diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 4d47402d..40dfc132 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -96356,7 +96356,9 @@ "20220701\t王伟叶" ], "same": [], - "related": [], + "related": [ + "024592" + ], "remark": "", "space": "", "unrelated": [] @@ -232438,7 +232440,8 @@ ], "same": [], "related": [ - "021474" + "021474", + "024585" ], "remark": "", "space": "4em", @@ -235195,7 +235198,8 @@ ], "same": [], "related": [ - "010232" + "010232", + "024590" ], "remark": "", "space": "4em", @@ -288498,7 +288502,8 @@ ], "same": [], "related": [ - "008226" + "008226", + "024590" ], "remark": "", "space": "4em", @@ -442216,7 +442221,8 @@ ], "same": [], "related": [ - "023787" + "023787", + "024592" ], "remark": "", "space": "4em", @@ -497683,7 +497689,9 @@ "20230705\t王伟叶" ], "same": [], - "related": [], + "related": [ + "024590" + ], "remark": "", "space": "4em", "unrelated": [] @@ -497795,7 +497803,9 @@ "20230705\t王伟叶" ], "same": [], - "related": [], + "related": [ + "024590" + ], "remark": "", "space": "4em", "unrelated": [] @@ -582902,7 +582912,8 @@ ], "same": [], "related": [ - "008119" + "008119", + "024585" ], "remark": "", "space": "", @@ -666955,6 +666966,200 @@ "space": "4em", "unrelated": [] }, + "024585": { + "id": "024585", + "content": "计算:\\\\\n(1) $6 \\cos 270^{\\circ}+10 \\sin 0^{\\circ}-4 \\tan 180^{\\circ}+5 \\cos 360^{\\circ}=$\\blank{50}.\\\\\n(2) $\\sin ^2 \\dfrac{\\pi}{4}-\\cos ^2 \\dfrac{\\pi}{2}+2 \\tan ^3 \\dfrac{\\pi}{4}=$\\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": [ + "021474", + "008119" + ], + "remark": "", + "space": "", + "unrelated": [] + }, + "024586": { + "id": "024586", + "content": "已知一个扇形的圆心角是 $\\dfrac{\\pi}{3}$, 弧长等于 $\\pi \\mathrm{cm}$, 则这个扇形的半径是\\blank{50}cm.", + "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": [] + }, + "024587": { + "id": "024587", + "content": "若 $\\dfrac{\\cos \\alpha}{\\sin \\alpha}=2$, 则 $\\dfrac{3 \\sin \\alpha-2 \\cos \\alpha}{2 \\sin \\alpha-3 \\cos \\alpha}$ 的值是\\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": [ + "040068" + ], + "remark": "", + "space": "", + "unrelated": [] + }, + "024588": { + "id": "024588", + "content": "若 $\\tan \\beta=-\\dfrac{\\sqrt{3}}{3}$, 且 $\\dfrac{\\pi}{2}<\\beta<\\pi$, 则 $\\beta=$\\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": [] + }, + "024589": { + "id": "024589", + "content": "若 $\\sin \\theta \\geq 0$ 且 $\\tan \\theta \\leq 0$, 请用集合的形式表达 $\\theta$ 的范围是\\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": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "024590": { + "id": "024590", + "content": "已知 $\\sin \\alpha+\\cos \\alpha=\\dfrac{1}{5}$, $0<\\alpha<\\pi$, 则 $\\sin \\alpha-\\cos \\alpha$ 的值是\\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": [ + "040067", + "018347", + "010232", + "018352", + "008226" + ], + "remark": "", + "space": "", + "unrelated": [] + }, + "024591": { + "id": "024591", + "content": "下面命题中, 正确的命题序号是\\blank{50}.\\\\\n\\textcircled{1} 与 $-\\dfrac{\\pi}{3}$ 终边相同的角的集合是 $\\{\\alpha | \\alpha=k \\cdot 360^{\\circ}-60^{\\circ}\\}$;\\\\\n\\textcircled{2} 若 $A=\\{\\alpha | \\alpha=\\dfrac{k \\pi}{6}, k \\in \\mathbf{Z}\\}$, $B=\\{\\beta | \\beta=\\dfrac{k \\pi}{2}$ 或 $\\dfrac{k \\pi}{2}\\pm \\dfrac{5 \\pi}{6}, k \\in \\mathbf{Z}\\}$, $C=\\{\\gamma | \\gamma=-\\dfrac{k \\pi}{6}, k \\in \\mathbf{Z}\\}$, 则 $A=B=C$;\\\\\n\\textcircled{3} 若 $\\alpha$ 在第三象限, 则 $\\dfrac{\\alpha}{3}$ 一定不在第三象限;\\\\\n\\textcircled{4} ``$\\alpha$ 是锐角''是``$2 \\alpha$ 是第一、二象限的角''的充分不必要条件.", + "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": [] + }, + "024592": { + "id": "024592", + "content": "已知角 $\\alpha$ 的顶点与坐标原点 $O$ 重合, 始边在 $x$ 轴的正半轴上, 终边上有一点 $P(1+m, 1-m)$, 并且 $\\sin \\alpha=-\\dfrac{\\sqrt{5}}{5}$. 求 $m$ 的值及 $\\cos \\alpha$、$\\tan\\alpha$ 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": [ + "015890", + "003058" + ], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "024593": { + "id": "024593", + "content": "已知 $\\tan ^2 \\alpha=2 \\tan ^2 \\beta+1$, 求 $\\sin ^2 \\beta-2 \\sin ^2 \\alpha-1$ 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": [], + "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