diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index a47823b3..9272df99 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -658813,7 +658813,9 @@ "id": "024274", "content": "设集合 $A=\\{x | x^2-3 x-4<0\\}$, $B=\\{x |-2=latex]\n\\draw [->] (-0.5,0) -0 (2.5,0) node [below] {$x$};\n\\draw [->] (0,-1.1) -- (0,1.1) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = {-1/6}:{11/6}, samples = 100] plot (\\x,{sin(180*\\x+30)});\n\\draw (0,0.5) node [above left] {$B$} coordinate (B);\n\\draw ({5/6},0) node [below left] {$A$} coordinate (A);\n\\draw ({-1/12},{sin(15)}) node [left] {$D$} coordinate (D);\n\\draw ($(D)!2!(A)$) node [right] {$C$} coordinate (C);\n\\draw (B)--(C)(D)--(C);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$-1$}{$-\\dfrac{5}{6}$}{$\\dfrac{5}{6}$}{$\\dfrac{5}{3}$}", "objs": [], - "tags": [], + "tags": [ + "第三单元", + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -658955,7 +658970,9 @@ "id": "024281", "content": "设集合 $A=(-1,3)$, $B=[0,4)$, 则 $A \\cup B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -658975,7 +658992,9 @@ "id": "024282", "content": "复数 $z$ 满足 $\\overline{z}=\\dfrac{1}{1+\\mathrm{i}}$ ($\\mathrm{i}$ 为虚数单位), 则 $|z|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -658997,7 +659016,9 @@ "id": "024283", "content": "已知一组数据 $6,7,8,8,9,10$, 则该组数据的标准差是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -659017,7 +659038,9 @@ "id": "024284", "content": "若向量 $\\overrightarrow{a}=(1,0,1)$, $\\overrightarrow{b}=(0,1,-1)$ , 则向量 $\\overrightarrow{a}$ 与 $\\overrightarrow{b}$ 的夹角为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -659041,7 +659064,9 @@ "id": "024285", "content": "在公差不为零的等差数列 $\\{a_n\\}$ 中, $a_3$ 是 $a_1$ 与 $a_9$ 的等比中项, 则 $\\dfrac{a_1+a_2+\\cdots+a_9}{a_9}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -659061,7 +659086,9 @@ "id": "024286", "content": "设函数 $f(x)=\\sin \\omega x $($0<\\omega<2$), 将 $f(x)$ 图像向左平移 $\\dfrac{2 \\pi}{3}$ 单位后所得函数图像与原函数图像的对称轴重合, 则 $\\omega=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -659084,7 +659111,9 @@ "id": "024287", "content": "设 $f(x)=x^\\alpha $($\\alpha \\in\\{-2,-1, \\dfrac{1}{3}, \\dfrac{1}{2}, 1,2\\}$), 则``$y=f(x)$ 图像经过点 $(-1,1)$''是``$y=f(x)$ 是偶函数''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件 }{充要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -659107,7 +659136,9 @@ "id": "024288", "content": "若曲线 $y=\\mathrm{e}^x-1$ 在 $x=x_0$ 处的切线方程为 $\\mathrm{e} x-y+t=0$, 则\\bracket{20}.\n\\fourch{$x_0=1$, $t=-1$}{$x_0=1$, $t=-\\mathrm{e}$}{$x_0=-1$, $t=-1$}{$x_0=-1$, $t=-\\mathrm{e}$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -659127,7 +659158,9 @@ "id": "024289", "content": "曲线 $y=\\dfrac{\\cos x}{x}$ 在 $(\\dfrac{\\pi}{2}, 0)$ 处的切线的斜率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -659147,7 +659180,9 @@ "id": "024290", "content": "设 $\\alpha: x>1$ 且 $y>2$, $\\beta: x+y>3$, 则 $\\alpha$ 是 $\\beta$ 成立的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -659167,7 +659202,9 @@ "id": "024291", "content": "已知四个数 $1,2,4, a$ 的平均数为 4 , 则这四个数的中位数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -659187,7 +659224,9 @@ "id": "024292", "content": "已知随机事件 $A$、$B$ 互相独立, 且 $P(A)=0.7$, $P(B)=0.4$, 则 $P(A \\cap \\overline{B})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -659209,7 +659248,9 @@ "id": "024293", "content": "若函数 $f(x)=\\mathrm{e}^x-(a-1) x+1$ 在 $(0,1)$ 上是严格减函数, 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": "", @@ -659229,7 +659270,9 @@ "id": "024294", "content": "将编号为 $1,2,3,4$ 的 4 个小球放入 3 个不同的盒子中, 每个盒子不空, 若放在同一盒子里的 2 个小球编号不相邻, 则共有\\blank{50}种不同的放法.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -659249,7 +659292,9 @@ "id": "024295", "content": "甲乙两人分别独立参加某高校自主招生考试, 若甲、乙能通过面试的概率都是 $\\dfrac{2}{3}$, 则面试结束后通过的人数 $X$ 的期望是\\bracket{20}.\n\\fourch{$\\dfrac{4}{3}$}{$\\dfrac{2}{3}$}{$\\dfrac{10}{9}$}{$\\dfrac{8}{9}$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -659269,7 +659314,9 @@ "id": "024296", "content": "若 $f(x)=x^2-2 x-4 \\ln x$, 则 $f'(x)>0$ 的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -659289,7 +659336,9 @@ "id": "024297", "content": "函数 $f(x)=(\\dfrac{2}{1+\\mathrm{e}^x}-1) \\sin x$ 图像的大致形状是\\bracket{20}.\n\\fourch{\\begin{tikzpicture}[>=latex, xscale = 0.23, yscale = 0.5]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -4:4, samples = 100] plot (\\x,{-(2/(1+exp(\\x))-1)*sin(\\x/pi*180)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, xscale = 0.23, yscale = 0.5]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -4:4, samples = 100] plot (\\x,{-(2/(1+exp(\\x))-1)*cos(\\x/pi*180)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, xscale = 0.23, yscale = 0.5]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -4:4, samples = 100] plot (\\x,{(2/(1+exp(\\x))-1)*sin(\\x/pi*180)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, xscale = 0.23, yscale = 0.5]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -4:4, samples = 100] plot (\\x,{(2/(1+exp(\\x))-1)*cos(\\x/pi*180)});\n\\end{tikzpicture}}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -659309,7 +659358,9 @@ "id": "024298", "content": "已知随机变量 $X$ 服从正态分布 $N(0, \\sigma^2)$, 且 $P(X<-1)=0.1$, 则 $P(0b$, 则下列结论中正确的是\\bracket{20}.\n\\fourch{$a^{-2}>b^{-2}$}{$a^{-1}>b^{-1}$}{$a^2>b^2$}{$a^3>b^3$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -659519,7 +659588,9 @@ "id": "024308", "content": "设 $70$ 且 $b>0$''是``$a+b>\\sqrt{a b}$''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充分必要条件}{既不充分也不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -659603,7 +659680,9 @@ "id": "024312", "content": "已知直线 $y=k x$ 是曲线 $y=\\ln x$ 的一条切线, 则 $k$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -659623,7 +659702,9 @@ "id": "024313", "content": "函数 $f(x)=\\mathrm{e}^x-x+1$ 在区间 $[-1,1]$ 上的最大值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -659643,7 +659724,9 @@ "id": "024314", "content": "$A$、$B$、$C$ 三位好友进行乒乓球擂台赛, $A$、$B$ 先进行一局决胜负, 负者下, 由 $C$ 挑战胜者, 继续进行一局决胜负, 负者下, 胜者接受第三人的挑战, 依次举行. 假设三人水平接近, 任意两人的对决胜负都是五五开, 已知三人共比赛了 3 局, 则三人各胜一局的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -659663,7 +659746,9 @@ "id": "024315", "content": "设集合 $A=(1,3)$, $B=(2,4)$, 则 $A \\cap B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -659685,7 +659770,9 @@ "id": "024316", "content": "曲线 $y=x^3$ 在点 $(1,1)$ 处的切线方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -659707,7 +659794,9 @@ "id": "024317", "content": "已知各项为正的等差数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 若 $a_5+a_7-a_6{}^2=0$, 则 $S_{11}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -659727,7 +659816,9 @@ "id": "024318", "content": "已知某社区的家庭年收入的频率分布如下表所示, 可以估计该社区内家庭的平均年收入为\\blank{50}万元.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline 家庭年收入(万元) &{$[4,5)$}&{$[5,6)$}&{$[6,7)$}&{$[7,8)$}&{$[8,9)$}&{$[9,10)$}\\\\\n\\hline 频率 $f$ & $0.2$ & $0.2$ & $0.2$ & $0.26$ & $0.07$ & $0.07$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -659747,7 +659838,9 @@ "id": "024319", "content": "经过点 $(2,4)$ 的抛物线 $y=a x^2$ 焦点坐标是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -659767,7 +659860,9 @@ "id": "024320", "content": "用 $0$、$1$ 两个数字编码, 码长为 $4$ 的二进制四位数 (首位可以是 $0$), 从所有码中任选一码, 则事件 $A$ :``码中至少有两个 1''发生的概率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -659787,7 +659882,9 @@ "id": "024321", "content": "如图, $PA \\perp$ 平面 $ABCD, ABCD$ 为矩形, 连接 $AC$、$BD$、$PB$、$PC$、$PD$, 下面各组向量中, 数量积不一定为零的是\\bracket{20} .\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (0,0,2) node [left] {$B$} coordinate (B);\n\\draw (3,0,2) node [right] {$C$} coordinate (C);\n\\draw (3,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,2,0) node [left] {$P$} coordinate (P);\n\\draw (P) -- (B) -- (C) -- (D) -- (P) (P) -- (C);\n\\draw [dashed] (A) -- (P) (A) -- (B) (A) -- (D) (A)--(C)(B)--(D);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\overrightarrow{PC}$ 与 $\\overrightarrow{BD}$}{$\\overrightarrow{PB}$ 与 $\\overrightarrow{DA}$}{$\\overrightarrow{PD}$ 与 $\\overrightarrow{AB}$}{$\\overrightarrow{PA}$ 与 $\\overrightarrow{CD}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -659807,7 +659904,10 @@ "id": "024322", "content": "下列选项中, $y$ 可表示为 $x$ 的函数是\\bracket{20}.\n\\fourch{$3^{|y|}-x^2=0$}{$x=y^{\\frac{2}{3}}$}{$x=\\sin y$}{$\\ln y=x^2$}", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -659827,7 +659927,9 @@ "id": "024323", "content": "已知集合 $A=\\{x|| x-1 |<1\\}$, $B=\\{1,2,3\\}$, 则 $A \\cap B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -659849,7 +659951,9 @@ "id": "024324", "content": "若复数 $z$ 满足 $z \\cdot(1+\\mathrm{i})=2$ ($\\mathrm{i}$ 为虚数单位), 则 $z=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -659869,7 +659973,9 @@ "id": "024325", "content": "已知 $k$ 为实数, 向量 $\\overrightarrow{a}=(4,-2)$, $\\overrightarrow{b}=(k, 2)$, 若 $\\overrightarrow{a}\\perp \\overrightarrow{b}$, 则 $k=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -659892,7 +659998,9 @@ "id": "024326", "content": "在 $(x+2)^6$ 的二项展开式中, 含 $x^3$ 项的系数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -659916,7 +660024,9 @@ "id": "024327", "content": "如图所示, 在平行六面体 $ABCD-A_1B_1C_1D_1$ 中, $A_1C_1 \\cap B_1D_1=F$, 若 $\\overrightarrow{AF}=x \\overrightarrow{AB}+y \\overrightarrow{AD}+z \\overrightarrow{AA_1}$,则 $x+y+z=$\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$A$} coordinate (A) --++ (2,0) node [below right] {$B$} coordinate (B) --++ (45:{2/2}) node [right] {$C$} coordinate (C)\n--++ (0.2,1.5) node [above right] {$C_1$} coordinate (C1)\n--++ (-2,0) node [above left] {$D_1$} coordinate (D1) --++ (225:{2/2}) node [left] {$A_1$} coordinate (A1) -- cycle;\n\\draw (A) ++ (2.2,1.5) node [right] {$B_1$} coordinate (B1) -- (B) (B1) --++ (45:{2/2}) (B1) --++ (-2,0) (D1) -- (B1);\n\\draw [dashed] (A) --++ (45:{2/2}) node [left] {$D$} coordinate (D) --++ (2,0) (D) --++ (0.2,1.5);\n\\draw (A1)--(C1);\n\\draw ($(A1)!0.5!(C1)$) node [above] {$F$} coordinate (F);\n\\draw [dashed, ->] (A)--(F);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -659936,7 +660046,9 @@ "id": "024328", "content": "曲线 $y=\\sin 3 x$ 在点 $P(\\dfrac{\\pi}{3}, 0)$ 处切线的斜率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -659956,7 +660068,9 @@ "id": "024329", "content": "已知随机变量 $X$ 服从的分布为 $\\begin{pmatrix}0 & 1 \\\\ \\dfrac{1}{3}& \\dfrac{2}{3}\\end{pmatrix}$, 则随机变量 $X$ 的方差 $D[X]=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -659976,7 +660090,9 @@ "id": "024330", "content": "经过点 $(1,1)$, 且法向量为 $(2,-1)$ 的直线方程是\\bracket{20}.\n\\fourch{$2 x-y-1=0$}{$2 x+y-3=0$}{$x-2 y+1=0$}{$x+2 y-3=0$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -659996,7 +660112,9 @@ "id": "024331", "content": "设 $\\alpha$、$\\beta$ 表示两个不同的平面, $l$ 表示一条直线, 且 $l \\subset \\alpha$, 则 $l \\parallel \\beta$ 是 $\\alpha \\parallel \\beta$ 的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -660018,7 +660136,9 @@ "id": "024332", "content": "已知函数 $f(x)=\\ln (1-x)$, 则其导函数 $f'(x)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660038,7 +660158,9 @@ "id": "024333", "content": "在 $(x+\\dfrac{1}{x})^5$ 的展开式中, 含 $x^3$ 项的系数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660062,7 +660184,9 @@ "id": "024334", "content": "从 $1,2,3,4,5$ 这五个数字中任意选取两个不同的数字组成一个两位数, 则这个两位数是偶数的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660084,7 +660208,9 @@ "id": "024335", "content": "若随机变量 $X$ 服从二项分布 $B(4, \\dfrac{2}{3})$, 则 $E[X]=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660104,7 +660230,9 @@ "id": "024336", "content": "函数 $f(x)=\\dfrac{1}{3}x^3-x$ 的单调增区间为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660124,7 +660252,9 @@ "id": "024337", "content": "在 $(2 x+1)^6$ 的展开式中, $x^2$ 的系数为\\blank{50}. (结果用数值表示)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660148,7 +660278,9 @@ "id": "024338", "content": "将两枚质地均匀的骰子各掷一次, 设事件 $A$ 表示两个点数互不相同, 事件 $B$ 表示至少出现一个 $4$ 点, 则 $P(B | A)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660171,7 +660303,10 @@ "id": "024339", "content": "从 $1 \\sim 9$ 这 $9$ 个数中任取 $4$ 个不同的数, 则这 $4$ 个不同的数的中位数为 $6$ 的概率为\\blank{50}.(结果用最简分数表示)", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660193,7 +660328,9 @@ "id": "024340", "content": "已知集合 $A=\\{1,2,3,4\\}$, 集合 $B=\\{2,3, m\\}$, 若 $A \\cap B=\\{2,3,4\\}$, 则 $m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660216,7 +660353,9 @@ "id": "024341", "content": "已知 $X$ 的分布是伯努利分布, 且 $P(X=1)-P(X=0)=0.2$, 则 $P(X=1)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660236,7 +660375,9 @@ "id": "024342", "content": "不等式 $\\dfrac{x}{1-x}\\geq 0$ 的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660258,7 +660399,9 @@ "id": "024343", "content": "函数 $f(x)=x^2 \\mathrm{e}^x$ 在 $[-1,1]$ 上的极小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660278,7 +660421,9 @@ "id": "024344", "content": "若 $\\dfrac{a+2 \\mathrm{i}}{\\mathrm{i}}=b+\\mathrm{i}$($a, b \\in \\mathbf{R}$, $\\mathrm{i}$ 为虚数单位 ), 则 $a+b=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660298,7 +660443,9 @@ "id": "024345", "content": "某圆锥的底面积为 $4 \\pi$, 侧面积为 $8 \\pi$, 则该圆锥的母线与底面所成角的大小为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660320,7 +660467,9 @@ "id": "024346", "content": "若正方形 $ABCD$ 的边长为 1 , 记 $\\overrightarrow{AB}=\\overrightarrow{a}$, $\\overrightarrow{BC}=\\overrightarrow{b}$, $\\overrightarrow{AC}=\\overrightarrow{c}$, 则 $|\\overrightarrow{a}+2 \\overrightarrow{b}-3 \\overrightarrow{c}|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660340,7 +660489,9 @@ "id": "024347", "content": "一个不透明的袋中装有 $5$ 个白球、$4$ 个红球 ($9$个球除颜色外其余完全相同), 经充分混合后, 从袋中随机摸出 $3$ 个球, 则摸出的 $3$ 个球中至少有一个是白球的概率为\\blank{50}. (结果用最简分数表示)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660362,7 +660513,9 @@ "id": "024348", "content": "函数 $y=2 \\cos 2 x$($x \\in \\mathbf{R}$) 的最小正周期为\\bracket{20}.\n\\fourch{$\\dfrac{\\pi}{2}$}{$\\pi$}{$2 \\pi$}{$4 \\pi$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -660382,7 +660535,9 @@ "id": "024349", "content": "$(1-2 x)^4$ 的二项展开式中 $x^3$ 项的系数为\\blank{50}. (结果用数字作答)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660406,7 +660561,9 @@ "id": "024350", "content": "函数 $f(x)=x-\\ln x$ 的极小值点为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660426,7 +660583,9 @@ "id": "024351", "content": "随机变量 $X$ 的取值为 $0,1,2$, 若 $P(X=0)=\\dfrac{1}{5}$, $E[X]=1$, 则 $P(X=1)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660448,7 +660607,9 @@ "id": "024352", "content": "直线 $l$ 的一个法向量 $\\overrightarrow{n}=(1,-1)$, 且经过圆 $(x-2)^2+(y-4)^2=4$ 的圆心, 则直线 $l$ 的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660468,7 +660629,9 @@ "id": "024353", "content": "函数 $y=x^3-3 x$ 的单调减区间是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660491,7 +660654,9 @@ "id": "024354", "content": "指数方程 $2^{x+3}=8^{x^2-9}$ 的解为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660511,7 +660676,9 @@ "id": "024355", "content": "将一枚骰子先后抛两次, 则向上的点数之积为 12 的概率为\\blank{50}. (结果用最简分数表示)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660533,7 +660700,9 @@ "id": "024356", "content": "已知随机变量 $X$ 的分布为 $\\begin{pmatrix}0 & 1 & m \\\\ 0.2 & n & 0.3\\end{pmatrix}$, 且 $E[X]=1.1$, 则 $D[X]=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660553,7 +660722,9 @@ "id": "024357", "content": "2022 年 2 月 4 日至 2 月 20 日春节期间,第 24 届冬奥会在北京市和张家口市联合举行. 共有 3 个冬奥村供运动员和代表队官员入住, 其中北京冬奥村的容量约为 $2250$ 人, 延庆冬奥村的容量约 $1440$ 人, 张家口冬奥村的容量约 $2610$ 人. 为了解各冬奥村服务质量, 现共准备了 $140$ 份调查问卷, 采用分层抽样的方法, 则需在延庆冬奥村投放的问卷数量是\\bracket{20}.\n\\fourch{$58$ 份}{$50$ 份}{$32$ 份}{$19$ 份}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -660573,7 +660744,9 @@ "id": "024358", "content": "已知集合 $A=\\{-1,0,1,2\\}$, $B=\\{x | x^2 \\leq 1\\}$, 则 $A \\cap B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660593,7 +660766,9 @@ "id": "024359", "content": "已知复数 $z$ 满足 $\\dfrac{1}{z-1}=\\mathrm{i}$ ($\\mathrm{i}$ 为虚数单位), 则 $|\\overline{z}|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660613,7 +660788,9 @@ "id": "024360", "content": "函数 $f(x)=\\dfrac{1}{3}x^3+x^2-2$ 在区间 $(-\\infty,-1]$ 上的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660635,7 +660812,9 @@ "id": "024361", "content": "若 $(x+\\dfrac{1}{x})^n$ 展开式的系数之和为 64 , 则展开式的常数项的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660659,7 +660838,9 @@ "id": "024362", "content": "设随机变量 $X$ 服从二项分布 $B(6, p)$, 且 $E[X]=2$, 则 $D[X]=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660681,7 +660862,9 @@ "id": "024363", "content": "以圆 $x^2+y^2+4 x+3=0$ 的圆心为焦点的拋物线标准方程为\\bracket{20}.\n\\fourch{$y^2=4 x$}{$y^2=-4 x$}{$y^2=-8 x$}{$y^2=8 x$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -660701,7 +660884,9 @@ "id": "024364", "content": "函数 $f(x)=x-\\ln x$ 的最小值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660721,7 +660906,9 @@ "id": "024365", "content": "设随机变量 $X \\sim N(30,6^2)$, 则 $D[X]=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660741,7 +660928,9 @@ "id": "024366", "content": "如果一个圆柱的底面积和侧面积分别为 $9 \\pi$ 和 $15 \\pi$, 则该圆柱的体积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660764,7 +660953,9 @@ "id": "024367", "content": "首项为 $1$, 公比为 $-\\dfrac{1}{2}$ 的无穷等比数列 $\\{a_n\\}$ 的前 $n$ 和的极限为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660788,7 +660979,9 @@ "id": "024368", "content": "甲乙两射手独立地射击同一目标, 他们的命中率分别为 0.8 和 0.9 , 则目标被击中的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660812,7 +661005,9 @@ "id": "024369", "content": "甲乙两工厂生产某种产品, 抽取连续 5 个月的产品生产产量 (单位: 件) 情况如下:\n甲: $80$、$70$、$100$、$50$、$90$;\n乙: $60$、$70$、$80$、$55$、$95$,\n则下列说法中正确的是\\bracket{20}.\n\\twoch{甲平均产量高, 甲产量稳定}{甲平均产量高, 乙产量稳定}{乙平均产量高, 甲产量稳定}{乙平均产量高, 乙产量稳定}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -660832,7 +661027,9 @@ "id": "024370", "content": "已知集合 $A=(-1,3)$, 集合 $B=(2,4)$, 则 $A \\cap B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660855,7 +661052,9 @@ "id": "024371", "content": "若复数 $-1+\\sqrt{3}\\mathrm{i}$ ($\\mathrm{i}$ 为虚数单位) 是关于 $x$ 的一元二次方程 $x^2+b x+c=0$($b, c \\in\\mathbf{R}$) 的一个根, 则此方程的另一个根为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660875,7 +661074,9 @@ "id": "024372", "content": "已知抛物线 $C$ 的方程为 $y^2=6 x$, 则 $C$ 的焦点与准线的距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660895,7 +661096,9 @@ "id": "024373", "content": "在 $(1+x)^8$ 的二项展开式中, $x^7$ 项的系数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660919,7 +661122,9 @@ "id": "024374", "content": "甲、乙两人下棋, 若甲获胜的概率是 $\\dfrac{1}{3}$, 和棋的概率是 $\\dfrac{1}{4}$, 则甲输的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660939,7 +661144,9 @@ "id": "024375", "content": "设 $f(x)=\\mathrm{e}^{-x}$, 则函数 $y=f(x)$($x \\in\\mathbf{R}$) 的导函数 $f'(x)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660959,7 +661166,9 @@ "id": "024376", "content": "已知 $ABCD$ 是正方形, 将正方形 $ABCD$ 绕直线 $AD$ 旋转一周所得到的几何体体积记为 $V_1$, 直角三角形 $ABD$ 绕直线 $AD$ 旋转一周所得到的几何体体积记为 $V_2$, 则 $\\dfrac{V_1}{V_2}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -660981,7 +661190,9 @@ "id": "024377", "content": "设无穷等比数列 $\\{x_n\\}$ 的公比为 $q$, 若 $\\displaystyle\\lim _{n \\to \\infty}(a_2+a_3+a_4+\\cdots+a_{n+1})=a_1+a_2$, 则 $q=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -661001,7 +661212,9 @@ "id": "024378", "content": "设 $x \\in\\mathbf{R}$, 则``$x<1$''是``$x^2<1$''的\\bracket{20} 条件\\bracket{20}.\n\\twoch{充分非必要}{必要非充分}{充要}{既非充分条件又非必要}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -661023,7 +661236,9 @@ "id": "024379", "content": "某工厂月产品的总成本 $y$ (单位: 万元) 与月产量 $x$ (单位: 万件)有如下一组数据, 从散点图分析可知 $y$ 与 $x$ 线性相关. 如果回归方程是 $y=x+3.5$, 那么表格中数据 $a$ 的值为 $\\bracket{20}$.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline$x /$ 万件 & 1 & 2 & 3 & 4 \\\\\n\\hline$y /$ 万元 & $3.8$ & $a$ & $6.4$ & $8.2$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\fourch{$4.4$}{$4.9$}{$5.4$}{$5.6$}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -661045,7 +661260,9 @@ "id": "024380", "content": "不等式 $|x+1|<2$ 的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -661069,7 +661286,9 @@ "id": "024381", "content": "若椭圆 $\\dfrac{x^2}{a^2}+y^2=1$($a>0$) 的右焦点的坐标为 $(1,0)$, 则 $a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -661091,7 +661310,9 @@ "id": "024382", "content": "数列 $\\{a_n\\}$ 是等差数列, $a_2=2$, $a_4=6$, 则 $a_1=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -661114,7 +661335,9 @@ "id": "024383", "content": "设 $k\\in \\mathbf{R}$, 向量 $\\overrightarrow{a}=(-k, 1)$, $\\overrightarrow{b}=(2,3 k-4)$. 若 $\\overrightarrow{a}\\perp \\overrightarrow{b}$, 则 $k=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -661137,7 +661360,9 @@ "id": "024384", "content": "在 $(1-x^2)(1+x)^6$ 的展开式中, $x^5$ 的系数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -661159,7 +661384,9 @@ "id": "024385", "content": "设 $a \\in \\mathbf{R}$, 若 $y=a x+\\ln (\\mathrm{e}^x+1)$ 是偶函数, 则 $a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -661181,7 +661408,9 @@ "id": "024386", "content": "设 $a \\neq 0$, $a, b\\in \\mathbf{R}$, $\\mathrm{i}$ 是虚数单位, 则``$-1+\\mathrm{i}$ 是一元二次方程 $a x^2+b x+2 a=0$的一个根''是``$b=2 a$''的\\bracket{20}条件.\n\\fourch{充分非必要}{必要非充分}{充要}{既非充分又非必要}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -661203,7 +661432,9 @@ "id": "024387", "content": "设 $X \\sim N(\\mu_1, \\sigma_1^2), Y \\sim N(\\mu_2, \\sigma_2^2)$, 这两个正态分布密度曲线如图所示, 下列结论中正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.6, yscale = 2]\n\\draw [->] (-5,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,-0) -- (0,1.2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\def\\s{0.4}\n\\def\\m{-0.5}\n\\draw [domain = -5:5, samples = 100, thick] plot (\\x,{1/sqrt(2)/sqrt(pi)/\\s*exp(-(\\x-\\m)*(\\x-\\m)/2/\\s/\\s)});\n\\def\\s{1}\n\\def\\m{0.3}\n\\draw [domain = -5:5, samples = 100, thin] plot (\\x,{1/sqrt(2)/sqrt(pi)/\\s*exp(-(\\x-\\m)*(\\x-\\m)/2/\\s/\\s)});\n\\draw (-5,0.5) node {$X$的正态分布密度曲线};\n\\draw (5,0.3) node {$Y$的正态分布密度曲线};\n\\end{tikzpicture}\n\\end{center}\n\\twoch{$P(Y \\geq \\mu_2) \\geq P$($Y \\geq \\mu_1$)}{$P(X \\leq \\sigma_2) \\leq P$($X \\leq \\sigma_1$)}{对任意正数 $t, P(X \\geq t) \\geq P$($Y \\geq t$)}{对任意正数 $t, P(X \\leq t) \\geq P$($Y \\leq t$)}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -661223,7 +661454,9 @@ "id": "024388", "content": "已知集合 $A=\\{x |-11$ 的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -661563,7 +661826,9 @@ "id": "024404", "content": "已知函数 $f(x)=x^{-1}$, 则 $y=f(x)$ 在 $x=2$ 处的导数 $f'(2)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -661585,7 +661850,9 @@ "id": "024405", "content": "已知复数 $z$ 的共轭复数为 $\\overline{z}$, 若 $\\mathrm{i}\\cdot \\overline{z}=3-4 \\mathrm{i}$ (其中 $\\mathrm{i}$ 为虚数单位), 则 $|z|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -661605,7 +661872,9 @@ "id": "024406", "content": "已知某球体的表面积为 $36 \\pi$, 则该球体的体积是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -661628,7 +661897,9 @@ "id": "024407", "content": "双曲线 $\\dfrac{y^2}{m^2}-\\dfrac{x^2}{1-m^2}=1$ 的离心率为 2 , 则正实数 $m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -661648,7 +661919,9 @@ "id": "024408", "content": "若离散型随机变量 $X$ 的分布为 $\\begin{pmatrix}0 & 1 & 2 \\\\ a & 2 a & \\dfrac{1}{3}\\end{pmatrix}$, 则 $X$ 的数学期望 $E[X]=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -661668,7 +661941,9 @@ "id": "024409", "content": "在 $(x-1)^7+(x+2)^6$ 的展开式中, $x^2$ 项的系数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -661688,7 +661963,9 @@ "id": "024410", "content": "已知空间直线 $l_1$ 和 $l_2$, 则``$l_1$、$l_2$ 无公共点''是``$l_1 \\parallel l_2$''的\\bracket{20}条件.\n\\fourch{充分非必要}{必要非充分}{充要}{既非充分又非必要}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -661710,7 +661987,9 @@ "id": "024411", "content": "某公司去年上半年的月收入 $x$ (单位: 万元)与月支出 $y$ (单位: 万元)的统计资料如表所示: \n\\begin{center}\\begin{tabular}{|c|c|c|c|c|c|c|}\\hline 月份 & 1 & 2 & 3 & 4 & 5 & 6 \\\\\n\\hline 月收入 $x$ (万元) & 12.3 & 14.5 & 15.0 & 17.0 & 19.8 & 20.6 \\\\\n\\hline 月支出 $y($ 万元 $)$ & 5.75 & 5.63 & 5.82 & 5.89 & 6.11 & 6.18 \\\\\n\\hline\n\\end{tabular}\\end{center}根据上述统计资料, 则\\bracket{20}.\n\\twoch{月收入的中位数是 $15, x$ 与 $y$ 有正相关关系}{月收入的中位数是 $15, x$ 与 $y$ 有负相关关系}{月收入的中位数是 $16, x$ 与 $y$ 有正相关关系}{月收入的中位数是 $16, x$ 与 $y$ 有负相关关系}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -661732,7 +662011,9 @@ "id": "024412", "content": "已知集合 $A=(-3,1)$, 则 $A \\cap \\mathbf{Z}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -661754,7 +662035,9 @@ "id": "024413", "content": "由 $1,2,3,4$ 这四个数字所组成的(数字可重复)三位数共有\\blank{50}个.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -661774,7 +662057,9 @@ "id": "024414", "content": "若某正四棱锥高为 $1$, 且侧棱与底面所成角为 $45^{\\circ}$, 则该四棱锥的体积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -661796,7 +662081,9 @@ "id": "024415", "content": "若随机变量 $X \\sim N(\\mu, \\sigma^2)$, 且 $P(X>5)=P(X<-1)=0.2$, 则 $P(2\\angle BAC$}{$\\angle BPC=\\angle BAC$}{$\\angle BPC$ 与 $\\angle BAC$ 的大小关系不确定}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -662029,7 +662336,9 @@ "id": "024426", "content": "相交成 $90^{\\circ}$ 的两条直线和一个平面所成的角分别是 $30^{\\circ}$ 和 $45^{\\circ}$, 且这两条直线在该平面内的射影所成的锐角为 $\\theta$, 则有\\bracket{20}.\n\\fourch{$\\sin \\theta=\\dfrac{\\sqrt{6}}{3}$}{$\\cos \\theta=\\dfrac{\\sqrt{6}}{3}$}{$\\tan \\theta=\\dfrac{\\sqrt{2}}{2}$}{$\\cos \\theta=-\\dfrac{\\sqrt{3}}{3}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -662049,7 +662358,9 @@ "id": "024427", "content": "不共面的四个定点到平面 $\\alpha$ 的距离都相等, 这样的平面 $\\alpha$ 共有\\blank{50}个.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -662073,7 +662384,9 @@ "id": "024428", "content": "四面体 $ABCD$ 中, $AC=BD$, $E, F$ 分别为 $AD, BC$ 的中点, 且 $EF=\\dfrac{\\sqrt{2}}{2}AC$, $\\angle BDC=90$, 求证: $BD \\perp$ 平面 $ACD$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -662095,7 +662408,9 @@ "id": "024429", "content": "已知正四棱锥 $P-ABCD, M$、$N$ 分别是 $PA$、$BD$ 上的点, 且 $PM: MA=BN: ND$, 求证: 直线 $MN \\parallel $ 平面 $PBC$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (-1,0,1) node [below] {$A$} coordinate (A);\n\\draw (1,0,1) node [below] {$B$} coordinate (B);\n\\draw (1,0,-1) node [right] {$C$} coordinate (C);\n\\draw (-1,0,-1) node [left] {$D$} coordinate (D);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\filldraw ($(P)!0.4!(A)$) circle (0.03) node [left] {$M$} coordinate (M);\n\\filldraw ($(B)!0.4!(D)$) circle (0.03) node [above] {$N$} coordinate (N);\n\\draw (P)--(A)--(B)--(C)--cycle(B)--(P);\n\\draw [dashed] (P)--(D)--(A)(D)--(C)(D)--(B)(M)--(N);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -662118,7 +662433,9 @@ "id": "024430", "content": "已知空间一个平面与一个正方体的 12 条棱的所成的角都等于 $\\alpha$, 则 $\\cos \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -662138,7 +662455,9 @@ "id": "024431", "content": "已知正四棱锥 $P-ABCD$ 的高为 $7$ , 且 $AB=2$, 则正四棱锥 $P-ABCD$ 的侧面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -662158,7 +662477,9 @@ "id": "024432", "content": "若正四棱柱 $ABCD-A_1B_1C_1D_1$ 的底面边长为 $2$, 高为 $4$, 则异面直线 $BD_1$ 与 $AD$ 所成角的大小为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -662182,7 +662503,9 @@ "id": "024433", "content": "已知正四棱锥的体积为 12 , 底面对角线的长为 $2 \\sqrt{6}$, 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": "", @@ -662204,7 +662527,9 @@ "id": "024434", "content": "在直三棱柱 $ABC-A_1 B_1 C_1$ 中, $\\angle ABC=90^{\\circ}$, $AB=BC=1$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,0) node [below right] {$B$} coordinate (B);\n\\draw (0,0,2) node [left] {$A$} coordinate (A);\n\\draw (A) --++ (0,{2*sqrt(2)}) node [left] {$A_1$} coordinate (A_1);\n\\draw [dashed] (B) --++ (0,{2*sqrt(2)}) node [above left] {$B_1$} coordinate (B_1);\n\\draw (C) --++ (0,{2*sqrt(2)}) node [right] {$C_1$} coordinate (C_1);\n\\draw (A) -- (C) (A_1) -- (B_1) -- (C_1) -- cycle (A_1) -- (C);\n\\draw [dashed] (B) -- (C) (A) -- (C) (A) -- (B);\n\\end{tikzpicture}\n\\end{center}\n(1) 求异面直线 $B_1 C_1$ 与 $AC$ 所成角的大小;\\\\\n(2) 若直线 $A_1 C$ 与平面 $ABC$ 所成角为 $45^{\\circ}$, 求三棱锥 $A_1-ABC$ 的体积.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -662226,7 +662551,9 @@ "id": "024435", "content": "下面是关于顶点在底面的投影在底面三角形内部的三棱锥的四个命题:\\\\\n\\textcircled{1} 底面是等边三角形, 且侧面与底面所成的二面角都相等的三棱锥是正三棱锥.\\\\\n\\textcircled{2} 底面是等边三角形, 侧面都是等腰三角形的三棱锥是正三棱锥.\\\\\n\\textcircled{3} 底面是等边三角形, 各侧面的面积都相等的三棱锥是正三棱锥.\\\\\n\\textcircled{4} 侧棱与底面所成的角相等, 且侧面与底面所成的二面角都相等的三棱锥是正三棱锥.\\\\\n其中, 真命题的编号是\\blank{50}. (写出所有真命题的编号)", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -662246,7 +662573,9 @@ "id": "024436", "content": "已知直角三角形的两直角边长分别为 $3 \\mathrm{cm}$ 和 $4 \\mathrm{cm}$, 则以斜边为轴旋转一周所得几何体的表面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -662270,7 +662599,9 @@ "id": "024437", "content": "如图, 在圆锥 $SO$ 中, 已知底面半径 $r=1$, 母线长 $l=4$, $M$ 为母线 $SA$ 上的一个点, 且 $SM=2$, 从点 $M$ 拉一根绳子, 围绕圆锥侧面转到点 $A$. 则绳子的最短长度为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (-1,0) node [left] {$A$} coordinate (A);\n\\draw (0,{sqrt(15)}) node [above] {$S$} coordinate (S);\n\\draw ($(S)!0.5!(A)$) node [left] {$M$} coordinate (M);\n\\draw (A) arc (180:360:1 and 0.25) -- (S)--cycle;\n\\draw [dashed] (A) arc (180:0:1 and 0.25) (A) -- (1,0) coordinate (B) (O)--(S);\n\\draw ($(S)!{sqrt(2)/3}!(B)$) coordinate (N);\n\\draw (A) .. controls ++ (1,0.5) and ($(S)!1.1!(N)$) .. (N);\n\\draw [dashed] (N) .. controls ($(S)!0.9!(N)$) and ++ (0.3,0.3) .. (M);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -662290,7 +662621,9 @@ "id": "024438", "content": "若圆台上、下底面的圆周都在一个直径为 10 的球面上, 其上、下底面半径分别为 4 和 5 , 则该圆台的体积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -662310,7 +662643,9 @@ "id": "024439", "content": "如图, 在三棱柱 $ABC-A_1B_1C_1$ 中, 侧面 $CBB_1C_1$ 是菱形, $\\angle C_1CB=60^{\\circ}$, 平面 $ABC \\perp$ 平面 $CBB_1C_1$, $M$ 为 $BB_1$ 的中点, $AC \\perp BC$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$C$} coordinate (C);\n\\draw (0,0,2) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$B$} coordinate (B);\n\\draw (A) ++ (1,{sqrt(3)},0) node [above] {$A_1$} coordinate (A_1);\n\\draw (C) ++ (1,{sqrt(3)},0) node [above] {$C_1$} coordinate (C_1);\n\\draw (B) ++ (1,{sqrt(3)},0) node [right] {$B_1$} coordinate (B_1);\n\\draw ($(B)!0.5!(B_1)$) node [right] {$M$} coordinate (M);\n\\draw (A)--(B)--(B_1)--(A_1)--cycle(A_1)--(C_1)--(B_1)(A_1)--(M);\n\\draw [dashed] (A)--(C)--(B)(C)--(A_1)(C)--(C_1)(C)--(M)--(C_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $CC_1 \\perp$ 平面 $A_1C_1M$;\\\\\n(2) 若 $CA=CB=2$, 求三棱锥 $C_1-A_1CM$ 的体积.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -662330,7 +662665,9 @@ "id": "024440", "content": "长方体 $ABCD-A_1B_1C_1D_1$ 中, $AB=2$, $BC=a$, $(a>0)$, $AA_1=2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{3}\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 (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\\end{tikzpicture}\n\\end{center}\n(1) 在 $BC$ 边上是否存在点 $Q$, 使得 $A_1Q \\perp QD$, 为什么?\\\\\n(2) 当存在 $BC$ 边上点 $Q$ 使 $A_1Q \\perp QD$ 时, 求 $a$ 的最小值, 并求出此时二面角 $A-A_1D-Q$ 的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -662350,7 +662687,9 @@ "id": "024441", "content": "北京大兴国际机场的显著特点之一是各种弯曲空间的运用, 刻画空间的弯曲性是几何研究的重要内容. 用曲率刻画空间弯曲性, 规定: 多面体顶点的曲率等于 $2 \\pi$与多面体在该点的面角之和的差(多面体的面的内角叫做多面体的面角, 角度用弧度制),多面体面上非顶点的曲率均为零, 多面体的总曲率等于该多面体各顶点的曲率之和. 例如: 正四面体在每个顶点有 3 个面角, 每个面角是 $\\dfrac{\\pi}{3}$, 所以正四面体在各顶点的曲率为 $2 \\pi-3 \\times \\dfrac{\\pi}{3}=\\pi$, 故其总曲率为 $4 \\pi$.\\\\\n(1) 求四棱锥的总曲率;\\\\\n(2) 若多面体满足: 顶点数 $-$ 棱数 $+$ 面数 $=2$, 证明: 这类多面体的总曲率是常数.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -662370,7 +662709,9 @@ "id": "024442", "content": "540 共有\\blank{50}个正约数.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -662394,7 +662735,9 @@ "id": "024443", "content": "四封不同的信投入三个不同的箱子, 共有\\blank{50}种不同的投法; 若每个箱子至少投一封信, 又有\\blank{50}种不同的投法.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -662418,7 +662761,9 @@ "id": "024444", "content": "$1 !+2 !+3 !+\\cdots+2013$ ! 的个位数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -662440,7 +662785,9 @@ "id": "024445", "content": "已知 $n \\in \\mathbf{N}$, $n \\geq 1$, 则 $\\mathrm{P}_1^1+2\\mathrm{P}_2^2+3\\mathrm{P}_3^3+\\cdots+n \\mathrm{P}_n^n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -662463,7 +662810,9 @@ "id": "024446", "content": "关于 $n$ 的不等式 $\\mathrm{C}_n^4>\\mathrm{C}_n^{n-6}$ 的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -662483,7 +662832,9 @@ "id": "024447", "content": "$\\mathrm{C}_3^3+\\mathrm{C}_4^3+\\mathrm{C}_5^3+\\cdots+C_{2012}^3=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -662506,7 +662857,9 @@ "id": "024448", "content": "数学小组有 10 名学生, 其中高二学生有 3 名, 高三有 7 名, 今派 5 人参加一次数学竞赛, 并且高二高三必须都要有, 共有种不同的选法.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -662526,7 +662879,9 @@ "id": "024449", "content": "(1) 让 8 个同学站成一排的排列数为 $m$, 让 8 个同学站成两排 (前排 3 人, 后排 5 人) 的排列数为 $n$, 试比较 $m$ 与 $n$ 的大小;\\\\\n(2) 让 8 个身高各不相同的同学站成两排 (前排 3 人, 后排 5 人), 且后排的任意一位同学都比前排的所有同学高, 共有多少种排法?", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -662549,7 +662904,9 @@ "id": "024450", "content": "甲、乙、丙、丁、戊、已、庚七人站成一排照相, 如果甲、乙、丙三人必须相邻, 那么共有\\blank{50}种站法.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -662569,7 +662926,9 @@ "id": "024451", "content": "甲、乙、丙、丁、戊、已、庚七位同学站成一排拍照, 要求甲在乙的左侧, 乙在丙的左侧. 若甲、乙、丙可以不相邻, 则有\\blank{50}种排法.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -662589,7 +662948,9 @@ "id": "024452", "content": "上海市的车辆牌照为``沪 A$* * * * *$'', 车辆管理所最近起用了个性化车牌, 规定``沪A''不变, 后面五位中, 第一位和第五位必须是阿拉伯数字, 中间三位中至少有一个英文大写字母 A、B、C、D, 其他也用阿拉伯数字, 则一共可以构造\\blank{50}个不同的个性化车牌.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -662609,7 +662970,9 @@ "id": "024453", "content": "全集 $U=\\{1,2, \\cdots, 10\\}$, 集合 $A=\\{a, b, c\\}$ 是 $U$ 的三元子集, 且 $a+b+c$ 是 3 的整数倍, 则所有满足条件的集合 $A$ 的个数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -662629,7 +662992,9 @@ "id": "024454", "content": "甲、乙、丙、丁四位同学每人写好了一张贺卡, 分别称为``甲卡''、``乙卡''、``丙卡''、``丁卡''. 现在把这四张贺卡都放在桌子上, 这四位同学每人从中取一张贺卡, 共有\\blank{50}种取法; 如果不能取自己写的贺卡, 那么共有\\blank{50}种取法.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -662649,7 +663014,9 @@ "id": "024455", "content": "用 $0$、$1$、$2$、$3$、$4$、$5$、$6$ 七个数字中的 5 个数字, 可以组成多少个符合下列条件的数?\\\\\n(1) 五位数;\\\\\n(2) 没有重复数字的五位数;\\\\\n(3) 没有重复数字的五位偶数;\\\\\n(4) 没有重复数字的五位数且能被 5 整除;\\\\\n(5) 没有重复数字的五位数且能被 3 整除;\\\\\n(6) 没有重复数字且万位上的数字是偶数的五位数.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -662674,7 +663041,9 @@ "id": "024456", "content": "(1)把 6 个不同的水果分给 3 个小孩, 每人 2 个水果, 问共有多少种分法?\\\\\n(2) 把 6 个不同的水果分成 3 堆, 每堆 2 个水果, 问共有多少种分法?\\\\\n(3) 把 7 个不同的水果分成 3 堆(要求 $7=1+3+3$ ), 问共有多少种分法?", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -662694,7 +663063,10 @@ "id": "024457", "content": "设 $[x]$ 表示不超 $x$ 的最大整数, (如 $[2]=2$, $[\\dfrac{5}{4}]=1$). 对于给定的 $n \\in \\mathbf{N}$, $n \\geq 1$, 定义 $\\mathrm{C}_n^x=\\dfrac{n(n-1)(n-2) \\cdots(n-[x]+1)}{x(x-1) \\cdots(x-[x]+1)}$, $x \\in[1,+\\infty)$.\\\\\n(1) 求 $\\mathrm{C}_8^{\\frac{3}{2}}$;\\\\\n(2) 当 $x \\in[2,3)$ 时, 求函数 $\\mathrm{C}_8^x$ 的值域.", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -662714,7 +663086,9 @@ "id": "024458", "content": "将一个各面均涂有油漆的正方体, 锯成 1000 个同样大小的小正方体, 若将这些小正方体均匀地搅拌在一起, 然后从中任取一个小正方体, 则恰好是一个具有两面漆的正方体的概率是\\bracket{20}.\n\\fourch{$\\dfrac{12}{125}$}{$\\dfrac{3}{25}$}{$\\dfrac{1}{10}$}{$\\dfrac{1}{12}$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -662736,7 +663110,9 @@ "id": "024459", "content": "若 $x_1$、$x_2$、$x_3$、$\\cdots$、$x_{314}$ 的标准差为 $2$, 那么 $3(x_1+5)$、$3(x_2+5)$、$\\cdots$、$3(x_{314}+5)$ 的标准差为\\bracket{20}.\n\\fourch{$18$}{$14$}{$6$}{$3$}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -662760,7 +663136,9 @@ "id": "024460", "content": "投掷一枚骰子, 下列事件中是对立事件的是\\bracket{20}.\n\\twoch{向上的点数是 1 与向上的点数是 5}{向上的点数小于 3 与向上的点数大于 3}{向上的点数是奇数与向上的点数是偶数}{向上的点数大于 3 与向上的点数小于 5}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -662780,7 +663158,9 @@ "id": "024461", "content": "高三年级有11名同学参加男子百米竞赛, 预赛成绩各不相同, 要取前 6 名参加决赛, 小明同学已经知道了自己的成绩, 为了判断自己是否能进入决赛, 他还需要知道11名同学成绩的\\bracket{20}.\n\\fourch{平均数}{众数}{中位数}{方差}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -662800,7 +663180,9 @@ "id": "024462", "content": "把 $10$ 本书随意地放在书架上, 则其中指定的 $3$ 本书放在一起的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -662823,7 +663205,9 @@ "id": "024463", "content": "将一颗质地均匀的骰子(一种各个面上分别标有 $1,2,3,4,5,6$ 个点的正方体玩具)先后抛掷 $2$ 次, 则出现向上的点数之和小于 $10$ 的概率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -662849,7 +663233,9 @@ "id": "024464", "content": "某甲、乙两人练习跳绳, 每人练习 $10$ 组, 每组不间断跳绳计数的茎叶图如图, 则下面结论中所有正确的序号是\\blank{50}.\n\\begin{center}\n\\begin{tabular}{cccccc|c|ccccc}\n\\multicolumn{6}{c|}{甲} & &\\multicolumn{5}{c}{乙}\\\\\n\\hline\n&&&&&8&0&9\\\\\n&&&&3&2&1&1&3&4&8&9\\\\\n7&6&5&4&2&0&2&0&1&1&3\\\\\n&&&&&7&3\n\\end{tabular}\n\\end{center}\n\\textcircled{1} 甲比乙的极差大;\\\\\n\\textcircled{2} 乙的中位数是 $18$;\\\\\n\\textcircled{3} 甲的平均数比乙的大;\\\\\n\\textcircled{4} 乙的众数是 $21$.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -662869,7 +663255,9 @@ "id": "024465", "content": "已知一个样本 $1,3,4,a,7$, 它的平均值是 $4$, 则这个样本的方差是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -662889,7 +663277,9 @@ "id": "024466", "content": "袋中装有标号为 $1,2,3,4$ 的四只球, 四人从中各取一只球, 其中甲不取 $1$ 号球, 乙不取 $2$ 号球, 丙不取 $3$ 号球, 丁不取 $4$ 号球的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -662909,7 +663299,9 @@ "id": "024467", "content": "甲、乙、丙三人 100 米跑的成绩 (互不影响) 合格的概率分别为 $\\dfrac{2}{5}$、$\\dfrac{3}{4}$、$\\dfrac{1}{3}$, 若对这三人进行一次 100 米跑检测, 则三人都合格的概率是\\blank{50}(结果用最简分数表示).", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -662929,7 +663321,9 @@ "id": "024468", "content": "若投掷 3 枚硬币, 则恰有二枚正面朝上的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -662952,7 +663346,9 @@ "id": "024469", "content": "随机投掷一枚均匀的硬币两次, 则两次都正面朝上的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -662975,7 +663371,9 @@ "id": "024470", "content": "某骰子为正方体, 六面分别印上 $1$、$2$、$3$、$4$、$5$、$6$ 的数字, 掷一次骰子出现``6''朝上的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -662995,7 +663393,9 @@ "id": "024471", "content": "设五个数值 $31,37,33, a, 35$ 的平均数是 34 , 则这组数据的方差是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -663015,7 +663415,9 @@ "id": "024472", "content": "若同时掷两颗骰子, 则出现两颗骰子的点数之和大于 $9$ 的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -663038,7 +663440,9 @@ "id": "024473", "content": "已知某社区的家庭年收入(单位: 万元)的频率分布直方图如图所示, 同一组中的数据用该组区间的中点值做代表, 则该社区内家庭的平均年收入的估计值是\\blank{50}万元.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.6, yscale = 10]\n\\draw [->] (3,0) -- (3.3,0) -- (3.4,0.02) -- (3.6,-0.02) -- (3.7,0) -- (11,0) node [below right] {收入/万元};\n\\draw [->] (3,0) -- (3,0.32) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (3,0) node [below left] {$O$};\n\\foreach \\i/\\j in {4/0.2,5/0.2,6/0.2,7/0.26,8/0.07,9/0.07}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (1,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {8/0.07,7/0.26,4/0.2}\n{\\draw [dashed] (\\i,\\j) -- (3,\\j) node [left] {$\\k$};};\n\\draw (10,0) node [below] {$10$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -663058,7 +663462,9 @@ "id": "024474", "content": "已知 10 件产品中有 2 件次品,\\\\\n(1) 任意取出 4 件产品检验, 求其中恰有 1 件次品的概率;\\\\\n(2) 为了保证使 2 件次品全部检验出的概率在 0.6 以上, 至少应抽取几件产品作检验?", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -663078,7 +663484,9 @@ "id": "024475", "content": "某教师为了了解本校高三学生一模考试的数学成绩情况, 将所教两个班级的数学成绩 (单位: 分)绘制成如图所示的茎叶图.\n\\begin{center}\n\\begin{tabular}{r|c|l}\n甲班 & & 乙班\\\\ \\hline\n& 8 & 8 \\\\\n9 \\ 9 \\ 8 \\ 7 \\ 6 \\ 5 \\ 5 \\ 4 \\ 3 \\ 1 \\ 1 \\ 0 & 9 & 0 \\ 1 \\ 1 \\ 2 \\ 2 \\ 2 \\ 2 \\ 4 \\ 6 \\ 7 \\ 8 \\ 8 \\ 9 \\\\\n9 \\ 8 \\ 8 \\ 7 \\ 7 \\ 6 \\ 5 \\ 5 \\ 4 \\ 3 \\ 3 \\ 3 \\ 3 \\ 0 & 10 & 0 \\ 1 \\ 1 \\ 1 \\ 1 \\ 2 \\ 2 \\ 5 \\ 6 \\ 6 \\ 7 \\ 9 \\ 9 \\\\\n9 \\ 7 \\ 5 \\ 5 \\ 5 \\ 2 \\ 2 \\ 1 \\ 0 & 11 & 0 \\ 3 \\ 4 \\ 5 \\ 5 \\ 6 \\ 7 \\ 9 \\\\\n9 \\ 8 \\ 6 \\ 5 \\ 3 \\ 3 \\ 1 \\ 0 \\ 0 & 12 & 0 \\ 1 \\ 2 \\ 2 \\ 3 \\ 3 \\ 5 \\ 8 \\ 9 \\\\\n6 \\ 2 \\ 1 \\ 1 & 13 & 0 \\ 1 \\ 8 \\\\\n6 \\ 0 & 14 & 3\n\\end{tabular}\n\\end{center}\n(1) 分别求出甲、乙两个班级数学成绩的中位数、众数;\\\\\n(2) 若规定成绩不小于 115 分为优秀, 分别求出两个班级数学成绩的优秀率.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -663098,7 +663506,9 @@ "id": "024476", "content": "$(1+\\sqrt{2})^7$ 展开式中有理项的项数是\\bracket{20}.\n\\fourch{4}{5}{6}{7}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -663118,7 +663528,9 @@ "id": "024477", "content": "已知 $(1+a x)(1+x)^5$ 的展开式中 $x^2$ 的系数为 5 , 则 $a=$\\bracket{20}.\n\\fourch{$-4$}{$-3$}{$-2$}{$-1$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -663138,7 +663550,9 @@ "id": "024478", "content": "$(1+x)^8(1+y)^4$ 的展开式中 $x^2 y^2$ 的系数是\\bracket{20}.\n\\fourch{$56$}{$84$}{$112$}{$168$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -663158,7 +663572,9 @@ "id": "024479", "content": "在 $(2 x-\\dfrac{1}{3}y)^7$ 展开式中, $x^5 y^2$ 的系数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -663178,7 +663594,9 @@ "id": "024480", "content": "$(\\sqrt[3]{5}+\\dfrac{1}{\\sqrt{5}})^{20}$ 的展开式中的有理项是展开式的第\\blank{50}项.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -663200,7 +663618,9 @@ "id": "024481", "content": "$(a+b)^n$ 的二项展开式中二项式系数的最大值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -663220,7 +663640,9 @@ "id": "024482", "content": "$\\mathrm{C}_{11}^1+\\mathrm{C}_{11}^3+\\mathrm{C}_{11}^5+\\mathrm{C}_{11}^7+\\mathrm{C}_{11}^9+\\mathrm{C}_{11}^{11}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -663240,7 +663662,9 @@ "id": "024483", "content": "$\\dfrac{\\mathrm{C}_{n+1}^1+\\mathrm{C}_{n+1}^2+\\cdots+\\mathrm{C}_{n+1}^n}{\\mathrm{C}_n^0+\\mathrm{C}_n^1+\\mathrm{C}_n^2+\\cdots+\\mathrm{C}_n^n}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -663260,7 +663684,9 @@ "id": "024484", "content": "求 $(x^2-3 x+2)^8$ 展开式中 $x^3$ 项的系数.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -663283,7 +663709,9 @@ "id": "024485", "content": "求 $55^{55}$ 被 8 除所得的余数.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -663309,7 +663737,9 @@ "id": "024486", "content": "求 $(\\sqrt[3]{2}+\\sqrt{3})^{12}$ 的展开式中的所有有理项的和.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -663329,7 +663759,10 @@ "id": "024487", "content": "数列 $\\{a_n\\}$ 的前 $n$ 项的和为 $S_n$, 且满足 $S_n=2 a_n-1$.\\\\\n(1) 求数列 $\\{a_n\\}$ 的通项公式;\\\\\n(2) 求和: $\\mathrm{C}_n^0S_1+\\mathrm{C}_n^1S_2+\\mathrm{C}_n^2S_3+\\cdots+\\mathrm{C}_n^n S_{n+1}$.", "objs": [], - "tags": [], + "tags": [ + "第四单元", + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -663349,7 +663782,9 @@ "id": "024488", "content": "利用二项式定理证明:\\\\\n(1) $2^n>n^2+n+1$($n \\in \\mathbf{N}^*$, $n \\geq 5$);\\\\\n(2) $2<(1+\\dfrac{1}{n})^n<3$($n \\geq 2$, $n \\in \\mathbf{N}^*$).", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -663370,7 +663805,9 @@ "id": "024489", "content": "设一个扇形的周长是 16 , 其面积是 12 , 则它的圆心角的大小为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -663395,7 +663832,9 @@ "id": "024490", "content": "已知点 $P(\\tan \\alpha, \\sin \\alpha)$ 在第三象限, 则角 $\\alpha$ 的终边在\\blank{50}第象限;", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -663415,7 +663854,9 @@ "id": "024491", "content": "化简 $\\dfrac{\\sin (\\pi-\\theta) \\cdot \\cos (\\dfrac{3 \\pi}{2}-\\theta) \\cdot \\tan \\theta}{\\cos (2 \\pi-\\theta) \\cdot \\cot (\\theta-\\dfrac{\\pi}{2}) \\cdot \\sin (\\dfrac{\\pi}{2}-\\theta)}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -663441,7 +663882,9 @@ "id": "024492", "content": "设 $\\sin (\\dfrac{\\pi}{4}+\\theta)=\\dfrac{1}{3}$, 则 $\\sin 2 \\theta=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -663463,7 +663906,9 @@ "id": "024493", "content": "在 $\\triangle ABC$ 中, 三边为 $a=2$, $b=3$, $c=4$, 则最大角的余弦值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -663486,7 +663931,9 @@ "id": "024494", "content": "在 $\\triangle ABC$ 中, 若 $A=30^\\circ$, $C=135^\\circ$, $a=1$, 则 $c=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -663506,7 +663953,9 @@ "id": "024495", "content": "函数 $y=\\sin x-\\sqrt{3}\\cos x$, $x \\in[-\\dfrac{\\pi}{6}, \\dfrac{\\pi}{2})$ 的值域是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -663530,7 +663979,9 @@ "id": "024496", "content": "函数 $y=\\sin ^2 x+2 \\cos x$ 的值域是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -663553,7 +664004,9 @@ "id": "024497", "content": "已知在 $\\triangle ABC$ 中, 角 $A, B, C$ 所对的边分别为 $a, b, c$. 若 $a=1$, $\\angle B=45^{\\circ}$, 三角形的面积 $S_{\\triangle ABC}=2$, 那么 $\\triangle ABC$ 的外接圆直径等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -663573,7 +664026,9 @@ "id": "024498", "content": "把函数 $y=\\sin (\\omega x+\\varphi)$, ($\\omega>0$, $|\\varphi|<\\pi$) 的图像向左平移 $\\dfrac{\\pi}{6}$ 个单位, 所得图像解析式为 $y=\\sin x$, 则$\\varphi=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -663593,7 +664048,9 @@ "id": "024499", "content": "在 $\\triangle ABC$ 中, 角 $A,B,C$ 的对边分别 $a,b,c$, 已知 $a+b=5$, $c=\\sqrt{7}$, 且 $\\sin ^22C+\\sin 2C \\cdot \\sin C+\\cos 2C=1$.\\\\\n(1) 求角 $C$ 的大小;\\\\\n(2) 求 $\\triangle ABC$ 的面积.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -663613,7 +664070,9 @@ "id": "024500", "content": "已知函数 $f(x)=\\cos (2 x-\\dfrac{\\pi}{3})+2 \\sin (x-\\dfrac{\\pi}{4}) \\sin (x+\\dfrac{\\pi}{4})$.\\\\\n(1) 求函数 $f(x)$ 的最小正周期和图像的对称轴方程;\\\\\n(2) 求函数 $f(x)$ 在区间 $[-\\dfrac{\\pi}{12}, \\dfrac{\\pi}{2}]$ 上的值域.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -663635,7 +664094,9 @@ "id": "024501", "content": "若非零向量 $\\overrightarrow{a}, \\overrightarrow{b}$ 满足 $|\\overrightarrow{a}+\\overrightarrow{b}|=|\\overrightarrow{a}-\\overrightarrow{b}|$, 则 $\\overrightarrow{a}$ 与 $\\overrightarrow{b}$ 所成角的大小为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -663657,7 +664118,9 @@ "id": "024502", "content": "已知 $|\\overrightarrow{a}|=|\\overrightarrow{b}|=2$, $\\overrightarrow{a}, \\overrightarrow{b}$ 的夹角为 $60^{\\circ}$, 则 $\\overrightarrow{a}+\\overrightarrow{b}$ 在 $\\overrightarrow{a}$ 方向上的投影为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -663680,7 +664143,9 @@ "id": "024503", "content": "如果 $\\overrightarrow{AP}=\\dfrac{1}{3}\\overrightarrow{PB}$, 且有 $\\overrightarrow{PB}=\\lambda \\overrightarrow{BA}$, 那么实数 $\\lambda$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -663703,7 +664168,9 @@ "id": "024504", "content": "已知点 $A$ 的坐标为 $(3,-2)$, 点 $B$ 在 $y$ 轴上, 且 $|\\overrightarrow{AB}|=3 \\sqrt{2}$, 则 $\\overrightarrow{AB}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -663723,7 +664190,9 @@ "id": "024505", "content": "已知 $z=m^2+2 m-15+\\dfrac{m^2-5 m+6}{m^2-25}\\mathrm{i}$.\\\\\n(1) 当且仅当实数 $m=$\\blank{50}时, $z$ 是实数;\\\\\n(2) 当且仅当实数 $m=$\\blank{50}时, $z$ 是纯虚数.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -663748,7 +664217,9 @@ "id": "024506", "content": "已知关于 $x$ 的方程 $x^2+k x+3=0$($k \\in \\mathbf{R}$) 有两个虚根 $\\alpha$ 和 $\\beta$, 且 $|\\alpha-\\beta|=2 \\sqrt{2}$, 则 $k=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -663772,7 +664243,9 @@ "id": "024507", "content": "计算: (1) $\\mathrm{i}+\\mathrm{i}^2+\\mathrm{i}^3+\\cdots +\\mathrm{i}^{200}=$\\blank{50};\\\\\n(2) $(1+\\mathrm{i})^{10}-(1-\\mathrm{i})^{10}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -663795,7 +664268,9 @@ "id": "024508", "content": "计算:\\\\\n(1) $|\\dfrac{(\\sqrt{5}-2 \\mathrm{i})(1+\\sqrt{3}\\mathrm{i})^2}{\\sqrt{13}+\\sqrt{23}\\mathrm{i}}|$\\\\\n(2) $(\\sqrt{3}+\\mathrm{i})^8$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -663817,7 +664292,9 @@ "id": "024509", "content": "已知 $\\overrightarrow{a}=(2,-3)$, $\\overrightarrow{b}$ 满足 $\\overrightarrow{b}\\parallel \\overrightarrow{a}$, 且 $\\overrightarrow{a}\\cdot \\overrightarrow{b}=-28$, 则 $\\overrightarrow{b}$ 的坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -663837,7 +664314,9 @@ "id": "024510", "content": "在直角坐标系中, 点 $A(-4,3)$ 为三角形 $\\triangle ABO$ 的直角顶点. 已知 $|\\overrightarrow{AB}|=2|\\overrightarrow{OA}|$, 且 $B$ 的纵坐标大于零, 则 $\\overrightarrow{AB}$ 的坐标是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -663857,7 +664336,9 @@ "id": "024511", "content": "如图所示, 平面内有三个向量 $\\overrightarrow{OA}, \\overrightarrow{OB}, \\overrightarrow{OC}$, 其中 $\\overrightarrow{OA}, \\overrightarrow{OB}$ 的夹角为 $120^{\\circ}$, $\\overrightarrow{OA}, \\overrightarrow{OC}$ 的夹角为 $30^{\\circ}$, 且 $|\\overrightarrow{OA}|=|\\overrightarrow{OB}|=1$, $|\\overrightarrow{OC}|=2 \\sqrt{3}$, 若 $\\overrightarrow{OC}=\\lambda \\overrightarrow{OA}+\\mu \\overrightarrow{OB}$($\\lambda, \\mu \\in \\mathbf{R}$), 则 $\\lambda+\\mu=$\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (1,0) node [below] {$A$} coordinate (A);\n\\draw (120:1) node [left] {$B$} coordinate (B);\n\\draw (30:{2*sqrt(3)}) node [right] {$C$} coordinate (C);\n\\foreach \\i in {A,B,C}\n{\\draw [->] (O)--(\\i);};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -663881,7 +664362,9 @@ "id": "024512", "content": "已知向量 $\\overrightarrow{OA}=(3,-4)$, $\\overrightarrow{OB}=(6,-3)$, $\\overrightarrow{OC}=(5-m,-4-m)$, 若 $\\triangle ABC$ 是直角三角形, 则数 $m$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -663903,7 +664386,9 @@ "id": "024513", "content": "已知关于 $x$ 的方程 $x^2+k x+k^2-2 k=0$($k \\in \\mathbf{R}$) 有一个模为 $1$ 的根, 求 $k$ 的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -663927,7 +664412,9 @@ "id": "024514", "content": "如图所示, $PQ$ 过 $\\triangle ABC$ 的重心 $G$, 设 $\\overrightarrow{AB}=\\overrightarrow{a}$, $\\overrightarrow{AC}=\\overrightarrow{b}$, 若 $\\overrightarrow{AP}=m \\overrightarrow{a}$, $\\overrightarrow{AQ}=n \\overrightarrow{b}$, 求证: $\\dfrac{1}{m}+\\dfrac{1}{n}=3$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$B$} coordinate (B);\n\\draw (2,0) node [below] {$C$} coordinate (C);\n\\draw (1.2,2) node [above] {$A$} coordinate (A);\n\\draw ($1/3*(A)+1/3*(B)+1/3*(C)$) node [below right] {$G$} coordinate (G);\n\\draw (A)--(B)--(C)--cycle(A)--($(A)!{3/2}!(G)$) node [below] {$D$} coordinate (D);\n\\draw ($(A)!{3/4}!(B)$) node [left] {$P$} coordinate (P);\n\\draw ($(A)!{3/5}!(C)$) node [right] {$Q$} coordinate (Q);\n\\draw (P)--(Q);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -663947,7 +664434,9 @@ "id": "024515", "content": "已知 $f(x)=\\ln \\dfrac{x+1}{x-1}$, 则 $f^{-1}(2)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -663967,7 +664456,9 @@ "id": "024516", "content": "函数 $f(x)=\\ln (x-2)+\\ln (x+2)$ 的奇偶性是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -663987,7 +664478,9 @@ "id": "024517", "content": "函数 $y=\\log _{(x+1)}(16-4^x)$ 的定义域是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -664007,7 +664500,9 @@ "id": "024518", "content": "函数 $y=4^x-2^{x+1}-1$, $x \\in[-1,2)$ 的值域是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -664027,7 +664522,9 @@ "id": "024519", "content": "若函数 $f(x)=2^{-|x|}-m$ 与 $x$ 轴有交点, 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": "", @@ -664047,7 +664544,9 @@ "id": "024520", "content": "将函数 $y=f(x)$ 的图像沿 $x$ 轴向左平移一个单位, 再沿 $y$ 轴翻折, 得到 $y=\\lg x$ 的图像, 则 $y=f(x)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -664067,7 +664566,9 @@ "id": "024521", "content": "已知集合 $A=\\{y | y=(\\dfrac{1}{3})^{|x|}\\}$, $B=\\{a | \\log _a(3 a-1)>0\\}$, 则 $A \\cap \\overline{B}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -664089,7 +664590,9 @@ "id": "024522", "content": "关于 $x$ 方程 $(\\dfrac{1}{2})^x=\\dfrac{2 a+3}{a-5}$ 有负根, 则实数 $a$ 取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -664113,7 +664616,9 @@ "id": "024523", "content": "设函数 $f(x)=\\dfrac{x^2+4 x+5}{x^2+4 x+4}$, 作出 $y=f(x)$ 的大致图像, 讨论 $y=f(x)$ 的性质, 并利用性质比较 $f(-\\pi)$ 与 $f(-\\dfrac{\\sqrt{2}}{2})$ 的大小.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -664135,7 +664640,9 @@ "id": "024524", "content": "(1) 已知关于 $x$ 的方程 $\\lg ^2 x+(m+4) \\lg x+4=0$ 有实数解, 求实数 $m$ 的取值范围;\\\\\n(2) 已知关于 $x$ 的方程 $9^x+(k+4) \\cdot 3^x+4=0$ 有实数解, 求实数 $k$ 的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -664157,7 +664664,9 @@ "id": "024525", "content": "已知 $f(x)=\\lg (x+1)$, $g(x)=2 \\lg (2 x+t)$.\\\\\n(1) 若 $t=-1$, 解不等式 $f(x) \\leq g(x)$;\\\\\n(2) 若对任意 $x \\in[0,1)$, 不等式 $f(x) \\leq g(x)$ 恒成立, 求实数 $t$ 的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -664177,7 +664686,9 @@ "id": "024526", "content": "设函数 $f(x)=\\log _a(x-3 a)$($a>0$, $a \\neq 1$), 当且仅当点 $P(x, y)$ 是函数 $y=f(x)$ 图像上的点时,点 $Q(x-2 a,-y)$ 是函数 $y=g(x)$ 的图像上的点.\\\\\n(1) 写出函数 $y=g(x)$ 的解析式;\\\\\n(2) 若当 $x \\in[a+2, a+3]$ 时, 恒有 $|f(x)-g(x)| \\leq 1$, 试确定 $a$ 的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -664197,7 +664708,9 @@ "id": "024527", "content": "若直线 $l$ 的斜率的取值范围为 $(-\\sqrt{2}, \\sqrt{3})$, 则其倾斜角的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -664220,7 +664733,9 @@ "id": "024528", "content": "已知点 $A(-2,3)$、$B(4,-7)$, 则线段 $AB$ 的垂直平分线的一般方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -664242,7 +664757,9 @@ "id": "024529", "content": "直线 $a x+(3-a) y+1=0$($a \\in \\mathbf{R}$) 恒过定点\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -664267,7 +664784,9 @@ "id": "024530", "content": "已知直线 $(1-a) x+3 a y+3=0$ 不经过第二象限, 则实数 $a$ 的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -664290,7 +664809,9 @@ "id": "024531", "content": "直线 $l$ 与直线 $y=1$, $x-y-7=0$ 分别交于 $P$、$Q$ 两点, 线段 $PQ$ 的中点为 $(1,-1)$, 则 $l$ 的倾斜角为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -664313,7 +664834,9 @@ "id": "024532", "content": "与直线 $2 x+3 y-6=0$ 关于点 $(1,-1)$ 对称的直线的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -664336,7 +664859,9 @@ "id": "024533", "content": "已知点 $A(0,5)$, $B(5,0)$, $C(-4,-3)$, 过点 $A$ 且斜率为 $k$ 的直线 $l$ 与射线 $BC$ 相交, 则 $k$ 的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -664362,7 +664887,9 @@ "id": "024534", "content": "直线 $4 x-3 y-8=0$ 的倾斜角平分线所在直线的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -664382,7 +664909,9 @@ "id": "024535", "content": "已知直线 $y=\\dfrac{1}{2}x+b$ 分别与 $x$ 轴、 $y$ 轴的交于点 $A$、$B$, $O$ 为坐标系原点, 且 $S_{\\triangle AOB}\\leq 2$, 则 $b$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -664402,7 +664931,9 @@ "id": "024536", "content": "若过点 $(1,4)$ 的直线 $l$ 在两坐标轴上的截距都是正数且截距之和最小, 则 $l$ 的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -664422,7 +664953,9 @@ "id": "024537", "content": "已知点 $A(2,0)$, 点 $B(-2,-4), P$ 为直线 $l: x-2 y+8=0$ 上一动点.\\\\\n(1) 求 $|PA|+|PB|$ 的最小值, 并求出此时点 $P$ 的坐标;\\\\\n(2) 求 $|PB|-|PA|$ 的最大值, 并求出此时点 $P$ 的坐标.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -664445,7 +664978,9 @@ "id": "024538", "content": "若直线 $l_1: a x+3 y+1=0$ 与 $l_2: 2 x+(a+1) y+1=0$ 互相平行, 则实数 $a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -664467,7 +665002,9 @@ "id": "024539", "content": "若直线 $(a+2) x+(1-a) y-3=0$ 与 $(a-1) x+(2 a+3) y+2=0$ 互相垂直, 则实数 $a =$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -664493,7 +665030,9 @@ "id": "024540", "content": "已知直线 $l$ 的方程为 $3 x+4 y-12=0$, 直线 $l$ 与 $l$ 垂直且 $l$ 与坐标轴围成的三角形面积为 6 , 则直线 $l'$ 的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -664515,7 +665054,9 @@ "id": "024541", "content": "已知直线 $x-\\sqrt{3}y+1=0$ 与直线 $x+k y+3=0$ 的夹角为 $60^{\\circ}$, 则实数 $k=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -664535,7 +665076,9 @@ "id": "024542", "content": "已知动点 $M(a, b)$ 在直线 $3 x+4 y-15=0$ 上, 则 $\\sqrt{a^2+b^2}$ 的最小值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -664559,7 +665102,9 @@ "id": "024543", "content": "直线 $l_1: 3 x-2 y-6=0$ 关于直线 $l_2: 2 x-3 y+1=0$ 对称的直线 $l_3$ 的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -664585,7 +665130,9 @@ "id": "024544", "content": "一直线过点 $P(1,2)$, 且被两条平行直线 $4 x+3 y+1=0$ 和 $4 x+3 y+6=0$ 截得线段长为 $\\sqrt{2}$, 则直线方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -664610,7 +665157,9 @@ "id": "024545", "content": "平面中两条直线 $l_1$ 和 $l_2$ 相交于点 $O$, 对于平面上任意一点 $M$, 若 $p$、$q$ 分别为 $M$ 到直线 $l_1$和 $l_2$ 的距离, 则称有序非负实数对 $(p, q)$ 是点 $M$ 的``距离坐标''. 根据上述定义,``距离坐标''是$(1,2)$的点的个数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -664630,7 +665179,9 @@ "id": "024546", "content": "过点 $M(-2,-1)$ 作直线 $l$, 使它与两点 $A(-3,-2)$、$B(5,-4)$ 的距离相等, 求直线 $l$ 的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -664653,7 +665204,9 @@ "id": "024547", "content": "如果原点在圆 $x^2+y^2+x+3 y-a=0$ 的外部, 则实数 $a$ 的取值范围是\\blank{50};", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -664675,7 +665228,9 @@ "id": "024548", "content": "若实数 $x, y$ 满足 $x^2+y^2-2 x+4 y=0$, 则 $x-y$ 的最大值为\\blank{50}, 最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -664700,7 +665255,9 @@ "id": "024549", "content": "对于圆 $x^2+(y+1)^2=1$ 上的任意一点, 恒有 $x+y+m>0$, 则实数 $m$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -664720,7 +665277,9 @@ "id": "024550", "content": "若 $A(a, a^2)$, $B(b, b^2)$($a \\neq b$) 两点的坐标满足方程 $a^2 \\sin \\theta+a \\cos \\theta=1$, $b^2 \\sin \\theta+b \\cos \\theta=1$, 圆 $C: x^2+y^2=1$, 则圆 $C$ 与直线 $AB$ 的位置关系是\\blank{50}(相交, 相切或者相离).", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -664740,7 +665299,9 @@ "id": "024551", "content": "过点 $A(1,4)$, 且与已知圆 $M: x^2+y^2-6 x-2 y+5=0$ 切于点 $B(1,2)$ 的圆 $C$ 的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -664762,7 +665323,9 @@ "id": "024552", "content": "圆心在直线 $x-y-1=0$ 上, 与直线 $4 x+3 y+14=0$ 相切, 且截直线 $3 x+4 y+10=0$ 所得的弦长等于 6 的圆的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -664782,7 +665345,9 @@ "id": "024553", "content": "``$D^2=4F$''是``圆 $x^2+y^2+D x+E y+F=0$ 与 $x$ 轴相切''的\\bracket{20}.\n\\fourch{充分非必要条件}{必要非充分条件}{充要条件}{非充分非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -664805,7 +665370,9 @@ "id": "024554", "content": "直线 $l$ 过点 $P(0,2)$, 且被圆 $x^2+y^2=4$ 所截得的线段长为 $2$, 那么直线 $l$ 的斜率为 \\bracket{20}.\n\\fourch{$\\sqrt{2}$ 或 $-\\sqrt{2}$}{$\\dfrac{\\sqrt{2}}{2}$ 或 $-\\dfrac{\\sqrt{2}}{2}$}{$\\sqrt{3}$ 或 $-\\sqrt{3}$}{$\\dfrac{\\sqrt{3}}{3}$ 或 $-\\dfrac{\\sqrt{3}}{3}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -664825,7 +665392,9 @@ "id": "024555", "content": "若直线 $y=x+\\log _2 t$ 与曲线 $y=\\sqrt{4-x^2}$ 恰有一个公共点, 求实数 $t$ 的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -664847,7 +665416,9 @@ "id": "024556", "content": "如图, 在平面直角坐标系中, 方程为 $x^2+y^2+D x+E y+F=0$ 的圆 $M$ 的内接四边形 $ABCD$的对角线 $AC$ 和 $BD$ 互相垂直, 且 $AC$ 和 $BD$ 分别在 $x$ 轴和 $y$ 轴上.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [name path = x,->] (-2,0) -- (2.5,0) node [below] {$x$};\n\\draw [name path = y,->] (0,-1) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [above left] {$O$} coordinate (O);\n\\draw [name path = circle] (0.5,0.9) circle (1.5) node [above] {$M$} coordinate (M);\n\\filldraw (M) circle (0.03);\n\\draw [name intersections = {of = circle and x, by = {A,C}}];\n\\draw (A) node [below left] {$A$};\n\\draw (C) node [below right] {$C$};\n\\draw [name intersections = {of = circle and y, by = {D,B}}];\n\\draw (B) node [below left] {$B$};\n\\draw (D) node [above left] {$D$};\n\\draw (A)--(B)--(C)--(D)--cycle;\n\\draw ($(C)!0.5!(D)$) node [above right] {$G$} coordinate (G);\n\\filldraw (G) circle (0.03);\n\\draw ($(A)!(O)!(B)$) node [below left] {$H$} coordinate (H);\n\\draw (O)--(H);\n\\filldraw (H) circle (0.03) (O) circle (0.03);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $F<0$;\\\\\n(2) 若四边形 $ABCD$ 面积为 $8$, 对角线 $AC$ 长为 $2$, 且 $\\overrightarrow{AB}\\cdot \\overrightarrow{AD}=0$, 求 $D^2+E^2-4F$ 的值;\\\\\n(3) 设四边形 $ABCD$ 的一条边 $CD$ 的中点为 $G$, $OH \\perp AB$ 且垂足为 $H$. 试用平面解析几何的研究方法判断点 $O$、$G$、$H$ 是否共线, 并说明理由.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -664867,7 +665438,9 @@ "id": "024557", "content": "如果向量 $\\overrightarrow{OA}=3 \\overrightarrow{i}+\\overrightarrow{j}-2 \\overrightarrow{k}$, $\\overrightarrow{OB}=-\\overrightarrow{i}+2 \\overrightarrow{j}-\\overrightarrow{k}$, $\\overrightarrow{OC}\\perp \\overrightarrow{OB}$, $\\overrightarrow{BC}\\parallel \\overrightarrow{OA}$, 那么向量 $\\overrightarrow{OC}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -664887,7 +665460,9 @@ "id": "024558", "content": "已知点 $G$ 是 $\\triangle ABC$ 的重心, $O$ 是空间任一点, 若 $\\overrightarrow{OA}+\\overrightarrow{OB}+\\overrightarrow{OC}=\\lambda \\overrightarrow{OG}$, 则 $\\lambda$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -664907,7 +665482,9 @@ "id": "024559", "content": "设 $A$、$B$、$C$、$D$ 是空间不共面的四点,\n且满足 $\\overrightarrow{AB}\\cdot \\overrightarrow{AC}=0$, $\\overrightarrow{AB}\\cdot \\overrightarrow{AD}=0$, $\\overrightarrow{AC}\\cdot \\overrightarrow{AD}=0$, 则 $\\triangle BCD$ 的形状为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -664927,7 +665504,9 @@ "id": "024560", "content": "异面直线 $l_1, l_2$ 的方向向量分别为 $\\overrightarrow{d}_1, \\overrightarrow{d}_2$, 若 $\\cos \\langle\\overrightarrow{d}_1$, $\\overrightarrow{d}_2\\rangle=-\\dfrac{1}{2}$, 则 $l_1, l_2$ 所成角的大小为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -664947,7 +665526,9 @@ "id": "024561", "content": "已知直线 $l_1, l_2$ 的方向向量分别为 $\\overrightarrow{d}_1, \\overrightarrow{d}_2$, 若 $\\overrightarrow{d_1}=(-1,0,1)$, $\\overrightarrow{d}_2=(1,1,-2)$, 则 $l_1, l_2$ 所成角的大小为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -664969,7 +665550,9 @@ "id": "024562", "content": "直线 $l$ 的方向向量为 $\\overrightarrow{d}=(-1,1,2)$, 平面 $\\alpha$ 的法向量 $\\overrightarrow{n}=(2,1,-1)$, 则 $l$ 与平面 $\\alpha$ 所成角的大小为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -664989,7 +665572,9 @@ "id": "024563", "content": "二面角 $\\alpha-l-\\beta$ 的平面角为锐角, $\\alpha, \\beta$ 的法向量分别 $\\overrightarrow{n_1}=(1,0,0)$, $\\overrightarrow{n_2}=(-1,2,2)$, 则二面角 $\\alpha-l-\\beta$ 的大小为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "",