From a76c028fa67b694d8110befb4a5e035d91652ed8 Mon Sep 17 00:00:00 2001 From: "weiye.wang" Date: Sun, 30 Jul 2023 21:50:40 +0800 Subject: [PATCH] =?UTF-8?q?=E9=80=9A=E8=BF=87=E5=AD=97=E7=AC=A6=E4=B8=B2?= =?UTF-8?q?=E6=AF=94=E5=AF=B9=E8=87=AA=E5=8A=A8=E8=B5=8B=E4=BA=88=E4=BA=86?= =?UTF-8?q?=E4=B8=80=E4=BA=9B=E9=A2=98=E7=9B=AE=E4=BB=A5=E5=8D=95=E5=85=83?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 题库0.3/Problems.json | 960 +++++++++++++++++++++++++++++++----------- 1 file changed, 720 insertions(+), 240 deletions(-) diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 1587efff..265b39e4 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -502364,7 +502364,9 @@ "id": "019665", "content": "已知函数 $f(x)=\\begin{cases}|x|,& x \\leq m,\\\\x^2-2 m x+4 m,& x>m,\\end{cases}$其中 $m>0$. 若存在实数 $b$, 使得关于 $x$ 的方程 $f(x)=b$ 有三个不同的根, 则 $m$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -502449,7 +502451,9 @@ "id": "019669", "content": "已知抛物线的方程为 $x^2=8 y$, 点 $F$ 是其焦点, 点 $A(-2,4)$, 在抛物线上求一点 $P$, 使 $\\triangle APF$ 的周长最小, 求此时点 $P$ 的坐标.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -502617,7 +502621,9 @@ "id": "019677", "content": "设函数 $f(x)=\\begin{cases}\\log _2(x+1),& x \\geq 0,\\\\\\sqrt{-x},& x<0,\\end{cases}$则满足 $f(x+1)<2$ 的 $x$ 的取值范围为\\bracket{20}.\n\\fourch{$(-4,3)$}{$(-5,2)$}{$(-3,4)$}{$(-\\infty,-3) \\cup(4,+\\infty)$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -502719,7 +502725,9 @@ "id": "019682", "content": "设等比数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 若 $S_3+S_6=2S_9$, 则数列的公比 $q$ 是\\bracket{20}.\n\\fourch{$-\\dfrac{\\sqrt[3]{3}}{2}$}{$\\dfrac{\\sqrt[3]{3}}{2}$}{$-\\dfrac{\\sqrt[3]{4}}{2}$}{$\\dfrac{\\sqrt[3]{4}}{2}$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -502824,7 +502832,9 @@ "id": "019687", "content": "已知 $M=\\{x | x-a=0\\}$, $N=\\{x | a x-1=0\\}$, 若 $M \\cap N=N$, 则实数 $a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -502844,7 +502854,9 @@ "id": "019688", "content": "若 $\\mathrm{C}_{10}^3=\\mathrm{C}_{10}^n$, 则正整数 $n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -502864,7 +502876,9 @@ "id": "019689", "content": "设数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n=3^n+2$, 则数列 $\\{a_n\\}$ 的通项公式为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -502924,7 +502938,9 @@ "id": "019692", "content": "已知函数 $f(x)=\\begin{cases}-\\log _2(3-x),& x<2\\\\2^{x-2}-1,& x \\geq 2\\end{cases}$, 若 $f(2-a)=1$, 则 $f(a)$ 等于\\bracket{20}.\n\\fourch{$-2$}{$-1$}{1}{2}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -502964,7 +502980,9 @@ "id": "019694", "content": "已知数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 对任意正整数 $n, a_{n+1}=3S_n$, 则下列关于 $\\{a_n\\}$ 的论断中正确的是\\bracket{20}.\n\\twoch{一定是等差数列}{一定是等比数列}{可能是等差数列, 但不会是等比数列}{可能是等比数列, 但不会是等差数列}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -503004,7 +503022,9 @@ "id": "019696", "content": "函数 $f(x)=[a x^2-(3 a+1) x+3 a+2] \\cdot \\mathrm{e}^x$ 在 $x=1$ 处取得极小值, 求 $a$ 的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -503044,7 +503064,9 @@ "id": "019698", "content": "在 $\\triangle ABC$ 中, 角 $A, B, C$ 的对边分别为 $a, b, c$, 已知 $2 \\sin C=\\tan A(1- 2 \\cos C)$, $c=2 b$, 则 $\\cos B$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -503084,7 +503106,9 @@ "id": "019700", "content": "在直角梯形 $ABCD$ 中, $AB=8$, $CD=4, AB \\parallel CD$, $AB \\perp AD$, $E$ 是 $BC$ 的中点, 则 $\\overrightarrow{AB}\\cdot(\\overrightarrow{AC}+\\overrightarrow{AE})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -503226,7 +503250,9 @@ "id": "019707", "content": "已知数列 $\\{a_n\\}$ 的通项公式为 $a_n=n \\cdot \\sin \\dfrac{n \\pi}{2}$, 前 $n$ 项和为 $S_n$, 求 $S_{2022}$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -503288,7 +503314,9 @@ "id": "019710", "content": "直线 $x+2 y-3=0$ 与直线 $a x+4 y+b=0$ 关于点 $A(1,0)$ 对称, 则 $b=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -503308,7 +503336,9 @@ "id": "019711", "content": "等差数列 $\\{a_n\\}$ 中,若 $a_4+a_6+a_8+a_{10}+a_{12}=120$, 则 $a_9-\\dfrac{1}{3}a_{11}$ 的值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -503348,7 +503378,9 @@ "id": "019713", "content": "已知函数 $f(x)=\\begin{cases}2-x^2,& x \\geq 0,\\\\-x,& x<0,\\end{cases}$ $x_1,x_2 \\in \\mathbf{R}$, $f(x_1)=f(x_2)=m$, 且 $x_1+x_2=0$, 则 $m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -503412,7 +503444,9 @@ "id": "019716", "content": "已知函数 $f(x)=\\begin{cases}x^2+4 x,& x \\geq 0,\\\\4 x-x^2,& x<0,\\end{cases}$ 若 $f(2-a^2)>f(a)$, 则实数 $a$ 的取值范围是\\bracket{20}.\n\\fourch{$(-\\infty,-1) \\cup(2,+\\infty)$}{$(-1,2)$}{$(-2,1)$}{$(-\\infty,-2) \\cup(1,+\\infty)$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -503432,7 +503466,9 @@ "id": "019717", "content": "已知 $f(x)$ 是定义在 $\\mathbf{R}$ 上的奇函数, 对任意 $x \\in \\mathbf{R}$, 恒有 $f(x)+f(x+2)=0$, 且当 $x \\in (0,1]$ 时 $f(x)=2^x+1$, 则 $f(0)+f(1)+f(2)+\\cdots+f(2021)=$\\bracket{20}.\n\\fourch{$1$}{$2$}{$3$}{$4$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -503492,7 +503528,9 @@ "id": "019720", "content": "已知数列 $\\{a_n\\}$ 与 $\\{b_n\\}$ 满足 $a_{n+1}-a_n=\\lambda(b_{n+1}-b_n)$ ($\\lambda$ 为非零常数), $n$ 为正整数.\\\\\n(1) 若 $\\{b_n\\}$ 是等差数列, 求证: 数列 $\\{a_n\\}$ 也是等差数列;\\\\\n(2) 若 $a_1=2$, $\\lambda=3$, $b_n=\\sin \\dfrac{n \\pi}{2}$, 求数列 $\\{a_n\\}$ 的前 $2021$ 项和;\\\\\n(3) 设 $a_1=b_1=\\lambda$, $b_2=\\dfrac{\\lambda}{2}$, $b_n=\\dfrac{b_{n-1}+b_{n-2}}{2}$($n \\geq 3$, $n \\in \\mathbf{N}$), 若对 $\\{a_n\\}$ 中的任意两项 $a_i$, $a_j$, $i, j$ 为正整数且 $i \\neq j$, $|a_i-a_j|<2$ 都成立, 求实数 $\\lambda$ 的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -503554,7 +503592,9 @@ "id": "019723", "content": "证明下列三角恒等式.\\\\\n(1) $\\dfrac{1+\\cos \\alpha}{\\sin \\alpha}=\\dfrac{\\sin \\alpha}{1-\\cos \\alpha}$;\\\\\n(2) $\\dfrac{\\sin ^2 \\alpha-\\sin ^2 \\beta}{\\tan ^2 \\alpha-\\tan ^2 \\beta}=\\cos ^2 \\alpha \\cos ^2 \\beta$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -503574,7 +503614,9 @@ "id": "019724", "content": "已知数列 $\\{a_n\\}$ 满足 $a_1=1$, $a_2=e$ ($e$ 是自然对数的底数), 且 $a_{n+2}=\\sqrt{a_{n+1}\\cdot a_n}$, 令 $b_n=\\ln a_n, n$ 为正整数.\\\\\n(1) 证明: $b_{n+2}>\\sqrt{b_{n+1}b_n}$;\\\\\n(2) 证明: $\\{\\dfrac{b_{n+2}-b_{n+1}}{b_{n+1}-b_n}\\}$ 是等比数列, 且 $\\{b_n\\}$ 的通项公式是 $b_n=\\dfrac{2}{3}[1-(-\\dfrac{1}{2})^{n-1}]$;\\\\\n(3) 是否存在常数 $t$, 对任意正整数 $n$ 均有 $b_{n+1}\\geq t b_n$ 成立? 若存在, 求 $t$ 的取值范围, 否则, 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -503594,7 +503636,9 @@ "id": "019725", "content": "证明: $2 \\pi$ 是函数 $f(x)=\\sin x$ 的最小正周期.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -503614,7 +503658,9 @@ "id": "019726", "content": "已知数列 $\\{a_n\\}$ 的通项公式为 $a_n=n+\\sqrt{3}$, 求证: 数列 $\\{a_n\\}$ 中的任意不同的三项不可能构成等比数列.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -503634,7 +503680,9 @@ "id": "019727", "content": "若函数 $f(x)=\\sqrt{x^2-1}+\\sqrt{a-x^2}$ 为偶函数且非奇函数, 则实数 $a$ 的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -503654,7 +503702,9 @@ "id": "019728", "content": "设 $\\triangle ABC$ 的内角 $A$、$B$、$C$ 所对的边分别为 $a$、$b$、$c$, 若 $b \\cos C+c \\cos B=a \\sin A$, 则 $\\triangle ABC$ 的形状为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -503776,7 +503826,9 @@ "id": "019734", "content": "已知数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 对任意正整数 $n, a_{n+1}=3S_n$, 则下列关于 $\\{a_n\\}$ 的论断中正确的是\\bracket{20}.\n\\twoch{一定是等差数列}{一定是等比数列}{可能是等差数列, 但不会是等比数列}{可能是等比数列, 但不会是等差数列}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -503816,7 +503868,9 @@ "id": "019736", "content": "如果 $a$、$b$ 都是正数, 且 $a \\neq b$, 求证: $\\dfrac{a}{\\sqrt{b}}+\\dfrac{b}{\\sqrt{a}}>\\sqrt{a}+\\sqrt{b}$.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -503856,7 +503910,9 @@ "id": "019738", "content": "定义在 $\\mathbf{R}$ 上的函数 $f(x)$ 满足: 对于任意的 $x_1, x_2 \\in \\mathbf{R}$, 当 $x_10$ 恒成立, 则正数 $a$ 的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -504163,7 +504225,9 @@ "id": "019753", "content": "已知函数 $f(x)=\\begin{cases}x^2+3 x,& x \\geq 0,\\\\3 x-x^2,& x<0.\\end{cases}$ 若 $f(a^2-3)+f(2 a)>0$, 则实数 $a$ 的取值范围为 3. 已知函数 $f(x)=|\\lg x|$, 若 $0=latex,scale = 1.5]\n\\def\\l{2}\n\\def\\m{1}\n\\def\\n{1}\n\\draw (0,0,0) node [below left] {$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$} coordinate (A_1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw ($(A)!0.5!(B)$) node [below] {$E$} coordinate (E);\n\\draw [dashed] (D)--(E)--(C)(A)--(D_1)--(C)(D_1)--(E);\n\\end{tikzpicture}\n\\end{center}\n(1) 求异面直线 $AD_1$ 与 $EC$ 所成角的大小;\\\\\n(2) 《九章算术》中, 将四个面都是直角三角形的四面体称为鳖臑. 试问四面体 $D_1CDE$ 是否为鳖臑? 并说明理由.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -601141,7 +601209,9 @@ "id": "031409", "content": "过双曲线 $\\dfrac{x^2}{9}-\\dfrac{y^2}{16}=1$ 的右焦点, 且平行于渐近线的直线方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -601204,7 +601274,9 @@ "id": "031412", "content": "若复数 $z=(\\sin \\theta-\\dfrac{3}{5})+(\\cos \\theta-\\dfrac{4}{5}) \\mathrm{i}$ 是纯虚数, 则 $\\tan (\\theta-\\dfrac{\\pi}{4})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -601470,7 +601542,9 @@ "id": "031425", "content": "已知双曲线的两个焦点为 $F_1(-\\sqrt{5}, 0), F_2(\\sqrt{5}, 0), P$ 是此双曲线上的一点, 且 $PF_1 \\perp PF_2$, $|PF_1| \\cdot|PF_2|=2$, 则该双曲线的方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -601490,7 +601564,9 @@ "id": "031426", "content": "若 $a_n=\\dfrac{1}{n+1}+\\dfrac{1}{n+2}+\\cdots+\\dfrac{1}{2 n}$ ($n$ 是正整数 $)$, 则 $a_{n+1}-a_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -601510,7 +601586,9 @@ "id": "031427", "content": "已知 $\\mathrm{C}_{10}^{2 x}-\\mathrm{C}_{10}^{x+1}=0$, 则 $x=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -601650,7 +601728,9 @@ "id": "031434", "content": "已知 $\\{a_n\\}$ 是等比数列, $a_1=2$, $a_3=18$, $\\{b_n\\}$ 是等差数列, $b_1=2$, $b_1+b_2+b_3+b_4=a_1+ a_2+a_3>20$. 则数列 $\\{b_n\\}$ 的前 $n$ 项和 $S_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -601730,7 +601810,9 @@ "id": "031438", "content": "在 $\\triangle ABC$ 中, 角 $A, B, C$ 的对边分别为 $a, b, c$, 且 $2 c \\cos ^2 \\dfrac{A}{2}=b+c$, 则 $\\triangle ABC$ 的形状是\\bracket{20}.\n\\fourch{正三角形}{直角三角形}{等腰三角形}{等腰直角三角形}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -601876,7 +601958,9 @@ "id": "031445", "content": "$y=4 x^2$ 的焦点到准线的距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -601896,7 +601980,9 @@ "id": "031446", "content": "若双曲线 $\\dfrac{x^2}{a^2}-\\dfrac{y^2}{9}=1$($a>0$) 的一条渐近线方程为 $x-2 y=0$, 则双曲线的离心率为 $e=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -602000,7 +602086,9 @@ "id": "031451", "content": "已知数列 $\\{a_n\\}$ 的前 $n$ 项和 $S_n=n^2+2 n-1$, 则 $a_1+a_3+a_5+\\cdots+a_{25}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -602020,7 +602108,9 @@ "id": "031452", "content": "$y=f(x)$ 是关于 $x=3$ 对称的奇函数, $f(1)=1$, $\\cos x-\\sin x=\\dfrac{3 \\sqrt{2}}{5}$,\n则 $f(\\dfrac{15 \\sin 2 x}{\\cos (x+\\dfrac{\\pi}{4})})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -602060,7 +602150,9 @@ "id": "031454", "content": "设 $\\mathrm{i}$ 是虚数单位,复数 $\\dfrac{1+a \\mathrm{i}}{2-\\mathrm{i}}$ 为纯虚数,则实数 $a$ 为\\bracket{20}.\n\\fourch{$2$}{$-2$}{$-\\dfrac{1}{2}$}{$\\dfrac{1}{2}$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -602100,7 +602192,9 @@ "id": "031456", "content": "设 $S_n$ 为等差数列 $\\{a_n\\}$ 的前 $n$ 项和, 若 $a_1=1$, 公差 $d=2$, $S_{k+2}-S_k=24$, 则 $k$ 等于\\bracket{20}.\n\\fourch{8}{7}{6}{5}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -602180,7 +602274,9 @@ "id": "031460", "content": "若 $z_1=a+2 \\mathrm{i}$, $z_2=3-4 \\mathrm{i}$, 且 $\\dfrac{z_1}{z_2}$ 为纯虚数, 则实数 $a$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -602222,7 +602318,9 @@ "id": "031462", "content": "已知抛物线 $x^2=3 y$ 上两点 $A, B$ 的横坐标恰是方程 $x^2+5 x+1=0$ 的两个实根, 则直线 $AB$ 的方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -602242,7 +602340,9 @@ "id": "031463", "content": "抛物线 $y^2=2 p x$($p>0$) 的准线经过双曲线 $x^2-y^2=1$ 的左焦点,则 $p=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -602262,7 +602362,9 @@ "id": "031464", "content": "在各项都为正数的等比数列 $\\{a_n\\}$ 中, 首项 $a_1=3$, 前三项和为 21 , 则 $a_3+a_4+a_5=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -602326,7 +602428,9 @@ "id": "031467", "content": "设 $0b>0$) 的右焦点为 $F(c, 0)$, 直线 $y=k(x-c)$ 与双曲线的右支有两个交点, 则\\bracket{20}.\n\\fourch{$|k|>\\dfrac{b}{a}$}{$|k|<\\dfrac{b}{a}$}{$|k|>\\dfrac{c}{a}$}{$|k|<\\dfrac{c}{a}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -602492,7 +602600,9 @@ "id": "031475", "content": "已知函数 $y=f(x)$ 是定义在 $\\mathbf{R}$ 上的严格增函数, 函数 $y=f(x-1)$ 的图像关于点 $(1,0)$ 对称. 若对任意的 $x, y \\in \\mathbf{R}$, $f(x^2-6 x+21)+f(y^2-8 y)<0$ 恒成立, 则当 $x>3$ 时, $x^2+y^2$ 的取值范围是\\bracket{20}.\n\\fourch{$(3,7)$}{$(9,25)$}{$(13,49)$}{$(9,49)$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -602572,7 +602682,9 @@ "id": "031479", "content": "若函数 $f(x)=\\log _a(x+\\sqrt{x^2+2 a^2})$ 是奇函数,则 $a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -602592,7 +602704,9 @@ "id": "031480", "content": "已知抛物线 $y^2=a x$ 的准线方程是 $x=-3$, 那么抛物线的焦点坐标是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -602714,7 +602828,9 @@ "id": "031486", "content": "设 $\\alpha$ 为第四象限的角, 若 $\\dfrac{\\sin 3 \\alpha}{\\sin \\alpha}=\\dfrac{13}{5}$, 则 $\\tan 2 \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -602734,7 +602850,9 @@ "id": "031487", "content": "设 $\\{a_n\\}$ 是正项数列, 其前 $n$ 项和 $S_n$ 满足 : $4S_n=(a_n-1)(a_n+3)$, 则数列 $\\{a_n\\}$ 的通项公式 $a_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -602754,7 +602872,9 @@ "id": "031488", "content": "圆 $x^2+y^2=1$ 与直线 $y=k x+2$ 有两个公共点的充要条件是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -602814,7 +602934,9 @@ "id": "031491", "content": "若函数 $f(x)=\\log _a(x^3-a x)$($a>0$, $a \\neq 1$) 在区间 $(-\\dfrac{1}{2}, 0)$ 是上严格增函数, 则 $a$ 的取值范围是\\bracket{20}.\n\\fourch{$[\\dfrac{1}{4}, 1)$}{$[\\dfrac{3}{4}, 1)$}{($\\dfrac{9}{4},+\\infty$)}{$(1, \\dfrac{9}{4})$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -602874,7 +602996,9 @@ "id": "031494", "content": "已知关于 $x$ 的不等式 $\\dfrac{(a+1) x-3}{x-1}<1$.\\\\\n(1) 当 $a=1$ 时,求该不等式的解集;\\\\\n(2) 当 $a>0$ 时,求该不等式的解集.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -602914,7 +603038,9 @@ "id": "031496", "content": "如果双曲线 $\\dfrac{x^2}{4}-\\dfrac{y^2}{2}=1$ 上一点 $P$ 到双曲线右焦点的距离是 $2$, 那么点 $P$ 到 $y$ 轴的距离是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -602934,7 +603060,9 @@ "id": "031497", "content": "若复数 $(1+b \\mathrm{i})(2+\\mathrm{i})$ 是纯虚数 ($\\mathrm{i}$ 是虚数单位, $b$ 是实数), 则 $b=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -602974,7 +603102,9 @@ "id": "031499", "content": "已知数列 $\\{a_n\\}$ 的前 $n$ 项和 $S_n=n^2-9 n$, 第 $k$ 项满足 $5\\sqrt{2}$) 的两条渐近线的夹角为 $\\dfrac{\\pi}{3}$, 则双曲线的离心率为 \\bracket{20}.\n\\fourch{$\\dfrac{2 \\sqrt{3}}{3}$}{$\\dfrac{2 \\sqrt{6}}{3}$}{$\\sqrt{3}$}{2}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -603276,7 +603416,9 @@ "id": "031514", "content": "已知集合 $M=\\{x|| x |<2\\}$, $N=\\{x | \\dfrac{x+1}{x-3}<0\\}$, 则集合 $M \\cap N$ 等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -603296,7 +603438,9 @@ "id": "031515", "content": "复数 $z=a+b \\mathrm{i}$, $a, b \\in \\mathbf{R}$, 且 $b \\neq 0$, 若 $z^2-4 b z$ 是实数, 则 $a, b$ 满足的条件是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -603316,7 +603460,9 @@ "id": "031516", "content": "在 $(x^2-\\dfrac{1}{x})^8$ 的展开式中, 含 $x$ 的项的系数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -603336,7 +603482,9 @@ "id": "031517", "content": "在等比数列 $\\{a_n\\}$ 中, 若 $a_1=1$, $a_4=\\dfrac{1}{8}$, 则该数列的前 10 项和为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -603356,7 +603504,9 @@ "id": "031518", "content": "若圆 $x^2+y^2-2 x-4 y=0$ 的圆心到直线 $x-y+a=0$ 的距离为 $\\dfrac{\\sqrt{2}}{2}$, 则 $a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -603376,7 +603526,9 @@ "id": "031519", "content": "经过圆 $x^2+2 x+y^2=0$ 的圆心 $C$, 且与直线 $x+y=0$ 垂直的直线方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -603396,7 +603548,9 @@ "id": "031520", "content": "已知向量 $\\overrightarrow{a}=(1,2)$ 和 $\\overrightarrow{b}=(x, 1)$, 若向量 $\\overrightarrow{a}+2 \\overrightarrow{b}$ 与 $2 \\overrightarrow{a}-\\overrightarrow{b}$ 平行, 则实数 $x=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -603416,7 +603570,9 @@ "id": "031521", "content": "若 $\\dfrac{\\cos 2 \\alpha}{\\sin (\\alpha-\\dfrac{\\pi}{4})}=-\\dfrac{\\sqrt{2}}{2}$, 则 $\\cos \\alpha+\\sin \\alpha$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -603456,7 +603612,9 @@ "id": "031523", "content": "如果函数 $f(x)=\\sin (\\pi x+\\theta)$($0<\\theta<2 \\pi$) 的最小正周期是 $T$, 且当 $x=2$ 时取得最大值,则 $T=$\\blank{50}, $\\theta=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -603518,7 +603676,9 @@ "id": "031526", "content": "平面 $\\alpha \\parallel $ 平面 $\\beta$ 的一个充分条件是\\bracket{20}.\n\\onech{存在一条直线 $a$, $a \\parallel \\alpha$, $a \\parallel \\beta$}{存在一条直线 $a$, $a \\subset \\alpha$, $a \\parallel \\beta$}{存在两条平行直线 $a$, $b$, $a \\subset \\alpha$, $b \\subset \\beta$, $a \\parallel \\beta$, $b \\parallel \\alpha$}{存在两条异面直线 $a$, $b$, $a \\subset \\alpha$, $b \\subset \\beta$, $a \\parallel \\beta$, $b \\parallel \\alpha$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -603538,7 +603698,9 @@ "id": "031527", "content": "如图, 在正方体 $ABCD-A_1B_1C_1D_1$ 中, $O$ 是底面正方形 $ABCD$ 的中心, $M$ 是 $DD_1$ 的中点, $N$ 是 $A_1B_1$ 上的动点, 则直线 $NO$、$AM$ 的位置关系是\\blank{50}.\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 (A_1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\filldraw ($(A)!0.5!(C)$) node [left] {$O$} coordinate (O) circle (0.03);\n\\draw ($(D)!0.5!(D_1)$) node [right] {$M$} coordinate (M);\n\\draw ($(A_1)!0.9!(B_1)$) node [above] {$N$} coordinate (N);\n\\draw [dashed] (A)--(M)(N)--(O);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{平行}{相交}{异面垂直}{异面不垂直}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -603600,7 +603762,9 @@ "id": "031530", "content": "已知函数 $y=\\sqrt{\\dfrac{1+x}{1-x}}+\\lg (3-4 x+x^2)$ 的定义域为 $M$.\\\\\n(1) 求 $M$;\\\\\n(2) 当 $x \\in M$ 时, 求 $f(x)=a \\cdot 2^{x+2}+3 \\times 4^x$($a>-3$) 的最小值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -603662,7 +603826,9 @@ "id": "031533", "content": "已知集合 $A=\\{x | x^2-3 x+2=0\\}$, $B=\\{x | \\log _x 4=2\\}$, 则 $A \\cup B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -603702,7 +603868,9 @@ "id": "031535", "content": "抛物线 $y^2=2 p x$($p>0$) 的准线经过双曲线 $x^2-y^2=1$ 的左焦点, 则 $p=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -603722,7 +603890,9 @@ "id": "031536", "content": "若函数 $y=x^3-2 m x^2+m^2 x$ 在 $x=1$ 处取得极小值, 则实数 $m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -603742,7 +603912,9 @@ "id": "031537", "content": "设等比数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 若 $S_{m-1}=5$, $S_m=-11$, $S_{m+1}=21$, 则 $m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -603762,7 +603934,9 @@ "id": "031538", "content": "已知 $\\tan \\alpha=2$, 则 $\\dfrac{\\sin (\\pi+\\alpha)-\\sin (\\dfrac{\\pi}{2}+\\alpha)}{\\cos (\\dfrac{3 \\pi}{2}+\\alpha)+\\cos (\\pi-\\alpha)}$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -603802,7 +603976,9 @@ "id": "031540", "content": "在 $\\triangle ABC$ 中, 已知内角 $A=\\dfrac{\\pi}{3}$, 边 $BC=2 \\sqrt{3}$, 则 $\\triangle ABC$ 的面积 $S$ 的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -603882,7 +604058,9 @@ "id": "031544", "content": "设 $a, b$ 是平面 $\\alpha$ 内两条不同的直线, $l$ 是平面 $\\alpha$ 外的一条直线,则``$l \\perp a$, $l \\perp b$''是``$l \\perp \\alpha$''的\\bracket{20}.\n\\twoch{充要条件}{充分而不必要的条件}{必要而不充分的条件}{既不充分也不必要的条件}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -603902,7 +604080,9 @@ "id": "031545", "content": "函数 $f(x)=\\ln (x+1)-\\dfrac{2}{x}$ 的零点所在的区间是\\bracket{20}.\n\\fourch{$(\\dfrac{1}{2}, 1)$}{$(1, \\mathrm{e}-1)$}{$(\\mathrm{e}-1,2)$}{$(2, \\mathrm{e})$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -603922,7 +604102,9 @@ "id": "031546", "content": "已知双曲线 $\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>b>0$) 的右焦点为 $F(c, 0)$, 直线 $y=k(x-c)$ 与双曲线的右支有两个交点, 则\\bracket{20}.\n\\fourch{$|k|>\\dfrac{b}{a}$}{$|k|<\\dfrac{b}{a}$}{$|k|>\\dfrac{c}{a}$}{$|k|<\\dfrac{c}{a}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -603942,7 +604124,9 @@ "id": "031547", "content": "已知函数 $y=f(x)$ 是定义在 $\\mathbf{R}$ 上的严格增函数, 函数 $y=f(x-1)$ 的图像关于点 $(1,0)$ 对称. 若对任意的 $x, y \\in \\mathbf{R}$, $f(x^2-6 x+21)+f(y^2-8 y)<0$ 恒成立, 则当 $x>3$ 时, $x^2+y^2$ 的取值范围是\\bracket{20}.\n\\fourch{$(3,7)$}{$(9,25)$}{$(13,49)$}{$(9,49)$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -603982,7 +604166,9 @@ "id": "031549", "content": "如图所示, 椭圆 $C: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$) 的一个焦点为 $F(1,0)$, 且过点 $(\\sqrt{2}, \\dfrac{\\sqrt{6}}{2})$.\\\\\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (-2.5,0) -- (4.5,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [name path = elli] (0,0) ellipse (2 and {sqrt(3)});\n\\draw (4,-2.5) -- (4,2.5);\n\\draw (4,0) node [above right] {$N$} coordinate (N);\n\\draw (80:2 and {sqrt(3)}) node [above right] {$A$} coordinate (A);\n\\draw (-80:2 and {sqrt(3)}) node [below right] {$B$} coordinate (B);\n\\draw (A)--(N) ($(A)!-0.2!(B)$) -- ($(B)!-0.2!(A)$);\n\\draw [name path = BN] (B)--(N);\n\\draw [name intersections = {of = BN and elli, by = {T,M}}];\n\\draw (A)--(M) node [below] {$M$};\n\\filldraw (1,0) node [above right] {$F$} coordinate (F) circle (0.05);\n\\end{tikzpicture}\n\\end{center}\n(1) 求椭圆 $C$ 的方程;\\\\\n(2) 已知 $A, B$ 为椭圆上的点, 且直线 $AB$ 垂直于 $x$ 轴, 直线 $l$ : $x=4$ 与 $x$ 轴交于点 $N$, 直线 $AF$ 与 $BN$ 交于点 $M$.\\\\\n(I) 求证: 点 $M$ 恒在椭圆 $C$ 上;\\\\\n(II) 求 $\\triangle AMN$ 面积的最大值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -604002,7 +604188,9 @@ "id": "031550", "content": "已知集合 $A=\\{x|| x |<2\\}$, $B=\\{x | \\dfrac{1}{x+1}>0\\}$, 则 $A \\cap B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -604106,7 +604294,9 @@ "id": "031555", "content": "若 $\\dfrac{\\sin \\alpha+\\cos \\alpha}{\\sin \\alpha-\\cos \\alpha}=\\dfrac{1}{2}$, 则 $\\tan 2 \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -604146,7 +604336,9 @@ "id": "031557", "content": "若椭圆 $\\dfrac{x^2}{25}+\\dfrac{y^2}{16}=1$ 上一点 $P$ 到焦点 $F_1$ 的距离为 6 , 则点 $P$ 到另一个焦点 $F_2$ 的距离是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -604230,7 +604422,9 @@ "id": "031561", "content": "正方体 $ABCD-A_1B_1C_1D_1$ 的棱长为 $1$, $E, F$ 分别为线段 $AA_1, B_1C$ 上的点, 则三棱锥 $D_1-EDF$ 的体积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -604312,7 +604506,9 @@ "id": "031565", "content": "设 $F_1F_2$ 是椭圆 $E: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$) 的左、右焦点, $P$ 为直线 $x=\\dfrac{3 a}{2}$ 上一点, $\\triangle F_2PF_1$ 是底角为 $30^{\\circ}$ 的等腰三角形, 则 $E$ 的离心率为\\bracket{20}.\n\\fourch{$\\dfrac{1}{2}$}{$\\dfrac{2}{3}$}{$\\dfrac{3}{4}$}{$\\dfrac{4}{5}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -604332,7 +604528,9 @@ "id": "031566", "content": "如图, 在棱长为 $a$ 的正方体 $ABCD-A_1B_1C_1D_1$ 中, $O$ 是 $AC, BD$ 的交点, $E, F$ 分别是 $AB$ 与 $AD$ 的中点.\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 (A_1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw ($(A)!0.5!(B)$) node [below] {$E$} coordinate (E);\n\\draw ($(A)!0.5!(D)$) node [left] {$F$} coordinate (F);\n\\draw ($(B)!0.5!(D)$) node [below] {$O$} coordinate (O);\n\\draw (A_1)--(C_1);\n\\draw [dashed] (E)--(F)(B)--(D)(A)--(C)(D_1)--(O);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: 直线 $OD_1$ 与直线 $A_1C_1$ 垂直;\\\\\n(2) 求异面直线 $EF$ 与 $A_1C_1$ 所成角的大小;\\\\\n(3) 求二面角 $B-AC-D_1$ 的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -604374,7 +604572,9 @@ "id": "031568", "content": "已知 $\\{a_n\\}$ 为等比数列, $a_4+a_7=2$, $a_5 a_6=-8$, 则 $a_1+a_{10}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -604416,7 +604616,9 @@ "id": "031570", "content": "已知函数 $f(x)=(a^2-1) x^2+2 x+a+1$ 是奇函数, 则实数 $a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -604518,7 +604720,9 @@ "id": "031575", "content": "若正四棱锥的底面对角线的长为 $2 \\sqrt{6}$, 体积为 $4 \\sqrt{3}$, 则侧面与底面所成的二面角等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -604770,7 +604974,9 @@ "id": "031587", "content": "已知函数 $f(x)=\\lg \\dfrac{1-x}{1+x}$, 若 $f(a)=b$, 则 $f(-a)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -604790,7 +604996,9 @@ "id": "031588", "content": "长方体 $ABCD-A_1B_1C_1D_1$ 中, 对角线 $AC_1$ 的长为 $l$, $\\angle DAC_1=45^{\\circ}$, $\\angle A_1AC_1=60^{\\circ}$, 则三棱锥 $C-B_1C_1D_1$ 的体积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -604830,7 +605038,9 @@ "id": "031590", "content": "已知定义在 $\\mathbf{R}$ 上的奇函数, $f(x)$ 满足 $f(x+2)=-f(x)$, 则 $f(-6)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -604850,7 +605060,9 @@ "id": "031591", "content": "椭圆 $\\dfrac{x^2}{4}+y^2=1$ 的两个焦点为 $F_1, F_2$, 过 $F_1$ 作垂直于 $x$ 轴的直线与椭圆相交, 一个交点为 $P$, 则 $|\\overrightarrow{PF_2}|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -604934,7 +605146,9 @@ "id": "031595", "content": "设直线 $l$ 过点 $(-2,0)$, 且与圆 $x^2+y^2=1$ 相切, 则 $l$ 的斜率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -604974,7 +605188,9 @@ "id": "031597", "content": "双曲线 $\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$) 的两个焦点为 $F_1, F_2$, 若 $P$ 为其上一点, 且 $|PF_1|= 2|PF_2|$, 则双曲线离心率的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -605014,7 +605230,9 @@ "id": "031599", "content": "在正方体 $ABCD-A_1B_1C_1D_1$ 中, $M, N$ 分别为棱 $A_1B_1$ 和 $BB_1$ 的中点, 那么异面直线 $AM$ 和 $CN$ 所成角的余弦值是\\bracket{20}.\n\\fourch{$\\dfrac{\\sqrt{3}}{2}$}{$\\dfrac{\\sqrt{10}}{2}$}{$\\dfrac{2}{5}$}{$-\\dfrac{2}{5}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -605114,7 +605332,9 @@ "id": "031604", "content": "已知 $\\mathrm{i}$ 是虚数单位, $\\dfrac{2-\\mathrm{i}}{z+\\mathrm{i}}=\\mathrm{i}$, 则 $|z|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -605178,7 +605398,9 @@ "id": "031607", "content": "抛物线 $y^2=2 p x$($p>0$) 的准线经过双曲线 $x^2-y^2=1$ 的左焦点, 则 $p=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -605218,7 +605440,9 @@ "id": "031609", "content": "设等比数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 若 $S_{m-1}=5$, $S_m=-11$, $S_{m+1}=21$, 则 $m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -605386,7 +605610,9 @@ "id": "031617", "content": "若 $f(x)=-\\dfrac{1}{2}x^2+b \\ln (x+2)$ 在 ($-1,+\\infty$) 上是减函数, 则 $b$ 的取值范围是\\bracket{20}.\n\\fourch{$[-1,+\\infty)$}{($-1,+\\infty$)}{$(-\\infty,-1]$}{($-\\infty,-1$)}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -605488,7 +605714,9 @@ "id": "031622", "content": "设集合 $A=\\{x |-2 \\leq x \\leq 2\\}$, $\\mathbf{Z}$ 为整数集, 则集合 $A \\cap \\mathbf{Z}$ 中含有 $2$ 的子集个数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -605597,7 +605825,9 @@ "id": "031627", "content": "已知直线 $y=k x$ 与圆 $(x-5)^2+y^2=9$ 有两个公共点, 则实数 $k$ 的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -605617,7 +605847,9 @@ "id": "031628", "content": "若 $\\tan \\theta=\\dfrac{1}{3}$, 则 $\\cos 2 \\theta=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -605702,7 +605934,9 @@ "id": "031632", "content": "在封闭的直三棱柱 $ABC-A_1B_1C_1$ 内有一个体积为 $V$ 的球. 若 $AB \\perp BC$, $AB=6$, $BC= 8$, $AA_1=3$, 则 $V$ 的最大值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -605870,7 +606104,9 @@ "id": "031640", "content": "已知 $\\tan (\\alpha-\\dfrac{5 \\pi}{4})=\\dfrac{1}{5}$, 则 $\\tan \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -605913,7 +606149,9 @@ "id": "031642", "content": "若双曲线 $\\dfrac{x^2}{a^2}-\\dfrac{y^2}{3}=1$($a>0$) 的一条渐近线被圆 $(x-2)^2+y^2=4$ 所截得的弦长为 $2$ , 则 $a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -605933,7 +606171,9 @@ "id": "031643", "content": "在长方体 $ABCD-A_1B_1C_1D_1$ 中, $E$ 为棱 $CC_1$ 的中点, 则异面直线 $AE$ 与 $CD$ 所成角的正切值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -605975,7 +606215,9 @@ "id": "031645", "content": "双曲线 $\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$) 的离心率为 $\\sqrt{3}$, 则其渐近线方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -606015,7 +606257,9 @@ "id": "031647", "content": "在直角坐标系 $xOy$ 中, 曲线 $C$ 的参数方程为 $\\begin{cases}x=2 \\cos \\theta,\\\\y=4 \\sin \\theta\\end{cases}$ ($\\theta$ 为参数), 直线 $l$ 的参数方程为 $\\begin{cases}x=1+t \\cos \\alpha,\\\\y=2+t \\sin \\alpha\\end{cases}$ ($t$ 为参数). 若曲线 $C$ 截直线 $l$ 所得线段的中点坐标为 $(1,2)$, 则直线 $l$ 的斜率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -606117,7 +606361,9 @@ "id": "031652", "content": "在等差数列 $\\{a_n\\}$ 中, 若 $a_3+a_8+a_{13}=C$, 则其前 $n$ 项和 $S_n$ 的值等于 $5C$ 的是\\bracket{20}.\n\\fourch{$S_{15}$}{$S_{17}$}{$S_7$}{$S_8$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -606263,7 +606509,9 @@ "id": "031659", "content": "函数 $y=\\sin ^4 x-\\sin ^2 x$ 的最小正周期是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -606283,7 +606531,9 @@ "id": "031660", "content": "从甲、乙等 $5$ 名学生中随机选出 $2$ 人, 则甲被选中的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -606325,7 +606575,9 @@ "id": "031662", "content": "已知等差数列 $\\{a_n\\}$ 中, $a_4+a_8=16$, $a_2=1$, 则 $a_{10}$ 的值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -606345,7 +606597,9 @@ "id": "031663", "content": "若不等式 $|x|+|x-1|>m$ 的解集是 $\\mathbf{R}$, 则实数 $m$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -606387,7 +606641,9 @@ "id": "031665", "content": "已知 $\\{a_n\\}$ 是等差数列, $\\{b_n\\}$ 是等比数列, 且 $b_2=3$, $b_3=9$, $a_1=b_1$, $a_{14}=b_4$. 设 $c_n=a_n+ b_n$, 则数列 $\\{c_n\\}$ 的前 $n$ 项和 $S_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -606507,7 +606763,9 @@ "id": "031671", "content": "函数 $f(x)=\\begin{cases}\\sin (\\pi x^2),&-1=latex]\n\\draw (0,0,0) node [above right] {$C$} coordinate (C);\n\\draw (-1,0,0) node [left] {$D$} coordinate (D);\n\\draw (0,0,1) node [left] {$A$} coordinate (A);\n\\draw (A) ++ (2,0,0) node [right] {$B$} coordinate (B);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\filldraw ($(A)!0.5!(B)$) node [below] {$E$} coordinate (E) circle (0.03);\n\\draw (P)--(D)--(A)--(B)--cycle(P)--(A);\n\\draw [dashed] (P)--(C)--(D)(C)--(A)(C)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $DC \\perp$ 平面 $PAC$;\\\\\n(2) 求证: 平面 $PAB \\perp$ 平面 $PAC$;\\\\\n(3) 设点 $E$ 为 $AB$ 的中点, 在棱 $PB$ 上是否存在点 $F$, 使得 $PA \\parallel $ 平面 $CEF$ ? 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -606587,7 +606849,9 @@ "id": "031675", "content": "已知椭圆 $C: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$ 过点 $A(2,0), B(0,1)$ 两点.\\\\\n(1) 求椭圆 $C$ 的方程及离心率;\\\\\n(2) 设 $P$ 为第三象限内一点且在椭圆 $C$ 上, 直线 $PA$ 与 $y$ 轴交于点 $M$, 直线 $PB$ 与 $x$ 轴交于点 $N$. 求证: 四边形 $ABNM$ 的面积为定值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -606647,7 +606911,9 @@ "id": "031678", "content": "若 $(a x-1)^5$ 的展开式中 $x^3$ 的系数是 $80$ , 则实数 $a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -606667,7 +606933,9 @@ "id": "031679", "content": "曲线 $y^2=4 x$ 关于直线 $x=2$ 对称的曲线方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -606687,7 +606955,9 @@ "id": "031680", "content": "若函数 $y=(\\dfrac{1}{2})^{|1-x|}+m$ 的图像与 $x$ 轴有公共点, 则实数 $m$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -606727,7 +606997,9 @@ "id": "031682", "content": "已知直线 $\\sqrt{3}x+y=0$ 和直线 $k x-y-1=0$, 若两直线的夹角为 $60^{\\circ}$, 则 $k=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -606747,7 +607019,9 @@ "id": "031683", "content": "设函数 $f(x)=\\begin{cases}x^3-3 x,& x \\leq a,\\\\-2 x,& x>a,\\end{cases}$ 若 $f(x)$ 无最大值, 则实数 $a$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -606789,7 +607063,9 @@ "id": "031685", "content": "已知 $\\sin 2 \\alpha=-\\dfrac{1}{3}$, 则 $\\dfrac{4 \\cos ^2 \\alpha}{\\cot \\dfrac{\\alpha}{2}-\\tan \\dfrac{\\alpha}{2}}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -606851,7 +607127,9 @@ "id": "031688", "content": "在复平面内, 复数 $\\dfrac{1+\\mathrm{i}}{(1-\\mathrm{i})^2}$ 对应的点位于\\bracket{20}.\n\\fourch{第一象限}{第二象限}{第三象限}{第四象限}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -607017,7 +607295,9 @@ "id": "031696", "content": "设数列 $\\{a_n\\}$ 是等差数列, 且 $a_2=-6$, $a_8=6, S_n$ 是数列 $\\{a_n\\}$ 的前 $n$ 项和, 则 $S_{10}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -607037,7 +607317,9 @@ "id": "031697", "content": "$(x-\\sqrt{2}y)^{10}$ 的展开式中 $x^6 y^4$ 项的系数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -607099,7 +607381,9 @@ "id": "031700", "content": "若过点 $A(4,0)$ 的直线 $l$ 与曲线 $(x-2)^2+y^2=1$ 有公共点, 则直线 $l$ 的斜率的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -607179,7 +607463,9 @@ "id": "031704", "content": "已知数列 $\\{a_n\\}$ 的前 $n$ 项和 $S_n$ 满足 $S_n=2 a_n+(-1)^n$, $n \\geq 1$. 则数列 $\\{a_n\\}$ 的前三项的和 $a_1+a_2+a_3=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -607221,7 +607507,9 @@ "id": "031706", "content": "函数 $y=\\sin (2 x+\\dfrac{\\pi}{3})$ 图像的对称轴方程可能是\\bracket{20}.\n\\fourch{$x=-\\dfrac{\\pi}{6}$}{$x=-\\dfrac{\\pi}{12}$}{$x=\\dfrac{\\pi}{6}$}{$x=\\dfrac{\\pi}{12}$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -607321,7 +607609,9 @@ "id": "031711", "content": "在长方体 $ABCD-A_1B_1C_1D_1$ 中, $AA_1=AD=2$, $E$ 是棱 $CD$ 上的一点.\\\\\n(1) 求证: $AD_1 \\perp$ 平面 $A_1B_1D$;\\\\\n(2) 求异面直线 $B_1E, AD_1$ 所成的角;\\\\\n(3) 若 $E$ 是棱 $CD$ 的中点, 在棱 $AA_1$ 上是否存在点 $P$, 使得 $DP \\parallel $ 平面 $B_1AE$ ? 若存在, 求出线段 $AP$ 的长; 若不存在, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -607341,7 +607631,9 @@ "id": "031712", "content": "不等式 $(|x|+1)(2 x-1)>0$ 的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -607361,7 +607653,9 @@ "id": "031713", "content": "已知圆 $x^2+y^2-6 x-7=0$ 与抛物线 $y^2=2 p x$($p>0$) 的准线相切, 则 $p=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -607381,7 +607675,9 @@ "id": "031714", "content": "若集合 $M=\\{(x, y) | y=2^{-x}\\}$, $P=\\{y | y=\\sqrt{x-1}\\}$, 则 $M \\cap P=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -607441,7 +607737,9 @@ "id": "031717", "content": "在 $\\triangle ABC$ 中, $\\angle A=60^{\\circ}$, $b=1$, 这个三角形的面积为 $\\sqrt{3}$, 则 $\\triangle ABC$ 外接圆的直径是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -607461,7 +607759,9 @@ "id": "031718", "content": "$(x^2-\\dfrac{1}{2 x})^9$ 展开式中 $x^9$ 的系数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -607501,7 +607801,9 @@ "id": "031720", "content": "抛物线 $y^2=2 p x$ 与直线 $a x+y-4=0$ 的一个交点是 $(1,2)$, 则抛物线的焦点到直线的距离是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -607561,7 +607863,9 @@ "id": "031723", "content": "已知 $f(x+1)=-f(x)$, 且 $f(x)=\\begin{cases}1,& -10,\\\\f(x+1)+1,& x \\leq 0,\\end{cases}$ 则 $f(\\dfrac{4}{3})+f(-\\dfrac{4}{3})$ 的值等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -608447,7 +608777,9 @@ "id": "031766", "content": "已知 $z=2+\\dfrac{1}{\\mathrm{i}}$, 则 $\\overline{z}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -608467,7 +608799,9 @@ "id": "031767", "content": "已知 $A=\\{-1,1, m\\}$, 集合 $B=\\{1,2\\}$, 若 $B \\subset A$, 则实数 $m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -608509,7 +608843,9 @@ "id": "031769", "content": "已知球 $O$ 的体积为 $36 \\pi$, 则该球的球面面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -608551,7 +608887,9 @@ "id": "031771", "content": "已知两条直线 $l_1: a x+3 y-3=0$, $l_2: 2 x+6 y+1=0$. 若 $l_1 \\parallel l_2$, 则直线 $l_1$ 与 $l_2$ 之间的距离 $d=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -608715,7 +609053,9 @@ "id": "031779", "content": "已知双曲线 $x^2-\\dfrac{y^2}{m}=1$ 的渐近线方程为 $y= \\pm \\sqrt{3}x$, 则该双曲线的离心率为\\bracket{20}.\n\\fourch{$\\dfrac{1}{2}$}{2}{$\\dfrac{\\sqrt{10}}{3}$}{$\\dfrac{3 \\sqrt{10}}{10}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -608735,7 +609075,9 @@ "id": "031780", "content": "如图, 在正方体 $ABCD-A_1B_1C_1D_1$ 中, $AB=a$, 任作平面 $\\alpha$ 与对角线 $AC_1$ 垂直, 使得 $\\alpha$ 与正方体的每个面都有公共点, 记这样得到的截面多边形的面积为 $S$, 周长为 $l$, 则\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = {(235:0.5cm)}]\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 (A_1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw [dashed] (A)--(C_1);\n\\draw (A_1)--(B) (D_1)--(B_1)--(C);\n\\draw [dashed] (A_1)--(D)--(B)(D_1)--(C);\n\\def\\lambda{0.4}\n\\draw ($(A_1)!\\lambda!(D_1)$) -- ($(A_1)!\\lambda!(B_1)$) -- ($(B)!\\lambda!(B_1)$) -- ($(B)!\\lambda!(C)$);\n\\draw [dashed] ($(B)!\\lambda!(C)$) -- ($(D)!\\lambda!(C)$) -- ($(D)!\\lambda!(D_1)$) -- ($(A_1)!\\lambda!(D_1)$);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$S$ 是定值, $l$ 不是定值}{$S$ 不是定值, $l$ 是定值}{$S$ 和 $l$ 都是定值}{$S$ 和 $l$ 都不是定值}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -608777,7 +609119,9 @@ "id": "031782", "content": "如图, 在直三棱柱 $ABC-A_1B_1C_1$ 中, $AB=BC=BB_1=4, M, N$ 分别为 $A_1B_1, AC$ 的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$B$} coordinate (B);\n\\draw (1.9,0,{sqrt(0.39)}) node [right] {$A$} coordinate (A);\n\\draw ({-sqrt(0.39)},0,1.9) node [left] {$C$} coordinate (C);\n\\draw (A) ++ (0,2,0) node [right] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,2,0) node [above] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,2,0) node [left] {$C_1$} coordinate (C_1);\n\\draw (C)--(A)--(A_1)--(C_1)--cycle(A_1)--(B_1)--(C_1);\n\\draw ($(A)!0.5!(C)$) node [below] {$N$} coordinate (N);\n\\draw ($(A_1)!0.5!(B_1)$) node [above] {$M$} coordinate (M);\n\\draw [dashed] (B)--(N)--(M)--cycle(C)--(B)--(A)(B)--(B_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $MN \\parallel $ 平面 $BCC_1B_1$;\\\\\n(2) 若 $AB \\perp MN$, 求异面直线 $MN$ 与 $A_1C_1$ 所成的角的余弦值.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -608837,7 +609181,9 @@ "id": "031785", "content": "已知函数 $f(x)=2^x$.\\\\\n(1) 求函数 $y=3 f(x)-f(-x)-2$ 的零点;\\\\\n(2) 证明: 当 $a \\leq 16$ 时, 函数 $F(x)=f(2 x)+a \\cdot f(-x)$ 是 ($1,+\\infty$) 上的严格递增函数;\\\\\n(3) 设 $g(x)=\\dfrac{1}{1+a \\cdot f(x)}-\\dfrac{1}{1+a \\cdot f(x-1)}$, 若对任意 $x \\in(-\\infty, 0]$, $g(x) \\geq g(0)$ 恒成立, 求正实数 $a$ 的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -608877,7 +609223,9 @@ "id": "031787", "content": "已知 $z=-1+2 \\mathrm{i}$, 则 $|\\overline{z}|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -608981,7 +609329,9 @@ "id": "031792", "content": "已知二项式 $(x-\\dfrac{a}{x})^5$ 的展开式中 $x^3$ 的系数为 10 , 则 $a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -609023,7 +609373,9 @@ "id": "031794", "content": "设 $\\{a_n\\}$ 是公比为 $q$ 的等比数列, $S_n$ 是它的前 $n$ 项和, 若 $\\{S_n\\}$ 是等差数列, 则 $q=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -609043,7 +609395,9 @@ "id": "031795", "content": "设 $\\triangle ABC$ 的内角 $A, B, C$ 的对边分别为 $a, b, c$, 且满足 $a \\cos B-b \\cos A=\\dfrac{3}{5}c$, 则 $\\dfrac{\\tan A}{\\tan B}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -609083,7 +609437,9 @@ "id": "031797", "content": "若实数 $a, b$ 满足 $a-4 \\sqrt{b}=2 \\sqrt{a-b}$, 且 $a, b$ 不同时为零, 则 $a$ 的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -609223,7 +609579,9 @@ "id": "031804", "content": "如图, 在直三棱柱 $ABC-A_1B_1C_1$ 中, 点 $D, E$ 分别为 $AC$ 和 $B_1C_1$ 的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [above right] {$D$} coordinate (D);\n\\draw ({-sqrt(2)},0,0) node [left] {$A$} coordinate (A);\n\\draw ({sqrt(2)},0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,{sqrt(2)}) node [below] {$B$} coordinate (B);\n\\draw (A) ++ (0,2,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,2,0) node [above] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,2,0) node [right] {$C_1$} coordinate (C_1);\n\\draw ($(B_1)!0.5!(C_1)$) node [below right] {$E$} coordinate (E);\n\\draw (A)--(B)--(C)--(C_1)--(A_1)--cycle(A_1)--(B_1)--(B)(B_1)--(C_1)(B)--(E);\n\\draw [dashed] (A)--(C)(A)--(E)--(D)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $DE \\parallel $ 平面 $ABB_1A_1$;\\\\\n(2) 若 $AB \\perp BC$, $AB=BC=AA_1=2$, 求点 $D$ 到平面 $ABE$ 的距离.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -609305,7 +609663,9 @@ "id": "031808", "content": "已知集合 $A=\\{x | x^2-x<0\\}$, $B=(0, a)$($a>0$), 若 $A \\subseteq B$, 则实数 $a$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -609325,7 +609685,9 @@ "id": "031809", "content": "设 $f(x)$ 是定义在 $\\mathbf{R}$ 上的奇函数, 若当 $x \\geq 0$ 时, $f(x)=\\log _3(3+x)$, 则 $f(-6)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -609345,7 +609707,9 @@ "id": "031810", "content": "已知双曲线 $y^2-\\dfrac{x^2}{m^2}=1$($m>0$) 的一条渐近线方程为 $x+\\sqrt{3}y=0$, 则 $m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -609385,7 +609749,9 @@ "id": "031812", "content": "已知等比数列 $\\{a_n\\}$ ($n$ 是正整数) 满足 $a_2 a_6=4(a_4-1)$, 则 $a_4=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -609425,7 +609791,9 @@ "id": "031814", "content": "已知 $\\alpha, \\beta$ 为锐角, 且 $\\cos (\\alpha+\\beta)=\\dfrac{\\sin \\alpha}{\\sin \\beta}$, 则 $\\tan \\alpha$ 的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -609445,7 +609813,9 @@ "id": "031815", "content": "已知关于 $x$ 的一元二次不等式 $a x^2+2 x+b>0$ 的解集为 $\\{x | x \\neq c\\}$, 则 $\\dfrac{a^2+b^2+7}{a+c}$(其中 $a+c \\neq 0$) 的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -609527,7 +609897,9 @@ "id": "031819", "content": "已知椭圆 $\\dfrac{x^2}{4}+\\dfrac{y^2}{3}=1$ 的左右顶点分别为 $A, B$, 过点 $C(0,1)$ 斜率为 $k$($k>1$) 的直线 $l$ 与椭圆交于 $M, N$, 记直线 $AM, BN$ 的斜率为 $k_1, k_2$, 且 $k_1=2 k_2$, 则 $k=$\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [name path = elli] (0,0) ellipse (2 and {sqrt(3)});\n\\draw (-2,0) node [below left] {$A$} coordinate (A);\n\\draw (2,0) node [below right] {$B$} coordinate (B);\n\\filldraw (-1,0) node [below] {$F_1$} coordinate (F_1) circle (0.03);\n\\filldraw (1,0) node [below] {$F_2$} coordinate (F_2) circle (0.03);\n\\filldraw (0,1) node [right] {$C$} coordinate (C) circle (0.03);\n\\draw [name path = MN] (C) ++ (0.8,1.2) --++ (-2.6,-3.9);\n\\draw [name intersections = {of = MN and elli, by = {M,N}}];\n\\draw (M)--(A)(N)--(B);\n\\draw (M) node [above] {$M$} (N) node [below] {$N$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -609547,7 +609919,9 @@ "id": "031820", "content": "下列函数中, 既不是奇函数, 也不是偶函数的是\\bracket{20}.\n\\fourch{$y=x+\\mathrm{e}^x$}{$y=x+\\dfrac{1}{x}$}{$y=2^x+\\dfrac{1}{2^x}$}{$y=\\sqrt{1+x^2}$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -609567,7 +609941,9 @@ "id": "031821", "content": "已知 $A$ 为 $\\triangle ABC$ 的一个内角, 且 $\\sin A+\\cos A=\\dfrac{\\sqrt{2}}{3}$, 则 $\\triangle ABC$ 的形状是\\bracket{20}.\n\\fourch{锐角三角形}{钝角三角形}{直角三角形}{不确定}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -609587,7 +609963,9 @@ "id": "031822", "content": "如图, 在正方体 $ABCD-A_1B_1C_1D_1$ 中, $E$ 是棱 $CC_1$ 的中点, $F$ 是侧面 $B_1BCC_1$ 上的动点, 并且 $A_1F \\parallel $ 平面 $AED_1$, 则动点 $F$ 的轨迹是\\bracket{20}.\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 (A_1);\n\\draw (B) ++ (0,\\l,0) node [above] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw ($(C)!0.5!(C_1)$) node [right] {$E$} coordinate (E);\n\\draw [dashed] (A)--(E)--(D_1)--cycle;\n\\filldraw (2,1.3,-0.3) node [right] {$F$} coordinate (F) circle (0.03);\n\\draw [dashed] (A_1)--(F);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{圆}{椭圆}{抛物线}{线段}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -609627,7 +610005,9 @@ "id": "031824", "content": "已知数列 $\\{a_n\\}$ 是等差数列, 且 $a_1=2$, $a_1+a_2+a_3=12$.\\\\\n(1) 求数列 $\\{a_n\\}$ 的通项公式;\\\\\n(2) 令 $b_n=2^{a_n}+9$, 数列 $\\{b_n\\}$ 前 $n$ 项和为 $S_n$, 若 $S_n \\geq 2022$, 求正整数 $n$ 的最小值.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -609647,7 +610027,9 @@ "id": "031825", "content": "如图, 在长方体 $ABCD-A_1B_1C_1D_1$ 中, $E, P$ 分别是 $BC, A_1D_1$ 的中点, $M, N$ 分别是 $AE, CD_1$ 的中点, $AD=AA_1=a$, $AB=2 a$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{4}\n\\def\\m{2}\n\\def\\n{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,-\\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 (A_1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw ($(B)!0.5!(C)$) node [right] {$E$} coordinate (E);\n\\draw ($(C)!0.5!(D_1)$) node [above] {$N$} coordinate (N);\n\\draw ($(A_1)!0.5!(D_1)$) node [above] {$P$} coordinate (P);\n\\draw ($(A)!0.5!(E)$) node [above left] {$M$} coordinate (M);\n\\draw [dashed] (A)--(E)--(P)--cycle(M)--(N)(C)--(D_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $MN \\parallel $ 面 $ADD_1A_1$;\\\\\n(2) 求三棱锥 $P-DEN$ 的体积.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -609687,7 +610069,9 @@ "id": "031827", "content": "已知函数 $f(x)=x^4+a x^3+2 x^2+b$($x \\in \\mathbf{R}$), 其中 $a, b \\in \\mathbf{R}$.\\\\\n(1) 当 $a=-\\dfrac{10}{3}$ 时, 讨论函数 $f(x)$ 的单调性;\\\\\n(2) 若函数 $f(x)$ 仅在 $x=0$ 处有极值, 求 $a$ 的取值范围;\\\\\n(3) 若对于任意的 $a \\in[-2,2]$, 不等式 $f(x) \\leq 1$ 在 $[-1,1]$ 上恒成立,求 $b$ 的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -609816,7 +610200,9 @@ "id": "031833", "content": "已知向量 $\\overrightarrow{a}=(2,-3)$, $\\overrightarrow{b}=(3, \\lambda)$, 若 $\\overrightarrow{a}\\parallel \\overrightarrow{b}$, 则 $\\lambda$ 等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -609858,7 +610244,9 @@ "id": "031835", "content": "已知等差数列 $\\{a_n\\}$ 满足 $a_{2022}=a_{20}+a_{22}=2$, 则 $a_1$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -609982,7 +610370,9 @@ "id": "031841", "content": "已知 $a\\dfrac{1}{b}$}{$a^2>b^2$}{$2-a>2-b$}{$2^a>2^b$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -610086,7 +610476,9 @@ "id": "031846", "content": "已知函数 $f(x)=a x+\\dfrac{1}{x-1}$, $a \\in \\mathbf{R}$.\\\\\n(1) 当 $a=2$ 时,求不等式 $f(x+2) \\leq f(x)+4$ 的解集;\\\\\n(2) 若函数 $f(x)$ 在区间 $[2,5]$ 上严格减, 求实数 $a$ 的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -610210,7 +610602,9 @@ "id": "031852", "content": "集合 $A=\\{x|| x-2 | \\leq 3, x \\in \\mathbf{R}\\}$, $B=\\{y | y=-x^2,-1 \\leq x \\leq 2\\}$, 则 $\\overline{A \\cap B}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -610252,7 +610646,9 @@ "id": "031854", "content": "已知圆 $x^2+y^2-6 x-7=0$ 与抛物线 $y^2=2 p x$($p>0$) 的准线相切, 则 $p=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -610292,7 +610688,9 @@ "id": "031856", "content": "在正四棱柱 $ABCD-A_1B_1C_1D_1$ 中, $AA_1=3$, 直线 $AC_1$ 与平面 $BCC_1B_1$ 所成角大小为 $30^{\\circ}$, 则该正四棱柱的外接球表面积大小为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -610312,7 +610710,9 @@ "id": "031857", "content": "等比数列 $\\{a_n\\}$ 的各项均为正数, 且 $a_{10}a_{11}+a_9 a_{12}=2 \\mathrm{e}^5$, 则 $\\ln a_1+\\ln a_2+\\cdots+\\ln a_{20}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -610372,7 +610772,9 @@ "id": "031860", "content": "已知数列 $\\{a_n\\}$ 满足: 当 $n \\geq 3$ 时, $a_n=2 a_{n-1}$ 或 $a_n=a_{n-1}+a_{n-2}$, 若 $a_1=1$, $a_2=2$, 则此数列前 2015 项中, 奇数项最多有\\blank{50}项.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -610392,7 +610794,9 @@ "id": "031861", "content": "设 $f(x)=\\begin{cases}2^x,& x \\leq 0,\\\\\\log _2 x,& x>0,\\end{cases}$ 若对任意 $y \\in$($2,+\\infty$), 都存在唯一的实数 $x$, 满足 $f(f(x))=2 a^2 y^2+a y$, 则正数 $a$ 的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -610472,7 +610876,9 @@ "id": "031865", "content": "$P$ 为双曲线 $\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$) 左支上一点, $F_1, F_2$ 为其左右焦点, 若 $\\dfrac{|PF_2|^2}{|PF_1|}$ 的最小值为 $10 a$, 则双曲线的离心率为\\bracket{20}.\n\\fourch{$4+\\sqrt{5}$}{$4-\\sqrt{5}$}{$4 \\pm \\sqrt{5}$}{4}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -610512,7 +610918,9 @@ "id": "031867", "content": "在直角 $\\triangle ABC$ 中, $\\angle ABC=90^{\\circ}$, $AC=2 \\sqrt{3}$, $AB=\\sqrt{3}, D, E$ 分别为 $AC, BD$ 的中点, 连结 $AE$ 并延长交 $BC$ 于点 $F$, 将 $\\triangle ABD$ 沿 $BD$ 折起, 使平面 $ABD \\perp$ 平面 $BCD$, 如图所示.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$B$} coordinate (B);\n\\draw (0,{sqrt(3)}) node [above] {$A$} coordinate (A);\n\\draw ({2*sqrt(3)},0) node [right] {$C$} coordinate (C);\n\\draw ($(A)!0.5!(C)$) node [above right] {$D$} coordinate (D);\n\\draw ($(B)!0.5!(D)$) node [above] {$E$} coordinate (E);\n\\draw (1,0) node [below] {$F$} coordinate (F);\n\\draw (A)--(B)--(C)--cycle(B)--(D)(A)--(F);\n\\end{tikzpicture}\n\\hspace*{3em}\n\\begin{tikzpicture}[>=latex]\n\\draw ({sqrt(3)},0,{-sqrt(3)/2}) node [above right] {$D$} coordinate (D);\n\\draw (0,0,0) node [left] {$B$} coordinate (B);\n\\draw ({2*sqrt(3)},0,0) node [right] {$C$} coordinate (C);\n\\draw (1,0,0) node [below] {$F$} coordinate (F);\n\\draw ($(B)!0.5!(D)$) node [above left] {$E$} coordinate (E);\n\\draw (E) ++ (0,{3/2},0) node [above] {$A$} coordinate (A);\n\\draw (A)--(B)--(C)--cycle;\n\\draw (A)--(F);\n\\draw [dashed] (B)--(D)--(C)(D)--(A)(A)--(E)--(F);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $AE \\perp CD$;\\\\\n(2) 求平面 $AEF$ 与平面 $ADC$ 所成二面角的正弦值.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -610594,7 +611002,9 @@ "id": "031871", "content": "已知复数 $z=-1+\\mathrm{i}$($\\mathrm{i}$ 为虚数单位), 计算: $\\dfrac{z \\cdot \\overline{z}}{z-\\overline{z}}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -610614,7 +611024,9 @@ "id": "031872", "content": "已知直线 $l$ 的方程为 $2 x-y-3=0$, 则直线 $l$ 的倾斜角为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -610657,7 +611069,9 @@ "id": "031874", "content": "已知等比数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 若 $a_2 a_8=2 a_3 a_6$, $S_5=-62$, 则 $a_1$ 的值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -610677,7 +611091,9 @@ "id": "031875", "content": "已知双曲线 $\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$ 的一条渐近线与圆 $x^2+y^2-6 x+4 y=0$ 相切, 则该双曲线的离心率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -610697,7 +611113,9 @@ "id": "031876", "content": "不等式``$|x-m|<1$''是不等式``$\\log _2 x>1$''成立的充分不必要条件, 则 $m$ 的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -610837,7 +611255,9 @@ "id": "031883", "content": "已知 $l, m$ 是两条不同的直线, $\\alpha$ 是一个平面, 有下列四个命题中真命题的是\\bracket{20}.\n\\onech{若 $l \\subset \\beta$, 且 $m \\perp l$, 则 $m \\perp \\alpha$}{若 $l \\perp \\alpha$, 且 $m \\parallel \\alpha$, 则 $l \\perp m$}{若 $l \\parallel m$,, 且 $m \\subset \\alpha$, 则 $l \\parallel \\alpha$}{若 $l, m$ 与平面 $\\alpha$ 所成的角相等, 则 $l \\parallel m$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -610919,7 +611339,9 @@ "id": "031887", "content": "如图, 四棱柱 $ABCD-A_1B_1C_1D_1$ 的底面 $ABCD$ 是平行四边形, 且 $AB=1$, $BC=2$, $\\angle ABC=60^{\\circ}$, $E$ 为 $BC$ 的中点, $AA_1 \\perp$ 平面 $ABCD$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{1}\n\\def\\n{{sqrt(2)}}\n\\draw (0,0,0) node [below left] {$B$} coordinate (B);\n\\draw (B) ++ (\\l,0,0) node [below right] {$C$} coordinate (C);\n\\draw (C) ++ ({1/2},0,{-sqrt(3)/2}) node [right] {$D$} coordinate (D);\n\\draw (B) ++ ({1/2},0,{-sqrt(3)/2}) node [left] {$A$} coordinate (A);\n\\draw (B) -- (C) -- (D);\n\\draw [dashed] (B) -- (A) -- (D);\n\\draw (B) ++ (0,\\n,0) node [left] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\n,0) node [right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\n,0) node [above right] {$D_1$} coordinate (D_1);\n\\draw (A) ++ (0,\\n,0) node [above left] {$A_1$} coordinate (A_1);\n\\draw (B_1) -- (C_1) -- (D_1) -- (A_1) -- cycle;\n\\draw (B) -- (B_1) (C) -- (C_1) (D) -- (D_1);\n\\draw [dashed] (A) -- (A_1);\n\\draw ($(B)!0.5!(C)$) node [below] {$E$} coordinate (E);\n\\draw [dashed] (A)--(E)--(D)--(A_1)(A_1)--(C)(A_1)--(E);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $DE \\perp$ 平面 $A_1AE$;\\\\\n(2) 若 $DE=A_1E$, 试求异面直线 $AE$ 与 $A_1D$ 所成角的余弦值.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -611019,7 +611441,9 @@ "id": "031892", "content": "已知全集 $U=\\mathbf{R}$, 集合 $A=\\{x | y=\\sqrt{1-\\dfrac{1}{x}}\\}$, 则 $\\overline{A}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -611061,7 +611485,9 @@ "id": "031894", "content": "在等差数列 $\\{a_n\\}$ 中,若 $a_1+a_7+a_8+a_{12}=12$, 则此数列的前 $13$ 项之和为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -611103,7 +611529,9 @@ "id": "031896", "content": "若椭圆 $\\dfrac{x^2}{5}+\\dfrac{y^2}{m}=1$ 的离心率为 $\\dfrac{\\sqrt{10}}{5}$, 则 $m$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -611187,7 +611615,9 @@ "id": "031900", "content": "若函数 $y=f(x-2)$ 的图像与函数 $y=\\log _3 \\sqrt{x}+2$ 的图像关于直线 $y=x$ 对称, 则 $f(x)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -611247,7 +611677,9 @@ "id": "031903", "content": "设 $f(x)$ 是定义在 $\\mathbf{R}$ 上的奇函数, 且 $f(2)=0$, 当 $x>0$ 时, $\\dfrac{f(x)}{x}$ 的导数小于零恒成立, 则不等式 $x^2 f(x)>0$ 的解集是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -611367,7 +611799,9 @@ "id": "031909", "content": "如图, 在四棱锥 $S-ABCD$ 中, 底面 $ABCD$ 为正方形, 侧棱 $SD \\perp$ 底面 $ABCD, E, F$ 分别是 $AB, SC$ 的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw (0,0,0) node [left] {$D$} coordinate (D);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,2) node [left] {$A$} coordinate (A);\n\\draw (2,0,2) node [below] {$B$} coordinate (B);\n\\draw ($(A)!0.5!(B)$) node [below] {$E$} coordinate (E);\n\\draw (0,4,0) node [above] {$S$} coordinate (S);\n\\draw ($(S)!0.5!(C)$) node [above right] {$F$} coordinate (F);\n\\draw (A)--(B)--(C)--(S)--cycle(S)--(B);\n\\draw [dashed] (A)--(F)--(E)(A)--(D)--(C)(D)--(S);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $EF \\parallel $ 平面 $SAD$;\\\\\n(2) 设 $SD=2CD$, 求二面角 $A-EF-D$ 的大小.\n第 18 题图\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -611427,7 +611861,9 @@ "id": "031912", "content": "在数列 $\\{a_n\\}$ 中, $a_1=1$, $a_2=2$, 且 $a_{n+1}=(1+q) a_n-q a_{n-1}$($n \\geq 2$, $q \\neq 0$).\\\\\n(1) 设 $b_n=a_{n+1}-a_n$($n \\in \\mathbf{N}$, $n \\geq 1$), 证明 $\\{b_n\\}$ 是等比数列;\\\\\n(2) 求数列 $\\{a_n\\}$ 的通项公式;\\\\\n(3) 若 $a_3$ 是 $a_6$ 与 $a_9$ 的等差中项, 求 $q$ 的值, 并证明: 对任意的 $n \\in \\mathbf{N}$, $n \\geq 1, a_n$ 是 $a_{n+3}$ 与 $a_{n+6}$ 的等差中项.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -611447,7 +611883,9 @@ "id": "031913", "content": "已知集合 $M=\\{3,2^a\\}$, $N=\\{a, b\\}$, 若 $M \\cap N=\\{1\\}$, 则 $M \\cup N=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -611489,7 +611927,9 @@ "id": "031915", "content": "$(x^3-\\dfrac{1}{x^2})^5$ 的二项展开式中的常数项为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -611509,7 +611949,9 @@ "id": "031916", "content": "已知 $|\\overrightarrow{a}|=2$, $|\\overrightarrow{b}|=\\sqrt{2}$, 且 $\\overrightarrow{a}$ 与 $\\overrightarrow{b}$ 的夹角为 $45^{\\circ}$, 要使 $\\lambda \\overrightarrow{b}-\\overrightarrow{a}$ 与 $\\overrightarrow{a}$ 垂直, 则 $\\lambda=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -611529,7 +611971,9 @@ "id": "031917", "content": "已知 $\\alpha \\in(0,2 \\pi)$, 若复数 $z=\\sin \\alpha \\cos \\alpha-(1-\\cos 2 \\alpha) \\mathrm{i}$ 是纯虚数, 则 $\\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -611549,7 +611993,9 @@ "id": "031918", "content": "方程 $\\log _2(x+14)+\\log _2(x+2)=3+\\log _2(x+6)$ 的解是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -611569,7 +612015,9 @@ "id": "031919", "content": "已知数列 $\\{a_n\\}$ 的首项 $a_1=2$, 其前 $n$ 项和为 $S_n$. 若 $S_{n+1}=2S_n+1$, 则 $a_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -611609,7 +612057,9 @@ "id": "031921", "content": "已知直线 $a, b$ 及平面 $\\alpha$, 下列命题中: \\textcircled{1} $\\begin{cases}a\\perp b,\\\\b \\perp \\alpha\\end{cases}\\Rightarrow a \\parallel \\alpha$; \\textcircled{2} $\\begin{cases}a\\perp b,\\\\b \\parallel \\alpha\\end{cases}\\Rightarrow a \\perp \\alpha$; \\textcircled{3} $\\begin{cases}a\\parallel b,\\\\b \\parallel \\alpha\\end{cases}\\Rightarrow a \\parallel \\alpha$; \n\\textcircled{4} $\\begin{cases}a\\parallel b,\\\\b \\perp \\alpha\\end{cases}\\Rightarrow a \\perp \\alpha$. 正确命题的序号为\\blank{50}(把你认为正确的序号都填上).", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -611751,7 +612201,9 @@ "id": "031928", "content": "已知函数 $f(x)=x^3-x+1$, 则\\bracket{20}.\n\\twoch{$f(x)$ 有三个极值点}{$f(x)$ 有三个零点}{点 $(0,1)$ 是曲线 $y=f(x)$ 的对称中心}{直线 $y=2 x$ 是曲线 $y=f(x)$ 的切线}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -611771,7 +612223,9 @@ "id": "031929", "content": "如图, 在直三棱柱 $ABC-A_1B_1C_1$ 中, $AB \\perp AC$, $AB=AC=AA_1=2, E$ 是 $BC$ 的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) coordinate (D);\n\\draw ({-sqrt(2)},0,0) node [left] {$A$} coordinate (A);\n\\draw ({sqrt(2)},0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,{sqrt(2)}) node [below] {$B$} coordinate (B);\n\\draw (A) ++ (0,2,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,2,0) node [above] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,2,0) node [right] {$C_1$} coordinate (C_1);\n\\draw ($(B)!0.5!(C)$) node [below right] {$E$} coordinate (E);\n\\draw (A)--(B)--(C)--(C_1)--(A_1)--cycle(A_1)--(B_1)--(B)(B_1)--(C_1)(B_1)--(C);\n\\draw [dashed] (A)--(C)(A)--(E)(A_1)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求四棱锥 $C-A_1B_1BA$ 的体积;\\\\\n(2) 求异面直线 $AE$ 与 $A_1C$ 所成的角.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -611791,7 +612245,9 @@ "id": "031930", "content": "已知 $a \\in \\mathbf{R}$, 函数 $f(x)=x|x-a|$.\\\\\n(1) 当 $a=2$ 时,求使 $f(x) \\geq x$ 成立的 $x$ 的集合;\\\\\n(2) 求函数 $y=f(x)$ 在区间 $[1,2]$ 上的最小值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -611915,7 +612371,9 @@ "id": "031936", "content": "若 $\\log _a 2 b=-1$, 则 $a+b$ 的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -611999,7 +612457,9 @@ "id": "031940", "content": "设无穷等比数列 $\\{a_n\\}$ 的公比是 $q$, 前 $n$ 项和为 $S_n$, 若 $3 a_1-\\displaystyle\\sum_{i=1}^{+\\infty}a_i=0$, 则 $q=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -612059,7 +612519,9 @@ "id": "031943", "content": "已知函数 $f(x)=\\begin{cases}x^{2023},& x \\leq a,\\\\x^{2022},& x>a,\\end{cases}$ 若存在实数 $b$, 使得函数 $g(x)= f(x)-b$ 有两个零点, 则 $a$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -612099,7 +612561,9 @@ "id": "031945", "content": "已知等差数列 $\\{a_n\\}$ 满足 $a_2=2$, 则 $\\dfrac{1}{a_1^2+1}+\\dfrac{1}{a_3^2+1}$ 的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -612287,7 +612751,9 @@ "id": "031954", "content": "已知数列 $\\{a_n\\}$, $a_1=4$, $a_2=p$, 且 $a_n+a_{n+1}+a_{n+2}=7 \\times(\\dfrac{1}{2})^{n-1}$.\\\\\n(1) 求 $a_4$;\\\\\n(2) 求数列 $\\{a_n\\}$ 的前 $n$ 项和 $S_n$;\\\\\n(3) 是否存在实数 $p$, 使得 $\\{a_n\\}$ 为严格减数列?", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -612307,7 +612773,9 @@ "id": "031955", "content": "已知 $A=\\{x | \\sqrt{x}<4\\}$, $B=\\{x | 3 x \\geq 1\\}$, 则 $A \\cap B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -612327,7 +612795,9 @@ "id": "031956", "content": "已知复数 $z$ 满足 $\\dfrac{2}{z-5}=1+\\mathrm{i}$, 则 $z=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -612369,7 +612839,9 @@ "id": "031958", "content": "在二项式 $(2 x-\\dfrac{1}{x})^5$ 的展开式中,含有 $x^3$ 项的系数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -612451,7 +612923,9 @@ "id": "031962", "content": "在等比数列 $\\{a_n\\}$ ($n$ 为正整数) 中, 若 $a_1=1$, $a_4=\\dfrac{1}{8}$, 则该数列的前 $10$ 项和为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -612514,7 +612988,9 @@ "id": "031965", "content": "椭圆 $\\Gamma: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$) 的左、右焦点分别为 $F_1, F_2$, 焦距为 $2 c$, 若直线 $y=\\sqrt{3}(x+c)$ 与椭圆 $\\Gamma$ 的一个交点 $M$ 满足 $\\angle MF_1F_2=2 \\angle MF_2F_1$, 则该椭圆的离心率等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -612554,7 +613030,9 @@ "id": "031967", "content": "已知直线的方程为 $4 x+2 y+c=0$, 则该直线的一个法向量为\\bracket{20}.\n\\fourch{$(2,-1)$}{$(2,1)$}{$(-1,2)$}{$(1,2)$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -612636,7 +613114,9 @@ "id": "031971", "content": "在直三棱柱 $ABC-A_1B_1C_1$ 中, $AB=AC=1$, $\\angle BAC=90^{\\circ}$, 且异面直线 $A_1B$ 与 $B_1C_1$ 所成的角等于 $60^{\\circ}$, 设 $AA_1=a$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,2) node [left] {$B$} coordinate (B);\n\\foreach \\i in {A,B,C}\n{\\draw (\\i) ++ (0,2,0) coordinate (\\i_1);};\n\\draw (A_1) node [above] {$A_1$} (B_1) node [left] {$B_1$} (C_1) node [right] {$C_1$};\n\\draw (B)--(C)--(C_1)--(A_1)--(B_1)--cycle(B_1)--(C_1);\n\\draw [dashed] (B)--(A_1)(B)--(A)--(C)(A)--(A_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求 $a$ 的值;\\\\\n(2) 求三棱锥 $B_1-A_1BC$ 的体积.\n第 17 题图\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "",