$${\text{If}} p(r,\theta ) {\text{is a point in polar coordinate system, then}} \theta {\text{is called}}$$

• A) $${\text{Acute angle of }}p$$
• B) $${\text{Reflex angle of }}p$$
• C) $${\text{Polar angle of }}p$$
• D) $${\text{Reflex angle of }}p$$

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

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

$$\int\limits_0^{\frac{\pi }{2}} {\int\limits_0^1 {r\,\,dr\,\,d\theta } } \,\,\, = \,\,\, - - - - - - -$$

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

$${\text{If}}\,\,\vec r(t) = 3{t^2}\hat i + 2t\,\hat j,\,\,{\text{then}}\,\,\int {\vec r(t)\,dt = \_\_\_\_\_\_\_\_\_.}$$

• A) $${t^3} + {t^2}\, + {C_1}$$
• B) $$6t\,\hat i + 2\,\hat j$$
• C) $${t^3}\hat i + {t^2}\,\hat j + {C_1}\hat i + {C_2}\hat j$$
• D) $${t^3}\hat i + {t^2}\,\hat j + {C_2}\hat j$$

${\text{Let }}\bar r(t)\,\, = \,\,2t\,\hat i\,\, + \,\,3t\,\,\hat j,\,\,{\text{then}}\,\,\bar r'(t)\,\, = \,\,2\,\hat i\,\, + \,\,3\,\hat j.$

• A) $False$
• B) $True$
• C)
• D)

A parametric curve C in 2-space or 3-space is called _________ if it is the graph of some smooth vector-valued function.

• A) discontinuous
• B) smooth
• C)
• D)

$${\text{ }}Equation{\text{ }}of{\text{ }}the{\text{ }}tangent{\text{ }}line{\text{ }}of{\text{ }}vector{\text{ }}valued{\text{ }}function~ r(t) ~ at~ r({t_0}) ~is$$

• A) $$r = r({t_0}) + r'({t_0})$$
• B) $$r = r({t_0}) + tr'({t_0})$$
• C) $$r = r({t_0})$$
• D) $$r = r({t_0}) + t$$

A single curve can be represented by

• A) Only one vector-valued function
• B) None of these
• C) Infinitely many vector-valued function
• D) Only two vector-valued function

$$\eqalign{ & {\text{Polar co - ordinates of a point are}} \left( {{\text{ - 3,}} \frac{{ - 3\pi }}{4}} \right){\text{. Which of the following is another possible polar }} \cr & {\text{co - ordinates representation of this point?}} \cr}$$

• A) $$\left( {{\text{ - 3,}} \frac{\pi }{3}} \right)$$
• B) $$\left( {{\text{ - 3,}} \frac{\pi }{4}} \right)$$
• C) $$\left( {{\text{ - 3,}} \frac{\pi }{2}} \right)$$
• D) $$\left( {{\text{ - 3,}} \frac{{3\pi }}{4}} \right)$$

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

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

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

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

$${\text{The relation between the polar coordinates }}(r,{\text{ }}\theta ){\text{ and the rectangular coordinates }}(x,{\text{ }}y)\,{\text{is given by - - - - - - - }}{\text{.}}$$

• A) $$x\, = \,r\,\cos \,\theta ,\,\,\,\,\,\,y\, = \,r\,\sin \,\theta$$
• B) $$x\, = \,r\,\sin \,\theta ,\,\,\,\,\,\,y\, = \,r\,\cos \,\theta$$
• C) $$x\, = \,r\,\sec \,\theta ,\,\,\,\,\,\,y\, = \,r\,\operatorname{cosec} \,\theta$$
• D) $$x\, = \,r\,\cos \,\theta ,\,\,\,\,\,\,y\, = \,r\,\sec \,\theta$$

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

• A) $$x = x(t),y = y(t), z = z(t){\text{ }}; \left( {a \leqslant t \leqslant b} \right)$$
• B) $$x = x'(t) + y'(t) + z'(t){\text{ }}; \left( {a \leqslant t \leqslant b} \right)$$
• C) $$x = x(t) + y(t) + z(t){\text{ }}; \left( {a \leqslant t \leqslant b} \right)$$
• D) $$x = x'(t),y = y'(t), z = z'(t){\text{ }}; \left( {a \leqslant t \leqslant b} \right)$$

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

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

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

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

$$\begin{gathered} {\text{In polar coordinate system, the equation }}r\, = \,2a\,\sin \,\theta {\text{ represents a circle passes through the origin,}}\, \ {\text{with center on}}\,{\text{ - - - - - - - }}{\text{.}} \\ \end{gathered}$$

• A) $$x - {\text{axis,}}\,\,{\text{right to the origin}}{\text{.}}$$
• B) $$x - {\text{axis,}}\,\,{\text{left to the origin}}{\text{.}}$$
• C) $$y - {\text{axis,}}\,\,{\text{above the origin}}{\text{.}}$$
• D) $$y - {\text{axis,}}\,\,{\text{below}}\,{\text{the origin}}{\text{.}}$$

$${\text{In the integration in polar coordinates, }}dx\,dy{\text{ is replaced by - - - - - - - - - - - }}{\text{.}}$$

• A) $$r\,dr\,d\theta$$
• B) $$r\,\,dr$$
• C) $$d\theta$$
• D) $$dr\,d\theta$$

$${\text{The}}\,\,{\text{vector}}\,\,\vec r\,\,{\text{is}}\,\,{\text{continuous}}\,\,{\text{at}}\,\,{t_0}\,\,{\text{if}}\,\,\_\_\_\_\_\_\_\_\_.$$

• A) $${\text{(c)}}\,\,\,\,\,\,\mathop {\lim }\limits_{t \to {t_0}} \vec r(t)\,\, = \vec r({t_0})\,.$$
• B) $${\text{(b)}}\,\,\,\,\,\,\mathop {\lim }\limits_{t \to {t_0}} \vec r(t)\,\,{\text{exist}}.$$
• C) $${\text{(a)}}\,\,\,\,\,\,\vec r({t_0})\,\,{\text{is}}\,\,{\text{defined}}.$$
• D) $${\text{(d)}}\,\,\,\,\,\,{\text{All}}\,\,{\text{(a),}}\,\,{\text{(b)}}\,\,{\text{and}}\,\,{\text{(c)}}\,{\text{.}}$$

$$\begin{gathered} {\text{If}}\,\,a > 0,\,\,{\text{then}}\,\,{\text{equations}}\,\,{\text{of}}\,\,{\text{the}}\,\,{\text{form:}} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{r^2} = {a^2}\cos 2\theta ,\,\,\,{r^2} = - {a^2}\cos 2\theta , \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{r^2} = {a^2}\sin 2\theta ,\,\,\,{r^2} = - {a^2}\sin 2\theta , \hfill \\ {\text{represent}}\,\,{\text{propeller - shaped}}\,\,{\text{curves}}\,\,{\text{called}}\,\,{\text{_________}}{\text{.}} \hfill \\\ \end{gathered}$$

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

$\int\limits_0^\pi {\int\limits_0^1 {{r^2}\,\,dr\,\,d\theta } } \,\, = \,\,\, - - - - - - - -$

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