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

• 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

\begin{array}{*{20}{l}} {{\text{Let }}f{\text{ be defined and continuous on a region }}S{\text{ in }}{\mathbb{R}^{\text{n}}}.\,{\text{Suppose that }}{X_1}{\text{ and }}{X_{\text{2}}}{\text{ are in }}S{\text{ and}}\,f\left( {{X_1}} \right){\text{ }} < {\text{ Y }} < {\text{ }}f\left( {{X_2}} \right){\text{. }}} \\\\ {{\text{Then }}f\left( X \right) = Y{\text{ - - - - - - - }}X{\text{ in }}S.} \\\\ {} \end{array}

• A) for all
• B) for some
• C)
• D)

\begin{gathered} {\text{Let }}u{\text{ and }}v{\text{ be functions of two variables with continuous second - order partial derivatives in a region }}S{\text{. }} \hfill \\\ {\text{Suppose that }}{u_x} = {v_y}{\text{ and }}{u_y} = - {v_x}{\text{ in }}S{\text{. Then, }}{u_{xx}} + {u_{yy}} - - - - . \hfill \\\ \hfill \\\\ \end{gathered}

• A) =0
• B) >0 but <1
• C) =1
• D) >0

If f(s) = \sqrt s and g(x,y) = 1 - {x^2} - {y^2} then domain of f and g are ______________________________.

• A) {D_g} = ( - \infty ,\infty ),{D_f} = {\mathbb{R}^2}
• B) {D_f} = [0,\infty ],{D_g} = {\mathbb{R}^2}
• C) {D_f} = [0,\infty ),{D_g} = {\mathbb{R}^2}
• D) {D_f} = [0,\infty ],{D_g} = \mathbb{R}

{\text{In}}\,{\mathbb{R}^2}{\text{, }}f\left( X \right) = {\left| X \right|^2}{\text{ and }}\Phi {\text{ = }}\left( {\frac{1}{{\sqrt 2 }},\frac{1}{{\sqrt 2 }}} \right),{\text{ then }}\frac{{\partial f\left( X \right)}}{{\partial \Phi }} =

• A) \frac{{{x_1} + {x_2}}}{2}
• B) {x_1} + {x_2}
• C) \sqrt 2 \left( {{x_1} + {x_2}} \right)
• D) \frac{{{x_1} + {x_2}}}{{\sqrt 2 }}

\begin{gathered} {\text{In }}{\mathbb{R}^n},{\text{ monotonicity, limits inferior and superior of sequences, and }} \hfill \\ {\text{divergence to }} \pm \infty \,\,{\text{are undefined for }}n{\text{ }} > {\text{ }}1\,\,{\text{because }}{\mathbb{R}^n}{\text{ is - - - - - - - - }}{\text{.}} \hfill \\\ \end{gathered}

• A) {\text{a Field}}
• B) {\text{not compact for }}n > 1
• C) {\text{not ordered for }}n > 1
• D) {\text{complete}}

\begin{gathered} {\text{The domain of the real valued function }}f:{\mathbb{R}^n} \to \mathbb{R}{\text{ defined and given by;}} \hfill \\ f\left( X \right){\text{ = }}{\left( {1 - x_1^2 - x_2^2 - \cdots - x_2^n} \right)^{ - 1}}{\text{ is - - - - - }}{\text{.}} \hfill \\\ \end{gathered}

• A) \left\{ {X|\left| X \right| = 1} \right\}
• B) \left\{ {X|\left| X \right| \ne 1} \right\}
• C)
• D)

{\text{If }}f{\text{ is continuous on a compact set }}S{\text{ in }}{\mathbb{R}^n},\,\,{\text{then }}f - - - - {\text{ on }}S.

• A) is also uniformly continuous
• B) attains all its bounds
• C) is also defined on all the limit points of “S”
• D) All above are equally valid

\begin{gathered} {\text{In }}{\mathbb{R}^2},f\left( {x,y} \right){\text{ = }}\left\{ \begin{gathered} \frac{{\sin \sqrt {1 - {x^2} - 2{y^2}} }}{{\sqrt {1 - {x^2} - 2{y^2}} }},\,\,{x^2} + 2{y^2} < 1 \hfill \\ 1,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{x^2} + 2{y^2} = 1 \hfill \\\ \end{gathered} \right. \hfill \\ {\text{is - - - - - - - - - - on }}{x^2} + 2{y^2} = 1{\text{ }}{\text{.}} \hfill \\\ \end{gathered}

• A) continuous
• B) discontinuous
• C)
• D)

\begin{gathered} {\text{If a function }}f{\text{ is }}continuous{\text{ on a }}compact{\text{ set }}S{\text{ in }}{\mathbb{R}^n}{\text{, and }}\alpha = \mathop {\inf }\limits_{X \in S} f\left( X \right),\,\beta = \mathop {\sup }\limits_{X \in S} f\left( X \right)\,, \hfill \\ {\text{then }}f\left( {{X_1}} \right){\text{ = }}\alpha {\text{ and }}f\left( {{X_2}} \right){\text{ = }}\beta \, - - - - - {X_1}{\text{ and }}{X_2}{\text{ in }}S. \hfill \\\ \end{gathered}

• A) for some
• B) for all
• C)
• D)

\begin{gathered} {\text{Let }}u{\text{ and }}v{\text{ be functions of two variables with continuous second - order partial derivatives in a region }}S{\text{. }} \hfill \\ {\text{Suppose that }}{u_x} = {v_y}{\text{ and }}{u_y} = - {v_x}{\text{ in }}S{\text{. Then, }}{v_{xx}} + {v_{yy}} - - - - . \hfill \\ \hfill \\\ \end{gathered}

• A) >0 but <1
• B) =0
• C) =1
• D) >0

\begin{gathered} {\text{Suppose in }}{\mathbb{R}^2}{\text{, }}f,{\text{ }}{f_x},{\text{ }}{f_y}{\text{ and }}{f_{xy}}{\text{ exist on neighborhood }}N{\text{ of }}\left( {{x_0},{y_0}} \right){\text{. Then }}{f_{yx}}\left( {{x_0},{y_0}} \right){\text{ exists, and}} \hfill \\\ {f_{yx}}\left( {{x_0},{y_0}} \right) = {f_{xy}}\left( {{x_0},{y_0}} \right){\text{ because if }}{f_{xy}}{\text{ is - - - - }}{\text{.}} \hfill \\\\ \end{gathered}

• A) differntiable
• B) partially differentiable
• C) bounded
• D) continuous

\begin{gathered} {\text{Let }}u{\text{ and }}v{\text{ be functions of two variables with continuous second - order partial derivatives in a region }}S{\text{. }} \hfill \\ {\text{Suppose that }}{u_x} = {v_y}{\text{ and }}{u_y} = - {v_x}{\text{ in }}S{\text{. Then, }}{u_{xx}} - {u_{yy}} - - - - . \hfill \\\ \end{gathered}

• A) none of these.
• B) =0
• C) =1
• D) >0