Question

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

• A. 4
• B. 9
• C. 1
• D. 0

Answer 1

Answer: (D)

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., $a=-2,a=2$ )
$\Rightarrow \text{ Sum }=2-2=0$