Let S be the set of all real values of

Question:

Let $S$ be the set of all real values of $\lambda$ such that a plane passing through the points $\left(-\lambda^{2}, 1,1\right)$, $\left(1,-\lambda^{2}, 1\right)$ and $\left(1,1,-\lambda^{2}\right)$ also passes through the point $(-1,-1,1)$. Then $S$ is equal to:

  1. $\{\sqrt{3}\}$

  2. $\{\sqrt{3}-\sqrt{3}\}$

  3. $\{1,-1\}$

  4. $\{3,-3\}$


Correct Option: , 2

Solution:

All four points are coplaner so

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

$\left(\lambda^{2}+1\right)^{2}\left(3-\lambda^{2}\right)=0$

$\lambda=\pm \sqrt{3}$

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