Question

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

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

Answer 1

The correct option is D a singleton

For no solution
$D=\left|\begin{array}{ccc}1& 1& 1\\ 1& a& 1\\ a& b& 1\end{array}\right|=0\Rightarrow a=1$
For $a=1,$ the equations become
$x+y+z=1x+y+z=1x+by+z=0$

The first two planes are coincident, for only $b=1$ the third plane becomes parallel to both the planes and system has no solution.
$\Rightarrow S$ is singleton