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

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

$$\eqalign{ & {\text{The arc length of the portation of the circular helix where }}(dx/dt) = - \sin t, (dy/dt) = \cos t \cr & {\text{and}} (dz/dt) = 1 {\text{and }}0 \leqslant t \leqslant \pi {\text{, then the arc lenght is}} \cr}$$

• A) $$L = \int\limits_0^\pi {\sqrt 2 } dx$$
• B) $$L = \int\limits_0^\pi {\sqrt 2 } dy$$
• C) $$L = \int\limits_0^\pi {\sqrt 2 } dt$$
• D) $$L = \int\limits_0^\pi { - \sqrt 2 } dt$$

$$\begin{array}{l} If ~x'(t) ~and~ y'(t)~ are ~continuous, ~then~ the ~curve~ given~ by ~the ~parametric ~equation\x = x(t) , y = y(t) ~has ~arc~ length \end{array}$$

• A) $$L = \int\limits_a^b {\sqrt {{{\left( {\frac{{dx}}{{dt}}} \right)}^2} + {{\left( {\frac{{dy}}{{dt}}} \right)}^2} } } dt$$
• B) $$L = \int\limits_a^b {\sqrt {{{\left( x \right)}^2} + {{\left( y \right)}^2} } } dxdy$$
• C) $$L = \int\limits_a^b {\sqrt {\frac{{dx}}{{dt}} + \frac{{dy}}{{dt}} } } dt$$
• D) None of these

$${\text{The}}\,\,{\text{equation,}}\,\,\,{r^2} = 4\cos 2\theta ,\,\,{\text{represents}}\,\,{\text{a}}\,\,\_\_\_\_\_\_\_\_\_.$$

• A) $${\text{cardioids}}$$
• B) $${\text{rose }}\,\,{\text{curve}}$$
• C) $${\text{spiral}}$$
• D) $${\text{lemniscate}}$$

$\int\limits_0^1 {\int\limits_0^1 {\int\limits_0^1 {xyz} } } \,dx\,\,dy\,\,dz = \,\,\, - - - - - - - -$

• A) $\frac{1}{{10}}$
• B) $\frac{1}{2}$
• C) $\frac{1}{8}$
• D) $\frac{1}{4}$

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

• 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) $${\text{vector}}\,\,{\text{product}}$$