$${\text{A vector valued function in 2 - D can be expressed as}}$$

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

$${\text{The}}\,\,{\text{curl}}\,\,{\text{operator,}}\,\,\nabla \times A,\,\,{\text{acts}}\,\,{\text{on}}\,\,{\text{a(an)}}\,\,{\text{_________}}\,\,{\text{and}}\,\,{\text{gives}}\,\,{\text{a}}\,\,{\text{vector}}\,\,{\text{as}}\,\,{\text{a}}\,\,{\text{result}}{\text{.}}$$

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

$${\text{If}} {\text{the}} {\text{integral}} {\text{is}} {\text{of}} {\text{the}} {\text{form}} \oint {(Pdx + Qdy)} {\text{ where }}P = - 5x - y{\text{ and }}Q = x - 2y{\text{ then}}$$

• A) $$- \iint\limits_R {\left( {\frac{{\partial P}}{{\partial x}} - \frac{{\partial Q}}{{\partial y}}} \right)}dxdy = 2 \iint\limits_R {dxdy}$$
• B) $$- \iint\limits_R {\left( {\frac{{\partial P}}{{\partial x}} - \frac{{\partial Q}}{{\partial y}}} \right)}dxdy = - 2\iint\limits_R {dxdy}$$
• C) $$- \iint\limits_R {\left( {\frac{{\partial P}}{{\partial x}} - \frac{{\partial Q}}{{\partial y}}} \right)}dxdy = \iint\limits_R {dxdy}$$
• D) $$- \iint\limits_R {\left( {\frac{{\partial P}}{{\partial x}} - \frac{{\partial Q}}{{\partial y}}} \right)}dxdy = - \iint\limits_R {dxdy}$$

$$I = \int\limits_C {f(x,y)dx = } \int\limits_C {f(x,y)\frac{{ds}}{{dx}}dx,\,\,{\text{where}}\,\,\frac{{ds}}{{dx}} = \_\_\_\_\_\_\_\_\_.}$$

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

In general, the value of the line integral depends on the particular path of integration.

• A) False
• B) True
• C)
• D)

$$\eqalign{ & {\text{Wallis sine formula when n is odd}} \cr & \int\limits_0^{\frac{\pi }{2}} {Co{s^7}x} dx = \cr}$$

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

Integration along two distinct paths joining the same two end points . . . . . . . . give the same results.

• A) always
• B) not necessarily
• C)
• D)

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

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

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

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

Sign of line integral is reversed when the direction of integration along the path is reversed.

• A) False
• B) True
• C)
• D)

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

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

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

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

$$\eqalign{ & {\text{Wallis sine formula when n is odd}} \cr & \int\limits_0^{\frac{\pi }{2}} {Si{n^7}xdx} = \cr}$$

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

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

• A) True
• B) False
• C)
• D)

$${\text{The}}\,\,{\text{line}}\,\,{\text{integral}}\,\,{\text{of}}\,\,F(r),\,\,\int\limits_C {\vec F.d\vec r\,,\,\,{\text{is}}\,\,{\text{a}}\,\,{\text{scalar}}\,\,{\text{because}}\,\,\vec F.d\vec r\,\,{\text{is}}\,\,{\text{a}}\,\,\_\_\_\_\_\_\_\_\_.}$$

• A) $${\text{scalar}}\,\,{\text{field}}$$
• B) $${\text{scalar}}\,\,{\text{product}}$$
• C) $${\text{vector}}\,\,{\text{field}}$$
• D) $${\text{vector}}\,\,{\text{product}}$$

$${\text{One of the line integral properties is}} \int\limits_c {Fds = }$$

• A) $$\int\limits_c {\left( {Pdx + Qdy} \right)}$$
• B) $$\int\limits_c {\left( {Pdx - Qdy} \right)}$$
• C) $$\int\limits_c {(P + Q)ds}$$
• D) $$\int\limits_c {\left( {Pdy + Qdx} \right)}$$

The integration taken round a closed curve is ------- provided $(Pdx+Qdy)$ is a(n) ------------ differential.

• A) zero, homogeneous
• B) one, homogeneous
• C) one, exact
• D) zero, exact

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

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

$${\text{One of the line integral properties is}} \int\limits_{AB} {Fds = } - \int\limits_{BA} {Fds}$$

• A) False
• B) True
• C)
• D)

If the integrand of the given integral is seen to be an exact differential, then the value of the line integral depends only on the _______.

• A) dimensions of the two end points
• B) mid points
• C) points of origin
• D) coordinates of the two end points