Question
Download Solution PDFIf \(x^{2} - 5x + 1 = 0,\) then the value of \(\frac{x^{6}+x^{4}+x^{2}+1}{5x^{3}}=?\)
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFGiven:
\(x^{2} - 5x + 1 = 0\)
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.
Last updated on Jul 17, 2025
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