Mathematics

The cistern normally fills in 9 hours, but due to the leak, it takes 10 hours. The leak causes the filling to slow down by 1 hour. Using the formula for combined work rates, the leak empties the cistern in 90 hours, which corresponds to option D.

When tossing four coins, the total number of outcomes is 2^4 = 16. The number of ways to get exactly two heads and two tails is the number of combinations of 4 coins taken 2 at a time, which is C(4,2) = 6. Therefore, the probability is 6/16 = 3/8.

Let the rate of pipe A be x (tank per hour). Then pipe B's rate is 2x, and pipe C's rate is 4x (since C is twice as fast as B, and B is twice as fast as A). The combined rate of all three pipes is x + 2x + 4x = 7x. They fill the tank together in 5 hours, so their combined rate is 1/5 tank per hour. Thus, 7x = 1/5, giving x = 1/35. Therefore, pipe A alone takes 35 hours to fill the tank. This matches option C.

The first pipe fills the tank in 18 minutes, so its rate is 1/18 tank per minute. The second pipe fills in 15 minutes, rate 1/15. The outlet empties in 45 minutes, rate -1/45. Combined rate = 1/18 + 1/15 - 1/45 = (5/90 + 6/90 - 2/90) = 9/90 = 1/10, so the tank fills in 10 minutes. However, recalculating carefully: 1/18 + 1/15 = (5/90 + 6/90) = 11/90; subtract 1/45 (2/90) gives 9/90 = 1/10. So the correct time is 10 minutes, which corresponds to option D. Therefore, the original correct_answer 'D' is correct. The explanation was missing and is now provided. Difficulty is medium due to rate calculations.

When three dice are thrown, the total outcomes are 6^3 = 216. To have exactly two dice showing the same number and the third different, choose the number for the pair (6 ways), choose which two dice show this number (3 ways), and choose a different number for the third die (5 ways). So favorable outcomes = 6 * 3 * 5 = 90. Probability = 90/216 = 5/12. However, this is the probability that exactly two dice show the same number, which matches option B (5/12). Therefore, the original correct answer B is correct, and the initial correction to A was a mistake. The correct answer is B (5/12).

Among the first 50 natural numbers, there are 50 numbers in total. The composite numbers are those that are not prime and not 1. Counting the composite numbers from 1 to 50 gives 42 composite numbers. Therefore, the probability is 42/50, which simplifies to 21/25. Hence, option A is correct.

The total number of ways to pick 3 fruits from 9 (5 apples + 4 oranges) is C(9,3) = 84. The event 'at least 2 apples' includes picking exactly 2 apples and 1 orange or exactly 3 apples. Number of ways to pick exactly 2 apples and 1 orange is C(5,2)*C(4,1) = 10*4 = 40. Number of ways to pick exactly 3 apples is C(5,3) = 10. Total favorable outcomes = 40 + 10 = 50. Therefore, the probability = 50/84 = 25/42, which matches option A.

The cistern normally fills in 8 hours, so its filling rate is 1/8 per hour. With the leak, it takes 10 hours, so the net filling rate is 1/10 per hour. The leak's emptying rate is the difference: 1/8 - 1/10 = 1/40 per hour, meaning the leak alone empties the cistern in 40 hours.

A leap year has 366 days, which is 52 full weeks plus 2 extra days. For the year to have 53 Sundays and 52 Mondays, one of the extra days must be Sunday and the other Monday. The possible pairs of extra days are (Sunday, Monday), (Monday, Tuesday), (Tuesday, Wednesday), (Wednesday, Thursday), (Thursday, Friday), (Friday, Saturday), and (Saturday, Sunday). Only one of these pairs includes Sunday and Monday in that order, so the probability is 1/7.

Let the time taken by the first pipe be x hours. Then the second pipe takes (x - 5) hours, and the third pipe takes (x - 9) hours (since the second pipe is 4 hours slower than the third). The first two pipes together fill the tank in the same time as the third pipe alone. Using the rates, 1/x + 1/(x-5) = 1/(x-9). Solving this equation gives x = 15 hours for the first pipe.

The quadratic equation 2x² + 5x + 2 = 0 factors as (2x + 1)(x + 2) = 0, giving roots x = -1/2 and x = -2, which corresponds to option A.

The volumes of the smaller cubes are 3³=27, 4³=64, and 5³=125, totaling 216 cm³. The large cube formed has a volume of 216 cm³, so its side length is 6 cm (since 6³=216). The total surface area of the smaller cubes is 6(3² + 4² + 5²) = 6(9 + 16 + 25) = 6 × 50 = 300 cm². The surface area of the large cube is 6 × 6² = 6 × 36 = 216 cm². Therefore, the ratio of total surface areas of smaller cubes to the large cube is 300:216, which simplifies to 25:18. However, this matches option C, but the question asks for the ratio of total surface areas of smaller cubes to the large cube, which is 25:18 (option C). So the original correct answer C is correct, and the explanation supports it.

The total number of ways to choose 5 people from 10 (5 couples) is $\left(\genfrac{}{}{0px}{}{10}{5}\right)=252$. To find the probability of selecting at least two couples, we count the number of 5-person groups that contain at least two complete couples. This involves combinatorial counting of groups with exactly two couples and groups with more than two couples (which is not possible here since 5 people cannot contain more than two full couples). The calculation shows the probability is $\frac{5}{21}$, confirming option A as correct.

The perfect cubes between 1 and 100 are 1³=1, 2³=8, 3³=27, 4³=64. There are 4 perfect cubes out of 100 numbers, so the probability is 4/100 = 1/25.

The surface area of a sphere is given by the formula 4πr². If the radius is doubled, the new radius becomes 2r. Substituting this into the formula gives the new surface area as 4π(2r)² = 4π × 4r² = 4 × (4πr²), which is four times the original surface area. Given the original surface area is 2464 cm², the new surface area will be 4 × 2464 = 9856 cm². Therefore, the correct answer is option C (9856 cm²).

The filling rate is 1/4 per hour and the emptying rate is 1/9 per hour. Net filling rate = 1/4 - 1/9 = (9 - 4)/36 = 5/36 per hour. Time to fill = 1 / (5/36) = 36/5 = 7.2 hours. However, the question asks for the time after which the cistern gets filled, so the correct answer is 7.2 hours, which corresponds to option D. Therefore, the original correct answer D is correct, but the explanation was missing and the options need clarification.

Solving the first quadratic equation 9a² + 18a + 5 = 0 using the quadratic formula gives roots a = -1/3 and a = -5/3 (approximately -0.333 and -1.667). Solving the second quadratic equation 2b² + 13b + 20 = 0 gives roots b = -5/2 and b = -4 (approximately -2.5 and -4). Comparing the roots, both roots of a are greater than the corresponding roots of b (since -0.333 > -2.5 and -1.667 > -4). Therefore, the correct comparison is a > b, which corresponds to option A.

Solving the first equation a^3 - 988 = 343 gives a^3 = 1331, so a = 11 (since 11^3 = 1331). Solving the second equation b^2 - 72 = 49 gives b^2 = 121, so b = ±11. Considering the positive root b = 11, we have a = b = 11. Therefore, the relation a ≥ b is true (since a = b). The option C (a < b) is incorrect. Given the options, the correct relation is B (a ≥ b).

Solving the first quadratic equation x² + 9x + 20 = 0 by factoring gives (x + 4)(x + 5) = 0, so the roots are x = -4 and x = -5. Solving the second quadratic equation y² + 5y + 6 = 0 by factoring gives (y + 2)(y + 3) = 0, so the roots are y = -2 and y = -3. Comparing the roots, both roots of x are less than both roots of y (since -5 < -3 and -4 < -2). Therefore, the correct relationship is x < y, which corresponds to option A.

The total number of ways to select 3 persons out of 10 is C(10,3) = 120. The number of ways to select 3 persons with no graduates (i.e., all non-graduates) is C(6,3) = 20. Therefore, the probability of selecting at least one graduate is 1 - (20/120) = 1 - 1/6 = 5/6.

When two dice are thrown, each die has 3 even numbers (2,4,6) and 3 odd numbers (1,3,5). The probability that one die shows an even number and the other shows an odd number is (3/6)*(3/6) + (3/6)*(3/6) = 2*(1/2)*(1/2) = 1/2.

Pipe A fills the tank in 15 minutes, so its rate is 1/15 tank per minute. Pipe B fills the tank in 20 minutes, so its rate is 1/20 tank per minute. Both pipes open for 4 minutes fill (1/15 + 1/20) * 4 = (4/60 + 4/80) = (1/15 + 1/20) * 4 = (4/60 + 3/60) = 7/60 of the tank. Remaining tank = 1 - 7/60 = 53/60. Pipe B alone fills the rest at 1/20 tank per minute, so time = (53/60) / (1/20) = (53/60) * 20 = 17.67 minutes total. Total time = 4 + 13.67 = 17.67 minutes, but this contradicts the options, so rechecking: Actually, the calculation is: Both pipes fill in 4 minutes: (1/15 + 1/20)*4 = (4/60 + 3/60) = 7/60 tank. Remaining = 1 - 7/60 = 53/60. Pipe B alone fills at 1/20 per minute, so time = (53/60) * 20 = 17.67 minutes. Total time = 4 + 17.67 = 21.67 minutes, which is not an option. Re-examining: The initial calculation was incorrect. Let's calculate carefully: Rate A = 1/15, Rate B = 1/20. Combined rate = 1/15 + 1/20 = (4/60 + 3/60) = 7/60 tank per minute. In 4 minutes, they fill 7/60 * 4 = 28/60 = 14/30 = 7/15 tank. Remaining tank = 1 - 7/15 = 8/15. Pipe B alone fills at 1/20 tank per minute, so time to fill remaining = (8/15) / (1/20) = (8/15) * 20 = 160/15 = 10.67 minutes. Total time = 4 + 10.67 = 14.67 minutes = 14 minutes 40 seconds. So the correct answer is D (14 min 40 sec). Explanation corrected accordingly.

The probability of drawing two clubs consecutively without replacement is (13/52) * (12/51) = (1/4) * (12/51) = 12/204 = 1/17. Thus, option B is correct.

The pump fills the tank in 2 hours, so its rate is 1/2 tank per hour. With the leak, it takes 7/3 hours (2 1/3 hours) to fill the tank, so the effective filling rate is 3/7 tank per hour. The leak's draining rate is the difference between the pump's rate and the effective rate: 1/2 - 3/7 = 1/14 tank per hour, meaning the leak alone can drain the tank in 14 hours. However, since the question asks 'The leak can drain all the water of the tank in?', the correct answer corresponds to 14 hours, which is option D. Therefore, the original correct_answer 'D' is correct, but the explanation was missing and has been added. The difficulty is medium as it involves fractional time and rate calculations.

The total number of marbles is 3 + 4 + 6 + 2 = 15. The number of marbles that are not blue is 4 + 6 + 2 = 12. The total ways to pick 4 marbles from 15 is C(15,4) = 1365. The ways to pick 4 marbles with none blue is C(12,4) = 495. Therefore, the probability is 495/1365, which simplifies to 33/91. However, 33/91 corresponds to option B, but the calculation shows 33/91 is correct. Rechecking: 495/1365 = 33/91, so option B is correct. The initial explanation was missing, so this explanation clarifies the correct answer is B.