There are two taps, T1 and T2, at the bottom of a water tank, either or both of which may be opened to empty the water tank, each at a constant rate. If T1 is opened keeping T2 closed, the water tank (initially full) becomes empty in half an hour. If both T1 and T2 are kept open, the water tank (initially full) becomes empty in 20 minutes. Then, the time (in minutes) it takes for the water tank (initially full) to become empty if T2 is opened while T1 is closed is ______

There are 5 parallel lines on the plane. On the same plane, there are \(n\) other lines which are perpendicular to the 5 parallel lines. If the number of distinct rectangles formed by these lines is 360, what is the value of \(n\)?

A class consists of 30 students. Each of them has registered for 5 courses. Each course instructor conducts an exam out of 200 marks. The average percentage marks of all 30 students across all courses they have registered for, is 80%. Two of them apply for revaluation in a course…

It is given that the sequence \(\{x_n\}\) satisfies \(x_1 = 0\), \(x_{n+1} = x_n + 1 + 2\sqrt{1 + x_n}\) for \(n = 1, 2, \dots\). Then \(x_{31}\) is ______

What is the minimum number of weights which enable us to weigh any integer number of grams of gold from 1 to 100 on a standard balance with two pans? (Weights can be placed only on the left pan.)

The sum up to 10 terms of the series \(1\cdot 3 + 5\cdot 7 + 9\cdot 11 + \dots\) is ______

If one of the lines given by the equation \(2x^2 + axy + 3y^2 = 0\) coincides with one of those given by \(2x^2 + bxy - 3y^2 = 0\), and the other lines represented by them are perpendicular, then \(a^2 + b^2\) is ______

The number of positive integers which divide \((1890) \cdot (130) \cdot (170)\) and are not divisible by 45 is ______

If a function \(f(a) = \max(a, 0)\), then the smallest integer value of \(x\) for which the equation \(f(x - 3) + 2f(x + 1) = 8\) holds true is ______