Complex Number Question 13
Question 13 - 2024 (31 Jan Shift 1)
If $\alpha$ denotes the number of solutions of $|1-i|^{x}=2^{x}$ and $\beta=\left(\frac{|z|}{\arg (z)}\right)$, where $z=\frac{\pi}{4}(1+i)^{4}\left(\frac{1-\sqrt{\pi} i}{\sqrt{\pi}+i}+\frac{\sqrt{\pi}-i}{1+\sqrt{\pi} i}\right), i=\sqrt{-1}$, then the distance of the point $(\alpha, \beta)$ from the line $4 x-3 y=7$ is
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Answer (3)
Solution
$$ \begin{aligned} & (\sqrt{2})^{x}=2^{x} \Rightarrow x=0 \Rightarrow \alpha=1 \\ & z=\frac{\pi}{4}(1+i)^{4}\left[\frac{\sqrt{\pi}-\pi i-i-\sqrt{\pi}}{\pi+1}+\frac{\sqrt{\pi}-i-\pi i-\sqrt{\pi}}{m^{2}}\right] \\ & =-\frac{\pi i}{2}\left(1+4 i+6 i^{2}+4 i^{3}+1\right) \\ & =2 \pi i \\ & \beta=\frac{2 \pi}{\frac{\pi}{2}}=4 \end{aligned} $$
Distance from $(1,4)$ to $4 x-3 y=7$
Will be $\frac{15}{5}=3$
