Let ABCD be a rectangle with AB = 72 cm and BC = 30 cm. A circle passing through points A and C cuts the side AB at P such that AP = 56 cm. The radius, in cm, of the circle is

Painter A can paint a building in 12 days while Painter B can paint it in 16 days. If A and B work on alternate days, and A starts the work on the first day, then the number of days required to paint the building is

If \(A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\), then the determinant of \(A + A^2 + A^3 + \cdots + A^{13}\) is

The number of integers \(n\) such that \(1 \leq n \leq 10^7\) and \(\gcd(n, 10^7) = 10^3\) is _____

Let the circle \(x^2 + y^2 = 2ax + 2by\) intersect the \(x\)-axis at point \(A(\alpha, 0)\) and \(y\)-axis at point \(B(0, \beta)\), where \(\alpha\beta \neq 0\). If the point \(C(p, q)\) lies on the chord AB, then \(\frac{p+\alpha}{a} + \frac{q+\beta}{b}\) equals

Two locations A and B are at diametrically opposite ends of a circular track. Rekha starts running along the track from location A in the clockwise direction. Sajal starts running simultaneously along the track in the anticlockwise direction from location B. If the length of the…

Three dice are thrown simultaneously and the sum of the three numbers appearing on the top faces of the dice is found to be 10. The probability that these three numbers are distinct, is

If \(n\) is an integer such that \(\frac{|n+6| - |n-3|}{\sqrt{100 - n^3}} \geq 0\), then the number of possible values of \(n\) is _______

If \(x\) is a real number such that \(\max(\min(x, 2-x), x-4, 2x-8) = \pi - 3\), then the number of possible values of \(x\) is