A cylindrical tank with a radius of 7 meters is being filled with water at a rate of 5 cubic meters per minute. A leak in the bottom of the tank causes water to drain out at a rate proportional to the square root of the water's height. If the tank is initially empty, and the leak rate proportionality constant is 0.1, what is the rate of change of the water height (in meters per minute) when the water level reaches 4 meters?

A biased coin has a probability 'p' of landing heads. You flip the coin 'n' times. Let X be the number of heads. If E[X] = 6 and Var[X] = 2, what is the value of 'n'?

A train travels from city A to city B at a speed of 60 km/h and returns from city B to city A at a speed of 80 km/h. However, during the return trip, it encounters a 30-minute delay. If the total time for the round trip is 6 hours (including the delay), what is the distance between city A and city B in kilometers?

A geometric progression has a first term of 3 and a common ratio of 2. An arithmetic progression has a first term of 5. If the sum of the first four terms of the geometric progression is equal to the sum of the first 'n' terms of the arithmetic progression, and the common difference of the arithmetic progression is an integer, find a possible value for the common difference.

A bag contains 5 red balls, 4 blue balls, and 3 green balls. Three balls are drawn at random without replacement. What is the probability that at least one of the balls is red, given that at least one of the balls is blue?

A function f(x) is defined as f(x) = x^3 + ax^2 + bx + c, where a, b, and c are constants. If f(1) = 2, f(2) = 5, and f(3) = 10, find the value of f(4).

A rectangular garden is surrounded by a path of uniform width. The dimensions of the garden are 10 meters by 20 meters. If the area of the path is equal to the area of the garden, what is the width of the path in meters?

Two pipes, A and B, can fill a tank in 20 minutes and 30 minutes respectively. A third pipe C can empty the tank in 40 minutes. If all three pipes are opened simultaneously, what percentage of the tank will be filled in 10 minutes?

A man invests Rs. P in a scheme that offers compound interest at a rate of 10% per annum, compounded annually. After 2 years, he withdraws Rs. X. After another year, the amount becomes Rs. 1.21P. What is the value of X?

A and B start a business with investments of Rs. 5000 and Rs. 4500 respectively. After 4 months, A withdraws half of his investment, and B withdraws one-third of his investment. At the end of the year, they make a profit of Rs. 8100. What is B's share of the profit?