1. The problem asks how many pairs of twin primes exist between 1 and 100. Twin primes are pairs of primes that differ by 2. The pairs are (3,5), (5,7), (11,13), (17,19), (29,31), (41,43), (59,61), (71,73). Counting these pairs, there are 8. 2. Find the missing term in the series 4, 12, 36, __, 324, 972. Notice the ratio between consecutive terms: 12/4 = 3, 36/12 = 3. So each term is multiplied by 3. Therefore, the missing term is 36 × 3 = 108. 3. Determine the largest among fractions 3/13, 2/15, 4/17. Convert to decimal: 3/13 ≈ 0.2307, 2/15 ≈ 0.1333, 4/17 ≈ 0.2353. Largest is 4/17. 4. Find HCF of 24, 30, and 42. Prime factors: 24 = 2³ × 3 30 = 2 × 3 × 5 42 = 2 × 3 × 7 Common prime factors: 2 × 3 = 6. Hence, HCF is 6. 5. Find LCM of 0.6, 9.6, and 0.12. Convert to whole numbers by multiplying all by 100: 0.6 → 60 9.6 → 960 0.12 → 12 Find LCM of 60, 960, and 12. Prime factors: 60 = 2² × 3 × 5 960 = 2⁶ × 3 × 5 12 = 2² × 3 LCM = 2⁶ × 3 × 5 = 64 × 3 × 5 = 960 Dividing by 100 to return to decimals, LCM = 9.6. 6. Cube root of -5832. Calculate cube root: 18³ = 5832. Since number is negative, cube root is -18. 7. Given 272 × 425 ÷ p² = 400, find p. Calculate 272 × 425 = 115600. So, 115600 ÷ p² = 400 → p² = 115600 ÷ 400 = 289. Therefore, p = √289 = 17. 8. Amount doubles with simple interest in 4 years, find annual interest rate. Formula: SI = (P × R × T)/100 Given final amount = 2P, SI = P. So, P = (P × R × 4)/100 → R = 100/4 = 25%. 9. Time taken by train to cover 200 km at 10 km/h. Time = Distance / Speed = 200 / 10 = 20 hours. 10. Length of the train passing a standing man in 90 meters. Length equals distance covered, so length = 90 m. 11. Upstream speed of boat with speed 8 km/h and stream 4 km/h. Upstream speed = Boat speed - Stream speed = 8 - 4 = 4 km/h. Final answers: 1: 8 2: 108 3: 4/17 4: 6 5: 9.6 6: -18 7: 17 8: 25% 9: 20h 10: 90m 11: 4 km/h