Sample Exam I
I. PROBLEM SOLVING.

1. There are ships with three masts and ships with 4 masts at the Tallship Exhibition. Millie counted a total of 30 masts on the ships she saw. How many of these ships had 4 masts? (8 points)

2. How many squares are there with corners at points on the following array? (8 points)

3. It takes 125 cubes of the same size to fill a cubical box that has no lid. How many of these cubes are not touching the sides or bottom of the box? (8 points)

4. When a teacher counted her students in groups of 4 she had 1 left over. When she counted them in groups of 5 she had 2 left over. If 20 of her students are girls and she had more girls than boys, how many boys did she have? (8 points)

5) Al decides to give half of his marbles to Bernice and 2 to Charlie. From what remains, he gives half to Denise and 2 more to Charlie. From what remains now, he gives one third to Fred and 20 more to Charlie. If he is finally left with 100 marbles, how many marbles did Al have at the start? (8 points)

II. CONCEPTS.

1. If you are given that f(x) = x2+4 and g(x) =5x+1, find the following: (2 points each)

a) g ( -2 )
b) f ( 2/3 )
c) f ( g (1) )
d) f ( a+b )

2. Which of the following arrow diagrams are functions? (2 points each)

a) . . .

b) . . .

c) . . .

3. If S is the set of Sophmores at Hard High, and B is the set of students who take Biology, describe the following sets in words? (2 points each)

a) . . .

b) . . .

c) . . .

d) . . .

e) . . .

4. Given the following sets:

W = {0, 1, 2, 3, 4, 5, 6 }

A = { 0, 2, 4, 6 }

B ={ 1, 2, 3 }

Compute each of the below. ( 2 points each)

a) . . .

b) . . .

c) . . .

d) . . .

e) . . .

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5. List all of the subsets of { m, n, o, p }. (6 points)

.

6. Shade the Venn Diagram to represent . . .

(6 points)

7. Identify the shaded portion of this Venn Diagram using set notation. (6 points) . . .

8. Consider the table below and explain each answer: (2 points each)

*ABC
AABC
BBCA
CCAB

a) What is C * B = ?

b) Is * a commutative operation?

c) If there is an identity for * , identify it.

d) Is the set { A, B, C } closed for the operation *?

9. Show by a counter example that - is not a commutative operation. (6 points)

10) Show all possible 1-1 correspondences between { 1, 2, 3 } and { a, b, c }. (6 points)

11. For each case below, write the number of the whole number property which justifiesthe statement. (2 points each)

a) 1 x 2 = 2 x 1 1)Closure +
b) 5 + 0 = 5 2)Closure x
c) 7 x (3+5) = (3+5 ) x 7 3)Commutativity +
d) 5 + 3 = 3 + 5 4)Commutativity x
e) 1 x 6 = 6 5)Associativity +
f) 3 x ( 7 + 5 ) = ( 3 x 7) + ( 3 X 5 ) 6)Associativity x
g) 1 + ( 2 + 3 ) = ( 1 + 2 ) + 3 7)Identity +
h) 7 + 2 is a whole number. 8)Identity x
9)Distributive Property

12) How many different 1-1 correspondences are there between two six element sets? (3 points)

13) Tell if the following sets are closed under addition. If not, give a counterexample. (3 points each)

a) { 0, 3, 6, 9, 12, .... }

b) { x | x 10 }

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III. COMPUTATION.

.

SHOW YOUR WORK WHENEVER POSSIBLE.

 1. Change MCMLXXIX to Hindu-Arabic numerals. (3 points)

2. Change 56 374 to each of the following systems. (6 points each) a) Myan

b) Babylonian

c) Egyptian

3. Write in Hindu Arabic numerals each of the following numbers: (6 points each)

a) Babylonian . . .

b) Myan . . .

c) Myan . . .

4) Construct truth tables for the following. (3 points each)

a) . . .

5) Using the following function machines, find all possible missing inputs or outputs: (3 points each)

a) . . .

b) . . .

c) . . .

d) . . .

6) Change 5492 to base eight notation. (4 points)

7) Change 2322(eight) to base ten notation. (4 points)