Let $f: \mathrm{R} \rightarrow \mathrm{R}$ be defined as
$f(x)=\left\{\begin{array}{cc}2 \sin \left(-\frac{\pi x}{2}\right), & \text { if } x<-1 \\ \left|a x^{2}+x+b\right|, & \text { if }-1 \leq x \leq 1 \\ \sin (\pi x), & \text { if } x>1\end{array}\right.$
If $f(\mathrm{x})$ is continuous on $\mathrm{R}$, then $\mathrm{a}+\mathrm{b}$ equals:
Correct Option: , 2
$f(\mathrm{x})$ is continuous on $\mathrm{R}$
$\Rightarrow f\left(1^{-}\right)=f(1)=f\left(1^{+}\right)$
$|a+1+b|=\lim _{x \rightarrow 1} \sin (\pi x)$
$|a+1+b|=0 \Rightarrow a+b=-1$...(1)
$\Rightarrow$ Also $f\left(-1^{-}\right)=f(-1)=f\left(-1^{+}\right)$
$\lim _{x \rightarrow-1} 2 \sin \left(\frac{-\pi x}{2}\right)=|a-1+b|$
$|a-1+b|=2$
Either $a-1+b=2$ or $a-1+b=-2$
$a+b=3 ..(2)$ or $a+b=-1$..(3)
from (1) and $(2) \Rightarrow a+b=3=-1$ (reject)
from (1) and (3) $\Rightarrow a+b=-1$
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