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Mathematics Induction and Binomial Theorem Online Test

Maan AliLast Updated: October 15, 2025

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

n/n+2

n/n+1

n/2(n+1)

2. 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

Positive integers n

Integers n

Real numbers n

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

-1≤x≤1

-1

-2≤x≤2

5. (1 + x)⁷ =

1-7x+21x²-35x³+35x⁴-21x⁵+7x⁶-x⁷

1+x+x²+x³+x⁴+x⁵+x⁶+x⁷

1+7x+21x²+35x³+35x⁴+21x⁵+7x⁶+x⁷

7+7x+21x²+35x³+35x⁴+21x⁵+7x⁶+x⁷

6. 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

30, 240

69, 760

None of these

99, 784

7. (1 + 2x)⁴ =

1 + 4x + 6x² + 4x³ + x⁴

1 – 4x + 6x² – 4x³ + x⁴

1 – 8x + 24x² – 32x³ + 16x⁴

1 + 8x + 24x² + 32x³ + 16x⁴

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

11th

13th

None of these

12th

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

28/243

-28/81

-28/243

28/81

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

1 – x

1 + x

1 – ½ x

1 + ½ x

11. 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

12. 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

1/2 – x/48

3/2 – 61x/48

3/2 – 48x/61

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

n ≤

n > 2

n ≥ 2

n < 2

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

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

9th term

8th term

6th term

7th term

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

27a³ – 36a²x

81a⁴ + 36a²x

27a³ + 36a²x

81a⁴ – 36a²x

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

n/2(n–3)

n/2(n+1)

n(n+3)

n/3(n–2)

18. 99th term of the series 2 + 7 + 14 + 23 + 34 + ……… is

9999

10000

9998

None of these

19. 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

924

2924

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

-2

-1

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

100000

1000

10000

100

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

3rd term

2nd term

4th term

Seven term

23. The middle term of (x + 1/x)¹⁰ is

None of these

¹⁰C₅

¹⁰C₄

¹⁰C₃

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

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

15th

16th

13th

14th

Page 2 of 4

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

2891

1771

3421

1563

27. (1 + x)ⁿ = 1 + nx + n(n – 1)/2! x² + n(n – 1)(n – 2)/3! x³ + …… is

Binomial series

Binomial theorem

Taylor series

None of these

28.

29. For (1+x)ⁿ the 2nd term is

n(n – 1)(n – 2)/2! x²

n(n – 1)(n – 2)/3! x³

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

31. The number of terms in the binomial expansion is

One less than the exponent

One more than the exponent

Equal to the exponent

Two less than the exponent

32. 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

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

r–q/q–p

p+q/r+q

None of these

p–r/p–q

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

35.

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

None of these

10 x 12

37.

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

-2

-1≤x≤1

-1

-2≤x≤2

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

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

None of these

Odd

Even

Prime

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

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

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

44.

45.

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

6th

5th

4th and 5th

4th

47. 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

3/2 and 6

2/3 and 9

2 and 9

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

504/259

None of these

405/256

450/263

49. 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.

50.

Page 3 of 4

51. 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 !

52. 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

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

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

4th term

5th term

3rd term

2nd term

55. The value of ¹⁵C₁₀ is equal to

15/11

5/10

15/10

5/11

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

2²⁻ⁿ

2ⁿ

2n – 1

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

58. (x – 1/x)³ =

x³ + 3x + 3/x + 1/x³

x³ – 3x + 3/x – 1/x³

None of these

x³ + x + 1/x + 1/x²

59.

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

Not a prime

An even number

Not an odd number

None of these

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

12a

-2a²

-8a

-4a

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

n + 1

n – 1

None of these

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

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

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

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

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

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

-1

None of these

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

n ≥ 5

n ≥ 3

n > 5

n > 3

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

n²

2ⁿ⁻²

2ⁿ⁻¹

2ⁿ

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

- 27/1024x

-4/x²

-1024

68. The exponent in (a-b)¹⁷ is

69. 3 + 3 a + 3 a² + ………… ∞ is equal to 45/8, a > 0 then a is

7/15

15/7

15/23

7/8

70. 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, 10)

C (12, 2)

None of these

P (12, 2)

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

None of these

n th term

N+2/3 th term

(n+1)th term

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

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

1024

74. 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

140

112

164

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

Page 4 of 4

76. The exponent in (a + b)⁷ is

77. (x – 1)³ =

x³ + 3x + 3x – 1

x³ – x² + x + 1

1 – 3x + 3x² – x³

x³ – 3x + 3x – 1

78. 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

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

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

180

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

(r – 1)th term

r th term

None of these

(r + 1)th term

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

|x|<2/3

|x|<1

|x|<1/2

|x|<1/3

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

p = 5/2, m =

None of these

p = – ½, m = 6

p = 1/2, m = 6

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

2ⁿ⁻¹

2ⁿ

2(n – 1)

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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%

Mathematics Induction and Binomial Theorem Online Test

Important Math Induction and Binomial Theorem Question and Answer

Math Induction and Binomial Theorem Practice Test Online With Explanation

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