Graph~ of~ c = {x_0}i + {y_0}j + {z_0}k

• A) is the line passing through the point ( {x_0}, {y_0}, {z_0})
• B) is the point ( {x_0}, {y_0}, {z_0})
• C) None of these
• D) is the curve passing through the point ( {x_0}, {y_0}, {z_0})

$$\eqalign{ & {\text{Wallis sine formula when n is even}} \cr & \int\limits_0^{\frac{\pi }{2}} {Si{n^n}x} dx = \cr} $$

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

\[{\text{The}}\,\,{\text{curl}}\,\,{\text{operator,}}\,\,\nabla \times A,\,\,{\text{acts}}\,\,{\text{on}}\,\,{\text{a(an)}}\,\,{\text{_________}}\,\,{\text{and}}\,\,{\text{gives}}\,\,{\text{a}}\,\,{\text{vector}}\,\,{\text{as}}\,\,{\text{a}}\,\,{\text{result}}{\text{.}}\]

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

\[\begin{array}{*{20}{l}} \begin{gathered} Ifx'(t),y'(t)\,and{\text{ }}z'(t)are{\text{ }}continuous,then\,the{\text{ }}curve{\text{ }}given{\text{ }}by{\text{ }}the \hfill \\ parametric{\text{ }}equation{\text{ }}x = x(t),{\text{ }}y = y(t),{\text{ }}z = z(t){\text{ }}has{\text{ }}arc{\text{ }}length \hfill \\\ \end{gathered} \end{array}\]

• A) \[L = \int\limits_a^b {\sqrt {{x^2} + {y^2} + {z^2}} } dxdydz\]
• B) \[L = \int\limits_a^b {\sqrt {\frac{{dx}}{{dt}} + \frac{{dy}}{{dt}} + \frac{{dz}}{{dt}}} } dt\]
• C) \[L = \int\limits_a^b {\sqrt {{{\left( {\frac{{dx}}{{dt}}} \right)}^2} + {{\left( {\frac{{dy}}{{dt}}} \right)}^2} + {{\left( {\frac{{dz}}{{dt}}} \right)}^2}} } dt\]
• D) None of these

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

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

$$\eqalign{ & {\text{Wallis sine formula when n is odd}} \cr & \int\limits_0^{\frac{\pi }{2}} {Co{s^7}x} dx = \cr} $$

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

\begin{gathered} {\text{Let }}\bar r(t)\,\, = \,\,2{t^2}\,\hat i\,\, + \,\,3{t^3}\,\,\hat j,\,\,{\text{then}}\,\,\bar r'(1)\,\, = \,\,4\,\hat i\,\, + \,\,9\,\hat j. \hfill \ \hfill \\ \end{gathered}

• A) False
• B) True
• C)
• D)

\[{\text{For}}\,\,{\text{a}}\,\,{\text{function}}\,\,\vec r(t) = x(t)\hat i + y(t)\hat j\,\,{\text{in}}\,\,{\text{2 - space}}\,\,{\text{we}}\,\,{\text{define}}\,\,\mathop {\lim }\limits_{t \to \alpha } \,\vec r(t) = \_\_\_\_\_\_\_\_\_.\]

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

{\text{The}}\,\,{\text{curl}}\,\,{\text{operator,}}\,\,\nabla \times A,\,\,{\text{acts}}\,\,{\text{on}}\,\,{\text{a(an)}}\,\,{\text{_________}}\,\,{\text{and}}\,\,{\text{gives}}\,\,{\text{a}}\,\,{\text{vector}}\,\,{\text{as}}\,\,{\text{a}}\,\,{\text{result}}{\text{.}}

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

$${\text{One of the line integral properties is}} \int\limits_{AB} {Fds = } - \int\limits_{BA} {Fds} $$

• A) False
• B) True
• C)
• D)

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

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

\[\begin{gathered} {\text{If}}\,\,Pdx + Qdy + Rdw\,\,{\text{is}}\,\,{\text{an}}\,\,{\text{exact}}\,\,{\text{differential}}\,\,{\text{equation}}\,\,{\text{then}}\,\,\int\limits_C {\left( {Pdx + Qdy + Rdw} \right)\,\,{\text{is}}} {\text{_________}}\,\,{\text{of}} \hfill \\\ {\text{the}}\,\,{\text{path}}\,\,{\text{of}}\,\,{\text{integration}}\,{\text{.}} \hfill \\\\ \end{gathered} \]

• A) \[{\text{dependent}}\]
• B) \[{\text{independent}}\]
• C)
• D)

$$\eqalign{ & {\text{Wallis sine formula when n is even}} \cr & \int\limits_0^{\frac{\pi }{2}} {Si{n^4}x} dx = \cr} $$

• A) $$\frac{4}{5} \cdot \frac{2}{3}$$
• B) $$\frac{4}{3} \cdot \frac{2}{1} \cdot \frac{\pi }{2}$$
• C) $$\frac{3}{4} \cdot \frac{1}{2} \cdot \frac{\pi }{2}$$
• D) $$\frac{3}{4} \cdot \frac{1}{2}$$

Which integral gives the arc length of the curve \[r(t) = \frac{1}{3}{t^3}i + tj + {t^2}k\] over the interval [1,3]

• A) \[\int\limits_1^3 {\sqrt {{{(\frac{1}{3} + t)}^2}} } dt\]
• B) \[\int\limits_1^3 {\sqrt {{{(1 + t)}^2}} } dt\]
• C) \[\int\limits_1^3 {\sqrt {{{(1 + {t^2})}^2}} } dt\]
• D) \[\int\limits_1^3 {\sqrt {1 + t} } dt\]

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

\eqalign{ & {\text{Wallis sine formula when n is even}} \cr & \int\limits_0^{\frac{\pi }{2}} {Si{n^4}x} dx = \cr}

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

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

• A) line parallel to the vector \[{\text{ i - 3j + 4k}}\]
• B) line parallel to the vector \[{\text{ i + 2k}}\]
• C) line Perpendicular to the vector \[{\text{ i + 2k}}\]
• D) line Perpendicular to the vector \[{\text{ i - 3j + 4k}}\]

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

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

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

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

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