Trial HSC 2004

2013 Trial HSC ExaminationGeneral Mathematics

2013

HIGHER SCHOOL CERTIFICATE

TRIAL EXAMINATION

General Mathematics

(General InstructionsReading Time - 5 minutesWorking Time 2 hoursWrite using a blue or black penBoard approved calculators may be usedA Formulae Sheet is provided at the back of this paper which may be detached and used throughout the paper)

(Total marks 100 25 marksAttempt Questions 1-25Answer on the Multiple Choice answer sheet providedAllow about 40 minutes for this section 75 marksAttempt questions 26 30Answer on the blank paper provided, unless otherwise instructed. Start a new sheet for each questionAllow about 1 hour & 50 minutes for this section)

(Section I)

(Section II)

Section I

Total marks (25)

Attempt Questions 1-25NAME: __________________________

Allow about 40 minutes for this sectionTEACHER:_______________________

Use the multiple choice answer sheet.

Select the alternative A, B, C or D that best answers the question. Fill in the response oval completely.

Sample:

2 + 4 =

(A) 2

(B) 6

(C) 8

(D) 9

A

B

C

D

If you think you have made a mistake, put a cross through the incorrect answer and fill in the new answer.

A

B

C

D

If you change your mind and have crossed out what you consider to be the correct answer, then indicate the correct answer by writing the word correct and drawing an arrow as follows.

A

B

C

D

Start

Here

1.

A

B

C

D

14.

A

B

C

D

2.

A

B

C

D

15.

A

B

C

D

3.

A

B

C

D

16.

A

B

C

D

4.

A

B

C

D

17.

A

B

C

D

5.

A

B

C

D

18.

A

B

C

D

6.

A

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D

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7.

A

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20.

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8.

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25.

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13.

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B

C

D

1. Expand and simplify fully:

(A)

(B)

(C)

(D)

2. Ben surveyed his soccer teams shoe sizes. He displayed the data on a dot plot.

8

9

10

11

12

13

14

Describe the skewness of Bens soccer teams shoe sizes.

(A) Positively Skewed

(B) Evenly Distributed

(C) Negatively skewed

(D) Normally Distributed

3. Gregs normal hourly rate is $23.50 for a 35 hours week and he is paid time and a half for all overtime. In one week he earns $1 175. How many hours of overtime did he do?

(A) 10 hours

(B) 15 hours

(C) 50 hours

(D) 8 hours

4. Simplify

(A)

(B)

(C)

(D)

5. Cassie is late for work and she still needs to get through three sets of traffic lights. If the probability of getting a green light is 0.37. What is the probability she will not get three green lights?

(A) 0.7

(B) 0.95

(C) 0.05

(D) 0.3

6. If there is a positive correlation between two variables then:

(A) There is no relationship between the two variables.

(B) As one variable increases, the other decreases.

(C) As one variable increases, the other increases as well.

(D) The line of best fit comparing the variables has a negative gradient.

7. Savannah wants to have $1 000 000 in her superannuation fund in 25 years for when she retires. If the best interest rate she can find is 6.8% p.a. compounded annually. Calculate the amount of each annual contribution she will need to make.

(A) $76167.95

(B) $193071.73

(C) $5179422.24

(D) $16270.19

8. The population of Australia is approximately people. If we continue to increase our population by 378 000 people each year. What will be the estimated population in 10 years? (Answer in scientific notation correct to 2 significant figures)

(A)

(B)

(C)

(D)

9. An IQ test is normally distributed with a mean IQ of 100 and a standard deviation of 18. Sheldon is tested and he has an IQ of 180, what does this mean?

(A) Sheldon has a z-score of 2.

(B) Sheldon is smarter than 99.7% of people.

(C) Sheldons z-score lies between -3 and 3.

(D) Sheldon failed the IQ test.

10. Frances and seven other students in her class nominate to be class captain and vicecaptain. What is the probability that Frances will get one of these jobs?

(A)

(B)

(C)

(D)

11. Gabriel wants to know how much her monthly repayment would be on a $10000 loan at 7.5% p.a. reducible interest over 4 years. Which of the following will give her the correct monthly repayment?

(A)

(B)

(C)

(D)

The following information is used to answer Q12 and Q13

The Australian netball squad has 12 players. The mean height of the 12 players is 1.75m and the standard deviation in the heights of the players is 0.2m. In the first game the shortest player, who is 1.6m tall breaks her leg and must withdraw from the squad. A replacement player is brought into the squad who is 1.73m tall.

12. After this replacement what is the new mean height of the Australian team?

(A) 1.74m

(B) 1.76m

(C) There is not enough information to work out the new mean.

(D) 1.75m

13. What effect will the replacement have upon the standard deviation?

(A) The standard deviation will increase.

(B) The standard deviation will decrease.

(C) The standard deviation will remain unchanged.

(D) The effect of the standard deviation cannot be calculated.

14. . Find the value of x, correct to 1 decimal place, in the right triangle below.

(x26m )

(A) 55.4m

(B) 18.4m

(C) 12.2m

(D) 23.0m

15. Lleyton Hewitt has a chance of beating Roger Federer. They have played each other 20 times. Based on this, how many times has Roger won?

(A) 18

(B) 6

(C) 20

(D) 14

16. The volume of a sphere with diameter (d) is given by the formula

The formula for the diameter would be given by:

(A)

(B)

(C)

(D)

17. The radius of Earth is approximately 6400km at the equator. If two cities lie on the equator and the angle subtended between them is 230. What is the shortest distance to the nearest kilometre between the two cities?

(A) 14521km

(B) 40212km

(C) 25691km

(D) 29042km

18. The speed limit on the M5 Motorway is 110km/h. This is equivalent to:

(A) 1.83m/s

(B) 30.6m/s

(C) 1833.3m/s

(D) 110m/s

19. The mean and population standard deviation of the following set of scores is:

stem

leaves

4

1111449

5

0289

6

1359

(A)

(B)

(C)

(D)

20. Jack takes out a simple interest loan for $2000 over 5 years. The simple interest is calculated monthly. The total amount he repays for this loan is $3600. The simple interest rate per year on this loan is closest to:

(A) 8.9%

(B) 16%

(C) 36%

(D) 5%

21. The following statistics were obtained from History and Geography tests.

Subject

Mean

Standard Deviation

History

55

12

Geography

66

8

What mark in History would be equivalent to a mark of 78 in Geography

(A) 67

(B) 71

(C) 73

(D) 75

22. Which of the following graphs could have the equation for x > 0

(A)

(B)

(C)

(D)

23. Kobi wants to buy shares in a medical research company. Over the past 10 years the share price in the company has risen by $6 per year for three of those years and fallen by $1.70 per year for the other seven years. Based upon these results if he purchased shares, what would his financial expectation be for the year ahead?

(A) $6.00

(B) -$1.70

(C) -$1.60

(D) $0.61

24. Matt and Indiana plan an overseas trip in 5 years time, they need $20 000 each. Matt pays $200 per month into an annuity that pays 6% p.a. interest compounded quarterly. Indiana wants to invest a single sum of money now under the same conditions as Matt. How much does Indiana need to invest to have the same amount of money as Matt in 5 years?

(A) $14 945.16

(B) $606.29

(C) $18 565.21

(D) $8 185.92

25. Penny is superstitious and can only live in a house with 2 digits in the address and the number must be greater than 35. If the only streets available have the possible digits 2, 3, 4 and 7. How many possible houses can Penny live in?

(A) 9

(B) 12

(C) 16

(D) 7

End of Section I

Section II

Total marks (75)

Attempt Questions 26-30

Allow about 1 hour & 50 minutes for this section

Answer all questions, starting each question on a new sheet of paper with your name and

question number at the top of the page. Do not write on the back of sheets.

Question 26 (15 marks) Use a separate sheet of paper

Marks

a) Solve the following equation:

3

b) Emma collected everyones phone before a test and placed them in a box at the front of the room. While the class was working on the test Miss Riley counted the phones and started drawing a probability tree for the phone brands on the board.

1st Phone Chosen

2nd Phone Chosen

Apple

Apple

Samsung

HTC

i) Copy and complete the tree diagram.

2

ii) Miss Riley hands back the phones at the end of the lesson. Calculate the probability the first 2 phones she selects are HTC.

1

iii) Calculate the probability that at least one of the first two phones selected is a Samsung.

2

Question 26 continues over the page

Question 26 continues

Marks

c) Georgie and James want to know the height of a building near their school.

James measured the length of the shadow cast by the building at a certain time and

also the shadow cast by a 1 metre ruler at the same time. His results are shown

below:

i) What is the height of the building to the nearest metre

1

ii) Georgie used a clinometer to measure the angle of elevation of the building from a distance of 10m from its base. Hence or otherwise using your answer from part i), what angle must have Georgies clinometer have given to the nearest minute?

2

d) A sailor in a boat is lost at sea and his emergency beacon has sent back its last known coordinates as

i)

A rescue helicopter must be sent, who is closer, one from Perth or Durban

2

ii)

After the call out it is realised a cruise ship returning from Antarctica is at and can travel at 40 knots. How long will it take to reach the ship? (If 1 on a great circle = 60M)

2

End Question 26

Question 27 (15 marks) Use a separate sheet of paper

Marks

a) Rhiannah collected 20 peoples mobile phone bills for the month of June. Ten people had their phone with Melstra and ten people used Doptus. Rhiannah created the following 5 number summaries for the data she collected.

Melstra

Doptus

Lower Extreme

10

Lower Extreme

50

Lower Quartile

50

Lower Quartile

65

Median

70

Median

80

Upper Quartile

80

Upper Quartile

90

Upper Extreme

95

Upper Extreme

100

i) Display this data in a box and whisker plot using the axis on page 19.

2

ii) Which company has the largest interquartile range and how much more is it?

2

iii) Describe the skewness of Melstras data set.

1

iv) Which company appears to have the higher bills? Give a reason why this may not be necessarily true.

2

b) Aziz pays a monthly contribution of 9% of her salary to her superannuation fund. Her employer also contributes an additional 3% of her salary to her superannuation fund. Azizs annual salary is $89500. The superannuation fund pays 5.4% p.a. compounded monthly.

i) What is the total amount contributed to her superannuation fund each month?

1

ii) If Aziz made her contributions at the end of every month starting in July 2013. What will her super fund balance be at the end of 2040, after she has deposited the last payment in June 2040?

2

c) Peter needs to create a new pass code for his bank account. The pass code must be 4 characters long. It can consist of any letter, digit or use the following 3 special characters ( * , ! , - )

i) How many different pass codes can Peter choose from?

1

ii) If Peter can only remember the first and last characters of his pass code

N _ _ *

What is the probability Peter can guess the correct pass code?

2

d) Simplify fully

2

End Question 27

Question 28 (15 marks) Use a separate sheet of paper

Marks

a)

Joanne is starting a new business and her sign is displayed above. The shape of the sign is an elliptical prism which is 4 cm thick; with 2 identical hexagonal prisms cut out of the solid. The sign is to be made of glass of negligible thickness and will be filled with a neon liquid gel.

i) Show the volume of the sign is 0.134 m3

2

ii) How much is this in litres if 1kL = 1m3

1

iii) If the neon gel costs $125/5L container. How much will the gel cost to fill the sign?

2

iv) How much gel will be left over?

1

b) Jayden and Stephen are bushwalking in the Blue Mountains. They park their car in the car park and take different walking tracks to 2 different lookouts. Jayden walks 3.7km on a bearing of 300 while Stephen walks 2.4km on a bearing of 73.

i) Draw a diagram to represent Jayden and Stephens positions

1

ii) How far apart are Jayden and Stephen when they reach their lookouts?

2

iii) The parks and wildlife department has just planted native trees in this enclosed triangular area, bounded by the car park and the 2 lookouts. What area to the nearest hectare have they covered with native trees?

3

iv) These trees were planted to increase the population of Black Cockatoos. While on their bushwalk Jayden and Stephen catch, tag and release 10 Black Cockatoos. Two weeks later they return and catch 8 Black Cockatoos two of which has a tag. They estimate by planting the native trees, in a years time the population of Black Cockatoos should be 27% more than now. What will the population be in a year?

2

c) Explain using calculations or otherwise why the number of ways that two people can be selected from ten for a committee is the same as the number of ways that eight people can be selected from ten for a committee.

1

End Question 28

Question 29 (15 marks) Use a separate sheet of paper

Marks

a) A football coach analyses his teams performance in wet and dry conditions during the season. He prepares a two way table to show the team the results.

Win

Lose

Total

Dry Conditions

38

4

42

Wet Conditions

10

8

18

Total

48

12

60

i) What percentage of games did the team lose?

1

ii) What percentage of games that were played in dry conditions resulted in a win?

1

iii) Do you think the weather had an impact on the teams performance? Explain your answer.

2

b) Jesse buys a new Nissan Micra which costs $21 000 and depreciates at 20% p.a.

i) What is the value of the car in 5 years time if she wishes to sell it?

1

ii) Estimate to the nearest year when Jesse needs to sell the car if she doesnt want to lose more than $11 000. (Use the guess and check method.)

2

c)

Lake Frosty is shown above. Using two applications of Simpsons Rule estimate the area of the lake.

2

d) Lachlan works as a TV presenter in Los Angles (120W). His show is televised at 9:00pm on Tuesday local time. His mum wants to watch him live on the internet, she lives in Sydney (150E). On what day and at what time must she watch Lachlans show?

2

Question 29 continues over the page

Question 29 continued.

Marks

e) Eli lives in Sydney and has a soccer ball signed by Sydney FC, which he must send to his cousin Jonathan for his birthday in Perth. The soccer ball is a sphere with diameter 30cm and weighs 800g.

i) What is the surface area of the soccer ball?

2

ii) Eli places the ball in a cubical box that fits it exactly in each direction. What is the volume of this box?

1

iii) Using the table below how much will it cost to send the soccer ball to Jonathon, if Perth is more than 1 000 km from Sydney?

Weight/ Distance

1000km

3kg

$18.20

$20.00

$22.90

1

End Question 29

Question 30 (15 marks) Use a separate sheet of paper

Marks

a) Better Marks High School wanted to know if studying had any effect on the marks gained in examinations. The table below shows the number of hours that each student in a particular class spent studying for a mathematics examination over a 2week period and the marks each student achieved.

Hours Studied

Mark

Hours Studied

Mark

1

40

12

77

3

58

15

100

7

65

19

90

8

69

20

85

10

72

i) On the graph provided on page 19. Draw a 3 median regression line.

2

ii) Show the gradient of this line is 2.

2

iii) Hence find the equation of this line.

1

iv) From your equation, what mark would you expect if you did no study?

1

v) What relationship does the gradient suggest? Could you guarantee a mark of 100? Explain your answer in terms of correlation.

2

b) The capacity of 600mL Moca Cola bottles is normally distributed with .

i) If I take a bottle of Moca Cola off the production line, what limits will the capacity almost certainly lie between?

1

ii) Moca Cola has a policy that states its bottles must lie between

598mL and 604mL. What percentage of bottles will be rejected?

2

iii) One of the production machines has been recalibrated. It has a . Would this machine be more or less efficient if Moca Colas policy still applies from part ii)? Explain your answer with calculations.

2

c) The time taken to complete a building project varies inversely with the number of workers. If the building can be completed in 150 days by 70 workers, find the number of workers needed to complete the building in 90 days.

2

End of Examination

NAME: ________________________ TEACHER:_____________________________

Question 27 (a) i)

Question 30 (a) i)

19

30

S ,

70

E

30

S ,

120

E

30

S ,

30

E

60

S ,

70

E

3

a

b

2

5

x

3

y

18

a

4

b

(

5

x

)

2

.

2

x

y

(

3

x

5

y

)

x

(

y

2

x

y

)

x

=

600

mL

and

s

n

=

2

mL

x

=

600

mL

and

s

n

=

1

mL

8

x

2

y

10

x

y

2

+

x

y

8

x

2

y

9

x

y

2

4

x

2

y

10

x

y

2

x

y

8

x

2

y

10

x

y

2

x

y

4

x

2

y

3

3

64

x

2

y

6

16

x

5

y

6

64

x

6

y

9

4

x

6

y

9

2.3

10

7

3.0

10

7

2.678

10

7

2.7

10

7

2.68

10

7

1

28

1

4

1

8

1

56

M

=

10000

0.075

1.074

4

1.074

4

1

M

=

10000

0.075

1.074

48

1.074

48

1

M

=

10000

0.006

1.006

4

1.006

4

1

M

=

10000

0.006

1.006

48

1.006

48

1

28

2

7

V

=

4

p

3

d

2

3

d

=

3

32

V

8

p

d

=

6

V

p

d

=

3

3

V

2

p

d

=

3

6

V

p

x

=

51.87

s

=

9.56

x

=

3.87

s

=

3.36

x

=

3.87

s

=

3.24

x

=

51.87

s

=

9.89

y

=

6

x

2

(

3

x

8

)

=

2

x

3

5

19

34

20

35

11

35