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) $${f_y}(x,y),{\text{ }}{f_x}(x,y)$$ are continuous everywhere.
• C) $${f_y}(x,y) = \frac{y}{{x - y}} + \frac{{{x^2} + {y^2}}}{{{{(x - y)}^2}}}.$$
• D) $${f_x}(x,y) = \frac{{2x}}{{x - y}} - \frac{{{x^2} + {y^2}}}{{{{(x + y)}^2}}}$$

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

• A) Bounded
• B) All of these
• C) Superemum
• D) Differentiable

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

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

$$\begin{gathered} {\text{The domain of the real valued function }}f:{\mathbb{R}^2} \to \mathbb{R}{\text{ defined and given by;}} \hfill \\\ f\left( X \right){\text{ = }}\frac{{\sin \sqrt {1 - {x^2} - 2{y^2}} }}{{\sqrt {1 - {x^2} - 2{y^2}} }}{\text{ is - - - - - of the region}}\,\,{\text{by the ellipse }}{x^2} + 2{y^2} = 1{\text{ }}{\text{.}} \hfill \\\\ \end{gathered}$$

• A) $${\text{exterior}}$$
• B) $${\text{interior and boundary}}$$
• C) $${\text{exterior}}\,\,{\text{and}}\,\,{\text{boundary}}$$
• D) $${\text{interior}}$$

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

• 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) $$\left( {{a^2} + 2{b^2}} \right) = 1$$

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

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

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

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

Identify the false statement(s)

• A) None of these
• B) The partial derivative is a special case of directional derivative.
• C) For a functions of n variables $$f({\mathbf{X}})$$ the partial derivative with respect to variable $${x_2}$$ is $$\frac{{\partial f({\mathbf{X}})}}{{\partial {{\mathbf{E}}_2}}}$$.
• D) The partial derivative of the function $$f(x,y,z) = 3xyz + 2{x^2} + {z^2}$$ with respect to third variable is $${f_z} = 2z$$

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) 0
• B) 1
• C) $$- \frac{1}{2}$$
• D) $$\frac{1}{2}$$

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

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

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

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

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

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

Let f be continuous on a region S in ${\mathbb{R}^n}$. Suppose that A and B are in S and _____________. Then $f\left( C \right) = u$ for some C in S.

• A) $$f(A){\rm{ }} < {\rm{ }}u{\rm{ }} < {\rm{ }}f(B)$$
• B) $$f(A){\rm{ }} \le {\rm{ }}u{\rm{ }} \le {\rm{ }}f(B)$$
• C) $$f(A){\rm{ }} > {\rm{ }}u{\rm{ }} > f(B)$$
• D) $$f(A){\rm{ }} \ge {\rm{ }}u{\rm{ }} \ge f(B)$$

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

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

$${\text{In }}{\mathbb{R}^n},{\text{ }}f\left( X \right) = \frac{1}{{\left| {X - {X_{_0}}} \right|}},and\mathop {\lim }\limits_{X \to {X_0}} f\left( X \right) = \infty ,{\text{ then }}\,f\left( X \right) > M > 0\,\, \Rightarrow {\text{ }}0 < \left| {X - {X_{_0}}} \right| < \delta = - - - - .$$

• A) $$\frac{1}{{\sqrt M }}$$
• B) $$\sqrt M$$
• C) $$M$$
• D) $$\frac{1}{M}\,$$

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

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

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) infinite
• B) $1$
• C) $0$
• D) $-1$

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

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

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

• A) False
• B) True
• C)
• D)

Let the function $f(s) = \left\{ \begin{array}{l}\frac{{\sin s}}{s},s \ne 0\\1,s = 0\end{array} \right.$ is continuous over domain _____________.

• A) (c) $$(0,\infty )$$
• B) (b) $$( - \infty ,\infty )$$
• C) Both a & b
• D) (a) $$\mathbb{R}$$