Question

If $S$ ids 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 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

These three equations will have no solution if derterminant of these equations is $0$.

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

$⟹a-b+a-1+b-2a=0$

$⟹-(a-1)2=0$

$⟹a=1$

If we put, $a=1$,

Eqaution (2) and (1), will become same.

Now, if we put $b=1$ in third equation, then it will become parallel to first two equations.

So, for this system to have no solutions, $b$ should be $1.$

Thus, $S$ is a singleton set containing one element $1$.