录入交大附中三模试题及答案
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#修改起始id,出处,文件名
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#修改起始id,出处,文件名
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starting_id = 17444
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starting_id = 17465
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raworigin = ""
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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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filename = r"C:\Users\weiye\Documents\wwy sync\临时工作区\自拟题目12.tex"
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editor = "20230602\t王伟叶"
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editor = "20230602\t王伟叶"
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@ -1,126 +1,63 @@
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usages
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ans
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000573
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17465
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p20230529 2023届高三03班 0.690
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$\{-1,1\}$
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17466
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$\dfrac{\sqrt{2}}{2}$
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000618
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17467
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p20230518 2023届高三11班 0.952
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$(-\infty,-3]\cup (1,+\infty)$
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p20230518 2023届高三12班 1.000
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p20230518 2023届高三01班 0.900
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p20230518 2023届高三06班 0.974
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p20230518 2023届高三09班 0.967
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17468
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$-\dfrac{1}{8}$
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000622
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17469
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p20230518 2023届高三11班 0.619
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$\pi$
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p20230518 2023届高三12班 0.739
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p20230518 2023届高三01班 0.900
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p20230518 2023届高三06班 0.921
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p20230518 2023届高三09班 0.767
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17470
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$x=4$
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000625
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17471
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p20230518 2023届高三11班 0.429
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$28$
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p20230518 2023届高三12班 0.565
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p20230518 2023届高三01班 0.933
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p20230518 2023届高三06班 0.921
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p20230518 2023届高三09班 0.467
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17472
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$8.5$
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000616
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17473
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p20230518 2023届高三12班 1.000
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$(-3,+\infty)$
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p20230518 2023届高三06班 0.947
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17474
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$95.5$
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000617
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17475
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p20230518 2023届高三12班 1.000
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$(-4,0)cup \{2\sqrt{2}-2\}$
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p20230518 2023届高三06班 1.000
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p20230518 2023届高三09班 1.000
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17476
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$1518.5$
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000624
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17477
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p20230518 2023届高三12班 0.870
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A
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p20230518 2023届高三09班 0.967
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17478
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D
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013307
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17479
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p20230519 2023届高三11班 0.636
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B
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p20230519 2023届高三12班 0.773
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p20230519 2023届高三08班 0.645
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p20230519 2023届高三09班 0.821
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17480
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B
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000637
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17481
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p20230519 2023届高三11班 0.905
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(1) $a_n=n$, $b_n=2^{n-1}$; (2) 证明略
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p20230519 2023届高三08班 0.923
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p20230519 2023届高三09班 0.929
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17482
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(1) $\dfrac{\pi}{4}$; (2) $\dfrac{\sqrt{3}}{3}$
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030154
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17483
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p20230519 2023届高三11班 0.286
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(1) $y=27\times (\dfrac{4}{3})^x$, $x\ge 0$; (2) $9$分钟
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p20230519 2023届高三12班 0.208
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p20230519 2023届高三09班 0.286
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000636
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p20230519 2023届高三12班 0.958
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p20230519 2023届高三01班 1.000
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p20230519 2023届高三06班 0.975
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p20230519 2023届高三09班 0.964
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000654
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p20230526 2023届高三11班 0.316
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p20230526 2023届高三12班 0.524
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p20230526 2023届高三01班 0.846
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p20230526 2023届高三06班 0.707
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p20230526 2023届高三08班 0.667
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p20230526 2023届高三09班 0.611
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000655
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p20230526 2023届高三11班 0.474
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p20230526 2023届高三12班 0.476
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p20230526 2023届高三08班 0.333
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p20230526 2023届高三09班 0.500
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030162
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p20230526 2023届高三12班 0.857
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p20230526 2023届高三08班 1.000
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p20230526 2023届高三09班 0.833
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000669
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p20230526 2023届高三12班 0.737
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p20230526 2023届高三01班 1.000
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p20230526 2023届高三06班 0.949
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p20230526 2023届高三09班 0.870
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000672
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p20230526 2023届高三12班 0.684
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p20230526 2023届高三09班 0.826
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000673
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p20230526 2023届高三12班 0.368
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p20230526 2023届高三01班 0.750
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p20230526 2023届高三06班 0.692
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p20230526 2023届高三09班 0.522
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000675
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p20230526 2023届高三12班 0.526
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p20230526 2023届高三01班 0.750
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p20230526 2023届高三06班 0.744
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p20230526 2023届高三09班 0.478
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000685
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p20230530 2023届高三12班 0.857
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p20230530 2023届高三01班 1.000
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p20230530 2023届高三06班 0.925
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p20230530 2023届高三08班 0.897
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17484
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(1) $6$; (2) $-4$; (3) $(2,0)$或$(-\dfrac{2}{7},-\dfrac{12}{7})$
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17485
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(1) 证明略; (2) $\dfrac{\mathrm{e}}{2}$; (3) $(-\dfrac{27}{\mathrm{e}^3},0)\cup (0,+\infty)$
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@ -451298,6 +451298,426 @@
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"space": "4em",
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"space": "4em",
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"unrelated": []
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"unrelated": []
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},
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},
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"017465": {
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"id": "017465",
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"content": "已知集合$A=\\{x \\| x | \\leq 1\\}, B=\\{-1,1,3,5\\}$, 则$A \\cap B=$\\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": "$\\{-1,1\\}$",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "2023届交大附中三模试题1",
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"edit": [
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"20230602\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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"unrelated": []
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},
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"017466": {
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"id": "017466",
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"content": "复数$z=\\dfrac{1-2 \\mathrm{i}}{3+\\mathrm{i}}$的模为\\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": "$\\dfrac{\\sqrt{2}}{2}$",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "2023届交大附中三模试题2",
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"edit": [
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"20230602\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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"unrelated": []
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},
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"017467": {
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"id": "017467",
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"content": "不等式$\\dfrac{x+3}{x-1} \\geq 0$的解集为\\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": "$(-\\infty,-3]\\cup (1,+\\infty)$",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "2023届交大附中三模试题3",
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"edit": [
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"20230602\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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"unrelated": []
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},
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"017468": {
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"id": "017468",
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"content": "已知幂函数$y=f(x)$的图像过点$(\\dfrac{1}{2}, 8)$, 则$f(-2)=$\\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": "$-\\dfrac{1}{8}$",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "2023届交大附中三模试题4",
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"edit": [
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"20230602\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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"unrelated": []
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},
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"017469": {
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"id": "017469",
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"content": "已知函数$f(x)=\\sin 2 x+2 \\sqrt{3} \\cos ^2 x$, 则函数$f(x)$的最小正周期是\\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": "$\\pi$",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "2023届交大附中三模试题5",
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"edit": [
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"20230602\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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"unrelated": []
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},
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"017470": {
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"id": "017470",
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"content": "由函数的观点, 不等式$2^x+\\log _4 x=17$的解是\\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": "$x=4$",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "2023届交大附中三模试题6",
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"edit": [
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"20230602\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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"unrelated": []
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},
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"017471": {
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||||||
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"id": "017471",
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"content": "$(\\dfrac{1}{x}-\\sqrt{x})^8$的展开式中含$x$项的系数为\\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": "$28$",
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||||||
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"solution": "",
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||||||
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"duration": -1,
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||||||
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"usages": [],
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||||||
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"origin": "2023届交大附中三模试题7",
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||||||
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"edit": [
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||||||
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"20230602\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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"unrelated": []
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},
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"017472": {
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"id": "017472",
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"content": "某单位为了解该单位党员开展学习党史知识活动情况, 随机抽取了部分党员, 对他们一周的党史学习时间进行了统计, 统计数据如下表所示:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline 党史学习时间(小时) & 7 & 8 & 9 & 10 & 11 \\\\\n\\hline 党员人数 & 6 & 10 & 9 & 8 & 7 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n则该单位党员一周学习党史时间的第$40$百分位数是\\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": "$8.5$",
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||||||
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"solution": "",
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||||||
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"duration": -1,
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||||||
|
"usages": [],
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||||||
|
"origin": "2023届交大附中三模试题8",
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"edit": [
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"20230602\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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"unrelated": []
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},
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"017473": {
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"id": "017473",
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"content": "若存在实数$a$, 使得$x=1$是方程$(x+a)^2=3 x+b$的解, 但不是方程$x+a=\\sqrt{3 x+b}$的解, 则实数$b$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": "$(-3,+\\infty)$",
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"solution": "",
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"duration": -1,
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"usages": [],
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|
"origin": "2023届交大附中三模试题9",
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||||||
|
"edit": [
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"20230602\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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"unrelated": []
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},
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"017474": {
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"id": "017474",
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"content": "若随机变量$X \\sim N(105,19^2)$, $Y \\sim N(100,9^2)$, 若$P(X \\leq A)=P(Y \\leq A)$, 那么实数$A$的值为\\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": "$95.5$",
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||||||
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"solution": "",
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||||||
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"duration": -1,
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||||||
|
"usages": [],
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||||||
|
"origin": "2023届交大附中三模试题10",
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||||||
|
"edit": [
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||||||
|
"20230602\t王伟叶"
|
||||||
|
],
|
||||||
|
"same": [],
|
||||||
|
"related": [],
|
||||||
|
"remark": "",
|
||||||
|
"space": "",
|
||||||
|
"unrelated": []
|
||||||
|
},
|
||||||
|
"017475": {
|
||||||
|
"id": "017475",
|
||||||
|
"content": "已知曲线$C_1: |y|=x+2$与曲线$C_2: (x-a)^2+y^2=4$恰有两个公共点, 则实数$a$的取值范围为\\blank{50}.",
|
||||||
|
"objs": [],
|
||||||
|
"tags": [],
|
||||||
|
"genre": "填空题",
|
||||||
|
"ans": "$(-4,0)cup \\{2\\sqrt{2}-2\\}$",
|
||||||
|
"solution": "",
|
||||||
|
"duration": -1,
|
||||||
|
"usages": [],
|
||||||
|
"origin": "2023届交大附中三模试题11",
|
||||||
|
"edit": [
|
||||||
|
"20230602\t王伟叶"
|
||||||
|
],
|
||||||
|
"same": [],
|
||||||
|
"related": [],
|
||||||
|
"remark": "",
|
||||||
|
"space": "",
|
||||||
|
"unrelated": []
|
||||||
|
},
|
||||||
|
"017476": {
|
||||||
|
"id": "017476",
|
||||||
|
"content": "函数$y=f(x)$是最小正周期为$4$的偶函数, 且在$x \\in[-2,0]$时, $f(x)=2 x+1$, 若存在$x_1$、$x_2$、$\\cdots$、$x_n$满足$0 \\leq x_1<x_2<\\cdots<x_n$, 且$|f(x_1)-f(x_2)|+|f(x_2)-f(x_3)|+\\cdots$$+|f(x_{n-1})-f(x_n)|=2023$, 则$n+x_n$的最小值为\\blank{50}.",
|
||||||
|
"objs": [],
|
||||||
|
"tags": [],
|
||||||
|
"genre": "填空题",
|
||||||
|
"ans": "$1518.5$",
|
||||||
|
"solution": "",
|
||||||
|
"duration": -1,
|
||||||
|
"usages": [],
|
||||||
|
"origin": "2023届交大附中三模试题12",
|
||||||
|
"edit": [
|
||||||
|
"20230602\t王伟叶"
|
||||||
|
],
|
||||||
|
"same": [],
|
||||||
|
"related": [],
|
||||||
|
"remark": "",
|
||||||
|
"space": "",
|
||||||
|
"unrelated": []
|
||||||
|
},
|
||||||
|
"017477": {
|
||||||
|
"id": "017477",
|
||||||
|
"content": "设$\\lambda \\in \\mathbf{R}$, 则``$\\lambda=1$''是``直线$3 x+(\\lambda-1) y=1$与直线$\\lambda x+(1-\\lambda) y=2$平行''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充要条件}{既不充分也不必要条件}",
|
||||||
|
"objs": [],
|
||||||
|
"tags": [],
|
||||||
|
"genre": "选择题",
|
||||||
|
"ans": "A",
|
||||||
|
"solution": "",
|
||||||
|
"duration": -1,
|
||||||
|
"usages": [],
|
||||||
|
"origin": "2023届交大附中三模试题13",
|
||||||
|
"edit": [
|
||||||
|
"20230602\t王伟叶"
|
||||||
|
],
|
||||||
|
"same": [],
|
||||||
|
"related": [],
|
||||||
|
"remark": "",
|
||||||
|
"space": "",
|
||||||
|
"unrelated": []
|
||||||
|
},
|
||||||
|
"017478": {
|
||||||
|
"id": "017478",
|
||||||
|
"content": "函数$y=f(x)$的导函数$y=f'(x)$的图像如图所示, 则函数$y=f(x)$的图像可能是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.4,0) -- (1.4,0) node [below] {$x$};\n\\draw [->] (0,-0.6) -- (0,0.6) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1:1.4] plot (\\x,{pow(\\x-0.2,3)-\\x+0.2});\n\\end{tikzpicture}\n\\end{center}\n\\fourch{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.4,0) -- (1.4,0) node [below] {$x$};\n\\draw [->] (0,-0.6) -- (0,0.6) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1:1.4] plot (\\x,{-(\\x+1)*(\\x-1.3)*(\\x+0.5)*(\\x-0.2)/1.5});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.4,0) -- (1.4,0) node [below] {$x$};\n\\draw [->] (0,-0.6) -- (0,0.6) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1:1.4] plot (\\x,{(\\x+1)*(\\x-1.3)*(\\x+0.5)*(\\x-0.2)/1.5});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.4,0) -- (1.4,0) node [below] {$x$};\n\\draw [->] (0,-0.6) -- (0,0.6) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1:1.4] plot (\\x,{-2.5*(0.05 + 0.192*\\x - 0.44* \\x*\\x - 0.2*pow(\\x,3)+ pow(\\x,4)/4)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.4,0) -- (1.4,0) node [below] {$x$};\n\\draw [->] (0,-0.6) -- (0,0.6) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1:1.4] plot (\\x,{2.5*(0.15 + 0.192*\\x - 0.44* \\x*\\x - 0.2*pow(\\x,3)+ pow(\\x,4)/4)});\n\\end{tikzpicture}}",
|
||||||
|
"objs": [],
|
||||||
|
"tags": [],
|
||||||
|
"genre": "选择题",
|
||||||
|
"ans": "D",
|
||||||
|
"solution": "",
|
||||||
|
"duration": -1,
|
||||||
|
"usages": [],
|
||||||
|
"origin": "2023届交大附中三模试题14",
|
||||||
|
"edit": [
|
||||||
|
"20230602\t王伟叶"
|
||||||
|
],
|
||||||
|
"same": [],
|
||||||
|
"related": [],
|
||||||
|
"remark": "",
|
||||||
|
"space": "",
|
||||||
|
"unrelated": []
|
||||||
|
},
|
||||||
|
"017479": {
|
||||||
|
"id": "017479",
|
||||||
|
"content": "已知函数$f(x)=a x^2+|x+a+1|$为偶函数, 则不等式$f(x)>0$的解集为\\bracket{20}.\n\\fourch{$\\varnothing$}{$(-1,0) \\cup(0,1)$}{$(-1,1)$}{$(-\\infty,-1) \\cup(1,+\\infty)$}",
|
||||||
|
"objs": [],
|
||||||
|
"tags": [],
|
||||||
|
"genre": "选择题",
|
||||||
|
"ans": "B",
|
||||||
|
"solution": "",
|
||||||
|
"duration": -1,
|
||||||
|
"usages": [],
|
||||||
|
"origin": "2023届交大附中三模试题15",
|
||||||
|
"edit": [
|
||||||
|
"20230602\t王伟叶"
|
||||||
|
],
|
||||||
|
"same": [],
|
||||||
|
"related": [],
|
||||||
|
"remark": "",
|
||||||
|
"space": "",
|
||||||
|
"unrelated": []
|
||||||
|
},
|
||||||
|
"017480": {
|
||||||
|
"id": "017480",
|
||||||
|
"content": "已知$n \\in \\mathbf{N}$, $n\\ge 1$, 集合$A=\\{\\sin (\\dfrac{k \\pi}{n}) | k \\in \\mathbf{N},\\ 0 \\leq k \\leq n\\}$, 若集合$A$恰有$8$个子集, 则$n$的可能值有\\bracket{20}个.\n\\fourch{$1$}{$2$}{$3$}{$4$}",
|
||||||
|
"objs": [],
|
||||||
|
"tags": [],
|
||||||
|
"genre": "选择题",
|
||||||
|
"ans": "B",
|
||||||
|
"solution": "",
|
||||||
|
"duration": -1,
|
||||||
|
"usages": [],
|
||||||
|
"origin": "2023届交大附中三模试题16",
|
||||||
|
"edit": [
|
||||||
|
"20230602\t王伟叶"
|
||||||
|
],
|
||||||
|
"same": [],
|
||||||
|
"related": [],
|
||||||
|
"remark": "",
|
||||||
|
"space": "",
|
||||||
|
"unrelated": []
|
||||||
|
},
|
||||||
|
"017481": {
|
||||||
|
"id": "017481",
|
||||||
|
"content": "已知$\\{a_n\\}$为等差数列, $\\{b_n\\}$为等比数列, $a_1=b_1=1$, $a_5=5(a_4-a_3)$, $b_5=4(b_4-b_3)$.\\\\\n(1) 求$\\{a_n\\}$和$\\{b_n\\}$的通项公式;\\\\\n(2) 记$\\{a_n\\}$的前$n$项和为$S_n$, 求证: $S_n S_{n+2}<S_{n+1}^2$($n \\in \\mathbf{N}$, $n \\ge 1$).",
|
||||||
|
"objs": [],
|
||||||
|
"tags": [],
|
||||||
|
"genre": "解答题",
|
||||||
|
"ans": "(1) $a_n=n$, $b_n=2^{n-1}$; (2) 证明略",
|
||||||
|
"solution": "",
|
||||||
|
"duration": -1,
|
||||||
|
"usages": [],
|
||||||
|
"origin": "2023届交大附中三模试题17",
|
||||||
|
"edit": [
|
||||||
|
"20230602\t王伟叶"
|
||||||
|
],
|
||||||
|
"same": [],
|
||||||
|
"related": [],
|
||||||
|
"remark": "",
|
||||||
|
"space": "4em",
|
||||||
|
"unrelated": []
|
||||||
|
},
|
||||||
|
"017482": {
|
||||||
|
"id": "017482",
|
||||||
|
"content": "如图, $PD \\perp$平面$ABCD$, 四边形$ABCD$为直角梯形, $AB\\parallel CD$, $\\angle ADC=90^{\\circ}$, $PD=CD=2AD=2AB=2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [above right] {$D$} coordinate (D);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw (0,0,1) node [left] {$A$} coordinate (A);\n\\draw (1,0,1) node [below] {$B$} coordinate (B);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (P)--(A)--(B)--(C)--cycle(P)--(B);\n\\draw [dashed] (A)--(D)--(C)(D)--(B)(D)--(P);\n\\end{tikzpicture}\n\\end{center}\n(1) 求异面直线$AB$与$PC$所成角的大小;\\\\\n(2) 求二面角$B-PC-D$的余弦值.",
|
||||||
|
"objs": [],
|
||||||
|
"tags": [],
|
||||||
|
"genre": "解答题",
|
||||||
|
"ans": "(1) $\\dfrac{\\pi}{4}$; (2) $\\dfrac{\\sqrt{3}}{3}$",
|
||||||
|
"solution": "",
|
||||||
|
"duration": -1,
|
||||||
|
"usages": [],
|
||||||
|
"origin": "2023届交大附中三模试题18",
|
||||||
|
"edit": [
|
||||||
|
"20230602\t王伟叶"
|
||||||
|
],
|
||||||
|
"same": [],
|
||||||
|
"related": [],
|
||||||
|
"remark": "",
|
||||||
|
"space": "4em",
|
||||||
|
"unrelated": []
|
||||||
|
},
|
||||||
|
"017483": {
|
||||||
|
"id": "017483",
|
||||||
|
"content": "流行性感冒简称流感, 是流感病毒引起的急性呼吸道感染, 也是一种传染性强、传播速度快的疾病, 了解引起流感的某些细菌、病毒的生存条件、繁殖习性等对于预防流感的传播有极其重要的意义, 某科研团队在培养基中放入一定是某种细菌进行研究. 经过$2$分钟菌落的覆盖面积为$48 \\text{mm}^2$, 经过$3$分钟覆盖面积为$64 \\text{mm}^2$, 后期其蔓延速度越来越快; 菌落的覆盖面积$y$(单位: $\\text{mm}^2$) 与经过时间$x$(单位: $\\text{min}$) 的关系现有三个函数模型: \\textcircled{1} $y=k a^x$($k>0$, $a>1$); \\textcircled{2} $y=\\log _b x$($b>1$); \\textcircled{3} $y=p \\sqrt{x}+q$($p>0$)可供选择.\\\\\n(1) 选出你认为符合实际的函数模型, 说明理由, 并求出该模型的解析式;\\\\\n(2) 在理想状态下, 至少经过多少分钟培养基中菌落的覆盖面积能超过$300 \\text{mm}^2$?(结果保留到整数)",
|
||||||
|
"objs": [],
|
||||||
|
"tags": [],
|
||||||
|
"genre": "解答题",
|
||||||
|
"ans": "(1) $y=27\\times (\\dfrac{4}{3})^x$, $x\\ge 0$; (2) $9$分钟",
|
||||||
|
"solution": "",
|
||||||
|
"duration": -1,
|
||||||
|
"usages": [],
|
||||||
|
"origin": "2023届交大附中三模试题19",
|
||||||
|
"edit": [
|
||||||
|
"20230602\t王伟叶"
|
||||||
|
],
|
||||||
|
"same": [],
|
||||||
|
"related": [],
|
||||||
|
"remark": "",
|
||||||
|
"space": "4em",
|
||||||
|
"unrelated": []
|
||||||
|
},
|
||||||
|
"017484": {
|
||||||
|
"id": "017484",
|
||||||
|
"content": "在平面直角坐标系$x O y$中, 已知椭圆$E: \\dfrac{x^2}{4}+\\dfrac{y^2}{3}=1$的左、右焦点分别为$F_1$、$F_2$, 点$A$在椭圆$E$上且在第一象限内, $AF_2 \\perp F_1F_2$, 直线$AF_1$与椭圆$E$相交于另一点$B$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2.5,0) -- (2.5,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\path [name path = elli,draw] (0,0) ellipse (2 and {sqrt(3)});\n\\draw (-1,0) node [above left] {$F_1$} coordinate (F_1);\n\\draw (1,0) node [below] {$F_2$} coordinate (F_2);\n\\path [name path = AF2] (F_2) --++ (0,2);\n\\path [name intersections = {of = AF2 and elli, by = A}];\n\\draw (A) node [above] {$A$} --(F_2); \n\\path [name path = AB] (A) -- ($(F_1)!-0.5!(A)$);\n\\path [name intersections = {of = AB and elli, by = B}];\n\\draw (A)--(B) node [left] {$B$} -- (O) -- cycle;\n\\end{tikzpicture}\n\\end{center}\n(1) 求$\\triangle AF_1F_2$的周长;\\\\\n(2) 在$x$轴上任取一点$P$, 直线$AP$与椭圆$E$的右准线相交于点$Q$, 求$\\overrightarrow{OP} \\cdot \\overrightarrow{QP}$的最小值;\\\\\n(3)设点$M$在椭圆$E$上, 记$\\triangle OAB$与$\\triangle MAB$的面积分别为$S_1$、$S_2$, 若$S_2=3S_1$, 求点$M$的坐标.",
|
||||||
|
"objs": [],
|
||||||
|
"tags": [],
|
||||||
|
"genre": "解答题",
|
||||||
|
"ans": "(1) $6$; (2) $-4$; (3) $(2,0)$或$(-\\dfrac{2}{7},-\\dfrac{12}{7})$",
|
||||||
|
"solution": "",
|
||||||
|
"duration": -1,
|
||||||
|
"usages": [],
|
||||||
|
"origin": "2023届交大附中三模试题20",
|
||||||
|
"edit": [
|
||||||
|
"20230602\t王伟叶"
|
||||||
|
],
|
||||||
|
"same": [],
|
||||||
|
"related": [],
|
||||||
|
"remark": "",
|
||||||
|
"space": "4em",
|
||||||
|
"unrelated": []
|
||||||
|
},
|
||||||
|
"017485": {
|
||||||
|
"id": "017485",
|
||||||
|
"content": "记$y=f'(x)$、$y=g'(x)$分别为函数$y=f(x)$、$y=g(x)$的导函数, 若存在$x_0 \\in \\mathbf{R}$, 满足$f(x_0)=g(x_0)$且$f'(x_0)=g'(x_0)$, 则称$x_0$为函数$f(x)$与$g(x)$的一个``$S$点''.\\\\\n(1) 证明: 函数$y=x$与$y=x^2+2 x-2$不存在``$S$点'';\\\\\n(2) 若函数$y=a x^2-1$与$y=\\ln x$存在``$S$点'', 求实数$a$的值;\\\\\n(3) 已知$f(x)=-x^2+a$, $g(x)=\\dfrac{b \\mathrm{e}^x}{x}$, 若存在实数$a>0$, 使函数$y=f(x)$与$y=g(x)$在区间$(0,+\\infty)$内存在``$S$点'', 求实数$b$的取值范围.",
|
||||||
|
"objs": [],
|
||||||
|
"tags": [],
|
||||||
|
"genre": "解答题",
|
||||||
|
"ans": "(1) 证明略; (2) $\\dfrac{\\mathrm{e}}{2}$; (3) $(-\\dfrac{27}{\\mathrm{e}^3},0)\\cup (0,+\\infty)$",
|
||||||
|
"solution": "",
|
||||||
|
"duration": -1,
|
||||||
|
"usages": [],
|
||||||
|
"origin": "2023届交大附中三模试题21",
|
||||||
|
"edit": [
|
||||||
|
"20230602\t王伟叶"
|
||||||
|
],
|
||||||
|
"same": [],
|
||||||
|
"related": [],
|
||||||
|
"remark": "",
|
||||||
|
"space": "4em",
|
||||||
|
"unrelated": []
|
||||||
|
},
|
||||||
"020001": {
|
"020001": {
|
||||||
"id": "020001",
|
"id": "020001",
|
||||||
"content": "判断下列各组对象能否组成集合, 若能组成集合, 指出是有限集还是无限集.\\\\\n(1) 上海市控江中学$2022$年入学的全体高一年级新生;\\\\\n(2) 中国现有各省的名称;\\\\\n(3) 太阳、$2$、上海市;\\\\\n(4) 大于$10$且小于$15$的有理数;\\\\\n(5) 末位是$3$的自然数;\\\\\n(6) 影响力比较大的中国数学家;\\\\\n(7) 方程$x^2+x-3=0$的所有实数解;\\\\ \n(8) 函数$y=\\dfrac 1x$图像上所有的点;\\\\ \n(9) 在平面直角坐标系中, 到定点$(0, 0)$的距离等于$1$的所有点;\\\\\n(10) 不等式$3x-10<0$的所有正整数解;\\\\\n(11) 所有的平面四边形.",
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"content": "判断下列各组对象能否组成集合, 若能组成集合, 指出是有限集还是无限集.\\\\\n(1) 上海市控江中学$2022$年入学的全体高一年级新生;\\\\\n(2) 中国现有各省的名称;\\\\\n(3) 太阳、$2$、上海市;\\\\\n(4) 大于$10$且小于$15$的有理数;\\\\\n(5) 末位是$3$的自然数;\\\\\n(6) 影响力比较大的中国数学家;\\\\\n(7) 方程$x^2+x-3=0$的所有实数解;\\\\ \n(8) 函数$y=\\dfrac 1x$图像上所有的点;\\\\ \n(9) 在平面直角坐标系中, 到定点$(0, 0)$的距离等于$1$的所有点;\\\\\n(10) 不等式$3x-10<0$的所有正整数解;\\\\\n(11) 所有的平面四边形.",
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Reference in New Issue