Question

If $a,b,c$ are in $A.P$, 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 $0$

Since $a,.b,c$ in AP, so $2b=a+c$

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

$R_2\to 2R_2-R_1-R_3$

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

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