Question

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

Answer 1

Answer: 2

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