Mathematics - Continuity - Differentiability Question with Solution | TestHub

$\begin{array}{rl}& \underset{x\to 0^{-}}{\lim }\frac{2}{\mathrm{x}}\left\{\sin \left({\mathrm{k}}_1+1\right)\mathrm{x}+\sin \left({\mathrm{k}}_2-1\right)\mathrm{x}\right\}=4\\ & \Rightarrow 2\left({\mathrm{k}}_1+1\right)+2\left({\mathrm{k}}_2-1\right)=4\\ & \Rightarrow {\mathrm{k}}_1+{\mathrm{k}}_2=2\\ & \Rightarrow \underset{\mathrm{x}\to 0^{+}}{\lim }\frac{2}{\mathrm{x}}\ln \left(\frac{2+{\mathrm{k}}_1\mathrm{x}}{2+{\mathrm{k}}_2\mathrm{x}}\right)=4\\ & \Rightarrow \underset{\mathrm{x}\to 0^{+}}{\lim }\frac{1}{\mathrm{x}}\ln \left(1+\frac{\left({\mathrm{k}}_1-{\mathrm{k}}_2\right)\mathrm{x}}{2+{\mathrm{k}}_2\mathrm{x}}\right)=2\\ & \Rightarrow \frac{{\mathrm{k}}_1-{\mathrm{k}}_2}{2}=2\\ & \Rightarrow {\mathrm{k}}_1-{\mathrm{k}}_2=4\\ & \therefore {\mathrm{k}}_1=3,{\mathrm{k}}_2=-1\\ & {\mathrm{k}}_1^2+{\mathrm{k}}_2^2=9+1=10\end{array}$