If n is a positive integer, then \((\sqrt{3}+1)^{2 \mathrm{n}}-(\sqrt{3}-1)^{2 \mathrm{n}}\) is

(D) a rational number other than positive integers

JEE Main · 2012 · 29 Apr 2012 · Offline · Binomial Theorem

If n is a positive integer, then

(3 + 1)^{2 n} - (3 - 1)^{2 n}

Answer: (A)

Square the brackets first:

(3 + 1)^{2 n} = (4 + 23)^{n} = 2^{n} (2 + 3)^{n}

and

(3 - 1)^{2 n} = 2^{n} (2 - 3)^{n}

. When

(2 + 3)^{n}

and

(2 - 3)^{n}

are expanded by the binomial theorem and subtracted, the even powers of

3

cancel while the odd powers double, leaving a surviving factor of

3

times an integer. The whole expression is therefore

2^{n + 1} 3 \times (integer)

, which is an irrational number.

JEE Main · 2012 · 29 Apr 2012 · Offline · Single correct option