Question

If a, b, c are in A.P., then the value of
$\left|\begin{array}{ccc}x+1& x+2& x+a\\ x+2& x+3& x+b\\ x+3& x+4& x+c\end{array}\right|$

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

Answer 1

The correct option is C $0$

Since, a,b,c are in A.P.
$\Rightarrow 2b=a+c$
$\left|\begin{array}{ccc}x+1& x+2& x+a\\ x+2& x+3& x+b\\ x+3& x+4& x+c\end{array}\right|$
$R_1\to R_1+R_3$
$=\left|\begin{array}{ccc}2(x+2)& 2(x+3)& 2x+a+c\\ x+2& x+3& x+b\\ x+3& x+4& x+c\end{array}\right|$
$=\left|\begin{array}{ccc}2(x+2)& 2(x+3)& 2x+2b\\ x+2& x+3& x+b\\ x+3& x+4& x+c\end{array}\right|$
$=2\left|\begin{array}{ccc}x+2& x+3& x+b\\ x+2& x+3& x+b\\ x+3& x+4& x+c\end{array}\right|$
$=0$