wangweiye
/
mathdeptv2
Archived
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

录入2025届高二周末卷03补充题目

This commit is contained in:
weiye.wang 2024-01-05 22:44:51 +08:00
parent 83093ec0b1
commit 66dacb3369
1 changed files with 220 additions and 0 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

220
题库0.3/Problems.json
View File

@ -620183,6 +620183,226 @@
"space": "4em",
"unrelated": []
},
"023142": {
"id": "023142",
"content": "点 $P$ 到 $\\triangle ABC$ 三个顶点的距离都等于 $4$, $O$ 为 $P$ 在平面 $ABC$ 上的射影, $\\triangle ABC$ 的三边长为 $3$、$4$、$5$, 则 $PO=$\\blank{50}.",
"objs": [],
"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "25届周末卷补充题目",
"edit": [
"20240105\t杨懿荔"
],
"same": [],
"related": [],
"remark": "",
"space": "",
"unrelated": []
},
"023143": {
"id": "023143",
"content": "已知 $A$、$B$ 是平面 $\\alpha$ 外的两点, 它们在平面 $\\alpha$ 内的射影分别是 $A_1$、$B_1$. 若 $AA_1=3 \\mathrm{cm}$, $BB_1=5 \\mathrm{cm}$, $A_1B_1=10 \\mathrm{cm}$, 则直线 $AB$ 与平面 $\\alpha$ 的所成角的大小是\\blank{50}.",
"objs": [],
"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "25届周末卷补充题目",
"edit": [
"20240105\t杨懿荔"
],
"same": [],
"related": [],
"remark": "",
"space": "",
"unrelated": []
},
"023144": {
"id": "023144",
"content": "棱长为 $2$ 的正方体 $ABCD-A_1B_1C_1D_1$ 中, 点 $P$ 是棱 $CC_1$ 的中点, 则直线 $AP$ 与平面 $BCC_1B_1$ 所成的角的大小是\\blank{50}.",
"objs": [],
"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "25届周末卷补充题目",
"edit": [
"20240105\t杨懿荔"
],
"same": [],
"related": [],
"remark": "",
"space": "",
"unrelated": []
},
"023145": {
"id": "023145",
"content": "如图, Rt $\\triangle ABC$ 的直角顶点 $A$ 在平面 $\\alpha$ 上, $BC \\parallel \\alpha$, $BC=6$, $AB$、$AC$ 与 $\\alpha$ 分别成 $30^{\\circ}$、$45^{\\circ}$ 角, 则 $BC$ 与平面 $\\alpha$ 的距离是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw (0,0,0) coordinate (S);\n\\draw (6,0,0) coordinate (T);\n\\draw (S)++ (0,{sqrt(6)},0) node [above] {$B$} coordinate (B);\n\\draw (T)++ (0,{sqrt(6)},0) node [above] {$C$} coordinate (C);\n\\draw (4,0,{sqrt(2)}) node [below] {$A$} coordinate (A);\n\\draw (A)--(S)--(B)--(C)--(T)--cycle(B)--(A)--(C);\n\\draw [dashed] (S)--(T);\n\\path [name path = BS] (B)--(S);\n\\path [name path = CT] (C)--(T);\n\\draw (S)++(-1,0,-2) coordinate (U) --++ (0,0,6) node [above right = 0 and 0.2] {$\\alpha$} --++ (8,0,0) --++ (0,0,-6) coordinate (V);\n\\path [name path = UV] (U)--(V);\n\\draw [name intersections = {of = UV and BS, by = M}];\n\\draw [name intersections = {of = UV and CT, by = N}];\n\\draw (U)--(M)(V)--(N);\n\\draw [dashed] (M)--(N);\n\\end{tikzpicture}\n\\end{center}",
"objs": [],
"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "25届周末卷补充题目",
"edit": [
"20240105\t杨懿荔"
],
"same": [],
"related": [],
"remark": "",
"space": "",
"unrelated": []
},
"023146": {
"id": "023146",
"content": "设 $PA, PB, PC$ 是从点 $P$ 引出的三条射线, 任意两条射线的夹角都等于 $60^{\\circ}$, 则直线 $PC$ 与平面 $APB$ 所成角的余弦值是\\blank{50}.",
"objs": [],
"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "25届周末卷补充题目",
"edit": [
"20240105\t杨懿荔"
],
"same": [],
"related": [],
"remark": "",
"space": "",
"unrelated": []
},
"023147": {
"id": "023147",
"content": "直线 $a$ 与平面 $\\alpha$ 所成的角的大小为 $20^{\\circ}$, 直线 $b$ 与平面 $\\alpha$ 所成的角的大小为 $50^{\\circ}$, 设直线 $a$ 与直线 $b$ 所成的角的大小为 $n^{\\circ}$, 则 $n$ 的取值范围为\\blank{50}.",
"objs": [],
"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "25届周末卷补充题目",
"edit": [
"20240105\t杨懿荔"
],
"same": [],
"related": [],
"remark": "",
"space": "",
"unrelated": []
},
"023148": {
"id": "023148",
"content": "正方体 $ABCD-A_1B_1C_1D_1$ 中.\\\\ \n(1) 若 $G$ 是 $CD$ 的中点, 求异面直线 $B_1C$ 与 $AG$ 所成角的正弦值; (2) 求直线 $BA_1$ 与平面 $ABC_1D_1$ 所成角大小.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "25届周末卷补充题目",
"edit": [
"20240105\t杨懿荔"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"023149": {
"id": "023149",
"content": "已知 $PC$ 垂直于三角形 $ABC$ 所在平面, 且 $AB=BC=CA=PC=1$, 求点 $B$ 到平面 $PAC$ 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": [
"20240105\t杨懿荔"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"023150": {
"id": "023150",
"content": "正方体 $ABCD-A_1B_1C_1D_1$, 点 $O$ 为棱 $BC$ 的中点, 过点 $O$ 的直线 $l$ 与直线 $AA_1$、$C_1D_1$ 分别交于 $M, N$ 两点, 求直线 $MN$ 与平面 $ADD_1A_1$ 所成角的正弦值.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "25届周末卷补充题目",
"edit": [
"20240105\t杨懿荔"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"023151": {
"id": "023151",
"content": "如图, 在底面是矩形的四棱锥 $P-ABCD$ 中, $PA \\perp$ 平面 $ABCD, PA=AB=1$, $BC=2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.5]\n\\draw (0,0,0) node [below] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,0,1) node [left] {$B$} coordinate (B);\n\\draw (2,0,1) node [right] {$C$} coordinate (C);\n\\draw (0,1,0) node [above] {$P$} coordinate (P);\n\\draw ($(P)!0.5!(D)$) node [above right] {$E$} coordinate (E);\n\\draw (B)--(C)--(D)--(P)--cycle(P)--(C);\n\\draw [dashed] (B)--(A)--(D)(A)--(E)(A)--(P);\n\\end{tikzpicture}\n\\end{center}\n(1) 求 $PC$ 与平面 $PAD$ 所成角的大小;\\\\\n(2) 若 $E$ 是 $PD$ 的中点, 求异面直线 $AE$ 与 $PC$ 所成角的大小;\\\\\n(3) 在 $BC$ 边上是否存在一点 $G$, 使得 $D$ 点到平面 $PAG$ 的距离为 $\\sqrt{2}$, 若存在, 求出 $BG$ 的值; 若不存在, 请说明理由.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "25届周末卷补充题目",
"edit": [
"20240105\t杨懿荔"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"023152": {
"id": "023152",
"content": "如图, 正方形 $ABCD$ 所在平面外有一点 $M$, 满足 $MD \\perp$ 平面 $ABCD$, $AB=\\sqrt{3}DM$. $P$ 为线段 $AD$ 上的动点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$D$} coordinate (D);\n\\draw (3,0,0) node [right] {$C$} coordinate (C);\n\\draw (3,0,3) node [right] {$B$} coordinate (B);\n\\draw (0,0,3) node [left] {$A$} coordinate (A);\n\\draw (0,{sqrt(3)},0) node [above] {$M$} coordinate (M);\n\\draw ($(A)!0.5!(D)$) node [below] {$P$} coordinate (P);\n\\draw ($(M)!0.25!(C)$) node [above right] {$E$} coordinate (E);\n\\draw (M)--(A)--(B)--(C)--cycle(M)--(B);\n\\draw [dashed] (A)--(D)--(C)(M)--(D)(M)--(P)--(B)(A)--(E)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $AD \\perp$ 平面 $CDM$;\\\\\n(2) 若 $P$ 为线段 $AD$ 的中点, 求直线 $MP$ 与平面 $BCM$ 所成角的大小;\\\\\n(3) 设 $E$ 为线段 $CM$ 上一点, 满足 $AE \\perp CM$, 问: 是否存在 $P$, 使得 $DE \\parallel $ 平面 $BMP$ ? 若存在, 求 $\\dfrac{AP}{DP}$ 的值; 若不存在, 说明理由.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "25届周末卷补充题目",
"edit": [
"20240105\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) ``$0<x<3$''是``$|x-1|<2$'' 的\\blank{50}条件.",