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1. $a - \frac{1}{a} = 5 \sqrt{3}, a^{2} + \frac{1}{a^{2}}$

Answer: Option A

Explanation:

a^{2} + \frac{1}{a^{2}} = {(a - \frac{1}{a})}^{2} + 2 a \frac{1}{a} = {(5 \sqrt{3})}^{2} + 2 = 5^{2} . \sqrt{3} . \sqrt{3} + 2 = 25 \times 3 + 2 = 75 + 2 = 77

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