Mathematics - Determinant Question with Solution | TestHub

${\lim }_{x\to 0}\left|\begin{array}{ccc}a+\frac{\sin x}{x}& 1& b\\ a& 1+\frac{\sin x}{x}& b\\ a& 1& b+\frac{\sin x}{x}\end{array}\right|=\lambda +\mu a+vb$ At $\lim x\to 0$, $\begin{array}{rl}& f(x)=\left|\begin{array}{ccc}a+1& 1& b\\ a& 1+1& b\\ a& 1& b+1\end{array}\right|=\lambda +\mu a+vb\\ & R_1\to R_1-R_2\\ & R_2\to R_2-R_3\\ & \left|\begin{array}{ccc}1& -1& 0\\ 0& 1& -1\\ a& 1& b+1\end{array}\right|=\lambda +\mu a+vb\\ & {\mathrm{C}}_2\to {\mathrm{C}}_1-{\mathrm{C}}_2\\ & \left|\begin{array}{ccc}1& 0& 0\\ 0& 1& -1\\ a& a+1& b+1\end{array}\right|=\lambda +\mu a+vb\\ & a+b+2=\lambda +\mu a+vb\\ & \lambda =2,\mu =1,v=1\\ & (\lambda +\mu +v)=(2+1+1{)}^2=16\end{array}$