From 0711413df5705796d08b5c8712f32de017ad0a05 Mon Sep 17 00:00:00 2001 From: "weiye.wang" Date: Thu, 23 Nov 2023 22:28:01 +0800 Subject: [PATCH] =?UTF-8?q?=E6=B7=BB=E5=8A=A02024=E5=B1=8A=E9=AB=98?= =?UTF-8?q?=E4=B8=89124=E5=88=86=E5=AE=88=E6=8A=A4=E5=8D=B73=E9=A2=98?= =?UTF-8?q?=E7=9B=AE?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 题库0.3/Problems.json | 360 ++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 360 insertions(+) diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index edf3d33f..4bea7566 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -604227,6 +604227,366 @@ "space": "4em", "unrelated": [] }, + "022822": { + "id": "022822", + "content": "若 $z(1+\\mathrm{i})=2 \\mathrm{i}$ ($\\mathrm{i}$ 是虚数单位), 则 $|z|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期124分守护卷题目", + "edit": [ + "20231123\t毛培菁" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022823": { + "id": "022823", + "content": "在 $\\triangle ABC$中, 若 $A=60^{\\circ}$, $AB=2$, $AC=2 \\sqrt{3}$, 则 $\\triangle ABC$ 的面积是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期124分守护卷题目", + "edit": [ + "20231123\t毛培菁" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022824": { + "id": "022824", + "content": "已知 $f(x)=x \\ln x$. 设函数 $y=f(x)$ 的导函数为 $y'=f'(x)$, 则 $f'(x)$ 的表达式为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期124分守护卷题目", + "edit": [ + "20231123\t毛培菁" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022825": { + "id": "022825", + "content": "圆锥的底面半径为 1 , 高为 2 , 则该圆锥的侧面积等于\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期124分守护卷题目", + "edit": [ + "20231123\t毛培菁" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022826": { + "id": "022826", + "content": "设向量 $\\overrightarrow{a}=(\\dfrac{3}{2}, \\sin \\alpha)$, $\\overrightarrow{b}=(\\cos \\alpha, \\dfrac{1}{3})$, 且 $\\overrightarrow{a}\\parallel \\overrightarrow{b}$, 则 $\\cos 2 \\alpha=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期124分守护卷题目", + "edit": [ + "20231123\t毛培菁" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022827": { + "id": "022827", + "content": "在 $(x^2+\\dfrac{2}{x})^5$ 的二项展开式中, $x$ 的一次项系数为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期124分守护卷题目", + "edit": [ + "20231123\t毛培菁" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022828": { + "id": "022828", + "content": "设 $a \\in \\mathbf{R}$. 已知方程 $x^2-a=0$ 有两个虚数根 $x_1, x_2$. 若 $|x_1-x_2|=1$, 则 $a=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期124分守护卷题目", + "edit": [ + "20231123\t毛培菁" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022829": { + "id": "022829", + "content": "若双曲线的渐近线方程为 $y= \\pm 3 x$, 它的焦距为 $2 \\sqrt{10}$, 实轴长大于虚轴长, 则该双曲线的标准方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期124分守护卷题目", + "edit": [ + "20231123\t毛培菁" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022830": { + "id": "022830", + "content": "若甲、乙两人从 6 门课程中各随机选修 3 门, 则甲、乙所选修的课程中恰有 1 门相同的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期124分守护卷题目", + "edit": [ + "20231123\t毛培菁" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022831": { + "id": "022831", + "content": "根据相关规定, 机动车驾驶人血液中的酒精含量大于 (等于) $20$ 毫克 $/ 100$ 毫升的行为属于饮酒驾车. 假设饮酒后, 血液中的酒精含量为 $p_0$ 毫克 $/ 100$ 毫升, 经过 $x$ 个小时, 酒精含量降为 $p$ 毫克 $/ 100$毫升, 且满足关系式 $p=p_0 \\cdot \\mathrm{e}^{r x}$ ($r$ 为常数). 若某人饮酒后血液中的酒精含量为 $89$ 毫克 $/ 100$ 毫升, $2$ 小时后, 测得其血液中酒精含量降为 $61$ 毫克 $/ 100$ 毫升, 则此人饮酒后需至少经过\\blank{50}小时方可驾车(精确到小时).", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期124分守护卷题目", + "edit": [ + "20231123\t毛培菁" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022832": { + "id": "022832", + "content": "如图, 在底面半径和高均为 $\\sqrt{2}$ 的圆锥中, $AB$、$CD$ 是底面圆 $O$ 的两条互相垂直的直径, $E$ 是母线 $PB$ 的中点. 已知过 $CD$ 与 $E$ 的平面与圆锥侧面的交线是以 $E$ 为顶点的抛物线的一部分, 则该抛物线的焦点到圆锥顶点 $P$ 的距离等于\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 2]\n\\def\\r{1}\n\\def\\h{1}\n\\draw ({-\\r},0,0) node [left] {$A$} coordinate (A) -- (0,\\h,0) node [above] {$P$} coordinate (P) -- (\\r,0,0) node [right] {$B$} coordinate (B);\n\\draw (0,0,0) node [below left] {$O$} coordinate (O);\n\\draw (A) arc (180:360:{\\r} and {\\r/4});\n\\draw [dashed] (A) arc (180:0:{\\r} and {\\r/4});\n\\draw [dashed] (A) -- (B) (O) -- (P);\n\\draw ($(P)!0.5!(B)$) node [above right] {$E$} coordinate (E);\n\\draw [dashed] (O) -- (E);\n\\draw ({\\r*cos(-70)},{\\r/4*sin(-70)}) node [below] {$C$} coordinate (C);\n\\draw ({\\r*cos(110)},{\\r/4*sin(110)}) node [below] {$D$} coordinate (D);\n\\draw (C) .. controls +({\\r/10},{\\r/10}) and +({\\r*cos(-70)/3},{\\r/4*sin(-70)/3}) .. (E);\n\\draw [dashed] (D) .. controls +({\\r/10},{\\r/10}) and +({-\\r*cos(-70)/3},{-\\r/4*sin(-70)/3}) .. (E) (C) -- (D);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期124分守护卷题目", + "edit": [ + "20231123\t毛培菁" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022833": { + "id": "022833", + "content": "从小到大排列的八个数据为: $55,57,57,60,61,61,63,64$, 则第 $75$ 百分位数为 \\bracket{20}.\n\\fourch{$56$}{$57$}{$61$}{$62$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期124分守护卷题目", + "edit": [ + "20231123\t毛培菁" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022834": { + "id": "022834", + "content": "实数构成的等比数列 $\\{a_n\\}$ 的公比不为 $1$. 若 $a_{11}, a_{12}, a_{13}, a_{14}$ 中有两项为 -3 , 则 $a_{11}+a_{12}+a_{13}+a_{14}=$\\bracket{20}.\n\\fourch{$0$}{$6$}{$8$}{$12$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期124分守护卷题目", + "edit": [ + "20231123\t毛培菁" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022835": { + "id": "022835", + "content": "设平面直角坐标中, $O$ 为原点, $M$、$N$ 为动点, 满足 $|\\overrightarrow{ON}|=6$, $\\overrightarrow{ON}=\\sqrt{5}\\overrightarrow{OM}$. 过点 $M$ 作 $MM_1 \\perp y$ 轴于 $M_1$, 过点 $N$ 作 $NN_1 \\perp x$ 轴于 $N_1, \\overrightarrow{M_1M}\\neq \\overrightarrow{0}$, $\\overrightarrow{N_1N}\\neq \\overrightarrow{0}$. 设 $\\overrightarrow{OT}=\\overrightarrow{M_1M}+\\overrightarrow{N_1N}$,动点 $T$ 的轨迹为 $\\Omega$, 若 $\\Omega$ 落在某一个圆锥曲线 $\\Gamma$ 上, 且 $\\Gamma$ 上恰有 $n$ 个点不在 $\\Omega$ 上, 则 $n=$\\bracket{20}.\n\\fourch{$1$}{$2$}{$4$}{$6$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期124分守护卷题目", + "edit": [ + "20231123\t毛培菁" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "022836": { + "id": "022836", + "content": "已知长方体 $ABCD-A_1B_1C_1D_1$ 中, $AB=2$, $BC=4$, $AA_1=4$,点 $M$ 是棱 $C_1D_1$ 上的中点. 如图.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\def\\l{2}\n\\def\\m{4}\n\\def\\n{4}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (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 ($(C_1)!0.5!(D_1)$) node [above] {$M$} coordinate (M);\n\\draw [dashed] (A_1)--(D)(B_1)--(D)(M)--(D);\n\\draw (A_1)--(M)--(B_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求三棱锥 $D-A_1B_1M$ 的体积;\\\\\n(2) 求直线 $AB$ 与平面 $DA_1M$ 所成角的大小.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期124分守护卷题目", + "edit": [ + "20231123\t毛培菁" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "022837": { + "id": "022837", + "content": "汽车智能辅助驾驶已开始得到应用, 其自动刹车的工作原理是用雷达测出车辆与前方障碍物之间的距离 (并结合车速转化为所需时间), 当此距离等于报警距离时就开始报警提醒, 等于危险距离时就自动刹车. 某种算法 (如下图所示) 将报警时间划分为 $4$ 段, 分别为准备时间 $t_0$ 、人的反应时间 $t_1$ 、系统反应时间 $t_2$ 、制动时间 $t_3$, 相应的距离分别为 $d_0$、$d_1$、$d_2$、$d_3$. 当车速为 $v$ (米/秒), 且 $v \\in[0,33.3]$时, 通过大数据统计分析得到下表 (其中系数 $k$ 随地面湿滑程度等路面情况而变化, $k \\in[0.5,0.9]$).\n\\begin{center}\n\\begin{tikzpicture}[>=latex,yscale = 0.5]\n\\draw (0,0) -- (9,0);\n\\draw [dashed] (1,-1.5) -- (1,3.1) (2,-1.5) -- (2,1.5) (4,-1.5) -- (4,2.3) (5.5,-1.5) -- (5.5,1.5) (8,-1.5) -- (8,3.1);\n\\draw (8,0) node [above right] {前车};\n\\draw (1,0) node [above left] {后车};\n\\draw [<->] (1,-0.75) -- (2,-0.75) node [midway, fill=white] {$t_0$};\n\\draw [<->] (2,-0.75) -- (4,-0.75) node [midway, fill=white] {$t_1$};\n\\draw [<->] (4,-0.75) -- (5.5,-0.75) node [midway, fill=white] {$t_2$};\n\\draw [<->] (5.5,-0.75) -- (8,-0.75) node [midway, fill=white] {$t_3$};\n\\draw [<->] (1,0.75) -- (2,0.75) node [midway, fill=white] {$d_0$};\n\\draw [<->] (2,0.75) -- (4,0.75) node [midway, fill=white] {$d_1$};\n\\draw [<->] (4,0.75) -- (5.5,0.75) node [midway, fill=white] {$d_2$};\n\\draw [<->] (5.5,0.75) -- (8,0.75) node [midway, fill=white] {$d_3$};\n\\draw [<->] (4,1.9) -- (8,1.9) node [midway, fill=white] {危险距离};\n\\draw [<->] (1,2.7) -- (8,2.7) node [midway, fill=white] {报警距离};\n\\end{tikzpicture}\n\\end{center}\n\\begin{center}\n\\begin{tabular}{|l|c|c|c|c|}\\hline 阶段 & 0、准备 & 1、人的反应 & 2、系统反应 & 3、制动 \\\\\n\\hline 时间 & $t_0$ & $t_1=0.8$ 秒 & $t_2=0.2$ 秒 & $t_3$ \\\\\n\\hline 距离 & $d_0=20$ 米 & $d_1$ & $d_2$ & $d_3=\\dfrac{1}{20 k}v^2$ 米 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1)请写出报警距离 $d$ (米) 与车速 $v$ (米/秒)之间的函数关系式 $d(v)$; 并求 $k=0.9$ 时, 若汽车达到报警距离时人和系统均不采取任何制动措施, 仍以此速度行驶, 则汽车撞上固定障碍物的最短时间. (精确到 $0.1$ 秒)\\\\\n(2) 若要求汽车不论在何种路面情况下行驶, 报警距离均小于 $80$ 米, 则汽车的行驶速度应限制在多少米/秒以下? 合多少千米/小时(精确到 $1$ 千米/小时)?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期124分守护卷题目", + "edit": [ + "20231123\t毛培菁" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "022838": { + "id": "022838", + "content": "设抛物线 $\\Gamma: y^2=4 x$, 直线 $l$ 经过点 $M(m, 0)$, 其中 $m>0, l$ 交 $\\Gamma$ 于两个不同的点 $A$ 和 $B$, 如图.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-0.5,0) -- (8,0) node [below] {$x$};\n\\draw [->] (0,{-sqrt(32)}) -- (0,{sqrt(32)}) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\filldraw (1,0) circle (0.05) node [below] {$F$} coordinate (F);\n\\draw [domain = {-sqrt(32)}:{sqrt(32)},name path= para] plot ({\\x*\\x/4},\\x);\n\\draw [name path=line] (1,5) -- (3,-5);\n\\draw [name intersections = {of = para and line, by = {B,A}}];\n\\draw (A) node [right] {$A$};\n\\draw (B) node [right] {$B$};\n\\draw (2,0) node [above right] {$M$};\n\\end{tikzpicture}\n\\end{center}\n(1) 若 $\\triangle AOM$ 是等边三角形, 求 $m$ 的值;\\\\\n(2) 若 $m=2$, 求证: 原点 $O$ 总在以线段 $AB$ 为直径的圆的内部.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高三上学期124分守护卷题目", + "edit": [ + "20231123\t毛培菁" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "022839": { + "id": "022839", + "content": "已知数列 $\\{a_n\\}$ 的每一项均为非负整数, 满足当 $n=2^k$ ($k$ 是正整数) 时, $a_n=\\dfrac{n}{2}$; 当 $n \\neq 2^k(k$是正整数) 时, $a_n| y|$''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