First slide

Theory of equations

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

Moderate

A

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

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

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

none of these

Solution

Given, $α, β$ are roots of equation
$x^{2} - 2 x + 3 = 0$
$\begin{array}{l} \Rightarrow α^{2} - 2 α + 3 = 0 (1) \\ and β^{2} - 2 β + 3 = 0 (2) \end{array}$
$\begin{array}{l} \Rightarrow α^{2} = 2 α - 3 or α^{3} = 2 α^{2} - 3 α \\ \Rightarrow P = (2 α^{2} - 3 α) - 3 α^{2} + 5 α - 2 \\ = - α^{2} + 2 α - 2 = 3 - 2 = 1 [Using (1)] \end{array}$
Similarly, we have Q = 2.
Now, sum of roots is 3 and product of roots is 2. Hence, the
required equation is $x^{2} - 3 x + 2 = 0$

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