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Geometric Progression

If α,β are ...

Question

If $α, β$ are the roots of the equation $x^{2} - 2 x + 3 = 0$ . Then the equation whose roots are
$P = α^{3} - 3 α^{2} + 5 α - 2$ and $Q = β^{3} - β^{2} + β + 5$ is

x^{2} + 3 x + 2 = 0

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x^{2} - 3 x - 2 = 0

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x^{2} - 3 x + 2 = 0

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None of the above

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Solution

The correct option is C $x^{2} - 3 x + 2 = 0$
Given $α, β$ are roots of $x^{2} - 2 x + 3 = 0$
$∴ α + β = 2, α β = 3$
$α^{2} - 2 α + 3 = 0, β^{2} - 2 β + 3 = 0$
$P = α^{3} - 3 α^{2} + 5 α - 2$
$= α^{3} - 2 α^{2} + 3 α - α^{2} + 2 α - 3 + 1$
$= α (α^{2} - 2 α + 3) - (α^{2} - 2 α + 3) + 1$
$∴ P = 1 (∵ α^{2} - 2 α + 3 = 0)$
$Q = β^{3} - β^{2} + β + 5$
$= β^{3} - 2 β^{2} + 3 β + β^{2} - 2 β + 3 + 2$
$= β (β^{2} - 2 β + 3) + (β^{2} - 2 β + 3) + 2$
$∴ Q = 2 (∵ β^{2} - 2 β + 3 = 0)$
Equation with P and Q as roots is $x^{2} - 3 x + 2 = 0$

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