diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 647f1288..28a3ffe1 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -602459,6 +602459,426 @@ "space": "", "unrelated": [] }, + "022767": { + "id": "022767", + "content": "设集合 $A=\\{1,2,3\\}$, $B=\\{1,2\\}$, 则 $A \\cup B=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期测验04试题1", + "edit": [ + "20231117\t毛培菁" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022768": { + "id": "022768", + "content": "若 $\\tan \\alpha=2$, 则 $\\tan (\\pi-\\alpha)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期测验04试题2", + "edit": [ + "20231117\t毛培菁" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022769": { + "id": "022769", + "content": "已知常数 $a \\in \\mathbf{R}$, 复数 $z=\\dfrac{1}{1+\\mathrm{i}}+a$ ($\\mathrm{i}$ 为虚数单位) 为纯虚数, 则 $a=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期测验04试题3", + "edit": [ + "20231117\t毛培菁" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022770": { + "id": "022770", + "content": "函数 $f(x)=\\sqrt{1-\\dfrac{1}{x}}$ 的定义域为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期测验04试题4", + "edit": [ + "20231117\t毛培菁" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022771": { + "id": "022771", + "content": "已知角 $\\alpha$ 的顶点在坐标原点, 始边与 $x$ 轴的正半轴重合, 终边与单位圆的交点坐标为 $(\\dfrac{3}{5},-\\dfrac{4}{5})$, 则 $\\sin 2 \\alpha=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期测验04试题5", + "edit": [ + "20231117\t毛培菁" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022772": { + "id": "022772", + "content": "已知圆锥的底面半径为 $1$, 侧面积为 $\\sqrt{2}\\pi$, 则该圆锥的体积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期测验04试题6", + "edit": [ + "20231117\t毛培菁" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022773": { + "id": "022773", + "content": "已知正数 $a, b$ 满足 $2 a b+1=b$, 则 $\\dfrac{a}{b}$ 的最大值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期测验04试题7", + "edit": [ + "20231117\t毛培菁" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022774": { + "id": "022774", + "content": "祖冲之是我国古代的数学家, 他估计圆周率 $\\pi$ 的范围是 $(3.1415926,3.1415927)$. 为纪念祖冲之在圆周率上的成就, 把 $3.1415926$ 称为``祖率''. 在 $8$ 张质地相同的卡片上分别写有数字 $3,1,4,1,5,9,2,6$, 从中随机抽取 $2$ 张, 则 $2$ 张卡片的数字之和大于 $8$ 的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期测验04试题8", + "edit": [ + "20231117\t毛培菁" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022775": { + "id": "022775", + "content": "已知函数 $y=f(x)$ 是定义域为 $\\mathbf{R}$ 的二次函数, 函数 $g(x)=(x^2+3 x+2) f(x)$ 是偶函数. 若关于 $x$ 的方程 $f(x)=1$ 在 $(1,2)$ 上有解, 则 $f(0)$ 的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期测验04试题9", + "edit": [ + "20231117\t毛培菁" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022776": { + "id": "022776", + "content": "在 $\\triangle ABC$ 中, $\\angle A=\\dfrac{2 \\pi}{3}$, $BC=\\sqrt{3}$, 则 $AB+\\dfrac{3}{2}AC$ 的最大值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期测验04试题10", + "edit": [ + "20231117\t毛培菁" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022777": { + "id": "022777", + "content": "设 $P$ 是边长为 2 的正六边形边上的动点, 长为 $2 \\sqrt{3}$ 的线段 $MN$ 是该正六边形外接圆的一条动弦, 则 $\\overrightarrow{PM}\\cdot \\overrightarrow{PN}$ 的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期测验04试题11", + "edit": [ + "20231117\t毛培菁" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022778": { + "id": "022778", + "content": "已知球 $O$ 的表面积为 $64 \\pi$, 球 $O$ 的两个小圆分别为圆 $I_1$, 圆 $I_2$, 半径分别为 $2 \\sqrt{2}, \\sqrt{7}$. 若圆心 $I_1$ 到圆 $I_2$ 所在平面的距离为 1 , 则圆 $I_1$ 与圆 $I_2$ 的公共弦长为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期测验04试题12", + "edit": [ + "20231117\t毛培菁" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022779": { + "id": "022779", + "content": "已知实数 $a, b$ 满足 $a>b$, 下列不等式恒成立的是\\bracket{20}.\n\\fourch{$a^2>b^2$}{$\\dfrac{1}{a}<\\dfrac{1}{b}$}{$\\mathrm{e}^a>\\mathrm{e}^b$}{$\\ln a>\\ln b$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期测验04试题13", + "edit": [ + "20231117\t毛培菁" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022780": { + "id": "022780", + "content": "要得到函数 $y=\\sin (2 x+\\dfrac{\\pi}{6})$ 的图像, 只要将函数 $y=\\sin 2 x$ 的图像\\bracket{20}.\n\\fourch{向右平移 $\\dfrac{\\pi}{6}$ 个单位}{向右平移 $\\dfrac{\\pi}{12}$ 个单位}{向左平移 $\\dfrac{\\pi}{6}$ 个单位}{向左平移 $\\dfrac{\\pi}{12}$ 个单位}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期测验04试题14", + "edit": [ + "20231117\t毛培菁" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022781": { + "id": "022781", + "content": "设 $z_1, z_2 \\in \\mathbf{C}$, 对于命题: \\textcircled{1} 若 $z_1-z_2>0$, 则 $z_1>z_2$; \\textcircled{2} 若 $|z_1|=|z_2|$, 则 $z_1^2=z_2^2$; \\textcircled{3} 若 $|\\dfrac{z_1}{z_2}|>1$, 则 $|z_1|>|z_2|$; \\textcircled{4} 若 $z_1^3+z_2^3=0$, 则 $z_1+z_2=0$, 真命题的个数为\\bracket{20}.\n\\fourch{$1$}{$2$}{$3$}{以上选项都不正确}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期测验04试题15", + "edit": [ + "20231117\t毛培菁" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022782": { + "id": "022782", + "content": "如图, 设二面角 $\\alpha-l-\\beta$ 的大小为 $\\theta_1$($90^{\\circ}<\\theta_1<180^{\\circ}$), 矩形 $ABCD$ 在半平面 $\\alpha$ 上, 点 $B, C$ 在 $l$ 上. 将矩形 $ABCD$ 绕直线 $CD$ 按逆时针方向旋转角度 $\\theta_2$($0^{\\circ}<\\theta_2<90^{\\circ}$), 得到矩形 $A' B' CD$,记矩形 $A' B' CD$ 所在平面与平面 $\\beta$ 所成的小于等于 $90^{\\circ}$ 的二面角的大小为 $\\theta_3$.对于命题: \\textcircled{1} 对任意的 $\\theta_1$ 和任意的 $\\theta_2$, 恒有 $\\theta_2<\\theta_3$; \\textcircled{2} 对任意的 $\\theta_1$ 和任意的 $\\theta_2$, 恒有 $\\sin \\theta_1<\\sin \\theta_3$, 下列判断正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [above] {$C$} coordinate (C);\n\\draw (2,0,0) coordinate (T);\n\\draw (2,0,2) coordinate (S);\n\\draw (0,0,2) node [below] {$B$} coordinate (B);\n\\draw ({-sqrt(2)},{sqrt(2)},0) node [right] {$D$} coordinate (D);\n\\draw (D) ++ (0,0,2) node [left] {$A$} coordinate (A);\n\\draw (A)--(B)--(S)--(T)--(C)--(D)--cycle(B)--(C) node [midway, left] {$l$};\n\\draw (S) node [above] {$\\beta$} (A) node [right] {$\\alpha$};\n\\draw [dashed] (C) -- ($(D)!1.2!(C)$);\n\\draw (D) -- ($(C)!1.2!(D)$) coordinate (P);\n\\draw [->] (P) ++ (270:0.3) arc (270:360:0.3);\n\\end{tikzpicture}\n\\end{center}\n\\twoch{\\textcircled{1}和\\textcircled{2}均为真命题}{\\textcircled{1}和\\textcircled{2}均为假命题}{\\textcircled{1}为真命题, \\textcircled{2}为假命题}{\\textcircled{1}为假命题, \\textcircled{2}为真命题}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期测验04试题16", + "edit": [ + "20231117\t毛培菁" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022783": { + "id": "022783", + "content": "已知平面向量 $\\overrightarrow{a}=(\\cos \\alpha, \\sin \\alpha)$, $\\overrightarrow{b}=(\\cos \\beta, \\sin \\beta)$, $\\overrightarrow{c}=(\\dfrac{1}{2},-\\dfrac{\\sqrt{3}}{2})$, $0 \\leq \\beta<\\alpha<2 \\pi$.\\\\\n(1) 若 $\\overrightarrow{a}=\\overrightarrow{c}$, $|\\overrightarrow{a}+2 \\overrightarrow{b}|=1$, 求 $\\beta$ 的值;\\\\\n(2) 若 $\\overrightarrow{a}+\\overrightarrow{b}=\\overrightarrow{c}$, 求 $\\alpha, \\beta$ 的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期测验04试题17", + "edit": [ + "20231117\t毛培菁" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "022784": { + "id": "022784", + "content": "已知常数 $a \\in \\mathbf{R}$, 函数 $f(x)=\\ln (\\sqrt{1+x^2}+a x)$.\\\\\n(1) 若 $a=-1$, 用单调性的定义证明函数 $y=f(x)$ 在 $[0,+\\infty)$ 上是严格减函数;\\\\\n(2) 若函数 $y=f(x)$ 的定义域为 $\\mathbf{R}$, 求 $a$ 的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期测验04试题18", + "edit": [ + "20231117\t毛培菁" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "022785": { + "id": "022785", + "content": "经销商经销某种农产品, 在一个销售季度内, 每售出 $1 \\mathrm{t}$ 该产品获利润 $500$ 元, 未售出的产品, 每 $1 \\mathrm{t}$ 亏损 $300$ 元. 根据历史资料, 得到销售季度内市场需求量的频率分布直方图, 如下图所示. 经销商为下一个销售季度购进了 $130 \\mathrm{t}$ 该农产品, 以 $X$ (单位: $\\mathrm{t}$, $100 \\leq X \\leq 150$)表示下一个销售季度内的市场需求量, $T$ (单位: 元)表示下一个销售季度内经销该农产品的利润.\\\\\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.06, yscale = 60]\n\\draw [->] (90,0) -- (92,0) -- (93.5,-0.003) -- (96.5,0.003) -- (98,0) -- (170,0) node [below] {需求量$/\\mathrm{t}$};\n\\draw [->] (90,0) -- (90,0.045) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (90,0) node [below left] {$O$};\n\\foreach \\i/\\j in {100/0.01,110/0.02,120/0.03,130/0.025,140/0.015}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (10,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {100/0.01,110/0.02,120/0.03,130/0.025,140/0.015}\n{\\draw [dashed] (\\i,\\j) -- (90,\\j) node [left] {$\\k$};};\n\\draw (150,0) node [below] {$150$};\n\\end{tikzpicture}\n\\end{center}\n(1) 将 $T$ 表示为 $X$ 的函数, 并根据直方图估计利润 $T$ 不少于 $57000$ 元的概率;\\\\\n(2) 在直方图的需求量分组中, 以各组的区间中点值代表该组的各个值, 并以需求量落入该区间的频率作为需求量取该区间中点的概率 (例如: 需求量 $X \\in[100,110$ ), 则取 $X=105$,且 $X=105$ 的概率等于需求量落入 $[100,110)$ 的频率), 求 $T$ 的数学期望.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期测验04试题19", + "edit": [ + "20231117\t毛培菁" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "022786": { + "id": "022786", + "content": "空间两两不同的 $5$ 个点 $A, B, C, O, P$ 满足 $A, B, P$ 不共线, $\\triangle POC$ 是正三角形, $OB \\perp$ 平面 $ABP$.\\\\\n(1) 若 $OB=\\sqrt{2}$, $OC=\\sqrt{3}$, $OC \\parallel $ 平面 $ABP$, 求异面直线 $OC$ 与 $BP$ 所成角的大小;\\\\\n(2) 设常数 $\\theta \\in(0, \\dfrac{\\pi}{2})$. 若 $AB \\perp BP$, $\\angle BOP=\\theta$, 点 $C$ 在平面 $AOB$ 上, 求 $\\theta$ 的取值范围;\\\\\n(3) 设线段 $BP$ 的中点为 $M$. 若 $AB=3$, $OM=\\sqrt{13}$, 平面 $POB \\perp$ 平面 $POC$, 求三棱锥 $O-ABC$ 体积的最大值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期测验04试题20", + "edit": [ + "20231117\t毛培菁" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "022787": { + "id": "022787", + "content": "已知函数 $y=f(x)$, $y=g(x)$ 的定义域分别为 $D_1, D_2$. 若存在常数 $C>0$, 同时满足: \\textcircled{1} 对任意 $x_0 \\in D_1$, 恒有 $x_0+C \\in D_1$, 且 $f(x_0) \\leq f(x_0+C)$; \\textcircled{2} 对任意 $x_0 \\in D_1$, 关于 $x$ 的不等式组 $f(x_0) \\leq g(x) \\leq g(x+C) \\leq f(x_0+C)$ 恒有解, 则称 $y=g(x)$ 为 $y=f(x)$ 的一个``嵌入常数为 $C$的嵌入函数''.\\\\\n(1) 设函数 $f(x)=\\begin{cases}-1,&0 \\leq x \\leq \\dfrac{1}{3},\\\\1,& x>\\dfrac{1}{3},\\end{cases}$ $g(x)=\\begin{cases}1,&0 \\leq x \\leq \\dfrac{1}{2},\\\\0,& x>\\dfrac{1}{2}.\\end{cases}$ 判断 $y=g(x)$ 是否为 $y=f(x)$ 的一个``嵌入常数为 $\\dfrac{1}{2}$ 的嵌入函数''(直接写出答案);\\\\\n(2) 设常数 $a \\in \\mathbf{R}$, 函数 $f(x)=-x^2+a \\mathrm{e}^x$($x \\in \\mathbf{R}$) (其中常数 $\\mathrm{e}$ 为自然对数的底数), $g(x)=2 x$($x \\in \\mathbf{R}$). 若 $y=g(x)$ 为 $y=f(x)$ 的一个``嵌入常数为 1 的嵌入函数'', 求 $a$ 的取值范围;\\\\\n(3) 设常数 $t>0$, 函数 $f(x)=x^2-4 x$($x \\geq 0)$, $g(x)=x+\\dfrac{2 t^2}{x}$($x>0$). 问: 是否存在 $t$,使得 $y=g(x)$ 为 $y=f(x)$ 的一个``嵌入常数为 $t$ 的嵌入函数''? 若存在, 求 $t$ 的取值范围; 若不存在, 说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期测验04试题21", + "edit": [ + "20231117\t毛培菁" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, "030001": { "id": "030001", "content": "若$x,y,z$都是实数, 则:(填写``\\textcircled{1} 充分非必要、\\textcircled{2} 必要非充分、\\textcircled{3} 充要、\\textcircled{4} 既非充分又非必要''之一)\\\\\n(1) ``$xy=0$''是``$x=0$''的\\blank{50}条件;\\\\\n(2) ``$x\\cdot y=y\\cdot z$''是``$x=z$''的\\blank{50}条件;\\\\\n(3) ``$\\dfrac xy=\\dfrac yz$''是``$xz=y^2$''的\\blank{50}条件;\\\\\n(4) ``$|x |>| y|$''是``$x>y>0$''的\\blank{50}条件;\\\\\n(5) ``$x^2>4$''是``$x>2$'' 的\\blank{50}条件;\\\\\n(6) ``$x=-3$''是``$x^2+x-6=0$'' 的\\blank{50}条件;\\\\\n(7) ``$|x+y|<2$''是``$|x|<1$且$|y|<1$'' 的\\blank{50}条件;\\\\\n(8) ``$|x|<3$''是``$x^2<9$'' 的\\blank{50}条件;\\\\\n(9) ``$x^2+y^2>0$''是``$x\\ne 0$'' 的\\blank{50}条件;\\\\\n(10) ``$\\dfrac{x^2+x+1}{3x+2}<0$''是``$3x+2<0$'' 的\\blank{50}条件;\\\\\n(11) ``$0