The number of real values of m for which the equation $z^{3} + (3 + i) z^{2} - 3 z - (m + i) = 0$ has at least one real root is

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Infinite

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Solution

The correct option is D
2

$Z^{3} + (3 + i) Z^{2} - 3 z - (m + i) = 0 (z^{3} + 3 z^{2} - 3 z - m) + i (z^{2} - 1) = 0$
If z is a real root, then
$z^{3} + 3 z^{2} - 3 z - m = 0$ and $z^{2} - 1 = 0$
$∴ z = \pm 1 Z = 1 \Rightarrow m = 1 Z = - 1 \Rightarrow m = 5$

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