$\mathrm{If} \mathrm{f}: \mathrm{R}\to \mathrm{R} \mathrm{defined} \mathrm{by}$ 

$\mathrm{f}\left(\mathrm{x}\right)=$ $\left\{\begin{array}{ll}\mathrm{a} {\cos }^2\mathrm{x} +{\mathrm{x}}^{2 }{\sin }^2 \mathrm{x} & , \mathrm{x}\le 0\\ {\mathrm{e}}^{\mathrm{ax}+{\mathrm{b}}^2}& , \mathrm{x}>0\end{array}\right.\mathrm{is} \mathrm{continuous}, \mathrm{then}$

f(x) is continuous at x=0 therefore

L.H.L=R.H.L

$$\begin{gathered} \underset{\mathrm{x}\to 0^{-}}{\lim }\mathrm{a} {\cos }^2 \mathrm{x}+{\mathrm{x}}^2 {\sin }^2 \mathrm{x} = \underset{\mathrm{x}\to 0^{+}}{\lim } {\mathrm{e}}^{\mathrm{ax}+{\mathrm{b}}^2} \\ \Rightarrow \mathrm{a}={\mathrm{e}}^{{\mathrm{b}}^2} \Rightarrow {\mathrm{b}}^2=\log \left|\mathrm{a}\right| \end{gathered}$$