ECAT Mathematics Induction and Binomial Theorem Online Test

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

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

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

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

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

7. The number of terms in the binomial expansion is

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

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

10. 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

11. (1 + x)ⁿ = 1 + nx + n(n – 1)/2! x² + n(n – 1)(n – 2)/3! x³ + …… is

12. 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

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

14. (1 + x)⁷ =

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

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

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

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

23. 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

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

25. 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

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

28. The exponent in (a-b)¹⁷ is

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

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

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

32. 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?

33. (1 + 2x)⁴ =

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

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

36. (x – 1)³ =

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

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

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

40. 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

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

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

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

44. 3 + 3 a + 3 a² + ………… ∞ is equal to 45/8, a > 0 then a is

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

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

47. The exponent in (a + b)⁷ is

48. 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

49. The middle term of (x + 1/x)¹⁰ is

50. For (1+x)ⁿ the 2nd term 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. In the expansion of (a-22b)³ the co-efficient of b² is

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

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

55.  

56. The value of ¹⁵C₁₀ is equal to

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

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

59. 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

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

61. 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

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

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

64. 99th term of the series 2 + 7 + 14 + 23 + 34 + ……… is

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

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

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

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

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

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

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

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

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

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

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

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

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

79. (x – 1/x)³ =

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

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

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

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

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