Q. If $\mathrm{\Delta}(x) = \left| \begin{matrix} f(x) + f( - x) & 0 & x^{4} \\ 3 & f(x) - f( - x) & \cos x \\ x^{4} & 2x & f(x)f( - x) \end{matrix} \right|$ (where $f(x)$ is a real valued function of $x),$ then the value of $\int_{- 2}^{2}{x^{4}\mathrm{\Delta}(x)}$
A) Depends upon the function $f(x)$
B) is 4
C) is $- 4$
D) is zero
Explanation
(d)
$\mathrm{\Delta}( - x) = \left| \begin{matrix}
f( - x) + f(x) & 0 & x^{4} \\
3 & f( - x) - f(x) & \cos x \\
x^{4} & - 2x & f( - x)f(x)
\end{matrix} \right|$
$\left| \begin{matrix}
f(x) + f( - x) & 0 & x^{4} \\
3 & f(x) - f( - x) & \cos x \\
x^{4} & - 2x & f(x)f( - x)
\end{matrix} \right| = - \mathrm{\Delta}(x)$
So, $\mathrm{\Delta}(x)$ is an odd function.
$\Rightarrow \ x^{4}\mathrm{\Delta}(x)$ is an odd function
$\Rightarrow \ \int_{- 2}^{2}{x^{4}\mathrm{\Delta}(x)dx = 0\ }$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.