1

FINAL EXAM REVIEW FOR MCR 3U FUNCTIONS (11 UNIVERSITY)

Overall Reminders: • To study for your exam your should redo all your past tests and quizzes • Write out all the formulas in the course to help you remember them for the exam. • Remember to bring your borrowed text book with you to the exam.

EXAM REVIEW: Rational Expressions

• Factoring (common, difference of squares, trinomial) • Simplifying Rational Expressions à Multiplication/ Division

à Addition/ Subtraction • Restrictions

1. Simplify.

a) ( ) ( )2 23 4 1 2 2x x x x+ − − − − b) ( ) ( )4 1 2 1 2 2 1 3y y y y y⎡ ⎤ ⎡ ⎤+ − + + −⎣ ⎦ ⎣ ⎦

c) ( )( )23 1 2 3 4z z z+ − − d) ( ) ( )( )22 2 3 2a b a b a b+ − − +

e) ( )22 5 3x x+ − f) ( ) ( ) ( ) ( )2 2 2 22 1 4 4x x x x+ − − − +

2. Simplify and state any restrictions.

a) 2 2

1 3

1812p qp q−−

b) 2

2 46 8t

t t+

+ + c) 2

2 44

xx

+

− d)

2

2

2 3 13 2 1x xx x− +

− −

e) 4 3 2

2 4 4

4 63 10x y x yx y x

−× f)

2

2

6 9 3 96 9 2 6

t t tt t t+ + −

×− + +

g) 2 2

2 2

2 5 2 2 3 22 3 9 2 3x x x xx x x x+ + + −

÷− − + −

h) ( )22 2

22 8 2x yx y

x x xy−−

÷−

i) 2

2 2

25 9 8 64 3 5 8 3

x xx x x x

− −×

+ − + + j)

2 2

2 2

3 10 2 1512 7 12

a a a aa a a a− − − −

÷− − + +

k) 2 2 2 2 2 2

2 2 2 2 2 2

6 8 5 6 8 129 20 6 2 15

a ab b a ab b a ab ba ab b a ab b a ab b

+ + + − + +× ÷

+ + − − + −

l) 4 53 1 1 3y y

++ −

m) 2 2

4 22 1 2 3m m m m

−− − + −

n) 2 2

3 212 16x x x

−− − −

o) 2

2a b a b a ba ab b b+ − +

− −+

2

EXAM REVIEW: Quadratics

• Solving quadratic equations à By factoring where possible àUsing Quadratic Formula when factoring is not possible

• Simplifying Radicals • Types of solutions (Nature of the Roots) 1. Simplify.

a) 54 b) 24− − c) 2 27+

d) 2 12 4 20 3 27 5 45+ − − e) ( )3 6 21+ f) ( )( )2 3 3 2 5 4 3− +

g) 8010

h) 2 53 2

i) 9 456

+ j) 4 202

− k) 2 32 3 3 2

−

+

2. Solve each of the following using the most efficient method. x C∈ (No decimals)

a) 23 7 4 0y y+ + = b) 2 4 11 0x x− − = c) 23 27 0x − = d) 22 3 3x x− =

e) 21 4 22y y= − f) 26 5 4x x− = g) 23 4 7x x+ = h) 23 48 0x + =

i) 25 4 20 0x x− + = j) 22 5 3 0x x+ − = k) 900 900 256x x

= ++

3. Solve each of the following inequalities.

a) 2 23 3 2x x x x+ ≤ − − b) ( ) ( )2 3 5 3 4 2m m− − > − + c) 1 52 3q q+ +

≥

4. Find the value(s) of k so that each function has: i. two x-intercepts ii. one x-intercept iii. no x-intercepts

a) ( ) 2 9 6f x kx x= − + b) ( ) 23 12f x x kx= + + c) ( ) 29 3f x x x k= + +

5. Determine the nature of the roots without solving the equation. (Type and number of solutions.) a) 2 5 7 0x x− + = b) 23 5 2 0x x− + = c) 24 20 25 0x x− + =

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EXAM REVIEW: Trigonometry

• Evaluate Expressions • Solving Trigonometric Equations à Special Triangles

à Cast Rule à Factoring à Using Radians

• Prove Identities 1. Evaluate the following: a) csc170.1o b) 7

4cot π c) ( )43sec π−

2. If 135secθ = , make a diagram and find the other five ratios, if ∠θ is acute.

3. If 221

cotθ = and ∠θ is obtuse, find the exact values of the other five ratios

and the approximate value of θ , 4. Determine the exact value.

a) sinπ b) 7cos6π c) 5tan

4π⎛ ⎞

−⎜ ⎟⎝ ⎠

d) 3 7sin cos tan tan4 4 6 3π π π π

+

e) 5 3 5 3cos cos sin sin3 2 6 2π π π π

− f) 5sec30°tan60°

5. If csc 2θ = − , find all possible values of 0° ≤ θ ≤ 360° .

6. Solve each of the following for 0 2x π≤ ≤ . Where necessary, round to one decimal place.

a) ( )sin 2cos 1 0x x + = b) 24cos 3 0x − = c) 2cot cotθ θ=

d) 22sin tan tan 0x x x− = e) 23

cscθ = − f) 215sin 7sin 2x x− =

g) 2cos sin cos 0x x x− = h) 24sin cos 2sinx x x= i) 22cos sin 2 0x x− − =

j) 26sin cos 5 0x x+ − = k) 26sec 5sec 4 0θ θ+ − =

6. Prove the following identities.

a) ( )( )2 21 sin 1 tan 1x x− + = b) sin tan cos tan 2cos tan tanx x x x x x x+ + = +

c) 2 4 2 4sin sin cos cosθ θ θ θ− = − d) 1 sin cos 2cos 1 sin cos

x xx x x

++ =

+

e) 2

22

tan sintan 1

θθ

θ=

+ f) 2

2 1 1sin 1 cos 1 cosx x x

= +− +

g) 2 2 22

1tan cos sincos

x x xx

+ + = h) sec cos sec sin 1 tanθ θ θ θ θ+ = +

Exact Values

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EXAM REVIEW: Transformations of Functions • Domain / Range • Amplitude, Period, Phase Shift, Vertical Translation, Reflection (Vertical and Horizontal) • Graphing Trigonometric Functions ( ( )siny a b x d c= − + or ( )cosy a b x d c= − + )

• Mapping Notation: ( ) 1, ,x y x d ay cb⎛ ⎞

→ + +⎜ ⎟⎝ ⎠

• Function Notation • Inverse Function y -1 or f –1(x) • Explanation of a transformation / Write an equation under given transformation 1. State the domain, range, period, amplitude, vertical translation and phase shift of each of the

following:

a) 1cos22

y x= b) 13sin2

y x= − c) 3sin 1y x= −

d) sin 34

y x π⎛ ⎞= − −⎜ ⎟

⎝ ⎠ e) ( )1cos 6 5

2y x π= − − f) 25cos 1

2 3xy π⎛ ⎞

= − − +⎜ ⎟⎝ ⎠

2. Sketch two cycles of each function in #2. 3. Describe how the graph of each of the following functions can be obtained from the graph

( )y f x= .

a) ( )5 3y f x= − + b) ( )3 4y f x= + − c) 14 22

y f x⎛ ⎞= +⎜ ⎟

⎝ ⎠

d) ( )( )7 5 1y f x= − + e) ( )( )8 7y f x= − + f) ( )1 2 6 93

y f x= − − −

4. Write the transformed equation if ( )y f x= undergoes the following transformation:

It is reflected across the y-axis, stretched vertically by a factor of 3, compressed horizontally by a factor of ¼, shifted 3 units down and translated 3 units to the left.

5. If ( ) 22 5 1,f x x x= − + find: a) ( )0f b) – ( )3f c) ( )3f x − d) x when ( ) 0f x =

6. State the domain and range of each of the following:

a) 3 4 6y x= − − + b) ( )22 3 5y x= + − c) 3 47 5

yx−

= −+

7. Given ( ) 3 8f x x= − + , determine: a) ( )4f b) x when ( ) 14f x = c) ( )1 6f − d) x when ( ) ( )1f x f x−=

8. Determine the inverse of each of the following functions:

a) ( ) 4 3f x x= + b) ( ) 4 8g x x= + c) ( ) 22 12 14h x x x= + − d) ( ) 2 75

k xx

= −+

9. If ( ) 22f x x x= − and ( ) 3g x x= − , determine:

a) ( )( )f g x b) ( )( )4g f − c) y when ( )cos 0f y = and 0 2y π≤ ≤

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EXAM REVIEW: Exponential Functions

• Simplifying using exponent rules (in power form, radical form, and rational form) • Solving exponential equations • Exponential Growth and Decay

1. Evaluate. (No decimals)

a) ( ) 34 −− b)

0 0

1

3 42−+ c)

1327

− d)

3416

−− e)

23125

8−⎛ ⎞⎜ ⎟⎝ ⎠

f) 2 3

2 1

5 34 4− −

−

−

2. Simplify: (leave your answer in power form)

a) 2( ) x y

x yaba b

+

b) 2 2 7x y x y xa a a+ + −× ÷ c) 4 4

3 3

( ) ( )a b

a b a bx xx x+ +

g

3. Simplify, and then evaluate for a = -1, b = 2: 4 2 2 5

4 2 3 6

30 ( 15 )(5 ) 45a b a bab a b

−×

4. Prove: 1 2(3 )(3 )(3 ) 2727

a a a

a

+ +

=

5. Simplify each of the following. Write the answer using positive exponents.

a) 1

1

1 21 2

pp

−

−

−+

b) 2 1

2 2

3 125 5

− −

−

⎛ ⎞ ⎛ ⎞×⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

c)

1 55 6 642m

m

−⎛ ⎞ ⎛ ⎞×⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠

6. Evaluate (without calculators):

a) 0

2 53 4 127 16

7− − ⎛ ⎞+ − ⎜ ⎟

⎝ ⎠ b) ( )

0.432 c) ( )

120.04 − d) ( )( )3 7 3 7

2 2 2 22 2 2 2− +

7. Solve. x C∈

a) 2 1 1636

x− = b) 3 15 5 600x x+ +− = c) 2 1464

x− = d) 22 93 243x x+ =

e) 2

320 12

8

xx + ⎛ ⎞

= ⎜ ⎟⎝ ⎠

f) ( )( )2 2 3 24 2 16x x− −= g) ( )( ) ( )2 2 3 2 14 2 162

x x− − ⎛ ⎞= ⎜ ⎟

⎝ ⎠

h) 23 6(3 ) 27 0x x− − = * i) 2 75 8 5x x+⋅ =

8. In 1976, a research hospital bought half of a gram of radium for cancer research. Assuming

the hospital still exists, how much of this radium will the hospital have in the year 6836,

if the half life of the radium is 1620 years?

9. There are 50 bacteria present initially in a culture. In 3min., the count is 204800.

What is the doubling period?

10. In a recent dig, a human skeleton was unearthed. It was later found that the amount of 14C

in it had decayed to ( )18 of its original amount. If 14C has a half life of 5760 years,

how old was the skeleton?

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EXAM REVIEW: Sequences and Series

• Arithmetic Sequences / Series

( )1nt a n d= + − ( )2 12nns a n d⎡ ⎤= + −⎣ ⎦

• Geometric Sequence / Series

1nnt ar −=

( )11

n

n

a rS

r−

=−

• Recursion Formula • Pascal’s Triangle 1. Find tn and t12 for each of the following sequences: a) 9, 15, 21, … b) –1, 2, –4, 8, … 2. Find the sum of each of the following series:

a) 251 + 243 + 235 +…–205 b) –4 – 12 – 36– … –8748

c) 21 + 23 + 25 + … + 43 d) 1280 – 640 +320 – … + 5

3. Write the first 4 term of each of the following sequences: a) tI = –6; tn = tn – 1 +5; n > 1 b) t1 = –2; t2 = –1; tn = tn – 1 x tn – 2 4. In an arithmetic sequence, the 3rd term is 25 and the 9th term is 43. How many terms are less

than 100? 5. The sum of the first 6 terms is 297 and the sum of the first 8 terms is 500. Determine the 5th term

if the sequence is arithmetic. 6. Insert 3 terms between 6 and 7776 so that you create a geometric sequence. 7. Insert 3 terms between 41 and 97 so that you create an arithmetic sequence. 8. For (1 – x)11 : a) find t3 b) how many terms are in the expansion? c) explain where the numerical coefficients of the expansion are coming from

8. a) Expand 41 3

4 4⎛ ⎞+⎜ ⎟⎝ ⎠

using binomial theorem; take the coefficients from Pascal triangle n = 4

b) Evaluate the sum.

9. Expand (Leave your final answer in power form): a) 5

32

123

zy

⎛ ⎞−⎜ ⎟

⎝ ⎠ b) ( )

333 d d−

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EXAM REVIEW: Finance

• Compound Interest: ( )1 nA P i= +

• Annuity: ( )1 1nR i

Ai

⎡ ⎤+ −⎣ ⎦=

• Present Value Annuity: ( )−⎡ ⎤− +

⎣ ⎦=1 1 nR i

Pi

1. In order to have $7250 for tuition in a year and a half, you plan to deposit money into a savings

account every three months. If that account pays 5.2% compounded quarterly, a) what will your regular deposits be? b) how much interest would you have earned? 2. Leo’s parents want him to be able to withdraw an allowance of $2700 every 6 months. How much

should they invest into an account today that pays 8.5% compounded semi-annually so that Leo can withdraw his allowance for the next 5 years?

3. Determine: a) the amount $8500 will grow to if invested at 8% compounded quarterly for ten years. b) the principal that must be invested now at 6% compounded annually to be worth $10 000 in 5

years. c) The total accumulated amount of $500 invested every month at 7% compounded monthly for 8

years. 4. How long will it take, to the nearest year, for $800 to grow to $1000 in an account that pays 4.5%

annually compounded semi-annually? 5. If a savings account compounds daily, what would the annual interest rate need to be for $500 to

grow to $650 in 3 years?

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MCR 3U1 Exam Review

1. Simplify and express with positive exponents: 3 2

1 5 2

1827a b ca b c

−

− −

2. Simplify: 3481

16

−⎛ ⎞⎜ ⎟⎝ ⎠

3. Simplify: (3 3 2 6)( 6 2 3)− +

4. Simplify: 8 82+

5. If 4 x y= then my x= for what value of m?

6. Find the lowest common multiple (stated in factored form is acceptable) for the

following expressions: 2 2 2( 3 2) ( 4 3) ( 5 6)x x and x x and x x+ + + + + +

7. A rectangular prism has the 3 sides measuring x–1, x–1 and 2x+2 centimeters. Create a simplified equation to represent R, the ratio of the surface area to volume. State any logical restrictions on x.

8. Simplify and state any restrictions on a: 2 2

2 2

6 8 63 4 9

a a a aa a a+ + + −

÷+ − −

9. State the domain and range (in good math form) of:

2 312 2( 1)y x= − + +

10. Given that 5 and -3 are the roots of a parabola and P(1,–32) is a point on the parabola,

what is the equation that models this, in standard form?

11. Find the solution(s)/point(s) of intersection of the following quadratic-linear system. Show/label steps appropriately: y + x² = – 6x –3 and x + y – 1 = 0. Solve using both an algebraic and graphical method.

12. Solve: 2 2 8 0x x− − = using both an algebraic and graphical method.

13. For the following equation, describe any and all transformations for

223 ( 5) 2y x= − + −

compared to y = x²

14. Current price is $100, current sales are 5000 units. For every increase in price of $1, sales decrease by 20 units. Find optimum revenue.

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15. Solve for p, given: 26 10 11p p= +

16. Height of an object in metres, based on time in seconds, is given as: 24.9 12 47h t t= − + + .

Determine time of landing of object, maximum height and when it reaches the maximum.

17. Height of object in metres, based on time in seconds, is given as: 25 39 135h t t= − + + , Determine time of landing of object, maximum height and when it reaches the maximum.

18. Given the diagram of f(x) at right, draw the graph of: f(-x), –f(x), f(x-2), 2f(x), f(0.5x), f(x)+3

19. Determine the inverse of the function: ( ) 4f x x= +

20. If 2( ) 3 5 2f x x x= − + and ( ) ( 2 ) 1g x f x= − − + ,

determine the new equation g(x) in terms of x.

21. Given: { (1,4) , (2,3) , (3,2) , (4,1) , (5,1) }, State Domain, Range, Is it a function or relation?, Inverse

22. Examine the given equations and determine: Domain,

Range and any vertical asymptotes. A sketch may help in your determinations.: 3( )1

f xx−

=+ and 2( ) 2 5xf x = +

23. If 23csc19cmcm

θ = , determine θ.

24. Given r, x, y, θ of a triangle, solve for the missing parts. a. r = 13, x =–5; b. x = 7, y = –11; c. θ = 133˚ and r = 44.

25. Given: ∠A is 20˚, b = 50 cm, for what possible range of lengths of side “a” would

rABC have two possible solutions (approximate to 2 decimal places)?

26. Given: ∠A is 114˚, b = 35cm, for what values of “side a” would yield no solutions at all?

27. Determine the exact trigonometric ratios for sine, cosine and tangent, given that the point P (–5, 3) is on the terminal arm. Find the principal angle, to 1 decimal place.

-­‐5

-­‐4

-­‐3

-­‐2

-­‐1

0

1

2

3

4

5

-­‐7 -­‐6 -­‐5 -­‐4 -­‐3 -­‐2 -­‐1 0 1 2 3 4 5 6 7

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28. Determine the exact trigonometric ratios for sine, cosine and tangent, given: θ = 210˚.

29. Determine the exact trigonometric ratios for sine, cosine and tangent, given: θ = 315˚.

30. Given: tan θ = –2.6, 0˚ ≤ θ ≤ 360˚ , solve for θ. 31. Given: sec θ = –6.4, 0˚ ≤ θ ≤ 360˚ , solve for θ.

32. Given that some function, f(x), is periodic, with a period of 3 and has the following known

values: f(1) =1 , f(2) = 2 , f(3)= 3. Determine f(1795), f(-13), f(0).

33. Given rABC has ∠A=79˚, ∠B=73˚, AC=21cm and AB=25cm, determine the length of BC.

34. Given rABC has ∠A = 53˚, b = 44cm and c = 59cm, determine a.

35. Sketch 2sin( 90) 1y x= − − + on top of this graph of sin( )y x=

36. Prove: csctan cotcosxx xx

+ = (using basic identities).

37. Determine the explicit formula for a) 5 2 7 8 3, , , , ...8 3 10 11 4 b) ,...,,,,, 10

1354

21

103

51

101

38. Determine the formula for the nth term of the sequence: 40, 20, 10, 5…

39. Determine the formula for the nth term of the sequence: 14, 27, 40, 53…

40. Given: t5 =29 and t9 =45, determine the sum of the first 15 terms of this arithmetic

series. 41. Determine the sum of the first 8 terms in the series: 8, – 4 + 2…

42. Given f(x)=2x, sketch g(x) = – f(2x) , h(x) = 3 f(x+1) , j(x) = f(x-3)+2, and determine

the y-intercept and horizontal asymptote of each.

43. Expand (2x–3)5 using Binomial Theorem.

44. What is the 7th term in the polynomial made by expanding (3x–2)8?

45. In Pascal’s triangle, if we examined the 17 row, and looked at the 11th diagonal column (counting 0th, 1st, 2nd, etc). What number would appear in that spot?

11

46. A country has a growth rate of 4.3%. Current population is 3 million, population in 15

years will be what?

47. A country has growth rate of –8.1%. Current population is 5 million, population in 12 years will be what?

48. Population doubles every 17 days. Starting population is 14. After 111 days, what is the

population?

49. An element has a half-life of 17 days. Starting amount of 24 473 grams. After 211 days, what amount remains?

50. A house increases in value by 11% per year. Values at $250 000 in 2005, what will be

the value in 2011?

51. A car depreciates by 18% per year. A $22 000 car will be worth how much after 9 years?

52. How much needs to be deposited, today, at 5% monthly interest, in order to have $2500

in 8 years?

53. You deposit $200 per month, for 25 years, at 9% annual interest compounded monthly, how much will you have?

54. You plan on withdrawing $2000 per month, for 25 years, at 9% annual interest

compounded monthly. How much will you need in the bank at the start of the 25 years?