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@ -4520,7 +4520,9 @@
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"20220624\t王伟叶, 余利成"
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],
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"same": [],
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"related": [],
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"related": [
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"030148"
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],
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"remark": "",
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"space": "12ex"
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@ -41730,7 +41732,7 @@
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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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"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}[scale = 0.5]\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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"K0611002B"
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],
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@ -42251,7 +42253,7 @@
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},
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"001641": {
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"id": "001641",
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"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.",
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"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$PAD$所成角的正切值;\\\\ \n(3) 求直线$PC$与平面$PAB$所成角的正切值.",
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"objs": [
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"K0610004B"
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],
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@ -42453,7 +42455,7 @@
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},
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"001649": {
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"id": "001649",
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"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}",
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"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}",
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"objs": [
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"K0612001B"
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],
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@ -42863,7 +42865,9 @@
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"20220625\t王伟叶"
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],
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"same": [],
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"related": [],
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"related": [
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"030146"
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],
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"remark": "",
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"space": ""
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},
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@ -105871,7 +105875,7 @@
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},
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"004283": {
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"id": "004283",
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"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{$[\\dfrac\\pi4,\\dfrac\\pi3]$}{$[\\arctan\\dfrac{\\sqrt{2}}2,\\arctan\\sqrt 2]$}{$[\\dfrac\\pi4,\\arctan\\sqrt 2]$}{$[\\arctan\\dfrac{\\sqrt{2}}2,\\dfrac\\pi2]$}",
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"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}[scale = 0.7]\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{$[\\dfrac\\pi4,\\dfrac\\pi3]$}{$[\\arctan\\dfrac{\\sqrt{2}}2,\\arctan\\sqrt 2]$}{$[\\dfrac\\pi4,\\arctan\\sqrt 2]$}{$[\\arctan\\dfrac{\\sqrt{2}}2,\\dfrac\\pi2]$}",
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"objs": [
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"K0610004B"
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],
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@ -105879,7 +105883,7 @@
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"第六单元"
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],
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"genre": "选择题",
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"ans": "",
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"ans": "C",
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"solution": "",
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"duration": -1,
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"usages": [
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@ -116412,7 +116416,7 @@
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},
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"004696": {
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"id": "004696",
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"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}",
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"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}[scale = 0.6]\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}",
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"objs": [
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"K0609003B",
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"K0611002B"
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@ -116421,7 +116425,7 @@
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"第六单元"
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],
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"genre": "选择题",
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"ans": "",
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"ans": "C",
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"solution": "",
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"duration": -1,
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"usages": [
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@ -217016,7 +217020,7 @@
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},
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"009144": {
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"id": "009144",
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"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}",
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"content": "如图, $E$、$F$分别是空间四边形$ABCD$的边$BC$、$AD$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}",
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"objs": [
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"K0608004B"
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],
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@ -217039,7 +217043,7 @@
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},
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"009145": {
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"id": "009145",
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"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}",
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"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$M$、$N$, 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}",
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"objs": [
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"K0608004B"
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],
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@ -217320,7 +217324,7 @@
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},
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"009158": {
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"id": "009158",
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"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$a\\parallel\\alpha$, $b\\parallel\\alpha$, $\\alpha \\parallel\\beta$, $b\\parallel\\beta$}",
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"content": "$\\alpha,\\beta$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$\\alpha$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$a \\parallel\\beta$, $b\\parallel\\beta$}{$a,b$是两条异面直线, 且$a\\parallel\\alpha$, $b\\parallel\\alpha$, $a \\parallel\\beta$, $b\\parallel\\beta$}",
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"objs": [
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"K0612002B"
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],
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@ -217467,7 +217471,9 @@
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"20220726\t王伟叶"
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],
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"same": [],
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"related": [],
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"related": [
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"030144"
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],
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"remark": "",
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"space": "12ex"
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},
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@ -228930,7 +228936,9 @@
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"20220730\t王伟叶"
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],
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"same": [],
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"related": [],
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"related": [
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"030149"
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],
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"remark": "",
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"space": "12ex"
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},
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@ -229023,7 +229031,7 @@
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},
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"009690": {
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"id": "009690",
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"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}",
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"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 = 0.8]\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}",
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@ -229107,7 +229115,9 @@
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"20220730\t王伟叶"
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@ -229183,7 +229193,7 @@
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},
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"009697": {
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"id": "009697",
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"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.",
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"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.",
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"objs": [
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@ -229201,7 +229211,9 @@
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"20220730\t王伟叶"
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@ -229270,7 +229282,9 @@
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"20220730\t王伟叶"
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],
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"030150"
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"remark": "",
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"space": "12ex"
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@ -229293,7 +229307,9 @@
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"20220730\t王伟叶"
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],
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"related": [
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"030145"
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"remark": "",
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"space": "12ex"
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@ -288079,7 +288095,7 @@
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"same": [],
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"related": [],
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"030093": {
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"id": "030093",
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@ -288121,7 +288137,7 @@
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"same": [],
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"related": [],
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"remark": "",
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"space": ""
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"space": "18ex"
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},
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"030095": {
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"id": "030095",
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@ -288164,7 +288180,7 @@
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"same": [],
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"related": [],
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"space": "18ex"
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},
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"030097": {
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"id": "030097",
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@ -288204,7 +288220,7 @@
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"same": [],
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"related": [],
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"remark": "",
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"space": ""
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"space": "18ex"
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},
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"030099": {
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"id": "030099",
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@ -288228,7 +288244,8 @@
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],
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"same": [],
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"related": [
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"009169"
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"009169",
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"030147"
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],
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"remark": "",
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"space": "24ex"
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@ -289114,5 +289131,223 @@
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"related": [],
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"remark": "",
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"space": ""
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},
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"030143": {
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"id": "030143",
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"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.\\\\\n(1) 如果$\\theta =45^\\circ$, $\\angle BOC=45^\\circ$, 求$\\angle AOC$;\\\\\n(2) 试证明: $\\angle AOC>\\theta$.\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 ($(O)!-0.5!(C)$) -- ($(O)!1.8!(C)$);\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 [scale = 1.1,draw,angle eccentricity = 1.7]{angle = C--O--B};\n\\end{tikzpicture}\n\\end{center}",
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"objs": [],
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"tags": [
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"第六单元"
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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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"duration": -1,
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"usages": [],
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"origin": "新教材必修第三册课堂练习-20221011修改",
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"edit": [
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"20220730\t王伟叶",
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"20221011\t余利成"
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],
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"same": [],
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"009693"
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"space": "12ex"
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},
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"030144": {
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"id": "030144",
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"content": "已知不共面的三条射线$a,b,c$均以点$P$为端点, 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: $\\alpha \\parallel\\beta$.",
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"objs": [],
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"tags": [
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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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"duration": -1,
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"20220726\t王伟叶",
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"20221011\t徐慧"
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],
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"same": [],
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"009164"
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],
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"remark": "",
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"space": "12ex"
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},
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"030145": {
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"id": "030145",
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"content": "如图, 已知$AB\\perp$平面$BCD$, $BC\\perp CD$, 有哪些平面互相垂直? 选择其中一对互相垂直的平面给出证明. \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}",
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"genre": "解答题",
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"ans": "",
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"edit": [
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"20220730\t王伟叶",
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"20221011\t徐慧"
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"009701"
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"030146": {
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"id": "030146",
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"content": "过$60^\\circ$的二面角$\\alpha-l-\\beta$的棱上一点$A$, 分别在$\\alpha,\\beta$内引两条射线, 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}.",
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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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"20220625\t王伟叶",
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"20221011\t徐慧"
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],
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"001665"
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"space": ""
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},
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"030147": {
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"id": "030147",
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"content": "下列命题是否为真命题? 如果是, 请回答``真''; 如果不是, 请说明理由:\\\\\n(1) 垂直于同一直线的两个平面平行;\\\\\n(2) 平行于同一平面的两条直线平行;\\\\\n(3) 垂直于同一平面的两条直线平行.",
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"genre": "解答题",
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"edit": [
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"20220726\t王伟叶",
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"20221002\t王伟叶",
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"20221011\t吴惠群, 余利成"
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],
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"same": [],
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"009169",
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"030099"
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"space": "24ex"
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},
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"030148": {
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"id": "030148",
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"content": "下列命题是否为真命题? 如果是, 请回答``真''; 如果不是, 请说明理由:\\\\\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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"tags": [
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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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"duration": -1,
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"usages": [],
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"origin": "教材复习题-20221011修改",
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"edit": [
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"20220624\t王伟叶, 余利成",
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"20221011\t吴惠群, 余利成"
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],
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"same": [],
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"related": [
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"000178"
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"remark": "",
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"space": "12ex"
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},
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"030149": {
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"id": "030149",
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"content": "下列命题是否为真命题? 如果是, 请回答``真''; 如果不是, 请说明理由:\\\\\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.",
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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": "新教材必修第三册课堂练习-20221011修改",
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"edit": [
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"20220730\t王伟叶",
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"20221011\t吴惠群, 余利成"
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],
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"same": [],
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"related": [
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"009685"
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],
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"remark": "",
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"space": "12ex"
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},
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"030150": {
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"id": "030150",
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"content": "已知平面$\\alpha\\perp$平面$\\beta$, 下列命题是否为真命题? 如果是, 请回答``真''; 如果不是, 请说明理由:\\\\ \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$l\\perp \\beta$.",
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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": "新教材必修第三册课堂练习-20221011修改",
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"edit": [
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"20220730\t王伟叶",
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"20221011\t吴惠群, 徐慧"
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],
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"same": [],
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"related": [
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"009700"
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],
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"remark": "",
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"space": "12ex"
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},
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"030151": {
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"id": "030151",
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"content": "下列命题是否为真命题? 如果是, 请回答``真''; 如果不是, 请说明理由:\\\\\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.",
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"objs": [],
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"tags": [
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"第六单元"
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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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"duration": -1,
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"usages": [],
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"origin": "新教材必修第三册课堂练习-20221011修改",
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"edit": [
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"20220730\t王伟叶",
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"20221011\t吴惠群, 徐慧"
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],
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"same": [],
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"related": [
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"009697"
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],
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"remark": "",
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"space": "12ex"
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}
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}
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