\[{\text{If}}\,\,P = 3{x^2} + 2{y^2}\,{\text{and}}\,\,Q = 4xy\,\,{\text{then}}\,\,\_\_\_\_\_\_\_\_\_.\]

• A) \[\frac{{\partial P}}{{\partial y}} = \frac{{\partial Q}}{{\partial x}}\]
• B) \[\frac{{\partial P}}{{\partial y}} \ne \frac{{\partial Q}}{{\partial x}}\]
• C) \[\frac{{\partial Q}}{{\partial x}} = 4xy\]
• D) \[\frac{{\partial P}}{{\partial y}} = 6x + 4y\]

{\text{One of the line integral properties is}} \int\limits_{AB} {Pdx + Qdy} = - \int\limits_{BA} {Pdx + Qdy}

• A) True
• B) False
• C)
• D)

r(t) = x(t)\,i + y(t)\,j

• A) is a real valued function
• B) is a vector valued function
• C)
• D)

$$\eqalign{ & {\text{In 2D - space the parametric equations }}x = {\text{ }}x(t),y{\text{ }} = {\text{ }}y(t){\text{ can be expressed in single vector }} \cr & {\text{equation}} {\text{as}} \cr} $$

• A) $$\vec r(t) = x(t) - y(t)$$
• B) $$\vec r(t) = x(t) + y(t)$$
• C) $$\vec r(t) = x(t)i + y(t)j$$
• D) $$\vec r(t) = x(t)j + y(t)i$$

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

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

{\text{If}}\,\,P = 3{x^2} + 2{y^2}\,{\text{and}}\,\,Q = 4xy\,\,{\text{then}}\,\,\_\_\_\_\_\_\_\_\_.

• A) \frac{{\partial P}}{{\partial y}} \ne \frac{{\partial Q}}{{\partial x}}
• B) \frac{{\partial P}}{{\partial y}} = \frac{{\partial Q}}{{\partial x}}
• C) \frac{{\partial P}}{{\partial y}} = 6x + 4y
• D) \frac{{\partial Q}}{{\partial x}} = 4xy

$$\eqalign{ & {\text{Wallis sine formula when n is odd}} \cr & \int\limits_0^{\frac{\pi }{2}} {Co{s^n}x} dx = \cr} $$

• A) $$\frac{{n - 1}}{n} \cdot \frac{{n - 3}}{{n - 2}} \cdot \frac{{n - 5}}{{n - 4}} \cdot \frac{{n - 7}}{{n - 6}} \cdot \cdot \cdot \frac{5}{6} \cdot \frac{3}{4} \cdot \frac{1}{2} \cdot \frac{\pi }{2}$$
• B) $$\frac{{n - 1}}{2} \cdot \frac{{n - 1}}{2} \cdot \frac{{n - 1}}{2} \cdot \frac{{n - 1}}{2} \cdot \cdot \cdot \frac{5}{6} \cdot \frac{3}{4} \cdot \frac{1}{2} \cdot \frac{\pi }{2}$$
• C) $$\frac{n}{2} \cdot \frac{{n - 2}}{2} \cdot \frac{{n - 4}}{2} \cdot \frac{{n - 6}}{2} \cdot \cdot \cdot \frac{6}{7} \cdot \frac{4}{5} \cdot \frac{2}{3}$$
• D) $$\frac{{n - 1}}{n} \cdot \frac{{n - 3}}{{n - 2}} \cdot \frac{{n - 5}}{{n - 4}} \cdot \frac{{n - 7}}{{n - 6}} \cdot \cdot \cdot \frac{6}{7} \cdot \frac{4}{5} \cdot \frac{2}{3}$$

{\text{The differential equation }}\,\,dz\, = \,8y\,dx\,\, + \,\,8x\,dy\,\,\,\,\,{\text{is}}\,{\text{an exact differential equation}}{\text{.}}

• A) {\text{True}}
• B) {\text{False}}
• C)
• D)

{\text{For a vector valued function }}\vec r(t){\text{ }} = {\text{ }}2t{\text{ }}i + (t - 1){\text{ }}j{\text{ then lenght of }}\vec r(t){\text{ }}

• A) \left\| {r(t)} \right\| = \sqrt {4{t^2} + {{(t - 1)}^2}}
• B) \left\| {r(t)} \right\| = \sqrt {4{t^2} - {{(t - 1)}^2}}
• C) \left\| {r(t)} \right\| = \sqrt {2{t^2} + {{(t - 1)}^2}}
• D) \left\| {r(t)} \right\| = \sqrt {{t^2} + {{(t - 1)}^2}}

\[\begin{gathered} {\text{If}}\,\,\vec r(t) = x(t)\hat i + y(t)\hat j\,\,{\text{is}}\,\,{\text{a}}\,\,{\text{vector - valued}}\,\,{\text{function}}\,\,{\text{in}}\,\,{\text{2 - space,}}\,\,{\text{and}}\,\,{\text{if}}\,\,x(t)\,\,{\text{and}}\,\,y(t)\,\,{\text{are}}\,\,{\text{differentiable,}}\,\, \hfill \\ {\text{then}}\,\,\,\frac{d}{{dt}}\,\left[ {\vec r(t)} \right] = \_\_\_\_\_\_\_\_\_. \hfill \\\ \end{gathered} \]

• A) \[x'(t)\hat i + y(t)\hat j\]
• B) \[x'(t)\hat i + y'(t)\hat j + z'(t)\hat k\]
• C) \[x'(t)\hat i + y'(t)\hat j\]
• D) \[x(t)\hat i + y'(t)\hat j\]

$$\eqalign{ & {\text{If }}x'(t), y'(t){\text{ and }}z'(t){\text{ are continuous for }}a \leqslant t \leqslant b{\text{, then the arc lenght for the given}} \cr & {\text{parametric equations }}x = x(t),y = y(t),z = z(t) {\text{ ;}} \left( {a \leqslant t \leqslant b} \right) {\text{is}} \cr} $$

• A) $$L = \int\limits_a^b {\sqrt {{{\left( {dx/dt} \right)}^2} + {{\left( {dy/dt} \right)}^2} + {{\left( {dz/dt} \right)}^2}} } dt$$
• B) $$L = \int\limits_a^b {\sqrt {{{\left( {dx/dt} \right)}^2} + {{\left( {dy/dt} \right)}^2} + {{\left( {dz/dt} \right)}^2}} } dx$$
• C) $$L = \int\limits_a^b {\sqrt {{{\left( {dx/dt} \right)}^2} - {{\left( {dy/dt} \right)}^2} - {{\left( {dz/dt} \right)}^2}} } dt$$
• D) $$L = \int\limits_a^b {\sqrt {{{\left( {dx/dt} \right)}^2} + {{\left( {dy/dt} \right)}^2} + {{\left( {dz/dt} \right)}^2}} } dy$$

\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 {{{\left( {\frac{{dx}}{{dt}}} \right)}^2} - {{\left( {\frac{{dy}}{{dt}}} \right)}^2}} \,\,dt
• 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 {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}}

{\text{The differential equation, }}{\kern 1pt} {\kern 1pt} dz{\kern 1pt} = {\kern 1pt} \left( {2{x^2} - 2xy + 3} \right){\kern 1pt} dx{\kern 1pt} {\kern 1pt} + {\kern 1pt} {\kern 1pt} \left( {6{y^2} - 2{x^2} + 1} \right){\kern 1pt} dy,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{is}}\,\,{\kern 1pt} {\text{an exact differential equation}}{\text{.}}

• A) {\text{True}}
• B) {\text{False}}
• C)
• D)

\[\begin{gathered} {\text{If}}\,\,\vec r(t) = x(t)\hat i + y(t)\hat j + z(t)\hat k\,\,{\text{is}}\,\,{\text{a}}\,\,{\text{vector - valued}}\,\,{\text{function}}\,\,{\text{in}}\,\,{\text{3 - space,}}\,\,{\text{and}}\,\,{\text{if}}\,\,x(t),\,\,y(t)\,\,{\text{and}}\,\,z(t)\,\,{\text{are}}\,\, \hfill \\ {\text{differentiable,}}\,\,{\text{then}}\,\,\,\frac{d}{{dt}}\,\left[ {\vec r(t)} \right] = \_\_\_\_\_\_\_\_\_. \hfill \\\ \end{gathered} \]

• A) \[x(t)\hat i + y'(t)\hat j + z'(t)\hat k\]
• B) \[x'(t)\hat i + y'(t)\hat j + z'(t)\hat k\]
• C) \[x'(t)\hat i + y'(t)\hat j + z(t)\hat k\]
• D) \[x'(t)\hat i + y'(t)\hat j\]

For exact differential equation of this form Pdx+Qdy=0, ---------

• A) \frac{{\partial P}}{{\partial y}}+\frac{{\partial Q}}{{\partial x}}=0
• B) \frac{{\partial P}}{{\partial y}}= \frac{{\partial Q}}{{\partial x}}
• C) \frac{{\partial P}}{{\partial y}}- \frac{{\partial Q}}{{\partial x}}=0
• D) \frac{{\partial P}}{{\partial y}} \ne \frac{{\partial Q}}{{\partial x}}

\[\begin{gathered} {\text{The}} ~ {\text{graph}}~ {\text{of}} ~ \hfill \ {\text{r = (1 + t)}} {\text{i + ( - 2}} {\text{ + }} {\text{3t)}} {\text{j - 4t}} {\text{k}} ~ \hfill \ {\text{is}} ~{\text{the}}~ \hfill \\ \end{gathered} \]

• A) line that passes through the point (2, 1, -4)
• B) line that passes through the point (1, -2, 0)
• C) None of these
• D) line that passes through the point (1, 3, -4)

\begin{gathered} {\text{Consider}}\,\,{\text{the}}\,\,{\text{two}}\,\,{\text{functions,}}\,\,P(x,y)\,\,{\text{and}}\,\,Q(x,y)\,\,{\text{have}}\,\,{\text{continuous}}\,\,{\text{partial}}\,\,{\text{derivatives}}\,\,{\text{in}}\,\,{\text{a}}\,\,{\text{certain}}\,\, \hfill \\ {\text{domain}}\,\,D\,\,{\text{(say)}}{\text{.}}\,\,{\text{The}}\,\,{\text{differential}}\,\,{\text{equation,}}\,\,P(x,y)dx + Q(x,y)dy = 0,\,\,{\text{is}}\,\,{\text{an}}\,\,{\text{exact}}\,\,{\text{differential}}\,\,{\text{equation}} \hfill \\ {\text{if}}\,\,{\text{and}}\,\,{\text{only}}\,\,{\text{if}}\,\,\_\_\_\_\_\_\_\_. \hfill \\\ \end{gathered}

• A) \frac{{\partial P}}{{\partial y}} \ne \frac{{\partial Q}}{{\partial x}}
• B) \frac{{\partial Q}}{{\partial x}}
• C) \frac{{\partial P}}{{\partial y}} = \frac{{\partial Q}}{{\partial x}}
• D) \frac{{\partial P}}{{\partial y}}

{\text{If the path of integration c joining A and B is divided into two parts AK and KB, then}}

• A) {I_c} = {I_{AK}} - {I_{KB}}
• B) {I_c} = - {I_{AK}} - {I_{KB}}
• C) - {I_c} = {I_{AK}} + {I_{KB}}
• D) {I_c} = {I_{AK}} + {I_{KB}}

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

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

\[{\text{If}}\,\,P = 1 + 4xy\,\,{\text{and}}\,\,Q = 5{x^2}\,\,{\text{then}}\,\,\_\_\_\_\_\_\_\_\_.\]

• A) \[\frac{{\partial P}}{{\partial y}} \ne \frac{{\partial Q}}{{\partial x}}\]
• B) \[\frac{{\partial Q}}{{\partial x}} = 5{x^2}\]
• C) \[\frac{{\partial P}}{{\partial y}} = \frac{{\partial Q}}{{\partial x}}\]
• D) \[\frac{{\partial P}}{{\partial y}} = 1 + 4x\]