Question

Number of positive integral value of '$p$' for which the equation $p{.2}^{e^x+e^{-x}}-{8.2}^{\frac{e^x+e^{-x}}{2}}+p=0$ has atleast one solution is

Answer 1

$2^{\frac{e^x+e^{-x}}{2}}=t\ge 2$
So, $p{t}^2-8t+p=0t+\frac{1}{t}=\frac{8}{p}$
Minimum value of $t+\frac{1}{t}\text{is}\frac{5}{2}\Rightarrow \frac{8}{p}\ge \frac{5}{2}\Rightarrow p\le \frac{16}{5}$
$p=1,2,3$