Question

If $\begin{vmatrix}x^{2}+x&x+1&x-2\\ 2x^{2}+3x-1&3x&3x-3\\ x^{2}+2x+3&2x-1&2x-1\end{vmatrix}=ax-12,$ then 'a' is equal to :

• A. 12
• B. 24
• C. - 12
• D. - 24

Answer 1

Answer: (B)

$\Delta=\begin{vmatrix}x^{2}+x & x+1 & x-2 \\ 2 x^{2}+3 x-1 & 3 x & 3 x-3 \\ x^{2}+2 x+3 & 2 x-1 & 2 x-1\end{vmatrix}=a x-12$
Operating $R_{2} \rightarrow R_{2}-\left(R_{1}+R_{3}\right)$ gives,
$\Delta= \begin{vmatrix}x^{2}+x & x+1 & x-2 \\ -4 & 0 & 0 \\ x^{2}+2 x+3 & 2 x-1 & 2 x-1\end{vmatrix}$
$\Rightarrow \Delta=-(-4)\begin{vmatrix}x+1 & x-2 \\ 2 x-1 & 2 x-1\end{vmatrix}=4(2 x-1)$
$\begin{vmatrix}x+1 & x-2 \\ 1 & 1\end{vmatrix}=4(2 x-1)(x+1-x+2)$
$=4(2 x-1)(3)=24 x-12=a x-12$ (given) $\Rightarrow a=24$