20221009 evening
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"metadata": {
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"kernelspec": {
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"language": "python",
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"name": "python3"
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"name": "python",
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"nbconvert_exporter": "python",
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"version": "3.8.8"
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"cells": [
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"execution_count": 1,
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"name": "stdout",
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"content": "判断下列命题的真假, 并说明理由:\\\\\n(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(2) 若直线$l\\perp\\text{平面}M$, 则$M$内不存在与$l$不垂直的直线;\\\\\n(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(4) 若直线$l\\parallel\\text{平面}M$, 则$M$内不存在与$l$不平行的直线.",
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"objs": [
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"K0610001B",
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"K0609001B"
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"K0609001B",
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"K0608002B"
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],
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"tags": [
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"第六单元"
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"content": "如图, 在正三棱柱$ABC-A_1B_1C_1$中, $|AB|=\\sqrt 2|AA_1|$, $D$是$A_1B_1$的中点, 点$E$在$A_1C_1$上, 且$DE\\perp AE$.\n\\begin{center}\n \\begin{tikzpicture}[thick]\n \\draw (0,0) node [left] {$A$} coordinate (A) -- (0,2) node [left] {$A_1$} coordinate (A1) --++ ({2*sqrt(2)},0) node [right] {$C_1$} coordinate (C1) --++ (0,-2) node [right] {$C$} coordinate (C);\n \\draw ({sqrt(2)},0) ++ (-45:{sqrt(6)/2}) node [below] {$B$} coordinate (B) --++ (0,2) node [left] {$B_1$} coordinate (B1) (B) -- (A) (B) -- (C) (B1) -- (A1) (B1) -- (C1);\n \\draw ($(A1)!0.5!(B1)$) node [above] {$D$} coordinate (D) -- ($(A1)!0.25!(C1)$) node [above] {$E$} coordinate (E);\n \\draw [dashed] (E) -- (A) -- (D) (A) -- (C1) (A) -- (C);\n \\end{tikzpicture}\n\\end{center}\n(1) 求证: $\\text{平面}ADE\\perp \\text{平面}ACC_1A_1$;\\\\\n(2) 求直线$AD$和平面$ABC_1$所成角的大小.",
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"objs": [
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"K0628004X",
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"K0630004X"
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"K0630004X",
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"K0613007B"
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],
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"tags": [
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"第六单元"
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"id": "000300",
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"content": "在长方体$ABCD-A_1B_1C_1D_1$中, $|AB|=|BC|=2$, $A_1D$与$BC_1$所成的角为$\\dfrac\\pi 2$. 求$BC_1$与平面$BB_1D_1D$所成角的大小.",
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"objs": [
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"K0630004X"
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"K0630004X",
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"K0610003B",
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"K0610004B"
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],
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"tags": [
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"第六单元"
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"content": "在正方体$ABCD-A_1B_1C_1D_1$中, $E$、$F$分别是$BC$、$CC_1$的中点.\\\\\n(1) 求证: 点$D_1$在平面$AEF$上;\\\\\n(2) 求平面$AEFD_1$与底面$ABCD$所成二面角的大小.",
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"objs": [
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"K0625003X",
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"K0631002X"
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"K0631002X",
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"K0613003B"
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],
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"tags": [
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"第六单元"
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"000749": {
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"id": "000749",
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"content": "长方体的对角线与过同一个顶点的三个表面所成的角分别为$\\alpha$, $\\beta$, $\\gamma$, 则$\\cos^2\\alpha+\\cos^2\\beta+\\cos^2\\gamma =$\\blank{50}.",
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"objs": [],
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"objs": [
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"K0610003B",
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"K0610004B"
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],
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"tags": [
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"第六单元"
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],
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"001611": {
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"id": "001611",
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"content": "在正方体$ABCD-A_1B_1C_1D_1$中, 如果$M$是$DD_1$的中点, 作图并证明: 直线$BD_1\\parallel$平面$MAC$.",
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"objs": [],
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"objs": [
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"K0608002B"
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],
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"tags": [
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"第六单元"
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],
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"001620": {
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"id": "001620",
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"content": "正方体$ABCD-A'B'C'D'$中, 求证:\\\\ \n(1) $D'B\\perp AC$;\\\\ \n(2) $D'B\\perp$平面$AB'C$. \n\\begin{center}\n\\begin{tikzpicture}\n \\draw (0,0) node [below left] {$A$} coordinate (A) --++ (3,0) node [below right] {$B$} coordinate (B) --++ (45:{3/2}) node [right] {$C$} coordinate (C)\n --++ (0,3) node [above right] {$C'$} coordinate (C1)\n --++ (-3,0) node [above left] {$D'$} coordinate (D1) --++ (225:{3/2}) node [left] {$A'$} coordinate (A1) -- cycle;\n \\draw (A) ++ (3,3) node [right] {$B'$} coordinate (B1) -- (B) (B1) --++ (45:{3/2}) (B1) --++ (-3,0);\n \\draw [dashed] (A) --++ (45:{3/2}) node [left] {$D$} coordinate (D) --++ (3,0) (D) --++ (0,3);\n \\draw (A) -- (B1) -- (C);\n \\draw [dashed] (A) -- (C) (B) -- (D1);\n\\end{tikzpicture}\n\\end{center}",
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"objs": [],
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"objs": [
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"K0611002B"
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],
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"tags": [
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"第六单元"
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],
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"001623": {
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"id": "001623",
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"content": "长方体$ABCD-A'B'C'D'$中, $AA'=2$, $AB=4$, 则$B'C'$到平面$A'BCD'$的距离为\\blank{60}.",
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"objs": [],
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"objs": [
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"K0609008B"
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],
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"tags": [
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"第六单元"
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],
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"001628": {
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"id": "001628",
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"content": "已知如图, 四面体$ABCD$中, $AB=AC=DB=DC=2$, $AD=BC=1$.\\\\ \n(1) 求证: $AD\\perp BC$;\\\\ \n(2) 求点$A$到面$BCD$的距离.\n\\begin{center}\n\\begin{tikzpicture}\n \\draw (0,0) node [left] {$B$} -- (1,-0.6) node [below] {$C$} -- (2.7,0.3) node [right] {$D$} -- (2,1.5) node [above] {$A$} -- cycle (1,-0.6) -- (2,1.5);\n \\draw [dashed] (0,0) -- (2.7,0.3);\n\\end{tikzpicture}\n\\end{center}",
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"objs": [],
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"objs": [
|
||||
"K0609008B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -41874,7 +41890,9 @@
|
|||
"001636": {
|
||||
"id": "001636",
|
||||
"content": "已知$PA$是$\\triangle ABC$所在平面$\\alpha$的斜线, 且$PA\\perp BC$, $\\angle ACB=90^\\circ$. 求证: 点$P$在平面$\\alpha$上的射影在直线$AC$上.",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0611002B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -41996,7 +42014,9 @@
|
|||
"001641": {
|
||||
"id": "001641",
|
||||
"content": "已知$PA\\perp$三角形$ABC$所在平面, 且$AB=AC=13$, $BC=10$, $PA=12$, $D$是$BC$中点.\\\\ \n(1) 求直线$PD$与平面$ABC$所成角的大小;\\\\ \n(2) 求直线$PC$与平面$PAD$所成角的正切;\\\\ \n(3) 求直线$PC$与平面$PAB$所成角的正切.",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0610004B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -42096,7 +42116,9 @@
|
|||
"001645": {
|
||||
"id": "001645",
|
||||
"content": "在正方体$ABCD-A'B'C'D'$中, $E,F,G$分别为$B'C'$, $A'D'$, $A'B'$的中点. 求证: 平面$EBD\\parallel$平面$FGA$.",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0612002B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -42120,7 +42142,9 @@
|
|||
"001646": {
|
||||
"id": "001646",
|
||||
"content": "证明: 如果两条异面直线都和两个平面平行, 那么这两个平面互相平行.",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0612002B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -42192,7 +42216,9 @@
|
|||
"001649": {
|
||||
"id": "001649",
|
||||
"content": "下列命题中不正确的是\\bracket{20}.\\\\ \n\\onech{垂直于同一条直线的两个平面平行}{垂直于同一个平面的两条直线相互平行}{若一个平面内有无数条直线都平行于另一个平面, 则这两个平面互相平行}{若两个平行平面分别和第三个平面相交, 则它们的交线互相平行}",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0612001B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -42432,7 +42458,9 @@
|
|||
"001659": {
|
||||
"id": "001659",
|
||||
"content": "已知$\\triangle ABC$为等边三角形, $PA\\perp $平面$ABC$, 且$PA=\\dfrac{1}{2}AC$, 则二面角$P-BC-A$为\\blank{60}.",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0613003B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -42578,7 +42606,9 @@
|
|||
"001665": {
|
||||
"id": "001665",
|
||||
"content": "如图, 过$60^\\circ$的二面角$\\alpha-l-\\beta$的棱上一点$A$, 分别在$\\alpha,\\beta$内$A$的同侧引两条射线, 使得它们与$l$都成$45^\\circ$角, 则这两条射线夹角的余弦值为\\blank{60}.",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0613002B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -42626,7 +42656,9 @@
|
|||
"001667": {
|
||||
"id": "001667",
|
||||
"content": "已知等边三角形$ABC$的边长为$1$, 沿$BC$边上的高将它折成直二面角后, 点$A$到$BC$的距离为\\blank{60}.",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0609007B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -42698,7 +42730,9 @@
|
|||
"001670": {
|
||||
"id": "001670",
|
||||
"content": "过正方形$ABCD$的顶点$A$作线段$AP\\perp$平面$ABCD$, 且$AP=AB$, 则面$ABP$与面$CDP$所成二面角的大小是\\blank{60}.",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0613003B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -42866,7 +42900,9 @@
|
|||
"001677": {
|
||||
"id": "001677",
|
||||
"content": "沿对角线$AC$将正方形$ABCD$折成直二面角后, $AB$与$CD$所在直线所成的角等于\\blank{50}.",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0613005B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -43354,7 +43390,9 @@
|
|||
"001697": {
|
||||
"id": "001697",
|
||||
"content": "高为$2$, 底面边长为$3$的正三棱锥底面中心到侧面的距离为\\blank{80}.",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0609007B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -43522,7 +43560,9 @@
|
|||
"001704": {
|
||||
"id": "001704",
|
||||
"content": "已知正四棱锥$S-ABCD$, 求证: 二面角$A-SB-C$的平面角一定为钝角.",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0613003B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -86757,7 +86797,9 @@
|
|||
"003499": {
|
||||
"id": "003499",
|
||||
"content": "在棱长为$2$的正方体$ABCD-A_1B_1C_1D_1$中, $O$为$BD$的中点.\\\\\n(1) 点$B$到平面$AB_1D_1$的距离为\\blank{50};\\\\\n(2) 直线$C_1O$和平面$AB_1D_1$的距离为\\blank{50};\\\\\n(3) 平面$AB_1D_1$和平面$C_1BD$的距离为\\blank{50};\\\\\n(4) 异面直线$BD$与$CC_1$的距离为\\blank{50}.",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0612006B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -95683,7 +95725,9 @@
|
|||
"003891": {
|
||||
"id": "003891",
|
||||
"content": "已知平面$\\alpha,\\beta$和直线$m$, 给出条件: \\textcircled{1} $m\\parallel \\alpha$; \\textcircled{2} $m\\perp \\alpha$; \\textcircled{3} $m\\subseteq \\alpha$; \\textcircled{4} $\\alpha\\perp \\beta$; \\textcircled{5} $\\alpha\\parallel\\beta$. 由给出的两个条件能推导出$m\\parallel \\beta$的是\\bracket{20}.\n\\fourch{\\textcircled{1}\\textcircled{4}}{\\textcircled{1}\\textcircled{5}}{\\textcircled{2}\\textcircled{4}}{\\textcircled{3}\\textcircled{5}}",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0608001B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -105123,7 +105167,9 @@
|
|||
"004283": {
|
||||
"id": "004283",
|
||||
"content": "在正方体$ABCD-A_1B_1C_1D_1$中, $P$、$Q$两点分别从点$B$和点$A_1$出发, 以相同的速度在棱$BA$和$A_1D_1$上运动至点$A$和点$D_1$, 在运动过程中, 直线$PQ$与平面$ABCD$所成角$\\theta$的变化范围为\\bracket{20}.\n\\begin{center}\n \\begin{tikzpicture}\n \\draw (0,0) node [below left] {$A$} coordinate (A) --++ (2,0) node [below right] {$B$} coordinate (B) --++ (45:{2/2}) node [right] {$C$} coordinate (C)\n --++ (0,2) node [above right] {$C_1$} coordinate (C1)\n --++ (-2,0) node [above left] {$D_1$} coordinate (D1) --++ (225:{2/2}) node [left] {$A_1$} coordinate (A1) -- cycle;\n \\draw (A) ++ (2,2) node [right] {$B_1$} coordinate (B1) -- (B) (B1) --++ (45:{2/2}) (B1) --++ (-2,0);\n \\draw [dashed] (A) --++ (45:{2/2}) node [left] {$D$} coordinate (D) --++ (2,0) (D) --++ (0,2);\n \\draw [dashed] ($(A1)!0.3!(D1)$) node [above left] {$Q$} -- ($(B)!0.3!(A)$) node [below] {$P$};\n \\end{tikzpicture}\n\\end{center}\n\\twoch{$[\\dfrac\\pi4,\\dfrac\\pi3]$}{$[\\arctan\\dfrac{\\sqrt{2}}2,\\arctan\\sqrt 2]$}{$[\\dfrac\\pi4,\\arctan\\sqrt 2]$}{$[\\arctan\\dfrac{\\sqrt{2}}2,\\dfrac\\pi2]$}",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0610004B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -115662,7 +115708,10 @@
|
|||
"004696": {
|
||||
"id": "004696",
|
||||
"content": "如图, 在正方体$ABCD-A_1B_1C_1D_1$中, 点$MN$分别在棱$AA_1CC_1$上, 则``直线$MN\\perp\\text{直线}C_1B$''是``直线$MN\\perp\\text{平面}C_1BD$''的\\bracket{20}.\n\\begin{center}\n \\begin{tikzpicture}\n \\draw (0,0) node [below left] {$A$} coordinate (A) --++ (2,0) node [below right] {$B$} coordinate (B) --++ (40:{2/2}) node [right] {$C$} coordinate (C)\n --++ (0,2) node [above right] {$C_1$} coordinate (C1)\n --++ (-2,0) node [above left] {$D_1$} coordinate (D1) --++ (220:{2/2}) node [left] {$A_1$} coordinate (A1) -- cycle;\n \\draw (A) ++ (2,2) node [right] {$B_1$} coordinate (B1) -- (B) (B1) --++ (40:{2/2}) (B1) --++ (-2,0);\n \\draw [dashed] (A) --++ (40:{2/2}) node [left] {$D$} coordinate (D) --++ (2,0) (D) --++ (0,2);\n \\draw ($(A)!0.8!(A1)$) node [left] {$M$} coordinate (M);\n \\draw ($(C1)!0.8!(C)$) node [right] {$N$} coordinate (N);\n \\draw [dashed] (M) -- (N) (C1) -- (D) -- (B);\n \\draw (B) -- (C1);\n \\end{tikzpicture}\n\\end{center}\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既不充分又不必要条件}",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0609003B",
|
||||
"K0611002B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -216043,7 +216092,9 @@
|
|||
"009137": {
|
||||
"id": "009137",
|
||||
"content": "判断题: (下列命题中, 是真命题的在横线上填入``$\\checkmark$''; 是假命题的在横线上填入``$\\times$'')\\\\\n(1) 一条直线在平面内的射影是一条直线.\\blank{20};\\\\\n(2) 在平面内射影是直线的图形一定是直线.\\blank{20};\\\\\n(3) 如果两条线段在同一平面内的射影长相等, 那么这两条线段的长相等.\\blank{20};\\\\\n(4) 如果两条斜线与平面所成的角相等, 那么这两条斜线互相平行.\\blank{20}.",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0610002B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -216152,7 +216203,9 @@
|
|||
"009142": {
|
||||
"id": "009142",
|
||||
"content": "在长方体$ABCD-A_1B_1C_1D_1$中, $AB=BC=4$, $A_1A=5$, $M$是$AB$的中点. 求直线$C_1M$与平面$ABCD$所成的角的大小.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$A$} coordinate (A) --++ (2,0) node [below right] {$B$} coordinate (B) --++ (45:{2/2}) node [right] {$C$} coordinate (C)\n--++ (0,2.5) node [above right] {$C_1$} coordinate (C1)\n--++ (-2,0) node [above left] {$D_1$} coordinate (D1) --++ (225:{2/2}) node [left] {$A_1$} coordinate (A1) -- cycle;\n\\draw (A) ++ (2,2.5) node [above] {$B_1$} coordinate (B1) -- (B) (B1) --++ (45:{2/2}) (B1) --++ (-2,0);\n\\draw [dashed] (A) --++ (45:{2/2}) node [left] {$D$} coordinate (D) --++ (2,0) (D) --++ (0,2.5);\n\\draw [dashed] ($(A)!0.5!(B)$) node [below] {$M$} -- (C1);\n\\end{tikzpicture}\n\\end{center}",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0610004B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -216194,7 +216247,9 @@
|
|||
"009144": {
|
||||
"id": "009144",
|
||||
"content": "如图, $EF$分别是空间四边形$ABCD$的边$BCAD$的中点, 过$EF$且平行于$AB$的平面与$AC$交于点$G$, 求证: $G$是$AC$的中点.\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 (2,-1) node [below] {$C$} coordinate (C);\n\\draw (1.5,2) node [above] {$A$} coordinate (A);\n\\draw ($(B)!0.5!(C)$) node [below left] {$E$} coordinate (E);\n\\draw ($(A)!0.5!(D)$) node [above right] {$F$} coordinate (F);\n\\draw ($(A)!0.5!(C)$) node [above left] {$G$} coordinate (G);\n\\draw (A) -- (B) -- (C) -- (D) -- cycle (A) -- (C) (E) -- (G) -- (F);\n\\draw [dashed] (E) -- (F) (B) -- (D);\n\\end{tikzpicture}\n\\end{center}",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0608004B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -216215,7 +216270,9 @@
|
|||
"009145": {
|
||||
"id": "009145",
|
||||
"content": "在长方体$ABCD-A_1B_1C_1D_1$中, 矩形$AA_1D_1D$和$D_1C_1CD$的中心分别为$MN$, 求证: $MN\\parallel$平面$ABCD$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$A$} coordinate (A) --++ (3,0) node [below right] {$B$} coordinate (B) --++ (45:{2/2}) node [right] {$C$} coordinate (C)\n--++ (0,2\n) node [above right] {$C_1$} coordinate (C1)\n--++ (-3,0) node [above left] {$D_1$} coordinate (D1) --++ (225:{2/2}) node [left] {$A_1$} coordinate (A1) -- cycle;\n\\draw (A) ++ (3,2\n) node [right] {$B_1$} coordinate (B1) -- (B) (B1) --++ (45:{2/2}) (B1) --++ (-3,0);\n\\draw [dashed] (A) --++ (45:{2/2}) node [left] {$D$} coordinate (D) --++ (3,0) (D) --++ (0,2\n);\n\\draw [dashed] ($(A)!0.5!(D1)$) node [above] {$M$} -- ($(C)!0.5!(D1)$) node [right] {$N$}; \n\\end{tikzpicture}\n\\end{center}",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0608004B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -216406,7 +216463,9 @@
|
|||
"009154": {
|
||||
"id": "009154",
|
||||
"content": "在正方体中$ABCD-A_1B_1C_1D_1$中, $E_1$为$A_1D_1$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$A$} coordinate (A) --++ (3,0) node [below right] {$B$} coordinate (B) --++ (45:{3/2}) node [right] {$C$} coordinate (C)\n--++ (0,3) node [above right] {$C_1$} coordinate (C1)\n--++ (-3,0) node [above left] {$D_1$} coordinate (D1) --++ (225:{3/2}) node [left] {$A_1$} coordinate (A1) -- cycle;\n\\draw (A) ++ (3,3) node [right] {$B_1$} coordinate (B1) -- (B) (B1) --++ (45:{3/2}) (B1) --++ (-3,0);\n\\draw [dashed] (A) --++ (45:{3/2}) node [left] {$D$} coordinate (D) --++ (3,0) (D) --++ (0,3);\n\\draw ($(A1)!0.5!(D1)$) node [above left] {$E_1$} coordinate (E1);\n\\draw (D1) -- (B1) -- (A);\n\\draw [dashed] (A) -- (E1) -- (B) (A) -- (D1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求二面角$E_1-AB-C$的大小;\\\\\n(2) 求二面角$C_1-B_1D_1-A$的大小.",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0613003B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -216448,7 +216507,9 @@
|
|||
"009156": {
|
||||
"id": "009156",
|
||||
"content": "两个平面分别经过两条平行直线, 这两个平面是否平行?",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0612001B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -216490,7 +216551,9 @@
|
|||
"009158": {
|
||||
"id": "009158",
|
||||
"content": "选择题:\n(1) $\\alpha,\\beta$是两个不重合的平面, $a,b$是两条不同的直线, 在下列条件中可判定$\\alpha \\parallel\\beta$的是\\bracket{20}.\n\\onech{平面$\\alpha,\\beta$都平行于直线$a,b$}{平面内有三个不共线的点到平面$\\beta$的距离相等}{$a,b$是平面$\\alpha$内的两条直线, 且$\\alpha \\parallel\\beta$, $b\\parallel\\beta$}{$a,b$是两条异面直线, 且$a\\parallel\\alpha$, $b\\parallel\\alpha$, $\\alpha \\parallel\\beta$, $b\\parallel\\beta$}",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0612002B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -216595,7 +216658,9 @@
|
|||
"009163": {
|
||||
"id": "009163",
|
||||
"content": "设$a,b$是异面直线, 直线$a$在平面$a$内, 直线$b$在平面$\\beta$内, 且$a\\parallel\\beta$, $b\\parallel a$, 求证: $a\\parallel\\beta$.",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0612002B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -216616,7 +216681,9 @@
|
|||
"009164": {
|
||||
"id": "009164",
|
||||
"content": "已知不共面的三条直线$a,b,c$相交于点$P$, 平面$\\alpha,\\beta$与直线$a,b,c$分别相交于$A,B,C$和$A_1,B_1,C_1$, 且$\\dfrac{PA}{PA_1}=\\dfrac{PB}{PB_1}=\\dfrac{PC}{PC_1}$, 求证: $\\alpha \\parallel\\beta$.",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0612002B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -216849,7 +216916,9 @@
|
|||
"009175": {
|
||||
"id": "009175",
|
||||
"content": "在长方体$ABCD-A_1B_1C_1D_1$中, $AB=4$, $BC=AA=3$, 分别求直线$BD_1$与平面$ABCD$、直线$BD_1$与平面$BB_1C_1C$所成的角的大小.",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0610004B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -216975,7 +217044,10 @@
|
|||
"009181": {
|
||||
"id": "009181",
|
||||
"content": "如果长方体的一条对角线与过同一个顶点的三个面所成的角分别是$\\alpha,\\beta,\\gamma$那么$\\sin ^2\\alpha +\\sin ^2\\beta +\\sin ^2\\gamma \\text=$\\blank{50}.",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0610003B",
|
||||
"K0610004B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -217080,7 +217152,9 @@
|
|||
"009186": {
|
||||
"id": "009186",
|
||||
"content": "已知$P$为平行四边形$ABCD$所在平面外一点, $M$为$PB$的中点, 求证: $PD\\parallel$平面$MAC$.",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0608002B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -228042,7 +228116,11 @@
|
|||
"009685": {
|
||||
"id": "009685",
|
||||
"content": "判断下列命题的真假, 并说明理由:\\\\\n(1) 若两直线$a$、$b$互相平行, 则$a$平行于经过$b$的任何平面;\\\\\n(2) 若直线$a$与平面$\\alpha$平行, 则$a$平行于$\\alpha$内的任何直线;\\\\\n(3) 若两直线$a$、$b$都与平面$\\alpha$平行, 则$a\\parallel b$;\\\\\n(4) 若直线$a$平行于平面$\\alpha$, 直线$b$在平面$\\alpha$上, 则$a\\parallel b$或者$a$与$b$为异面直线.",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0608001B",
|
||||
"K0608002B",
|
||||
"K0608004B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -228063,7 +228141,10 @@
|
|||
"009686": {
|
||||
"id": "009686",
|
||||
"content": "证明: 若不在给定平面上的两条平行直线中的一条平行于给定平面, 则另一条直线也平行于给定平面.",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0608002B",
|
||||
"K0608004B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -228147,7 +228228,9 @@
|
|||
"009690": {
|
||||
"id": "009690",
|
||||
"content": "如图, 已知$PA$垂直于平面$\\alpha$, $PB$垂直于平面$\\beta$, $A$、$B$为相应的垂足, 且$l$为平面$\\alpha$与平面$\\beta$的交线. 求证: $l\\perp$平面$PAB$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.5]\n\\draw (0,0,0) coordinate (O) (2,0,0) coordinate (R) (-1,1,0) coordinate (L);\n\\draw (O) ++ (0,0,2) coordinate (O1) (R) ++ (0,0,2) coordinate (R1) (L) ++ (0,0,2) coordinate (L1);\n\\draw (L) -- (L1) -- (O1) -- (R1) -- (R);\n\\path [name path = OR] (O) -- (R);\n\\path [name path = OL] (O) -- (L);\n\\draw [name path = AP] (1.5,0,1) node [right] {$A$} coordinate (A) --++ (0,2,0) node [right] {$P$} coordinate (P);\n\\draw [name path = BP] (A) --++ (-1.5,0,0) coordinate (O2) --++ (-0.7,0.7,0) node[above] {$B$} coordinate (B) -- (P);\n\\draw [name intersections = {of = AP and OR, by = T}];\n\\draw [name intersections = {of = BP and OL, by = S}];\n\\draw (O2) -- (O1) (T) -- (R) (S) -- (L);\n\\draw [dashed] (O) -- (O1) (T) -- (O) -- (S);\n\\draw (1.9,0,2) node [above] {$\\alpha$} (L1) ++ (0.2,-0.2,0) node [above] {$\\beta$} (O1) node [above] {$l$};\n\\end{tikzpicture}\n\\end{center}",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0609003B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -228189,7 +228272,9 @@
|
|||
"009692": {
|
||||
"id": "009692",
|
||||
"content": "在正方体$ABCD-A_1B_1C_1D_1$中, $E$是边$A_1D_1$的中点.\\\\\n(1) 求$A_1C$和底面$ABCD$所成角的大小;\\\\\n(2) 求$EB$和底面$A_1B_1C_1D_1$所成角的大小.",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0610004B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -228210,7 +228295,9 @@
|
|||
"009693": {
|
||||
"id": "009693",
|
||||
"content": "如图, 平面$\\alpha$上的斜线$l$与平面$\\alpha$所成的角为$\\theta$, $l'$是$l$在平面$\\alpha$上的投影, $O$是$l$与平面$\\alpha$的交点, 点$B$是$l$上一点$A$在$\\alpha$上的投影, $OC$是$\\alpha$上的任意一条直线. 如果$\\theta =45^\\circ$, $\\angle BOC=45^\\circ$, 求$\\angle AOC$, 并验证$\\angle AOC>\\theta$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (-1,0,0) -- (3,0,0) node [right] {$l'$} (2,2,0) node [above] {$A$} coordinate (A) -- (2,0,0) coordinate (B) node [below] {$B$} (2.5,2.5,0) node [right] {$l$} -- (0,0,0) coordinate (O) node [below] {$O$};\n\\draw (1,0,1) node [below] {$C$} coordinate (C);\n\\draw ($(O)!-0.5!(C)$) -- ($(O)!1.8!(C)$) node [right] {$l''$};\n\\draw [name path = edge] (-1.5,0,-2.5) coordinate (L) -- (-1.5,0,2.5) --++ (5,0,0) --++ (0,0,-5) coordinate (R);\n\\path [name path = LR] (L) -- (R);\n\\path [name path = OA] (O) -- (A);\n\\path [name path = AB] (A) -- (B);\n\\path [name intersections = {of = OA and LR, by = A1}];\n\\path [name intersections = {of = AB and LR, by = B1}];\n\\draw (L) -- (A1) (B1) -- (R);\n\\draw [dashed] (A1) -- (B1);\n\\path [name path = down] ($(O)!-0.6!(A)$) -- (O);\n\\path [name intersections = {of = down and edge, by = T}];\n\\draw (T) -- ($(O)!-0.6!(A)$);\n\\draw [dashed] (T) -- (O);\n\\draw (O) pic [\"$\\theta$\",draw,angle eccentricity = 1.5] {angle = B--O--A};\n\\draw (O) pic [\"$45^\\circ$\",scale = 1.1,draw,angle eccentricity = 1.7]{angle = C--O--B};\n\\end{tikzpicture}\n\\end{center}",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0610005B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -228231,7 +228318,9 @@
|
|||
"009694": {
|
||||
"id": "009694",
|
||||
"content": "过$\\triangle ABC$所在平面$\\alpha$外的一点$P$, 作$PO\\perp \\alpha$, 垂足为$O$, 连接$PA$、$PB$及$PC$.\\\\\n(1) 若$PA=PB=PC$, 则点$O$是$\\triangle ABC$的\\blank{50}心;\\\\\n(2) 若$PA=PB=PC$, $\\angle ACB=90^\\circ$, 则点$O$是边$AB$的\\blank{50}点;\\\\\n(3) 若$PA\\perp PB$, $PB\\perp PC$, $PC\\perp PA$, 则点$O$是$\\triangle ABC$的\\blank{50}心.",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0610002B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -228273,7 +228362,12 @@
|
|||
"009696": {
|
||||
"id": "009696",
|
||||
"content": "如图, 已知$ABCD$是矩形, $PA\\perp$平面$ABCD$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$A$} coordinate (A) -- (2,0,0) node [below] {$B$} coordinate (B) -- (2,0,-2) node [right] {$C$} coordinate (C) -- (0,2,0) node [above] {$P$} coordinate (P) -- (A) (P) -- (B);\n\\draw [dashed] (0,0,-2) node [left] {$D$} coordinate (D) -- (P) (D) --(A) (D) --(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $\\angle PBC=90^\\circ$;\\\\\n(2) 若$PC\\perp BD$, 求证: 四边形$ABCD$是正方形.",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0609001B",
|
||||
"K0609002B",
|
||||
"K0611002B",
|
||||
"K0611003B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -228294,7 +228388,10 @@
|
|||
"009697": {
|
||||
"id": "009697",
|
||||
"content": "判断下列命题的真假, 并说明理由:\\\\\n(1) 若一个平面内的两条直线均平行于另一个平面, 则这两个平面平行;\n(2) 若一个平面内两条不平行的直线都平行于另一个平面, 则这两个平面平行;\n(3) 若两个平面平行, 则其中一个平面中的任何直线都平行于另一个平面;\n(4) 平行于同一个平面的两个平面平行;\n(5) 若一个平面内的任何一条直线都平行于另一个平面, 则这两个平面平行.",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0612002B",
|
||||
"K0612004B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -228315,7 +228412,11 @@
|
|||
"009698": {
|
||||
"id": "009698",
|
||||
"content": "如图, 已知正方体$ABCD-A_1B_1C_1D_1$的棱长为$a$, 求:\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$A$} coordinate (A) --++ (2,0) node [below right] {$B$} coordinate (B) --++ (45:{2/2}) node [right] {$C$} coordinate (C)\n--++ (0,2) node [above right] {$C_1$} coordinate (C1)\n--++ (-2,0) node [above left] {$D_1$} coordinate (D1) --++ (225:{2/2}) node [left] {$A_1$} coordinate (A1) -- cycle;\n\\draw (A) ++ (2,2) node [right] {$B_1$} coordinate (B1) -- (B) (B1) --++ (45:{2/2}) (B1) --++ (-2,0);\n\\draw [dashed] (A) --++ (45:{2/2}) node [left] {$D$} coordinate (D) --++ (2,0) (D) --++ (0,2);\n\\draw (B1) -- (D1);\n\\end{tikzpicture}\n\\end{center}\n(1) 点$A_1$到直线$BC$的距离;\\\\\n(2) 点$A$到平面$B_1BCC_1$的距离;\\\\\n(3) $B_1D_1$到平面$ABCD$的距离;\\\\\n(4) 平面$B_1BCC_1$到平面$A_1ADD_1$的距离.",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0612005B",
|
||||
"K0609007B",
|
||||
"K0609008B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -228357,7 +228458,9 @@
|
|||
"009700": {
|
||||
"id": "009700",
|
||||
"content": "已知平面$\\alpha\\perp$平面$\\beta$, 判断下列命题是否正确, 并说明理由:\\\\ \n(1) 平面$\\alpha$上的任意一条直线都垂直于平面$\\beta$上的任意一条直线;\\\\\n(2) 平面$\\alpha$上的任意一条直线都垂直于平面$\\beta$上的无数条直线;\\\\\n(3) 平面$\\alpha$上的任意一条直线都垂直于平面$\\beta$;\\\\\n(4) 过平面$\\alpha$上任意一点作平面$\\alpha$与$\\beta$交线的垂线$l$, 则$l\\perp \\beta$.",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0613009B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -228378,7 +228481,9 @@
|
|||
"009701": {
|
||||
"id": "009701",
|
||||
"content": "如图, 已知$AB\\perp$平面$BCD$, $BC\\perp CD$, 有哪些平面互相\n垂直? 为什么? \n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$B$} coordinate (B) -- (2.4,0,1.6) node [below] {$C$} coordinate (C) -- (3.6,0,0) node [right] {$D$} coordinate (D) -- (0,2,0) node [above] {$A$} coordinate (A);\n\\draw (A) -- (B) (A) -- (C);\n\\draw [dashed] (B) -- (D); \n\\end{tikzpicture}\n\\end{center}",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0613007B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -228399,7 +228504,9 @@
|
|||
"009702": {
|
||||
"id": "009702",
|
||||
"content": "证明: 如果两个平面垂直, 那么过第一个平面上一点且垂直于第二个平面的直线, 必在第一个平面上.",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0613009B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -245506,7 +245613,9 @@
|
|||
"010464": {
|
||||
"id": "010464",
|
||||
"content": "证明: 两条平行直线和同一个平面所成的角相等.",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0610003B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -286617,7 +286726,9 @@
|
|||
"030091": {
|
||||
"id": "030091",
|
||||
"content": "叙述直线与平面平行的判定定理.",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0608001B"
|
||||
],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
|
|
@ -286636,7 +286747,9 @@
|
|||
"030092": {
|
||||
"id": "030092",
|
||||
"content": "叙述并证明直线与平面平行的性质定理.",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0608003B"
|
||||
],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
|
|
@ -286655,7 +286768,9 @@
|
|||
"030093": {
|
||||
"id": "030093",
|
||||
"content": "叙述直线与平面垂直的判定定理.",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0609002B"
|
||||
],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
|
|
@ -286674,7 +286789,9 @@
|
|||
"030094": {
|
||||
"id": "030094",
|
||||
"content": "叙述并证明直线与平面垂直的性质定理.",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0609004B"
|
||||
],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
|
|
@ -286693,7 +286810,9 @@
|
|||
"030095": {
|
||||
"id": "030095",
|
||||
"content": "叙述平面与平面平行的判定定理.",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0612001B"
|
||||
],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
|
|
@ -286712,7 +286831,10 @@
|
|||
"030096": {
|
||||
"id": "030096",
|
||||
"content": "叙述并证明平面与平面平行的性质定理.",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0612003B",
|
||||
"K0613006B"
|
||||
],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
|
|
@ -286731,7 +286853,9 @@
|
|||
"030097": {
|
||||
"id": "030097",
|
||||
"content": "叙述平面与平面垂直的判定定理.",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0613008B"
|
||||
],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
|
|
@ -286769,7 +286893,10 @@
|
|||
"030099": {
|
||||
"id": "030099",
|
||||
"content": "回答下列问题, 并说明理由:\\\\\n(1) 垂直于同一直线的两个平面是否平行?\\\\\n(2) 平行于同一平面的两条直线是否平行?\\\\\n(3) 垂直于同一平面的两条直线是否平行?",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0609004B",
|
||||
"K0609005B"
|
||||
],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
|
|
@ -286793,7 +286920,9 @@
|
|||
"030100": {
|
||||
"id": "030100",
|
||||
"content": "已知$\\alpha$, $\\beta$是两个平行平面, 点$P\\in \\alpha$, 直线$l\\subset \\beta$. 证明: 过点$P$所作的$l$的平行线一定在平面$\\alpha$上.",
|
||||
"objs": [],
|
||||
"objs": [
|
||||
"K0612004B"
|
||||
],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "证明略",
|
||||
|
|
|
|||
Reference in New Issue