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Domain and Range of Trigonometric Ratios

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 these

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Solution

The correct option is C $x^{2} - 3 x + 2 = 0$
$P$ can be written as
$α (α^{2} - 2 α + 3) - 1 (α^{2} - 2 α + 3) + 1$
and similarly $Q$ can be written as
$β (β^{2} - 2 β + 3) + 1 (β^{2} - 2 β + 3) + 2.$
Since, $α$ and $β$ are the roots of the given equation therefore $P = 1$ and $Q = 2$ .
Now, sum of roots $P + Q = 3$ and product of roots $P Q = 2$ .
So, required equation is $x^{2} - 3 x + 2 = 0$ .
Hence, option 'C' is correct.

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