# Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board chapter 4 - Methods of Induction and Binomial Theorem [Latest edition]

## Chapter 4: Methods of Induction and Binomial Theorem

Exercise 4.1Exercise 4.2Exercise 4.3Exercise 4.4Exercise 4.5Miscellaneous Exercise 4
Exercise 4.1 [Pages 73 - 74]

### Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board Chapter 4 Methods of Induction and Binomial Theorem Exercise 4.1 [Pages 73 - 74]

Exercise 4.1 | Q 1 | Page 73

Prove by method of induction, for all n ∈ N:

2 + 4 + 6 + ..... + 2n = n (n+1)

Exercise 4.1 | Q 2 | Page 73

Prove by method of induction, for all n ∈ N:

3 + 7 + 11 + ..... + to n terms = n(2n+1)

Exercise 4.1 | Q 3 | Page 73

Prove by method of induction, for all n ∈ N:

12 + 22 + 32 + .... + n2 = ("n"("n" + 1)(2"n" + 1))/6

Exercise 4.1 | Q 4 | Page 73

Prove by method of induction, for all n ∈ N:

12 + 32 + 52 + .... + (2n − 1)2 = "n"/3 (2"n" − 1)(2"n" + 1)

Exercise 4.1 | Q 5 | Page 73

Prove by method of induction, for all n ∈ N:

13 + 33 + 53 + .... to n terms = n2(2n2 − 1)

Exercise 4.1 | Q 6 | Page 73

Prove by method of induction, for all n ∈ N:

1.2 + 2.3 + 3.4 + ..... + n(n + 1) = "n"/3 ("n" + 1)("n" + 2)

Exercise 4.1 | Q 7 | Page 73

Prove by method of induction, for all n ∈ N:

1.3 + 3.5 + 5.7 + ..... to n terms = "n"/3(4"n"^2 + 6"n" - 1)

Exercise 4.1 | Q 8 | Page 73

Prove by method of induction, for all n ∈ N:

1/(1.3) + 1/(3.5) + 1/(5.7) + ... + 1/((2"n" - 1)(2"n" + 1)) = "n"/(2"n" + 1)

Exercise 4.1 | Q 9 | Page 74

Prove by method of induction, for all n ∈ N:

1/(3.5) + 1/(5.7) + 1/(7.9) + ... to n terms = "n"/(3(2"n" + 3))

Exercise 4.1 | Q 10 | Page 74

Prove by method of induction, for all n ∈ N:

(23n − 1) is divisible by 7

Exercise 4.1 | Q 11 | Page 74

Prove by method of induction, for all n ∈ N:

(24n−1) is divisible by 15

Exercise 4.1 | Q 12 | Page 74

Prove by method of induction, for all n ∈ N:

3n − 2n − 1 is divisible by 4

Exercise 4.1 | Q 13 | Page 74

Prove by method of induction, for all n ∈ N:

5 + 52 + 53 + .... + 5n = 5/4(5^"n" - 1)

Exercise 4.1 | Q 14 | Page 74

Prove by method of induction, for all n ∈ N:

(cos θ + i sin θ)n = cos (nθ) + i sin (nθ)

Exercise 4.1 | Q 15 | Page 74

Prove by method of induction, for all n ∈ N:

Given that tn+1 = 5tn + 4, t1 = 4, prove that tn = 5n − 1

Exercise 4.1 | Q 16 | Page 74

Prove by method of induction, for all n ∈ N:

[(1, 2),(0, 1)]^"n" = [(1, 2"n"),(0, 1)] ∀ n ∈ N

Exercise 4.2 [Page 77]

### Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board Chapter 4 Methods of Induction and Binomial Theorem Exercise 4.2 [Page 77]

Exercise 4.2 | Q 1. (i) | Page 77

Expand: (sqrt(3) + sqrt(2))^4

Exercise 4.2 | Q 1. (ii) | Page 77

Expand: (sqrt(5) - sqrt(2))^5

Exercise 4.2 | Q 2. (i) | Page 77

Expand: (2x2 + 3)4

Exercise 4.2 | Q 2. (ii) | Page 77

Expand: (2x - 1/x)^6

Exercise 4.2 | Q 3. (i) | Page 77

Find the value of (sqrt(3) + 1)^4- (sqrt(3) - 1)^4

Exercise 4.2 | Q 3. (ii) | Page 77

Find the value of (2 + sqrt(5))^5 + (2 - sqrt(5))^5

Exercise 4.2 | Q 4. (i) | Page 77

Prove that (sqrt(3) + sqrt(2))^6 + (sqrt(3) - sqrt(2))^6 = 970

Exercise 4.2 | Q 4. (ii) | Page 77

Prove that (sqrt(5) + 1)^5 - (sqrt(5) - 1)^5 = 352

Exercise 4.2 | Q 5. (i) | Page 77

Using binomial theorem, find the value of (102)4

Exercise 4.2 | Q 5. (ii) | Page 77

Using binomial theorem, find the value of (1.1)5

Exercise 4.2 | Q 6. (i) | Page 77

Using binomial theorem, find the value of (9.9)3

Exercise 4.2 | Q 6. (ii) | Page 77

Using binomial theorem, find the value of (0.9)4

Exercise 4.2 | Q 7. (i) | Page 77

Without expanding, find the value of (x + 1)4 − 4(x + 1)3 (x − 1) + 6 (x + 1)2 (x − 1)2 − 4(x + 1) (x − 1)3 + (x − 1)4

Exercise 4.2 | Q 7. (ii) | Page 77

Without expanding, find the value of (2x − 1)4 + 4(2x − 1)3 (3 − 2x) + 6(2x − 1)2 (3 − 2x)2 + 4(2x − 1)1 (3 − 2x)3 + (3 − 2x)4

Exercise 4.2 | Q 8 | Page 77

Find the value of (1.02)6, correct upto four places of decimal

Exercise 4.2 | Q 9 | Page 77

Find the value of (1.01)5, correct up to three places of decimals.

Exercise 4.2 | Q 10 | Page 77

Find the value of (0.9)6, correct upto four places of decimal

Exercise 4.3 [Page 80]

### Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board Chapter 4 Methods of Induction and Binomial Theorem Exercise 4.3 [Page 80]

Exercise 4.3 | Q 1. (i) | Page 80

In the following expansion, find the indicated term.

(2x^2 + 3/(2x))^8, 3rd term

Exercise 4.3 | Q 1. (ii) | Page 80

In the following expansion, find the indicated term.

(x^2 - 4/(x^3))^11, 5th term

Exercise 4.3 | Q 1. (iii) | Page 80

In the following expansion, find the indicated term.

((4x)/5 - 5/(2x))^9, 7th term

Exercise 4.3 | Q 1. (iv) | Page 80

In the following expansion, find the indicated term.

(1/3 + "a"^2)^12, 9th term

Exercise 4.3 | Q 1. (v) | Page 80

In the following expansion, find the indicated term.

(3"a" + 4/"a")^13, 10th term

Exercise 4.3 | Q 2. (i) | Page 80

In the following expansion, find the indicated coefficient.

x3 in (x^2 + (3sqrt(2))/x)^9

Exercise 4.3 | Q 2. (ii) | Page 80

In the following expansion, find the indicated coefficient.

x8 in (2x^5 - 5/x^3)^8

Exercise 4.3 | Q 2. (iii) | Page 80

In the following expansion, find the indicated coefficient.

x9 in (1/x + x^2)^18

Exercise 4.3 | Q 2. (iv) | Page 80

In the following expansion, find the indicated coefficient.

x–3 in (x - 1/(2x))^5

Exercise 4.3 | Q 2. (v) | Page 80

In the following expansion, find the indicated coefficient.

x–20 in (x^3 - 1/(2x^2))^15

Exercise 4.3 | Q 3. (i) | Page 80

Find the constant term (term independent of x) in the expansion of (2x + 1/(3x^2))^9

Exercise 4.3 | Q 3. (ii) | Page 80

Find the constant term (term independent of x) in the expansion of (x - 2/x^2)^15

Exercise 4.3 | Q 3. (iii) | Page 80

Find the constant term (term independent of x) in the expansion of (sqrt(x) - 3/x^2)^10

Exercise 4.3 | Q 3. (iv) | Page 80

Find the constant term (term independent of x) in the expansion of (x^2 - 1/x)^9

Exercise 4.3 | Q 3. (v) | Page 80

Find the constant term (term independent of x) in the expansion of (2x^2 - 5/x)^9

Exercise 4.3 | Q 4. (i) | Page 80

Find the middle term in the expansion of (x/y + y/x)^12

Exercise 4.3 | Q 4. (ii) | Page 80

Find the middle terms in the expansion of (x^2 + 1/x)^7

Exercise 4.3 | Q 4. (iii) | Page 80

Find the middle term in the expansion of (x^2 - 2/x)^8

Exercise 4.3 | Q 4. (iv) | Page 80

Find the middle term in the expansion of (x/"a" - "a"/x)^10

Exercise 4.3 | Q 4. (v) | Page 80

Find the middle terms in the expansion of (x^4 - 1/x^3)^11

Exercise 4.3 | Q 5 | Page 80

In the expansion of (k + x)8, the coefficient of x5 is 10 times the coefficient of x6. Find the value of k.

Exercise 4.3 | Q 6 | Page 80

Find the term containing x6 in the expansion of (2 − x) (3x + 1)9

Exercise 4.3 | Q 7 | Page 80

The coefficient of x2 in the expansion of (1 + 2x)m is 112. Find m

Exercise 4.4 [Page 82]

### Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board Chapter 4 Methods of Induction and Binomial Theorem Exercise 4.4 [Page 82]

Exercise 4.4 | Q 1. (i) | Page 82

State, by writing first four terms, the expansion of the following, where |x| < 1

(1 + x)−4

Exercise 4.4 | Q 1. (ii) | Page 82

State, by writing first four terms, the expansion of the following, where |x| < 1

(1 - x)^(1/3)

Exercise 4.4 | Q 1. (iii) | Page 82

State, by writing first four terms, the expansion of the following, where |x| < 1

(1 – x2)–3

Exercise 4.4 | Q 1. (iv) | Page 82

State, by writing first four terms, the expansion of the following, where |x| < 1

(1 + x)^(-1/5)

Exercise 4.4 | Q 1. (v) | Page 82

State, by writing first four terms, the expansion of the following, where |x| < 1

(1 + x2)–1

Exercise 4.4 | Q 2. (i) | Page 82

State, by writing first four terms, the expansion of the following, where |b| < |a|

(a − b)−3

Exercise 4.4 | Q 2. (ii) | Page 82

State, by writing first four terms, the expansion of the following, where |b| < |a|

(a + b)−4

Exercise 4.4 | Q 2. (iii) | Page 82

State, by writing first four terms, the expansion of the following, where |b| < |a|

("a" + "b")^(1/4)

Exercise 4.4 | Q 2. (iv) | Page 82

State, by writing first four terms, the expansion of the following, where |b| < |a|

("a" - "b")^(-1/4)

Exercise 4.4 | Q 2. (v) | Page 82

State, by writing first four terms, the expansion of the following, where |b| < |a|

("a" + "b")^(-1/3)

Exercise 4.4 | Q 3. (i) | Page 82

Simplify first three terms in the expansion of the following

(1 + 2x)–4

Exercise 4.4 | Q 3. (ii) | Page 82

Simplify first three terms in the expansion of the following

(1 + 3x)^(-1/2)

Exercise 4.4 | Q 3. (iii) | Page 82

Simplify first three terms in the expansion of the following

(2 - 3x)^(1/3)

Exercise 4.4 | Q 3. (iv) | Page 82

Simplify first three terms in the expansion of the following

(5 + 4x)^(-1/2)

Exercise 4.4 | Q 3. (v) | Page 82

Simplify first three terms in the expansion of the following

(5 - 3x)^(-1/3)

Exercise 4.4 | Q 4. (i) | Page 82

Use binomial theorem to evaluate the following upto four places of decimal

sqrt(99)

Exercise 4.4 | Q 4. (ii) | Page 82

Use binomial theorem to evaluate the following upto four places of decimal

root(3)(126)

Exercise 4.4 | Q 4. (iii) | Page 82

Use binomial theorem to evaluate the following upto four places of decimal

root(4)(16.08)

Exercise 4.4 | Q 4. (iv) | Page 82

Use binomial theorem to evaluate the following upto four places of decimal

(1.02)–5

Exercise 4.4 | Q 4. (v) | Page 82

Use binomial theorem to evaluate the following upto four places of decimal

(0.98)–3

Exercise 4.5 [Page 84]

### Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board Chapter 4 Methods of Induction and Binomial Theorem Exercise 4.5 [Page 84]

Exercise 4.5 | Q 1 | Page 84

Show That C0 + C1 + C2 + .... C8 = 256

Exercise 4.5 | Q 2 | Page 84

Show That C0 + C1 + C2 + .... C9 = 512

Exercise 4.5 | Q 3 | Page 84

Show That C1 + C2 + C3 + .... C7 = 127

Exercise 4.5 | Q 4 | Page 84

Show That C1 + C2 + C3 + .... C6 = 63

Exercise 4.5 | Q 5 | Page 84

Show That C0 + C2 + C4 + C6 + C8 = C1 + C3 + C5 + C7 = 128

Exercise 4.5 | Q 6 | Page 84

Show That C1 + C2 + C3 + .... Cn = 2n − 1

Exercise 4.5 | Q 7 | Page 84

Show That C0 + 2C1 + 3C2 + 4C3 + ... + (n + 1)Cn = (n + 2)2n−1

Miscellaneous Exercise 4 [Page 85]

### Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board Chapter 4 Methods of Induction and Binomial Theorem Miscellaneous Exercise 4 [Page 85]

Miscellaneous Exercise 4 | Q I. (1) | Page 85

Select the correct answer from the given alternatives.

The total number of terms in the expression of (x + y)100 + (x − y)100 after simplification is:

• 50

• 51

• 100

• 202

Miscellaneous Exercise 4 | Q I. (2) | Page 85

Select the correct answer from the given alternatives.

The middle term in the expansion of (1 + x)2n will be :

• (n − 1)th

• nth

• (n + 1)th

• (n + 2)th

Miscellaneous Exercise 4 | Q I. (3) | Page 85

Select the correct answer from the given alternatives.

In the expansion of (x2 − 2x)10, the coefficient of x16 is

• −1680

• 1680

• 3360

• 6720

Miscellaneous Exercise 4 | Q I. (4) | Page 85

Select the correct answer from the given alternatives.

The term not containing x in expansion of (1 - x)^2 (x + 1/x)^10 is

• 11C

• 10C5

• 10C

• 10C

Miscellaneous Exercise 4 | Q I. (5) | Page 85

Select the correct answer from the given alternatives.

The number of terms in expansion of (4y + x)8 − (4y − x)8

• 4

• 5

• 8

• 9

Miscellaneous Exercise 4 | Q I. (6) | Page 85

Select the correct answer from the given alternatives.

The value 14C1 + 14C3 + 14C5 + ..... + 14C11 is

• 214 − 1

• 214 − 14

• 212

• 213 − 14

Miscellaneous Exercise 4 | Q I. (7) | Page 85

Select the correct answer from the given alternatives.

The value 11C2 + 11C4 + 11C6 + 11C8 is equal to

• 210 − 1

• 210 − 11

• 210 + 12

• 210 − 12

Miscellaneous Exercise 4 | Q I. (8) | Page 85

Select the correct answer from the given alternatives.

In the expansion of (3x + 2)4, the coefficient of the middle term is

• 36

• 54

• 81

• 216

Miscellaneous Exercise 4 | Q I. (9) | Page 85

Select the correct answer from the given alternatives.

The coefficient of the 8th term in the expansion of (1 + x)10 is:

• 7

• 120

• 10C8

• 210

Miscellaneous Exercise 4 | Q I. (10) | Page 85

Select the correct answer from the given alternatives.

If the coefficient of x2 and x3 in the expansion of (3 + ax)9 are the same, then the value of a is

• -7/9

• -9/7

• 7/9

• 9/7

Miscellaneous Exercise 4 [Pages 85 - 86]

### Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board Chapter 4 Methods of Induction and Binomial Theorem Miscellaneous Exercise 4 [Pages 85 - 86]

Miscellaneous Exercise 4 | Q II. (1) (i) | Page 85

Prove, by method of induction, for all n ∈ N

8 + 17 + 26 + … + (9n – 1) = "n"/2(9"n" + 7)

Miscellaneous Exercise 4 | Q II. (1) (ii) | Page 85

Prove, by method of induction, for all n ∈ N

12 + 42 + 72 + ... + (3n − 2)2 = "n"/2 (6"n"^2 - 3"n" - 1)

Miscellaneous Exercise 4 | Q II. (1) (iii) | Page 85

Prove, by method of induction, for all n ∈ N

2 + 3.2 + 4.22 + ... + (n + 1)2n–1 = n.2n

Miscellaneous Exercise 4 | Q II. (1) (iv) | Page 85

Prove, by method of induction, for all n ∈ N

1/(3.4.5) + 2/(4.5.6) + 3/(5.6.7) + ... + "n"/(("n" + 2)("n" + 3)("n" + 4)) = ("n"("n" + 1))/(6("n" + 3)("n" + 4))

Miscellaneous Exercise 4 | Q II. (2) | Page 85

Given that tn+1 = 5tn − 8, t1 = 3, prove by method of induction that tn = 5n−1 + 2

Miscellaneous Exercise 4 | Q II. (3) | Page 85

Prove by method of induction

[(3, -4),(1, -1)]^"n" = [(2"n" + 1, -4"n"),("n", -2"n" + 1)], ∀  "n" ∈ "N"

Miscellaneous Exercise 4 | Q II. (4) | Page 85

Expand (3x2 + 2y)5

Miscellaneous Exercise 4 | Q II. (5) | Page 85

Expand ((2x)/3 - 3/(2x))^4

Miscellaneous Exercise 4 | Q II. (6) | Page 85

Find third term in the expansion of (9x^2 - y^3/6)^4

Miscellaneous Exercise 4 | Q II. (7) | Page 85

Find tenth term in the expansion of (2x^2 + 1/x)^12

Miscellaneous Exercise 4 | Q II. (8) (i) | Page 85

Find the middle term (s) in the expansion of ((2"a")/3 - 3/(2"a"))^6

Miscellaneous Exercise 4 | Q II. (8) (ii) | Page 85

Find the middle term (s) in the expansion of (x - 1/(2y))^10

Miscellaneous Exercise 4 | Q II. (8) (iii) | Page 85

Find the middle term (s) in the expansion of (x2+ 2y2)

Miscellaneous Exercise 4 | Q II. (8) (iv) | Page 85

Find the middle term (s) in the expansion of ((3x^2)/2 - 1/(3x))^9

Miscellaneous Exercise 4 | Q II. (9) (i) | Page 86

Find the coefficients of x6 in the expansion of (3x^2 - 1/(3x))^9.

Miscellaneous Exercise 4 | Q II. (9) (ii) | Page 86

Find the coefficients of x60 in the expansion of (1/x^2 + x^4)^18

Miscellaneous Exercise 4 | Q II. (10) (i) | Page 86

Find the constant term in the expansion of ((4x^2)/3 + 3/(2x))^9

Miscellaneous Exercise 4 | Q II. (10) (ii) | Page 86

Find the constant term in the expansion of (2x^2 - 1/x)^12

Miscellaneous Exercise 4 | Q II. (11) (i) | Page 86

Prove by method of induction loga xn = n logax, x > 0, n ∈ N

Miscellaneous Exercise 4 | Q II. (11) (ii) | Page 86

Prove by method of induction 152n–1 + 1 is divisible by 16, for all n ∈ N.

Miscellaneous Exercise 4 | Q II. (11) (iii) | Page 86

Prove by method of induction 52n − 22n is divisible by 3, for all n ∈ N

Miscellaneous Exercise 4 | Q II. (12) | Page 86

If the coefficient of x16 in the expansion of (x2 + ax)10 is 3360, find a

Miscellaneous Exercise 4 | Q II. (13) | Page 86

If the middle term in the expansion of (x + "b"/x)^6 is 160, find b

Miscellaneous Exercise 4 | Q II. (14) | Page 86

If the coefficient of x2 and x3 in the expansion of (3 + kx)9 are equal, find k

Miscellaneous Exercise 4 | Q II. (15) | Page 86

If the constant term in the expansion of (x^3 + "k"/x^8)^11 is 1320, find k

Miscellaneous Exercise 4 | Q II. (16) | Page 86

Show that there is no term containing x6 in the expansion of (x^2 - 3/x)^11

Miscellaneous Exercise 4 | Q II. (17) | Page 86

Show that there is no constant term in the expansion of (2x - x^2/4)^9

Miscellaneous Exercise 4 | Q II. (18) | Page 86

State, first four terms in the expansion of (1 - (2x)/3)^(-1/2)

Miscellaneous Exercise 4 | Q II. (19) | Page 86

State, first four terms in the expansion of (1 - x)^(-1/4)

Miscellaneous Exercise 4 | Q II. (20) | Page 86

State, first three terms in the expansion of (5 + 4x) ^(-1/2)

Miscellaneous Exercise 4 | Q II. (21) | Page 86

Using binomial theorem, find the value of root(3)(995) upto four places of decimals

Miscellaneous Exercise 4 | Q II. (22) | Page 86

Find approximate value of 1/4.08 upto four places of decimals

Miscellaneous Exercise 4 | Q II. (23) | Page 86

Find the term independent of x in the in expansion of (1 - x^2) (x + 2/x)^6

Miscellaneous Exercise 4 | Q II. (24) | Page 86

(a + bx) (1 − x)6 = 3 − 20x + cx2 + ..... then find a, b, c

Miscellaneous Exercise 4 | Q II. (25) | Page 86

The 3rd term of (1 + x)n is 36x2. Find 5th term

Miscellaneous Exercise 4 | Q II. (26) | Page 86

Suppose (1 + kx)n = 1 − 12x + 60x2 − .... find k and n.

## Chapter 4: Methods of Induction and Binomial Theorem

Exercise 4.1Exercise 4.2Exercise 4.3Exercise 4.4Exercise 4.5Miscellaneous Exercise 4

## Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board chapter 4 - Methods of Induction and Binomial Theorem

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Concepts covered in Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board chapter 4 Methods of Induction and Binomial Theorem are Principle of Mathematical Induction, Binomial Theorem for Positive Integral Index, General Term in Expansion of (a + b)n, Middle term(s) in the expansion of (a + b)n, Binomial Theorem for Negative Index Or Fraction, Binomial Coefficients.

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