ECAT Mathematics Induction and Binomial Theorem Online Test

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

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

3. The exponent in (a + b)⁷ is

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

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

6. (1 + x)⁷ =

7. 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?

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

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

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

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

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

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

14. (1 + x)ⁿ = 1 + nx + n(n – 1)/2! x² + n(n – 1)(n – 2)/3! x³ + …… is

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

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

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

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

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

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

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

25. The number of terms in the binomial expansion is

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

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

28. The value of ¹⁵C₁₀ is equal to

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

31. 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

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

33. For (1+x)ⁿ the 2nd term is

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

35.  

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

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

38. (x – 1)³ =

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

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

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

42. 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

43. The middle term of (x + 1/x)¹⁰ is

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

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

46. 3 + 3 a + 3 a² + ………… ∞ is equal to 45/8, a > 0 then a is

47. 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

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

49. The exponent in (a-b)¹⁷ is

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

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

52. 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

53. 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

54. (x – 1/x)³ =

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

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

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

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

60. 99th term of the series 2 + 7 + 14 + 23 + 34 + ……… is

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

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

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

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

65. 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

66. (1 + 2x)⁴ =

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

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

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

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

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

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

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

75. 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

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

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

79. 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

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

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

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

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

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