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ECAT Mathematics Induction and Binomial Theorem Online Test

Maan AliLast Updated: April 18, 2025

Mathematics Induction and Binomial Theorem MCQS Online Test

ECAT Math Induction and Binomial Theorem Chapter Online Test

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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ECAT Math Induction and Binomial Theorem Chapter Test

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Page 1 of 4

Math

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

n ≥ 2

n < 2

n ≤

n > 2

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

10000

100000

100

1000

3. (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

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

4th

5th

4th and 5th

6th

5. The middle term of (x + 1/x)¹⁰ is

None of these

¹⁰C₃

¹⁰C₄

¹⁰C₅

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

- 27/1024x

-1024

-4/x²

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

n–2/2

n–1/2

n²–1/2

n²+1/2

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

n > 5

n > 3

n ≥ 3

n ≥ 5

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

10. For (1+x)ⁿ the 2nd term is

n(n – 1)(n – 2)/3! x³

n(n – 1)(n – 2)/2! x²

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

n/3(n–2)

n/2(n–3)

n(n+3)

n/2(n+1)

12. 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

Integers n

Real numbers n

Positive integers n

Positive real numbers n

13. The exponent in (a + b)⁷ is

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

14th

16th

13th

15th

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

-1

-1≤x≤1

-2≤x≤2

16. 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 !

4 ! 3 !

5 (3 !)

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

18. 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 and 9

2/3 and 9

3 and 2

3/2 and 6

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

5th term

4th term

2nd term

3rd term

20. (1 + 2x)⁴ =

1 – 8x + 24x² – 32x³ + 16x⁴

1 + 8x + 24x² + 32x³ + 16x⁴

1 – 4x + 6x² – 4x³ + x⁴

1 + 4x + 6x² + 4x³ + x⁴

21. 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

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

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

2ⁿ⁻¹

2ⁿ⁻²

2ⁿ

n²

24. 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

69, 760

30, 240

25. 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

Page 2 of 4

26.

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

p = 1/2, m = 6

None of these

p = – ½, m = 6

p = 5/2, m =

28.

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

-2

-1

-2≤x≤2

-1≤x≤1

30.

31.

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

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

n – 1

None of these

n + 1

34.

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

-1

-2

36. 99th term of the series 2 + 7 + 14 + 23 + 34 + ……… is

9999

9998

None of these

10000

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

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

39. 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, 10)

C (12, 2)

P (12, 2)

40. 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

200

100

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

6th term

7th term

8th term

9th term

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

A.P.

H.P.

None of these

G.P.

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

None of these

10 x 12

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

None of these

-1

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

924

792

1594

2924

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

1024

47. The exponent in (a-b)¹⁷ is

48.

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

50. 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 – 48x/61

1/2 – x/48

3/2 – 61x/48

2/3 – 61x/48

Page 3 of 4

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

An even number

Not a prime

Not an odd number

None of these

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

1563

2891

3421

1771

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

2nd term

Seven term

3rd term

4th term

54. 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

(√a – √b/√2)²

2a b/2+b

a–b/a

a+b/2

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

11th

13th

None of these

12th

56. 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

112

164

None of these

57.

58.

59. 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 th term

(n+1)th term

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

61. 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

p–r/p–q

None of these

r–q/q–p

62. 3 + 3 a + 3 a² + ………… ∞ is equal to 45/8, a > 0 then a is

15/7

7/8

7/15

15/23

63. (x – 1)³ =

x³ – 3x + 3x – 1

x³ – x² + x + 1

x³ + 3x + 3x – 1

1 – 3x + 3x² – x³

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

None of these

(r + 1)th term

r th term

(r – 1)th term

65. The value of ¹⁵C₁₀ is equal to

15/11

15/10

5/11

5/10

66. (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

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

|x|<1/3

|x|<1/2

|x|<1

|x|<2/3

68. (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

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

-28/243

28/243

28/81

-28/81

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

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

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

180

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

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

-8a

-4a

12a

-2a²

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

2ⁿ⁻¹

2(n – 1)

2ⁿ

Page 4 of 4

76. 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)

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

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

81a⁴ – 36a²x

27a³ + 36a²x

27a³ – 36a²x

81a⁴ + 36a²x

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

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

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

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

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

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

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

None of these

Odd

Even

Prime

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

405/256

None of these

504/259

450/263

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

2ⁿ

2n – 1

2²⁻ⁿ

84. 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

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Maan AliLast Updated: April 18, 2025

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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%

ECAT Mathematics Induction and Binomial Theorem Online Test

Mathematics Induction and Binomial Theorem MCQS Online Test

ECAT Math Induction and Binomial Theorem Chapter Online Test

ECAT Math Induction and Binomial Theorem Chapter Test

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