Mathematics

The total number of groups of 3 boys from 5 boys (A, B, C, D, E) is $\left(\genfrac{}{}{0ex}{}{5}{3}\right)=10$. The groups that contain both C and D are those that include C, D, and one of the other boys (A, B, or E), which are 3 groups: ACD, BCD, and CDE. Excluding these 3 groups leaves 10 - 3 = 7 groups that do not contain both C and D together. Therefore, the maximum number of such different groups is 7, which corresponds to option C.

Vessel P has 62.5% alcohol, so in 2 litres, alcohol = 2 × 0.625 = 1.25 litres. Vessel Q has 87.5% alcohol, so in 4 litres, alcohol = 4 × 0.875 = 3.5 litres. Total alcohol = 1.25 + 3.5 = 4.75 litres. Total volume = 2 + 4 = 6 litres, so water = 6 - 4.75 = 1.25 litres. The ratio of alcohol to water = 4.75 : 1.25 = 19 : 5. However, the options given do not include 19:5 as a correct ratio for alcohol to water; option D is 19:5 but the question asks for the ratio of alcohol to water, which matches option D. Therefore, the original correct answer D is correct. On rechecking, the ratio 19:5 is correct for alcohol to water. The initial calculation confirms option D is correct.

The digits given are {2, 3, 5, 1, 7, 9}. To form a four-digit even number without repetition, the last digit must be even. The only even digit available is 2, so the last digit is fixed as 2. For the first digit, we can choose any of the remaining 5 digits (3, 5, 1, 7, 9). For the second digit, we can choose any of the remaining 4 digits, and for the third digit, any of the remaining 3 digits. Therefore, the total number of such numbers is 5 × 4 × 3 × 1 = 60. This matches option A. The original answer D=240 is incorrect, and the explanation in the question was confusing but ultimately supports 60 as the correct count.

Let Arslan have A marbles and Bilal have B marbles. From Arslan's statement: if Bilal gives x marbles to Arslan, both have equal marbles, so A + x = B - x. From Bilal's statement: if Arslan gives twice as many marbles (2x) to Bilal, then Bilal has 30 more than Arslan, so B + 2x = A - 2x + 30. Solving these equations gives x = 4.

A runs 1 km in 4.5 minutes, so speed of A = 1000/4.5 = 222.22 m/min. B runs 1 km in 5 minutes, so speed of B = 1000/5 = 200 m/min. To finish together, A must give B a start distance 'd' such that time taken by A to run 1000 m equals time taken by B to run (1000 - d) m. So, 1000/222.22 = (1000 - d)/200, solving gives d = 125 m.

Let the three consecutive odd integers be x, x+2, and x+4. According to the problem, the sum of the last two integers is 13 more than the first integer: (x+2) + (x+4) = x + 13. Simplifying, 2x + 6 = x + 13, so x = 7. Therefore, the integers are 7, 9, and 11, which corresponds to option D. However, checking the sum: 9 + 11 = 20, and 7 + 13 = 20, so the condition holds true for option D, confirming it is correct.

The grocer wants to gain 10% by selling the mixture at Rs. 68.20 per kg. First, calculate the cost price of the mixture: Cost Price = Selling Price / (1 + Gain %) = 68.20 / 1.10 = Rs. 62 per kg. Using the allegation method to find the ratio of the two teas worth Rs. 60 and Rs. 65 per kg to get a mixture costing Rs. 62 per kg: Ratio = (65 - 62) : (62 - 60) = 3 : 2. Therefore, the grocer must mix the two varieties of tea in the ratio 3:2. This matches option A, confirming that the correct answer is A.

Each question has 2 possible answers (True or False), so for 10 questions, the total number of different answer sequences is 2^10 = 1024. This ensures no two candidates have identical sequences if the number of candidates is at most 1024.

Since there are 12 people and one dinner is hosted each month, each person will host exactly one dinner per year (12 dinners / 12 people = 1 dinner per person).

Initially, the mixture contains 20% water, so the volume of water is 20% of 150 liters = 30 liters. Let x be the amount of water to add. The new total volume is 150 + x liters, and the new water volume is 30 + x liters. We want the water to be 25% of the new mixture, so (30 + x) / (150 + x) = 0.25. Solving this equation: 30 + x = 0.25(150 + x) → 30 + x = 37.5 + 0.25x → x - 0.25x = 37.5 - 30 → 0.75x = 7.5 → x = 10 liters. Therefore, 10 liters of water should be added, which corresponds to option C.