1. sec 3 theta / 2 = -2 BOOK answer: {4pi/9, 8pi/9, 16pi/9} 2. cos(2theta- pi / 2) = -1 BOOK answer: {3pi/4, 7pi/4} Solution I am using theta = x to make it clearly visible, sec (3x/2) = -2 write it in cos, cos(3x/2) = -1/2 3x/2 = 4pi/3, 2pi/3, 8pi/3 x = 8pi/9 , 4pi/9, 16 pi/9.

1. 1.: Set Theory
Look up a roulette wheel diagram. The following sets are defined:
A = the set of red numbers
B = the set of black numbers
C = the set of green numbers
D = the set of even numbers
E = the set of odd numbers
F = {1,2,3,4,5,6,7,8,9,10,11,12}
From these, determine each of the following:
A?B
A?D
B?C
C?E
B?F
E?F
2. Relations and Functions
The implementation of the program that runs the game involves testing. One of the necessary
tests is to see if the simulated spins are random. Create an n-ary relation, in table form, that
depicts possible results of 10 trials of the game. Include the following results of the game:
Number
Color
Odd or even (note: 0 and 00 are considered neither even nor odd.)
Also include a primary key. What is the value of n in this n-ary relation?
3. Graphs and Trees
Create a tree that models the following scenario. A player decides to play a maximum of 4 times,

2. betting on red each time. The player will quit after losing twice. In the tree, any possible last
plays will be an ending point of the tree. Branches of the tree should indicate the winning or
losing, and how that affects whether a new play is made.
4. Combinatorics and Probability
In the roulette game, what is the probability of an outcome of:
Any odd number
Any green number
Any number that is red or green
Any number that is red and even
You are asked by a state government official to investigate whether under a proposed license
plate system, there will be enough license plate codes for the state. There are 2 schemes
proposed:
Each license plate is to show 3 letters, followed by 3 numbers, with no restrictions on repeated
characters or their order. The only restriction is that the letters I and O are not to be used.
The second scheme is the same as above, except no letter or number can appear twice in the
same license plate.
Under each scheme, how many different license plates can be produced?
Under which scheme is it more likely that there will be enough license plates? Explain.
5. Algorithm Analysis
Consider searching algorithms on the following array of data:
[22 21 9 4 16 2 10 14 20 31 26 19 17 28 8 13]
Suppose you want to implement a searching algorithm to see if the data set contains the number
19. Demonstrate how the search would go if you used:
A sequential search
A binary search
State the runtime for each of the searches, in this example, and for general data sets of size n.
Address the issue of the order of the data in binary searching.

3. Suppose an algorithm that processes a data set of size 8 has a runtime of 72. The same algorithm
has a runtime of 110 when applied to a data set of size 10; and when applied to a data set of size
20, it has a runtime of 420. Using big-O notation, state the runtime for this algorithm for the
general case of a data set of size n.
Suppose you develop an algorithm that processes the first element of an array (length of n), then
processes the first 2 elements, then the first 3 elements, and so on, until the last iteration of a
loop, when it processes all elements. Thus, if n = 4, the runtime would be 1 + 2 + 3 + 4 = 10.
Create a table that depicts the runtime for arrays of length 1 to 10. Would you expect the general
runtime to be O(n), O(n2), O(n3), or some other function of n? Explain.
Solution
ANS A = set of red numbers = {1, 27, 25, 12, 19, 18, 21, 16, 23, 14, 9, 30, 7, 32, 5,
34, 3, 36} B = set of black numbers = {10, 29, 8, 31, 6, 33, 4, 35, 2, 28, 26, 11, 20, 17, 22, 15,
24, 13} C = set of green numbers = {00, 0} D = set of even numbers = {10, 12, 8, 18, 6, 16, 4,
14, 2, 28, 26, 30, 20, 32, 22, 34, 24, 36, 0, 00} E = set of odd numbers = {1, 27, 25, 29, 19, 31,
21, 33, 23, 35, 9, 11, 7, 17, 5, 15, 3, 13} To describe how the notion of union and intersection
apply to retrieving records in a database, I’ll use an example of two sets contained in an auto
dealer database. Set A and B represent garages that contain the make of vehicles parked inside.
Set A (known as garage A) contains the members (or makes of vehicles parked inside) which are
Chevy, Audi, Ford, Saab, Lincoln, and Chrysler. Set A is expressed below: A = {Chevy, Audi,
Ford, Saab, Lincoln, Chrysler} Set B (known as garage B) contains the members (or makes of
vehicles parked inside) which are Mercury, BMW, Ford, Saab, Dodge, and Audi. Set B is
expressed below: B = {Mercury, BMW, Ford, Saab, Dodge, Audi}