Question

If $\alpha ,\beta$ are the roots of the equation $x^2-px+q=0$ then the quadratic equation whose roots are
$(\alpha -\beta {)}^2(\alpha +\beta \left)\right({\alpha }^2+{\beta }^2+\alpha \beta )$ and $({\alpha }^3{\beta }^2+{\alpha }^2{\beta }^3)$ is where $S=p[p^4-5p^{2}q+5q^2],P=p^2{q}^2(p^4-5p^{2}q+4q^2)$

• A. $x^2-Sx+P=0$
• B. $x^2+Sx+P=0$
• C. $x^2+Sx-P=0$
• D. None of these

Answer 1

The correct option is A $x^2-Sx+P=0$

$CONVENTIONALAPPROACH$
$\alpha +\beta =p,\alpha ,\beta =q$
$A=(\alpha -\beta {)}^2(\alpha +\beta \left)\right({\alpha }^2+{\beta }^2+\alpha \beta )$
$=(p^2-4q)p(p^2-q)=p[p^4-5p^{2}q+4q^2]$
$B={\alpha }^2{\beta }^2(\alpha +\beta )=q^{2}p$
$\therefore Sumoftheroots=A+B=p[p^4-5p^{2}q+5q^2]$
$Productoftheroots=p^2{q}^2(p^4-5p^{2}q+4q^2)$
The required equation is $x^2-Sx+P=0$