Question

If a, b, c are in A.P., then the determinant
\[\begin{vmatrix}x + 2 & x + 3 & x + 2a \\ x + 3 & x + 4 & x + 2b \\ x + 4 & x + 5 & x + 2c\end{vmatrix}\]

Options

• 0

• 1

• x

• 2x

Answer 1

\[\begin{vmatrix} x + 2 & x + 3 & x + 2a\\x + 3 & x + 4 & x + 2b\\x + 4 & x + 5 & x + 2c \end{vmatrix}\]
\[ = \begin{vmatrix} 0 & 0 & 2\left( a + c - 2b \right)\\x + 3 & x + 4 & x + 2b\\x + 4 & x + 5 & x + 2c \end{vmatrix} \left[\text{ Applying }R_1 \to R_1 + R_3 - R_2 , R_1 \to R_1 - R_2 \right]\]
\[ = \begin{vmatrix} 0 & 0 & 0\\x + 3 & x + 4 & x + 2b\\x + 4 & x + 5 & x + 2c \end{vmatrix} \left[ \because\text{ a, b, c are in A . P . }\right]\]
\[ = 0\]