录入2025届高二上学期校本作业部分题库外题目
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"space": "4em",
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"unrelated": []
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},
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"019858": {
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"id": "019858",
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"content": "试用集合符号表示:\\\\\n(1) 点 $A$ 在直线 $l$ 上, 点 $B$ 不在直线 $l$ 上: \\blank{100};\\\\\n(2) 点 $A$ 在平面 $\\alpha$ 内, 点 $B$ 不在平面 $\\beta$ 内: \\blank{100};\\\\\n(3) 直线 $l$ 平行于平面 $\\alpha$: \\blank{100};\\\\\n(4) 直线 $l$ 在平面 $\\alpha$ 上: \\blank{100}.",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "自拟题目(2025届高二上学期校本作业用)",
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"edit": [
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"20230930\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"019859": {
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"id": "019859",
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"content": "判断以下命题是否正确, 并说明理由:\\\\\n(1) 如果一条直线与另两条直线都相交, 那么这三条直线必共面;\\\\\n(2) 如果三条直线相互平行, 那么这三条直线在同一个平面上;\\\\\n(3) 如果两个平面有无数个公共点, 那么这两个平面重合;\\\\\n(4) 如果梯形 $ABCD$ 中的两条边在平面 $\\alpha$ 内, 那么它的另外两条边也在平面 $\\alpha$ 内.",
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"objs": [],
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"tags": [],
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"genre": "解答题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "自拟题目(2025届高二上学期校本作业用)",
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"edit": [
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"20230930\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "4em",
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"unrelated": []
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},
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"019860": {
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"id": "019860",
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"content": "如图, 已知正方体 $ABCD-A_1B_1C_1D_1$, 点 $E$ 是棱 $DC$ 的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A_1$} coordinate (A_1);\n\\draw (A_1) ++ (\\l,0,0) node [below right] {$B_1$} coordinate (B_1);\n\\draw (A_1) ++ (\\l,0,-\\l) node [right] {$C_1$} coordinate (C_1);\n\\draw (A_1) ++ (0,0,-\\l) node [left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1);\n\\draw [dashed] (A_1) -- (D_1) -- (C_1);\n\\draw (A_1) ++ (0,\\l,0) node [left] {$A$} coordinate (A);\n\\draw (B_1) ++ (0,\\l,0) node [right] {$B$} coordinate (B);\n\\draw (C_1) ++ (0,\\l,0) node [above right] {$C$} coordinate (C);\n\\draw (D_1) ++ (0,\\l,0) node [above left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C) -- (D) -- cycle;\n\\draw (A_1) -- (A) (B_1) -- (B) (C_1) -- (C);\n\\draw [dashed] (D_1) -- (D);\n\\filldraw ($(C)!0.5!(D)$) node [above] {$E$} coordinate (E) circle (0.03);\n\\end{tikzpicture}\n\\hspace*{3em}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A_1$} coordinate (A_1);\n\\draw (A_1) ++ (\\l,0,0) node [below right] {$B_1$} coordinate (B_1);\n\\draw (A_1) ++ (\\l,0,-\\l) node [right] {$C_1$} coordinate (C_1);\n\\draw (A_1) ++ (0,0,-\\l) node [left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1);\n\\draw [dashed] (A_1) -- (D_1) -- (C_1);\n\\draw (A_1) ++ (0,\\l,0) node [left] {$A$} coordinate (A);\n\\draw (B_1) ++ (0,\\l,0) node [right] {$B$} coordinate (B);\n\\draw (C_1) ++ (0,\\l,0) node [above right] {$C$} coordinate (C);\n\\draw (D_1) ++ (0,\\l,0) node [above left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C) -- (D) -- cycle;\n\\draw (A_1) -- (A) (B_1) -- (B) (C_1) -- (C);\n\\draw [dashed] (D_1) -- (D);\n\\filldraw ($(C)!0.5!(D)$) node [above] {$E$} coordinate (E) circle (0.03);\n\\end{tikzpicture}\n\\end{center}\n(1) 画出由点 $E$、$A$、$B_1$ 确定的平面 $\\alpha$ 截正方体所得的截面, 并写出作图步骤. \\\\\n(2) 画出由点 $E$、$A$、$C_1$ 确定的平面 $\\beta$ 截正方体所得的截面, 并写出作图步骤.",
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"objs": [],
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"tags": [],
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"genre": "解答题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "自拟题目(2025届高二上学期校本作业用)",
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"edit": [
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"20230930\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "4em",
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"unrelated": []
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},
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"019861": {
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"id": "019861",
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"content": "正方体 $ABCD-A_1B_1C_1D_1$ 中, 直线 $BA_1$ 与 $CC_1$ 所成角的大小为\\blank{50}; 直线 $BA_1$ 与 $B_1C$ 所成角的大小为\\blank{50}.",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "自拟题目(2025届高二上学期校本作业用)",
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"edit": [
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"20230930\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"019862": {
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"id": "019862",
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"content": "如图, 在四面体$ABCD$中, $AC=8$, $BD=6$, $M$、$N$分别为$AB$、$CD$的中点, 并且异面直线$AC$与$BD$所成的角为$90^\\circ$. 则$MN$的长为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$B$} coordinate (B);\n\\draw (3,0) node [right] {$D$} coordinate (D);\n\\draw (1.4,2.3) node [above] {$A$} coordinate (A);\n\\draw (2,-0.8) node [below] {$C$} coordinate (C);\n\\draw ($(D)!0.5!(C)$) node [below right] {$N$} coordinate (N);\n\\draw ($(A)!0.5!(B)$) node [above left] {$M$} coordinate (M);\n\\draw (A) -- (C) (B) -- (C) -- (D) (D) -- (A) -- (B);\n\\draw [dashed] (M) -- (N) (B) -- (D);\n\\end{tikzpicture}\n\\end{center}",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "自拟题目(2025届高二上学期校本作业用)",
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"edit": [
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"20230930\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"019863": {
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"id": "019863",
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"content": "已知 $A$ 是 $\\triangle BCD$ 平面外的一点, $E$、$F$ 分别是 $BC$、$AD$ 的中点. 若 $AC \\perp BD$, 且 $AC=BD$, 求 $EF$ 与 $BD$ 所成的角.",
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"objs": [],
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"tags": [],
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"genre": "解答题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "自拟题目(2025届高二上学期校本作业用)",
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"edit": [
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"20230930\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "4em",
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"unrelated": []
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},
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"019864": {
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"id": "019864",
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"content": "在正方体 $ABCD-A_1B_1C_1D_1$ 中, $P, Q$ 分别是 $AD_1, BD$ 上的点, 且 $AP=BQ$, 求证: $PQ \\parallel $ 平面 $DCC_1D_1$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,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 ($(B)!0.3!(D)$) node [above] {$Q$} coordinate (Q);\n\\draw ($(A)!0.3!(D_1)$) node [left] {$P$} coordinate (P);\n\\draw [dashed] (P)--(Q)(B)--(D)(A)--(D_1);\n\\end{tikzpicture}\n\\end{center}",
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"objs": [],
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"tags": [],
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"genre": "解答题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "自拟题目(2025届高二上学期校本作业用)",
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"edit": [
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"20230930\t王伟叶"
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],
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"same": [],
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"remark": "",
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"space": "4em",
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"unrelated": []
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},
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"020001": {
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"id": "020001",
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"content": "判断下列各组对象能否组成集合, 若能组成集合, 指出是有限集还是无限集.\\\\\n(1) 上海市控江中学$2022$年入学的全体高一年级新生;\\\\\n(2) 中国现有各省的名称;\\\\\n(3) 太阳、$2$、上海市;\\\\\n(4) 大于$10$且小于$15$的有理数;\\\\\n(5) 末位是$3$的自然数;\\\\\n(6) 影响力比较大的中国数学家;\\\\\n(7) 方程$x^2+x-3=0$的所有实数解;\\\\ \n(8) 函数$y=\\dfrac 1x$图像上所有的点;\\\\ \n(9) 在平面直角坐标系中, 到定点$(0, 0)$的距离等于$1$的所有点;\\\\\n(10) 不等式$3x-10<0$的所有正整数解;\\\\\n(11) 所有的平面四边形.",
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