Given $f(x)=\left|\begin{array}{ccc}x & x+1 & x+3 \\ x+2 & x+4 & x+7 \\ x+6 & x+9 & x+13\end{array}\right|$
Put $x=5$ in $f(x)$
$$
f(5)=\left|\begin{array}{ccc}
5 & 6 & 8 \\
7 & 9 & 12 \\
11 & 14 & 18
\end{array}\right|
$$
Expand w.r.t. .... $\mathrm{R}_1$.
$$
\begin{aligned}
& =5(18 \times 9-14 \times 12)-6(18 \times 7-11 \times 12)+ \\
& 8(7 \times 14-11 \times 9) \\
& =5(162-168)-6(126-132)+8(98-99) \\
& =5 \times(-6)-6 \times(-6)+8 \times(-1) \\
& =-30+36-8 \\
& =-38+36=-2
\end{aligned}
$$
Therfore, option (c) is correct.