Question

Find the term independent of x in the expansion of the following expressions:

(i) (3/2 x² – 1/3x)⁹

(ii) (2x + 1/3x²)⁹

(iii) (2x² – 3/x³)²⁵

(iv) (3x – 2/x²)¹⁵

(v) ((√x/3) + √3/2x²)¹⁰

Answer 1

(i) (3/2 x² – 1/3x)⁹

Given as

(3/2 x² – 1/3x)⁹

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

Now, we have:

T_r+1 = ⁿC_r x^n-r a^r

For this term to be independent of x, we must have

18 – 3r = 0

3r = 18

r = 18/3

= 6

Therefore, the required term is 7^th term.

We have,

T₇ = T₆₊₁

= ⁹C₆ × (3^9-12)/(2^9-6)

= (9 × 8 × 7)/(3 × 2) × 3^-3 × 2^-3

= 7/18

Thus, the term independent of x is 7/18.

(ii) (2x + 1/3x²)⁹

Given as

(2x + 1/3x²)⁹

If (r + 1)^th term in the given expression is independent of x.
Now, we have:

T_r+1 = ⁿC_r x^n-r a^r

For this term to be independent of x, we must have

9 – 3r = 0

3r = 9

r = 9/3

= 3

Therefore, the required term is 4^th term.

We have,

T₄ = T₃₊₁

= ⁹C₃ × (2⁶)/(3³)

= ⁹C₃ × 64/27

Thus, the term independent of x is ⁹C₃ × 64/27.

(iii) (2x² – 3/x³)²⁵

Given as

(2x² – 3/x³)²⁵

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

Now, we have:

T_r+1 = ⁿC_r x^n-r a^r

= ²⁵C_r (2x²)^25-r (-3/x³)^r

= (-1)^r 25C_r × 2^25-r × 3^r x^50-2r-3r

For this term to be independent of x, we must have

50 – 5r = 0

5r = 50

r = 50/5

= 10

Therefore, the required term is 11^th term.

We have,

T₁₁ = T₁₀₊₁

= (-1)^10 25C₁₀ × 2^25-10 × 3¹⁰

= ²⁵C₁₀ (2¹⁵ × 3¹⁰)

Thus, the term independent of x is ²⁵C₁₀ (2¹⁵ × 3¹⁰).

(iv) (3x – 2/x²)¹⁵

Given as

(3x – 2/x²)¹⁵

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

Now, we have:

T_r+1 = ⁿC_r x^n-r a^r

= ¹⁵C_r (3x)^15-r (-2/x²)^r

= (-1)^r 15C_r × 3^15-r × 2^r x^15-r-2r

For this term to be independent of x, we must have

15 – 3r = 0

3r = 15

r = 15/3

= 5

Therefore, the required term is 6^th term.

We have,

T₆ = T₅₊₁

= (-1)^5 15C₅ × 3^15-5 × 2⁵

= -3003 × 3¹⁰ × 2⁵

Thus, the term independent of x is -3003 × 3¹⁰ × 2⁵.

(v) ((√x/3) + √3/2x²)¹⁰

Given as

((√x/3) + √3/2x²)¹⁰

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

Now, we have:

T_r+1 = ⁿC_r x^n-r a^r

For this term to be independent of x, we must have

(10-r)/2 – 2r = 0

10 – 5r = 0

5r = 10

r = 10/5

= 2

Therefore, the required term is 3^rd term.

We have,

T₃ = T₂₊₁

Thus, the term independent of x is 5/4.