Question

If $\alpha ,\beta$ are the roots of the equation $x^2-2x+3=0$, then the equation whose roots are $P={\alpha }^3-3{\alpha }^2+5\alpha -2$ and $Q={\beta }^3-{\beta }^2+\beta +5$ is

• A. $x^2+3x+2=0$
• B. $x^2-5x+4=0$
• C. $x^2-3x+2=0$
• D. $x^2+5x+4=0$

Answer 1

The correct option is C $x^2-3x+2=0$

Given $\alpha ,\beta$ are roots of the equation
$x^2-2x+3=0$
$\Rightarrow {\alpha }^2-2\alpha +3=0\cdots \left(1\right)$
and ${\beta }^2-2\beta +3=0\cdots \left(2\right)$
$\Rightarrow {\alpha }^2=2\alpha -3\Rightarrow {\alpha }^3=2{\alpha }^2-3\alpha$
$\Rightarrow P=(2{\alpha }^2-3\alpha )-3{\alpha }^2+5\alpha -2$​​​​$\Rightarrow P=-{\alpha }^2+2\alpha -2$
$\Rightarrow P=3-2=1\text{[Using (1)]}$

Similarly, we have $Q=2$.

$P+Q=3$ and $PQ=2$
Hence, the required equation is $x^2-3x+2=0$