Name: Date: Final Exam Review Mrs. Boughen · 2019-01-08 · Final Exam Review Mrs. Boughen 1....

Name: ________________________

Date: _________________________

Pre-Calculus 30

Final Exam Review

Mrs. Boughen

Multiple Choice

Identify the choice that best completes the statement or answers the question.

____ 1. The two functions in the graph shown are reflections of each other. Select the type of reflection(s).

A a reflection in the line y = x C a reflection in the y-axis

B a reflection in the x-axis D a reflection in the x-axis and the y-axis

____ 2. Compared to the graph of the base function f(x) = x 2 , the graph of the function g(x) = x − 4( )2

+ 9 is

translated

A 4 units to the left and 9 units down C 9 units to the left and 4 units down

B 4 units to the right and 9 units up D 9 units to the right and 4 units up

Name: ________________________ ID: A

____ 3. When the function f x( ) = x| | is transformed to f x( ) = −4 x + 3| | + 2, the graph looks like

Name: ________________________ ID: A

____ 4. In the following graph, which transformations must be applied to the blue curve to obtain the red curve?

A a reflection in the x-axis, a vertical translation of 3 units up, and a horizontal translation

of 4 units to the right

B a reflection in the x-axis, a vertical translation of 4 units up, and a horizontal translation

of 3 units to the right

C a reflection in the x-axis, a vertical translation of 3 units down, and a horizontal

translation of 4 units to the right

D a reflection in the x-axis, a vertical translation of 4 units up, and a horizontal translation

of 3 units to the left

Name: ________________________ ID: A

____ 5. Which graph represents the inverse of the graph shown?

____ 6. Which of the following functions is the correct inverse for the function f(x) = 3x + 5?

A f −1(x) = 1

3C f −1(x)= −

B f −1(x) = 1

3D f −1(x) = −

____ 7. Which of the following functions is the correct inverse for the function f(x) = x − 2 , {x | x ≥ 0, x ∈ R}?

A f −1(x) = x 2 + 2 C f −1(x) = x + 2

B f −1(x) = x − 2( )2

D f −1(x) = x + 2

Name: ________________________ ID: A

____ 8. Which is the graph of the square root of the function f(x) = (x − 5)2

− 2?

____ 9. What is the solution to the radical equation 0 = 2 2(x + 4) − 8?

A –4 C 12

B 4 D 128

Name: ________________________ ID: A

____ 10. Which graph represents an odd-degree polynomial function with two x-intercepts?

____ 11. Which of the following is a polynomial function?

A g x( ) = x + 4 C y = −4x4

− 7x2

B f x( ) = −4x

− 7 D y =−4x + 9

____ 12. How many x-intercepts are possible for the polynomial function h x( ) = ax5

+ cx3?

A 4 C 3

B 1 D 5

____ 13. What is the remainder when x4

− 8x2

+ 9x + 8 is divided by x + 6?

A 1070 C –962

B –1070 D 962

____ 14. If −2x3

− 6x2

+ 5x − 7 is divided by x − 7 to give a quotient of −2x2

− 20x − 135 and

a remainder of –952, then which of the following is true?

A x − 7( )(−2x2

− 20x − 135) = 952

B x − 7( )(−2x2

− 20x − 135) = –952

C −2x3

− 6x2

+ 5x − 7 = x − 7( )(−2x2

− 20x − 135) – 952

D −2x3

− 6x2

+ 5x − 7= x − 7( )(−2x2

− 20x − 135) + 952

Name: ________________________ ID: A

____ 15. Which of the following graphs of polynomial functions corresponds to a polynomial equation with zeros –6

(multiplicity of 2) and –1 (multiplicity of 2)?

Name: ________________________ ID: A

____ 16. Which of the following graphs of polynomial functions corresponds to a cubic polynomial equation with

roots 4, 1, and 3?

____ 17. Which of the following angles, in degrees, is coterminal with, but not equal to, 6

5π radians?

A 486° C 576°

B 396° D 216°

____ 18. Determine the arc length of a circle with radius 5.5 cm if it is subtended by a central angle of 5

2π radians.

Round your answer to one decimal place.

A 6.9 cm C 4.4 cm

B 43.2 cm D 1.4 cm

____ 19. If the angle θ is 1600° in standard position, in which quadrant does it terminate?

A quadrant II C quadrant IV

B quadrant III D quadrant I

Name: ________________________ ID: A

____ 20. Which graph represents an angle in standard position with a measure of 5

8π rad?

____ 21. 4

3π radians is equal to how many degrees?

A 150° C 330°

B 240° D 420°

____ 22. Which point on the unit circle corresponds to tan θ = 3?

A (−1

2, 3 ) C (−

B (−1

2) D ( 3 ,−

____ 23. Identify the point on the unit circle corresponding to an angle of 300° in standard position.

A ( − 3 , −3

2) C (

2, −

2, − 3 ) D (−

Name: ________________________ ID: A

____ 24. The amplitude and period (in degrees) of y = −2sin 5x are

A amplitude = −1

period = 90°

C amplitude = –2

period = 90°

B amplitude = 2

period = 72°

D amplitude = 1

period = 72°

____ 25. Which function, where x is in radians, is represented by the graph shown below?

A y = cos x C y = sin x

B y = −sin x D y = −cos x

____ 26. Give an equation for a transformed sine function with an amplitude of 5

8, a period of

5, a phase shift of

rad to the right, and a vertical translation of 9 units down.

A y = 5

2π x + 3 / 2( ) – 9 C y =

2π x − 3 / 2( ) – 9

B y = 5

8 sin 5 / 2 x − 3 / 2( )

ÈÎÍÍÍ

˘˚˙̇˙ − 9 D y =

8 sin 5 / 2 x + 3 / 2( )

ÈÎÍÍÍ

˘˚˙̇˙ − 9

____ 27. The graph of y = sin x can be obtained by translating the graph of y = cos x

3 units to the right C π units to the right

4 units to the right D

2 units to the right

____ 28. What are the solutions for sin2x −

2 = 0 in the interval 0° ≤ x ≤ 360°?

A x = 30° and 210° and 135° C x = 60° and 240° and 45°

B x = 45° and 225° and 315° and 135° D x = 90° and 270° and 225°

____ 29. What does the expression cot θsinθcsc θ simplify to?

A cot θ C tanθ

B sinθ D cos θ

____ 30. Which expression is equivalent to sec 2θ sin2 θ?

A 1 C sin4 θ

B cos2θ D tan2θ

Name: ________________________ ID: A

____ 31. Determine the exact value of cos 15°.

A 2 2 C6 − 2

____ 32. A colony of ants has an initial population of 750 and triples every day. Which function can be used to model

the ant population, p, after t days?

A p t( ) = 3 750( )t

C p t( ) =1

3750( )

B p t( ) = 7501

ÁÁÁÁÁÁ

˜̃˜̃˜̃

D p t( ) = 750 3( )t

____ 33. Which function results when the graph of the function y = 4x is reflected in the y-axis, compressed vertically

by a factor of 1

5, and shifted 2 units down?

A y =1

x+ 2 C y =

−x+ 2

B y =1

x− 2 D y =

−x− 2

____ 34. To the nearest year, how long would an investment need to be left in the bank at 5%, compounded annually,

for the investment to triple?

A 15 years C 26 years

B 23 years D 28 years

____ 35. Another way of writing 55

= 3125 is

A log53125 = 5 C log5 5 = 3125

B log3125

5 = 5 D log55 = 3125

____ 36. Evaluate log465 536.

A 0.13 C 4096

B 5.33 D 8

____ 37. What is the equation for the asymptote of the function f(x) = − log7[−5(x + 2)] − 3?

A x = –5 C x = –2

B x = –3 D x = 2

____ 38. Which of the following is equivalent to the expression log4s + 7log4v + log4 z?

A log4 sv7z C 7log4 svz

B log4 sz + log28v D log4 7svz

Name: ________________________ ID: A

____ 39. Solve log3 x = log32 + log3 3.

A x = 2

3C x = 6

B x = 3

2D x = 729

____ 40. A graduate student determines that the relationship between the length, s, in metres, of the skull and the body

mass, m, in kilograms, of a particular species can be expressed using the equation

3.6045logs = logm − 3.4425. Determine the body mass of an animal with a skull size of 0.56 m. Round your

answer to the nearest kilogram.

A 547 kg C 376 kg

B 343 kg D 313 kg

____ 41. Solve 102x − 5

= 7x + 4

. Round your answer to two decimal places.

A 3.06 C 7.26

B 2.95 D –1.40

____ 42. Which graph represents the function f(x) =−4

x − 9− 5?

Name: ________________________ ID: A

____ 43. Which rational function has the following characteristics?

• a vertical asymptote with equation x = 2

• a horizontal asymptote with equation y = 8

• a point of discontinuity at (3, 9)

A f x( ) =x − 3

(x − 3)(x − 2)+ 8 C f x( ) =

(x + 2)(x + 3)+ 8

B f x( ) =x − 2

(x − 3)(x + 2)+ 8 D f x( ) = −

x − 3

(x + 3)(x − 2)− 8

____ 44. Which graph of a rational function has the following characteristics?

• a vertical asymptote with equation x = −1

• a horizontal asymptote with equation y = −1

• a point of discontinuity at (−2, − 2)

Name: ________________________ ID: A

____ 45. Which function has a point of discontinuity at x = 3?

A f(x) =x − 3

− 6x − 12C f(x) =

− 6x − 12

B f(x) =x − 3

− 2x − 12D f(x) =

− 6x + 9

____ 46. Given the functions f x( ) = x2

− 3 and g x( ) = −9 − x , determine the equation for the combined function

y = f x( ) + g x( ) .

A y = x2

− x + 6 C y = x2

− 27x − 12

B y = x2

+ 27x + 6 D y = x2

− x − 12

− 8 and g x( ) = −5 − x , determine an equation for the combined function

h x( ) = f(g x( )).

A y = x2

+ 10x + 17 C y = −x2

− 13

B y = x2

− 10x + 3 D y = −x2

− 4 and g x( ) = x − 4, a graph of the combined function h x( ) =f x( )

g x( ) most

likely resembles

Name: ________________________ ID: A

− 81x and g x( ) = x + 9, determine the simplified equation for the combined

function y =f x( )

g x( ).

A x x − 9( ) , x ≠ −9 Cx

x − 9, x ≠ −9, 9

x + 9, x ≠ −9 D x x + 9( ) , x ≠ −9

____ 50. Given the function f x( ) = x2

− 5, determine the value of f(f(−1)).

A –4 C 16

B 11 D –8

____ 51. How many three-digit numbers with no repeating digits can be formed using the digits 0, 1, 2, 8, and 9?

A 100 C 60

B 125 D 48

____ 52. Evaluate 15 P8 .

A 32 432 400 C 163 459 296 000

B 6435 D 259 459 200

____ 53. The number of different ways that 9 bikes can be locked in a bike rack is

A 362 880 C 20 160

B 40 320 D 3 628 800

____ 54. For a mock United Nations, 6 boys and 7 girls are to be chosen. If there are 12 boys and 9 girls to choose

from, how many groups are possible?

A 846 720 C 33 264

B 960 D 120 708 403 200

____ 55. The leadership committee at a high school has 4 grade 10 students, 2 grade 11 students, and 6 grade 12

students. This year, 12 grade 10, 8 grade 11, and 10 grade 12 students applied for the committee. How many

ways are there to select the committee?

A 2 910 600 C 733

B 100 590 336 000 D 163 136

____ 56. Rachael has a digital music player that holds 800 songs. She has 1500 songs in her music library. She decides

that her 50 favourite songs must be on the player. Which expression can be used to calculate the number of

ways can she load songs on to the MP3 player so that it is full?

A 1450C750 C 800C50

B 1500C800 D 1500C50

____ 57. How many terms are there in the expansion of (2x − 4y)12

A 6 C 12

B 11 D 13

____ 58. Determine the 5th term in the expansion of (x − 2)12

A 7920x8

C −7920x5

B 202 752x12

D −7920x8

Name: ________________________ ID: A

____ 59. Determine the coefficient, a, for the term ax5y

7 of the binomial expansion of (x + y)

A 3 991 680 C 792

B 60 D 35

Matching

Match each graph of a rational function with its equation.

A f(x) =9

− 4E f(x) =

−3x − 9

B f(x) =1

x + 4F f(x) =

+ 6x + 8

C f(x) = −1

x + 4G f(x) =

(x + 4)2

D f(x) =1

+ 4H f(x) =

−3(x + 4)

____ 1.

____ 2.

Name: ________________________ ID: A

____ 3.

____ 4.

____ 5.

Name: ________________________ ID: A

Short Answer

1. Given the graph of a function, sketch the resulting graph after a reflection in the y-axis

2. Using each graph of y = f(x), sketch the graph of y = f(x) .

Name: ________________________ ID: A

3. a) Use long division to divide x3 + 3x2 – 7 by x + 2. Express the result in quotient form.

b) Identify any restrictions on the variable.

c) Write the corresponding statement that can be used to check the division.

d) Verify your answer.

4. Without using a calculator, determine two angles between 0° and 360° that have a cosecant of −2

Include an explanation of how you determined the two angles.

5. Prove the identity tan2 θ − sin2θ = sin2θ tan2 θ .

6. Find the exact value of cos2π

18+ sin

7. For the function y =1

x − 2,

a) describe the transformations of the function when compared to the function y = 3x

b) sketch the graph of the given function and y = 3x on the same set of axes using transformations.

c) state the domain, the range, and the equation of the asymptote

8. Solve for n: 9n − 1

ÁÁÁÁÁÁ

˜̃˜̃˜̃

4n − 1

9. Solve for x.2log4 (x + 4) − log4 (x + 12) = 1

Name: ________________________ ID: A

10. Consider the function f(x) =1

+ 2x − 8.

a) Determine the key features of the function:

i) domain and range

ii) intercepts

iii) equations of any asymptotes

b) Sketch the graph of the function.

Name: ________________________ ID: A

11. Consider the function f(x) =3

4x − 5.

a) Determine the key features of the function:

i) domain and range

ii) intercepts

iii) equations of any asymptotes

b) Sketch the graph of the function.

12. Sun Eui and Pattie are arguing about the number of ways the letters in PERMUTATION and

COMBINATION can be rearranged. Sun Eui believes that there are more ways to rearrange

PERMUTATION, while Pattie thinks that COMBINATION has more unique arrangements. Which person is

correct?

13. Use the binomial theorem to expand a

ÁÁÁÁÁÁ

˜̃˜̃˜̃

Problem

1. Factor 2x4 – 7x3 – 41x2 – 53x – 21 fully.

2. The point (–5, 7) is located on the terminal arm of ∠A in standard position.

a) Determine the primary trigonometric ratios for ∠A.

b) Determine the measures of ∠A, to the nearest degree.

Name: ________________________ ID: A

3. Consider the function f x( ) =1

2sin [3 x − 30°( )] + 4.

a) Determine the amplitude, the period, the phase shift, and the vertical shift of the function with respect to

y = sinx .

b) What are the minimum and maximum values of the function?

c) Determine the horizontal increments of the function in the interval 0° ≤ x ≤ 360°.

d) Graph one period of the function

4. Prove the identity 1 + tanθ

1 + cotθ=

1 − tanθ

cotθ − 1.

5. Prove the identity 1

1 + cos θ+

1 − cos θ=

sin2θ

6. Prove that the equation 2csc2x =

1 + cos x+

1 − cos x is true for all values of x.

7. Prove the identity cos

2θ − sin

cos2θ + sinθ cos θ

= 1 − tanθ.

8. Prove the identity 1 − cos 2θ + sin2θ

1 + cos 2θ + sin2θ= tanθ.

9. Solve 4sin4x + 3sin

2x − 1 = 0 over the domain 0° ≤ x ≤ 360°.

10. A radioactive sample with an initial mass of 72 mg has a half-life of 10 days.

a) Write a function to relate the amount remaining, A, in milligrams, to the time, t, in days.

b) What amount of the radioactive sample will remain after 20 days?

c) What amount of the radioactive sample was there 30 days ago?

d) How long, to the nearest day, will it take for there to be 0.07 mg of the initial sample remaining?

11. The magnitude of an earthquake is defined as M = logA

ÁÁÁÁÁÁ

˜̃˜̃˜̃ , where A is the amplitude of the ground motion

and A0 is the amplitude corrected for the distance from the actual earthquake that would be expected for a

“standard earthquake.” On March 2, 2012, an earthquake with an amplitude 105.1

times A0 was recorded in

Norman Wells, Northwest Territories.

a) What was the earthquake’s magnitude on the Richter scale?

b) How does the earthquake in Norman Wells compare to the earthquake off Vancouver Island in 1946 that

measured 7.3 on the Richter scale?

Name: ________________________ ID: A

12. Given the graphs of the functions f x( ) = x2

− 3 and g x( ) = −2x + 3, sketch the graph of the combined

function y = (f + g)(x) and state its domain and range.

13. Seven students (Catherine, Jim, Jerome, Lucia, Lisa, Melinda, and Zyoji) are entered in a debate contest.

a) In how many ways can they give their arguments if the following conditions must be met?

i) Lucia must go second.

ii) Lisa must go first and Zyoji must go second to last.

iii) All of the boys go first followed by the girls.

iv) All of the girls go first followed by the boys.

v) Females and males alternate with a female going first.

14. Ms. Jackson’s class of 28 students all submitted projects for a recent math fair. The principal asked Ms.

Jackson to prepare a display for the upcoming parent–teacher evening.

a) In how many ways can 20 of the projects be chosen?

b) The principal asks to set the 4 regional winners out of the 20 selected near the entrance of the room. In

how many ways can the projects be displayed? Leave your answer in factorial form.

15. A standard deck of playing cards contains 4 suits (spades, clubs, diamonds, and hearts), each with 13 cards.

a) How many different 5-card hands are possible?

b) How many different 5-card hands with only black cards are possible?

c) How many different 5-card hands are possible containing at least 3 black cards?

Answer Section

MULTIPLE CHOICE

1. ANS: B PTS: 1 DIF: Average OBJ: Section 1.2

NAT: RF5 TOP: Reflections and Stretches KEY: reflection

2. ANS: B PTS: 1 DIF: Easy OBJ: Section 1.3

NAT: RF4 TOP: Combining Transformations

KEY: horizontal translation | vertical translation

3. ANS: A PTS: 1 DIF: Average OBJ: Section 1.3

NAT: RF4 | RF5 TOP: Combining Transformations

KEY: graph | vertical translation | horizontal translation | stretch | reflection

NAT: RF4 | RF5 TOP: Combining Transformations

KEY: graph | horizontal translation | vertical translation | reflection

5. ANS: D PTS: 1 DIF: Average OBJ: Section 1.4

NAT: RF6 TOP: Inverse of a Relation KEY: graph | inverse of a function

6. ANS: A PTS: 1 DIF: Easy OBJ: Section 1.4

NAT: RF6 TOP: Inverse of a Relation

KEY: inverse of a function | function notation

7. ANS: A PTS: 1 DIF: Easy OBJ: Section 2.1 | Section 2.2

NAT: RF13 TOP: Radical Functions and Transformations | Square Root of a Function

KEY: inverse of a radical function

NAT: RF13 TOP: Square Root of a Function KEY: graph

9. ANS: B PTS: 1 DIF: Difficult OBJ: Section 2.3

NAT: RF13 TOP: Solving Radical Equations Graphically

KEY: algebraic solution

NAT: RF12 TOP: Characteristics of Polynomial Functions

KEY: odd-degree | x-intercepts

11. ANS: C PTS: 1 DIF: Easy OBJ: Section 3.1

KEY: polynomial function

12. ANS: D PTS: 1 DIF: Easy OBJ: Section 3.1

KEY: x-intercepts

NAT: RF11 TOP: The Remainder Theorem KEY: remainder theorem | remainder

14. ANS: C PTS: 1 DIF: Average OBJ: Section 3.2

NAT: RF11 TOP: The Remainder Theorem KEY: quotient | remainder

NAT: RF12 TOP: Equations and Graphs of Polynomial Functions

KEY: polynomial equation | zeros | graph | multiplicity

NAT: RF12 TOP: Equations and Graphs of Polynomial Functions

KEY: polynomial equation | roots | graph

NAT: T1 TOP: Angles and Angle Measure KEY: radians | degrees

NAT: T1 TOP: Angles and Angle Measure KEY: rotations | standard position

NAT: T1 TOP: Angles and Angle Measure KEY: radians

NAT: T2 TOP: Trigonometric Ratios

KEY: Unit Circle | exact value | tangent ratio

NAT: T2 TOP: Trigonometric Ratios KEY: exact value | unit circle

NOT: tan90 and tan 270 changed to remove undefined

NAT: T4 TOP: Graphing Sine and Cosine Functions

KEY: amplitude | period | sinusoidal function

NAT: T4 TOP: Graphing Sine and Cosine Functions

KEY: graph | sinusoidal function

26. ANS: C PTS: 1 DIF: Difficult OBJ: Section 5.2

NAT: T4 TOP: Transformations of Sinusoidal Functions

KEY: transformations | equation | properties | sinusoidal function

NAT: T4 TOP: Transformations of Sinusoidal Functions

KEY: translation | primary trigonometric function

28. ANS: B PTS: 1 DIF: Difficult + OBJ: Section 5.4

NAT: T4 TOP: Equations and Graphs of Trigonometric Functions

KEY: quadratic trigonometric equation

NAT: T6 TOP: Reciprocal, Quotient, and Pythagorean Identities

KEY: quotient identity | reciprocal identity

NAT: T6 TOP: Reciprocal, Quotient, and Pythagorean Identities

KEY: trigonometric identity

NAT: T6 TOP: Sum, Difference, and Double-Angle Identities

KEY: difference identities

NAT: RF9 TOP: Transformations of Exponential Functions

KEY: modelling | exponential growth

NAT: RF9 TOP: Transformations of Exponential Functions

KEY: transformations of exponential functions

NAT: RF10 TOP: Solving Exponential Equations KEY: compound interest

NAT: RF7 TOP: Understanding Logarithms KEY: logarithm | exponential function

NOT: Draft

NAT: RF7 TOP: Understanding Logarithms KEY: logarithm

NAT: RF8 TOP: Transformations of Logarithmic Functions

KEY: horizontal translation | asymptote

NAT: RF9 TOP: Laws of Logarithms KEY: product law | laws of logarithms

NAT: RF10 TOP: Logarithmic and Exponential Equations

KEY: logarithmic equation

KEY: exponential equation | logarithmic equation

KEY: logarithmic equation | exponential equation

NAT: RF14 TOP: Exploring Rational Functions Using Transformations

KEY: reciprocal of linear function | graph from function

43. ANS: A PTS: 1 DIF: Difficult OBJ: Section 9.2

NAT: RF14 TOP: Analysing Rational Functions

KEY: quadratic denominator | function from characteristics

KEY: quadratic denominator | graph from characteristics

KEY: reciprocal of quadratic function | vertical asymptote

NAT: RF1 TOP: Sums and Differences of Functions

KEY: add functions | subtract functions

NAT: RF1 TOP: Sums and Differences of Functions

KEY: add functions | subtract functions

NAT: RF1 TOP: Products and Quotients of Functions

KEY: divide functions | graph

NAT: RF1 TOP: Products and Quotients of Functions

KEY: divide functions

NAT: RF1 TOP: Composite Functions KEY: composite functions | evaluate

NAT: PC2 TOP: Permutations KEY: fundamental counting principle

NAT: PC2 TOP: Permutations KEY: permutations

NAT: PC1 TOP: Permutations KEY: fundamental counting principle

54. ANS: C PTS: 1 DIF: Difficult OBJ: Section 11.2

NAT: PC3 TOP: Combinations KEY: combinations

NAT: PC3 TOP: Combinations

KEY: combinations | fundamental counting principle

NAT: PC3 TOP: Combinations KEY: combinations

NAT: PC4 TOP: The Binomial Theorem

KEY: Pascal's triangle | binomial expansion | binomial theorem

KEY: binomial expansion | binomial theorem

MATCHING

1. ANS: F PTS: 1 DIF: Average OBJ: Section 9.2

NAT: RF14 TOP: Analysing Rational Functions KEY: rational functions | graphs

4. ANS: H PTS: 1 DIF: Average OBJ: Section 9.2

5. ANS: E PTS: 1 DIF: Average OBJ: Section 9.2

SHORT ANSWER

1. ANS:

PTS: 1 DIF: Difficult + OBJ: Section 1.2 NAT: RF3 | RF5

TOP: Reflections and Stretches KEY: graph | reflection

2. ANS:

The graph of y = f(x) is shown in black, and the graph of y = f(x) is shown in blue.

PTS: 1 DIF: Average OBJ: Section 2.2 NAT: RF13

TOP: Square Root of a Function KEY: graph | square root of a function

3. ANS:

a) x + 2 x3

+ 0x − 7

− 2x − 7

−2x − 4

+ x − 2

x + 2= x

2+ x − 2 −

b) x ≠ –2

c) x3 + 3x2 – 7 = (x + 2)(x2 + x – 2) – 3

d) Expand the expression in part c) to verify.

(x + 2)(x2

+ x − 2) − 3 = x3

− 2x + 2x2

+ 2x − 4 − 3

TOP: The Remainder Theorem KEY: long division | restriction | quotient form | verify

4. ANS:

Since csc θ = −2

3, sinθ = −

2. Since sin60° =

2, the reference angle is 60°. The ratio is negative in

quadrants III and IV.This means that the angle can be found by looking for reflections of 60° that lie in these

quadrants.

quadrant III: 180° + 60° = 240°

quadrant IV: 360° – 60° = 300°

PTS: 1 DIF: Average OBJ: Section 4.3 NAT: T2

TOP: Trigonometric Ratios

KEY: reference angle | reciprocal trigonometric ratios | unit circle

5. ANS:

L.S. = tan2θ − sin2 θ

=sin2 θ

cos2θ− sin2θ

=sin2θ − sin2 θ cos2θ

cos2 θ

=sin2θ 1 − cos2 θ( )

cos2 θ

=sin2θ sin2θ( )

cos2 θ

= sin2θ tan2θ

R.S . = sin2θ tan2 θ

L.S. = R.S.

PTS: 1 DIF: Average OBJ: Section 6.1 | Section 6.3

NAT: T6 TOP: Reciprocal, Quotient, and Pythagorean Identities | Proving Identities

KEY: Pythagorean identities | trigonometric identity

6. ANS:

cos2π

18+ sin

18= cos

ÁÁÁÁÁÁ

˜̃˜̃˜̃

= cos4π

ÁÁÁÁÁÁ

˜̃˜̃˜̃

= cos3π

ÁÁÁÁÁÁ

˜̃˜̃˜̃

= cosπ

ÁÁÁÁÁÁ

˜̃˜̃˜̃

TOP: Sum, Difference, and Double-Angle Identities KEY: difference identities | exact value

7. ANS:

a) a vertical compression by a factor of 1

2 and a translation of 2 units to the right

b) The graph of y = 3x is shown in blue and the graph of y =1

x − 2 is shown in red.

c) domain {x| x ∈ R}, range {y| y > 0, y ∈ R}, y = 0

TOP: Transformations of Exponential Functions

KEY: graph | transformations of exponential functions

8. ANS:

9n − 1

ÁÁÁÁÁÁ

˜̃˜̃˜̃

4n − 1

ËÁÁÁ

ˆ¯˜̃̃

n − 1

= 3−1Ê

ËÁÁÁ

ˆ¯˜̃̃

4n − 1

32n − 2

= 31 − 4n

Equate the exponents:

2n − 2 = 1 − 4n

6n = 3

TOP: Solving Exponential Equations KEY: change of base

9. ANS:

2log4 (x + 4) − log4(x + 12) = 1

log4 (x + 4)2

− log4(x + 12) = 1

(x + 4)2

(x + 12)= 1

=(x + 4)

(x + 12)

(x + 4)2

= 4x + 48

+ 8x + 16 = 4x + 48

+ 4x − 32 = 0

(x + 8)(x − 4) = 0

x = −8, x = 4

Since x = –8 is an extraneous root, the solution is x = 4.

TOP: Logarithmic and Exponential Equations

KEY: logarithmic equation | laws of logarithms NOT: Draft

10. ANS:

a) i) {x| x ≠ 2,−4, x ∈ R}, {y| y ≠ 0, y ∈ R}

ii) x-intercept: none, y-intercept: −1

iii) x = 2, x = −4, y = 0

PTS: 1 DIF: Difficult OBJ: Section 9.2 | Section 9.3

NAT: RF14 TOP: Analysing Rational Functions | Connecting Graphs and Rational Equations

KEY: reciprocal of quadratic function | key features | graph

11. ANS:

a) i) x| x ≠5

4, x ∈ R

ÔÔÔÔÔÔÔÔ

ÔÔÔÔÔÔÔÔ, {y| y ≠ 0, y ∈ R}

ii) x-intercept: none, y-intercept: −3

iii) x =5

4, y = 0

NAT: RF14 TOP: Analysing Rational Functions | Connecting Graphs and Rational Equations

KEY: reciprocal of linear function | key features | graph

12. ANS:

Let P represent the number of ways the letters in PERMUTATION can be arranged, and C the number of

ways the letters in COMBINATION can be arranged.

P =11!

= 19 958 400

C =11!

2!2!2!

= 4 989 600

There are more arrangements for PERMUTATION, so Sun Eui is correct.

PTS: 1 DIF: Average OBJ: Section 11.1 NAT: PC1

TOP: Permutations KEY: fundamental counting principle

13. ANS:

ÁÁÁÁÁÁ

˜̃˜̃˜̃

ÁÁÁÁÁÁ

˜̃˜̃˜̃

ÁÁÁÁÁÁ

˜̃˜̃˜̃

ÁÁÁÁÁÁ

˜̃˜̃˜̃

ÁÁÁÁÁÁ

˜̃˜̃˜̃

ÁÁÁÁÁÁ

˜̃˜̃˜̃

ÁÁÁÁÁÁ

˜̃˜̃˜̃

ÁÁÁÁÁÁ

˜̃˜̃˜̃

ÁÁÁÁÁÁ

˜̃˜̃˜̃

ÁÁÁÁÁÁ

˜̃˜̃˜̃

ÁÁÁÁÁÁ

˜̃˜̃˜̃

ÁÁÁÁÁÁ

˜̃˜̃˜̃

ÁÁÁÁÁÁ

˜̃˜̃˜̃

ÁÁÁÁÁÁ

˜̃˜̃˜̃

ÁÁÁÁÁÁ

˜̃˜̃˜̃

ÁÁÁÁÁÁ

˜̃˜̃˜̃

ÁÁÁÁÁÁ

˜̃˜̃˜̃

ÁÁÁÁÁÁ

˜̃˜̃˜̃

ÁÁÁÁÁÁ

˜̃˜̃˜̃

ÁÁÁÁÁÁ

˜̃˜̃˜̃

ÁÁÁÁÁÁ

˜̃˜̃˜̃

TOP: The Binomial Theorem KEY: binomial expansion | binomial theorem

PROBLEM

1. ANS:

Let P(x) = 2x4 – 7x3 – 41x2 – 53x – 21.

Test x = –1 in the factor theorem.

P(x) = 2x4

− 7x3

− 41x2

− 53x − 21

P(−1) = 2(−1)4

− 7(−1)3

− 41(−1)2

− 53(−1) − 21

= 2 + 7 − 41 + 53 − 21

P(−1) = 0

Thus, x + 1 is a factor. Divide to determine another factor.

1 2 −7 −41 −53 −21

− 2 −9 −32 −21

× 2 −9 −32 −21 0

P(x) = (x + 1)(2x3

− 9x2

− 32x − 21)

Now factor the cubic. Test x = –1 in the factor theorem. Let Q(x) = 2x3

− 9x2

− 32x − 21.

Q(x) = 2x3

− 9x2

− 32x − 21

Q(−1) = 2(−1)3

− 9(−1)2

− 32(−1) − 21

= −2 − 9 + 32 − 21

Q(−1) = 0

Thus, x + 1 is a factor. Divide to determine another factor.

1 2 −9 −32 −21

− 2 −11 −21

× 2 −11 −21 0

P(x) = (x + 1)(x + 1)(2x2

− 11x − 21)

= (x + 1)2(2x

2− 14x + 3x − 21)

= (x + 1)2[2x(x − 7) + 3(x − 7)]

= (x + 1)2(x − 7)(2x + 3)

TOP: The Factor Theorem KEY: factor theorem | integral zero theorem | factor

2. ANS:

∠A is in quadrant II. Therefore, only the sine ratio will be positive.

Use the Pythagorean theorem.

= −5( )2

= 25 + 49

r = 74

Therefore, sin A =7 74

74, cos A = −

74, and tan A = −

b) The reference angle for ∠A is:

ref∠A = sin−1 7 74

ref∠A ≈ 54°

∠A = 180° − 54°

= 126°

NAT: T2 | T3 TOP: Unit Circle | Trigonometric Ratios

KEY: trigonometric ratios | reference angle | unit circle

3. ANS:

a) amplitude: 1

2; period:

3 or 120°; phase shift: 30° to the right; vertical shift: 4 units up

b) minimum: −1

2+ 4 or

2, maximum:

2+ 4 or

c) horizontal increments: 30°, 60°, 90°,120°,150°

d) Graph:

TOP: Transformations of Sinusoidal Functions

KEY: amplitude | period | shifts | transformations | sine function

4. ANS:

L.S. =1 + tanθ

1 + cotθ

1 +sinθ

cos θ

1 +cos θ

cos θ + sinθ

cos θ

sinθ + cos θ

=cos θ + sinθ

cos θ

ÁÁÁÁÁÁ

˜̃˜̃˜̃

cos θ + sinθ

ÁÁÁÁÁÁ

˜̃˜̃˜̃

=sinθ

cos θ

R.S. =1 − tanθ

cot θ − 1

1 −sinθ

cos θ

sinθ− 1

cos θ − sinθ

cos θ

cos θ − sinθ

=cos θ − sinθ

cos θ

ÁÁÁÁÁÁ

˜̃˜̃˜̃

cos θ − sinθ

ÁÁÁÁÁÁ

˜̃˜̃˜̃

=sinθ

cos θ

L.S. = R.S.

KEY: quotient identities | proof

5. ANS:

L.S. =1

1 + cos θ+

1 − cos θ

=1 − cos θ + 1 + cos θ

1 + cos θ( ) 1 − cos θ( )

1 − cos2θ

sin2θ

R.S. =2

sin2θ

L.S. = R.S.

KEY: Pythagorean identities | proof

6. ANS:

L.S. = 2csc2 x R.S. =1

1 + cos x+

1 − cos x

=1 − cos x( ) + 1 + cos x( )

1 + cos x( ) 1 − cos x( )

1 − cos2x

sin2 x

= 2csc 2 x

L.S. = R.S.

This proves that the expression is valid for all values of x.

KEY: reciprocal trigonometric ratios | proof

7. ANS:

L.S. =cos

2θ − sin

cos2θ + sinθ cos θ

=cos θ − sinθ( ) cos θ + sinθ( )

cos θ cos θ + sinθ( )

=cos θ − sinθ

cos θ

R.S. = 1 − tanθ

= 1 −sinθ

cos θ

=cos θ − sinθ

cos θ

L.S. = R.S.

TOP: Proving Identities KEY: trigonometric identity | proof

8. ANS:

L.S. =1 − cos 2θ + sin2θ

1 + cos 2θ + sin2θ

=1 − (1 − 2sin

2θ) + 2sinθ cos θ

1 + (2cos2θ − 1) + 2sinθ cos θ

2θ + 2sinθ cos θ

2cos2θ + 2sinθ cos θ

=2sinθ sinθ + cos θ( )

2cos θ sinθ + cos θ( )

=sinθ

cos θ

= tanθ

L.S. = tanθ

L.S. = R.S

Therefore, 1 − cos 2θ + sin2θ

1 + cos 2θ + sin2θ= tanθ.

NAT: T6 TOP: Sum, Difference, and Double-Angle Identities | Proving Identities

KEY: double-angle identities | proof

9. ANS:

4sin4x + 3sin2 x − 1 = 0

(4sin2 x − 1)(sin2x + 1) = 0 sin2x = −1 ** No sol'n: You can't take the square root of a negative

4sin2 x − 1 = 0

sinx = ±1

sin x is positive in quadrants I and II and negative in quadrants III and IV. The solution is

x = 30°,150°,210°,330° .

TOP: Solving Trigonometric Equations Using Identities

KEY: general solutions | quadratic equation

10. ANS:

a) A = 721

ÁÁÁÁÁÁ

˜̃˜̃˜̃

b) A = 721

ÁÁÁÁÁÁ

˜̃˜̃˜̃

ÁÁÁÁÁÁ

˜̃˜̃˜̃

ÁÁÁÁÁÁ

˜̃˜̃˜̃

There will be 18 mg remaining after 20 days.

c) A = 721

ÁÁÁÁÁÁ

˜̃˜̃˜̃

ÁÁÁÁÁÁ

˜̃˜̃˜̃

ÁÁÁÁÁÁ

˜̃˜̃˜̃

There was 576 mg 30 days ago.

d) A = 721

ÁÁÁÁÁÁ

˜̃˜̃˜̃

0.07 = 721

ÁÁÁÁÁÁ

˜̃˜̃˜̃

ÁÁÁÁÁÁ

˜̃˜̃˜̃

Use logarithmes

0.000 972 =1

ÁÁÁÁÁÁ

˜̃˜̃˜̃

log0.000972 = log1

ÁÁÁÁÁÁ

˜̃˜̃˜̃

log0.000972 =t

ÁÁÁÁÁÁ

˜̃˜̃˜̃

10log0.000972

ÁÁÁÁÁÁ

˜̃˜̃˜̃

t = 100.07

** Check your answer using a graphing calculator.

It will take approximately 100 days for there to be 0.07 mg remaining.

NAT: RF9 | RF10 TOP: Transformations of Exponential Functions | Solving Exponential Equations

KEY: modelling | evaluate exponential functions

11. ANS:

a) M = log105.1 A0

ÁÁÁÁÁÁÁÁ

˜̃˜̃˜̃˜̃

M = log105.1

M = 5.1

The earthquake in Norman Wells measured 5.1 on the Richter scale.

b) Vancouver Island amplitude

Norman Wells amplitude=

≈ 158

The earthquake off Vancouver Island was about 158 times as strong as the earthquake off Norman Wells.

TOP: Understanding Logarithms KEY: logarithm NOT: Draft

12. ANS:

TOP: Composite Functions KEY: composite functions | graph | transformations

13. ANS:

a) i) Assume the order is ____, Lucia, ____, ____, ____, ____, ____, where the only spot taken is the second

position by Lucia. The number of choices is determined by the number of students remaining for each

position:

6, Lucia (1), 5, 4, 3, 2, 1

Thus, there are 6(5)(4)(3)(2)(1) or 720 ways to organize the students.

ii) Assume the order is Lisa, ____, ____, ____, ____, Zyoji, ____, where the only spots taken are the first

position by Lisa and the second-last position by Zyoji. The number of choices is determined by the number of

students remaining for each position:

Lisa (1), 5, 4, 3, 2, Zyoji (1), 1

Thus, there are 120 ways to organize the students.

iii) There are (3)(2)(1)(4)(3)(2)(1) or 144 different ways if all the boys go first.

iv) There are (4)(3)(2)(1)(3)(2)(1) or 144 different ways if all the girls go first

v) There are (4)(3)(3)(2)(2)(1)(1) or 144 different ways if boys and girls alternate.

TOP: Permutations KEY: fundamental counting principle | conditions

14. ANS:

a) 28 C20 =28!

= 3 108 105

There are 3 108 105 possible ways to choose the 20 projects.

b) There are 4! ways to arrange the 4 regional winners, and 16! ways to arrange the other 16. Thus, there are

(4!)(16!) ways to display the projects.

TOP: Permutations KEY: permutations

15. ANS:

a) 52 C5 = 2 598 960

b) There are 26 black cards (spades and clubs), so there are 26 C5 or 65 780 possible hands with only black

cards.

c) There are 3 possible situations:

Case 1: all 5 cards are black:

26 C5 = 65 780

Case 2: 4 black cards, 1 red card

P(black) × P(red) = (26 C4 )(26 C1)

= 388 700

Case 3: 3 black cards, 2 red card

P(black) × P(red) = (26 C3 )(26 C2)

= 845 000

The total number of different 5-card hands containing at least 3 black cards is

65 780 + 388 700 + 845 000 or 1 299 480.

PTS: 1 DIF: Difficult OBJ: Section 11.2 NAT: PC3

TOP: Combinations KEY: combinations

Name: Date: Final Exam Review Mrs. Boughen · 2019-01-08 · Final Exam Review Mrs. Boughen 1....

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