\[\begin{array}{1 1}(1)purely \; imaginary& (2)purely\; real \\(3)0&(4) neither \;real\;nor\;imaginary\end{array}\]

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The complex number purely imaginary

Let the complex number be $z= x+iy$

Given amplitudeof $z= \large\frac{\pi}{2}$

$\tan ^{-1} \bigg( \large\frac{y}{x} \bigg) =\large\frac{\pi}{2}$

$\large\frac{y}{x} $$ =\tan \large\frac{\pi}{2}$

$\large\frac{y}{x} $$ =\infty$

$\quad= \large\frac{1}{0}$

$0=x$

Hence 1 is the correct answer.

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