{\text{In }}{\mathbb{R}^2},{\text{ for the function }}f\left( {x,y} \right) = {x^2} + 2xy,\,\,{\text{the }}\left( {{d_{\left( {2,1} \right)}}f} \right)\left( {X - \left( {2,1} \right)} \right) = - - - - .

• A) 4\left( {x - 2} \right) + 6\left( {y - 1} \right)
• B) 6\left( {x - 2} \right) + 2\left( {y - 1} \right)
• C) 2\left( {x - 2} \right) + 6\left( {y - 1} \right)
• D) 6\left( {x - 2} \right) + 4\left( {y - 1} \right)\,

If f(x,y) is continuous at ({x_0},{y_0}) and {f_{yx}}({x_0},{y_0}) exists. Then,

• A) None of these
• B) {f_{yx}}({x_0},{y_0}) = {f_{xy}}({x_0},{y_0})
• C) {f_x}({x_0},{y_0}) = {f_y}({x_0},{y_0})
• D) {f_{xx}}({x_0},{y_0}) = {f_{yy}}({x_0},{y_0})

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

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

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

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

\begin{array}{*{20}{l}} {{\text{Let }}f{\text{ be defined on an interval }}I{\text{ in }}{\mathbb{R}^2}.\,{\text{Suppose that }}{x_1}{\text{ and }}{x_{\text{2}}}{\text{ are in }}I{\text{ and}}\,f\left( {{x_1}} \right){\text{ }} < {\text{ }}y{\text{ }} < {\text{ }}f\left( {{x_2}} \right){\text{. }}} \\\ {{\text{Then }}f\left( x \right) = y{\text{ for some }}x{\text{ in }}I.} \\\ {} \end{array}

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

\begin{gathered} {\text{In }}{\mathbb{R}^n},{\text{ if the first order partial derivative of a function }}f\left( X \right){\text{ exits at }}{X_0}{\text{, }} \hfill \\ {\text{then it is essentially continuous at }}{X_0}. \hfill \\\ \end{gathered}

• A) False
• B) True
• C)
• D)

\[\begin{gathered} {\text{In }}{\mathbb{R}^2},{\text{ for the function }}g\left( {x,y} \right){\text{ = }}\left\{ \begin{gathered} \frac{{xy}}{{{x^2} + {y^2}}},\,\,\,\,\left( {x,y} \right) \ne \left( {0,0} \right) \hfill \\ 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {x,y} \right) = \left( {0,0} \right) \hfill \\\ \end{gathered} \right., \hfill \\ \frac{{\partial g}}{{\partial x}}\left( {0,0} \right) = \frac{{\partial g}}{{\partial y}}\left( {0,0} \right),\,{\text{and }}g\left( {x,y} \right){\text{ is - - - - - - - at }}\left( {0,0} \right). \hfill \\\ \end{gathered} \]

• A) Continuous
• B) Discontinuous
• C)
• D)

{\text{In }}{\mathbb{R}^n}{\text{, }}\mathop {\lim }\limits_{X \to {X_0}} f\left( X \right) = - \infty ,\,\,{\text{if}}

• A) \mathop {\lim }\limits_{X \to {X_0}} \left( { - f} \right)\left( X \right) = \infty
• B) \mathop {\lim }\limits_{X \to {X_0}} f\left( { - X} \right) = \infty
• C) \mathop {\lim }\limits_{X \to - {X_0}} f\left( X \right) = \infty
• D) \mathop {\lim }\limits_{ - X \to {X_0}} f\left( X \right) = \infty

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

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

For the function \[f(x,y) = \frac{{xy}}{{{x^2} + {y^2}}},\] the limit of \[f(x,y)\] as \[(x,y) \to (0,0)\] along the line \[y = - x\] is

• A) \[ - \frac{1}{2}\]
• B) \[\frac{1}{2}\]
• C) 1
• D) 0

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

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

A function f is ______________ on a subset S of its domain in {\mathbb{R}^n} if for every \varepsilon > 0 there is a \delta > 0 such that \left| {f(X) - f(X')} \right| < \varepsilon whenever \left| {X - X'} \right| < \delta and X,X' \in S

• A) All of these
• B) Differentiable
• C) Uniformly continuous
• D) Compact

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

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

Period of the function Sin z is _____________ degree.

• A) 90
• B) 0
• C) 180
• D) 360

The complex exponential function ez is onto function.

• A) True
• B) False
• C)
• D)

The function ez is an entire function.

• A) False
• B) True
• C)
• D)

Period of the function Cos z is _____________ degree.

• A) 0
• B) 180
• C) 360
• D) 90

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