Question

Let $S$ be the set of all $\lambda$ $\in R$ for which the system of linear equations
$2x-y+2z=2$
$x-2y+\lambda z=-4$
$x+\lambda y+z=4$
has no solution. Then the set $S$

• A. is an empty set.
• B. is a singleton.
• C. contains more than two elements.
• D. contains exactly two elements.

Answer 1

The correct option is D contains exactly two elements.

For no solution

$\Delta =0\&{\Delta }_1|{\Delta }_2|{\Delta }_3\ne 0$

$\Delta =\left|\begin{array}{ccc}2& -1& 2\\ 1& -2& \lambda \\ 1& \lambda & 1\end{array}\right|=0$

$\Rightarrow 2{(-2-\lambda )}^2+1(1-\lambda )+2(\lambda +2)=0$
$\Rightarrow -4-2{\lambda }^2+1=0$
$\Rightarrow -2{\lambda }^2+\lambda +1=0$
$\Rightarrow 2{\lambda }^2-\lambda -1=0\Rightarrow \lambda =1,\frac{-1}{2}$

For two values of $\lambda$ equations has no solution