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

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

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

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

From the passage: Directions (Q.41-Q.45): A pharmaceutical company has tested five drugs...

Organism R can be treated with

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 w…

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

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?

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