\begin{gathered} {\text{If}}\,\,{\text{the}}\,\,{\text{region}}\,\,{\text{is}}\,\,{\text{bounded}}\,\,{\text{below}}\,\,{\text{and}}\,\,{\text{above}}\,\,{\text{by}}\,\,{\text{the}}\,\,{\text{horizontal}}\,\,{\text{lines}}\,\,y = c\,\,{\text{and}}\,\,y = d\,\,{\text{and}}\,\,{\text{is}}\,\,{\text{bounded}}\,\, \hfill \\ {\text{on}}\,\,{\text{the}}\,\,{\text{left}}\,\,{\text{and}}\,\,{\text{right}}\,\,{\text{by}}\,\,{\text{the}}\,\,{\text{continuous}}\,\,{\text{curves}},\,\,x = {h_1}(y)\,\,{\text{and}}\,\,x = {h_2}(y),\,\,{\text{satisfying}}\,\,{h_1}(y)\,\, \leqslant {h_2}(y),\, \hfill \\ {\text{for}}\,\,c \leqslant y \leqslant d.\,\,{\text{Then}}\,\,\iint\limits_R {f(x,y)\,dx\,dy = }\,\,\_\_\_\_\_\_\_\_\_. \hfill \\\ \end{gathered}

• A) \int\limits_c^d {f(x,y)\,dx\,dy}
• B) \int\limits_{{h_1}(y)}^{{h_2}(y)} {f(x,y)\,dx\,dy\,}
• C) \int\limits_c^d {\int\limits_{{h_1}(y)}^{{h_2}(y)} {f(x,y)\,dx\,dy\,} }
• D) \int {f(x,y)\,dy\,dx}

$\int\limits_0^\pi {\int\limits_0^1 {{r^2}\,\,dr\,\,d\theta } } \,\, = \,\,\, - - - - - - - - $

• A) $\frac{\pi }{4}$
• B) $\frac{\pi }{5}$
• C) $\frac{\pi }{3}$
• D) $\frac{\pi }{2}$

\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)

\[\begin{gathered} {\text{In polar coordinate system, the equation }}r\, = \,-2a\,\sin \,\theta {\text{ represents a circle passes through the origin,}}\, \ {\text{with center on}}\,{\text{ - - - - - - - }}{\text{.}} \\ \end{gathered} \]

• A) \[x - {\text{axis,}}\,\,{\text{right to the origin}}{\text{.}}\]
• B) \[y - {\text{axis,}}\,\,{\text{below}}\,{\text{the origin}}{\text{.}}\]
• C) \[y - {\text{axis,}}\,\,{\text{above the origin}}{\text{.}}\]
• D) \[x - {\text{axis,}}\,\,{\text{left to the origin}}{\text{.}}\]

\int\limits_0^{\frac{\pi }{2}} {\int\limits_0^1 {r\,\,dr\,\,d\theta } } \,\,\, = \,\,\, - - - - - - -

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

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

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

$${{\text{If a point }}p(r,\theta ) {\text{in polar coordinate system, then }}r{\text{ is the distance of }}p{\text{ from the }}}$$

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

\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

\begin{gathered} {\text{The}}\,\,{\text{equations}}\,\,{\text{of}}\,\,{\text{the}}\,\,{\text{form:}} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,r = a + b\,\sin \theta ,\,\,\,r = a - b\,\sin \theta , \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,r = a + b\,\cos \theta ,\,\,\,r = a - b\,\sin \theta , \hfill \\ {\text{produce}}\,\,{\text{polar}}\,\,{\text{curves}}\,\,{\text{called}}\,\,{\text{_________}}{\text{.}} \hfill \\\ \end{gathered}

• A) {\text{limacons}}
• B) {\text{rose}}\,\,{\text{curve}}
• C) {\text{spiral}}
• D) {\text{straight line}}

\eqalign{ & {\text{Given the integral }}\iint\limits_R {f(x,y)dxdy}{\text{,}} {\text{can}} {\text{be}} {\text{expressed}} {\text{in}} {\text{ploar coordinates}} {\text{as}} ..............{\text{,}} \cr & {\text{where}} a \leqslant \theta \leqslant b {\text{and}} c \leqslant r \leqslant d. \cr}

• A) \int_a^b {\int_c^d {f(r,\theta )} } drd\theta
• B) \int_a^b {\int_c^d {f(r,\theta )} } rdrd\theta
• C) \int_a^b {\int_c^d {f(r,\theta )} } rd\theta dr
• D) \int_a^c {\int_b^d {f(r,\theta )} } rd\theta dr

Can we evaluate the following integral in given order of integration {\kern 1pt} \int\limits_0^{\frac{1}{2}} {\int\limits_{2x}^1 {{e^{{y^2}}}} } dydx

• A) No
• B) Yes
• C)
• D)

\int\limits_0^1 {\int\limits_0^1 {\int\limits_0^1 {xyz} } } \,dx\,\,dy\,\,dz = \,\,\, - - - - - - - -

• A) \frac{1}{{10}}
• B) \frac{1}{4}
• C) \frac{1}{2}
• D) \frac{1}{8}

{\text{The}}\,\,{\text{equation,}}\,\,\,{r^2} = 4\cos 2\theta ,\,\,{\text{represents}}\,\,{\text{a}}\,\,\_\_\_\_\_\_\_\_\_.

• A) {\text{cardioids}}
• B) {\text{rose }}\,\,{\text{curve}}
• C) {\text{spiral}}
• D) {\text{lemniscate}}

{\text{The}}\,\,{\text{curve}},\,\,r = \theta \,\,(\theta \geqslant 0),\,\,{\text{is}}\,\,{\text{an}}\,\,{\text{equation}}\,\,{\text{of}}\,\,\_\_\_\_\_\_\_\_\_.

• A) {\text{roses}}
• B) {\text{spiral}}\,\,{\text{with}}\,\,a = 1
• C) {\text{cardioids}}
• D) {\text{limacons}}

\eqalign{ & {\text{Polar co - ordinates of a point are}} \left( {{\text{ - 1,}} \frac{{ - 3\pi }}{4}} \right){\text{. Which of the following is another possible polar }} \cr & {\text{co - ordinates representation of this point?}} \cr}

• A) \left( {{\text{ - 1,}} \frac{{3\pi }}{4}} \right)
• B) \left( {{\text{ - 1,}} \frac{\pi }{2}} \right)
• C) \left( {{\text{ - 1,}} \frac{\pi }{4}} \right)
• D) \left( {{\text{ - 1,}} \frac{\pi }{3}} \right)

{\rm{The}}\,{\rm{value}}\,{\rm{of}}\,{\rm{integral }}\int\limits_0^1 {\int\limits_0^{x^2 } {(x^2 + y^2 )dydx} } \,\,{\rm{is}}

• A) 2 \over 105
• B) 1 \over 105
• C) 26 \over 105
• D) 26 \over 5

\begin{gathered} {\text{In polar coordinate system, the equation }}r\, = \,2a\,\sin \,\theta {\text{ represents a circle passes through the origin,}}\, \ {\text{with center on}}\,{\text{ - - - - - - - }}{\text{.}} \\ \end{gathered}

• A) x - {\text{axis,}}\,\,{\text{left to the origin}}{\text{.}}
• B) y - {\text{axis,}}\,\,{\text{below}}\,{\text{the origin}}{\text{.}}
• C) x - {\text{axis,}}\,\,{\text{right to the origin}}{\text{.}}
• D) y - {\text{axis,}}\,\,{\text{above the origin}}{\text{.}}

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

\int\limits_0^\pi {\int\limits_0^1 {{r^2}\,\,dr\,\,d\theta } } \,\, = \,\,\, - - - - - - - -

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

{\text{The equation }}r = \,a\,\theta {\text{ represents - - - - - - - - - , where }}a{\text{ is positive}}{\text{.}}

• A) {\text{lemniscate}}
• B) {\text{cardioid}}
• C) {\text{rose curve}}
• D) {\text{spiral}}