List of practice Questions

Consider the function \( f : \mathbb{R}^2 \to \mathbb{R}^2 \) given by \[ f(x, y) = (e^{2\pi x} \cos 2\pi y, e^{2\pi x} \sin 2\pi y). \] Then, which of the following is/are TRUE?

Let \( \mathbb{R}^1 \) and \( \mathbb{R}^2 \) be provided with the respective Euclidean topologies, and let \[ S^1 = \{ (x_1, x_2) \in \mathbb{R}^2 : x_1^2 + x_2^2 = 1 \} \] be assigned the subspace topology induced from \( \mathbb{R}^2 \). If \( f: S^1 \to \mathbb{R}^1 \) is a non-constant continuous function, then which of the following is/are TRUE?

Let \( X \) be a random variable with the probability density function \[ f(x) = \begin{cases} c(x - \lfloor x \rfloor), & 0<x<3, \\ 0, & \text{elsewhere}, \end{cases} \] where \( c \) is a constant and \( \lfloor x \rfloor \) denotes the greatest integer less than or equal to \( x \). If \[ A = \left[ \frac{1}{2}, 2 \right], \text{ then } P(X \in A) \, \text{(rounded off to two decimal places) is equal to} \, ________. \]