\begin{array}{*{20}{l}} \begin{gathered} Ifx'(t),y'(t)\,and{\text{ }}z'(t)are{\text{ }}continuous,then\,the{\text{ }}curve{\text{ }}given{\text{ }}by{\text{ }}the \hfill \\ parametric{\text{ }}equation{\text{ }}x = x(t),{\text{ }}y = y(t),{\text{ }}z = z(t){\text{ }}has{\text{ }}arc{\text{ }}length \hfill \\\ \end{gathered} \end{array}

• A) L = \int\limits_a^b {\sqrt {\frac{{dx}}{{dt}} + \frac{{dy}}{{dt}} + \frac{{dz}}{{dt}}} } dt
• B) None of these
• C) L = \int\limits_a^b {\sqrt {{x^2} + {y^2} + {z^2}} } dxdydz
• D) L = \int\limits_a^b {\sqrt {{{\left( {\frac{{dx}}{{dt}}} \right)}^2} + {{\left( {\frac{{dy}}{{dt}}} \right)}^2} + {{\left( {\frac{{dz}}{{dt}}} \right)}^2}} } dt

{\text{If}} {\text{the}} {\text{integral}} {\text{is}} {\text{of}} {\text{the}} {\text{form}} \oint {(Pdx + Qdy)} {\text{ where }}P = - 5x - y{\text{ and }}Q = x - 2y{\text{ then}}

• A) - \iint\limits_R {\left( {\frac{{\partial P}}{{\partial x}} - \frac{{\partial Q}}{{\partial y}}} \right)}dxdy = - \iint\limits_R {dxdy}
• B) - \iint\limits_R {\left( {\frac{{\partial P}}{{\partial x}} - \frac{{\partial Q}}{{\partial y}}} \right)}dxdy = - 2\iint\limits_R {dxdy}
• C) - \iint\limits_R {\left( {\frac{{\partial P}}{{\partial x}} - \frac{{\partial Q}}{{\partial y}}} \right)}dxdy = \iint\limits_R {dxdy}
• D) - \iint\limits_R {\left( {\frac{{\partial P}}{{\partial x}} - \frac{{\partial Q}}{{\partial y}}} \right)}dxdy = 2 \iint\limits_R {dxdy}

\eqalign{ & {\text{In 2D - space the parametric equations }}x = {\text{ }}x(t),y{\text{ }} = {\text{ }}y(t){\text{ can be expressed in single vector }} \cr & {\text{equation}} {\text{as}} \cr}

• A) \vec r(t) = x(t) + y(t)
• B) \vec r(t) = x(t) - y(t)
• C) \vec r(t) = x(t)i + y(t)j
• D) \vec r(t) = x(t)j + y(t)i

\eqalign{ & {\text{If }}x'(t), y'(t){\text{ and }}z'(t){\text{ are continuous for }}a \leqslant t \leqslant b{\text{, then the arc lenght for the given}} \cr & {\text{parametric equations }}x = x(t),y = y(t),z = z(t) {\text{ ;}} \left( {a \leqslant t \leqslant b} \right) {\text{is}} \cr}

• A) L = \int\limits_a^b {\sqrt {{{\left( {dx/dt} \right)}^2} + {{\left( {dy/dt} \right)}^2} + {{\left( {dz/dt} \right)}^2}} } dx
• B) L = \int\limits_a^b {\sqrt {{{\left( {dx/dt} \right)}^2} + {{\left( {dy/dt} \right)}^2} + {{\left( {dz/dt} \right)}^2}} } dy
• C) L = \int\limits_a^b {\sqrt {{{\left( {dx/dt} \right)}^2} - {{\left( {dy/dt} \right)}^2} - {{\left( {dz/dt} \right)}^2}} } dt
• D) L = \int\limits_a^b {\sqrt {{{\left( {dx/dt} \right)}^2} + {{\left( {dy/dt} \right)}^2} + {{\left( {dz/dt} \right)}^2}} } dt

\eqalign{ & {\text{Wallis sine formula when n is odd}} \cr & \int\limits_0^{\frac{\pi }{2}} {Co{s^7}x} dx = \cr}

• A) \frac{6}{7} \cdot \frac{4}{5} \cdot \frac{2}{3}
• B) \frac{6}{7} \cdot \frac{4}{5} \cdot \frac{2}{3} \cdot \frac{\pi }{2}
• C) \frac{7}{6} \cdot \frac{5}{4} \cdot \frac{3}{2} \cdot \frac{\pi }{2}
• D) \frac{7}{6} \cdot \frac{5}{4} \cdot \frac{3}{2}

{\text{For}}\,\,{\text{a}}\,\,{\text{function}}\,\,\vec r(t) = x(t)\hat i + y(t)\hat j\,\,{\text{in}}\,\,{\text{2 - space}}\,\,{\text{we}}\,\,{\text{define}}\,\,\mathop {\lim }\limits_{t \to \alpha } \,\vec r(t) = \_\_\_\_\_\_\_\_\_.

• A) x(t)\hat i + \left( {\mathop {\lim }\limits_{t \to \alpha } y(t)} \right)\hat j
• B) \left( {\mathop {\lim }\limits_{t \to \alpha } x(t)} \right)\hat i + y(t)\hat j
• C) \left( {\mathop {\lim }\limits_{t \to \alpha } x(t)} \right)\hat i + \left( {\mathop {\lim }\limits_{t \to \alpha } y(t)} \right)\hat j
• D) \left( {\mathop {\lim }\limits_{t \to \alpha } x(t)} \right)\hat i + \left( {\mathop {\lim }\limits_{t \to \alpha } y(t)} \right)\hat j + \left( {\mathop {\lim }\limits_{t \to \alpha } z(t)} \right)\hat k

\begin{gathered} {\text{If}}\,\,\vec r(t) = x(t)\hat i + y(t)\hat j + z(t)\hat k\,\,{\text{is}}\,\,{\text{a}}\,\,{\text{vector - valued}}\,\,{\text{function}}\,\,{\text{in}}\,\,{\text{3 - space,}}\,\,{\text{and}}\,\,{\text{if}}\,\,x(t),\,\,y(t)\,\,{\text{and}}\,\,z(t)\,\,{\text{are}}\,\, \hfill \\ {\text{differentiable,}}\,\,{\text{then}}\,\,\,\frac{d}{{dt}}\,\left[ {\vec r(t)} \right] = \_\_\_\_\_\_\_\_\_. \hfill \\\ \end{gathered}

• A) x'(t)\hat i + y'(t)\hat j
• B) x'(t)\hat i + y'(t)\hat j + z(t)\hat k
• C) x(t)\hat i + y'(t)\hat j + z'(t)\hat k
• D) x'(t)\hat i + y'(t)\hat j + z'(t)\hat k

I = \int\limits_C {f(x,y)dx = } \int\limits_C {f(x,y)\frac{{ds}}{{dx}}dx,\,\,{\text{where}}\,\,\frac{{ds}}{{dx}} = \_\_\_\_\_\_\_\_\_.}

• A) \sqrt {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}}
• B) \sqrt {1 + \frac{{dy}}{{dx}}}
• C) \sqrt {1 - {{\left( {\frac{{dy}}{{dx}}} \right)}^2}}
• D) \sqrt {{{\left( {\frac{{dy}}{{dx}}} \right)}^2}}

{\text{The differential equation }}\,\,dz\, = \,\left( {1 + 2y} \right)\,dx\,\, + \,\,\left( {1 + 2x} \right)\,dy\,\,\,\,\,{\text{is}}\,{\text{an exact differential equation}}{\text{.}}

• A) {\text{True}}
• B) {\text{False}}
• C)
• D)

{\text{If}}\,\,P = 1 + 4xy\,\,{\text{and}}\,\,Q = 5{x^2}\,\,{\text{then}}\,\,\_\_\_\_\_\_\_\_\_.

• A) \frac{{\partial Q}}{{\partial x}} = 5{x^2}
• B) \frac{{\partial P}}{{\partial y}} \ne \frac{{\partial Q}}{{\partial x}}
• C) \frac{{\partial P}}{{\partial y}} = \frac{{\partial Q}}{{\partial x}}
• D) \frac{{\partial P}}{{\partial y}} = 1 + 4x

{\text{If}}\,\,Pdx + Qdy + Rdw\,\,{\text{is}}\,\,{\text{an}}\,\,{\text{exact}}\,\,{\text{differential}}\,\,{\text{equation}}\,\,{\text{then}}\,\,\_\_\_\_\_\_\_.

• A) {\text{(a)}}\,\,\,\,\,\frac{{\partial P}}{{\partial y}} = \frac{{\partial Q}}{{\partial x}}
• B) {\text{(d)}}\,\,\,\,\,{\text{All}}\,\,{\text{(a),}}\,\,{\text{(b)}}\,\,{\text{and}}\,\,{\text{(c)}}{\text{.}}
• C) {\text{(c)}}\,\,\,\,\,\frac{{\partial R}}{{\partial y}} = \frac{{\partial Q}}{{\partial w}}
• D) {\text{(b)}}\,\,\,\,\,\frac{{\partial P}}{{\partial w}} = \frac{{\partial R}}{{\partial x}}

\eqalign{ & {\text{If }}x'(t){\text{ and }}y'(t){\text{ are continuous for }}a \leqslant t \leqslant b{\text{, then the given}} \cr & {\text{parametric equations are}} \cr}

• A) x = x(t),y = y(t){\text{ ;}} \left( {a \leqslant t \leqslant b} \right)
• B) x = x'(t),y = y'(t){\text{ ;}} \left( {a \leqslant t \leqslant b} \right)
• C) x = x(t) + y(t){\text{ ;}} \left( {a \leqslant t \leqslant b} \right)
• D) x = x'(t) + y'(t){\text{ ;}} \left( {a \leqslant t \leqslant b} \right)

\int\limits_0^{\frac{\pi }{2}} {{{\sin }^2}x} dx = \frac{1}{2}\left| {\frac{\pi }{2} - \frac{{\sin \pi }}{2}} \right| =

• A) \frac{\pi }{4}
• B) \frac{\pi }{2}
• C) \frac{\pi }{3}
• D) \frac{{3\pi }}{4}

\begin{gathered} {\text{If}}\,\,\vec r(t) = x(t)\hat i + y(t)\hat j\,\,{\text{is}}\,\,{\text{a}}\,\,{\text{vector - valued}}\,\,{\text{function}}\,\,{\text{in}}\,\,{\text{2 - space,}}\,\,{\text{and}}\,\,{\text{if}}\,\,x(t)\,\,{\text{and}}\,\,y(t)\,\,{\text{are}}\,\,{\text{differentiable,}}\,\, \hfill \\ {\text{then}}\,\,\,\frac{d}{{dt}}\,\left[ {\vec r(t)} \right] = \_\_\_\_\_\_\_\_\_. \hfill \\\ \end{gathered}

• A) x(t)\hat i + y'(t)\hat j
• B) x'(t)\hat i + y'(t)\hat j
• C) x'(t)\hat i + y(t)\hat j
• D) x'(t)\hat i + y'(t)\hat j + z'(t)\hat k

{\text{The differential equation, }}{\kern 1pt} {\kern 1pt} dz{\kern 1pt} = {\kern 1pt} 4xy{\kern 1pt} dx{\kern 1pt} {\kern 1pt} + {\kern 1pt} {\kern 1pt} \left( {2{x^2} + 3{y^2}} \right){\kern 1pt} dy,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{is}}\,\,{\kern 1pt} {\text{an exact differential equation}}{\text{.}}

• A) {\text{True}}
• B) {\text{False}}
• C)
• D)

\begin{gathered} {\text{The}} ~ {\text{graph}}~ {\text{of}} ~ \hfill \ {\text{r = (1 + t)}} {\text{i + ( - 2}} {\text{ + }} {\text{3t)}} {\text{j - 4t}} {\text{k}} ~ \hfill \ {\text{is}} ~{\text{the}}~ \hfill \\ \end{gathered}

• A) line that passes through the point (1, 3, -4)
• B) line that passes through the point (2, 1, -4)
• C) line that passes through the point (1, -2, 0)
• D) None of these

If r is a vector-valued function in 2-space or 3- space, then {\frac{d}{{dt}}\left[ {\int {r(t)dt} } \right]}

• A) None of above
• B) {r(t)}
• C) {\int {r(t)dt} }
• D) d\int {r(t)}

{\text{A vector valued function in 2 - D can be expressed as}}

• A) \vec r(t) = x(t) - y(t)
• B) \vec r(t) = x(t) + y(t)
• C) \vec r(t) = x(t)j + y(t)i
• D) \vec r(t) = x(t)i + y(t)j

\eqalign{ & {\text{If }}x'(t){\text{ and }}y'(t){\text{ are continuous for }}a \leqslant t \leqslant b{\text{, then the}} {\text{arc}} {\text{lenght}} {\text{for}} {\text{the given}} \cr & {\text{parametric}} {\text{equations}} x = x(t),y = y(t){\text{ ;}} \left( {a \leqslant t \leqslant b} \right) {\text{is}} \cr}

• A) L = \int\limits_a^b {\sqrt {{{\left( {dx/dt} \right)}^2} + {{\left( {dy/dt} \right)}^2}} } dt
• B) L = \int\limits_a^b {\sqrt {{{\left( {dx/dt} \right)}^2} - {{\left( {dy/dt} \right)}^2}} } dt
• C) L = \int\limits_a^b {\sqrt {{{\left( {dx/dt} \right)}^2} + {{\left( {dy/dt} \right)}^2}} } dx
• D) L = \int\limits_a^b {\sqrt {{{\left( {dx/dt} \right)}^2} + {{\left( {dy/dt} \right)}^2}} } dy

{\text{Green's Theorem states that}} \iint\limits_R {\left( {\frac{{\partial P}}{{\partial x}} - \frac{{\partial Q}}{{\partial y}}} \right)}dxdy

• A) \oint {(Pdx + Qdy)}
• B) - \oint {(Pdx - Qdy)}
• C) - \oint {(Pdx + Qdy)}
• D) \oint {(Pdx - Qdy)}