If α denotes the number of solutions of |1 − i|^x = 2x and β = (|z| / arg(z))

Question

If \(\alpha\) denotes the number of solutions of \(|1-\mathrm{i}|^{\mathrm{x}}=2^{\mathrm{x}}\) and \(\beta=\left(\frac{|z|}{\arg (z)}\right)\), where \(\mathrm{z}=\frac{\pi}{4}(1+\mathrm{i})^{4}\left(\frac{1-\sqrt{\pi} \mathrm{i}}{\sqrt{\pi}+\mathrm{i}}+\frac{\sqrt{\pi}-\mathrm{i}}{1+\sqrt{\pi} \mathrm{i}}\right), \mathrm{i}=\sqrt{-1}\), then the distance of the point \((\alpha, \beta)\) from the line \(4 x-3 y=7\) is ____.
 

Answer 1

Correct answer: 3

\((\sqrt{2})^{x}=2^{x} \Rightarrow x=0 \Rightarrow \alpha=1\)

\(\mathrm{z}=\frac{\pi}{4}(1+\mathrm{i})^{4}\left[\frac{\sqrt{\pi}-\pi \mathrm{i}-\mathrm{i}-\sqrt{\pi}}{\pi+1}+\frac{\sqrt{\pi}-\mathrm{i}-\pi \mathrm{i}-\sqrt{\pi}}{1+\pi}\right]\)

\(=-\frac{\pi \mathrm{i}}{2}\left(1+4 \mathrm{i}+6 \mathrm{i}^{2}+4 \mathrm{i}^{3}+1\right)\)

\(=2 \pi \mathrm{i}\)

\(\beta=\frac{2 \pi}{\frac{\pi}{2}}=4\)

Distance from \((1,4)\) to \(4 x-3 y=7\)

Will be \(\frac{15}{5}=3\)