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

Answer 1

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

Given: $x^2-px+q=0$ with roots $\alpha ,\beta$.

$\alpha +\beta =p,\alpha \beta =q$

Let $A=(\alpha -\beta {)}^2(\alpha +\beta \left)\right({\alpha }^2+{\beta }^2+\alpha \beta )$

$=(p^2-4q)p(p^2-q)=p[p^4-5p^{2}q+4q^2]$

Let $B={\alpha }^2{\beta }^2(\alpha +\beta )=q^{2}p$

$\therefore$ Sum of the roots $=A+B=p[p^4-5p^{2}q+5q^2]$

Product of the roots $=p^2{q}^2(p^4-5p^{2}q+4q^2)$

The required equation is $x^2-Sx+P=0$