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

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

The\,value\,of\,integral\,\int\limits_0^1 {\int\limits_0^x \, } dydx\,\,is

• A) {{1} \over 2}
• B) 1
• C) 2
• D) 3

Which is not the property of double integral?

• A) additivity
• B) commutativity
• C) area property
• D) linearity

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

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

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

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

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

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

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

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

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

• A) 16
• B) 8
• C) 12
• D) 4

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

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

If the value of the integral $\int_0^3 {\int_0^1 {(x + 1)dydx} } $ =15/2 , then what will be the value of the integral $\int_0^1 {\int_0^3 {(x + 1)dxdy} } $?

• A) -15/2
• B) 15/2
• C) 0
• D) 9/2

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

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

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

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

{\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

{\rm{The}}\,{\rm{value}}\,{\rm{of}}\,{\rm{integral }}\int\limits_0^\pi {\int\limits_0^x {\frac{{\sin x}}{x}dydx} } \,\,{\rm{is}}

• A) 2
• B) -2
• C) -1
• D) 1

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

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

$\int\limits_0^{\frac{\pi }{2}} {{{\sin }^2}\,\,\theta \,\,\,d\theta } \,\, = \,\,\, - - - - - - - - $

• A) $\frac{1}{2}$
• B) $\frac{1}{3}.\frac{\pi }{2}$
• C) $\frac{1}{2}.\frac{\pi }{2}$
• D) $\frac{2}{3}$

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

\[{\text{The relation between the polar coordinates }}(r,{\text{ }}\theta ){\text{ and the rectangular coordinates }}(x,{\text{ }}y)\,{\text{is given by - - - - - - - }}{\text{.}}\]

• A) \[x\, = \,r\,\sin \,\theta ,\,\,\,\,\,\,y\, = \,r\,\cos \,\theta \]
• B) \[x\, = \,r\,\cos \,\theta ,\,\,\,\,\,\,y\, = \,r\,\sec \,\theta \]
• C) \[x\, = \,r\,\sec \,\theta ,\,\,\,\,\,\,y\, = \,r\,\operatorname{cosec} \,\theta \]
• D) \[x\, = \,r\,\cos \,\theta ,\,\,\,\,\,\,y\, = \,r\,\sin \,\theta \]

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

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