NUMBER SYSTEM (CDS)

Total Questions: 101

61. The digit in the unit's place of the resulting number of the expression (234)¹⁰⁰ + (234)¹⁰¹, is [2015 (II) Evening Shift ]

Correct Answer: (d) 0
Solution:We have,

(234)¹⁰⁰ + (234)¹⁰¹
= (234)¹⁰⁰ (1 + 234)
= 235 (234)¹⁰⁰

We know that square of any number having 4 at unit place is a number in which 6 at unit place. Any exponent of a number 6 at unit place is always 6 is at unit place.

∵ (235) (...6) = ... 30
Resulting number have 0 at unit place.

62. The value of [2015 (II) Evening Shift ]

Correct Answer: (b)
Solution:We have,


63. A number when divided by 7 leaves a remainder 3 and the resulting quotient, when divided by 11 leaves a remainder 6. If the same number when divided by 11 leaves a remainder m and the resulting quotient when divided by 7 leaves a remainder n. What are the values of m and n, respectively? [2015 (II) Evening Shift ]

Correct Answer: (a) 1 and 4
Solution:This is an example of successive division. Let the number be N. The number and successive quotients, the successive divisors and the corresponding remainders are tabulated below.

64. The seven digit number 876p37q is divisible by 225. The values of p and q can be respectively [2015 (II) Evening Shift ]

Correct Answer: (e) (0, 5) and (9, 5)
Solution:Seven digits number 876p37q is divisible by 225, then this number is also divisible by 9 and 25.
If this number is divisible by 9.
∴ Sum of its digits is divisible by 9. Now, sum of digits
= 8 + 7 + 6 + p + 3 + 7 + q
= 31 + p + q
∴ p + q = 5 or p + q = 14 q must be 5, if q = 5 p = 0 and 9.
Options (c) and (d) both are correct.

65. Let x and y be positive integers such that x > y The expressions 3x + 2y and 2x + 3y when divided by 5 leave remainders 2 and 3, respectively. What is the remainder when (x - y) is divided by 5? [2015 (II) Evening Shift ]

Correct Answer: (a) 4
Solution:We have, 3x + 2y is divided by 5 remainder is 2.

66. The sum of first 47 terms of the series [2015 (II) Evening Shift ]

Correct Answer: (b)
Solution:The sum of first 47 terms of the series

67. Consider all positive two-digit numbers each of which when divided by 7 leaves a remainder 3. What is their sum? [2015 (II) Evening Shift ]

Correct Answer: (c) 676
Solution:Here, required numbers are 10, 17, 24, ..., 94

68. How many right angled triangles can be formed by joining the vertices of a cuboid ? [2015 (II) Evening Shift ]

Correct Answer: (d) None of these
Solution:A cuboid has 8 vertices, 12 edges and 6 faces. By selectively choosing and 3 vertices of the cuboid, we can form 2 types of right angled triangles.

69. Let x and y be positive integers such that x is prime and y is composite. Which of the following statements are correct? [2015 (II) Evening Shift ]

1. (y - x) can be an even integer.
2. xy can be an even integer.
3. 0.5(x + y) can be an even integer.

Select the correct answer using the code given below.

Correct Answer: (d) 1, 2 and 3
Solution:x is prime and y is composite number. Since, x is prime number

70. The last digit in the expansion of 17²⁵⁶ is [2015 (I) Morning Shift ]

Correct Answer: (d) 1
Solution: