यदि \(x^{2} - 5x + 1 = 0\) है, तो \(\frac{x^{6}+x^{4}+x^{2}+1}{5x^{3}}\) का मान क्या होगा?

Question

Question

This question was previously asked in

SSC CGL 2022 Tier-I Official Paper (Held On : 01 Dec 2022 Shift 1)

Answer 1

दिया गया है:

\(x^{2} - 5x + 1 = 0\)

प्रयुक्त अवधारणा:

यदि a + 1/a = b, तो

a3 + 1/a3 = b3 - 3b

गणना:

x2 - 5x + 1 = 0

⇒ x2 + 1 = 5x

⇒ x + 1/x = 5

अब,

\(\frac{x^{6}+x^{4}+x^{2}+1}{5x^{3}}\)

⇒ \(\frac{x^{3}+x+\frac{1}{x}+\frac{1}{x^3}}{5}\) [\(\frac{1}{x^3}\) से विभाजित करने पर]

⇒ \(\frac{5^3 -3\times5+5}{5}\)

⇒ \(\frac{125 - 15 +5}{5}\)

⇒ \(\frac{115}{5}\)

⇒ 23

∴ अभीष्ट उत्तर 23 है।