ECAT Mathematics Induction and Binomial Theorem Online Test

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

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

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

4. 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

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

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

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

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

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

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

12. (x – 1/x)³ =

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

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

15. The exponent in (a + b)⁷ is

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

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

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

20. (x – 1)³ =

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

22. The exponent in (a-b)¹⁷ is

23. 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

25. 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

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

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

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

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

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

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

32.  

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

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

35. For (1+x)ⁿ the 2nd term is

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

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

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

39. 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

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

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

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

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

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

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

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

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

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

50. The number of terms in the binomial expansion is

51. 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

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

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

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

56. (1 + x)⁷ =

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

59. 3 + 3 a + 3 a² + ………… ∞ is equal to 45/8, a > 0 then a is

60. (1 + x)ⁿ = 1 + nx + n(n – 1)/2! x² + n(n – 1)(n – 2)/3! x³ + …… is

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

62. 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

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

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

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

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

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

69. The middle term of (x + 1/x)¹⁰ is

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

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

72. The value of ¹⁵C₁₀ is equal to

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

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

76. 99th term of the series 2 + 7 + 14 + 23 + 34 + ……… is

77. 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?

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

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

80. 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

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

82. 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

83. (1 + 2x)⁴ =

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