Question 63

If $$x-3+[\frac{1}{(x-3)}]=4$$, what is the value of $$(x-3)^3+[\frac{1}{(x-3)^3}]$$ ?

Solution

Given : $$x-3+[\frac{1}{(x-3)}]=4$$ ------------(i)

Cubing both sides, we get :

=> $$(x-3+[\frac{1}{(x-3)}])^3=(4)^3$$

=> $$(x-3)^3+(\frac{1}{x-3})^3+3(x-3)(\frac{1}{x-3})[(x-3)+(\frac{1}{x-3})]=64$$

=> $$(x-3)^3+(\frac{1}{x-3})^3+3(4)=64$$ [Using equation (i)]

=> $$(x-3)^3+[\frac{1}{(x-3)^3}]=64-12=52$$

=> Ans - (C)

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