Explanation

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