If $\begin{vmatrix}2 & 3 & 2\\x & x & x\\4 & 9 & 1\end{vmatrix}+3=0$ then the value of x is :

-1

Given determinant equation:

$\left| \begin{matrix} 2 & 3 & 2 \\ x & x & x \\ 4 & 9 & 1 \end{matrix} \right| + 3 = 0$

Compute the determinant using cofactor expansion along the first row:

$|A| = 2 \begin{vmatrix} x & x \\ 9 & 1 \end{vmatrix} - 3 \begin{vmatrix} x & x \\ 4 & 1 \end{vmatrix} + 2 \begin{vmatrix} x & x \\ 4 & 9 \end{vmatrix}$

Compute 2×2 determinants:

$\begin{vmatrix} x & x \\ 9 & 1 \end{vmatrix} = x*1 - x*9 = -8x$

$\begin{vmatrix} x & x \\ 4 & 1 \end{vmatrix} = x*1 - x*4 = -3x$

$\begin{vmatrix} x & x \\ 4 & 9 \end{vmatrix} = x*9 - x*4 = 5x$

Substitute back:

$|A| = 2(-8x) - 3(-3x) + 2(5x) = -16x + 9x + 10x = 3x$

Set $|A| + 3 = 0$:

$3x + 3 = 0 \Rightarrow 3x = -3 \Rightarrow x = -1$

Answer: $x = -1$