Question

If $f(x)=\begin{vmatrix}x-3 & 2 x^2-18 & 3 x^3-81 \\ x-5 & 2 x^2-50 & 4 x^3-500 \\ 1 & 2 & 3\end{vmatrix}$ then $f(1) f(3)+f(3) f(5)+f(5) f(1)$ is equal to

• A. $f(1)+f(5)$
• B. $f(3)+f(5)$
• C. $f(1)+f(3)$
• D. $f(1)+f(3)+f(5)$

Answer 1

Answer: (B)

Clearly $f(3)=\begin{vmatrix}0 & 0 & 0 \\ -2 & -32 & -392 \\ 1 & 2 & 3\end{vmatrix}=0$ and $f(5)=\begin{vmatrix}2 & 32 & 294 \\ 0 & 0 & 0 \\ 1 & 2 & 3\end{vmatrix}=0$, where $f(1) \neq 0$
$\therefore f (1) f (3)+ f (3) f (5)+ f (5) f (1)=0= f (3)+ f (5)$