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

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

{\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}} \int\limits_0^1 {{e^{{y^2}}}dx} =

• A) e - 1
• B) {e^{{y^2}}}
• C) 0
• D) e + 1

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) Yes
• B) No
• C)
• D)

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

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

\eqalign{ & {\text{Polar co - ordinates of a point are}} \left( {{\text{ - 3,}} \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{ - 3,}} \frac{\pi }{2}} \right)
• B) \left( {{\text{ - 3,}} \frac{\pi }{3}} \right)
• C) \left( {{\text{ - 3,}} \frac{{3\pi }}{4}} \right)
• D) \left( {{\text{ - 3,}} \frac{\pi }{4}} \right)

\[ {\text{After}}\,{\text{reversing the order of limits of }}\,\int\limits_0^2 {\int\limits_x^2 {f(x,\,y)\,dy\,dx} } ,\,{\text{we}}\,{\text{get}}\, - - - - - - \]

• A) \[ \int\limits_y^2 {\int\limits_x^2 {f(x,\,y)\,dx\,dy} } \]
• B) \[ \int\limits_x^2 {\int\limits_0^y {f(x,\,y)\,dy\,dx} } \]
• C) \[ \int\limits_0^2 {\int\limits_0^y {f(x,\,y)\,dx\,dy} } \]
• D) \[ \int\limits_0^2 {\int\limits_x^2 {f(x,\,y)\,dx\,dy} } \]

\[\begin{gathered} {\text{If}}\,\,a > 0,\,\,{\text{then}}\,\,{\text{equations}}\,\,{\text{of}}\,\,{\text{the}}\,\,{\text{form:}} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{r^2} = {a^2}\cos 2\theta ,\,\,\,{r^2} = - {a^2}\cos 2\theta , \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{r^2} = {a^2}\sin 2\theta ,\,\,\,{r^2} = - {a^2}\sin 2\theta , \hfill \\ {\text{represent}}\,\,{\text{propeller - shaped}}\,\,{\text{curves}}\,\,{\text{called}}\,\,{\text{_________}}{\text{.}} \hfill \\\ \end{gathered} \]

• A) \[{\text{lemniscates}}\]
• B) \[{\text{rose }}\,\,{\text{curve}}\]
• C) \[{\text{cardioids}}\]
• D) \[{\text{spiral}}\]

{\rm{The}}\,{\rm{value}}\,{\rm{of}}\,{\rm{integral }}\int\limits_0^1 {\int\limits_0^1 {e^{x + y} dxdy} } \,\,{\rm{is}}

• A) e-1
• B) e^2-1 \over 2
• C) e^2-1 \over 4
• D) (e-1)^2

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

• A) \(e^2-1 \over 2\)
• B) \(e-1\)
• C) \((e-1)^2\)
• D) \(e^2-1 \over 4\)

\[{\text{In polar coordinates, }}\iint\limits_R {f(x,\,\,y)\,\,dx\,dy\,\, = \,\, - - - - - - }\,\]

• A) \[\iint\limits_G {f(r\,\sin \,\theta ,\,\,r\,\cos \theta )\,rdr\,d\theta }\]
• B) \[\iint\limits_G {f(r\,\cos \theta ,\,r\,\sin \,\theta )\,dr\,d\theta }\,\,\]
• C) \[\int\limits_G {f(r\,\cos \theta ,\,r\,\sin \,\theta )\,rdr\,d\theta } \]
• D) \[\iint\limits_G {f(r\,\cos \theta ,\,r\,\sin \,\theta )\,rdr\,d\theta }\,\]

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

• A) $$e + 1$$
• B) $$0$$
• C) $$e - 1$$
• D) $${e^{{y^2}}}$$

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

{\text{While changing Cartesian Integral into Polar integral, we change }}x{\text{ by - - - - - - - - - and }}y{\text{ by - - - - - - - - - - - - }}{\text{.}}

• A) Sec\,\,\theta ,\,\,\,\,\,\,Co\sec \,\,\theta
• B) Sin\,\,\theta ,\,\,\,\,\,\,Cos\,\,\theta
• C) Cos\,\,\theta ,\,\,\,\,\,Sin\,\,\theta
• D) Co\sec \,\,\theta ,\,\,\,\,\,\,Sec\,\,\theta

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

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

\[ \int\limits_0^1 {\,\int\limits_0^{\ln \,y} {x\,} } dy\,dx\,\, = \,\, - - - - - - - - - \]

• A) \[ \frac{{\ln \,y}} {2} \]
• B) \[ \frac{{\ln \,x}} {2} \]
• C) \[ \ln \,y \]
• D) \[ \frac{{\left( {\ln \,x} \right)^2 }} {2} \]

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

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

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

• A) $$\int\limits_0^1 {\int\limits_{\frac{y}{2}}^{4x} {{e^{{x^2}}}dydx} } $$
• B) $$\int\limits_0^1 {\int\limits_0^{{x^2}} {{e^{{x^2}}}dy} } dx$$
• C) $$\int\limits_0^1 {\int\limits_0^x {{e^{{x^2}}}dy} } dx$$
• D) $$\int\limits_0^1 {\int\limits_0^{2x} {{e^{{x^2}}}dydx} } $$

\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}{{27}}
• C) \frac{1}{9}
• D) \frac{1}{3}

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

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