\(The\,value\,of\,integral\,\int\limits_0^1 {\int\limits_{x^2 }^x {x\,} } dydx\,\,is \)

• A) \(0\)
• B) \(1\)
• C) \({{1} \over 12}\)
• D) \(12\)

The\,point\,(\,\,3,\,\,{189^0}\,\,)\,\,{\text{and the point - - - - - - - - - - - - - - }}\,{\text{are the same in polar system}}{\text{.}}

• A) (\,\, - 3,\,\,{189^0}\,\,)
• B) (\,\,3,\,\,{99^0}\,)
• C) (\,\, - 3,\,\,{9^0}\,\,)
• D) ( - 3,\,\,{279^0}\,\,)

$${\text{The}} \int\limits_0^1 {2x{e^{{x^2}}}dx} = $$

• A) $$e - 1$$
• B) $$e + 1$$
• C) $${e^{{x^2}}}$$
• D) $$e$$

\[{\text{In polar coordinate system, the equation }}r\, = \,a{\text{ represents a circle with center at}}\,{\text{ - - - - - - - }}{\text{.}}\]

• A) \[{\text{None of these}}{\text{.}}\]
• B) \[{\text{Origin}}\]
• C) \[x - {\text{axis}}\,\,{\text{and passes through the origin}}{\text{.}}\]
• D) \[y - {\text{axis}}\,\,{\text{and passes through the origin}}{\text{.}}\]

If the value of the integral \int_0^3 {\int_0^1 {(x + 1)dydx} } =15/2 , then what will be the value of the integral \int_0^1 {\int_0^3 {(x + 1)dxdy} }?

• A) -15/2
• B) 15/2
• C) 0
• D) 9/2

The\,value\,of\,integral\,\int\limits_0^1 {\int\limits_{x^2 }^x {x\,} } dydx\,\,is

• A) 12
• B) {{1} \over 12}
• C) 1
• D) 0

\(The\,value\,of\,integral\,\int\limits_0^4 {\int\limits_0^{\frac{1}{4}y} {5\,} } dxdy\,is \)

• A) 10
• B) 5
• C) 4
• D) 0

\({\rm{The}}\,{\rm{value}}\,{\rm{of}}\,{\rm{integral }}\int\limits_0^{\frac{\pi }{2}} {\int\limits_0^\pi {\cos (x + y)dxdy} } \,\,{\rm{is}} \)

• A) 0
• B) 2
• C) 4
• D) -2

$${{\text{In polar coordinate system, x - axis is also called}}}$$

• A) $${{\text{None of these}}}$$
• B) $${{\text{Pole}}}$$
• C) $${{\text{Imaginary axis}}}$$
• D) $${{\text{Polar axis}}}$$

\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{independent}}
• B) {\text{dependent}}
• C)
• D)

{\text{The arc length of the curve }}r(t) = {t^2}i - tj{\text{ }};{\text{ }}0 \leqslant t \leqslant 1{\text{, can be written as }}

• A) L = \int\limits_0^1 {\sqrt {2{t^2} - 1} } dt
• B) L = \int\limits_0^1 {\sqrt {4{t^2} - 1} } dt
• C) L = \int\limits_0^1 {\sqrt {4{t^2} + 1} } dt
• D) L = \int\limits_0^1 {\sqrt {2{t^2} + 1} } dt

\[r(t) = x(t)\,i + y(t)\,j\]

• A) is a real valued function
• B) is a vector valued function
• C)
• D)

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

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

{\text{For}}\,\,{\text{a}}\,\,{\text{function}}\,\,\vec r(t) = x(t)\hat i + y(t)\hat j + z(t)\hat k\,\,{\text{in}}\,\,{\text{3 - 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 + \left( {\mathop {\lim }\limits_{t \to \alpha } y(t)} \right)\hat j + \left( {\mathop {\lim }\limits_{t \to \alpha } z(t)} \right)\hat k
• 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
• C) \left( {\mathop {\lim }\limits_{t \to \alpha } x(t)} \right)\hat i + \left( {\mathop {\lim }\limits_{t \to \alpha } y(t)} \right)\hat j + z(t)\hat k
• D) x(t)\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

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

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

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

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

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

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

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

• A) True
• B) False
• C)
• D)

\[I = \int\limits_C {f(x,y)dx = } \int\limits_C {f(x,y)\frac{{ds}}{{dx}}dx,\,\,{\text{where}}\,\,\frac{{ds}}{{dx}} = \_\_\_\_\_\_\_\_\_.} \]

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