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

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 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

If \(\log_4 x = a\) and \(\log_{25} x = b\) then \(\log_x 10\) 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 \(|x+1| + (y+2)^2 = 0\) and \(ax - 3ay = 1\), then the value 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

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

For some non-zero real values of a, b and c, it is given that \(\left|\frac{c}{a}\right| = 4\), \(\left|\frac{a}{b}\right| = \frac{1}{3}\) and \(\frac{b}{c} = -\frac{3}{4}\). If ac > 0, then \(\left(\frac{b+c}{a}\right)\) equals