\[\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {a,b} \right)} \frac{{\sin \sqrt {1 - {x^2} - 2{y^2}} }}{{\sqrt {1 - {x^2} - 2{y^2}} }} = 1,{\text{ if - - - - - - }}{\text{.}}\]

• A) \[\left( {{a^2} + 2{b^2}} \right)\mathop > \limits_ < 1\]
• B) \[\left( {{a^2} + 2{b^2}} \right) = 1\]
• C) \[\left( {{a^2} + 2{b^2}} \right) > 1\]
• D) \[\left( {{a^2} + 2{b^2}} \right) < 1\]

\[\begin{gathered} {\text{If }}g\left( {x,y} \right) = \sqrt {1 - {x^2} - 2{y^2}} ,{\text{ }}\,f\left( t \right){\text{ = }}\left\{ \begin{gathered} \frac{{\sin t}}{t},\,t \ne 0\,\, \hfill \\ 1,\,\,\,\,\,\,\,\,\,t = 0 \hfill \\\ \end{gathered} \right., \hfill \\ {\text{then domain of }}f \circ g = - - - - - . \hfill \\\ \end{gathered} \]

• A) \[\mathbb{R}\]
• B) \[{\mathbb{R}^2}\]
• C) \[\left\{ {\left( {x,y} \right):{x^2} - 2{y^2} \geqslant 1} \right\}\]
• D) \[\left\{ {\left( {x,y} \right):{x^2} - 2{y^2} \leqslant 1} \right\}\]

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

• 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) \[\mathop {\lim }\limits_{ - X \to {X_0}} f\left( X \right) = \infty \]

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

• A) 9
• B) 6
• C) 8
• D) 3

\begin{gathered} {\text{If }}g\left( {x,y} \right) = \sqrt {1 - {x^2} - 2{y^2}} ,{\text{ }}\,f\left( t \right){\text{ = }}\left\{ \begin{gathered} \frac{{\sin t}}{t},\,t \ne 0\,\, \hfill \\ 1,\,\,\,\,\,\,\,\,\,t = 0 \hfill \\\ \end{gathered} \right., \hfill \\ {\text{then domain of }}f \circ g = - - - - - . \hfill \\\ \end{gathered}

• A) \left\{ {\left( {x,y} \right):{x^2} - 2{y^2} \leqslant 1} \right\}
• B) {\mathbb{R}^2}
• C) \left\{ {\left( {x,y} \right):{x^2} - 2{y^2} \geqslant 1} \right\}
• D) \mathbb{R}

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

• A) unbounded
• B) bounded above
• C) bounded below
• D) bounded

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

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

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

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

\[{\text{If }}f\left( t \right) = \sqrt t {\text{, }}g\left( {x,y} \right) = 1 - {x^2} - 2{y^2}{\text{, then the domain of }}f \circ g = {\text{ - - - - - - }}{\text{.}}\]

• A) \[{\mathbb{R}^2}\]
• B) \[\left\{ {\left( {x,y} \right):{x^2} - 2{y^2} \geqslant 1} \right\}\]
• C) \[\mathbb{R}\]
• D) \[\left\{ {\left( {x,y} \right):{x^2} - 2{y^2} \leqslant 1} \right\}\]

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

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

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

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

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

{\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)