যদি \(\rm x-\frac{1}{x}=-6\) হয়, তাহলে \(\rm x^5-\frac{1}{x^5}\) এর মান কত?

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This question was previously asked in

SSC CGL 2023 Tier-I Official Paper (Held On: 18 Jul 2023 Shift 2)

Answer 1

প্রদত্ত:

x - (1/x) = (- 6)

অনুসৃত সূত্র:

যদি x - (1/x) = P হয়, তাহলে

x + (1/x) = √(P² + 4)

যদি x + (1/x) = P হয়, তাহলে

x3 + (1/x3) = (P³ - 3P)

x⁵ - (1/x⁵) = {x³ + (1/x³)} × {x² - 1/x²} + {x - (1/x)}

গণনা:

x - (1/x) = (- 6)

x + (1/x) = √{(- 6)2 + 4} = √40 = 2√10

x3 + (1/x3) = (√40)³ - 3√40

⇒ 40√40 - 3√40 = 37 × 2√10 = 74√10

অতএব,

x5 - (1/x5) = {x3 + (1/x3)} × {x2 - 1/x2} + {x - (1/x)}

⇒ {74√10 × x + (1/x) × x - (1/x)} + (- 6)

⇒ {74√10 × 2√10 × (-6)} - 6

⇒ 74√10 × { (- 12√10)} - 6

⇒ (- 8880) - 6 = - 8886

∴ নির্ণেয় উত্তর হল - 8886

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