Solve this:

Dear Student,

$$\begin{gathered} f\left(x\right)=\left|\begin{array}{ccc}mx& mx-p& mx+p\\ n& n+p& n-p\\ mx+2n& mx+2n+p& mx+2n-p\end{array}\right| \\ Applying C_2\to C_2+C_3 we get \\ \Rightarrow f\left(x\right)=\left|\begin{array}{ccc}mx& 2mx& mx+p\\ n& 2n& n-p\\ mx+2n& 2mx+4n& mx+2n-p\end{array}\right| \\ \Rightarrow f\left(x\right)=2\left|\begin{array}{ccc}mx& mx& mx+p\\ n& n& n-p\\ mx+2n& mx+2n& mx+2n-p\end{array}\right| \\ Since, two columns are same \\ \Rightarrow f\left(x\right)=2\times 0=0 \end{gathered}$$
$\therefore y=f\left(x\right)=0$ must be a straight line parallel to x-axis.
Regards,