Question

If $2^\alpha$ and $2^\beta$ are the roots of the equation $4 x^2-64 x+p=0$ 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)
$\left(2^\alpha\right)^2+\left(2^\beta\right)^2=2^{\alpha+\beta+1}$
$\Rightarrow \alpha=\beta=3, p=2^\beta$
$\alpha+\beta+p=262$