录入2025届测验补充题目（测验1）
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"023082": {
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"id": "023082",
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"content": "如图, 正方体 $ABCD-A_1B_1C_1D_1$ 中, $CC_1$ 与平面 $ACD_1$ 所成的角为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{1.5}\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 (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,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] (A)--(C)--(D1)--cycle;\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": "25届测验补充题目",
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"20240105\t杨懿荔"
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"023083": {
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"id": "023083",
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"content": "如图, 已知 $\\triangle ABC$ 是边长为 $2 a$ 的正三角形, 那么它的平面直观图 $\\triangle A' B' C'$ 的面积为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0,0) -- (2,0,0) node [below] {$x'$};\n\\draw [->] (0,0,1) -- (0,0,-3) node [left] {$y'$};\n\\draw (0,0) node [below] {$O$};\n\\draw (-1,0,0) node [below] {$A'$} coordinate (A');\n\\draw (1,0,0) node [below] {$B'$} coordinate (B');\n\\draw (0,0,{-sqrt(3)}) node [above left] {$C'$} coordinate (C');\n\\draw (A')--(C')--(B');\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": "25届测验补充题目",
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"20240105\t杨懿荔"
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},
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"023084": {
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"id": "023084",
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"content": "已知命题:\\\\\n\\textcircled{1} 若直线 $l$ 与平面 $M$ 斜交, 则 $M$ 内不存在与 $l$ 垂直的直线;\\\\\n\\textcircled{2} 若直线 $l \\perp$ 平面 $M$, 则 $M$ 内不存在与 $l$ 不垂直的直线;\\\\\n\\textcircled{3} 若直线 $l$ 与平面 $M$ 斜交, 则 $M$ 内不存在与 $l$ 平行的直线;\\\\\n\\textcircled{4} 若直线 $l \\parallel$ 平面 $M$, 则 $M$ 内不存在与 $l$ 不平行的直线.\\\\\n以上 4 个命题中正确的序号是\\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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"023085": {
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"id": "023085",
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"content": "在矩形 $ABCD$ 中, $AB=\\sqrt{3}$, $BC=1$, 将 $\\triangle ABC$ 与 $\\triangle ADC$ 沿 $AC$ 所在的直线进行任意翻折, 在翻折过程中直线 $AD$ 与直线 $BC$ 所成角的取值范围是\\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": "25届测验补充题目",
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"20240105\t杨懿荔"
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},
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"023086": {
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"id": "023086",
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"content": "在长方体 $ABCD-A_1B_1C_1D_1$ 中, $AB=8$, $BC=3$, $AA_1=2$, $P, Q$ 分别为 $AB, CC_1$ 的中点, $R$ 为线段 $A_1D_1$ 上的点, $A_1R=1$, 则平面 $PQR$ 截该长方体的侧面 $BCC_1B_1$ 得到的线段长为\\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": "25届测验补充题目",
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"20240105\t杨懿荔"
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"023087": {
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"id": "023087",
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"content": "在空间, 我们作平面, 使得位置确定的等边三角形的三个顶点到所作平面的距离均为 1 . 如果该等边三角形可以作出的平面超过 3 个, 那么称这样的等边三角形为``魔三角形''. 在边长分别为 $1$、$2$、$3$ 的三个确定的等边三角形中,``魔三角形''的个数为\\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": "25届测验补充题目",
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"20240105\t杨懿荔"
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},
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"023088": {
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"id": "023088",
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"content": "在 $\\text{Rt}\\triangle ABC$ 中, $AC \\perp BC$, $\\angle A=30^{\\circ}$, $BC=1$, $D$ 为斜边 $AB$ 的中点. 将 $\\triangle ACD$ 沿直线 $CD$ 折起至 $\\triangle A' CD$,使得直线 $A' C$ 与平面 $ABC$ (原来的 $\\triangle ABC$ 所在平面) 所成角的正弦值为 $\\dfrac{\\sqrt{3}}{4}$. 若 $AA'>1$, 则 $AA'=$\\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": "25届测验补充题目",
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"20240105\t杨懿荔"
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},
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"023089": {
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"id": "023089",
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"content": "下列平面几何的结论在空间不一定成立的是\\bracket{20}.\n\\onech{两组对边分别平行的四边形是平行四边形}{过已知直线外的定点作该直线的平行线, 有且只有一条直线与这条直线平行}{若两条直线分别与第三条直线垂直, 则这两条直线平行}{若两条直线分别与第三条直线平行, 则这两条直线平行}",
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"objs": [],
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"tags": [],
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"genre": "选择题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"origin": "25届测验补充题目",
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"20240105\t杨懿荔"
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},
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"023090": {
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"id": "023090",
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"content": "已知直线 $a, b$ 为平面 $\\alpha$ 的两条斜线, 设 $a, b$ 在 $\\alpha$ 上的射影为分别直线 $a', b'$, 对于命题: \\textcircled{1} 若 $a' \\perp b'$,则 $a \\perp b$; \\textcircled{2} 若 $a', b'$ 所成角为 $30^{\\circ}$, 则 $a, b$ 所成角一定小于 $30^{\\circ}$, 下列判断正确的是\\bracket{20}.\n\\twoch{\\textcircled{1}、\\textcircled{2}均为真命题}{\\textcircled{1}、\\textcircled{2}均为假命题}{\\textcircled{1}为真命题, \\textcircled{2}为假命题}{\\textcircled{1}为假命题、\\textcircled{2}为真命题}",
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"objs": [],
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"tags": [],
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"genre": "选择题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"origin": "25届测验补充题目",
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"20240105\t杨懿荔"
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},
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"023091": {
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"id": "023091",
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"content": "如图, 在四面体 $P-ABC$ 中, $AB=BC=2 \\sqrt{2}$, $PA=PB=PC=AC=4$, $O$ 为 $AC$ 的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (-1,0,0) node [left] {$A$} coordinate (A);\n\\draw (1,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,1) node [below] {$B$} coordinate (B);\n\\draw (0,0,0) node [above right] {$O$} coordinate (O);\n\\draw (0,{sqrt(3)},0) node [above] {$P$} coordinate (P);\n\\draw (A)--(B)--(C)--(P)--cycle(P)--(B);\n\\draw [dashed] (A)--(C)(B)--(O)--(P);\n\\filldraw ($(B)!{1/3}!(C)$) circle (0.03) node [below right] {$M$} coordinate (M);\n\\end{tikzpicture}\n\\hspace*{5em}\n\\begin{tikzpicture}[>=latex]\n\\draw (-1,0,0) node [left] {$A$} coordinate (A);\n\\draw (1,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,1) node [below] {$B$} coordinate (B);\n\\draw (0,0,0) node [above right] {$O$} coordinate (O);\n\\draw (0,{sqrt(3)},0) node [above] {$P$} coordinate (P);\n\\draw (A)--(B)--(C)--(P)--cycle(P)--(B);\n\\draw [dashed] (A)--(C)(B)--(O)--(P);\n\\filldraw ($(B)!{1/2}!(C)$) circle (0.03) node [below right] {$M$} coordinate (M);\n\\filldraw ($(B)!{1/2}!(A)$) circle (0.03) node [below left] {$N$} coordinate (N);\n\\draw [dashed] (M)--(N);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $PO \\perp$ 平面 $ABC$;\\\\\n(2) 若点 $M$ 在棱 $BC$ 上, 且 $MC=2MB$, 求点 $C$ 到平面 $POM$ 的距离.\\\\\n(3) 若点 $M, N$ 分别是棱 $BC, AB$ 上的中点, 求半平面 $PMN$和半平面 $PAC$ 所成的二面角的大小;",
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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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"20240105\t杨懿荔"
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"space": "4em",
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},
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"023092": {
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"id": "023092",
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"content": "如图, 在棱长为 $4$ 的正方体 $ABCD-A_1B_1C_1D_1$ 中, 点 $O$ 是 $A_1C_1$ 与 $B_1D_1$ 的交点.\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 [above] {$B$} 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\\draw ($(B_1)!0.5!(D_1)$) node [below] {$O$} coordinate (O);\n\\draw [dashed] (A_1)--(C_1)(B_1)--(D_1)(O)--(C);\n\\draw (A)--(C)(B_1)--(C)(B)--(C_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: 直线 $OC$ 与 $BC_1$ 是异面直线.\\\\\n(2) 求直线 $OC$ 与 $BC_1$ 所成的角的大小.\\\\\n(3) 求证: 平面 $A_1BD \\parallel $ 平面 $B_1CD_1$, 并求两个平面的距离.",
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"objs": [],
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"genre": "解答题",
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"space": "4em",
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},
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"030001": {
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"id": "030001",
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"id": "030001",
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"content": "若$x,y,z$都是实数, 则:(填写``\\textcircled{1} 充分非必要、\\textcircled{2} 必要非充分、\\textcircled{3} 充要、\\textcircled{4} 既非充分又非必要''之一)\\\\\n(1) ``$xy=0$''是``$x=0$''的\\blank{50}条件;\\\\\n(2) ``$x\\cdot y=y\\cdot z$''是``$x=z$''的\\blank{50}条件;\\\\\n(3) ``$\\dfrac xy=\\dfrac yz$''是``$xz=y^2$''的\\blank{50}条件;\\\\\n(4) ``$|x |>| y|$''是``$x>y>0$''的\\blank{50}条件;\\\\\n(5) ``$x^2>4$''是``$x>2$'' 的\\blank{50}条件;\\\\\n(6) ``$x=-3$''是``$x^2+x-6=0$'' 的\\blank{50}条件;\\\\\n(7) ``$|x+y|<2$''是``$|x|<1$且$|y|<1$'' 的\\blank{50}条件;\\\\\n(8) ``$|x|<3$''是``$x^2<9$'' 的\\blank{50}条件;\\\\\n(9) ``$x^2+y^2>0$''是``$x\\ne 0$'' 的\\blank{50}条件;\\\\\n(10) ``$\\dfrac{x^2+x+1}{3x+2}<0$''是``$3x+2<0$'' 的\\blank{50}条件;\\\\\n(11) ``$0<x<3$''是``$|x-1|<2$'' 的\\blank{50}条件.",
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"content": "若$x,y,z$都是实数, 则:(填写``\\textcircled{1} 充分非必要、\\textcircled{2} 必要非充分、\\textcircled{3} 充要、\\textcircled{4} 既非充分又非必要''之一)\\\\\n(1) ``$xy=0$''是``$x=0$''的\\blank{50}条件;\\\\\n(2) ``$x\\cdot y=y\\cdot z$''是``$x=z$''的\\blank{50}条件;\\\\\n(3) ``$\\dfrac xy=\\dfrac yz$''是``$xz=y^2$''的\\blank{50}条件;\\\\\n(4) ``$|x |>| y|$''是``$x>y>0$''的\\blank{50}条件;\\\\\n(5) ``$x^2>4$''是``$x>2$'' 的\\blank{50}条件;\\\\\n(6) ``$x=-3$''是``$x^2+x-6=0$'' 的\\blank{50}条件;\\\\\n(7) ``$|x+y|<2$''是``$|x|<1$且$|y|<1$'' 的\\blank{50}条件;\\\\\n(8) ``$|x|<3$''是``$x^2<9$'' 的\\blank{50}条件;\\\\\n(9) ``$x^2+y^2>0$''是``$x\\ne 0$'' 的\\blank{50}条件;\\\\\n(10) ``$\\dfrac{x^2+x+1}{3x+2}<0$''是``$3x+2<0$'' 的\\blank{50}条件;\\\\\n(11) ``$0<x<3$''是``$|x-1|<2$'' 的\\blank{50}条件.",
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Reference in New Issue