Question

Complete set of values of $k$ for which the equation $4^x-(k+2)2^x+2k=0,$ has exactly one positive root is

• A. $R$
• B. $(-\infty ,1]\cup \left\{2\right\}$
• C. $(\frac{1}{2},\infty )$
• D. $(0,\infty )$

Answer 1

The correct option is B $(-\infty ,1]\cup \left\{2\right\}$

Let $2^x=t,\alpha \&\beta$ be the roots of the given equation
$\alpha \le 0<\beta$
$\Rightarrow 2^{\alpha }\le 1<2^{\beta }$
Let $f\left(t\right)=t^2-(k+2)t+2k$
So $f\left(1\right)\le 0,$
$1-k-2+2k\le 0$
$k-1\le 0$
$k\le 1$

For $\Delta =0\Rightarrow k=2$