A person standing at the center of an open ground first walks 32 meters towards the east, takes a right turn and walks 16 meters, takes another right turn and walks 8 meters, and so on. How far will the person be from the original starting point after an infinite number of such walks in this pattern?

Which of the following straight lines are both tangent to the circle \(x^2 + y^2 - 6x + 4y - 12 = 0\)

The equation \(x^2 + y^2 - 2x - 4y + 5 = 0\) represents

Let \(a_1, a_2, a_3\) be three distinct real numbers in geometric progression. If the equations \(a_1x^2 + 2a_2x + a_3 = 0\) and \(b_1x^2 + 2b_2x + b_3 = 0\) have a common root, then which of the following is necessarily true?

If \(\cos\alpha + \cos\beta = 1\), then the maximum value of \(\sin\alpha - \sin\beta\) is

A rabbit is sitting at the base of a staircase which has 10 steps. It proceeds to the top of the staircase by climbing either one step at a time or two steps at a time. The number of ways it can reach the top is

Let p be a positive integer such that the unit digit of \(p^3\) is 4. What are the possible unit digits of \((p+3)^3\)?

In a chess tournament there are 5 contestants. Each player plays against all the others exactly once. No game results in a draw. The winner in a game gets one point and the loser gets zero point. Which of the following sequences cannot represent the scores of the five players?

The probability that a randomly chosen positive divisor of \(10^{2023}\) is an integer multiple of \(10^{2001}\) is