Find the term independent of x in the expansion of the following expressions: \begin{array}{l} (i){(3/2{x^2} - 1/3x)^9}\\ (ii)(2x + 1/3{x^2}){}^9\\ \end{array}
Find the term independent of x in the expansion of the following expressions: \begin{array}{l} (i){(3/2{x^2} - 1/3x)^9}\\ (ii)(2x + 1/3{x^2}){}^9\\ \end{array}

Answers:

(i) 

If (r + 1)th term in the given expression is independent of x.

Tr+1 = nCr xn-r ar

RD Sharma Solutions for Class 11 Maths Chapter 18 – Binomial Theorem image - 57

For this term to be independent of x,

18 – 3r = 0

3r = 18

r = 18/3

r = 6

The term is 7th term.

T7 = T6+1

9C6 × (39-12)/(29-6)

= (9×8×7)/(3×2) × 3-3 × 2-3

= 7/18

Term independent of x is 7/18.

(ii) 

If (r + 1)th term in the given expression is independent of x.

Tr+1 = nCr xn-r ar

RD Sharma Solutions for Class 11 Maths Chapter 18 – Binomial Theorem image - 58

For this term to be independent of x,

9 – 3r = 0

3r = 9

r = 9/3

r = 3

The term is 4th term.

T4 = T3+1

9C3 × (26)/(33)

9C3 × 64/27

Term independent of x is 9C3 × 64/27.