Question

The number of distinct real values of $\lambda$, for which the vectors $-{\lambda }^2\hat{i}+\hat{j}+\hat{k}$ , $\hat{i}-{\lambda }^2\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}-{\lambda }^2\hat{k}$ are coplanar, is

• A. Zero
• B. One
• C. Two
• D. Three

Answer 1

The correct option is C Two

For the given vectors to be coplanar,

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

$\Rightarrow {\lambda }^6-3{\lambda }^2-2=0$
$\Rightarrow (1+{\lambda }^2{)}^2({\lambda }^2-2)=0$
$\Rightarrow \lambda =\pm \sqrt{2}$.