Mathematics

The original total sum of observations is 50 × 36 = 1800. Since the observation 48 was wrongly recorded as 23, the sum was understated by 48 - 23 = 25. Adding this difference to the original sum gives 1800 + 25 = 1825. The corrected mean is then 1825 ÷ 50 = 36.5. Therefore, the correct answer is option C (36.5).

The total runs scored in the first 20 matches is 20 × 40 = 800. In the next 10 matches, the batsman scored an average of 13 runs, so total runs = 10 × 13 = 130. Total runs in all 30 matches = 800 + 130 = 930. Therefore, the average runs in 30 matches = 930 ÷ 30 = 31. Hence, the correct answer is A (31).

Let the original number of boys be n. The total weight is 30n. After adding a boy weighing 35 kg, the average becomes 31 kg with n+1 boys. So, (30n + 35) / (n + 1) = 31. Solving gives n = 6.

The production forms an arithmetic progression starting at 5000 tonnes in January, increasing by 100 tonnes each month. The total production for the year is the sum of 12 terms: S = 12/2 * (2*5000 + (12-1)*100) = 6 * (10000 + 1100) = 6 * 11100 = 66600 tonnes. The average monthly production is 66600 / 12 = 5550 tonnes. However, since the options do not include 5550 as the average but as an option, rechecking the calculation shows the sum is 66600, average 5550, which matches option C. Therefore, the correct answer is C.

First, find the radii from the circumferences: r = C/(2π). For the smaller circle, r = 264/(2π) = 42 meters; for the larger, r = 352/(2π) = 56 meters. Then, calculate areas: A = πr². Smaller area = π×42² = 1764π; larger area = π×56² = 3136π. The difference is 3136π - 1764π = 1372π ≈ 4304 sq m.

The area of a triangle can be found using the formula Area = inradius × semiperimeter. Given the perimeter is 28 cm, the semiperimeter is 14 cm. Multiplying by the inradius 2.5 cm gives an area of 35 cm². However, since 35 cm² is not listed, the closest correct calculation is 2.5 × 14 = 35 cm², so the correct answer should be 35 cm². Since 35 cm² is not an option, the options are incorrect. The best approach is to correct the options to include 35 cm² and select it as correct. Alternatively, if options cannot be changed, then 'None of these' would be correct. But here, options do not include 35 cm², so the correct answer is 'None of these'.

Given the quadratic equation x² – 9x + 20 = 0, the sum of roots a + b = 9 and the product ab = 20. Using the identity a² + b² = (a + b)² - 2ab, we get a² + b² = 9² - 2×20 = 81 - 40 = 41. Therefore, a² + b² + ab = 41 + 20 = 61. Hence, the correct answer is option C.

The area of a trapezium is calculated using the formula: Area = 1/2 × (sum of parallel sides) × height. Here, (20 + 18) = 38 cm, and height = 15 cm, so area = 1/2 × 38 × 15 = 285 cm². However, the calculation shows 285 cm², which corresponds to option C, so the original correct answer C is correct. The explanation was missing and has been added. Difficulty is easy given the straightforward calculation.

The total age of 15 students is 15 × 15 = 225 years. The total age of 5 students is 5 × 14 = 70 years, and the total age of 9 students is 9 × 16 = 144 years. The 15th student's age = 225 - (70 + 144) = 11 years. However, since the sum of 5 and 9 is 14, not 15, the question likely meant 5 students, 9 students, and the 15th student separately. The 15th student's age is 225 - (70 + 144) = 11 years, which corresponds to option A. But option A is 11 years, so correct answer is A, not D. Correction: The sum of 5 and 9 is 14, so the 15th student is the one left. So the age is 225 - (70 + 144) = 11 years. Therefore, correct answer is A.

Let the marks obtained in Physics be P. The total marks in Physics, Chemistry, and Mathematics is P + 150. Therefore, the sum of marks in Chemistry and Mathematics is (P + 150) - P = 150. The average mark in Chemistry and Mathematics is 150 ÷ 2 = 75.