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If S is the set of all real values of 'a' such that a plane passing through the points $(- a^{2}, 1, 1), (1, - a^{2}, 1), (1, 1, - a^{2})$ also passes through the point $(- 1, - 1, 1),$ then S=

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${1, - 1}$

${\sqrt{3}}$

${\sqrt{3}, - \sqrt{3}}$

${3, - 3}$

answer is B.

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Detailed Solution

$∴ |\begin{array}{ccc} - 1 + a^{2} & - 2 & 0 \\ - 2 & - 1 + a^{2} & 0 \\ - 2 & - 2 & 1 + a^{2} \end{array}| = 0$

$\Rightarrow (1 + a^{2}) [{(a^{2} - 1)}^{2} - 4] = 0$

$\Rightarrow {(a^{2} + 1)}^{2} (a^{2} - 3) = 0$

$a = \pm \sqrt{3}$

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