diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 4c2aa34d..a407bd04 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -599137,6 +599137,1266 @@ "space": "4em", "unrelated": [] }, + "022697": { + "id": "022697", + "content": "已知集合 $A=\\{-1,0\\}$, 集合 $B=\\{2, a\\}$, 若 $A \\cap B=\\{0\\}$, 则 $a=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2026届高一上学期期中考试试题1", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022698": { + "id": "022698", + "content": "``$x>1$ 或 $y>1$''的否定形式为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2026届高一上学期期中考试试题2", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022699": { + "id": "022699", + "content": "不等式 $\\dfrac{x-2}{x-1}<0$ 的解集为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2026届高一上学期期中考试试题3", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022700": { + "id": "022700", + "content": "已知幂函数 $y=x^a$ 的图像经过点 $(\\sqrt[4]{2}, 2)$, 则实数 $a$ 的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2026届高一上学期期中考试试题4", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022701": { + "id": "022701", + "content": "关于 $x$ 的方程 $x^2-4 x+1=0$ 的两根为 $x_1, x_2$, 则 $x_1^2+x_2^2$ 的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2026届高一上学期期中考试试题5", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022702": { + "id": "022702", + "content": "已知 $\\log _23=a$, 用 $a$ 表示 $\\log _46=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2026届高一上学期期中考试试题6", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022703": { + "id": "022703", + "content": "已知正实数 $a, b$ 满足 $a+2 b=1$, 则 $a b$ 的最大值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2026届高一上学期期中考试试题7", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022704": { + "id": "022704", + "content": "对于任意实数 $x$, 不等式 $a x^2+a x+1>0$ 恒成立, 则实数 $a$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2026届高一上学期期中考试试题8", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022705": { + "id": "022705", + "content": "对任意 $x \\leq 1$, 指数函数 $y=a^x$ 的值总大于 $\\dfrac{1}{2}$, 则实数 $a$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2026届高一上学期期中考试试题9", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022706": { + "id": "022706", + "content": "关于 $x$ 的不等式 $|x-1|+|x-a| \\geq a$ 的解集是 $\\mathbf{R}$, 则实数 $a$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2026届高一上学期期中考试试题10", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022707": { + "id": "022707", + "content": "已知正实数 $a, b$ 满足 $a^{\\lg b}=2$, $a^{\\lg a}b^{\\lg b}=5$, 则 $(a b)^{\\lg (a b)}$ 的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2026届高一上学期期中考试试题11", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022708": { + "id": "022708", + "content": "若``对于任意的实数 $a$, 关于 $x$ 的不等式 $|2^x+a| \\geq m$ 在区间 $[0,1]$ 上总有解''是真命题, 则实数 $m$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2026届高一上学期期中考试试题12", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022709": { + "id": "022709", + "content": "函数 $y=x^{\\frac{2}{3}}$ 的图像是\\bracket{20}.\n\\fourch{\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1.8) -- (0,1.8) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0.0001:2] plot (\\x,{pow(\\x,0.6)}) plot ({-\\x},{-pow(\\x,0.6)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-0.6) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0.0001:2] plot (\\x,{pow(\\x,1.5)}) plot ({-\\x},{pow(\\x,1.5)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-0.6) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0.55:2] plot (\\x,{pow(\\x,-1.5)}) plot ({-\\x},{pow(\\x,-1.5)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-0.6) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0.0001:2] plot (\\x,{pow(\\x,2/3)}) plot ({-\\x},{pow(\\x,2/3)});\n\\end{tikzpicture}}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2026届高一上学期期中考试试题13", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022710": { + "id": "022710", + "content": "已知实数 $a$ 满足 $0b$, 则 $a^3>b^3$; \\textcircled{2} 若 $a>b>1$, 则 $\\log _a 2>\\log _b 2$; \\textcircled{3} 若 $ab d$; \\textcircled{4} 若 $10\\}$ (其中常数 $a>0$, $a \\neq 1$), $B=\\{y | y=x^k, x \\in A\\}$ (其中 $k$ 是常数), 则``$k<0$''是``$A \\cap B=\\varnothing$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充分必要条件}{既非充分又非必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2026届高一上学期期中考试试题16", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022713": { + "id": "022713", + "content": "已知函数 $y=\\log _2(9-x^2)$ 的定义域为集合 $A$, 集合 $B=[a-2, a+2]$.\\\\\n(1) 当 $a=2$ 时, 求 $A \\cup B$;\\\\\n(2) 若 $A \\cap B=B$, 求实数 $a$ 的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2026届高一上学期期中考试试题17", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "022714": { + "id": "022714", + "content": "已知 $a>0$, $b>0$.\\\\\n(1) 比较 $a^3+b^3$ 与 $a^2 b+b^2 a$ 的大小;\\\\\n(2) 若 $a+b=1$, 求 $\\dfrac{a^2}{b}+\\dfrac{b^2}{a}$ 的最小值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2026届高一上学期期中考试试题18", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "022715": { + "id": "022715", + "content": "《上海市生活垃圾管理条例》于 2019 年 7 月 1 日正式实施. 某小区全面实施垃圾分类处理. 已知该小区每月垃圾分类处理量不超过 $300$ 吨, 每月垃圾分类处理成本 $y$ (元) 与每月分类处理量 $x$ (吨) 之间的函数关系式可近似表示为 $y=x^2-200 x+40000$, 而分类处理一吨垃圾小区也可以获得 $300$ 元的收益.\\\\\n(1) 该小区每月分类处理多少吨垃圾, 才能使得每吨垃圾分类处理的平均成本最低?\\\\\n(2) 要保证该小区每月的垃圾分类处理不亏损, 每月的垃圾分类处理量应控制在什么范围?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2026届高一上学期期中考试试题19", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "022716": { + "id": "022716", + "content": "已知 $a$ 为实常数, 函数 $y=2^x+\\dfrac{a}{2^x}$.\\\\\n(1) 当 $a=-3$ 时, 求所有满足 $y=2$ 的 $x$ 的值;\\\\\n(2) 若对任意的 $x \\in \\mathbf{R}$, 都有 $y \\geq 3$ 成立, 求实数 $a$ 的取值范围;\\\\\n(3) 若方程 $2^x+\\dfrac{a}{2^x}=6$ 有两个不相等的实数根 $x_1, x_2$, 且 $|x_1-x_2| \\leq 1$, 求实数 $a$ 的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2026届高一上学期期中考试试题20", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "022717": { + "id": "022717", + "content": "集合 $A=\\{a_1, a_2, \\cdots, a_n\\}$ 是由 $n$($n \\geq 3$) 个正整数组成的集合, 如果任意去掉其中一个元素 $a_i$($i=1,2, \\cdots, n$)之后, 剩余的所有元素组成的集合都能分为两个交集为空的集合, 且这两个集合的所有元素之和相等, 就称集合 $A$ 为``可分集合''.\\\\\n(1) 判断集合 $\\{1,2,3,4\\}$, $\\{1,3,5,7,9,11,13\\}$ 是否为``可分集合''(不用说明理由);\\\\\n(2) 求证: 五个元素的集合 $A=\\{a_1, a_2, a_3, a_4, a_5\\}$ 一定不是``可分集合'';\\\\\n(3) 若集合 $A=\\{a_1, a_2, \\cdots, a_n\\}$ 是``可分集合'', 证明 $n$ 是奇数.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2026届高一上学期期中考试试题21", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "022718": { + "id": "022718", + "content": "已知球的半径为 $2$, 则球的表面积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二上学期期中考试试题1", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022719": { + "id": "022719", + "content": "空间中, 两条异面直线所成角的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二上学期期中考试试题2", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022720": { + "id": "022720", + "content": "已知圆柱的侧面展开图是边长为 $3$ 的正方形, 则圆柱的体积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二上学期期中考试试题3", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022721": { + "id": "022721", + "content": "已知正三棱柱 $ABC-A_1B_1C_1$ 的侧棱长与底面边长均相等, 则直线 $BC_1$ 与平面 $ABB_1A_1$ 所成角的正弦值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二上学期期中考试试题4", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022722": { + "id": "022722", + "content": "已知正四棱锥 $P-ABCD$ 的底面边长和侧棱长均为 $2$, 则该四棱锥的侧面与底面所成的二面角的大小为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二上学期期中考试试题5", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022723": { + "id": "022723", + "content": "已知数列 $\\{a_n\\}$ 满足 $a_{n+1}=2 a_n+3$($n \\in \\mathbf{N}$, $n \\geq 1$), 且 $a_1=2$, 则 $\\{a_n\\}$ 的通项公式\n$a_n=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二上学期期中考试试题6", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022724": { + "id": "022724", + "content": "有一块多边形花圃, 它的水平放置的平面图对应的斜二测直观图是一个直角梯形 (如图所示). 已知 $\\angle ABC=45$, $AB=\\sqrt{2}$ 千米, $AD=1$ 千米, $AD \\parallel BC$, $DC \\perp BC$, 则这块花圃的实际面积为\\blank{50}平方千米.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1,0) -- (3,0) node [below] {$x'$};\n\\draw [->] (45:-1) -- (45:2.5) node [left] {$y'$};\n\\draw (0,0) node [above] {$B$} coordinate (B);\n\\draw (2,0) node [below] {$C$} coordinate (C);\n\\draw (2,1) node [right] {$D$} coordinate (D);\n\\draw (1,1) node [left] {$A$} coordinate (A);\n\\draw (C)--(D)--(A);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二上学期期中考试试题7", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022725": { + "id": "022725", + "content": "圆锥的侧面积与表面积之比为 $3: 5$, 用通过圆锥的轴的平面截此圆锥, 则截面三角形的顶角为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二上学期期中考试试题8", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022726": { + "id": "022726", + "content": "已知等比数列 $\\{a_n\\}$ 中每一项均为正数, 且 $a_1$、$\\dfrac{1}{2}a_3$、$2 a_2$ 成等差数列, 则 $\\dfrac{a_{2025}+a_{2022}}{a_{2023}+a_{2020}}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二上学期期中考试试题9", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022727": { + "id": "022727", + "content": "已知 $\\angle ACB=60^\\circ$, 点 $P$ 为平面 $ABC$ 外一点, $PC=4$, 点 $P$ 到 $\\angle ACB$ 的两边 $AC$、$BC$ 的距离均为 $2 \\sqrt{3}$, 那么点 $P$ 到平面 $ABC$ 的距离为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二上学期期中考试试题10", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022728": { + "id": "022728", + "content": "已知正方体 $ABCD-A_1B_1C_1D_1$ 的棱长为 $2$, $E$ 为正方体表面上 (包含棱、顶点) 的一个动点. 若三棱锥 $A-EBC$ 的体积为 $\\dfrac{2}{3}$, 则 $|ED_1|$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二上学期期中考试试题11", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022729": { + "id": "022729", + "content": "如图所示, 在四面体 $ABCD$ 中, $AB=CD=\\sqrt{10}$, $AC=BD=\\sqrt{5}$, $AD=BC=\\sqrt{13}$, $E$ 和 $F$分别是 $AD$ 和 $BC$ 的中点. 若用一个与 $EF$ 垂直且与四面体的每个面都相交的平面 $\\alpha$ 去截该四面体, 由此得到一个多边形截面, 则该多边形截面面积的最大值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (3,0,1) node [above] {$B$} coordinate (B);\n\\draw (3,-2,0) node [below] {$D$} coordinate (D);\n\\draw (0,-2,1) node [below] {$C$} coordinate (C);\n\\draw (0,0,0) node [above] {$A$} coordinate (A);\n\\draw ($(B)!0.5!(C)$) node [below] {$F$} coordinate (F);\n\\draw ($(A)!0.5!(D)$) node [above] {$E$} coordinate (E);\n\\draw (A)--(B)--(D)--(C)--cycle(B)--(C);\n\\draw [dashed] (A)--(D)(E)--(F);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二上学期期中考试试题12", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022730": { + "id": "022730", + "content": "下列命题中, 正确的是\\bracket{20} .\n\\twoch{平行于同一条直线的两个平面平行}{垂直于同一条直线的两条直线平行}{平行于同一个平面的两条直线平行}{垂直于同一个平面的两条直线平行}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二上学期期中考试试题13", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022731": { + "id": "022731", + "content": "用数学归纳法证明``当 $n$ 为正奇数时, $x^n+y^n$ 能被 $x+y$ 整除''的第二步是\\bracket{20}(注: 以下四个选项中均有 $k \\in \\mathbf{N}$).\n\\onech{假设 $n=2 k+1$ 时命题正确, 再推 $n=2 k+3$ 时命题正确}{假设 $n=2 k-1$ 时命题正确, 再推 $n=2 k+1$ 时命题正确}{假设 $n=k$ 时命题正确, 再推 $n=k+1$ 时命题正确}{假设 $n \\leq k$($k \\geq 1$) 时命题正确, 再推 $n=k+2$ 时命题正确}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二上学期期中考试试题14", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022732": { + "id": "022732", + "content": "若无穷等比数列 $\\{a_n\\}$ 的首项为 $\\dfrac{1}{2}$, 公比为 $\\dfrac{5}{2}-a$, 且 $\\{a_n\\}$ 的各项的和为 $a-1$, 则实数 $a$ 的值为\\bracket{20}.\n\\fourch{$2$ 或 $\\dfrac{1}{2}$}{$2$}{$\\dfrac{1}{2}$}{$3$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二上学期期中考试试题15", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022733": { + "id": "022733", + "content": "如图, 已知正方形 $ABCD$ 的边长为 $2$, $E$ 和 $F$ 分别是 $AB$ 和 $BC$ 的中点, 将 $\\triangle ADE$、$\\triangle EFB$ 、 $\\triangle FCD$ 分别沿 $DE$、$EF$、$FD$ 折起, 使得 $A$、$B$、$C$ 三个点重合于 $A'$. 现有空间中一点 $G$, 它与四面体 $A' DEF$ 的四个顶点在同一个球面上, 则以 $\\triangle DEF$ 为底面的三棱锥 $G-DEF$ 的高 $h$ 的最大值为 $\\bracket{20}$\\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] {$C$} coordinate (C);\n\\draw (2,2) node [above right] {$D$} coordinate (D);\n\\draw (0,2) node [above left] {$A$} coordinate (A);\n\\draw ($(A)!0.5!(B)$) node [left] {$E$} coordinate (E);\n\\draw ($(B)!0.5!(C)$) node [below] {$F$} coordinate (F);\n\\draw (A)--(B)--(C)--(D)--cycle(E)--(F)--(D)--cycle;\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex]\n\\draw (2,0,-2) node [right] {$D$} coordinate (D);\n\\draw (0,0,-1) node [left] {$E$} coordinate (E);\n\\draw (1,0,0) node [below] {$F$} coordinate (F);\n\\draw ({2/3},{2/3},{-2/3}) node [above] {$A'$} coordinate (A');\n\\draw (A')--(E)--(F)--(D)--cycle(A')--(F);\n\\draw [dashed] (D)--(E);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\dfrac{\\sqrt{6}}{2}+\\dfrac{5 \\sqrt{2}}{6}$}{$\\dfrac{\\sqrt{6}}{2}+\\dfrac{1}{3}$}{$\\sqrt{6}-\\dfrac{5 \\sqrt{2}}{6}$}{$\\sqrt{6}-\\dfrac{1}{3}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二上学期期中考试试题16", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022734": { + "id": "022734", + "content": "如图, 在长方体 $ABCD-A_1B_1C_1D_1$ 中, $2AB=2BC=AA_1$, $E$、$F$ 分别为 $AB$、$AA_1$ 的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{4}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\filldraw ($(A)!0.5!(B)$) circle (0.05) node [below] {$E$} coordinate (E);\n\\filldraw ($(A)!0.5!(A1)$) circle (0.05) node [left] {$F$} coordinate (F);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $E$、$C$、$D_1$、$F$ 四点共面;\\\\\n(2) 求二面角 $D-A_1C_1-D_1$ 的大小.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二上学期期中考试试题17", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "022735": { + "id": "022735", + "content": "如图, 已知直三棱柱 $ABE-DCG$ 的顶点分别都在圆柱的上下底面的圆周上, 四边形 $ABCD$ 是过圆柱轴的一个截面, 且 $ABCD$ 为正方形.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (-1,0) node [left] {$A$} coordinate (A);\n\\draw (1,0) node [right] {$B$} coordinate (B);\n\\draw (1,2) node [right] {$C$} coordinate (C);\n\\draw (-1,2) node [left] {$D$} coordinate (D);\n\\draw (A) arc (180:360:1 and 0.25) (C) arc (0:360:1 and 0.25);\n\\draw [dashed] (A) arc (180:0:1 and 0.25);\n\\filldraw (0,0) circle (0.03) (0,2) circle (0.03);\n\\draw (A)--(D)--(C)--(B);\n\\draw [dashed] (A)--(B);\n\\draw (-100:1 and 0.25) node [below] {$E$} coordinate (E);\n\\draw (E) ++ (0,2) node [below right] {$G$} coordinate (G);\n\\draw (C)--(G)--(D)(G)--(E);\n\\draw [dashed] (A)--(E)--(B)(E)--(D)(B)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $CG \\parallel $ 平面 $BDE$;\\\\\n(2) 若该圆柱与三棱柱 $ABE-DCG$ 的体积之比为 $\\pi: 1$, 求直线 $DE$ 与平面 $ABCD$ 所成角的大小.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二上学期期中考试试题18", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "022736": { + "id": "022736", + "content": "由市场调查得知: 某公司生产的一种食品, 如果不做广告宣传且每千克盈利 $a$ 元, 那么销售量为 $a_0$ 千克; 如果做广告宣传且每件售价不变, 那么广告费用为 $n \\times 1000$ 元时的销售量比广告费用为 $(n-1) \\times 1000$ 元时的销售量多 $a_0 \\times \\dfrac{1}{2^n}$ 千克 ($n>0$ 且 $n \\in \\mathbf{Z}$).\\\\\n(1) 设广告费用为 $n \\times 1000$ 元时的销售量为 $a_n$, 求销售量 $a_n$ 关于 $n$ 的代数表达式;\\\\\n(2) 当 $a=10$, $a_0=4000$ 时, 公司应做几千元广告, 才能使得去掉广告费用后的获利最大? 此时销售量为多少千克?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二上学期期中考试试题19", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "022737": { + "id": "022737", + "content": "如图, 已知正方体 $ABCD-A_1B_1C_1D_1$ 的棱长为 $2 \\sqrt{3}$, $M$、$N$ 为体对角线 $BD_1$ 的三等分点, 动点 $P$ 在 $\\triangle ACB_1$ 内(包括边界).\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw [dashed] (A)--(C)(B)--(D1);\n\\draw ({1+sqrt(2)/4},-1) node {第(1)题图};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw [dashed] (A)--(C)(B)--(D1);\n\\filldraw ($(A)!0.5!(C)$) circle (0.03) node [below] {$P$} coordinate (P);\n\\filldraw ($(B)!{2/3}!(D1)$) circle (0.03) node [above right] {$M$} coordinate (M);\n\\filldraw ($(B)!{1/3}!(D1)$) circle (0.03) node [above right] {$N$} coordinate (N);\n\\draw [dashed] (A1)--(M)(A1)--(N)(A1)--(P)(M)--(P)(P)--(N);\n\\draw ({1+sqrt(2)/4},-1) node {第(2)题图};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw [dashed] (A)--(C)(B)--(D1);\n\\draw (A)--(B1)--(C);\n\\filldraw ($(B)!{2/3}!(D1)$) circle (0.03) node [above right] {$M$} coordinate (M);\n\\filldraw ($(B)!{1/3}!(D1)$) circle (0.03) node [below left] {$N$} coordinate (N);\n\\filldraw ($(N)!{sqrt(3)/3}!(C)$) circle (0.03) node [below] {$P$} coordinate (P);\n\\draw [dashed] (M)--(P)--(N);\n\\draw ({1+sqrt(2)/4},-1) node {第(3)题图};\n\\end{tikzpicture}\n\\end{center}\n(1) 求异面直线 $AC$ 与 $BD_1$ 所成角的大小;\\\\\n(2) 若 $P$ 为 $AC$ 的中点, 求三棱锥 $A_1-MNP$ 的体积;\\\\\n(3) 若 $\\triangle PMN$ 的面积 $S_{\\triangle PMN}=\\dfrac{2 \\sqrt{6}}{3}$, 求点 $P$ 生成的轨迹的长度.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二上学期期中考试试题20", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "022738": { + "id": "022738", + "content": "如图, 在三棱柱 $ABC-A_1B_1C_1$ 中, 各个侧面均是边长为 $2$ 的正方形, $D$ 为线段 $AC$ 的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$B$} coordinate (B);\n\\draw (1,0,{-sqrt(3)}) node [above left] {$C$} coordinate (C);\n\\draw (A) ++ (0,2,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,2,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,2,0) node [above] {$C_1$} coordinate (C_1);\n\\draw ($(A)!0.5!(C)$) node [above left] {$D$} coordinate (D);\n\\draw (A)--(B)--(B_1)--(C_1)--(A_1)--cycle(A_1)--(B_1);\n\\draw [dashed] (B)--(D)--(C_1)--cycle(A)--(C)--(B)(C)--(C_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $BD \\perp$ 平面 $ACC_1A_1$;\\\\\n(2) 求 $BB_1$ 与平面 $BDC_1$ 所成角的大小;\\\\\n(3) 设 $M$ 为线段 $BC_1$ 上的任意一点, 在 $\\Delta BC_1D$ 内的平面区域(包括边界) 是否存在点 $E$, 使得 $CE \\perp DM$? 若存在, 请求出点 $E$ 的位置; 若不存在, 请说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二上学期期中考试试题21", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "022739": { + "id": "022739", + "content": "不等式 $\\dfrac{x-2}{x+1}<0$ 的解集为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期期中考试试题1", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022740": { + "id": "022740", + "content": "函数 $y=\\lg (x^2-5 x+4)$ 的定义域为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期期中考试试题2", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022741": { + "id": "022741", + "content": "复数 $\\dfrac{2+4 \\mathrm{i}}{1+\\mathrm{i}}$ (其中 $\\mathrm{i}$ 为虚数单位) 的虚部为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期期中考试试题3", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022742": { + "id": "022742", + "content": "已知 $a$ 为实数. 若关于 $x$ 的方程 $x^2-4 x+a=0$ 有一个根为 $2+\\mathrm{i}$ (其中 $\\mathrm{i}$ 为虚数单位), 则 $a$ 的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期期中考试试题4", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022743": { + "id": "022743", + "content": "已知 $a$ 为实数. 若数据 $1,2, a, 6$ 的平均数为 3 , 则这组数据的标准差为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期期中考试试题5", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022744": { + "id": "022744", + "content": "若 $\\tan (\\alpha+\\beta)=3$, $\\tan (\\alpha+\\dfrac{\\pi}{4})=-3$, 则 $\\tan (\\beta-\\dfrac{\\pi}{4})$ 的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期期中考试试题6", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022745": { + "id": "022745", + "content": "已知 $a$ 为实数. 若 $y=f(x)$ 是定义在 $\\mathbf{R}$ 上的偶函数, 且它在区间 $[0,+\\infty)$ 上是严格增函数, 则使得 $f(a) \\geq f(3)$ 成立的 $a$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期期中考试试题7", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022746": { + "id": "022746", + "content": "某工厂生产 $A$、$B$ 两种型号的不同产品, 产品数量之比为 $2: 3$. 现用分层抽样的方法抽出一个样本容量为 $n$ 的样本, 则其中 $A$ 种型号的产品有 10 件. 现从样本中抽出两件产品, 此时含有 $A$ 型号产品的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期期中考试试题8", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022747": { + "id": "022747", + "content": "已知 $\\varphi \\in(-\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2})$. 若函数 $f(x)=\\sin (3 x+\\varphi)$ 的图像关于直线 $x=\\dfrac{3 \\pi}{5}$ 对称, 则 $\\varphi$ 的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期期中考试试题9", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022748": { + "id": "022748", + "content": "如图所示, 两块斜边长均等于 $\\sqrt{2}$ 的直角三角板拼在一起,则 $\\overrightarrow{OD}\\cdot \\overrightarrow{BA}$ 的值为\\blank{50}.\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 (0,2) node [left] {$B$} coordinate (B);\n\\draw (A) ++ (45:{sqrt(6)}) node [right] {$D$} coordinate (D);\n\\draw ($(A)!0.5!(B)$) node [below left] {$C$} coordinate (C);\n\\draw (B)--(O)--(A)--(D)--(C)(A)--(B);\n\\draw pic [draw,scale = 0.5] {right angle = A--O--B};\n\\draw pic [draw,scale = 0.5,\"$45^\\circ$\", angle eccentricity = 2.5] {angle = B--A--O};\n\\draw pic [draw,scale = 0.5,\"$60^\\circ$\", angle eccentricity = 2.5] {angle = A--C--D};\n\\draw (O)--(D);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期期中考试试题10", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022749": { + "id": "022749", + "content": "在棱长为 1 的正方体 $ABCD-A_1B_1C_1D_1$ 中, 点 $P_1$、$P_2$ 分别是线段 $AB$、$BD_1$ (不包括端点) 上的动点, 且线段 $P_1P_2$ 平行于平面 $A_1ADD_1$. 若 $\\overrightarrow{AP_1}=2 \\overrightarrow{P_1B}$, 则四面体 $P_1P_2AB_1$ 的体积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期期中考试试题11", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022750": { + "id": "022750", + "content": "已知 $f(x)=x^3+2023 x$. 若实数 $a, b \\in$($0,+\\infty$) 且 $f(\\dfrac{1}{2}-3 a)+f(\\dfrac{1}{2}-b)=0$, 则 $\\dfrac{a^2+b^2+a}{a b}$的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期期中考试试题12", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022751": { + "id": "022751", + "content": "对任意向量 $\\overrightarrow{a}$、$\\overrightarrow{b}$, 下列关系式中不恒成立的是\\bracket{20} .\n\\twoch{$(\\overrightarrow{a}+\\overrightarrow{b})^2=|\\overrightarrow{a}+\\overrightarrow{b}|^2$}{$(\\overrightarrow{a}+\\overrightarrow{b}) \\cdot(\\overrightarrow{a}-\\overrightarrow{b})=\\overrightarrow{a}^2-\\overrightarrow{b}^2$}{$|\\overrightarrow{a}\\cdot \\overrightarrow{b}| \\leq|\\overrightarrow{a}| \\cdot|\\overrightarrow{b}| ; $}{$|\\overrightarrow{a}-\\overrightarrow{b}| \\leq|| \\overrightarrow{a}|-| \\overrightarrow{b}||$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期期中考试试题13", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022752": { + "id": "022752", + "content": "一个直角三角形的两条直角边长分别为 1 和 $\\sqrt{3}$, 将该三角形分别绕其两条直角边所在直线旋转一周得到两个圆锥, 则这两个圆锥的体积的比值为\\bracket{20}.\n\\fourch{$1$}{$\\sqrt{3}$}{$3$}{$3 \\sqrt{3}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期期中考试试题14", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022753": { + "id": "022753", + "content": "已知函数 $y=f(x)$, $x \\in \\mathbf{R}$. 若 $f(1)0\\}$, 则集合 $A$ 的元素个数为\\bracket{20}.\n\\fourch{$1011$}{$1012$}{$2022$}{$2023$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期期中考试试题16", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022755": { + "id": "022755", + "content": "如图, 已知圆锥的顶点为 $P$, 底面圆心为 $O$, 高为$3$, 底面半径为 $2$.\n\\begin{center}\n\\begin{tikzpicture}\n\\node (0,0) [left] {$O$} coordinate (O);\n\\draw (-1.5,0) arc (180:360:1.5 and {1.5/3}) node [right] {$B$} coordinate (B);\n\\draw [dashed] (1.5,0) arc (0:180:1.5 and {1.5/3}) coordinate (C);\n\\draw (C) -- (0,2.25) node [above] {$P$} coordinate (P) -- (B); \n\\coordinate (A) at ({1.5*cos(250)},{0.5*sin(250)});\n\\draw [dashed] (A) node [below left] {$A$} -- (O) -- (B) -- cycle;\n\\coordinate (M) at ($(A)!0.5!(B)$);\n\\draw [dashed] (O) -- (P) -- (M) node [shift = {(-45:0.5)}] {$M$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求该圆锥的侧面积;\\\\\n(2) 设 $OA$、$OB$ 为该圆锥的底面半径, 且 $\\angle AOB=90^{\\circ}, M$ 为线段 $AB$ 的中点, 求直线 $PM$ 与直线 $OB$ 所成的角的大小.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期期中考试试题17", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "022756": { + "id": "022756", + "content": "已知 $a$ 为实数. 设 $f(x)=x^2+|x-a|$.\\\\\n(1) 若 $a=1$, 求函数 $y=f(x)$, $x \\in \\mathbf{R}$ 的最小值;\\\\\n(2) 判断函数 $y=f(x)$, $x \\in \\mathbf{R}$ 的奇偶性, 并说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期期中考试试题18", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "022757": { + "id": "022757", + "content": "如图所示, 某市郊外景区内一条笔直的公路 $a$ 经过三个景点 $A, B, C$. 景区管委会又开发了风景优美的景点 $D$. 经测量景点 $D$ 位于景点 $A$ 的北偏东 $30^{\\circ}$ 方向 $16 \\mathrm{km}$ 处, 位于景点 $B$ 的正北方向,还位于景点 $C$ 的北偏西 $75^{\\circ}$ 方向上. 已知 $AB=10 \\mathrm{km}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.25]\n\\draw [->] (12,3) --++ (2,0) node [right] {东};\n\\draw [->] (12,3) --++ (0,2) node [above] {北};\n\\draw [->] (0,0) node [below] {$A$} coordinate (A) -- (0,8) node [left] {北} coordinate (N);\n\\draw (A) --++ (60:8) node [above] {$D$} coordinate (D);\n\\draw (4,3) node [below] {$B$} coordinate (B) -- (D);\n\\draw [name path = linea] (A) -- ($(A)!2.2!(B)$) node [right] {$a$} coordinate (a);\n\\path [name path = DC] (D) --++ (-15:4);\n\\path [name intersections = {of = linea and DC, by = C}];\n\\draw (D) -- (C) node [below] {$C$};\n\\draw (60:2) arc (60:90:2);\n\\draw (75:4) node {$30^\\circ$};\n\\draw [dashed] (C) --++ (0,3) coordinate (T);\n\\draw pic [draw, scale = 0.5, angle eccentricity = 1.7, \"$75^\\circ$\"] {angle = T--C--D};\n\\end{tikzpicture}\n\\end{center}\n(1) 景区管委会准备由景点 $D$ 向景点 $B$ 修建一条笔直的公路. 求线段 $BD$ 的长度(长度单位精确到 $0.1 \\mathrm{km})$;\\\\\n(2) 求线段 $AC$ 的长度 (长度单位精确到 $0.1 \\mathrm{km}$).", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期期中考试试题19", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "022758": { + "id": "022758", + "content": "已知 $k$ 为实数. $f(x)=2 \\sin ^2(\\dfrac{\\pi}{4}+x)-k \\cdot \\cos 2 x$.\\\\\n(1) 若 $k=0$, 求关于 $x$ 的方程 $f(x)=1$ 在 $[0, \\pi]$ 上的解;\\\\\n(2) 若 $k=\\sqrt{3}$, 求函数 $y=f(x)$, $x \\in \\mathbf{R}$ 的单调减区间;\\\\\n(3) 已知 $a$ 为实数且 $k=\\sqrt{3}$. 若关于 $x$ 的不等式 $|f(x)-a|<2$ 在 $x \\in[\\dfrac{\\pi}{4}, \\dfrac{\\pi}{2}]$ 时恒成立, 求 $a$ 的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期期中考试试题20", + "edit": [ + "20231109\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "022759": { + "id": "022759", + "content": "已知实数 $a>0$. 设 $f(x)=-\\dfrac{2}{3}a x^3+x^2$.\\\\\n(1) 若 $a=3$, 求函数 $y=f(x)$, $x \\in \\mathbf{R}$ 的图像在点 $(1,-1)$ 处的切线方程;\\\\\n(2) 若 $a=\\dfrac{1}{3}$, 求函数 $y=f(x)$, $x \\in(2,+\\infty$) 的值域;\\\\\n(3) 若对于任意的 $x_1 \\in$($2,+\\infty$), 总存在 $x_2 \\in$($1,+\\infty$), 使得 $f(x_1) \\cdot f(x_2)=1$, 求 $a$ 的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期期中考试试题21", + "edit": [ + "20231109\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