If $\alpha$ and $\beta$ be the coefficients of $x^{4}$ and $x^{2}$ respectively in the expansion of $\left(x+\sqrt{x^{2}-1}\right)^{6}+\left(x-\sqrt{x^{2}-1}\right)^{6}$, then:

JEE Main · 2020 · 08 Jan 2020 · Shift 2 · Binomial Theorem

α

and

β

be the coefficients of

x^{4}

and

x^{2}

respectively in the expansion of

(x + x^{2} - 1)^{6} + (x - x^{2} - 1)^{6}

, then:

Answer: (C)

Only the even-power terms survive when the two expansions are added, so

(x + x^{2} - 1)^{6} + (x - x^{2} - 1)^{6} = 2 [x^{6} + 15 x^{4} (x^{2} - 1) + 15 x^{2} (x^{2} - 1)^{2} + (x^{2} - 1)^{3}]

. Collecting the coefficient of

x^{4}

inside the bracket gives

- 15 - 30 - 3 = - 48

and that of

x^{2}

gives

15 + 3 = 18

, so

α = 2 (- 48) = - 96

and

β = 2 (18) = 36

. Hence

α - β = - 96 - 36 = - 132

JEE Main · 2020 · 08 Jan 2020 · Shift 2 · Single correct option