9 @@ "20220726\t王伟叶" ], "same": [], - "related": [], + "related": [ + "024576" + ], "remark": "", "space": "4em", "unrelated": [] @@ -288219,7 +288232,9 @@ "20220806\t王伟叶" ], "same": [], - "related": [], + "related": [ + "024568" + ], "remark": "", "space": "4em", "unrelated": [] @@ -288287,7 +288302,9 @@ "20220806\t王伟叶" ], "same": [], - "related": [], + "related": [ + "024573" + ], "remark": "", "space": "", "unrelated": [] @@ -288335,7 +288352,9 @@ "20220806\t王伟叶" ], "same": [], - "related": [], + "related": [ + "024583" + ], "remark": "", "space": "4em", "unrelated": [] @@ -442563,7 +442582,9 @@ "20230503\t王伟叶" ], "same": [], - "related": [], + "related": [ + "024583" + ], "remark": "", "space": "4em", "unrelated": [] @@ -497706,7 +497727,9 @@ "20230705\t王伟叶" ], "same": [], - "related": [], + "related": [ + "024577" + ], "remark": "", "space": "4em", "unrelated": [] @@ -535730,7 +535753,8 @@ ], "same": [], "related": [ - "013011" + "013011", + "024583" ], "remark": "", "space": "", @@ -582107,7 +582131,9 @@ "20230209\t王伟叶" ], "same": [], - "related": [], + "related": [ + "024569" + ], "remark": "", "space": "", "unrelated": [] @@ -582753,7 +582779,9 @@ "20230209\t王伟叶" ], "same": [], - "related": [], + "related": [ + "024575" + ], "remark": "", "space": "", "unrelated": [] @@ -583727,7 +583755,9 @@ "20230209\t王伟叶" ], "same": [], - "related": [], + "related": [ + "024580" + ], "remark": "", "space": "4em", "unrelated": [] @@ -583817,7 +583847,9 @@ "20230209\t王伟叶" ], "same": [], - "related": [], + "related": [ + "024582" + ], "remark": "", "space": "4em", "unrelated": [] @@ -648506,7 +648538,8 @@ ], "same": [], "related": [ - "015890" + "015890", + "024575" ], "remark": "", "space": "", @@ -666531,6 +666564,397 @@ "space": "4em", "unrelated": [] }, + "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.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "自拟题目", + "edit": [ + "20240217\t王伟叶" + ], + "same": [], + "related": [ + "040527" + ], + "remark": "", + "space": "", + "unrelated": [] + }, + "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}$\\}$.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "自拟题目", + "edit": [ + "20240217\t王伟叶" + ], + "same": [], + "related": [ + "005879", + "008110", + "010224" + ], + "remark": "", + "space": "", + "unrelated": [] + }, + "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, + "usages": [], + "origin": "自拟题目", + "edit": [ + "20240217\t王伟叶" + ], + "same": [], + "related": [ + "021448" + ], + "remark": "", + "space": "", + "unrelated": [] + }, + "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": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "自拟题目", + "edit": [ + "20240217\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "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.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "自拟题目", + "edit": [ + "20240217\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "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": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "自拟题目", + "edit": [ + "20240217\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "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": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "自拟题目", + "edit": [ + "20240217\t王伟叶" + ], + "same": [], + "related": [ + "010226" + ], + "remark": "", + "space": "", + "unrelated": [] + }, + "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}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "自拟题目", + "edit": [ + "20240217\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "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": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "自拟题目", + "edit": [ + "20240217\t王伟叶" + ], + "same": [], + "related": [ + "023787", + "021470" + ], + "remark": "", + "space": "", + "unrelated": [] + }, + "024576": { + "id": "024576", + "content": "已知 $\\sin \\alpha=-\\dfrac{1}{4}$, 且角 $\\alpha$ 是第三象限角, $\\tan \\alpha=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "自拟题目", + "edit": [ + "20240217\t王伟叶" + ], + "same": [], + "related": [ + "000448", + "008141" + ], + "remark": "", + "space": "", + "unrelated": [] + }, + "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}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "自拟题目", + "edit": [ + "20240217\t王伟叶" + ], + "same": [], + "related": [ + "003087", + "018349" + ], + "remark": "", + "space": "", + "unrelated": [] + }, + "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}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "自拟题目", + "edit": [ + "20240217\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "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).", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "自拟题目", + "edit": [ + "20240217\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "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.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "自拟题目", + "edit": [ + "20240217\t王伟叶" + ], + "same": [], + "related": [ + "021502" + ], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "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" + ], + "same": [], + "related": [ + "003080", + "005957" + ], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "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 $\\cos \\alpha-\\sin \\alpha$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "自拟题目", + "edit": [ + "20240217\t王伟叶" + ], + "same": [], + "related": [ + "021505", + "003089" + ], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "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": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "自拟题目", + "edit": [ + "20240217\t王伟叶" + ], + "same": [], + "related": [ + "010228", + "019951", + "015906" + ], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "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?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "自拟题目", + "edit": [ + "20240217\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, "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