JEE Mains · Maths · STD 11 - 4.1 complex nubers
If \(\alpha\) denotes the number of solutions of \(|1-i|^x=2^x\) and \(\beta=\left(\frac{|z|}{\arg (z)}\right)\), where \(z=\frac{\pi}{4}(1+i)^4\left(\frac{1-\sqrt{\pi i}}{\sqrt{\pi}+i}+\frac{\sqrt{\pi}-i}{1+\sqrt{\pi} \mathrm{i}}\right), i=\sqrt{-1}\), then the distance of the point \((\alpha, \beta)\) from the line \(4 x-3 y=7\) is
- A \(2\)
- B \(3\)
- C \(4\)
- D \(5\)
Answer & Solution
Correct Answer
(B) \(3\)
Step-by-step Solution
Detailed explanation
\((\sqrt{2})^x=2^x \Rightarrow x=0 \Rightarrow \alpha=1\) \(z=\frac{\pi}{4}(1+i)^4\left[\frac{\sqrt{\pi}-\pi i-i-\sqrt{\pi}}{\pi+1}+\frac{\sqrt{\pi}-i-\pi i-\sqrt{\pi}}{1+\pi}\right]\) \(=-\frac{\pi i}{2}\left(1+4 i+6 i^2+4 i^3+1\right)\) \(=2 \pi i\)…
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