Maharashtra State BoardHSC Science (General) 11th
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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]

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Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board - Shaalaa.com

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

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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

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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

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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 

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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

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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`

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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

Answer the following:

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

Answer the following:

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

Answer the following:

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

Answer the following:

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

Answer the following:

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

Answer the following:

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

Answer the following:

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

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

Answer the following:

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

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

Answer the following:

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

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

Answer the following:

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

Answer the following:

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

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

Answer the following:

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

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

Answer the following:

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

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

Answer the following:

x6 in the expansion of `(3x^2 - 1/(3x))^9`

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

Answer the following:

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

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

Answer the following:

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

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

Answer the following

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

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

Answer the following:

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

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

Answer the following:

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

Answer the following:

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

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

Answer the following:

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

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

Answer the following:

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

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

Answer the following:

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

Answer the following:

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

Answer the following:

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

Answer the following:

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

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

Answer the following:

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

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

Answer the following:

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

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

Answer the following:

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

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

Answer the following:

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

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

Answer the following:

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

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

Answer the following:

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

Answer the following:

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

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

Answer the following:

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

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

Answer the following:

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

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Chapter 4: Methods of Induction and Binomial Theorem

Exercise 4.1Exercise 4.2Exercise 4.3Exercise 4.4Exercise 4.5Miscellaneous Exercise 4
Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board - Shaalaa.com

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

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board chapter 4 (Methods of Induction and Binomial Theorem) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the Maharashtra State Board Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board solutions in a manner that help students grasp basic concepts better and faster.

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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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