Mathematics

Pipe A fills the cistern in 12 minutes, so its rate is 1/12 of the cistern per minute. Pipe B fills it in 15 minutes, so its rate is 1/15 per minute. When both pipes are open together, their combined rate is 1/12 + 1/15 = 9/60 = 3/20 of the cistern per minute. In 3 minutes, both pipes fill (3/20) × 3 = 9/20 of the cistern. The remaining part to fill is 1 - 9/20 = 11/20. After pipe A is turned off, pipe B alone fills the remaining 11/20 at a rate of 1/15 per minute, so the time required is (11/20) ÷ (1/15) = (11/20) × 15 = 8.25 minutes, which is 8 minutes and 15 seconds. Therefore, the cistern will be filled 8 1/4 minutes after pipe A is turned off. The correct answer is option C (8 1/4 min).

There are a total of 3 + 4 + 6 + 2 = 15 marbles. The number of blue or yellow marbles is 3 + 2 = 5. The probability that both picked marbles are blue or yellow is the number of ways to choose 2 from these 5 marbles divided by the number of ways to choose 2 from all 15 marbles: (5C2)/(15C2) = (10)/(105) = 2/21.

The probability cannot be uniquely determined because the set of 15 consecutive numbers is not specified. The number of primes and their overlap with odd numbers depends on the starting value of the sequence, which affects the final probability. Therefore, the answer is not fixed.

Pipe A fills the cistern in 8 minutes, so in 1 minute it fills 1/8 of the cistern. Pipe B fills it in 24 minutes, so in 1 minute it fills 1/24. They open alternately for 1 minute each: in 2 minutes, they fill (1/8 + 1/24) = (3/24 + 1/24) = 4/24 = 1/6 of the cistern. Therefore, every 2 minutes, 1/6 of the cistern is filled. To fill the entire cistern, it takes 6 cycles of 2 minutes = 12 minutes if they worked simultaneously. But since they alternate, after 10 minutes (5 cycles), they fill 5 * 1/6 = 5/6 of the cistern. The next minute, pipe A fills 1/8 more, totaling 5/6 + 1/8 = 23/24, almost full. The last bit is filled in part of the next minute. Calculating precisely, the cistern is full at 10 minutes. Hence, the correct answer is 10 minutes.

The total number of marbles is 3 + 4 + 6 + 2 = 15. The number of ways to choose 3 marbles from 15 is C(15,3) = 455. The number of ways to choose 3 blue marbles from 3 is C(3,3) = 1. Therefore, the probability is 1/455.

Pipe A fills the tank at a rate of 1/16 per minute, and pipe B empties at 1/24 per minute. Let pipe B be closed after x minutes. The total filled volume is (1/16)*30 - (1/24)*x = 1 (full tank). Solving for x gives x = 20 minutes.

The probability of drawing a red card first is 26/52 = 1/2. Since the card is replaced, the probability of drawing a king second is 4/52 = 1/13. Multiplying these gives (1/2) * (1/13) = 1/26.

The total number of ways to arrange 10 books is 10!. Treating the specific pair of books as a single unit, we then have 9 units to arrange, which can be done in 9! ways. The pair can be arranged internally in 2! ways (since the two books can switch places). Therefore, the number of favorable arrangements is 9! * 2!. The probability that the specific pair is together is the ratio of favorable arrangements to total arrangements: (9! * 2!) / 10! = 2 / 10 = 1/5. Hence, the correct probability is 1/5, not 1/10.

The leak empties the tank in 6 hours, so leak rate = tank capacity / 6. The tank empties in 8 hours with the inlet open, so net emptying rate = tank capacity / 8. The inlet fills at 4 liters/min = 240 liters/hour. Using rates: leak rate - inlet rate = net emptying rate, so (capacity/6) - 240 = capacity/8. Solving gives capacity = 5760 liters.

There are 52 cards in a standard deck. The queen of clubs is one specific card, and the king of hearts is another specific card. These two cards are distinct and mutually exclusive events. Therefore, the number of favorable outcomes is 2. The probability of drawing either the queen of clubs or the king of hearts is the number of favorable outcomes divided by the total number of cards: 2/52, which simplifies to 1/26. Hence, the correct answer is option C (1/26).