Q55 / 155 · WBJEE 2020 · Mathematics

The number of complex numbers $\mathrm{p}$ such that $|\mathrm{p}|=1$ and imaginary part of $\mathrm{p}^{4}$ is 0, is

  • Correct answer
    A
    4
  • B
    2
  • C
    8
  • D
    infinitely many

Solution

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.

Hence, the correct answer is option: 'A'.

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