Find the value
If \[ x = \sqrt{6} + \sqrt{5} \]
(a) \(2\sqrt{6}\) \quad (b) \(2\sqrt{5}\) \quad (c) 24 \quad (d) 20
Solution:
\[ \frac{1}{x} = \sqrt{6} – \sqrt{5} \]
\[ x + \frac{1}{x} = (\sqrt{6} + \sqrt{5}) + (\sqrt{6} – \sqrt{5}) = 2\sqrt{6} \]
\[ x^2 + \frac{1}{x^2} – 2 = (x + \frac{1}{x})^2 – 4 \]
\[ = (2\sqrt{6})^2 – 4 = 24 – 4 = 20 \]
\[ \therefore \text{Correct Answer: } 20 \ (\text{Option d}) \]