year 9 extension mathematics semester 2 exam preparation ...€¦ · sketching parabolas of the...

© John Wiley & Sons Australia, Ltd 2012

Year 9 Extension Mathematics Semester 2 Exam Preparation Overview

Reading: 15 minutes writing: 90 Minutes Questions: 4 sections:

Structure of examination:

Materials: Not allowed: digital electronic devices, formula sheet and dictionary. Permitted: writing materials, 1xA4 double sided hand written notes, ruler, CAS calculator.

Semester 2 Topics Covered: CHAPTERS -7 Linear and non-linear graphs CHAPTERS - 16 Quadratic algebra CHAPTER - 17 Quadratic functions CHAPTERS - 13 Probability

EXAM STRUCTURE:

A: Vocabulary This section contains questions that require students to know the meaning of commonly used terms from the Semester 2 topics covered.

B: Multiple Choice This section requires students to record answers in a multi-choice grid, and does not require that working be shown. It is highly recommended, however, that students complete working out of their solutions. Soft pencil should be used to fill the circle in front of correct answer on the answer sheet.

C: Short Answer This section contains short answer questions that test the student’s knowledge and skills. Working out MUST BE SHOWN in this section in order to get full marks.

D: Analysis Questions This section is likely to require use of the CAS calculator; working out MUST BE SHOWN in order to get full marks.


Study Design Key Knowledge and Key Skills: Summary — Topic 7: Linear and non Linear Graphs

• Plotting linear graphs • Substitute values of x to find the coordinate of y. • Complete a table of values. • Find the equation of a straight line. • From the equation of a straight line, identify the y- intercept and gradient. • Sketching linear graphs, using the x and y intercept method and/or the gradient and y intercept

method. • Determining linear rules, from 2 coordinates given, from a linear graph, or if the gradient and a

coordinate is given. • Practical applications of linear graphs • Find the Midpoint of a line segment and distance between two points • Non-linear relations (parabolas, hyperbolas, circles) and identify their key features.

Summary — Chapter 16: Quadratic Equations

• Factorisation patterns • Factorising monic quadratics • Factorising non-monic quadratic • Simplifying algebraic fractions • Quadratic equations • The Null Factor Law • Solving the quadratic equation ax2 + bx + c = 0 • Solving quadratic equations with two terms • Applications problem solving.

Summary — Chapter 17: Quadratic Functions Plotting parabolas

• Produce a table of values by substituting each integer value of x into the equation. • Plot a graph by drawing and labelling a set of axes, plotting the points from the table and joining the

points to form a smooth curve. • The axis of symmetry is the line that divides the parabola exactly in half. • The turning point is the point where the graph changes direction or turns. • The turning point is a maximum if it is the highest point on the graph and a minimum if it is the

lowest point on the graph. • The x-intercepts are the x-coordinates of the points where the graph crosses the x-axis. • The y-intercept is the y-coordinate of the point where the graph crosses the y-axis.

Sketching parabolas

• If the graph of 2y x= is translated c units vertically, the equation becomes 2y x c= + . • If the graph of 2y x= is translated h units horizontally, the equation becomes ( )2y x h= − .


• If the graph of 2y x= is dilated by factor a , the graph becomes narrower if 1a > and wider if 0 1a< < .

• If the 2x term is positive, the graph is upright. If there is a negative sign in front of the 2x term, the graph is inverted.

• Invariant points are points that do not change under a transformation. Sketching parabolas in turning point form

• If the equation of a parabola is in turning point form, ( )2y a x h k= − + , then the turning point is

( ),h k . • If a is positive, the graph is upright with a minimum turning point. • If a is negative, the graph is inverted with a maximum turning point. • If the magnitude of a is greater than 1, the graph is narrower than the graph of 2y x= . • If the magnitude of a is between 0 and 1, the graph is wider than the graph of 2y x= . • To find the y-intercept, substitute x = 0 into the equation. • To find the x-intercepts, substitute y = 0 into the equation and solve for x.

Sketching parabolas of the form y = ax2 + bx + c

• If the equation is in the form 2y ax bx c= + + , the coordinates of the turning point can be found by: o using the completing the square method to change the equation into turning point form o finding the x-coordinate of the point exactly halfway between the two x-intercepts (this is the

x-coordinate of the turning point), then substituting the x-value into the equation to find the y-coordinate.

o using 2bxa

= − , then substitute the x-value into the equation to find the

y-coordinate. • The graph should also show both the y-intercept and the x-intercepts of the parabola if they exist.

Summary — Topic 13: Probability Review of probability

• Probabilities can be expressed as percentages, fractions or decimals in the range 0 to 1 (inclusive).

• Experimental probability number of times an event has occurredtotal number of trials

= .

• Relative frequency of a score frequency of the score or total sum of frequencies

f

f=

∑

• The theoretical probability that an event, E, will occur is ( )P( )( )

n EEn ζ

=

where n(E) = number of times or ways an event, E, can occur and ( )n ζ = the total number of ways all outcomes can occur.

• P( ) 1ζ = • Venn diagrams provide a diagrammatic representation of sample spaces.

Complementary and mutually exclusive events

• Complementary events have no common elements and together make up the universal set. • If A and A′ are complementary events, then P(A) + P(A′) = 1. This may be rearranged to:

o P(A′) = 1 − P(A) or P(A) = 1 − P(A′). • Mutually exclusive events have no common elements and cannot occur simultaneously. • If events A and B are not mutually exclusive, then:


o P(A or B) = P(A) + P(B) − P(A and B)or

o P( ) P( ) P( ) P( )A B A B A B∪ = + − ∩ , where P( )A B∩ is the probability of the intersection ofsets A and B, or the common elements in sets A and B. This is the Addition Law ofProbability.

• If events A and B are mutually exclusive, then:o P(A or B) = P(A) + P(B)

oro P( ) P( ) P( )A B A B∪ = + , since P( ) 0A B∩ = .

• Mutually exclusive events may or may not be complementary events.• Complementary events are always mutually exclusive.

Tree diagrams • Tree diagrams are useful in working out the sample space and calculating probabilities of various

events, especially if there is more than one event. On each branch of a tree diagram, the probabilityassociated with the branch is listed. The products of the probabilities given on the branches are takento calculate the probability for an outcome.

• The probabilities of all outcomes add to 1.Independent and dependent events

• Events are independent if the occurrence of one event does not affect the occurrence of the other.• If events A and B are independent, then P(A )B∩ = P( )A ×P(B) . This is the Multiplication Law of

Probability. Conversely, if P( )A P(× )B P(= A∩ B) , then events A and B are independent.• Dependent events affect the probability of occurrence of one another.• mutually exclusive events.

Ch 7: Linear 1 Using the following table of values, plot the

points and join them with a straight line.

3 2 1 0 1 2 3 48 7 6 5 4 3 2 1

xy

− − −


2 For the rule y = 3x, complete the following table of values, plot the points and then join them to form a linear graph.

4 3 2 1 0 1 2 3 4xy

− − −


3 (a) Calculate the gradient of the line shown.

(b) Calculate the gradient of the line shown.


4 State the gradient of the following lines shown. (a)

(b)

5 Find the gradient (m) of the line joining the points (1, 4) and (5, 9).

6 Find the gradient (m) of the line joining the points (−3, 4) and (5, −6).


7 Find the gradient of the rule y = 7x + 2.

8 Find the gradient of the rule 3x – 5y = 10.

9 Find the y-intercept of the rule y = 4x – 3.

10 Find the gradient and y-intercept for the rule 2x + 3y – 6 = 0.


11 For the rule y = 2x + 2, complete the following table of values, plot the points and then join them to form a linear graph.

5 4 3 2 1 0 1 2 3 4xy

− − − − −

12 Sketch the graph of y = –4x – 9 using the gradient–intercept method.


13 Without completing a table of values, draw the graph of y = −4x – 9, for x = −20 to +20.

14 Sketch the graph of y = 5x – 10 using the x- and y-intercept method.

15 Sketch the graph of y = 6.


16 Sketch the graph of x = −1.

17 Determine the rule for the straight line whose x-intercept = 10 and y-intercept = –5.

18 Determine the rule for the straight line with a gradient = −1.7 and which passes through the point (0.2, −1.8).


19 Determine the rule for the straight line passing through (−4, 10) and (10, 38).

20 Determine the linear rule for the line passing through the origin and through the point (−5, 12).

Ch16: Quadratic Equations

1 Identify the following equations as linear, quadratic or other. (a) x + 3x2 = 0 (b) x + 3x2 = x3 (c) x + 3 = 4x – 2 (d) 4y2 – 2y + 4 = 17y2


2 Identify the following equations as linear, quadratic or other.

(a) 22 53 x

xx =+

(b) 0 = p2 – 2p + 1

(c) 2

2 372x

xx =−+

(d) 6(x – 2) = 5(x + 3) – 4(x – 3)

3 Rearrange each of the following quadratic equations so that they are in general/standard form, that is, ax2 + bx + c = 0. (a) 2 − 4x2 = 5x

(b) 3(x − 4) = x(x + 2)

(c) x(3 − 2x) = 2 + x(3x − 4)

4 Solve the quadratic equation: (x – 2)(x + 3) = 0.


5 Solve the quadratic equation: (2x – 5)(3x + 8) = 0.

6 Solve the quadratic equation: (x + 4)2 = 0.

7 Solve the quadratic equation: p(−2p – 7) = 0.

8 Solve the quadratic equation: (4y – 5)(4y + 5) = 0.


9 Solve the quadratic equation: (2x − 3)2 = 0.

10 A ball thrown upwards follows the path of the function h = (t + 0.75)(6 − 1.5t), where h metres is the height of the ball t seconds after it has been thrown. Find how long it takes for the ball to reach the ground after it has been thrown.

21 Identify the following equations as linear, quadratic or other. (e) 5x4 + 6 = 0(f) 8x – 2x2 = x(g) 8x + 6 = 4x – 2(h) 4y2 – 2y + 4 = 17y2


22 Rearrange the following quadratic equations so that they are in general/standard form; that is, ax2 + bx + c = 0. (a) 2 − 3x = 4x2 − 6

(b) x(2 − x) + 3 = 3(x2 + 5)

23 Solve the quadratic equation: (3x – 6)(2x + 9) = 0.

24 Solve the quadratic equation: −2x2 + 18 = 0.

25 Solve the quadratic equation: 8x2 – 32x = 0.


26 Solve the quadratic equation: 5x2 + 50x = 0.

27 Solve the quadratic equation: x2 + 5x – 14 = 0.

28 Solve the quadratic equation: x2 − 5x − 24 = 0.

29 Solve the quadratic equation: −2(x + 3)2 + 50 = 0.


30 Solve the quadratic equation: 0.2(x + 1.1)2 – 7.2 = 0.

Ch 17: Quadratic Graphs

1 How is a quadratic graph affected by placing a coefficient of: (a) 3 in front of the x2?

(b) 21 in front of the x2?

2 How does a negative sign in front of the x2, in aquadratic equation, affect the graph?

3 Consider the graph y = x2 – 7x + 18. When x = 2, what will be the value of y?


4 For the graph drawn below, state:

(a) the turning point(b) the equation of the axis of symmetry.

5 Give an example of an equation whose graph would be: (a) upright

(b) inverted (turned upside down).

6 Sketch the graph of y = x2 and y =21 x2 on the

axes provided.

7 Sketch the graph of y = −x2 and y = −2x2 on the set of axes below.


8 How does adding a constant value to a quadratic equation move the graph?

9 Sketch the graph of y = x2 – 2 on the axes provided. Label the turning point.

10 The vertical cross-section through the top of Devil’s Tower mountain can be approximated by the graph y = −x2 + 3.

(a) Sketch the graph of y = −x2 + 3 on theaxes provided. Label the turning point.

(b) If the x-axis represents sea level, andboth x and y are in kilometres, find themaximum height of the mountain.

11 For the graph of the equation y = ax2 what is the effect of a value of:

(a) a = 41

? (b) a = −3?


12 For the graph of the equation y = x2 + c what is the effect of a value of: (a) c > 0?

(b) c < 0?

13 For the graph of the equation y = (x – b) what is the effect of a value of: (a) b > 0?

(b) b < 0?

14 (a) Sketch the graph of y = (x + 2) on the axes provided.

(b) Where is the turning point and the axis ofsymmetry?

(a)

2

2


15 (a) Sketch the graph of y = (x – 2) on the axes provided.

(b) Where is the turning point and the axisof symmetry?

16 For the equation y = –(x + 3) state: (a) the co-ordinates of the turning point(b) if it is a maximum or minimum turning

point(c) the equation of the axis of symmetry.

17 (a) Sketch the graph of y = (x – 3) + 4 on the axes provided.

(b) Label the turning point and the axis ofsymmetry.

2

2

2

18 (a) Sketch the graph of y = −(x + 2) − 3 on the axes provided.

(b) Label the turning point, the y-interceptand the axis of symmetry.

19 For y = (x + 1) − 2 state: (a) the co-ordinates of the turning point(b) if it is a maximum or minimum turning

point(c) the equation of the axis of symmetry.

20 State a possible equation of a quadratic function with a minimum turning point of (1, −4).

21 On the set of axes provided sketch the graphs of: (a) y = x2

(b) y = −x2.

2

2


(b) y =41 x2

(c) y = 4x2.


(b) y = x2 + 2(c) y = x2 – 1.


(b) y = (x – 2)2

(c) y = (x + 1)2.

25 On the set of axes provided sketch the graph of y = (x – 3)2 – 2. Label the turning point and axis of symmetry.

26 Factorise the quadratic equation: y = x2 + 6x − 72.

27 What are the two x-intercepts of the graph: y = (x + 4)(x – 3)?

28 Find the turning point of the graph: y = (x – 1)(x + 5).


29 Sketch the graph y = (x – 5)(x + 3) by finding the x-intercepts and turning point first.


30 Sketch y = x2 – 6x + 8 by: (a) factorising to express it in intercept form (b) finding the x-intercepts finding the turning point.

Ch 13 Probability


1 Rank the following events on a scale from least likely (probability close to 0) to most likely (probability close to 1). (a) The sun will rise tomorrow. (b) Your favourite football team will win

the Grand Final. (c) A pig will fly. (d) It will rain tomorrow. (e) It will rain 40 days in a row. (f) The answer to a question in your

textbook is printed correctly.

2 A bag contains 5 red cards, 5 yellow cards and 1 black card. Two cards are drawn at random. List the sample space for this experiment.

3 A six-sided die is tossed and two coins are tossed. List the sample space for this experiment.


4 Which of the following events are equally likely: (a) a coin landing heads or tails? (b) a football team winning or losing the

Grand Final? (c) selecting a picture card or a non picture

card from a standard deck of cards?

5 Give an example of an event that has a probability of: (a) 0

(b) 21

(c) 1.

6 A factory produces 10 000 light bulbs each day. A certain number are defective. How can the factory determine the proportion of defective light bulbs?

7 A car manufacturer produces a certain number of cars each day, in various colours, as shown in the table below.

1256402334NumberblackwhitebluelimeredColour

Find the probability that a car is either black or white.


8 A cereal manufacturer offers up to 3 prizes in each packet of cereal. Each day, the number of boxes with the various prizes is determined by the following table.

25 150600400boxes ofNumber 3210prizes ofNumber

Find the probability that a box contains 1 prize.

9 Using the data from question 8, determine the probability of obtaining more than 1 prize.

10 A 1 kg packet of lasagne may contain more or less than exactly 1 kg. The manufacturer weighs a number of packets and produces the following table.

334370380390370320Number1100105010251000975950)(gWeight

Determine the proportion of packets that are overweight.


11 Which of the following are equally likely events? (a) Getting a Head when tossing 1 coin. (b) Getting at least one Head when tossing

2 coins. (c) Getting a 3 when tossing a single die. (d) Getting an even number when tossing a

single die. (e) Getting 1 Head and 1 Tail when tossing

2 coins.

12 A card is drawn from a standard deck. Find the probability of each of the following: (a) selecting an ace (b) selecting a spade (c) selecting the ace of spades (d) selecting a black ace (e) selecting a black picture card.

13 The heights (in cm) of a class of Year 9 students are recorded in the following table.

1034532Number162160158156154152150Height

Find the probability that a student is between 153 and 159 cm tall.


14 Using the data from question 3, what could you say about the number of Year 9 students who are shorter than 151 cm in a school where there are 198 Year 9 students in total?

15 A pharmacy is able to fill the prescriptions for

2019

of its customers. If the pharmacy expects 300 customers today, how many prescriptions can it expect to fill?

16 Consider the following table that shows the number of different brands of mobile phones owned by males and females in a youth group.

Males FemalesApple 56 120Nokia 34 148Sony 12 136

Samsung 10 108LG 68 224

Determine the relative frequency of Samsung mobiles in the group.


17 Using the data from question 6, a mobile gets lost at the rate of 1 per day. If this mobile can be considered to be randomly selected from the entire group, what is the probability that the phone that is lost is a Nokia phone owned by a male?

18 At a football match 20 Collingwood supporters were asked what other team they disliked most. Their responses were:

• 18 said Carlton • 12 said Essendon.

Show this information in a Venn diagram.

19 A Collingwood supporter is chosen at random. Based on the Venn diagram in question 8. What is the probability that this supporter: (a) dislikes both Carlton and Essendon? (b) dislikes only Carlton? (c) dislikes Essendon?


20 Mr Venn is a mathematics teacher who asks each student in his class if they study geography, history or economics. Here is the data that Mr Venn gathered.

• 10 study geography only • 5 study history only • 8 study economics only • 1 studies all three subjects • 3 study geography and history only • 7 study geography and economics only • None study economics and history only

(a) Display the information in a Venn

diagram. (b) Use the diagram to determine the number

of students from Mr Venss’s maths class in each of the three classes.

(c) How many students are in Mr Venn’s

mathematics class?

21 In the Tattslotto draw there are 45 numbered

balls. Find the probability that the first number drawn is: (a) a 1 (b) a 45 (c) even (d) odd (e) greater than 40.

22 A bag contains 3 white balls, 4 red balls, 5 black balls, 6 green balls and 7 yellow balls. A single ball is drawn from the bag. What is the probability that it is neither white nor black?


23 Consider the following table, showing the number of cars repaired by 5 different mechanics during a day.

162212158fixed Cars54321Mechanic

A customer returns his car because it was not repaired properly. Without knowing which mechanic worked on it, determine the probability that it was mechanic 1.

24 In a survey of voters in Australia, 43% were in favour of increasing the size of the army. If the survey asked 6500 people, how many were not in favour of increasing the size?

25 Consider the following table showing voter preferences in all 6 Australian states.

Labor Liberal GreensQLD 340 620 190NSW 560 442 219VIC 618 589 218TAS 307 419 167SA 478 462 226WA 712 423 178

Determine the relative frequency of Greens supporters.


26 Using the table in question 25, if a voter who participated in the survey is chosen at random, find the probability (as a decimal correct to 2 decimal places) that: (a) they supported the Labor party

(b) they supported the Liberal party (c) they are a Greens supporter from

Victoria.

27 In a class of 40 students, 12 liked both fish and meat, and 6 liked neither. If there were a total of 25 who liked meat, construct a Venn Diagram of this situation.


28 In a gambling game, you win if you draw a ‘winning card’ from a normal deck of 52 playing cards. Winning cards are:

• any spade • king of hearts • queen of diamonds • jack of clubs • any diamond in the range 2 to 10.

Is this a ‘fair’ game? (Do you have approximately the same chance of winning as of losing?)

29 Two dice are rolled, and the outcome is a pair of numbers. Determine the probability that the sum of the two dice is 8 given that their total is greater than 6.

30 In a survey, the probability that a person drinks

coffee is 1725

. The probability that a person

drinks tea, given that they also drink coffee, is 1334

. What is the probability that a person

selected at random drinks both coffee and tea?

1 R E V I S I O N

STUDENT NAME:

SUBJECT TEACHER: TUTOR GROUP:

Year 9

Extension Mathematics Semester 2 -2016 Written examination

Reading time: 10 minutes Writing time: 90 Minutes

QUESTION AND ANSWER BOOK Structure of book

Section Number of questions to be answered

Number of marks

Suggested time

per section A: Vocabulary 5 5 6 mins B: Multiple Choice 20 20 25 mins C: Short Answer 10 38 47 mins D: Analysis 1 10 12 mins

Total = 73 marks

Total = 90 mins

INSTRUCTIONS TO STUDENTS • Students are permitted to bring into the examination room: pens, pencils, highlighters, erasers,

sharpeners and rulers.• Students are NOT permitted to bring into the examination room: blank sheets of paper and/or

white out liquid/tape.• Calculators are allowed in this examination. A formula sheet and multiple choice answer sheet

are provided.Materials supplied • Additional space is available at the end of the book if you need extra paper to complete an answer.

Instructions • Write your student name, tutor group & subject teacher’s name in the space provided above

on this page.• All written responses must be in English.

Students are NOT permitted to bring mobile phones, mp3 players and/or any other unauthorised electronic devices into the examination room.

2 R E V I S I O N

SECTION A: VOCABULARY KNOWLEDGE (5 marks)

Choose the correct word from the list below.

1. When the equation of a straight line is written in the form y = mx + c, the c represents the _____________.

2. If a quadratic trinomial is in the form ax² + bx + c, and a = 1, then it is called a

___________________quadratic trinomial.

3. If two events cannot both occur at the same time then it is said the two events are ________.

.

4. Adding a constant, c, to the rule y = ax² translates the graph ____________________.

5. To find a ____________, one quantity is divided by another.

Word list

horizontally rate y-intercept

x-intercept gradient mutually exclusive

proportion monic dependent

non-monic polynomial vertically

END OF SECTION A

3 R E V I S I O N

SECTION B: MULTIPLE CHOICE QUESTIONS (20 marks)

Circle the most correct choice on the answer sheet provided.

All questions are worth 1 mark each.

Question 1

The gradient of the line joining the points (-2, 3) and (3, 3) is:

A. 0

B. 15

C. 1

D. 5

E. 10

Question 2

The y-intercept of the graph of the linear equation y + 2x = 3 is:

A. -3

B. -2

C. 2

D. 3

E. 23

4 R E V I S I O N

Question 3

The equation of the unlabelled parabola shown below is:

A. y = x² + 2

B. y = x² − 2

C. y = (x − 2)²

D. y = (x + 2)²

E. y = ((x + 2)² + 2

Question 4

A card is drawn from a pack of 52 playing cards. Determine the probability of not drawing a

diamond.

A. 152

B. 113

C. 14

D. 34

E. 1213

5 R E V I S I O N

Question 5

The following Venn diagram shows some characteristics of teacher’s at Sea Star College. If one

teacher is selected at random, what is the probability that he or she is over 30 years of age, plays

golf and has a child?

answer= 722

Question 6

The letters A, B, C, D and E are written on identical pieces of card and placed in a box. A letter is

drawn at random from the box. After replacing the first card, a second card is drawn. Find the

probability that B is drawn both times.

A. 19

B. 110

C. 125

D. 120

E. 15

Question 7

To obtain the graph of y = x2 - 5 we translate the graph y = x2 :

A. 5 units to the right

B. 5 units to the left

C. 5 units down

D. 5 units up

E. 2 units to the right

A. 722

B. 717

C. 522

D. 1422

E. 1322

6 R E V I S I O N

Question 8

The expression 16a² + 20b factorises to:

A. ab(16 + 20) B. 16a(a + 20b) C. 4a(4 + 5b) D. 4(4a² + 5b) E. 4(5b² + 4a)

Question 9

The expression x² − 7x + 10 factorises to:

A. (x-7)(x+10) B. (x-5)(x-2) C. (x+5)(x+2) D. (x+5)(x-2) E. (x-5)(x+2)

Question 10

The solution to the quadratic equation 2x² + 8x + 6 = 0 is:

A. -3 and -1 B. 3 and -1 C. 6 and 1 D. -3 and 1 E. -3 and 3

Question 11

The solution to the equation (x + 3)(x + 1)(2x − 5) = 0 is:

A. 3, 1 and 5 B. -3, -1 and - 5

2

C. -3, -1 and 52

D. 3, -1 and - 25

E. -3, 1 and 2

7 R E V I S I O N

Question 12

The axis of symmetry for the following quadratic graph is:

A. y = -1 B. x = 0 C. x = -2 D. y = 2 E. x = 2

Question 13

The coordinate of the turning point for the following quadratic graph is:

A. (-5, 0) B. (0, 15) C. (-4, -1) D. (-3, 0) E. (-1, -4)

8 R E V I S I O N

Question 14

When the dilation of factor of a quadratic graph is greater than 0 but less than 1, when compared to the standard y=x² graph, the graph will be:

A. wider B. narrower C. inverted D. upside down E. exactly the same

Question 15

A group of 80 boys consists of 54 footballers and 35 cricketers. Each member of the group is either a footballer, cricketer or both. You may wish to use the Venn Diagram to assist you. The probability that a randomly selected student is a footballer given that he is also a cricketer is closest to:

A. 0.26

B. 0.44

C. 0.68

D. 0.17

E. 0.63

Question 16

If a ∝ b, and a = 3 when b = 9, the rule linking a and b is:

A. a = b B. b = 2a C. 2a = b D. a = 4b E. b = 3a

9 R E V I S I O N

Question 17

y is inversely proportional to x and y = 20 when x = 4. The constant of proportionality, k, is:

A. 5 B. 40 C. 80 D. 24 E. 16

Question 18

If P(A) = 45 and P(B|A) = 2

3, P(A ∩ B) will be:

A. 810

B. 34

C. 65

D. 23

E. 815

Question 19

The equation of the following linear graph is:

A. y = 3x + 2 B. y = -2x + 3 C. y = 2x + 3 D. y = -2x +4 E. y = 2x - 3

Question 20

A metal bolt of volume 30cm³ has a mass of 90g. Its density (mass per unit of volume) is:

A. 30 g/cm³ B. 90 g/cm³ C. 2700 g/cm³ D. 3 g/cm³ E. 120 g/cm³

END OF SECTION B

10 R E V I S I O N

SECTION C: SHORT ANSWER QUESTIONS (38 marks)

Question 1

For the set of points (2, 3) and (4, 7), find the:

(a) Gradient between the points:

(2 marks)

(b) Distance between the points:

(2 marks)

Question 2

Use the y-intercept and the gradient to sketch the graph of y = 2x + 1.

(2 marks)

11 R E V I S I O N

Question 3

Solve the following equations using the Null Factor Law:

(a) 2x² + 4x = 0

(2 marks)

(b) x² - 8x + 16 = 0

(2 marks) (c) 7x² − 33x + 20 = 0

(2 marks)

12 R E V I S I O N

Question 4

Sketch the graph with the equation y = x² + 5 clearly showing the turning point.

(2 marks)

Question 5

Five coins were flipped a different number of times each and the results recorded in the table below:

(a) What was the relative frequency of heads for Coin 3?

(2 marks)

(b) For which coin was the relative frequency of tails the greatest?

13 R E V I S I O N

(3 marks)

Question 6

Ann-Marie has rolled two dice and is interested in the sum of the two numbers showing uppermost.

(a) List the sample space showing all possible outcomes.

(2 marks)

(b) Using the sample space, find:

(i) Pr(total of 7)

(1 mark)

(ii) Pr(total is less than 6)

(1 mark)

Question 7

Thirty students were asked which subjects they enjoy — Maths, Religion or Science. Five students chose all three subjects. Six students chose Maths and Religion, 7 students chose Maths and Science while 9 chose Religion and Science. Fifteen students chose Maths, 14 students chose Religion and 18 students chose Science.

a) Enter the given information in the Venn diagram below:

14 R E V I S I O N

If a student is selected at random, determine the probability of selecting a student who:

b) Likes Maths

c) Likes all three subjects

d) Does not like Science

e) Does not like Science or Religion

f) Find P[ (Religion ∪ Science) Maths′ ]

(2 + 1 + 1 + 1 + 1 + 1 = 7 marks)

15 R E V I S I O N

Question 8

Using the same set of axes, sketch the graphs of y = −x²+ 2 and y = −3x²+ 2, marking

the coordinates of the turning point and the intercepts. State which graph is narrower.

(3 marks)

Question 9

Peter works part time and is paid a fixed rate per hour. If he earns $140 for 8 hours work, how much would he earn in 12 hours?

(3 marks)

Question 10

Find the new equation when the following transformations are performed on the graph of y = x2.

(a) The graph is dilated by a factor of 7 and translated 8 units up.

16 R E V I S I O N

(b) The graph is dilated by a factor of 13 and translated down 2 units.

(2 marks)

END OF SECTION C

17 R E V I S I O N

SECTION D: ANALYSIS QUESTION (10 marks)

The graph below shows the path of the feet of a diver, diving from a high board. The y-values give the height above the water (metres). The x-values give the horizontal distance from a point below the end of the board (metres).

(a) Is this relationship linear or quadratic? How do you know?

(2 marks)

(b) When the equation of the graph is be written as y = – x2 + x+ a. What is the value of a? How do you know?

(2 marks)

(c) Solve the equation and find the x-intercepts of the graph. What does this solution mean in terms of the diver?

(3 marks)

18 R E V I S I O N

d) Calculate the highest point reached by the diver’s feet.

(2 marks)

(f) Explain why you cannot find a solution to the equation y = 3.

(1 mark)

END OF SECTION D

END OF EXAM

of 22

STUDENT NAME:

SUBJECT TEACHER: TUTOR GROUP:

Year 9

Extension Mathematics Written examination

Reading time: (15 minutes) Writing time: (90 minutes)

Semester Two, 2018

QUESTION AND ANSWER BOOK Structure of book

Section Number of questions

Number of questions to be answered

Number of marks Suggested time per section

A Vocabulary 5 5 5 5 mins B Multiple

Choice 20 20 20 20 mins

C Short Answer 10 10 55 55 mins

D Analysis 2 2 10 10 mins Total = 90 marks Total = 90 mins

INSTRUCTIONS TO STUDENTS • Students are permitted to bring into the examination room: pens, pencils, highlighters, erasers,

sharpeners, rulers and one page A4 double-sided handwritten notes. • Students are NOT permitted to bring into the examination room: blank sheets of paper and/or

white out liquid/tape. • CAS calculator is allowed in this examination. Materials supplied • Additional space is available at the end of the book if you need extra paper to complete an answer. Instructions • Write your student name, tutor group & subject teacher’s name in the space provided above

on this page. • All written responses must be in English. Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the examination room

of 22

SECTION A: VOCABULARY KNOWLEDGE (Total = 5 marks) Choose the correct word from the list below.

1. When the equation of a straight line is written in the form y = mx + c, the c represents the

___________________.

2. When a linear graph has no rise it is said it will have ____________ gradient.

3. A parabola with a minimum turning point will have a coefficient on x².

4. When considering the event A in probability, the event of A is is

given by A’.

5. The ______________________________ rule is known as 𝑎𝑎2 − 𝑏𝑏2 = (𝑎𝑎 − 𝑏𝑏)(𝑎𝑎 + 𝑏𝑏)

Word list horizontally null factor y-intercept difference of two squares gradient mutually exclusive maximum complement independent negative positive zero

END OF SECTION A

of 22

SECTION B: MULTIPLE CHOICE QUESTIONS (Total = 20 marks)

Circle the most correct choice on the answer sheet provided.

All questions are worth 1 mark each.

Question 1

The rule for a line whose gradient is 5 and y-intercept = –2 is:

A y = –5x + 2

B y = –2x + 5

C y = 5x – 2

D y = 2x – 5

E y = 3x – 2

Question 2

For the rule y = 3x – 17, what is y when x = 4?

A –2

B 3

C –17

D –29

E –5

Question 3

The x-intercept of the rule y = 3x – 18 is:

A y = –5

B x = –6

C x = 2

D x = 6

E x = –5

Continue Next Page

of 22

Question 4

For the rule y = 7x – 2, the gradient and y intercepts, respectively are:

A –7 and –2

B 7 and –2

C 2 and –7

D –2 and –7

E 0 and –2

Question 5

A straight line passes through the points (0, 4) and (3, 10). Its rule is:

A y = 2x

B y = 2x + 4

C y = 4x + 10

D y = 3x + 10

E y = 6x + 3

Question 6

What is the gradient of the line represented by the equation 3𝑦𝑦 = −15𝑥𝑥 + 9?

A 15

B –15

C 9

D –5

E 1

Question 7

When factorised and simplified, 𝑥𝑥2+3𝑥𝑥4𝑥𝑥2+12𝑥𝑥

equals:

A 81

B 74

C 41

D 1

E 4

of 22

Question 8

The equivalent of (2𝑥𝑥 − 1)2 is:

A 2𝑥𝑥2 − 1

B 2𝑥𝑥2 + 1

C 4𝑥𝑥2 − 1

D 4𝑥𝑥2 + 1

E 4𝑥𝑥2 − 4𝑥𝑥 + 1

Question 9

The solutions to (3x – 2)(x + 1) = 0 are:

A x = 0, x = –1

B x = 1, x = –2

C x = –1, x = 23

D x = –2, x = –1

E x = 0

Question 10

The solutions to 6x2 – 54 = 0 are:

A x = 6, x = –54

B x = 3

C x = –3

D x = 9, x = –9

E x = 3, x = –3

Question 11

Factorising 18x2 –12x becomes:

A 2x(9x –12)

B 3x(6x – 18)

C 6x(3x – 2)

D 6x

E x(18x – 12)

of 22

Question 12

Compared with the graph y = x2, the graph 𝑦𝑦 = −5𝑥𝑥2 will be:

A upright and moved up

B upright and moved down

C inverted and moved up

D inverted and narrower

E inverted down and wider

Question 13

The x-intercepts of the graph y = x2 – 9 are:

A 9 and 0

B −9 and 9

C 3 and −3

D −9 and 0

E 0 and 1

Question 14

The turning point of the graph y = (x – 4)(x + 2) will be at:

A (−4, 2)

B (−1, −9)

C (4, −2)

D (1, –9)

E (1, 9)

Continue Next Page

of 22

Question 15

The graph with equation y = −x2 + 3 could look like:

A

B

C

D

E

Continue Next Page

of 22

The following is for use in Questions 16 and 17

Consider the following table of 180 students who played sport in 2018.

Year 9

Football 60

Soccer 40

Cricket 35

Basketball 25

Table Tennis 20

Question 16

If a student is chosen at random, then the probability that the student chosen played soccer or

table tennis is:

A 92

B 31

C 91

D 61

E 21

Continue Next Page

of 22

Question 17

The probability that a student chosen at random did not play cricket is:

A 9029

B 3629

C 351

D 21

E 0

Question 18

In the figure below, the number of elements in set A∩B is equal to:

A 25

B 36

C 11

D 16

E 27

Continue Next Page

of 22

Question 19

What is the complementary probability to getting a factor of 20 when rolling a 10 fair sided

regular die?

A 21

B 61

C 51

D 65

E 1

Question 20

In picking a random card from a regular deck of 52 cards, what is the probability of drawing

a red heart?

A 135

B 41

C 131

D 265

E 0

Continue Next Page

END OF SECTION B

of 22

SECTION C: SHORT ANSWER QUESTIONS (Total = 55 marks)

Question 1 (3 marks)

Sketch the graph of the line with equation y – 2x – 8 = 0 using x and y Intercept Method.

Make sure you clearly show how you obtained the two intercepts and label appropriately.

Continue Next Page

of 22

Question 2 (2+2+2+2+2 marks)

For the set of points A and B with coordinates (-2, 5) and (4, 13), find the:

a) Gradient of the line passing through points A and B.

b) Equation of the line passing through the points A and B.

c) Distance between the points A and B, correct to one decimal place.

d) Midpoint of the points A and B.

e) The equation of the line parallel to the line AB and passing through the point (0, − 4)

of 22

Question 3 (3 + 3 marks)

Fully factorise the following:

a) 𝑥𝑥2 − 3𝑥𝑥 − 28

b) 2𝑥𝑥2 − 14𝑥𝑥 − 16


Factorise and simplify

𝑥𝑥2 + 7𝑥𝑥 + 10𝑥𝑥2 − 6𝑥𝑥 + 8

÷ 2𝑥𝑥 + 102𝑥𝑥 − 8

Continue Next Page

of 22


Fully factorise the following by using the Difference of Two Squares rule

49𝑥𝑥2 − 64𝑦𝑦2

Question 6 (2+2 marks)

Factorise first and then solve the following equations using the Null Factor Law:

a) 30𝑥𝑥 − 6𝑥𝑥2 = 0

b) 𝑥𝑥2 − 2𝑥𝑥 − 15 = 0

Continue Next Page

of 22

Question 7 (1+1+2+2+1+2 marks)

For the equation 𝑦𝑦 = (𝑥𝑥 − 3)2 − 1

a) Using the case of y = x², state what is the dilation in 𝑦𝑦 = (𝑥𝑥 − 3)2 − 1

b) Using the case of y = x², state what is the translation in 𝑦𝑦 = (𝑥𝑥 − 3)2 − 1

c) Find the y intercept for the above equation.

d) Find x intercepts for the above equation (if any).

e) State the coordinate of the Turning Point and if it is a maximum or minimum.

Continue Next Page

of 22

f) Sketch the graph of 𝑦𝑦 = (𝑥𝑥 − 3)2 − 1, clearly showing key points and appropriate

labelling.

Question 8 (1+2+2+1+2 marks)

For the equation 𝑦𝑦 = 𝑥𝑥2 − 6𝑥𝑥 + 5

a) Find the y intercept.

b) Find the x intercepts (if any).

Continue Next Page

of 22

c) Find the coordinate of the Turning Point and state if it’s a maximum or minimum.

d) State equation of the Axis of Symmetry.

e) Sketch the graph clearly showing all key points and labelled appropriately

Continue Next Page

of 22

Question 9 (1+1+1 marks)

A bag contains 24 balls of which 8 are red, 6 are blue and the rest are black.

Find:

a) n(not black)

b) P(not black)

c) P(red or blue)


A group of students were surveyed on which of two chocolates, X or Y, they preferred. The

results are shown in the Venn diagram below.

a) How many students were surveyed?

Continue Next Page

of 22

b) What is the probability that a student preferred chocolate only?

c) What is the probability that a student preferred neither chocolate?

Continue Next Page

END OF SECTION C

of 22

SECTION D: ANALYSIS QUESTION (Total = 10 marks)

Question 1 (2+1+1+1 marks)

100 Year 9 students are survey on game console satisfaction.

50 students like Playstation

42 students like Xbox

6 students liked both consoles

a) Draw a Venn diagram of this situation.

b) Find the probability that a student liked Playstation only.

c) Find the probability that a student liked neither console.

d) Find the probability that a student liked both consoles.

of 22


David found a small photo frame and decided to put one of his favourite photos in it.

The width of the metal frame surrounding the photo is x cm.

a) Write expressions for the height and width of the photo (white part).

b) Show that an equation for the area of the photo is A = 270 34 4x x− + cm2.

c) Calculate the area of the photo if its perimeter is 16cm, correct to two decimal places.

END OF SECTION D

END OF EXAM

of 22

THIS PAGE HAS BEEN LEFT BLANK FOR EXTRA WORKING

year 9 extension mathematics semester 2 exam preparation ...€¦ · sketching parabolas of the...

Documents