Mathematics

Pipe A fills 1/5 of the tank per hour, pipe B fills 1/10, and pipe C fills 1/30. Combined, they fill 1/5 + 1/10 + 1/30 = (6 + 3 + 1)/30 = 10/30 = 1/3 of the tank per hour. Therefore, the tank will be filled in 3 hours. However, the calculation shows 3 hours, but the sum is 1/3, so the time is 3 hours, which corresponds to option C. Rechecking the sum: 1/5=0.2, 1/10=0.1, 1/30≈0.0333; sum=0.3333, so time=1/0.3333=3 hours. So the original correct answer C is correct. The explanation is now provided and difficulty is set to medium.

Solving the first equation a² + 8a + 16 = 0 can be rewritten as (a + 4)² = 0, which gives a single repeated root a = -4. Solving the second equation b² - 4b + 3 = 0 factors as (b - 1)(b - 3) = 0, giving two roots b = 1 and b = 3. Comparing the roots, a = -4 is less than both b = 1 and b = 3, so the relationship is a < b. Therefore, the correct answer is option A: If a < b.

The external dimensions of the cistern are 3.3 m by 2.6 m by 1.1 m, with walls 5 cm thick. Converting to centimeters: length = 330 cm, width = 260 cm, height = 110 cm. The walls reduce the internal length and width by twice the wall thickness (5 cm on each side), so internal length = 330 - 10 = 320 cm, internal width = 260 - 10 = 250 cm. The internal volume is given as 8000 liters, which is 8,000,000 cubic centimeters. Using the volume formula: 320 × 250 × internal height = 8,000,000, solving gives internal height = 100 cm. The external height is 110 cm, so the thickness of the bottom is 110 - 100 = 10 cm, which is 1 dm. Therefore, the correct answer is option B (1 dm).

Pipe P fills 1/12 of the cistern per minute, and pipe Q fills 1/15 per minute. Together, they fill (1/12 + 1/15) = 9/60 = 3/20 of the cistern per minute. After 3 minutes, they fill 3 * 3/20 = 9/20 of the cistern. Remaining is 11/20. Pipe Q alone fills 1/15 per minute, so time to fill remaining is (11/20) / (1/15) = 11/20 * 15 = 8.25 minutes or 8 1/4 minutes. Adding the initial 3 minutes gives total time 11 1/4 minutes, so the cistern takes 8 1/4 minutes longer after pipe P is turned off.

Solving the first quadratic equation a² - 13a + 42 = 0 by factoring gives roots a = 6 and a = 7. Solving the second quadratic equation b² - 15b + 56 = 0 by factoring gives roots b = 7 and b = 8. Comparing the roots, the possible values of a are 6 or 7, and the possible values of b are 7 or 8. Therefore, a can be less than or equal to b, but a > b is not always true. Hence, the correct inequality is a ≤ b, which corresponds to option D.

The quadratic equation x² + x - 42 = 0 can be factored as (x + 7)(x - 6) = 0. Setting each factor equal to zero gives the roots x = -7 and x = 6. These roots correspond exactly to option D (-7, 6). Therefore, the correct answer is D, not A.

The combined filling rate of the two pipes is (1/20 + 1/30) = 1/12 tank per hour. Due to the leak, only two-thirds of this water remains, so the effective filling rate is (2/3) × (1/12) = 1/18 tank per hour. Therefore, the total time to fill the tank is 18 hours.

Pipe A fills the tank at a rate of 1/5 tank per hour, Pipe B fills at 1/6 tank per hour, and Pipe C empties at 1/12 tank per hour. The combined filling rate when all three pipes are open is (1/5 + 1/6 - 1/12) = (12/60 + 10/60 - 5/60) = 17/60 tank per hour. Therefore, the time taken to fill the tank is the reciprocal of the combined rate, which is 60/17 hours, approximately 3.529 hours or 3 hours 31 minutes. This matches option C (3 9/17 hours). The initially given correct answer B is incorrect based on this calculation.

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.

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.