ECAT Mathematics Induction and Binomial Theorem Online Test

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

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

3. (1 + 2x)⁴ =

4. The value of ¹⁵C₁₀ is equal to

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

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

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

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

9. For (1+x)ⁿ the 2nd term is

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

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

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

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

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

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

17. The number of terms in the binomial expansion is

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

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

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

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

23. The middle term of (x + 1/x)¹⁰ is

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

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

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

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

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

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

31. 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

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

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

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

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

36. 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

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

39. 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

40. 3 + 3 a + 3 a² + ………… ∞ is equal to 45/8, a > 0 then a is

41. 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

42. (x – 1)³ =

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

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

45. 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

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

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

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

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

50. (1 + x)ⁿ = 1 + nx + n(n – 1)/2! x² + n(n – 1)(n – 2)/3! x³ + …… is

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

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

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

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

57. The exponent in (a-b)¹⁷ is

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

59. The exponent in (a + b)⁷ is

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

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

62. 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

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

64. 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

65. (1 + x)⁷ =

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

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

69. (x – 1/x)³ =

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

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

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

73. 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

74.  

75. 99th term of the series 2 + 7 + 14 + 23 + 34 + ……… is

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

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

78. 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?

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

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

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

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

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

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