录入建平高二期中考试试题及答案
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#修改起始id,出处,文件名
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starting_id = 40605
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starting_id = 40626
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raworigin = ""
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filename = r"C:\Users\weiye\Documents\wwy sync\临时工作区\自拟题目12.tex"
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editor = "20230423\t王伟叶"
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@ -1,65 +1,65 @@
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ans
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40605
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$6$
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40606
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$90$
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40626
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$3$
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40607
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$10$
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40627
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$462$
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40608
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$15$
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40628
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$1$
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40609
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$24$
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40629
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$240$
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40610
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$(0,\dfrac{9\sqrt{2}-2}4]$
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40630
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$\dfrac{31}3$
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40611
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$540$
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40631
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$2$或$3$
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40612
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$\dfrac{16\pi}3$
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40632
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$\mathrm{e}$
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40613
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$35$
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40633
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$2449440x^6$
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40614
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\textcircled{2}\textcircled{4}
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40634
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$\dfrac{15}{56}$
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40615
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$[6,+\infty)$
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40635
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$(-\infty,1)$
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40616
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$\log_2 \mathrm{e}$
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40636
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$6790$
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40617
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C
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40637
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$\dfrac{8\sqrt{3}}9$
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40618
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40638
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A
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40619
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D
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40620
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40639
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B
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40621
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(1) $\dfrac{\sqrt{10}}{10}$; (2) $\dfrac{3\sqrt{5}}{5}$
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40640
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C
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40622
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(1) $80$; (2) $1$或$-3$
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40641
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B
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40623
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(1) $\pm \sqrt{3}$; (2) 存在, 点$M$的坐标为$(-3,0)$
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40642
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(1) $y=-2x+2$; (2) $[3-\sqrt{6},3+\sqrt{6}]$
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40624
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(1) $\dfrac{x^2}4+y^2=1$; (2) $k=\pm 2$; (3) 定值为$\dfrac 23$, 证明略
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40643
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(1) $\dfrac 13$; (2) $\dfrac{17}{81}$
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40625
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(1) $y=-2$; (2) $k=0$或$4$; (3) $\dfrac 34-\ln 2$
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40644
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(1) $\theta=\dfrac\pi 4$时, $S=\dfrac{2-\sqrt{2}+\sqrt{3}}2R^2$, ``得地率''约为$74\%$; (2) $\theta = \arcsin\dfrac{1+\sqrt{33}}8$时, ``得地率''最大
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40645
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(1) $b=\sqrt{3}$; (2) $a=2$; (3) $C$的坐标为$(-\dfrac 43,-\dfrac 13)$或$(-\dfrac 43,\dfrac 13)$
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40646
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(1) $f(x)$是``线性控制函数'', $g(x)$不是``线性控制函数'', 理由略; (2) 是真命题, 理由略; (3) 证明略
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@ -475468,5 +475468,404 @@
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"related": [],
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"remark": "",
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"space": "12ex"
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},
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"040626": {
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"id": "040626",
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"content": "已知$\\mathrm{C}_9^x=\\mathrm{C}_9^{2 x}$, 则正整数$x=$\\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": "$3$",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "202304建平高二期中考试试题1",
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"edit": [
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"20230423\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": ""
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},
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"040627": {
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"id": "040627",
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"content": "$\\mathrm{C}_4^4+\\mathrm{C}_5^4+\\mathrm{C}_6^4+\\mathrm{C}_7^4+\\mathrm{C}_8^4+\\mathrm{C}_9^4+\\mathrm{C}_{10}^4=$\\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": "$462$",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "202304建平高二期中考试试题2",
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"edit": [
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"20230423\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": ""
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},
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"040628": {
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"id": "040628",
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"content": "函数$f(x)=\\dfrac{1}{2} x^2-2 x+\\ln x$的驻点为\\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": "$1$",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "202304建平高二期中考试试题3",
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"edit": [
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"20230423\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": ""
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},
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"040629": {
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"id": "040629",
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"content": "$(x+\\dfrac{2}{\\sqrt{x}})^6$的二项展开式中常数项是\\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": "$240$",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "202304建平高二期中考试试题4",
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"edit": [
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"20230423\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": ""
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},
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"040630": {
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"id": "040630",
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"content": "函数$f(x)=\\dfrac{1}{3} x^3+3 x^2+5 x+2$的极大值为\\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": "$\\dfrac{31}3$",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "202304建平高二期中考试试题5",
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"edit": [
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"20230423\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": ""
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},
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"040631": {
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"id": "040631",
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"content": "已知男、女学生共有$8$人, 若从男生中任选$2$人, 从女生中任选$1$人, 共有$30$种不同的选法, 则女生的总人数为\\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": "$2$或$3$",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "202304建平高二期中考试试题6",
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"edit": [
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"20230423\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": ""
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},
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"040632": {
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"id": "040632",
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"content": "已知函数$f(x)=(x-1) \\cdot \\mathrm{e}^x$, 则$\\displaystyle\\lim _{x \\to 1} \\dfrac{f(x)-f(1)}{x-1}=$\\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": "$\\mathrm{e}$",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "202304建平高二期中考试试题7",
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"edit": [
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"20230423\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": ""
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},
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"040633": {
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"id": "040633",
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"content": "$(2+3 x)^{10}$的二项展开式中系数最大的项为\\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": "$2449440x^6$",
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"solution": "",
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"duration": -1,
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"usages": [],
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||||
"origin": "202304建平高二期中考试试题8",
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"edit": [
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"20230423\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": ""
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},
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"040634": {
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"id": "040634",
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"content": "一场晩会共有$5$个唱歌节目和$3$个舞蹈节目, 随机排序形成一个节目单, 则节目单中前$3$个节目有$2$个舞蹈节目的概率为\\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": "$\\dfrac{15}{56}$",
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"solution": "",
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"duration": -1,
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"usages": [],
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||||
"origin": "202304建平高二期中考试试题9",
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"edit": [
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||||
"20230423\t王伟叶"
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],
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||||
"same": [],
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||||
"related": [],
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||||
"remark": "",
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||||
"space": ""
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},
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"040635": {
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"id": "040635",
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"content": "已知关于$x$的不等式$x-\\ln x-a>0$对任意$x \\in(0,+\\infty)$恒成立, 则实数$a$的取值范围是\\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": "$(-\\infty,1)$",
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||||
"solution": "",
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"duration": -1,
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"usages": [],
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||||
"origin": "202304建平高二期中考试试题10",
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"edit": [
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"20230423\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": ""
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},
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"040636": {
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"id": "040636",
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"content": "若$(1+x)^8+(2+x)^8=a_0+a_1(1-x)^1+a_2(1-x)^2+\\cdots+a_8(1-x)^8$对任意$x \\in \\mathbf{R}$恒成立, 则$a_4=$\\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": "$6790$",
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"solution": "",
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"duration": -1,
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"usages": [],
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||||
"origin": "202304建平高二期中考试试题11",
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"edit": [
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"20230423\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": ""
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},
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"040637": {
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"id": "040637",
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"content": "已知$A(a, 1-a^2)$, $B(b, 1-b^2)$, 其中$a b<0$, 过$A$、$B$分别作二次函数$y=1-x^2$的切线, 则两条切线与$x$轴围成的三角形面积的最小值为\\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": "$\\dfrac{8\\sqrt{3}}9$",
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"solution": "",
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"duration": -1,
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"usages": [],
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||||
"origin": "202304建平高二期中考试试题12",
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"edit": [
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||||
"20230423\t王伟叶"
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||||
],
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||||
"same": [],
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||||
"related": [],
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"remark": "",
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"space": ""
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},
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"040638": {
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"id": "040638",
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"content": "在古典概率模型中, $\\Omega$是样本空间, $x$是样本点, $A$是随机事件, 则下列表述正确的\\bracket{20}.\n\\fourch{$x \\in \\Omega$}{$x \\subseteq \\Omega$}{$A \\in \\Omega$}{$\\Omega \\subseteq A$}",
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"objs": [],
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||||
"tags": [],
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||||
"genre": "选择题",
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"ans": "A",
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"solution": "",
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"duration": -1,
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||||
"usages": [],
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||||
"origin": "202304建平高二期中考试试题13",
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"edit": [
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||||
"20230423\t王伟叶"
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||||
],
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||||
"same": [],
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"related": [],
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||||
"remark": "",
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||||
"space": ""
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},
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"040639": {
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"id": "040639",
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||||
"content": "已知$A$、$B$为两个随机事件, 则``$A$、$B$为互斥事件''是``$A$、$B$为对立事件''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{非充分非必要条件}",
|
||||
"objs": [],
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||||
"tags": [],
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||||
"genre": "选择题",
|
||||
"ans": "B",
|
||||
"solution": "",
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||||
"duration": -1,
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||||
"usages": [],
|
||||
"origin": "202304建平高二期中考试试题14",
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||||
"edit": [
|
||||
"20230423\t王伟叶"
|
||||
],
|
||||
"same": [],
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||||
"related": [],
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||||
"remark": "",
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||||
"space": ""
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||||
},
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||||
"040640": {
|
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"id": "040640",
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||||
"content": "下列关于排列数$\\mathrm{P}_n^{m-1}$和组合数$\\mathrm{C}_n^{m-1}$的计算中正确的是\\bracket{20}.\n\\twoch{$\\mathrm{P}_n^{m-1}=\\dfrac{n !}{(m-1) !}$}{$\\mathrm{P}_n^{m-1}=\\dfrac{n !}{(n-m-1) !}$}{$\\mathrm{C}_n^{m-1}=\\dfrac{n !}{(m-1) !(n-m+1) !}$}{$\\mathrm{C}_n^{m-1}=\\dfrac{n !}{(m-1) !(n-m-1) !}$}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "选择题",
|
||||
"ans": "C",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "202304建平高二期中考试试题15",
|
||||
"edit": [
|
||||
"20230423\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": ""
|
||||
},
|
||||
"040641": {
|
||||
"id": "040641",
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||||
"content": "已知$x \\in \\mathbf{N}$, $y \\in \\mathbf{N}$, $x<y$, 则方程$x^y=y^x$的解的组数为\\bracket{20}.\n\\fourch{$0$}{$1$}{$2$}{无穷多个}",
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||||
"objs": [],
|
||||
"tags": [],
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||||
"genre": "选择题",
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||||
"ans": "B",
|
||||
"solution": "",
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||||
"duration": -1,
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||||
"usages": [],
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||||
"origin": "202304建平高二期中考试试题16",
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||||
"edit": [
|
||||
"20230423\t王伟叶"
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||||
],
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||||
"same": [],
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||||
"related": [],
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||||
"remark": "",
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||||
"space": ""
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||||
},
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"040642": {
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||||
"id": "040642",
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||||
"content": "已知函数$f(x)=-x^3+a x^2+(1-2 a) x+a$.\\\\\n(1) 求函数$f(x)$在$x=1$处的切线方程;\\\\\n(2) 若函数$f(x)$在$\\mathbf{R}$上严格减, 求实数$a$的取值范围.",
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||||
"objs": [],
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||||
"tags": [],
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||||
"genre": "解答题",
|
||||
"ans": "(1) $y=-2x+2$; (2) $[3-\\sqrt{6},3+\\sqrt{6}]$",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "202304建平高二期中考试试题17",
|
||||
"edit": [
|
||||
"20230423\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "12ex"
|
||||
},
|
||||
"040643": {
|
||||
"id": "040643",
|
||||
"content": "甲、乙两人进行乒乓球决赛, 采用五局三胜制, 对于每局比赛, 甲获胜的概率为$\\dfrac{2}{3}$, 乙获胜的概率为$\\dfrac{1}{3}$, 且每局比赛的结果互相独立.\\\\\n(1) 在乒乓球比赛中, 如果一方全胜最终获得比赛的胜利, 那么将其形象地称之为``剃光头''. 求甲、乙的这场乒兵球决赛``剃光头''的概率;\\\\\n(2) 在乒乓球比赛中, 如果实力较弱的一方最终获得比赛的胜利, 那么将其称之为``爆冷门'', 求甲、乙的这场乒乓球决赛``爆冷门''的概率.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "(1) $\\dfrac 13$; (2) $\\dfrac{17}{81}$",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "202304建平高二期中考试试题18",
|
||||
"edit": [
|
||||
"20230423\t王伟叶"
|
||||
],
|
||||
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|
||||
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|
||||
"remark": "",
|
||||
"space": "12ex"
|
||||
},
|
||||
"040644": {
|
||||
"id": "040644",
|
||||
"content": "``得地率''是指可供人活动的区域的占地面积与总占地面积之比. ``得地率''越高, 也就意味着人们可活动的区域更大, 因此在设计活动场地时, 通常会将``得地率''作为一个重要的指标进行考虑. 上海某大型购物商场欲将地下一层的一块半圆形空地改建为亲子乐园, 建造一个供亲子游玩的海洋球池和两个供人们休息和娱乐, 且大小完全相同的休息区. 除海洋球池和休息区外的剩余空地全部用气垫筑起``高墙'', 以保护亲子乐园中的人们. 如图所示, 设半圆形空地的圆心为$O$, 半径为$R$, $MN$为直径, 矩形海洋球池$ABCD$的顶点$A$、$B$在$MN$上, 顶点$C$、$D$在半圆的圆周上, 矩形休息区$BEFG$和$AHIJ$的顶点$E$、$H$在$MN$上, 顶点$F$、$I$在半圆的圆周上, 顶点$G$、$J$分别在线段$BC$、$AD$上. 已知$\\angle EOF=\\dfrac{\\pi}{6}$, 设$\\angle BOC=\\theta$, 其中$\\theta \\in[\\dfrac{\\pi}{4}, \\dfrac{\\pi}{3}]$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\t{50}\n\\draw (3,0) node [below] {$N$} coordinate (N);\n\\draw (-3,0) node [below] {$M$} coordinate (M);\n\\filldraw (0,0) circle (0.03) node [below] {$O$} coordinate (O);\n\\draw (N) arc (0:180:3) -- cycle;\n\\draw (30:3) node [above right] {$F$} coordinate (F);\n\\draw (150:3) node [above left] {$I$} coordinate (I);\n\\draw (F) -- ($(M)!(F)!(N)$) node [below] {$E$} coordinate (E);\n\\draw (I) -- ($(M)!(I)!(N)$) node [below] {$H$} coordinate (H);\n\\draw (\\t:3) node [above right] {$C$} coordinate (C);\n\\draw ({180-\\t}:3) node [above left] {$D$} coordinate (D);\n\\draw (C) -- ($(M)!(C)!(N)$) node [below] {$B$} coordinate (B);\n\\draw (D) -- ($(M)!(D)!(N)$) node [below] {$A$} coordinate (A);\n\\draw (C)--(D);\n\\draw (I) -- ($(A)!(I)!(D)$) node [right] {$J$} coordinate (J);\n\\draw (F) -- ($(B)!(F)!(C)$) node [left] {$G$} coordinate (G);\n\\draw ($(A)!0.5!(C)$) node {海洋球池};\n\\draw ($(A)!0.5!(I)$) node {息};\n\\draw ($(A)!0.5!(I)$) ++ (0,0.4) node {休};\n\\draw ($(A)!0.5!(I)$) ++ (0,-0.4) node {区};\n\\draw ($(B)!0.5!(F)$) node {息};\n\\draw ($(B)!0.5!(F)$) ++ (0,0.4) node {休};\n\\draw ($(B)!0.5!(F)$) ++ (0,-0.4) node {区};\n\\end{tikzpicture}\n\\end{center}\n(1) 求当$\\theta=\\dfrac{\\pi}{4}$时该亲子乐园可供人活动的区域面积$S$, 并求出此时的``得地率''(结果精确到$1 \\%)$;\\\\\n(2) 求当$\\theta$为多大时, 该亲子乐园的``得地率''最大?",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "(1) $\\theta=\\dfrac\\pi 4$时, $S=\\dfrac{2-\\sqrt{2}+\\sqrt{3}}2R^2$, ``得地率''约为$74\\%$; (2) $\\theta = \\arcsin\\dfrac{1+\\sqrt{33}}8$时, ``得地率''最大",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "202304建平高二期中考试试题19",
|
||||
"edit": [
|
||||
"20230423\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "12ex"
|
||||
},
|
||||
"040645": {
|
||||
"id": "040645",
|
||||
"content": "已知椭圆$\\Gamma: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的左、右焦点分别为$F_1$、$F_2$. 椭圆$\\Gamma$上有互异的且不在$x$轴上的三点$A$、$B$、$C$满足直线$AC$经过$F_1$, 直线$BC$经过$F_2$.\\\\\n(1) 若椭圆$\\Gamma$的长轴长为 $4$ , 离心率为$\\dfrac{1}{2}$, 求$b$的值;\\\\\n(2) 若点$C$的坐标为$(0,1)$, $\\triangle ABC$的面积$S=\\dfrac{64}{49} \\sqrt{3}$, 求$a$的值;\\\\\n(3) 若$a=\\sqrt{2}$, $b=1$, 直线$AB$经过点$(\\dfrac{3}{2}, 0)$, 求$C$的坐标.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "(1) $b=\\sqrt{3}$; (2) $a=2$; (3) $C$的坐标为$(-\\dfrac 43,-\\dfrac 13)$或$(-\\dfrac 43,\\dfrac 13)$",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "202304建平高二期中考试试题20",
|
||||
"edit": [
|
||||
"20230423\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "12ex"
|
||||
},
|
||||
"040646": {
|
||||
"id": "040646",
|
||||
"content": "已知定义在$\\mathbf{R}$上的函数$f(x)$的导函数为$f'(x)$, 若$|f'(x)| \\leq 1$对任意$x \\in \\mathbf{R}$恒成立, 则称函数$f(x)$为``线性控制函数''.\\\\\n(1) 判断函数$f(x)=\\sin x$和$g(x)=\\mathrm{e}^x$是否为``线性控制函数'', 并说明理由;\\\\\n(2) 若函数$f(x)$为``线性控制函数'', 且$f(x)$在$\\mathbf{R}$上严格增, 设$A$、$B$为函数$f(x)$图像上互异的两点, 设直线$AB$的斜率为$k$, 判断命题``$0<k \\leq 1$''的真假, 并说明理由;\\\\\n(3) 若函数$f(x)$为``线性控制函数'', 且$f(x)$是以$T$($T>0$)为周期的周期函数, 证明: 对任意$x_1$、$x_2$都有$|f(x_1)-f(x_2)| \\leq T$.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "(1) $f(x)$是``线性控制函数'', $g(x)$不是``线性控制函数'', 理由略; (2) 是真命题, 理由略; (3) 证明略",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "202304建平高二期中考试试题21",
|
||||
"edit": [
|
||||
"20230423\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "12ex"
|
||||
}
|
||||
}
|
||||
Reference in New Issue