01. Math Practice Exam

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CHAPTER 5

Praxis Study Guide for the Mathematics: Content Knowledge Test 43

NOTATIONS

a b,( ) x a x b: < <{ }

a b,[ ) x a x b: ≤ <{ }

a b,( ] x a x b: < ≤{ }

a b,[ ] x a x b: ≤ ≤{ }

gcd ,m n( ) greatest common divisor of two integers m and n

lcm ,m n( ) least common multiple of two integers m and n

x[ ] greatest integer m such that m x≤

m k n≡ ( )mod m and k are congruent modulo n (m and k have the same remainder when divided by n, or equivalently, m k− is a multiple of n)

f −1 inverse of an invertible function f; (not to be read as 1f )

limx a

f x→ +

( ) right-hand limit of f x( ) ; limit of f x( ) as x approaches a from the right

limx a

f x→ −

( )

left-hand limit of f x( ) ; limit of f x( ) as x approaches a from the left

∅ the empty set

x S∈ x is an element of set S

S T⊂ set S is a proper subset of set T

S T⊆ either set S is a proper subset of set T or S = T

S complement of set S; the set of all elements not in S that are in some specified universal set

T S\ relative complement of set S in set T, i.e., the set of all elements of T that are not elements of S

S T∪ union of sets S and T

S T∩ intersection of sets S and T

DEFINITIONS

A relation ℜ on a set S is

reflexive if x xℜ for all x S∈

symmetric if x y y xℜ ⇒ ℜ for all x y S, ∈

transitive if x y y z x zℜ ℜ( )⇒ ℜ and for all x y z S, , ∈

antisymmetric if x y y x x yℜ ℜ( )⇒ = and for all x y S, ∈

An equivalence relation is a reflexive, symmetric, and transitive relation.

This ebook was issued to Mike Larson, order #18562186828. Unlawful distribution of this ebook is prohibited.

Customer name Mike Larson , Order Id: 18562186828


CHAPTER 5

44 Praxis Study Guide for the Mathematics: Content Knowledge Test

FORMULAS

Sum

sin sin cos cos sinx y x y x y±( )= ±

cos cos cos sin sinx y x y x y±( )= ∓

tan tan tan

tan tanx y x yx y±( )= ±

1∓

Half-angle (sign depends on the quadrant of θ2 ))

sin cosθ θ

21

2=± −

cos cosθ θ

21

2=± +

Range of Inverse Trigonometric Functions

sin−1 x −

π π2 2,

cos−1 x 0, π[ ]

tan−1 x −

π π2 2,

Law of Sines

sin Aa

Bb

Cc= =

sin sin

Law of Cosines

c a b ab C2 2 2 2= + − ( )cos

DeMoivre’s Theorem

cos sin cos sinθ θ θ θ+( ) = ( )+ ( )i k i kk

B

CA b

c a




CHAPTER 5


Coordinate Transformation

Rectangular x y,( ) to polar r, θ( ) : r x y2 2 2= + ; tan θ= yx if x≠ 0

Polar r, θ( ) to rectangular x y,( ) : x r= cos ;θ y r= sin θ

Distance from point ( , )x y1 1 to line Ax By C+ + = 0

d

Ax By C

A B=

+ +

+1 1

2 2

Volume

Sphere with radius r: V r= 43

3π

Right circular cone with height h and base of radius r: V r h= 13

2π

Right circular cylinder with height h and base of radius r: V r h= π 2

Pyramid with height h and base of area B: V Bh= 13

Right prism with height h and base of area B: V Bh=

Surface Area

Sphere with radius r: A r= 4 2π

Lateral area of a right circular cone with radius r and slant height s: A rs= π

Differentiation

f x g x f x g x f x g x( ) ( )( )′ = ′( ) ( )+ ( ) ′( )

f g x f g x g x( )( )( )′ = ′ ( )( ) ′( )

f xg x

f x g x f x g x

g x

( )( )

′=′( ) ( )− ( ) ′( )

( )( )2 if g x( )≠ 0

Integration by Parts

u dv uv vdu= −∫∫




CHAPTER 5


MATHEMATICS: CONTENT KNOWLEDGE

Recommended time—125 minutes

50 Questions

In this chapter, questions 1 – 37, 40 – 44, 47 – 48 and 50 are followed by four suggested answers or completions. Select the one that is best in each case. For questions 38 – 39, 45 – 46 and 49, enter your answer in the appropriate answer boxes.

20 3

10

4

1. The area of a rectangular region with adjacent sides of lengths a and b, respectively, can be used to represent the product a b. What product is represented by the shaded rectangular region in the figure above?

(A) 4 3×(B) 4 20×(C) 10 4×(D) 10 20×

2. José is considering what sandwich to buy for lunch. He has a choice of 2 different types of bread, 3 different types of cheese, and 4 different types of meat. For any sandwich, he must choose one type of bread and either one type of cheese or one type of meat, or he must choose one type of bread and one type of cheese and one type of meat. How many different sandwiches can he choose?

(A) 14(B) 24(C) 38(D) 48




CHAPTER 5


x

y

O

3. If the graph of the function y f x is shown above, which of the following could be the graph of 1 2 ?y f x

(A)

x

y

O

(B)

x

y

O

(C)

x

y

O

(D)

x

y

O




CHAPTER 5


1

5

10

15

2 3Number of Days

NUMBER OF DAYS TAKEN TO COMPLETE A CERTAIN ASSIGNMENT

Num

ber

of S

tude

nts

4 5

4. An assignment was given to 50 middle-school students to complete. The graph above shows a frequency distribution of the number of days it took the students to complete the assignment. What is the mean number of days it took for a student to complete the assignment?

(A) 2.8(B) 3.0(C) 3.1(D) 3.3

5. The graph of the axis of symmetry of the parabola 22 11 3y x x is best described by which of the following equations?

(A) x = 2(B) x = 2 25.(C) x = 2 5.(D) x = 2 75.

6. In the figure above, lines l and m are parallel. What is the value of y ?

(A) 19

(B) 1375

(C) 2235

(D) 1373




CHAPTER 5


x

y

8

4

2 1

E

O

7. In the figure above, what is the distance from point E to the origin?

(A) 10

(B) 15

(C) 5 5

(D) 5 102

8. If tan x 23

and x 32

, then

cosx

(A) 313

(B) 213

(C) 213

(D) 313

9. If A is 2 2× matrix and Axy

xy

=

3

for all real x and y, then A =

(A) 3 00 3

(B) 3 00 0

(C) 3 33 3

(D) 0 33 0




CHAPTER 5


x

y

–5 5

–5

5

10. The graph of the integrable function y f x is shown above. Which of the following is the best approximation of

f x dx( ) ?−∫ 2

4

(A) 7(B) 5(C) 2(D) 0

11. In a league that has 5 teams, each team plays each of the other teams twice in a season. What is the total number of games played during the season?

(A) 10(B) 12(C) 20(D) 24

R SO

T

50°

12. In the circle above with center O, what is the degree measure of arc R T ?

(A) 40°(B) 50°(C) 80°(D) 90°

13. Beginning at time t = 0, the number

of rabbits in a certain population at

time t years is modeled by the function

0.510,000

10 50 tf te

. According to

this model, which of the following best

describes how the size of the population

changes over time?

(A) It increases without bound.(B) It increases for several years and

then levels off.(C) It increases for several years and

then decreases to zero.(D) It increases and decreases in cycles.




CHAPTER 5


14. The Richter scale is a base 10 logarithmic scale used to measure the magnitude of earthquakes; i.e., an earthquake measuring 7 is ten times as strong as an earthquake measuring 6. An earthquake that measures 6.8 on the Richter scale has a magnitude that is approximately what percent of that of an earthquake measuring 6.6 ?

(A) 103%(B) 120%(C) 158%(D) 200%

15. The surface area of a sphere is approximately 1,500 square inches. What is the approximate volume of the sphere, in cubic inches?

(A) 500(B) 1,500(C) 3,500(D) 5,500

40 ft

50 ft

16. A rabbit is hopping around in a fenced-off flat rectangular field with dimensions as shown above. If the position of the rabbit is uniformly random throughout the field, what is the probability that the rabbit is 10 feet or more away from the fence at any given time?

(A) 0.80(B) 0.75(C) 0.60(D) 0.30




CHAPTER 5


17. Which of the following sets is NOT closed under division?

(A) Integers, excluding 0(B) Rational numbers, excluding 0(C) Real numbers, excluding 0(D) Complex numbers, excluding 0

18. Given f x x g x x4 12, , and

h x x1 , for what value or values

of x is h f g x 14?

(A) 0

(B) 32

(C) 32

(D) i 1515

19. If 37 is expressed in decimal form, what

digit is in the 19th decimal place?

(A) 1(B) 2(C) 4(D) 5

20. In the figure above, ∠ACB is a right

angle and CD is perpendicular to AB . If

AD = 9 and DB = 3 , then CA CD CB+ + =

(A) 6 9 3(B) 6 3 9(C) 2 2 7(D) 4 3 2

21. When solving an equation with real coefficients, which of the following procedures can result in an equation that yields a real root that does not satisfy the original equation?

(A) Subtracting the same number from both sides of the equation

(B) Raising both sides of the equation to the third power

(C) Squaring both sides of the equation(D) Dividing both sides of the equation

by a nonzero number




CHAPTER 5


22. If the real-valued function f x xn , where n is a positive integer, has neither a local maximum nor a local minimum, which of the following must be true?

(A) n is even.(B) n is odd.(C) n is a multiple of 3.(D) n can be any positive integer.

h

d

23. A glass window is composed of a

rectangle that is h feet long by d feet

wide and a semicircle with diameter d,

as shown in the figure above. If the area

of the semicircular region is 13 of the

total area of the window, what is the

ratio of h to d ?

(A) to 2(B) to 3(C) to 4(D) It cannot be determined from

the information given.

24. Which of the following is the solution set of the matrix equation 1 1

21 3

x x

y y ?

(A)

00

(B)

0 1,

0 1

(C)

, where is any real numberc

cc

(D) There are no solutions to the matrix equation.

25. Karen opened a bank account for her son on his 1st birthday with a $100 deposit. After that, $50 was deposited into the account on each birthday. No withdrawals and no other deposits were made until his 11th birthday. The bank pays 8% interest per year, compounded annually. Which of the following recursive sequences models the amount of money in the account after n years, 1 10n ?

(A) AA n A n

0 1000 08 1 50

( ) =( ) = −( ) +{ .

(B) AA n A n

0 1001 08 1 50

( ) =( ) = −( ) +{ .

(C) AA n A n

0 1000 08 1 50

( ) =( ) = −( ) +[ ]

.

(D) AA n A n

0 01 08 1 100 50

( ) =( ) = −( ) + +

.




CHAPTER 5


a b

a b a

b a a

26. The table above defines an operation on the set S a b, . Which of the

following statements about is true?

(A) It is neither associative nor commutative.

(B) It is associative but not commutative.

(C) It is commutative but not associative.

(D) It is both associative and commutative.

27. In the x y-plane, what is the radius

of the circle described by the

equation x x y2 22 0?

(A) 1(B) 2(C) 3(D) 4

28. What is the period of the function

y x= +

13

12 3

sin ?

(A) 13

(B) 6

(C)

(D)

4




CHAPTER 5


2

4

6

8

10

Freq

uenc

y

Below60

60-69

SCORES ON AN ALGEBRA I TEST

Scores

70-79 80-89 90-99 100

29. Which of the following statistics about the Algebra I test scores can be determined using only the information above?

I. The range of the scores I I. The median score I I I. The average (arithmetic mean)

score

(A) None(B) I only(C) I I only(D) I I I only

30. The function f x x3 1 is one-to-one; therefore, the inverse

function y f x1 exists. Which of the

following is an equation for y f x1 ?

(A) yx

113

(B)

y x 13

(C)

y x3 1

(D) yx1

13

31. Consider the set S of 2 2 matrices, all of whose entries are nonzero real numbers. Which of the following properties is satisfied by S under multiplication?

I. S is closed. I I. S is commutative. I I I. S contains an identity.

(A) None(B) I only(C) I and I I only(D) I and I I I only

32. For which of the following values

of x is f x x

x x x

2

3 24

5 3 undefined?

I. 3 I I. 2 I I I. 1

(A) I only(B) I I only(C) I and I I I only(D) I, I I, and I I I




CHAPTER 5


x

y

O

33. The triangle shown in the plane above is to be transformed in the plane by being reflected about the line y x and then rotated 90° clockwise about the origin. Which of the following could be the result of this transformation?

(A)

x

y

O

(B)

x

y

O

(C)

x

y

O

(D)

x

y

O




CHAPTER 5


34. A new temperature scale T is designed so that a temperature of 0° T is equivalent to a temperature of 24°C on the Celsius scale. If every increase of 5 degrees on the T scale is equivalent to an increase of 8 degrees on the Celsius scale, what is the temperature on the T scale that is equivalent to a temperature of 80°C?

(A) 40° T(B) 48° T(C) 65° T(D) 80° T

35. At how many points in the x y-plane do the graphs of y x x x x0 25 0 4 12 0 75 0 254 3 2. . . . . and y x0 5 2. intersect?

(A) None(B) Two(C) Three(D) Four

36. A six-sided number cube is weighted so that the probabilities of throwing 2, 3, 4, 5, or 6 are equal and the probability of throwing a 1 is twice the probability of throwing a 2. If the number cube is thrown twice, what is the probability that the sum of the numbers thrown will be 4 ?

(A)

112

(B)

549

(C)

37

(D)

511

37. The function ,y f x g x where

f x x and 1 if 00 if 0,1 if 0

x

g x x

x

is equivalent to which of the

following functions?

(A) y f x

(B) y f x

(C) y f x

(D) y f x




CHAPTER 5


For the following question, enter your answers in the appropriate answer boxes.

18 3 36 3 3 22x x x a x b− − = +( ) −( ) for all values of x, where a and b are positive integers.

38. What are the values of the positive integers a and b ?

a =

b =

For the following question, enter your answer in the answer box.

39. The width of a rectangle is 8 inches and remains constant, while the length of the rectangle increases at a constant rate of 3 inches per minute. When the length of the rectangle is 6 inches, what is the rate of change, in inches per minute, of the length of a diagonal in the rectangle?

inches per minute




CHAPTER 5


40. A certain company sent out leaflets describing its product. Of the 8,000 people who received the leaflets, 12% requested more information about the product. Of those who requested more information, 60% actually purchased the product. How many of the people who requested more information actually purchased the product?

(A) 576(B) 720(C) 960(D) 4,800

p xx

q xx

( )−

= ( ) +−32

3 for all values

x ≠ 3.

41. In the equation above, p x( ) and q x( ) are

polynomial functions. What is the value

of p 3( ) ?

(A) −3(B) − 2(C) 2(D) Undefined

42. Juanita is 4 feet tall. Which of the following could be used to calculate her height in centimeters? 1 2 54inch centemeters= .( )

(A) 4 112

12 54ft ft

inin

cm× × .

(B) 4 112

2 541ft ft

incm

in× × .

(C) 4 121

12 54ft in

ftin

cm× × .

(D) 4 121

2 541ft in

ftcm

in× × .

43. ddx u

dux 1

11

3

+

=∫

(A) ln 1 3+( )x

(B) 1

1 3+ x

(C) 3

1 3+ x

(D) 1

3 9+ x

44. What is the complex conjugate of the complex number − −3 2i ?

(A) 6i(B) − +3 2i(C) 3 2− i(D) 3 2+ i




CHAPTER 5



45. What are the coefficients in the binomial expansion of the polynomial x y+( )4 ?

x y x x y x y xy y+ = + + + +( )4 4 3 2 2 3 4


46. The function f x x x x x( ) = + − + −6 75 254 3 2 has four distinct zeros. Two of the zeros are

x = − 52

and x = 13

. What are the two other zeros?

x i= ±

47. Which of the following is equivalent to the expression 3 22 3 3 2

x x− −( ) ( ) for all positive values of x ?

(A) 1108 12x

(B) 136 12x

(C) 612x

(D) 274 12x

48. A stone is projected with a slingshot vertically upward. The height h, in feet, of the stone t seconds after it is projected is given by the equation h t t= − + +16 80 52 . What is the average rate of change, in feet per second, of the height of the stone for the 2-second time interval from t = 3 seconds to t = 5 seconds?

(A) −56 5.(B) −48 0.(C) 52 5.(D) 64 0.




CHAPTER 5


For the following question, enter your answer in the appropriate answer boxes.

49. What is the value of limtancos

?θ

θθ θ→

( )( )0

23 4

Give your answer as a fraction.

50. To estimate the Fahrenheit temperature (F) equivalent to a given Celsius temperature (C), a tourist adds 30 degrees to twice the Celsius temperature. In a country where the temperature range is from 0°C to 30°C, what is the greatest possible error the tourist can make using this estimation technique correctly? (The formula for converting Celsius temperature to Fahrenheit temperature is F C= +1 8 32. .)

(A) 2°F(B) 3°F(C) 4°F(D) 5°F



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01. Math Practice Exam

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