Question

If $a,b$ and $c$ are in A.P., then the value of $\left|\begin{array}{ccc}x+2& x+3& x+a\\ x+4& x+5& x+b\\ x+6& x+7& x+c\end{array}\right|$ is

• A. $x-(a+b+c)$
• B. $9x^2+a+b+c$
• C. $0$
• D. $a+b+c$

Answer 1

Answer: (C)

Let $\Delta =\left|\begin{array}{ccc}x+2& x+3& x+a\\ x+4& x+5& x+b\\ x+6& x+7& x+c\end{array}\right|$
Applying $R_1\to R_1-R_2$ and $R_2\to R_2-R_3$, we get
$\Delta =\left|\begin{array}{ccc}-2& -2& a-b\\ -2& -2& b-c\\ x+6& x+7& x+c\end{array}\right|$
Since, $a,b$ and $c$ are in $AP$.
$\therefore a-b=b-c$
Thus, Rows $R_1$ and $R_2$ are identical.
Hence, $\Delta =0$