$$\begin{gathered} {\text{In }}{\mathbb{R}^n}{\text{, if }}{S_1},{S_2},{S_3} \ldots \,\,{\text{are non - empty closed subsets such that }}{S_1} \supset {S_2} \supset \cdots \; \supset {S_r} \supset \cdots {\text{ and }} \hfill \\ {\text{sup}}\left\{ {\left| {{X_r} - {Y_r}} \right|:{X_r},{Y_r} \in {S_r},r \geqslant 1} \right\} \to 0{\text{ as }}r \to \infty {\text{,then the order of set }}\mathop \cap \limits_{r = 1}^\infty {S_r} = - - - . \hfill \\\ \end{gathered}$$

• A) $$\operatorname{infinite}$$
• B) $${\text{one}}$$
• C) $${\text{finite but more than one}}$$
• D) $${\text{zero}}$$

Let $f\left( {x,y} \right) = \frac{{xy}}{{{x^2} + {y^2}}}$ then the limit of $f$ along the line $y=-x$ as $(x,y)$ approach $(0,0)$ is ______.

• A) $-1/2$
• B) $1/2$
• C) $0$
• D) undefined

${\text{An open disc: }}\left\{ {\left( {x,y} \right):{x^2} + {y^2} < 1} \right\}{\text{ in }}{\mathbb{R}^2}{\text{ is - - - - - - }}{\text{.}}$

• A) ${\text{none of these}}{\text{.}}$
• B) ${\text{connected polygonally}}$
• C) ${\text{disconnected polygonally}}$
• D) ${\text{diconnected}}$

$${\text{In }}{\mathbb{R}^n},{\text{which of the following is true about }}\phi = \left\{ {} \right\}{\text{ and }}A = \left\{ {\left( {{a_1},{a_2}, \ldots ,{a_n}} \right),{a_i} \in \mathbb{R},1 \leqslant i \leqslant n,i \in \mathbb{N}} \right\}?$$

• A) $${\text{Both are Disconnected}}$$
• B) $${\text{Both are Connected}}$$
• C) $$\phi {\text{ is Connected and }}A{\text{ is Disconnected}}$$
• D) $$\phi {\text{ is Disconnected and }}A{\text{ is Connected}}$$

If $\mathop {\lim }\limits_{X \to {X_0}} f\left( X \right)$ exists, then it is _____.

• A) 0
• B) 1
• C) none of these
• D) unique

${\text{A compact set in }}{\mathbb{R}^n}{\text{ is - - - - - - - - - - }}{\text{.}}$

• A) ${\text{closed and bounded}}$
• B) ${\text{open and bounded}}$
• C) ${\text{closed and unbounded}}$
• D) ${\text{open and unbounded}}$

$${\text{Intervals }}\left( {{\text{0,1}}} \right){\text{ and }}\left( {{\text{1,2}}} \right){\text{ are example of disconnected sets in }}\mathbb{R}{\text{ because - - - - - - - }}{\text{.}}$$

• A) $$\left( {{\text{0,1}}} \right) \cap \left( {{\text{1,2}}} \right) = \phi$$
• B) $$\left\{ {{\text{closure of}}\left( {{\text{0,1}}} \right)} \right\} \cap \left( {{\text{1,2}}} \right) = \phi {\text{ and}}\left\{ {{\text{closure of}}\left( {{\text{1,2}}} \right)} \right\} \cap \left( {{\text{0,1}}} \right) = \phi$$
• C)
• D)

$\begin{gathered} {\text{For a non empty closed and bounded subset }}S{\text{ in }}{\mathbb{R}^n},{\text{ if }}{\rm H}{\text{ is the collection open sets such that }} \hfill \\ S \subset \cup \left\{ {H:H \in {\rm H}} \right\},{\text{then by Heine - Borel theorem,}} \hfill \\\ \end{gathered}$

• A) $S \subset \mathop \cup \limits_{\alpha = 1}^n \left\{ {{H_\alpha }:{H_\alpha } \in {\rm H}} \right\}$
• B) $S \subset \mathop \cup \limits_{\lambda \in \Lambda } \left\{ {{H_\lambda }:{H_\lambda } \in {\rm H}} \right\}$
• C) $S \subset \mathop \cup \limits_{\alpha = 1}^\infty \left\{ {{H_\alpha }:{H_\alpha } \in {\rm H}} \right\}$
• D) ${\text{All above are equally valid}}$

$${\text{Set of isolated point(s) of the complement of set }}\left\{ {\left( {x,y} \right): - n < x,y < n,\left( {x,y} \right) \ne \left( {0,0} \right),n \in \mathbb{N}} \right\}{\text{ in }}{\mathbb{R}^{\text{2}}},{\text{is}} - - - - .$$

• A) $$\left\{ {\left( {x,y} \right) \in {\mathbb{R}^2}:x = y = \left| n \right|,n \in \mathbb{N}} \right\}$$
• B) $$\left\{ {\left( {x,y} \right) \in {\mathbb{R}^2}:x = y = \left| n \right|,n \in \mathbb{N}} \right\} \cap \left\{ {\left( {0,0} \right)} \right\}$$
• C) $$\left\{ {\left( {0,0} \right)} \right\}$$
• D) $$\left\{ {\left( {x,y} \right) \in {\mathbb{R}^2}:x = y = \left| n \right|,n \in \mathbb{N}} \right\} \cup \left\{ {\left( {0,0} \right)} \right\}$$

${\text{The set }}\left\{ {\left( {x,y} \right): - n < x,y < n,\left( {x,y} \right) \ne \left( {0,0} \right),n \in \mathbb{N}} \right\}{\text{ is - - - - - - in }}{\mathbb{R}^{\text{2}}}.$

• A) neither open nor closed
• B) open
• C) closed
• D) both open or closed

$${\text{In }}{\mathbb{R}^2},{\text{ the set }}\left\{ {\left( {x,y} \right):\left( {{x^2} + {y^2} \leqslant a} \right)\,\, \vee \,\left( {{x^2} + {y^2} \geqslant b} \right),a < b} \right\}\,{\text{is a region}}{\text{.}}$$

• A) True
• B) False
• C)
• D)

The limit of the function $f(x) = \frac{{xy}}{{{x^2} + {y^2}}}$ by letting $\left( {x,{\rm{ }}y} \right)$approach $\left( {0,{\rm{ }}0} \right)$along the line $y{\rm{ }} = {\rm{ }}x$ is ________.

• A) Undefined
• B) $$- \frac{1}{2}$$
• C) Finite
• D) $$\frac{1}{2}$$

$${\text{If }}{\mathbb{R}^n}{\text{ is connected, such that }}{\mathbb{R}^n} = A \cup B{\text{ with }}\bar A \cap B = A \cap \bar B = \phi ,{\text{then - - - - - }}{\text{.}}$$

• A) $${\text{All above can be concluded}}$$
• B) $${\text{Either }}A = {\mathbb{R}^n},B = \phi {\text{ or }}B = {\mathbb{R}^n},A = \phi$$
• C) $$A{\text{ and }}B{\text{ are both open and closed}}$$
• D) $${\text{ }}\bar A \subset A,\bar B \subset B$$

A set “ S ” is polygonally connected if, ------- pair of points in S can be connected by a polygonal path lying -------- in “ S ”.

• A) every, partially
• B) some, entirely
• C) every , entirely
• D) some, partially

${\text{If }}\phi \ne S \subseteq {\mathbb{R}^n},{\text{ then the set }}S{\text{ is bounded if - - - - - - - }}{\text{.}}$

• A) ${\text{sup}}\left\{ {\left| {X - Y} \right|:X,Y \in S} \right\} < \infty \,$
• B) $\inf \left\{ {\left| {X - Y} \right|:X,Y \in S} \right\} < \infty$
• C) $\inf \left\{ {\left| {X - Y} \right|:X,Y \in S} \right\} = \infty$
• D) ${\text{sup}}\left\{ {\left| {X - Y} \right|:X,Y \in S} \right\} = \infty$

$\begin{gathered} {\text{In }}{\mathbb{R}^3},{\text{ the lines }}{{\text{L}}_{\text{1}}}:X = \left( {2, - 1,5} \right) + \alpha \left( {2, - 1,3} \right){\text{ and }}{{\text{L}}_2}:X = \left( {2, - 1,5} \right) + \beta \left( { - 5,\frac{5}{2}, - \frac{{15}}{2}} \right){\text{ are traversed}} \hfill \\ {\text{in - - - - - - - - - directions, where }} - \infty < \alpha ,\beta < \infty . \hfill \\\ \end{gathered}$

• A) ${\text{opposite}}$
• B) ${\text{perpendicular}}$
• C) ${\text{oblique}}$
• D) ${\text{same}}$

$${\text{Which of the following non - empty subset on Real line }}\mathbb{R}{\text{ is taken as }}region\,?$$

• A) $${\text{Range of pointwise or uniform real valued convergent sequences}}$$
• B) $${\text{Natural numbers }}\mathbb{N}$$
• C) $${\text{Rationals }}\mathbb{Q}{\text{ or Irrationals }}{\mathbb{Q}^c}$$
• D) $${\text{Intervals (open, closed, semi open or closed)}}$$

${\text{In }}{\mathbb{R}^n},\,{\text{the interior }}{S^0}{\text{ of a non - empty set S is - - - - - - }}{\text{.}}$

• A) ${\text{open - connected (region)}}$
• B) ${\text{open - disconnected}}$
• C)
• D)

$${\text{Which of the following non - empty subset on Real line }}\mathbb{R}{\text{ is taken as }}region\,?$$

• A) $${\text{Rationals }}\mathbb{Q}{\text{ or Irrationals }}{\mathbb{Q}^c}$$
• B) $${\text{Range of pointwise or uniform real valued convergent sequences}}$$
• C) $${\text{Natural numbers }}\mathbb{N}$$
• D) $${\text{Intervals (open, closed, semi open or closed)}}$$

$${\text{Set of isolated point(s) of the complement of set }}\left\{ {\left( {x,y} \right): - n < x,y < n,\left( {x,y} \right) \ne \left( {0,0} \right),n \in \mathbb{N}} \right\}{\text{ in }}{\mathbb{R}^{\text{2}}},{\text{is}} - - - - .$$

• A) $$\left\{ {\left( {x,y} \right) \in {\mathbb{R}^2}:x = y = \left| n \right|,n \in \mathbb{N}} \right\} \cup \left\{ {\left( {0,0} \right)} \right\}$$
• B) $$\left\{ {\left( {x,y} \right) \in {\mathbb{R}^2}:x = y = \left| n \right|,n \in \mathbb{N}} \right\}$$
• C) $$\left\{ {\left( {x,y} \right) \in {\mathbb{R}^2}:x = y = \left| n \right|,n \in \mathbb{N}} \right\} \cap \left\{ {\left( {0,0} \right)} \right\}$$
• D) $$\left\{ {\left( {0,0} \right)} \right\}$$