Let `( 5 + 2 sqrt(6))^(n) = I + f ` , where n, ` I in N ` and ` 0 lt f lt 1`, then
the value of ` f^(2) - f + I * f - I ` . Is

Options

A. `(1)/(f) -f`
B. `(1)/(1 +f) -f`
C. `(1)/(1 - f)- f`
D. `(1)/(1 - f) +f`

Answer

c

Explanation

Let G = ` (5 - 2 sqrt(6))^(6)`. Then
`I + f + G = (5 + 2 sqrt(6))^(n) + (5 - 2 sqrt(6))^(n)`
`rArr I + f + G = 2 { ""^(n)C_(0) 5^(n) (2 sqrt(6))^(0)+ ""^(n)C_(2) 5^(n-2) (2 sqrt(6))^(2) +...}`
`rArr I + f + G ` = An even integer
`rArr f + G = 1`
Now,
`I + f = (5 + 2 sqrt(6))^(n)`
`rArr I = m (5 + 2 sqrt(6))^(n) -f`
`rArr I = (1)/((5 - 2 sqrt(6))^(n))-f`
`rArr f = (1)/(1-f) - f [{:(because (5 - 2 sqrt(6))^(n) = G and f + G = 1 ),(therefore (5 - 2 sqrt(6))^(n) = 1 = f):}]`s