The number of real values of λ for which the system of linear equations, 2 x + 4 y - λ z = 0 , 4 x…

For Infinite many solution $\mathrm{\Delta }=0$

i.e , $\left|\begin{array}{ccc}2& 4& -\lambda \\ 4& \lambda & 2\\ \lambda & 2& 2\end{array}\right|=0$

$\Rightarrow 2\left(2-2\lambda \right)-4\left(8-2\lambda \right)-\lambda \left(8-{\lambda }^2\right)=0$

$\Rightarrow {\lambda }^3+4\lambda -40=\mathrm{0 }$

$f\left(x\right)=x^3+4x-40$ is strictly increasing

i.e., equation has only one real root,

Also, ${∆}_1=0$, ${∆}_2=0$, ${∆}_3=0$

$\therefore$ Only one value of $\lambda$ is possible