Algebra II / DeBord, Graham, Hansen

Name: _____________________________

May 1999 Practice Final Exam (2 hours)

  

The purpose of this practice exam is to help you review by showing you the wording and types of problems on the actual exam. The actual exam is somewhat shorter and contains different problems. Although most of these practice problems are similar to real ones, not all types of problems are represented here. For example, you are expected to be familiar with all the conic sections we studied (circle, ellipse, parabola, hyperbola), as well as all 6 of the formulas for nth term and sum of arithmetic and geometric series (see blue boxes on pp. 567, 583, 584, and 591).

No notes or scratch paper
You cannot use notes or scratch paper on the actual exam. Ample room to write will be provided. However, to save paper, the practice exam has been written with the problems all scrunched together. You will definitely need some extra paper to write out your answers to the practice exam.

Calculator
You may use your calculator on all parts of the exam. However, since your work is what will be graded, not your answer, in most cases you should use your calculator only to check your work. Decimal answers are acceptable only when the problem says "round to ____ places" or "round to the nearest _____ ."

Simplifying
The word "simplify" means to leave everything in simplest form (e.g., write 2Ö 2 instead of Ö 8 in your final answer), with no radicals in the denominator, and in a + bi format in the case of complex numbers.

Concise directions (as printed on actual exam)
For full credit show all work. Reduce fractions to simplest form and rationalize all denominators. Give exact answers (no decimals) unless otherwise requested. Use units of measure where appropriate.

1.

When poker players say "straight flush," they usually exclude the case of a royal flush. In other words, the events are to be treated as being disjoint. Using the formulas P(royal flush) = 4/(52C5) and P(straight flush) = 9P(royal flush), compute the probability of a straight flush or royal flush dealt in 5-card poker.

2.

Simplify each of the following.

(a)

(b)

(c)

(d)

3.

Consider the relation x = 2y2 – 3y – 1.

(a)

Express the relation in standard form, x = a(y – k)2 + h, by completing the square.

(b)

Sketch the graph.

(c)

Mark and label the coordinates of the vertex.

(d)

Mark and label the coordinates of the x- and y-intercepts.

(e)

Make a dotted line to indicate the axis of symmetry, and label it with its equation.

4.

Find a cubic equation with real-number coefficients if one solution is 3 and another is –5 – 2i.

5.

Express each of the following in standard form and then sketch, indicating the coordinates of the vertices, center, and x- and y-intercepts, if appropriate.

(a)

x2 – 8y2 = 512

(b)

y2 = 6 – x2

6.

Solve the following system of equations over the set of complex numbers:
y = –3x2 + 2
y2 – 6x2 = 4

7.

Solve each equation separately over  :

(a)

(b)

(c)

8.

Because of economies of scale, the cost per km of building a certain expressway varies inversely with the 0.2 power of the length of the expressway.

(a)

Initial plans for the expressway were for it to be 50 km long and cost $10 million per km. Write an equation expressing cost per km in terms of the length.

(b)

To the nearest dollar, what was the initial total planned cost of the expressway?

(c)

To the nearest dollar, what will be the cost per km, as well as the total cost, if the expressway is extended to 60 km?

(d)

The maximum funding available from the state legislature is $350 million. To the nearest 0.1 km, how long an expressway can be built for this amount of money?

9.

For this problem, we have 4 boys and 6 girls in a classroom.

(a)

In how many ways can the 10 students be arranged in a line if the girls must be kept together as a group?

(b)

In how many ways can 3 boys and 4 girls be selected from this class?

10.

From an urn containing 7 green and 14 orange balls, 4 balls are drawn at random without replacement. Compute the probability of each of the following events:

(a)

drawing exactly 4 green balls

(b)

drawing 2 green balls and 2 orange balls

(c)

drawing at least 1 orange ball

11.

Repeat problem 10 (parts a, b, and c) if the draws are made with replacement after each ball is drawn.

12.

From a deck of playing cards, we remove 4 red cards and 4 black cards, and we throw the rest of the deck in the trash. In how many ways can the 8 cards be arranged face up, left to right, if red and black must alternate? (Ignore the fact that some cards, such as the 9 of spades, look different depending on how they are oriented. Consider only the number of arrangements of the cards themselves.)

13.

Simplify. Express answers exactly (no decimals).

(a)

(b)

14.

To the nearest 0.0001 radian, compute a first-quadrant value x such that x = csc–1 7.6438.

15.

Evaluate to the nearest tenth. (Note: For a simpler alternative, change 1.008a to 1.008a.)

16.

Write a formula for the nth term of the sequence

17.(a)

(Two parts.) Using sigma notation, write S400 for the series

and for the series

(b)

(Two parts.) Compute S400 for each of the series given in part (a).

18.

If Sn = 979.8, t1 = 20.8, and t7 = 19.6, find n if this is an arithmetic series.

19.

Find the value(s) to which a geometric series converges if t1 = 55 and t3 = 2.2.

20.(a)

In Titanic there is a scene in which Leonardo DiCaprio holds Kate Winslet as she dangerously hangs out over the bow of the ship. If Kate’s nose is 62 feet above water level, if the distance from the tip of Kate’s nose to the top of the first smokestack is 318 feet (as the crow flies), and if Kate would have to look backward over her shoulder at an angle of 17° to see the top of the first smokestack, how high (to the nearest inch) is the smokestack above sea level? (The angle of 17° is for Kate’s line of sight relative to the horizontal deck of the ship.)

(b)

Extra credit: Name the only recent movie that is longer than this exam. (Remember, though, that the actual exam is somewhat shorter.)

21.

Expand: (x – 2y)7