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

p+q/r+q

None of these

r–q/q–p

p–r/p–q

2. 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

2/3 and 9

2 and 9

3 and 2

3/2 and 6

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

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

16th

13th

14th

15th

5. 3 + 3 a + 3 a² + ………… ∞ is equal to 45/8, a > 0 then a is

7/15

7/8

15/23

15/7

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

2ⁿ⁻²

n²

2ⁿ⁻¹

2ⁿ

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

10 x 12

None of these

8. 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

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

11. (x – 1)³ =

1 – 3x + 3x² – x³

x³ – x² + x + 1

x³ – 3x + 3x – 1

x³ + 3x + 3x – 1

12. The middle term of (x + 1/x)¹⁰ is

¹⁰C₃

¹⁰C₄

¹⁰C₅

None of these

13. 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

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

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

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

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

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

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

27a³ – 36a²x

81a⁴ – 36a²x

81a⁴ + 36a²x

27a³ + 36a²x

16. 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

None of these

69, 760

99, 784

30, 240

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

-28/243

28/243

-28/81

28/81

18.

19. 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?

4 ! 3 !

5 (3 !)

5 ! 3 !

5 !

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

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

-1024

-4/x²

- 27/1024x

22. The number of terms in the binomial expansion is

Two less than the exponent

One more than the exponent

One less than the exponent

Equal to the exponent

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

2nd term

3rd term

Seven term

4th term

24. (1 + x)ⁿ = 1 + nx + n(n – 1)/2! x² + n(n – 1)(n – 2)/3! x³ + …… is

Taylor series

None of these

Binomial theorem

Binomial series

25.

Page 2 of 4

26. 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)

P (12, 2)

None of these

C (12, 2)

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

2(n – 1)

2ⁿ⁻¹

2ⁿ

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

6th

4th and 5th

4th

5th

29. (1 + x)⁷ =

7+7x+21x²+35x³+35x⁴+21x⁵+7x⁶+x⁷

1+7x+21x²+35x³+35x⁴+21x⁵+7x⁶+x⁷

1+x+x²+x³+x⁴+x⁵+x⁶+x⁷

1-7x+21x²-35x³+35x⁴-21x⁵+7x⁶-x⁷

30. For (1+x)ⁿ the 2nd term is

n(n – 1)(n – 2)/3! x³

n(n – 1)(n – 2)/2! x²

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

n ≥ 5

n > 5

n ≥ 3

n > 3

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

1024

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

None of these

11th

13th

12th

34. (1 + 2x)⁴ =

1 – 8x + 24x² – 32x³ + 16x⁴

1 + 4x + 6x² + 4x³ + x⁴

1 + 8x + 24x² + 32x³ + 16x⁴

1 – 4x + 6x² – 4x³ + x⁴

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

2²⁻ⁿ

2ⁿ

2n – 1

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

37. 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

G.P.

A.P.

H.P.

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

-4a

-2a²

-8a

12a

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

1594

924

792

2924

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

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

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

9th term

6th term

8th term

7th term

43.

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

Not an odd number

Not a prime

None of these

An even number

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

n/2(n–3)

n/3(n–2)

n(n+3)

n/2(n+1)

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

47.

48. 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

Real numbers n

Integers n

Positive integers n

Positive real numbers n

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

50. The exponent in (a-b)¹⁷ is

Page 3 of 4

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

180

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

-2≤x≤2

-1

-1≤x≤1

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

n – 1

None of these

n + 1

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

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

56.

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

-2

-1

58. 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

140

None of these

112

164

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

405/256

450/263

504/259

None of these

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

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

100

100000

10000

1000

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

63. 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)²

a–b/a

a+b/2

64. The exponent in (a + b)⁷ is

65. 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

3/2 – 61x/48

2/3 – 61x/48

3/2 – 48x/61

1/2 – x/48

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

n > 2

n ≥ 2

n < 2

n ≤

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

|x|<1/3

|x|<1

|x|<1/2

|x|<2/3

68. 99th term of the series 2 + 7 + 14 + 23 + 34 + ……… is

9998

None of these

9999

10000

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

Odd

Prime

Even

None of these

70. (x – 1/x)³ =

x³ + x + 1/x + 1/x²

x³ + 3x + 3/x + 1/x³

x³ – 3x + 3/x – 1/x³

None of these

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

None of these

p = – ½, m = 6

p = 5/2, m =

p = 1/2, m = 6

72.

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

1771

1563

3421

2891

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

N+2/3 th term

None of these

(n+1)th term

n th term

75.

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76. 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

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

2nd term

4th term

5th term

3rd term

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

-1

None of these

79. 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

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

(r + 1)th term

None of these

(r – 1)th term

r th term

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

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

-1

-2

-1≤x≤1

-2≤x≤2

83. The value of ¹⁵C₁₀ is equal to

15/11

5/10

15/10

5/11

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

n/2(n+1)

n/n+2

n/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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