Math 17 SAMPLEX

1st Long Exam

True or False:

1.) Composite numbers have prime factors.2.) Let

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A <U , then

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n(A0) = n(U) − n(A).3.) If

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A = {w,s,0} and

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B = {y,z,0} , then

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A∩B = 0.4.) The set of irrational numbers is closed under addition.5.) The numbers 15 and 81 are relatively prime.6.) It can never be true that

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a +b = an +bn .

7.) The cardinality of

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A = {0,{0},{{0}}} is 1.8.) Every real number has a multiplicative inverse.9.)

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−2 ⋅ −8 = 6.10.)The union of the set of natural numbers and their additive inverses is the set of integers.11.)Any real number has a square root.12.) If A has exactly two subsets and B has exactly one subset, then

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A∩Bhas at least one element.

13.)For all real numbers

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a ,

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(a +b)1/ 2 ≠ a1/ 2 +b1/ 2 .14.)The set of counting numbers relatively prime to 2 is closed under multiplication.

Problems:

1.) Find all the subsets of G={a,b,c} which contain a but do not contain c.

For numbers 2 to 3:

Jules has 15 classmates. 7 are tall, 10 are chinitas, 3 don’t have the qualities of being tall nor pretty. 10 have at least 2 of the 3 qualities. 7 are ugly. Two are tall but neither a chinita nor pretty.

2.) Draw the Venn diagram to summarize these data.3.) How many are pretty, tall, and chinita?

For numbers 4 to 12:

If

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A = {1,2,3,4,5,} ,

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B = {2,3,4} , and

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C = {2,4,5} , which of the following are true:4.)

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A⊂B5.)

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A⊂C6.)

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B⊂ A7.)

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B⊂C8.)

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C ⊂ A9.)

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C ⊂B10.)

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C ⊂C11.)

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0⊂B12.)

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B⊂C

13.)Factor completely:

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a3x 2 − a3y 2 +b3x 2 −b3y 2

14.)What is the value of

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i + i 2 + i 3 + i 4 + ...+ i101?15.)Factor completely:

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24a3 + 27b−8a3b−81

16.)Several students were asked what subject they enjoyed most. Out of the 50 respondents, 29 said they like Math while 22 prefer English. There are 13 students who like both Math and Philo. 14 of those asked said they do not like either English or Philo. 4 respondents

don’t like any of these subjects, while 27 do not like Philo. If there are 8 who like all of these subjects, then how many enjoy only Philo? Draw a Venn diagram to present this data.

17.)

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3

x − y+

2x

xy − y 2−

6y

x 2 − y 2

18.)

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(2i 5 − i 51)2

1− 2i (in standard form)

ANSWER KEY:

1.) True2.) True3.) False:

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A∩B = {0}.4.) False5.) False6.) False7.) False: every separate brace indicates another cardinality.8.) False: Zero has no multiplicative inverse9.) False:

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−2 ⋅ −8 = 6

i 2 ⋅i 8 = 6

i 2 36 = 6

(−1)(6) = 6

−6 ≠ 610.)False11.)True12.)False.13.)False, i.e. if a and b equal zero.14.)True

1.) {a} and {a,b}2.) :

3.) None4.) False5.) False

6.) True

7.) False8.) True9.) False10.)True11.)False: {0} c B12.)False13.) :

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=a3x 2 − a3y 2 +b3x 2 −b3y 2

= a3 (x 2 − y 2) +b3 (x 2 − y 2)

= (a3 +b3)(x 2 − y 2)

= (a +b)(a2 − ab+b2)(x + y)(x − y)14.)Recall: i=i, i2=-1, i3=-I, i4=1. Every four values of I, the sum is 0. So we can divide the

exponent of I, which is 101, by 4 and the remainder would be used to identify the value. Therefore, the answer is i.

15.)

22

3

3 5

ChinitaTall

Pretty

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=24a3 + 27b−8a3b−81

= (8a3)(3−b) − (27)(3−b)

= (8a3 − 27)(3−b)

= (2a − 3)(4a2 + 6a + 9)(3−b)16.) :

17.) :

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=3

x − y+

2x

xy + y 2−

6y

x 2 − y 2

=3

x − y+

2x

y(x + y)−

6y

(x + y)(x − y)

=3y(x + y) + 2x(x − y) − 6y 2

y(x + y)(x − y)

=3xy + 3y 2 + 2x 2 − 2xy − 6y 2

y(x + y)(x − y)

=2x 2 + xy − 3y 2

y(x + y)(x − y)

=2x + 3y

y(x + y)18.) :

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(2i 5 − i 51)2

(1− 2i)=(2i 4i − i 48i 3)2

(1− 2i)

=(2i + i)2

(1− 2i)

=9i 2

(1− 2i)⋅1+ 2i

1+ 2i

=−9−18i

1− 4i 2

=−9−18i

5

105

9

6 1

PhiloMath

English

8

7