Algebra 2 with Trig Exam Ebook

AALLGGEEBBRRAA 22 WWIITTHH TTRRIIGGOONNOOMMEETTRRYY

eeMMAATTHHIINNSSTTRRUUCCTTIIOONN,, RREEDD HHOOOOKK,, NNYY 1122557711,, ©© 22000099

AALLGGEEBBRRAA 22

WWIITTHH TTRRIIGGOONNOOMMEETTRRYY

VVEERRSSIIOONN 22..00

KKIIRRKK WWEEIILLEERR

eeMMAATTHHIINNSSTTRRUUCCTTIIOONN..CCOOMM

AALLGGEEBBRRAA 22 WWIITTHH TTRRIIGGOONNOOMMEETTRRYY,, SSUUPPPPLLEEMMEENNTTSS –– RREEFFEERREENNCCEE SSHHEEEETT


AALLGGEEBBRRAA 22 WWIITTHH TTRRIIGGOONNOOMMEETTRRYY FFOORRMMUULLAA RREEFFEERREENNCCEE SSHHEEEETT

Area of a Triangle

1sin

2K ab C=

Functions of the Sum of Two Angles

( )

( )

( )

sin sin cos cos sin

cos cos cos sin sin

tan tantan

1 tan tan

A B A B A B

A B A B A B

A BA B

A B

+ = +

+ = -

++ =

-

Functions of the Difference of Two Angles

( )

( )

( )

sin sin cos cos sin

cos cos cos sin sin

tan tantan

1 tan tan

A B A B A B

A B A B A B

A BA B

A B

- = -

- = +

-- =

+

Law of Sines

sin sin sin

a b c

A B C= =

Sum of a Finite Arithmetic Series

( )1

2

n

n

n a aS

+=

Binomial Theorem

( )

( )

0 1 1 2 2 0

0 1 2

0

n n n n n

n n n n n

nn n r r

n r

r

a b C a b C a b C a b C a b

a b C a b

- -

-

=

+ = + + + × × × +

+ =å

Law of Cosines

2 2 2 2 cosa b c bc A= + -

Functions of the Double Angle

2 2

2

2

2

sin 2 2sin cos

cos2 cos sin

cos2 2cos 1

cos2 1 2sin

2 tantan 2

1 tan

A A A

A A A

A A

A A

AA

A

=

= -

= -

= -

=-

Functions of the Half Angle

1 1 cossin

2 2

1 1 coscos

2 2

1 1 costan

2 1 cos

AA

AA

AA

A

-= ±

+= ±

-= ±

+

Sum of a Finite Geometric Series

( )1 1

1

n

n

a rS

r

-=

-

0.1%

The Normal Distribution

Based on Standard Deviation

19.1% 19.1%

15.0%

9.2%

4.4%

1.7%

15.0%

9.2%

4.4%

1.7%

0.1% 0.5% 0.5%

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3



AALLGGEEBBRRAA 22 WWIITTHH TTRRIIGGOONNOOMMEETTRRYY,, SSAAMMPPLLEE EEXXAAMMSS


Algebra 2 with Trigonometry

Sample Test #1

Answer all 39 questions in this examination. Write your answers to the Part I multiple-choice questions

on the separate answer sheet. No partial credit will be allowed on the multiple-choice section.

For Parts II, III, and IV, clearly indicate the necessary steps, including appropriate formula substitutions,

diagrams, graphs, charts, etc. For all questions in these parts, a correct numerical answer with no work shown

will receive only 1 credit.

A reference sheet that you may need to answer some questions in this examination is included.

Scrap paper is not permitted for any part of this examination, but you may use the blank spaces in this

examination as scrap paper. Scrap graph paper is provided at the end of this examination for any question for

which graphing may be helpful but is not required. Any work done on this sheet of scrap graph paper will not be

scored. Write all your work in pen, except graphs and drawings, which should be done in pencil.

Note: A graphing calculator and a straightedge (ruler) must be available for you to use while taking this

examination.



Part I

Answer all 27 questions in this part. Each correct answer will receive 2 credits. No partial credit will

be allowed. For each question, write on the separate answer sheet the numeral preceding the word or

expression that best completes the statement or answers the question. [54]

1. If 2

sin5

d ? then * +cos 2d =

(1) 1

5 (3)

11

25

(2) 17

25 (4)

3

5/

2. If the sum of 4 2i/ and 7 8i/ - were plotted in the complex plane, the result would fall in which of the

following quadrants?

(1) I (3) III

(2) II (4) IV

3. The solution set to the quadratic inequality 24 7 2 0x x- / ~ is

(1) } ’1| 24

x x/ ~ ~ (3) } ’| 3 or 1x x x> / @

(2) } ’| 3 1x x/ > > (4) } ’1| 2 or 4

x x x~ / ‡

4. The largest solution, to the nearest degree, of the equation 25sin 9sin 2 0x x- / ? on the interval

0 360xfl fl~ ~ is which of the following?

(1) 12fl (3) 168fl

(2) 278fl (4) 212fl

5. The smallest root of 4 32 8 0x x- / ? is approximately

(1) 2.5/ (3) 3.8/

(2) 1.7 (4) 4.6



6. A quadratic function of the form 2y ax bx c? - - has a leading coefficient, a, that is positive and a turning

point at * +6, 2/ / . Which of the following represents the range of the quadratic function?

(1) ] +2,/ ¢ (3) * +, 2/¢ /

(2) ] +6,/ ¢ (4) * +6,/ ¢

7. Eight finalists for the New York State math championship are competing. Different prizes are awarded for

the top four finishers. Assuming no ties are possible, in how many ways can these prizes be given out?

(1) 8! (3) 8 4C

(2) 4! (4) 8 4P

8. Which of the following represents the inverse of 5 30y x? / ?

(1) 5 30y x? / - (3) 1

305

y x? -

(2) 1

65

y x? / / (4) 1

65

y x? -

9. The value of sin100fl can be expressed equivalently as

(1) 2sin50fl (3) 2sin50 cos50fl fl

(2) sin50 cos50fl fl- (4) 2 2cos 50 sin 50fl fl/

10. The interquartile range of the data set } ’3, 7,13,14,17, 20 is

(1) 17 (3) 12

(2) 13.5 (4) 10



11. Which of the following quadratics would have roots that sum to 3/ ?

(1) 23 2 1 0x x/ - ? (3)

25 15 2 0x x- / ?

(2) 2 3 8 0x x/ - ? (4)

22 8 1 0x x- / ?

12. Which of the following represents the range of the function * +4sin 6y x? / - ?

(1) * +2,10 (3) ] _10, 2/

(2) ] _2,10 (4) * +10, 2/

13. A function, * +y f x? , has a y-intercept of 7. What is the y-intercept of the function * +3 10y f x? / ?

(1) 11 (3) 9/

(2) 2/ (4) 10/

14. An angle is drawn in standard position. If its terminal ray lies in the lies in the second quadrant and

intersects the unit circle at an x-coordinate of 1

4x ? / , then the y-coordinate of intersection is

(1) 3

4y ? (3)

15

4y ?

(2) 3

2y ? / (4)

1

2y ? /

15. In the triangle shown below, what is the measure of the smallest angle, to the nearest degree?

(1) 29fl (3) 76fl

(2) 47fl (4) 22fl

24

18 12



16. On a multiple choice test with 4 choices per question, which of the following gives the probability of getting

exactly seven out of ten questions correct?

(1)

7 3

10 7

1 3

4 4PÃ Ô Ã ÔÄ Õ Ä ÕÅ Ö Å Ö

(3)

1 3

10 7

7 3

10 10C

Ã Ô Ã ÔÄ Õ Ä ÕÅ Ö Å Ö

(2)

7 3

10 7

1 3

4 4C

Ã Ô Ã ÔÄ Õ Ä ÕÅ Ö Å Ö

(4)

3 7

10 7

1 3

4 4PÃ Ô Ã ÔÄ Õ Ä ÕÅ Ö Å Ö

17. For an angle A where 90 180Afl fl> > it is given that 5

sin3

A ? . Which of the following is the value of

cos A ?

(1) 4

9 (3)

3

2/

(2) 2

3/ (4)

3

3

18. Which of the following sets gives the x-coordinates where the parabola 2 4 50y x x? - / and the line

7 10y x? / intersect when drawn in the coordinate plane?

(1) } ’5, 8/ (3) } ’10, 4/

(2) } ’2,12/ (4) } ’0, 25

19. In which of the following situations would it be most appropriate to use an entire population instead of a

sample?

(1) when determining the average age of teachers in a school district

(2) when determining the average weight of freshmen in American high schools

(3) when determining the average speed of drivers on an interstate highway

(4) when determining the average thickness of the ozone layer surrounding the planet

20. In MNPF , 30m N flæ ? , 10MN ? and 6MP ? . Given this information angle P could be

(1) acute only (3) acute or obtuse

(2) right only (4) obtuse only



21. The complex fraction

1 3

21 1

2 2

x

x

-

/ can be simplified as

(1) 6 1x

x

- (3)

6

1

x

x

-/

(2) 6 x

x

- (4)

3x

x

-

22. For an arithmetic sequence whose third term is 14 and whose common difference is four, which of the

following gives the value of the 20th

term?

(1) 90 (3) 86

(2) 76 (4) 82

23. Which of the following is the value of 3

1

2 j

j?/Â ?

(1) 12 (3) 3

48

(2) 1

215 (4) 1

25

24. If the graph of * +5sin 3y x r? / was compared to the graph of * +5sin 3y x? then the phase shift would be

which of the following?

(1) r (3) 3

r

(2) 5r (4) 6

r/

25. Which of the following values of x solves: 2 16 36x x/ /? ?

(1) 4/ (3) 14

(2) 13

(4) 4



26. Which of the following inequalities represents the graph shown below?

(1) 2 4y x@ / (3) 24y x~ /

(2) 24y x> / (4) 2 4y x‡ -

27. The solution set to the equation 10

3xx

/ ? is

(1) } ’2, 5/ (3) } ’0, 6

(2) } ’1,10/ (4) } ’4, 2/

Part II

Answer all 8 questions in this part. Each correct answer will receive 2 credits. Clearly indicate the

necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all

questions in this part, a correct numerical answer with no work shown will receive only 1 credit. [16]

28. Benjamin is choosing three jelly filled donut holes out of a box. If the box contains eight strawberry and six

blueberry filled donuts, find the probability that Benjamin will choose two strawberry and one blueberry by

randomly choosing his three donuts.

29. Find the following product in simplest form:

* +* +* +* +5 5 2 2x x x x- / - /

y

x



30. Solve the equation shown below for all values of x in simplest radical form.

2 2 6 0x x/ / ?

31. Find the sum of the complex numbers 4 i/ - and 2 3i- . Express your answer in a bi- form and then plot

the result on the complex plane below.

32. Explain why the triangle shown below cannot exist.

33. For a particular real number a and base b it is known that log 2.75b a ? . Determine the value of * +3logb a .

Imaginary

Real

18 8

30fl



34. The graph below can be described by an equation of the form * +cosy A x B? - . Determine the values of A

and B and show the calculations that lead to your answers.

35. A drug company found that a sample of 140 people who took their new diet pill lost an average of 5.75

pounds over a two week period. If their results were normally distributed with a standard deviation of 2

pounds, then approximately how many people lost more than 8.75 pounds?

Part III




36. In quadrilateral ABCD it is known that 16AB ? , 24BC ? , 36CD ? , 32m C flæ ? and 64m ABD flæ ? .

Determine, to the nearest tenth the length of AD .

y

x

3 2r/ 2r/ 2r/r/ 2r 3 2r 2r r

D

B

C

A

36

32fl

64fl

24

16



37. Combine and simplify the following subtraction.

2 2

8 6

2 3 3x x x x/

- / -

38. Solve the following equation for all value(s) of x: 3 9 2x x- / ? .

Part IV

Answer the question from this part. The correct answer will receive 6 credits. Clearly indicate the

necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. A correct

numerical answer with no work shown will receive only 1 credit. [6]

39. Algebraically determine the intersection point(s) of the two logarithmic functions given below.

* +3 3log 6 and 3 logy x y x? / ? /




Area of a Triangle

1sin

2K ab C=


( )

( )

( )

sin sin cos cos sin

cos cos cos sin sin

tan tantan

1 tan tan

A B A B A B

A B A B A B

A BA B

A B

+ = +

+ = -

++ =

-


( )

( )

( )

sin sin cos cos sin

cos cos cos sin sin

tan tantan

1 tan tan

A B A B A B

A B A B A B

A BA B

A B

- = -

- = +

-- =

+

Law of Sines

sin sin sin

a b c

A B C= =


( )1

2

n

n

n a aS

+=

Binomial Theorem

( )

( )

0 1 1 2 2 0

0 1 2

0

n n n n n

n n n n n

nn n r r

n r

r


a b C a b

- -

-

=

+ = + + + × × × +

+ =å

Law of Cosines

2 2 2 2 cosa b c bc A= + -


2 2

2

2

2

sin 2 2sin cos

cos2 cos sin

cos2 2cos 1

cos2 1 2sin

2 tantan 2

1 tan

A A A

A A A

A A

A A

AA

A

=

= -

= -

= -

=-


1 1 cossin

2 2

1 1 coscos

2 2

1 1 costan

2 1 cos

AA

AA

AA

A

-= ±

+= ±

-= ±

+


( )1 1

1

n

n

a rS

r

-=

-

0.1%



19.1% 19.1%

15.0%

9.2%

4.4%

1.7%

15.0%

9.2%

4.4%

1.7%

0.1% 0.5% 0.5%

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3



ALGEBRA 2 WITH TRIGONOMETRY PRACTICE EXAM

ANSWER SHEET

Student ……………………………………………………………….

Teacher ………………………………………………………………. School ……………………………………

Your answers to Part I should be recorded on this answer sheet.

Part I

Answer all 27 questions in this part.

1 …………….….. 8 ………….…….. 15 ……………….. 22 ………………..

2 ……………….. 9 ……………….. 16 ……………….. 23 ………………..

3 ……………….. 10 ……………….. 17 ……………….. 24 ………………..

4 ……………….. 11 ……………….. 18 ……………….. 25 ………………..

5 ……………….. 12 ……………….. 19 ……………….. 26 ………………..

6 ……………….. 13 ……………….. 20 ……………….. 27 ………………..

7 ……………….. 14 ……………….. 21 ………………..

Your answers for Parts II, III, and IV should be written in the test booklet.




Sample Test #2

Answer all 39 questions in this examination. Write your answers to the Part I multiple-choice

questions on the separate answer sheet. No partial credit will be allowed on the multiple-choice section.

For Parts II, III, and IV, clearly indicate the necessary steps, including appropriate formula

substitutions, diagrams, graphs, charts, etc. For all questions in these parts, a correct numerical answer

with no work shown will receive only 1 credit.


Scrap paper is not permitted for any part of this examination, but you may use the blank spaces in

this examination as scrap paper. Scrap graph paper is provided at the end of this examination for any

question for which graphing may be helpful but is not required. Any work done on this sheet of scrap

graph paper will not be scored. Write all your work in pen, except graphs and drawings, which should be

done in pencil.

Note: A graphing calculator and a straightedge (ruler) must be available for you to use while

taking this examination.



Part I




1. A geometric sequence is defined by 1 2a ? and 3r ? . Which term has a value of 39,366?

(1) 8th

(3) 9

th

(2) 10th

(4) 11

th

2. Written in simplest radical form 147/ is equal to

(1) 7 3/ (3) 3 7i

(2) 3 7/ (4) 7 3i

3. A circle whose diameter is 4 inches has a central angle measuring 8

r radians that intersects its

circumference. Which of the following gives the length of the arc that is subtended by this angle?

(1) 4

r inches (3)

2

r inches

(2) 4

r inches (4)

32

r inches

4. The rational expression 2

2

2 9 5

3 13 10

x x

x x

/ // /

can be simplified to

(1) 2 3

7

x

x

-/

(3) 5

3 1

x

x

/-

(2) 2 5

3 2

x

x

/-

(4) 2 1

3 2

x

x

--

5. Which of the following is the equation of the line that passes through * +2, 7/ / and * +6, 5 ?

(1) 3

42

y x? / (3) 2

53

y x? / -

(2) 2

73

y x? / (4) 3

12

y x? -



6. A survey of 550 adult male weights had a normal distribution with a mean of 210 pounds and a standard

deviation of 24 pounds. Approximately what number of weights were between 198 and 246 pounds?

(1) 425 (3) 216

(2) 343 (4) 298

7. Which of the following models would best fit the data in the table below if it was fit by a power regression?

(1) 4y a x? (3) 2y ax?

(2) a

yx

? (4) y a x?

8. For the sequence defined by 12 1n na a /? - with

1 3a ? , which of the following represents the value of 4a ?

(1) 15 (3) 63

(2) 31 (4) 9

9. Which expression is equivalent to 1 1

1x x/

/?

(1) 2

1

x

x / (3)

2

1

x x

//

(2) 1

2 1x / (4)

2

2 1

1

x

x

//

10. The x-intercepts of the parabola 23 17 6y x x? - / are given by

(1) } ’1 , 76

/ (3) } ’4, 8/

(2) } ’33,2

/ (4) } ’16,3

/

x 6 10 26 42

y 16 18 21 26



11. Which of the following quadratics would have the roots 1 2‒ ?

(1) 2 3 6 0x x- / ? (3)

22 1 0x x- - ?

(2) 2 2 1 0x x- - ? (4)

2 2 1 0x x/ / ?

12.If the probability that it will rain on any given day this week is 80%, which of the following is closest to the

probability it will rain exactly 5 out of the 7 days of this week?

(1) 71% (3) 34%

(2) 28% (4) 12%

13. For * + * +sin 2 cosh x x x? - what is the value of * +30h fl ?

(1) 1 (3) 3

(2) 2 (4) 2 3

14. Which correlation coefficient corresponds to the scatterplot shown below?

(1) 0r ? (3) 1r ? /

(2) 0.8r ? / (4) 0.8r ?

15. Upon completing the square, the trinomial 2 10 2x x- - would be written as

(1) * +25 4x / / (3) * +2

10 78x - /

(2) * +210 5x / / (4) * +2

5 23x - /

y

x



16. When the fractions 5

3x / and

15

3 x/ are summed, the result is

(1) 2

15

9x / (3) 6/

(2) 10

3x

//

(4) 5

17. How many different arrangements are there of all of the letters in the word BOOHOO?

(1) 360 (3) 180

(2) 720 (4) 30

18. In DEFF , 18EF ? , 75m D flæ ? and 20m F flæ ? . To the nearest tenth, the length of DE is

(1) 4.8 (3) 6.4

(2) 5.0 (4) 7.8

19. Which of the following is the exact value of csc60fl?

(1) 1

2 (3)

2 3

3

(2) 5

2 (4)

3

2

20. Written in factored form, the expression 3 2 25 25x x x/ / - is

(1) * +* +* +3 5 1x x x/ - / (3) * +* +* +5 5 1x x x/ - /

(2) * +* +* +25 1 1x x x- - / (4) * +* +* +5 5 1x x x- - -

D

F 18

20fl

E

75fl



21. Which of the following lines is parallel to 3 8y x/ ? and passes through the point * +5, 9/ ?

(1) * +9 3 5y x- ? / - (3) * +9 3 5y x/ ? -

(2) * +9 3 5y x/ ? / - (4) * +9 3 5y x- ? -

22. Which of the following represents the third term of the expansion of

7

2

1x

x

Ã Ô-Ä ÕÅ Ö

?

(1) 3

7

x (3)

4

21

x

(2) 37x (4) 21x

23. In ABCF , 4, 2a b? ? and 6

sin7

A ? . Which of the following must be the value of sin B ?

(1) 3

7 (3)

2

3

(2) 1

2 (4)

12

7

24. Which of the following values of x solves: * +3log 2 5 2x / ? ?

(1) 7 (3) 5/

(2) 6.5 (4) 11/

25. Which of the following sets represents all solutions to the equation 5 7 30x / ? ?

(1) } ’1‒ (3) } ’2,11/

(2) } ’8‒ (4) } ’1,13



26. A quickly decaying radioactive substance loses 12% of its radioactive mass each hour. If a sample of the

substance originally contains 500 grams of radioactive mass, after how many hours, to the nearest hour, will

the sample contain only 50 grams of radioactive mass?

(1) 9 hours (3) 14 hours

(2) 11 hours (4) 18 hours

27. The sum 1 2 3 4

2 3 4 5- - - can be expressed using sigma notation as

(1) 5

2

1

i i?Â (3)

5

1 1i

i

i? /Â

(2) 4

2

2

3i

i

i? -Â (4) 4

1 1i

i

i? -Â

Part II




28. What value(s) of x are not in the domain of * +2

6

2 1

xh x

x x

-?

- /? Justify your answer.

29. Find the larger root of 22 11 3 0x x/ - ? to the nearest tenth.



30. The function * +y f x? is shown below. On the same axes, sketch an accurate graph of * +1

2y f x? .

31. Given that 7

sin3

s ? determine the value of * +cos 2s in simplest form.

32. Given below is the domain } ’5, 0, 5/ . Show how this set gets mapped to a range by the function * + 2f x x?

and explain why this function is not one-to-one.

y

x

-5

0

5

Domain of f Range of f



33. Determine the equation for the inverse of 3

97

y x? - in simplest y ax b? - form.

34. Express the following complex calculation in simplest a bi- form. Show the work that leads to your

answer.

* +* + * +2 3 4 2 4i i i i- / / / -

35. Determine, in simplest form, the period of the equation 4

sin5

y xrÃ Ô? Ä Õ

Å Ö.

Part III




36. In the diagram shown below, points D, E, and F are collinear. If 22m D flæ ? , 52m CEF flæ ? and 25DE ?

then find the length of CF to the nearest tenth.

F D 25 E

C

22fl52fl



37. Giovanni would like to create a flower bouquet for his girlfriend. He visits a florist who has ten red roses,

eight yellow roses, and twelve white roses. If Giovanni chooses three roses at random from the 30 the

florist has, determine the probability that they are all of the same color.

38. Find all complex solutions, in simplest a bi- form, to the following equation.

2

19 81

x x- ?

Part IV




39. Evin is designing a circular garden whose radius is 10 yards. She is planning it out on a coordinate grid, so

places the center of the circle at the origin. She would like to place a path through the garden whose

equation is 2y x? - . First, find an equation of the circle that encloses her garden. Then, algebraically

determine the intersection points of her path and the enclosing circle.




Area of a Triangle

1sin

2K ab C=


( )

( )

( )

sin sin cos cos sin

cos cos cos sin sin

tan tantan

1 tan tan

A B A B A B

A B A B A B

A BA B

A B

+ = +

+ = -

++ =

-


( )

( )

( )

sin sin cos cos sin

cos cos cos sin sin

tan tantan

1 tan tan

A B A B A B

A B A B A B

A BA B

A B

- = -

- = +

-- =

+

Law of Sines

sin sin sin

a b c

A B C= =


( )1

2

n

n

n a aS

+=

Binomial Theorem

( )

( )

0 1 1 2 2 0

0 1 2

0

n n n n n

n n n n n

nn n r r

n r

r


a b C a b

- -

-

=

+ = + + + × × × +

+ =å

Law of Cosines

2 2 2 2 cosa b c bc A= + -


2 2

2

2

2

sin 2 2sin cos

cos2 cos sin

cos2 2cos 1

cos2 1 2sin

2 tantan 2

1 tan

A A A

A A A

A A

A A

AA

A

=

= -

= -

= -

=-


1 1 cossin

2 2

1 1 coscos

2 2

1 1 costan

2 1 cos

AA

AA

AA

A

-= ±

+= ±

-= ±

+


( )1 1

1

n

n

a rS

r

-=

-

0.1%



19.1% 19.1%

15.0%

9.2%

4.4%

1.7%

15.0%

9.2%

4.4%

1.7%

0.1% 0.5% 0.5%

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3




ANSWER SHEET

Student ……………………………………………………………….

Teacher ………………………………………………………………. School ……………………………………


Part I


1 …………….….. 8 ………….…….. 15 ……………….. 22 ………………..

2 ……………….. 9 ……………….. 16 ……………….. 23 ………………..

3 ……………….. 10 ……………….. 17 ……………….. 24 ………………..

4 ……………….. 11 ……………….. 18 ……………….. 25 ………………..

5 ……………….. 12 ……………….. 19 ……………….. 26 ………………..

6 ……………….. 13 ……………….. 20 ……………….. 27 ………………..

7 ……………….. 14 ……………….. 21 ………………..





Sample Test #3

Answer all 39 questions in this examination. Write your answers to the Part I multiple-choice

questions on the separate answer sheet. No partial credit will be allowed on the multiple-choice section.

For Parts II, III, and IV, clearly indicate the necessary steps, including appropriate formula

substitutions, diagrams, graphs, charts, etc. For all questions in these parts, a correct numerical answer

with no work shown will receive only 1 credit.


Scrap paper is not permitted for any part of this examination, but you may use the blank spaces in

this examination as scrap paper. Scrap graph paper is provided at the end of this examination for any

question for which graphing may be helpful but is not required. Any work done on this sheet of scrap

graph paper will not be scored. Write all your work in pen, except graphs and drawings, which should be

done in pencil.

Note: A graphing calculator and a straightedge (ruler) must be available for you to use while

taking this examination.



Part I




1. For 0x @ , the expression

32

14

x

x/

can be rewritten as

(1) 5x (3) 4 7x

(2) 6 2x (4)

3 9x

2. The center of the circle 2 28 4 10x x y y- - / ? is

(1) * +4, 2/ (3) * +8, 4/

(2) * +2, 14/ / (4) * +6, 2/

3. Which of the following is not equal to sin 20fl?

(1) 21 cos 20fl/ (3) 2sin10fl

(2) 2sin10 cos10fl fl (4) sin15 cos5 sin5 cos15fl fl fl fl-

4. A graph of a semicircle is shown below. Which of the following represents the domain of this relationship?

(1) * +5, 3/ (3) * +2, 2/

(2) ] _2, 2/ (4) ] _5, 3/

5. If the first three terms of an arithmetic sequence are given in the list 8, 3, 2,.../ , then the sixth term of the

sequence is

(1) 8 (3) 22/

(2) 7/ (4) 17/

y

x



6. If 2loga x? then the expression * +6

2log 8x can be rewritten as

(1) * +2 4 3a - (3) * +3 2 1a -

(2) 6 8a - (4)

6 3a -

7. In a classroom of 24 students, 14 of whom are boys and 10 of whom are girls, what is the probability that 6

of them selected at random will all be boys?

(1) 39

1748 (3)

13

2185

(2) 49

973 (4)

23

1437

8. In which of the following sets of ordered pairs is y not a function of x?

(1) * + * + * +} ’3, 7 , 5, 9 , 1 1, 7 (3) * + * + * +} ’2, 5 , 5,1 , 2, 7/ /

(2) * + * + * +} ’3, 9 , 0, 0 , 3, 9/ (4) * + * + * +} ’5, 9 , 1, 3 , 7,15/ /

9. An angle that measures 5 radians has a measure, in degrees, most close to which of the following?

(1) 50fl (3) 286fl

(2) 900fl (4) 195fl

10. When the equation 2 72 40 0x b/ ? is solved for x in terms of b, where 0b ‡ , then x ?

(1) 420b b‒ (3)

32 5b b‒

(2) 34 2b b‒ (4) 2 32 10b b‒



11. Which of the four scatterplots shown below would have a correlation coefficient closest to zero?

(1) (3)

(2) (4)

12. Written in its simplest form, the fraction

1 1

8 21

16

xx

x

/

/ is equal to

(1) 2

4x

/-

(3) 4

4x

//

(2) 8

8x / (4)

4

4

x

x

/-

13. An isosceles triangle has legs with lengths of 24 inches and base angles that measure 56fl. To the nearest

square inch, which of the following is the area of this isosceles triangle?

(1) 184 (3) 239

(2) 267 (4) 108

14. Which of the following sets of data will have the smallest variance?

(1) } ’8, 9, 9, 9,10 (3) } ’1, 4, 7,10,14

(2) } ’2, 2, 5, 7, 9/ (4) } ’3, 5, 7, 9,15

y

x

y

x

y

x

y

x



15. Which equation below best describes the graph shown?

(1) 2y x? (3) 2xy ?

(2) * +12

x

y ? (4) 1y x? -

16. A sequence is defined recursively by the formula 1 1n n nt t t- /? © . If

1 21 and 3t t? ? then the value of 5t is


(1) 81 (3) 27

(2) 9 (4) 48

17. Which of the following is not a solution to the equation * +sin cos 1 0x x / ? ?

(1) 0fl (3) 180fl

(2) 90fl (4) 360fl

18. A sinusoidal equation of the form * +siny A x C? - has a maximum y-value of 14 and a minimum y-value of

6/ . Which of the following is the value of C?

(1) 10 (3) 2/

(2) 8/ (4) 4

19. The solution to 2 9 0x / @ can be represented on a number line by

(1) (3)

(2) (4)

y

x

-5 0 5

-5 0 5

-5 0 5

-5 0 5



20. When the imaginary base, i, is raised to the 103rd

power the result is

(1) 1 (3) 1/

(2) i/ (4) i

21. An angle is drawn in standard position with its terminal ray intersecting the circle 2 2 169x y- ? at the point

* +12, 5/ . Which of the following represents the cosine of this angle?

(1) 12

169 (3)

5

13/

(2) 12

13 (4)

5

169/

22. If the parabola 2 4y x? / was shifted two units to the right and seven units down its new equation would be

(1) * +22 11y x? / / (3) * +2

7 8y x? - /

(2) * +22 7y x? - / (4) * +2

7 6y x? / /

23. In an ideal gas, pressure and volume are inversely related. If a container containing an ideal gas has an

initial pressure of 42 pounds per square inch and a volume of 200 cubic inches, what is the gas’s new

volume if the pressure is increased to 56 pounds per square inch?

(1) 150 cubic inches (3) 225 cubic inches

(2) 75 cubic inches (4) 175 cubic inches

24. The summation 5

1

sin2n

nr?

Ã ÔÄ ÕÅ Ö

Â has a value of

(1) 1 (3) 3

(2) 1/ (4) 0



25. Expressed in a bi- form, the quotient 2

3 5i- is equivalent to

(1) 2 2

3 5i- (3)

3 5

17 17i/

(2) 3 5

34 34i- (4)

3 5

2 2i/

26. What is the coefficient of the second term of the expansion of * +53 2x y- ?

(1) 162 (3) 810

(2) 30 (4) 6

27. If * + * +8 and 4 9f x x g x x? / ? - then * +* +2g f ? ?

(1) 15/ (3) 9

(2) 33 (4) 11/

Part II




28. At a local fair, Maxwell is playing a game where he has a 14

probability of winning. If he plays the game

5 times, find the probability that he will win the game exactly once.

29. Solve for all value(s) of x: 2 1 5 0x / / ? .



30. Given the functions * + 2

3

xf x

x?

/ and * + 2g x x? / find a formula, in simplest form, for * +* +f g xc .

31. Solve the equation: * +62 1125

25

xx

// ? .

32. Solve the equation * +4sin 3 0s - ? for all values of s on the interval 0 360sfl fl~ ~ . Round your answers to

the nearest degree.

33. Express the following sum using sigma notion. Use k as your index variable.

1 11 3 9 27

9 3- - - - -



34. Completely factor the following expression:

3 24 12 9 27x x x- / /

35. Find the value of a such that the quadratic shown below will have two equal, rational roots. Justify your

answer. 2 6 1 0ax x- - ?

Part III




36. Solve the inequality shown below. Represent your solution both in set-builder notation and on a number

line.

2 7 17x - ‡



37. Find all complex solutions, in simplest a bi- form, to the following equation.

2

19 81

x x- ?

38. For two angles A and B it is known that 0 90Afl fl> > and 180 270Bfl fl> > . If 5

cos3

A ? and 3

sin5

B ? /

then find the exact value of * +sin A B- .

Part IV




39. Given the right triangle shown below, determine the value for each of the six fundamental trigonometric

functions in simplest form.

sinc ?

cosc ?

tanc ?

secc ?

cscc ?

cotc ?

c

4

3

5




Area of a Triangle

1sin

2K ab C=


( )

( )

( )

sin sin cos cos sin

cos cos cos sin sin

tan tantan

1 tan tan

A B A B A B

A B A B A B

A BA B

A B

+ = +

+ = -

++ =

-


( )

( )

( )

sin sin cos cos sin

cos cos cos sin sin

tan tantan

1 tan tan

A B A B A B

A B A B A B

A BA B

A B

- = -

- = +

-- =

+

Law of Sines

sin sin sin

a b c

A B C= =


( )1

2

n

n

n a aS

+=

Binomial Theorem

( )

( )

0 1 1 2 2 0

0 1 2

0

n n n n n

n n n n n

nn n r r

n r

r


a b C a b

- -

-

=

+ = + + + × × × +

+ =å

Law of Cosines

2 2 2 2 cosa b c bc A= + -


2 2

2

2

2

sin 2 2sin cos

cos2 cos sin

cos2 2cos 1

cos2 1 2sin

2 tantan 2

1 tan

A A A

A A A

A A

A A

AA

A

=

= -

= -

= -

=-


1 1 cossin

2 2

1 1 coscos

2 2

1 1 costan

2 1 cos

AA

AA

AA

A

-= ±

+= ±

-= ±

+


( )1 1

1

n

n

a rS

r

-=

-

0.1%



19.1% 19.1%

15.0%

9.2%

4.4%

1.7%

15.0%

9.2%

4.4%

1.7%

0.1% 0.5% 0.5%

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3




ANSWER SHEET

Student ……………………………………………………………….

Teacher ………………………………………………………………. School ……………………………………


Part I


1 …………….….. 8 ………….…….. 15 ……………….. 22 ………………..

2 ……………….. 9 ……………….. 16 ……………….. 23 ………………..

3 ……………….. 10 ……………….. 17 ……………….. 24 ………………..

4 ……………….. 11 ……………….. 18 ……………….. 25 ………………..

5 ……………….. 12 ……………….. 19 ……………….. 26 ………………..

6 ……………….. 13 ……………….. 20 ……………….. 27 ………………..

7 ……………….. 14 ……………….. 21 ………………..





Sample Test #4



For Parts II, III, and IV, clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in these parts, a correct numerical answer with no work shown will receive only 1 credit.


Scrap paper is not permitted for any part of this examination, but you may use the blank spaces in this examination as scrap paper. Scrap graph paper is provided at the end of this examination for any question for which graphing may be helpful but is not required. Any work done on this sheet of scrap graph paper will not be scored. Write all your work in pen, except graphs and drawings, which should be done in pencil.

Note: A graphing calculator and a straightedge (ruler) must be available for you to use while taking this examination.



Part I




1. Which of the following is not a solution to the equation 24sin 1x ? ?

(1) 30fl (3) 150fl

(2) 210fl (4) 120fl

2. Which of the following represents the middle term of the expansion of * +44x / ?

(1) 296x (3) 416x/

(2) 296x/ (4) 416x 3. Which of the following linear equations best fits the data set shown in the table below?

(1) 2.7 7.1y x? / (3) 4.5 1.2y x? -

(2) 3.2 9.4y x? / (4) 1.8 10.5y x? /

4. All values of x that solve the inequality 2 5x - ‡ are given by the set

(1) } ’| 3 3x x/ ~ ~ (3) } ’| 7 3x x/ ~ ~

(2) } ’| 3 7x x/ > > (4) } ’| 7 or 3x x x~ / ‡

5. When the complex number 4 3i/ is divided by its conjugate, the result can be expressed as

(1) 1/ (3) 1 i/

(2) 7 24

25 25i/ (4)

241

7i/

x 3/ 1 5 8 10

y 20/ 3/ 4 15 25



6. For what value of k will the equation * + * +2 2log 4 log 3 5k / ? be satisfied?

(1) 12 (3) 19 (2) 9 (4) 24

7. The x-intercepts of the parabola 2 10 23y x x? / - are given in exact form by

(1) 5 2‒ (3) 4 3‒

(2) 10 2‒ (4) 7 3‒

8. Which of the following gives the common ratio of the geometric sequence starting with 9, 6, 4, ... ?

(1) 23

r ? (3) 32

r ?

(2) 12

r ? (4) 3r ? /

9. After a reflection in the x-axis, the parabola 2 10y x? / would have the equation

(1) 2 10y x? - (3) 210y x? /

(2) 2 10y x x? / (4) * +210 1y x? / /

10. Accurate to the nearest degree, the third quadrant solution of 5cos 2 0x - ? is

(1) 224fl (3) 114fl

(2) 66fl (4) 246fl



11. On a tropical island, it reliably rains on 60% of all days. Which of the following gives the probability, to the nearest percent, that it will rain exactly 5 out of 7 days when tourists visit it?

(1) 32% (3) 71% (2) 43% (4) 26%

12. A sequence is defined recursively by the formula 15 20k kb b /? / , where 1 2b ? . Based on this definition,

the summation 4

1

k

k

b?Â =?

(1) 370/ (3) 220/

(2) 250 (4) 448/

13. Which of the following represents the solution set of the inequality 2 1

04

x

x

/~

-?

(1) * 14,2×/ Ú (3) * + +1, 4 ,

2Ç/¢ / ̌ ¢É

(2) 14,2

Ç ×/É Ú (4) * _ +1, 4 ,2

Ç/¢ / ̌ ¢É

14. An angle whose radian measure is 8

3

r is drawn in standard position. In what quadrant does its terminal ray

lie?

(1) I (3) III (2) II (4) IV 15. Which of the following functions, when mapping the real numbers to the real numbers, would be considered

one-to-one?

(1) 2y x? (3) y x?

(2) siny x? (4) 2

xy ?



16. Which of the following is not a factor of 4 213 36x x/ - ?

(1) 2x / (3) 3x -

(2) 3x / (4) 6x - 17. Of Mr. Weiler’s 16 math team members, he must pick five of them to be on his A-Team, where the order in

which he selects the five is not important. Which of the following gives the number of different five member teams Mr. Weiler can form?

(1) 5! (3) 16!

(2) 16 5C (4)

16 5P

18. If a quadratic has two roots that are rational and unequal, then its discriminant could be (1) 16 (3) 10 (2) 12/ (4) 0

19. If * + * +5 10 and 2 1f x x g x x? / - ? / then * +* +4f g / ?c ?

(1) 59 (3) 55

(2) 35/ (4) 75/

20. For a given angle c it is known that 1

cos3

c ? . Which of the following could be the value of tan2

c?

(1) 2

3 (3)

2

2

(2) 1

2 (4)

2

3

21. If an angle drawn in standard position intersects the unit circle at * +0.28, 0.96/ then which of the following

represents the sine of this angle?

(1) 0.96 (3) 0.28/

(2) 1.25/ (4) 0.68



22. A triangle whose area is 15 has two sides of length 6 and 10. If the angle created by these two sides is obtuse, then its measure must be

(1) 120fl (3) 30fl

(2) 135fl (4) 150fl

23. Which of the following represents the value of * +lna be ?

(1) b

a (3)

a

b

(2) b

a (4)

bae

24. The expression a b

a b

//

, for a b” , is the same as

(1) 2a ab b- / (3) a b-

(2) a b/ (4) 3a b-

25. Which of the following correlation coefficients shows the best linear fit between two variables?

(1) 0.85r ? (3) 0r ?

(2) 0.55r ? (4) 0.98r ? /

26. Which of the following points falls in the solution set to 2 12y x x> / ?

(1) * +1,13/ (3) * +2, 26/

(2) * +0, 0 (4) * +7, 30/

27. Which of the following equations represents the inverse of * +5log 1y x? / ?

(1) * +5log 1y x? / - (3) 5 1xy ? -

(2) * + 11

5

x

y-

? (4) * +15

log 1y x? -



Part II




28. The following mapping diagram shows how the set A is mapped to set B using the function * + 2f x x? .

Explain why this mapping is one-to-one but not onto. 29. Mr. Richmond’s traffic engineering class is trying to determine people’s attitudes towards their evening

commute. Students in his class decide to stop drivers on their way home to conduct this survey. Why would this survey method introduce bias into their results?

30. Solve the equation below for all values of x. State your answers in simplest a bi- form.

2 10 29 0x x/ - ? 31. Realtors in a Florida community have found that the average price of a four bedroom house varies inversely

with the distance that the house lies away from the beach. If, on average, a four bedroom house located 2 miles from the beach costs $750,000, what is the average cost of a house that lies 5 miles from the beach?

1

2

1

2

3

4

Set A Set B



32. A radar is tracking the location of a plane. Initially the plane is at a distance of 6.5 miles from the radar.

After the radar rotates through an angle of 115fl , the plane is now at a distance of 3.2 miles from the radar. What straight line distance, to the nearest tenth of a mile, has the plane flown between these two readings?

33. If a geometric sequence is defined by a first term of 6/ and a common ratio of 2/ then find the sum of the first ten terms of this sequence.

34. A particularly virulent flu strain is spreading around a town such that the number of people infected doubles

every three days. When the health department first starts to track the disease, only five people were

infected. If the number of people infected, I, is given by the equation * + 35 2d

I ? , where d is the number of

days since the first five were detected, then determine the number of days until 2560 people are infected.

115fl

6.5 miles 3.2 miles



35. A quadratic equation has roots that sum to 8/ and have a product of 3 . Write a quadratic equation that fits this description if its leading coefficient is equal to 2.

Part III




36. At a local PTA meeting, a sample of parents were surveyed to determine how many children they currently

had attending school. Their results are shown in the frequency table below: Determine the mean, median, and standard deviation for this sample. Round

any non-integer answers to the nearest tenth. Determine how many of the 55 families surveyed have a number of children

that was within one standard deviation of the mean.

37. Find all solutions, on the interval 0 360xfl fl~ ~ , that solve 22sin sin 1 0x x/ / ? .

Number of

Children

Number of

Families

1 16

2 24

3 8

4 3

5 2

7 2



38. The following graph shows the function * +y h x? . A new function is defined by the formula

* + * + 2g x h x? / - . Sketch a graph of * +y g x? on the same axes and explain the transformations that have

occurred to the graph of * +h x to produce the graph of * +g x . Specify both the type and the order.

Part IV




39. Simplify the following expression.

2 2 2

2 2 2

4 9 1 2

3 7 2 5 6 4 24

x x x x

x x x x x x

/ / /© „

- - - / -

y

x




Area of a Triangle

1sin

2K ab C=


( )

( )

( )

sin sin cos cos sin

cos cos cos sin sin

tan tantan

1 tan tan

A B A B A B

A B A B A B

A BA B

A B

+ = +

+ = -

++ =

-


( )

( )

( )

sin sin cos cos sin

cos cos cos sin sin

tan tantan

1 tan tan

A B A B A B

A B A B A B

A BA B

A B

- = -

- = +

-- =

+

Law of Sines

sin sin sin

a b c

A B C= =


( )1

2

n

n

n a aS

+=

Binomial Theorem

( )

( )

0 1 1 2 2 0

0 1 2

0

n n n n n

n n n n n

nn n r r

n r

r


a b C a b

- -

-

=

+ = + + + × × × +

+ =å

Law of Cosines

2 2 2 2 cosa b c bc A= + -


2 2

2

2

2

sin 2 2sin cos

cos2 cos sin

cos2 2cos 1

cos2 1 2sin

2 tantan 2

1 tan

A A A

A A A

A A

A A

AA

A

=

= -

= -

= -

=-


1 1 cossin

2 2

1 1 coscos

2 2

1 1 costan

2 1 cos

AA

AA

AA

A

-= ±

+= ±

-= ±

+


( )1 1

1

n

n

a rS

r

-=

-

0.1%



19.1% 19.1%

15.0%

9.2%

4.4%

1.7%

15.0%

9.2%

4.4%

1.7%

0.1% 0.5% 0.5%

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3




ANSWER SHEET

Student ……………………………………………………………….

Teacher ………………………………………………………………. School ……………………………………


Part I


1 …………….….. 8 ………….…….. 15 ……………….. 22 ………………..

2 ……………….. 9 ……………….. 16 ……………….. 23 ………………..

3 ……………….. 10 ……………….. 17 ……………….. 24 ………………..

4 ……………….. 11 ……………….. 18 ……………….. 25 ………………..

5 ……………….. 12 ……………….. 19 ……………….. 26 ………………..

6 ……………….. 13 ……………….. 20 ……………….. 27 ………………..

7 ……………….. 14 ……………….. 21 ………………..





Sample Test #5












examination.



Part I




1. The tip of a pendulum travels through an arc length of 2.5 feet. If the pendulum has a length of 5 feet,

through what radian angle does it rotate?

(1) 2

r (3)

4

r

(2) 1

2 (4)

1

4

2. The area of an equilateral triangle, in simplest radical form, whose side lengths measure 10 is

(1) 50 2 (3) 25 3

(2) 22 7 (4) 100 2

3. Which of the following equations is graphed on the grid shown below?

(1) 3

yx

? (3) 3 1xy ? -

(2) * +1 23

x

y ? - (4) * +14

x

y ?

4. Which of the following values of x solves the equation 23 75 0x - ? ?

(1) 10x ? (3) 5x i?

(2) 5x ? (4) 25x i?

5. If * + 2f x x? and * + 2 1g x x? / then * +* +f g x ?c ?

(1) 22 1x - (3)

24 4 1x x/ -

(2) 22 1x / (4)

24 1x /

y

x



6. The value of sin 228fl can be expressed as a function of a positive, acute angle by

(1) sin 42fl (3) sin 42fl/

(2) sin 48fl (4) sin 48fl/

7. Which equation below describes the graph shown?

(1) 5 1y x? / /

(2) 5 1y x? - /

(3) 5 1y x? / -

(4) 5 1y x? - -

8. Frederick has 8 coins in his pocket, consisting of five quarters and three nickels. If Frederick pulls two

coins out without replacement, what is the probability that they will both be quarters or they will both be

nickels?

(1) 30

28 (3)

13

70

(2) 15

392 (4)

13

28

9. Which of the following is not a factor of the polynomial 3 210 13 24x x x/ - - ?

(1) 3x / (3) 8x /

(2) 6x - (4) 1x -

10. What phase shift would be necessary to map cosy x? onto siny x? if the domain variable is measured in

radians?

(1) r (3) 2

r

(2) 2

r/ (4) r/

y

x



11. Which of the following sets gives the x-coordinates where the parabola 2 4 50y x x? - / and the line

7 10y x? / intersect when drawn in the coordinate plane?

(1) } ’5, 8/ (3) } ’10, 4/

(2) } ’2,12/ (4) } ’0, 25

12. Which of the following sets represents all solutions to the equation 1

9 27x - ? ?

(1) } ’2,1/ (3) } ’32

‒

(2) } ’1‒ (4) } ’12

‒

13. For a given angle c , where 0 360cfl fl~ > , it is known that 2

sin2

c ? and 2

cos2

c ? / . Which of the

following is the exact value of c ?

(1) 45fl (3) 135fl

(2) 225fl (4) 120fl

14. A geometric sequence is defined by 1 2a ? and 3r ? . Which term has a value of 39,366?

(1) 8th

(3) 9

th

(2) 10th

(4) 11

th

15. Which of the following functions represents an onto mapping from the set of real numbers to the set of real

numbers?

(1) * + 2f x x? (3) * + 2f x x?

(2) * +f x x? (4) * + 2xf x ?



16. Upon completing the square, the trinomial 2 10 2x x- - would be written as

(1) * +25 4x / / (3) * +2

10 78x - /

(2) * +210 5x / / (4) * +2

5 23x - /

17. The fraction 2

2

2

4

x x

x

//

reduces to which of the following?

(1) 2

x

x

/-

(3) 2

x

x /

(2) 2

2

x/ (4)

2

x

x

//

18. Which of the following represents the 3rd * +6

4x y/ term of the expansion if written in standard form?

(1) 4 23840x y (3) 3 31280x y/

(2) 4 2256x y/ (4) 3 364x y

19. When drawn on the same grid, the graphs of y x? and 2 3y x? - will intersect at

(1) * +1,1/ (3) * +3, 3

(2) * +12,2

(4) * +4, 4/ /

20. The sum 1 2 3 4

2 3 4 5- - - can be expressed using sigma notation as

(1) 5

2

1

i i?Â (3)

5

1 1i

i

i? /Â

(2) 4

2

2

3i

i

i? -Â (4) 4

1 1i

i

i? -Â



21. The expression 8

3 5/ can be written equivalently as

(1) 8 8 53/ (3) 6 2 5-

(2) 4 3 5/ (4) 5 3 53-

22. If * +2

xf x ? and * + sing x x? then * +* +90g f fl ?c

(1) 2

2 (3)

1

2

(2) 0 (4) 3

2

23. Which of the following equations represents a circle that is centered at the origin and that passes through the

point * +3, 4/ ?

(1) * + * +2 23 4 25x y- - / ? (3) 2 2 25x y- ?

(2) * + * +2 23 4 5x y/ - - ? (4) 2 2 5x y- ?

24. A car’s value decreases by about 2% for every thousand miles that it is driven. If a car has an initial price

given by 0P and has been driven m miles, which of the following equations gives its current value, V?

(1) * + /1000

0 0.98m

V P? (3) 0 0.02V P m? /

(2) * +0 0.98m

V P? (4) * + /1000

0 1.02m

V P?

25. All values of x that solve the inequality 2 5x - ‡ are given by the set

(1) } ’| 3 3x x/ ~ ~ (3) } ’| 7 3x x/ ~ ~

(2) } ’| 3 7x x/ > > (4) } ’| 7 or 3x x x~ / ‡



26. Given the diagram below, then the value of x, to the nearest tenth, is

(1) 32.8 (3) 30.1

(2) 13.4 (4) 18.7

27. The sum * +25

1

3 1n

n?

-Â has a value of

(1) 550 (3) 225

(2) 1000 (4) 325

Part II




28. Solve the following equation for all value(s) of x: 3 4 2 16x - / ? .

29. Explain why the equation 2cos 3 0s / ? fails to have any solutions.

12

22 x

32fl



30. If six runners enter a race, in how many different ways can the three top awards be given out assuming there

can be no ties?

31. Find the values of the second and third terms of a sequence, which is defined recursively by the definition

given below. 2

1 1 13 and 2k k kc c c c/ /? / ? -

32. Maria is sitting on a Ferris wheel such that she rotates around a circular path whose radius is 22 feet.

Through what radian angle does Maria rotate if she travels a distance of 552 feet? Express your answer to

the nearest radian.

33. The division 3 3

3 3

/-

can be placed in the form 3a / . Determine the value of a.



34. Find one value of c such that the following quadratic will have two unequal, rational roots.

2 8 0x x c- - ?

35. Simplify the following complex fraction.

2

1 1

361 1

36 6

x

x

/

-

Part III




36. On the grid below, sketch the function * + 3 5g x x? - / . Then, state the domain and range of this function

using interval notation.

Domain:

Range:

y

x



37. Solve the rational inequality shown below. Represent your solution as a set and on a number line.

22 13 15

07

x x

x

/ -‡

-

38. If a standard six sided die is rolled seven times, find the probability of rolling a number greater than four at

least six times.

Part IV




39. Solve the equation 2cos 2 2cos cosx x x? / for all values of x on the interval 0 360xfl fl~ ~ . Round any non-

integer answers to the nearest degree.




Area of a Triangle

1sin

2K ab C=


( )

( )

( )

sin sin cos cos sin

cos cos cos sin sin

tan tantan

1 tan tan

A B A B A B

A B A B A B

A BA B

A B

+ = +

+ = -

++ =

-


( )

( )

( )

sin sin cos cos sin

cos cos cos sin sin

tan tantan

1 tan tan

A B A B A B

A B A B A B

A BA B

A B

- = -

- = +

-- =

+

Law of Sines

sin sin sin

a b c

A B C= =


( )1

2

n

n

n a aS

+=

Binomial Theorem

( )

( )

0 1 1 2 2 0

0 1 2

0

n n n n n

n n n n n

nn n r r

n r

r


a b C a b

- -

-

=

+ = + + + × × × +

+ =å

Law of Cosines

2 2 2 2 cosa b c bc A= + -


2 2

2

2

2

sin 2 2sin cos

cos2 cos sin

cos2 2cos 1

cos2 1 2sin

2 tantan 2

1 tan

A A A

A A A

A A

A A

AA

A

=

= -

= -

= -

=-


1 1 cossin

2 2

1 1 coscos

2 2

1 1 costan

2 1 cos

AA

AA

AA

A

-= ±

+= ±

-= ±

+


( )1 1

1

n

n

a rS

r

-=

-

0.1%



19.1% 19.1%

15.0%

9.2%

4.4%

1.7%

15.0%

9.2%

4.4%

1.7%

0.1% 0.5% 0.5%

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3




ANSWER SHEET

Student ……………………………………………………………….

Teacher ………………………………………………………………. School ……………………………………


Part I


1 …………….….. 8 ………….…….. 15 ……………….. 22 ………………..

2 ……………….. 9 ……………….. 16 ……………….. 23 ………………..

3 ……………….. 10 ……………….. 17 ……………….. 24 ………………..

4 ……………….. 11 ……………….. 18 ……………….. 25 ………………..

5 ……………….. 12 ……………….. 19 ……………….. 26 ………………..

6 ……………….. 13 ……………….. 20 ……………….. 27 ………………..

7 ……………….. 14 ……………….. 21 ………………..





Practice Exam #1 – Answer Key












examination.



Part I




1. If 2

sin5

b = then ( )cos 2b =

(1) 1

5 (3)

11

25

(2) 17

25 (4)

3

5-

2. If the sum of 4 2i- and 7 8i- + were plotted in the complex plane, the result would fall in which of the

following quadrants?

(1) I (3) III

(2) II (4) IV

3. The solution set to the quadratic inequality 24 7 2 0x x+ - £ is

(1) { }1| 24

x x- £ £ (3) { }| 3 or 1x x x< - >

(2) { }| 3 1x x- < < (4) { }1| 2 or 4

x x x£ - ³

4. The largest solution, to the nearest degree, of the equation 25sin 9sin 2 0x x+ - = on the interval

0 360x° °£ £ is which of the following?

(1) 12° (3) 168°

(2) 278° (4) 212°

5. The smallest root of 4 32 8 0x x+ - = is approximately

(1) 2.5- (3) 3.8-

(2) 1.7 (4) 4.6

(2)

( )

( ) ( )

2

2

cos 2 1 2sin

2 41 2 1 25 25

8 17125 25

b b= -

= - = -

= - =

( )

4 2 7 8 3 6

3, 6 Quadrant II

i i i- + - + = - +

- Þ (2)

( ) ( )4 1 2 0

1 or 24

Now test values to finish.

x x

x x

- + =

= = -

(1)

This problem is most easily solved by

graphing 2

5sin 9sin 2y x x= + - and

then finding its largest x-intercept

using your graphing calculator.

(3)

(1) Graph the function

4 32 8y x x= + - and use your

calculator to find the smallest zero of this

function, which is approximately 2.507...-



6. A quadratic function of the form 2y ax bx c= + + has a leading coefficient, a, that is positive and a turning

point at ( )6, 2- - . Which of the following represents the range of the quadratic function?

(1) [ )2,- ¥ (3) ( ), 2-¥ -

(2) [ )6,- ¥ (4) ( )6,- ¥

7. Eight finalists for the New York State math championship are competing. Different prizes are awarded for

the top four finishers. Assuming no ties are possible, in how many ways can these prizes be given out?

(1) 8! (3) 8 4C

(2) 4! (4) 8 4P

8. Which of the following represents the inverse of 5 30y x= - ?

(1) 5 30y x= - + (3) 1

305

y x= +

(2) 1

65

y x= - - (4) 1

65

y x= +

9. The value of sin100° can be expressed equivalently as

(1) 2sin50° (3) 2sin50 cos50° °

(2) sin50 cos50° °+ (4) 2 2cos 50 sin 50° °-

10. The interquartile range of the data set { }3, 7,13,14,17, 20 is

(1) 17 (3) 12

(2) 13.5 (4) 10

A leading coefficient that is

positive indicates that the

parabola opens upwards and has

a minimum y-value of 2- but no

maximum y-value.

(1)

( ) ( ) ( ) ( )

8 4

1 2 3 4

8 7 6 5

st nd rd th

Tn n n n n

P

= × × ×

= × × × =

(4)

( )

5 30 30 5

1 130 6

5 5

x y x y

y x x

= - Þ + =

= + = + (4)

(3) ( )

( ) ( )

Using sin 2 2sin cos we have:

sin 100 sin 2 50 2sin 50 cos 50

A A A

° ° ° °

=

= × =

1

3

3 1

7 found using the calculator

17 found using calculator

IQR 10

Q

Q

Q Q

=

=

= - =

(4)



11. Which of the following quadratics would have roots that sum to 3- ?

(1) 23 2 1 0x x- + = (3)

25 15 2 0x x+ - =

(2) 2 3 8 0x x- + = (4)

22 8 1 0x x+ - =

12. Which of the following represents the range of the function ( )4sin 6y x= - + ?

(1) ( )2,10 (3) [ ]10, 2-

(2) [ ]2,10 (4) ( )10, 2-

13. A function, ( )y f x= , has a y-intercept of 7. What is the y-intercept of the function ( )3 10y f x= - ?

(1) 11 (3) 9-

(2) 2- (4) 10-

14. An angle is drawn in standard position. If its terminal ray lies in the lies in the second quadrant and

intersects the unit circle at an x-coordinate of 1

4x = - , then the y-coordinate of intersection is

(1) 3

4y = (3)

15

4y =

(2) 3

2y = - (4)

1

2y = -

15. In the triangle shown below, what is the measure of the smallest angle, to the nearest degree?

(1) 29° (3) 76°

(2) 47° (4) 22°

24

18 12

15sum 3

5

b

a

- -= = = - (3)

[ ]

min

max

6 4 2

6 4 10

Range: 2, 10

y

y

= - =

= + = (2)

( )

( )

( )

0 7

3 0 10

3 7 10 11

f

y f

y

=

= -

= - =

(1)

(3)

2

2 2

2

1 11 1

4 16

15 15 15

16 16 4

15 due to Quad. II

4

y y

y y

y

æ ö- + = Þ + =ç ÷è ø

= Þ = ± = ±

=

( ) ( )2 2 212 24 18 2 24 18 cos

144 900 864cos

756cos 29

864

x

x

x x°

= + -

= -

-= Þ »-

(1)



16. On a multiple choice test with 4 choices per question, which of the following gives the probability of getting

exactly seven out of ten questions correct?

(1)

7 3

10 7

1 3

4 4Pæ ö æ öç ÷ ç ÷è ø è ø

(3)

1 3

10 7

7 3

10 10Cæ ö æ öç ÷ ç ÷è ø è ø

(2)

7 3

10 7

1 3

4 4Cæ ö æ öç ÷ ç ÷è ø è ø

(4)

3 7

10 7

1 3

4 4Pæ ö æ öç ÷ ç ÷è ø è ø

17. For an angle A where 90 180A° °< < it is given that 5

sin3

A = . Which of the following is the value of

cos A?

(1) 4

9 (3)

3

2-

(2) 2

3- (4)

3

3

18. Which of the following sets gives the x-coordinates where the parabola 2 4 50y x x= + - and the line

7 10y x= - intersect when drawn in the coordinate plane?

(1) { }5, 8- (3) { }10, 4-

(2) { }2,12- (4) { }0, 25

19. In which of the following situations would it be most appropriate to use an entire population instead of a

sample?

(1) when determining the average age of teachers in a school district

(2) when determining the average weight of freshmen in American high schools

(3) when determining the average speed of drivers on an interstate highway

(4) when determining the average thickness of the ozone layer surrounding the planet

20. In MNPD , 30m N °Ð = , 10MN = and 6MP = . Given this information angle P could be

(1) acute only (3) acute or obtuse

(2) right only (4) obtuse only

( ) ( )37

10 7

Binomial probability with:

31 , , 10, and 74 4

314 4

p q n r

C

= = = =

(2)

(2)

2

2

2 2

5cos 1

3

5 4cos 1 cos

9 9

4 2cos

9 3

2Quad II cos

3

A

A A

A

A

æ ö+ =ç ÷ç ÷è ø

+ = Þ =

= ± = ±

Þ = -

( ) ( )

2

2

4 50 7 10

3 40 0

8 5 0

8 or 5

x x x

x x

x x

x x

+ - = -

- - =

- + =

= = -

(1)

(1)

It is only appropriate to use the entire population when we can be certain that every member of the population will be

surveyed. In all other choices, surveying the entire population is not possible because of its size.

sin sin 30sin 0.833...

10 6

56.4 or 123.6

Both work with 30 .

PP

P

°

° °

°

= Þ =

» (3)



21. The complex fraction

1 3

21 1

2 2

x

x

+

- can be simplified as

(1) 6 1x

x

+ (3)

6

1

x

x

+

-

(2) 6 x

x

+ (4)

3x

x

+

22. For an arithmetic sequence whose third term is 14 and whose common difference is four, which of the

following gives the value of the 20th

term?

(1) 90 (3) 86

(2) 76 (4) 82

23. Which of the following is the value of 3

1

2 j

j=-å ?

(1) 12 (3) 3

48

(2) 1

215 (4) 1

25

24. If the graph of ( )5sin 3y x p= - was compared to the graph of ( )5sin 3y x= then the phase shift would be


(1) p (3) 3

p

(2) 5p (4) 6

p-

25. Which of the following values of x solves: 2 16 36x x- -= ?

(1) 4- (3) 14

(2) 13

(4) 4

(3)

( )

( )

1 32

62

1 1 12

2 2

xxx

xx

x

æ ö+ç ÷ +è ø= =

-æ ö-ç ÷

è ø

( ) ( )

2 1

20 1

14 4 10 10 4 6

19 6 19 4 82

a a

a a d

= - = Þ = - =

= + = + =

(4)

1 0 1 2 3

12

2 2 2 2 2

1 1 2 4 8 152

-= + + + +

= + + + + =

(2)

( )5sin 3

5sin 33

y x

x

p

p

= -

æ öæ ö= -ç ÷ç ÷

è øè ø

(3)

(3)

( )2 1 2 2 1 26 6 6 6

12 1 2 4 14

xx x x

x x x x

-- - -= Þ =

- = - Þ = Þ =



26. Which of the following inequalities represents the graph shown below?

(1) 2 4y x> - (3) 24y x£ -

(2) 24y x< - (4) 2 4y x³ +

27. The solution set to the equation 10

3xx

- = is

(1) { }2, 5- (3) { }0, 6

(2) { }1,10- (4) { }4, 2-

Part II




28. Benjamin is choosing three jelly filled donut holes out of a box. If the box contains eight strawberry and six

blueberry filled donuts, find the probability that Benjamin will choose two strawberry and one blueberry by

randomly choosing his three donuts.

29. Find the following product in simplest form:

( )( )( )( )5 5 2 2x x x x+ - + -

Since the parabola opens

downward 0a < . As well,

since the curve is dashed,

the equality is not

included.

(2)

y

x

(1)

( )

( ) ( )

2 2

103

10 3 3 10 0

2 5 0 2 or 5

x x xx

x x x x

x x x x

æ ö- =ç ÷

è ø

- = Þ - - =

+ - = Þ = - =

( ) ( ) ( )( )

8 2 6 1

14 3

two strawberry one blueberry 28 6 168 6two strawberry and one blueberry or

total sets of 3 donuts 364 364 13

n n C CP

n C

× × ×= = = =

( ) ( )2 2

4 2 2

4 2

25 4

4 25 100

29 100

x x

x x x

x x

= - -

= - - +

= - +



30. Solve the equation shown below for all values of x in simplest radical form.

2 2 6 0x x- - =

31. Find the sum of the complex numbers 4 i- + and 2 3i+ . Express your answer in a bi+ form and then plot

the result on the complex plane below.

32. Explain why the triangle shown below cannot exist.

33. For a particular real number a and base b it is known that log 2.75b a = . Determine the value of ( )3logb a .

Imaginary

Real

18 8

30°

( ) ( ) ( ) ( )( )

22 2 4 1 6 2 28

2 1 2

22 4 7 2 2 7

2 2

x- - ± - - - ±

= =

± × ±= = =

( )1 7

2

±

1 7= ±

( )

4 2 3

2 4

2, 4

i i

i

= - + + +

= - +

-

sin sin 30

18 8

18sin 30sin 1.125

8

But 1 sin 1.

Thus this triangle cannot exist.

x

x

x

°

°

=

= =

- £ £

( ) ( )3log 3log 3 2.75 8.25

b ba a= = =



34. The graph below can be described by an equation of the form ( )cosy A x B= + . Determine the values of A

and B and show the calculations that lead to your answers.

35. A drug company found that a sample of 140 people who took their new diet pill lost an average of 5.75

pounds over a two week period. If their results were normally distributed with a standard deviation of 2

pounds, then approximately how many people lost more than 8.75 pounds?

Part III




36. In quadrilateral ABCD it is known that 16AB = , 24BC = , 36CD = , 32m C °Ð = and 64m ABD °Ð = .

Determine, to the nearest tenth the length of AD .

y

x

3 2p- 2p- 2p-p- 2p 3 2p 2p p

D

B

C

A

36

32°

64°

24

16

( )

min max

max min

min max

1 and 4

4 12.5

2 2

1 41.5

2 2

y y

y yA

y yB

= - =

- --= = =

+ - += = =

( ) ( )

8.75 5.751.5 's above the mean

2

Percent greater 4.4 1.7 0.5 0.1 6.7%

Number of people 0.067 140 9.38 9

s-

=

= + + + =

= = »

( ) ( )

( ) ( )

2 2 2

2

2 2 2

2

36 24 2 36 24 cos 32

406.572... 406.572... 20.163...

16 2 16 cos 64

379.719... 379.719... 19.486... 19.5

x

x x

y x x

y y

°

°

= + -

= Þ = =

= + -

= Þ = = »

x y



37. Combine and simplify the following subtraction.

2 2

8 6

2 3 3x x x x-

+ - +

38. Solve the following equation for all value(s) of x: 3 9 2x x+ - = .

Part IV




39. Algebraically determine the intersection point(s) of the two logarithmic functions given below.

( )3 3log 6 and 3 logy x y x= - = -

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )( )

8 6

3 1 3

8 1 6

3 1 1 3

8 6 6

3 1 3 1

2 32 6

3 1

x x x x

x x

x x x x x x

x x

x x x x x x

xx

x x x

= -+ - +

-= × - ×

+ - - +

-= -

+ - + -

++= =

+ - ( )3x x + ( ) ( )2

11 x xx=

--

( )

3 2 9

3 2 9

3 2 9 or 3 2 9

3 12 or 6

4 or 6

x x

x x

x x x x

x x

x x

+ = +

+ = ± +

+ = - - + = +

= - - =

= - = -

Be sure to check and reject!

( ) ( )

( )( ) ( )

( ) ( )

3 3 3 3

3

3

2 2

log 6 3 log log 6 log 3

log 6 3 6 3

6 27 6 27 0

3 9 0 3

x x x x

x x x x

x x x x

x x x

- = - Þ - + =

- = Þ - =

- = Þ - - =

+ - = Þ = - or 9

3 is an extraneous root and must be rejected

x

x

=

= -

( ) ( )

( )

3 3

We still need the coordinate of intersection:

log 9 6 log 3 1

Intersection at: 9, 1

y

y

-

= - = =




ANSWER SHEET

Student ……………………………………………………………….

Teacher ………………………………………………………………. School ……………………………………


Part I


1 …………….….. 8 ………….…….. 15 ……………….. 22 ………………..

2 ……………….. 9 ……………….. 16 ……………….. 23 ………………..

3 ……………….. 10 ……………….. 17 ……………….. 24 ………………..

4 ……………….. 11 ……………….. 18 ……………….. 25 ………………..

5 ……………….. 12 ……………….. 19 ……………….. 26 ………………..

6 ……………….. 13 ……………….. 20 ……………….. 27 ………………..

7 ……………….. 14 ……………….. 21 ………………..


(2)

(2)

(1)

(3)

(1)

(1)

(4)

(4)

(3)

(4)

(3)

(2)

(1)

(3)

(1)

(2)

(2)

(1)

(1)

(3)

(3)

(4)

(2)

(3)

(3)

(2)

(1)

Answer Key

(3)

Documents

Algebra 2 with Trig Exam Ebook