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)

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