NUMBER SYSTEM (CDS)

Total Questions: 101

51. What is the difference between the sum of the cubes and that of squares of first ten natural numbers? [2016 (II) Evening Shift ]

Correct Answer: (b) 2640
Solution:Sum of cubes of n natural numbers

52. What is the unit digit of 7¹³⁹ ? [2016 (II) Evening Shift ]

Correct Answer: (d) 3
Solution:The unit digit of 7¹ = 7

53. Let A and B be finite non-empty sets with the number of elements in A = m and number of elements in B = n . Let m > n If for some integer k ≥ 1 the number of non-empty subsets of A = 2ᵏ + 2ⁿ the number of non-empty subsets of B, then which one of the following is correct? [2016 (II) Evening Shift ]

Correct Answer: (b) m = n + 1
Solution:According to the question,

54. 7¹⁰ - 5¹⁰ is divisible by [2016 (II) Evening Shift ]

Correct Answer: (d) 11
Solution:7¹⁰ - 5¹⁰ = (7⁵)² - (5⁵)²

55. If m and n are distinct natural numbers, then which of the following is/are integer(s) ? [2016 (II) Evening Shift ]

Correct Answer: (b) Only 2
Solution:

56. What is the maximum value of m, if the number N = 90 × 42 × 324 × 55 is divisible by 3ᵐ? [2016 (I) Morning Shift ]

Correct Answer: (b) 7
Solution:We have, N = 90 × 42 × 324 × 55

57. Consider the following statements [2016 (I) Morning Shift ]

1. Every natural number is a real number.
2. Every real number is an irrational number.
3. Every integer is a real number.
4. Every rational number is a real number.

Which of the above statements are correct?

Correct Answer: (b) 1, 3 and 4
Solution:1. Every natural number is a real number, which is correct.
2. Every real number is an irrational number, which is incorrect as real number can be rational or irrational.
3. Every integer is a real number, which is correct.
4. Every rational number is a real number, which is correct.

58. Consider the following statements in respect of the [2016 (I) Morning Shift ]

Correct Answer: (a) Only 1
Solution:

59. Let S be a set of first fourteen natural numbers. The possible number of pairs (a, b), where a, b ∈ S and a ≠ b such that ab leaves remainder I when divided by 15, is [2016 (I) Morning Shift ]

Correct Answer: (d) None of the above
Solution:The possible set of pairs (a, b) such that ab leaves remainder 1 when divided by 15, are (2,8), (8, 2), (7,13) and (13,7).

∴ Number of possible set of pairs = 4

60. The largest natural number which divides every natural number of the form (n³ - n) (n - 2) where in is a natural number greater than 2, is [2015 (II) Evening Shift ]

Correct Answer: (c) 24
Solution:Let x = (n³ - n) (n - 2) where n > 2