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Q.
The sum of all possible real values of a so that range of function $f(x)=4 \sin ^2 x+4 \sin x+a^2-3$ is [0,9] $\forall x \in R$, is equal to
Relations and Functions - Part 2
Solution:
We have, $f(x)=4 \sin ^2 x+4 \sin x+a^2-3=(2 \sin x+1)^2+\left(a^2-4\right)$ for range to $\in[0,9]$
$a ^2-4=0 \Rightarrow a =2 \text { or }-2$
Clearly, discriminant of $f ( x )$ must be equal to zero, so
$16=16\left( a ^2-3\right) $
$\Rightarrow a ^2=4 $
$\text { so, } a = \pm 2$
Hence, there are two values of a (i.e., $a=-2, a=2$ )
$\Rightarrow \text { Sum }=2-2=0 $