From 92faa9cf4fe80c961af5bb6528e8626ae1de306a Mon Sep 17 00:00:00 2001 From: "weiye.wang" Date: Sat, 20 Jan 2024 10:59:35 +0800 Subject: [PATCH] =?UTF-8?q?=E5=BD=95=E5=85=A52024=E5=B1=8A=E4=B9=9D?= =?UTF-8?q?=E7=9C=81=E8=81=94=E8=80=83=E8=AF=95=E9=A2=98?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 题库0.3/Problems.json | 380 ++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 380 insertions(+) diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index ea00a286..3c11d779 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -638248,6 +638248,386 @@ "space": "4em", "unrelated": [] }, + "023608": { + "id": "023608", + "content": "样本数据 $16,24,14,10,20,30,12,14,40$ 的中位数为\\bracket{20}.\n\\fourch{$14$}{$16$}{$18$}{$20$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届九省联考试题1", + "edit": [ + "20240120\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "023609": { + "id": "023609", + "content": "椭圆 $\\dfrac{x^2}{a^2}+y^2=1$($a>1$) 的离心率为 $\\dfrac{1}{2}$, 则 $a=$\\bracket{20}.\n\\fourch{$\\dfrac{2 \\sqrt{3}}{3}$}{$\\sqrt{2}$}{$\\sqrt{3}$}{2}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届九省联考试题2", + "edit": [ + "20240120\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "023610": { + "id": "023610", + "content": "记等差数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, $a_3+a_7=6$, $a_{12}=17$, 则 $S_{16}=$\\bracket{20}.\n\\fourch{$120$}{$140$}{$160$}{$180$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届九省联考试题3", + "edit": [ + "20240120\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "023611": { + "id": "023611", + "content": "设 $\\alpha, \\beta$ 是两个平面, $m, l$ 是两条直线, 则下列命题为真命题的是\\bracket{20}.\n\\twoch{若 $\\alpha \\perp \\beta$, $m \\parallel \\alpha$, $l \\parallel \\beta$, 则 $m \\perp l$}{若 $m \\subset \\alpha, l \\subset \\beta, m \\parallel l$, 则 $\\alpha \\parallel \\beta$}{若 $\\alpha \\cap \\beta=m$, $l \\parallel \\alpha$, $l \\parallel \\beta$, 则 $m \\parallel l$}{若 $m \\perp \\alpha$, $l \\perp \\beta$, $m \\parallel l$, 则 $\\alpha \\perp \\beta$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届九省联考试题4", + "edit": [ + "20240120\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "023612": { + "id": "023612", + "content": "甲、乙、丙等 $5$ 人站成一排, 且甲不在两端, 乙和丙之间恰有 $2$ 人, 则不同排法共有\\bracket{20}.\n\\fourch{$20$ 种}{$16$ 种}{$12$ 种}{$8$ 种}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届九省联考试题5", + "edit": [ + "20240120\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "023613": { + "id": "023613", + "content": "已知 $Q$ 为直线 $l: x+2 y+1=0$ 上的动点, 点 $P$ 满足 $\\overrightarrow{QP}=(1,-3)$, 记 $P$ 的轨迹为 $E$, 则\\bracket{20}.\n\\twoch{$E$ 是一个半径为 $\\sqrt{5}$ 的圆}{$E$ 是一条与 $l$ 相交的直线}{$E$ 上的点到 $l$ 的距离均为 $\\sqrt{5}$}{$E$ 是两条平行直线}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届九省联考试题6", + "edit": [ + "20240120\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "023614": { + "id": "023614", + "content": "已知 $\\theta \\in(\\dfrac{3 \\pi}{4}, \\pi)$, $\\tan 2 \\theta=-4 \\tan (\\theta+\\dfrac{\\pi}{4})$, 则 $\\dfrac{1+\\sin 2 \\theta}{2 \\cos ^2 \\theta+\\sin 2 \\theta}=$\\bracket{20}.\n\\fourch{$\\dfrac{1}{4}$}{$\\dfrac{3}{4}$}{1}{$\\dfrac{3}{2}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届九省联考试题7", + "edit": [ + "20240120\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "023615": { + "id": "023615", + "content": "设双曲线 $C: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$) 的左、右焦点分别为 $F_1, F_2$, 过坐标原点的直线与 $C$ 交于 $A, B$ 两点, $|F_1B|=2|F_1A|$, $\\overrightarrow{F_2A}\\cdot \\overrightarrow{F_2B}=4 a^2$, 则 $C$ 的离心率为\\bracket{20}.\n\\fourch{$\\sqrt{2}$}{$2$}{$\\sqrt{5}$}{$\\sqrt{7}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届九省联考试题8", + "edit": [ + "20240120\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "023616": { + "id": "023616", + "content": "已知函数 $f(x)=\\sin (2 x+\\dfrac{3 \\pi}{4})+\\cos (2 x+\\dfrac{3 \\pi}{4})$, 则\\blank{50}.\\\\\n\\textcircled{1} 函数 $f(x-\\dfrac{\\pi}{4})$ 为偶函数; \\textcircled{2} 曲线 $y=f(x)$ 的对称轴为 $x=k \\pi$, $k \\in \\mathbf{Z}$; \\textcircled{3} $f(x)$ 在区间 $(\\dfrac{\\pi}{3}, \\dfrac{\\pi}{2})$ 单调递增; \\textcircled{4} $f(x)$ 的最小值为 $-2$.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届九省联考试题9", + "edit": [ + "20240120\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "023617": { + "id": "023617", + "content": "已知复数 $z, w$ 均不为 $0$ , 则\\blank{50}.\\\\\n\\textcircled{1} $z^2=|z|^2$; \\textcircled{2} $\\dfrac{z}{\\overline{z}}=\\dfrac{z^2}{|z|^2}$; \\textcircled{3} $\\overline{z-w}=\\overline{z}-\\overline{w}$; \\textcircled{4} $|\\dfrac{z}{w}|=\\dfrac{|z|}{|w|}$.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届九省联考试题10", + "edit": [ + "20240120\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "023618": { + "id": "023618", + "content": "已知函数 $f(x)$ 的定义域为 $\\mathbf{R}$, 且 $f(\\dfrac{1}{2}) \\neq 0$, 若 $f(x+y)+f(x) f(y)=4 x y$, 则\\blank{50}.\\\\\n\\textcircled{1} $f(-\\dfrac{1}{2})=0$; \\textcircled{2} $f(\\dfrac{1}{2})=-2$; \\textcircled{3} 函数 $f(x-\\dfrac{1}{2})$ 是偶函数; \\textcircled{4} 函数 $f(x+\\dfrac{1}{2})$ 是减函数.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届九省联考试题11", + "edit": [ + "20240120\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "023619": { + "id": "023619", + "content": "已知集合 $A=\\{-2,0,2,4\\}$, $B=\\{x|| x-3 | \\leq m\\}$, 若 $A \\cap B=A$, 则 $m$ 的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届九省联考试题12", + "edit": [ + "20240120\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "023620": { + "id": "023620", + "content": "已知轴截面为正三角形的圆锥 $MM'$ 的高与球 $O$ 的直径相等, 则圆锥 $MM'$ 的体积与球 $O$ 的体积的比值是\\blank{50}, 圆锥 $MM'$ 的表面积与球 $O$ 的表面积的比值是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届九省联考试题13", + "edit": [ + "20240120\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "023621": { + "id": "023621", + "content": "以 $\\max M$ 表示数集 $M$ 中最大的数. 设 $0=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [below] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (-1,{sqrt(2)},1) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (-1,{sqrt(2)},1) node [above] {$B_1$} coordinate (B1);\n\\draw (C) ++ (-1,{sqrt(2)},1) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (-1,{sqrt(2)},1) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw [dashed] (A)--(C)(B)--(D);\n\\draw ($(A)!0.5!(C)$) node [below] {$O$} coordinate (O);\n\\draw [dashed] (C1)--(O);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $C_1O \\perp$ 平面 $ABCD$;\\\\\n(2) 求二面角 $B-AA_1-D$ 的正弦值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届九省联考试题17", + "edit": [ + "20240120\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "023625": { + "id": "023625", + "content": "已知抛物线 $C: y^2=4 x$ 的焦点为 $F$, 过 $F$ 的直线 $l$ 交 $C$ 于 $A, B$ 两点, 过 $F$ 与 $l$ 垂直的直线交 $C$ 于 $D, E$ 两点, 其中 $B, D$ 在 $x$ 轴上方, $M, N$ 分别为 $AB, DE$ 的中点.\\\\\n(1) 证明: 直线 $MN$ 过定点;\\\\\n(2) 设 $G$ 为直线 $AE$ 与直线 $BD$ 的交点, 求 $\\triangle GMN$ 面积的最小值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届九省联考试题18", + "edit": [ + "20240120\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "023626": { + "id": "023626", + "content": "离散对数在密码学中有重要的应用. 设 $p$ 是素数, 集合 $X=\\{1,2, \\cdots, p-1\\}$, 若 $u, v \\in X$, $m \\in \\mathbf{N}$, 记 $u \\otimes v$ 为 $u v$ 除以 $p$ 的余数, $u^{m, \\otimes}$ 为 $u^m$ 除以 $p$ 的余数; 设 $a \\in X$ , $1, a, a^{2, \\otimes}, \\cdots, a^{p-2, \\otimes}$ 两两不同, 若 $a^{n, \\otimes}=b$($n \\in\\{0,1, \\cdots, p-2\\}$), 则称 $n$ 是以 $a$ 为底 $b$ 的离散对数, 记为 $n=\\log (p)_a b$.\\\\\n(1) 若 $p=11$, $a=2$, 求 $a^{p-1, \\otimes}$;\\\\\n(2) 对 $m_1, m_2 \\in\\{0,1, \\cdots, p-2\\}$, 记 $m_1 \\oplus m_2$ 为 $m_1+m_2$ 除以 $p-1$ 的余数 (当 $m_1+m_2$ 能被 $p-1$ 整除时, $m_1 \\oplus m_2=0$). 证明: $\\log (p)_a(b \\otimes c)=\\log (p)_a b \\oplus \\log (p)_a c$, 其中 $b, c \\in X$;\\\\\n(3) 已知 $n=\\log (p)_a b$. 对 $x \\in X$, $k \\in\\{1,2, \\cdots, p-2\\}$, 令 $y_1=a^{k, \\otimes}$, $y_2=x \\otimes b^{k, \\otimes}$. 证明: $x=y_2 \\otimes y_1^{n(p-2), \\otimes}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届九省联考试题19", + "edit": [ + "20240120\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