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Algebra of Complex Numbers

If α, β∈ R ar...

Question

If $α, β \in R$ are such that $1 - 2 i$ (here $i^{2} = - 1$ ) is a root of $z^{2} + α z + β = 0,$ then $(α - β)$ is equal to

A

7

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B

- 3

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C

3

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D

- 7

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Solution

The correct option is D $- 7$
$1 - 2 i$ is a root of $z^{2} + α z + β = 0.$
So, $(1 - 2 i)^{2} + α (1 - 2 i) + β = 0$
$\Rightarrow 1 - 4 - 4 i + α - 2 i α + β = 0$
$\Rightarrow (α + β - 3) - i (4 + 2 α) = 0$
$\Rightarrow α + β - 3 = 0$ and $4 + 2 α = 0$
So, $α = - 2, β = 5$
$∴ α - β = - 7$

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