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

Since, given vectors are coplanar

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

$\Rightarrow {\lambda }^6-{3\lambda }^2-2=0$

$\Rightarrow {(1+{\lambda }^2)}^2({\lambda }^2-2)=0$

$\Rightarrow \lambda =\pm \sqrt{2.}$