录入2024届高一上学期期中考试答案及解答
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20231103 2024届高三01班 0.771
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22697
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$0$.
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20231103 2024届高三01班 0.514
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20231103 2024届高三01班 0.886
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22698
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$x \leq 1$ 且 $y \leq 1$.
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030356
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20231103 2024届高三01班 0.971 0.857 0.943
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004009
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20231103 2024届高三01班 0.657
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22699
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$(1,2)$.
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030418
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20231103 2024届高三01班 0.914
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030419
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20231103 2024届高三01班 0.943
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22700
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$4$.
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030421
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20231103 2024届高三01班 0.714
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20231103 2024届高三01班 0.914 0.929
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22701
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$14$.
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030382
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20231103 2024届高三01班 0.829
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030379
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20231103 2024届高三11班 0.900
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22702
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$\dfrac{a+1}{2}$.
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030407
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20231103 2024届高三11班 0.400
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030415
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20231103 2024届高三11班 0.933
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22703
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$\dfrac{1}{8}$.
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030356
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20231103 2024届高三11班 0.933 0.867 0.867
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004009
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20231103 2024届高三11班 0.533
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22704
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$[0,4)$.
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030418
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20231103 2024届高三11班 1.000
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030419
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20231103 2024届高三11班 0.933
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22705
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$(\dfrac{1}{2}, 1)$.
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030421
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20231103 2024届高三11班 0.767
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030369
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20231103 2024届高三11班 0.900 0.867
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22706
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$(-\infty, \dfrac{1}{2}]$.
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030382
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20231103 2024届高三11班 0.900
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030379
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20231103 2024届高三03班 0.840
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22707
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$20$.
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030407
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20231103 2024届高三03班 0.560
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030415
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20231103 2024届高三03班 0.840
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22708
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$(-\infty, \dfrac{1}{2}]$.
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030356
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20231103 2024届高三03班 0.720 0.500 0.540
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004009
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20231103 2024届高三03班 0.420
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22709
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D
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030418
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20231103 2024届高三03班 0.880
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030419
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20231103 2024届高三03班 0.960
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22710
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A
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030421
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20231103 2024届高三03班 0.680
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030369
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20231103 2024届高三03班 0.880 0.820
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22711
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C
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030382
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20231103 2024届高三03班 0.900
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20231103 2024届高三04班 0.844
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22712
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A
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030407
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20231103 2024届高三04班 0.938
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030415
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20231103 2024届高三04班 0.938
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22713
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(1) $(-3,4]$; (2) $(-1,1)$
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030356
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20231103 2024届高三04班 0.938 0.844 0.703
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004009
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20231103 2024届高三04班 0.469
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22714
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(1) 当 $a=b$ 时, $a^3+b^3=a^2 b+a b^2$; 当 $a \neq b$ 时, $a^3+b^3>a^2 b+a b^2$; (2) 最小值为$1$
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20231103 2024届高三04班 0.969
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030419
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20231103 2024届高三04班 0.969
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22715
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(1) $200$吨; (2) $100$至$300$吨(含$100$吨与$300$吨)
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030421
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20231103 2024届高三04班 0.656
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030369
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20231103 2024届高三04班 0.812 0.719
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22716
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(1) $\log_2 3$; (2) $[\dfrac{9}{4},+\infty)$; (3) $[8,9)$
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030382
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20231103 2024届高三04班 0.906
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030379
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20231103 2024届高三05班 0.900
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22717
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(1) $\{1,2,3,4\}$ 不是``可分集合'', $\{1,3,5,7,9,11,13\}$ 是``可分集合''; (2) 证明略; (3) 证明略
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030407
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20231103 2024届高三05班 0.567
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030415
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20231103 2024届高三05班 0.800
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solution
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030356
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20231103 2024届高三05班 0.883 0.800 0.833
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22713
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(1) $A=(-3,3)$, $B=[0,4]$, 故 $A \cup B=(-3,4]$.\\
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(2) $A \cap B=B$ 等价于 $B \subseteq A$, 故 $a-2>-3$ 且 $a+2<3$, 故 $a$ 的取值范围是 $(-1,1)$.
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004009
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20231103 2024届高三05班 0.783
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030418
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20231103 2024届高三05班 0.933
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22714
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(1) $a^3+b^3-a^2 b-a b^2=(a-b)^2(a+b)$, 故当 $a=b$ 时, $a^3+b^3=a^2 b+a b^2$; 当 $a \neq b$ 时, $a^3+b^3>a^2 b+a b^2$.\\
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(2) 由 (1) 知 $a^3+b^3 \geq a^2 b+a b^2$, 又由于 $a>0$, $b>0$, 故 $\dfrac{a^3+b^3}{a b}\geq \dfrac{a^2 b+a b^2}{a b}$, 即 $\dfrac{a^2}{b}+\dfrac{b^2}{a}\geq a+b=1$,当 $a=b=\dfrac{1}{2}$ 时等号成立. 故 $\dfrac{a^2}{b}+\dfrac{b^2}{a}$ 的最小值是 $1$.
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030419
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20231103 2024届高三05班 0.833
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030421
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20231103 2024届高三05班 0.633
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22715
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(1) 该小区每月处理 $x$ 吨垃圾, 则每吨垃圾的平均成本: $w=\dfrac{x^2-2 x+40000}{x}=x+\dfrac{40000}{x}-200 \geq 2 \sqrt{40000}-200=200$. 当且仅当 $x=200$ 时等号成立. 故该小区每月分类处理 200 吨垃圾, 才能使得每吨垃圾分类处理的平均成本最低.\\
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(2) 题意即: $300 x \geq x^2-200 x+40000$. 解集为 $[100,400]$, 又 $x \leq 300$, 故 $x \in[100,300]$. 所以该小区每月垃圾分类处理量 $x$ 的取值范围是 $[100,300]$.
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030369
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20231103 2024届高三05班 0.650 0.783
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030382
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20231103 2024届高三05班 0.700
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22716
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(1) $2^x-\dfrac{3}{2^x}=2 \Leftrightarrow(2^x)^2-2(2^x)-3=0 \Leftrightarrow 2^x=3$ 或 $2^x=-1$ (舍), 故 $x=\log _23$.\\
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(2) 题意即: 对任意 $x \in \mathbf{R}$, $2^x+\dfrac{a}{2^x}\geq 3$ 恒成立 $\Leftrightarrow a \geq-(2^x)^2+3(2^x)=-(2^x-\dfrac{3}{2})^2+\dfrac{9}{4}$ 恒成立.
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又 $-(2^x-\dfrac{3}{2})^2+\dfrac{9}{4}\leq \dfrac{9}{4}$, 当 $x=\log _2 \dfrac{3}{2}$ 时等号成立.
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故 $-(2^x-\dfrac{3}{2})^2+\dfrac{9}{4}$ 的最大值是 $\dfrac{9}{4}$, 故实数 $a$ 的取值范围是 $[\dfrac{9}{4},+\infty)$.\\
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(3) 令 $t=2^x \in$($0,+\infty$), 故 $2^x+\dfrac{a}{2^x}=6$ 有两个不同的实数根 $\Leftrightarrow t+\dfrac{a}{t}-6=0$ 有两个不同的正实数根 $\Leftrightarrow t^2-6 t+a=0$ 有两个不同的正实数根.
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故 $\begin{cases}\Delta=36-4 a>0,\\6>0,\\a>0 ;\end{cases}$, 解得 $a \in(0,9)$.
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又 $|x_1-x_2|=|\log _2 t_1-\log _2 t_2|=|\log _2 \dfrac{t_1}{t_2}| \leq 1$,
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故 $\dfrac{1}{2}\leq \dfrac{t_1}{t_2}\leq 2$, 即 $\dfrac{1}{2}\leq \dfrac{3-\sqrt{9-a}}{3+\sqrt{9+a}}\leq 2$, 即 $\sqrt{9-a}\leq 1$, 即 $a \in[8,9]$.
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综上, $a$ 的取值范围是 $[8,9)$.
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030379
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20231103 2024届高三06班 0.944
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030407
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20231103 2024届高三06班 0.639
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22717
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(1) 集合 $\{1,2,3,4\}$ 不是``可分集合'', $\{1,3,5,7,9,11,13\}$ 是``可分集合''.\\
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(2) 反证法: 假设集合 $A$ 是``可分集合'', 不妨设 $a_1<a_2<a_3<a_4<a_5$.
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去掉``$a_1$''后, 剩下的元素可分为两个集合, 可能分别有两个元素, 可能一个集合有 1 个元素, 另一个集合有 3 个元素, 故 $a_2+a_5=a_3+a_4$\textcircled{1} 或 $a_5=a_2+a_3+a_4$\textcircled{2}.\\
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去掉``$a_2$''后, 得到 $a_1+a_5=a_3+a_4$\textcircled{3} 或 $a_5=a_1+a_3+a_4$\textcircled{4}. 由\textcircled{1}\textcircled{3}知, $a_1=a_2$, 与集合元素的互异性矛盾, 由\textcircled{1}\textcircled{4}知, $a_1+a_2=0$, 与 $a_1, a_2$ 是正整数矛盾, 由\textcircled{2}\textcircled{3}知, $a_1+a_2=0$, 与 $a_1, a_2$ 是正整数矛盾, 由\textcircled{2}\textcircled{4}知, $a_1=a_2$, 与集合元素的互异性矛盾. 故假设不成立, 所以集合 $A$ 不是``可分集合''.\\
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(3) 记 $S=a_1+a_2+\cdots+a_n$, 因集合 $A$ 是``可分集合'', 故 $S-a_1, S-a_2, \cdots, S-a_n$ 是偶数, 故 $a_1, a_2, \cdots, a_n, S$奇偶性相同.\\
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情况 1: 当 $a_1$ 是奇数时, 有 $a_1, a_2, \cdots, a_n, S$ 都是奇数, 又 $S=a_1+a_2+\cdots+a_n$ 是 $n$ 个奇数的和, 故 $n$ 是奇数.情况 2: 当 $a_2$ 是偶数时, 因集合 $A$ 是``可分集合'', 故去掉``$a_i$''($i=1,2, \cdots, n$) 后, 剩余的所有元素组成的集合能分为两个交集为空的集合 $B_i, C_i$, 且这两个集合的所有元素之和相等, 记为 $S_{B_i}=S_{C_i}$.\\
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下证 $A_1=\{\dfrac{a_1}{2}, \dfrac{a_2}{2}, \cdots, \dfrac{a_n}{2}\}$ 是``可分集合''.
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因 $a_1, a_2, \cdots, a_n, S$ 是正偶数, 故 $\dfrac{a_1}{2}, \dfrac{a_2}{2}, \cdots, \dfrac{a_n}{2}$ 均为正整数. 去掉 $\dfrac{a_i}{2}$ 后, 集合 $B_i$ 中的元素分别乘以 $\dfrac{1}{2}$ 后组成集合 $B_i'$, 集合 $C_i$ 中的元素分别乘以 $\dfrac{1}{2}$ 后组成集合 $C_i'$, 因集合 $B_i \cap C_i=\varnothing$, 故 $B_i' \cap C_i'=\varnothing$. 因 $S_{B_i}=S_{C_i}$, 故 $S_{B_i'}=S_{C_i'}=\dfrac{1}{2}S_{B_i}=\dfrac{1}{2}S_{C_i}$.
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即 $A_1$ 去掉 $\dfrac{a_i}{2}$ 后的元素组成的集合能可分为两个交集为空的集合, 且这两个集合的所有元素之和相等, 故 $A_1$是``可分集合''.\\
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这个步骤可以一直进行下去, 直至 $m$ 次后, $\dfrac{a_1}{2^m}$ 是奇数, 此时因``可分集合''中元素的奇偶性相同, 故由情况 $1$ 可知 $n$ 是奇数.
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030415
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20231103 2024届高三06班 0.917
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030356
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20231103 2024届高三06班 0.889 0.861 0.778
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004009
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20231103 2024届高三06班 0.653
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030418
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20231103 2024届高三06班 1.000
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030419
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20231103 2024届高三06班 0.917
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030421
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20231103 2024届高三06班 0.917
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030369
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20231103 2024届高三06班 0.694 0.653
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030382
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20231103 2024届高三06班 0.833
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030379
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20231103 2024届高三07班 0.946
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030407
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20231103 2024届高三07班 0.703
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030415
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20231103 2024届高三07班 0.946
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030356
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20231103 2024届高三07班 0.851 0.689 0.703
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004009
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20231103 2024届高三07班 0.608
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030418
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20231103 2024届高三07班 0.892
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030419
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20231103 2024届高三07班 0.784
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030421
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20231103 2024届高三07班 0.622
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030369
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20231103 2024届高三07班 0.770 0.689
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030382
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20231103 2024届高三07班 0.689
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003225
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20231109 2024届高三11班 0.933
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009876
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20231109 2024届高三11班 1.000 0.833 0.967
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001749
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20231109 2024届高三11班 0.933 0.933
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001767
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20231109 2024届高三11班 0.867
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001789
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20231109 2024届高三11班 0.733
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008417
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20231109 2024届高三11班 0.983
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001785
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20231109 2024届高三11班 1.000
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000321
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20231109 2024届高三11班 0.533
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000312
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20231109 2024届高三11班 0.867
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003238
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20231109 2024届高三11班 0.900
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030473
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20231109 2024届高三11班 0.533
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001771
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20231109 2024届高三11班 0.733
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003244
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20231109 2024届高三11班 1.000 1.000
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003243
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20231109 2024届高三11班 0.900
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003253
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20231109 2024届高三11班 0.800
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001781
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20231109 2024届高三11班 0.200
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006793
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20231109 2024届高三11班 0.267
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008456
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20231109 2024届高三11班 0.650
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009887
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20231109 2024届高三11班 0.833
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001839
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20231109 2024届高三11班 1.000 0.733
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006892
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20231109 2024届高三11班 0.933
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003216
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20231109 2024届高三11班 0.133
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001783
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20231109 2024届高三11班 1.000 0.200
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003225
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20231109 2024届高三05班 0.931
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009876
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20231109 2024届高三05班 1.000 0.966 0.931
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001749
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20231109 2024届高三05班 0.966 0.966
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001767
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20231109 2024届高三05班 0.690
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001789
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20231109 2024届高三05班 0.828
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008417
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20231109 2024届高三05班 0.879
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001785
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20231109 2024届高三05班 0.724
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000321
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20231109 2024届高三05班 0.138
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000312
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20231109 2024届高三05班 0.759
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003238
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20231109 2024届高三05班 0.828
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030473
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20231109 2024届高三05班 0.586
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001771
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20231109 2024届高三05班 0.862
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003244
|
||||
20231109 2024届高三05班 1.000 1.000
|
||||
|
||||
003243
|
||||
20231109 2024届高三05班 0.828
|
||||
|
||||
003253
|
||||
20231109 2024届高三05班 0.586
|
||||
|
||||
001781
|
||||
20231109 2024届高三05班 0.552
|
||||
|
||||
006793
|
||||
20231109 2024届高三05班 0.379
|
||||
|
||||
008456
|
||||
20231109 2024届高三05班 0.345
|
||||
|
||||
009887
|
||||
20231109 2024届高三05班 0.897
|
||||
|
||||
001839
|
||||
20231109 2024届高三05班 1.000 0.448
|
||||
|
||||
006892
|
||||
20231109 2024届高三05班 0.828
|
||||
|
||||
003216
|
||||
20231109 2024届高三05班 0.172
|
||||
|
||||
001783
|
||||
20231109 2024届高三05班 1.000 0.121
|
||||
|
||||
003225
|
||||
20231109 2024届高三04班 0.933
|
||||
|
||||
009876
|
||||
20231109 2024届高三04班 0.967 0.933 0.933
|
||||
|
||||
001749
|
||||
20231109 2024届高三04班 0.900 0.900
|
||||
|
||||
001767
|
||||
20231109 2024届高三04班 0.733
|
||||
|
||||
001789
|
||||
20231109 2024届高三04班 0.833
|
||||
|
||||
008417
|
||||
20231109 2024届高三04班 1.000
|
||||
|
||||
001785
|
||||
20231109 2024届高三04班 0.900
|
||||
|
||||
000321
|
||||
20231109 2024届高三04班 0.333
|
||||
|
||||
000312
|
||||
20231109 2024届高三04班 0.900
|
||||
|
||||
003238
|
||||
20231109 2024届高三04班 0.933
|
||||
|
||||
030473
|
||||
20231109 2024届高三04班 0.567
|
||||
|
||||
001771
|
||||
20231109 2024届高三04班 0.800
|
||||
|
||||
003244
|
||||
20231109 2024届高三04班 0.967 0.967
|
||||
|
||||
003243
|
||||
20231109 2024届高三04班 0.867
|
||||
|
||||
003253
|
||||
20231109 2024届高三04班 0.733
|
||||
|
||||
001781
|
||||
20231109 2024届高三04班 0.233
|
||||
|
||||
006793
|
||||
20231109 2024届高三04班 0.183
|
||||
|
||||
008456
|
||||
20231109 2024届高三04班 0.533
|
||||
|
||||
009887
|
||||
20231109 2024届高三04班 0.867
|
||||
|
||||
001839
|
||||
20231109 2024届高三04班 0.967 0.567
|
||||
|
||||
006892
|
||||
20231109 2024届高三04班 0.900
|
||||
|
||||
003216
|
||||
20231109 2024届高三04班 0.067
|
||||
|
||||
001783
|
||||
20231109 2024届高三04班 1.000 0.133
|
||||
|
||||
003225
|
||||
20231109 2024届高三07班 0.950
|
||||
|
||||
009876
|
||||
20231109 2024届高三07班 0.950 0.975 1.000
|
||||
|
||||
001749
|
||||
20231109 2024届高三07班 0.975 0.950
|
||||
|
||||
001767
|
||||
20231109 2024届高三07班 0.875
|
||||
|
||||
001789
|
||||
20231109 2024届高三07班 0.750
|
||||
|
||||
008417
|
||||
20231109 2024届高三07班 0.950
|
||||
|
||||
001785
|
||||
20231109 2024届高三07班 0.925
|
||||
|
||||
000321
|
||||
20231109 2024届高三07班 0.300
|
||||
|
||||
000312
|
||||
20231109 2024届高三07班 0.637
|
||||
|
||||
003238
|
||||
20231109 2024届高三07班 0.875
|
||||
|
||||
030473
|
||||
20231109 2024届高三07班 0.650
|
||||
|
||||
001771
|
||||
20231109 2024届高三07班 0.800
|
||||
|
||||
003244
|
||||
20231109 2024届高三07班 1.000 1.000
|
||||
|
||||
003243
|
||||
20231109 2024届高三07班 0.850
|
||||
|
||||
003253
|
||||
20231109 2024届高三07班 0.675
|
||||
|
||||
001781
|
||||
20231109 2024届高三07班 0.512
|
||||
|
||||
006793
|
||||
20231109 2024届高三07班 0.525
|
||||
|
||||
008456
|
||||
20231109 2024届高三07班 0.225
|
||||
|
||||
009887
|
||||
20231109 2024届高三07班 0.787
|
||||
|
||||
001839
|
||||
20231109 2024届高三07班 0.925 0.425
|
||||
|
||||
006892
|
||||
20231109 2024届高三07班 0.950
|
||||
|
||||
003216
|
||||
20231109 2024届高三07班 0.275
|
||||
|
||||
001783
|
||||
20231109 2024届高三07班 0.887 0.250
|
||||
|
||||
030428
|
||||
20231107 2024届高三11班 0.968
|
||||
|
||||
030449
|
||||
20231107 2024届高三11班 0.903
|
||||
|
||||
030450
|
||||
20231107 2024届高三11班 1.000
|
||||
|
||||
030431
|
||||
20231107 2024届高三11班 0.935
|
||||
|
||||
030432
|
||||
20231107 2024届高三11班 0.968
|
||||
|
||||
030433
|
||||
20231107 2024届高三11班 1.000
|
||||
|
||||
030434
|
||||
20231107 2024届高三11班 0.968
|
||||
|
||||
030435
|
||||
20231107 2024届高三11班 0.935
|
||||
|
||||
030436
|
||||
20231107 2024届高三11班 0.806
|
||||
|
||||
030437
|
||||
20231107 2024届高三11班 0.871
|
||||
|
||||
030438
|
||||
20231107 2024届高三11班 0.903
|
||||
|
||||
030439
|
||||
20231107 2024届高三11班 0.323
|
||||
|
||||
030440
|
||||
20231107 2024届高三11班 1.000
|
||||
|
||||
030441
|
||||
20231107 2024届高三11班 0.871
|
||||
|
||||
030442
|
||||
20231107 2024届高三11班 1.000
|
||||
|
||||
030443
|
||||
20231107 2024届高三11班 0.710
|
||||
|
||||
030444
|
||||
20231107 2024届高三11班 1.000 0.903
|
||||
|
||||
030445
|
||||
20231107 2024届高三11班 0.968 0.903
|
||||
|
||||
030446
|
||||
20231107 2024届高三11班 1.000 0.855
|
||||
|
||||
012074
|
||||
20231107 2024届高三11班 0.935 0.161
|
||||
|
||||
030428
|
||||
20231107 2024届高三04班 1.000
|
||||
|
||||
030449
|
||||
20231107 2024届高三04班 0.839
|
||||
|
||||
030450
|
||||
20231107 2024届高三04班 0.935
|
||||
|
||||
030431
|
||||
20231107 2024届高三04班 0.903
|
||||
|
||||
030432
|
||||
20231107 2024届高三04班 0.968
|
||||
|
||||
030433
|
||||
20231107 2024届高三04班 0.935
|
||||
|
||||
030434
|
||||
20231107 2024届高三04班 0.968
|
||||
|
||||
030435
|
||||
20231107 2024届高三04班 0.839
|
||||
|
||||
030436
|
||||
20231107 2024届高三04班 0.903
|
||||
|
||||
030437
|
||||
20231107 2024届高三04班 0.645
|
||||
|
||||
030438
|
||||
20231107 2024届高三04班 0.710
|
||||
|
||||
030439
|
||||
20231107 2024届高三04班 0.387
|
||||
|
||||
030440
|
||||
20231107 2024届高三04班 0.935
|
||||
|
||||
030441
|
||||
20231107 2024届高三04班 0.839
|
||||
|
||||
030442
|
||||
20231107 2024届高三04班 0.806
|
||||
|
||||
030443
|
||||
20231107 2024届高三04班 0.935
|
||||
|
||||
030444
|
||||
20231107 2024届高三04班 1.000 0.871
|
||||
|
||||
030445
|
||||
20231107 2024届高三04班 0.935 0.935
|
||||
|
||||
030446
|
||||
20231107 2024届高三04班 0.935 0.742
|
||||
|
||||
012074
|
||||
20231107 2024届高三04班 0.935 0.145
|
||||
|
||||
030428
|
||||
20231107 2024届高三06班 1.000
|
||||
|
||||
030449
|
||||
20231107 2024届高三06班 0.971
|
||||
|
||||
030450
|
||||
20231107 2024届高三06班 0.941
|
||||
|
||||
030431
|
||||
20231107 2024届高三06班 0.912
|
||||
|
||||
030432
|
||||
20231107 2024届高三06班 1.000
|
||||
|
||||
030433
|
||||
20231107 2024届高三06班 0.971
|
||||
|
||||
030434
|
||||
20231107 2024届高三06班 0.971
|
||||
|
||||
030435
|
||||
20231107 2024届高三06班 0.824
|
||||
|
||||
030436
|
||||
20231107 2024届高三06班 0.853
|
||||
|
||||
030437
|
||||
20231107 2024届高三06班 0.647
|
||||
|
||||
030438
|
||||
20231107 2024届高三06班 0.794
|
||||
|
||||
030439
|
||||
20231107 2024届高三06班 0.471
|
||||
|
||||
030440
|
||||
20231107 2024届高三06班 0.971
|
||||
|
||||
030441
|
||||
20231107 2024届高三06班 0.941
|
||||
|
||||
030442
|
||||
20231107 2024届高三06班 0.912
|
||||
|
||||
030443
|
||||
20231107 2024届高三06班 0.912
|
||||
|
||||
030444
|
||||
20231107 2024届高三06班 0.941 0.912
|
||||
|
||||
030445
|
||||
20231107 2024届高三06班 0.868 0.868
|
||||
|
||||
030446
|
||||
20231107 2024届高三06班 0.985 0.647
|
||||
|
||||
012074
|
||||
20231107 2024届高三06班 0.941 0.471
|
||||
|
||||
|
|
|
|||
|
|
@ -599643,7 +599643,7 @@
|
|||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"ans": "$0$.",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [
|
||||
|
|
@ -599676,7 +599676,7 @@
|
|||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"ans": "$x \\leq 1$ 且 $y \\leq 1$.",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [
|
||||
|
|
@ -599709,7 +599709,7 @@
|
|||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"ans": "$(1,2)$.",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [
|
||||
|
|
@ -599742,7 +599742,7 @@
|
|||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"ans": "$4$.",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [
|
||||
|
|
@ -599775,7 +599775,7 @@
|
|||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"ans": "$14$.",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [
|
||||
|
|
@ -599808,7 +599808,7 @@
|
|||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"ans": "$\\dfrac{a+1}{2}$.",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [
|
||||
|
|
@ -599841,7 +599841,7 @@
|
|||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"ans": "$\\dfrac{1}{8}$.",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [
|
||||
|
|
@ -599874,7 +599874,7 @@
|
|||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"ans": "$[0,4)$.",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [
|
||||
|
|
@ -599907,7 +599907,7 @@
|
|||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"ans": "$(\\dfrac{1}{2}, 1)$.",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [
|
||||
|
|
@ -599940,7 +599940,7 @@
|
|||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"ans": "$(-\\infty, \\dfrac{1}{2}]$.",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [
|
||||
|
|
@ -599973,7 +599973,7 @@
|
|||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"ans": "$20$.",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [
|
||||
|
|
@ -600006,7 +600006,7 @@
|
|||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"ans": "$(-\\infty, \\dfrac{1}{2}]$.",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [
|
||||
|
|
@ -600039,7 +600039,7 @@
|
|||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"ans": "D",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [
|
||||
|
|
@ -600072,7 +600072,7 @@
|
|||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"ans": "A",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [
|
||||
|
|
@ -600105,7 +600105,7 @@
|
|||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"ans": "C",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [
|
||||
|
|
@ -600138,7 +600138,7 @@
|
|||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"ans": "A",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [
|
||||
|
|
@ -600171,8 +600171,8 @@
|
|||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"ans": "(1) $(-3,4]$; (2) $(-1,1)$",
|
||||
"solution": "(1) $A=(-3,3)$, $B=[0,4]$, 故 $A \\cup B=(-3,4]$.\\\\\n(2) $A \\cap B=B$ 等价于 $B \\subseteq A$, 故 $a-2>-3$ 且 $a+2<3$, 故 $a$ 的取值范围是 $(-1,1)$.",
|
||||
"duration": -1,
|
||||
"usages": [
|
||||
"20231108\t2026届高一01班\t0.968\t0.940",
|
||||
|
|
@ -600204,8 +600204,8 @@
|
|||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"ans": "(1) 当 $a=b$ 时, $a^3+b^3=a^2 b+a b^2$; 当 $a \\neq b$ 时, $a^3+b^3>a^2 b+a b^2$; (2) 最小值为$1$",
|
||||
"solution": "(1) $a^3+b^3-a^2 b-a b^2=(a-b)^2(a+b)$, 故当 $a=b$ 时, $a^3+b^3=a^2 b+a b^2$; 当 $a \\neq b$ 时, $a^3+b^3>a^2 b+a b^2$.\\\\\n(2) 由 (1) 知 $a^3+b^3 \\geq a^2 b+a b^2$, 又由于 $a>0$, $b>0$, 故 $\\dfrac{a^3+b^3}{a b}\\geq \\dfrac{a^2 b+a b^2}{a b}$, 即 $\\dfrac{a^2}{b}+\\dfrac{b^2}{a}\\geq a+b=1$,当 $a=b=\\dfrac{1}{2}$ 时等号成立. 故 $\\dfrac{a^2}{b}+\\dfrac{b^2}{a}$ 的最小值是 $1$.",
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"duration": -1,
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"usages": [
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"20231108\t2026届高一01班\t0.926\t0.782",
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@ -600237,8 +600237,8 @@
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"objs": [],
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"tags": [],
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"genre": "解答题",
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"ans": "",
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"solution": "",
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"ans": "(1) $200$吨; (2) $100$至$300$吨(含$100$吨与$300$吨)",
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"solution": "(1) 该小区每月处理 $x$ 吨垃圾, 则每吨垃圾的平均成本: $w=\\dfrac{x^2-2 x+40000}{x}=x+\\dfrac{40000}{x}-200 \\geq 2 \\sqrt{40000}-200=200$. 当且仅当 $x=200$ 时等号成立. 故该小区每月分类处理 200 吨垃圾, 才能使得每吨垃圾分类处理的平均成本最低.\\\\\n(2) 题意即: $300 x \\geq x^2-200 x+40000$. 解集为 $[100,400]$, 又 $x \\leq 300$, 故 $x \\in[100,300]$. 所以该小区每月垃圾分类处理量 $x$ 的取值范围是 $[100,300]$.",
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"duration": -1,
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"usages": [
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"20231108\t2026届高一01班\t0.571\t0.921",
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@ -600270,8 +600270,8 @@
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"genre": "解答题",
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"ans": "(1) $\\log_2 3$; (2) $[\\dfrac{9}{4},+\\infty)$; (3) $[8,9)$",
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"solution": "(1) $2^x-\\dfrac{3}{2^x}=2 \\Leftrightarrow(2^x)^2-2(2^x)-3=0 \\Leftrightarrow 2^x=3$ 或 $2^x=-1$ (舍), 故 $x=\\log _23$.\\\\\n(2) 题意即: 对任意 $x \\in \\mathbf{R}$, $2^x+\\dfrac{a}{2^x}\\geq 3$ 恒成立 $\\Leftrightarrow a \\geq-(2^x)^2+3(2^x)=-(2^x-\\dfrac{3}{2})^2+\\dfrac{9}{4}$ 恒成立.\n又 $-(2^x-\\dfrac{3}{2})^2+\\dfrac{9}{4}\\leq \\dfrac{9}{4}$, 当 $x=\\log _2 \\dfrac{3}{2}$ 时等号成立.\n故 $-(2^x-\\dfrac{3}{2})^2+\\dfrac{9}{4}$ 的最大值是 $\\dfrac{9}{4}$, 故实数 $a$ 的取值范围是 $[\\dfrac{9}{4},+\\infty)$.\\\\\n(3) 令 $t=2^x \\in$($0,+\\infty$), 故 $2^x+\\dfrac{a}{2^x}=6$ 有两个不同的实数根 $\\Leftrightarrow t+\\dfrac{a}{t}-6=0$ 有两个不同的正实数根 $\\Leftrightarrow t^2-6 t+a=0$ 有两个不同的正实数根.\n故 $\\begin{cases}\\Delta=36-4 a>0,\\\\6>0,\\\\a>0 ;\\end{cases}$, 解得 $a \\in(0,9)$.\n又 $|x_1-x_2|=|\\log _2 t_1-\\log _2 t_2|=|\\log _2 \\dfrac{t_1}{t_2}| \\leq 1$,\n故 $\\dfrac{1}{2}\\leq \\dfrac{t_1}{t_2}\\leq 2$, 即 $\\dfrac{1}{2}\\leq \\dfrac{3-\\sqrt{9-a}}{3+\\sqrt{9+a}}\\leq 2$, 即 $\\sqrt{9-a}\\leq 1$, 即 $a \\in[8,9]$.\n综上, $a$ 的取值范围是 $[8,9)$.",
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"duration": -1,
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"usages": [
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"20231108\t2026届高一01班\t0.952\t0.683\t0.420",
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@ -600303,8 +600303,8 @@
|
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"objs": [],
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"genre": "解答题",
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||||
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|
||||
"solution": "",
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||||
"ans": "(1) $\\{1,2,3,4\\}$ 不是``可分集合'', $\\{1,3,5,7,9,11,13\\}$ 是``可分集合''; (2) 证明略; (3) 证明略",
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"solution": "(1) 集合 $\\{1,2,3,4\\}$ 不是``可分集合'', $\\{1,3,5,7,9,11,13\\}$ 是``可分集合''.\\\\\n(2) 反证法: 假设集合 $A$ 是``可分集合'', 不妨设 $a_1<a_2<a_3<a_4<a_5$.\n去掉``$a_1$''后, 剩下的元素可分为两个集合, 可能分别有两个元素, 可能一个集合有 1 个元素, 另一个集合有 3 个元素, 故 $a_2+a_5=a_3+a_4$\\textcircled{1} 或 $a_5=a_2+a_3+a_4$\\textcircled{2}.\\\\\n去掉``$a_2$''后, 得到 $a_1+a_5=a_3+a_4$\\textcircled{3} 或 $a_5=a_1+a_3+a_4$\\textcircled{4}. 由\\textcircled{1}\\textcircled{3}知, $a_1=a_2$, 与集合元素的互异性矛盾, 由\\textcircled{1}\\textcircled{4}知, $a_1+a_2=0$, 与 $a_1, a_2$ 是正整数矛盾, 由\\textcircled{2}\\textcircled{3}知, $a_1+a_2=0$, 与 $a_1, a_2$ 是正整数矛盾, 由\\textcircled{2}\\textcircled{4}知, $a_1=a_2$, 与集合元素的互异性矛盾. 故假设不成立, 所以集合 $A$ 不是``可分集合''.\\\\\n(3) 记 $S=a_1+a_2+\\cdots+a_n$, 因集合 $A$ 是``可分集合'', 故 $S-a_1, S-a_2, \\cdots, S-a_n$ 是偶数, 故 $a_1, a_2, \\cdots, a_n, S$奇偶性相同.\\\\\n情况 1: 当 $a_1$ 是奇数时, 有 $a_1, a_2, \\cdots, a_n, S$ 都是奇数, 又 $S=a_1+a_2+\\cdots+a_n$ 是 $n$ 个奇数的和, 故 $n$ 是奇数.情况 2: 当 $a_2$ 是偶数时, 因集合 $A$ 是``可分集合'', 故去掉``$a_i$''($i=1,2, \\cdots, n$) 后, 剩余的所有元素组成的集合能分为两个交集为空的集合 $B_i, C_i$, 且这两个集合的所有元素之和相等, 记为 $S_{B_i}=S_{C_i}$.\\\\\n下证 $A_1=\\{\\dfrac{a_1}{2}, \\dfrac{a_2}{2}, \\cdots, \\dfrac{a_n}{2}\\}$ 是``可分集合''.\n因 $a_1, a_2, \\cdots, a_n, S$ 是正偶数, 故 $\\dfrac{a_1}{2}, \\dfrac{a_2}{2}, \\cdots, \\dfrac{a_n}{2}$ 均为正整数. 去掉 $\\dfrac{a_i}{2}$ 后, 集合 $B_i$ 中的元素分别乘以 $\\dfrac{1}{2}$ 后组成集合 $B_i'$, 集合 $C_i$ 中的元素分别乘以 $\\dfrac{1}{2}$ 后组成集合 $C_i'$, 因集合 $B_i \\cap C_i=\\varnothing$, 故 $B_i' \\cap C_i'=\\varnothing$. 因 $S_{B_i}=S_{C_i}$, 故 $S_{B_i'}=S_{C_i'}=\\dfrac{1}{2}S_{B_i}=\\dfrac{1}{2}S_{C_i}$.\n即 $A_1$ 去掉 $\\dfrac{a_i}{2}$ 后的元素组成的集合能可分为两个交集为空的集合, 且这两个集合的所有元素之和相等, 故 $A_1$是``可分集合''.\\\\\n这个步骤可以一直进行下去, 直至 $m$ 次后, $\\dfrac{a_1}{2^m}$ 是奇数, 此时因``可分集合''中元素的奇偶性相同, 故由情况 $1$ 可知 $n$ 是奇数.",
|
||||
"duration": -1,
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"usages": [
|
||||
"20231108\t2026届高一01班\t0.845\t0.417\t0.036",
|
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Reference in New Issue