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

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

\[\begin{gathered} {\text{The }}\,\,{\text{orientation}}\,\,{\text{ of }}\,\,{\text{the }}\,\,{\text{rose}}\,\,{\text{ relative}}\,\,{\text{ to}}\,\,{\text{ the}}\,\,{\text{ polar }}\,\,{\text{axis }}\,\,{\text{depends}}\,\,{\text{ on}}\,\,{\text{ the }}\,\,{\text{sign}}\,\,{\text{ of }}\,\,{\text{the }}\,\, \hfill \\ {\text{constant }}\,\,a{\text{ }}\,\,{\text{and }}\,\,{\text{whether}}\,\,{\text{ _________ }}\,\,{\text{appears }}\,\,{\text{in }}\,\,{\text{the}}\,\,{\text{ equation}}{\text{.}} \hfill \\\ \end{gathered} \]

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

{\text{Area}}\,\,{\text{of}}\,\,{\text{double}}\,\,{\text{integral}}\,\,{\text{can}}\,\,{\text{be}}\,\,{\text{calculated}}\,\,{\text{by}}\,\,{\text{the}}\,\,{\text{formula:}}\,{\text{area}}\,\,{\text{of}}\,\,R = {\text{_________}}{\text{.}}

• A) \iint\limits_R {dA}
• B) \frac{1}{{{\text{area}}\,\,{\text{of}}\,\,R}}\iint\limits_R {dA}
• C) \iint\limits_R {f(x,y)\,dy}
• D) \int\limits_R {f(x,y)\,dx\,dy}

$\begin{gathered} {\text{Let G be the rectangular box defined by the inequalities }}a \leqslant x \leqslant b,\,\,\,c \leqslant y \leqslant d,\,\,\,\,\,e \leqslant z \leqslant f. \hfill \ {\text{If }}f\,\,{\text{is continuous on G, then}}\,\,\int\limits_a^b {\int\limits_c^d {\int\limits_e^f {f(x,\,\,y,\,\,z)} } } \,dz\,\,dy\,\,dx = \,\,\, - - - - - - - - \hfill \\\ \end{gathered} $

• A) $\int\limits_c^d {\int\limits_a^b {\int\limits_e^f {f(x,\,\,y,\,\,z)} } } \,dz\,\,dx\,\,dy\,$
• B) $\int\limits_e^f {\int\limits_a^b {\int\limits_c^d {f(x,\,\,y,\,\,z)} } } \,dy\,\,dx\,\,dz$
• C) ${\text{All}}\,{\text{three}}\,{\text{options}}\,{\text{are}}\,{\text{true}}{\text{.}}$
• D) $\int\limits_a^b {\int\limits_e^f {\int\limits_c^d {f(x,\,\,y,\,\,z)} } } \,dy\,\,dz\,\,dx$

The\,point\,( - \,42,\,\,{35^0}\,\,)\,\,{\text{and the point - - - - - - - - - - - - - - }}\,{\text{are the same in polar system}}{\text{.}}

• A) (\,42,\,\,{215^0}\,)
• B) (\,\,42,\,\,{35^0}\,\,)
• C) (\,122,\,\,{215^0}\,\,)
• D) (122,\,\,{35^0}\,\,)

\[The\,point\,(\,\,3,\,\,{189^0}\,\,)\,\,{\text{and the point - - - - - - - - - - - - - - }}\,{\text{are the same in polar system}}{\text{.}}\]

• A) \[(\,\,3,\,\,{99^0}\,)\]
• B) \[( - 3,\,\,{279^0}\,\,)\]
• C) \[(\,\, - 3,\,\,{9^0}\,\,)\]
• D) \[(\,\, - 3,\,\,{189^0}\,\,)\]

\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) (a)\,\,\,\sin 2\theta
• D) (c)\,\,\,\tan \theta

$${{\text{In polar coordinate system, x - axis is also called}}}$$

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

{\text{After}}\,\,{\text{reversing}}\,\,{\text{the}}\,\,{\text{order}}\,{\text{of}}\,\,{\text{limits}}\,\,{\text{of}}\,\,\int\limits_0^{\frac{1}{2}} {\int\limits_{2x}^1 {{e^{{y^2}}}dy\,dx,\,} } \,{\text{we}}\,\,{\text{get}}\,\,{\text{___________}}{\text{.}}

• A) \,\int\limits_0^1 {\int\limits_{\frac{1}{2}}^{2x} {{e^{{y^2}}}dx\,dy\,\,} }
• B) \,\int\limits_0^1 {\int\limits_{2x}^{\frac{1}{2}} {{e^{{y^2}}}dx\,dy\,\,} }
• C) \,\int\limits_0^1 {\int\limits_{\frac{y}{2}}^0 {{e^{{y^2}}}dx\,dy\,\,} }
• D) \,\int\limits_0^1 {\int\limits_0^{\frac{y}{2}} {{e^{{y^2}}}dx\,dy\,\,} }

The\,point\,(\,\,3,\,\,{189^0}\,\,)\,\,{\text{and the point - - - - - - - - - - - - - - }}\,{\text{are the same in polar system}}{\text{.}}

• A) (\,\, - 3,\,\,{9^0}\,\,)
• B) (\,\,3,\,\,{99^0}\,)
• C) ( - 3,\,\,{279^0}\,\,)
• D) (\,\, - 3,\,\,{189^0}\,\,)

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

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

\[{\text{The graph of the equation }}r\, = \,a(1 - \cos \,\theta ){\text{ is symmetric about - - - - - - - - - }}{\text{.}}\]

• A) \[{\text{None of these}}{\text{.}}\]
• B) \[{\text{pole}}\]
• C) \[{\text{initial line}}\]
• D) \[{\text{y - axis}}\]

{\text{In polar coordinate system, the equation }}r\, = \,a{\text{ represents a circle with center at}}\,{\text{ - - - - - - - }}{\text{.}}

• A) {\text{None of these}}{\text{.}}
• B) {\text{Origin}}
• C) x - {\text{axis}}\,\,{\text{and passes through the origin}}{\text{.}}
• D) y - {\text{axis}}\,\,{\text{and passes through the origin}}{\text{.}}

\begin{gathered} {\text{Let G be the rectangular box defined by the inequalities }}a \leqslant x \leqslant b,\,\,\,c \leqslant y \leqslant d,\,\,\,\,\,e \leqslant z \leqslant f. \hfill \ {\text{If }}f\,\,{\text{is continuous on G, then}}\,\,\int\limits_a^b {\int\limits_c^d {\int\limits_e^f {f(x,\,\,y,\,\,z)} } } \,dz\,\,dy\,\,dx = \,\,\, - - - - - - - - \hfill \\\ \end{gathered}

• A) {\text{All}}\,{\text{three}}\,{\text{options}}\,{\text{are}}\,{\text{true}}{\text{.}}
• B) \int\limits_a^b {\int\limits_e^f {\int\limits_c^d {f(x,\,\,y,\,\,z)} } } \,dy\,\,dz\,\,dx
• C) \int\limits_e^f {\int\limits_a^b {\int\limits_c^d {f(x,\,\,y,\,\,z)} } } \,dy\,\,dx\,\,dz
• D) \int\limits_c^d {\int\limits_a^b {\int\limits_e^f {f(x,\,\,y,\,\,z)} } } \,dz\,\,dx\,\,dy\,

{\text{If}}\,\,Pdx + Qdy + Rdw\,\,{\text{is}}\,\,{\text{an}}\,\,{\text{exact}}\,\,{\text{differential}}\,\,{\text{equation}}\,\,{\text{then}}\,\,\oint\limits_C {\left( {Pdx + Qdy + Rdw} \right)} \,\,{\text{is}}\,\,{\text{__________}}{\text{.}}\,\,

• A) - 1
• B) {\text{infinite}}
• C) {\text{finite}}
• D) {\text{zero}}

If a vector field F(r) exist for all points of the curve C, then we can form -----------------F for each element of arc.

• A) vector Field
• B) scalar field
• C)
• D)

\begin{gathered} {\text{If}}\,\,{\text{a}}\,\,\_\_\_\_\_\_\_\_\_\,\,V(r)\,\,{\text{exists}}\,\,{\text{for}}\,\,{\text{all}}\,\,{\text{points}}\,\,{\text{on}}\,\,{\text{the}}\,{\text{curve,}}\,\,{\text{then}}\,\,\sum\limits_{p = 1}^n {V(r)\,d{r_p}} \,\,{\text{with}}\,\,dr \to 0\,\,{\text{defines}}\,\, \hfill \\ {\text{the}}\,\,{\text{line}}\,\,{\text{integral}}\,\,{\text{of}}\,\,V\,\,{\text{i}}{\text{.e}}{\text{.}}\,\,{\text{line}}\,\,{\text{integral}} = \int\limits_C {V(r)\,dr.} \hfill \\\ \end{gathered}

• A) {\text{scalar}}\,\,{\text{field}}
• B) {\text{vector}}\,\,{\text{quantity}}
• C) {\text{vector}}\,\,{\text{field}}
• D) {\text{vector}}\,\,{\text{space}}

The differential dz of the function \[z = {x^2} + {y^2}\] is ------------

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

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

{\text{The}}\,\,{\text{differential}}\,\,{\text{equation,}}\,\,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,\,\,{\text{is}}\,\,{\kern 1pt} {\text{an exact differential equation}}{\text{.}}

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