If $i z^{3} + z^{2} - z + i = 0$ , then find the value of $| z |$ .

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Solution

The correct option is A $1$
$i z^{3} + z^{2} - z + i = 0$
$z^{3} - i z^{2} + i z + 1 = 0$
$z^{2} (z - i) + i (z - i) = 0$
$(z - i) (z^{2} + i) = 0$
$z = i, z^{2} = - i$
Now,
$z = i ⟹ | z | = | i | = 1$
and, $z^{2} = - i$
$| z^{2} | = | - i | = 1$
$| z^{2} | = 1 ⟹ | z | = 1$
Hence, in either case, we have $| z | = 1$ .

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