Question

The term independent of x in the expansion of `[(x + 1)/(x^(2/3) - x^(1/3) + 1) - (x - 1)/(x - x^(1/2))]^10`, x ≠ 1 is equal to ______.

पर्याय

• 210

• 220

• 230

• 240

Answer 1

The term independent of x in the expansion of `[(x + 1)/(x^(2/3) - x^(1/3) + 1) - (x - 1)/(x - x^(1/2))]^10`, x ≠ 1 is equal to 210.

Explanation:

Given expansion :

`[(x + 1)/(x^(2/3) - x^(1/3) + 1) - (x - 1)/(x - x^(1/2))]^10`

= `[((x^(1/3) + 1)(x^(2/3) - x^(1/3) + 1))/((x^(2/3) - x^(1/3) + 1)) - ((x^(1/2) - 1)(x^(1/2 + 1)))/(x^(1/2)(x^(1/2) - 1))]^10`  ...[∵ a³ + b³ = (a + b)(a² – ab + b²)]

= `[(x^(1/3) + 1) - (1 + x^(1/2))]^10`

= `[x^(1/3) - x^(-1/2)]^10`

General term :

T_r+1 = `""^10C_r[x^(1/3)]^((10 - r))[x^(-1/2)]^r`

For independent of x :

`(10 - r)/3 - r/2` = 0

⇒ r = 4

∴ Term independent of x = ¹⁰C₄ = 210.