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Sample for the Second Examination
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1. \ Let \ $r=-2+3i$ \ and $\ s=4-i$. \ Find each of the following:
a) \ $r+s$ \qquad b) $\ r-s$ \qquad c) \ $rs$ \qquad d) \ $\frac{r}{s}$ \qquad
e) \ $\frac{s}{r}$ \qquad f) \ $\frac{r}{s}\cdot \frac{s}{r}$ \qquad
g) \ $\overline{r}$ \qquad h) \ $r\overline{r}$ \qquad i) \ $\overline{s}$ \qquad
j)\ $s\overline{s}$ \qquad k) $\ rs-\overline{r}$ \qquad l) \ $r\overline{s}$
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\noindent
2. \ Repeat Problem 1, with $r=5-3i$ \ and $\ s=-2i$.
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\noindent
3. \ Repeat Problem 1, with $r=2$ \ and $\ s=3i$.
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\noindent
4. \ Solve for all real values of $x$:
a) \ $\frac{3}{x-2}+\frac{7}{x+4}=2$ \qquad b) \ $\frac{x}{x+1}+\frac{x-1}{x+2}=1$
\qquad c) \ $\frac{1}{x-2}+\frac{3}{x-4}=\frac{x-1}{x^{2}-6x+8}$ \qquad
d) \ $\frac{x+4}{x+3}-\frac{x-1}{x-4}=1$ \qquad e) \ $\sqrt{2x+6}+x=0$ \qquad
f) \ $\sqrt{12x+1}+5=0$ \qquad
g) \ $\sqrt{8+2x}+\sqrt{4x+9}=9$ \qquad h) \ $x^{7}-127x^{7/2}=128$ \qquad
i) \ $x^{6}+9x^{3}-8=0$ \qquad j) \ $x^{4}-x^{2}-12=0$ \qquad
k) \ $x^{2}+4x-12\leq 0$ \qquad
l) \ $x^{2}+5x-24>0$ \qquad m) \ $x^{2}+6x+9\leq 0$ \qquad
n) \ $x^{2}+x-3\geq 0$ \qquad
o) \ $x^{2}-3x+3\geq 0$ \qquad p) \ $x^{2}+3x+5<0$
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\noindent
5. \ Find all roots (real or complex) of the following quadratic equations.
(Simplify your answers. \ Complex roots should be expressed in the form \
$a+bi$ ):
a) \ $2x^{2}=7x-5$ \qquad b) \ $2x^{2}=7x+5$ \qquad c) \ $x^{2}+x-5=0$ \qquad
d) \ $x^{2}+x+5=0$
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\noindent
6. \ Find the sum of all roots (real or complex) of the following pair of
quadratic equations:
a) \ $3x^{2}+10x=-7$ \ and \ \ $x^{2}+7x-8=0$ \qquad
b) \ $2x^{2}+7x=-3$ \ and \ \ $x^{2}+7x-8=0$
c) \ $x^{2}+9x+6$ \ and \ \ $x^{2}+x+3=0$
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\noindent
7. \ Find a quadratic equation with integer coefficients which has the
given roots
a) $\ x=4$ and $x=-2$ \qquad b) \ $x=\frac{5}{2}$ and $x=-\frac{1}{6}$ \qquad
c) \ $x=-\frac{3}{2}$ and $x=-4$
d) \ $x=3+\sqrt{5}$ and $x=3-\sqrt{5}$ \qquad
e) \ $-4+3i$ \ and \
$-4-3i$ \qquad f) \ $\sqrt{7}i$ \ and \ $-\sqrt{7}i$
g) \ only \ $-11$ \qquad h) \ only \ $\frac{5}{7}$ \qquad
\vspace{12pt}
\newpage
\noindent
8. \ a) \ One number is 13 more than another, and their product is 30. \
Write an equation which expresses this statement algebraically. \ Then find
the numbers.
b) \ Three consecutive integers have the property that three
times the product of the smaller two is 58 more than twice the
square of the larger number.
c) \ Two consecutive even integers have the property that their product is
eight less than four time their sum.
d) \ Two trains travel on parallel tracks from one station to another which
is 240 miles away. \ \ They depart at the same time, but\ one travels 24
miles an hour faster than the other and arrives at their destination 30
minutes sooner. \ How fast do the two trains travel?
e) \ Two trains travel on parallel tracks from one station to another. \ \
The first train departs at 1:00 PM and arrives at its destination at 3:30
PM, which the other train departs at 4:00 PM and arrives at 8:00PM. If the
first train travels 30 miles an hour faster than the second. \ How far apart
are the stations?
f) \ If John can complete a job in15 minutes less than it would take Sally
to complete the same job and if it would take the two of them working
together 5 minutes less than it would take John alone, How long would it
take John to complete the job if he works alone?
g) \ A rock is thrown off the roof of a building which is 240 feet high with
an upwards initial velocity of 32 feet per second.
\qquad i) \ How long does it take to hit the ground?
\qquad ii) \ How far above the ground is it at its highest point?
\qquad iii) \ How fast is it moving at its highest point?
\qquad iv) \ How fast is it moving when it hits the ground?
\qquad v) \ How long does it take to pass a window which is 192 feet above
the ground? \ How fast is it moving at that time?
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\newpage
\noindent
9. \ Describe the graph of each of the following equations as a circle, an
imaginary circle, a point circle, a parabola, a straight line or none of the
above. \ If it is a circle, give the center and radius. \ If it is a
parabola, give the vertex and axis of symmetry and indicate in which
direction it opens. \ If it is a straight line, give the slope and
y-intercept.
a) \ $x^{2}+y^{2}-8y+12=0$
b) \ $x^{2}+3y^{2}-4x+8y-5=x^{2}+3y^{2}+x-y+5$
c) \ $3x^{2}-18x-4y+15=0$
d) \ $2y^{2}-4x+12y+7=0$
e) \ $x^{2}+y^{2}-6x+5=0$
f) \ $x^{2}+y^{2}-6x=-9$
g) \ $x^{2}+y^{2}-6x+14=0$
h) \ $x^{3}+y^{3}+3x-5y+1=0$
i) \ \ $x^{4}-y^{2}+4x=x^{4}+x^{2}+6y$
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\noindent
10. \ Find the equation of each of the following curves:
a) \ \ The circle with center at $(1,5)$ which passes through the point $(4,1)$.
b) \ \ The circle with center $(0,1)$ which is tangent to the line $y=x+7$.
c) \ \ The circle with center $(0,-4)$ which passes through the point $(4,-1)$
d) \ The circle with center at $(-3,7)$ and radius $5$.
e) \ \ The parabola with directrix $x=2$ \ and focus $(-4,1)$.
f) \ The parabola with vertex $(1,-3)$ and directrix $x=5$
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\noindent
11. \ Find each of the following:
a) \ The center and radius of the circle $3x^{2}+3y^{2}-6x+12y=15$
b) \ The center and radius of the circle $x^{2}+y^{2}-5x-y=6$
c) \ The vertex, focus, directrix, axis of symmetry and intercepts of the
parabola $\left( y+3\right) ^{2}=12\left( x-7\right) $
d) \ The vertex, focus, directrix, axis of symmetry and
intercepts of the parabola $x^{2}-4x+16y=28$
e) \ The vertex, focus, directrix, axis of symmetry and
intercepts of the parabola $2x-y^{2}+5y-1=0$
f) \ The slope and y-intercept of the line $3x-4y-12=0$
\vspace{12pt}
\newpage
12. \ Find the implied domain of $(f+g)(x)$
a) \ $f(x)=\frac{\sqrt{x-7}}{x^{2}-8x+16}$,\qquad $g(x)=\sqrt{2x^{2}-32}$
b) \ $f(x)=\sqrt{x^{2}+4x-21}$, \qquad $g(x)=\frac{1}{\sqrt{x^{2}+4x-21}}$
c) \ $f(x)=\frac{1}{x^{2}+4x+5}$, \qquad $g(x)=\sqrt[3]{x^{2}-8x+15}$
d) \ $f(x)=\frac{1}{\sqrt[3]{x^{2}-6x+8}}$, \qquad $g(x)=\sqrt{8-x^{2}}$
\vspace{12pt}
\noindent
13. \ Describe the function $(h\circ g\circ f)(x)$ where
a) $\ \ f(x)=x^{3}-1$, \ $g(x)=\frac{1}{x+1}$, \ $h(x)=\sqrt[3]{x}$. \
b) $\ \ f(x)=x^{3}-1$, \ $g(x)=\frac{1}{x+1}$, \ $h(x)=\sqrt[3]{x+1}$. \
c) \ $\ f(x)=x^{2}-2$, \ $g(x)=\frac{1}{x+3}$, \ $h(x)=\sqrt{x+2}$.
d) \ $\ f(x)=x^{2}-2$, \ $g(x)=\sqrt{x+2}$, $h(x)=\frac{1}{x+3}$. \
e) \ $f(x)=\sqrt{x+2}$, $\ g(x)=x^{2}-2$, \ $h(x)=\frac{1}{x+3}$. \
f) \ $\ f(x)=\frac{1}{x+3}$, $g(x)=\sqrt{x+2}$, $\ g(x)=x^{2}-2$.
\vspace{12pt}
\noindent
14. \ Express each of the following functions as a composite of two or more
of the following functions: \ $f(x)=(x+1)^{3}$, \ $g(x)=\frac{1}{\sqrt[3]{x}}
$, and $h(x)=2x-1$
a) \ $\frac{1}{x+1}$
b) \ $8x^{3}$
c) \ $\frac{1}{\sqrt[3]{2x^{3}+6x^{2}+6x+1}}$
d) \ $\frac{1}{\sqrt[3]{4x^{2}-4x+1}}$ \ \
e) \ $4x-3$
f) \ $\sqrt[9]{2x-1}$
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\noindent
15. \ For each of the following functions, determine whether or
not it has an inverse. \ If it does have an inverse, find the inverse
function as well as its domain and range. \ If it doesn't have an inverse
you must explain why not.
a) \ \ $f(x)=\frac{2x+1}{2}$
b) \ $g(x)=\sqrt[3]{x^{3}-8}$
c) \ $g(x)=\sqrt{x^{3}-8}\qquad $
d) \ \ $h(x)=x^{2}-5x+4$ \ \ \ \qquad \qquad \qquad\ \ \ \ \
e) \ $k(x)=\frac{3x-5}{x+3}$
f) \ \ $f(x)=x^{3}+1$ \ \ \ \qquad \qquad \qquad \qquad
g) \ $f(x)=\frac{1}{1-x}$ \ \ \ \
h) \ $k(x)=4-\frac{1}{x}$
\vspace{12pt}
\newpage
\noindent
16. \ \ Sketch the graph of \
a) $\ P(x)=\left\{
\begin{array}{ccc}
1+x & \textstyle{if} & x<-2 \\
x^{3} & \textstyle{if} & -2\leq x\leq 2 \\
8 & \textstyle{if} & 2