If $(1 - i)$ is a root of the equation, $z^{3} - 2 (2 - i) z^{2} + (4 - 5 i) z - 1 + 3 i = 0$ , then find the other two roots.

1, (3 - i)

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1, (2 - i)

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0, (2 - i)

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None of these

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Solution

The correct option is B $1, (2 - i)$
$∴ [z^{2} + (i - 3) z + 2 - i] (z - 1 + i) = 0$
$\Rightarrow z^{2} + (i - 3) z - (i - 2) = 0$
$\Rightarrow z^{2} + {(i - 2) - 1} z - (i - 2) = 0$
$\Rightarrow z^{2} + (i - 2) z - z - (i - 2) = 0$
$\Rightarrow z [z + i - 2)] - 1 [z + i - 2)] = 0$
$\Rightarrow z = 1, 2 - i$

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