Question

For the set of linear equations $\gamma x-3y+z=0$ , $x+\gamma y+3z=1$ , $3x+y+5z=2$ the value of $\gamma$ for which Ihe equations does not have a unique solution.

• A. $-1,\frac{11}{5}$
• B. $-1,-\frac{11}{5}$
• C. $\frac{11}{5},1$
• D. $1,\frac{11}{5}$

Answer 1

The correct option is A $-1,\frac{11}{5}$

As system have no unique solution.
$\Rightarrow$ system either have no solution or infinite solution .
$\therefore \left|A\right|=0$ for both cases.

$\Rightarrow \left|\begin{array}{ccc}\gamma & -3& 1\\ 1& \gamma & 3\\ 3& 1& 5\end{array}\right|=0$
$\Rightarrow 5{\gamma }^2-6\gamma -11=0$
$(5\gamma -11)(\gamma +1)=0$
$\Rightarrow \gamma =\frac{11}{5},\gamma =-1$