- Home
- Standard 12
- Mathematics
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 $S$
contains more than two elements.
is a singleton.
contains exactly two elements.
is an empty set.
Solution
$2 x-y+2 z=2$
$x-2 y+\lambda z=-4$
$x+\lambda y+z=4$
For no solution :
$\mathrm{D}=\left|\begin{array}{ccc}2 & -1 & 2 \\ 1 & -2 & \lambda \\ 1 & \lambda & 1\end{array}\right|=0$
$\Rightarrow 2\left(-2-\lambda^{2}\right)+1(1-\lambda)+2(\lambda+2)=0$
$\Rightarrow-2 \lambda^{2}+\lambda+1=0$
$\Rightarrow \lambda=1,-\frac{1}{2}$
$D_{x}=\left|\begin{array}{ccc}2 & -1 & 2 \\ -4 & 2 & \lambda \\ 4 & \lambda & 1\end{array}\right|=2\left|\begin{array}{ccc}1 & -1 & 2 \\ -2 & -2 & \lambda \\ \lambda & \lambda & 1\end{array}\right|$
$=2(1+\lambda)$
whichis not equal to zero for
$\lambda=1,-\frac{1}{2}$
