录入25届周末卷15补充题目并添加related
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@ -24736,7 +24736,9 @@
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"20220624\t朱敏慧, 王伟叶"
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],
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@ -140124,7 +140126,8 @@
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"000800",
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"000873",
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"004250",
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"011628"
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"011628",
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"023322"
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],
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"remark": "",
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"space": "",
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@ -301894,7 +301897,9 @@
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"20220806\t王伟叶"
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],
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"related": [
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"023319"
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"remark": "",
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"space": "4em",
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@ -338396,7 +338401,9 @@
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"004558"
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"023322"
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@ -345906,7 +345913,9 @@
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"20221212\t王伟叶"
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],
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@ -346877,7 +346886,9 @@
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"20221214\t王伟叶"
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],
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@ -361241,7 +361252,9 @@
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"20230118\t王伟叶"
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],
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@ -372009,7 +372022,9 @@
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"20230123\t王伟叶"
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],
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"023332"
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@ -464561,7 +464576,8 @@
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],
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"023279"
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"023279",
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"023320"
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@ -471560,7 +471576,9 @@
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"20230602\t王伟叶"
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],
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"same": [],
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"023339"
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@ -477465,7 +477483,9 @@
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"20230606\t王伟叶"
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],
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"related": [],
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"related": [
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"023320"
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@ -508961,7 +508981,9 @@
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"20230707\t王伟叶"
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],
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"related": [],
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"related": [
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"023329"
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"remark": "",
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"space": "4em",
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@ -525298,7 +525320,8 @@
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],
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"023279"
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"023279",
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@ -623893,6 +623916,787 @@
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"space": "4em",
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"unrelated": []
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},
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"023318": {
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"id": "023318",
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"content": "已知$n\\in \\mathbf{N}$, 且$n\\ge 6$. 若 $\\mathrm{C}_n^6=\\mathrm{C}_n^4$, 则 $n=$\\blank{50}.",
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"objs": [],
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"tags": [],
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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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"usages": [],
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"edit": [
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"20240107\t杨懿荔"
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"remark": "",
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"unrelated": []
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"023319": {
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"id": "023319",
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"content": "用 $1,2,3,4,5$ 组合成没有重复数字的三位数, 从中随机地取一个, 取得的数为偶数的概率是\\blank{50}.",
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"objs": [],
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"tags": [],
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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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"usages": [],
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"origin": "25届周末卷补充题目",
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"edit": [
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"20240107\t杨懿荔"
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"same": [],
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"010871"
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"remark": "",
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"space": "",
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"023320": {
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"id": "023320",
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"content": "总体是由编号为 $01$、$02$、$\\cdots$、$29$、$30$ 的 30 个个体组成. 利用下面的随机数表选取 5 个个体, 选取方法是从随机数表第 1 行的第 5 列和第 6 列数字开始由左到右依次选取两个数字, 则选出来的第 5 个个体的编号为\\blank{50}.\n\\begin{center}\n\\begin{tabular}{llllllll}\n7816 & 1572 & 0802 & 6315 & 0216 & 4319 & 9714 & 0198 \\\\\n3104 & 9234 & 4936 & 8200 & 3623 & 4869 & 6938 & 7181\n\\end{tabular}\n\\end{center}",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "",
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"duration": -1,
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"017725",
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"019851"
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"remark": "",
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"space": "",
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"unrelated": []
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"023321": {
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"id": "023321",
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"content": "记一个正方体的表面积为 $S_1$, 正方体的内切球的表面积为 $S_2$, 则 $\\dfrac{S_1}{S_2}=$\\blank{50}.",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "",
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"duration": -1,
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"same": [],
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"remark": "",
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"space": "",
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},
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"023322": {
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"id": "023322",
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"content": "$(x+\\dfrac{1}{\\sqrt{x}})^9$ 的二项展开式中的常数项为\\blank{50}.",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "",
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"duration": -1,
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"usages": [],
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"012209",
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"012511"
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"remark": "",
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"023323": {
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"id": "023323",
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"content": "6 名同学排队站成一排, 要求甲、乙两人不相邻, 共有\\blank{50}种不同的排法.",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "",
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"duration": -1,
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"edit": [
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"20240107\t杨懿荔"
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"same": [],
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"remark": "",
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"space": "",
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"023324": {
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"id": "023324",
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"content": "正方体 $ABCD-A_1B_1C_1D_1$ 的棱长为 $a$, $E$ 是棱 $DD_1$ 的中点, 则异面直线 $AB$ 与 $CE$ 的距离为\\blank{50}.",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "",
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"duration": -1,
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"edit": [
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"20240107\t杨懿荔"
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"same": [],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"023325": {
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"id": "023325",
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"content": "若一个圆锥的母线长为 $2$, 母线与旋转轴的夹角大小为 $30^{\\circ}$ , 则这个圆锥的侧面积为\\blank{50}.",
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"objs": [],
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"tags": [],
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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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"usages": [],
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"edit": [
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"same": [],
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"000668",
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"012484"
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"remark": "",
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"unrelated": []
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},
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"023326": {
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"id": "023326",
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"content": "如图, 已知一个二面角的平面角为 $120^{\\circ}$, 它的棱上有两个点、 $B$, 线段 $AC$、$BD$ 分别在这个二面角的两个面内, 并且都垂直于棱 $AB, AC=\\sqrt{2}$, $BD=2 \\sqrt{2}$, $CD=3 \\sqrt{2}$, 则线段 $AB$ 的长为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw (0,0,0) coordinate (P) (4,0,0) coordinate (Q);\n\\foreach \\i in {P,Q}\n{\\draw (\\i) ++ (0,0,4) coordinate (\\i_1) (\\i) ++ (0,{1.3*sqrt(3)},-1.3) coordinate (\\i_2);};\n\\draw (P)--(Q)--(Q_2)--(P_2)--cycle(P)--(P_1)--(Q_1)--(Q);\n\\draw (1,0,0) node [above left] {$A$} coordinate (A);\n\\draw (3,0,0) node [above] {$B$} coordinate (B);\n\\draw (A) ++ (0,{sqrt(3)},-1) node [left] {$C$} coordinate (C);\n\\draw (B) ++ (0,0,{2*sqrt(2)}) node [right] {$D$} coordinate (D);\n\\draw (C)--(D)(A)--(C)(B)--(D);\n\\end{tikzpicture}\n\\end{center}",
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"objs": [],
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"tags": [],
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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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"usages": [],
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"edit": [
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"20240107\t杨懿荔"
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],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"023327": {
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"id": "023327",
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"content": "某位同学参加物理、化学、生物科目的等级考, 依据以往成绩估算该同学在物理、化学、生物科目等级中达到 A$+$的概率分别为 $\\dfrac{5}{6}, \\dfrac{3}{4}, \\dfrac{3}{5}$ , 假设各门科目考试的结果互不影响, 则该同学等级考至多有 1 门学科没有获得 A$+$的概率为\\blank{50}.",
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"objs": [],
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"tags": [],
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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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"usages": [],
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"origin": "25届周末卷补充题目",
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},
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"023328": {
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"id": "023328",
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"content": "如图所示, 一个灯笼由一根提竿 $PQ$ 和一个圆柱组成, 提竿平行与圆柱的底面, 在圆柱上下底面圆周上分别有两点 $A$、$B$, $AB$ 与圆柱的底面不垂直, 则在圆绕着其旋转轴旋转一周的过程中, 直线 $PQ$ 与直线 $AB$ 垂直的次数为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (-1,0) arc (180:360:1 and 0.25);\n\\draw (-1,0) --++ (0,2) (1,0) --++ (0,2);\n\\draw (-1,2) arc (180:540:1 and 0.25);\n\\draw [dashed] (-1,0) arc (180:0:1 and 0.25);\n\\filldraw (0,2) circle (0.03) coordinate (T);\n\\draw (T) --++ (0,0.5) node [above right] {$Q$} coordinate (Q) --++ (-2,0) node [left] {$P$} coordinate (P);\n\\draw (0,2) ++ (-60:1 and 0.25) node [above] {$A$} coordinate (A);\n\\draw (0,0) ++ (-120:1 and 0.25) node [below] {$B$} coordinate (B);\n\\draw [dashed] (A)--(B);\n\\end{tikzpicture}\n\\end{center}",
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"objs": [],
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"tags": [],
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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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"usages": [],
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},
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"023329": {
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"id": "023329",
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"content": "有 8 本不同的书, 其中数学书 3 本, 外文书 2 本, 其他书 3 本, 若将这些书连排排成一列放在书架上, 则数学书恰好排在一起, 外文书也恰好排成一起的排法有\\blank{50}种.",
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"objs": [],
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"tags": [],
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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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"usages": [],
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},
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"023330": {
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"id": "023330",
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"content": "从 $(3 x+1)^5$ 的展开式各项的系数中任取两个, 其和为奇数的概率是\\blank{50}.",
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"objs": [],
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"tags": [],
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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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},
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"023331": {
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"id": "023331",
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"content": "已知各顶点都在一个球面上的正三棱柱的高为 $2$, 这个球的体积为 $\\dfrac{20 \\sqrt{5}}{3}\\pi$, 则这个正三棱柱的体积为\\blank{50}.",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "",
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||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "25届周末卷补充题目",
|
||||
"edit": [
|
||||
"20240107\t杨懿荔"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "",
|
||||
"unrelated": []
|
||||
},
|
||||
"023332": {
|
||||
"id": "023332",
|
||||
"content": "如果两个球的表面积之比为 $4: 9$, 那么这两个球的体积之比为\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "25届周末卷补充题目",
|
||||
"edit": [
|
||||
"20240107\t杨懿荔"
|
||||
],
|
||||
"same": [],
|
||||
"related": [
|
||||
"013381"
|
||||
],
|
||||
"remark": "",
|
||||
"space": "",
|
||||
"unrelated": []
|
||||
},
|
||||
"023333": {
|
||||
"id": "023333",
|
||||
"content": "正方体的 $6$ 个面无限延展后把空间分成\\blank{50}个部分.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "25届周末卷补充题目",
|
||||
"edit": [
|
||||
"20240107\t杨懿荔"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "",
|
||||
"unrelated": []
|
||||
},
|
||||
"023334": {
|
||||
"id": "023334",
|
||||
"content": "某电池厂有 $A, B$ 两条生产线, 现从 $A$ 生产线中取出产品 $8$ 件, 测得它们的可充电次数的平均值为 $210$ , 方差为 $4$ ; 从 $B$ 生产线中取出产品 $12$ 件, 测得它们的可充电次数的平均值为 $200$ , 方差为 $4$ . 则 $20$ 件产品组成的总样本的方差为\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "25届周末卷补充题目",
|
||||
"edit": [
|
||||
"20240107\t杨懿荔"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "",
|
||||
"unrelated": []
|
||||
},
|
||||
"023335": {
|
||||
"id": "023335",
|
||||
"content": "甲、乙两人组成``星队''参加猜谜活动, 每轮活动由甲、乙各猜一个, 已知甲每轮猜对的概率为 $\\dfrac{3}{4}$, 乙每轮猜对的概率为 $\\dfrac{2}{3}$. 在每轮活动中, 甲和乙猜对与否互不影响, 各轮结果也互不影响, 则``星队''在两轮活动中猜对 $3$ 个的概率为\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "25届周末卷补充题目",
|
||||
"edit": [
|
||||
"20240107\t杨懿荔"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "",
|
||||
"unrelated": []
|
||||
},
|
||||
"023336": {
|
||||
"id": "023336",
|
||||
"content": "在一个棱长为 $6 \\mathrm{cm}$ 的密封正方体盒子中, 放一个半径为 $1 \\mathrm{cm}$ 的小球, 无论怎样摇动盒子, 小球在盒子中不能达到的空间体积是\\blank{50} $\\mathrm{cm}^3$.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "25届周末卷补充题目",
|
||||
"edit": [
|
||||
"20240107\t杨懿荔"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "",
|
||||
"unrelated": []
|
||||
},
|
||||
"023337": {
|
||||
"id": "023337",
|
||||
"content": "在由正整数构成的无穷数列 $\\{a_n\\}$ 中, 对任意的正整数, 都有 $a_n \\leq a_{n+1}$ 且对任意的整数 $k$,数列 $\\{a_n\\}$ 中恰有 $k$ 个 $k$, 则 $a_{2023}=$\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "25届周末卷补充题目",
|
||||
"edit": [
|
||||
"20240107\t杨懿荔"
|
||||
],
|
||||
"same": [],
|
||||
"related": [
|
||||
"012992"
|
||||
],
|
||||
"remark": "",
|
||||
"space": "",
|
||||
"unrelated": []
|
||||
},
|
||||
"023338": {
|
||||
"id": "023338",
|
||||
"content": "棱长为 $1$ 的正方体 $ABCD-A_1B_1C_1D_1$, 点 $P$ 沿正方形 $ABCD$ 按 $ABCDA$ 的方向做匀速运动, 点 $Q$ 沿正方形 $B_1C_1CB$ 按 $B_1C_1CBB_1$ 的方向以同样的速度做匀速运动, 且点 $P$ 、 $Q$ 分别从点 $A$ 与点 $B_1$ 同时出发, 则 $PQ$ 的中点的轨迹所围成图形的面积大小是\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "25届周末卷补充题目",
|
||||
"edit": [
|
||||
"20240107\t杨懿荔"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "",
|
||||
"unrelated": []
|
||||
},
|
||||
"023339": {
|
||||
"id": "023339",
|
||||
"content": "若干个能确定一个立体图形的体积的量称为该立体图形的``基本量'', 已知长方体 $ABCD-A_1B_1C_1D_1$ , 下列四组量中, 不能作为该长方体的``基本量''的是\\bracket{20}.\n\\twoch{$AB$、$AD$、$AA_1$ 的长度}{$AB_1$、$AC$、$AD_1$ 的长度}{$AB$、$BA_1$、$BD_1$ 的长度}{$AB$、$AC_1$、$B_1C$ 的长度}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "25届周末卷补充题目",
|
||||
"edit": [
|
||||
"20240107\t杨懿荔"
|
||||
],
|
||||
"same": [],
|
||||
"related": [
|
||||
"017458"
|
||||
],
|
||||
"remark": "",
|
||||
"space": "",
|
||||
"unrelated": []
|
||||
},
|
||||
"023340": {
|
||||
"id": "023340",
|
||||
"content": "从某中学甲、乙两班各随机抽取 $10$ 名同学, 测量他们的身高 (单位: $\\mathrm{cm}$), 所得数据用茎叶图表示如下, 由此可估计甲、乙两班同学的身高情况, 则下列结论正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tabular}{ccc|c|ccccc}\n\\multicolumn{3}{r|}{甲班} & & \\multicolumn{5}{l}{乙班} \\\\\n\\hline\n& 2 & 1 & 18 & 2 \\\\\n8 & 2 & 0 & 17 & 1 & 2 & 6 & 8 & 9 \\\\\n6 & 5 & 3 & 16 & 2 & 4 & 7 \\\\\n& 8 & 7 & 15 & 9\n\\end{tabular}\n\\end{center}\n\\twoch{甲乙两班同学身高的极差不相等}{甲班同学身高的平均值较大}{甲班同学的身高的中位数较大}{甲班同学身高在 $175 \\mathrm{cm}$ 以上的人数较多}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "25届周末卷补充题目",
|
||||
"edit": [
|
||||
"20240107\t杨懿荔"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "",
|
||||
"unrelated": []
|
||||
},
|
||||
"023341": {
|
||||
"id": "023341",
|
||||
"content": "如图, 样本 $A$ 和 $B$ 分别取自两个不同的总体, 它们的平均数分别为 $\\overline{x_A}$ 和 $\\overline{x_B}$, 标准差分别为 $s_A$ 和 $s_B$, 则 \\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = {3/7}, yscale = {3/17.5}]\n\\draw [->] (0,0) -- (7,0) node [below] {$n$};\n\\draw [->] (0,0) -- (0,17.5) node [left] {$x_n$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {1,2,...,6}\n{\\draw [dashed] (\\i,0) node [below] {$\\i$} -- (\\i,17.5);};\n\\foreach \\i in {2.5,5,7.5,10,12.5,15,17.5}\n{\\draw [dashed] (0,\\i) --++ (7,0);};\n\\draw [dashed] (7,0) -- (7,17.5);\n\\foreach \\i in {5,10,15}\n{\\draw (0,\\i) node [left] {$\\i$};};\n\\draw (4.5,16.25) node {$A$};\n\\draw [dashed] (1,2.5) -- (2,10) -- (3,5) -- (4,7.5) -- (5,2.5) -- (6,10);\n\\foreach \\i in {(1,2.5),(2,10),(3,5),(4,7.5),(5,2.5),(6,10)}\n{\\filldraw \\i circle (0.14 and 0.35);};\n\\end{tikzpicture}\n\\hspace*{3em}\n\\begin{tikzpicture}[>=latex, xscale = {3/7}, yscale = {3/17.5}]\n\\draw [->] (0,0) -- (7,0) node [below] {$n$};\n\\draw [->] (0,0) -- (0,17.5) node [left] {$x_n$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {1,2,...,6}\n{\\draw [dashed] (\\i,0) node [below] {$\\i$} -- (\\i,17.5);};\n\\foreach \\i in {2.5,5,7.5,10,12.5,15,17.5}\n{\\draw [dashed] (0,\\i) --++ (7,0);};\n\\draw [dashed] (7,0) -- (7,17.5);\n\\foreach \\i in {5,10,15}\n{\\draw (0,\\i) node [left] {$\\i$};};\n\\draw (4.5,16.25) node {$B$};\n\\draw [dashed] (1,15) -- (2,10) -- (3,12.5) -- (4,10) -- (5,12.5) -- (6,10);\n\\foreach \\i in {(1,15),(2,10),(3,12.5),(4,10),(5,12.5),(6,10)}\n{\\filldraw \\i circle (0.14 and 0.35);};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\overline{x_A}>\\overline{x_B}$, $s_A>s_B$}{$\\overline{x_A}<\\overline{x_B}$, $s_A>s_B$}{$\\overline{x_A}>\\overline{x_B}$, $s_A<s_B$}{$\\overline{x_A}<\\overline{x_B}$, $s_A<s_B$}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "25届周末卷补充题目",
|
||||
"edit": [
|
||||
"20240107\t杨懿荔"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "",
|
||||
"unrelated": []
|
||||
},
|
||||
"023342": {
|
||||
"id": "023342",
|
||||
"content": "``杨辉三角''是中国古代数学文化的瑰宝之一, 它揭示了二项式展开式中的组合数在三角形数表中的一种几何排列规律, 如图所示, 则下列关于``杨辉三角''的结论正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,xscale = 0.3, yscale = 0.5]\n\\foreach \\i in {0,1,...,8}\n{\\draw (-12,{-\\i}) node {第$\\i$行};};\n\\draw (-12,-9) node {$\\cdots$};\n\\draw (0,0) node {$1$};\n\\draw (-1,-1) node {$1$} (1,-1) node {$1$};\n\\draw (-2,-2) node {$1$} (0,-2) node {$2$} (2,-2) node {$1$};\n\\foreach \\i/\\j in {0/1,1/3,2/3,3/1}\n{\\draw ({2*\\i-3},-3) node {$\\j$};};\n\\foreach \\i/\\j in {0/1,1/4,2/6,3/4,4/1}\n{\\draw ({2*\\i-4},-4) node {$\\j$};};\n\\foreach \\i/\\j in {0/1,1/5,2/10,3/10,4/5,5/1}\n{\\draw ({2*\\i-5},-5) node {$\\j$};};\n\\foreach \\i/\\j in {0/1,1/6,2/15,3/20,4/15,5/6,6/1}\n{\\draw ({2*\\i-6},-6) node {$\\j$};};\n\\foreach \\i/\\j in {0/1,1/7,2/21,3/35,4/35,5/21,6/7,7/1}\n{\\draw ({2*\\i-7},-7) node {$\\j$};};\n\\foreach \\i/\\j in {0/1,1/8,2/28,3/56,4/70,5/56,6/28,7/8,8/1}\n{\\draw ({2*\\i-8},-8) node {$\\j$};};\n\\draw (0,-9) node {$\\cdots$};\n\\draw (0,1) node {杨辉三角};\n\\end{tikzpicture}\n\\end{center}\n\\onech{$\\mathrm{C}_3^2+\\mathrm{C}_4^2+\\mathrm{C}_5^2+\\cdots+\\mathrm{C}_{10}^2=165$}{在第 2022 行中第 1011 个数最大}{第 6 行的第 7 个数、第 7 行的第 7 个数及第 8 行的第 7 个数之和等于第 9 行的第 8 个数}{第 34 行中第 15 个数与第 16 个数之比为 $2: 3$}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "25届周末卷补充题目",
|
||||
"edit": [
|
||||
"20240107\t杨懿荔"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "",
|
||||
"unrelated": []
|
||||
},
|
||||
"023343": {
|
||||
"id": "023343",
|
||||
"content": "一个质地均匀的正四面体木块的四个面上分别标有数字 $1$、$2$、$3$、$4$. 连续抛掷这个正四面体木块两次, 并记录每次正四面体木块朝下的面上的数字, 记事件 $A$ 为``第一次向下的数字为 2 或 3'', 事件 $B$ 为``两次向下的数字之和为奇数'', 则下列结论正确的是\\bracket{20}.\n\\twoch{$P(A)=\\dfrac{1}{4}$}{事件 $A$ 与事件 $B$ 互斥}{事件 $A$ 与事件 $B$ 互相独立}{$P(A \\cup B)=\\dfrac{1}{2}$}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "25届周末卷补充题目",
|
||||
"edit": [
|
||||
"20240107\t杨懿荔"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "",
|
||||
"unrelated": []
|
||||
},
|
||||
"023344": {
|
||||
"id": "023344",
|
||||
"content": "如图, 三棱柱 $ABC-A_1B_1C_1$ 满足棱长都相等, 且 $AA_1 \\perp$ 平面 $ABC$, $D$ 是棱 $CC_1$ 的中点, $E$ 是棱 $AA_1$ 上的动点, 设 $AE=x$, 随着 $x$ 的增大, 平面 $BDE$ 与底面 $ABC$ 所成锐二面角的平面角是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\h{2}\n\\draw ({-\\l/2},0,0) node [left] {$A$} coordinate (A);\n\\draw (0,0,{\\l/2*sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw ({\\l/2},0,0) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,\\h) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\h) node [below right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\h) node [right] {$C_1$} coordinate (C_1);\n\\draw (A) -- (B) -- (C) (A) -- (A_1) (B) -- (B_1) (C) -- (C_1) (A_1) -- (B_1) -- (C_1) (A_1) -- (C_1);\n\\draw [dashed] (A) -- (C);\n\\draw ($(C)!0.5!(C_1)$) node [right] {$D$} coordinate (D);\n\\draw ($(A)!0.3!(A_1)$) node [left] {$E$} coordinate (E);\n\\draw (E)--(B)--(D);\n\\draw [dashed] (E)--(D);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{先增大再减小}{减小}{增大}{先减小再增大}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "25届周末卷补充题目",
|
||||
"edit": [
|
||||
"20240107\t杨懿荔"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "",
|
||||
"unrelated": []
|
||||
},
|
||||
"023345": {
|
||||
"id": "023345",
|
||||
"content": "抛掷一颗均匀的骰子, 设事件 $A$ 表示``点数为奇数'', 事件 $B$ 表示``点数不超过 2''.\\\\\n(1) 求 $P(A \\cup B)$\\\\\n(2) 再抛掷一次骰子, 设事件 $C$ 表示``两次点数的差的绝对值不小于 4'', 求 $P(C)$.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "25届周末卷补充题目",
|
||||
"edit": [
|
||||
"20240107\t杨懿荔"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"023346": {
|
||||
"id": "023346",
|
||||
"content": "若 $(1-x-2 x^2)^5=a_0+a_1 x+a_2 x^2+\\cdots+a_{10}x^{10}$.\\\\\n(1) 求 $a_0+a_1+a_2+\\cdots+a_{10}$ 的值.\\\\\n(2) 求 $a_2+a_4+a_6+a_8+a_{10}$ 的值.\\\\\n(3) 求 $a_1$ 的值.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "25届周末卷补充题目",
|
||||
"edit": [
|
||||
"20240107\t杨懿荔"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"023347": {
|
||||
"id": "023347",
|
||||
"content": "某高校承办了奥运会的志愿者选拔面试工作, 现随机抽取了 100 名候选者的面试成绩并分成五组: 第一组 $[45,55)$ , 第二组 $[55,65)$ , 第三组 $[65,75)$ , 第四组 $[75,85)$ , 第五组 $[85,95) , $ 绘制成如图所示的频率分布直方图, 已知第三、四、五组的频率之和为 $0.7$, 第一组和第五组的频率相同.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.05, yscale = 60]\n\\draw [->] (35,0) -- (38,0) -- (39,0.003) -- (41,-0.003) -- (42,0)-- (105,0) node [below] {分数};\n\\draw [->] (35,0) -- (35,0.06) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (35,0) node [below left] {$O$};\n\\foreach \\i/\\j in {45/0.005,55/0.025,65/0.045,75/0.020,85/0.005}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (10,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {55/0.025/b,65/0.045,75/0.020,85/0.005/a}\n{\\draw [dashed] (\\i,\\j) -- (35,\\j) node [left] {$\\k$};};\n\\draw (95,0) node [below] {$95$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求 $a, b$ 的值;\\\\\n(2) 估计这个 100 名候选者面试成绩的平均数和第 60 百分位数 (精确到 $0.1$);\\\\\n(3) 在第四、五两组志愿者中, 按比例分层抽样抽取 5 人, 然后再从这 5 人中选出 2 人, 求选出的两人来自同一组的概率.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "25届周末卷补充题目",
|
||||
"edit": [
|
||||
"20240107\t杨懿荔"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"023348": {
|
||||
"id": "023348",
|
||||
"content": "将一个边长为 $2$ 的正六边形 $ABCDEF$ (图 1) 沿 $CF$ 对折, 形成如图 2 所示的五面体, 其中, 底面 $ABDE$ 是正方形.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\foreach \\i/\\j/\\k in {-120/A/below left,-60/B/below right,0/C/right,60/D/above right,120/E/above left,180/F/left}\n{\\draw (\\i:1) node [\\k] {$\\j$} coordinate (\\j);};\n\\draw (A)--(B)--(C)--(D)--(E)--(F)--cycle;\n\\draw ($(A)!0.5!(B)$) ++ (0,-0.7) node {图1};\n\\end{tikzpicture}\n\\hspace*{3em}\n\\begin{tikzpicture}[>=latex]\n\\draw (-0.5,0,0.5) node [below left] {$A$} coordinate (A);\n\\draw (0.5,0,0.5) node [below right] {$B$} coordinate (B);\n\\draw (-0.5,0,-0.5) node [above] {$E$} coordinate (E);\n\\draw (0.5,0,-0.5) node [right] {$D$} coordinate (D);\n\\draw (-1,{sqrt(2)/2},0) node [left] {$F$} coordinate (F);\n\\draw (F) ++ (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (F)--(A)--(B)--(C)--cycle(C)--(D)--(B);\n\\draw [dashed] (F)--(E)--(D)(E)--(A);\n\\draw ($(A)!0.5!(B)$) ++ (0,-0.7) node {图2};\n\\end{tikzpicture}\n\\hspace*{3em}\n\\begin{tikzpicture}[>=latex]\n\\draw (-0.5,0,0.5) node [below left] {$A$} coordinate (A);\n\\draw (0.5,0,0.5) node [below right] {$B$} coordinate (B);\n\\draw (-0.5,0,-0.5) node [above] {$E$} coordinate (E);\n\\draw (0.5,0,-0.5) node [right] {$D$} coordinate (D);\n\\draw (-1,{sqrt(2)/2},0) node [left] {$F$} coordinate (F);\n\\draw (F) ++ (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (F)--(A)--(B)--(C)--cycle(C)--(D)--(B);\n\\draw [dashed] (F)--(E)--(D)(E)--(A);\n\\draw ($(E)!0.5!(D)$) node [above] {$H$} coordinate (H);\n\\draw ($(A)!0.3!(B)$) node [below] {$G$} coordinate (G);\n\\draw (F)--(G);\n\\draw [dashed] (G)--(H)--(F);\n\\draw ($(A)!0.5!(B)$) ++ (0,-0.7) node {图3};\n\\end{tikzpicture}\n\\end{center}\n(1) 求二面角 $A-FC-D$ 的大小;\\\\\n(2) 如图 3, 点 $G$、$H$ 分别为棱 $AB$、$ED$ 上的动点, 求 $\\Delta FGH$ 周长的最大值.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "25届周末卷补充题目",
|
||||
"edit": [
|
||||
"20240107\t杨懿荔"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"023349": {
|
||||
"id": "023349",
|
||||
"content": "设 $S_n$ 是等差数列 $\\{a_n\\}$ 的前 $n$ 项和, 数列 $\\{b_n\\}$ 满足 $b_n=n-(-1)^n S_n$, $a_1+b_1=3$, $a_2-b_2=5$.\\\\\n(1) 求数列 $\\{b_n\\}$ 的通项公式;\\\\\n(2) 设数列 $\\{b_n\\}$ 的前 $n$ 项和为 $T_n$,\\\\\n\\textcircled{1} 求 $T_{10}$ ; \\\\\n\\textcircled{2} 若集合 $A=\\{n | n \\leq 100$ 且 $T_n \\leq 100$, $n \\in \\mathbb{N}, n \\geq 1\\}$ , 求集合 $A$ 中所有元素的和.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "25届周末卷补充题目",
|
||||
"edit": [
|
||||
"20240107\t杨懿荔"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"023350": {
|
||||
"id": "023350",
|
||||
"content": "从 $1$、$2$、$3$、$\\cdots$、$99$ 这 $99$ 个自然数中, 每次任取 $5$ 个不同的数, 若 $5$ 个数能成等差数列, 则这样的等差数列共有\\blank{50}个.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "25届周末卷补充题目",
|
||||
"edit": [
|
||||
"20240107\t杨懿荔"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "",
|
||||
"unrelated": []
|
||||
},
|
||||
"023351": {
|
||||
"id": "023351",
|
||||
"content": "化简: $1\\mathrm{C}_{100}^1+2\\mathrm{C}_{100}^2+3\\mathrm{C}_{100}^3+\\cdots+50\\mathrm{C}_{100}^{50}=$\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "25届周末卷补充题目",
|
||||
"edit": [
|
||||
"20240107\t杨懿荔"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "",
|
||||
"unrelated": []
|
||||
},
|
||||
"023352": {
|
||||
"id": "023352",
|
||||
"content": "已知三棱锥 $P-ABC$ 的顶点 $P$ 在底面的射影 $O$ 与 $\\triangle ABC$ 的垂心重合, 且 $\\dfrac{S_{\\triangle ABC}}{S_{\\triangle PBC}}=\\dfrac{S_{\\triangle PBC}}{S_{\\triangle OBC}}$ , 若三棱锥 $P-ABC$ 的外接球半径为 $3$ , 则 $S_{\\triangle PAB}+S_{\\triangle PBC}+S_{\\triangle PCA}$ 的最大值为\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "25届周末卷补充题目",
|
||||
"edit": [
|
||||
"20240107\t杨懿荔"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "",
|
||||
"unrelated": []
|
||||
},
|
||||
"023353": {
|
||||
"id": "023353",
|
||||
"content": "等差数列 $\\{a_n\\}$ 的通项公式是 $a_n=3 n-1$ , 等比数列 $\\{b_n\\}$ 满足 $b_1=a_p$, $b_2=a_q$, 其中 $q>p>1$, 且 $n, p, q$ 均为正整数. 有关数列 $\\{b_n\\}$, 有如下四个命题:\\\\\n\\textcircled{1} 存在 $p, q$, 使得数列 $\\{b_n\\}$ 的所有项均在数列 $\\{a_n\\}$ 中; \\\\\n\\textcircled{2} 存在 $p, q$ , 使得数列 $\\{b_n\\}$ 仅有有限项(至少 $1$ 项)不在数列 $\\{a_n\\}$ 中; \\\\\n\\textcircled{3} 存在 $p, q$, 使得数列 $\\{b_n\\}$ 的某一项的值为 $2023$;\\\\\n\\textcircled{4} 存在 $p, q$, 使得数列 $\\{b_n\\}$ 的前若干项的和为 $2023$.\\\\\n其中正确的命题个数是\\bracket{20}个.\n\\fourch{0}{1}{2}{3}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "25届周末卷补充题目",
|
||||
"edit": [
|
||||
"20240107\t杨懿荔"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "",
|
||||
"unrelated": []
|
||||
},
|
||||
"023354": {
|
||||
"id": "023354",
|
||||
"content": "已知正方体 $ABCD-A_1B_1C_1D_1$ 中, $AB=6$, 点 $P$ 在平面 $AB_1D_1$ 内, $A_1P=3 \\sqrt{2}$, 求点 $P$ 到 $BC_1$ 距离的最小值.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "25届周末卷补充题目",
|
||||
"edit": [
|
||||
"20240107\t杨懿荔"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"023355": {
|
||||
"id": "023355",
|
||||
"content": "有一种投掷骰子走跳棋的游戏: 棋盘上标有第 1 站、第 2 站、第 3 站、... 第 10 站,共 10 站, 设棋子跳到第 $n$ 站的概率为 $p_n$ , 若一枚棋子开始在第 1 站, 棋手每次投掷骰子一次, 棋子向前跳动一次, 若骰子点数小于等于 $3$ , 棋子向前跳一站; 否则, 棋子向前跳两站, 直到棋子跳到第 9 站(失败)或者第 10 站(获胜)时, 游戏结束, 求该棋手获胜的概率.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "25届周末卷补充题目",
|
||||
"edit": [
|
||||
"20240107\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.",
|
||||
|
|
|
|||
Reference in New Issue