#### Chapters

Chapter 2: Relations and Functions

Chapter 3: Trigonometric Functions

Chapter 4: Principle of Mathematical Induction

Chapter 5: Complex Numbers and Quadratic Equations

Chapter 6: Linear Inequalities

Chapter 7: Permutations and Combinations

Chapter 8: Binomial Theorem

Chapter 9: Sequences and Series

Chapter 10: Straight Lines

Chapter 11: Conic Sections

Chapter 12: Introduction to Three Dimensional Geometry

Chapter 13: Limits and Derivatives

Chapter 14: Mathematical Reasoning

Chapter 15: Statistics

Chapter 16: Probability

## Chapter 8: Binomial Theorem

### NCERT solutions for Mathematics Exemplar Class 11 Chapter 8 Binomial TheoremSolved Examples [Pages 132 - 142]

#### Short Answer

Find the r^{th} term in the expansion of `(x + 1/x)^(2r)`

Expand the following (1 – x + x^{2})^{4}

Find the 4^{th} term from the end in the expansion of `(x^3/2 - 2/x^2)^9`

Evaluate: `(x^2 - sqrt(1 - x^2))^4 + (x^2 + sqrt(1 - x^2))^4`

Find the coefficient of x^{11} in the expansion of `(x^3 - 2/x^2)^12`

Determine whether the expansion of `(x^2 - 2/x)^18` will contain a term containing x^{10}?

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

Find the middle term in the expansion of `(2ax - b/x^2)^12`.

Find the middle term (terms) in the expansion of `(p/x + x/p)^9`.

Show that `2^(4n + 4) - 15n - 16`, where n ∈ N is divisible by 225.

#### Long Answer

Find numerically the greatest term in the expansion of (2 + 3x)^{9}, where x = `3/2`.

If n is a positive integer, find the coefficient of x^{–1} in the expansion of `(1 + x)^2 (1 + 1/x)^n`

Which of the following is larger? 99^{50 }+ 100^{50 } or 101^{50 }

Find the coefficient of x^{50} after simplifying and collecting the like terms in the expansion of (1 + x)^{1000 }+ x(1 + x)^{999} + x^{2}(1 + x)^{998} + ... + x^{1000 }.

If a_{1}, a_{2}, a_{3} and a_{4} are the coefficient of any four consecutive terms in the expansion of (1 + x)^{n}, prove that `(a_1)/(a_1 + a_2) + (a_3)/(a_3 + a_4) = (2a_2)/(a_2 + a_3)`

#### Objective Type Questions

The total number of terms in the expansion of (x + a)^{51} – (x – a)^{51} after simplification is ______.

102

25

26

None of these

If the coefficients of x^{7} and x^{8} in `2 + x^n/3` are equal, then n is ______.

56

55

45

15

If (1 – x + x^{2})^{n} = a_{0} + a_{1} x + a_{2} x^{2} + ... + a_{2n} x^{2n} , then a_{0} + a_{2} + a_{4} + ... + a_{2n} equals ______.

`(3^"n" + 1)/2`

`(3^"n" - 1)/2`

`(1 - 3^"n")/2`

`3^"n" + 1/2`

The coefficient of x^{p} and x^{q} (p and q are positive integers) in the expansion of (1 + x)^{p + q} are ______.

Equal

Equal with opposite signs

Reciprocal of each other

None of these

The number of terms in the expansion of (a + b + c)^{n}, where n ∈ N is ______.

`((n + 1)(n + 2))/2`

n + 1

n + 2

(n + 1)n

The ratio of the coefficient of x^{15} to the term independent of x in `x^2 + 2^15/x` is ______.

12:32

1:32

32:12

32:1

If z = `sqrt(3)/2 + i^5/2 + sqrt(3)/2 - i^5/2`, then ______.

Re (z) = 0

I

_{m}(z) = 0Re (z) > 0, I

_{m}(z) > 0Re (z) > 0, I

_{m}(z) < 0

### NCERT solutions for Mathematics Exemplar Class 11 Chapter 8 Binomial TheoremExercise [Pages 142 - 146]

#### Short Answer

Find the term independent of x, x ≠ 0, in the expansion of `((3x^2)/2 - 1/(3x))^15`

If the term free from x in the expansion of `(sqrt(x) - k/x^2)^10` is 405, find the value of k.

Find the coefficient of x in the expansion of (1 – 3x + 7x^{2})(1 – x)^{16}.

Find the term independent of x in the expansion of `(3x - 2/x^2)^15`

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

Find the middle term (terms) in the expansion of `(3x - x^3/6)^9`

Find the coefficient of x^{15} in the expansion of (x – x^{2})^{10}.

Find the coefficient of `1/x^17` in the expansion of `(x^4 - 1/x^3)^15`

Find the sixth term of the expansion `(y^(1/2) + x^(1/3))^"n"`, if the binomial coefficient of the third term from the end is 45.

Find the value of r, if the coefficients of (2r + 4)^{th} and (r – 2)^{th} terms in the expansion of (1 + x)^{18} are equal.

If the coefficient of second, third and fourth terms in the expansion of (1 + x)^{2n} are in A.P. Show that 2n^{2} – 9n + 7 = 0.

Find the coefficient of x^{4} in the expansion of (1 + x + x^{2} + x^{3})^{11}.

#### Long Answer

If p is a real number and if the middle term in the expansion of `(p/2 + 2)^8` is 1120, find p.

Show that the middle term in the expansion of `(x - 1/x)^(2x)` is `(1 xx 3 xx 5 xx ... (2n - 1))/(n!) xx (-2)^n`

Find n in the binomial `(root(3)(2) + 1/(root(3)(3)))^n` if the ratio of 7^{th} term from the beginning to the 7^{th} term from the end is `1/6`

In the expansion of (x + a)^{n} if the sum of odd terms is denoted by O and the sum of even term by E. Then prove that O^{2} – E^{2} = (x^{2} – a^{2})^{n}

In the expansion of (x + a)^{n} if the sum of odd terms is denoted by O and the sum of even term by E. Then prove that 4OE = (x + a)^{2n} – (x – a)^{2n}

If x^{p} occurs in the expansion of `(x^2 + 1/x)^(2n)`, prove that its coefficient is `(2n!)/(((4n - p)/3)!((2n + p)/3)!)`

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

#### Objective Type Questions from 18 to 24

The total number of terms in the expansion of (x + a)^{100} + (x – a)^{100} after simplification is ______.

50

202

51

None of these

Given the integers r > 1, n > 2, and coefficients of (3r)^{th} and (r + 2)^{nd} terms in the binomial expansion of (1 + x)^{2n} are equal, then ______.

n = 2r

n = 3r

n = 2r + 1

None of these

The two successive terms in the expansion of (1 + x)^{24} whose coefficients are in the ratio 1:4 are ______.

3

^{rd}and 4^{th}4

^{th}and 5^{th}5

^{th}and 6^{th}6

^{th}and 7^{th}

The coefficient of x^{n} in the expansion of (1 + x)^{2n} and (1 + x)^{2n–1} are in the ratio ______.

1 : 2

1 : 3

3 : 1

2 : 1

If the coefficients of 2^{nd}, 3^{rd} and the 4^{th} terms in the expansion of (1 + x)^{n} are in A.P., then value of n is ______.

2

7

11

14

If A and B are coefficient of x n in the expansions of (1 + x)^{2n} and (1 + x)^{2n–1} respectively, then `A/B` equals ______.

1

2

`1/2`

`1/"n"`

If the middle term of `(1/x + x sin x)^10` is equal to `7 7/8`, then value of x is ______.

`2npi + pi/6`

`npi + pi/6`

`npi + (-1)^n pi/6`

`npi + (-1)^n pi/3`

#### Fill in the blanks 25 to 33.

The largest coefficient in the expansion of (1 + x)^{30} is ______.

The number of terms in the expansion of (x + y + z)^{n} ______.

In the expansion of `(x^2 - 1/x^2)^16`, the value of constant term is ______.

If the seventh terms from the beginning and the end in the expansion of `(root(3)(2) + 1/(root(3)(3)))^n` are equal, then n equals ______.

The coefficient of a^{–6}b^{4} in the expansion of `(1/a - (2b)/3)^10` is ______.

Middle term in the expansion of (a^{3} + ba)^{28} is ______.

The ratio of the coefficients of x^{p} and x^{q} in the expansion of (1 + x)^{p + q} is ______.

The position of the term independent of x in the expansion of `(sqrt(x/3) + 3/(2x^2))^10` is ______.

If 25^{15} is divided by 13, the reminder is ______.

#### State whether the following is True or False: 34 to 40

The sum of the series `sum_(r = 0)^10 ""^20C_r` is `2^19 + (""^20C_10)/2`

True

False

The expression 7^{9} + 9^{7} is divisible by 64.

True

False

The number of terms in the expansion of [(2x + y^{3})^{4}]^{7} is 8.

True

False

The sum of coefficients of the two middle terms in the expansion of (1 + x)^{2n–1} is equal to ^{2n–1}C_{n}.

True

False

The last two digits of the numbers 3^{400} are 01.

True

False

If the expansion of `(x - 1/x^2)^(2n)` contains a term independent of x, then n is a multiple of 2.

True

False

Number of terms in the expansion of (a + b)^{n} where n ∈ N is one less than the power n.

True

False

## Chapter 8: Binomial Theorem

## NCERT solutions for Mathematics Exemplar Class 11 chapter 8 - Binomial Theorem

NCERT solutions for Mathematics Exemplar Class 11 chapter 8 (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 CBSE Mathematics Exemplar Class 11 solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Mathematics Exemplar Class 11 chapter 8 Binomial Theorem are Binomial Theorem for Positive Integral Indices, General and Middle Terms, Introduction of Binomial Theorem, Proof of Binomial Therom by Pattern, Proof of Binomial Therom by Combination, Rth Term from End, Simple Applications of Binomial Theorem.

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