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 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

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

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 difference between the maximum real root and the minimum real root of the equation \((x^2-5)^4 + (x^2-7)^4 = 16\) 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 \(\theta\) is the angle between the pair of tangents drawn from the point \(A\left(0, \frac{7}{2}\right)\) to the circle \(x^2 + y^2 - 14x + 16y + 88 = 0\), then \(\tan\theta\) 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

The numbers \(2^{2024}\) and \(5^{2024}\) are expanded and their digits are written out consecutively on one page. The total number of digits written on the page is