收录高二寒假作业2
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20240203-221900 高二寒假作业1(9-16题)
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019519,024425:024427,031901,023142,024428:024430,022083
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20240204-000056 高二寒假作业2
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024431:024432,016870,024433,016886,023213,023215,016827,024434:024435,016865,024436:024437,014695,024438:024441
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"20220624\t王伟叶, 余利成"
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@ -661891,6 +661903,238 @@
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"024431": {
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"id": "024431",
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"content": "已知正四棱锥 $P-ABCD$ 的高为 $7$ , 且 $AB=2$, 则正四棱锥 $P-ABCD$ 的侧面积为\\blank{50}.",
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"objs": [],
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"genre": "填空题",
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"ans": "",
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"024432": {
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"id": "024432",
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"content": "若正四棱柱 $ABCD-A_1B_1C_1D_1$ 的底面边长为 $2$, 高为 $4$, 则异面直线 $BD_1$ 与 $AD$ 所成角的大小为\\blank{50}.",
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"024433": {
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"id": "024433",
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"content": "已知正四棱锥的体积为 12 , 底面对角线的长为 $2 \\sqrt{6}$, 则侧面与底面所成的二面角等于\\blank{50}.",
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"024434": {
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"id": "024434",
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"content": "在直三棱柱 $ABC-A_1 B_1 C_1$ 中, $\\angle ABC=90^{\\circ}$, $AB=BC=1$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,0) node [below right] {$B$} coordinate (B);\n\\draw (0,0,2) node [left] {$A$} coordinate (A);\n\\draw (A) --++ (0,{2*sqrt(2)}) node [left] {$A_1$} coordinate (A_1);\n\\draw [dashed] (B) --++ (0,{2*sqrt(2)}) node [above left] {$B_1$} coordinate (B_1);\n\\draw (C) --++ (0,{2*sqrt(2)}) node [right] {$C_1$} coordinate (C_1);\n\\draw (A) -- (C) (A_1) -- (B_1) -- (C_1) -- cycle (A_1) -- (C);\n\\draw [dashed] (B) -- (C) (A) -- (C) (A) -- (B);\n\\end{tikzpicture}\n\\end{center}\n(1) 求异面直线 $B_1 C_1$ 与 $AC$ 所成角的大小;\\\\\n(2) 若直线 $A_1 C$ 与平面 $ABC$ 所成角为 $45^{\\circ}$, 求三棱锥 $A_1-ABC$ 的体积.",
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"genre": "解答题",
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"ans": "",
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"024435": {
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"id": "024435",
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"content": "下面是关于顶点在底面的投影在底面三角形内部的三棱锥的四个命题:\\\\\n\\textcircled{1} 底面是等边三角形, 且侧面与底面所成的二面角都相等的三棱锥是正三棱锥.\\\\\n\\textcircled{2} 底面是等边三角形, 侧面都是等腰三角形的三棱锥是正三棱锥.\\\\\n\\textcircled{3} 底面是等边三角形, 各侧面的面积都相等的三棱锥是正三棱锥.\\\\\n\\textcircled{4} 侧棱与底面所成的角相等, 且侧面与底面所成的二面角都相等的三棱锥是正三棱锥.\\\\\n其中, 真命题的编号是\\blank{50}. (写出所有真命题的编号)",
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"genre": "填空题",
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"024436": {
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"id": "024436",
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"content": "已知直角三角形的两直角边长分别为 $3 \\mathrm{cm}$ 和 $4 \\mathrm{cm}$, 则以斜边为轴旋转一周所得几何体的表面积为\\blank{50}.",
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"024437": {
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"id": "024437",
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"content": "如图, 在圆锥 $SO$ 中, 已知底面半径 $r=1$, 母线长 $l=4$, $M$ 为母线 $SA$ 上的一个点, 且 $SM=2$, 从点 $M$ 拉一根绳子, 围绕圆锥侧面转到点 $A$. 则绳子的最短长度为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (-1,0) node [left] {$A$} coordinate (A);\n\\draw (0,{sqrt(15)}) node [above] {$S$} coordinate (S);\n\\draw ($(S)!0.5!(A)$) node [left] {$M$} coordinate (M);\n\\draw (A) arc (180:360:1 and 0.25) -- (S)--cycle;\n\\draw [dashed] (A) arc (180:0:1 and 0.25) (A) -- (1,0) coordinate (B) (O)--(S);\n\\draw ($(S)!{sqrt(2)/3}!(B)$) coordinate (N);\n\\draw (A) .. controls ++ (1,0.5) and ($(S)!1.1!(N)$) .. (N);\n\\draw [dashed] (N) .. controls ($(S)!0.9!(N)$) and ++ (0.3,0.3) .. (M);\n\\end{tikzpicture}\n\\end{center}",
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"objs": [],
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"genre": "填空题",
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"ans": "",
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"024438": {
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"id": "024438",
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"content": "若圆台上、下底面的圆周都在一个直径为 10 的球面上, 其上、下底面半径分别为 4 和 5 , 则该圆台的体积为\\blank{50}.",
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"024439": {
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"id": "024439",
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"content": "如图, 在三棱柱 $ABC-A_1B_1C_1$ 中, 侧面 $CBB_1C_1$ 是菱形, $\\angle C_1CB=60^{\\circ}$, 平面 $ABC \\perp$ 平面 $CBB_1C_1$, $M$ 为 $BB_1$ 的中点, $AC \\perp BC$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$C$} coordinate (C);\n\\draw (0,0,2) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$B$} coordinate (B);\n\\draw (A) ++ (1,{sqrt(3)},0) node [above] {$A_1$} coordinate (A_1);\n\\draw (C) ++ (1,{sqrt(3)},0) node [above] {$C_1$} coordinate (C_1);\n\\draw (B) ++ (1,{sqrt(3)},0) node [right] {$B_1$} coordinate (B_1);\n\\draw ($(B)!0.5!(B_1)$) node [right] {$M$} coordinate (M);\n\\draw (A)--(B)--(B_1)--(A_1)--cycle(A_1)--(C_1)--(B_1)(A_1)--(M);\n\\draw [dashed] (A)--(C)--(B)(C)--(A_1)(C)--(C_1)(C)--(M)--(C_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $CC_1 \\perp$ 平面 $A_1C_1M$;\\\\\n(2) 若 $CA=CB=2$, 求三棱锥 $C_1-A_1CM$ 的体积.",
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"genre": "解答题",
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"024440": {
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"id": "024440",
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"content": "长方体 $ABCD-A_1B_1C_1D_1$ 中, $AB=2$, $BC=a$, $(a>0)$, $AA_1=2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{3}\n\\def\\n{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,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,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\\end{tikzpicture}\n\\end{center}\n(1) 在 $BC$ 边上是否存在点 $Q$, 使得 $A_1Q \\perp QD$, 为什么?\\\\\n(2) 当存在 $BC$ 边上点 $Q$ 使 $A_1Q \\perp QD$ 时, 求 $a$ 的最小值, 并求出此时二面角 $A-A_1D-Q$ 的大小.",
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"id": "024441",
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"content": "北京大兴国际机场的显著特点之一是各种弯曲空间的运用, 刻画空间的弯曲性是几何研究的重要内容. 用曲率刻画空间弯曲性, 规定: 多面体顶点的曲率等于 $2 \\pi$与多面体在该点的面角之和的差(多面体的面的内角叫做多面体的面角, 角度用弧度制),多面体面上非顶点的曲率均为零, 多面体的总曲率等于该多面体各顶点的曲率之和. 例如: 正四面体在每个顶点有 3 个面角, 每个面角是 $\\dfrac{\\pi}{3}$, 所以正四面体在各顶点的曲率为 $2 \\pi-3 \\times \\dfrac{\\pi}{3}=\\pi$, 故其总曲率为 $4 \\pi$.\\\\\n(1) 求四棱锥的总曲率;\\\\\n(2) 若多面体满足: 顶点数 $-$ 棱数 $+$ 面数 $=2$, 证明: 这类多面体的总曲率是常数.",
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"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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@ -691399,7 +691643,9 @@
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],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
|
|
|
|||
Reference in New Issue