# NCERT solutions for Mathematics Exemplar Class 11 chapter 8 - Binomial Theorem [Latest edition]

## Chapter 8: Binomial Theorem

Solved ExamplesExercise
Solved Examples [Pages 132 - 142]

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

Solved Examples | Q 1 | Page 132

Find the rth term in the expansion of (x + 1/x)^(2r)

Solved Examples | Q 2 | Page 132

Expand the following (1 – x + x2)4

Solved Examples | Q 3 | Page 132

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

Solved Examples | Q 4 | Page 132

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

Solved Examples | Q 5 | Page 133

Find the coefficient of x11 in the expansion of (x^3 - 2/x^2)^12

Solved Examples | Q 6 | Page 133

Determine whether the expansion of (x^2 - 2/x)^18 will contain a term containing x10?

Solved Examples | Q 7 | Page 134

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

Solved Examples | Q 8 | Page 134

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

Solved Examples | Q 9 | Page 135

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

Solved Examples | Q 10 | Page 135

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

Solved Examples | Q 11 | Page 135

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

Solved Examples | Q 12 | Page 136

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

Solved Examples | Q 13 | Page 137

Which of the following is larger? 9950 + 10050  or 10150

Solved Examples | Q 14 | Page 137

Find the coefficient of x50 after simplifying and collecting the like terms in the expansion of (1 + x)1000 + x(1 + x)999 + x2(1 + x)998 + ... + x1000 .

Solved Examples | Q 15 | Page 138

If a1, a2, a3 and a4 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

Solved Examples | Q 16 | Page 139

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

• 102

• 25

• 26

• None of these

Solved Examples | Q 17 | Page 139

If the coefficients of x7 and x8 in 2 + x^n/3 are equal, then n is ______.

• 56

• 55

• 45

• 15

Solved Examples | Q 18 | Page 140

If (1 – x + x2)n = a0 + a1 x + a2 x2 + ... + a2n x2n , then a0 + a2 + a4 + ... + a2n equals ______.

• (3^"n" + 1)/2

• (3^"n" - 1)/2

• (1 - 3^"n")/2

• 3^"n" + 1/2

Solved Examples | Q 19 | Page 140

The coefficient of xp and xq (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

Solved Examples | Q 20 | Page 141

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

Solved Examples | Q 21 | Page 141

The ratio of the coefficient of x15 to the term independent of x in x^2 + 2^15/x is ______.

• 12:32

• 1:32

• 32:12

• 32:1

Solved Examples | Q 22 | Page 142

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

• Re (z) = 0

• Im (z) = 0

• Re (z) > 0, Im (z) > 0

• Re (z) > 0, Im (z) < 0

Exercise [Pages 142 - 146]

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

Exercise | Q 1 | Page 142

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

Exercise | Q 2 | Page 142

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

Exercise | Q 3 | Page 142

Find the coefficient of x in the expansion of (1 – 3x + 7x2)(1 – x)16.

Exercise | Q 4 | Page 142

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

Exercise | Q 5.(i) | Page 142

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

Exercise | Q 5.(ii) | Page 142

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

Exercise | Q 6 | Page 143

Find the coefficient of x15 in the expansion of (x – x2)10.

Exercise | Q 7 | Page 143

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

Exercise | Q 8 | Page 143

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.

Exercise | Q 9 | Page 143

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.

Exercise | Q 10 | Page 143

If the coefficient of second, third and fourth terms in the expansion of (1 + x)2n are in A.P. Show that 2n2 – 9n + 7 = 0.

Exercise | Q 11 | Page 143

Find the coefficient of x4 in the expansion of (1 + x + x2 + x3)11.

Exercise | Q 12 | Page 143

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

Exercise | Q 13 | Page 143

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

Exercise | Q 14 | Page 143

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

Exercise | Q 15.(i) | Page 143

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 O2 – E2 = (x2 – a2)n

Exercise | Q 15.(ii) | Page 143

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

Exercise | Q 16 | Page 144

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

Exercise | Q 17 | Page 144

Find the term independent of x in the expansion of (1 + x + 2x3) (3/2 x^2 - 1/(3x))^9

#### Objective Type Questions from 18 to 24

Exercise | Q 18 | Page 144

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

• 50

• 202

• 51

• None of these

Exercise | Q 19 | Page 144

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

Exercise | Q 20 | Page 144

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

• 3rd and 4th

• 4th and 5th

• 5th and 6th

• 6th and 7th

Exercise | Q 21 | Page 144

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

• 1 : 2

• 1 : 3

• 3 : 1

• 2 : 1

Exercise | Q 22 | Page 144

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

• 2

• 7

• 11

• 14

Exercise | Q 23 | Page 145

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"

Exercise | Q 24 | Page 145

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.

Exercise | Q 25 | Page 145

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

Exercise | Q 26 | Page 145

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

Exercise | Q 27 | Page 145

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

Exercise | Q 28 | Page 145

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 ______.

Exercise | Q 29 | Page 146

The coefficient of a–6b4 in the expansion of (1/a - (2b)/3)^10 is ______.

Exercise | Q 30 | Page 146

Middle term in the expansion of (a3 + ba)28 is ______.

Exercise | Q 31 | Page 146

The ratio of the coefficients of xp and xq in the expansion of (1 + x)p + q is ______.

Exercise | Q 32 | Page 146

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

Exercise | Q 33 | Page 146

If 2515 is divided by 13, the reminder is ______.

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

Exercise | Q 34 | Page 146

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

• True

• False

Exercise | Q 35 | Page 146

The expression 79 + 97 is divisible by 64.

• True

• False

Exercise | Q 36 | Page 146

The number of terms in the expansion of [(2x + y3)4]7 is 8.

• True

• False

Exercise | Q 37 | Page 146

The sum of coefficients of the two middle terms in the expansion of (1 + x)2n–1 is equal to 2n–1Cn

• True

• False

Exercise | Q 38 | Page 146

The last two digits of the numbers 3400 are 01.

• True

• False

Exercise | Q 39 | Page 146

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

• True

• False

Exercise | Q 40 | Page 146

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

Solved ExamplesExercise

## 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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