In a class, 60% and 68% of students passed their Physics and Mathematics examinations respectively. Then at least ______ percentage of students passed both their Physics and Mathematics examinations.

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 ______

Suppose that a real-valued function \(f(x)\) of real numbers satisfies \(f(x + xy) = f(x) + f(xy)\) for all real \(x, y\), and that \(f(2020) = 1\). Compute \(f(2021)\).

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 ______

Suppose that \(\log_2[\log_3(\log_4 a)] = \log_3[\log_4(\log_2 b)] = \log_4[\log_2(\log_3 c)] = 0\). Then the value of \(a + b + c\) 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.)

Let \(S_n\) be the sum of the first \(n\) terms of an A.P. \(\{a_n\}\). If \(S_5 = S_9\), what is the ratio of \(a_3 : a_5\)?

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…

If \(A\), \(B\) and \(A + B\) are non-singular matrices and \(AB = BA\), then \(2A - B - A(A + B)^{-1}A + B(A + B)^{-1}B\) equals