Question

If I is the integral part of (2 + √3)ⁿ and f is the fractional part. Then (I + f) (1 –  f) is equal to

(a)  0

(b)  1

(c)  n

(d)  None of these

Answer 1

Correct option  (b) 1 

Explanation:

Let (2 + √3)ⁿ = I + f..........................(1) (0 ≤ f < 1)

and (2– √3)ⁿ = F.............................(2)

(1) + (2) gives

2(ⁿC₀ .2ⁿ + ⁿC₂.2^{n - 2}(√3)² + .....) = I + f + F

(adding 0 < f + F < 2)

⇒ I + f + F is an even integer

⇒ I + F is an integer

⇒ f + F = 1      (∴0 < f + F < 2)

F = 1 – f

(I + F)(1 – f) = (2 + √3)ⁿ (2 – √3)ⁿ = (4 – 3)ⁿ = 1