Question

The sum of the co-efficients of all even degree terms in x in the expansion of `(x + sqrt(x^3 - 1))^6 + (x - sqrt(x^3 - 1))^6, (x > 1)` is equal to ______.

Options

• 26

• 24

• 32

• 29

Answer 1

The sum of the co-efficients of all even degree terms in x in the expansion of `(x + sqrt(x^3 - 1))^6 + (x - sqrt(x^3 - 1))^6, (x > 1)` is equal to 24.

Explanation:

`(x + sqrt(x^3 - 1))^6 + (x - sqrt(x^3 - 1))^6 = ""^6C_0x^6 + ""^6C_1x^5sqrt(x^3 - 1) + ""^6C_2x^4(sqrt(x^3 - 1))^2 + ... + ""^6C_6(sqrt(x^3 - 1))^6 + ""^6C_0x^6 - ""^6C_1x^5sqrt(x^3 - 1) + ""^6C_2x^4(sqrt(x^3 - 1))^2 .... + ""^6C_6(sqrt(x^3 - 1))^6`

= `2(""^6C_0 + ""^6C_2x^4(sqrt(x^3 - 1))^2 + ... + ""^6C_6(sqrt(x^3 - 1))^6)`

= `2{""^6C_0 + ""^6C_2x^4(x^3 - 1)^1 + ""^6C_4x^2(x^3 - 1)^2 + ""^6C_6(x^3 - 1)^3`

= `2{""^6C_0 + ""^6C_2x^4(x^3 - 1) + ""^6C_4x^2(x^6 - 2x^3 + 1) + ""^6C_6(x^9 - 3x^6 + 3x^3 - 1)`

Sum of even power

2(⁶C₀ – ⁶C₂ + ⁶C₄ + ⁶C₄ – ⁶C₆ × 3 – ⁶C₆) = 24