Excel Success One HSC General Mathematics SAMPLE 2014

SUCCESS ONE HSC ®

*

MATHEMATICS GENERAL 2 Past HSC General Mathematics Papers & Worked Answers 2001–2013

Free-to-download HSC Exam with answers

CHAPTER 12 • 2012 HSC EXAMINATION PAPER

CHAPTER 12

2012 HSC Examination Paper

2012 H 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

© PASCAL PRESS 2014 ISBN 978 1 74125 470 9

General Mathematics

General Instructions • Reading time – 5 minutes • Working time – 2 –12 hours • Write using black or blue pen Black pen is preferred • Calculators may be used • A formulae sheet is provided at the back of this paper • In Questions 26–30, show relevant mathematical reasoning and/or calculations • Write your Centre Number and Student Number on the Question 28 Writing Booklet 348

Total marks – 100 Section I

Pages 2–13

25 marks • Attempt Questions 1–25 • Allow about 35 minutes for this section Section II

Pages 14–27

75 marks • Attempt Questions 26–30 • Allow about 1 hour and 55 minutes for this section

Excel 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

2012 HSC EXAMINATION PAPER • QUESTIONS

Section I 25 marks Attempt Questions 1–25 Allow 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. 5 6 7 8 9

3 2 7 2 1

4 6 7 4 3

7 8

9

5

7

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

3

© PASCA L PRESS 2014 I SBN 978 1 74125 470 9

(D) Systematic 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


SuccessOne HSC MathsGeneral2 BOOK 2014Ed.indb 349

349

5/12/13 12:21 PM


4

50° x

29

NOT TO SCALE

40°

Which expression could be used to calculate the value of x in this triangle? (A) 29 × cos 40 °

5

(B)

29 × cos 50 °

(C)

cos 40 ° 29

(D)

cos 50 ° 29

has intercepts p and q, where p and q are positive integers.

The line

y p

O

q


What is the gradient of line ?

350

(A) −

p q

(B)

−

q p

(C)

p q

(D)

q p



x


6 0.05 m 4m

NOT TO SCALE

30 cm

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

The Pi Company has two bakeries. The radar chart displays the monthly sales for the bakeries. Monthly sales ($ ’000s) January 35 30 25 20

June

February

15 10

KEY

5

Monthly sales for Bakery 1 Monthly sales for Bakery 2

May

March


7

April

What was the difference in sales in June between the two bakeries? (A) $7.50 (B)

$17.50

(C)

$7500

(D) $17 500 Excel 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

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8

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

Shape 1

Shape 2

Shape 3

The number of dots used in each shape is recorded in the table. Shape (S)

1

2

3

Number of dots (N)

6

8

10

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. Compounded values of $1


Period

Interest 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

Which figure from the table should Tracy use to calculate the value of her investment at the end of 2 years? (A) 1.020 (B)

1.082

(C)

1.083

(D) 1.369 352



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10 8.8 m

NOT TO SCALE

9.9 m 57° 50° 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.

(A)

1 6

(B)

11 36

(C)

25 36

(D)

5 6

© PASCAL P RESS 2014 I SBN 978 1 74125 470 9

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



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13

Conversion graphs can be used to convert from one currency to another. 100

Euros

80 60 40 20 0 0

20

40

0

20

40

60 80 100 Australian dollars

120

140

100

120

140

New Zealand dollars

100 80 60 40 20 0 60 80 Euros


Sarah converted 60 Australian dollars into Euros. She then converted all of these Euros into New Zealand dollars. How much money, in New Zealand dollars, should Sarah have? (A) $26 (B)

$45

(C)

$78

(D) $135

354



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14

Which of the following expresses 2x 2 (5 – x) – x (x – 2) in its simplest form? (A) –2x 3 + 9x 2 + 2x (B)

–2x 3 – 9x 2 – 2x

(C)

9x 2 – x + 2

(D) 9x 2 – x – 2

15

The time taken to complete a journey varies inversely with the speed of a car. A car takes 6 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

A machine was bought for $25 000. Which graph best represents the salvage value of the machine over 10 years using the declining balance method of depreciation? (A)

(B) 25 000

0

Time (years)

0

10

(C)

Time (years)

10

Time (years)

10


Salvage value ($)

Salvage value ($)

25 000

(D) 25 000

25 000 Salvage value ($)

Salvage value ($)

16

0

Time (years)

10

0


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17

A spinner with different coloured sectors is spun 40 times. The results are recorded in the table. Colour obtained Red Yellow Blue Orange Green Purple

Frequency 2 4 6 10 12

What is the relative frequency of obtaining the colour orange? (A)

3 20

(B)

1 5

(C)

6

(D) 8

18

Jo qualifies for both Rent Assistance and Youth Allowance and receives a fortnightly payment 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 the dollar 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

356



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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. 12

Weeks of training

10 8 6 4 2 0 20

70 30 40 50 60 80 Resting pulse (beats per minute)

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

10

10 Weeks of training

(B) 12

8 6 4 2

8 6 4 2

0

0 20


(D) 12

10

10 Weeks of training

(C) 12

8 6 4 2

20


20


8 6 4 2

0

0 20




Weeks of training

(A) 12

Weeks of training

✗

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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. N B NOT TO SCALE

59 km 80 km C 31 km A What is the bearing of Town C from Town B? (A) 019° (B)

122°

(C)

161°

(D) 341°


21

358

Which of the following correctly expresses c as the subject of E = mc2 + p? (A) c = ±

E−p m

(B)

c=±

E−p m

(C)

c=±

E −p m

(D) c = ±

E −p m



5/12/13 12:21 PM


✗

22

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

6.3 cm

4.8 cm

6.3 cm

NOT TO SCALE

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 and other 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, compounded monthly. Which equation should the club use to calculate the amount, M, it will need to deposit each month in order to have $12 000 at the end of 10 years? ⎧ 1 + 0 . 06 10 − 1 ⎫ ) ⎪ ⎪( (A) 12 000 = M ⎨ ⎬ 0 .06 ⎪⎩ ⎪⎭ ⎧ 1 + 0 . 06 10 − 1 ⎫ ) ⎪( ⎪ (B) 12 000 = M ⎨ 10 ⎬ ⎪⎩ 0 . 06 (1 + 0 . 06 ) ⎪⎭ ⎧ 1 + 0 . 005 120 − 1 ⎫ ) ⎪ ( ⎪ (C) 12 000 = M ⎨ 120 ⎬ ⎪⎩ 0 . 005 (1 + 0 . 005) ⎪⎭


✗

⎧ 1 + 0 . 005 120 − 1 ⎫ ) ⎪( ⎪ (D) 12 0 0 0 = M ⎨ ⎬ 0 . 005 ⎪⎩ ⎪⎭ Question 23 continues on following page


359

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

Loan balance ($)

24

400 000 350 000 300 000 250 000 200 000 150 000 100 000 50 000 0

Monthly repayments Fortnightly repayments

0

5

10 15 20 Time (years)

25

30

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 of repayment, 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.


21 cm

NOT TO SCALE

8 cm

What is the total surface area of the solid, to the nearest square centimetre? (A) 628 cm2 (B) (C)

679 cm2 729 cm2

(D) 829 cm2 360



5/12/13 12:22 PM


Section II 75 marks Attempt Questions 26–30 Allow 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/or calculations.

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?

1

(ii)

Peta remembers that the first two digits of a town’s postcode are 2 and then 4. She is unable to remember the rest of the postcode.

1

2

4

?

?

What is the probability that Peta guesses the correct postcode?

(b)

Jim buys a photocopier for $22 000. Its value is depreciated using the declining balance method at the rate of 15% per annum.

2

What is its value at the end of 3 years?

Heather used her credit card to purchase a plane ticket valued at $1990 on 28 January 2011. She made no other purchases on her credit card account in January. She paid the January account in full on 19 February 2011.

2


(c)

The credit card account has no interest free period. Simple interest is charged daily at the rate of 20% per annum, including the date of purchase and the date the account is paid. How much interest did she pay, to the nearest cent? Question 26 continues on following page


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Question 26 (continued) (d)

Greg needs to conduct a statistical inquiry into how much time people aged 18–25 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 NOT in order. A

Writing a report

B

Posing questions

C

Organising data

D

Analysing data and drawing conclusions

E

Collecting data

F

Summarising and displaying data

(i)

Using the letters A, B, C, D, E and F, list the steps in the most appropriate order for Greg to conduct his statistical inquiry.

2

(ii)

Greg conducts his statistical inquiry.

1

At which step in the process would he have drawn this graph? Time spent accessing social media websites (in hours)

16 hours or more


11–15 hours

5 hours or less 6–10 hours

26 continues on page 16 QuestionQuestion 26 continues on following page

362

E N E–R A L 2 E x c e 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 –G15


5/12/13 12:22 PM


Question 26 (continued)

(e)

The dot plot shows the number of push-ups that 13 members of a fitness class can do in one minute.

36

(f)

37

38

39

40

41

42

(i)

What is the probability that a member selected at random from the class can do more than 38 push-ups in one minute?

1

(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 probability calculated in part (e) (i)? Justify your answer.

1

The capture-recapture technique was used to estimate a population of seals in 2012.

2

• 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 estimated population for 2008. What was the estimated population for 2008?

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

2

Weekly Use

Pool Chlorine 35 kg

3 cups on 1st day of each week 1 cup daily for the rest of the week

Weekly Use 3 cups on 1st day of each week 1 cup daily for the rest of the week


(g)

1 cup = 250 g

1 cup = 250 g

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 Excel 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

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Question 27 (15 marks) Use the Question 27 Writing Booklet. (a)

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

3

• 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°. 13 cm 230°

2

NOT TO SCALE

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.

1

What is the actual distance between the two mountain peaks, in kilometres?


(ii)

Two cities are 75 km apart.

1

How far apart are the two cities on the map, in centimetres? Question 27 continues on following page

364



5/12/13 12:22 PM



(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°.

3

NOT TO SCALE

13 cm Ramp

30 cm 13 cm 30 cm

13 cm 5° d Calculate the extra distance, d, that the ramp will extend beyond the bottom step. Give your answer to the nearest centimetre.

A box contains 33 scarves made from two different fabrics. There are 14 scarves made 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.

2

S 14 33

19 33

S W

S W W

(ii)

Question 25 continues on following page are made from Calculate the probability that the two scarves selected silk.

1

(iii)

Calculate the probability that the two scarves selected are made from different fabrics.

2



(e)

365

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Question 28 (15 marks) Use the Question 28 Writing Booklet. [See page 529 of this book.]

✗

(a)

The solid shown in the diagram on page 1 of the Question 28 Writing Booklet is a cube.

2

Complete a sketch of the cube using the vanishing points V1 and V2 . Leave all construction lines on your diagram and label the vertices.

18ab

c

2

(b)

Simplify fully

(c)

Jacques and a flagpole both cast shadows on the ground. The difference between the lengths of their shadows is 3 metres.

3a

2

×

b

.

3

NOT TO SCALE

4m 1.5 m 3m

Shadow of flagpole

dm

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


Question 28 continues on following page

366



5/12/13 12:22 PM



(d)

The test results in English and Mathematics for a class were recorded and displayed in the box-and-whisker plots. English Mathematics

0

20

30

40

50

60

70

80

90

100

(i)

What is the interquartile range for English?

1

(ii)

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

3

Matthew bought a laptop priced at $2800. He paid a 10% deposit and made monthly repayments of $95.20 for 3 years.

4

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

End of Question 28


(e)

10


367

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Question 29 (15 marks) Use the Question 29 Writing Booklet. (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 over a 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. Steam eruptions

Time to next eruption (minutes)

100 90 80 70 60 50 40


Geyser eruption

1.5

2.0

2.5 3.0 3.5 4.0 4.5 Duration of eruption (minutes)

5.0

(i)

Tony sees an eruption that lasts 4 minutes. Based on the data in the graph, what is the minimum time that he can expect to wait for the next eruption?

1

(ii)

Julia saw two consecutive eruptions, one hour apart. Based on the data in the graph, what was the longest possible duration of the first eruption that she saw?

1

(iii)

What does the graph suggest about the relationship between the duration of an eruption and the time to the next eruption?

1

Question 29 continues on following page

368



5/12/13 12:22 PM



(b)

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

2

To confirm the setting of the machine, three nails are randomly selected. In one sample 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 more nails 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 with suitable calculations.

(c)

Raj cycles around a course. The course starts at E, passes through F, G and H and finishes at E. The distances EH and GH are equal. G 64 km F 139°

NOT TO SCALE

82 km 31°

(d)

H

(i)

What is the length of EF, to the nearest kilometre?

2

(ii)

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

3

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

✗

(i)

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

2

(ii)

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

3

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



E

369

5/12/13 12:21 PM


Question 30 (15 marks) Use the Question 30 Writing Booklet.

✗

(a)

3

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 of 30 knots to reach the ship. How long will it take the rescue boat to reach the ship? (You may assume that 1° on a great circle equals 60 nautical miles.) (b)

A golf ball is hit from point A to point B, which is on the ground as shown. Point A is 30 metres above the ground and the horizontal distance from point A to point B is 300 m.

Path of the ball A NOT TO SCALE

30 m

Ground

B

300 m

The path of the golf ball is modelled using the equation h = 30 + 0.2d – 0.001d 2 © PASCAL PRESS 2014 ISBN 978 1 74125 470 9

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. Question 30 continues on following page

370



5/12/13 12:22 PM



The graph of this equation is drawn below. h

40

30

20

10

–200

–100

O

100

200

300

400 d

(i)

What is the maximum height the ball reaches above the ground?

1

(ii)

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

1

(iii)

What is the height of the ball above the ground when it still has to travel a horizontal distance of 50 metres to hit the ground at point B?

1

(iv)

Only part of the graph applies to this model.

2

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

Question 30 continues on following Question 30 continues on page 26page



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

371

5/12/13 12:21 PM


Question 30 (continued) (c)

In 2010, the city of Thagoras modelled the predicted population of the city using the equation P = A(1.04)n.

That year, the city introduced a policy to slow its population growth. The new predicted population 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. P

P = A (1.04)n

8 000 000

7 000 000

6 000 000

P = A (b)n

Population

5 000 000

4 000 000

3 000 000

2 000 000


1 000 000

0 0 (2010)

10 (2020)

20 (2030)

30 (2040)

n

Years Predicted population if the policy had not been introduced Predicted population with the policy introduced

Question 30 continues on following Question 30 continues on page 27page 372



5/12/13 12:22 PM



(i)

Use the graph to find the predicted population of Thagoras in 2030 if the population policy had NOT been introduced.

1

(ii)

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

1

What does A represent?

(iii)

(iv)

The guess-and-check method is to be used to find the value of b, in P = A(b)n. (1) Explain, with or without calculations, why 1.05 is not a suitable first estimate for b.

1

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

2

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

2

End of paper


✗


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Extract from Question 28 answer booklet Question 28 (a) The solid shown is a cube. R S

P

N

Q

M O

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

P


O

374



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2012 HSC EXAMINATION PAPER • ANSWERS

2012 HSC Examination Paper Worked Answers Section I 1. 6. 11. 16. 21.

D C B A A

2. 7. 12. 17. 22.

D D B A B

(Total 25 marks) 3. 8. 13. 18. 23.

C A C D D

4. 9. 14. 19. 24.

A C A B B

5. 10. 15. 20. 25.

A C D C B

5.

y p

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

5 6 7 8 9

3 2 7 2 1

4 6 7 4 3

O 7 8

9

5

7

x

vertical change in position horizontal change in position 2p = q p = 2 q Answer A

gradient =

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. 635 Number of pairs = [or 6C2] 231 = 15 Answer C 4.

q

6.

4 m = 400 cm 0.05 m = 5 cm 5 cm

4400 m cm 30 cm cm

NOT TO SCALE

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

50°

The volume is 60 000 cm3.

29

Answer C 7. June sales for Bakery 1 = $17 500

40°

cos 408 =

NOT TO SCALE


x

June sales for Bakery 2 = $35 000 x 29

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

x = 29 3 cos 408 Answer A


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5/12/13 12:21 PM


Monthly sales ($ ’000s) January 35 30 25 20

June

February

15 10

KEY

5

Monthly sales for Bakery 1 Monthly sales for Bakery 2

May

March

April

8.

Shape (S)

1

2

3

Number of dots (N)

6

8

10

= 41.656 635 17…

The rule is N = 2S 1 4 When S = 156,

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

N = 2 3 156 1 4 = 316

Answer C

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 figure is 1.083.


12. On one die:

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

73°

1 6 5 P(not rolling a 6) = 6 On two dice: P(rolling a 6) =

8.8 m

5 5 3 6 6 25 = 36 25 P(at least one 6) = 1 2 36 11 = 36 Answer B P(neither showing 6) =

9.9 m 57°

376

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

Answer C

50°

1 A = ab sin C 2 1 = 3 9.9 3 8.8 3 sin 738 2

NOT TO SCALE



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

Euros

40

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.

20

(A)

0

20

40 60 Aust Dollars

80

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

Salvage value ($)

25 000

0

0

Answer A 60

17. Number of spins = 40 Total frequencies shown in table = 2 1 4 1 6 1 10 1 12 = 34

40

20

Frequency for orange = 40 2 34 =6

0 0

20

40

60

80

Relative frequency of orange =

6 40

=

3 20

Euros

Answer C Answer A 14.

2x2(5

2 x) 2 x(x 2 2) = 10x2 2 2x3 2 x2 1 2x = 22x3 1 9x2 1 2x

18. Income above $236 = $300 2 $236 = $64 Reduction = 0.5 3 $64 = $32

Answer A 15.

10

Time (years)

D S D 6= 60 D = 6 3 60 = 360 360 So T = S When S = 100, 360 T= 100 = 3.6 T=

Jo’s youth allowance = $402.70 2 $32 = $370.70 Total payment = $370.70 1 $119.40 = $490.10 Answer D

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



NZ Dollars

80

377

5/12/13 12:21 PM


22. Circle: radius = 2.4 cm

(B) 12

Ellipse: a = 6.3 1 2.4 = 8.7

Weeks of training

10

b = 2.4

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]

8 6 4

Answer B

2

23. A = 12 000

0 20


6% pa = 0.5% per month So r = 0.005 10 years = 120 months

Answer B 20. Using the cosine rule in nABC, 59 2 1 80 2 2 31 2 cos B = 2 3 59 3 80 = 0.944 9152…. B = 19.105 8926…8 = 198 [nearest degree]

So n = 120 (1 1 r) n 2 1 f r (1 1 0.005) 120 2 1 12 000 = M e f 0.005 Answer D A=Me

24. Monthly repayments:

N

$2796.86 over 30 years

B u

Total repaid = $2796.86 3 30 3 12 = $1 006 869.60 Fortnightly repayments:

19° 59 km 80 km

Total repaid = $1404.76 3 23 3 26 = $840 046.48 C 31 km

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

A

25. Hemisphere: A = 2pr2

u = 1808 2 198 = 1618 © PASCAL PRESS 2014 ISBN 978 1 74125 470 9

$1404.76 over 23 years

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

The bearing of C from B is 1618.

Total surface area:

Answer C

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]

21. E = mc2 1 p mc2 1 p = E mc2 = E 2 p E2p c2 = m c= 6

Answer B

E2p Å m

Answer A 378



Section II 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 1 P(correct guess) = (1 mark) 100 (b) V0 = 22 000 r = 0.15

(ii) Yes, the new member does change the probability. There are now 14 members. 7 P(more than 38 push-ups) = 14 =

(1 mark)

(f) The fraction of caught seals that had been 30 tagged = 120

n=3

=

S = V0(1 2 r) = 22 000(1 2 0.15)3 = 13 510.75

1 2

n

1 4

1 of the population in 2012 was 60 seals. 4 Population in 2012 = 60 3 4 = 240

So

The value after 3 years is $13 510.75. (2 marks)

So 240 is 89% of 2008 population.

(c) Number of days = 4 1 19 = 23 P = $1990 r = 0.2 n = I = Prn

Now 100% 2 11% = 89%. 89% = 240 1% = 240 4 89 = 2.696 629 213…

23 365

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

23 = $1990 3 0.2 3 365 = $25.079 452… = $25.08 [nearest cent]

The estimated population in 2008 was 270.

Heather paid $25.08 interest, to the nearest cent.

(2 marks)

(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)

(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….

(e)

(2 marks)

36

37

38

39

40

41

42

QUESTION 27 (i) 7 members can do more than 38 pushups in one minute. 7 P(more than 38 push-ups) = 13 (1 mark)

(a) Total deductions = $296.40 1 $24.50 1 $15.80 = $336.70 Net wage = $1024 2 $336.70 = $687.30



The container will last 15 full weeks.

379

5/12/13 12:21 PM


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

Let x cm be the horizontal length of the ramp. 39 tan 58 = x 39 x= tan 5° = 445.772 0398…

Percentage of net wage 70 = 3 100% 687.30 = 10.184 78… % = 10.2% [1 decimal place] (b)

= 446 [nearest unit]

(3 marks)

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

13 cm

d 1 60 cm = 446 cm

NOT TO SCALE

230°

Arc length: l =

d = 386 cm

(3 marks) First Scarf

(e) (i)

u 2pr 360

14 ___ 33

S

19 ___ 33

13 ___ 32

S

19 ___ 14 32 ___ 32

230 = 3 2 3 p 3 13 360 = 52.185 3446… = 52 cm [nearest cm]

Second Scarf

18 ___ 32

91 528 (iii) P(different fabrics) = P(SW) 1 P(WS)

(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.

W

(2 marks)

14 13 3 33 32

(ii) P(SS) =

(2 marks)

S

W

Perimeter = 52 cm 1 2 3 13 cm = 78 cm The perimeter of the sector is 78 cm, to the nearest centimetre.

W

2 cm represents 10 km.

=

14 19 19 14 3 1 3 33 32 33 32

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

=

133 264

(1 mark)

(2 marks)

(ii) 1 cm represents 5 km.

QUESTION 28

Now 75 4 5 = 15 © PASCAL PRESS 2014 ISBN 978 1 74125 470 9

So 15 cm would represent 75 km. The two towns will be 15 cm apart on the map.

(1 mark)

(a) V1

V2

R S

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

P

N

Q M

O

(2 marks)

39 cm 5° d

60 cm

(b)

c 18abc 18ab = 2 3 3a b 3a 2b

x cm

380


=

6c a

(2 marks)


(c)

NOT TO SCALE

4m

1.5 m

3m

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

The triangle formed by the flagpole and its shadow and the triangle formed by Jacques and his shadow are equiangular, so they are similar. 31d d 5 So 1.5 4

(e) Deposit = 0.1 3 $2800 = $280 Balance owing = $2800 2 $280 = $2520

4d = 4.5 1 1.5d 2.5d = 4.5 d = 1.8

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

(3 marks)

Interest = $3427.20 2 $2520 = $907.20

(d)

I = $907.20

English

P = $2520

n=3

I = Prn Mathematics

10

20

30

40

50

60

70

80

90

(i) English: lower quartile = 50

100

r = $907.20 4 $7560 = 0.12 The annual rate of interest was 12%.

upper quartile = 80 interquartile range = 80 2 50 = 30

(4 marks) (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.

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 first 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 first eruption, the longer the wait until the next eruption. (1 mark)



0

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

381

5/12/13 12:21 PM


(b) x = 6.000 s = 0.040

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

x 1 s = 6.000 1 0.040 = 6.040

EH = GH = x km

x 2 s = 6.000 2 0.040 = 5.960

x2 1 x2 = 822

By Pythagoras’ theorem in nGEH, 2x2 = 6724

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

x2 = 3362 x = 57.982 756… [x > 0] The length of GH is 58 km to the nearest kilometre.

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)

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

G

(c)

64 km F 139° NOT TO SCALE

82 km

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

31° E

Contribution = 0.05 3 $6800 = $340

H

(i) /FGE = 1808 2 (139 1 31)8 = 108

n = 13 3 12 = 156 r = 0.066 4 12 = 0.0055

82 sin 10° EF = sin 139°

A=Me

= 21.704 069… = 22 km [nearest km] EF 64 {Or: 5 sin 10° sin 31°


(1 1 r) n 2 1 f r

= $340 e

(1 1 0.0055) 156 2 1 f 0.0055

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

64 sin 10° sin 31°

= 21.577 984… = 22 km [nearest km]}

(2 marks)

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

By the sine rule in nEFG, EF 82 5 sin 10° sin 139°

EF =

(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:]

(3 marks) (2 marks)

Or: By the cosine rule in ΔEFG,

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

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

M = 0.05 3 $81 600 = $4080

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

A=Me (2 marks)

(1 1 r) n 2 1 f r

= $4080 e

(1 1 0.0055) 156 2 1 f 0.0055

= $1 003 606.701… = $1 003 606.70 [nearest cent]} 382



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) h

40

30

20

10

–100

O

100

(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)

200

300

400 d

(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)

(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) Excel 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


–200

383

5/12/13 12:21 PM


(c)

P

P = A (1.04)

n

8 000 000

7 000 000

6 000 000

P = A (b)n

Population

5 000 000

4 000 000

3 000 000

2 000 000

1 000 000

0 0 (2010)

20 (2030)

10 (2020)

n

30 (2040)

Years Predicted population if the policy had not been introduced Predicted population with the policy introduced

(i) In 2030, n = 20

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

From the graph, when n = 20,

Try b = 1.01

P = 6 600 000

P = 3 000 000(1.01)20 = 3 660 570.12…

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

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

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

Try b = 1.02 P = 3 000 000(1.02)20 = 4 457 842.188… = 4 460 000 [3 significant figures]

(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)

So the value of b should be 1.02 to 2 decimal places. (2 marks) (iv) In 2050, n = 40


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

P = 3 000 000(1.02)40 = 6 624 118.991…

P = 3 000 000(b)20 Try b = 1.03

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

P = 3 000 000(1.03)20 = 5 418 333.704….

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. 384


Excel Success One HSC General Mathematics SAMPLE 2014

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