Question

If $\lambda >0$ such that the system of equations
$x+y-2z=0$,
$\lambda x-2y+z=0$,
$x+3y-\lambda z=0$,
has infinite number of solution then $\lambda$ is

• A. $\sqrt{10}+2$
• B. $\sqrt{10}-2$
• C. 2
• D. 1

Answer 1

Answer: (A)

$\sqrt{10}+2$

Given system of equation can be written as $AX=O$
where $A=\left[\begin{array}{ccc}1& 1& -2\\ \lambda & -2& 1\\ 1& 3& -\lambda \end{array}\right]$, $X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right],O=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$
For infinitely many solution
$D=0$
$\left|\begin{array}{ccc}1& 1& -2\\ \lambda & -2& 1\\ 1& 3& -\lambda \end{array}\right|=0$
$\Rightarrow {\lambda }^2-4\lambda -6=0$
$\Rightarrow \lambda =2\pm \sqrt{10}$
So, option A