Mathematics

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.

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'.

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.

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.

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 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.

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 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.

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 number series follows the pattern where each term is obtained by multiplying the previous term by 3 and then subtracting 3. Specifically, aₙ₊₁ = 3 × aₙ - 3. Starting with 3: 3 × 3 - 3 = 6, 3 × 6 - 3 = 15, 3 × 15 - 3 = 42, 3 × 42 - 3 = 123, and 3 × 123 - 3 = 366. Therefore, the missing number in the series is 42 (option C).

The pattern in the series is that the differences between consecutive terms increase by 4 each time: 5-2=3, 12-5=7, 23-12=11, 38-23=15, 57-38=19. The next difference should be 23 (19 + 4), so the next number is 57 + 23 = 80. Therefore, the correct answer is C (80).

The total price of the land is Rs. 6 crore 75 lakhs, which equals Rs. 6.75 crore or Rs. 67,500,000. A 2% commission on this amount is 0.02 × 67,500,000 = Rs. 1,350,000, which corresponds to option A.

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 percentage is 25%, so the new cost price must satisfy: cost price × 1.25 = 150, giving cost price = 120. The cost price increased from 100 to 120, which is a 20% increase. However, this contradicts the options, so let's re-check: The new profit is 25%, so new cost price × 1.25 = new selling price = 150. Thus, new cost price = 150 / 1.25 = 120. The increase in cost price is from 100 to 120, which is 20%. But the options list 20% as A, so correct answer is A. Therefore, correct answer is A (20%).

If 10 men take 12 days to complete the work, the total work is 10 × 12 = 120 man-days. With 12 men working, the time taken is 120 ÷ 12 = 10 days. However, the calculation shows 10 days, which matches option B, so let's recheck carefully. Actually, 10 men × 12 days = 120 man-days. For 12 men, days = 120 ÷ 12 = 10 days. So the correct answer is B, not C. The original correct_answer was 'X', which is invalid. The correct answer is B (10 days).

The total age of A, B, C, D, and E is 40 × 5 = 200 years. The total age of A and B is 35 × 2 = 70 years, and the total age of C and D is 42 × 2 = 84 years. Therefore, the age of E = 200 - (70 + 84) = 46 years. However, since 46 is option B, the correct answer is B, not A. Correction: The age of E is 46 years, which corresponds to option B.

Let Arif's share be x. Hafiz's share is 4x, and Arif's share is one-third of Haider's, so Haider's share is 3x. Total share: x + 4x + 3x = 8x = 2560, so x = 2560/8 = 320. Therefore, Arif's share is Rs 320.

If 10% of the workers were fired, then 90% remained. Since 540 workers survived, 90% corresponds to 540. Therefore, the total number of workers originally = 540 ÷ 0.9 = 600.

A and B together complete the work in 15 days, so their combined work rate is 1/15 per day. B alone completes the work in 20 days, so B's work rate is 1/20 per day. Therefore, A's work rate is (1/15 - 1/20) = (4/60 - 3/60) = 1/60 per day, meaning A alone can complete the work in 60 days. However, since 60 is option A, the correct answer should be A, not D. Rechecking the calculation: A's rate = 1/15 - 1/20 = (4/60 - 3/60) = 1/60, so A alone takes 60 days. Therefore, correct answer is A.

Let the four consecutive two-digit odd numbers be x, x+2, x+4, and x+6. Their sum is 4x + 12. When this sum is divided by 10, the quotient must be a perfect square. We want (4x + 12)/10 = k, where k is a perfect square integer. Thus, 4x + 12 = 10k. Since x is odd and two-digit, testing perfect squares for k shows that k can be 16 or 36. For k=16, sum = 160, so 4x + 12 = 160 => 4x = 148 => x = 37, giving numbers 37, 39, 41, 43. For k=36, sum = 360, so 4x + 12 = 360 => 4x = 348 => x = 87, giving numbers 87, 89, 91, 93. Among the options given (21, 25, 41, 67), only 41 appears in the valid set (37, 39, 41, 43). Therefore, option C (41) is the correct answer.

Twelve men take 6 hours to finish the work, so total work = 12 × 6 = 72 man-hours. After working for 1 hour, 12 men complete 12 man-hours of work, leaving 60 man-hours remaining. Now, 20 men work to complete the remaining 60 man-hours. Time required = 60 ÷ 20 = 3 hours. Therefore, 20 men will take 3 more hours to finish the remaining work, making the correct answer option A (3 hours).

The expression 'ab/c/cd/a' is ambiguous as written, but if interpreted as (ab/c) ÷ (cd/a), it simplifies to (ab/c) × (a/cd) = a^2 b / c^2 d, which matches option B. However, the original expression is unclear due to missing parentheses and ambiguous notation, so the safest choice is 'None of these' unless clarified.

The volume of a sphere is given by the formula V = (4/3)πr³. To find the radius r in terms of the volume V, rearrange the formula: r³ = (3V) / (4π). Taking the cube root of both sides gives r = ³√(3V / 4π). This corresponds exactly to option C. Therefore, the correct answer is C, not B.

The least common multiple (LCM) of 70 and 80 is the smallest number divisible by both. The prime factors of 70 are 2, 5, and 7; for 80, they are 2^4 and 5. Taking the highest powers, LCM = 2^4 × 5 × 7 = 16 × 5 × 7 = 560.

Let the two numbers be 5x and 7x. Their sum is 12x. According to the problem, subtracting 5 from the sum gives 79, so 12x - 5 = 79. Solving for x, we get 12x = 84, so x = 7. Therefore, the numbers are 5*7 = 35 and 7*7 = 49, which corresponds to option D.

To find the sales tax, multiply the item cost by the tax rate: $2000 × 3% = $2000 × 0.03 = $60. Therefore, the correct tax amount is $60, which corresponds to option B.