$${\text{In }}{\mathbb{R}^n},{\text{ the function }}f\left( X \right){\text{ is continuous at }}{X_0}{\text{, if }}\mathop {\lim }\limits_{X \to {X_0}} f\left( X \right) = f\left( {{X_0}} \right),{\text{ then - - - - - - }}{\text{.}}$$

• A) $${X_0}\,{\text{is in the domian }}{D_f}{\text{ and limit point of }}{D_f}$$
• B) $${\text{neither}}\,{X_0}\,{\text{is in the domian }}{D_f}\,{\text{nor the limit point of }}{D_f}$$
• C) $${X_0}\,{\text{is in the domian }}{D_f}{\text{ but not a limit point of }}{D_f}$$
• D) $${X_0}\,{\text{is not in the domian }}{D_f}{\text{ but a limit point of }}{D_f}$$

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

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

$$\begin{gathered} {\text{If }}{z_0} = f\left( {{x_0},{y_0}} \right){\text{ then }}z\left( t \right) = f\left( {x + \phi t,y + \phi t} \right){\text{ represents a - - - - - - - - - - through }}\left( {{x_0},{y_0},{z_0}} \right){\text{ in the plane }} \hfill \ {\text{determined by the unit vectors }}\hat \Phi {\text{ and }}\hat k{\text{.}} \hfill \\\ \end{gathered}$$

• A) Sub-Surface
• B) Straight line
• C) Curve
• D) Pair of lines

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

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

$$\begin{gathered} {\text{If }}{z_0} = f\left( {{x_0},{y_0}} \right){\text{ then }}z\left( t \right) = f\left( {x + \phi t,y + \phi t} \right){\text{ represents a curve through }}\left( {{x_0},{y_0},{z_0}} \right){\text{ in the plane }} \hfill \\ {\text{determined by the unit vectors }} - - - - {\text{.}} \hfill \\\ \end{gathered}$$

• A) $$\hat \Phi {\text{ and reciprocal vector of }}\hat k$$
• B) $$\hat \Phi {\text{ and }}\hat j$$
• C) $$\hat \Phi {\text{ and }}\hat k$$
• D) $$\hat \Phi {\text{ and }}\hat i$$

$${\text{In }}{\mathbb{R}^2},{\text{ }}\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {2,2} \right)} \frac{{\sin \left( {x - y} \right)}}{{\sqrt {x - y} }} = - - - - .$$

• A) $$1$$
• B) $$\sqrt 2$$
• C) $$\infty$$
• D) $$0\,$$

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

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

For the functions $f(x) = 3{x^2} + 5{x^3},$ the differential at x = - 1 exists because

• A) $$\mathop {\lim }\limits_{x \to - 1} \frac{{f(x) + 2 - 9(x + 1)}}{{x + 1}} = 1.$$
• B) $$\mathop {\lim }\limits_{x \to - 1} \frac{{f(x)}}{{x + 1}} = \infty .$$
• C) $$\mathop {\lim }\limits_{x \to - 1} \frac{{f(x) + 2 - 9(x + 1)}}{{x + 1}} = 0.$$
• D) $$\mathop {\lim }\limits_{x \to - 1} \frac{{9(x + 1)}}{{x + 1}} = 0.$$

$$\begin{gathered} {\text{Analogous to the derivative of a function of one variable in }}{\mathbb{R}^2}{\text{, the directional derivative of a function}} \hfill \\ f{\text{ at }}{X_0}{\text{ in }}{\mathbb{R}^n}{\text{ has - - - - - - value(s)}}{\text{. }} \hfill \\\ \end{gathered}$$

• A) integral
• B) multiple
• C) unique
• D) infinite many

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

• A) for some
• B) for all
• C)
• D)

A sufficient condition for a function of several variables to be differentiable at point is

• A) only the partial derivative should exists at that point.
• B) only limit at that point should exists.
• C) All partial derivatives exists and are continuous at that point
• D) None of these

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

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

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

{\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)

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

{\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\} \cap \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\} \cup \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\}

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) undefined
• B) 0
• C) -1/2
• D) 1/2

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

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

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

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