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

• 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) $$x'(t)\hat i + y'(t)\hat j + z(t)\hat k$$

$${\text{The arc length of the curve}} r(t) = {t^3} i + t j + \frac{1}{2}\sqrt 6 {{\text{t}}^2}k {\text{;}} 0 \leqslant t \leqslant 2{\text{, can be written as }}$$

• A) $$L = \int\limits_0^2 {\sqrt {9{t^4} + 1 - 6{t^2}} } dt$$
• B) $$L = \int\limits_0^2 {\sqrt {9{t^4} + 1 - 3{t^2}} } dt$$
• C) $$L = \int\limits_0^2 {\sqrt {9{t^4} + 1 + 6{t^2}} } dt$$
• D) $$L = \int\limits_0^2 {\sqrt {9{t^4} + 1 + 3{t^2}} } dt$$

If $$z = f(x, y)$$ and $$dz = Pdx + Qdy$$ then dz is exact differential when

• A) $$\frac{{\partial P}}{{\partial x}} = \frac{{\partial Q}}{{\partial y}}$$
• B) $$\frac{{\partial P}}{{\partial y}} = \frac{{\partial Q}}{{\partial x}}$$
• C) $$P = Q$$
• D) $$\frac{{{\partial ^2}P}}{{\partial x\partial y}} = \frac{{{\partial ^2}Q}}{{\partial x\partial y}}$$

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

• A) $${\text{a straight line}}$$
• B) $${\text{cardioid}}$$
• C) $${\text{rose curve}}$$
• D) $${\text{lemniscate}}$$

$${\text{The graph of the equation }}r\, = \,a(1 + \sin \,\theta ){\text{ is symmetric about - - - - - - - - - }}{\text{.}}$$

• A) $${\text{pole}}$$
• B) $${\text{y - axis}}$$
• C) $${\text{initial line}}$$
• D) $${\text{None of these}}{\text{.}}$$

$${\text{For}}\,\,{\text{a}}\,\,{\text{function}}\,\,\vec r(t) = x(t)\hat i + y(t)\hat j + z(t)\hat k\,\,{\text{in}}\,\,{\text{3 - space}}\,\,{\text{we}}\,\,{\text{define}}\,\,\mathop {\lim }\limits_{t \to \alpha } \,\vec r(t) = \_\_\_\_\_\_\_\_\_.$$

• A) $$x(t)\hat i + \left( {\mathop {\lim }\limits_{t \to \alpha } y(t)} \right)\hat j + \left( {\mathop {\lim }\limits_{t \to \alpha } z(t)} \right)\hat k$$
• B) $$\left( {\mathop {\lim }\limits_{t \to \alpha } x(t)} \right)\hat i + \left( {\mathop {\lim }\limits_{t \to \alpha } y(t)} \right)\hat j + z(t)\hat k$$
• C) $$\left( {\mathop {\lim }\limits_{t \to \alpha } x(t)} \right)\hat i + \left( {\mathop {\lim }\limits_{t \to \alpha } y(t)} \right)\hat j$$
• D) $$\left( {\mathop {\lim }\limits_{t \to \alpha } x(t)} \right)\hat i + \left( {\mathop {\lim }\limits_{t \to \alpha } y(t)} \right)\hat j + \left( {\mathop {\lim }\limits_{t \to \alpha } z(t)} \right)\hat k$$