Given,
$$
\begin{aligned}
& f(x)=\left\{\begin{array}{cl}
\sin x, & \text { if } x \leq 0 \\
x^2+a^2, & \text { if } 0 < x < 1 \\
b x+2, & \text { if } 1 \leq x \leq 2 \\
0, & \text { if } x>2
\end{array}\right. \text { is continuous on IR, } \\
& \therefore \lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{+}} f(x) \\
& \text { and } \lim _{x \rightarrow 2^{-}} f(x)=\lim _{x \rightarrow 2^{+}} f(x) \\
& \Rightarrow \lim _{x \rightarrow 0^{-}} \sin x=\lim _{x \rightarrow 0^{-}} x^2+a^2 \\
& \text { and } \lim _{x \rightarrow 2^{-}} b x+2=\lim _{x \rightarrow 2^{+}} 0 \\
& \Rightarrow \quad 0=0+a^2 \text { and } 2 b+2=0 \\
& \Rightarrow \quad a=0 \text { and } b=-1 \\
& \text { Now, } a+b+a b=0+(-1)+0=-1 \\
&
\end{aligned}
$$