添加一些单元标签
This commit is contained in:
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1de1d923ab
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4351715033
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@ -1,188 +1,302 @@
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ans
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tags
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024729
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第三单元
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024730
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第三单元
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024731
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第三单元
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024732
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第三单元
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024733
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第三单元
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024734
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第三单元
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024735
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第二单元
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第三单元
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024736
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第三单元
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024737
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第三单元
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024738
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第三单元
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024739
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$\{x|x\ne \dfrac{\pi}{6}+k\pi, \ k\in \mathbf{Z}\}$
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第三单元
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024740
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$[-1,\dfrac{5}{4}]$
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第三单元
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024742
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充分非必要
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024741
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$\dfrac{5\pi}{6}$
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第三单元
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024742
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第三单元
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024743
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$0$
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第三单元
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024744
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$[\dfrac{13\pi}{6},+\infty)$
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第三单元
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011448
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(1) $[0,\dfrac{\pi}{3}]$; (2) $\dfrac{24}{25}$
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004308
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(1) $AN=\dfrac{4\sqrt{3}}{3}\sin\theta$, $AM=\dfrac{4\sqrt{3}}{3}\sin(\theta+\dfrac{\pi}{3})$, $\theta\in (0,\dfrac{2\pi}{3})$; (2) 使$AM=AN=2\text{km}$, 才能使工厂产生的噪声对居民的影响最小
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018453
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(1) 最大值是$6$, 当且仅当$x=\pi+2k\pi$, $k\in \mathbf{Z}$时取到; 最小值是$-2$, 当且仅当$x=2k\pi$, $k\in \mathbf{Z}$时取到;\\
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(2) 最大值是$1$, 当且仅大哥$x=0$时取到; 最小值是$-\dfrac{1}{2}$, 当且仅当$x=-\dfrac{4\pi}{3}$时取到
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018454
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最小正周期为$\pi$, 单调增区间是$[k\pi-\dfrac{\pi}{3}k\pi+\dfrac{\pi}{6}]$, $k\in \mathbf{Z}$
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018456
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(1) $\{x|\dfrac{\pi}{2}+2k\pi<x<\dfrac{3\pi}{2}+2k\pi, \ k\in \mathbf{Z}\}$; (2) $\{x|-\dfrac{3\pi}{4}+2k\pi<x<\dfrac{3\pi}{4}+2k\pi, \ k\in \mathbf{Z}\}$
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009604
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$\dfrac{1}{2}$
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009605
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(1) 奇函数, 理由略; (2) 奇函数, 理由略; (3) 既不是奇函数, 又不是偶函数, 理由略
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009606
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最小正周期是$4\pi$, 单调减区间为$[\dfrac{\pi}{3}+4k\pi,\dfrac{7\pi}{3}+4k\pi]$, $k\in \mathbf{Z}$, 单调增区间为$[-\dfrac{5\pi}{3}+4k\pi,\dfrac{\pi}{3}+4k\pi]$, $k\in \mathbf{Z}$
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010289
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(1) \begin{tikzpicture}[>=latex, scale = 0.6]
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\draw [->] (-0.5,0) -- (7,0) node [below] {$x$};
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\draw [->] (0,-3.5) -- (0,1.5) node [left] {$y$};
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\draw (0,0) node [below left] {$O$};
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\draw [domain = 0:2*pi, samples = 100] plot (\x,{cos(\x/pi*180)*2-1});
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\draw [dashed] (2*pi,1) -- (0,1) node [left] {$1$};
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\draw [dashed] (2*pi,1) -- (2*pi,0) node [below] {$2\pi$};
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\draw [dashed] (pi,0) node [above] {$\pi$} -- (pi,-3) -- (0,-3) node [left] {$-3$};
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\end{tikzpicture}\\
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(2) \begin{tikzpicture}[>=latex]
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\draw [->] (-7,0) -- (7,0) node [below] {$x$};
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\draw [->] (0,-0.5) -- (0,1.5) node [left] {$y$};
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\draw (0,0) node [below left] {$O$};
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\draw [domain = -pi/2:pi/2, samples = 100] plot (\x,{cos(\x/pi*180)});
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\draw [domain = pi/2:3*pi/2, samples = 100] plot (\x,{-cos(\x/pi*180)});
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\draw [domain = 3*pi/2:7, samples = 100] plot (\x,{cos(\x/pi*180)});
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\draw [domain = -3*pi/2:-pi/2, samples = 100] plot (\x,{-cos(\x/pi*180)});
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\draw [domain = -7:-3*pi/2, samples = 100] plot (\x,{cos(\x/pi*180)});
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\draw (pi/2,0) node [below] {$\frac{\pi}{2}$};
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\draw (0,1) node [above left] {$1$};
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\end{tikzpicture}
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023606
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$\dfrac{\pi}{6}$, $\pi$
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010291
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(1) 最大值为$3$, 取得最大值时$x$的集合为$\{x|x=k\pi, \ k\in \mathbf{Z}\}$; 最小值为$\dfrac{1}{3}$, 取得最小值时$x$的集合为$\{x|x=\dfrac{\pi}{2}+k\pi, \ k\in \mathbf{Z}\}$;\\
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(2) 最大值为$1$, 取得最大值时$x$的集合为$\{x|x=2k\pi, \ k\in \mathbf{Z}\}$; 最小值为$-\dfrac{5}{4}$, 取最小值时$x$的集合为$\{x|x=\dfrac{2\pi}{3}+2k\pi\text{ 或 }\dfrac{4\pi}{3}+2k\pi, \ k\in \mathbf{Z}\}$
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023607
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在$[-\dfrac{\pi}{6},\dfrac{\pi}{3}]$上严格减, 在$[\dfrac{\pi}{3},\dfrac{2\pi}{3}]$上严格增, 值域为$[-1,1]$
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018457
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\begin{tikzpicture}[>=latex]
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\draw [->] (-7,0) -- (7,0) node [below] {$x$};
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\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};
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\draw (0,0) node [below right] {$O$};
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\draw [domain = -7:7, samples = 200, ultra thick] plot (\x,{2*sin(\x/pi*180)});
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\draw [domain = -7:7, samples = 200, thick] plot (\x,{1*sin(2*\x/pi*180)});
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\draw [domain = -7:7, samples = 200, dashed, ultra thick] plot (\x,{sin(\x/pi*180+90)});
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\draw [domain = -7:7, samples = 200, dashed] plot (\x,{sin(\x/pi*180)});
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\draw [dashed] (-6,-3) -- (-5.5,-3) node [right] {$y=\sin x$};
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\draw [ultra thick] (-3,-3) -- (-2.5,-3) node [right] {$y=2\sin x$};
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\draw [thick] (0,-3) -- (0.5,-3) node [right] {$y=\sin 2x$};
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\draw [ultra thick, dashed] (3,-3) -- (3.5,-3) node [right] {$y=\sin (x+\dfrac{\pi}{2})$};
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\end{tikzpicture}
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018458
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大致图像: \begin{tikzpicture}[>=latex, scale = 0.5]
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\draw [->] (-5,0) -- (5,0) node [below] {$x$};
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\draw [->] (0,-3.5) -- (0,3.5) node [left] {$y$};
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\draw (0,0) node [below right] {$O$};
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\draw [domain = -5:5, samples = 100] plot (\x,{3*sin(2*\x/pi*180+45)});
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\foreach \i in {-3,-2,-1,1,2,3}
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{\draw (0.1,\i) -- (0,\i) node [left] {$\i$};};
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\foreach \i in {{-pi/8},{pi/8},{3*pi/8},{5*pi/8},{7*pi/8}}
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{\draw (\i,0.2) -- (\i,0);};
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\draw (-pi/8,0) node [above left] {$-\frac{\pi}{8}$};
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\draw (pi/8,0) node [above] {$\frac{\pi}{8}$};
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\draw (3*pi/8,0) node [above right] {$\frac{3\pi}{8}$};
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\draw (5*pi/8,0) node [below] {$\frac{5\pi}{8}$};
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\draw (7*pi/8,0) node [below right] {$\frac{7\pi}{8}$};
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\end{tikzpicture}\\
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振幅为$3$, 频率为$\dfrac{1}{\pi}$, 初始相位为$\dfrac{\pi}{4}$
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018459
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周期$T=0.02\text{s}$, 频率$f=50\text{Hz}$, 电流的最大值为$10\text{A}$; $I_0=10$, $\omega = 100\pi$, $\varphi = 0$
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009607
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(1) \begin{tikzpicture}[>=latex]
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\draw [->] (-5,0) -- (5,0) node [below] {$x$};
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\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};
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\draw (0,0) node [below left] {$O$};
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\draw [domain = -5:5, samples = 100] plot (\x,{sin(\x/pi*180+30)});
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\draw [dashed] (pi/3,0) node [below] {$\frac{\pi}{3}$}-- (pi/3,1) -- (0,1) node [left] {$1$};
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\draw [dashed] (-2*pi/3,0) node [below] {$-\frac{2\pi}{3}$}-- (-2*pi/3,-1) -- (0,-1) node [right] {$-1$};
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\end{tikzpicture}\\
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(2) \begin{tikzpicture}[>=latex, scale = 0.5]
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\draw [->] (-5,0) -- (5,0) node [below] {$x$};
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\draw [->] (0,-4) -- (0,4) node [left] {$y$};
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\draw (0,0) node [below left] {$O$};
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\draw [domain = -5:5, samples = 100] plot (\x,{3*sin(2*\x/pi*180-60)});
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\draw [dashed] (5*pi/12,0) node [below] {$\frac{5\pi}{12}$} --++ (0,3) -- (0,3) node [left] {$3$};
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\draw [dashed] (-pi/12,0) node [above] {$\frac{\pi}{12}$} --++ (0,-3) -- (0,-3) node [right] {$-3$};
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\end{tikzpicture}
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009608
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D
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009609
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$y=\dfrac{3}{2}\sin (2x+\dfrac{\pi}{6})$
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018460
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$\dfrac{4\pi}{3}$
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018461
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$\varphi=\dfrac{\pi}{2}$, $\omega = \dfrac{3}{4}$
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010298
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振幅为$\sqrt{2}$, 频率为$30\pi$, 初始相位为$-\dfrac{\pi}{12}$
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010301
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\begin{tikzpicture}[>=latex]
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\draw [->] (-0.5,0) -- (3.5,0) node [below] {$x$};
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\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};
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\draw (0,0) node [below left] {$O$};
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\draw [domain = 0:3.5, samples = 100] plot (\x,{2*sin(\x*180+45)});
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\draw [dashed] (0.25,0) node [below] {$\frac{1}{4}$} --++ (0,2) -- (0,2) node [left] {$2$};
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\draw [dashed] (1.25,0) node [above] {$\frac{5}{4}$} --++ (0,-2) -- (0,-2) node [left] {$-2$};
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\end{tikzpicture}\\
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(1) 开始振动的位置在平衡位置上方$\sqrt{2}\text{cm}$处; (2) 最高点和最低点与平衡位置的距离都是$2$; (3) 经过$2\text{s}$小球往复振动一次; (4) 每秒小球往复振动$0.5$次
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010303
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$y=2\cos(\dfrac{1}{2} x+\dfrac{5\pi}{3})$
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018462
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定义域为$\{x|x\in \mathbf{R}, \ x\ne 6k+1, \ k\in \mathbf{Z}\}$, 单调增区间为$(6k-5,6k+1)$, $k\in \mathbf{Z}$
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018463
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$\dfrac{\pi}{2}$
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018464
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$2\sqrt{2}$
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009610
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$\{\alpha|\alpha = \dfrac{\pi}{3}+k\pi, \ k\in \mathbf{Z}\}$
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009611
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(1) $\tan (-\dfrac 27\pi )>\tan (-\dfrac 25\pi )$, 理由略; (2) $\cot 231^\circ>\cot 237^\circ$, 理由略; (3) $\tan (k\pi -\dfrac\pi 3)<\tan (k\pi +\dfrac\pi 3)$, 理由略
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018465
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$n=3$, 在$(-\dfrac{\pi}{18}+\dfrac{k\pi}{3}, \dfrac{5\pi}{18}+\dfrac{k\pi}{3})$, $k\in \mathbf{Z}$上是严格增函数
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010308
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(1) 奇函数, 理由略; (2) 偶函数, 理由略; (3) 奇函数, 理由略; (4) 偶函数, 理由略
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024747
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$\pi$
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第三单元
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024748
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面积$S=2\sqrt{2}\sin\theta\sin(\dfrac{\pi}{4}-\theta)(=\sqrt{2}\sin(2\theta+\dfrac{\pi}{4})-1)$, $\theta\in (0,\dfrac{\pi}{4})$, 面积的最大值为$\sqrt{2}-1$
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第三单元
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024750
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第三单元
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032150
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第五单元
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032151
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第五单元
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032152
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第七单元
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032153
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第九单元
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032154
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第四单元
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032155
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032156
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032157
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032158
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032159
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032160
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032161
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032162
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032163
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032164
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032165
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032166
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032167
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032168
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032169
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第七单元
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032170
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第七单元
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032171
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第七单元
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032172
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032173
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032174
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第六单元
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032175
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第六单元
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032176
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032177
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032178
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032179
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032180
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032181
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第一单元
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032182
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032183
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第八单元
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032184
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032185
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第五单元
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032186
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第五单元
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032187
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第一单元
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032188
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第九单元
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032189
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第一单元
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第三单元
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第七单元
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032190
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第六单元
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032191
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第三单元
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032192
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第九单元
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第八单元
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032193
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第七单元
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032194
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第二单元
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第四单元
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041088
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第七单元
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041089
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第七单元
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041090
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第七单元
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041091
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第七单元
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041092
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第九单元
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041093
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第九单元
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041094
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第七单元
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041095
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第七单元
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041096
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第七单元
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@ -690040,7 +690040,9 @@
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"id": "024729",
|
||||
"content": "对于函数$y=\\sin x$, 当$x=\\dfrac{7\\pi}{6}$时, $\\sin(x+\\dfrac{2\\pi}{3})=\\sin x$是否成立? 如果成立, 那么$\\dfrac{2\\pi}{3}$是不是$y=\\sin x$的周期? 为什么?",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第三单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "不是, 如$x=\\dfrac{\\pi}{3}$时, $\\sin (x+\\dfrac{2\\pi}{3})\\ne \\sin x$",
|
||||
"solution": "",
|
||||
|
|
@ -690062,7 +690064,9 @@
|
|||
"id": "024730",
|
||||
"content": "已知$y=\\sin (\\dfrac{k}{2}x+\\dfrac{\\pi}{3})$的最小正周期为$2$, 则实数 $k$ 的值为\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第三单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "$\\pm 2\\pi$",
|
||||
"solution": "",
|
||||
|
|
@ -690091,7 +690095,9 @@
|
|||
"id": "024731",
|
||||
"content": "函数$y=\\sin x+\\sqrt{3}\\cos x$, $x \\in(-\\dfrac{\\pi}{3}, \\dfrac{\\pi}{2})$ 的值域为\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第三单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "$(0,2]$",
|
||||
"solution": "",
|
||||
|
|
@ -690122,7 +690128,9 @@
|
|||
"id": "024732",
|
||||
"content": "函数$y=\\sin x \\cos x$ 的严格增区间是\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第三单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "$[-\\dfrac{\\pi}{4}+k\\pi, \\dfrac{\\pi}{4}+k\\pi]$, $k\\in \\mathbf{Z}$",
|
||||
"solution": "",
|
||||
|
|
@ -690151,7 +690159,9 @@
|
|||
"id": "024733",
|
||||
"content": "函数$y=1-2\\cos^2(x-\\dfrac{\\pi}{4})$是\\blank{50}函数.(填写奇偶性)",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第三单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "奇",
|
||||
"solution": "",
|
||||
|
|
@ -690181,7 +690191,9 @@
|
|||
"id": "024734",
|
||||
"content": "已知$\\varphi \\in [0,2\\pi]$, 若函数 $y=\\sin (55 x+\\varphi-\\dfrac{\\pi}{3})$ 是偶函数, 则 $\\varphi=$\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第三单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "$\\dfrac{5\\pi}{6}$或$\\dfrac{11\\pi}{6}$",
|
||||
"solution": "",
|
||||
|
|
@ -690214,7 +690226,10 @@
|
|||
"id": "024735",
|
||||
"content": "函数 $y=\\ln (2 \\sin x-\\sqrt{3})$ 的定义域是\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第二单元",
|
||||
"第三单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "$\\{x|\\dfrac{\\pi}{3}+2k\\pi<x<\\dfrac{2\\pi}{3}+2k\\pi, \\ k\\in \\mathbf{Z}\\}$",
|
||||
"solution": "",
|
||||
|
|
@ -690246,7 +690261,9 @@
|
|||
"id": "024736",
|
||||
"content": "已知 $x \\in(0, \\dfrac{5 \\pi}{6})$, 若关于 $x$ 的方程 $a \\sin x-9=0$ 有且仅有一解, 则实数 $a$ 的取值范围是\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第三单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "$\\{9\\}\\cup [18,+\\infty)$",
|
||||
"solution": "",
|
||||
|
|
@ -690275,7 +690292,9 @@
|
|||
"id": "024737",
|
||||
"content": "已知函数 $y=f(x)$, $x \\in \\mathbf{R}$ 的周期是 $2$, 当 $x \\in[0,2)$ 时, $f(x)=3^x$ ,则当 $x \\in[4,6)$ 时, $f(x)=$\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第三单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "$3^{x-4}$",
|
||||
"solution": "",
|
||||
|
|
@ -690308,7 +690327,9 @@
|
|||
"id": "024738",
|
||||
"content": "用五点法做出函数 $y=3 \\sin (x+\\dfrac{\\pi}{6})+1$ 的大致图像.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第三单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-2,0) -- (8,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -2:8, samples = 200] plot (\\x,{3*sin(\\x/pi*180+30)+1});\n\\draw [dashed] (pi/3,0) --++ (0,4) -- (0,4) (4*pi/3,0) --++ (0,-2) -- (0,-2);\n\\draw (pi/3,0) node [below] {$\\frac{\\pi}{3}$} (0,4) node [left] {$4$} (4*pi/3,0) node [above] {$\\frac{4\\pi}{3}$} (0,-2) node [left] {$-2$};\n\\end{tikzpicture}",
|
||||
"solution": "",
|
||||
|
|
@ -690341,7 +690362,9 @@
|
|||
"id": "024739",
|
||||
"content": "$f(x)=\\tan (x+\\dfrac{\\pi}{3})$ 的定义域是\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第三单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "$\\{x|x\\ne \\dfrac{\\pi}{6}+k\\pi, \\ k\\in \\mathbf{Z}\\}$",
|
||||
"solution": "",
|
||||
|
|
@ -690363,7 +690386,9 @@
|
|||
"id": "024740",
|
||||
"content": "$f(x)=\\sin ^2 x+\\cos x$ 的值域是\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第三单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "$[-1,\\dfrac{5}{4}]$",
|
||||
"solution": "",
|
||||
|
|
@ -690383,7 +690408,9 @@
|
|||
"id": "024741",
|
||||
"content": "函数 $f(x)=\\sin (2 x+\\theta)-\\sqrt{3}\\cos (2 x+\\theta)$ 是偶函数. 则最小正实数 $\\theta=$\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第三单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "$\\dfrac{5\\pi}{6}$",
|
||||
"solution": "",
|
||||
|
|
@ -690403,7 +690430,9 @@
|
|||
"id": "024742",
|
||||
"content": "已知 $\\omega \\in \\mathbf{R}, $``$\\omega=\\pi$''是``函数 $f(x)=\\sin \\omega x \\cdot \\cos \\omega x$ 的最小正周期是 1''的\\blank{50}\n条件.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第三单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "充分非必要",
|
||||
"solution": "",
|
||||
|
|
@ -690426,7 +690455,9 @@
|
|||
"id": "024743",
|
||||
"content": "已知函数 $y=f(x)$,$x \\in \\mathbf{R}$, 满足$f(x+\\pi)=f(x)+\\sin x$, 且当$0 \\leq x<\\pi$ 时, $f(x)=0$.\n则 $f(\\dfrac{17 \\pi}{6})=$\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第三单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "$0$",
|
||||
"solution": "",
|
||||
|
|
@ -690446,7 +690477,9 @@
|
|||
"id": "024744",
|
||||
"content": "已知 $x \\in(\\dfrac{\\pi}{2}, \\theta)$, 若对于任意 $a \\in (18,+\\infty)$. 关于 $x$ 的方程 $a \\sin x-9=0$ 至少有两个不同解, 则实数 $\\theta$ 的取值范围是\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第三单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "$[\\dfrac{13\\pi}{6},+\\infty)$",
|
||||
"solution": "",
|
||||
|
|
@ -690519,7 +690552,9 @@
|
|||
"id": "024747",
|
||||
"content": "函数$y=\\sin^2 x$的最小正周期为\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第三单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "$\\pi$",
|
||||
"solution": "",
|
||||
|
|
@ -690544,7 +690579,9 @@
|
|||
"id": "024748",
|
||||
"content": "如图, 圆心在$O$处, 半径为$1$, 圆心角为$\\dfrac{\\pi}{2}$的扇形以直线$OP$为对称轴($P$在圆弧上). 设$A,B,A',B'$是扇形边界上的四点, 其中$A,A'$在扇形的两条半径上, $B,B'$在圆弧上, $AB\\parallel OP$, $AA'\\perp OP$, $BB'\\perp OP$. 用$\\theta = \\angle BOP$表示矩形$AA'B'B$的面积, 并求面积的最大值.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) -- (2,0) arc (0:90:2) -- cycle;\n\\draw (0,0) node [below] {$O$} coordinate (O) (45:2) node [right] {$P$} coordinate (P);\n\\draw [dashed] ($(O)!-0.2!(P)$) -- ($(O)!1.2!(P)$);\n\\draw (30:2) node [right] {$B$} coordinate (B) (60:2) node [above] {$B'$} coordinate (B');\n\\draw ({sin(15)/sin(45)*2},0) node [below] {$A$} coordinate (A) (0,{sin(15)/sin(45)*2}) node [left] {$A'$} coordinate (A');\n\\draw (A)--(B)--(B')--(A')--cycle;\n\\draw [dashed] (O)--(B);\n\\end{tikzpicture}\n\\end{center}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第三单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "面积$S=2\\sqrt{2}\\sin\\theta\\sin(\\dfrac{\\pi}{4}-\\theta)(=\\sqrt{2}\\sin(2\\theta+\\dfrac{\\pi}{4})-1)$, $\\theta\\in (0,\\dfrac{\\pi}{4})$, 面积的最大值为$\\sqrt{2}-1$",
|
||||
"solution": "",
|
||||
|
|
@ -690590,7 +690627,9 @@
|
|||
"id": "024750",
|
||||
"content": "函数$y=(\\sin x+1)(\\cos x+1)$的值域是\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第三单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -750058,7 +750097,9 @@
|
|||
"id": "032150",
|
||||
"content": "已知向量 $\\overrightarrow{a}=(1,2)$, $\\overrightarrow{b}=(2,-1)$, $\\overrightarrow{c}=(1, \\lambda)$, 若 $\\overrightarrow{c}\\perp(\\overrightarrow{a}+\\overrightarrow{b})$, 则 $\\lambda=$\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第五单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -750085,7 +750126,9 @@
|
|||
"id": "032151",
|
||||
"content": "已知复平面上一个动点 $Z$ 对应复数 $z$, 若 $|z-4 \\mathrm{i}| \\leq 2$, 其中 $\\mathrm{i}$ 是虚数单位, 则向量 $\\overrightarrow{OZ}$ 扫过的面积为\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第五单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -750112,7 +750155,9 @@
|
|||
"id": "032152",
|
||||
"content": "已知直线 $l_1: a x+y+1=0$ 与直线 $l_2: x+a y-2=0$, 则``$a=1$''是``$l_1 \\parallel l_2$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -750156,7 +750201,9 @@
|
|||
"id": "032153",
|
||||
"content": "两位跳水运动员甲和乙, 某次比赛中的得分如下表所示, 则正确的选项为\\bracket{20}.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline & 第一跳 & 第二跳 & 第三跳 & 第四跳 & 第五跳 \\\\\n\\hline 甲 & 85.5 & 96 & 86.4 & 75.9 & 94.4 \\\\\n\\hline 乙 & 79.5 & 80 & 95.7 & 94.05 & 86.4 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\onech{甲和乙的中位数相等, 甲的均分小于乙}{甲的均分大于乙, 甲的方差大于乙}{甲的均分大于乙, 甲的方差等于乙}{甲的均分大于乙, 甲的方差小于乙}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第九单元"
|
||||
],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -750183,7 +750230,10 @@
|
|||
"id": "032154",
|
||||
"content": "已知等差数列 $\\{a_n\\}$, 公差为 $d, f(x)=|x-a_1|+|x-a_2|$, 则下列命题正确的是\\bracket{20}.\n\\onech{函数 $y=f(x) $($x \\in \\mathbf{R}$) 可能是奇函数}{若函数 $y=f(x)$($x \\in \\mathbf{R}$) 是偶函数, 则 $d=0$}{若 $d=0$, 则函数 $y=f(x)$($x \\in \\mathbf{R}$) 是偶函数}{若 $d \\neq 0$, 则函数 $y=f(x)$($x \\in \\mathbf{R}$) 的图像是轴对称图形}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第二单元",
|
||||
"第四单元"
|
||||
],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -750210,7 +750260,9 @@
|
|||
"id": "032155",
|
||||
"content": "已知三角形 $ABC, \\overrightarrow{CA}\\cdot \\overrightarrow{CB}=-1$, 三角形的面积 $S=\\dfrac{1}{2}$,\\\\\n(1) 求角 $C$ 的值;\\\\\n(2) 若 $\\sin A \\cos A=\\dfrac{\\sqrt{3}}{4}$, 求 $c$.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第三单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -750237,7 +750289,9 @@
|
|||
"id": "032156",
|
||||
"content": "交通拥堵指数 (TPI) 是表征交通拥堵程度的客观指标, 用 TPI 表示, TPI 越大代表拥堵程度越高. 某平台计算 TPI 的公式为: TIP $=\\dfrac{\\text{实际行程时间}}{\\text{畅通行程时间}}$, 并按 TPI 的大小将城市道路拥堵程度划分如下表所示的 4 个等级:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline TPI &{$[1,1.5)$}&{$[1.5,2)$}&{$[2,4)$}& 不低于 $4$ \\\\\n\\hline 拥堵等级 & 畅通 & 缓行 & 拥堵 & 严重拥堵 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n某市 2023 年元旦及前后共 7 天与 2022 年同期的交通高峰期城市道路 TPI 的统计数据如下图:\\\\\n\\begin{center}\n\\begin{tikzpicture}[>=latex,xscale = 0.85, yscale = 1.5]\n\\foreach \\i in {0,0.5,1,1.5,2,2.5}\n\\draw (0,\\i) -- (14,\\i);\n\\foreach \\i in {0,2,...,14}\n\\draw (\\i,0) -- (\\i,-0.05);\n\\draw (1,0) node [below] {12月29日};\n\\draw (3,0) node [below] {12月30日};\n\\draw (5,0) node [below] {12月31日};\n\\draw (7,0) node [below] {1月1日};\n\\draw (9,0) node [below] {1月2日};\n\\draw (11,0) node [below] {1月3日};\n\\draw (13,0) node [below] {1月4日};\n\\draw [dashed] (4,-0.7) -- (5,-0.7);\n\\filldraw (4.5,-0.7) circle ({0.03/0.85} and 0.02);\n\\draw (6,-0.7) node {2023年};\n\\draw (8,-0.7) -- (9,-0.7);\n\\filldraw (8.5,-0.7) ++ ({-0.03/0.85},-0.02) rectangle++ ({0.06/0.85},0.04);\n\\draw (10,-0.7) node {2022年};\n\\foreach \\i/\\j/\\k in {1/2.055/above,3/2.393/above,5/1.529/above,7/1.302/above,9/1.642/above,11/1.837/below,13/1.755/below}\n{\\filldraw (\\i,\\j) ++ ({-0.03/0.85},-0.02) rectangle++ ({0.06/0.85},0.04);\n\\draw (\\i,\\j) node [\\k] {$\\j$};}\n\\draw (1,2.055) -- (3,2.393) -- (5,1.529) -- (7,1.302) -- (9,1.642) -- (11,1.837) -- (13,1.755);\n\\foreach \\i/\\j/\\k in {1/1.908/below,3/2.081/below,5/1.331/below,7/1.202/below,9/1.271/below,11/2.256/above,13/2.012/above}\n{\\filldraw (\\i,\\j) circle ({0.03/0.85} and 0.02);\n\\draw (\\i,\\j) node [\\k] {$\\j$};}\n\\draw [dashed] (1,1.908) -- (3,2.081) -- (5,1.331) -- (7,1.202) -- (9,1.271) -- (11,2.256) -- (13,2.012);\n\\draw (-1,3) rectangle (15,-1);\n\\end{tikzpicture}\n\\end{center}\n(1) 从 2022 年元旦及前后共 7 天中任取 1 天, 求这一天交通高峰期城市道路拥堵程度为``拥堵''的概率;\\\\\n(2) 从 2023 年元旦及前后共 7 天中任取 3 天, 将这 3 天中交通高峰期城市道路 TPI 比 2022 年同日 TPI 高的天数记为 $X$, 求所有 $X$ 的可能值及其发生的概率.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第九单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -750264,7 +750318,9 @@
|
|||
"id": "032157",
|
||||
"content": "已知抛物线 $\\Gamma_1: y^2=4 x$, $\\Gamma_2: y^2=2 x$, 直线 $l$ 交抛物线 $\\Gamma_1$ 于点 $A$、$D$, 交抛物线 $\\Gamma_2$ 于点 $B$、$C$,其中点 $A$、$B$ 位于第一象限.\\\\\n(1) 若点 $A$ 到抛物线 $\\Gamma_1$ 焦点的距离为 2 , 求点 $A$ 的坐标;\\\\\n(2) 若点 $A$ 的坐标为 $(4,4)$, 且线段 $AC$ 的中点在 $x$ 轴上, 求原点 $O$ 到直线 $I$ 的距离;\\\\\n(3) 若 $\\overrightarrow{AB}=2 \\overrightarrow{CD}$, 求 $\\triangle AOD$ 与 $\\triangle BOC$ 的面积之比.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -750284,7 +750340,9 @@
|
|||
"id": "032158",
|
||||
"content": "已知函数 $f(x)=\\dfrac{x}{\\mathrm{e}^x}$, $g(x)=f(\\ln x)$.\\\\\n(1) 写出函数 $y=g(x)$ 的解析式, 并求函数 $y=f(x)$、$y=g(x)$ 的单调区间和极值;\\\\\n(2) 请严格证明曲线 $y=f(x)$、$y=g(x)$ 有唯一交点;\\\\\n(3) 对于常数 $a \\in(0, \\dfrac{1}{\\mathrm{e}})$, 若直线 $y=a$ 和曲线 $y=f(x)$、$y=g(x)$ 共有三个不同交点 $(x_1, a)$、$(x_2, a)$、$(x_3, a)$, 其中 $x_1<x_2<x_3$, 求证: $x_1$、$x_2$、$x_3$ 成等比数列.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第二单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -750311,7 +750369,9 @@
|
|||
"id": "032159",
|
||||
"content": "设集合 $A=\\{1,2,3\\}$, $B=\\{x | 1<x<3\\}$, 则 $A \\cap B=$\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第一单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -750347,7 +750407,9 @@
|
|||
"id": "032160",
|
||||
"content": "函数 $y=\\ln (x^2-1)$ 的定义域为\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第二单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -750382,7 +750444,9 @@
|
|||
"id": "032161",
|
||||
"content": "某小组成员的年龄分布茎叶图如图所示, 则该小组成员年龄的第 75 百分位数是\\blank{50}.\\begin{center}\n\\begin{tabular}{l|lllll}\n2 & 7 & 8 & & & \\\\\n3 & 2 & 3 & 6 & 6 & 8 \\\\\n4 & 0 & 5 & & & \\\\\n5 & 2 & 4 & 8 & &\n\\end{tabular}\n\\end{center}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第九单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -750417,7 +750481,9 @@
|
|||
"id": "032162",
|
||||
"content": "设复数 $z_1, z_2$ 在复平面上对应的点分别为 $Z_1(2,1), Z_2(1,-2)$, 则 $z_1 \\cdot z_2=$\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第五单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -750450,7 +750516,9 @@
|
|||
"id": "032163",
|
||||
"content": "已知 $P$ 为抛物线 $y^2=4 x$ 上的点, 且 $P$ 的纵坐标为 3 , 则 $P$ 到该抛物线焦点的距离为\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -750483,7 +750551,9 @@
|
|||
"id": "032164",
|
||||
"content": "在 $(x-\\dfrac{1}{\\sqrt{x}})^7$ 的二项展开式中, $x$ 的一次项的系数为\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第八单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -750518,7 +750588,9 @@
|
|||
"id": "032165",
|
||||
"content": "已知 $\\cos \\alpha=-\\dfrac{3}{5}, \\alpha$ 为第二象限角, 则 $\\tan 2 \\alpha=$\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第三单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -750551,7 +750623,9 @@
|
|||
"id": "032166",
|
||||
"content": "已知等比数列 $\\{a_n\\}$ 的公比为 $2$, 设其前 $n$ 项和为 $S_n$, 则使得 $\\dfrac{S_k}{a_k}>\\dfrac{4047}{2024}$ 成立的最小正整数 $k=$\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第四单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -750584,7 +750658,9 @@
|
|||
"id": "032167",
|
||||
"content": "已知常数 $a>0$ 且 $a \\neq 1$, 函数 $f(x)=\\dfrac{a^x-1}{2^x+1}$ 为奇函数, 则 $y=f(x)$ 的值域为\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第二单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -750619,7 +750695,9 @@
|
|||
"id": "032168",
|
||||
"content": "端午节吃粽子是我国的传统习俗. 一盘中放有 10 个外观完全相同的粽子, 其中豆沙粽 5个, 肉粽 3 个, 白米粽 2 个. 现从盘子任意取出 3 个, 则取到白米粽的个数的数学期望为\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第八单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -750654,7 +750732,10 @@
|
|||
"id": "032169",
|
||||
"content": "在平面直角坐标系 $x O y$ 中, $A, B$ 为直线 $x+y=1$ 上的两点, $|AB|=\\dfrac{2}{3}$, 且存在 $\\lambda \\in \\mathbf{R}$,使得 $|\\lambda \\overrightarrow{OA}-\\overrightarrow{OB}|=\\dfrac{1}{3}$, 则 $|OA|$ 的取值范围为\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第五单元",
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -750687,7 +750768,9 @@
|
|||
"id": "032170",
|
||||
"content": "已知 $x, y, \\alpha \\in \\mathbf{R}$, 满足 $4 x^2-y^2+75=10 y \\cdot \\sin \\alpha+40 x \\cdot \\cos \\alpha$, 则 $y^2+10 y \\cdot \\sin \\alpha$ 的最小值为\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -750720,7 +750803,9 @@
|
|||
"id": "032171",
|
||||
"content": "直线 $y=2 x+1$ 的一个法向量可以是\\bracket{20}.\n\\fourch{$(1,2)$}{$(1,-2)$}{$(2,1)$}{$(2,-1)$}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -750755,7 +750840,9 @@
|
|||
"id": "032172",
|
||||
"content": "某同学上学的路上有 3 个相互独立的红绿灯, 他走到每个红绿灯路口遇到绿灯的概率都为 $\\dfrac{2}{3}$, 则该同学在上学的路上至少遇到 2 次绿灯的概率为\\bracket{20}.\n\\fourch{$\\dfrac{4}{9}$}{$\\dfrac{20}{27}$}{$\\dfrac{22}{27}$}{$\\dfrac{8}{9}$}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第八单元"
|
||||
],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -750790,7 +750877,9 @@
|
|||
"id": "032173",
|
||||
"content": "已知函数 $y=f(x)$ 满足对任意 $x_1 \\in (-\\infty, 0)$, 总存在 $x_2 \\in (0,+\\infty)$, 使得 $f(x_1)-f(x_2)<-1$,则 $y=f(x)$ 可以是\\bracket{20}.\n\\fourch{$y=-x^2$}{$y=2 \\sin x$}{$y=-\\mathrm{e}^{-x}+2$}{$y=-x^4+\\dfrac{1}{2}x$}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第二单元"
|
||||
],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -750823,7 +750912,9 @@
|
|||
"id": "032174",
|
||||
"content": "已知三棱锥 $S-ABC$ 满足 $AB=BC=CA$, $\\angle ASB=\\angle BSC=\\angle CSA$. 对于命题: \\textcircled{1} 存在三棱锥 $S-ABC$, 使得线段 $SA, SB, SC$ 任意 2 条的长度都不相等; \\textcircled{2} 任意三棱锥 $S-ABC$, $\\triangle SAB, \\triangle SBC, \\triangle SCA$ 都是锐角三角形, 下列判断正确的是\\bracket{20}.\n\\twoch{\\textcircled{1}和\\textcircled{2}均为真命题}{\\textcircled{1}和\\textcircled{2}均为假命题}{\\textcircled{1}为真命题, \\textcircled{2}为假命题}{\\textcircled{1}为假命题, \\textcircled{2}为真命题}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -750856,7 +750947,9 @@
|
|||
"id": "032175",
|
||||
"content": "如图, 圆锥的顶点为 $S$, 底面圆心为 $O$, 母线 $SA=2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (0,0) node [left] {$O$} coordinate (O);\n\\draw (1,0) node [right] {$A$} coordinate (A);\n\\draw ({1*cos(-120)},{0.25*sin(-120)}) node [below left] {$B$} coordinate (B);\n\\draw (0,2) node [above] {$S$} coordinate (S);\n\\draw ($(A)!0.5!(B)$) node [below] {$M$} coordinate (M);\n\\draw (A) arc (0:-180:1 and 0.25);\n\\draw [dashed] (A) arc (0:180:1 and 0.25);\n\\draw (S)--(-1,0) (S)--(A) (S)--(B);\n\\draw [dashed] (O)--(A) (O)--(S) (O)--(B) (A)--(B) (S)--(M);\n\\end{tikzpicture}\n\\end{center}\n(1) 若圆锥的侧面积为 $2 \\sqrt{2}\\pi$, 求圆锥的体积;\\\\\n(2) 若圆锥的底面半径为 $1, B$ 是底面圆周上的点, 满足 $OA \\perp OB, M$ 为线段 $AB$ 的中点, 求直线 $SM$ 与平面 $SOA$ 所成角的大小.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -750889,7 +750982,9 @@
|
|||
"id": "032176",
|
||||
"content": "设 $\\triangle ABC$ 的三个内角分别为 $A, B, C$.\\\\\n(1) 若 $\\sin ^2A=\\sin ^2B+\\sin ^2C-\\sqrt{3}\\sin B \\sin C$, 求角 $A$ 的大小;\\\\\n(2) 若 $A=\\dfrac{2 \\pi}{3}, D$ 在线段 $BC$ 上, 满足 $AC \\perp AD$. 设 $\\angle ADC=\\alpha$, 用 $\\alpha$ 表示 $\\dfrac{AD}{BD}+\\dfrac{3AD}{CD}$,并求当 $\\alpha$ 变化时, $\\dfrac{AD}{BD}+\\dfrac{3AD}{CD}$ 的最大值.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第三单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -750922,7 +751017,9 @@
|
|||
"id": "032177",
|
||||
"content": "电解电容是常见的电子元件之一. 检测组在 $85^{\\circ}\\mathrm{C}$ 的温度条件下对电解电容进行质量检测,按检测结果将其分为次品、正品, 其中正品分合格品、优等品两类.\\\\\n(1) 铝箔是组成电解电容必不可少的材料. 现检测组在 $85^{\\circ}\\mathrm{C}$ 的温度条件下, 对铝箔质量与电解电容质量进行测试, 得到如下统计表. 规定显著性检验水平为 $0.05$, 判断电解电容质量与铝箔质量是否有关;\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline & 电解电容为次品 & 电解电容为正品 \\\\\n\\hline 铝箔为次品 & 174 & 76 \\\\\n\\hline 铝篚为正品 & 108 & 142 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(2) 电解电容经检验为正品后才能装箱. 已知两箱电解电容 (每箱各 50 个), 第一箱和第二箱分别有优等品 8 个和 9 个. 现用户从两箱中随机挑选一箱, 并从该箱中先后随机抽取两个元件, 求在第一次取出的是优等品的条件下, 第二次取出的是合格品的概率.\\\\\n附: $P(\\chi^2 \\geq 3.841) \\approx 0.05$, $\\chi^2=\\dfrac{n(a d-b c)^2}{(a+b)(c+d)(a+c)(b+d)}$, 其中 $n=a+b+c+d$.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第九单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -750957,7 +751054,9 @@
|
|||
"id": "032178",
|
||||
"content": "设椭圆 $\\Gamma: \\dfrac{x^2}{4}+\\dfrac{y^2}{3}=1$ 的右焦点为 $F$, 右顶点为 $M$. 动点 $P$ 在 $\\Gamma$ 上, 且 $P$ 在 $x$ 轴的上方.\\\\\n(1) 若 $\\cos \\angle PFM=\\dfrac{4}{5}$, 求 $|PF|$ 的值;\\\\\n(2) 设 $N$ 为线段 $PM$ 的中点, 射线 $FN$ 与 $\\Gamma$ 交于点 $Q$, 满足 $2|QN|=|FN|$, 求直线 $PM$ 的斜率;\\\\\n(3) 过点 $F$ 作直线 $l \\perp PF, l$ 与 $\\Gamma$ 相交于 $A, B$ 两点. 问: 是否存在点 $P$, 使得 $\\cos \\angle APB= -\\dfrac{7}{25}$ ? 若存在, 求出所有这样的点 $P$; 若不存在, 说明理由.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -750990,7 +751089,9 @@
|
|||
"id": "032179",
|
||||
"content": "若函数 $y=f(x)$ 满足: 存在公差不为零的无穷等差数列 $\\{a_n\\}$, 使得对任意正整数 $n$, 恒有 $a_{n+1}=f(a_n)$, 则称 $y=f(x)$ 具有性质 $\\mathbf{P}$.\\\\\n(1) 判断函数 $y=x+1$ 是否具有性质 $\\mathbf{P}$, 说明理由;\\\\\n(2) 设常数 $a \\in \\mathbf{R}$, 函数 $y=2 x-|x+a|$ 具有性质 $\\mathbf{P}$, 求 $a$ 的取值范围;\\\\\n(3) 已知函数 $y=f(x)$ 的定义域为 $[0,+\\infty)$, 满足: \\textcircled{1} 当 $x \\in[0,2)$ 时, $f(x)=-4 x^2+9 x-2$; \\textcircled{2} 对任意 $x \\in[0,+\\infty)$, 恒有 $f(x+2)=f(x)+2$. 问: $y=f(x)$ 是否具有性质 $\\mathbf{P}$ ? 若具有, 求所有可能的 $\\{a_n\\}$ 的公差; 若不具有, 给出证明.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第二单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -751023,7 +751124,9 @@
|
|||
"id": "032180",
|
||||
"content": "数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n=3^n$, 则 $a_n=$\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第四单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -751049,7 +751152,9 @@
|
|||
"id": "032181",
|
||||
"content": "关于 $x$ 的不等式 $a x^2-2 x+1 \\leq 0$ 在 $(0,2]$ 上有解, 则实数 $a$ 的取值范围是\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第一单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -751069,7 +751174,9 @@
|
|||
"id": "032182",
|
||||
"content": "三个男生三个女生站成一排, 已知其中女生甲不在两端, 则有且只有两个女生相邻的概率是\\bracket{20} . \n\\fourch{$\\dfrac{2}{5}$}{$\\dfrac{3}{10}$}{$\\dfrac{9}{20}$}{$\\dfrac{3}{5}$}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第八单元"
|
||||
],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -751089,7 +751196,10 @@
|
|||
"id": "032183",
|
||||
"content": "2023 年 9 月 23 日至 10 月 8 日, 第 19 届亚运会在杭州成功举办, 杭州亚运会的志愿者被称为``小青荷''. 某运动场馆内共有小青荷 36 名, 其中男生 12 名, 女生 24 名, 这些小青荷中会说日语和会说韩语的人数统计如下:\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline & 男生小青荷 & 女生小青荷 \\\\\n\\hline 会说日语 & 8 & 12 \\\\\n\\hline 会说韩语 & $m$ & $n$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n其中 $m$、$n$ 均为正整数, $6 \\leq m \\leq 8$.\\\\\n(1) 从这36名小青荷中随机抽取两名作为某活动主持人, 求抽取的两名小青荷中至少有一名会说日语的概率;\\\\\n(2) 从这些小青荷中随机抽取一名去接待外宾, 用 $A$ 表示事件``抽到的小青荷是男生'', 用 $B$ 表示事件``抽到的小青荷会说韩语''. 试给出一组 $m$、$n$ 的值, 使得事件 $A$ 与 $B$ 相互独立,并说明理由.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第九单元",
|
||||
"第八单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -751109,7 +751219,9 @@
|
|||
"id": "032184",
|
||||
"content": "设数列 $\\{a_n\\}$ 是首项为 1 , 公比为 $-\\dfrac{1}{2}$ 的无穷等比数列, 则数列 $\\{a_n\\}$ 的所有项的和为\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第四单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -751133,7 +751245,9 @@
|
|||
"id": "032185",
|
||||
"content": "已知平面向量 $\\overrightarrow{a}=(2,1), \\overrightarrow{b}$ 为单位向量, 且 $(\\overrightarrow{a}+2 \\overrightarrow{b}) \\perp(\\overrightarrow{a}-\\overrightarrow{b})$, 则向量 $\\overrightarrow{b}$ 在向量 $\\overrightarrow{a}$ 上的投影向量的坐标为\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第五单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -751153,7 +751267,9 @@
|
|||
"id": "032186",
|
||||
"content": "若平面上的三个单位向量 $\\overrightarrow{a}$、$\\overrightarrow{b}$、$\\overrightarrow{c}$ 满足 $|\\overrightarrow{a}\\cdot \\overrightarrow{b}|=\\dfrac{1}{2}$, $|\\overrightarrow{a}\\cdot \\overrightarrow{c}|=\\dfrac{\\sqrt{3}}{2}$, 则 $\\overrightarrow{b}\\cdot \\overrightarrow{c}$ 的所有可能的值组成的集合为\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第五单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -751173,7 +751289,9 @@
|
|||
"id": "032187",
|
||||
"content": "已知 $a$、$b \\in \\mathbf{R}$, $a>b$, 则下列不等式中不一定成立的是\\bracket{20}.\n\\fourch{$a+2>b+2$}{$2 a>2 b$}{$a^2>b^2$}{$2^a>2^b$}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第一单元"
|
||||
],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -751193,7 +751311,9 @@
|
|||
"id": "032188",
|
||||
"content": "用最小二乘法求回归方程是为了使\\bracket{20}.\n\\fourch{$\\displaystyle\\sum_{i=1}^n(y_i-\\hat{y}_i)^2$ 最小}{$\\displaystyle\\sum_{i=1}^n(y_i-\\hat{y}_i)$ 最小}{$\\displaystyle\\sum_{i=1}^n(y_i-\\overline{y})=0$}{$\\displaystyle\\sum_{i=1}^n(y_i-\\hat{y}_i)=0$}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第九单元"
|
||||
],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -751215,7 +751335,11 @@
|
|||
"id": "032189",
|
||||
"content": "若直线 $a x+b y=2$ 经过点 $M(\\cos \\theta, \\sin \\theta)$, 其中 $\\theta \\in \\mathbf{R}$, 则\\bracket{20}.\n\\fourch{$a^2+b^2 \\leq 4$}{$a^2+b^2 \\geq 4$}{$\\dfrac{1}{a^2}+\\dfrac{1}{b^2}\\leq 4$}{$\\dfrac{1}{a^2}+\\dfrac{1}{b^2}\\geq 4$}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第一单元",
|
||||
"第三单元",
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -751237,7 +751361,9 @@
|
|||
"id": "032190",
|
||||
"content": "如图, 在四棱锥 $P-ABCD$ 中, 底面 $ABCD$ 是边长为 $a$ 的正方形, 侧面 $PAD \\perp$ 底面 $ABCD$,且 $PA=PD=\\dfrac{\\sqrt{2}}{2}a$, 设 $E$、$F$ 分别为 $PC$、$BD$ 的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.5]\n\\draw (0,0,1) node [below] {$A$} coordinate (A);\n\\draw (2,0,1) node [below] {$B$} coordinate (B);\n\\draw (2,0,-1) node [right] {$C$} coordinate (C);\n\\draw (0,0,-1) node [left] {$D$} coordinate (D);\n\\draw (0,1,0) node [above] {$P$} coordinate (P);\n\\draw ($(B)!0.5!(D)$) node [below] {$F$} coordinate (F);\n\\draw ($(C)!0.5!(P)$) node [above] {$E$} coordinate (E);\n\\draw (A)--(B)--(C)--(P)--cycle(P)--(B);\n\\draw [dashed] (A)--(D)--(C)(P)--(D)--(B)(E)--(F);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明:直线 $EF \\parallel $ 平面 $PAD$;\\\\\n(2) 求直线 $PB$ 与平面 $ABCD$ 所成的角的正切值.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第六单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -751257,7 +751383,9 @@
|
|||
"id": "032191",
|
||||
"content": "在 $\\triangle ABC$ 中, 角 $A$、$B$、$C$ 所对边的边长分别为 $a$、$b$、$c$, 且 $a-2 c \\cos B=c$.\\\\\n(1) 若 $\\cos B=\\dfrac{1}{3}$, $c=3$, 求 $b$ 的值;\\\\\n(2) 若 $\\triangle ABC$ 为锐角三角形, 求 $\\sin C$ 的取值范围.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第三单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -751277,7 +751405,10 @@
|
|||
"id": "032192",
|
||||
"content": "某无人飞机研发中心最近研发了一款新能源无人飞机, 在投放市场前对 100 架新能源无人飞机进行了单次最大续航里程的测试现对测试数据进行分析, 得到如图所示的频率分布直方图:\\\\\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.0143, yscale = 333.3333]\n\\draw [->] (130,0) -- (146.67,0) -- (150.83,0.0006) -- (159.17,-0.0006) -- (163.33,0)-- (480,0) node [below right] {单次最大续航里程(千米)};\n\\draw [->] (130,0) -- (130,0.012) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (130,0) node [below] {$0$};\n\\foreach \\i/\\j in {180/0.002,230/0.004,280/0.009,330/0.004,380/0.001}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (50,0) --++ (0,-\\j);};\n\\draw (430,0) node [below] {$430$};\n\\foreach \\i/\\j/\\k in {180/0.002/0.002,330/0.004/0.004,280/0.009/0.009,380/0.001/0.001}\n{\\draw [dashed] (\\i,\\j) -- (130,\\j) node [left] {$\\k$};};\n\\end{tikzpicture}\n\\end{center}\n(1) 估计这 100 架新能源无人飞机的单次最大续航里程的平均值 $\\overline{x}$ (同一组中的数据用该组区间的中点值代表); \\\\\n(2) 经计算第 (1) 问中样本标准差 $s$ 的近似值为 50 , 根据大量的测试数据, 可以认为这款新能源无人飞机的单次最大续航里程 $X$ 近似地服从正态分布 $N(\\mu, \\sigma^2)$ (用样本平均数 $\\overline{x}$ 和标准差 $s$ 分别作为 $\\mu$ 和 $\\sigma$ 的近似值), 现任取一架新能源无人飞机, 求它的单次最大续航里程 $X \\in[250,400]$ 的概率;\n(参考数据: 若随机变量 $X \\sim N(\\mu, \\sigma^2)$, 则 $P(\\mu-\\sigma \\leq X \\leq \\mu+\\sigma) \\approx 0.6827$, $P(\\mu-2 \\sigma \\leq X \\leq \\mu+2 \\sigma) \\approx 0.9545$, $P(\\mu-3 \\sigma \\leq X \\leq \\mu+3 \\sigma) \\approx 0.9973$)\n(3) 该无人飞机研发中心依据新能源无人飞机的载重量和续航能力分为卓越 $A$ 型、卓越 $B$ 型和卓越 $C$ 型, 统计分析可知卓越 $A$ 型、卓越 $B$ 型和卓越 $C$ 型的分布比例为 $3: 2: 1$, 研发中心在投放市场前决定分别按卓越 $A$ 型、卓越 $B$ 型和卓越 $C$ 型的分布比例分层随机共抽取 6 架, 然后再从这 6 架中随机抽取 3 架进行综合性能测试, 记随机变量 $Y$ 是综合性能测试的 3 架中卓越 $A$ 型的架数, 求随机变量 $Y$ 的分布和数学期望.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第九单元",
|
||||
"第八单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -751297,7 +751428,9 @@
|
|||
"id": "032193",
|
||||
"content": "椭圆 $C$ 的方程为 $x^2+3 y^2=4$, $A$、$B$ 为椭圆的左右顶点, $F_1$、$F_2$ 为左右焦点, $P$ 为椭圆上的动点.\\\\\n(1) 求椭圆的离心率;\\\\\n(2) 若 $\\triangle PF_1F_2$ 为直角三角形, 求 $\\triangle PF_1F_2$ 的面积;\\\\\n(3) 若 $Q$、$R$ 为椭圆上异于 $P$ 的点, 直线 $PQ$、$PR$ 均与圆 $x^2+y^2=\\dfrac{1}{4}$ 相切, 记直线 $PQ$、$PR$ 的斜率分别为 $k_1$、$k_2$, 是否存在位于第一象限的点 $P$, 使得 $k_1 k_2=1$ ? 若存在, 求出点 $P$ 的坐标, 若不存在, 请说明理由.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -751319,7 +751452,10 @@
|
|||
"id": "032194",
|
||||
"content": "已知 $a \\in \\mathbf{R}$, $f(x)=(a-2) x^3-x^2+5 x+(1-a) \\ln x$.\\\\\n(1) 若 1 为函数 $y=f(x)$ 的驻点, 求实数 $a$ 的值;\\\\\n(2) 若 $a=0$, 试问曲线 $y=f(x)$ 是否存在切线与直线 $x-y-1=0$ 互相垂直? 说明理由;\\\\\n(3) 若 $a=2$, 是否存在等差数列 $x_1, x_2, x_3$($0<x_1<x_2<x_3$), 使得曲线 $y=f(x)$ 在点 $(x_2, f(x_2))$处的切线与过两点 $(x_1, f(x_1))$、$(x_3, f(x_3))$ 的直线互相平行? 若存在, 求出所有满足条件的等差数列; 若不存在, 说明理由.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第二单元",
|
||||
"第四单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -781387,7 +781523,9 @@
|
|||
"id": "041088",
|
||||
"content": "过抛物线 $y^2=4 x$ 的焦点作一条直线与抛物线相交于 $A, B$ 两点, 它们的横坐标之和为 5 , 则这样的直线方程是\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -781419,7 +781557,9 @@
|
|||
"id": "041089",
|
||||
"content": "过点 $P(1,0)$ 的直线 $l$ 与抛物线 $y^2=2 x$ 交于两点 $M, N, O$ 为原点, 若直线 $OM$ 与直线 $ON$ 的斜率之和为 1 , 则直线 $l$ 的方程为\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -781451,7 +781591,9 @@
|
|||
"id": "041090",
|
||||
"content": "若点 $A$ 的坐标为 $(3,2), F$ 为抛物线 $y^2=2 x$ 的焦点, 点 $P$ 是抛物线上的动点. 当 $|PA|+|PF|$ 取到最小值时,点 $P$ 的坐标为\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -781487,7 +781629,9 @@
|
|||
"id": "041091",
|
||||
"content": "已知直线 $l: 4 x-3 y+6=0$. 抛物线 $C: y^2=4 x$ 上的一个动点 $P$ 到直线 $l$ 与 $y$ 轴的距离之和的最小值为\\blank{50}, 取到最小值时, 点 $P$ 的坐标为\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -781517,7 +781661,9 @@
|
|||
"id": "041092",
|
||||
"content": "在一次为期 30 天的博览会上, 主办方统计了每天的参观人数 (单位: 千人), 得到样本的茎叶图 (如右图), 则该样本的第 70 百分位数是\\blank{50}.\n\\begin{center}\n\\begin{tabular}{c|ccccccccc}\n2&1 &1&3&6&8\\\\\n3&0&2&2&4&7&8&8&9 \\\\\n4&1&1&1&3&3&6&7&7&9 \\\\\n5&0&0&1&2&2&2&3&5 \n\\end{tabular}\n\\end{center}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第九单元"
|
||||
],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -781547,7 +781693,9 @@
|
|||
"id": "041093",
|
||||
"content": "空气质量指数 AQI 是反映空气质量状况的指数, 其对应关系如下表:\\\\\n\\begin{center}\n\\begin{tabular}{|l|l|l|l|l|l|l|}\n\\hline$AQI$ 指数值 & $0 \\sim 50$ & $51 \\sim 100$ & $101 \\sim 150$ & $151 \\sim 200$ & $201 \\sim 300$ & $>300$ \\\\\n\\hline 空气质量 & 优 & 良 & 轻度污染 & 中度污染 & 重度污染 & 严重污染 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n为监测某化工厂排放废气对周边空气质量指数的影响, 某科学兴趣小组在工厂附近某处测得 10 月 1 日至 20 日 AQI 的数据并绘成折线图如下:\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.5, yscale = 0.015]\n\\draw [->] (0,0) -- (21,0) node [below right] {日期};\n\\draw [->] (0,0) -- (0,350) node [left] {AQI};\n\\foreach \\i in {50,100,...,300}\n{\\draw (20,\\i) -- (0,\\i) node [left] {$\\i$};};\n\\foreach \\i in {1,2,...,20}\n{\\draw (\\i,5) -- (\\i,0) node [below] {$\\i$};};\n\\draw (0,0) node [left] {$0$};\n\\filldraw (1,141) circle (0.06 and 2) coordinate (P); \n\\foreach \\i/\\j in {2/135,3/63,4/29,5/38,6/35,7/61,8/63,9/93,10/133,11/165,12/136,13/209,14/237,15/279,16/125,17/163,18/75,19/27,20/40}\n{\\draw (P) -- (\\i,\\j) coordinate (P);\n\\filldraw (P) circle (0.06 and 2);};\n\\end{tikzpicture}\n\\end{center}\n下列叙述正确的是\\bracket{20}.\n\\onech{这 20 天中 AQI 的中位数略大于 150}{10 月 4 日到 10 月 11 日, 空气质量越来越好}{这 20 天中的空气质量为优的天数占 $25 \\%$}{10 月上旬 AQI 的极差大于中旬 AQI 的极差}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第九单元"
|
||||
],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -781577,7 +781725,9 @@
|
|||
"id": "041094",
|
||||
"content": "如图, 直线 $y=\\dfrac{1}{2}x$ 与抛物线 $y=\\dfrac{1}{8}x^2-4$ 交于 $A, B$ 两点, 线段 $AB$ 的垂直平分线与直线 $y=-5$ 交于 $Q$ 点, $O$是原点. 当 $P$ 为抛物线上位于线段 $AB$ 下方(含 $A, B$ )的动点时, 求三角形 $OPQ$ 的面积 $S$ 的最大值.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.2]\n\\draw [->] (-10,0) -- (10,0) node [below] {$x$};\n\\draw [->] (0,-10) -- (0,10) node [left] {$y$};\n\\draw (0,0) node [above left] {$O$};\n\\draw [name path = line, domain = -8:10] plot (\\x,{\\x/2});\n\\draw [name path = para, domain = -9:9] plot (\\x,{pow(\\x,2)/8-4});\n\\draw (-10,-5) -- (10,-5);\n\\draw (0,-5) node [below left] {$-5$};\n\\draw (-4,-2) node [below] {$A$} coordinate (A);\n\\draw (8,4) node [below right] {$B$} coordinate (B);\n\\filldraw (6,{36/8-4}) circle (0.1) node [right] {$P$};\n\\draw (-2,9) coordinate (T) -- (6,-7) (5,-5) node [below left] {$Q$};\n\\draw (2,1) coordinate (V);\n\\draw (V) pic [draw, scale = 0.3] {right angle = B--V--T};\n\\end{tikzpicture}\n\\end{center}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -781609,7 +781759,9 @@
|
|||
"id": "041095",
|
||||
"content": "抛物线 $y^2=16 x$ 上有三点 $A(1,4), B(x_B, y_B)$ 和 $C(x_C, y_C)$, 其中 $x_B<x_C$, 且三角形 $ABC$ 的重心是焦点 $F$.\\\\\n(1) 求 $x_B+x_C$ 与 $y_B+y_C$;\\\\\n(2) 求直线 $BC$ 的方程;\\\\\n(3) 求点 $B, C$ 的坐标.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
@ -781639,7 +781791,9 @@
|
|||
"id": "041096",
|
||||
"content": "设 $C_1$ 是以 $F$ 为焦点的抛物线 $y^2=2 p x(p>0), C_2$ 是以直线 $2 x-\\sqrt{3}y=0$ 与 $2 x+\\sqrt{3}y=0$ 为渐近线, 以 $(0, \\sqrt{7})$ 为一个焦点的双曲线.\\\\\n(1) 写出双曲线 $C_2$ 的标准方程;\\\\\n(2) 若 $C_1$ 与 $C_2$ 在第一象限内有两个不同的公共点 $A$ 和 $B$, 求 $p$ 的取值范围, 并求 $\\overrightarrow{FA}\\cdot \\overrightarrow{FB}$ 的最大值;\\\\\n(3) 条件同(2), 若三角形 $FAB$ 的面积 $S$ 满足 $S=\\dfrac{2}{3}\\overrightarrow{FA}\\cdot \\overrightarrow{FB}$, 求 $p$ 的值.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"tags": [
|
||||
"第七单元"
|
||||
],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
|
|
|
|||
Reference in New Issue