Question

If $S$ is the set of distinct values of $b$ for which the following system of linear equations
$x+y+z=1$
$x+ay+z=1$
$ax+by+z=0$
has no solution, then $S$ is :

• A. An infinite set
• B. A finite set containing two or more elements
• C. A singleton
• D. An empty set

Answer 1

Answer: (C)

$\begin{vmatrix}1&1&1\\ 1&a&1\\ a&b&1\end{vmatrix}= 0$
$\Rightarrow -\left(1 - a\right)^{2} = 0$
$\Rightarrow a = 1$
For a = 1
Eq. (1) & (2) are identical i.e.,x + y + z = 1
To have no solution with x + by + z = 0.
$b = 1$