Question

If a, b, c are in AP, then $\left|\begin{array}{ccc}x+1& x+2& x+a\\ x+2& x+3& x+b\\ x+3& x+4& x+c\end{array}\right|$ equals

• A. a+b+c
• B. x+a+b+c
• C. 0
• D. none of these

Answer 1

The correct option is C 0

Let $X=\left|\begin{array}{ccc}x+1& x+2& x+a\\ x+2& x+3& x+b\\ x+3& x+4& x+c\end{array}\right|$

Replace $R_2\to 2R_2-(R_1+R_3)$

$X=\left|\begin{array}{ccc}x+1& x+2& x+a\\ 2(x+2)-(2x+4)& 2(x+3)-(2x+6)& 2(x+b)-(2x+c+a)\\ x+3& x+4& x+c\end{array}\right|$

$=\left|\begin{array}{ccc}x+1& x+2& x+a\\ 0& 0& 2b-(a+c)\\ x+3& x+4& x+c\end{array}\right|$

Also given $a,b,cϵ$ A.P $\Rightarrow 2b=a+c$

$\Rightarrow X=0$