Question

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

(i) (x – 1/x²)³ⁿ

(ii) (1/2 x^1/3 + x^-1/5)⁸

(iii) (1 + x + 2x³) (3/2x² – 3/3x)⁹

(iv) (³√x + 1/2 ³√x)¹⁸, x > 0

(v) (3/2x² – 1/3x)⁶

Answer 1

(i) (x – 1/x²)³ⁿ

Given as

(x – 1/x²)³ⁿ

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

= ³ⁿC_r x^3n-r (-1/x²)^r

= (-1)^r 3nC_r x^3n-r-2r

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

3n – 3r = 0

r = n

Therefore, the required term is (n + 1)^th term.

We have,

(-1)^n 3nC_n

Thus, the term independent of x (-1)^n 3nC_n

(ii) (1/2 x^1/3 + x^-1/5)⁸

Given as

(1/2 x^1/3 + x^-1/5)⁸

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

(8-r)/3 – r/5 = 0

(40 – 5r – 3r)/15 = 0

40 – 5r – 3r = 0

40 – 8r = 0

8r = 40

r = 40/8

= 5

Therefore, the required term is 6^th term.

We have,

T₅ = T₅₊₁

= ⁸C₅ × 1/(2^8-5)

= (8 × 7 × 6)/(3 × 2 × 8)

= 7

Thus, the term independent of x is 7.

(iii) (1 + x + 2x³) (3/2x² – 3/3x)⁹

Given as

(1 + x + 2x³) (3/2x² – 3/3x)⁹

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

Now, we have:

(1 + x + 2x³) (3/2x² – 3/3x)⁹ =

= 7/18 – 2/27

= (189 – 36)/486

= 153/486 (divide by 9)

= 17/54

Thus, the term independent of x is 17/54.

(iv) (³√x + 1/2 ³√x)¹⁸, x > 0

Given as

(∛x + 1/2∛x)¹⁸, x > 0

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

Then, we have:

T_r+1 = ⁿC_r x^n-r a^r

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

(18 - r)/3 – r/3 = 0

(18 – r – r)/3 = 0

18 – 2r = 0

2r = 18

r = 18/2

= 9

Therefore, the required term is 10^th term.

We have,

T₁₀ = T₉₊₁

= ¹⁸C₉ × 1/2⁹

Thus, the term independent of x is ¹⁸C₉ × 1/2⁹.

(v) (3/2x² – 1/3x)⁶

Given as

(3/2x² – 1/3x)⁶

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 r, we must have

12 – 3r = 0

3r = 12

r = 12/3

= 4

Therefore, the required term is 5^th term.

We have,

T₅ = T₄₊₁

Thus, the term independent of x is 5/12.