Question

If $a,b,c$ are in A.P, then the determinant
$\left|\begin{array}{ccc}x+2& x+3& x+2a\\ x+3& x+4& x+2b\\ x+4& x+5& x+2c\end{array}\right|$ is

• A. $0$
• B. $1$
• C. $x$
• D. $2x$

Answer 1

The correct option is A $0$

Given $a,b,c$ are in A.P.

$\Rightarrow 2b=a+c$ .....(1)

$\left|\begin{array}{ccc}x+2& x+3& x+2a\\ x+3& x+4& x+2b\\ x+4& x+5& x+2c\end{array}\right|$

$R_1\to R_1+R_3$

$\left|\begin{array}{ccc}2(x+3)& 2(x+4)& 2x+2(a+c)\\ x+3& x+4& x+2b\\ x+4& x+5& x+2c\end{array}\right|$

$\left|\begin{array}{ccc}2(x+3)& 2(x+4)& 2x+4b\\ x+3& x+4& x+2b\\ x+4& x+5& x+2c\end{array}\right|$ (by (1))

$=2\left|\begin{array}{ccc}x+3& x+4& x+2b\\ x+3& x+4& x+2b\\ x+4& x+5& x+2c\end{array}\right|$

Since, $R_1,R_2$ are identical, the value of determinant is $0$.