If \(x^{2} - 5x + 1 = 0,\) then the value of \(\frac{x^{6}+x^{4}+x^{2}+1}{5x^{3}}=?\)

Question

Question

This question was previously asked in

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

Answer 1

Given:

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

Concept used:

If a + 1/a = b, then

a³ + 1/a³ = b³- 3b

Calculation:

x² - 5x + 1 = 0

⇒ x2 + 1 = 5x

⇒ x + 1/x = 5

Now,

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

⇒ \(\frac{x^{3}+x+\frac{1}{x}+\frac{1}{x^3}}{5}\) [By dividing \(\frac{1}{x^3}\)]

⇒ \(\frac{5^3 -3\times5+5}{5}\)

⇒ \(\frac{125 - 15 +5}{5}\)

⇒ \(\frac{115}{5}\)

⇒ 23

∴ The required answer is 23.

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