MCQ Bank
\[\begin{gathered} {\text{In }}{\mathbb{R}^2},{\text{ for the function }}g\left( {x,y} \right){\text{ = }}\left\{ \begin{gathered} \frac{{xy}}{{{x^2} + {y^2}}},\,\,\,\,\left( {x,y} \right) \ne \left( {0,0} \right) \hfill \\ 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {x,y} \right) = \left( {0,0} \right) \hfill \\\ \end{gathered} \right., \hfill \\ \frac{{\partial g}}{{\partial x}}\left( {0,0} \right) = \frac{{\partial g}}{{\partial y}}\left( {0,0} \right),\,{\text{and }}g\left( {x,y} \right){\text{ is - - - - - - - at }}\left( {0,0} \right). \hfill \\\ \end{gathered} \]
- 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) \[\frac{{{x_1} + {x_2}}}{2}\]