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

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

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

{\rm{The}}\,{\rm{value}}\,{\rm{of}}\,{\rm{integral }}\int\limits_0^{\frac{\pi }{2}} {\int\limits_0^\pi {\cos (x + y)dxdy} } \,\,{\rm{is}}

• A) -2
• B) 2
• C) 4
• D) 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The\,value\,of\,integral\,\int\limits_0^4 {\int\limits_0^{\frac{1}{4}y} {5\,} } dxdy\,is

• A) 0
• B) 10
• C) 4
• D) 5

\begin{gathered} {\text{If}}\,\,{\text{the}}\,\,{\text{region}}\,\,{\text{is}}\,\,{\text{bounded}}\,\,{\text{the}}\,\,{\text{left}}\,\,{\text{and}}\,\,{\text{right}}\,\,{\text{by}}\,\,{\text{vertical}}\,\,{\text{lines}}\,\,x = a\,\,{\text{and}}\,\,x = b\,\,{\text{and}}\,\,{\text{is}}\,\,{\text{bounded}}\,{\text{below}}\,\,{\text{and}}\,\, \hfill \\ {\text{above}}\,\,{\text{by}}\,\,{\text{curves}},\,\,y = {g_1}(x)\,\,{\text{and}}\,\,y = {g_2}(x),\,\,{\text{where}}\,\,{g_1}(x)\,\, \leqslant {g_2}(x)\,\,{\text{for}}\,\,a \leqslant x \leqslant b.\,\,{\text{Then}}\,\,\iint\limits_R {f(x,y)\,dA = }\,\,\_\_\_\_\_\_\_. \hfill \\\ \end{gathered}

• A) \int\limits_a^b {f(x,y)\,dx\,dy}
• B) \int\limits_{{g_1}(x)}^{{g_2}(x)} {f(x,y)\,dy\,dx}
• C) \int\limits_a^b {\int\limits_{{g_1}(x)}^{{g_2}(x)} {f(x,y)\,dy\,dx} }
• D) \int {f(x,y)\,dy\,dx}

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

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

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

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

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

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

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

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

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

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

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

{\text{Let }}\bar r(t)\,\, = \,\,2t\,\hat i\,\, + \,\,3t\,\,\hat j,\,\,{\text{then}}\,\,\bar r'(t)\,\, = \,\,2\,\hat i\,\, + \,\,3\,\hat j.

• A) False
• B) True
• C)
• D)