\eqalign{ & {\text{Wallis sine formula when n is odd}} \cr & \int\limits_0^{\frac{\pi }{2}} {Si{n^7}xdx} = \cr}

• A) \frac{6}{7} \cdot \frac{4}{5} \cdot \frac{2}{3}
• B) \frac{6}{7} \cdot \frac{4}{5} \cdot \frac{2}{3} \cdot \frac{\pi }{2}
• C) \frac{7}{6} \cdot \frac{5}{4} \cdot \frac{3}{2}
• D) \frac{7}{6} \cdot \frac{5}{4} \cdot \frac{3}{2} \cdot \frac{\pi }{2}

{\text{To eveluate a line integral, the integrand is expressed in terms of }}x,y,z{\text{ with }}

• A) dr = x i + y j + z k
• B) dr = dx + dy + dz
• C) dr = dx i + dy j + dz k
• D) dr = x + y + z

\eqalign{ & {\text{Wallis sine formula when n is odd}} \cr & \int\limits_0^{\frac{\pi }{2}} {Co{s^n}x} dx = \cr}

• A) \frac{{n - 1}}{n} \cdot \frac{{n - 3}}{{n - 2}} \cdot \frac{{n - 5}}{{n - 4}} \cdot \frac{{n - 7}}{{n - 6}} \cdot \cdot \cdot \frac{6}{7} \cdot \frac{4}{5} \cdot \frac{2}{3}
• B) \frac{{n - 1}}{n} \cdot \frac{{n - 3}}{{n - 2}} \cdot \frac{{n - 5}}{{n - 4}} \cdot \frac{{n - 7}}{{n - 6}} \cdot \cdot \cdot \frac{5}{6} \cdot \frac{3}{4} \cdot \frac{1}{2} \cdot \frac{\pi }{2}
• C) \frac{n}{2} \cdot \frac{{n - 2}}{2} \cdot \frac{{n - 4}}{2} \cdot \frac{{n - 6}}{2} \cdot \cdot \cdot \frac{6}{7} \cdot \frac{4}{5} \cdot \frac{2}{3}
• D) \frac{{n - 1}}{2} \cdot \frac{{n - 1}}{2} \cdot \frac{{n - 1}}{2} \cdot \frac{{n - 1}}{2} \cdot \cdot \cdot \frac{5}{6} \cdot \frac{3}{4} \cdot \frac{1}{2} \cdot \frac{\pi }{2}

\eqalign{ & {\text{The arc length of the portation of the circular helix where }}(dx/dt) = - \sin t, (dy/dt) = \cos t \cr & {\text{and}} (dz/dt) = 1 {\text{and }}0 \leqslant t \leqslant \pi {\text{, then the arc lenght is}} \cr}

• A) L = \int\limits_0^\pi {\sqrt 2 } dx
• B) L = \int\limits_0^\pi {\sqrt 2 } dt
• C) L = \int\limits_0^\pi {\sqrt 2 } dy
• D) L = \int\limits_0^\pi { - \sqrt 2 } dt

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

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

\eqalign{ & {\text{In 3D - space the parametric equations }}x = {\text{ }}x(t),y{\text{ }} = {\text{ }}y(t),{\text{ }}z = z(t){\text{ can be expressed in single vector }} \cr & {\text{equation as}} \cr}

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

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

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

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

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

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

• A) {I_{{c_1}}} - {I_{{c_2}}} = 0
• B) {I_{{c_2}}} + {I_{{c_2}}} = 0
• C) {I_{{c_1}}} \times {I_{{c_2}}} = 0
• D) {I_{{c_1}}} + {I_{{c_2}}} = 0

{\text{For a vector valued function }}\vec r(t){\text{ }} = {\text{ }}3{\text{ }}i + 4j - 0k{\text{ then lenght of }}\vec r(t){\text{ }}

• A) \left\| r \right\| = 5
• B) \left\| r \right\| = 25
• C) \left\| r \right\| = 4
• D) \left\| r \right\| = 9

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

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

{\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{finite}}
• C) {\text{zero}}
• D) - 1

{\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 + 2\hat j
• C) \,\hat i + 2t\,\hat j
• D) \,\hat i + \hat j

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

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

\[{\text{The}}\,\,{\text{grad}}\,\,{\text{operator}}\,\,\nabla \,\,{\text{acts}}\,\,{\text{on}}\,\,{\text{a(an)}}\,\,{\text{_________}}\,\,{\text{and}}\,\,{\text{gives}}\,\,{\text{a}}\,\,{\text{vector}}{\text{.}}\]

• A) \[{\text{scalar}}\]
• B) \[{\text{constant}}\]
• C) \[{\text{vector}}\]
• D) \[{\text{unit}}\,\,{\text{vector}}\]

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

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

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

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

If r is a vector-valued function in 2-space or 3- space, then \[{\frac{d}{{dt}}\left[ {\int {r(t)dt} } \right]}\]

• A) \[{r(t)}\]
• B) \[{\int {r(t)dt} }\]
• C) \[d\int {r(t)} \]
• D) None of above

{\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) \,\,\hat i + t\,\hat j + 3\,\hat k
• C) \,\,\hat i + \hat j + \,\hat k
• D) \,2\,t\,\hat i + t\,\hat j + 3\,\hat k