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) $$\sum\limits_{\rho = 0}^n {V(r)d{r_\rho }}$$

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

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

Unless we are directed otherwise, we always proceed round the closed boundary in a (an) _________ manner.

• A) clockwise
• B) anticlockwise
• C)
• D)

The value of the line integral is independent of the path taken and depends only on the ……………. if the integrand is seen to be an exact differential.

• A) Coordinates of the y-axis
• B) Coordinates of the two starting points
• C) Coordinates of the x-axis
• D) Coordinates of the two end points

$$\int\limits_0^{\frac{\pi }{2}} {Co{s^2}x} dx = \frac{1}{2}\left| {\frac{\pi }{2} + \frac{{\sin \pi }}{2}} \right| =$$

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

If the integrand of the given integral is seen to be a (an) _________ differential, then the value of the line integral is independent of the path taken.

• A) total
• B) partial
• C) exact
• D) non-exact

The value of the integral is --------- independent of the path of integration taken.

• A) may or may not
• B) always
• C)
• D)

$$\begin{gathered} {\text{Consider}}\,\,{\text{a}}\,\,{\text{vector}}\,\,A\,\,\,{\text{for}}\,\,{\text{which}}\,\,\nabla .A = 0\,\,{\text{at}}\,\,{\text{all}}\,\,{\text{points,}}\,\,{\text{i,e}}{\text{.}}\,\,{\text{for}}\,\,{\text{all}}\,\,{\text{values}}\,\,{\text{of}}\,\,x,\,y,\,z,\,\,{\text{is}}\,\,{\text{called}}\,\, \hfill \\ {\text{a(an)}}\,\,\_\_\_\_\_\_\_\_\_\,. \hfill \\\ \end{gathered}$$

• A) $${\text{constant}}$$
• B) $${\text{unit}}\,\,{\text{vector}}$$
• C) $${\text{solenoid}}\,\,{\text{vector}}$$
• D) $${\text{scalar}}$$

$${\text{If}}\,\,\vec F = {F_1}\,\hat i + {F_2}\,\hat j + {F_3}\,\hat k\,\,{\text{and}}\,\,d\vec r = dx\,\hat i + dy\,\hat j + dz\,\hat k.\,\,{\text{Then,}}\,\,\vec F.d\vec r = \_\_\_\_\_\_\_\_.$$

• A) $$(b)\,\,\,\,\,\,\int\limits_C {\left( {{F_1}\,dx + {F_2}\,dy + {F_3}\,dz} \right)}$$
• B) $$(c)\,\,\,\,\,\,\left( {{F_1}\,\hat i + {F_2}\,\hat j + {F_3}\,\hat k} \right)$$
• C) $$(d)\,\,\,\,\,\,{\text{Both}}\,\,{\text{(a)}}\,\,{\text{and}}\,\,{\text{(b)}}{\text{.}}$$
• D) $$(a)\,\,\,\,\,\,\left( {{F_1}\,\hat i + {F_2}\,\hat j + {F_3}\,\hat k} \right).\left( {dx\,\hat i + dy\,\hat j + dz\,\hat k} \right)$$

$${\text{If}}\,\,Pdx + Qdy + Rdw\,\,{\text{is}}\,\,{\text{an}}\,\,{\text{exact}}\,\,{\text{differential}}\,\,{\text{equation}}\,\,{\text{then}}\,\,\oint\limits_C {\left( {Pdx + Qdy + Rdw} \right)} \,\,{\text{is}}\,\,{\text{__________}}{\text{.}}\,\,$$

• A) $${\text{infinite}}$$
• B) $${\text{zero}}$$
• C) $${\text{finite}}$$
• D) $$- 1$$

$${\text{If}} I = \int\limits_{AB} {Pdx + Qdy} {\text{ and }}(Pdx + Qdy){\text{ is an exact differential}} {\text{then}}$$

• A) $${I_{{c_2}}} = {I_{{c_2}}}$$
• B) $${I_{{c_1}}} = - {I_{{c_2}}}$$
• C) $${I_{{c_1}}} = 2{I_{{c_1}}}$$
• D) $${I_{{c_1}}} = {I_{{c_2}}}$$

$$\int\limits_0^{\frac{\pi }{2}} {{{\sin }^2}x} dx = \frac{1}{2}\left| {\frac{\pi }{2} - \frac{{\sin \pi }}{2}} \right| =$$

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