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. The number of times they meet each other 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

If \(a_1, a_2, \ldots, a_8\) are the roots of the equation \(x^8 + x^7 + \cdots + x + 1 = 0\), then the value of \(a_1^{2025} + a_2^{2025} + \cdots + a_8^{2025}\) is

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

Which of the following numbers is divisible by \(3^{10} + 2\) ?

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

A circle touches the \(y\)-axis at \((0,4)\) and passes through the point \((-2,0)\). Then the radius of the circle is

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

If \(y = a + b\log_e x\) then which of the following is true?