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%Sets a problem. 
%\prob {Problem #}
%{text of problem}
%\ans
%{number1}{answer1}  	number1 is where you want answer1 to 
%go
%{number2}{answer2}
%{number3}{answer3}
%{number4}{answer4}
%{answer5}				no number. This one goes into 
%remaining slot.
%If you repeat a number, the second time you get a blank.
%Shouldn't break a page in the middle of a problem & answer unless 
%you have several paragraphs in the problem or the answers.


\prob {1}{0} 
{Evaluate $\int\!\!\int\limits_V\!\!\int z\,dV $  over the solid lying
inside the sphere
 $x^2 + y^2 + z^2  = a^2$ and above the plane $z = 0$.}
\ans
{5}{\displaystyle \frac{\pi a^4}{4}}
{1}{\displaystyle \frac{\pi a^4}{5}}
{2}{\displaystyle \frac{\pi a^4}{7}}
{3}{\displaystyle \frac{2\pi a^4}{5}}
{\displaystyle \frac{3\pi a^4}{7}}

\prob{2}{100} 
{Find the mass of the solid inside the cylinder
$x^2 + y^2  = 9$ lying above $z = 0$
and below $z = 5$ if the mass density at a point is equal to the 
distance from 
the point to the $z${--}axis.}
\ans
{4}{90\pi}
{5}{80\pi}
{1}{70\pi}
{2}{60\pi}
{100\pi}

\prob{3}{-100}
{Find the coordinates of the centroid of the region enclosed
by the cardioid \hspace*{\fill}\break
$r = 1 + \cos \theta$.\hspace*{\fill}\break
\break
Hint:  $\int^{2\pi}_0 \cos ^4 \theta  d\theta  = \frac{3}{4}\pi$, 
$\int^{2\pi}_0 \cos ^3 \theta  d\theta  = 0$  and
$\int^{2\pi}_0 \cos ^2 \theta  d\theta  = \pi$.}
\ans
{1}{(\displaystyle \frac{5}{6},0)}
{2}{(\displaystyle \frac{1}{2},0)}
{3}{(\displaystyle \frac{1}{4},0)}
{4}{(\displaystyle \frac{\sqrt{3}}{2},0)}
{(\displaystyle \frac{\sqrt{2}}{2},0)}


\prob {4}{100}
{Find the length of the curve parameterized by ${\bf r}(t) =
\cos (\pi t){\bf i} +\sin (\pi t){\bf j} + \pi t{\bf k}$,
$0 \leq t \leq 3$.}
\ans
{2}{3\pi \sqrt{2}}
{3}{1}
{4}{\pi}
{5}{\sqrt{3}\pi}
{\displaystyle\frac{9}{2}\pi}

\prob {5}{100}
{Let ${\bf F}(x,y,z) = \ln (y){\bf i} + y^3z{\bf k}$.  
Compute $\int_C {\bf F\cdot}\,dr$  where $C$ is the curve 
\hspace*{\fill}\linebreak
${\bf r}(t) = t{\bf i} + e^{-t}{\bf j} + e^t{\bf k}$, $0 \leq t \leq 1$.}
\ans
{3}{\displaystyle {\frac {1}{2} -\frac{1}{e}}}
{4}{1 + e}
{5}{1 - e}
{1}{-1}
{0}

\prob {6}{-120}
{Let ${\bf F}(x,y,z) = y\cos (x){\bf i} +x\cos (z){\bf j} + z\cos (y){\bf 
k}$.
Find $\mbox{curl}\;\bf F$ at the point $(0,0,\pi)$.}
\ans
{5}{-2{\bf k}}
{1}{\vec{0}}
{2}{\pi {\bf k}}
{3}{1}
{-{\bf i} + 2{\bf k}}

\prob {7}{100}
{Compute the line integral
$$\int_C 2xye^{^{\displaystyle x^2}}\,dx + e^{^{\displaystyle 
x^2}}\,dy$$
\noindent
where $C$ is the boundary of the region defined by:
$0 \leq x \leq 1$, $0 \leq y \leq x$, and $C$ is oriented 
counterclockwise.}
\ans
{1}{0}
{2}{1}
{3}{\displaystyle\frac{1}{2}}
{4}{e - 1}
{\displaystyle\frac {e}{2}}

\prob {8}{100}
{Evaluate $\displaystyle\int\!\!\int\limits_{\textstyle\Sigma } 
e^{^{\displaystyle z^2}}\,dS$  where
$\Sigma$ is the portion of the cone
$z  =  \sqrt{x^2 + y^2}$  for which \linebreak $1 \leq x^2 + y^2 \leq 
2$.}
\ans
{4}{\sqrt{2}\pi(e^2 - e)\;\;\;}
{5}{\pi}
{1}{\sqrt{2}}
{2}{\sqrt{2} \pi (e - e^2)\;\;\;}
{\pi (e - 1)}

\prob {9}{-100}
{Compute $\displaystyle \int\!\!\int\limits_{\textstyle \Sigma }
{\bf F\cdot n}\;dS$
where $\Sigma$ is the unit cube
bounded by the planes $x = 1$, $y = 1$, $z = 1$ and the coordinate
planes, $\bf n$ is the outward unit normal and
${\bf F} = (x +e^{^{\displaystyle y^2}}){\bf i} 
\hspace*{\fill} \break
+ (2y + \arctan(\sqrt{(x^2 - z^2})){\bf j} 
+ (3z + \sin x + \cos y){\bf k}$.}
\ans
{4}{6}
{1}{0}
{2}{e^2 + \sec ^2 1}
{3}{10}
{2}


\prob {10}{100}
{Evaluate $\displaystyle\int\!\!\int\limits_{\textstyle\Sigma } 
\mbox{curl}\;{\bf F}{\bf\cdot }{\bf n}\,dS$ 
where $\Sigma$ is the sphere $x^2 + y^2 + z^2  = 25$, ${\bf n}$ is 
the outward unit normal and 
${\bf F} = xz{\bf i} + xy^2z{\bf j} - e^{2y}{\bf k}$.}
\ans
{3}{0}
{4}{23\sqrt{\pi }}
{5}{1}
{1}{-13\pi}
{-2}

\prob {11}{120}
{Evaluate $\displaystyle\int\!\!\int\limits_{\textstyle\Sigma }
{\bf F}{\bf\cdot }{\bf n}\, dS$ where
$\Sigma$ is the portion of the paraboloid 
$z = 2(x^2 + y^2)$ below the plane $z = 2$, ${\bf n}$ is downward  and
${\bf F} = ye^{^{\displaystyle z^3}}{\bf i} - xe^{^{\displaystyle 
z^3}}{\bf j} - {\bf k}$.}
\ans
{1}{\pi}
{2}{1}
{3}{-2\pi}
{4}{3\pi}
{-2}

\prob {12}{-100}
{Evaluate $\displaystyle\int\!\!\int\limits_{\textstyle\Sigma } 
\mbox {curl}\;{\bf F}{\bf\cdot }{\bf n}\,dS$  
where $\Sigma$ is the lower hemisphere $x^2 + y^2 + z^2  = 1$, $z 
\leq 0$,
${\bf n}$ is downward and
${\bf F} = ye^{^{\displaystyle z^3}}{\bf i} + z^{23}\tan (xy^3){\bf j} -
\sin (x^2y^2)e^{^{\displaystyle xz^3}}{\bf k}$.}
\ans
{2}{\pi}
{3}{0}
{4}{-2\pi}
{5}{\displaystyle\frac{\pi}{2}}
{\displaystyle\frac{-3\pi}{2}}


\prob {13}{100}
{Which of the following is {\bf not }  a linear differential equation 
?}
\ans
{5}{2y^{\prime\prime } + 3 y^{\prime } + 4y = y^2}
{1}{e^xy^{\prime } + (\ln x)y - 17x^2 =  0}
{2}{xy^{\prime\prime } + {\displaystyle\frac{1}{x}} - y = x^3}
{3}{y = xy^{\prime\prime }}
{(\sin x)y^{\prime\prime } + (\cos x)y = 0}

\prob {14}{100}
{The general solution of the differential equation
$y\,dx - x\,dy = 0$ is}
\ans
{5}{y = Cx}
{1}{y = 2x + C}
{2}{y = \displaystyle\frac{C}{x}}
{3}{x^2 - y^2 = C}
{y^2x - x^2y = C}

\prob {15}{-100}
{Solve the equation 
$x^2y^{\prime } + 2xy = 3x^2$  subject to the initial condition $y(1) = 
5$
and evaluate the solution at $x = 2$.}
\ans
{3}{3}
{4}{2}
{5}{0}
{1}{-3}
{5}


\prob {16}{100}
{Which of the following pairs of functions is independent ?}
\ans
{1}{\displaystyle \frac{e^x + e^{-x}}{2}, e^x - e^{-x}}
{2}{\displaystyle \frac{e^x + e^{-x}}{2}, e^x + e^{-x}}
{3}{1 + \cos 2x, \cos^2 x}
{4}{x^2, 15x^2}
{x^3 + x, 2x^3 + 2x}

\prob {17}{100}
{If $y(x)$ is the solution (for $x > 0$) to the initial value problem
$\displaystyle\frac{dy}{dx} = -\frac{y^2 + xy + x^2}{x^2}$,
$y(1) = -\displaystyle\frac{1}{2}$, then $y(e)$ equals.}
\ans
{2}{-1}
{3}{e^2}
{4}{-\displaystyle\frac{2}{3}e}
{5}{\displaystyle\frac{1}{3}}
{3}

\prob {18}{-100}
{Find the solution to
$$\frac{dy}{dx} = \frac {y\cos x + \cos y}{x\sin y - \sin x}$$}
\ans
{2}{y\sin x + x\cos y  = C}
{3}{y\sin x +  \sin y  = C}
{4}{x\sin y + y\cos x  = C}
{5}{x\sin y +  \sin x  = C}
{\sin x + \cos y = C}

\prob {19}{100}
{The differential equation
$$y\,dx + (2x - ye^y)\,dy = 0$$}
\ans
{5}{\mbox{ has an integrating factor which is a function of $y$ 
alone}}
{1}{\mbox{ is exact}}
{2}{\mbox{ is homogeneous}}
{3}{\mbox{ is separable}}
{\mbox{ has an integrating factor which is a function of $x$ alone}}

\prob {20}{100}
{A ball of mass 2 kg.  is thrown upward  from the ground
with an initial velocity of
20 m/sec.  Ignore air resistance.  Find the maximum
height above the ground the ball reaches if you
suppose the gravitational acceleration is 10 m/sec.$^2$.}
\ans
{4}{20 \mbox{m.}}
{5}{200 \mbox{m.}}
{1}{10 \mbox{m.}}
{2}{40 \mbox{m.}}
{400 \mbox{m.}}


\prob{21}{-100}
{The function $y_1(x) = e^x$ is one solution to the equation
$xy^{\prime \prime} - (x+1)y^{\prime} + y = 0$. Which of the 
functions below
is a second solution to this equation, $y_2(x)$, so that
$y_1(x)$ and $y_2(x)$ form a basis for the space of solutions to this 
equation?}
\ans
{1}{x+1}
{2}{e^{-x}}
{3}{2e^x}
{4}{xe^x}
{\sin x}

\prob {22}{100}
{Which of the functions below is a solution to the equation
$2y^{\prime \prime} + 3y^{\prime} + 2y = 0$ ?}
\ans
{2}{e^{^{\frac{\scriptstyle -3x}{\scriptstyle 4}}}
\sin \left(\frac{\sqrt{7}}{4} x\right)}
{3}{e^{^{\frac{\scriptstyle \sqrt{7}x}{\scriptstyle 4}}}
\cos \left(\frac{-3}{4}x\right)}
{4}{e^{^{\frac{\scriptstyle -\sqrt{7}x}{\scriptstyle 4}}}
\sin \left(\frac{3}{4}x\right)}
{5}{e^{^{\frac{\scriptstyle {3x}}{\scriptstyle 4}}}
\cos \left(\frac{\sqrt{7}}{4}x\right)}
{e^{^{\frac{\scriptstyle -3x}{\scriptstyle 4}}}
\sin \left(\frac{1}{2}x\right)}

\prob {23}{100}
{Which function below is a solution to the equation
$y^{\prime \prime } -2y^{\prime } + y = e^x$ ?}
\ans
{3}{\displaystyle \frac{x^2e^x}{2} + xe^x}
{4}{x^2 -2x + 1}
{5}{e^x}
{1}{\displaystyle \frac{(x-2) e^x}{3}}
{\displaystyle \frac{(x^2 -2x +1) e^x}{4}}

\prob {24}{-100}
{Consider the equation
$$2x^2y^{\prime \prime } + 3xy^{\prime } - y = 2x^{\frac{3}{2}}$$
A basis for the space of solutions to the associated homogeneous 
equation is
$y_1(x) = \displaystyle \frac {1}{x}$  and $y_2(x) = \sqrt{x}$.
Which of the functions below is a solution to the given equation ?}
\ans
{2}{\displaystyle \frac{2}{5} x^{\frac {3}{2}}}
{3}{\displaystyle \frac{2}{x} + 3\sqrt{x} + x}
{4}{\displaystyle \frac{2}{x} - 3\sqrt{x}}
{5}{x^{\frac{3}{2}} -\displaystyle \frac{1}{x} + \sqrt{x}}
{x^2 -3x + 2 + \displaystyle \frac{2}{x} - 3\sqrt{x}}

\prob {25}{100}
{Which of the equations below has solutions which exhibit resonance 
?}
\ans
{4}{4\ddot{u} + u = 3\cos (t/2)}
{5}{4\ddot{u} + u = 2\cos (t/4)}
{1}{4\ddot{u} + u = 5\cos (t/3)}
{2}{4\ddot{u} + u = 6\sin (2t/7)}
{4\ddot{u} + u = 7\sin (t/5)}


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\end{document}
