Mathematics

Initially, the mixture has 80% milk, so in 180 g, milk = 0.8 × 180 = 144 g. Adding 36 g of pure milk gives total milk = 144 + 36 = 180 g. Total mixture weight = 180 + 36 = 216 g. Percentage of milk = (180/216) × 100 = 83.33%, which rounds to 84%.

The word MEADOWS has 7 letters with vowels E, A, O (3 vowels) and consonants M, D, W, S (4 consonants). The even positions in a 7-letter word are positions 2, 4, and 6 (3 places). The vowels must occupy these 3 even places, so they can be arranged in 3! = 6 ways. The consonants occupy the remaining 4 positions and can be arranged in 4! = 24 ways. Total arrangements = 6 × 24 = 144.

The word 'LOGARITHMS' contains 10 distinct letters. To form 4-letter words without repetition, we calculate the number of permutations of 10 letters taken 4 at a time, which is given by 10P4 = 10 × 9 × 8 × 7 = 5040. Therefore, the correct answer is 5040, corresponding to option C.

The word 'MATHEMATICS' has 11 letters with vowels A, A, E, I (4 vowels) and consonants M, T, H, M, T, C, S (7 consonants). Treating the vowels as a single unit, we have 7 consonants + 1 vowel block = 8 units. The number of arrangements of these 8 units, considering repeated consonants M (2 times) and T (2 times), is 8! / (2! * 2!) = 10080. The vowels inside the block can be arranged in 4! / 2! = 12 ways (since A repeats twice). Total arrangements = 10080 * 12 = 120960.

The problem involves arranging 6 balls (2 red, 2 black, 2 white) in 6 cells so that no two balls of the same color are adjacent. By using combinatorial methods and inclusion-exclusion principle, the total number of valid arrangements is 18.

Using the allegation method, the ratio is calculated as (6.30 - 5.70) : (7.20 - 6.30) = 0.60 : 0.90 = 2 : 3. Hence, the rice priced at Rs.7.20 should be mixed with rice priced at Rs.5.70 in the ratio 2:3.

Using the alligation method, the ratio of the two varieties of sugar costing Rs.18 per kg and Rs.24 per kg to get a mixture costing Rs.20 per kg is calculated as (24 - 20) : (20 - 18) = 4 : 2 = 2 : 1. This means 2 parts of the Rs.18 sugar should be mixed with 1 part of the Rs.24 sugar to get the desired mixture cost. Therefore, the correct ratio is 2:1, which corresponds to option D.

First, arrange the six boys in 6! ways. Then, place the six girls in the 7 possible gaps between and at the ends of the boys so that no two girls sit together. The number of ways to choose and arrange the girls in these 7 positions is 7P6. Therefore, total ways = 6! * 7P6.

Let the present ages of the father, mother, and daughter be F, M, and D respectively. The average age is 35, so (F + M + D)/3 = 35, which gives F + M + D = 105. Fifteen years from now, the mother's age will be M + 15. According to the problem, M + 15 = F + D. Using the first equation, F + D = 105 - M. Substituting, M + 15 = 105 - M, which simplifies to 2M = 90, so M = 45 years.

Initially, the vessel contains 90 litres of pure milk. When 9 litres of milk are removed and replaced with water, the amount of milk left is 90 - 9 = 81 litres. The total volume remains 90 litres, so the milk concentration is now 81/90 = 0.9 (90%). Next, 9 litres of this milk-water mixture is removed. Since the mixture is 90% milk, the amount of milk removed is 9 × 0.9 = 8.1 litres. Therefore, the amount of milk left after the second removal is 81 - 8.1 = 72.9 litres. After replacing the removed 9 litres with water, the total volume remains 90 litres, but the pure milk quantity is 72.9 litres. Hence, the correct answer is option B (72.9 litres).