If in a moderately asymmetrical distribution mode and mean of the data are 6 and 9 respectively, then median is

$\sin ^{-1}\left(\sum_{i=1}^{\infty} x^{i+1}-x \sum_{i=1}^{\infty}\left(\frac{x}{2}\right)^i\right)=\frac{\pi}{2}-\cos ^{-1}\left(\sum_{i=1}^{\infty}\left(-\frac{x}{2}\right)^i-\sum_{i=1}^{\infty}(-x)^i\right)$

lying in the interval $\left(-\frac{1}{2}, \frac{1}{2}\right)$ is. . . . .

(Here, the inverse trigonometric functions $\sin ^{-1} x$ and $\cos ^{-1} x$ assume values in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ and $[0, \pi]$, respectively.)

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Let $S=S_1 \cap S_2 \cap S_3$, where

$S_1=\{z \in C:|z|<4\}, S_2=\left\{z \in C: \operatorname{Im}\left[\frac{z-1+\sqrt{3} i}{1-\sqrt{3} i}\right]>0\right\} \text { and } $

$S_3:\{z \in C: \operatorname{Re} z>0\} .$

$1.$ Area of $S=$

$(A)$ $\frac{10 \pi}{3}$ $(B)$ $\frac{20 \pi}{3}$ $(C)$ $\frac{16 \pi}{3}$ $(D)$ $\frac{32 \pi}{3}$

$2.$ $\min _{z \in S}|1-3 i-z|=$

$(A)$ $\frac{2-\sqrt{3}}{2}$ $(B)$ $\frac{2+\sqrt{3}}{2}$ $(C)$ $\frac{3-\sqrt{3}}{2}$$(D)$ $\frac{3+\sqrt{3}}{2}$

Give the answer question $1$ and $2$

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Let the curve $C$ be the mirror image of the parabola $y^2=4 x$ with respect to the line $x+y+4=0$. If $A$ and $B$ are the points of intersection of $C$ with the line $y=-5$, then the distance between $A$ and $B$ is

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