UPSC NDA Math Model Paper 5

$px^{2}+qx+r=0$ Let α and β be the two roots of the quadratic equation. The sum of the roots of a quadratic equation is given by: $\alpha+\beta=-\frac{q}{p}$ The product of the roots is given by: $\alpha\beta=\frac{r}{p}$ Given: Both the roots are positive Therefore, the sum of roots and products of roots is also positive. $\Rightarrow$ Sum of roots $>0$ $\Rightarrow-\frac{q}{p}>0$ Multiplying both side by $p^{2}$, we get $\Rightarrow -qp>0$ $\therefore pq<0$ Now, products of roots $>0$ $\Rightarrow\frac{r}{p}>0$ Multiplying both sides by $p^{2}$, we get $\Rightarrow pr>0$ Hence correct answer is $pq<0, pr>0$ Multiplying each side of an inequality by a positive number does not change the direction of the inequality symbol