Mathematics

The average speed for a round trip is given by the formula 2 * (speed1 * speed2) / (speed1 + speed2). Here, it is 2 * (100 * 150) / (100 + 150) = 120 km/hr.

Let the son's current age be x. The father says he was x years old when the son was born, so the father's current age is x + x = 2x. Given the father is 36 now, 2x = 36, so x = 18 (son's current age). Five years ago, the son was 18 - 5 = 13 years old. This matches option A.

The expression (1000)^9 divided by 10^(24) simplifies by converting 1000 to 10^3, so (10^3)^9 = 10^(27). Then 10^(27) / 10^(24) = 10^(27-24) = 10^3 = 1000. However, since option A is 10,000 and option B is 1000, the correct answer is 1000 (option B). The original options had a comma in 10,000 which is 10^4, so the correct answer is B, 1000.

The total work is calculated as 12 men × 6 hours = 72 man-hours. After working for 1 hour, the 12 men complete 12 man-hours of work, leaving 60 man-hours remaining. When 8 more men join, making 20 men in total, the remaining work of 60 man-hours will be completed in 60 ÷ 20 = 3 hours. Therefore, the correct answer is A (3 hours).

The power set of a set with n elements contains 2^n elements. Since the set (1,2,3) has 3 elements, its power set has 2^3 = 8 elements.

Five years ago, the total age of A, B, C, and D was 4 × 45 = 180 years. Now, each has aged 5 years, so their total age is 180 + (4 × 5) = 200 years. The average age of all five now is 49, so total age of all five is 5 × 49 = 245 years. Therefore, E's age is 245 - 200 = 45 years. However, this contradicts the options, so let's re-check: Actually, the total age of A, B, C, and D now is 180 + 20 = 200 years. Total age of all five now is 5 × 49 = 245 years. So, E's age is 245 - 200 = 45 years. The correct answer is 45 years, which corresponds to option C, not D. Therefore, correct_answer is C.

Let the original length be L and width be 2L. Original area = L × 2L = 2L². After increase, length = 1.3L and width = 1.2 × 2L = 2.4L. New area = 1.3L × 2.4L = 3.12L². Percentage increase = ((3.12L² - 2L²) / 2L²) × 100% = (1.12 / 2) × 100% = 56%. None of the given options match this, so the correct answer is 'None of these'.

The sum of the first n positive odd integers is n squared. For n=50, the sum is 50² = 2500. However, the options given do not include 2500 as the correct sum. Actually, option B is 2500, which matches the correct sum. Therefore, the correct answer is B, not A. The original correct_answer was B, which is correct. The explanation was missing, so it has been added.

The number of diagonals in a polygon with n sides is given by the formula n(n-3)/2. For an 8-sided polygon (octagon), substituting n=8 gives 8*(8-3)/2 = 8*5/2 = 20 diagonals. Therefore, the correct answer is option B (20), not D (30).

Given the function P(Z) = Z - A, substituting Z = A gives P(A) = A - A = 0. Therefore, the correct answer is D (0).