\[{\text{The graph of the equation }}r\, = \,a(1 - \cos \,\theta ){\text{ is symmetric about - - - - - - - - - }}{\text{.}}\]

• A) \[{\text{None of these}}{\text{.}}\]
• B) \[{\text{pole}}\]
• C) \[{\text{initial line}}\]
• D) \[{\text{y - axis}}\]

{\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{.}}

\begin{gathered} {\text{Let G be the rectangular box defined by the inequalities }}a \leqslant x \leqslant b,\,\,\,c \leqslant y \leqslant d,\,\,\,\,\,e \leqslant z \leqslant f. \hfill \ {\text{If }}f\,\,{\text{is continuous on G, then}}\,\,\int\limits_a^b {\int\limits_c^d {\int\limits_e^f {f(x,\,\,y,\,\,z)} } } \,dz\,\,dy\,\,dx = \,\,\, - - - - - - - - \hfill \\\ \end{gathered}

• A) {\text{All}}\,{\text{three}}\,{\text{options}}\,{\text{are}}\,{\text{true}}{\text{.}}
• B) \int\limits_a^b {\int\limits_e^f {\int\limits_c^d {f(x,\,\,y,\,\,z)} } } \,dy\,\,dz\,\,dx
• C) \int\limits_e^f {\int\limits_a^b {\int\limits_c^d {f(x,\,\,y,\,\,z)} } } \,dy\,\,dx\,\,dz
• D) \int\limits_c^d {\int\limits_a^b {\int\limits_e^f {f(x,\,\,y,\,\,z)} } } \,dz\,\,dx\,\,dy\,

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

If a vector field F(r) exist for all points of the curve C, then we can form -----------------F for each element of arc.

• A) vector Field
• B) scalar field
• C)
• D)

\begin{gathered} {\text{If}}\,\,{\text{a}}\,\,\_\_\_\_\_\_\_\_\_\,\,V(r)\,\,{\text{exists}}\,\,{\text{for}}\,\,{\text{all}}\,\,{\text{points}}\,\,{\text{on}}\,\,{\text{the}}\,{\text{curve,}}\,\,{\text{then}}\,\,\sum\limits_{p = 1}^n {V(r)\,d{r_p}} \,\,{\text{with}}\,\,dr \to 0\,\,{\text{defines}}\,\, \hfill \\ {\text{the}}\,\,{\text{line}}\,\,{\text{integral}}\,\,{\text{of}}\,\,V\,\,{\text{i}}{\text{.e}}{\text{.}}\,\,{\text{line}}\,\,{\text{integral}} = \int\limits_C {V(r)\,dr.} \hfill \\\ \end{gathered}

• A) {\text{scalar}}\,\,{\text{field}}
• B) {\text{vector}}\,\,{\text{quantity}}
• C) {\text{vector}}\,\,{\text{field}}
• D) {\text{vector}}\,\,{\text{space}}

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

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

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

• A) line that passes through the point (2, 1, -4)
• B) line that passes through the point (1, 3, -4)
• C) line that passes through the point (1, -2, 0).
• D) None of these

{\text{The}}\,\,{\text{differential}}\,\,{\text{equation,}}\,\,dz{\kern 1pt} = {\kern 1pt} 4xy{\kern 1pt} dx{\kern 1pt} {\kern 1pt} + {\kern 1pt} {\kern 1pt} \left( {2{x^2} + 3{y^2}} \right){\kern 1pt} dy,\,\,{\text{is}}\,\,{\kern 1pt} {\text{an exact differential equation}}{\text{.}}

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

{\text{To}}\,\,{\text{evaluate}}\,\,{\text{the}}\,\,{\text{line}}\,\,{\text{integral,}}\,\,\int\limits_C {V(r)\,dr\,{\text{,}}\,\,{\text{the}}\,\,{\text{integrand}}\,\,{\text{is}}\,\,{\text{expressed}}\,\,{\text{in}}\,\,{\text{terms}}\,\,{\text{of}}\,\,x,\,\,y,\,\,z\,\,{\text{with}}\,\,d\vec r = \_\_\_\_\_\_\_\_\_.}

• A) dx\,\hat i + dy\,\hat j
• B) dx\,\hat i
• C) dx\,\hat i + dy\,\hat j + dz\,\hat k
• D) \sqrt {dx\,\hat i + dy\,\hat j + dz\,\hat k}

\[{\text{The}}\,\,{\text{differential}}\,\,{\text{equation,}}\,\,dz{\kern 1pt} = {\kern 1pt} 4xy{\kern 1pt} dx{\kern 1pt} {\kern 1pt} + {\kern 1pt} {\kern 1pt} \left( {2{x^2} + 3{y^2}} \right){\kern 1pt} dy,\,\,{\text{is}}\,\,{\kern 1pt} {\text{an exact differential equation}}{\text{.}}\]

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

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

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

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

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

The line integral \int\limits_C {V(r)dr} representing the area of the --------- surface between the end points of the curve.

• A) smooth
• B) curved
• C) plane
• D) none of these

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

The division by zero is allowed in mathematics.

• A) True
• B) False
• C)
• D)

In the proof by contradiction, we lead the assumption to --------------

• A) contingency
• B) None of these.
• C) tautology
• D) Absurdity

In inductive property the first iteration of the loop is solved for?

• A) I(0)
• B) I(k + 1)
• C) None
• D) I(k)