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

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

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

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

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

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

{\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

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

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

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

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

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

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

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

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

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

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

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

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

• 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) \[\left\{ {\left( {0,0} \right)} \right\}\]
• D) \[\left\{ {\left( {x,y} \right) \in {\mathbb{R}^2}:x = y = \left| n \right|,n \in \mathbb{N}} \right\} \cap \left\{ {\left( {0,0} \right)} \right\}\]

{\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)

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

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

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

Which statement(s) is(are) true about the function f(x,y) = \frac{{{x^2} + {y^2}}}{{x - y}},

• A) f is differentiable every where except at the points where y = x
• B) {f_y}(x,y) = \frac{y}{{x - y}} + \frac{{{x^2} + {y^2}}}{{{{(x - y)}^2}}}.
• C) {f_y}(x,y),{\text{ }}{f_x}(x,y) are continuous everywhere.
• D) {f_x}(x,y) = \frac{{2x}}{{x - y}} - \frac{{{x^2} + {y^2}}}{{{{(x + y)}^2}}}

{\text{How many }}third{\text{ order partial derivatives of }}g\left( {x,y} \right) = xy + {x^2}{y^3}{\text{ exist in }}{\mathbb{R}^3}?

• A) 3
• B) 9
• C) 6
• D) 8

If f\left( {x,y,z} \right) = \cos \left( {\frac{1}{{{x^2} + 2{y^2} + {z^2}}}} \right) then \mathop {\lim }\limits_{\left| X \right| \to \infty } f\left( X \right) = \_\_\_\_\_

• A) 0
• B) infinite
• C) 1
• D) -1