Mathematics

The first pipe fills the tank at a rate of 1/20 tank per minute, and the second at 1/24 tank per minute. Let the tank capacity be x gallons. The waste pipe empties 3 gallons per minute. Working together, the net filling rate is x/15 tanks per minute. Setting up the equation: (x/20 + x/24) - 3 = x/15. Solving for x gives 180 gallons.

If three pipes fill the tank in 8 hours, the rate of one pipe is 1/(3*8) = 1/24 of the tank per hour. Two pipes will fill at a rate of 2/24 = 1/12 per hour, so the time taken is 12 hours. However, the calculation shows 12 hours, so option B is correct. Rechecking: One pipe fills in 24 hours, so two pipes fill in 12 hours. Therefore, the correct answer is B, not C. The original correct_answer was B, which is correct. The explanation was missing and is now added. Difficulty is set to easy based on the problem type.

The first quadratic equation x² + 5x + 6 = 0 factors as (x + 2)(x + 3) = 0, giving roots x = -2 and x = -3. The second quadratic equation y² + 9y + 14 = 0 factors as (y + 7)(y + 2) = 0, giving roots y = -7 and y = -2. Both equations share the root -2, but the other roots differ (-3 for x and -7 for y). Since the roots overlap and the other roots do not have a consistent inequality relationship, the relationship between x and y cannot be definitively established. Therefore, the correct answer is D: "If x = y or the relationship between x and y cannot be established."

Tap A fills the tank at a rate of 1/12 per hour, B at 1/15 per hour, and C at 1/20 per hour. Tap A is open continuously, while taps B and C are open alternately for one hour each. In every two-hour cycle, the tank is filled by 2*(1/12) + 1/15 + 1/20 = 1/6 + 1/15 + 1/20 = 17/60 of the tank. After three such cycles (6 hours), the tank is 3 * 17/60 = 51/60 full, leaving 9/60 (or 3/20) of the tank to be filled. In the next hour, tap A and tap B are open together, filling 1/12 + 1/15 = 9/60 = 3/20 of the tank, which completes the filling. Therefore, the total time to fill the tank is 6 + 1 = 7 hours, corresponding to option C.

Pipe A fills the tank in 6 hours, so its filling rate is 1/6 tank per hour. With the leak, it takes 9 hours, so the net filling rate is 1/9 tank per hour. The leak's emptying rate is the difference: 1/6 - 1/9 = 1/18 tank per hour, meaning the leak alone empties the tank in 18 hours.

The cistern normally fills in 8 hours, but with the leak, it takes 10 hours. The leak causes the extra 2 hours delay. Using the formula for combined work rates, the leak empties the cistern in 40 hours.

Let the slower pipe fill the tank in x minutes. Then the faster pipe fills it in x/3 minutes. Their combined rate is 1/x + 3/x = 4/x tanks per minute. Given they fill the tank together in 36 minutes, their combined rate is 1/36. So, 4/x = 1/36, which gives x = 144 minutes. However, this contradicts the initial calculation, so let's re-express carefully: slower pipe rate = 1/x, faster pipe rate = 1/(x/3) = 3/x. Together rate = 1/36, so 1/x + 3/x = 4/x = 1/36, thus x = 144 minutes. Therefore, the slower pipe alone takes 144 minutes to fill the tank. The correct answer is C (144 min).

Solving the first quadratic equation x² + 11x + 30 = 0 gives roots x = -5 and x = -6. Solving the second quadratic y² + 15y + 56 = 0 gives roots y = -7 and y = -8. Since the largest root of x (-5) is greater than the largest root of y (-7), it follows that x > y, making option B correct.

The total number of ways to choose 4 bulbs out of 9 is C(9,4) = 126. The number of ways to choose 4 defective bulbs out of 4 defective is C(4,4) = 1. Therefore, the probability that all four chosen bulbs are defective is 1/126. However, the option 1/126 corresponds to D, but the correct probability is 1/126, so option D is correct. But the explanation shows 1/126, so the correct answer is D, not C. Rechecking: C(9,4)=126, C(4,4)=1, probability=1/126, which matches option D. So the original correct_answer 'D' is correct. The explanation was missing and is now added. The difficulty is easy, and tags added accordingly.

The probability of winning at least one prize when buying 4 tickets is 1 minus the probability of winning no prizes. The probability of no prize on a single ticket is 0.8, so for 4 tickets it's 0.8^4 = 0.4096. Thus, the probability of at least one prize is 1 - 0.4096 = 0.5904. However, the option closest to this correct calculation is B (0.5904), so the original correct_answer B is correct. The initial calculation in explanation mistakenly matched option A's value, but option B is correct.

There are 10 even numbers (2,4,6,8,10,12,14,16,18,20) and 8 prime numbers (2,3,5,7,11,13,17,19) among the first 20 natural numbers. Since 2 is both even and prime, total favorable numbers = 10 + 8 - 1 = 17. Therefore, probability = 17/20 = 0.85 = 4/5.

Pipe A fills the tank at a rate of 1/10 tank per minute, and pipe B empties it at a rate of 1/6 tank per minute. When both pipes are open, the net rate is (1/10 - 1/6) = -1/15 tank per minute, indicating the tank is being emptied. The tank is initially two-fifths full, which is equivalent to 6/15 of the tank. The time to empty the tank is therefore (6/15) ÷ (1/15) = 6 minutes. Hence, the tank will empty in 6 minutes, matching option A.

The single tap fills the tank in 6 hours, so its rate is 1/6 tank per hour. Half the tank is filled in 3 hours. Then, with 4 taps open, the rate is 4 × (1/6) = 2/3 tank per hour. The remaining half tank takes (1/2) ÷ (2/3) = 3/4 hour = 45 minutes. Total time = 3 hours + 45 minutes = 3 hours 45 minutes. However, the question asks for total time to fill the tank completely, which is 3 hours (half tank) + 45 minutes (remaining half) = 3 hours 45 minutes. The correct answer is B.

The total number of socks is 4 + 3 + 5 = 12. The total ways to pick 2 socks is C(12,2) = 66. The ways to pick 2 white socks is C(4,2) = 6, 2 blue socks is C(3,2) = 3, and 2 grey socks is C(5,2) = 10. So, total favorable ways = 6 + 3 + 10 = 19. Therefore, the probability is 19/66.

The quadratic equation 3x² – 7x – 6 = 0 can be solved using the quadratic formula or factoring. Factoring gives (3x + 2)(x - 3) = 0, so the roots are x = -2/3 and x = 3, which corresponds to option B.

A standard deck has 52 cards with 13 cards in each suit. There are 13 spades and 4 kings in total. Since the king of spades is counted in both groups, it should only be counted once. Therefore, the total favorable outcomes are 13 (spades) + 4 (kings) - 1 (king of spades) = 16. The probability is 16/52, which simplifies to 4/13. This matches option D.

Pipe A fills the cistern in 37.5 minutes, so its rate is 1/37.5 per minute; pipe B fills it in 45 minutes, so its rate is 1/45 per minute. Let pipe B be turned off after x minutes. Then, total filled = (rate of A + rate of B) * x + (rate of A) * (30 - x) = 1. Solving for x gives 10 minutes.

The prime factorization of 252 is 2 × 2 × 3 × 3 × 7, which can also be written as 2² × 3² × 7. This matches option A. The previously given answer B (2 × 2 × 2 × 3 × 7) is incorrect because it represents 168, not 252.

The volume of earth dug from the rectangular field equals the volume of the platform formed, adjusted by the increase factor (10/5 = 2). Calculating the platform volume and dividing by the field area gives the required depth. The correct depth is 12 cm, not 10 cm.

The curved surface area (CSA) of a cylinder is given by the formula CSA = 2πrh, where r is the radius and h is the height. Given the radius as 2r and height as 3r, substituting these values gives CSA = 2 × π × 2r × 3r = 12πr². Therefore, the correct curved surface area is 12πr², which corresponds to option C.

The greatest common divisor (GCD) of 80, 144, and 368 is 16. This means each can should contain 16 liters of a single drink to maximize the volume per can without mixing drinks. The number of cans required is calculated by dividing each drink quantity by 16 and summing the results: (80/16) + (144/16) + (368/16) = 5 + 9 + 23 = 37 cans. Therefore, the least number of cans required is 37, which corresponds to option B.

To find the G.C.D of decimal numbers, convert them to integers by multiplying by a power of 10 that removes the decimals. Here, multiply each by 100: 1.08 = 108, 0.36 = 36, 0.9 = 90. The G.C.D of 108, 36, and 90 is 18. Dividing back by 100 gives 0.18. Therefore, the G.C.D of 1.08, 0.36, and 0.9 is 0.18, which corresponds to option C.

The volume of the original sphere is $\frac{4}{3}\pi (10{)}^3=\frac{4000}{3}\pi$. Dividing this volume equally into 8 smaller spheres, each smaller sphere has volume $\frac{4000}{3}\pi ÷8=\frac{500}{3}\pi$. Using the volume formula for a sphere, $\frac{4}{3}\pi r^3=\frac{500}{3}\pi$, solving for $r$ gives $r=5$. The surface area of each smaller sphere is $4\pi r^2=4\pi (5{)}^2=100\pi$.

The volume of soil extracted is calculated by multiplying the length, width, and depth of the pond: 20 m × 10 m × 5 m = 1000 cubic meters.

The volume of a cone is given by the formula (1/3) × base area × height. Given the base area is 30 cm² and the height is 6 cm, the volume = (1/3) × 30 × 6 = (1/3) × 180 = 60 cm³. Therefore, the correct answer is D, 60 cm³.