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Q.
If the equation $2^{ x }+2^{-2}=2 k$ has exactly one real solution, then sum of all integral values of $k$ in $[-100,100]$ is equal to
Complex Numbers and Quadratic Equations
Solution:
We have
$2^x=2 k-\frac{1}{4}$
$\therefore $ For exactly one real solution,
$\qquad 2 k-\frac{1}{4}>0 \Rightarrow k>\frac{1}{8}\left(\text { As } 2^x>0\right) $
$\text { Hence sum }=1+2+\ldots \ldots . .+100=5050$
[Note: There is no integral value of $k$ for which the given equation has exactly one integral solution. Because $2^{x+2}=8 k-1$ (Not Possible Because left hand side is even and right hand side is Odd.)
