KVPY Sample Paper KVPY Stream-SX Model Paper-20

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

A) \[\left\{ \sqrt{3} \right\}\]

B) \[\left\{ \sqrt{3},-\sqrt{3} \right\}\]

C) \[\left\{ 1,-1 \right\}\]

D) \[\left\{ 3,-3 \right\}\]

Correct Answer: B

Solution :

All four points are coplanar so

\[\left| \begin{matrix} 1-{{\lambda }^{2}} & 2 & 0 \\ 2 & -{{\lambda }^{2}}+1 & 0 \\ 2 & 2 & -{{\lambda }^{2}}-1 \\ \end{matrix} \right|=0\]

\[{{({{\lambda }^{2}}+1)}^{2}}(3-{{\lambda }^{2}})=0\]

\[\lambda =\pm \sqrt{3}.\]