Vector valued function has ---------

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

$$\begin{gathered} {\text{If}}\,\,\vec r(t) = x(t)\hat i + y(t)\hat j\,\,{\text{is}}\,\,{\text{a}}\,\,{\text{vector - valued}}\,\,{\text{function}}\,\,{\text{in}}\,\,{\text{2 - space,}}\,\,{\text{and}}\,\,{\text{if}}\,\,x(t)\,\,{\text{and}}\,\,y(t)\,\,{\text{are}}\,\,{\text{differentiable,}}\,\, \hfill \\ {\text{then}}\,\,\,\frac{d}{{dt}}\,\left[ {\vec r(t)} \right] = \_\_\_\_\_\_\_\_\_. \hfill \\\ \end{gathered}$$

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

$\begin{gathered} {\text{Let G be the rectangular box defined by the inequalities }}a \leqslant x \leqslant b,\,\,\,c \leqslant y \leqslant d,\,\,\,\,\,e \leqslant z \leqslant f. \hfill \ {\text{If }}f\,\,{\text{is continuous on G, then}}\,\,\int\limits_a^b {\int\limits_c^d {\int\limits_e^f {f(x,\,\,y,\,\,z)} } } \,dz\,\,dy\,\,dx = \,\,\, - - - - - - - - \hfill \\\ \end{gathered}$

• A) ${\text{All}}\,{\text{three}}\,{\text{options}}\,{\text{are}}\,{\text{true}}{\text{.}}$
• B) $\int\limits_a^b {\int\limits_e^f {\int\limits_c^d {f(x,\,\,y,\,\,z)} } } \,dy\,\,dz\,\,dx$
• C) $\int\limits_e^f {\int\limits_a^b {\int\limits_c^d {f(x,\,\,y,\,\,z)} } } \,dy\,\,dx\,\,dz$
• D) $\int\limits_c^d {\int\limits_a^b {\int\limits_e^f {f(x,\,\,y,\,\,z)} } } \,dz\,\,dx\,\,dy\,$

$$\begin{gathered} {\text{In}}\,\,{\text{3 - space,}}\,\,\vec r(t) = x(t)\hat i + y(t)\hat j + z(t)\hat k,\,\,{\text{is}}\,\,{\text{smooth}}\,\,{\text{function}}\,\,{\text{of}}\,\,t\,\,{\text{if}}\,\,x'(t),\,\,\,y'(t)\,\,{\text{and}}\,\,z'(t)\,\,{\text{are}}\,\, \hfill \\ {\text{_________}}\,\,{\text{and}}\,\,{\text{there}}\,\,{\text{is}}\,\,{\text{no}}\,\,{\text{value}}\,\,{\text{of}}\,\,{\text{t}}\,\,{\text{at}}\,\,{\text{which}}\,\,{\text{all}}\,\,{\text{three}}\,\,{\text{derivatives}}\,\,{\text{are}}\,\,{\text{zero}}{\text{.}} \hfill \\\ \end{gathered}$$

• A) $${\text{continuous}}$$
• B) $${\text{discontinuous}}$$
• C)
• D)

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

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

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

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

$$\begin{gathered} {\text{The}}\,\,{\text{lemniscates}}\,\,{\text{are}}\,\,{\text{centered}}\,\,{\text{at}}\,\,{\text{the}}\,\,{\text{origin,}}\,\,{\text{but}}\,\,{\text{the}}\,\,{\text{position}}\,\,{\text{relative}}\,\,{\text{to}}\,{\text{the}}\,\,{\text{polar}}\,\,{\text{axis}}\,\, \hfill \\ {\text{depends}}\,\,{\text{on}}\,\,{\text{the}}\,\,{\text{sign}}\,\,{\text{preceding}}\,\,{\text{the}}\,\,{a^2}\,\,{\text{and}}\,\,{\text{whether}}\,\,{\text{_________}}\,\,{\text{appears}}\,\,{\text{in}}\,\,{\text{the}}\,\,{\text{equation}}{\text{.}} \hfill \\\ \end{gathered}$$

• A) $${\text{(d)}}\,\,\,{\text{Both}}\,\,{\text{(a)}}\,\,{\text{or}}\,\,{\text{(b)}}{\text{.}}$$
• B) $$(b)\,\,\,\cos 2\theta$$
• C) $$(c)\,\,\,\tan \theta$$
• D) $$(a)\,\,\,\sin 2\theta$$

$${\text{A vector valued function in 3 - D can be expressed as}}$$

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

$${\text{If}}\,\,P = 1 + 4xy\,\,{\text{and}}\,\,Q = 5{x^2}\,\,{\text{then}}\,\,\_\_\_\_\_\_\_\_\_.$$

• A) $$\frac{{\partial P}}{{\partial y}} \ne \frac{{\partial Q}}{{\partial x}}$$
• B) $$\frac{{\partial Q}}{{\partial x}} = 5{x^2}$$
• C) $$\frac{{\partial P}}{{\partial y}} = \frac{{\partial Q}}{{\partial x}}$$
• D) $$\frac{{\partial P}}{{\partial y}} = 1 + 4x$$

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

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

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

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

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

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

The differential dz of the function $$z = {x^2} + {y^2}$$ is ------------

• A) $$dz = 2xdx + 2ydy$$
• B) $$dz = 2x + 2y$$
• C) $$dz = 2dx + 2dy$$
• D) $$dz = (2x + 2y)dz$$

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) $$\frac{{{\partial ^2}P}}{{\partial x\partial y}} = \frac{{{\partial ^2}Q}}{{\partial x\partial y}}$$

Natural domain of a vector valued function is --------

• A) union of the natural domains of its components
• B) intersection of the natural domains of its components
• C) none of these
• D) not dependent on domains of its components

A vector-valued function is continuous at some point t0 if -----------

• A) all of its component should be continuous every where in the domain.
• B) each of its component is continuous at that point.
• C) some of its component is continuous at that point.
• D) each of its component is differentiable at that point.

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

• A) $False$
• B) $True$
• C)
• D)

$${\text{Green's Theorem states that}} \iint\limits_R {\left( {\frac{{\partial P}}{{\partial x}} - \frac{{\partial Q}}{{\partial y}}} \right)}dxdy$$

• A) $$- \oint {(Pdx + Qdy)}$$
• B) $$- \oint {(Pdx - Qdy)}$$
• C) $$\oint {(Pdx - Qdy)}$$
• D) $$\oint {(Pdx + Qdy)}$$

$${\text{Path}}\,\,{\text{of}}\,\,{\text{integration}}\,\,{\text{parallel}}\,\,{\text{to}}\,\,\_\_\_\_\_\_\_\_\_,\,\,dx = 0.\,\,\,\,\,\therefore \,\,{I_C} = \int\limits_C {Q\,dy} .$$

• A) $$dz = 0$$
• B) $$x{\text{ - axis}}$$
• C) $$z{\text{ - axis}}$$
• D) $$y{\text{ - axis}}$$

$$\eqalign{ & {\text{If a scalar field }}V(r){\text{ exists for all points on the curve, the }}.........{\text{ with }}dr \to 0{\text{,}} \cr & {\text{defines the line integral of }}V {\text{i}}{\text{.e, line integral = }} \int\limits_c {V(r)dr} \cr}$$

• A) $$\sum\limits_{\rho = 1}^\infty {V(r)d{r_\rho }}$$
• B) $$\sum\limits_{\rho = 0}^\infty {V(r)d{r_\rho }}$$
• C) $$\sum\limits_{\rho = 1}^n {V(r)d{r_\rho }}$$
• D) $$\sum\limits_{\rho = 0}^n {V(r)d{r_\rho }}$$