diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index c9cc42fb..d308c8d2 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -137547,7 +137547,9 @@ "20220709\t王伟叶" ], "same": [], - "related": [], + "related": [ + "011402" + ], "remark": "", "space": "", "unrelated": [] @@ -316809,7 +316811,9 @@ "20220818\t王伟叶" ], "same": [], - "related": [], + "related": [ + "004479" + ], "remark": "", "space": "", "unrelated": [] @@ -499598,7 +499602,9 @@ "20230706\t王伟叶" ], "same": [], - "related": [], + "related": [ + "023190" + ], "remark": "", "space": "4em", "unrelated": [] @@ -620990,6 +620996,249 @@ "space": "4em", "unrelated": [] }, + "023181": { + "id": "023181", + "content": "以下两个命题中, 正确的有\\blank{50}.\\\\\n\\textcircled{1} 底面为正多边形的棱柱是正棱柱;\\\\\n\\textcircled{2} 侧面均是全等矩形的棱柱为正棱柱.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "25届周末卷补充题目", + "edit": [ + "20240106\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "023182": { + "id": "023182", + "content": "若$P$是全体长方体组成的集合, $M$是全体正四棱柱组成的集合, $N$是全体直四棱柱组成的集合, $Q$是全体直平行六面体组成的集合, 则四个集合的关系为\\bracket{20}.\n\\fourch{$M \\subseteq P \\subseteq N \\subseteq Q$}{$M \\subseteq P \\subseteq Q \\subseteq N$}{$P \\subseteq M \\subseteq N \\subseteq Q$}{$P \\subseteq M \\subseteq Q \\subseteq N$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "25届周末卷补充题目", + "edit": [ + "20240106\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "023183": { + "id": "023183", + "content": "斜五棱柱的侧面最多可以有\\blank{50}个面是矩形.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "25届周末卷补充题目", + "edit": [ + "20240106\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "023184": { + "id": "023184", + "content": "一个棱柱至少有\\blank{50}个面.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "25届周末卷补充题目", + "edit": [ + "20240106\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "023185": { + "id": "023185", + "content": "已知长方体 $ABCD-A_1B_1C_1D_1$, 交于点 $A$ 的三个面的面积分别是 $6$、$12$、$8$, 则它的对角线长是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "25届周末卷补充题目", + "edit": [ + "20240106\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "023186": { + "id": "023186", + "content": "长方体的一条对角线与过同一顶点的三个面中的两个面所成角为 $30^\\circ$ 和 $45^\\circ$, 则这条对角线与第三个面所成角的大小是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "25届周末卷补充题目", + "edit": [ + "20240106\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "023187": { + "id": "023187", + "content": "若斜三棱柱的侧棱长是 15 , 侧棱与底面的夹角为 $60^\\circ$, 则此棱柱的高是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "25届周末卷补充题目", + "edit": [ + "20240106\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "023188": { + "id": "023188", + "content": "如图, 点 $P$ 在正方体 $ABCD-A_1B_1C_1D_1$ 的面对角线 $BC_1$ 上运动, 则下列四个结论正确的是\\blank{50}.\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 [above] {$B_1$} 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)--(C_1);\n\\filldraw ($(B)!0.7!(C_1)$) node [below right = 0 and -0.2] {$P$} coordinate (P) circle (0.03);\n\\draw [dashed] (A)--(C)--(D_1)--cycle;\n\\end{tikzpicture}\n\\end{center}\n\\textcircled{1} $P$ 到平面 $ACD_1$ 的距离不变;\\\\\n\\textcircled{2} $A_1P \\parallel $ 平面 $ACD_1$;\\\\\n\\textcircled{3} $DP \\perp BC_1$;\\\\\n\\textcircled{4} 平面 $PDB_1 \\perp$ 平面 $ACD_1$.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "25届周末卷补充题目", + "edit": [ + "20240106\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "023189": { + "id": "023189", + "content": "已知正三棱柱 $ABC-A_1B_1C_1$, $AB=2AA_1$, $D$ 是 $AB$ 的中点.\\\\\n(1) 求证: $B_1C \\perp A_1D$;\\\\\n(2) 求 $CB_1$ 与平面 $A_1DC$ 所成角的大小.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "25届周末卷补充题目", + "edit": [ + "20240106\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "023190": { + "id": "023190", + "content": "如图, 在斜三棱柱 $ABC-A_1B_1C_1$ 中, $\\angle A_1AC=\\angle ACB=\\dfrac{\\pi}{2}$, $\\angle AA_1C=\\dfrac{\\pi}{6}$, 侧棱 $BB_1$ 与底面所成的角为 $\\dfrac{\\pi}{3}$, $AA_1=4 \\sqrt{3}$, $BC=4$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.3]\n\\draw ({-2*sqrt(2)},0,0) node [below] {$A$} coordinate (A);\n\\draw ({2*sqrt(2)},0,0) node [below] {$B$} coordinate (B);\n\\draw (0,0,{-2*sqrt(2)}) node [above right] {$C$} coordinate (C);\n\\draw (A) -- (B);\n\\draw [dashed] (A) -- (C) -- (B);\n\\draw (A) ++ ({-sqrt(6)},6,{-sqrt(6)}) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ ({-sqrt(6)},6,{-sqrt(6)}) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ ({-sqrt(6)},6,{-sqrt(6)}) node [above] {$C_1$} coordinate (C1);\n\\draw [dashed] (C) -- (C1) (A1) -- (C);\n\\draw (A) -- (A1) (B) -- (B1) (A1) -- (B1) (A1) -- (C1) -- (B1); \n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $AC \\perp$ 面 $BCC_1B_1$;\\\\\n(2) 求斜三棱柱 $ABC-A_1B_1C_1$ 的高.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "25届周末卷补充题目", + "edit": [ + "20240106\t杨懿荔" + ], + "same": [], + "related": [ + "018719", + "030489" + ], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "023191": { + "id": "023191", + "content": "在棱长为 $1$ 的正方体 $ABCD-A_1B_1C_1D_1$ 中, 过 $AC$ 作与底面成 $\\alpha$ 角的截面 ($0<\\alpha \\leq \\dfrac{\\pi}{2}$), 求此截面的面积.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "25届周末卷补充题目", + "edit": [ + "20240106\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "023192": { + "id": "023192", + "content": "如图, 在平面四边形 $ABCP$ 中, $D$ 为 $PA$ 的中点, $PA \\perp AB$, $CD \\parallel AB$, 且 $PA=CD=2AB =4$ . 将此平面四边形 $ABCP$ 沿 $CD$ 折起, 且平面 $PDC \\perp$ 平面 $DCB$, 连接 $PA, PB, BD$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw (0,0) node [left] {$D$} coordinate (D);\n\\draw (4,0) node [right] {$C$} coordinate (C);\n\\draw (0,-2) node [below left] {$A$} coordinate (A);\n\\draw (2,-2) node [below] {$B$} coordinate (B);\n\\draw (0,2) node [left] {$P$} coordinate (P);\n\\draw (P)--(A)--(B)--(C)--cycle(D)--(C);\n\\end{tikzpicture}\n\\hspace*{3em}\n\\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw (0,0,0) node [left] {$D$} coordinate (D);\n\\draw (4,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,2) node [below left] {$A$} coordinate (A);\n\\draw (2,0,2) node [below] {$B$} coordinate (B);\n\\draw (0,2,0) node [left] {$P$} coordinate (P);\n\\draw (P)--(A)--(B)--(C)--cycle(P)--(B);\n\\draw [dashed] (A)--(D)--(C)(D)--(P)(D)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: 平面 $PBD \\perp$ 平面 $PBC$;\\\\\n(2) 求点 $D$ 到平面 $PBC$ 的距离.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "25届周末卷补充题目", + "edit": [ + "20240106\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, "030001": { "id": "030001", "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