ECAT Mathematics Induction and Binomial Theorem Online Test

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

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

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

7. The middle term of (x + 1/x)¹⁰ is

8. 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

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

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

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

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

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

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

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

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

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

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

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

20. 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

21. The exponent in (a-b)¹⁷ is

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

23. The number of terms in the binomial expansion is

24. The value of ¹⁵C₁₀ is equal to

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

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

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

28. 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?

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

30. 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

31.  

33. 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

34. 3 + 3 a + 3 a² + ………… ∞ is equal to 45/8, a > 0 then a is

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

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

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

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

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

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

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

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

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

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

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

46. (x – 1)³ =

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

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

49. 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

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

51. The exponent in (a + b)⁷ is

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

53. For (1+x)ⁿ the 2nd term is

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

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

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

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

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

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

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

62. (1 + x)ⁿ = 1 + nx + n(n – 1)/2! x² + n(n – 1)(n – 2)/3! x³ + …… is

63. 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

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

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

66. 99th term of the series 2 + 7 + 14 + 23 + 34 + ……… is

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

68. 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

69. (x – 1/x)³ =

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

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

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

74. (1 + x)⁷ =

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

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

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

79. 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

80. (1 + 2x)⁴ =

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

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

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

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