Question -
Answer -
(i) (3/2 x2 –1/3x)9
Given:
(3/2 x2 –1/3x)9
If (r + 1)th termin the given expression is independent of x.
Then, we have:
Tr+1 = nCr xn-r ar
For this term to beindependent of x, we must have
18 – 3r = 0
3r = 18
r = 18/3
= 6
So, the required termis 7th term.
We have,
T7 = T6+1
= 9C6 ×(39-12)/(29-6)
= (9×8×7)/(3×2) × 3-3 ×2-3
= 7/18
Hence, the termindependent of x is 7/18.
(ii) (2x + 1/3x2)9
Given:
(2x + 1/3x2)9
If (r + 1)th termin the given expression is independent of x.
Then, we have:
Tr+1 = nCr xn-r ar
For this term to beindependent of x, we must have
9 – 3r = 0
3r = 9
r = 9/3
= 3
So, the required termis 4th term.
We have,
T4 = T3+1
= 9C3 ×(26)/(33)
= 9C3 ×64/27
Hence, the termindependent of x is 9C3 × 64/27.
(iii) (2x2 –3/x3)25
Given:
(2x2 –3/x3)25
If (r + 1)th termin the given expression is independent of x.
Then, we have:
Tr+1 = nCr xn-r ar
= 25Cr (2x2)25-r (-3/x3)r
= (-1)r 25Cr ×225-r × 3r x50-2r-3r
For this term to beindependent of x, we must have
50 – 5r = 0
5r = 50
r = 50/5
= 10
So, the required termis 11th term.
We have,
T11 =T10+1
= (-1)10 25C10 ×225-10 × 310
= 25C10 (215 ×310)
Hence, the termindependent of x is 25C10 (215 ×310).
(iv) (3x – 2/x2)15
Given:
(3x – 2/x2)15
If (r + 1)th termin the given expression is independent of x.
Then, we have:
Tr+1 = nCr xn-r ar
= 15Cr (3x)15-r (-2/x2)r
= (-1)r 15Cr ×315-r × 2r x15-r-2r
For this term to beindependent of x, we must have
15 – 3r = 0
3r = 15
r = 15/3
= 5
So, the required termis 6th term.
We have,
T6 = T5+1
= (-1)5 15C5 ×315-5 × 25
= -3003 × 310 ×25
Hence, the termindependent of x is -3003 × 310 × 25.
(v) ((√x/3) + √3/2x2)10
Given:
((√x/3) + √3/2x2)10
If (r + 1)th termin the given expression is independent of x.
Then, we have:
Tr+1 = nCr xn-r ar
For this term to beindependent of x, we must have
(10-r)/2 – 2r = 0
10 – 5r = 0
5r = 10
r = 10/5
= 2
So, the required termis 3rd term.
We have,
T3 = T2+1
Hence, the termindependent of x is 5/4.
(vi) (x – 1/x2)3n
Given:
(x – 1/x2)3n
If (r + 1)th termin the given expression is independent of x.
Then, we have:
Tr+1 = nCr xn-r ar
= 3nCr x3n-r (-1/x2)r
= (-1)r 3nCr x3n-r-2r
For this term to beindependent of x, we must have
3n – 3r = 0
r = n
So, the required termis (n+1)th term.
We have,
(-1)n 3nCn
Hence, the termindependent of x is (-1)n 3nCn
(vii) (1/2 x1/3 +x-1/5)8
Given:
(1/2 x1/3 +x-1/5)8
If (r + 1)th termin the given expression is independent of x.
Then, we have:
Tr+1 = nCr xn-r ar
For this term to beindependent 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
So, the required termis 6th term.
We have,
T6 = T5+1
= 8C5 ×1/(28-5)
= (8×7×6)/(3×2×8)
= 7
Hence, the termindependent of x is 7.
(viii) (1 + x + 2x3)(3/2x2 – 3/3x)9
Given:
(1 + x + 2x3)(3/2x2 – 3/3x)9
If (r + 1)th termin the given expression is independent of x.
Then, we have:
(1 + x + 2x3)(3/2x2 – 3/3x)9 =
= 7/18 – 2/27
= (189 – 36)/486
= 153/486 (divide by9)
= 17/54
Hence, the termindependent of x is 17/54.
(ix) (∛x + 1/2∛x)18, x > 0
Given:
(∛x + 1/2∛x)18, x > 0
If (r + 1)th termin the given expression is independent of x.
Then, we have:
Tr+1 = nCr xn-r ar
For this term to beindependent 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
So, the required termis 10th term.
We have,
T10 =T9+1
= 18C9 ×1/29
Hence, the termindependent of x is 18C9 × 1/29.
(x) (3/2x2 –1/3x)6
Given:
(3/2x2 –1/3x)6
If (r + 1)th termin the given expression is independent of x.
Then, we have:
Tr+1 = nCr xn-r ar
For this term to beindependent of r, we must have
12 – 3r = 0
3r = 12
r = 12/3
= 4
So, the required termis 5th term.
We have,
T5 = T4+1
Hence, the termindependent of x is 5/12.