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Mathematics Induction and Binomial Theorem Online Test

Maan AliLast Updated: October 15, 2025

Start your preparation for Mathematics Induction and Binomial Theorem Online Test. This online test contains important MCQs from past papers. Attempting this test will help you prepare for Annual Exams and Entry Tests. Read the questions carefully and select the correct option below to see your result.

Important Math Induction and Binomial Theorem Question and Answer

Total Questions:	84
Time Allowed:	40 Minutes
Total Marks:	84
How to Attempt Test:	Please choose the right Option in MCQ-type question.
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Math Induction and Binomial Theorem Practice Test Online With Explanation

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Page 1 of 4

Math

1. If the pth, qth and rth terms of an A.P. are in G.P:., then the common ratio of the G.P. is

p–r/p–q

None of these

p+q/r+q

r–q/q–p

2. The expansion of (1 – 3x)⅔ is valid if

|x|<1/2

|x|<2/3

|x|<1

|x|<1/3

4. The first three terms in the expansion of (1+ax)ⁿ (n ≠ 0) are 1, 6x and 16x², then the values of a and n are respectively

3 and 2

2 and 9

2/3 and 9

3/2 and 6

5. If n is not a natural number, then expansion of (1+x)³ is

-1≤x≤1

-2≤x≤2

-1

6. The co-efficient of x⁴ in the expansion of (x/2 – 3/x²)¹⁰ is

450/263

405/256

None of these

504/259

7. (x – 1)³ =

x³ – 3x + 3x – 1

x³ – x² + x + 1

x³ + 3x + 3x – 1

1 – 3x + 3x² – x³

8. (1 + 2x)⁴ =

1 + 8x + 24x² + 32x³ + 16x⁴

1 – 8x + 24x² – 32x³ + 16x⁴

1 – 4x + 6x² – 4x³ + x⁴

1 + 4x + 6x² + 4x³ + x⁴

9. If exponent in the binomial expansion is 6, then the middle term is

2nd term

5th term

3rd term

4th term

10.

11. If C (n, 10) = C (n, 12), then n is equal to

None of these

10 x 12

12. The number of term in the expansion of (a + b)ⁿ is

None of these

n + 1

n – 1

13. The nth term of the series 1 + 1+2/2 + 1+2+3/3 + …………. Is

n–1/2

n–2/2

n²+1/2

n²–1/2

14. The greatest integer which divides the number 101¹⁰⁰ – 1 is

1000

10000

100000

100

15. The sum of odd co-efficient in the expansion (1+x)⁴ is

16. 99th term of the series 2 + 7 + 14 + 23 + 34 + ……… is

9999

None of these

9998

10000

17. Ten different letters of an alphabet are given. Words with five letters are formed from these given letters. Then the number of words which have at least one letter repeated is

99, 784

None of these

30, 240

69, 760

18. If |x|<1, then the expansion of (1-x)½ up to two terms is

1 + ½ x

1 + x

1 – x

1 – ½ x

19. 3 + 3 a + 3 a² + ………… ∞ is equal to 45/8, a > 0 then a is

15/7

7/8

7/15

15/23

20. If the 4th term is the expansion of (px + x⁻¹)ᵐ is 2.5 for all x ϵ R, then

p = – ½, m = 6

p = 1/2, m = 6

p = 5/2, m =

None of these

21. In a college examination, a candidate is required to attempt 6 out of 10 question, which are divided into two sections each containing 5 question. Further, the candidate is not permitted to attempt more than 4 questions from either of the section. The number of ways in which he can make up a choice of 6 questions is

100

200

22.

23. The 3rd term in the expansion of (1 + 6/x)⅓ = ?

-4/x²

- 27/1024x

-1024

24. In the expansion of (a-22b)³ the co-efficient of b² is

-4a

-2a²

-8a

12a

25. The sum of exponents of a and b in every term of the expansion (a+b)ⁿ is

Page 2 of 4

26. If ‘n’ is even them the middle term in the expansion of (a+b)ⁿ will be

None of these

(n+1)th term

N+2/3 th term

n th term

27. Sum of even coefficient is equal to ………. In the binomial expansion of (1 + x)ⁿ

2n – 1

2²⁻ⁿ

2ⁿ

28. The number 11 ……………1 (91 times) is

An even number

None of these

Not a prime

Not an odd number

29. If the middle term in the expansion (a + b)ⁿ is n/2 +1, then n is

None of these

Prime

Even

Odd

30. The expansion of (3a – x/3a)⁴ up to two terms is

81a⁴ – 36a²x

27a³ + 36a²x

81a⁴ + 36a²x

27a³ – 36a²x

31. If a+bx/a–bx = b+cx/b–cx = c+dx/c–dx (x ≠ 0), then a, b, c, d are in

None of these

A.P.

G.P.

H.P.

32. If in the expansion of (1 + x)ⁿ, co-efficients of 2nd, 3rd and 4th terms are in A.P., then x =

33. There are 4 letters and 4 direct envelopes. The number of ways in which every letter be put into a wrong envelope is

34. The exponent in (a-b)¹⁷ is

35. If (1 + x – 2x²)⁴ = 1 + a₁x + a₂x² + a₃x³ + …… then the value of a₂ + a₄ + a₆ + ……. + a₁₂ will be

1024

36. The term independent of x in the expansion of (x² – 1/3x)⁹ is equal to

28/81

-28/81

-28/243

28/243

37. A family of 4 brothers and 3 sisters is to be arranged in a row for a photograph. In how many ways can they be seated if all sisters sit together?

5 ! 3 !

5 !

5 (3 !)

4 ! 3 !

38. If exponent I binomial expansion is 6, then the no. of terms are

3rd term

Seven term

4th term

2nd term

39. The positive integer just greater than (1+0.0001)¹⁰⁰⁰⁰ is

40. The number of terms in the expansion (x² – 1/x²)⁷ is

41.

42. The exponent in (a + b)⁷ is

43. The value of ¹⁵C₁₀ is equal to

5/11

15/10

5/10

15/11

44. A lady gives a dinner party for six guests. The number of ways in which they may be selected from among ten friends if two of the friends will not attend the party together is

164

112

140

None of these

45. The middle term in the expansion of (a + b)¹² is

13th

11th

None of these

12th

46. The nth term of the series 3, √3, 1, ……. Is 1/243, then n is:

47. Digit in the unit place of the number 183!+3¹⁸³

48. If ‘x’ be so small that is its square and higher powers may be neglected them (1 – x)½(9 – 4x)½/(8+3x)⅓

2/3 – 61x/48

3/2 – 48x/61

1/2 – x/48

3/2 – 61x/48

49. The middle term of (x + 1/x)¹⁰ is

¹⁰C₃

None of these

¹⁰C₅

¹⁰C₄

50. The number of handshakes that can be exchanged among a party of 12 students if every student strikes hands once with every other student is

None of these

P (12, 2)

P (12, 10)

C (12, 2)

Page 3 of 4

51. The coefficient of 21st term in the expansion of (a + b)²³ is

1771

1563

3421

2891

52. If the sum of co-efficient in the expansion of (a + b)ⁿ is 4096, then the greatest co-efficient in the expansion is

2924

1594

924

792

53. The middle tern in the expansion of (x + y)³⁰ us

16th

15th

14th

13th

54. (1 + x)⁷ =

1-7x+21x²-35x³+35x⁴-21x⁵+7x⁶-x⁷

7+7x+21x²+35x³+35x⁴+21x⁵+7x⁶+x⁷

1+7x+21x²+35x³+35x⁴+21x⁵+7x⁶+x⁷

1+x+x²+x³+x⁴+x⁵+x⁶+x⁷

55. pth term of an H.P. is qr and qth term is pr then a₅ is

-1

-2

56. Six identical coins are arranged in a row. The number of ways in which the number of tails is equal to number of heads is

120

57. For (1+x)ⁿ the 2nd term is

n(n – 1)(n – 2)/3! x³

n(n – 1)(n – 2)/2! x²

58. The sum of the coefficients in the expansion (1 + x)ⁿ is

n²

2ⁿ⁻¹

2ⁿ

2ⁿ⁻²

59. The number of terms in the binomial expansion is

One more than the exponent

Two less than the exponent

One less than the exponent

Equal to the exponent

60. The sum of co-efficient in (1 + x – 3x²)⁴¹⁶³ is

-1

None of these

61.

62. The no of terms in the expansion of (a-b)¹⁷ is

63. The middle term in the expansion (a + b)¹²

9th term

8th term

7th term

6th term

64. If a statement P(n) is true for n = 1 and the truth of P(n) for n = k implies the truth of P(n) for n = k + 1, then P(n) is true for all

Positive integers n

Integers n

Real numbers n

Positive real numbers n

65. (1 + x)ⁿ = 1 + nx + n(n – 1)/2! x² + n(n – 1)(n – 2)/3! x³ + …… is

None of these

Taylor series

Binomial theorem

Binomial series

66. The sum of even co-efficient in the expansion (1+x)ⁿ is valid for

-2

-1≤x≤1

-1

-2≤x≤2

67. The sum of the coefficients in the expansion (1 + x)ⁿ is

68. If n is any positive integer, then 1²+2²+3²+…..+n² =

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

(n + 1)(2n + 1)/3

n(n + 1)(2n +)/3

n(n + 1)(2n +)/6

69. If n is any positive integer, then n! > 3ⁿ⁻¹ is true for all

n > 5

n ≥ 5

n ≥ 3

n > 3

70.

71. The number of triangles that can be formed with 6 points on a circle is

72. (x – 1/x)³ =

x³ + 3x + 3/x + 1/x³

x³ – 3x + 3/x – 1/x³

x³ + x + 1/x + 1/x²

None of these

73. If P, Q, R be the A.M., G.M., H.M. respectively between any two rational numbers a and b, then P – Q is

2a b/2+b

a+b/2

a–b/a

(√a – √b/√2)²

74. Sum of odd coefficients in the binomial expansion is equal to ……….

2ⁿ

2(n – 1)

2ⁿ⁻¹

75. If in the expansion of (1 + x)²⁰, then co-efficient of rth ad (r + 4)th term are equal, then r is

Page 4 of 4

76.

77.

78. The general term in the expansion (a + x)ⁿ is

r th term

(r + 1)th term

None of these

(r – 1)th term

79. The number of diagonals in an n sides figure is equal to

n/3(n–2)

n(n+3)

n/2(n–3)

n/2(n+1)

80. If n is any positive integer, then 4ⁿ > 3ⁿ + 4 is true for all

n < 2

n ≥ 2

n > 2

n ≤

81. The sum of the odd coefficients in the expansion (1+x)⁵ is

82. The number of ways in which a mixed double tennis game be arranged between 10 players consisting of 6 men and 4 women?

180

83. If n is any positive integer, then 1/1.2 + 1/2.3 + 1/3.4 + …….. + 1/n(n+1) =

n/n+1

n/n+2

n/2(n+1)

84. The greatest term in the expansion of (3 + 2x)⁹, when x = 1 is

5th

6th

4th

4th and 5th

How to Score High in Mathematics Induction and Binomial Theorem Online Test?

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Concepts First: Ensure you understand the underlying theories before attempting the MCQs.
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Maan AliLast Updated: October 15, 2025

3 Comments

Ummar says:

October 6, 2021 at 9:58 am

There must be reasons for every question that how this answer /why this answer, mention at the end of the tests.

Ayan Raza says:

September 21, 2017 at 7:28 pm

92%

umair ahmed says:

March 19, 2017 at 1:28 pm

80%

Important Math Induction and Binomial Theorem Question and Answer

Math Induction and Binomial Theorem Practice Test Online With Explanation

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