RD Sharma Solutions Class 11 Maths Chapter 18 – Free PDF Download
RD Sharma Solutions for Class 11 Maths Chapter 18 Binomial Theorem are provided here to boost in-depth knowledge of concepts among students. An algebraic expression containing two terms is called a binomial expression. The RD Sharma Class 11 Maths Solutions are developed by our experienced faculty team at BYJU’S for the purpose of clarifying students’ doubts immediately. These solutions also provide guidance to students to solve problems confidently, which in turn, helps in improving their problem-solving skills, which is very important from the examination point of view.
Chapter 18 – Binomial Theorem contains two exercises, and RD Sharma Solutions provide accurate solutions to the questions present in each exercise. Students who wish to learn the right steps for solving problems with ease can make use of RD Sharma Solutions in PDF format updated for the 2023-24 syllabus from the links given below. Now, let us have a look at the concepts discussed in this chapter.
- Binomial theorem for positive integral index.
- Some important conclusions from the binomial theorem.
- General terms and middle terms in a binomial expansion.
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EXERCISE 18.1 PAGE NO: 18.11
1. Using the binomial theorem, write down the expressions of the following:
Solution:
(i) (2x + 3y) 5
Let us solve the given expression
(2x + 3y) 5 = 5C0 (2x)5 (3y)0 + 5C1 (2x)4 (3y)1 + 5C2 (2x)3 (3y)2 + 5C3 (2x)2 (3y)3 + 5C4 (2x)1 (3y)4 + 5C5 (2x)0 (3y)5
= 32x5 + 5 (16x4) (3y) + 10 (8x3) (9y)2 + 10 (4x)2 (27y)3 + 5 (2x) (81y4) + 243y5
= 32x5 + 240x4y + 720x3y2 + 1080x2y3 + 810xy4 + 243y5
(ii) (2x – 3y) 4
Let us solve the given expression
(2x – 3y) 4 = 4C0 (2x)4 (3y)0 – 4C1 (2x)3 (3y)1 + 4C2 (2x)2 (3y)2 – 4C3 (2x)1 (3y)3 + 4C4 (2x)0 (3y)4
= 16x4 – 4 (8x3) (3y) + 6 (4x2) (9y2) – 4 (2x) (27y3) + 81y4
= 16x4 – 96x3y + 216x2y2 – 216xy3 + 81y4
(iv) (1 – 3x) 7
Let us solve the given expression
(1 – 3x) 7 = 7C0 (3x)0 – 7C1 (3x)1 + 7C2 (3x)2 – 7C3 (3x)3 + 7C4 (3x)4 – 7C5 (3x)5 + 7C6 (3x)6 – 7C7 (3x)7
= 1 – 7 (3x) + 21 (9x)2 – 35 (27x3) + 35 (81x4) – 21 (243x5) + 7 (729x6) – 2187(x7)
= 1 – 21x + 189x2 – 945x3 + 2835x4 – 5103x5 + 5103x6 – 2187x7
(viii) (1 + 2x – 3x2)5
Let us solve the given expression
Let us consider (1 + 2x) and 3x2 as two different entities and apply the binomial theorem.
(1 + 2x – 3x2)5 = 5C0 (1 + 2x)5 (3x2)0 – 5C1 (1 + 2x)4 (3x2)1 + 5C2 (1 + 2x)3 (3x2)2 – 5C3 (1 + 2x)2 (3x2)3 + 5C4 (1 + 2x)1 (3x2)4 – 5C5 (1 + 2x)0 (3x2)5
= (1 + 2x)5 – 5(1 + 2x)4 (3x2) + 10 (1 + 2x)3 (9x4) – 10 (1 + 2x)2 (27x6) + 5 (1 + 2x) (81x8) – 243x10
= 5C0 (2x)0 + 5C1 (2x)1 + 5C2 (2x)2 + 5C3 (2x)3 + 5C4 (2x)4 + 5C5 (2x)5 – 15x2 [4C0 (2x)0 + 4C1 (2x)1 + 4C2 (2x)2 + 4C3 (2x)3 + 4C4 (2x)4] + 90x4 [1 + 8x3 + 6x + 12x2] – 270x6(1 + 4x2 + 4x) + 405x8 + 810x9 – 243x10
= 1 + 10x + 40x2 + 80x3 + 80x4 + 32x5 – 15x2 – 120x3 – 3604 – 480x5 – 240x6 + 90x4 + 720x7 + 540x5 + 1080x6 – 270x6 – 1080x8 – 1080x7 + 405x8 + 810x9 – 243x10
= 1 + 10x + 25x2 – 40x3 – 190x4 + 92x5 + 570x6 – 360x7 – 675x8 + 810x9 – 243x10
(x) (1 – 2x + 3x2)3
Let us solve the given expression
2. Evaluate the following:
Solution:
Let us solve the given expression
= 2 [5C0 (2√x)0 + 5C2 (2√x)2 + 5C4 (2√x)4]
= 2 [1 + 10 (4x) + 5 (16x2)]
= 2 [1 + 40x + 80x2]
Let us solve the given expression
= 2 [6C0 (√2)6 + 6C2 (√2)4 + 6C4 (√2)2 + 6C6 (√2)0]
= 2 [8 + 15 (4) + 15 (2) + 1]
= 2 [99]
= 198
Let us solve the given expression
= 2 [5C1 (34) (√2)1 + 5C3 (32) (√2)3 + 5C5 (30) (√2)5]
= 2 [5 (81) (√2) + 10 (9) (2√2) + 4√2]
= 2√2 (405 + 180 + 4)
= 1178√2
Let us solve the given expression
= 2 [7C0 (27) (√3)0 + 7C2 (25) (√3)2 + 7C4 (23) (√3)4 + 7C6 (21) (√3)6]
= 2 [128 + 21 (32)(3) + 35(8)(9) + 7(2)(27)]
= 2 [128 + 2016 + 2520 + 378]
= 2 [5042]
= 10084
Let us solve the given expression
= 2 [5C1 (√3)4 + 5C3 (√3)2 + 5C5 (√3)0]
= 2 [5 (9) + 10 (3) + 1]
= 2 [76]
= 152
Let us solve the given expression
= (1 – 0.01)5 + (1 + 0.01)5
= 2 [5C0 (0.01)0 + 5C2 (0.01)2 + 5C4 (0.01)4]
= 2 [1 + 10 (0.0001) + 5 (0.00000001)]
= 2 [1.00100005]
= 2.0020001
Let us solve the given expression
= 2 [6C1 (√3)5 (√2)1 + 6C3 (√3)3 (√2)3 + 6C5 (√3)1 (√2)5]
= 2 [6 (9√3) (√2) + 20 (3√3) (2√2) + 6 (√3) (4√2)]
= 2 [√6 (54 + 120 + 24)]
= 396 √6
= 2 [a8 + 6a6 – 6a4 + a4 + 1 – 2a2]
= 2a8 + 12a6 – 10a4 – 4a2 + 2
3. Find (a + b) 4 – (a – b) 4. Hence, evaluate (√3 + √2)4 – (√3 – √2)4.
Solution:
Firstly, let us solve the given expression
(a + b) 4 – (a – b) 4
The above expression can be expressed as,
(a + b) 4 – (a – b) 4 = 2 [4C1 a3b1 + 4C3 a1b3]
= 2 [4a3b + 4ab3]
= 8 (a3b + ab3)
Now,
Let us evaluate the expression
(√3 + √2)4 – (√3 -√2)4
So consider, a = √3 and b = √2 we get,
(√3 + √2)4 – (√3 -√2)4 = 8 (a3b + ab3)
= 8 [(√3)3 (√2) + (√3) (√2)3]
= 8 [(3√6) + (2√6)]
= 8 (5√6)
= 40√6
4. Find (x + 1) 6 + (x – 1) 6. Hence, or otherwise, evaluate (√2 + 1)6 + (√2 – 1)6.
Solution:
Firstly, let us solve the given expression
(x + 1) 6 + (x – 1) 6
The above expression can be expressed as,
(x + 1) 6 + (x – 1) 6 = 2 [6C0 x6 + 6C2 x4 + 6C4 x2 + 6C6 x0]
= 2 [x6 + 15x4 + 15x2 + 1]
Now,
Let us evaluate the expression
(√2 + 1)6 + (√2 – 1)6
So consider x = √2 then, we get,
(√2 + 1)6 + (√2 – 1)6 = 2 [x6 + 15x4 + 15x2 + 1]
= 2 [(√2)6 + 15 (√2)4 + 15 (√2)2 + 1]
= 2 [8 + 15 (4) + 15 (2) + 1]
= 2 [8 + 60 + 30 + 1]
= 198
5. Using the binomial theorem, evaluate each of the following:
(i) (96)3
(ii) (102)5
(iii) (101)4
(iv) (98)5
Solution:
(i) (96)3
We have,
(96)3
Let us express the given expression as two different entities and apply the binomial theorem.
(96)3 = (100 – 4)3
= 3C0 (100)3 (4)0 – 3C1 (100)2 (4)1 + 3C2 (100)1 (4)2 – 3C3 (100)0 (4)3
= 1000000 – 120000 + 4800 – 64
= 884736
(ii) (102)5
We have,
(102)5
Let us express the given expression as two different entities and apply the binomial theorem.
(102)5 = (100 + 2)5
= 5C0 (100)5 (2)0 + 5C1 (100)4 (2)1 + 5C2 (100)3 (2)2 + 5C3 (100)2 (2)3 + 5C4 (100)1 (2)4 + 5C5 (100)0 (2)5
= 10000000000 + 1000000000 + 40000000 + 800000 + 8000 + 32
= 11040808032
(iii) (101)4
We have,
(101)4
Let us express the given expression as two different entities and apply the binomial theorem.
(101)4 = (100 + 1)4
= 4C0 (100)4 + 4C1 (100)3 + 4C2 (100)2 + 4C3 (100)1 + 4C4 (100)0
= 100000000 + 4000000 + 60000 + 400 + 1
= 104060401
(iv) (98)5
We have,
(98)5
Let us express the given expression as two different entities and apply the binomial theorem.
(98)5 = (100 – 2)5
= 5C0 (100)5 (2)0 – 5C1 (100)4 (2)1 + 5C2 (100)3 (2)2 – 5C3 (100)2 (2)3 + 5C4 (100)1 (2)4 – 5C5 (100)0 (2)5
= 10000000000 – 1000000000 + 40000000 – 800000 + 8000 – 32
= 9039207968
6. Using binomial theorem, prove that 23n – 7n – 1 is divisible by 49, where n ∈ N.
Solution:
Given:
23n – 7n – 1
So, 23n – 7n – 1 = 8n – 7n – 1
Now,
8n – 7n – 1
8n = 7n + 1
= (1 + 7) n
= nC0 + nC1 (7)1 + nC2 (7)2 + nC3 (7)3 + nC4 (7)2 + nC5 (7)1 + … + nCn (7) n
8n = 1 + 7n + 49 [nC2 + nC3 (71) + nC4 (72) + … + nCn (7) n-2]
8n – 1 – 7n = 49 (integer)
So now,
8n – 1 – 7n is divisible by 49
Or
23n – 1 – 7n is divisible by 49.
Hence proved.
EXERCISE 18.2 PAGE NO: 18.37
1. Find the 11th term from the beginning and the 11th term from the end in the expansion of (2x – 1/x2)25.
Solution:
Given:
(2x – 1/x2)25
The given expression contains 26 terms.
So, the 11th term from the end is the (26 − 11 + 1) th term from the beginning.
In other words, the 11th term from the end is the 16th term from the beginning.
Then,
T16 = T15+1 = 25C15 (2x)25-15 (-1/x2)15
= 25C15 (210) (x)10 (-1/x30)
= – 25C15 (210 / x20)
Now, we shall find the 11th term from the beginning.
T11 = T10+1 = 25C10 (2x)25-10 (-1/x2)10
= 25C10 (215) (x)15 (1/x20)
= 25C10 (215 / x5)
2. Find the 7th term in the expansion of (3x2 – 1/x3)10.
Solution:
Given:
(3x2 – 1/x3)10
Let us consider the 7th term as T7
So,
T7 = T6+1
= 10C6 (3x2)10-6 (-1/x3)6
= 10C6 (3)4 (x)8 (1/x18)
= [10×9×8×7×81] / [4×3×2×x10]
= 17010 / x10
∴ The 7th term of the expression (3x2 – 1/x3)10 is 17010 / x10.
3. Find the 5th term in the expansion of (3x – 1/x2)10.
Solution:
Given:
(3x – 1/x2)10
The 5th term from the end is the (11 – 5 + 1)th, is., 7th term from the beginning.
So,
T7 = T6+1
= 10C6 (3x)10-6 (-1/x2)6
= 10C6 (3)4 (x)4 (1/x12)
= [10×9×8×7×81] / [4×3×2×x8]
= 17010 / x8
∴ The 5th term of the expression (3x – 1/x2)10 is 17010 / x8.
4. Find the 8th term in the expansion of (x3/2 y1/2 – x1/2 y3/2)10.
Solution:
Given:
(x3/2 y1/2 – x1/2 y3/2)10
Let us consider the 8th term as T8
So,
T8 = T7+1
= 10C7 (x3/2 y1/2)10-7 (-x1/2 y3/2)7
= -[10×9×8]/[3×2] x9/2 y3/2 (x7/2 y21/2)
= -120 x8y12
∴ The 8th term of the expression (x3/2 y1/2 – x1/2 y3/2)10 is -120 x8y12.
5. Find the 7th term in the expansion of (4x/5 + 5/2x) 8.
Solution:
Given:
(4x/5 + 5/2x) 8
Let us consider the 7th term as T7
So,
T7 = T6+1
∴ The 7th term of the expression (4x/5 + 5/2x) 8 is 4375/x4.
6. Find the 4th term from the beginning and 4th term from the end in the expansion of (x + 2/x) 9.
Solution:
Given:
(x + 2/x) 9
Let Tr+1 be the 4th term from the end.
Then, Tr+1 is (10 − 4 + 1)th, i.e., 7th, the term from the beginning.
7. Find the 4th term from the end in the expansion of (4x/5 – 5/2x) 9.
Solution:
Given:
(4x/5 – 5/2x) 9
Let Tr+1 be the 4th term from the end of the given expression.
Then, Tr+1 is (10 − 4 + 1)th term, i.e., the 7th term, from the beginning.
T7 = T6+1
∴ The 4th term from the end is 10500/x3.
8. Find the 7th term from the end in the expansion of (2x2 – 3/2x) 8.
Solution:
Given:
(2x2 – 3/2x) 8
Let Tr+1 be the 4th term from the end of the given expression.
Then, Tr+1 is (9 − 7 + 1)th term, i.e., the 3rd term, from the beginning.
T3 = T2+1
∴ The 7th term from the end is 4032 x10.
9. Find the coefficient of:
(i) x10 in the expansion of (2x2 – 1/x)20
(ii) x7 in the expansion of (x – 1/x2)40
(iii) x-15 in the expansion of (3x2 – a/3x3)10
(iv) x9 in the expansion of (x2 – 1/3x)9
(v) xm in the expansion of (x + 1/x)n
(vi) x in the expansion of (1 – 2x3 + 3x5) (1 + 1/x)8
(vii) a5b7 in the expansion of (a – 2b)12
(viii) x in the expansion of (1 – 3x + 7x2) (1 – x)16
Solution:
(i) x10 in the expansion of (2x2 – 1/x)20
Given:
(2x2 – 1/x)20
If x10 occurs in the (r + 1)th term in the given expression.
Then, we have:
Tr+1 = nCr xn-r ar
(ii) x7 in the expansion of (x – 1/x2)40
Given:
(x – 1/x2)40
If x7 occurs at the (r + 1) th term in the given expression.
Then, we have:
Tr+1 = nCr xn-r ar
40 − 3r =7
3r = 40 – 7
3r = 33
r = 33/3
= 11
(iii) x-15 in the expansion of (3x2 – a/3x3)10
Given:
(3x2 – a/3x3)10
If x−15 occurs at the (r + 1)th term in the given expression.
Then, we have:
Tr+1 = nCr xn-r ar
(iv) x9 in the expansion of (x2 – 1/3x)9
Given:
(x2 – 1/3x)9
If x9 occurs at the (r + 1)th term in the above expression.
Then, we have:
Tr+1 = nCr xn-r ar
For this term to contain x9, we must have
18 − 3r = 9
3r = 18 – 9
3r = 9
r = 9/3
= 3
(v) xm in the expansion of (x + 1/x)n
Given:
(x + 1/x)n
If xm occurs at the (r + 1)th term in the given expression.
Then, we have:
Tr+1 = nCr xn-r ar
(vi) x in the expansion of (1 – 2x3 + 3x5) (1 + 1/x)8
Given:
(1 – 2x3 + 3x5) (1 + 1/x)8
If x occurs at the (r + 1)th term in the given expression.
Then, we have:
(1 – 2x3 + 3x5) (1 + 1/x)8 = (1 – 2x3 + 3x5) (8C0 + 8C1 (1/x) + 8C2 (1/x)2 + 8C3 (1/x)3 + 8C4 (1/x)4 + 8C5 (1/x)5 + 8C6 (1/x)6 + 8C7 (1/x)7 + 8C8 (1/x)8)
So, ‘x’ occurs in the above expression at -2x3.8C2 (1/x2) + 3x5.8C4 (1/x4)
∴ Coefficient of x = -2 (8!/(2!6!)) + 3 (8!/(4! 4!))
= -56 + 210
= 154
(vii) a5b7 in the expansion of (a – 2b)12
Given:
(a – 2b)12
If a5b7 occurs at the (r + 1)th term in the given expression.
Then, we have:
Tr+1 = nCr xn-r ar
(viii) x in the expansion of (1 – 3x + 7x2) (1 – x)16
Given:
(1 – 3x + 7x2) (1 – x)16
If x occurs at the (r + 1)th term in the given expression.
Then, we have:
(1 – 3x + 7x2) (1 – x)16 = (1 – 3x + 7x2) (16C0 + 16C1 (-x) + 16C2 (-x)2 + 16C3 (-x)3 + 16C4 (-x)4 + 16C5 (-x)5 + 16C6 (-x)6 + 16C7 (-x)7 + 16C8 (-x)8 + 16C9 (-x)9 + 16C10 (-x)10 + 16C11 (-x)11 + 16C12 (-x)12 + 16C13 (-x)13 + 16C14 (-x)14 + 16C15 (-x)15 + 16C16 (-x)16)
So, ‘x’ occurs in the above expression at 16C1 (-x) – 3x16C0
∴ Coefficient of x = -(16!/(1! 15!)) – 3(16!/(0! 16!))
= -16 – 3
= -19
10. Which term in the expansion of 
contains x and y to one and the same power?
Solution:
Let us consider Tr+1 th term in the given expansion contains x and y to one and the same power.
Then we have,
Tr+1 = nCr xn-r ar
11. Does the expansion of (2x2 – 1/x) contain any term involving x9?
Solution:
Given:
(2x2 – 1/x)
If x9 occurs at the (r + 1)th term in the given expression.
Then, we have:
Tr+1 = nCr xn-r ar
For this term to contain x9, we must have
40 – 3r = 9
3r = 40 – 9
3r = 31
r = 31/3
It is not possible since r is not an integer.
Hence, there is no term with x9 in the given expansion.
12. Show that the expansion of (x2 + 1/x)12 does not contain any term involving x-1.
Solution:
Given:
(x2 + 1/x)12
If x-1 occurs at the (r + 1)th term in the given expression.
Then, we have
Tr+1 = nCr xn-r ar
For this term to contain x-1, we must have
24 – 3r = -1
3r = 24 + 1
3r = 25
r = 25/3
It is not possible since r is not an integer.
Hence, there is no term with x-1 in the given expansion.
13. Find the middle term in the expansion of:
(i) (2/3x – 3/2x)20
(ii) (a/x + bx)12
(iii) (x2 – 2/x)10
(iv) (x/a – a/x)10
Solution:
(i) (2/3x – 3/2x)20
We have,
(2/3x – 3/2x)20 where n = 20 (even number)
So the middle term is (n/2 + 1) = (20/2 + 1) = (10 + 1) = 11. ie., the 11th term
Now,
T11 = T10+1
= 20C10 (2/3x)20-10 (3/2x)10
= 20C10 210/310 × 310/210 x10-10
= 20C10
Hence, the middle term is 20C10.
(ii) (a/x + bx)12
We have,
(a/x + bx)12 where n = 12 (even number)
So the middle term is (n/2 + 1) = (12/2 + 1) = (6 + 1) = 7. ie., 7th term
Now,
T7 = T6+1
= 924 a6b6
Hence, the middle term is 924 a6b6.
(iii) (x2 – 2/x)10
We have,
(x2 – 2/x)10 where n = 10 (even number)
So the middle term is (n/2 + 1) = (10/2 + 1) = (5 + 1) = 6, i.e., the 6th term
Now,
T6 = T5+1
Hence, the middle term is -8064x5.
(iv) (x/a – a/x)10
We have,
(x/a – a/x) 10 where n = 10 (even number)
So the middle term is (n/2 + 1) = (10/2 + 1) = (5 + 1) = 6, i.e., the 6th term
Now,
T6 = T5+1
Hence, the middle term is -252.
14. Find the middle terms in the expansion of:
(i) (3x – x3/6)9
(ii) (2x2 – 1/x)7
(iii) (3x – 2/x2)15
(iv) (x4 – 1/x3)11
Solution:
(i) (3x – x3/6)9
We have,
(3x – x3/6)9 where n = 9 (odd number)
So the middle terms are ((n+1)/2) = ((9+1)/2) = 10/2 = 5 and
((n+1)/2 + 1) = ((9+1)/2 + 1) = (10/2 + 1) = (5 + 1) = 6
The terms are 5th and 6th.
Now,
T5 = T4+1
Hence, the middle terms are 189/8 x17 and -21/16 x19.
(ii) (2x2 – 1/x)7
We have,
(2x2 – 1/x)7 where n = 7 (odd number)
So the middle terms are ((n+1)/2) = ((7+1)/2) = 8/2 = 4 and
((n+1)/2 + 1) = ((7+1)/2 + 1) = (8/2 + 1) = (4 + 1) = 5
The terms are 4th and 5th.
Now,
Hence, the middle terms are -560x5 and 280x2.
(iii) (3x – 2/x2)15
We have,
(3x – 2/x2)15 where n = 15 (odd number)
So the middle terms are ((n+1)/2) = ((15+1)/2) = 16/2 = 8 and
((n+1)/2 + 1) = ((15+1)/2 + 1) = (16/2 + 1) = (8 + 1) = 9
The terms are 8th and 9th.
Now,
Hence, the middle terms are (-6435×38×27)/x6 and (6435×37×28)/x9.
(iv) (x4 – 1/x3)11
We have,
(x4 – 1/x3)11
Where n = 11 (odd number)
So the middle terms are ((n+1)/2) = ((11+1)/2) = 12/2 = 6 and
((n+1)/2 + 1) = ((11+1)/2 + 1) = (12/2 + 1) = (6 + 1) = 7
The terms are 6th and 7th.
Now,
T7 = T6+1
Hence, the middle terms are -462x9 and 462x2.
15. Find the middle terms in the expansion of:
(i) (x – 1/x)10
(ii) (1 – 2x + x2)n
(iii) (1 + 3x + 3x2 + x3)2n
(iv) (2x – x2/4)9
(v) (x – 1/x)2n+1
(vi) (x/3 + 9y)10
(vii) (3 – x3/6)7
(viii) (2ax – b/x2)12
(ix) (p/x + x/p)9
(x) (x/a – a/x)10
Solution:
(i) (x – 1/x)10
We have,
(x – 1/x)10 where n = 10 (even number)
So the middle term is (n/2 + 1) = (10/2 + 1) = (5 + 1) = 6 i.e., the 6th term
Now,
T6 = T5+1
Hence, the middle term is -252.
(ii) (1 – 2x + x2)n
We have,
(1 – 2x + x2)n = (1 – x)2n where n is an even number.
So the middle term is (2n/2 + 1) = (n + 1)th term.
Now,
Tn = Tn+1
= 2nCn (-1)n (x)n
= (2n)!/(n!)2 (-1)n xn
Hence, the middle term is (2n)!/(n!)2 (-1)n xn.
(iii) (1 + 3x + 3x2 + x3)2n
We have,
(1 + 3x + 3x2 + x3)2n = (1 + x)6n where n is an even number.
So the middle term is (n/2 + 1) = (6n/2 + 1) = (3n + 1)th term.
Now,
T2n = T3n+1
= 6nC3n x3n
= (6n)!/(3n!)2 x3n
Hence, the middle term is (6n)!/(3n!)2 x3n.
(iv) (2x – x2/4)9
We have,
(2x – x2/4)9 where n = 9 (odd number)
So the middle terms are ((n+1)/2) = ((9+1)/2) = 10/2 = 5 and
((n+1)/2 + 1) = ((9+1)/2 + 1) = (10/2 + 1) = (5 + 1) = 6
The terms are 5th and 6th.
Now,
T5 = T4+1
And,
T6 = T5+1
Hence, the middle term is 63/4 x13 and -63/32 x14.
(v) (x – 1/x)2n+1
We have,
(x – 1/x)2n+1 where n = (2n + 1) is an (odd number)
So the middle terms are ((n+1)/2) = ((2n+1+1)/2) = (2n+2)/2 = (n + 1) and
((n+1)/2 + 1) = ((2n+1+1)/2 + 1) = ((2n+2)/2 + 1) = (n + 1 + 1) = (n + 2)
The terms are (n + 1)th and (n + 2)th.
Now,
Tn = Tn+1
And,
Tn+2 = Tn+1+1
Hence, the middle term is (-1)n.2n+1Cn x and (-1)n+1.2n+1Cn (1/x).
(vi) (x/3 + 9y)10
We have,
(x/3 + 9y)10 where n = 10 is an even number.
So the middle term is (n/2 + 1) = (10/2 + 1) = (5 + 1) = 6 i.e., the 6th term.
Now,
T6 = T5+1
Hence, the middle term is 61236x5y5.
(vii) (3 – x3/6)7
We have,
(3 – x3/6)7 where n = 7 (odd number).
So the middle terms are ((n+1)/2) = ((7+1)/2) = 8/2 = 4 and
((n+1)/2 + 1) = ((7+1)/2 + 1) = (8/2 + 1) = (4 + 1) = 5
The terms are 4th and 5th.
Now,
T4 = T3+1
= 7C3 (3)7-3 (-x3/6)3
= -105/8 x9
And,
T5 = T4+1
= 9C4 (3)9-4 (-x3/6)4
Hence, the middle terms are -105/8 x9 and 35/48 x12.
(viii) (2ax – b/x2)12
We have,
(2ax – b/x2)12 where n = 12 is an even number.
So the middle term is (n/2 + 1) = (12/2 + 1) = (6 + 1) = 7, i.e., the 7th term.
Now,
T7 = T6+1
Hence, the middle term is (59136a6b6)/x6.
(ix) (p/x + x/p)9
We have,
(p/x + x/p)9 where n = 9 (odd number).
So the middle terms are ((n+1)/2) = ((9+1)/2) = 10/2 = 5 and
((n+1)/2 + 1) = ((9+1)/2 + 1) = (10/2 + 1) = (5 + 1) = 6
The terms are 5th and 6th.
Now,
T5 = T4+1
And,
T6 = T5+1
= 9C5 (p/x)9-5 (x/p)5
Hence, the middle terms are 126p/x and 126x/p.
(x) (x/a – a/x)10
We have,
(x/a – a/x) 10 where n = 10 (even number)
So the middle term is (n/2 + 1) = (10/2 + 1) = (5 + 1) = 6, i.e., the 6th term
Now,
T6 = T5+1
Hence, the middle term is -252.
16. 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
(vi) (x – 1/x2)3n
(vii) (1/2 x1/3 + x-1/5)8
(viii) (1 + x + 2x3) (3/2x2 – 3/3x)9
(ix) (∛x + 1/2∛x)18, x > 0
(x) (3/2x2 – 1/3x)6
Solution:
(i) (3/2 x2 – 1/3x)9
Given:
(3/2 x2 – 1/3x)9
If (r + 1)th term in the given expression is independent of x.
Then, we have
Tr+1 = nCr xn-r ar
For this term to be independent of x, we must have
18 – 3r = 0
3r = 18
r = 18/3
= 6
So, the required term is the 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 term independent of x is 7/18.
(ii) (2x + 1/3x2)9
Given:
(2x + 1/3x2)9
If (r + 1)th term in the given expression is independent of x.
Then, we have
Tr+1 = nCr xn-r ar
For this term to be independent of x, we must have
9 – 3r = 0
3r = 9
r = 9/3
= 3
So, the required term is the 4th term.
We have,
T4 = T3+1
= 9C3 × (26)/(33)
= 9C3 × 64/27
Hence, the term independent of x is 9C3 × 64/27.
(iii) (2x2 – 3/x3)25
Given:
(2x2 – 3/x3)25
If (r + 1)th term in 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 be independent of x, we must have
50 – 5r = 0
5r = 50
r = 50/5
= 10
So, the required term is the 11th term.
We have,
T11 = T10+1
= (-1)10 25C10 × 225-10 × 310
= 25C10 (215 × 310)
Hence, the term independent of x is 25C10 (215 × 310).
(iv) (3x – 2/x2)15
Given:
(3x – 2/x2)15
If (r + 1)th term in 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 be independent of x, we must have
15 – 3r = 0
3r = 15
r = 15/3
= 5
So, the required term is the 6th term.
We have,
T6 = T5+1
= (-1)5 15C5 × 315-5 × 25
= -3003 × 310 × 25
Hence, the term independent of x is -3003 × 310 × 25.
(v) ((√x/3) + √3/2x2)10
Given:
((√x/3) + √3/2x2)10
If (r + 1)th term in the given expression is independent of x.
Then, we have:
Tr+1 = nCr xn-r ar
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
So, the required term is the 3rd term.
We have,
T3 = T2+1
Hence, the term independent of x is 5/4.
(vi) (x – 1/x2)3n
Given:
(x – 1/x2)3n
If (r + 1)th term in 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 be independent of x, we must have
3n – 3r = 0
r = n
So, the required term is (n+1)th term.
We have,
(-1)n 3nCn
Hence, the term independent 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 term in the given expression is independent of x.
Then, we have:
Tr+1 = nCr xn-r ar
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
So, the required term is the 6th term.
We have,
T6 = T5+1
= 8C5 × 1/(28-5)
= (8×7×6)/(3×2×8)
= 7
Hence, the term independent 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 term in 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 by 9)
= 17/54
Hence, the term independent 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 term in the given expression is independent of x.
Then, we have
Tr+1 = nCr xn-r ar
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
So, the required term is the 10th term.
We have,
T10 = T9+1
= 18C9 × 1/29
Hence, the term independent of x is 18C9 × 1/29.
(x) (3/2x2 – 1/3x)6
Given:
(3/2x2 – 1/3x)6
If (r + 1)th term in the given expression is independent of x.
Then, we have:
Tr+1 = nCr xn-r ar
For this term to be independent of r, we must have
12 – 3r = 0
3r = 12
r = 12/3
= 4
So, the required term is the 5th term.
We have,
T5 = T4+1
Hence, the term independent of x is 5/12.
17. If the coefficients of (2r + 4)th and (r – 2)th terms in the expansion of (1 + x)18 are equal, find r.
Solution:
Given:
(1 + x)18
We know the coefficient of the r term in the expansion of (1 + x)n is nCr-1
So, the coefficients of the (2r + 4) and (r – 2) terms in the given expansion are 18C2r+4-1 and 18Cr-2-1
For these coefficients to be equal, we must have
18C2r+4-1 = 18Cr-2-1
18C2r+3 = 18Cr-3
2r + 3 = r – 3 (or) 2r + 3 + r – 3 = 18 [Since, nCr = nCs => r = s (or) r + s = n]
2r – r = -3 – 3 (or) 3r = 18 – 3 + 3
r = -6 (or) 3r = 18
r = -6 (or) r = 18/3
r = -6 (or) r = 6
∴ r = 6 [since r should be a positive integer.]
18. If the coefficients of (2r + 1)th term and (r + 2)th term in the expansion of (1 + x)43 are equal, find r.
Solution:
Given:
(1 + x)43
We know the coefficient of the r term in the expansion of (1 + x)n is nCr-1
So, the coefficients of the (2r + 1) and (r + 2) terms in the given expansion are 43C2r+1-1 and 43Cr+2-1
For these coefficients to be equal, we must have
43C2r+1-1 = 43Cr+2-1
43C2r = 43Cr+1
2r = r + 1 (or) 2r + r + 1 = 43 [Since, nCr = nCs => r = s (or) r + s = n]
2r – r = 1 (or) 3r + 1 = 43
r = 1 (or) 3r = 43 – 1
r = 1 (or) 3r = 42
r = 1 (or) r = 42/3
r = 1 (or) r = 14
∴ r = 14 [since value ‘1’ gives the same term]
19. Prove that the coefficient of (r + 1)th term in the expansion of (1 + x)n + 1 is equal to the sum of the coefficients of rth and (r + 1)th terms in the expansion of (1 + x)n.
Solution:
We know, the coefficients of (r + 1)th term in (1 + x)n+1 is n+1Cr
So, the sum of the coefficients of the rth and (r + 1)th terms in (1 + x)n is
(1 + x)n = nCr-1 + nCr
= n+1Cr [since, nCr+1 + nCr = n+1Cr+1]
Hence proved.
Also, Access Exercises of RD Sharma Solutions for Class 11 Maths Chapter 18 – Binomial Theorem
Frequently Asked Questions on RD Sharma Solutions for Class 11 Maths Chapter 18
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Name the concepts covered in Chapter 18 of RD Sharma Solutions for Class 11 Maths.
The concepts covered in Chapter 18 of RD Sharma Solutions for Class 11 Maths are as listed:
- Binomial theorem for positive integral index.
- Some important conclusions from the binomial theorem.
- General terms and middle terms in a binomial expansion.
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