QJEEM21Made6d: JEE Main 2021 Mathematics - Class 11 JEEM21Made6d Solution: Let p and q be two positive numbers such that p + q = 2 and p 4 …

Question

Verified Solution • 2025–26 Session moderate

Let p and q be two positive numbers such that p + q = 2 and p⁴+q⁴ = 272. Then p and q are roots of the equation :

Accepted Answer

{p^2} + {q^2} = {(p + q)^2} - 2pq

= 4 - 2pq

Now,

{\left( {{p^2} + {q^2}} \right)^2} = {p^4} + {q^4} + 2{p^2}{q^2}

\Rightarrow {\left( {4 - 2pq} \right)^2} = 272 + 2{p^2}{q^2}

\Rightarrow 16 + 4{p^2}{q^2} - 16pq = 272 + 2{p^2}{q^2}

\Rightarrow 2{p^2}{q^2} - 16pq - 256 = 0

\Rightarrow {p^2}{q^2} - 8pq - 128 = 0

\Rightarrow (pq - 16)(pq + 8) = 0

\Rightarrow pq = 16, - 8

Here, pq = - 8 is not possible as p and q are positive.

\therefore

pq = 16

Now, the equation whose roots are p and q is

{x^2} - 2x + 16 = 0

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