Number System (4 Questions)

Q8. What is the remainder when 33^{27} is divided by 4?

1. 0
2. 1
3. 2
4. 3

Q9. The sum of the first 50 odd natural numbers is:

1. 2,500
2. 2,550
3. 2,600
4. 2,700

Q10. A number when divided by 6 leaves a remainder of 4. What will be the remainder when its square is divided by 6?

1. 0
2. 1
3. 2
4. 4

Q11. How many integers between 1 and 1000 are divisible by both 3 and 5?

1. 66
2. 100
3. 133
4. 200

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Missing Numbers (2 Questions)

Q12. Find the missing number in the series: 3, 7, 15, 31, 63, ___.

1. 127
2. 126
3. 129
4. 130

Q13. What should come in place of the question mark in the series: 2, 6, 12, 20, ?

Geometry (1 Question)

Q14. The area of a triangle is 60 sq. units, and its base is 10 units. What is its height?

1. 10 units
2. 12 units
3. 15 units
4. 20 units

Solutions Number System

Q8 Solution: \(3^{27} \mod 4: \text{Cycle of } 3^n \mod 4 = 3, 1\). Since \(27\) is odd, remainder = \(3\).

Q9 Solution: Sum of first \(n\) odd numbers = \(n^2\). For \(n = 50\): \(50^2 = 2500\).

Q10 Solution: Let number = \(6k + 4\). Square = \( (6k + 4)^2 = 36k^2 + 48k + 16 \). Remainder = \(16 \mod 6 = 4\).

Q11 Solution: Numbers divisible by 15 (LCM of 3 and 5) = \( \lfloor \frac{1000}{15} \rfloor = 66\).

Missing Numbers

Q12 Solution: Series pattern: \(2^n – 1\). \[ 63 = 2^6 – 1 \implies 127 = 2^7 – 1. \]

Q13 Solution: Series difference: \(4, 6, 8, 10…\). Next difference = \(12\). Next term = \(20 + 12 = 32\).

Geometry

Q14 Solution: Area = \( \frac{1}{2} \times \text{base} \times \text{height} \). \[ 60 = \frac{1}{2} \times 10 \times h \implies h = 12. \]