$$\begin{gathered} {\text{The}}\,\,{\text{equations}}\,\,{\text{of}}\,\,{\text{the}}\,\,{\text{form:}} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,r = a\,\sin \,n\theta ,\,\,\, \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,r = a\cos n\theta , \hfill \\ {\text{represent}}\,\,{\text{flower - shaped}}\,\,{\text{curves}}\,\,{\text{called}}\,\,{\text{_________}}{\text{.}} \hfill \\\ \end{gathered}$$

• A) $${\text{cardioids}}$$
• B) $${\text{spiral}}$$
• C) $${\text{roses}}$$
• D) $${\text{limacons}}$$

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

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

$${\text{For a vector valued function }}\vec r(t){\text{ }} = {\text{ }}3{\text{ }}i + 4j - 0k{\text{ then lenght of }}\vec r(t){\text{ }}$$

• A) $$\left\| r \right\| = 25$$
• B) $$\left\| r \right\| = 5$$
• C) $$\left\| r \right\| = 9$$
• D) $$\left\| r \right\| = 4$$

A smooth vector-valued function has a _________ line at every point on its graph.

• A) tangent
• B) straight
• C) secant
• D) curved

Natural domain of a vector valued function is

• A) intersection of the natural domains of its components
• B) not dependent on domains of its components
• C) union of the natural domains of its components
• D) none of these

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

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

$${\text{The point }}p(r,\theta ){\text{ in polar coordinate system lies on}}$$

• A) $${\text{Polar axis}}$$
• B) $${\text{y - axis}}$$
• C) $${\text{None of these}}$$
• D) $${\text{Pole}}$$

$${\text{The equation }}r\, = \,a(1 + \cos \,\theta ){\text{ represents - - - - - - - - - }}{\text{.}}$$

• A) $${\text{a straight line}}$$
• B) $${\text{rose curve}}$$
• C) $${\text{lemniscate}}$$
• D) $${\text{cardioid}}$$

$$\eqalign{ & {\text{Polar co - ordinates of a point are}} \left( {{\text{5,}} \frac{{ - \pi }}{4}} \right){\text{. Which of the following is another possible polar }} \cr & {\text{co - ordinates representation of this point?}} \cr}$$

• A) $$\left( {{\text{5,}} \frac{{ - 3\pi }}{4}} \right)$$
• B) $$\left( {{\text{5,}} \frac{{3\pi }}{4}} \right)$$
• C) $$\left( {{\text{ - 5,}} \frac{{ - 3\pi }}{4}} \right)$$
• D) $$\left( {{\text{ - 5,}} \frac{{3\pi }}{4}} \right)$$

The rose curve has n-equally spaced petals or loops if n is ________.

• A) even
• B) odd
• C)
• D)

The differential dz of the function $$z = {x^2} + {y^2}$$ is

• A) $$dz = 2dx + 2dy$$
• B) $$dz = 2x + 2y$$
• C) $$dz = (2x + 2y)dz$$
• D) $$dz = 2xdx + 2ydy$$

$$\begin{gathered} {\text{The}}\,\,{\text{lemniscates}}\,\,{\text{are}}\,\,{\text{centered}}\,\,{\text{at}}\,\,{\text{the}}\,\,{\text{origin,}}\,\,{\text{but}}\,\,{\text{the}}\,\,{\text{position}}\,\,{\text{relative}}\,\,{\text{to}}\,{\text{the}}\,\,{\text{polar}}\,\,{\text{axis}}\,\, \hfill \\ {\text{depends}}\,\,{\text{on}}\,\,{\text{the}}\,\,{\text{sign}}\,\,{\text{preceding}}\,\,{\text{the}}\,\,{a^2}\,\,{\text{and}}\,\,{\text{whether}}\,\,{\text{__________}}\,\,{\text{appears}}\,\,{\text{in}}\,\,{\text{the}}\,\,{\text{equation}}{\text{.}} \hfill \\\ \end{gathered}$$

• A) $$(b)\,\,\,\cos 2\theta$$
• B) $${\text{(d)}}\,\,\,{\text{Both}}\,\,{\text{(a)}}\,\,{\text{or}}\,\,{\text{(b)}}{\text{.}}$$
• C) $$(c)\,\,\,\tan \theta$$
• D) $$(a)\,\,\,\sin 2\theta$$

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

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

$$\begin{gathered} {\text{The }}\,\,{\text{position }}\,\,{\text{of }}\,\,{\text{the}}\,\,{\text{ limacon }}\,\,{\text{relative }}\,\,{\text{to}}\,\,{\text{ the}}\,\,{\text{ polar}}\,\,{\text{ axis}}\,\,{\text{ depends}}\,\,{\text{ on }}\,\,{\text{whether}}\,\,{\text{ _________ }}\,\, \hfill \\ {\text{appears}}\,\,{\text{ in}}\,\,{\text{ the}}\,\,{\text{ equation }}\,\,{\text{and}}\,\,{\text{ whether}}\,\,\, + {\text{ }}\,\,{\text{or}}\,\,{\text{ }} - \,\,{\text{occurs}}{\text{.}} \hfill \\\ \end{gathered}$$

• A) $${\text{(d)}}\,\,\,{\text{Both}}\,\,{\text{(a)}}\,\,{\text{or}}\,\,{\text{(b)}}{\text{.}}$$
• B) $$(a)\,\,\,\sin \theta$$
• C) $$(c)\,\,\,\tan \theta$$
• D) $$(b)\,\,\,\cos \theta$$

A vector-valued function is continuous at some point t0 if

• A) all of its component should be continuous every where in the domain
• B) some of its component is continuous at that point
• C) each of its component is differentiable at that point
• D) each of its component is continuous at that point

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

• A) line parallel to the vector $${\text{ i - 3j + 4k}}$$
• B) line parallel to the vector $${\text{ i + 2k}}$$
• C) line Perpendicular to the vector $${\text{ i + 2k}}$$
• D) line Perpendicular to the vector $${\text{ i - 3j + 4k}}$$

$\int\limits_0^1 {\int\limits_0^1 {\int\limits_0^1 {{x^2}{y^2}{z^2}} } } \,dx\,\,dy\,\,dz = \,\,\, - - - - - - - -$

• A) $\frac{1}{{30}}$
• B) $\frac{1}{3}$
• C) $\frac{1}{9}$
• D) $\frac{1}{{27}}$

Which integral gives the arc length of the curve $$r(t) = \frac{1}{3}{t^3}i + tj + {t^2}k$$ over the interval [1,3]

• A) $$\int\limits_1^3 {\sqrt {{{(1 + t)}^2}} } dt$$
• B) $$\int\limits_1^3 {\sqrt {1 + t} } dt$$
• C) $$\int\limits_1^3 {\sqrt {{{(\frac{1}{3} + t)}^2}} } dt$$
• D) $$\int\limits_1^3 {\sqrt {{{(1 + {t^2})}^2}} } dt$$

$$Graph~ of~ c = {x_0}i + {y_0}j + {z_0}k$$

• A) is the line passing through the point $$( {x_0}, {y_0}, {z_0})$$
• B) is the point $$( {x_0}, {y_0}, {z_0})$$
• C) None of these
• D) is the curve passing through the point $$( {x_0}, {y_0}, {z_0})$$

$$\begin{array}{l} If ~x'(t) ~and~ y'(t)~ are ~continuous, ~then~ the ~curve~ given~ by ~the ~parametric ~equation\x = x(t) , y = y(t) ~has ~arc~ length \end{array}$$

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