Question

If \( a \), \( b \), and \( c \) are in AP, then determinant \[ \begin{vmatrix} x+2 & x+3 & x+4
x+4 & x+5 & x+6
x+7 & x+8 & x+9 \end{vmatrix} \]

• A. 0
• B. 1
• C. \( x \)
• D. \( 2x \)

Hint:

For matrices where rows or columns are in arithmetic progression, the determinant is often zero.

Answer 1

The Correct Option is A

Step 1: Analyzing the determinant.
Since the rows and columns of the determinant are linear, we know that the determinant of the matrix will be zero when \( a \), \( b \), and \( c \) are in arithmetic progression.

Step 2: Conclusion.
The determinant equals zero, corresponding to option (1).