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

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

A smooth function is a function that has continuous derivatives up to some desired order over some domain

• A) No
• B) Yes
• 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) $$y - {\text{axis,}}\,\,{\text{below}}\,{\text{the origin}}{\text{.}}$$
• C) $$x - {\text{axis,}}\,\,{\text{left to the origin}}{\text{.}}$$
• D) $$y - {\text{axis,}}\,\,{\text{above the origin}}{\text{.}}$$

$${\text{The differential equation, }}{\kern 1pt} {\kern 1pt} dz{\kern 1pt} = {\kern 1pt} 4xy{\kern 1pt} dx{\kern 1pt} {\kern 1pt} + {\kern 1pt} {\kern 1pt} \left( {2{x^2} + 3{y^2}} \right){\kern 1pt} dy,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{is}}\,\,{\kern 1pt} {\text{an exact differential equation}}{\text{.}}$$

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

$$\begin{gathered} {\text{Consider}}\,\,{\text{the}}\,\,{\text{two}}\,\,{\text{functions,}}\,\,P(x,y)\,\,{\text{and}}\,\,Q(x,y)\,\,{\text{have}}\,\,{\text{continuous}}\,\,{\text{partial}}\,\,{\text{derivatives}}\,\,{\text{in}}\,\,{\text{a}}\,\,{\text{certain}}\,\, \hfill \\ {\text{domain}}\,\,D\,\,{\text{(say)}}{\text{.}}\,\,{\text{The}}\,\,{\text{differential}}\,\,{\text{equation,}}\,\,P(x,y)dx + Q(x,y)dy = 0,\,\,{\text{is}}\,\,{\text{an}}\,\,{\text{exact}}\,\,{\text{differential}}\,\,{\text{equation}} \hfill \\ {\text{if}}\,\,{\text{and}}\,\,{\text{only}}\,\,{\text{if}}\,\,\_\_\_\_\_\_\_\_. \hfill \\\ \end{gathered}$$

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

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

• A) $$L = \int\limits_a^b {\sqrt {{{\left( {dx/dt} \right)}^2} - {{\left( {dy/dt} \right)}^2} - {{\left( {dz/dt} \right)}^2}} } dt$$
• B) $$L = \int\limits_a^b {\sqrt {{{\left( {dx/dt} \right)}^2} + {{\left( {dy/dt} \right)}^2} + {{\left( {dz/dt} \right)}^2}} } dy$$
• C) $$L = \int\limits_a^b {\sqrt {{{\left( {dx/dt} \right)}^2} + {{\left( {dy/dt} \right)}^2} + {{\left( {dz/dt} \right)}^2}} } dx$$
• D) $$L = \int\limits_a^b {\sqrt {{{\left( {dx/dt} \right)}^2} + {{\left( {dy/dt} \right)}^2} + {{\left( {dz/dt} \right)}^2}} } dt$$

$$\eqalign{ & {\text{In 2D - space the parametric equations }}x = {\text{ }}x(t),y{\text{ }} = {\text{ }}y(t){\text{ can be expressed in single vector }} \cr & {\text{equation}} {\text{as}} \cr}$$

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

$$\eqalign{ & {\text{Polar co - ordinates of a point are}} \left( {{\text{ - 4,}} \frac{{ - 3\pi }}{2}} \right){\text{. Which of the following is another possible polar }} \cr & {\text{co - ordinates representation of this point?}} \cr}$$

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

$${\text{If}}\,\,\vec r(t) = {t^2}\,\hat i + t\,\hat j + 3\,t\,\hat k,\,\,{\text{then}}\,\,\,\frac{d}{{dt}}\,\left[ {\vec r(t)} \right] = \_\_\_\_\_\_\_\_\_.$$

• A) $$\,2\,t\,\hat i + \hat j + 3\,\hat k$$
• B) $$\,2\,t\,\hat i + t\,\hat j + 3\,\hat k$$
• C) $$\,\,\hat i + \hat j + \,\hat k$$
• D) $$\,\,\hat i + t\,\hat j + 3\,\hat k$$

Vector valued function has

• A) none of these
• B) domain consists of vectors and range consists of real numbers
• C) domain consists of real numbers and range consists of vectos
• D) domain and range consists of real numbers

$$\begin{gathered} {\text{The}} ~ {\text{graph}}~ {\text{of}} ~ \hfill \ {\text{r = (1 + t)}} {\text{i + ( - 2}} {\text{ + }} {\text{3t)}} {\text{j - 4t}} {\text{k}} ~ \hfill \ {\text{is}} ~{\text{the}}~ \hfill \\ \end{gathered}$$

• A) line that passes through the point (1, -2, 0)
• B) line that passes through the point (2, 1, -4)
• C) None of these
• D) line that passes through the point (1, 3, -4)

${\text{The differential equation }}\,\,dz\, = \,\left( {1 + 2y} \right)\,dx\,\, + \,\,\left( {1 + 2x} \right)\,dy\,\,\,\,\,{\text{is}}\,{\text{an exact differential equation}}{\text{.}}$

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

${\text{The differential equation }}\,\,dz\, = \,{x^2}\,dx\,\, + \,\,{y^2}\,dy\,\,\,\,\,{\text{is}}\,{\text{an exact differential equation}}{\text{.}}$

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

$${\text{The equation }}r = \,a\,\theta {\text{ represents - - - - - - - - - , where }}a{\text{ is positive}}{\text{.}}$$

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

$$\begin{gathered} {\text{A}}\,\,{\text{curve}}\,\,{\text{that}}\,\,'{\text{winds}}\,\,{\text{around}}\,\,{\text{the}}\,\,{\text{origin'}}\,\,{\text{infinitely}}\,\,{\text{many}}\,\,{\text{times}}\,\,{\text{in}}\,\,{\text{such}}\,\,{\text{a}}\,\,{\text{way}}\,\,{\text{that}}\,\,r\,\,{\text{increases}}\,\, \hfill \\ \left( {{\text{or}}\,\,{\text{decreases}}} \right)\,\,{\text{steadily}}\,\,{\text{as}}\,\,\theta \,\,\,{\text{increases}}\,\,{\text{is}}\,\,{\text{called}}\,\,{\text{a}}\,\,\_\_\_\_\_\_\_\_. \hfill \\\ \end{gathered}$$

• A) $${\text{roses}}$$
• B) $${\text{limacons}}$$
• C) $${\text{cardioids}}$$
• D) $${\text{spiral}}$$

$$\begin{gathered} {\text{The}}\,\,{\text{equations}}\,\,{\text{of}}\,\,{\text{the}}\,\,{\text{form:}} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,r = a + b\,\sin \theta ,\,\,\,r = a - b\,\sin \theta , \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,r = a + b\,\cos \theta ,\,\,\,r = a - b\,\sin \theta , \hfill \\ {\text{produce}}\,\,{\text{polar}}\,\,{\text{curves}}\,\,{\text{called}}\,\,{\text{_________}}{\text{.}} \hfill \\\ \end{gathered}$$

• A) $${\text{limacons}}$$
• B) $${\text{spiral}}$$
• C) $${\text{straight line}}$$
• D) $${\text{rose}}\,\,{\text{curve}}$$