Sample for the Second Examination

1. Let $r=-2+3i$ and $\ s=4-i$. Find each of the following:
a) $r+s$          b) $\ r-s$          c) $rs$          d) $\frac{r}{s}$          e) $\frac{s}{r}$          f) $\frac{r}{s}\cdot \frac{s}{r}$         
g) $\overline{r}$          h) $r\overline{r}$          i) $\overline{s}$          j) $s\overline{s}$          k) $\ rs-\overline{r}$         l) $r\overline{s}$

2. Repeat Problem 1, with $r=5-3i$ and $\ s=-2i$.

3. Repeat Problem 1, with $r=2$ and $\ s=3i$.

4. Solve for all real values of $x$:
a) $\frac{3}{x-2}+\frac{7}{x+4}=2$          b) $\frac{x}{x+1}+\frac{x-1}{x+2}=1$          c) $\frac{1}{x-2}+\frac{3}{x-4}=\frac{x-1}{x^{2}-6x+8}$
d) $\frac{x+4}{x+3}-\frac{x-1}{x-4}=1$          e) $\sqrt{2x+6}+x=0$          f) $\sqrt{12x+1}+5=0$
g) $\sqrt{8+2x}+\sqrt{4x+9}=9$          h) $x^{7}-127x^{7/2}=128$          i) $x^{6}+9x^{3}-8=0$
j) $x^{4}-x^{2}-12=0$          k) $x^{2}+4x-12\leq 0$          l) $x^{2}+5x-24>0$
m) $x^{2}+6x+9\leq 0$          n) $x^{2}+x-3\geq 0$          o) $x^{2}-3x+3\geq 0$         p) $x^{2}+3x+5<0$

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) $2x^{2}=7x-5$          b) $2x^{2}=7x+5$          c) $x^{2}+x-5=0$          d) $x^{2}+x+5=0$

6. Find the sum of all roots (real or complex) of the following pair of quadratic equations:
a) $3x^{2}+10x=-7$ and $x^{2}+7x-8=0$          b) $2x^{2}+7x=-3$ and $x^{2}+7x-8=0$
c) $x^{2}+9x+6$ and $x^{2}+x+3=0$

7. Find a quadratic equation with integer coefficients which has the given roots
a) $\ x=4$ and $x=-2$          b) $x=\frac{5}{2}$ and $x=-\frac{1}{6}$          c) $x=-\frac{3}{2}$ and $x=-4$
d) $x=3+\sqrt{5}$ and $x=3-\sqrt{5}$          e) $-4+3i$ and $-4-3i$         
f) $\sqrt{7}i$ and $-\sqrt{7}i$          g) only $-11$          h) only $\frac{5}{7}$         

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.
        i) How long does it take to hit the ground?
        ii) How far above the ground is it at its highest point?
        iii) How fast is it moving at its highest point?
        iv) How fast is it moving when it hits the ground?
        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?

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) $x^{4}-y^{2}+4x=x^{4}+x^{2}+6y$

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 $x=5$

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 $3x-4y-12=0$

12. Find the implied domain of $(f+g)(x)$ where
a) $f(x)=\frac{\sqrt{x-7}}{x^{2}-8x+16}$,         $g(x)=\sqrt{2x^{2}-32}$
b) $f(x)=\sqrt{x^{2}+4x-21}$,          $g(x)=\frac{1}{\sqrt{x^{2}+4x-21}}$
c) $f(x)=\frac{1}{x^{2}+4x+5}$,          $g(x)=\sqrt[3]{x^{2}-8x+15}$
d) $f(x)=\frac{1}{\sqrt[3]{x^{2}-6x+8}}$,          $g(x)=\sqrt{8-x^{2}}$

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) $\ f(x)=\frac{1}{x+3}$,        $g(x)=\sqrt{x+2}$,        $\ h(x)=x^{2}-2$.

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) $\sqrt[9]{2x-1}$

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) $h(x)=x^{2}-5x+4$                         
e) $k(x)=\frac{3x-5}{x+3}$         f) $f(x)=x^{3}+1$          g) $f(x)=\frac{1}{1-x}$         h) $k(x)=4-\frac{1}{x}$

16. Sketch the graph of
a) $\ P(x)=\left\{ \begin{array}{ccc} 1+x & \textstyle{if} & x<-2 \\ x^{3} & \... ...tyle{if} & -2\leq x\leq 2 \\ 8 & \textstyle{if} & 2<x \end{array} \right\}$
b) $P(x)=\left\{ \begin{array}{ccc} x^{2}+x-1 & \textstyle{if} & x\leq 2\;\;\;\;... ...4 \\ \left\vert 5-x\right\vert & \textstyle{if} & x>4 \end{array} \right\}$

17. Discuss the graph of each of the following functions, with respect to
         i) symmetry
         ii) intercepts
         iii) asymptotes
         iv) its graph
         v) whether or not it has an inverse
a) $p(x)=\allowbreak x^{3}+x^{2}-8x-12$          b) $p(x)=\allowbreak x^{3}+3x^{2}-9x+5$         
c) $p(x)=\allowbreak x^{3}+6x^{2}+3x-10$          d) $q(x)=\frac{x^{2}-7x+6}{x^{2}-4}$         
e) $q(x)=\frac{x^{2}-7x+10}{x^{2}-9}$          f) $t\left( x\right) =\sqrt{x+3}$          g) $f(x)=\sqrt{\frac{2x^{2}-8}{x^{2}-9}}$