Number System
कुल प्रश्न: 77 | पेज 7
61. The sum of two numbers is 37 and the difference of their squares is 185. What is the difference between the two numbers?
सही उत्तर: [A]
हल (Solution):
Correct Answer: A) 5
Solution:
a^2 - b^2 = 185 -> (a+b)(a-b) = 185
37 * (a-b) = 185 -> (a-b) = 185 / 37 = 5.
Correct Answer: A) 5
Solution:
a^2 - b^2 = 185 -> (a+b)(a-b) = 185
37 * (a-b) = 185 -> (a-b) = 185 / 37 = 5.
62. Find the unit digit of 4^11 + 4^12 + 4^13 + 4^14.
सही उत्तर: [A]
हल (Solution):
Correct Answer: A) 0
Solution:
4^(odd) ends in 4, 4^(even) ends in 6.
4 + 6 + 4 + 6 = 20. Unit digit is 0.
Correct Answer: A) 0
Solution:
4^(odd) ends in 4, 4^(even) ends in 6.
4 + 6 + 4 + 6 = 20. Unit digit is 0.
63. Which is the largest fraction among 2/3, 4/5, 6/7, 7/8?
सही उत्तर: [D]
हल (Solution):
Correct Answer: D) 7/8
Solution:
The difference between numerator and denominator is 1 in all cases. In such cases, the fraction with the largest numerator is the largest.
Correct Answer: D) 7/8
Solution:
The difference between numerator and denominator is 1 in all cases. In such cases, the fraction with the largest numerator is the largest.
64. The product of two rational numbers is -9/16. If one of the numbers is -4/3, what is the other number?
सही उत्तर: [B]
हल (Solution):
Correct Answer: B) 27/64
Solution:
Let the other number be x.
x * (-4/3) = -9/16
x = (-9/16) * (-3/4) = 27/64.
Correct Answer: B) 27/64
Solution:
Let the other number be x.
x * (-4/3) = -9/16
x = (-9/16) * (-3/4) = 27/64.
65. Find the sum of all natural numbers between 100 and 200 which are multiples of 3.
सही उत्तर: [B]
हल (Solution):
The sequence is 102, 105, ..., 198.
Number of terms (n) = [(198 - 102) / 3] + 1 = 33.
Sum = (n/2) * (first + last) = (33/2) * (102 + 198) = 16.5 * 300 = 4950.
The sequence is 102, 105, ..., 198.
Number of terms (n) = [(198 - 102) / 3] + 1 = 33.
Sum = (n/2) * (first + last) = (33/2) * (102 + 198) = 16.5 * 300 = 4950.
66. Find the total number of prime factors of the product: (8)^20 x (15)^24 x (17)^15.
सही उत्तर: [C]
हल (Solution):
Correct Answer: C) 123
Solution:
Break into prime bases: (2^3)^20 x (3 x 5)^24 x (17)^15.
= 2^60 x 3^24 x 5^24 x 17^15.
Total prime factors = 60 + 24 + 24 + 15 = 123.
Correct Answer: C) 123
Solution:
Break into prime bases: (2^3)^20 x (3 x 5)^24 x (17)^15.
= 2^60 x 3^24 x 5^24 x 17^15.
Total prime factors = 60 + 24 + 24 + 15 = 123.
67. If a number is multiplied by 25% of itself, the result is 144. What is the number?
सही उत्तर: [B]
हल (Solution):
Correct Answer: B) 24
Solution:
Let the number be x. x * (25x/100) = 144.
x^2 / 4 = 144 -> x^2 = 576 -> x = 24.
Correct Answer: B) 24
Solution:
Let the number be x. x * (25x/100) = 144.
x^2 / 4 = 144 -> x^2 = 576 -> x = 24.
68. What will be the remainder when 2^31 is divided by 5?
सही उत्तर: [C]
हल (Solution):
Correct Answer: C) 3
Solution:
2^1 = 2 (rem 2)
2^2 = 4 (rem 4)
2^3 = 8 (rem 3)
2^4 = 16 (rem 1). The cycle is of 4.
31 / 4 gives remainder 3. So, 2^3 yields remainder 3.
Correct Answer: C) 3
Solution:
2^1 = 2 (rem 2)
2^2 = 4 (rem 4)
2^3 = 8 (rem 3)
2^4 = 16 (rem 1). The cycle is of 4.
31 / 4 gives remainder 3. So, 2^3 yields remainder 3.
69. The sum of the digits of a two-digit number is 9. If 27 is added to the number, its digits reverse. Find the number.
सही उत्तर: [B]
हल (Solution):
Correct Answer: B) 36
Solution:
Let number be 10x+y. x+y = 9.
10x+y + 27 = 10y+x -> 9y - 9x = 27 -> y - x = 3.
Solving x+y=9 and y-x=3, we get y=6, x=3. Number is 36.
Correct Answer: B) 36
Solution:
Let number be 10x+y. x+y = 9.
10x+y + 27 = 10y+x -> 9y - 9x = 27 -> y - x = 3.
Solving x+y=9 and y-x=3, we get y=6, x=3. Number is 36.
70. A boy was asked to multiply a number by 5/3. By mistake, he divided the number by 5/3. His answer was 16 less than the correct answer. The number was:
सही उत्तर: [A]
हल (Solution):
Correct Answer: A) 15
Solution:
Let number be x. Correct: 5x/3, Incorrect: 3x/5.
5x/3 - 3x/5 = 16 -> (25x - 9x)/15 = 16 -> 16x/15 = 16 -> x = 15.
Correct Answer: A) 15
Solution:
Let number be x. Correct: 5x/3, Incorrect: 3x/5.
5x/3 - 3x/5 = 16 -> (25x - 9x)/15 = 16 -> 16x/15 = 16 -> x = 15.