Mathematics

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.

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.

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.

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.

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 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.

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.

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.

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.

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.