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

• 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

{\text{One of the line integral properties is}} \int\limits_c {Fds = }

• A) \int\limits_c {\left( {Pdx - Qdy} \right)}
• B) \int\limits_c {\left( {Pdx + Qdy} \right)}
• C) \int\limits_c {(P + Q)ds}
• D) \int\limits_c {\left( {Pdy + Qdx} \right)}

{\text{If}}\,\,\vec r(t) = 3{t^2}\hat i + 2t\,\hat j,\,\,{\text{then}}\,\,\int {\vec r(t)\,dt = \_\_\_\_\_\_\_\_\_.}

• A) {t^3} + {t^2}\, + {C_1}
• B) {t^3}\hat i + {t^2}\,\hat j + {C_1}\hat i + {C_2}\hat j
• C) 6t\,\hat i + 2\,\hat j
• D) {t^3}\hat i + {t^2}\,\hat j + {C_2}\hat j

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

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

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

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

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

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

$$\eqalign{ & {\text{If }}g{\text{ is a real valued function, then substituting }}t = g\left( u \right){\text{ for this change in parameter from }}r\left( t \right){\text{ to }}g\left( u \right) \cr & g{\text{ will satisfies the following conditions}} \cr & g{\text{ is differentiable}}{\text{.}} \cr & g{\text{ is continuous}}{\text{.}} \cr & g'(u) \ne 0 {\text{for any }}u{\text{ in the domain in }}g{\text{.}} \cr & {\text{the range of }}g{\text{ is the domain of }}r{\text{.}} \cr} $$

• A) True
• B) False
• C)
• D)

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

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

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

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

\[{\text{The}}\,\,{\text{vector}}\,\,\vec r\,\,{\text{is}}\,\,{\text{continuous}}\,\,{\text{at}}\,\,{t_0}\,\,{\text{if}}\,\,\_\_\_\_\_\_\_\_\_.\]

• A) \[{\text{(d)}}\,\,\,\,\,\,{\text{All}}\,\,{\text{(a),}}\,\,{\text{(b)}}\,\,{\text{and}}\,\,{\text{(c)}}\,{\text{.}}\]
• B) \[{\text{(c)}}\,\,\,\,\,\,\mathop {\lim }\limits_{t \to {t_0}} \vec r(t)\,\, = \vec r({t_0})\,.\]
• C) \[{\text{(b)}}\,\,\,\,\,\,\mathop {\lim }\limits_{t \to {t_0}} \vec r(t)\,\,{\text{exist}}.\]
• D) \[{\text{(a)}}\,\,\,\,\,\,\vec r({t_0})\,\,{\text{is}}\,\,{\text{defined}}.\]

\[\begin{gathered} {\text{For}}\,\,{\text{line}}\,\,{\text{integral}}\,\,{\text{with}}\,\,{\text{respect}}\,\,{\text{to}}\,\,{\text{arc}}\,\,{\text{length,}}\,\,\,{\text{when}}\,\,x\,\,{\text{and}}\,\,y\,\,{\text{are}}\,\,{\text{expressed}}\,\,{\text{in}}\,\,{\text{parametric}}\,\,{\text{form,}}\,\, \hfill \\\ I = \int\limits_C {f(x,y)ds = \int\limits_{{t_1}}^{{t_2}} {f(x,y)ds} } {\text{,}}\,\,{\text{where}}\,\,ds = {\text{_______}}{\text{.}} \hfill \\\\ \end{gathered} \]

• A) \[\sqrt {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}} \]
• B) \[\sqrt {{{\left( {\frac{{dx}}{{dt}}} \right)}^2} + {{\left( {\frac{{dy}}{{dt}}} \right)}^2}} \,\,dt\]
• C) \[\sqrt {{{\left( {\frac{{dx}}{{dt}}} \right)}^2} + {{\left( {\frac{{dy}}{{dt}}} \right)}^2}} \]
• D) \[\sqrt {{{\left( {\frac{{dx}}{{dt}}} \right)}^2} - {{\left( {\frac{{dy}}{{dt}}} \right)}^2}} \,\,dt\]

\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 = y'(t), z = 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(t) + 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)

If \[z = f(x, y)\] and \[dz = Pdx + Qdy\] then dz is exact differential when

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

{\text{For a vector valued function }}\vec r(t){\text{ }} = {\text{ }}\sqrt 2 t{\text{ }}i + (t + 1){\text{ }}j - k{\text{ then lenght of }}\vec r(t){\text{ }}

• A) \left\| {r(t)} \right\| = \sqrt {4{t^2} - {{(t + 1)}^2} - 1}
• B) \left\| {r(t)} \right\| = \sqrt {{t^2} + {{(t + 1)}^2} + 1}
• C) \left\| {r(t)} \right\| = \sqrt {4{t^2} + {{(t + 1)}^2} - 1}
• D) \left\| {r(t)} \right\| = \sqrt {2{t^2} + {{(t + 1)}^2} + 1}

{\text{For a vector valued function }}\vec r(t){\text{ }} = {\text{ }}\sqrt {2t} {\text{ }}i + (t + 1){\text{ }}j{\text{ then lenght of }}\vec r(t){\text{ }}

• A) \left\| {r(t)} \right\| = \sqrt {2t + {{(t - 1)}^2}}
• B) \left\| {r(t)} \right\| = \sqrt {2{t^2} + {{(t - 1)}^2}}
• C) \left\| {r(t)} \right\| = \sqrt {4t + {{(t - 1)}^2}}
• D) \left\| {r(t)} \right\| = \sqrt {4{t^2} + {{(t - 1)}^2}}

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

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