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

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

\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

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

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

\(The\,value\,of\,integral\,\int\limits_0^\pi {\int\limits_0^\pi {d\theta d\varphi } } \,\,is \)

• A) \(\pi ^2 \)
• B) \(0\)
• C) \(\frac{\pi }{2} \)
• D) \(1\)

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

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

\({\rm{The}}\,{\rm{value}}\,{\rm{of}}\,{\rm{integral }}\int\limits_0^3 {\int\limits_0^2 {x^2 dxdy} } \,\,{\rm{is}} \)

• A) 8
• B) 16
• C) 12
• D) 4

{\text{The}} \int\limits_0^1 {2x{e^{{x^2}}}dx} =

• A) e
• B) e - 1
• C) {e^{{x^2}}}
• D) e + 1

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

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

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

\[{\text{In the integration in polar coordinates, }}dx\,dy{\text{ is replaced by - - - - - - - - - - - }}{\text{.}}\]

• A) \[r\,\,dr\]
• B) \[r\,dr\,d\theta \]
• C) \[dr\,d\theta \]
• D) \[d\theta \]

$${\text{Reversing the order of integration}} \int\limits_0^1 {\int\limits_{4x}^4 {{e^{ - {y^2}}}dydx = } } $$

• A) $$\int\limits_0^4 {\int\limits_0^{\frac{y}{2}} {{e^{ - {y^2}}}dx} } dy$$
• B) $$\int\limits_0^4 {\int\limits_0^{3y} {{e^{ - {y^2}}}dx} } dy$$
• C) $$\int\limits_0^4 {\int\limits_0^{\frac{y}{4}} {{e^{ - {y^2}}}dx} } dy$$
• D) $$\int\limits_0^4 {\int\limits_0^{2y} {{e^{ - {y^2}}}dx} } dy$$

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

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

\begin{gathered} {\text{A}}\,\,{\text{curve}}\,\,{\text{that}}\,\,'{\text{winds}}\,\,{\text{around}}\,\,{\text{the}}\,\,{\text{origin'}}\,\,{\text{infinitely}}\,\,{\text{many}}\,\,{\text{times}}\,\,{\text{in}}\,\,{\text{such}}\,\,{\text{a}}\,\,{\text{way}}\,\,{\text{that}}\,\,r\,\,{\text{increases}}\,\, \hfill \\ \left( {{\text{or}}\,\,{\text{decreases}}} \right)\,\,{\text{steadily}}\,\,{\text{as}}\,\,\theta \,\,\,{\text{increases}}\,\,{\text{is}}\,\,{\text{called}}\,\,{\text{a}}\,\,\_\_\_\_\_\_\_\_. \hfill \\\ \end{gathered}

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

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

• A) \(1 \over 2\)
• B) \(-1 \over 3\)
• C) \(-1 \over 2\)
• D) \(1 \over 3\)

The expression $\int_a^b {f(x,y)dy} $ is a function of …..

• A) None of these
• B) Both x and y
• C) x
• D) y

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

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

$$\eqalign{ & {\text{Polar co - ordinates of a point are}} \left( {{\text{ - 4,}} \frac{{ - 3\pi }}{2}} \right){\text{. Which of the following is another possible polar }} \cr & {\text{co - ordinates representation of this point?}} \cr} $$

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

\[\begin{gathered} {\text{A}}\,\,{\text{curve}}\,\,{\text{that}}\,\,'{\text{winds}}\,\,{\text{around}}\,\,{\text{the}}\,\,{\text{origin'}}\,\,{\text{infinitely}}\,\,{\text{many}}\,\,{\text{times}}\,\,{\text{in}}\,\,{\text{such}}\,\,{\text{a}}\,\,{\text{way}}\,\,{\text{that}}\,\,r\,\,{\text{increases}}\,\, \hfill \\ \left( {{\text{or}}\,\,{\text{decreases}}} \right)\,\,{\text{steadily}}\,\,{\text{as}}\,\,\theta \,\,\,{\text{increases}}\,\,{\text{is}}\,\,{\text{called}}\,\,{\text{a}}\,\,\_\_\_\_\_\_\_\_. \hfill \\\ \end{gathered} \]

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

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

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

{\text{After}}\,{\text{reversing the order of limits of }}\,\int\limits_0^3 {\int\limits_{x^2 }^9 {y\,\cos \,x\,\,dy\,dx} } ,\,{\text{we}}\,{\text{get}}\, - - - - - -

• A) \int\limits_0^3 {\int\limits_0^{\sqrt x } {y\,\cos \,x\,\,dxdy} }
• B) \int\limits_{x^2 }^6 {\int\limits_0^3 {y\,\cos \,x\,\,dxdy} }
• C) \int\limits_0^9 {\int\limits_0^{\sqrt y } {y\,\cos \,x\,\,dxdy} }
• D) \int\limits_0^9 {\int\limits_0^{\sqrt y } {y\,\cos \,x\,\,dydx} }