Question

The coefficient of $x$ in $f\left(x\right)=\begin{vmatrix} x & 1+sinx & cosx \\ 1 & log\left(1 + x\right) & 2 \\ x^{2} & 1+x^{2} & 0 \end{vmatrix},-1 < x\leq 1$ , is $p$ then $\left|p\right|$ is

Answer 1

Given, $f\left(x\right)=\begin{vmatrix} x & 1+sinx & cosx \\ 1 & log\left(1 + x\right) & 2 \\ x^{2} & 1+x^{2} & 0 \end{vmatrix}$
$=x\left\{- 2 \left(1 + x^{2}\right)\right\}-\left(1 + sin x\right)\left(- 2 x^{2}\right)+cosx\left\{1 + x^{2} - x^{2} log \left(1 + x\right)\right\}$
$=-2x-2x^{3}+2x^{2}+2x^{2}sinx+cosx\left\{1 + x^{2} - x^{2} log \left(1 + x\right)\right\}$
$\therefore $ Coefficient of $x$ in $f\left(x\right)=-2$
$\therefore \left|p\right|=2$