Integration along two separate paths joining the same two end points does not necessarily give identical results.

• A) False
• B) True
• C)
• D)

If a vector field exists for all points of the curve C, then for each element of arc we can form the ________.

• A) scalar filed
• B) vector field
• C) vector product
• D) scalar product

If the integration is carried out along the path of a particular curve, such an integral is called . . . . . . . . .

• A) Surface integral.
• B) Line integral.
• C) Volume integral.
• D) Indefinite integral.

$${\text{To eveluate a line integral, the integrand is expressed in terms of }}x,y,z{\text{ with }}$$

• A) $$dr = x i + y j + z k$$
• B) $$dr = dx i + dy j + dz k$$
• C) $$dr = x + y + z$$
• D) $$dr = dx + dy + dz$$

If a vector field $F(r)$ exist for all points of the curve $C$, then we can form -----------------$F$ for each element of arc.

• A) scalar field
• B) vector Field
• C)
• D)

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

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

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

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

Line integral is used to calculate --------

• A) length
• B) force
• C) area
• D) volume

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

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

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

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

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

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

Evaluate $\int lnxdx=--------$

• A) $xlnx-\frac{1}{x}+c$
• B) $xlnx-x+c$
• C) $lnx-\frac{1}{x}+c$
• D) $lnx-x+x$

Path of integration must be _________.

• A) (b) discontinuous
• B) (a) continuous
• C) (d) Both (a) and (c).
• D) (c) single-valued

A line integral naturally involved --------- independent variables in space.

• A) four
• B) two
• C) three
• D) five

$$\eqalign{ & {\text{Wallis sine formula when n is even}} \cr & \int\limits_0^{\frac{\pi }{2}} {Si{n^4}x} dx = \cr}$$

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

The total work done in moving a particle along the curve is given by -----------

• A) $limit_{\delta->{\infty}}\sum F_{t} \delta_{s}=\int F_{t}ds$
• B) $limit_{\delta->{0}}\sum F_{t} \delta_{s}=\int F_{t}ds$
• C) $limit_{\delta->{0}}\sum W_{t} \delta_{s}=\int F_{t}ds$
• D) $limit_{\delta->{\infty}}\sum W_{t} \delta_{s}=\int F_{t}ds$

Integration along two separate paths joining the same two end points--------- give(s) identical results.

• A) Always
• B) Partially
• C) Necessarily
• D) Does not necessarily

$$\eqalign{ & {\text{Wallis sine formula when n is even}} \cr & \int\limits_0^{\frac{\pi }{2}} {Co{s^4}x} dx = \cr}$$

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

$${\text{The}}\,\,{\text{div}}\,\,{\text{operator}}\,\,\nabla \,\,{\text{acts}}\,\,{\text{on}}\,\,{\text{a(an)}}\,\,{\text{_________}}\,\,{\text{and}}\,\,{\text{gives}}\,\,{\text{a}}\,\,{\text{scalar}}{\text{.}}$$

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

Applications of Green’s Theorem are meant to be in -------

• A) two-dimensional
• B) One-dimensional
• C) three-dimensional
• D) four-dimensional