Praxis II 0061 Practice Exam

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Listening. Learning. Leading!>

STM

The Official

Practice Test Mathematics:

Content Knowledge Test Code: 0061

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MATHEMATICS: CONTENT KNOW LEDGE 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 ques tions 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 scori ng 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.

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Use only a pencil with soft black lead (No. 2 or HB) to complete this answer sheet. Be sure to fill in completely the oval that corresponds to the proper letter or number. Completely erase any errors or stray marks.

Answer

PRAXIS.

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Jan. Feb.

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July Aug. Sept.

TELEPHONE NUMBER:

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Nov.

SIGNATURE:

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5. CANDIDATE ID NUMBER

4. SOCIAL SECURITY NUMBER

Day

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I 8. TEST BOOK SERIAL NUMBER

6. TEST CENTER I REPORTING LOCATION

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Center Number

19.T£ST FORM Center

Name

City

10. TEST NAME

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State or Province

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SIGNATURE: - - - - - - - - - - - - - - - - - - - - - -

BE SURE EACH MARK IS DARK AND COMPLETELY FILLS THE INTENDED SPACE AS ILLUSTRATED HERE: 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 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

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Version 5

TEST DIRECTIONS TestGroupG Test Book o Match the 4-digit test code number and title on the cover of this book with the information printed on your o

admission ticket. If the code and title do not match, raise your hand. 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.



NOTATION

[a, b]

{x: a< x < b} {x:a:s;x
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(modn)

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)

(a, b)

[a, b) (a, b]

inverse of an invertible function! (not the same as

~)

lim J(x)

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

lim f(x)

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

x~a+

x~a-

0

the empty set

XES

x is an element of set S set S is a proper subset of set T

SeT S<;;.T SuT SnT

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 A relation 9t on a set S is 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

antisymmetric if (x 9t y andy 9t x) => x = y for all x, y

S E

S

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

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FORMULAS Half-angle (sign depends on the quadrant of sin(x ± y) = sinx cosy± cosx sin y cos(x ± y) = cosx cosy+ sinx siny tanx±tany tan( x ±Y) = 1 + tanxtany

. §_ _ + /1 - cos B sm2--v 2 cos §_ -_ +P _ +cosO 2 2

Range of Inverse Trigonometric Functions sin- 1 x

[-~' ~]

COS- IX

[0, 1r]

tan- 1 x

(- ~.;)

Law of Sines B

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 Rectangular (x, y) to polar (r, B) : r2 = x 2 + Polar (r, B) to rectangular (x, y) :

I Ax!

+ Byl +

tanB =

l. if x X

x = r cos B; y = r sin B

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

l;

Cl

~A2+ B2 -7-

:t;

0

~)

Volume V =

Sphere with radius r :

4 3 -f!r 3

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 :

v = lsh

Right prism with height h and base of area B :

V = Bh

3

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))' ( g(x)

=

f'(x)g(x) - f(x)g'(x) if g(x) (g(x))2

-:t:-

0

Integration by Parts

Ju dv = uv - Jv du

-8-

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

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?

1ft 1 in (A) 4 ft x 12in x 2.54cm 2.54 em 1 ft (B) 4 ft x -12· x 1· lll lll

(A) 0 (B) 1

(C) 5 (D) 8

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

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

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12 in

x~x

2.54 em 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)

y

(B)

y I

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-~-~-~-1--1-·

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1 X

0

1

(C)

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I

·t-- - - t - -1- -1-· I I I I I -t--1--1-· I I I

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I I I lw l -t--f--f- --1-· I I I lu I -f-1--1-·

y

X

1

y

(D)

I

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I I IW I I -t--f-- -1--1-· I I I I - - -t--1--1-·

u

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0

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

1 X

0

1

-1o-

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

X

1

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Contribution to Course Grade

y

Percent

Average (arithmetic mean) of chapter test scores

50%

Project score

20%

Final exam score

30%

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--~ -4-- : -~-- + --~-4--~-

-~--~-+ 5

-- -- ~ --}-~--+--~-~--}-

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-3 -2 -1 0

-4--~-

L __ L

L-~

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~-

~

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--~-~--~-~--+--~-1---

I __ri -~-;I1--TI~~~~Ir-~1--T I

- ;1 --T I __r l ~lr-- x I 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

82.2 82.8 83. 1 83.7

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

I

-~--~-t 4 --~ -~-- ~ -~-- ; --~- ~ --~-

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) (B) (C) (D)

I

-~--~-+ 7 --~-~--~-~-- : --~-~--~-·

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

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3 3.5 4 4.5

GO ON TO THE NEXT PAGE.

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

y = x2 2

:

-

4 intersects the ellipse with equation

2

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 ?

+ Y = 1 in how many points? 4 6

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

R

None

One

1 (A) 10

Two Four

(B)

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

.!. 8

(C)

1 5 1 (D) 4

(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-

GO ON TO THE NEXT PAGE.

y

Other ~r-----+---------x

0

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?

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?

f"(c) (B) f"(c) (C) f"(c) (D) f"(c)

(A)

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

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

c

-13-

= 0 and f'(c) > 0 = 0 and f'(c) < 0

> 0 and f'(c) = 0 < 0 and f'(c) = 0

GO ON TO THE NEXT PAGE.

11. The matrix representation for the linear transformation 3

3

T R -> R d

(A)

(B)

(C)

(D)

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

=

(~ ~

nm ff v =

ilien T(v)

=

m m m m

-14-

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

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

Point P lies on .e and has coordinates (-2, 2). If .e is rotated counterclockwise 45° about the

graph of y = 4sin 2x cos (- 2x) would coincide with the graph of which of the following? (A) (B) (C) (D)

y y y y

= = = =

origin, what will be the coordinates of the image

2sin2x 2sin4x 4sin4x - 2cos4x

of P under this rotation?

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

13. Event B happened after event A and before

(C) (0, 2)

event C. The elapsed time between event A and

(D) (0, 2F2")

9

event C was 10 seconds, and the elapsed time 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?

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

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

(C) 106 (D) 103

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-15-

15 20 25 30

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60m

Jar 1

Jar 2

Red

4

5

20m

Black

6

5

p 100m

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?

0

(A) 1 3

(B)

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?

(C) 1 2

(D) ;

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

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

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A

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 8

(A) - 2

1

(B)

2

(C)

2

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?

1

(D) 13

(A)

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

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

61 3

(B) 10

(C) 13.!_ 3 (D) 15

(0, 0) (0, 1) (1, 0)

(1, 1)

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

~b

2

-

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?

4ac = 0, which of the following

statements about the roots of the equation ax 2

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

be true? (A) (B) (C) (D)

The equation has two distinct real roots. The equation has two distinct nonreal roots. The equation has one real root. The equation has one nonreal root.

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

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y

a 11 x + a12 y + a 13 z = b1 a21X +a22Y + ~3Z = b2 a31x +a32Y + ~3Z = b3 24. Let A be the m atrix 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

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

following matrix products? (Note: A1 denotes the transpose of A, and

A- 1 denotes the inverse of A.)

F(t) =

J:f(t)dt.

(A) BA- 1

At which of the labeled points is F decreasing?

(B) BtA- 1

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

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

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c

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

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) 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

(A) a = b

d

c

(B) a = b

d

(C)

c

a+c

c

=

b+d

(D) a + b = c

c

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-20-

d

+d b

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y

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?

4

3

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

-3 -2 -1 0 -1

-2 -3

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

29. The graphs of the linear functions

f

and g

are shown in the xy-plane above. The linear

h(x) = f(g(x)) for all x. What is the value of h(2) ?

function his defmed by

(A) - 4

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-21-

(B)

0

(C) (D)

1 2

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

P(t) = 60- 25cos ( ;

1

31. The integer n has the form shown above, where both p and 2P - 1 are prime numbers

t ).

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

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)

(B) 2p

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

(C) P2 (D) 2p-2

(D) 6 P.M.

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p

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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.

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33. The operation ® is defined on the set

z+

of

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?

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

I. g(x) = 2 +cos x, where x is a real number II. h(x) = ln(l + x), where x > -1

operation ® ? (A) (B) (C) (D)

1 is a right identity but not a left identity. 1 is a left identity but not a right identity. 1 is both a right and a left identity. 1 is neither a right nor a left identity.

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

Neither I nor II I only II only Both I and II

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

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v

8 6

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

4 2

2

4

6

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

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) (B)

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

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-25-

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0 90 H as t he fl u

Has a high temperature

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

Does not have a high temperature

<: 12

Does not. have the t1u

Has a high temperature

the relationship between c and k ?

Does not have

a high temperature

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

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) (B) (C) (D)

(B) k = 5.53e-o.l1c

(C) k = 5.53(1.1tt (D) k = e

(

2 c-5.53) O.Il

0.470 0.572 0.685 0.770

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-26-

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Krishna

y I I I

I I I

I

i5x=1

I

X= - 1<_:

I

I

I

I I I I

I I I

I

I I

-----------j-1

:

y=l

: rV -t----------

Note: Figure not drawn to scale.

1

-4

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

4

40. The .xy-plane above shows the three branches of the graph of y = f(x ) =

ax2 +b , where a, b, 2

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

is the value of c ?

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

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

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-Z7-

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

y

G arden

Note: Figure not drawn to scale. 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?

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) ?

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) (A)

(A) 2p 2 (B) p2 2

(C)

p2 4 2

(D) f!_

8

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c

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)

(_13' 25) 5

(C)

(±~) 3' 9

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) (B) (C) (D)

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-29-

130° 143° 150° 153°

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TEMPERATURE IN A REG ION Jan. --c==:r=:::::J~ Feb. Mar. ---{:::::I:::J~ Apr. ~ May June --cr::::JJuly -{::I:J-

--cn--

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

Aug.

Sept. Oct. Nov. Dec.

Jan. Feb.

Mar. Apr. May June July Aug.

Sept. Oct. Nov. Dec.

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.

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G

47. The graph of y = xex on the .xy-plane has (A) (B) (C) (D)

a vertical asymptote positive slope for all real x a relative maximum an absolute minimum

F

D

E

Given: ( 1 ) ABC is a right triangle. (2) AFGB and ACDE are squares. Prove: FC :BE

Statement -

Reason

-

1. FA ::BA

All sides of a square are congruent.

2. AC :: AE

All sides of a square are 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) (B) (C) (D)

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-31-

SSS SAS SAA ASA

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

(8, 5000)

.s g

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

(1.)

of k for which bk < 0.001 ?

~

~ ~

(A) 30

o ~------------­

o

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

Number of Years

Note: Figure not drawn to scale. 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 gr aph 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

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

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-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.)

Table 1- Answers and Content Categories for the Mathematics: Content Knowledge Practice Test 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 r aw score =

Correct Answer

Content Category

Sequence Number

Correct Answer

Content Category

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

B B

II

c

I

A

Ill

c

Ill

B

I

D

II

D

v

B

IV

D

Ill

D

I I IV IV

D

c c A A B A A B

Ill

c

IV

B

II

c c c

II Ill

B

II

c

I IV

B

v II II II

v

I

D

v

c

Ill

B

I

c

II

A

I

D

v

c

Ill

D

Ill

A B A A B B

v IV IV Ill II Ill

D

Ill

A B

Ill II

D

IV

D

Ill

B

II

c

Ill

D

v

(the number of correct answers) .

-33-

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-Scor e Conversion Table Raw Score

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

Scaled Score 100 100 100

100 100 100 100 100 100 100 100 100 100 100 102 105 109 112 115 118 122 125 128

Your scaled scor e =

Raw Score 23 24

Scaled Score 131 134

Raw Score 46 47

25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

138 140 143 146 149 151 154 156 158 160 162 165 167 169 170 172 174 176 178 180 182

48 49 50

Scaled Score 184 186 189

192 194

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 scor e (Subtract) Your raw score Additional cor rect answers you would have needed to r each the passing scor e

-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 Category

Number of Correct Answers Possible

I

8

II

12

III

14

IV

8

v

8

Number of Incorrect Answers

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 Cor rect Answers for the Mathematics: Content Knowledge Test Sequence Number

Percentage of Examinees Choosing Correct Answer

Sequence Number

Percentage of Examinees 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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