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

• 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) $\frac{\pi }{4}$

$${\text{The}}\,\,{\text{curve}},\,\,r = \theta \,\,(\theta \geqslant 0),\,\,{\text{is}}\,\,{\text{an}}\,\,{\text{equation}}\,\,{\text{of}}\,\,\_\_\_\_\_\_\_\_\_.$$

• A) $${\text{cardioids}}$$
• B) $${\text{roses}}$$
• C) $${\text{spiral}}\,\,{\text{with}}\,\,a = 1$$
• D) $${\text{limacons}}$$

$$\eqalign{ & {\text{Polar co - ordinates of a point are}} \left( {{\text{ - 1,}} \frac{{ - 3\pi }}{4}} \right){\text{. Which of the following is another possible polar }} \cr & {\text{co - ordinates representation of this point?}} \cr}$$

• A) $$\left( {{\text{ - 1,}} \frac{\pi }{4}} \right)$$
• B) $$\left( {{\text{ - 1,}} \frac{{3\pi }}{4}} \right)$$
• C) $$\left( {{\text{ - 1,}} \frac{\pi }{2}} \right)$$
• D) $$\left( {{\text{ - 1,}} \frac{\pi }{3}} \right)$$

$${\text{The differential equation, }}{\kern 1pt} {\kern 1pt} dz{\kern 1pt} = {\kern 1pt} \left( {2{x^2} - 2xy + 3} \right){\kern 1pt} dx{\kern 1pt} {\kern 1pt} + {\kern 1pt} {\kern 1pt} \left( {6{y^2} - 2{x^2} + 1} \right){\kern 1pt} dy,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{is}}\,\,{\kern 1pt} {\text{an exact differential equation}}{\text{.}}$$

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

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

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

$$\eqalign{ & {\text{Given the integral }}\iint\limits_R {f(x,y)dxdy}{\text{,}} {\text{can}} {\text{be}} {\text{expressed}} {\text{in}} {\text{ploar coordinates}} {\text{as}} ..............{\text{,}} \cr & {\text{where}} a \leqslant \theta \leqslant b {\text{and}} c \leqslant r \leqslant d. \cr}$$

• A) $$\int_a^c {\int_b^d {f(r,\theta )} } rd\theta dr$$
• B) $$\int_a^b {\int_c^d {f(r,\theta )} } drd\theta$$
• C) $$\int_a^b {\int_c^d {f(r,\theta )} } rd\theta dr$$
• D) $$\int_a^b {\int_c^d {f(r,\theta )} } rdrd\theta$$

$${\text{If}}\,\,P = 3{x^2} + 2{y^2}\,{\text{and}}\,\,Q = 4xy\,\,{\text{then}}\,\,\_\_\_\_\_\_\_\_\_.$$

• A) $$\frac{{\partial P}}{{\partial y}} \ne \frac{{\partial Q}}{{\partial x}}$$
• B) $$\frac{{\partial Q}}{{\partial x}} = 4xy$$
• C) $$\frac{{\partial P}}{{\partial y}} = \frac{{\partial Q}}{{\partial x}}$$
• D) $$\frac{{\partial P}}{{\partial y}} = 6x + 4y$$

$$\eqalign{ & {\text{If }}g{\text{ is a real valued function, then substituting }}t = g\left( u \right){\text{ for this change in parameter from }}r\left( t \right){\text{ to }}g\left( u \right) \cr & g{\text{ will satisfies the following conditions}} \cr & g{\text{ is differentiable}}{\text{.}} \cr & g{\text{ is continuous}}{\text{.}} \cr & g'(u) \ne 0 {\text{for any }}u{\text{ in the domain in }}g{\text{.}} \cr & {\text{the range of }}g{\text{ is the domain of }}r{\text{.}} \cr}$$

• A) False
• B) True
• C)
• D)

The rose curve has 2n-equally spaced petals of loops if n is _______.

• A) odd
• B) even
• C)
• D)

$${\text{In polar coordinate system, the equation }}r\, = \,a{\text{ represents a circle with center at}}\,{\text{ - - - - - - - }}{\text{.}}$$

• A) $$y - {\text{axis}}\,\,{\text{and passes through the origin}}{\text{.}}$$
• B) $$x - {\text{axis}}\,\,{\text{and passes through the origin}}{\text{.}}$$
• C) $${\text{Origin}}$$
• D) $${\text{None of these}}{\text{.}}$$

$${\text{In the integration of polar coordinates}} dxdy {\text{is replaced by}}$$

• A) $$rdrd\theta$$
• B) $$rdr$$
• C) $$d\theta$$
• D) $$drd\theta$$

Graph of a vector-valued function can fail to have a tangent vector at a point because

• A) b) derivative is zero at that point
• B) a) derivative does not exists at that point
• C) c) Both a) and b)
• D) d) Neither a) nor b)

$$The\,point\,(\,\,3,\,\,{189^0}\,\,)\,\,{\text{and the point - - - - - - - - - - - - - - }}\,{\text{are the same in polar system}}{\text{.}}$$

• A) $$(\,\, - 3,\,\,{189^0}\,\,)$$
• B) $$(\,\, - 3,\,\,{9^0}\,\,)$$
• C) $$(\,\,3,\,\,{99^0}\,)$$
• D) $$( - 3,\,\,{279^0}\,\,)$$

$\int\limits_0^{\frac{\pi }{2}} {{{\sin }^2}\,\,\theta \,\,\,d\theta } \,\, = \,\,\, - - - - - - - -$

• A) $\frac{2}{3}$
• B) $\frac{1}{3}.\frac{\pi }{2}$
• C) $\frac{1}{2}.\frac{\pi }{2}$
• D) $\frac{1}{2}$

$${\text{The arc length of the curve }}r(t) = {t^2}i - tj{\text{ }};{\text{ }}0 \leqslant t \leqslant 1{\text{, can be written as }}$$

• A) $$L = \int\limits_0^1 {\sqrt {2{t^2} - 1} } dt$$
• B) $$L = \int\limits_0^1 {\sqrt {2{t^2} + 1} } dt$$
• C) $$L = \int\limits_0^1 {\sqrt {4{t^2} - 1} } dt$$
• D) $$L = \int\limits_0^1 {\sqrt {4{t^2} + 1} } dt$$

$$\begin{gathered} {\text{If}}\,\,\vec r(t) = x(t)\hat i + y(t)\hat j + z(t)\hat k\,\,{\text{is}}\,\,{\text{a}}\,\,{\text{vector - valued}}\,\,{\text{function}}\,\,{\text{in}}\,\,{\text{3 - space,}}\,\,{\text{and}}\,\,{\text{if}}\,\,x(t),\,\,y(t)\,\,{\text{and}}\,\,z(t)\,\,{\text{are}}\,\, \hfill \\ {\text{differentiable,}}\,\,{\text{then}}\,\,\,\frac{d}{{dt}}\,\left[ {\vec r(t)} \right] = \_\_\_\_\_\_\_\_\_. \hfill \\\ \end{gathered}$$

• A) $$x'(t)\hat i + y'(t)\hat j + z'(t)\hat k$$
• B) $$x(t)\hat i + y'(t)\hat j + z'(t)\hat k$$
• C) $$x'(t)\hat i + y'(t)\hat j$$
• D) $$x'(t)\hat i + y'(t)\hat j + z(t)\hat k$$