Question

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

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

Answer 1

The correct option is C singleton set

$Here$

$D=\left|\begin{array}{ccc}1& 1& 1\\ 1& a& 1\\ a& b& 1\end{array}\right|=0\Rightarrow a=1$

For a= 1, the equations become

$X+Y+Z=1$

$X+Y+Z=1$

$X+bY+Z=0$

These equation give no solution for b=1

Therefore, S is $singleton$ $set.$