{\text{To}}\,\,{\text{evaluate}}\,\,{\text{the}}\,\,{\text{line}}\,\,{\text{integral,}}\,\,\int\limits_C {V(r)\,dr\,{\text{,}}\,\,{\text{the}}\,\,{\text{integrand}}\,\,{\text{is}}\,\,{\text{expressed}}\,\,{\text{in}}\,\,{\text{terms}}\,\,{\text{of}}\,\,x,\,\,y,\,\,z\,\,{\text{with}}\,\,d\vec r = \_\_\_\_\_\_\_\_\_.}

• A) dx\,\hat i + dy\,\hat j
• B) dx\,\hat i
• C) dx\,\hat i + dy\,\hat j + dz\,\hat k
• D) \sqrt {dx\,\hat i + dy\,\hat j + dz\,\hat k}

\[{\text{The}}\,\,{\text{differential}}\,\,{\text{equation,}}\,\,dz{\kern 1pt} = {\kern 1pt} 4xy{\kern 1pt} dx{\kern 1pt} {\kern 1pt} + {\kern 1pt} {\kern 1pt} \left( {2{x^2} + 3{y^2}} \right){\kern 1pt} dy,\,\,{\text{is}}\,\,{\kern 1pt} {\text{an exact differential equation}}{\text{.}}\]

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

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

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

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

• A) line parallel to the vector {\text{ i - 3j + 4k}}
• B) line parallel to the vector {\text{ i + 2k}}
• C) line Perpendicular to the vector {\text{ i - 3j + 4k}}
• D) line Perpendicular to the vector {\text{ i + 2k}}

\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)

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

The line integral \int\limits_C {V(r)dr} representing the area of the --------- surface between the end points of the curve.

• A) smooth
• B) curved
• C) plane
• D) none of these

If z = f(x, y) and dz = Pdx + Qdy then dz is exact differential when ----------

• A) P = Q
• B) \frac{{\partial P}}{{\partial x}} = \frac{{\partial Q}}{{\partial y}}
• C) \frac{{{\partial ^2}P}}{{\partial x\partial y}} = \frac{{{\partial ^2}Q}}{{\partial x\partial y}}
• D) \frac{{\partial P}}{{\partial y}} = \frac{{\partial Q}}{{\partial x}}