If ` x= (7 + 4 sqrt(3))^(2n) = [x] + f ` , where ` n in ` N and ` 0 le f lt 1`, then x (1-f) is equal to

Options

A. `(7 - 4 sqrt(3))^(2n)`
B. `(1)/(1 +f) -f`
C. 1
D. none of these

Answer

c

Explanation

Let G = `(7 - 4 sqrt(3))^(2n)`. Then
`R + G = (7 + 4 sqrt(3))^(2n) + (7 - 4 sqrt(3))^(2n)`
`rArr R + G = ` An even integer
`rArr I + f + G` = An even integer
`rArr f + G` = An even integer
`rArr f + G= `An integer
`rArr f + G = 1 [because 0 le flt 1 , 0 lt G lt 1 therefore - lt f + G lt 2 rArr f + G = 1]`
`rArr G = 1 - f `
`therefore R (1 - f) = RG = (7 + 4 sqrt(3))^(2n) (7 - 4sqrt(3))^(2n) = 1`