Q. If $f(x) = \left| \begin{matrix} x - 3 & 2x^{2} - 18 & 3x^{3} - 81 \\ x - 5 & 2x^{2} - 50 & 4x^{3} - 500 \\ 1 & 2 & 3 \end{matrix} \right|,$then $f(1).f(3) + f(3).f(5) + f(5).f(1)$ is equal to
A) $f(1)$
B) $f(3)$
C) $f(1) + f(3)$
D) $f(1) + f(5)$
Explanation
(b)
$f(1) = \left| \begin{matrix}
- 2 & - 16 & - 78 \\
- 4 & - 48 & - 496 \\
1 & 2 & 3
\end{matrix} \right| = 2928$
$f(3) = \left| \begin{matrix}
0 & 0 & 0 \\
- 2 & - 32 & - 392 \\
1 & 2 & 3
\end{matrix} \right| = 0$
and $f(5) = \left| \begin{matrix}
2 & 32 & 294 \\
0 & 0 & 0 \\
1 & 2 & 3
\end{matrix} \right| = 0$
$\therefore\ \ f(1).f(3) + f(3).f(5) + f(5).f(1)$
$= f(1).0 + 0 + f(1).0 = 0 = f(3)\$or$\ f\ (5)$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.