SUCCESS ONE HSC...2012 HSC EXAMIN A TION P APE R • QUESTION S Excel SUCCES S ONE HSC • M A THEM...

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HSC*

MATHEMATICSGENERAL 2

2012 HSC Examination Paper

General Mathematics

2012H I G H E R S C H O O L C E R T I F I C AT E

E X A M I N AT I O N

General Instructions

• Reading time – 5 minutes

• Working time – hours

• Write using black or blue penBlack pen is preferred

• Calculators may be used

• A formulae sheet is provided atthe back of this paper

• In Questions 26–30, showrelevant mathematical reasoningand/or calculations

• Write your Centre Number andStudent Number on theQuestion 28 Writing Booklet

2 1–2

Total marks – 100

Pages 2–13

25 marks

• Attempt Questions 1–25

• Allow about 35 minutes for this section

Pages 14–27

75 marks

• Attempt Questions 26–30

• Allow about 1 hour and 55 minutes for this section

Section II

Section I

348

C H A P T E R 1 2 • 2 0 1 2 H S C E X A M I N A T I O N P A P E R

Exce l S U C C E S S O N E H S C • M A T H E M A T I C S G E N E R A L 2

C H A P T E R 1 2

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Section I

25 marksAttempt Questions 1–25Allow about 35 minutes for this section

Use the multiple-choice answer sheet for Questions 1–25.

1 A set of 15 scores is displayed in a stem-and-leaf plot.

What is the median of these scores?

(A) 7

(B) 8

(C) 77

(D) 78

2 Handmade chocolates are checked for size and shape. Every 30th chocolate is sampled.

Which term best describes this type of sampling?

(A) Census

(B) Random

(C) Stratified

(D) Systematic

3 A pair of players is to be selected from 6 people.

How many different pairs of players can be selected?

(A) 6

(B) 12

(C) 15

(D) 30

56789

3 42 6 77 7 8 92 41 3 5 7

349

2 0 1 2 H S C E X A M I N A T I O N P A P E R • Q U E S T I O N S


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Which expression could be used to calculate the value of x in this triangle?

5 The line has intercepts p and q, where p and q are positive integers.

What is the gradient of line ?

(A)

(B)

(C)

(D)

29 40

29 50

40

29

50

29

× °

× °

°

°

cos

cos

cos

cos

y

xO q

p

40°

50°

NOT TOSCALE

x29

4

(A)

(B)

(C)

(D)

−

−

p

q

q

p

p

q

q

p

350



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What is the volume of this rectangular prism in cubic centimetres?

(A) 6 cm3

(B) 600 cm3

(C) 60 000 cm3

(D) 6 000 000 cm3

7 The Pi Company has two bakeries. The radar chart displays the monthly sales for thebakeries.

What was the difference in sales in June between the two bakeries?

(A) $7.50

(B) $17.50

(C) $7500

(D) $17 500

NOT TOSCALE

0.05 m

4 m

30 cm

6

5

10

15

20

25

30

35

5

10

15

20

25

30

35

June

May

April

Monthly sales ($ ’000s)

March

February

January

Monthly salesfor Bakery 1


KEY

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8 Dots were used to create a pattern. The first three shapes in the pattern are shown.

The number of dots used in each shape is recorded in the table.

How many dots would be required for Shape 156?

(A) 316

(B) 520

(C) 624

(D) 936

9 Tracy invests some money for 2 years at 4% per annum, compounded quarterly.

Which figure from the table should Tracy use to calculate the value of her investment atthe end of 2 years?

(A) 1.020

(B) 1.082

(C) 1.083

(D) 1.369

Shape 3Shape 2Shape 1

Shape (S) 1 2 3

Number of dots (N) 6 8 10

Compounded values of $1

PeriodInterest rate per period

1% 2% 3% 4% 5%

1 1.010 1.020 1.030 1.040 1.050

2 1.020 1.040 1.061 1.082 1.103

3 1.030 1.061 1.093 1.125 1.158

4 1.041 1.082 1.126 1.170 1.216

5 1.051 1.104 1.159 1.217 1.276

6 1.062 1.126 1.194 1.265 1.340

7 1.072 1.149 1.230 1.316 1.407

8 1.083 1.172 1.267 1.369 1.477

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What is the area of this triangle, to the nearest square metre?

(A) 33 m2

(B) 37 m2

(C) 42 m2

(D) 44 m2

11 Which of the following relationships would most likely show a negative correlation?

(A) The population of a town and the number of hospitals in that town.

(B) The hours spent training for a race and the time taken to complete the race.

(C) The price per litre of petrol and the number of people riding bicycles to work.

(D) The number of pets per household and the number of computers per household.

12 Two unbiased dice, each with faces numbered 1, 2, 3, 4, 5 and 6, are rolled.

What is the probability of a 6 appearing on at least one of the dice?

NOT TOSCALE8.8 m

50°

57°

9.9 m

10

(A)

(B)

(C)

(D)

16

1136

2536

56

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13 Conversion graphs can be used to convert from one currency to another.

Sarah converted 60 Australian dollars into Euros. She then converted all of these Eurosinto New Zealand dollars.

How much money, in New Zealand dollars, should Sarah have?

(A) $26

(B) $45

(C) $78

(D) $135

0010 20 40 60 80 120 140

Eur

os

0

20

40

60

80

100

Australian dollars

0010 20 40 60 80 120 140

New

Zea

land

dol

lars

0

20

40

60

80

100

Euros

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14 Which of the following expresses 2x2 (5 – x) – x(x – 2) in its simplest form?

(A) –2x3 + 9x2 + 2x

(B) –2x3 – 9x2 – 2x

(C) 9x2 – x + 2

(D) 9x2 – x – 2

15 The time taken to complete a journey varies inversely with the speed of a car. A car takes6 hours to complete a journey when travelling at 60 km/h.

How long would the same journey take if the car were travelling at 100 km/h?

(A) 36 minutes

(B) 1 hour and 40 minutes

(C) 3 hours and 6 minutes

(D) 3 hours and 36 minutes

16 A machine was bought for $25 000.

Which graph best represents the salvage value of the machine over 10 years using thedeclining balance method of depreciation?

(B)

10

25 000

Time (years)

Sal

vage

val

ue (

$)

(C)

10

25 000

Time (years)

Sal

vage

val

ue (

$)

0

0

(A)

100

25 000

Time (years)

Sal

vage

val

ue (

$)

(D)

10

25 000

Time (years)

Sal

vage

val

ue (

$)

0

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17 A spinner with different coloured sectors is spun 40 times. The results are recorded in thetable.

What is the relative frequency of obtaining the colour orange?

18 Jo qualifies for both Rent Assistance and Youth Allowance and receives a fortnightlypayment from the government.

Rent Assistance is $119.40 per fortnight.

The maximum Youth Allowance is $402.70 per fortnight. It is reduced by 50 cents in thedollar for any income earned over $236 per fortnight.

Jo earns $300 per fortnight from a part-time job.

What is the total payment Jo receives each fortnight from the government?

(A) $370.70

(B) $372.10

(C) $458.60

(D) $490.10

Colourobtained

Frequency

RedYellowBlueOrangeGreenPurple

246

1012

(A)

(B)

(C)

(D)

320

15

6

8

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19 A coach compares the resting pulse and the number of weeks of training for nine cyclists.The information is graphed in order to draw a median regression line.

Which of the following graphs best shows the median regression line for the data?

20 7030 40 50 60 80

Wee

ks o

f tra

inin

g

0

2

4

6

8

10

12

Resting pulse (beats per minute)

20 7030 40 50 60 80

Wee

ks o

f tra

inin

g

0

2

4

6

8

10

12(A)


20 7030 40 50 60 80

Wee

ks o

f tra

inin

g

0

2

4

6

8

10

12(C)


20 7030 40 50 60 80

Wee

ks o

f tra

inin

g

0

2

4

6

8

10

12(B)


20 7030 40 50 60 80

Wee

ks o

f tra

inin

g

0

2

4

6

8

10

12(D)


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20 Town B is 80 km due north of Town A and 59 km from Town C.Town A is 31 km from Town C.

What is the bearing of Town C from Town B?

(A) 019°

(B) 122°

(C) 161°

(D) 341°

21 Which of the following correctly expresses c as the subject of E = mc2 + p?

N

C

NOT TOSCALE

31 km

59 km

80 km

A

B

(A)

(B)

(C)

(D)

c E pm

cE pm

c Em p

c Em p

= ±−

= ±−

= ± −

= ± −

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22 A label is designed with a circle inside an ellipse as shown.

What is the area of the shaded part of the label, to the nearest square centimetre?

(A) 29 cm2

(B) 48 cm2

(C) 113 cm2

(D) 244 cm2

23 A football club knows that at the end of 10 years it will need to replace goal posts andother equipment. It is estimated that the replacement cost will be $12 000. For 10 years,the club will invest an amount at the end of each month at 6% per annum, compoundedmonthly.

Which equation should the club use to calculate the amount, M, it will need to depositeach month in order to have $12 000 at the end of 10 years?

4.8 cm4.8 cm6.3 cm6.3 cm 6.3 cm6.3 cm NOT TOSCALE

(A)

(B)

12 0001 0 06 1

0 06

12 0001

10

=+( ) −⎧

⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

=

M

M

.

.

++( ) −

+( )⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

=

0 06 1

0 06 1 0 06

12 0001

10

10

.

. .

(C) M++( ) −

+( )⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

0 005 1

0 005 1 0 005

12 0

120

120

.

. .

(D) 0001 0 005 1

0 005

120

=+( ) −⎧

⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪M

.

.

Question 23 continues on following page

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24 A $400 000 loan can be repaid by making either monthly or fortnightly repayments. Thegraph shows the loan balances over time using these two different methods of repayment.

The monthly repayment is $2796.86 and the fortnightly repayment is $1404.76.

What is the difference in the total interest paid using the two different methods ofrepayment, to the nearest dollar?

(A) $51 596

(B) $166 823

(C) $210 000

(D) $234 936

25 The solid shown is made of a cylinder with a hemisphere (half a sphere) on top.

What is the total surface area of the solid, to the nearest square centimetre?

(A) 628 cm2

(B) 679 cm2

(C) 729 cm2

(D) 829 cm2

400 000350 000300 000250 000200 000150 000100 000

50 000

Time (years)

Loan

bal

ance

($)

0300 5 10 15 20 25

Monthly repayments

Fortnightly repayments

8 cm

21 cmNOT TOSCALE

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Section II

75 marksAttempt Questions 26–30Allow about 1 hour and 55 minutes for this section

Answer each question in the appropriate writing booklet. Extra writing booklets are available.

In Questions 26–30, your responses should include relevant mathematical reasoning and/orcalculations.

Question 26 (15 marks) Use the Question 26 Writing Booklet.

(a) Postcodes in Australia are made up of four digits eg 2040.

(i) How many different postcodes beginning with a 2 are possible?

(ii) Peta remembers that the first two digits of a town’s postcode are 2 andthen 4. She is unable to remember the rest of the postcode.

What is the probability that Peta guesses the correct postcode?

(b) Jim buys a photocopier for $22 000. Its value is depreciated using the decliningbalance method at the rate of 15% per annum.

What is its value at the end of 3 years?

(c) Heather used her credit card to purchase a plane ticket valued at $1990 on28 January 2011. She made no other purchases on her credit card account inJanuary. She paid the January account in full on 19 February 2011.

The credit card account has no interest free period. Simple interest is chargeddaily at the rate of 20% per annum, including the date of purchase and the datethe account is paid.

How much interest did she pay, to the nearest cent?

1

42 ??

1

2

2


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Question 26 (continued)

(d) Greg needs to conduct a statistical inquiry into how much time people aged18–25 years have spent accessing social media websites in the last two weeks.He has decided to survey a sample of students from his university.

The process of statistical inquiry includes the following steps, which are NOTin order.

(i) Using the letters A, B, C, D, E and F, list the steps in the most appropriateorder for Greg to conduct his statistical inquiry.

(ii) Greg conducts his statistical inquiry.

At which step in the process would he have drawn this graph?

Question 26 continues on page 16

A Writing a report

B Posing questions

C Organising data

D Analysing data and drawing conclusions

E Collecting data

F Summarising and displaying data

2

16 hoursor

more

11–15hours

6–10hours

5 hoursor

less

Time spent accessing social media websites (in hours)

1

– 15 –


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(e) The dot plot shows the number of push-ups that 13 members of a fitness classcan do in one minute.

(i) What is the probability that a member selected at random from the classcan do more than 38 push-ups in one minute?

(ii) A new member who can do 32 push-ups in one minute joins the class.Does the addition of this new member to the class change the probabilitycalculated in part (e) (i)? Justify your answer.

(f) The capture-recapture technique was used to estimate a population of sealsin 2012.

• 60 seals were caught, tagged and released.

• Later, 120 seals were caught at random.

• 30 of these 120 seals had been tagged.

The estimated population of seals in 2012 was 11% less than the estimatedpopulation for 2008.

What was the estimated population for 2008?

(g) Bhawana purchases pool chlorine in a new container which holds 35 kg.

She begins using this new container on the first day of a week.

How many full weeks should this container last?

End of Question 26

1

3736 38 39 40 41 42

2

Pool Chlorine35 kg

Weekly Use3 cups on 1st day of each week1 cup daily for the rest of the week

1 cup = 250 g

Weekly Use

3 cups on 1st day of each week

1 cup daily for the rest of the week

1 cup = 250 g

1

2

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(a) Tai earns a gross weekly wage of $1024. Each week her deductions are:

• tax instalment of $296.40• health fund contribution of $24.50• union fees of $15.80.

She also pays $3640 over the year as her share of the household expenses.

What percentage of her net wage does Tai pay for household expenses?

(b) The sector shown has a radius of 13 cm and an angle of 230°.

What is the perimeter of the sector to the nearest centimetre?

(c) A map has a scale of 1: 500 000.

(i) Two mountain peaks are 2 cm apart on the map.

What is the actual distance between the two mountain peaks, inkilometres?

(ii) Two cities are 75 km apart.

How far apart are the two cities on the map, in centimetres?

1

1

13 cmNOT TOSCALE

230°

2

3


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(d) A disability ramp is to be constructed to replace steps, as shown in the diagram.The angle of inclination for the ramp is to be 5°.

Calculate the extra distance, d, that the ramp will extend beyond the bottom step.Give your answer to the nearest centimetre.

(e) A box contains 33 scarves made from two different fabrics. There are 14 scarvesmade from silk (S) and 19 made from wool (W).

Two girls each select, at random, a scarf to wear from the box.

(i) Copy and complete the probability tree diagram in your answer booklet.

(ii) Calculate the probability that the two scarves selected are made fromsilk.

(iii) Calculate the probability that the two scarves selected are made fromdifferent fabrics.

End of Question 27

3

dd

13 cm

30 cm13 cm

30 cm

Ramp

NOT TOSCALE

5°13 cm

2

S

W

S

W

S

W

1433

1933

2

1

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(a) The solid shown in the diagram on page 1 of the Question 28 Writing Bookletis a cube.

Complete a sketch of the cube using the vanishing points V1 and V2.

Leave all construction lines on your diagram and label the vertices.

(b) Simplify fully .

(c) Jacques and a flagpole both cast shadows on the ground. The difference betweenthe lengths of their shadows is 3 metres.

What is the value of d, the length of Jacques’ shadow?

2183 2

aba

cb

×

d m3 mShadow of flagpole

NOT TOSCALE4 m

1.5 m

2

3


[See page 529 of this book.]

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(d) The test results in English and Mathematics for a class were recorded anddisplayed in the box-and-whisker plots.

(i) What is the interquartile range for English?

(ii) Compare and contrast the two data sets by referring to the skewness ofthe distributions and the measures of location and spread.

(e) Matthew bought a laptop priced at $2800. He paid a 10% deposit and mademonthly repayments of $95.20 for 3 years.

What annual flat rate of interest was Matthew charged? Justify your answer withsuitable calculations.

End of Question 28

1

3

100 20 30 40 50 60 70 80 90 100

Mathematics

English

4

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(a) Tourists visit a park where steam erupts from a particular geyser.

The brochure for the park has a graph of the data collected for this geyser overa period of time.

The graph shows the duration of an eruption and the time until the next eruption,timed from the end of one eruption to the beginning of the next.

(i) Tony sees an eruption that lasts 4 minutes. Based on the data in thegraph, what is the minimum time that he can expect to wait for the nexteruption?

(ii) Julia saw two consecutive eruptions, one hour apart. Based on the data inthe graph, what was the longest possible duration of the first eruptionthat she saw?

(iii) What does the graph suggest about the relationship between the durationof an eruption and the time to the next eruption?

1

1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Tim

e to

nex

t eru

ptio

n (m

inut

es)

40

50

60

70

80

90

100

Duration of eruption (minutes)

Steam eruptions

Geyser eruption

1

1


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(b) A machine produces nails. When the machine is set correctly, the lengths of thenails are normally distributed with a mean of 6.000 cm and a standard deviationof 0.040 cm.

To confirm the setting of the machine, three nails are randomly selected. In onesample the lengths are 5.950, 5.983 and 6.140.

The setting of the machine needs to be checked when the lengths of two or morenails in a sample lie more than 1 standard deviation from the mean.

Does the setting on the machine need to be checked? Justify your answer withsuitable calculations.

(c) Raj cycles around a course. The course starts at E, passes through F, G and Hand finishes at E. The distances EH and GH are equal.

(i) What is the length of EF, to the nearest kilometre?

(ii) What is the total distance that Raj cycles, to the nearest kilometre?

(d) Su-Lin pays a monthly contribution of 5% of her salary into a superannuationfund. Her salary is $81 600 per annum. The fund pays interest of 6.6% perannum, compounded monthly.

(i) What is the amount of Su-Lin’s monthly contribution?

(ii) Su-Lin made a contribution at the end of each month, starting at the endof January 2000.

What will be the accumulated value of her superannuation at the end ofDecember 2012 after making the December contribution?

End of Question 29

2

3

2

139°

31°

64 km

82 km

NOT TOSCALE

E H

F

G

3

2

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(a) A ship located at 4°N 160°E needs assistance.

A rescue boat leaves Honiara 9°S 160°E and travels due north at a speed of30 knots to reach the ship.

How long will it take the rescue boat to reach the ship? (You may assume that1° on a great circle equals 60 nautical miles.)

3

(b) A golf ball is hit from point A to point B, which is on the ground as shown. Point A is30 metres above the ground and the horizontal distance from point A to point B is 300 m.

The path of the golf ball is modelled using the equation

h = 30 + 0.2d – 0.001d2

where

h is the height of the golf ball above the ground in metres, and

d is the horizontal distance of the golf ball from point A in metres.

Ground

Path of the ball

NOT TOSCALE

A

30 m

B

300 m


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The graph of this equation is drawn below.

(i) What is the maximum height the ball reaches above the ground?

(ii) There are two occasions when the golf ball is at a height of 35 metres.

What horizontal distance does the ball travel in the period between thesetwo occasions?

(iii) What is the height of the ball above the ground when it still has to travela horizontal distance of 50 metres to hit the ground at point B?

(iv) Only part of the graph applies to this model.

Find all values of d that are not suitable to use with this model, andexplain why these values are not suitable.


h

d

1010

2020

3030

4040

OO 100100–200–200 200200 400400–100–100 300300

1

1

2

1


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(c) In 2010, the city of Thagoras modelled the predicted population of the city using theequation

P = A(1.04)n.

That year, the city introduced a policy to slow its population growth. The new predictedpopulation was modelled using the equation

P = A(b)n.

In both equations, P is the predicted population and n is the number of years after 2010.

The graph shows the two predicted populations.


0(2010)

20(2030)

n

P

10(2020)

30(2040)

Pop

ulat

ion

0

1 000 000

2 000 000

3 000 000

4 000 000

5 000 000

6 000 000

7 000 000

8 000 000

Years

Predicted population if the policy had not been introduced

Predicted population with the policy introduced

P = A(b)n

P = A(1.04)n


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(i) Use the graph to find the predicted population of Thagoras in 2030 ifthe population policy had NOT been introduced.

(ii) In each of the two equations given, the value of A is 3 000 000.

What does A represent?

(iii) The guess-and-check method is to be used to find the value of b, inP = A(b)n.

(1) Explain, with or without calculations, why 1.05 is not a suitable firstestimate for b.

(2) With n = 20 and P = 4 460 000, use the guess-and-check method andthe equation P = A(b)n to estimate the value of b to two decimalplaces. Show at least TWO estimate values for b, includingcalculations and conclusions.

(iv) The city of Thagoras was aiming to have a population under 7 000 000in 2050. Does the model indicate that the city will achieve this aim?Justify your answer with suitable calculations.

End of paper

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✗

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Question 28 (a)

The solid shown is a cube.

Complete a sketch of this cube using the vanishing points V1 and V2.

The edge OP has been drawn for you.

Leave all construction lines on your diagram and label the vertices.

V1 V2

O

P

O

P

N M

S Q

R

Extract from Question 28 answer booklet

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375Exce l S U C C E S S O N E H S C • M A T H E M A T I C S G E N E R A L 2

2 0 1 2 H S C E X A M I N A T I O N P A P E R • A N S W E R S

2012 HSC Examination PaperWorked Answers

(Total 25 marks)

1. D 2. D 3. C 4. A 5. A 6. C 7. D 8. A 9. C 10. C 11. B 12. B 13. C 14. A 15. D 16. A 17. A 18. D 19. B 20. C 21. A 22. B 23. D 24. B 25. B

1. The median of 15 scores is the 8th score.

5

6

7

8

9

3 4

2 6 7

7 7 8 9

2 4

1 3 5 7

The 8th score is 78.

Answer D

2. It is an example of systematic sampling.

Answer D

3. Unordered selection of 2 players from 6 players.

Number of pairs = 6 3 52 3 1

[or 6C2]

= 15

Answer C

4.

40°

50°

x29

cos 408 = x29

x = 29 3 cos 408

Answer A

5.

gradient = vertical change in position

horizontal change in position

= 2pq

= 2pq

Answer A

6. 4 m = 400 cm

0.05 m = 5 cm

4 m

30 cm

V = lbh= 400 3 30 3 5= 60 000

The volume is 60 000 cm3.

Answer C

7. June sales for Bakery 1 = $17 500

June sales for Bakery 2 = $35 000

Difference = $35 000 2 $17 500= $17 500

Answer D

Section I

NOT TOSCALE

y

�

xO q

p

NOT TOSCALE

400 cm

5 cm

30 cm

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376 Exce l S U C C E S S O N E H S C • M A T H E M A T I C S G E N E R A L 2


5

10

15

20

25

30

35

5

10

15

20

25

30

35

June

May

April

Monthly sales ($ ’000s)

March

February

January



KEY

8. Shape (S) 1 2 3

Number of dots (N) 6 8 10

The rule is N = 2S 1 4

When S = 156,

N = 2 3 156 1 4= 316

Answer A

9. 4% pa = 1% per quarter

So the interest rate per period is 1%.

2 years = 8 quarters

So the number of periods is 8.

From the table, the required fi gure is 1.083.

Answer C

10. Third angle = 1808 2 (50 1 57)8= 738

8.8 m

50°

57°

9.9 m

NOT TOSCALE

73°

A = 12

ab sin C

= 12

3 9.9 3 8.8 3 sin 738

= 41.656 635 17…

The area is 42 m2, to the nearest square metre.

Answer C

11. For a relationship to show negative correlation one part should increase as the other decreases. This happens in option B. As the hours spent training increase, the time to complete the race should decrease. This relationship is most likely to show negative correlation.

Answer B

12. On one die:

P(rolling a 6) = 16

P(not rolling a 6) = 56

On two dice:

P(neither showing 6) = 56

356

= 2536

P(at least one 6) = 1 2 2536

= 1136

Answer B© P

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13. From the first graph, 60 Australian dollars is about 46 Euros.

Eur

os

Aust Dollars

20

2000

40

40 60 80

From the second graph 46 Euros is about 78 NZ dollars.

NZ

Dol

lars

Euros

20

2000

40

60

80

40 60 80

Answer C

14. 2x2(5 2 x) 2 x(x 2 2)= 10x2 2 2x3 2 x2 1 2x= 22x3 1 9x2 1 2x

Answer A

15. T = DS

6 = D60

D = 6 3 60= 360

So T = 360S

When S = 100,

T = 360100

= 3.6

So the journey takes 3.6 hours or 3 hours and 36 minutes.

Answer D

16. [The declining balance method of depreciation is used, not the straight-line method, so options B and D are not correct. The machine will still have some value after 10 years so option C is not correct.]

The graph which best represents the salvage value of the machine is this graph.

(A)

100

25 000

Time (years)

Sal

vage

val

ue (

$)

Answer A

17. Number of spins = 40

Total frequencies shown in table= 2 1 4 1 6 1 10 1 12= 34

Frequency for orange = 40 2 34= 6

Relative frequency of orange = 640

= 320

Answer A

18. Income above $236 = $300 2 $236= $64

Reduction = 0.5 3 $64= $32

Jo’s youth allowance = $402.70 2 $32= $370.70

Total payment = $370.70 1 $119.40= $490.10

Answer D

19. [The median of each group of 3 points is the middle one of the 3 as you go across the graph in each case. First join the median of the first group to the median of the last group, then slide this line one-third of the way up towards the median of the middle group.]

This graph best shows the median regression line.

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20 7030 40 50 60 80

Wee

ks o

f tra

inin

g

0

2

4

6

8

10

12(B)


Answer B

20. Using the cosine rule in nABC,

cos B = 592 1 802 2 312

2 3 59 3 80

= 0.944 9152….

B = 19.105 8926…8

= 198 [nearest degree]

N

u

19° 59 km

31 km

80 km

A

B

C

u = 1808 2 198

= 1618

The bearing of C from B is 1618.

Answer C

21. E = mc2 1 p

mc2 1 p = E

mc2 = E 2 p

c2 = E 2 p

m

c = 6 ÅE 2 pm

Answer A

22. Circle: radius = 2.4 cm

Ellipse: a = 6.3 1 2.4 b = 2.4= 8.7

Shaded area= area of ellipse 2 area of circle= pab 2 pr2

= p 3 8.7 3 2.4 2 p 3 2.42

= 47.500 8809…= 48 cm2 [nearest square centimetre]

Answer B

23. A = 12 000

6% pa = 0.5% per month

So r = 0.005

10 years = 120 months

So n = 120

A = M e (1 1 r)n 2 1r

f 12 000 = M e (1 1 0.005)120 2 1

0.005f

Answer D

24. Monthly repayments:

$2796.86 over 30 years

Total repaid = $2796.86 3 30 3 12= $1 006 869.60

Fortnightly repayments:

$1404.76 over 23 years

Total repaid = $1404.76 3 23 3 26= $840 046.48

Difference = $1 006 869.60 2 $840 046.48= $166 823.12= $166 823 [nearest dollar]

Answer B

25. Hemisphere: A = 2pr2

Cylinder open at one end: A = 2prh 1 pr2

Total surface area:

A = 2prh 1 3pr2

= 2 3 p 3 4 3 21 1 3 3 p 3 42

= 678.584 0132…= 679 cm2 [nearest square centimetre]

Answer B

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QUESTION 26

(a) (i) Number of postcodes starting with 2= 1 3 10 3 10 3 10= 1000 (1 mark)

(ii) Number of postcodes starting with 24= 1 3 1 3 10 3 10= 100

P(correct guess) = 1

100 (1 mark)

(b) V0 = 22 000 r = 0.15 n = 3

S = V0(1 2 r)n

= 22 000(1 2 0.15)3

= 13 510.75

The value after 3 years is $13 510.75.

(2 marks)

(c) Number of days = 4 1 19= 23

P = $1990 r = 0.2 n = 23

365I = Prn

= $1990 3 0.2 3 23365

= $25.079 452…= $25.08 [nearest cent]

Heather paid $25.08 interest, to the nearest cent. (2 marks)

(d) (i) B, E, C, F, D, A (2 marks)

(ii) A graph is a data display.

This would be done in step F. (1 mark)

(e)

3736 38 39 40 41 42

(i) 7 members can do more than 38 push-ups in one minute.

P(more than 38 push-ups) = 7

13 (1 mark)

(ii) Yes, the new member does change the probability. There are now 14 members.

P(more than 38 push-ups) = 7

14

= 12

(1 mark)

(f) The fraction of caught seals that had been

tagged = 30

120

= 14

So 14

of the population in 2012 was 60 seals.

Population in 2012 = 60 3 4= 240

Now 100% 2 11% = 89%.

So 240 is 89% of 2008 population.

89% = 240

1% = 240 4 89= 2.696 629 213…

100% = 2.696 629 213… 3 100= 269.662 9213…= 270 [nearest unit]

The estimated population in 2008 was 270.

(2 marks)

(g) Cups used each week = 3 1 6= 9

Mass used per week = 9 3 250 g= 2250 g= 2.25 kg

Number of weeks = 35 4 2.25= 15.555 555….

The container will last 15 full weeks.

(2 marks)

QUESTION 27

(a) Total deductions = $296.40 1 $24.50 1 $15.80= $336.70

Net wage = $1024 2 $336.70= $687.30

Section II

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Weekly share of household expenses= $3640 4 52= $70

Percentage of net wage

= 70

687.30 3 100%

= 10.184 78… %= 10.2% [1 decimal place] (3 marks)

(b) 13 cm

230°

Arc length: l = u

3602pr

= 230360

3 2 3 p 3 13

= 52.185 3446…= 52 cm [nearest cm]

Perimeter = 52 cm 1 2 3 13 cm= 78 cm

The perimeter of the sector is 78 cm, to the nearest centimetre. (2 marks)

(c) (i) Scale 1:500 000

1 cm represents 500 000 cm

But 500 000 cm = 5000 m = 5 km

So 1 cm represents 5 km.

2 cm represents 10 km.

The actual distance between the two mountain peaks is 10 km. (1 mark)

(ii) 1 cm represents 5 km.

Now 75 4 5 = 15

So 15 cm would represent 75 km.

The two towns will be 15 cm apart on the map. (1 mark)

(d) Vertical height of the ramp= 3 3 13 cm= 39 cm

5°

39 cm

60 cm dx cm

Let x cm be the horizontal length of the ramp.

tan 58 = 39x

x = 39

tan 5°

= 445.772 0398…

= 446 [nearest unit]

So the horizontal length of the ramp is 446 cm, to the nearest centimetre.

d 1 60 cm = 446 cm

d = 386 cm (3 marks)

(e) (i) FirstScarf

SecondScarf

S

W

14___33

19___33

S

W

14___32

18___32

S

W

13___32

19___32

(2 marks)

(ii) P(SS) = 1433

31332

= 91

528 (1 mark)

(iii) P(different fabrics) = P(SW) 1 P(WS)

= 1433

31932

1 1933

31432

= 133264

(2 marks)

QUESTION 28

(a)

(2 marks)

(b) 18ab3a2 3

cb

= 18abc3a2b

= 6ca

(2 marks)

NOT TOSCALE

O

P

R

S Q

N M

V1 V2

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(c)

The triangle formed by the fl agpole and its shadow and the triangle formed by Jacques and his shadow are equiangular, so they are similar.

So d

1.55

3 1 d4

4d = 4.5 1 1.5d

2.5d = 4.5

d = 1.8 (3 marks)

(d)

(i) English: lower quartile = 50

upper quartile = 80

interquartile range = 80 2 50= 30 (1 mark)

(ii) [Different answers are possible.]

Both data sets are negatively skewed.

The spread of marks for English is much greater than for Mathematics. It is more than double. (The range for English is 85 while the range for Maths is 40.) The interquartile range for English (30) is also double that for Maths (15).

The highest mark in English (95) was higher than the highest mark in Maths (90), but the upper quartiles are the same (80). The median (70 for English, 75 for Maths), lower quartile (50 for English, 65 for Maths) and lower extreme (10 for English, 50 for Maths) are all higher for Maths than English.

d md m3 m3 m

Shadow of flagpoleShadow of flagpole

NOT TOSCALE4 m4 m

1.5 m1.5 m

100 20 30 40 50 60 70 80 90 100

Mathematics

English

25% of students scored marks in English that were less than any marks scored in Maths. (3 marks)

(e) Deposit = 0.1 3 $2800= $280

Balance owing = $2800 2 $280= $2520

Total repayments = $95.20 3 3 3 12= $3427.20

Interest = $3427.20 2 $2520= $907.20

I = $907.20 P = $2520 n = 3

I = Prn

$907.20 = $2520 3 r 3 3= $7560r

r = $907.20 4 $7560= 0.12

The annual rate of interest was 12%.

(4 marks)

QUESTION 29

(a) (i) Using the graph, for an eruption of 4 minutes, the minimum time to wait for the next eruption is 70 minutes.

(1 mark)

(ii) For eruptions one hour apart, the longest possible duration of the fi rst eruption, based on the data, is 3 minutes.

(1 mark)

(iii) The graph suggests a strong positive relationship between the duration of an eruption and the length of time until the next eruption. The longer the duration of the fi rst eruption, the longer the wait until the next eruption. (1 mark)

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(b) x = 6.000 s = 0.040

x 1 s = 6.000 1 0.040= 6.040

x 2 s = 6.000 2 0.040= 5.960

So the lengths of two or more of the 3 nails should be between 5.960 cm and 6.040 cm.

Now 5.950 < 5.960

and 6.140 > 6.040

So the lengths of two nails are more than one standard deviation from the mean.

Yes, the machine does need to be checked.

(2 marks)

(c)

(i) /FGE = 1808 2 (139 1 31)8

= 108

By the sine rule in nEFG,EF

sin 10°5

82sin 139°

EF = 82 sin 10°sin 139°

= 21.704 069…= 22 km [nearest km]

{Or: EF

sin 10°5

64sin 31°

EF = 64 sin 10°

sin 31°

= 21.577 984…= 22 km [nearest km]} (2 marks)

Or: By the cosine rule in ΔEFG,

EF 2 = 642 1 822 2 2 3 64 3 82 3 cos 10°= 483.457 82…

EF = 21.987 674… (EF > 0)= 22 km [nearest km]}

(2 marks)

(ii) Let the length of GH be x km.

EH = GH = x km

By Pythagoras’ theorem in nGEH,

x2 1 x2 = 822

2x2 = 6724

x2 = 3362

x = 57.982 756… [x > 0]

The length of GH is 58 km to the nearest kilometre.

Total distance = (2 3 57.982… 1 64 1 21.704…) km= 201.669 58… km= 202 km [nearest kilometre]

(3 marks)

(d) [Note: This question is ambiguous because it doesn’t specify what salary (monthly or annual) is being used. If you assumed that the monthly contribution is 5% of monthly salary:]

(i) Monthly salary = $81 600 4 12= $6800

Contribution = 0.05 3 $6800= $340 (2 marks)

(ii) From January 2000 until December 2012 is 13 years.

n = 13 3 12= 156

r = 0.066 4 12= 0.0055

A = M e (1 1 r)n 2 1r

f= $340 e (1 1 0.0055)156 2 1

0.0055f

= $83 633.891 79…= $83 633.89 [nearest cent]

(3 marks)

{[If the monthly contribution is 5% of annual salary then:]

M = 0.05 3 $81 600= $4080

A = M e (1 1 r)n 2 1r

f= $4080 e (1 1 0.0055)156 2 1

0.0055f

= $1 003 606.701…= $1 003 606.70 [nearest cent]}

139°

31°

64 km

82 km

E H

F

G

NOT TOSCALE

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QUESTION 30

(a) Degrees difference = 48 1 98

= 138

Distance = 13 3 60 nautical miles= 780 nautical miles

Speed = 30 knots= 30 nautical miles per hour

Time = (780 4 30) hours= 26 hours (3 marks)

(b)

(i) From the graph, the maximum height reached by the ball is 40 m. (1 mark)

(ii) The ball is at a height of 35 m when it is at a horizontal distance of 30 m and again when it is at a horizontal distance of 170 m.

Difference = 170 2 30= 140 m

The horizontal distance travelled by the ball in that time is 140 m.

(1 mark)

(iii) The ball hits the ground at a distance of 300 m.

50 m short of that distance is 250 m.

The height of the ball is 17.5 m. (1 mark)

h

d

1010

2020

3030

4040

OO 100100–200–200 200200 400400–100–100 300300

(iv) If d < 0 the graph is not suitable.

The ball was hit forwards and negative values of d apply to distances behind where the ball was hit.

If d > 300 the graph is not suitable. The ball hits the ground at a horizontal distance of 300 m and the graph suggests the height will then be negative but the ball will not go below the ground.

(2 marks)

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(c)

(i) In 2030, n = 20

From the graph, when n = 20,

P = 6 600 000

If the policy is not introduced the predicted population in 2030 is 6 600 000.

(1 mark)

(ii) A = 3 000 000 represents the population of Thagoras in 2010. (1 mark)

(iii) (1) The population growth was slower using P = A(b)n. So the value of b must be smaller than in the original equation in order for P to be smaller. That is b < 1.04. So a value of 1.05 is not suitable.

(1 mark)(2) When n = 20, P = 4 460 000

P = 3 000 000(b)20

Try b = 1.03

P = 3 000 000(1.03)20

= 5 418 333.704….

0(2010)

20(2030)

n

P

10(2020)

30(2040)

Pop

ulat

ion

0

1 000 000

2 000 000

3 000 000

4 000 000

5 000 000

6 000 000

7 000 000

8 000 000

Years

Predicted population if the policy had not been introduced

Predicted population with the policy introduced

P = A(b)nP = A(b)n

P = A(1.04)nP = A(1.04)n

This value of P is too high so b needs to be smaller.

Try b = 1.01

P = 3 000 000(1.01)20

= 3 660 570.12…

This value of P is too small so b needs to be larger.

Try b = 1.02

P = 3 000 000(1.02)20

= 4 457 842.188…= 4 460 000 [3 signifi cant fi gures]

So the value of b should be 1.02 to 2 decimal places. (2 marks)

(iv) In 2050, n = 40

P = 3 000 000(1.02)40

= 6 624 118.991…

This is less than 7 000 000 so the model does suggest that the city will achieve its goal. (2 marks)

Don’t forget!To practise for the new HSC Mathematics General 2 Examination:• Use the 2014 Formulae and Data Sheet on page 422.• When you see:

✗ , you can ignore this question

, turn to page viii to see the required additional information for this question.© P

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SUCCESS ONE HSC...2012 HSC EXAMIN A TION P APE R • QUESTION S Excel SUCCES S ONE HSC • M A THEM...

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