Question

[(√3 + 1)⁶] = .......................; where [ ] is integer part function.

(a) 415 

(b) 416 

(c) 417 

(d) 418

Answer 1

Correct option: (b) 416

Explanation:

Let (√3 + 1)⁶ = I + F where I ∈ N & 0 < F < 1.

Let (√3 – 1)⁶ = G then I + F + G = (√3 + 1)⁶ + (√3 – 1)⁶ 

= 2[⁶c₀ (√3)⁶ + ⁶c₂ (√3)^6–2 + -----] = even integer

∴  F + G = 1.

Now

I + F + G   = 2[⁶c₀ (√3)⁶ + ⁶c₂ (√3)⁴ + ⁶c₄ (√3)² + ⁶c₆ (√3)⁰]

= 2[27 + 135 + 45 + 1]

= 416