ECAT Mathematics Induction and Binomial Theorem Online Test

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

2. The exponent in (a + b)⁷ is

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

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

5. 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

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

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

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

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

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

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

15. 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

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

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

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

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

20.  

21. 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

22. 99th term of the series 2 + 7 + 14 + 23 + 34 + ……… is

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

24. (x – 1)³ =

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

26. 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

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

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

29. For (1+x)ⁿ the 2nd term is

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

31. (1 + x)⁷ =

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

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

34. 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

35. The middle term of (x + 1/x)¹⁰ is

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

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

38. 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

39. 3 + 3 a + 3 a² + ………… ∞ is equal to 45/8, a > 0 then a is

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

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

42. The number of terms in the binomial expansion is

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

45. 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

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

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

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

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

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

52. 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

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

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

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

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

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

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

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

60. (x – 1/x)³ =

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

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

63. The value of ¹⁵C₁₀ is equal to

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

65. (1 + x)ⁿ = 1 + nx + n(n – 1)/2! x² + n(n – 1)(n – 2)/3! x³ + …… is

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

68. 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

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

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

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

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

74. 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?

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

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

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

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

79. The exponent in (a-b)¹⁷ is

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

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

82. (1 + 2x)⁴ =

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

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