Question:
If 16x^4 + 1/(16x^4) = 14159, then find the value of 2x + 1/(2x).
यदि 16x^4 + 1/(16x^4) = 14159 है, तो 2x + 1/(2x) का मान ज्ञात कीजिए।
Options:
- a) 12
- b) 11
- c) 15
- d) 9
- a) 12
- b) 11
- c) 15
- d) 9
correct answer : b)
Explanation:
Let y = 2x + 1/(2x). Squaring both sides: y^2 = 4x^2 + 1/(4x^2) + 2 => 4x^2 + 1/(4x^2) = y^2 – 2. Squaring again: (4x^2 + 1/(4x^2))^2 = 16x^4 + 1/(16x^4) + 2 => (y^2 – 2)^2 = 14159 + 2 = 14161. Taking square root: y^2 – 2 = sqrt(14161) = 119 => y^2 = 121 => y = 11. Thus, 2x + 1/(2x) = 11.
मान लीजिए y = 2x + 1/(2x)। दोनों पक्षों का वर्ग करने पर: y² = 4x² + 1/(4x²) + 2 => 4x² + 1/(4x²) = y² – 2। पुनः वर्ग करने पर: (4x² + 1/(4x²))² = 16x⁴ + 1/(16x⁴) + 2 => (y² – 2)² = 14159 + 2 = 14161। वर्गमूल लेने पर: y² – 2 = sqrt(14161) = 119 => y² = 121 => y = 11। इस प्रकार, 2x + 1/(2x) = 11।