# If function $f(x)$ is given by $\small\left|\begin {array} {c,c,c} -cos(x+x^2) & -sin(x+x^2) & cos(x+x^2)\\sin(x^2-x) & -cos(x^2-x) & sin(x^2-x)\\sin2x &0 &sin2x^2 \end {array}\right|$, then find $f(0)$

$\begin{array}{1 1} 2 \\ 1 \\ 0 \\ -2 \end{array}$

Put $x=0$ in $f(x)$, then we get
$f(0)=\left|\begin {array}{ccc}-1 & 0 &1 \\0 & -1 & 0\\0 & 0 & 0\end {array}\right|$
Since all the entries of $R_3$ of the deterrminent are $0$,
the determinent =0
$f(0)=0$