MH SET 2023 Paper 2 Mathematical Science Question Paper (26-Mar-2023)

Question 1

Consider the sets given by$X = \{2m + 1 / m \in \mathbb{Z}\}$ and$Y = \{2n + 3 / n \in \mathbb{Z}\}$.Then :

Accepted Answer

X = Y

Answer

X is a proper subset of Y

Answer

Y is a proper subset of X

Answer

Either X is finite or Y is finite

Question 2

The series$1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \ldots\ldots$

Accepted Answer

converges to $\log_e^{2}$

Answer

is divergent

Answer

converges to zero

Answer

converges to $\log_e^{1}$

Question 3

The improper integral $\int_0^{\infty} \frac{x^{s-1}}{1 - x}$ is convergent if and only if :

Accepted Answer

$0 < s < 1$

Answer

$0 \le s \le 1$

Answer

$0 < s \le 1$

Answer

$0 \le s < 1$

Question 4

Let X and Y be metric spaces and $f, g : X \to Y$ be continuous. Then $\{x \mid f(x) = g(x)\}$ is :

Accepted Answer

Closed

Answer

Open

Answer

Connected

Answer

Compact

Question 5

Which of the following pair X, Y of sets do not have same cardinality, i.e. $|X| = |Y|$ ?

Accepted Answer

X = the set of irrationals, Y = the set of algebraic numbers

Answer

$X = [0, 1]$, $Y = [0, 1] \times [0, 1]$

Answer

$X = \mathbb{R}$, $Y = \mathbb{C}$

Answer

X = a bounded interval $[x, y]$ with $x < y$ in $\mathbb{R}$, $Y = (-\infty, \infty) = \mathbb{R}$