Mathematics

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.

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.

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.

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.

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.

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 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.

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.

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 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.