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

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…

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

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

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

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

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

Let \(A(1,3)\) and \(B(5,1)\) be two points. If a line with slope \(m\) intersects AB at an angle of \(45^\circ\), then the possible values of \(m\) are