TNSET 2024 Mathematical Sciences Paper 2 Question Paper (09-Mar-2025) (Shift-2)

Question 1

Define $f(x) = \sum_{n=0}^{\infty} \frac{x^2}{(1+x^2)^n}$ $\forall x \in R, n=0, 1, 2, ......$ then which of the following is true ?

Accepted Answer

$f(x) = \begin{cases} 0 & x = 0 \ 1 & X 
eq 0 \end{cases}$

Answer

$f(x) = 1+x^2 \forall x$

Answer

$f(x) = \begin{cases} 0 & x = 0 \ \frac{1}{1+x^2} & X 
eq 0 \end{cases}$

Answer

$f(x) = 0 \forall x$

Question 2

$\forall x \in [0,1]$ define $f_n (x)=x^n, n=1, 2, ......$ and $f(x) = \begin{cases} 0, & \forall x \in [0,1) \ 1, & x = 1 \end{cases}$Then which of the following is true ?

Accepted Answer

{ $f_n$ } converges pointwise to $f$

Answer

{ $f_n$ } uniformly converges to $f$

Answer

{ $f_n$ } diverges

Answer

{ $f_n$ } Converges pointwise but not to $f$

Question 3

Mean value theorem can be written as follows :$\frac{f(x+h)-f(x)}{h} = f'(x + 	heta h)$ where

Accepted Answer

$0

Answer

$1

Answer

$-\infty

Answer

$-\infty

Question 4

Define $f(x) = \begin{cases} 0, & 	ext{if x is irrational} \ \frac{1}{n}, & 	ext{if x is rational } x = \frac{m}{n}, 	ext{gcd m,n)=1} \end{cases}$Then

Accepted Answer

$f$ is continuous at every irrational point

Answer

$f$ is continuous at every real number

Answer

$f$ is continuous at every rational point

Answer

$f$ is not continuous at any real number

Question 5

Let $R^n$ and $C^n$ denote the set of all n-tuples of real and complex number. Then

Accepted Answer

Both $R^n$ and $C^n$ can be made into normed linear spaces in an infinite number of ways.

Answer

Neither $R^n$ nor $C^n$ is a normed linear space.

Answer

$R^n$ is a normed linear space but $C^n$ is not

Answer

Both $R^n$ and $C^n$ can be made into normed linear spaces in a unique way