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录入2024届松江区一模试题

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题库0.3/Problems.json
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"space": "4em",
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"031976": {
"id": "031976",
"content": "已知全集为 $\\mathbf{R}$, 集合 $P=\\{x | x \\geq 1\\}$, 则集合 $\\overline{P}=$\\blank{50}.",
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"genre": "填空题",
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"031977": {
"id": "031977",
"content": "双曲线 $\\dfrac{x^2}{3}-y^2=1$ 的右焦点坐标是\\blank{50}.",
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"genre": "填空题",
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"031978": {
"id": "031978",
"content": "已知复数 $z=2+\\mathrm{i}$ (其中 $\\mathrm{i}$ 是虚数单位), 则 $|\\overline{z}|=$\\blank{50}.",
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"genre": "填空题",
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"031979": {
"id": "031979",
"content": "已知向量 $\\overrightarrow{a}=(1,2)$, $\\overrightarrow{b}=(4,3)$, 则 $\\overrightarrow{a}\\cdot(2 \\overrightarrow{a}-\\overrightarrow{b})=$\\blank{50}.",
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"genre": "填空题",
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"031980": {
"id": "031980",
"content": "已知 $\\sin \\theta=\\dfrac{3}{5}$, $\\theta \\in(0, \\dfrac{\\pi}{2})$, 则 $\\tan (\\theta-\\dfrac{\\pi}{4})$ 的值为\\blank{50}.",
"objs": [],
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"genre": "填空题",
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},
"031981": {
"id": "031981",
"content": "已知 $\\lg a+\\lg b=1$, 则 $a+2 b$ 的最小值为\\blank{50}.",
"objs": [],
"tags": [],
"genre": "填空题",
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"031982": {
"id": "031982",
"content": "在二项式 $(3+x)^n$ 的展开式中, $x^2$ 项的系数是常数项的 $5$ 倍, 则 $n=$\\blank{50}.",
"objs": [],
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"genre": "填空题",
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"031983": {
"id": "031983",
"content": "有 $5$ 名同学报名参加暑期区科技馆志愿者活动, 共服务两天, 每天需要两人参加活动, 则恰有 $1$ 人连续参加两天志愿者活动的概率为\\blank{50}.",
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"genre": "填空题",
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"031984": {
"id": "031984",
"content": "在 $\\triangle ABC$ 中, 设角 $A$、$B$ 及 $C$ 所对边的边长分别为 $a$、$b$ 及 $c$, 若 $a=3$, $c=5$, $B=2A$,则边长 $b=$\\blank{50}.",
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"genre": "填空题",
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"031985": {
"id": "031985",
"content": "已知函数 $f(x)=-x^2+6 x+m$, $g(x)=2 \\sin (2 x+\\dfrac{\\pi}{3})$. 对任意 $x_0 \\in[0, \\dfrac{\\pi}{4}]$, 存在 $x_1, x_2 \\in[-1,3]$, 使得 $f(x_1) \\leq g(x_0) \\leq f(x_2)$, 则实数 $m$ 的取值范围是\\blank{50}.",
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"031986": {
"id": "031986",
"content": "若函数 $y=f(x)$ 是定义在 $\\mathbf{R}$ 上的不恒为零的偶函数, 且对任意实数 $x$ 都有 $x \\cdot f(x+2)=(x+2) \\cdot f(x)+2$, 则 $f(2023)=$\\blank{50}.",
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"genre": "填空题",
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"solution": "",
"duration": -1,
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"origin": "2024届松江区一模试题11",
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],
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"remark": "",
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},
"031987": {
"id": "031987",
"content": "已知正四面体 $A-BCD$ 的棱长为 $2 \\sqrt{2}$, 空间内任意点 $P$ 满足 $|\\overrightarrow{PB}+\\overrightarrow{PC}|=2$, 则 $\\overrightarrow{AP}\\cdot \\overrightarrow{AD}$ 的取值范围是\\blank{50}.",
"objs": [],
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"genre": "填空题",
"ans": "",
"solution": "",
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"origin": "2024届松江区一模试题12",
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],
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"031988": {
"id": "031988",
"content": "英国数学家哈利奥特最先使用``$<$''和``$>$''符号, 并逐渐被数学界接受, 不等号的引入对不等式的发展影响深远. 对于任意实数 $a$、$b$、$c$、$d$, 下列命题是真命题的是\\bracket{20}.\n\\twoch{若 $a^2<b^2$, 则 $a<b$}{若 $a<b$, 则 $a c<b c$}{若 $a<b$, $c<d$, 则 $a c<b d$}{若 $a<b$, $c<d$, 则 $a+c<b+d$}",
"objs": [],
"tags": [],
"genre": "选择题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "2024届松江区一模试题13",
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"031989": {
"id": "031989",
"content": "如图所示的茎叶图记录了甲、乙两支篮球队各 $6$ 名队员某场比赛的得分数据 (单位: 分).则下列说法正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tabular}{ll|l|ll}\n\\multicolumn{2}{c|}{甲队} & & \\multicolumn{2}{c}{乙队}\\\\\n\\hline & 7 & 0 & 8 & 9 \\\\\n2 & 6 & 1 & 9 & 7 \\\\\n0 & 2 & 2 & 7 & 8 \\\\\n& 1 & 3 & \n\\end{tabular}\n\\end{center}\n\\twoch{甲队数据的中位数大于乙队数据的中位数}{甲队数据的平均值小于乙队数据的平均值}{甲队数据的标准差大于乙队数据的标准差}{乙队数据的第 $75$ 百分位数为 $27$}",
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"origin": "2024届松江区一模试题14",
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"031990": {
"id": "031990",
"content": "函数 $y=f(x)$ 的图像如图所示, $y=f'(x)$ 为函数 $y=f(x)$ 的导函数, 则不等式 $\\dfrac{f'(x)}{x}<0$ 的解集为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-4,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-4,2) .. controls (-3.8,1) and (-3.2,0.5) .. (-3,0.5) .. controls (-2.6,0.5) and (-1.8,1.5) .. (-1,1.5) .. controls (-0.6,1.5) and (0.6,0.3) .. (1,0.3) .. controls (1.3,0.3) and (2.7,1.5) .. (3,2.3);\n\\draw [dashed] (-3,0.5) -- (-3,0) node [below] {$-3$};\n\\draw [dashed] (-1,1.5) -- (-1,0) node [below] {$-1$};\n\\draw [dashed] (1,0.3) -- (1,0) node [below] {$1$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$(-3,-1)$}{$(0,1)$}{$(-3,-1) \\cup(0,1)$}{$(-\\infty,-3) \\cup$($1,+\\infty$)}",
"objs": [],
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"genre": "选择题",
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"origin": "2024届松江区一模试题15",
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"031991": {
"id": "031991",
"content": "关于曲线 $M: x^{\\frac{1}{2}}+y^{\\frac{1}{2}}=1$, 有下述两个结论: \\textcircled{1} 曲线 $M$ 上的点到坐标原点的距离最小值是 $\\dfrac{\\sqrt{2}}{2}$; \\textcircled{2} 曲线 $M$ 与坐标轴围成的图形的面积不大于 $\\dfrac{1}{2}$, 则下列说法正确的是\\bracket{20}.\n\\fourch{\\textcircled{1}、\\textcircled{2}都正确}{\\textcircled{1}正确 \\textcircled{2}错误}{\\textcircled{1}错误 \\textcircled{2}正确}{\\textcircled{1}、\\textcircled{2}都错误}",
"objs": [],
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"genre": "选择题",
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"duration": -1,
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"origin": "2024届松江区一模试题16",
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"031992": {
"id": "031992",
"content": "如图, 在四棱锥 $P-ABCD$ 中, $PA \\perp$ 底面 $ABCD$, $AB \\perp AD$, 点 $E$ 在线段 $AD$ 上, 且 $CE \\parallel AB$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (3,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,0,1.5) node [left] {$B$} coordinate (B);\n\\draw (2,0,1.5) node [below] {$C$} coordinate (C);\n\\draw (2,0,0) node [above] {$E$} coordinate (E);\n\\draw (0,1.5,0) node [above] {$P$} coordinate (P);\n\\draw (P)--(B)--(C)--(D)--cycle(P)--(C);\n\\draw [dashed] (B)--(A)--(D)(A)--(P)(P)--(E)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $CE \\perp$ 平面 $PAD$;\\\\\n(2) 若四棱锥 $P-ABCD$ 的体积为 $\\dfrac{5}{6}$, $AB=1$, $AD=3$, $CD=\\sqrt{2}$, $\\angle CDA=45^{\\circ}$, 求二面角 $P-CE-A$ 的大小.",
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"space": "4em",
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"031993": {
"id": "031993",
"content": "已知数列 $\\{a_n\\}$ 为等差数列, $\\{b_n\\}$ 是公比为 $2$ 的等比数列, 且 $a_2-b_2=a_3-b_3=b_4-a_4$.\\\\\n(1) 证明: $a_1=b_1$ :\\\\\n(2) 若集合 $M=\\{k | b_k=a_m+a_1, 1 \\leq m \\leq 50\\}$, 求集合 $M$ 中的元素个数.",
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"duration": -1,
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"origin": "2024届松江区一模试题18",
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"remark": "",
"space": "4em",
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"031994": {
"id": "031994",
"content": "为了鼓励居民节约用气, 某市对燃气收费实行阶梯计价, 普通居民燃气收费标准如下:第一档: 年用气量在 $0$-$310$ (含) 立方米, 价格为 $a$ 元/立方米;\n第二档: 年用气量在 $310$-$520$ (含) 立方米, 价格为 $b$ 元/立方米;\n第三档: 年用气量在 $520$ 立方米以上, 价格为 $c$ 元/立方米.\\\\\n(1) 请写出普通居民的年度燃气费用 (单位: 元) 关于年度的燃气用量 (单位: 立方米)的函数解析式 (用含 $a$、$b$、$c$ 的式子表示);\\\\\n(2) 已知某户居民 2023 年部分月份用气量与缴费情况如下表, 求 $a$、$b$、$c$ 的值.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\\hline 月份 & 1 & 2 & 3 & 4 & 5 & 9 & 10 & 12 \\\\\n\\hline 当月燃气用量 (立方米) & 56 & 80 & 66 & 58 & 60 & 53 & 55 & 63 \\\\\n\\hline 当月燃气费 (元) & 168 & 240 & 198 & 174 & 183 & 174.9 & 186 & 264.6 \\\\\n\\hline\n\\end{tabular}\n\\end{center}",
"objs": [],
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"031995": {
"id": "031995",
"content": "已知椭圆 $\\Gamma: \\dfrac{y^2}{a^2}+\\dfrac{x^2}{b^2}=1$($a>b>0$) 的离心率为 $\\dfrac{\\sqrt{2}}{2}$, 其上焦点 $F$ 与抛物线 $K: x^2=4 y$ 的焦点重合.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1.6) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [name path = elli] (0,0) ellipse (1 and {sqrt(2)});\n\\filldraw (0,1) circle (0.03) node [below right] {$F$} coordinate (F);\n\\draw [name path = para, domain = -1.4:1.4] plot (\\x,{\\x*\\x});\n\\draw [name path = line] (2,1.9) -- (-2,0.1);\n\\draw [name intersections = {of = line and para, by = {D,C}}];\n\\draw [name intersections = {of = line and elli, by = {A,B}}];\n\\foreach \\i in {A,B,C,D}\n{\\filldraw (\\i) circle (0.03);};\n\\draw (A) node [above] {$A$};\n\\draw (B) node [left] {$B$};\n\\draw (C) node [below] {$C$};\n\\draw (D) node [right] {$D$};\n\\draw (0,-1.6) node [below] {图1};\n\\end{tikzpicture}\n\\hspace*{3em}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1.6) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [name path = elli] (0,0) ellipse (1 and {sqrt(2)});\n\\filldraw (0,1) circle (0.03) node [below right] {$F$} coordinate (F);\n\\draw [name path = para, domain = -1.4:1.4] plot (\\x,{\\x*\\x});\n\\draw [name path = lineEG] (2,1.9) -- (-2,0.1);\n\\draw [name path = lineAB] (-0.5,1.95) -- (1,-0.9);\n\\draw [name intersections = {of = lineEG and para, by = {E,G}}];\n\\draw [name intersections = {of = lineAB and elli, by = {A,B}}];\n\\foreach \\i in {A,B,E,G}\n{\\filldraw (\\i) circle (0.03);};\n\\draw (A)--(E)--(B)--(G)--cycle;\n\\draw (F) pic [draw, scale = 0.2] {right angle = A--F--E};\n\\draw (A) node [above left] {$A$};\n\\draw (B) node [right] {$B$};\n\\draw (E) node [left] {$E$};\n\\draw (G) node [below right] {$G$};\n\\draw (0,-1.6) node [below] {图2};\n\\end{tikzpicture}\n\\end{center}\n(注, 图仅作为示意, 大小和形状可能不正确)\\\\\n(1) 求椭圆 $\\Gamma$ 的方程;\\\\\n(2) 若过点 $F$ 的直线交椭圆 $\\Gamma$ 于点 $A$、$B$, 同时交抛物线 $K$ 于点 $C$、$D$ (如图 1 所示, 点 $B,D,A,C$在直线$l$上按从左至右的顺序排列), 试比较线段 $AC$ 与 $BD$ 长度的大小, 并说明理由;\\\\\n(3) 若过点 $F$ 的直线交椭圆 $\\Gamma$ 于点 $A$、$B$, 过点 $F$ 与直线 $AB$ 垂直的直线 $EG$ 交抛物线 $K$于点 $E$、$G$ (如图 2 所示), 试求四边形 $AEBG$ 面积的最小值.",
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"031996": {
"id": "031996",
"content": "已知函数 $y=f(x)$, 记 $f(x)=x+\\sin x$, $x \\in D$.\\\\\n(1) 若 $D=[0,2 \\pi]$, 判断函数的单调性;\\\\\n(2) 若 $D=(0, \\dfrac{\\pi}{2}]$, 不等式 $f(x)>k x$ 对任意 $x \\in D$ 恒成立, 求实数 $k$ 的取值范围;\\\\\n(3) 若 $D=\\mathbf{R}$, 则曲线 $y=f(x)$ 上是否存在三个不同的点 $A$、$B$、$C$, 使得曲线 $y=f(x)$在 $A$、$B$、$C$ 三点处的切线互相重合? 若存在, 求出所有符合要求的切线的方程; 若不存在, 请说明理由.",
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"040001": {
"id": "040001",
"content": "参数方程$\\begin{cases}x=3 t^2+4, \\\\ y=t^2-2\\end{cases}$($0 \\leq t \\leq 3$)所表示的曲线是\\bracket{20}.\n\\fourch{一支双曲线}{线段}{圆弧}{射线}",