Question

If a, b, c are in A. P., then the value of
$\left|\begin{array}{ccc}x+1& x+2& x+a\\ x+2& x+3& x+b\\ x+3& x+4& x+c\end{array}\right|$ 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 , $\left|\begin{array}{ccc}x+1& x+2& x+a\\ x+2& x+3& x+b\\ x+3& x+4& x+c\end{array}\right|$
[ Applying $R_2\to 2R_2$]
$=\frac{1}{2}\left|\begin{array}{ccc}x+1& x+2& x+a\\ 2x+4& 2x+6& 2x+2b\\ x+3& x+4& x+c\end{array}\right|$
$=\frac{1}{2}\left|\begin{array}{ccc}x+1& x+2& x+a\\ 2x+4& 2x+6& 2x+\left(a+c\right)\\ x+3& x+4& x+c\end{array}\right|$
[using equation (1)]
$=\frac{1}{2}\left|\begin{array}{ccc}x+1& x+2& x+a\\ 0& 0& 0\\ x+3& x+4& x+c\end{array}\right|$
[Applying $R_2\to R_2-\left(R_1+R_3\right)$]
$=\frac{1}{2}.0=0$