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

9th term

8th term

7th term

6th term

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

3. (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

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

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

27a³ + 36a²x

81a⁴ – 36a²x

81a⁴ + 36a²x

27a³ – 36a²x

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

3rd term

2nd term

4th term

5th term

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

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

-2

-1

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

10. 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

None of these

112

164

140

11. 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 real numbers n

Integers n

Real numbers n

Positive integers n

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

n th term

None of these

N+2/3 th term

(n+1)th term

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

n < 2

n > 2

n ≥ 2

n ≤

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

2²⁻ⁿ

2ⁿ

2n – 1

15. $Math 8. 04$

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

n > 5

n ≥ 5

n ≥ 3

n > 3

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

180

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

(r – 1)th term

(r + 1)th term

None of these

r th term

19. The value of ¹⁵C₁₀ is equal to

5/11

5/10

15/10

15/11

20. 99th term of the series 2 + 7 + 14 + 23 + 34 + ……… is

None of these

9998

10000

9999

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

4th term

Seven term

3rd term

2nd term

22. 3 + 3 a + 3 a² + ………… ∞ is equal to 45/8, a > 0 then a is

15/7

15/23

7/15

7/8

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

12a

-2a²

-4a

-8a

24. The middle term of (x + 1/x)¹⁰ is

None of these

¹⁰C₄

¹⁰C₃

¹⁰C₅

25. $Math 8. 06$

Page 2 of 4

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

27. The number of terms in the binomial expansion is

One less than the exponent

Two less than the exponent

One more than the exponent

Equal to the exponent

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

None of these

10 x 12

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

n²+1/2

n²–1/2

n–2/2

n–1/2

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

-1

-1≤x≤1

-2≤x≤2

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

504/259

405/256

None of these

450/263

32. 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

69, 760

None of these

99, 784

30, 240

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

n + 1

None of these

n – 1

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

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

3421

1563

2891

1771

36. 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

H.P.

G.P.

A.P.

37. $Math 8. 02$

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

Prime

Even

None of these

Odd

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

40. (1 + x)⁷ =

7+7x+21x²+35x³+35x⁴+21x⁵+7x⁶+x⁷

1+x+x²+x³+x⁴+x⁵+x⁶+x⁷

1+7x+21x²+35x³+35x⁴+21x⁵+7x⁶+x⁷

1-7x+21x²-35x³+35x⁴-21x⁵+7x⁶-x⁷

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

1 – x

1 + ½ x

1 – ½ x

1 + x

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

1000

100

10000

100000

43. 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

3/2 and 6

2 and 9

2/3 and 9

3 and 2

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

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

1024

46. 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 !

5 (3 !)

4 ! 3 !

5 ! 3 !

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

-2≤x≤2

-1≤x≤1

-1

-2

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

49. $Math 8. 01$

50. $Math 8. 03$

Page 3 of 4

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

n/n+2

n/2(n+1)

n/n+1

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

15th

16th

14th

13th

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

54. $Math 8. 08$

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

n²

2ⁿ⁻²

2ⁿ⁻¹

2ⁿ

56. 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 = 5/2, m =

p = 1/2, m = 6

None of these

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

4th and 5th

4th

5th

6th

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

|x|<1

|x|<2/3

|x|<1/2

|x|<1/3

59. 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

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

792

1594

2924

924

61. (x – 1/x)³ =

None of these

x³ – 3x + 3/x – 1/x³

x³ + x + 1/x + 1/x²

x³ + 3x + 3/x + 1/x³

62. $Math 8. 05$

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

64. For (1+x)ⁿ the 2nd term is

n(n – 1)(n – 2)/3! x³

n(n – 1)(n – 2)/2! x²

65. The exponent in (a + b)⁷ is

66. 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

P (12, 2)

None of these

C (12, 2)

P (12, 10)

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

None of these

12th

11th

13th

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

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

70. $Math 8. 07$

71. 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

r–q/q–p

p+q/r+q

None of these

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

2ⁿ⁻¹

2ⁿ

2(n – 1)

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

None of these

Not an odd number

Not a prime

An even number

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

-1

None of these

75. 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

1/2 – x/48

2/3 – 61x/48

3/2 – 61x/48

3/2 – 48x/61

Page 4 of 4

76. 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

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

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

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

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

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

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

n/2(n+1)

n/3(n–2)

n/2(n–3)

n(n+3)

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

28/243

28/81

-28/81

-28/243

80. (x – 1)³ =

x³ + 3x + 3x – 1

x³ – 3x + 3x – 1

x³ – x² + x + 1

1 – 3x + 3x² – x³

81. The exponent in (a-b)¹⁷ is

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

-4/x²

- 27/1024x

-1024

83. (1 + 2x)⁴ =

1 + 8x + 24x² + 32x³ + 16x⁴

1 – 4x + 6x² – 4x³ + x⁴

1 + 4x + 6x² + 4x³ + x⁴

1 – 8x + 24x² – 32x³ + 16x⁴

84. 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

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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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