Mathematics

If 10% of the workers were fired, then 90% remained. Given that 540 workers survived, we can set up the equation 0.9 × total_workers = 540. Solving for total_workers gives 540 ÷ 0.9 = 600 workers originally.

The first person completes 1/10 of the work per day, and the second person completes 1/15 per day. Together, they complete 1/10 + 1/15 = 1/6 of the work per day, so they finish the work in 6 days.

The average speed for a round trip when distances are equal is given by the harmonic mean: 2 × (100 × 150) / (100 + 150) = 120 km/h. Hence, option D is correct.

Twelve men take 6 hours, so total work = 12 × 6 = 72 man-hours. After 1 hour, 12 men complete 12 man-hours, leaving 60 man-hours. Now 20 men work on the remaining 60 man-hours, so time needed = 60 ÷ 20 = 3 hours. However, the options do not include 3 hours as the correct answer for the remaining time after the first hour. Rechecking: After 1 hour, work done = 12 men × 1 hour = 12 man-hours. Remaining work = 72 - 12 = 60 man-hours. With 20 men working, time = 60 ÷ 20 = 3 hours. The correct answer is 3 hours, which corresponds to option A, not C. Therefore, correct_answer should be A.

The number series follows the pattern where each term is obtained by multiplying the previous term by 3 and then subtracting 3. Specifically: 3 × 3 - 3 = 6, 6 × 3 - 3 = 15, 15 × 3 - 3 = 42, 42 × 3 - 3 = 123, and 123 × 3 - 3 = 366. Therefore, the missing number in the series is 42, which corresponds to option C.

Let the original cost price be 100. With 20% profit, the original selling price is 120. After a 25% increase, the new selling price is 120 × 1.25 = 150. The new profit is 25%, so the new cost price must satisfy: new cost price × 1.25 = 150, giving new cost price = 120. The increase in cost price is from 100 to 120, which is a 20% increase. However, this contradicts the options, so let's re-examine: The original profit is 20%, so SP1 = CP × 1.20. After a 25% increase in selling price, SP2 = SP1 × 1.25 = CP × 1.20 × 1.25 = CP × 1.5. The new profit is 25%, so SP2 = new CP × 1.25. Equate: new CP × 1.25 = CP × 1.5 → new CP = (CP × 1.5)/1.25 = CP × 1.2, which is a 20% increase in cost price. Therefore, the percentage increase in cost price is 20%. The correct answer is A, not B. Correction made accordingly.

Let the son's present age be x years. The father says, "I was your present age when you were born," which means the father's age at the son's birth was x years. Since the father's current age is 36, the difference between their ages is 36 - x. This difference is constant and equals the son's current age x. So, 36 - x = x, which gives 2x = 36, hence x = 18 years. Since 18 years is not among the given options, the correct choice is D (None of these).

(1000)^0 equals 1 because any non-zero number raised to the power of zero is 1. 10^24 is a very large number (1 followed by 24 zeros). Adding 1 to 10^24 does not match any of the given numeric options, so the correct choice is 'None of these'.

The total price of the land is Rs. 6 crore 75 lakhs, which is Rs. 6,75,00,000. A 2% commission on this amount is calculated as (2/100) × 6,75,00,000 = Rs. 13,50,000. Hence, option A is correct.

The moment M about point p is calculated by the cross product of the position vector r (from p to Q) and the force vector F. Here, r = i + j + k and F = 2i - j - k. The cross product r × F = (i + j + k) × (2i - j - k) = 3j - 3k.