Question

The number of complex numbers $p$ such that $|p|=1$ and imaginary part of $p^{4}$ is 0 , is
(A) 4
(B) 2
(C) 8
(D) infinitely many

Answer 1

Ans : (A)
Hint : Let $p=x+i y, p^{2}=\left(x^{2}-y^{2}\right)+2 i x y, p^{4}=\left(x^{2}-y^{2}\right)^{2}-4 x^{2} y^{2}+4 i x y\left(x^{2}-y^{2}\right)$
Now, $x y\left(x^{2}-y^{2}\right)=0, \quad x=\pm y \Rightarrow y^{2}=\frac{1}{2} \Rightarrow y=\pm \frac{1}{\sqrt{2}}$
Four complex numbers.

Answer 2

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