Let \(X\) be a connected metric space. Then which of the following is true ?

Let \(X, Y\) be metric spaces. Suppose \(f, g : X \to Y\) are continuous functions and \(D = \{x \mid f(x) = g(x)\}\). Then \(f \equiv g\) if \(D\) is :

Which of the following statements is not true in any metric space ?

The series :

\(\frac{2}{1} + \frac{3}{4} + \frac{4}{9} + \ldots + \frac{n+1}{n^2} + \ldots\)

For any two sequences \(\{x_n\}\) and \(\{y_n\}\), it is known that,

\(\liminf x_n + \liminf y_n \le \liminf (x_n + y_n) \le \limsup (x_n + y_n) \le \limsup x_n + \limsup y_n \quad (*)\)

Suppose that \(x_n = \begin{cases} 0 & \text{if } n \text{ is even} \\ 1 & \text{if } n \text{ is odd} \end{cases}\) and \(y_n = 1 - x_n\), for all \(n \ge 1\).

Then, which of the following statements is/are true for \(\{x_n\}\) and \(\{y_n\}\) ?

\(S_1\) : strict inequalities hold in \((*)\) above.

\(S_2\) : \(\{x_n + y_n\}\) converges, while \(\{x_n\}\) and \(\{y_n\}\) do not.

Suppose that \(\{f_n\}\) is a sequence of functions defined by :

\(f_n(x) = \frac{x^n}{1 + x^n}, \ 0 \le x \le 1.\)

Then, which of the following statements is true ?

\(S_1 : f_n \to f \equiv 0\) uniformly on \([0, 1]\)

\(S_2 : \sup_{0 \le x \le \frac{1}{2}} f_n(x) \to F(x) \equiv 0\) on \([0, \frac{1}{2}]\)

\(S_3 : f_n \to f = 0\) uniformly on \([0, \frac{1}{2}]\)

Suppose that \(f\) is a function defined by :

\(f(x) = \frac{x}{3 - 2x}, \ 0 < x < 1\)

If \(g(x) = f''(x) + f'(x) - [f'(x)]^2\), then \(g(1)\) is equal to :

Suppose that \(f\) is a function on \(\mathbb{R}^2\) defined by :

\(f(x, y) = \begin{cases} \frac{xy}{x^2 + y^2}, & (x, y) \neq (0, 0) \\ 0, & (x, y) = (0, 0) \end{cases}\)

Then, which of the following statements is/are true ?

\(S_1\) : \(f\) is continuous at \((0, 0)\).

\(S_2\) : Both partial derivatives exist at \((0, 0)\).

Consider the function \(f\) defined by :

\(f(x) = \begin{cases} \frac{\lambda \cos(x)}{\pi/2 - x}, & x \neq \pi/2 \\ 1, & x = \pi/2 \end{cases}\)

What should be the value of \(\lambda\) so that \(f\) would be continuous at \(x = \pi/2\) ?

.................... is a client server application that identifies each host on the internet with a unique user-friendly name.

Every OS has a ................ or .......... which primarily resides in the main memory of the computer to perform some of the basic functions of the OS.

(i) Kernel

(ii) Interface

(iii) Nucleus

(iv) Compiler

Codes:

Which of the following are referred to as low level language?

(i) Machine language

(ii) Assembly language

(iii) High level language

(iv) Natural language

Codes:

Match the following:

Codes: