If \(\log_{25}\left[5\log_3\left(1+\log_3\left(1+2\log_2 x\right)\right)\right] = \frac{1}{2}\) then \(x\) is

Let \(P(x)\) be a quadratic polynomial such that \(\begin{vmatrix} P(0) & P(1) \\ P(0) & P(2) \end{vmatrix} = 0\). Let \(P(0) = 2\) and \(P(1) + P(2) + P(3) = 14\). Then \(P(4)\) equals

The area of the triangle, formed by the straight lines \(y = 0\), \(12x - 5y = 0\), and \(3x + 4y = 7\), is

If \(8x^2 - 2kx + k = 0\) is a quadratic equation in \(x\), such that one of its roots is \(p\) times the other, and \(p, k\) are positive real numbers, then \(k\) equals

Consider a triangle with side lengths 4 meters, 6 meters, and 9 meters. A dog runs around the triangle in such a way that the shortest distance of the dog from the triangle is exactly 1 meter. The total distance covered (in meters) by the dog in one round is

Let \(S_1 = \{100, 105, 110, 115, \ldots\}\) and \(S_2 = \{100, 95, 90, 85, \ldots\}\) be two series in arithmetic progression. If \(a_k\) and \(b_k\) are the \(k\)-th terms of \(S_1\) and \(S_2\), respectively, then \(\sum_{k=1}^{20} a_k b_k\) equals

Given that \(1 + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots = \frac{\pi^2}{6}\), the value of \(1 + \frac{1}{3^2} + \frac{1}{5^2} + \frac{1}{7^2} + \cdots\) is

Anindita invests a total of 1 lakh rupees distributed across three schemes A, B and C for a period of two years. These schemes offer an interest rate of 10%, 8% and 12% per annum, respectively, each compounded annually. If the initial investment amount in scheme A is 30000 rupees…

Two swimmers, Ankit and Bipul, start swimming from the opposite ends of a swimming pool at the same time. Ankit can cover the length of the pool once in 10 minutes. Bipul can cover the length of the pool once in 15 minutes. They swim back and forth for 80 minutes without stopping…