$$\begin{gathered} {\text{If}}\,\,Pdx + Qdy + Rdw\,\,{\text{is}}\,\,{\text{an}}\,\,{\text{exact}}\,\,{\text{differential}}\,\,{\text{equation}}\,\,{\text{then}}\,\,\int\limits_C {\left( {Pdx + Qdy + Rdw} \right)\,\,{\text{is}}} {\text{__________}}\,\,{\text{of}} \hfill \\\ {\text{the}}\,\,{\text{path}}\,\,{\text{of}}\,\,{\text{integration}}\,{\text{.}} \hfill \\\\ \end{gathered}$$

• A) $${\text{independent}}$$
• B) $${\text{dependent}}$$
• C)
• D)

$$\begin{gathered} {\text{For}}\,\,{\text{line}}\,\,{\text{integral}}\,\,{\text{with}}\,\,{\text{respect}}\,\,{\text{to}}\,\,{\text{arc}}\,\,{\text{length,}}\,\,\,{\text{when}}\,\,x\,\,{\text{and}}\,\,y\,\,{\text{are}}\,\,{\text{expressed}}\,\,{\text{in}}\,\,{\text{parametric}}\,\,{\text{form,}}\,\, \hfill \\\ I = \int\limits_C {f(x,y)ds = \int\limits_{{t_1}}^{{t_2}} {f(x,y)ds} } {\text{,}}\,\,{\text{where}}\,\,ds = {\text{__________}}{\text{.}} \hfill \\\\ \end{gathered}$$

• A) $$\sqrt {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}}$$
• B) $$\sqrt {{{\left( {\frac{{dx}}{{dt}}} \right)}^2} + {{\left( {\frac{{dy}}{{dt}}} \right)}^2}}$$
• C) $$\sqrt {{{\left( {\frac{{dx}}{{dt}}} \right)}^2} - {{\left( {\frac{{dy}}{{dt}}} \right)}^2}} \,\,dt$$
• D) $$\sqrt {{{\left( {\frac{{dx}}{{dt}}} \right)}^2} + {{\left( {\frac{{dy}}{{dt}}} \right)}^2}} \,\,dt$$

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

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

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

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

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

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

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

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

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

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

Expression for the double integral for the volume of a solid bounded above by z=8-x-y and below by rectangle R=$0 \leqslant x \leqslant 4,\,5 \leqslant y \leqslant 8$ is........

• A) $V = \int_5^8 {\int_0^4 {(8 - x - y)dx} } $
• B) $V = \int_5^8 {\int_0^4 {(8 - x - y)dxdy} } $
• C) $V = \int_0^4 {\int_5^8 {(8 - x - y)dxdy} } $
• D) \[V = \int_5^8 {\int_0^4 {(8 - x - y)dydx} } \]

{\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_0^x {{e^{{x^2}}}dy} } dx
• B) \int\limits_0^1 {\int\limits_{\frac{y}{2}}^{4x} {{e^{{x^2}}}dydx} }
• C) \int\limits_0^1 {\int\limits_0^{2x} {{e^{{x^2}}}dydx} }
• D) \int\limits_0^1 {\int\limits_0^{{x^2}} {{e^{{x^2}}}dy} } dx

\[\begin{gathered} {\text{The}}\,\,{\text{equations}}\,\,{\text{of}}\,\,{\text{the}}\,\,{\text{form:}} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,r = a\,\sin \,n\theta ,\,\,\, \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,r = a\cos n\theta , \hfill \\ {\text{represent}}\,\,{\text{flower - shaped}}\,\,{\text{curves}}\,\,{\text{called}}\,\,{\text{_________}}{\text{.}} \hfill \\\ \end{gathered} \]

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

\[\begin{gathered} {\text{The }}\,\,{\text{orientation}}\,\,{\text{ of }}\,\,{\text{the }}\,\,{\text{rose}}\,\,{\text{ relative}}\,\,{\text{ to}}\,\,{\text{ the}}\,\,{\text{ polar }}\,\,{\text{axis }}\,\,{\text{depends}}\,\,{\text{ on}}\,\,{\text{ the }}\,\,{\text{sign}}\,\,{\text{ of }}\,\,{\text{the }}\,\, \hfill \\ {\text{constant }}\,\,a{\text{ }}\,\,{\text{and }}\,\,{\text{whether}}\,\,{\text{ _________ }}\,\,{\text{appears }}\,\,{\text{in }}\,\,{\text{the}}\,\,{\text{ equation}}{\text{.}} \hfill \\\ \end{gathered} \]

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

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

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

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

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

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

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

\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) (a)\,\,\,\sin 2\theta
• D) (c)\,\,\,\tan \theta

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

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

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

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

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

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

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