$${\text{Path}}\,\,{\text{of}}\,\,{\text{integration}}\,\,{\text{parallel}}\,\,{\text{to}}\,\,\_\_\_\_\_\_\_\_\_,\,\,dy = 0.\,\,\,\,\,\therefore \,\,{I_C} = \int\limits_C {P\,dx} .$$

• A) $$x{\text{ - axis}}$$
• B) $$dz = 0$$
• C) $$z{\text{ - axis}}$$
• D) $$y{\text{ - axis}}$$

Unless we are directed otherwise, we always proceed round the closed boundary in a (an) _________ manner.

• A) clockwise
• B) anticlockwise
• C)
• D)

The value of the line integral is independent of the path taken and depends only on the ……………. if the integrand is seen to be an exact differential.

• A) Coordinates of the y-axis
• B) Coordinates of the two starting points
• C) Coordinates of the x-axis
• D) Coordinates of the two end points

$$\int\limits_0^{\frac{\pi }{2}} {Co{s^2}x} dx = \frac{1}{2}\left| {\frac{\pi }{2} + \frac{{\sin \pi }}{2}} \right| =$$

• A) $$\frac{\pi }{4}$$
• B) $$\frac{\pi }{2}$$
• C) $$\frac{\pi }{3}$$
• D) $$\frac{{3\pi }}{4}$$

If the integrand of the given integral is seen to be a (an) _________ differential, then the value of the line integral is independent of the path taken.

• A) total
• B) partial
• C) exact
• D) non-exact

The value of the integral is --------- independent of the path of integration taken.

• A) may or may not
• B) always
• C)
• D)

$$\begin{gathered} {\text{Consider}}\,\,{\text{a}}\,\,{\text{vector}}\,\,A\,\,\,{\text{for}}\,\,{\text{which}}\,\,\nabla .A = 0\,\,{\text{at}}\,\,{\text{all}}\,\,{\text{points,}}\,\,{\text{i,e}}{\text{.}}\,\,{\text{for}}\,\,{\text{all}}\,\,{\text{values}}\,\,{\text{of}}\,\,x,\,y,\,z,\,\,{\text{is}}\,\,{\text{called}}\,\, \hfill \\ {\text{a(an)}}\,\,\_\_\_\_\_\_\_\_\_\,. \hfill \\\ \end{gathered}$$

• A) $${\text{constant}}$$
• B) $${\text{unit}}\,\,{\text{vector}}$$
• C) $${\text{solenoid}}\,\,{\text{vector}}$$
• D) $${\text{scalar}}$$

$${\text{If}}\,\,\vec F = {F_1}\,\hat i + {F_2}\,\hat j + {F_3}\,\hat k\,\,{\text{and}}\,\,d\vec r = dx\,\hat i + dy\,\hat j + dz\,\hat k.\,\,{\text{Then,}}\,\,\vec F.d\vec r = \_\_\_\_\_\_\_\_.$$

• A) $$(b)\,\,\,\,\,\,\int\limits_C {\left( {{F_1}\,dx + {F_2}\,dy + {F_3}\,dz} \right)}$$
• B) $$(c)\,\,\,\,\,\,\left( {{F_1}\,\hat i + {F_2}\,\hat j + {F_3}\,\hat k} \right)$$
• C) $$(d)\,\,\,\,\,\,{\text{Both}}\,\,{\text{(a)}}\,\,{\text{and}}\,\,{\text{(b)}}{\text{.}}$$
• D) $$(a)\,\,\,\,\,\,\left( {{F_1}\,\hat i + {F_2}\,\hat j + {F_3}\,\hat k} \right).\left( {dx\,\hat i + dy\,\hat j + dz\,\hat k} \right)$$

$${\text{If}}\,\,Pdx + Qdy + Rdw\,\,{\text{is}}\,\,{\text{an}}\,\,{\text{exact}}\,\,{\text{differential}}\,\,{\text{equation}}\,\,{\text{then}}\,\,\oint\limits_C {\left( {Pdx + Qdy + Rdw} \right)} \,\,{\text{is}}\,\,{\text{__________}}{\text{.}}\,\,$$

• A) $${\text{infinite}}$$
• B) $${\text{zero}}$$
• C) $${\text{finite}}$$
• D) $$- 1$$

$${\text{If}} I = \int\limits_{AB} {Pdx + Qdy} {\text{ and }}(Pdx + Qdy){\text{ is an exact differential}} {\text{then}}$$

• A) $${I_{{c_2}}} = {I_{{c_2}}}$$
• B) $${I_{{c_1}}} = - {I_{{c_2}}}$$
• C) $${I_{{c_1}}} = 2{I_{{c_1}}}$$
• D) $${I_{{c_1}}} = {I_{{c_2}}}$$

$$\int\limits_0^{\frac{\pi }{2}} {{{\sin }^2}x} dx = \frac{1}{2}\left| {\frac{\pi }{2} - \frac{{\sin \pi }}{2}} \right| =$$

• A) $$\frac{\pi }{4}$$
• B) $$\frac{{3\pi }}{4}$$
• C) $$\frac{\pi }{2}$$
• D) $$\frac{\pi }{3}$$

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

• 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