Question

If a, b, c are in A. P., then the value of
$\begin{vmatrix}x+1&x+2&x+a\\ x+2&x+3&x+b\\ x+3&x+4&x+c\end{vmatrix}$ is :

• A. 3
• B. -3
• C. 0
• D. none of these

Answer 1

Answer: (C)

Given a, b, c are in A.P.
$\therefore \ 2b = a +c $ ....(1)
Now , $\begin{vmatrix}x+1&x+2&x+a\\ x+2&x+3&x+b\\ x+3&x+4&x+c\end{vmatrix} $
[ Applying $R_{2} \to2R_{2} $]
$= \frac{1}{2} \begin{vmatrix}x+1&x+2&x+a\\ 2x+4&2x+6&2x+2b\\ x+3&x+4&x+c\end{vmatrix} $
$= \frac{1}{2} \begin{vmatrix}x+1&x+2&x+a\\ 2x+4&2x+6&2x+\left(a+c\right)\\ x+3&x+4&x+c\end{vmatrix} $
[using equation (1)]
$= \frac{1}{2} \begin{vmatrix}x+1&x+2&x+a\\ 0&0&0\\ x+3&x+4&x+c\end{vmatrix} $
[Applying $R_{2} \to R_{2} - \left(R_{1}+R_{3}\right)$]
$ = \frac{1}{2} . 0 = 0 $