यदि x + \(\frac{1}{x}\) = 1 है, तो x¹² + x⁹ + x⁶ + x³ + 1 का मान क्या है?

Question

Question

This question was previously asked in

SSC CGL 2022 Tier-I Official Paper (Held On : 06 Dec 2022 Shift 1)

Answer 1

दिया गया:

x12 + x9 + x6 + x3 + 1 का मान

if x + (1/x) = 1

x2 + 1 = x

x2 - x + 1 = 0

(x + 1)(x2 - x + 1) = 0 (x + 1)

(x3 + 13) = 0

x3 = -1

प्रयुक्त सूत्र:

x + 1/x = 1 है, तो x³ = -1

गणना:

प्रश्न के अनुसार,

x12 + x9 + x6 + x3 + 1

⇒ (x³)⁴ + (x³)³ + (x³)² + (x3) + 1

⇒ (−1)⁴ + (-1)³ + (-1)² + (-1) + 1

⇒ 1 - 1 + 1 - 1 + 1

⇒ 3 - 2 = 1

∴ x12 + x9 + x6 + x3 + 1 का मान 1 है।