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

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

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 \(S_1 = \{100, 105, 110, 115, \ldots\}\) 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

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

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

Monica, who is 18 years old, is one-third the age of her father. The age at which she will be half the age of her father is ______

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