Let ABC be an equilateral triangle, with each side of length k. If a circle is drawn with diameter AB, then the area of the portion of the triangle lying inside the circle is

If \(\log_4 x = a\) and \(\log_{25} x = b\) then \(\log_x 10\) is

Let ABC be a triangle with AB = AC and D be a point on BC such that \(\angle BAD = 30^\circ\). If E is a point on AC such that AD = AE, then \(\angle CDE\) equals

If \(|x+1| + (y+2)^2 = 0\) and \(ax - 3ay = 1\), then the value of a is

If 5 boys and 3 girls randomly sit around a circular table, the probability that there will be at least one boy sitting between any two girls, is

The angle of elevation of the top of a pole from a point A on the ground is \(30^\circ\). The angle of elevation changes to \(45^\circ\), after moving 20 metres towards the base of the pole. Then the height of the pole, in metres, is

The side AB of a triangle ABC is c. The median BD is of length k. If \(\angle BDA = \theta < 90^\circ\), then the area of triangle ABC is

If \(A = \begin{bmatrix} x_1 & x_2 & 7 \\ y_1 & y_2 & y_3 \\ z_1 & 8 & 3 \end{bmatrix}\) is a matrix such that the sum of all three elements along any row, column or diagonal are equal to each other, then the value of determinant of A is ______

Let \(a = \frac{(\log_7 4)(\log_7 5 - \log_7 2)}{(\log_7 25)(\log_7 8 - \log_7 4)}\). Then the value of \(5^a\) is