#### Chapters

## 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 Exercise 4.1 [Pages 73 - 74]

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

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

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

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

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

1^{2} + 2^{2} + 3^{2} + .... + n^{2} = `("n"("n" + 1)(2"n" + 1))/6`

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

1^{2} + 3^{2} + 5^{2} + .... + (2n − 1)^{2} = `"n"/3 (2"n" − 1)(2"n" + 1)`

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

1^{3} + 3^{3} + 5^{3} + .... to n terms = n^{2}(2n^{2} − 1)

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

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

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)`

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)`

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))`

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

(2^{3n} − 1) is divisible by 7

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

(2^{4n}−1) is divisible by 15

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

3^{n} − 2n − 1 is divisible by 4

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

5 + 5^{2} + 5^{3} + .... + 5^{n} = `5/4(5^"n" - 1)`

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

(cos θ + i sin θ)^{n} = cos (nθ) + i sin (nθ)

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

Given that t_{n+1} = 5t_{n} + 4, t_{1} = 4, prove that t_{n} = 5^{n} − 1

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

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

### 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]

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

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

Expand: (2x^{2} + 3)^{4}

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

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

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

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

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

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

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

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

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

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}

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}

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

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

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

### 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]

In the following expansion, find the indicated term.

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

In the following expansion, find the indicated term.

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

In the following expansion, find the indicated term.

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

In the following expansion, find the indicated term.

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

In the following expansion, find the indicated term.

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

In the following expansion, find the indicated coefficient.

x^{3} in `(x^2 + (3sqrt(2))/x)^9`

In the following expansion, find the indicated coefficient.

x^{8} in `(2x^5 - 5/x^3)^8`

In the following expansion, find the indicated coefficient.

x^{9} in `(1/x + x^2)^18`

In the following expansion, find the indicated coefficient.

x^{–3} in `(x - 1/(2x))^5`

In the following expansion, find the indicated coefficient.

x^{–20} in `(x^3 - 1/(2x^2))^15`

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

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

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

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

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

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

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

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

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

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

In the expansion of (k + x)^{8}, the coefficient of x^{5} is 10 times the coefficient of x^{6}. Find the value of k.

Find the term containing x^{6} in the expansion of (2 − x) (3x + 1)^{9}

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

### 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]

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

(1 + x)^{−4}

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

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

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

(1 – x^{2})^{–3}

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

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

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

(1 + x^{2})^{–1}

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

(a − b)^{−3}

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

(a + b)^{−4}

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

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

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

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

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

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

Simplify first three terms in the expansion of the following

(1 + 2x)^{–4}

Simplify first three terms in the expansion of the following

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

Simplify first three terms in the expansion of the following

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

Simplify first three terms in the expansion of the following

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

Simplify first three terms in the expansion of the following

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

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

`sqrt(99)`

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

`root(3)(126)`

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

`root(4)(16.08)`

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

(1.02)^{–5}

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

(0.98)^{–3}

### 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]

Show That C_{0} + C_{1} + C_{2} + .... C_{8} = 256

Show That C_{0} + C_{1} + C_{2} + .... C_{9} = 512

Show That C_{1} + C_{2} + C_{3} + .... C_{7} = 127

Show That C_{1} + C_{2} + C_{3} + .... C_{6} = 63

Show That C_{0} + C_{2} + C_{4} + C_{6} + C_{8} = C_{1} + C_{3} + C_{5} + C_{7} = 128

Show That C_{1} + C_{2} + C_{3} + .... C_{n} = 2^{n} − 1

Show That C_{0} + 2C_{1} + 3C_{2} + 4C_{3} + ... + (n + 1)C_{n} = (n + 2)2^{n−1}

### 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]

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

Select the correct answer from the given alternatives.

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

(n − 1)

^{th}n

^{th}(n + 1)

^{th}(n + 2)

^{th}

Select the correct answer from the given alternatives.

In the expansion of (x^{2} − 2x)^{10}, the coefficient of x^{16} is

−1680

1680

3360

6720

Select the correct answer from the given alternatives.

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

^{11}C_{5 }^{10}C_{5}^{10}C_{4 }^{10}C_{7 }

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

Select the correct answer from the given alternatives.

The value ^{14}C_{1} + ^{14}C_{3} + ^{14}C_{5} + ..... + ^{14}C_{11} is

2

^{14 }− 12

^{14}− 142

^{12}2

^{13}− 14

Select the correct answer from the given alternatives.

The value ^{11}C_{2} + ^{11}C_{4} + ^{11}C_{6} + ^{11}C_{8} is equal to

2

^{10}− 12

^{10}− 112

^{10}+ 122

^{10}− 12

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

Select the correct answer from the given alternatives.

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

7

120

^{10}C_{8}210

Select the correct answer from the given alternatives.

If the coefficient of x^{2} and x^{3} in the expansion of (3 + ax)^{9} are the same, then the value of a is

`-7/9`

`-9/7`

`7/9`

`9/7`

### 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]

Answer the following:

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

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

Answer the following:

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

1^{2} + 4^{2} + 7^{2} + ... + (3n − 2)^{2} = `"n"/2 (6"n"^2 - 3"n" - 1)`

Answer the following:

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

2 + 3.2 + 4.2^{2} + ... + (n + 1)2^{n–1} = n.2^{n}

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))`

Answer the following:

Given that t_{n+1} = 5t_{n} − 8, t_{1} = 3, prove by method of induction that t_{n} = 5^{n−1} + 2

Answer the following:

Prove by method of induction

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

Expand (3x^{2} + 2y)^{5}

Answer the following:

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

Answer the following:

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

Answer the following:

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

Answer the following:

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

Answer the following:

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

Answer the following:

Find the middle term (s) in the expansion of (x^{2}+ 2y^{2})^{7 }

Answer the following:

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

Answer the following:

x^{6} in the expansion of `(3x^2 - 1/(3x))^9`

Answer the following:

x^{60} in the expansion of `(1/x^2 + x^4)^18`

Answer the following:

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

Answer the following

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

Answer the following:

Prove by method of induction log_{a} x^{n} = n log_{a}x, x > 0, n ∈ N

Answer the following:

Prove by method of induction 15^{2n–1} + 1 is divisible by 16, for all n ∈ N.

Answer the following:

Prove by method of induction 5^{2n} − 2^{2n} is divisible by 3, for all n ∈ N

Answer the following:

If the coefficient of x^{16} in the expansion of (x^{2} + ax)^{10} is 3360, find a

Answer the following:

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

Answer the following:

If the coefficient of x^{2} and x^{3} in the expansion of (3 + kx)^{9} are equal, find k

Answer the following:

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

Answer the following:

Show that there is no term containing x^{6} in the expansion of `(x^2 - 3/x)^11`

Answer the following:

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

Answer the following:

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

Answer the following:

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

Answer the following:

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

Answer the following:

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

Answer the following:

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

Answer the following:

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

Answer the following:

(a + bx) (1 − x)^{6} = 3 − 20x + cx^{2} + ..... then find a, b, c

Answer the following:

The 3^{rd} term of (1 + x)^{n} is 36x^{2}. Find 5^{th} term

Answer the following:

Suppose (1 + kx)^{n} = 1 − 12x + 60x^{2} − .... find k and n.

## 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

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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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