Discussion Forum

 If a, b, c are in A. P, then the value of $\begin{vmatrix}x+1& x+2 & x+a \\ x+2 & x+3 & x+b \\ x+3 & x+4 & x+c \end{vmatrix}$ is?

Answer:

Explanation:

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

.'. 2b = a+c

Now,$\begin{vmatrix}x+1& x+2 & x+a \\ x+2 & x+3 & x+b \\ x+3 & x+4 & x+c \end{vmatrix}$

[Applying R₂ → 2R₂]

= $\frac{1}{2}\begin{vmatrix}x+1& x+2 & x+a \\ 2x+4 & 2x+6 & 2x+2b \\ x+3 & x+4 & x+c \end{vmatrix}$

= $\frac{1}{2}\begin{vmatrix}x+1& x+2 & x+a \\ 2x+4 & 2x+6 & 2x+(a+c) \\ x+3 & x+4 & x+c \end{vmatrix}$

[Using equation(i)]

=$\frac{1}{2}\begin{vmatrix}x+1& x+2 & x+a \\ 0 & 0 & 0 \\ x+3 & x+4 & x+c \end{vmatrix}$

=$\frac{1}{2}.0$ =0

[Applying R₂ → R₂ - (R₁+R₃)]