Mathematics

To gain 20% by selling at cost price, the spirit is effectively sold at 80% of its original price due to dilution. Let the ratio of water to spirit be x:1. The cost price of the mixture is the cost of spirit only, but the selling price is the cost price of the mixture. Setting up the equation: (1 volume spirit) / (1 + x volume mixture) = 80%, solving gives x = 0.4 or 2:5 water to spirit.

Let the number of rabbits be R and peacocks be P. Each animal has one head, so R + P = 60. Rabbits have 4 legs and peacocks have 2 legs, so 4R + 2P = 192. Solving these equations gives R = 30 and P = 30. Therefore, the number of rabbits is 30.

Each die has faces numbered from 1 to 6, so total outcomes for three dice are 6^3 = 216. The number of outcomes with no dice showing 2 is 5^3 = 125 (since each die can show any of the 5 numbers except 2). Therefore, outcomes with at least one die showing 2 is 216 - 125 = 91. However, the options do not include 91 correctly; the closest correct calculation is 91, which corresponds to option C. Therefore, the correct answer is C.

The total number of sequences for 7 players is 7! = 5040. The number of sequences where the youngest player is last is 6! = 720. Therefore, sequences where the youngest player is not last = 5040 - 720 = 4320. However, 4320 is option C, so the original correct answer C is correct. The explanation was missing and has been added. The topic is permutations, and difficulty is medium due to factorial calculations.

The lock has three rings, each with 6 different letters. The total number of possible combinations is 6 × 6 × 6 = 216. Since only one combination opens the lock, the maximum number of distinct unsuccessful attempts is total combinations minus one, which is 216 - 1 = 215. Therefore, option C (215) is the correct answer.

The word 'MESMERISE' has 9 letters with repetitions: M appears 2 times, E appears 3 times, and S appears 2 times. The number of distinct permutations is calculated using the formula for permutations of multiset: total factorial divided by the factorial of each repeated letter count. Thus, the number of distinct permutations is 9! / (2! * 3! * 2!) = 9! / (2!)^2 * 3!. Option C correctly represents this formula.

Initially, the mixture has 70 litres with 10% water, so water volume is 7 litres. Let x be the litres of water added. The new total volume is 70 + x litres, and the water volume is 7 + x litres. We want the water percentage to be 12.5% (or 1/8), so (7 + x)/(70 + x) = 1/8. Solving gives x = 5 litres.

Each problem has 3 internal choices plus the option to skip it, so there are 4 choices per problem. For 5 problems, the total number of ways to choose is 4^5 = 1024. Since the candidate must attempt one or more problems, the case where all problems are skipped (1 way) is excluded. Therefore, the total number of ways to attempt one or more problems is 1024 - 1 = 1023. Hence, the correct answer is option C (1023).

Let Zahid's present age be Z and Shahid's present age be S. Eight years ago, Z-8 = (4/3)(S-8). Eight years hence, Z+8 = (6/5)(S+8). Solving these two equations gives Z = 40 years. Hence, the correct answer is 40 years.

Let the number of one rupee coins be x and 50 paise coins be y. We have x + y = 93 and 1*x + 0.5*y = 56. Solving these equations gives y = 60, the number of 50 paise coins.