praxis ii 0061 practice exam

@ THEPRAXIS Listening. Learning. Leading!>

s E R I E STM

The Official

Practice Test

Mathematics: Content Knowledge

Test Code: 0061

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MATHEMATICS: CONTENT KNOWLEDGE

The supervisor will tell you when to begin work on the test and when to stop. if you finish the test before time is called, go back and check your work on it.

SHO ULD YOU GUESS? Your test score is based on the number of questions you answer correctly. Skipped ques tions count as wrong answers, so trv to answer every question even if you have to guess. Do not spend too much time puzzling over a question that seems difficult. Answer the easier questions first, then return to the harder ones.

Where necessary, you may use blank spaces in the test book for scratch paper. Do not use any other paper or the margins or back of the answer sheet to do scratch work. For reference, pages 6 8 of this booklet contain a list of selected notational conventions , definitions, and formulas .

You may use a graphing calculator while taking this test. You may not use a calculator with a QWERTY (typewriter) keyboard.

There are times when a calculator can give incorrect or misleading results . When using results provided by your calculator, use your knowledge of mathematics to determine if the results provided by the calculator are correct.

YOU ARE TO INDICATE ALL OF YOUR ANSWERS ON T HE SEPARATE ANSWER SHEET. No credit will be given for anything written in this examination book. After you ha\e decided which of the suggested answers is best, fil l in the corresponding space on the answer sheet. BE SURE THAT EACH MARK IS HEAV Y AND DARK AND COMPLETELY FILLS THE ANSWER SPACE. Light or partial marks may not be read by the scoring machine. Give only one answer to each question. If you change an answer, be sure that the previous mark is erased completely. Incomplete erasures may be read as intended answers.

This test may include one or more questions that do not count toward your score .

Time 120 minutes

50 Questions

DO NOT BREAK THE SEAL UNTIL YOU ARE TOLD TO DO SO.

tm\ Educational ~ Testing Service

EDUCATIONAL T EST! NO SERVICE, ETS, ETS logo and the modernized ETS logo arc registered trademarks of Educational Testing Service.

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Version 5

Test Book

TEST DIRECTIONS

TestGroupG

o Match the 4-digit test code number and title on the cover of this book with the information printed on your admission ticket. If the code and title do not match, raise your hand.

o Print your full name, last name first, on the front cover. Then read the directions on the back cover.

Answer Sheet Complete items 1-6 on page 1. Make certain that you completely fill in the appropriate ovals.

o In item 3, add a zero before any single digit. o In item 4, enter your Social Security number. While ETS cannot require your Social Security number, many state

licensing agencies and institutions of higher education use Social Security numbers to process certification records. If you choose not to provide this number, your certification could be delayed.

o In item 5, enter your candidate ID number from your admission ticket. o In item 6, print the information for this test center as posted.

Items 7-10 are crucial for the correct scoring of your test. Use the front cover of your test book to complete the information requested.

o In item 7, write the test code/form code in the spaces provided and fill in the appropriate ovals. o For item 8, the serial number is printed in the upper right comer of the cover of your test book. • In item 9, list the alphanumeric test form. • In item 10, write the test name located on the front cover of this test book. • Now turn your answer sheet to page 2. Copy the certification statement as directed; then sign your name and enter

today's date.

Put your admission ticket away. Your photo identification must remain on your desk during the test administration.

The testing staff cannot answer questions about test content at any time, but if you have other questions once you have started or find something wrong with your test book or answer sheet, raise your hand. If you find a question that contains a typographical error or that you think is otherwise faulty, inform the staff inunediately after the test. You can circle any errors in your test book.

There will be no rest break. If you need to leave the room during the test, raise your hand so your test book and answer sheet can be collected. The time you are out of the testing room cannot be made up. You may not use a telephone or cell phone.

The testing staff will keep the official timing for the test. They will post the time remaining after 30 minutes and announce when 10 minutes remain. You will have two hours to complete the test. You must remain until the end of the testing session.

When you are finished reading these instructions, close your test book and look up.

(a, b)

[a, b)

(a, b]

[a, b]

gcd(m, n)

lcm(m, n)

[x] m = k(modn)

lim J(x) x~a+

lim f(x) x~a-

{x: a< x < b} {x:a:s;x<b} {x:a<x:s;b} {x:a:s;x:s;b}

NOTATION

greatest common divisor of two integers m and n

least common multiple of two integers m and n

greatest integer m such that m ::; x

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)

inverse of an invertible function! (not the same as ~)

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

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

the empty set 0 XES

SeT S<;;.T

SuT SnT

x is an element of set S

A relation 9t on a set S is

set S is a proper subset of set T

either set S is a proper subset of set Tor S = T

union of sets S and T

intersection of sets S and T

DEFINITIONS

reflexive if x 9t x for all x E S symmetric if x 9t y => y 9t x for all x, y E S transitive if (x 9t y andy 9t z) => x 9t z for all x, y, z E S antisymmetric if (x 9t y andy 9t x) => x = y for all x, y E S

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

-6-

sin(x ± y) = sinx cosy± cosx sin y

cos(x ± y) = cosx cosy+ sinx siny

( ±) tanx±tany

tan x Y = 1 + tanxtany

Range of Inverse Trigonometric Functions

sin- 1 x [-~' ~]

COS- IX [0, 1r]

tan- 1 x (- ~.;)

Law of Sines

sinA sinB sine = =

a b c

Law of Cosines

DeMoivre' s Theorem

(cos B + i sin B)k = cos(kB) + i sin (kB)

Coordinate Transformation

FORMULAS

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

. §_ _ + /1 - cos B sm2--v 2

§_ _ +P +cosO cos2

- _ 2

B

Rectangular (x, y) to polar (r, B) : r2 = x2 + l; tanB = l. if x :t; 0 X

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

Distance from point ( x1, y1 ) to line Ax + By + C = 0

d = I Ax! + Byl + Cl ~A2+ B2

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Volume

Sphere with radius r :

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

Right circular cylinder with height h and base of radius r :

Pyramid with height h and base of area B :

Right prism with height h and base of area B :

Surface Area

Sphere with radius r :

Right circular cone with radius r and slant height s :

Differentiation

(j(x)g(x))' = f'(x)g(x) + f(x)g'(x)

(f(g(x)))' = f'(g(x))g'(x)

(f(x))' = f'(x)g(x) - f(x)g'(x) if g(x) -:t:- 0 g(x) (g(x))2

Integration by Parts

J u dv = uv - J v du

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4 3 V = -f!r 3

v = lsh 3

V = Bh

MATHEMATICS: CONTENT KNOWLEDGE

Time-120 minutes

50 Questions

Directions: Each of the questions or incomplete statements below is followed by four suggested answers or completions. Select the one that is best in each case and then fill in the corresponding lettered space on the answer sheet with a heavy, dark mark so that you cannot see the letter.

1.8984375 E - 20

1. Shown above is a number displayed in scientific notation on a calculator. What is the 20th digit to the right of the decimal point when the number is expressed in decimal notation?

(A) 0 (B) 1 (C) 5 (D) 8

Unauthorized copyin!;J or reuse of any part of this page 1s illegal. -9-

2. Juanita is 4 feet tall. Which of the following could be used to calculate her height in centimeters? (1 inch= 2.54 centimeters)

1ft 1 in (A) 4 ft x 12in x 2.54cm

(B) 4 ft 1 ft 2.54 em

x -12· x 1· lll lll

(C) 4 ft x 12 in x 1 in 1ft 2.54 em

(D) 4 ft 12 in 2.54 em x~x lin

GO ON TO THE NEXT PAGE.

3. Which of the following figures shows vectors u, v, and win the .xy-plane such that w = u + v?

(A)

(C)

Unauthorized copyin~ or reuse of any part of this page 1s illegal.

y

y

1

0

I I I I I -~-~-~-1--1-·

I I I lw l -t--f--f- --1-·

lu I I I I - -f- -1--1-·

I I I I ·t-- --t--1--1-· I I I I I

-t--1--1-· I I I

1

1

(B) y

1 X

0

(D) y

1 X

0

-1o-

I I I I I -~-~-~-1--1-·

I I I I I - t--f--f- -1--1-·

- -~~-:--:--I I I I

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I I IW I I -t--f-- -1--1-· u I I I I

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f- -t- - 1- - I-. I I I I

-t--1--1-· I I I

X

1


Contribution to Course Grade Percent

Average (arithmetic mean) 50% of chapter test scores

Project score 20%

Final exam score 30%

4. In determining course grades, Ms. Rose calculates a weighted average as shown in the table above. If Frank's chapter test scores are 82, 7 5, and 79, his project score is 90, and his final exam score is 85, what is Frank's course grade, to the nearest tenth?

(A) 82.2 (B) 82.8 (C) 83. 1 (D) 83.7


y

I I I I I I I I I I I

-~--~-+ 7 --~-~--~-~-- : --~-~--~-· I I I I I I I I I I

-~--~-+ 6 --~-4-- : -~-- +--~-4--~-1 I I I I I I I I I

-~--~-+ 5 -- -- ~--}-~--+--~-~--}-1 I I I I I I I I I I

-~--~-t 4 --~-~--~-~-- ; --~-~--~-1 I I I I I I I I I I

-~--~-~ 3 -4--~-~-- ~-- -4--~-1 I I I I I I I I I I ~ _L_~ _ L-~ L ~- ~ L __ L

I I I I I I I I I I I I I I I I I I I I I

-~-- h -- 1 --~-~--~-~--+--~-1---I , I I I I I I I I I

-~-;1--TI~~~~r-~1--TI __ ri -;1--TI __ rl ~lr--x I

0 I I I 1

-3 -2 -1 1 2 3 4 5 6 7 8

5. The graph of the piecewise-linear function f is shown in the xy-plane above. Each of the following numbers is in the range off EXCEPT

(A) 3 (B) 3.5 (C) 4 (D) 4.5


6. In the xy-plane, the parabola with equation

y = x2 - 4 intersects the ellipse with equation

2 2 :6

+ Y4

= 1 in how many points?

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

7. In the xy-plane, which of the following points lies inside the circular region of radius 3 centered at (1,-2)?

(A) (-1,1)

(B) (1,2) (C) (0,-5) (D) (3,0)

Unauthorized copyin~ or reuse of any part of this page 1s illegal. -12-

R s - 6 14

8. On the number line above, a point between points RandS will be selected at random. What is the probability that the point selected will be within 2 units of the point with coordinate 8 ?

1 (A) 10

(B) .!. 8

(C) 1 5

(D) 1 4


Other

9. The circle graph above shows how Maria spent her allowance in one year. She spent $200 on recreation. Of the amount she spent on clothing, $80 was spent on sweaters. What percent of the amount that she spent on clothing was the amount that she spent on sweaters?

(A) 8% (B) 25% (C) 32% (D) 40%


y

~r-----+---------x 0 c

10. In the xy-plane above, point P( c, f( c)) is a point

of inflection for the graph of the polynomial

function f In which of the following pairs of

statements are both statements true?

(A) f"(c) = 0 and f'(c) > 0

(B) f"(c) = 0 and f'(c) < 0

(C) f"(c) > 0 and f'(c) = 0

(D) f"(c) < 0 and f'(c) = 0


11. The matrix representation for the linear transformation

Unauthorized copyin~ or reuse of any part of this page 1s illegal.

T R3

-> R3

d = (~ ~ n ff v = m ilien T(v) =

(A) m (B) m (C) m (D) m

-14-GO ON TO THE NEXT PAGE.

12. On a graphing calculator's viewing screen, the

graph of y = 4sin 2x cos (- 2x) would coincide

with the graph of which of the following?

(A) y = 2sin2x

(B) y = 2sin4x

(C) y = 4sin4x

(D) y = - 2cos4x

13. Event B happened after event A and before

event C. The elapsed time between event A and

event C was 109 seconds, and the elapsed time

between event B and event C was 103 seconds.

Which of the following is closest to the elapsed

time, in seconds, between event A and event B ?

(A) 109

(B) 108

(C) 106

(D) 103


14. In the xy-plane, line .e has equation y = - x.

Point P lies on .e and has coordinates ( -2, 2).

If .e is rotated counterclockwise 45° about the

origin, what will be the coordinates of the image

of P under this rotation?

(A) ( -2F2", 0) (B) (-2, 0)

(C) (0, 2)

(D) ( 0, 2F2")

15. A new drug was tested on a group of 200 patients with high blood pressure. By the end of the trial period, 90 patients showed improvement from the drug, 35 patients developed side effects, and 95 patients showed no improvement and had no side effects. How many patients showed improvement and had side effects?

(A) 15 (B) 20 (C) 25 (D) 30


Jar 1 Jar 2

Red 4 5

Black 6 5

16. The table above shows the distribution of red and black chips in two jars. One chip is to be selected at random from each jar. What is the probability that both chips will be of the same color?

(A) 1 3

(B) ~

(C) 1 2

(D) ;

17. Changing the value of which of the positive constants a, b, c, and d will change the period of the function y =a sin(bx +c)+ d?

(A) a (B) b (C) c (D) d


60m

20m

p

100m

0

Note: Figure not drawn to scale.

18. In a lawn game, Jill walked from point 0 to point P, as shown in the figure above. She walked north for 100 meters, then east for 60 meters, and finally south for 20 meters. What is the distance between Jill's starting point and her ending point?

(A) 80 m (B) 90 m (C) 100m (D) 120m


19. Definition: The normal line to the curve y = f ( x) at the point P is the line perpendicular

to the tangent line to the curve at P.

If an equation of the normal line to the graph of the function y = f(x) at the point (-3, 13) is

y = - 2x+7, then f'(- 3) equals

(A) - 2

1 (B) 2

1 (C) 2

(D) 13

20. Which of the following complex numbers is the multiplicative identity for any complex number (a, b)?

(A) (0, 0)

(B) (0, 1)

(C) (1, 0)

(D) (1, 1)


A

8

21. In triangle ABD above, angle D is a right angle and triangular regions ABC and ACD have area 180 and 90, respectively. If BC = 20 = DX, what is the length of XC?

(A) 61 3

(B) 10

(C) 13.!_ 3

(D) 15


22. If a, b, and c are real numbers such that

~ b2 - 4ac = 0, which of the following

statements about the roots of the equation

ax2 + bx + c = 0, where a :;:. 0, must

be true?

(A) The equation has two distinct real roots. (B) The equation has two distinct nonreal roots. (C) The equation has one real root. (D) The equation has one nonreal root.


23. For a normally distributed population, approximately 68 percent of the population lies within 1 standard deviation of the mean and approximately 95 percent of the population lies within 2 standard deviations of the mean. In a certain environmental study, the weights of 10,000 fish are normally distributed with a mean of 12.5 ounces and a standard devialion of 4 .2 ounces. Approximately what percent of the 10,000 fish have weights between 16.7 and 20.9 ounces?

(A) 7% (B) 14% (C) 27% (D) 34%


a11 x + a12y + a13z = b1

a21X +a22Y + ~3Z = b2

a31x +a32Y + ~3Z = b3

24. Let A be the matrix of coefficients aij of the

system of equations above, and let B be the

column vector whose entries are b1, b2 , and b3 .

If the system of equations has a unique solution,

then the solution is given by which of the

following matrix products?

(Note: A1 denotes the transpose of A, and

A- 1 denotes the inverse of A.)

(A) BA- 1

(B) BtA- 1

(C) A1B

(D) A-1B


y

25. The graph of the continuous function f is shown in the xy-plane above. The function F (graph not shown) is defined by

F(t) = J: f(t)dt.

At which of the labeled points is F decreasing?

(A) A (B) B (C) c (D) D


26. In determining which of the following quantities is it LEAST appropriate to use the figure above as a model?

(A) The number of games scheduled for 5 teams when each team is scheduled to play each of the other teams exactly once

(B) The number of different ways that 5 people can sit at a table having 5 sides, if one person sits at each side

(C) The number of 2-element subsets of a set of 5 elements

(D) The number of diagonals in a regular pentagon


27. In the figure above, the segments of each chord of the circle have the lengths shown. Which of the following must be true?

(A) a = b c d

(B) a = b d c

(C) a+c b+d = c d

(D) a + b = c + d c b


28. In order to be included on the ballot in a certain city, a candidate must get at least 12.5% of the registered voters in each of three or more of the city's five districts to sign a petition. The ratio of the numbers of registered voters in the five districts is 6:5:5:4:5. In order for a candidate to be included on a ballot, what is the minimum percent of the total number of registered voters that must sign a petition?

(A) 2% (B) 5% (C) 7% (D) 10%


4

3

y

----r-+-~-r-+~~+-;--r~x

-3 -2 -1 0 -1 -2 -3

29. The graphs of the linear functions f and g

are shown in the xy-plane above. The linear

function his defmed by h(x) = f(g(x)) for

all x. What is the value of h(2) ?

(A) - 4

(B) 0 (C) 1 (D) 2


30. The daily power requirement P, in megawatts, for the town of Kupinville can be modeled by the equation

P(t) = 60- 25cos (1; t ).

where t is the number of hours that have elapsed since midnight. At what time of the day is Kupinville's power requirement greatest?

(A) Midnight (B) 6 A.M.

(C) Noon (D) 6 P.M.


31. The integer n has the form shown above,

where both p and 2P - 1 are prime numbers

and p > 2. What is the total number of positive divisors of n, including 1 and n, in terms of p?

(A) p

(B) 2p

(C) P2

(D) 2p-2


Stage 1 Stage 2 Stage 3

32. The figures above show the first three stages in the construction of a fractal. The construction starts with

an unshaded equilateral triangle (not shown). At stage 1, the triangle with vertices at the midpoints of this

triangle is shaded. At each subsequent stage, the process is repeated for each unshaded triangle in the previous

stage. Which of the following statements is true about the perimeter Pn and area An of the shaded portion of

the figure at stage n as n ~ oo ?

(A) Both Pn and An approach a finite limit.

(B) Only Pn approaches a finite limit.

(C) Only An approaches a finite limit.

(D) Neither Pn nor An approaches a finite limit.



33. The operation ® is defined on the set z+ of

positive integers by x ® y = xY for all x and

y in z+. Which of the following statements

is true about the number 1 with respect to the

operation ® ?

(A) 1 is a right identity but not a left identity. (B) 1 is a left identity but not a right identity. (C) 1 is both a right and a left identity. (D) 1 is neither a right nor a left identity.

34. Three riders enter a train car that has 5 empty seats. Each rider will choose a different empty seat. How many possible arrangements are there ofthe 3 riders among the 5 seats?

(A) 10 (B) 15 (C) 30 (D) 60


35. Let f(x) =sin x, where x is a real number. The

graphs of which of the following functions in the

xy-plane intersect the graph off in one or more

points?

I. g(x) = 2 +cos x, where x is a real number

II. h(x) = ln(l + x), where x > -1

(A) Neither I nor II (B) I only (C) II only (D) Both I and II


8

6

4

2

v

2 4 6 8

36. The velocity of an object, v(t), in feet

per second, is given by the equation t2

v(t) = 8- 2 ,

where t is time, in seconds. The graph of v is

shown in the coordinate system above. To the

nearest foot, how far does the object travel in the

first 2 seconds?

(A) 6ft (B) 7ft (C) 13ft (D) 15ft


37. The linear transformation of the plane defined by the matrix equation above is a dilation of the plane. Under this transformation, the line y = 3x + 2 is mapped onto a line with slope

(A) 3 (B) 4 (C) 8 (D) 12


0 90 Has a high

H h fl temperature

as t e u

Does not have a high temperature

<:12 Has a high

D . temperature

oes not have the t1u

Does not have a high temperature

38. The probability diagram above represents the incidence of high temperature among the students at a school who have and do not have the flu. For example, 0.90 is the probability that a student who has the flu also has a high temperature. What is the probability that a student has neither the flu nor a high temperature?

(A) 0.470 (B) 0.572 (C) 0.685 (D) 0.770


39. For a given scatterplot of data points (c, k ), a

student transformed the data using a log scale on

the vertical axis. The student then determined that

a good fit for the transformed data points (c, ln k) was the regression line y = 5.53- O.llc, where

y = ln k. Based on the student's regression line,

which of the following equations approximates

the relationship between c and k ?

(A) k = 252e-o.!lc

(B) k = 5.53e-o.l1c

(C) k = 5.53(1.1tt

(c-5.53)

2

(D) k = e O.Il


I I I I I X= -1<_: I I I I I I I I -----------j--1

-4

y

I I I

i5x=1 I I I I I

: y = l : rV -t----------1

4

40. The .xy-plane above shows the three branches of

ax2 +b the graph of y = f(x ) = 2

, where a, b, x +c

and c are constants. If lim f(x ) = +oo, what x~I-

is the value of c ?

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

Unauthorized copyin!;J or reuse of any part of this page 1s illegal. -Z7-

Krishna


41. Standing on a bridge, Krishna sees two boats, as shown above. The angle, 0, between his line

of sight to each boat is 58°, and the boats are

43 meters and 72 meters away, respectively, along

his line of sight. What is the distance between the

two boats?

(A) 41.69 m (B) 61.25 m (C) 83.86 m (D) 101.56 m


y

42. The graph of the function y = f(x) is shown in

the xy-plane above. Which of the following could

be an equation for y = f(x) ?

(A) y=(x - a)(x -b) (B) y = x(x - a)(x- b) (C) y = x(x -a)(x+b) (D) y = x(x + a)(x +b)


---;.---------.-Wall

Garden


43. Claire wants to fence in a rectangular plot of land along a wall to make a garden. She has p feet of fencing, which will extend along three sides of the garden, with the wall forming the fourth side, as shown in the figure above. Which of the following expressions in p is equal to the maximum possible area, in square feet, of Claire's garden?

(A) 2p2

(B) p2 2

p2 (C) 4

2 (D) f!_

8


44. The graph of the function f ( x) = x2 in the

.xy-plane is translated three units to the right and one unit up to create a graph of the function g. At what point do the graphs off and g intersect?

(A) (t· 2:)

(B) (_1 25) 3' 5

(C) (±~) 3 ' 9


c

45. In the figure above, AB = BC = 2, point E is the

midpoint of BC, and point F is the midpoint of

AB. What is the measure, to the nearest degree,

of LFDE?

(A) 130° (B) 143° (C) 150° (D) 153°


��

Jan.

Feb.

Mar.

Apr.

May

June

July

Aug.

Sept. Oct. Nov. Dec.

TEMPERATURE IN A REGION

--c==:r=:::::J~

---{:::::I:::J~

~ --cn--

--cr::::J--{::I:J-

-cr::::::::J--

0 10 20 30 40 50 60 70 80 90 100

Temperature (degrees)

46. The graph above shows boxplots of daily high temperatures in a certain region for each of twelve consecutive months. Based on these boxplots, which of the following statements about the temperature in the region during the twelve-month period is true?

(A) It was colder in the region in June than in September. (B) The range of the daily high temperatures for August was

greater than the corresponding range for January. (C) The daily high temperature in February never exceeded

that in April. (D) The median daily high temperature for the months of

April through October was above 50 degrees.

Jan.

Feb.

Mar. Apr.

May

June

July

Aug.

Sept. Oct. Nov. Dec.

Unauthorized copying or reuse of any part of this page Is Illegal. GO ON TO THE NEXT PAGE.

47. The graph of y = xex on the .xy-plane has

(A) a vertical asymptote (B) positive slope for all real x (C) a relative maximum (D) an absolute minimum


G

F

E D

Given: ( 1 ) ABC is a right triangle.

(2) AFGB and ACDE are squares. - -

Prove: FC :BE

Statement Reason - - All sides of a square are 1. FA ::BA

congruent. - - All sides of a square are 2. AC :: AE

congruent.

3. LFAB:: LCAE All right angles are congruent.

4. LBAC :: LBAC Reflexive property of congruence

5. LFAC:: LBAE Angle addition postulate

6. !:J.FAC :: !:J.BAE ?

- -7. FC :BE Corresponding parts of

congruent triangles are congruent.

48. The figure above consists of a right triangle and

squares on two of the sides. The table gives all

but one of the reasons for the statements used to --

prove that FC :: BE. Which of the following is the missing reason for statement 6 ?

(A) SSS (B) SAS (C) SAA (D) ASA


.s (1.)

g ~

~ ~

y

(8, 5000)

o~------------o

Number of Years


49. An investor deposited a certain amount of money

in a new account at an annual interest rate of r,

compounded continuously. The figure above

shows the graph of y = p(t), where p(t) is the

balance, in dollars, of the account t years after

the investor made the deposit and t ;::: 0. If no

other transactions were made in the account,

which of the following is closest to the amount,

in dollars, the investor deposited in the account

at time t = 0? (Note: p(t) = Po en) (A) $540 (B) $1,000 (C) $1,160 (D) $1,270

50. In the geometric sequence above, b1 = 1, 000 and

bn = jbn- I for all n ;::: 2. What is the least value

of k for which bk < 0.001 ?

(A) 30 (B) 32 (C) 34 (D) 36

STOP If you finish before time is called, you may check your work on this test.

Unauthorized copying or reuse of any part of this page Is illegal. -32-

Scoring Your Practice Test

First, determine your raw score. To determ ine your raw score, check your answers against the answers shown in Table 1, and then count how many of the questions you answered correctly. (A description of the content categories represented by the Roman numerals in Table 1 can be found on page 35.)

Sequence Number

1 2 3 4

5 6 7 8

9 10 11

12 13

14

15 16 17 18 19 20 21 22 23 24

25

Your raw score =

Table 1- Answers and Content Categories for the Mathematics: Content Knowledge Practice Test

Correct Content Sequence Correct Answer Category Number Answer

B I 26 B

D II 27 B

D v 28 c B IV 29 A

D Ill 30 c D I 31 B

D I 32 c c IV 33 A

c IV 34 D

A Ill 35 c A v 36 D

B II 37 A

A II 38 B

A II 39 A

B v 40 A

c IV 41 B

B II 42 B

c II 43 D

c Ill 44 A

c I 45 B

B II 46 D

c I 47 D

B IV 48 B

D v 49 c c Ill 50 D

(the number of correct answers) .

-33-

Content Category

v II

I Ill

Ill

I II

I v Ill

Ill

v IV IV Ill

II

Ill

Ill

Ill

II

IV Ill

II

Ill

v

Next, determin e your scaled score. Use Table 2 to fmd the scaled score corresponding to your raw score. You can compare your scaled score to the passing score required by your state. (You can fmd a list of the minimum passing scores at www.ets.org/praxis.)

Table 2-Score Conversion Table

Raw Scaled Raw Scaled Raw Scaled Score Score Score Score Score Score

0 100 23 131 46 184

1 100 24 134 47 186

2 100 25 138 48 189

3 100 26 140 49 192 4 100 27 143 50 194

5 100 28 146

6 100 29 149

7 100 30 151

8 100 31 154

9 100 32 156

10 100 33 158

11 100 34 160

12 100 35 162

13 100 36 165 14 102 37 167

15 105 38 169

16 109 39 170

17 112 40 172

18 115 41 174

19 118 42 176

20 122 43 178

21 125 44 180

22 128 45 182

Your scaled score = Your state's passing score=

If your score was not high enough for you to pass, determine how many additional correct answers you would have needed to reach the required passing score.

The r aw score corresponding to your state's passing score

(Subtract) Your raw score

Additional cor rect answers you would have needed to reach the passing score

-34-

Next, assess your strengths in the five content categories. Use the content category information in Table 1 to determine whether you need to prepare more intensely in any of the five areas covered by the test. The content categories are represented in Table 1 by Roman numerals, which correspond to the following descriptions:

I. Algebra and Number Theory

II. Measurement, Geometry, and Trigonometry

III. Functions and Calculus

IV. Data Analysis and Statistics and Probability

V. Matrix Algebra and Discrete Mathematics

Fill in Table 3 to see where you have the most room for improvement.

Table 3- Assessment of Strengths in Each Category

Content Number of Number of Category Correct Answers Incorrect Answers

Possible

I 8

II 12

III 14

IV 8

v 8

Focus on the content area or areas where there are the most incorrect answers (the rightmost column in Table 3) . The Mathematics: Content Knowledge Study Guide can help you review these content areas. (The guide can be purchased at www.ets.org/store.)

-35-

Measure of Question Difficulty

The values in the fourth column of Table 4 are the percentage of examinees that answered each question correctly. This is used as a measure of question difficulty. In general, the higher the percentage the easier the question.

Table 4-Percentage of Examinees Choosing Correct Answers for the Mathematics: Content Knowledge Test

Sequence Percentage of Examinees Sequence Percentage of Examinees Number Choosing Correct Answer Number Ch oosing Correct Answer

1 56% 26 38%

2 91% 27 27%

3 61% 28 51%

4 90% 29 56%

5 77% 30 63%

6 47% 31 42%

7 65% 32 28%

8 41% 33 32%

9 80% 34 53%

10 57% 35 71%

11 66% 36 38%

12 75% 37 44%

13 35% 38 66%

14 38% 39 33%

15 60% 40 47%

16 55% 41 73%

17 63% 42 60%

18 83% 43 28%

19 31% 44 67%

20 20% 45 40%

21 56% 46 56%

22 58% 47 32%

23 59% 48 74%

24 39% 49 48%

25 32% 50 33%

NOTE: Percentages are based on the test records of 2,299 examinees who took the 60-rninute version of the Mathematics: Content Knowledge test in November 2008.

In general, questions may be considered as easy, average, or difficult based on the following percentages:

Easy questions= 75 % or more answered correctly Average questions = 55%-74% answered correctly Difficult questions = less than 55% answered correctly

-36-

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Practice Test

$13.95 USA

Listening. Learning. Leading.®

www.ets.org

Cop)<ight C> 2009 by Educational Testi1g Se<vice. AI rilt\IS reseMld. ETS, the ETS logo, LISlENING. LEARNING. LEADING., and PPST are regstered trOOemar1<s of ECI.Jcational Testing Se<vice (ETS) in the Uited States and other countries. THE PRAXIS SERIES is a trOOemar1< of ETS. 10212

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