If $f(x)=\begin{vmatrix}x+λ& x & x\\x & x+λ & x\\x & x & x+λ\end{vmatrix},$ then $f(3x)-f(x)=$

$6x\, λ^2$

The correct answer is option (2) : $6x\, λ^2$

We have,

$f(x)= \begin{vmatrix}x+λ& x & x\\x & x+λ & x\\x & x & x+λ\end{vmatrix}$

$⇒f(x)=\begin{vmatrix}3x+λ& x & x\\3x+λ & x+λ & x\\3x+λ & x & x+λ\end{vmatrix}$         $\begin{bmatrix} Applying \\ C_1→C_1+C_2+C_3\end{bmatrix}$

$⇒f(x) = (3x+λ)\begin{vmatrix}1 & x & x\\1 & x+λ & x\\1 & x & x+λ\end{vmatrix}$

$⇒f(x) = (3x+λ)\begin{vmatrix}1 & x & x\\0 & λ & 0\\0 & 0 & λ\end{vmatrix}$      $\begin{bmatrix} Applying \, R_2→R_2-R_1\\R_3→R_3-R_1\end{bmatrix}$

$⇒f(x)=(3x+\lambda )\lambda^2 $

$∴f(3x) - f(x) = (9x+\lambda)\lambda^2 - (3x + \lambda ) \lambda^2 = 6x \lambda^2 $.