If the angles \(A, B, C\) of a triangle are in arithmetic progression such that \(\sin(2A + B) = \frac{1}{2}\), then \(\sin(B + 2C)\) is equal to

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

The unit digit in \((743)^{85} - (525)^{37} + (987)^{96}\) is

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 \(a_3 : a_5\)?

The set of all real values of \(p\) for which the equation \(3\sin^2 x + 12\cos x - 3 = p\) has at least one solution 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

ABCD is a quadrilateral whose diagonals AC and BD intersect at O. If triangles AOB and COD have areas 4 and 9 respectively, then the minimum area that ABCD can have 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 \(f(2021)\).

The highest possible value of the ratio of a four digit number and the sum of its four digits is: