Question

If a system of equation $- ax + y + z = 0$
$x - by + z = 0$
$x + y - cz = 0 (a, b, c \ne -1)$
has a non-zero solution then $\frac{1}{1+a} + \frac{1}{1+b} + \frac{1}{1+c} = $

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

Answer: (B)

$\Delta = \begin{vmatrix}-a&1&1\\ 1&-b&1\\ 1&1&-c\end{vmatrix} = 0$ for non-zero solution
$\Rightarrow abc - a - b - c - 2 = 0$
$\Rightarrow abc = a + b + c + 2$
Now, $\frac{1}{1+a} + \frac{1}{1+b} + \frac{1}{1+c}$
$= \frac{3+2\left(a+b+c\right)+\left(ab+bc+ac\right)}{1+\left(a+b+c\right)+\left(ab+bc+ac\right)+abc}$
$= \frac{3+2\left(a+b+c\right)+\left(ab+bc+ac\right)}{1+2\left(a+b+c\right)+2+ab+bc+ac} = 1$