Question

If $2^{\alpha }$ and $2^{\beta }$ are the roots of the equation $4x^2-64x+p=0$ such that $2^{2\alpha }+2^{2\beta }=2^{\alpha +\beta +1}$ then $(\alpha +\beta +p)$ equals

• A. 32
• B. 64
• C. 262
• D. 512

Answer 1

Answer: (C)

$2^{\alpha }+2^{\beta }=16$ .....(i)
$2^{\alpha }{2}^{\beta }=\frac{p}{4}$ ....(ii)
${\left(2^{\alpha }\right)}^2+{\left(2^{\beta }\right)}^2=2^{\alpha +\beta +1}$
$\Rightarrow \alpha =\beta =3,p=2^{\beta }$
$\alpha +\beta +p=262$