Question:
If \(x = 3 + 2\sqrt{2}\), find the value of \(\sqrt{x} – \frac{1}{\sqrt{x}}\).
यदि \(x = 3 + 2\sqrt{2}\) है, तो \(\sqrt{x} – \frac{1}{\sqrt{x}}\) का मान क्या होगा?
Options:
- 2
- \(\sqrt{2}\)
- 4
- \(2\sqrt{2}\)
- 2
- \(\sqrt{2}\)
- 4
- \(2\sqrt{2}\)
Correct Answer: a)
Explanation:
\(x = 2 + 1 + 2\sqrt{2} = (\sqrt{2} + 1)^2 \implies \sqrt{x} = \sqrt{2} + 1\).
\(\frac{1}{\sqrt{x}} = \frac{1}{\sqrt{2} + 1} = \sqrt{2} – 1\).
\(\sqrt{x} – \frac{1}{\sqrt{x}} = (\sqrt{2} + 1) – (\sqrt{2} – 1) = 2\).
\(\frac{1}{\sqrt{x}} = \frac{1}{\sqrt{2} + 1} = \sqrt{2} – 1\).
\(\sqrt{x} – \frac{1}{\sqrt{x}} = (\sqrt{2} + 1) – (\sqrt{2} – 1) = 2\).
\(\sqrt{x} = \sqrt{2} + 1\), \(1/\sqrt{x} = \sqrt{2} – 1\)। घटाने पर मान 2 प्राप्त होता है।