做了目前为止题目的单元挂钩
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@ -734407,7 +734407,9 @@
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"id": "032089",
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"content": "设不等式 $a \\leq|x-5|+|x-3|$ 对所有实数 $x$ 恒成立, 则实数 $a$ 的取值范围为\\blank{50}.",
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"objs": [],
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"tags": [],
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"tags": [
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"第一单元"
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],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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@ -734427,7 +734429,9 @@
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"id": "032090",
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"content": "如图, 在四棱锥 $P-ABCD$ 中, $PA \\perp$ 平面 $ABCD$,底面 $ABCD$ 是矩形, $|AP|=|AB|=2$, $|AD|=4$, $E$ 是 $BC$ 上的点, 直线 $PB$ 与平面 $PDE$ 所成的角是 $\\arcsin \\dfrac{\\sqrt{3}}{6}$, 则 $BE$ 的长为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (4,0,0) node [right] {$D$} coordinate (D);\n\\draw (4,0,2) node [below] {$C$} coordinate (C);\n\\draw (0,0,2) node [below] {$B$} coordinate (B);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw ($(B)!0.5!(C)$) node [below] {$E$} coordinate (E);\n\\draw (P)--(B)--(C)--(D)--cycle(P)--(C);\n\\draw [dashed] (B)--(A)--(D)(P)--(A);\n\\draw (P)--(E);\n\\draw [dashed] (E)--(D);\n\\end{tikzpicture}\n\\end{center}",
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"objs": [],
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"tags": [],
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"tags": [
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"第六单元"
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],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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@ -734447,7 +734451,9 @@
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"id": "032091",
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"content": "不等式 $\\log _2 x+\\dfrac{x}{2}<4$ 的解集为\\blank{50}.",
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"objs": [],
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"tags": [],
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"tags": [
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"第二单元"
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],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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@ -734467,7 +734473,9 @@
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"id": "032092",
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"content": "在国家开发西部的号召下, 某西部企业得到了一笔 400 万元的无息贷款用做设备更新. 据预测, 该企业设备更新后, 第 1 个月收入为 20 万元, 在接下来的 5 个月中, 每月收入都比上个月增长 $20 \\%$, 从第 7 个月开始, 每个月的收入都比前一个月增加 2 万元. 则从新设备使用开始计算, 该企业用所得收入偿还 400 万无息贷款只需\\blank{50}个月. (结果取整)",
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"objs": [],
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"tags": [],
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"tags": [
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"第二单元"
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],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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@ -734487,7 +734495,9 @@
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"id": "032093",
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"content": "记 $f(x)=\\ln x+x^2-2 k x+k^2$, 若存在实数 $a$、$b$, 满足 $\\dfrac{1}{2}\\leq a<b \\leq 2$, 使得函数 $y=f(x)$ 在区间 $[a, b]$ 上是严格增函数, 则实数 $k$ 的取值范围是\\blank{50}.",
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"objs": [],
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"tags": [],
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"tags": [
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"第二单元"
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],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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@ -734507,7 +734517,9 @@
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"id": "032094",
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"content": "记点 $P$ 到图形 $C$ 上每一个点的距离的最小值称为点 $P$ 到图形 $C$ 的距离, 那么平面内到定圆 $C$ 的距离与到定点 $A$ 的距离相等的点的轨迹不可能是\\bracket{20}.\n\\fourch{直线}{圆}{椭圆}{双曲线的一支}",
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"objs": [],
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"tags": [],
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"tags": [
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"第七单元"
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],
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"genre": "选择题",
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"ans": "",
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"solution": "",
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@ -734529,7 +734541,9 @@
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"id": "032095",
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"content": "记 $f(x)=\\sin ^2 x-\\cos ^2 x+2 \\sqrt{3}\\sin x \\cos x+\\lambda$($x \\in \\mathbf{R}$), 其中 $\\lambda$ 为实常数.\\\\\n(1) 求函数 $y=f(x)$ 的最小正周期;\\\\\n(2) 若函数 $y=f(x)$ 的图像经过点 $(\\dfrac{\\pi}{2}, 0)$, 求其在区间 $[0, \\dfrac{2}{3}\\pi]$ 上的最大值和最小值.",
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"objs": [],
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"tags": [],
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"tags": [
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"第三单元"
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],
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"genre": "解答题",
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"ans": "",
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"solution": "",
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@ -734549,7 +734563,9 @@
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"id": "032096",
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"content": "如果函数 $y=f(x)$ 满足以下两个条件``(1) 对任意的 $x \\in(0,1)$, 总有 $f(x)>0$;\\\\\n(2) 当 $x_1>0$, $x_2>0$, $x_1+x_2<1$ 时, 总有 $f(x_1+x_2)<f(x_1)+f(x_2)$ 成立'', 我们就称 $y=f(x)$ 为 $L$ 型函数:\\\\\n(1) 记 $g(x)=x^2+\\dfrac{1}{2}$, 求证: $y=g(x)$ 为 $L$ 型函数;\\\\\n(2) 设 $b \\in \\mathbf{R}$, 记 $p(x)=\\ln (x+b)$, 若 $y=p(x)$ 是 $L$ 型函数, 求 $b$ 的取值范围;\\\\\n(3) 是否存在 $L$ 型函数 $y=r(x)$ 满足: 对于任意的 $m \\in(0,4)$, 都存在 $x_0 \\in(0,1)$, 使得等式 $r(x_0)=m$ 成立? 请说明理由.",
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"objs": [],
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"tags": [],
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"tags": [
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"第二单元"
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],
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"genre": "解答题",
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"ans": "",
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"solution": "",
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@ -734569,7 +734585,9 @@
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"id": "032097",
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"content": "已知三个互不相同的实数 $a$、$b$、$c$ 满足 $a+b+c=1$, $a^2+b^2+c^2=3$, 则 $a b c$ 的取值范围为\\blank{50}.",
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"objs": [],
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"tags": [],
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"tags": [
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"第一单元"
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],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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@ -734589,7 +734607,9 @@
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"id": "032098",
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"content": "已知椭圆 $\\Gamma$ 的离心率是 $\\dfrac{1}{2}$, 长轴长 $4$, 椭圆的中心是坐标原点, 焦点在 $x$ 轴上.\n \\begin{center}\n \\begin{tikzpicture}[>=latex]\n \\draw [->] (-2.5,0) -- (2.5,0) node [below] {$x$};\n \\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\draw (0,0) ellipse (2 and {sqrt(3)});\n \\filldraw (-1,0) circle (0.03) (1,0) circle (0.03) node [below] {$F$};\n \\end{tikzpicture}\n \\end{center}\n (1) 求椭圆 $\\Gamma$ 的标准方程;\\\\\n (2) 已知 $A, B, C$ 是椭圆 $\\Gamma$ 上三个不同的点, $F$ 是椭圆 $\\Gamma$ 的右焦点, 若原点 $O$ 是 $\\triangle ABC$ 的重心, 求 $|FA|+|FB|+|FC|$ 的值;\\\\\n (3) 已知 $T(1,1)$, 椭圆 $\\Gamma$ 四个动点 $M, N, P, Q$ 满足 $\\overrightarrow{MT}=3 \\overrightarrow{TQ}$, $\\overrightarrow{NT}=3 \\overrightarrow{TP}$,求直线 $MN$ 的方程.",
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"objs": [],
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"tags": [],
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"tags": [
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"第七单元"
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],
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"genre": "解答题",
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"ans": "",
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"solution": "",
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@ -759501,7 +759521,9 @@
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"id": "040971",
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"content": "椭圆 $\\dfrac{x^2}{16}+\\dfrac{y^2}{9}=1$ 的焦距是\\blank{50}, 焦点坐标是\\blank{50}.",
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"objs": [],
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"tags": [],
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"tags": [
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"第七单元"
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],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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@ -759533,7 +759555,9 @@
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"id": "040972",
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"content": "已知 $A(0,3)$、$B(0,-3)$ 两点. 若动点 $P$ 满足 $|PA|+|PB|=6$, 则点 $P$ 的轨迹方程是\\blank{50}.",
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"objs": [],
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"tags": [],
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"tags": [
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"第七单元"
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],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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@ -759563,7 +759587,9 @@
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"id": "040973",
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"content": "已知 $A(0,3)$、$B(0,-3)$ 两点. 若动点 $P$ 满足 $|PA|+|PB|=8$, 则点 $P$ 的轨迹为\\bracket{20}.\n\\fourch{焦点在 $x$ 轴上的椭圆}{焦点在 $y$ 轴上的椭圆}{$x$ 轴上的线段}{$y$ 轴上的线段}",
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"objs": [],
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"tags": [],
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"tags": [
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"第七单元"
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],
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"genre": "选择题",
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"ans": "",
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"solution": "",
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@ -759593,7 +759619,9 @@
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"id": "040974",
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"content": "已知 $A(0,3)$、$B(0,-3)$ 两点. 若动点 $P$ 满足 $|PA|+|PB|=8$, 则点 $P$ 的轨迹方程是\\blank{50}.",
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"objs": [],
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"tags": [],
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"tags": [
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"第七单元"
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],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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@ -759623,7 +759651,9 @@
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"id": "040975",
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"content": "``$m>2$''是``方程 $\\dfrac{x^2}{m-2}+\\dfrac{y^2}{5-m}=1$ 表示的曲线是椭圆''的\\bracket{20}.\n\\fourch{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}",
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"objs": [],
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"tags": [],
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"tags": [
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"第七单元"
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],
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"genre": "选择题",
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"ans": "",
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"solution": "",
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@ -759653,7 +759683,9 @@
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"id": "040976",
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"content": "椭圆的中心在原点, 焦距为 6 , 且经过点 $(0,4)$, 则它的标准方程是\\blank{50}.",
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"objs": [],
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"tags": [],
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"tags": [
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"第七单元"
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],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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@ -759683,7 +759715,9 @@
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"id": "040977",
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"content": "写出分别满足下列条件的椭圆的标准方程:\\\\\n(1) 焦点坐标为 $(-6,0)$、$(6,0)$, 且椭圆经过点 $(0,8)$.\\\\\n(2) 椭圆经过 $(0,-2)$、$(\\sqrt{6}, 0)$ 两点.",
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"objs": [],
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"tags": [],
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"tags": [
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"第七单元"
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],
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"genre": "解答题",
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"ans": "",
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"solution": "",
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@ -759715,7 +759749,9 @@
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"id": "040978",
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"content": "(1) 已知方程 $\\dfrac{x^2}{k-4}+\\dfrac{y^2}{10-k}=1$ 表示圆, 求实数 $k$ 的值;\\\\\n(2) 已知方程 $\\dfrac{x^2}{k-4}+\\dfrac{y^2}{10-k}=1$ 表示焦点在 $y$ 轴上的椭圆, 求实数 $k$ 的取值范围.",
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"objs": [],
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"tags": [],
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"tags": [
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"第七单元"
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],
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"genre": "解答题",
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"ans": "",
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"solution": "",
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@ -759745,7 +759781,9 @@
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"id": "040979",
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"content": "若椭圆的方程为 $16 x^2+25 y^2=400$, 则此椭圆的长半轴长为\\blank{50}, 短轴长为\\blank{50}, 焦距为\\blank{50}, 离心率为\\blank{50}.",
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"objs": [],
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"tags": [],
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"tags": [
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"第七单元"
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],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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@ -759777,7 +759815,9 @@
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"id": "040980",
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"content": "已知动点 $M$ 到定点 $A(-\\dfrac{9}{4}, 0)$ 与 $B(\\dfrac{9}{4}, 0)$ 的距离的和是 $\\dfrac{25}{2}$, 则点 $M$ 的轨迹方程是\\blank{50}.",
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"objs": [],
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"tags": [],
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"tags": [
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"第七单元"
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],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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@ -759809,7 +759849,9 @@
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"id": "040981",
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"content": "已知椭圆 $C$ 的两个焦点分别为 $F_1(-3,0)$、$F_2(3,0)$, 若其离心率为\\blank{50}, 则椭圆 $C$ 的方程为 $\\dfrac{x^2}{25}+\\dfrac{y^2}{16}=1$.",
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"objs": [],
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"tags": [],
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"tags": [
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"第七单元"
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],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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@ -759841,7 +759883,9 @@
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"id": "040982",
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"content": "焦距为 6 的椭圆上一点到其两个焦点的距离之和为 10 , 则此椭圆的标准方程为\\blank{50}.",
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"objs": [],
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"tags": [],
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"tags": [
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"第七单元"
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],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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@ -759873,7 +759917,9 @@
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"id": "040983",
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"content": "若椭圆 $\\dfrac{x^2}{25}+\\dfrac{y^2}{9}=1$ 的两个焦点分别为 $F_1$、$F_2$, 点 $P$ 为此椭圆上的任意一点, 则 $\\triangle PF_1F_2$的周长为\\blank{50}.",
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"objs": [],
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"tags": [],
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"tags": [
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"第七单元"
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],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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@ -759905,7 +759951,9 @@
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"id": "040984",
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"content": "如果椭圆 $5 x^2+k y^2=5$ 的一个焦点是 $(0,2)$, 那么 $k=$\\blank{50}.",
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"objs": [],
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"tags": [],
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"tags": [
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"第七单元"
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],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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@ -759935,7 +759983,9 @@
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"id": "040985",
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"content": "已知点 $P$ 是椭圆 $\\dfrac{x^2}{36}+\\dfrac{y^2}{20}=1$ 上一个动点, $F_1$ 是椭圆的左焦点,\\\\\n(1) 求 $|PF_1|$ 的最大值;\\\\\n(2) 求 $|PF_1|$ 的最小值.",
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"objs": [],
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"tags": [],
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"tags": [
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"第七单元"
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],
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"genre": "解答题",
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"ans": "",
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"solution": "",
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@ -759967,7 +760017,9 @@
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"id": "040986",
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"content": "写出分别满足下列条件的椭圆的标准方程:\\\\\n(1) 焦距等于 4 , 且经过点 $P(\\dfrac{2 \\sqrt{6}}{3},-\\dfrac{2 \\sqrt{6}}{3})$.\\\\\n(2) 过 $(-\\dfrac{3}{2}, \\dfrac{5}{2})$ 与 $(\\sqrt{3}, \\sqrt{5})$ 两点.\\\\\n(3) 长轴长是短轴长的 2 倍, 且过点 $(-2,-4)$.\\\\\n(4) 椭圆的一个顶点和一个焦点分别是直线 $x+3 y-6=0$ 与两坐标轴的交点.\\\\\n(5) 已知椭圆在 $x$ 轴上的一个焦点 $F$ 与短轴 $B_1B_2$ 两端点的连线互相垂直, 且点 $F$ 和长轴上较近的端点 $A$ 的距离是 $\\sqrt{10}-\\sqrt{5}$.",
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"objs": [],
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"tags": [],
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"tags": [
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"第七单元"
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],
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"genre": "解答题",
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"ans": "",
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"solution": "",
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@ -759997,7 +760049,9 @@
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"id": "040987",
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"content": "在平面直角坐标系中, 椭圆 $C$ 的中心为原点, 焦点 $F_1$、$F_2$ 在 $x$ 轴上, 且 $\\dfrac{c}{a}=\\dfrac{\\sqrt{2}}{2}$, 如果经过 $F_1$ 的直线 $l$ 交椭圆 $C$ 于 $A$、$B$ 两点, 且 $\\triangle ABF_2$ 的周长为 16 , 那么椭圆 $C$ 的方程为\\blank{50}.",
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"objs": [],
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"tags": [],
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"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760027,7 +760081,9 @@
|
|||
"id": "040988",
|
||||
"content": "已知两点 $B(-3,0)$、$C(3,0)$. 若 $\\triangle ABC$ 的周长为 20 , 则顶点 $A$ 的轨迹方程为\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760057,7 +760113,9 @@
|
|||
"id": "040989",
|
||||
"content": "已知椭圆 $\\dfrac{x^2}{9}+\\dfrac{y^2}{2}=1$ 的焦点为 $F_1$、$F_2$, 点 $P$ 在椭圆上, 若 $|PF_1|=4$, 则 $|PF_2|=\\blank{50}, \\angle F_1PF_2$ 的大小为\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760089,7 +760147,9 @@
|
|||
"id": "040990",
|
||||
"content": "在 $\\triangle ABC$ 中, 已知 $A(-1,0)$、$C(1,0)$. 若 $a>b>c$, 且满足 $2 \\sin B=\\sin A+\\sin C$, 则顶点 $B$ 的轨迹的方程是\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760121,7 +760181,9 @@
|
|||
"id": "040991",
|
||||
"content": "已知椭圆 $\\dfrac{x^2}{12}+\\dfrac{y^2}{3}=1$ 的焦点为 $F_1$、$F_2$, 点 $P$ 在此椭圆上.若线段 $PF_1$ 的中点 $M$ 恰在 $y$轴上, 则 $|PF_1|$ 是 $|PF_2|$ 的\\bracket{20}.\n\\fourch{7 倍}{5 倍}{4 倍}{3 倍}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760151,7 +760213,9 @@
|
|||
"id": "040992",
|
||||
"content": "$\\triangle ABC$ 的两个顶点 $A$、$B$ 的坐标分别是 $(-6,0)$、$(6,0), AC$、$BC$ 边所在直线的斜率之积等于 $-\\dfrac{4}{9}$, 求顶点 $C$ 的轨迹方程.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760183,7 +760247,9 @@
|
|||
"id": "040993",
|
||||
"content": "已知 $P$ 是椭圆 $\\dfrac{x^2}{25}+\\dfrac{y^2}{16}=1$ 上的一点, $F_1, F_2$ 是椭圆的两个焦点, 且 $\\angle F_1PF_2=30^{\\circ}$,求 $\\Delta F_1PF_2$ 的面积.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760213,7 +760279,9 @@
|
|||
"id": "040994",
|
||||
"content": "椭圆 $\\dfrac{x^2}{25}+\\dfrac{y^2}{9}=1$ 上一点 $M$ 到左焦点 $F_1$ 的距离为 $2, N$ 是 $MF_1$ 的中点, $O$ 是坐标原点,则 $|ON|$ 的长为\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760235,7 +760303,9 @@
|
|||
"id": "040995",
|
||||
"content": "已知椭圆 $\\dfrac{x^2}{2}+y^2=1$.\\\\\n(1) 求过点 $M(\\dfrac{1}{2}, \\dfrac{1}{2})$ 且被 $M$ 平分的弦所在直线方程;\\\\\n(2) 求斜率为 2 的平行弦中点 $P$ 的轨迹方程;\\\\\n(3) 过椭圆的左焦点 $F_1$ 引椭圆的割线, 求截得的弦的中点 $Q$ 的轨迹方程.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760257,7 +760327,9 @@
|
|||
"id": "040996",
|
||||
"content": "直线 $y=k x-2$ 交椭圆 $\\dfrac{x^2}{80}+\\dfrac{y^2}{20}=1$ 于不同两点 $P$ 和 $Q$, 若弦 $PQ$ 的中点的横坐标等于 2 ,则弦长 $|PQ|=$\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760279,7 +760351,9 @@
|
|||
"id": "040997",
|
||||
"content": "若 $AB$ 为过椭圆 $\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$) 中心的弦, $F_1$ 为此椭圆的焦点, 则 $\\Delta F_1AB$ 的面积的最大值为\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760301,7 +760375,9 @@
|
|||
"id": "040998",
|
||||
"content": "已知 $F_1$、$F_2$ 分别为椭圆 $\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$) 的左、右焦点, 点 $P$ 在椭圆上, $O$ 为坐标原点, $\\triangle POF_2$ 是面积为 $4 \\sqrt{3}$ 的正三角形, 则 $b^2$ 的值为\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760323,7 +760399,9 @@
|
|||
"id": "040999",
|
||||
"content": "填空题:\\\\\n(1) 若 $P$ 是双曲线 $x^2-y^2=9$ 的左支上一点, $F_1$、$F_2$ 分别是双曲线的左、右焦点, 则 $|PF_1|-|PF_2|=$\\blank{50}.\\\\\n(2) 设 $a$ 是正实数. 已知点 $P$ 与 $A(-a, 0)$、$B(a, 0)$ 两定点的连线的斜率之积为定值 $t$($t>0$), 则点 $P$ 的轨迹方程是\\blank{50}.\\\\\n(3) 若椭圆 $\\dfrac{x^2}{4}+\\dfrac{y^2}{2 k^2}=1$ 与双曲线 $\\dfrac{x^2}{k^2}-\\dfrac{y^2}{2}=1$ 的焦点相同, 则正数 $k$ 的值是\\blank{50}.\\\\\n(4) 若椭圆 $\\dfrac{x^2}{4}+\\dfrac{y^2}{2 k^2}=1$ 与双曲线 $\\dfrac{x^2}{k^2}-\\dfrac{y^2}{2}=1$ 的焦距相同, 则正数 $k$ 的值是\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760343,7 +760421,9 @@
|
|||
"id": "041000",
|
||||
"content": "选择题:\\\\\n(1) 已知 $F_1(0,-5)$、$F_2(0,5)$ 两点, 若动点 $P$ 满足 $|PF_1|-|PF_2|=8$, 则点 $P$ 的轨迹是\\bracket{20}.\n\\fourch{双曲线}{双曲线靠近 $F_1$ 的一支}{双曲线靠近 $F_2$ 的一支}{一条射线}\\\\\n(2) 在 $\\triangle ABC$ 中, 已知 $A(-4,0)$、$B(4,0)$ 两点. 若 $\\sin A-\\sin B=\\dfrac{1}{2}\\sin C$, 则顶点 $C$ 的轨迹方程是\\bracket{20}.\n\\fourch{$\\dfrac{x^2}{4}-\\dfrac{y^2}{12}=1$($x<-2$)}{$\\dfrac{x^2}{4}-\\dfrac{y^2}{12}=1$($x>2$)}{$\\dfrac{x^2}{12}-\\dfrac{y^2}{4}=1$($x>2 \\sqrt{3}$)}{$\\dfrac{x^2}{12}-\\dfrac{y^2}{4}=1$($y \\neq 0$)}\\\\\n(3) 已知椭圆 $\\dfrac{x^2}{m^2}+\\dfrac{y^2}{n^2}=1$($|m|>|n|$) 和双曲线 $\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$ 相同的两焦点 $F_1$、$F_2$, 若点 $P$ 为两曲线的一个交点, 则 $|PF_1| \\cdot|PF_2|$ 等于\\bracket{20}.\n\\fourch{$m^2-a^2$}{$\\dfrac{1}{2}(m^2-a^2)$}{$m-n$}{$a^2-m^2$}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760363,7 +760443,9 @@
|
|||
"id": "041001",
|
||||
"content": "填空题:\\\\\n(1) 如果中心在原点, 对称轴在坐标轴上的等轴双曲线的一个焦点为 $F_1(0,-6)$, 那么此双曲线的标准方程是\\blank{50}, 离心率为\\blank{50}.\\\\\n(2) 双曲线 $2 x^2-y^2=8$ 的焦点坐标是\\blank{50}, 两条渐近线的夹角为\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760385,7 +760467,9 @@
|
|||
"id": "041002",
|
||||
"content": "选择题:\\\\\n(1) 若双曲线的中心在坐标原点, 它的一个焦点的坐标是 $(-5,0)$, 两个顶点间的距离为 6 , 则此双曲线的方程是\\bracket{20}.\n\\fourch{$\\dfrac{x^2}{9}-\\dfrac{y^2}{16}=1$}{$\\dfrac{x^2}{36}-\\dfrac{y^2}{11}=1$}{$\\dfrac{x^2}{16}-\\dfrac{y^2}{9}=1$ }{$\\dfrac{x^2}{11}-\\dfrac{y^2}{36}=1$}\\\\\n(2) 在下列双曲线中, 以 $y= \\pm \\dfrac{1}{2}x$ 为渐近线的是\\bracket{20}.\n\\fourch{$\\dfrac{x^2}{16}-\\dfrac{y^2}{4}=1$ ,}{$\\dfrac{x^2}{4}-\\dfrac{y^2}{16}=1$ }{$\\dfrac{x^2}{2}-y^2=1$ ,}{$x^2-\\dfrac{y^2}{2}=1$}\\\\\n(3) 若方程 $4 x^2+k y^2=4 k$ 表示双曲线, 则此双曲线的虚轴长等于\n\\fourch{$2 \\sqrt{k}$}{$2 \\sqrt{-k}$}{$\\sqrt{k}$}{$\\sqrt{-k}$}\\\\\n(4) 已知双曲线 $\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$) 的两焦点为 $F_1$、$F_2$, 若弦 $AB$ 经过点 $F_1$, 且\n$A$、$B$ 均在此双曲线的左支上, $|AB|=l$, 则 $\\triangle ABF_2$ 的周长为\\bracket{20}.\n\\fourch{$4 a+l$ }{$4 a+2 l$}{$2 a+l$ }{$4 a-l$}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760405,7 +760489,9 @@
|
|||
"id": "041003",
|
||||
"content": "(1)求以椭圆 $\\dfrac{x^2}{8}+\\dfrac{y^2}{5}=1$ 的焦点为顶点, 以椭圆的顶点为焦点的双曲线的方程.\\\\\n(2) 已知双曲线的虚轴的长为 6 , 一条渐近线的方程为 $3 x-y=0$, 求此双曲线的标准方程.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760427,7 +760513,9 @@
|
|||
"id": "041004",
|
||||
"content": "填空题:\\\\\n(1) 已知双曲线 $C$ 的中心在坐标原点, 若它的一个焦点为 $(3,0)$, 一条渐近线的方程为 $2 x-3 y=0$, 则此双曲线的方程为\\blank{50}.\\\\\n(2) 若双曲线的两渐近线的夹角为 $\\dfrac{\\pi}{3}$, 则焦距与实轴长之比 $\\dfrac{c}{a}=$\\blank{50}.\\\\\n(3) 已知 $F_1(-3,0)$、$F_2(3,0)$ 两点, 满足条件 $|MF_1|+|MF_2|=2 m+1$ 的动点 $M$ 的轨迹是椭圆, 满足条件 $|NF_1|-|NF_2|=2 m-1$ 的动点 $N$ 的轨迹是双曲线, 则实数 $m$ 的取值范围是\\blank{50}.\\\\\n(4) 若双曲线 $\\dfrac{x^2}{4}-\\dfrac{y^2}{m}=1$ 的渐近线方程为 $y= \\pm \\dfrac{\\sqrt{3}}{2}x$, 则此双曲线的焦点坐标是\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760447,7 +760535,9 @@
|
|||
"id": "041005",
|
||||
"content": "选择题:\\\\\n(1) 若 $a b \\neq 0$ 且 $a \\neq b$, 则 $a x-y+b=0$ 和 $b x^2+a y^2=a b$ 所表示的曲线只可能是下图中的\\bracket{20}.\n\\fourch{\\begin{tikzpicture}[>=latex,scale = 0.4]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw (0,0) ellipse (1 and 2);\n\\draw (-2.5,3) -- (1.5,-3);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.4]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -3:3] plot (\\x,{sqrt(1+\\x*\\x/2)});\n\\draw [domain = -3:3] plot (\\x,{-sqrt(1+\\x*\\x/2)});\n\\draw (-3,-0.5) -- (0.5,3);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.4]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,0) ellipse (2 and 1);\n\\draw (-1.5,3) -- (2.5,-3);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.4]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -3:3] plot ({sqrt(0.5+0.25*\\x*\\x)},\\x);\n\\draw [domain = -3:3] plot ({-sqrt(0.5+0.25*\\x*\\x)},\\x);\n\\draw (-2,-3) -- (3,-0.5);\n\\end{tikzpicture}}\\\\\n(2) 已知曲线 $\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$) 的两条渐近线均和圆 $C: x^2+y^2-6 x+5=0$ 相切, 且双曲线的右焦点为圆 $C$ 的圆心, 则该双曲线的方程为\\bracket{20}.\n\\fourch{$\\dfrac{x^2}{5}-\\dfrac{y^2}{4}=1$}{$\\dfrac{x^2}{4}-\\dfrac{y^2}{5}=1$ }{$\\dfrac{x^2}{3}-\\dfrac{y^2}{6}=1$}{$\\dfrac{x^2}{6}-\\dfrac{y^2}{3}=1$}\n(3) 已知 $F_1(-8,3)$、$F_2(2,3)$ 两点, 动点 $P$ 满足 $|PF_1|-|PF_2|=2 a$. 当 $a$ 分别为 3 和 5 时, 点 $P$ 的轨迹分别为\\bracket{20}.\n\\twoch{双曲线和一条直线}{双曲线和一条射线}{双曲线的一支和一条直线}{双曲线的一支和一条射线}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760469,7 +760559,9 @@
|
|||
"id": "041006",
|
||||
"content": "已知双曲线方程为 $x^2-\\dfrac{y^2}{4}=1$, 过 $P(1,0)$ 的直线 $l$ 与双曲线只有一个公共点, 则 $l$ 的条数共\\blank{50}条.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760491,7 +760583,9 @@
|
|||
"id": "041007",
|
||||
"content": "rep[021273,000669,018851,021319,016575] 根据下列条件, 写出抛物线的标准方程:\\\\\n(1) 准线方程是 $x=\\dfrac{1}{4}$;\\\\\n(2) 焦点到准线的距离是 2 ;\\\\\n(3) 过点 $(-3,4)$;\\\\\n(4) 过焦点且与 $x$ 轴垂直的弦长是 16 ;\\\\\n(5) 焦点在直线 $3 x-4 y-12=0$ 上的抛物线标准方程.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760511,7 +760605,9 @@
|
|||
"id": "041008",
|
||||
"content": "求抛物线 $y=a x^2$ 的焦点坐标和准线方程.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760534,7 +760630,9 @@
|
|||
"id": "041009",
|
||||
"content": "动圆 $M$ 经过点 $A(3,0)$ 且与直线 $l: x=-3$ 相切, 则动圆圆心 $M$ 的轨迹方程\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760554,7 +760652,9 @@
|
|||
"id": "041010",
|
||||
"content": "抛物线 $y^2=2 x$ 上的 $A$、$B$ 两点到焦点 $F$ 的距离之和是 5 , 则线段 $AB$ 的中点的横坐标是\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760576,7 +760676,9 @@
|
|||
"id": "041011",
|
||||
"content": "已知抛物线的顶点在原点, 对称轴为 $x$ 轴, 抛物线上的点 $M(-3, m)$ 到焦点的距离等于 5 , 求抛物线的方程和 $m$ 的值.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760596,7 +760698,9 @@
|
|||
"id": "041012",
|
||||
"content": "一椭圆过点 $A(1,-\\dfrac{\\sqrt{2}}{2})$, 它的中心在抛物线 $y^2=-4 x$ 的顶点上, 且椭圆的左焦点与抛物线的焦点重合, 试求: (1) 抛物线的焦点坐标与准线方程; (2) 椭圆方程; (3) 抛物线与椭圆的交点坐标.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760616,7 +760720,9 @@
|
|||
"id": "041013",
|
||||
"content": "已知点 $M$ 为抛物线 $y^2=4 x$ 上一动点, $F$ 为抛物线的焦点, 定点 $P(3,1)$, 求 $|MP|+|MF|$ 的最小值, 并求此时点 $M$ 的坐标.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760638,7 +760744,9 @@
|
|||
"id": "041014",
|
||||
"content": "动圆 $M$ 与定直线 $y=2$ 相切, 且与定圆 $C: x^2+(y+3)^2=1$ 相外切, 求动圆圆心 $M$ 的轨迹方程.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760660,7 +760768,9 @@
|
|||
"id": "041015",
|
||||
"content": "点 $P$ 到点 $F(2,0)$ 的距离比它到直线 $x+4=0$ 的距离小 2 , 求动点 $P$ 的轨迹方程.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760682,7 +760792,9 @@
|
|||
"id": "041016",
|
||||
"content": "$P$ 为抛物线 $y^2=2 p x$($p>0$) 上任意一点, $F$ 为焦点, 判断以 $|PF|$ 为直径的圆与 $y$ 轴的位置关系, 并说明理由.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760704,7 +760816,9 @@
|
|||
"id": "041017",
|
||||
"content": "(1) 抛物线 $y^2=8 x$ 的动弦 $AB$ 的长为 16 , 求弦 $AB$ 的中点 $M$ 到 $y$ 轴的最短距离;\\\\\n(2) 抛物线 $y^2=8 x$ 的动弦 $AB$ 的长为 1 , 求弦 $AB$ 的中点 $M$ 到 $y$ 轴的最短距离.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760724,7 +760838,9 @@
|
|||
"id": "041018",
|
||||
"content": "过抛物线 $x^2=4 y$ 的焦点 $F$ 的直线交抛物线于 $A(x_1, y_1), B(x_2, y_2)$ 两点, 如果 $y_1+y_2=6$, 那么 $|AB|=$\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760746,7 +760862,9 @@
|
|||
"id": "041019",
|
||||
"content": "已知抛物线 $y^2=2 p x$($p>0$) 的一条经过焦点的弦被焦点分成长为 $m$、$n$ 的两部分.求证: $\\dfrac{1}{m}+\\dfrac{1}{n}$ 为定值.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760768,7 +760886,9 @@
|
|||
"id": "041020",
|
||||
"content": "对于抛物线 $C: y^2=4 x$, 我们称满足 $y_0{}^2<4 x_0$ 的点 $M(x_0, y_0)$ 在抛物线的内部, 若点 $M(x_0, y_0)$ 在抛物线的内部, 则直线 $l: y_0 y=2(x+x_0)$ 与抛物线 $C$\\bracket{20}.\n\\fourch{恰有一个公共点}{恰有二个公共点}{有一个公共点也可能有二个公共点}{没有公共点}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760788,7 +760908,9 @@
|
|||
"id": "041021",
|
||||
"content": "如图, 过抛物线 $y^2=2 p x$($p>0$) 上一定点 $P(x_0, y_0)$($y_0>0$), 作两条直线分别交抛物线于 $A(x_1, y_1), B(x_2, y_2)$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, line cap = round, line join = round, scale = 1]\n\\draw [->] (-0.5,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$}; \n\\draw [domain = -1.8:1.8, samples = 1000] plot (\\x*\\x, \\x);\n\\draw (1.44,1.2) node [above] {$P$};\n\\draw (0.49,-0.7) node [below] {$A$} -- (1.44,1.2) -- (2.89,-1.7) node [below] {$B$} -- cycle;\n\\end{tikzpicture}\n\\end{center}\n(1) 求该抛物线上纵坐标为 $\\dfrac{p}{2}$ 的点到其焦点 $F$ 的距离;\\\\\n(2) 当 $PA$ 与 $PB$ 的斜率存在且倾斜角互补时,求 $\\dfrac{y_1+y_2}{y_0}$ 的值, 并证明直线 $AB$ 的斜率是非零常数.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760808,7 +760930,9 @@
|
|||
"id": "041022",
|
||||
"content": "方程 $y=-\\sqrt{x^2-2 x+1}$ 的图形是下图中的\\bracket{20}.\n\\fourch{}{}{}{}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760828,7 +760952,9 @@
|
|||
"id": "041023",
|
||||
"content": "``点 $M(a, b)$ 在曲线 $y^2=x$ 上''是``点 $M(a, b)$ 在曲线 $y=\\sqrt{x}$ 上''的\\blank{50}条件.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760850,7 +760976,9 @@
|
|||
"id": "041024",
|
||||
"content": "直线 $y=k x-3$ 与曲线 $|x-1|+y=0$ 的公共点最多有\\bracket{20}.\n\\fourch{0 个}{1 个}{2 个}{3 个}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760870,7 +760998,9 @@
|
|||
"id": "041025",
|
||||
"content": "(1) 已知两条曲线的方程是 $F_1(x, y)=0$ 和 $F_2(x, y)=0$, 它们的交点是 $M(x_0, y_0)$. 求证: 当 $\\lambda \\in \\mathbf{R}$ 时, 方程 $F_1(x, y)+\\lambda F_2(x, y)=0$ 的曲线也经过点 $M(x_0, y_0)$.\\\\\n(2) 已知两条曲线 $x^2+y^2-3 x+y=0$ 和 $3 x^2+3 y^2+4 x-y=0$ 有两个交点, 求经过这两交点的直线的方程.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760890,7 +761020,9 @@
|
|||
"id": "041026",
|
||||
"content": "求证: 无论 $m$ 取何实数, 方程 $(1+m) x^2+(m^2-1) x+(m^2-2 m-3) y-(2 m^2-m-3)=0$所表示的曲线总是经过一个定点.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760910,7 +761042,9 @@
|
|||
"id": "041027",
|
||||
"content": "已知方程 $y=\\sqrt{1-x^2}$ 与方程 $y=k x+2$ 有且仅有一个公共解, 求实数 $k$ 的取值范围.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760932,7 +761066,9 @@
|
|||
"id": "041028",
|
||||
"content": "已知椭圆 $\\begin{cases}x=3 \\cos \\theta\\\\y=2 \\sin \\theta\\end{cases}$ ($\\theta$ 为参数), 则\\\\\n(1) $\\theta=\\dfrac{\\pi}{6}$ 时对应的点 $P$ 的坐标是\\blank{50}; (2) 直线 $OP$ 的倾斜角是\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760954,7 +761090,9 @@
|
|||
"id": "041029",
|
||||
"content": "若圆 $C$ 的方程为 $(x-a)^2+(y-b)^2=r^2$, 写出圆 $C$ 的一个参数方程.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760974,7 +761112,9 @@
|
|||
"id": "041030",
|
||||
"content": "设 $P$、$Q$ 是椭圆 $\\dfrac{x^2}{4}+y^2=1$ 上相异的两点. 设 $A(2,0)$、$B(0,1)$.\n命题甲: 若 $|AP|=|AQ|$ , 则 $P$ 与 $Q$ 关于 $x$ 轴对称;\n命题乙: 若 $|BP|=|BQ|$, 则 $P$ 与 $Q$ 关于 $y$ 轴对称.\n关于这两个命题的真假, 以下四个论述中, 正确的是\\bracket{20}.\n\\fourch{甲和乙都是真命题}{甲是真命题, 乙是假命题}{甲是假命题, 乙是真命题}{甲和乙都是假命题}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -760994,7 +761134,9 @@
|
|||
"id": "041031",
|
||||
"content": "如图, 点 $A$ 是曲线 $y=\\sqrt{x^2+2}$($y \\leq 2$) 上的任意一点, $P(0,-2), Q(0,2)$, 射线 $QA$ 交曲线 $y=\\dfrac{1}{8}x^2$ 于 $B$ 点, $BC$ 垂直于直线 $y=3$, 垂足为点 $C$. 则下列判断:\\\\\n\\textcircled{1} $|AP|-|AQ|$ 为定值 $2 \\sqrt{2}$;\\\\\n\\textcircled{2} $|QB|+|BC|$ 为定值 5 . 其中正确的说法是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-5,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,6) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [name path = hline] (-5,3) -- (5,3);\n\\filldraw (0,-2) circle (0.06) node [left] {$P$} coordinate (P);\n\\filldraw (0,2) circle (0.06) node [left] {$Q$} coordinate (Q);\n\\draw [domain = {-sqrt(2)}:{sqrt(2)}, samples = 100] plot (\\x,{sqrt(\\x*\\x+2)});\n\\draw [name path = para, domain = -5:5, samples = 100] plot (\\x, {\\x*\\x/8});\n\\filldraw (1,{sqrt(3)}) circle (0.06) node [above] {$A$} coordinate (A);\n\\path [name path = QA] (Q)--($(Q)!4!(A)$);\n\\draw [name intersections = {of = QA and para, by = B}];\n\\draw (Q)--(B) node [below] {$B$};\n\\path [name path = BC] (B)--++ (0,2);\n\\draw [name intersections = {of = hline and BC, by = C}];;\n\\draw (B)--(C) node [above] {$C$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{\\textcircled{1}\\textcircled{2}都正确}{\\textcircled{1}\\textcircled{2}都错误}{\\textcircled{1}正确, \\textcircled{2}错误}{\\textcircled{1}都错误, \\textcircled{2}正确}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -761014,7 +761156,9 @@
|
|||
"id": "041032",
|
||||
"content": "已知椭圆 $\\dfrac{x^2}{9}+\\dfrac{y^2}{4}=1$ 的焦点为 $F_1, F_2$, 椭圆上的动点 $P$ 坐标 $(x_0, y_0)$, 且 $\\angle F_1PF_2$ 为锐角, $x_0$ 的取值范围为\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -761036,7 +761180,9 @@
|
|||
"id": "041033",
|
||||
"content": "设 $P$ 是双曲线 $x^2-\\dfrac{y^2}{15}=1$ 的右支上一点, 过点 $P$ 分别作圆 $(x+4)^2+y^2=4$ 和 $(x-4)^2+y^2=1$的切线, 切点分别为 $M, N$, 则 $|PM|^2-|PN|^2$ 的最小值为\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -761056,7 +761202,9 @@
|
|||
"id": "041034",
|
||||
"content": "点 $P$ 在圆 $C: x^2+(y-2)^2=\\dfrac{1}{9}$ 上移动, 点 $Q$ 在椭圆 $x^2+4 y^2=4$ 上移动, 则 $|PQ|$ 的最大值为\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -761078,7 +761226,9 @@
|
|||
"id": "041035",
|
||||
"content": "直线 $x=1$ 上有动点 $P, O$ 为坐标原点, 等腰直角 $\\triangle OPQ$ 中, $\\angle POQ=\\dfrac{\\pi}{2}$, 动点 $Q$ 的轨迹方程为\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -761098,7 +761248,9 @@
|
|||
"id": "041036",
|
||||
"content": "设动点 $\\mathrm{A}$ 的轨迹为抛物线 $y^2=4 x$, 点 $B(2,0)$ 为定点. 若线段 $AB$ 的中点为点 $P$, 则点 $P$ 的轨迹方程为\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -761118,7 +761270,9 @@
|
|||
"id": "041037",
|
||||
"content": "设 $P$ 是双曲线 $\\dfrac{x^2}{3}-\\dfrac{y^2}{6}=1$ 上的一点, $F_1$、$F_2$ 是该双曲线的左、右焦点. 若 $(\\overrightarrow{F_1P}+\\overrightarrow{F_2P}) \\cdot(\\overrightarrow{F_1P}-\\overrightarrow{F_2P})=72$, 则 $|\\overrightarrow{F_1P}|=$\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -761138,7 +761292,9 @@
|
|||
"id": "041038",
|
||||
"content": "已知曲线 $C_1:\\begin{cases}x=-4+\\cos t,\\\\y=3+\\sin t,\\end{cases}$ ($t$ 为参数), $C_2:\\begin{cases}x=8 \\cos \\theta,\\\\y=3 \\sin \\theta,\\end{cases}$ ($\\theta$ 为参数).\\\\\n(1) 化 $C_1, C_2$ 的方程为普通方程, 并说明它们分别表示什么曲线;\\\\\n(2) 若 $C_1$ 上的点 $P$ 对应的参数为 $t=\\dfrac{\\pi}{2}$, 点 $Q$ 为 $C_2$ 上的动点, 求 $PQ$ 中点 $M$ 到直线 $C_3:\\begin{cases}x=3+2 t,\\\\y=-2+t\\end{cases}$ ($t$ 为参数) 距离的最小值.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -761158,7 +761314,9 @@
|
|||
"id": "041039",
|
||||
"content": "设双曲线 $\\Gamma$ 的方程为 $x^2-\\dfrac{y^2}{4}=1$.\\\\\n(1) 设 $l$ 是经过点 $M(1,1)$ 的直线, 且和 $\\Gamma$ 有且仅有一个公共点, 求 $l$ 的方程;\\\\\n(2) 设 $l_1$ 是 $\\Gamma$ 的一条渐近线, $A$、$B$ 是 $l_1$ 上相异的两点. 若点 $P$ 是 $\\Gamma$ 上的一点, $P$ 关于点 $A$ 的对称点记为 $Q, Q$ 关于点 $B$ 的对称点记为 $T$. 试判断点 $T$ 是否可能在 $\\Gamma$ 上, 并说明理由.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -761178,7 +761336,9 @@
|
|||
"id": "041040",
|
||||
"content": "已知长方体$ABCD-A_1B_1C_1D_1$的上、下底面都是边长为$3$的正方形, 长方体的高为$4$, 如图建立空间直角坐标系. 下列空间向量分别是图中连接哪两个顶点的直线的一个方向向量?\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\def\\l{3}\n\\def\\m{3}\n\\def\\n{4}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below ] {$B$} 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\\draw [dashed] (C)--(A)(D)--(B)($(A)!0.5!(C)$)--($(A1)!0.5!(C1)$);\n\\draw [->] (A) -- ($(A)!-0.5!(C)$) node [below] {$x$} coordinate (x);\n\\draw [->] (B) -- ($(B)!-0.5!(D)$) node [right] {$y$} coordinate (y);\n\\draw [->] ($(A1)!0.5!(C1)$) --++ (0,1.5,0) node [right] {$z$} coordinate (z);\n\\end{tikzpicture}\n\\end{center}\n(1) $\\overrightarrow{a}=(1,0,0)$.\\\\\n(2) $\\overrightarrow{b}=(0,1,0)$.\\\\\n(3) $\\overrightarrow{c}=(3 \\sqrt{2}, 0,4)$.\\\\\n(4) $\\overrightarrow{d}=(0,3 \\sqrt{2}, 8)$.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -761198,7 +761358,9 @@
|
|||
"id": "041041",
|
||||
"content": "已知长方体$ABCD-A_1B_1C_1D_1$的上、下底面都是边长为$3$的正方形, 长方体的高为$4$, 如图建立空间直角坐标系. 分别求下列平面的一个法向量.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\def\\l{3}\n\\def\\m{3}\n\\def\\n{4}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below ] {$B$} 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\\draw [dashed] (C)--(A)(D)--(B)($(A)!0.5!(C)$)--($(A1)!0.5!(C1)$);\n\\draw [->] (A) -- ($(A)!-0.5!(C)$) node [below] {$x$} coordinate (x);\n\\draw [->] (B) -- ($(B)!-0.5!(D)$) node [right] {$y$} coordinate (y);\n\\draw [->] ($(A1)!0.5!(C1)$) --++ (0,1.5,0) node [right] {$z$} coordinate (z);\n\\end{tikzpicture}\n\\end{center}\n(1) 平面 $AA_1D_1D$.\\\\\n(2) 平面 $BB_1D_1D$.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -761218,7 +761380,9 @@
|
|||
"id": "041042",
|
||||
"content": "已知曲线 $C: f(x)=x^3+x$.\\\\\n (1) 求曲线 $C$ 在点 $(1,2)$ 处切线的方程:\\\\\n (2) 设曲线上任意一点处切线的侧斜角为 $\\alpha$, 求 $\\alpha$ 的取值范围.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第二单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -761238,7 +761402,9 @@
|
|||
"id": "041043",
|
||||
"content": "曲线 $y=x \\ln x$ 上的点到直线 $x-y-2=0$ 的最短距离为\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第二单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -761284,7 +761450,9 @@
|
|||
"id": "041044",
|
||||
"content": "已知奇函数, $f(x)=\\begin{cases}\\dfrac{\\mathrm{e}^x}{x}-1 & (x>0),\\\\h(x) & (x<0),\\end{cases}$ 则函数 $h(x)$ 的最大值为\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第二单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -761304,7 +761472,9 @@
|
|||
"id": "041045",
|
||||
"content": "已知函数 $f(x)=a x-\\ln x$, 当 $x \\in(0, \\mathrm{e})(\\mathrm{c}$ 为自然常数 $)$ 时, 函数 $f(x)$ 的最小值为 3 , 则 $a$ 的值为\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第二单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -761338,7 +761508,9 @@
|
|||
"id": "041046",
|
||||
"content": "设函数 $f(x)=t x^2+2 t^2 x+t-1$($x \\in \\mathbf{R}$, $t>0$).\\\\\n (1) 求 $f(x)$ 的最小值 $h(t)$;\\\\\n (2) 若 $h(t)<-2 t+m$ 对 $t \\in(0,2)$ 恒成立, 求实数 $m$ 的取值萢围.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第二单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -761366,7 +761538,9 @@
|
|||
"id": "041047",
|
||||
"content": "已知离散型随机变量 $X$ 的分布列为:\n$\\begin{pmatrix}\n0 & 1 & 2 \\\\\n0.5 & 1-2 q & q^2\\end{pmatrix}$, 则常数 $q=$\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第八单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -761388,7 +761562,9 @@
|
|||
"id": "041048",
|
||||
"content": "已知 $X$ 的分布列为: $\\begin{pmatrix}\n-1 & 0 & 1 \\\\\n\\dfrac{1}{2}& \\dfrac{1}{3}& \\dfrac{1}{6}\n\\end{pmatrix}$, 两个随机变量 $X, Y$ 满足 $X+2Y=4$, 则 $E[X]=$\\blank{50}, $E[Y]=$\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第八单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -761410,7 +761586,9 @@
|
|||
"id": "041049",
|
||||
"content": "已知 $X$ 的分布列为: $\\begin{pmatrix}\n-1 & 0 & 1 \\\\\n\\dfrac 12 & \\dfrac 13 & \\dfrac 16 \n\\end{pmatrix}$. \n两个随机变量 $X, Y$ 满足 $X+2Y=4$, 则 $D[X]=$\\blank{50}, $D[Y]=$\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第八单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
|
|||
Reference in New Issue