ECAT Mathematics Induction and Binomial Theorem Online Test

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

3. For (1+x)ⁿ the 2nd term is

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

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

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

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

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

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

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

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

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

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

16. 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

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

18. 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

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

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

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

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

24. The exponent in (a-b)¹⁷ is

25. The exponent in (a + b)⁷ is

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

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

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

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

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

32. 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

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

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

35. The number of terms in the binomial expansion is

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

37. 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

38. (1 + 2x)⁴ =

39. (x – 1/x)³ =

40.  

41. 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

42. 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

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

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

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

46. 3 + 3 a + 3 a² + ………… ∞ is equal to 45/8, a > 0 then a is

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

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

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

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

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

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

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

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

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

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

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

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

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

60. 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

61. 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?

62. (1 + x)⁷ =

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

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

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

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

67. 99th term of the series 2 + 7 + 14 + 23 + 34 + ……… is

68. (1 + x)ⁿ = 1 + nx + n(n – 1)/2! x² + n(n – 1)(n – 2)/3! x³ + …… is

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

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

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

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

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

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

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

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

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

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

81. (x – 1)³ =

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

83. The middle term of (x + 1/x)¹⁰ is

84. The value of ¹⁵C₁₀ is equal to