ECAT Mathematics Induction and Binomial Theorem Online Test

1. 99th term of the series 2 + 7 + 14 + 23 + 34 + ……… is

2. (x – 1)³ =

3. 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

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

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

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

7. (1 + 2x)⁴ =

8. (1 + x)⁷ =

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

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

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

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

13. The number of terms in the binomial expansion is

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

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

17. For (1+x)ⁿ the 2nd term is

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

19. 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

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

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

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

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

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

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

26. (x – 1/x)³ =

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

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

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

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

31. 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

34. 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

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

36. The exponent in (a-b)¹⁷ is

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

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

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

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

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

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

43.  

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

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

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

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

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

49. 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

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

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

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

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

55. 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

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

57. 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

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

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

60. 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

61. 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

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

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

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

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

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

68. The middle term of (x + 1/x)¹⁰ is

69. (1 + x)ⁿ = 1 + nx + n(n – 1)/2! x² + n(n – 1)(n – 2)/3! x³ + …… is

70. The value of ¹⁵C₁₀ is equal to

71. 3 + 3 a + 3 a² + ………… ∞ is equal to 45/8, a > 0 then a is

72. The exponent in (a + b)⁷ is

73. 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?

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

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

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

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

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

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

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

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

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