Let S be the set of all lambda in mathrm{R} f | vedclass

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Question

$2 x-y+2 z=2$

$x-2 y+\lambda z=-4$

$x+\lambda y+z=4$

For no solution :

$\mathrm{D}=\left|\begin{array}{ccc}2 & -1 & 2 \ 1 & -

Vedclass · Accepted Answer

contains exactly two elements.

Vedclass · Answer

$2 x-y+2 z=2$

$x-2 y+\lambda z=-4$

$x+\lambda y+z=4$

For no solution :

$\mathrm{D}=\left|\begin{array}{ccc}2 & -1 & 2 \ 1 & -2 & \lambda \ 1 & \lambda & 1\end{array}ight|=0$

$\Rightarrow 2\left(-2-\lambda^{2}ight)+1(1-\lambda)+2(\lambda+2)=0$

$\Rightarrow-2 \lambda^{2}+\lambda+1=0$

$\Rightarrow \lambda=1,-\frac{1}{2}$

$D_{x}=\left|\begin{array}{ccc}2 & -1 & 2 \ -4 & 2 & \lambda \ 4 & \lambda & 1\end{array}ight|=2\left|\begin{array}{ccc}1 & -1 & 2 \ -2 & -2 & \lambda \ \lambda & \lambda & 1\end{array}ight|$

$=2(1+\lambda)$

whichis not equal to zero for

$\lambda=1,-\frac{1}{2}$

Let $S$ be the set of all $\lambda \in \mathrm{R}$ for which the system of linear equations

$2 x-y+2 z=2$

$x-2 y+\lambda z=-4$

$x+\lambda y+z=4$

has no solution. Then the set $S$

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