Question

If $f^{\prime }\left(x\right)=\left|\begin{array}{ccc}mx& mx-p& mx+p\\ n& n+p& n-p\\ mx-2n& mx+2n+p& mx+2n-p\end{array}\right|$, they $y=f\left(x\right)$ represents

• A. a straight line parallel to x-axis
• B. a straight line parallel to y-axis
• C. parabola
• D. a straight line with negative slope

Answer 1

The correct option is A a straight line parallel to x-axis

$f^{\prime }\left(x\right)=\left|\begin{array}{ccc}mx& mx-p& mx+p\\ n& n+p& n-p\\ mx-2n& mx+2n+p& mx+2n-p\end{array}\right|$

Applying $R_3\to R_3-R_1-2R_2,$ we get

$f^{\prime }\left(x\right)=\left|\begin{array}{ccc}mx& mx-p& mx+p\\ n& n+p& n-p\\ 0& 0& 0\end{array}\right|=0$

$\Rightarrow f\left(x\right)=c$

$\Rightarrow$ a straight line parallel to x-axis.