Correct Answer: Option A

Explanation

\((\sqrt{3} - 5\sqrt{2})(\sqrt{3} + \sqrt{2}) = p + q\sqrt{6}\)

Expanding the left-hand side:

\(\sqrt{3}(\sqrt{3} + \sqrt{2}) - 5\sqrt{2}(\sqrt{3} + \sqrt{2})\)

Simplifying: \(\sqrt{9} + \sqrt{6} - 5\sqrt{6} - 5\sqrt{4}\)

\(3 - 4\sqrt{6} - 10\)

\(-7 - 4\sqrt{6}\)

Comparing with the right-hand side:

\(-7 - 4\sqrt{6} = p + q\sqrt{6}\)

From this, we have:

\(-4\sqrt{6} = q\sqrt{6}\)

This leads to: q = -4

And similarly: -7 = p \(\quad \Rightarrow \quad\) p = -7
 

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