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

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

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

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

Consider the polynomials \(f(x) = ax^2 + bx + c\), where \(a > 0\), \(b, c\) are real, and \(g(x) = -2x\). If \(f(x)\) cuts the x-axis at \((-2, 0)\) and \(g(x)\) passes through \((a, b)\), then the minimum value of \(f(x) + 9a + 1\) is

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

In a city, 50% of the population can speak in exactly one language among Hindi, English and Tamil, while 40% of the population can speak in at least two of these three languages. Moreover, the number of people who cannot speak in any of these three languages is twice the number o…

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

A train left point A at 12 noon. Two hours later, another train started from point A in the same direction. It overtook the first train at 8 PM. It is known that the sum of the speeds of the two trains is 140 km/hr. Then, at what time would the second train overtake the first tra…