Question

If the equation $2^x+2^{-2}=2k$ has exactly one real solution, then sum of all integral values of $k$ in $[-100,100]$ is equal to

• A. 5050
• B. 10100
• C. 0
• D. -5050

Answer 1

Answer: (A)

We have
$2^x=2k-\frac{1}{4}$
$\therefore$ For exactly one real solution,

$2k-\frac{1}{4}>0\Rightarrow k>\frac{1}{8}\left(\text{ As }{2}^x>0\right)$
$\text{ Hence sum }=1+2+\dots \dots ..+100=5050$
[Note: There is no integral value of $k$ for which the given equation has exactly one integral solution. Because $2^{x+2}=8k-1$ (Not Possible Because left hand side is even and right hand side is Odd.)