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

Answer 1

Answer: (B)

$f(x)=(x-3)(x-5)\begin{vmatrix}1 & 2(x+3) & 3\left(x^{2}+9+3 x\right) \\ 1 & 2(x+5) & 4\left(x^{2}+25+5 x\right) \\ 1 & 2 & 3\end{vmatrix}$
$\Rightarrow f(3)=f(5)=0$
So, $f(1) f(3)+f(3) \cdot f(5)+f(5) f(1)=0=f(3)$