Question

Let $p$ and $q$ be two positive numbers such that $p + q =2$ and $p ^{4}+ q ^{4}=272 .$ Then $p$ and $q$ are roots of the equation :

• A. $x^{2}-2 x+2=0$
• B. $x^{2}-2 x+8=0$
• C. $x^{2}-2 x+136=0$
• D. $x^{2}-2 x+16=0$

Answer 1

Answer: (D)

Consider $\left( p ^{2}+ q ^{2}\right)^{2}-2 p ^{2} q ^{2}=272$
$\left(( p + q )^{2}-2 pq \right)^{2}-2 p ^{2} q ^{2}=272$
$16-16 pq +2 p ^{2} q ^{2}=272$
$( pq )^{2}-8 pq -128=0$
$( pq )^{2}-8 pq -128=0$
$pq =\frac{8 \pm 24}{2}=16,-8$
$\therefore pq =16$
$\therefore $ Required equation
$: x ^{2}-(2) x +16=0$