The sum of the co-efficient of all even degree terms in $x$ in the expansion of $\left(x+\sqrt{x^{3}-1}\right)^{6}+\left(x-\sqrt{x^{3}-1}\right)^{6},(x>1)$ is equal to:

JEE Main · 2019 · 08 Apr 2019 · Shift 1 · Binomial Theorem

The sum of the co-efficient of all even degree terms in

x

in the expansion of

(x + x^{3} - 1)^{6} + (x - x^{3} - 1)^{6}, (x > 1)

is equal to:

Answer: (B)

Adding the two expansions cancels the odd powers of the surd, leaving

2 [(0 6) x^{6} + (2 6) x^{4} (x^{3} - 1) + (4 6) x^{2} (x^{3} - 1)^{2} + (6 6) (x^{3} - 1)^{3}]

, that is

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

. Collecting the terms whose power of

x

is even, the inner coefficients are

1 - 3 = - 2

for

x^{6}

- 15

for

x^{4}

15

for

x^{8}

15

for

x^{2}

and

- 1

for the constant, summing to

- 2 - 15 + 15 + 15 - 1 = 12

. Doubling gives the sum of even-degree coefficients as

24

JEE Main · 2019 · 08 Apr 2019 · Shift 1 · Single correct option