Question 35

If $$x = 5 - \frac{1}{x} $$, then what is the value of $$x^{5} + \frac{1}{x^{5}}$$?

Solution

Given : $$x=5-\frac{1}{x}$$

=> $$x+\frac{1}{x}=5=k$$

Now, $$x^5+\frac{1}{x^5}=[(x^3+\frac{1}{x^3})\times(x^2+\frac{1}{x^2})]-(x+\frac{1}{x})$$

= $$[(x+\frac{1}{x})^3-3(x+\frac{1}{x})\times(x+\frac{1}{x})^2-2(x)(\frac{1}{x})]-(x+\frac{1}{x})$$

= $$[(k^3-3k)\times(k^2-2)]-(k)$$

= $$[(125-15)\times(25-2)]-(5)$$

= $$(110\times23)-5$$

= $$2530-5=2525$$

=> Ans - (C)

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