practice hsc papers general 2

CHAPTER 19 ample HSC Examination 1

time: .2112 hours Total marks: 100

Marks:

Attempt Questions 1 to 25. ,. Allow about 35 minutes for this section Each question is worth 1 mark. Oll Circle only ONE option.

1 Which of these relationships shows a strong positive correlation? .-------~

A B C

"

D

2 The figure drawn is a closed cylinder. Which of the following expressions will give the surface area of the cylinder? A 1t X 82 + 2 X 1t X 8 X 25 B 2 X 1t X 82 + 2 X 1t X 8 X 25

D 1t X 42 + 2 X 1t X 4 X 25

3 Energy is used at the rate of 20 joules per second. The power developed is: A 100 W B 25 W C 20 W D 5 W

..

~ 8 em --J>-

4 A die is thrown and a coin is tossed at the same time. Find the probability of throwing a number less than 4 and a head.

1 A

2 1 3

c 1 4

5 As shown in the figure, a square is drawn in a circle of radius 6 em. The area of the shaded region is given by: A nx 122 -122 B nx 62 -122

D nx62 -12x6

D 1 6

t 25 em

t

6 A car that was purchased for $39 900 is worth $20 400 after 5 years. The annual amount of depreciation, using the straight Line method, is: A $3900 B $4080 C $7980 D $12 060

7 3x

Solve 3(x- 2) = 2 A )( = -4 B )( = LJ. c X== -2 D x=2

8 Find angle e correct to the nearest degree. L3 A 37 B 6,10 c 53 D 49

4

9 2.5 g of a medicine is used to make a 200-mL solution. The concentration of this medicine, in mg/ml, is: A 0.125 B 1.25 C 12.5 D 125

Chapter 19-Sample HSC Examinations

Sam pi HSC Examination l 10 Stamp duty on new vehicles is given.

Car value Stamp duty $59 113 or less 3% ($6 for every $200 or part thereof) Over $59113 5% ($10 for every $200 or part thereof)

Calculate the stamp duty payable on a car worth $29 850. A $895.50 B $897 c $900 D $1773.40

11 Use the sine rule to find the length of the side x correct to one significant figure. A A 7 em B 5 em $

'\ C 4 em D 8 em v,

60 50 B X

12 A bathroom tap generally runs at 8 L of water per minute. Tim leaves the tap running while he brushes his teeth for 3lf2 minutes each day. The water he has wasted in a week is about: A 160 L B 180 L C 200 L D 220 L

13 Lauren's z-score on a particular test was 1.5. H the class average and standard deviation were 64 and 8 respectively, Lauren's mark was: A 52 B 65.5 C 76 D 86

14 Which of the following equals 220 bytes? A 1 kilobyte B 1 megabyte C 1 gigabyte D 1 terabyte

c

15 Saccio earns commission selling musical instruments and equipment. He is paid a retainer of $100 per week as well as 5% commission on sales up to $10 000 and 7% commission on additional sales over $10 000. What does he earn in a week where he sells $12 500 worth of goods? A $675 B $775 C $875 D $975

16 David buys a sound system for $2849 including GST of 10%. How much GST was paid by David? A $28.49 B $284.09 C $284.90 D $259

17 The cost of hiring a venue involves a fixed cost of $300 then varies directly with the number of people in attendance. If it costs $450 for 30 people, how much would it cost to hire the venue for 40 people?

A $480 B $490 c $500 D $550

18 The gradient of the line l is: 1 y

A B -5 5 1

c +- D +5 5

238 Excel ESSENTIAL SKILLS HSC Mathematics General 2 Revision & Exam Workbook

sample HSC Examination 1 ----

Luanne bought two tickets in a raffle. The tickets cost $5 each. 500 tickets are sold and there is one prize of $1000. Luanne's financial expectation is: A -$10 B -$6 C -$1 D $4 A new laptop was purchased for $1800 on 15 June using a credit card. Simple interest was charged at a rate of 23.42% p.a. for purchases using the credit card. No other purchases were made and there is no interest-free period. Interest is charged on both the date of purchase and date of payment. What was the total amount owing on 7 July the same year? A $1826.56 B $1827.72 C $2656.41 D $2771.90

21 A plank is 8 em wide to the nearest centimetre. The percentage error is: A 12.5% B 8% C 6.25% D 4%

22 Consider the following chart. 180 170 160 150 140 130

01 120 ::::.- 110 +-' -, 100 'iii 90 3

80 70 60 50 40 30

1.4 1.5 1.6 1.7 1.8 1.9 2.0 Height (m)

A 1.8-m tall man has a weight of 120 kg. What is the least amount of weight he needs to lose so that he is no Longer in the obese category? A 10 kg B 20 kg C 30 kg D 40 kg

Which of the following is a linear function? C y = 3x- 7

In the following diagram, what is the true bearing of P from Q? A 015 B 075

D 285

12 D y=-

x

N

p

N

s

Emiko used the capture/recapture technique to estimate the number of frogs living around a pond. Initially, she caught, marked and released 26 frogs. When she returned a week later, only 16 of the 24 frogs she caught were marked. What is the approximate population of frogs around the pond? A 17 B 34 C 39 D 42


ample HSC Examinati n 1

Sample HSC Examination 1 Question 26 continued

d Solve the following equations.

7 x - 2 0 = 3x + 16

ii Vx+3 = 8

e A radial survey of a field is given. D 320

0

c 205

Find the size of LAOB.

ii Calculate the area of LAOB to the nearest square metre.

m Find the distance from A to B to the nearest metre.


A 060

8 130

2 marks

I.

2 marks

1 mark

2 marks

2 marks

tion 1 Question 27 a i Solomon invests $25 000 in an account earning 6% p.a. interest, compounded quarterly.

How much is his investment worth, to the nearest whole dollar, at the end of 5 years? 2

ii Sally invests $3000 at the end of every 6 months in an account that earns 8% p.a. interest, compounded 6-monthly. Using the table below, calculate the amount her investment is worth, to the nearest dollar, at the end of 5 years. 2

future value of $1 Interest rate per period

Period 3% 4% 5/o 6/o 10/o 1 1.0000 1.0000 1.0000 1.0000 1.0000 2 2.0300 2.0400 2.0500 2.0600 2.1000 3 3.0909 3.1216 3 .. 1525 3.1836 3.3100 4 4.1836 4.2465 4.3101 4.3746 4.6410 5 5.3091 5.4163 5.5256 5.6371 6.1051

10 11.4639 12.0061 12.5779 13.1808 15.9374 20 26.8704 29.7781 33.0660 36.7856 57.2750

iii How much more would Solomon need to have invested to have the same amount as Sally at the end of the 5 years? Give your answer to the nearest dollar. 2 marks



b A mobile phone plan has a 25-cent connection fee then charges 31 cents for each 1/2 minute block. Calculate the cost for a 4-min 39-s call. 2 marks

c A delegation of three people is to be chosen from a group of eight people.

How many different delegations are possible? 2 marks

ii Each of the members of the delegation must make a speech at the meeting they are to attend. The order in which the three members will speak will be listed in the agenda for the meeting. In how many ways can three chosen members of the delegation be Listed to speak? 2 marks

d A scientist has developed a new screening test for a virus. The test was trialled on people who had been exposed to the virus but before any symptoms had time to develop. A positive result meant that the test showed the person had the virus. After several days it was possible to see which people actually had the virus and hence whether the test results were correct. The results are shown in the two-way table below.

Test positive Test negative Total Had virus 102 37 139 Did not have virus 26 135 161 Total 128 172 300

How many people had a false result to their test? 1 mark

ii Find the probability, as a percentage, that the test was accurate. 1 mark

What is the probability that a person with a positive test actually had the virus? 1

Chapter 19-Samp!e HSC Examinations

s tion l Question

a Angela borrowed $30 000 at 9.5% p.a. simple interest so that she could furnish her home unit. She repaid the loan, plus interest, with equal monthly instalments over 5 years.

Calculate the amount of interest charged. 2

ii How much was each monthly instalment? 1

b The table gives an average water use per person per day.

Use litresjday Bath/shower 72 Sink 18 Toilet 35 Washing clothes 45 Washing dishes 29 Cooking 8 Miscellaneous 6

Total

Complete the table to insert the value for total. 1 mark

ii Some people will look at this table and argue that they don't use this much water. How can you respond? 1 mark

iii How much water is used in the sink over a year? 1 mark

iv How much water (in kilolitres) does the average person use in a year? 1 mark


sample HSC Examination 1 Question 28 continued

v One of these areas uses 21.1% of the daily water use. Which is it? 1 mark

vi What percentage of water is spent on washing dishes? 1 mark

vii A water-efficient toilet can save about 20% on a person's water use of that device. A What is the daily saving in this area? 1 mark

B What is the annual saving in this area? 1 mark

c The distance, d metres, that an object falls is proportional to the square of the time, t seconds. An object dropped from a plane falls 45 m in 3 seconds. How far will it fall in 8 seconds? 2 marks

weight (kg) d The body mass index can be calculated from BMI = h . h ( ) h . h ( )

e1g t m X e1g t m A man has a BMI of 29 and a height of 1 m 80 em. Calculate his weight. 2 marks

Chapter 19-Sampie HSC Examinations

ample H C Examination 1 Question 29

a John leaves Sydney (34S, 150E) at 2 pm Wednesday by plane on a 7-hour flight to Singapore (1 N, 104E).

What is the time difference between Sydney and Singapore?

ii What time is it in Singapore when the plane leaves Sydney?

2 marks

1 marks

Hi If another flight leaves Singapore at 10 am (local time) Wednesday when does it arrive in Sydney? (The flight time is 7 hours.) 2 marks

b A and Bare two locations on the Equator with longitudes 60E and 20W respectively. Find, to the nearest kilometre, the distance along the Equator from A to B. (The radius of the Earth is 6400 km.) 2 marks

c A class of Year 12 students recorded their bodyweight and displayed the data in the back-to-back stem-and-leaf plot shown:

Boys Girls 4 9

6 2 5 3 4 6 7 8 9 8 7 4 2 1 0 6 0 0 2 3 8

5 5 4 0 7 2 5 1 8 7 9

Determine. the upper and lower quartiles for the boys' data. 2 marks



ii Show if there is an outlier in the boys' data. 2 marks

d A company manufactures cylindrical pipes all of Length 70 em. The approximate capacity, in Litres, of the pipes for different diameters is shown by the graph.

Capacity (litres)

600

550

500

450

400

350

300

250

200

150

100

50

10 20 30 40 50 60 70 80 90 100 110 120 Diameter (em)

What would be the diameter of a pipe with capacity 100 Litres?

ii Carmel predicted that a pipe with diameter 120 em would have a capacity of 600 Litres. Do you agree with this prediction? Briefly comment.

iii If the capacity can be approximated using the formula C = 0.055d 2, find the capacity when d = 120.


1 mark

2 marks

1 mark

247

' '-(

I I I

S mple H n 1 Question 30

a Twenty-five swimmers participated in an interstate diving competition. Each dive was awarded points on a scale of 1 to 10. The distribution of the point scores is shown in the following table.

Point score 4 5 6 7 8 9 10 Number of swimmers 2 1 3 5 6 5 3

Find the mean. 1

ii Find the standard deviation correct to two decimal places. 1

iii How many of the scores lie within two standard deviations of the mean? 1

iv If one of the 25 swimmers is chosen at random, what is the probability that this swimmer's dive score is less than 7? 2

b Two red and eight blue pens are in a box. Leah selects two pens at random, without replacement.

Complete the tree diagram, by writing the probabilities on each of the branches.

2nd pen

1stp~R ~R--B

~ -----R B~ B

ii What is the probability that the pens are different colours?

1

1


s mple HSC Examination 1 Question 30 continued

What is the probability that at least one of the pens is red?

c A water reservoir is constructed as shown. 60 m

30m

33m

One end is semicircular, while the other end is rectangular. The length of the reservoir is 60 m, with rectangular cross-sections at regular intervals as shown.

Calculate the area of the semicircular end.

ii Calculate the area of the opposite rectangular end.

iii Using two applications of Simpson's rule, calculate the area of the tank, correct to the nearest tenth of a cubic metre.

How many litres of water can the tank hold?


1 mark

2 marks

1 mark

3 marks

1 mark

Sample HSC Examination 2 Total ti~1U:!: 2V:z hours Total marks: 100

SECTION I Marl

Sample HSC Examination 4 Each division represents 10 m.

Calculate the approximate area of this Lake.

C 3900 m2 D 39 000 m2

5 John is driving a car at a constant speed of 120 km/h. What distance will he cover in 1 hour and 15 minutes? A 96km B 125 km D 150 km

6 For the set of scores 30, 60, 70, 30, 50 find the di-Fference between the mode and the median. A 10 B 20 C 30 D 40

7 The graph shows a typical household's water use.

A typical family uses 240 kl water each year. From this graph, approximately how much water was used in the garden? A 80000 L B 100000 L C 120000 L D 140000 L

8 For the formula V = u +at, find the value of V if u = 32, a= 3.5 and t = 6. A 11 B 53 C 213 D 672

weight (kg) x adult dosage 9 Clark's formula for calculating the medicinal dosage for children is dosage = ---=--_.______:_-=-: _____ :::_

. 70 What weight should a child be to receive one-quarter the adult dosage of the medicine? A 15kg B 17.5kg c 20kg D 22.5kg

10 Simplify 16x- 12 + 4x + 2.

A 12x- 10 B 12x + 10 C 20x + 10 D 20x- 10 .I

11 How many files of size 4.8 MB can be stored on a hard drive of 4GB?

A 4 X 1024 + 4.8 B 4 X 1024 X 4.8 C 4 X 4.8 + 1024 D 1024 + (4 X 4.8) '

Chapter 19-Sample HSC Examinations 251

Sam'ple HSC Examination 2 12 Which of the following formulae can be used to find the length x?

l

A

c 1 . A= -ab smC 2

b2 + c2 - a2 B cosA=----

2bc

D a

sin A b

sin B

13 Find the interquartile range for the stem-and-leaf plot shown.

A 20 B 20.5

c 25 D 38

211368

3 23459

412468

5 34589

p

14 Anne buys 2 tickets in a raffle in which 50 tickets are sold and there are 2 prizes. What is the probability of Anne winning both prizes?

2 1 2 1 2 1 2 1 A -+- B -X- C -+- D -X-

50 50 50 50 50 49 50 49

15 A concrete pipe has an inner diameter of 20 em and is 3 em thick. What volume of concrete is used to make a pipe section 15 m long?

16

B 0.43 m3 C 3251.55 m3 D 6078.98 m3

By applying one application of Simpson's rule, which of the following expressions will give the area of the field?

40 A 3(55 + 280 + 35)

40 c -(55+ 70 + 35)

3

B 80 -(55 + 280 + 35) 3 80

D -(55+ 70 + 35) 3

55 m

40 m

- 70 m 35 m

40 m

17 Life expectancy in China in 1960 was 43.5 years. In 2010 it was 73.3 years. The percentage increase in these 50 years is:

18

19

A 29.8% B 40.7% c 54.4% D 68.5%

In the triangle drawn, the value of y is given by: A y = 9 cos 63

9 c y=

cos 63

32xV -7- 20xy4 = 8 8x3

A -x7i B 5 5y

B y = 9 sin 63 /9 9 D y= sin 63 o 8 8x1o

c -x3y D -5 5y12

) Excell ESSENTIAL SI

sample HSC Examination 2 In the solution of the equation below, which lines do not follow correctly from the previous lines?

3 (x - 5) = 5x + 9 3x- 5 = 5x + 9 Line 1

5=2x+9 Line 2 -4 = 2x Line 3 -2 =X Line 4

A Lines 2 and .4 B Lines 1 and 2 c Lines 1 and 3 D Lines 2 and 3

\, Increase $5000 by 20% and then decrease the final amount by 20%. The answer is:

A $4800 B $5000 c $5400 D $6000

Matthew took out a Loan of $9000 at the flat interest rate of 8% p.a. over a term of 36 months. How much will he have to repay?

A $6840 B $11337 c $11160 D $2160

y The equation of this graph could be:

y=x2 +3 y=x3 +3

A B 1 X

3 Y = 3x c y=-

D X

f 24 The mean of a set of 6 scores is 35. What is the new mean of a set of scores after a score of 56 is added?

A 39 B 3.75 c 38

25 Find the volume of the triangular prism.

A 1560 cm 3 B 1200 cm3

C 300 cm3 D 600 cm 3

Chapter 19-Sampte HSC Examinations

D 15

p

E u N

'"'

s ' ' ' ' ' ' ' ' ' : ' y] _______ u

/

R

253

I .I

Sample HSC Examination 2 SECTION U Marks:

/ Instructions c Attempt Questiohs 26 to 30.

Allow about 1 hour 55 minutes for this section. .. Show all relevant mathematical reasoning and/or ca,tc;ulations.

Question 26

a i . Giveh that x3 = 2.16 x 108, find the value of x. 1 mark

. 2 ii Fmd the exact value of -(m3 - m2 + m) when m = -3. 3 . 1 mark

iii Solve for x: 4(x- 3) = 9 - 3(x + 2). 2 marks

1 3 A car travels at 90 km/h for 24 hours and at 110 km/h for 14 hours. Find the total distance travelled. 1 mark

ii If this distance is travelled on 50 L of petrol how far would the car travel on 35 L of petrol? " 1 mark

Excel ESSENTIAL SKILLS HSC Mathematics General 2 Revision & Exam Workbook

mple HSC Exami ation continued

c The graph shows the line of best fit in an experiment where the burning time of a candle was measured. 28

26

24

22

20 \ ~

+-' 18 ::J c

.E 16 ~

QJ E 14 ~ 01 c 12 " ::J 10 o::J

8

6

4

2

Name the dependent variable.

ii What is they-intercept of this line?

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Length of candle (em)

iii What is the significance of they-intercept?

iv What is the gradient of the line, correct to one decimal place?

v Write the equation of the line in the form y = mx +bin terms of the given variables.

d If the average peak rate for electricity is $0.25 per kilowatt-hour, calculate the cost of running a 200-watt TV for 7 hours and a 180-watt computer for 8 hours during peak time..,


1 mark

1 mark

1 mark

2 marks

2 marks

2 marks

/

255

Sample HSC Examination Ouestior1 27

a Penny made up a game to be played by tossing a die. If the uppermost face shows a 6 the player wins $4, if it shows a 4 the player wins $3 and if it shows a 2 the player wins $2. If the player tosses an odd number, she or he Loses $1.

What is the financial expectation when playing this game? 2 marks

ii If Penny charged $1 to play the game, would you recommend someone to play the game? Justify your answer. 2 marks

b The following survey is a field book entry when a block of Land was surveyed. All dimensions are in metres.

B 70

D 25 50 15 12 c 0 A

Draw a sketch of the block of land. 2 marks

ii Calculate the area of the block of land. 2 marks

ExceB ESSENTIAL SKILLS HSC Mathematics General 2

Question 27 continued

m Find the length of the boundary ADB, to the nearest metre.

c Cairo (30N, 31 E) and Durban (305, 31 E) lie on the same great circle.

What is the angular distance between these two cities along the great circle?

ii Find the distance between the two cities along the great circle, in kilometres. (You may assume the radius of the Earth is 6400 km.)

d For calculating medicinal doses, age of child (in years) x adult dosage

Young's formula (children 1-12 years): Dosage= f h'ld (' ) age o c 1 m years + 12

weight in kg X adult dosage Clark's formula: Dosage=--=---=---------=---70

If the usual adult dose of a drug is 60 mg, use Young's formula to find the dose for a child of 7 years, weighing 28 kg.

ii Use Clarke's formula to calculate the dose for a child of 7 years, weighing 28 kg.

iii Suggest a reason for why the doses calculated above using the two formulae are different.

Chapter 19-Samp!e HSC Examinations

2 marks

1 mark

1 mark

1 mark

1 mark

1 mark

j

Sample HSC Examination Question 23

a The weight of the contents of cans of a particular brand of dog food is normally distributed with mean 757 g and standard deviation 3.5 g. There are 24 cans per carton. In a shipment of 20 cartons of this dog food, how many cans would you expect to have less than the labelled weight of 750 g? 3 marks

b The graph shows the life expectancy of people from Cambodia and Myanmar (formerly known as Burma).

~ "' QJ 2J u c

"' -

Examination 2

c The points P, Q and Ron the diagram represent three towns. Q is due west of P. The bearing of R from Pis 240 and the bearing of R from Q is 205. The distance from Q to R is 58 km.

a r-----------;r P

R

What is the size of LPQR? 2 marks

ii What is the size of LQPR? 2 marks

iii Find the distance from P to R, to the nearest kilometre. 2 marks

iv What is the bearing of P from R? 1 mark

/


Sample HSC Examination 2 Questiotjl 29

a Ben invests $5000 into a savings fund. Interest is paid at the rate of 6.5% p.a. compounded monthly on this amount. How much will the investment be worth at the end of 25 years? (Give your answer to the nearest dollar.)

b In a certain state the stamp duty for a new private car is as shown.

Car value Stamp duty $600 or less $20 $601 to $35 000 3% ($3 for every $100 or part thereof) $35001 to $40000 $1050 + 11% for the proportion over $35 000 $40 001 and over 4% ($4 for every $100 or part thereof)

How much more in stamp duty is paid for a vehicle costing $40 000 than one costing $20 000?

c Tim wants to fence a small rectangular yard for his dog. He intends to use the back wall of his shed as one side and an existing fence as another side so that he only needs to fence two sides as shown in the diagram. He has enough materials to fence 8 m. The length of one side is x m.

Show that the area A m2 of the yard is given by A = 8x- x2

The graph of A = 8x- x2 is a parabola as shown below. A

0

X

Shed

8 X

2 marks

2 marks

Existing fence

2 marks

Ex

Sam pi n2 Question 29 continued

ii What value of x will make the area a maximum? 1

iii What is the largest possible area of the yard? 1

iv What shape is the yard when the area is a maximum? 1 mark

v Tim decides his dog needs a bigger yard. He has worked out the maximum area for different lengths of fence. These results are shown in the table.

length of fence (l) 10 12 14 16 18 20 22 Maximum area (M) 25 36 49 64 81 100 121

Write an equation that describes the relationship between M and I.

d A survey was taken of the number of phone calls made in a business office over a 24-hour period.

How many calls were made altogether?

u c Q.J :::l 0" 2:'

Ll...

50

40

30

20

10

0 E 0..

N ,--<

I E ro

00

ii What percentage of calls were made between 12 pm and 4 pm?

E E E E 0.. 0.. ro ro

..;- 00 N ..;-I I ,--< I E E I E 0.. 0.. E ro

N ..;- 0.. N ,--< 00 ,--<

Time of day

24

1~-4

2 marks

E ro

00 I E ro

..;-

1 mark

1 mark

iii Offpeak calls occur between 8 pm and 8 am. If the total cost fen these calls is $11, calculate the charge for each call. 2 marks


Sample HSC Examination t

Question 30

a Calculate the difference in the surface area of a tennis ball (radius 3.5 em) and a soccer ball (radius 10 em). Give your answer correct to two significant figures. 3 marks

b Students in two classes recorded how many sit-ups they could complete in one minute. The results are shown on the box-and-whisker plots below.

Class 1

Class 2

5 10 15 20 25 30 35 40 45 50 55 60 65 70

Describe the general shape of the data for class 1.

ii What percentage of students in class 2 completed between 35 and 50 sit-ups?

iii Given that there is an equal number of students in both classes, compare and contrast the results.

1 mark

1 mark

2 marks

c The angle of depression made by a Ladder against a wall is 70. If a 3m Ladder positioned at the same spot reaches 1.5 m Lower down the wall, calculate its angle of elevation to the nearest degree. 2 marks

A ----------

1.5m

D

3m B L-------" C

lEJiUCell ESSENTIAL SKILLS HSC Mathematics General 2 Revision & Exam Workbook

d Michelle and Holly decide to set aside regular sums of money to save for future expenses. Use the table below to answer the following questions.

Future value of $1 Interest rate per period

Period 3% 4% 5% 6% 10% 1 1.0000 1.0000 1.0000 1.0000 1.0000 2 2.0300 ' 2.0400 2.0500 2.0600 2.1000 3 3.0909 3.1216 3.1525 3.1836 3.3100 4 4.1836 4.2465 Ld101 4.3746 4.6410 5 5.3091 5.4163 5.5256 5.6371 6.1051

10 11.4639 12.0061 12.5779 13.1808 15.9374 20 26.8704 29.7781 33.0660 36.7856 57.2750

Michelle chooses to set aside $600 every 6 months. If she earns 10% interest per annum, how much will she have in 5 years? 2 marks

ii If Holly saves $1200 per year, how much will her investment amount to in 5 years, if interest is charged at the same rate? 1 mark

iii Both girls have a savings goal of $10 000. How much more would they each have to put into their savings account to reach their goal in 5 years? Answer to the nearest dollar. 3 marks


Answers PAGE 231 1 a 0.8 kW x 3.5 hours= 2.8 kWh b 2.8 kWh x 0.24 dollars per kWh= $0.67 c 1.2 x 2 = 2.4 kWh d 2.4 X 0.24 = $0.58 e 0.5 X 4 = 2.0 kWh f 2.0 X 0.24 = $0.48 g 3.0 X 40 = 120 kWh h 120 X 0.24 = $28.80 i 0.54 x 168 = 90.72 kWh j 90.72 x 0.24 = $21.77 k 0.028 x 1000 = 28 kWh L 28 x 0.24 = $6.72 m 0.15 x 42 = 6.3 kWh 11 6.3 X 0.24 = $1.51 2 a i 202 X 10 X 0.21 = $424.20 ii 221 X 10 X 0.21 = $464.10 iii 160 X 10 X 0.21 = $336.00 iv 240 x 10 x 0.21 = $504.00 b Not a strong relationship. Sets W and Z are almost the same size yet there is around an $80 difference in their running costs over 10 years. c 24 x 365 = 8760 hours d 0.25 W = 0.000 25 kW, or 2.5 x 10-4 kW so energy used for the year is 0.000 25 x 8760 = 2.19 kWh. Cost= 2.19 x 0.21 = $0.46 e $0.46 x 5 = $2.30 PAGE 232 1 a 0.51 b 1 star is pegged at 1; 2 stars is 0.8 x 1 = 0.8; 3 stars is 0.8 x 0.8 x 1 = 0.64; and so 4 stars is 0.8 x 0.8 x 0.8 x 1 = 0.512 which is the value on the graph. c From th~~~raph this is 0.45 d energy efficient e False. Between 1 and 2 stars there is a 0.2 drop. Between, say, 4 and 5 sfars the drop is 0.51- 0.41 = 0.1 drop. 2 a i 0.77 X 0.77 = 0.5929 il 0.77 X 0.77x 0.77 = 0.4565 m (0.77)4 = 0.3515 iv (0.77) 5 = 0.2707 b From part (a) you can see a formula developing. CEC = 0.77star '!Lting- 1 . Using this, the comparative energy consumption is 0.777 = 0.1605.

PAGE 233 1 a

250 240 230 220 210 200 190 180

::2 170 ~ 160 ~ 150 -iJ 140 :g 130 ';:., 120

Estimated costs of producing electricity in Australia

+-' ~ ::: 1-.~----;;-----, -~- -~90 ~>Wlnd ~~ ' ~ nuclear 60

l!Ffj~i~l-d:-1 ;:;: l:~~~igg) 2010 2015 2020 2025 2030 2035 2040 2045 2050 2055

Year

PAGE 234 1 A 2 B 3 C 4 A 5 C 6 D 7 A 8 B 9 D

b 2027 c In 2040 the cost is $184 per MWh. This is 18 400 cents per 1000 kWh. Hence the cost is 18.411:/kWh. d Except for existing coal, the costs of generating electricity by various means will slowly decrease over the decades. e i Existing coal ii Solar f Coal g Nuclear h No, although the trend appears to be heading in that direction. We have no idea whether the trend will continue well past 2050. It may stabilise at some point above coal. i Coal is a non-renewable resource. This means at some time (several hundred years from now) the amount of coal still available to be burned will be far less. Costs for coal-generated electricity will rise and coal use will ultimately cease altogether. 2 a 100- (76.7 + 1.0 + 15.0 + 4.7 + 1.5) = 1.1% b Solar, tidal, geothermal etc. c Coal, oil, natural gas d 76.7 + 1.0 + 15.0 = 92.7% e These non-renewable resources will eventually run out. (It is more likely it will become too expensive to remove the few remaining isolated remnants from the grounds.) Also if we are to have an impact on reducing C0 2 emissions, we need to develop alternative C0 2-neutral sources of energy.

PAGES 235-236 10 a 346 x 7.5 = 2595 Wh = 2.595 kWh b 0.346 kW are used each hour, and this costs $0.0590. So 1 kWh= $0.0590/0.346 = $0.170 or 1711:/kWh c $0.0590 x 7.5 x 365 = $161.51 (Or, if using 365.25 days, $0.0590 x 7.5 x 365.25 = $161.62.) 11 a 3 months: June, July, August

295.22 ii -- = 4970 iii 5103 x 0.0396 = $202.08 iv 437.76 + 295.22 + 202.08 = $935.06 0.0594

437.76 b i -- = $0.0014 312 688

12 a i To walk 10 km takes 2 h (2 x 60 x 60 = 7200 seconds). Energy= 100 x 7200 = 720 000 J = 720 kJ ii To ride 10 km takes 10 h = ~ = Z4 minutes= 24 x 60 = 1440 seconds. Energy= 100 x 1440 = 144 000 J = 144 kJ b The energy required:; cove/this distance by bike is 144 =~of the energy used in walking. So riding a bike is about 5 times as

ffi . lk" 720 5 e c1ent as wa mg. 13 a 4 stars b Values are discrete, not continuous. You can have, say, 2 stars, 2112 stars, 3 stars etc. but you can't have 23/4 stars.

3 c No, the points do not line up in a straight line. The drop decreases with higher star ratings. d Around - of the BEC 14 Electricity used= 2.6 x 8 x 30 = 624 kWh. Cost= 624 x 0.22 = $137.28 10

CHAPTER 19-Sample HSC Examinations PAGES 237-249 1 A 2 C 3 C 4 C 5 D 6 A 7 B 8 C 9 C 10 C 11 A 12 C 13 C ( 1.5 = x- 64 , therefore x = 8 x 1.5 + 64, x = 76) 14 B 15 B (earnings= 100 + 0.05 x 10000 + 0.07 x 2500, earnings= $775)

8 16 D 17 C (variable cost= (450- 300) + 30, variable cost= $5 per person. Total cost= 300 + 5 x 40, total cost= $500) 18 A

. 23 . 19 B 20 A (Amount ow1ng = 1800 + 1800 x 0.2342 x-, amount owmg = $1826.56) 21 C 22 B 23 C 24 D 25 C 365

291 Answers

sw,ers 26 a 1000 b i gradient= (62- 42)/50 = 0.4 ii Around 1977 the life expectancy was about 33 years. iii Perhaps war, famine, a brutal dictatorship. c 37.8 km/h d i x = 9 ii x = 61 e i 70 ii 3361 m2 iii '100m / ( 6%)5 X4 . 27 a i FV = 25 000 1 + 4 , FV = $33 671 (to the nearest dollar) ii FV = 3000 x 12.0061 , FV = $36 018 (to the nearest dollar) 1ii 36 018 = PV(1.015) 20 , PV =, PV = $26 743 (to the nearest dollar) Therefore, he would need to have invested 26 743 - 25 000 = $1743 more to have the same amount as Sally at the end of 5 years. b In 4 min 39 s there are 10 half-minute blocks. Cost= 25 + 10 x 31 = 335 cents= $3.35 c i 56 ii 6 d i 63 ii 79% iii ~ 28 a i $14 250 ii $737.50

64 b i 213 ii While they may not use much water, others do. This table is an average taken from many water users.

28 a 12 cans b i 1995 ii Cambodians reached this life expectancy age in 2005, around 10 years later. (64 -43)

iii gradient= ( ) = 0.42 iv Each decade, life expectancy increases by an average of 4.2 years. 2010 -1960

c i 115 o ii 30 iii 105 km iv 060

( 6. 5%)25 XlZ 29 a FV = 5000 1 +}2 , FV = 25 280.989 22 ... , FV = $25 281 b $40 000 vehicle: $1050 + 0.11 x 5000 = $1600; $20 000 vehicle: 0.03 x 20 000 = $600. Difference= $1600 - $600 = $1000 c i If the length is x m, the breadth is (8- x) m. The area is x(8- x) m2 ii x = 4 iii 16 m2 iv Square v M =I 2 or M = (tY d i Total calls= 25 +50+ 40 + 25 + 1D + 15 = 165 calls

11 costs - = $0.22 or 22c. 50 .

50 ii - X 100 = 30.3% 165

iii 50 calls were made in this period, so each call

30 a 41T x 102 - 41T x 3.5 2 = 1102.6990 ... = 1100 cm 2 (to 3sf) b i The data for Class 1 is symmetrical ii 25% iii Both classes have the same median (35), while the data for class 2 is more spread out at both ends .

. _11.5sin20 . . c Angle BCD= 70 -Angle ACD. If angle ACD = sm , then the new angle of elevatwn 1s 60 to the nearest degree.

3 d i 600 X 12.5779 = $7546.74 ii 1200 X 6.1051 = $7326.12 iii Michelle: 10000 + 12.5779 = 795.05, 795.05- 600 = 195.05 Michelle needs to save $195 more per 6 months. Holly: 10000 + 6.1051 = 1637.97,1637.97- 1200 = 437.97 Holly needs to save $438 more per year.

Answers 293

I I j

1

Reading time- 5 minutes : Working time- 2112 hours , 21fz hours Write using black or blue pen, 1 Calculators may be used,

A formulae and data sheet is provided on pp. ~-00-402.

the correct answer (A, B, 35 mirnutes

1-25. (Each is worth Orie made)

How many ordered selections of three people ffom five are possible? r~ 10 B 15 c 20 D 60

1 2 1 If x = 4 andy= 5, what is the value of -xy ? 2

-- ~ ii 12.5 B 50 c 100 D 200

1 Mary's z-score in a test was 1.4. Ifthe mean was 69% and the standard deviation 5%, what was Mary's raw

score? A 74% c 76%

B 75% D 77%

If it is 2 pm inN arrabri (30S, 150E), what is the time - in St Petersburg ( 60N, 30E), ignoring time zones?

A 10 pm B 6 am C 8 am D 8pm

Which is closest to the cost to run an 1800 W pump for 9 hours a day for 30 days if electricity costs 28.6 c/kWh? A $14 c $139

E $17 D $154

Riley has a phone plan where there is a connection fee of 40 cents for calls and they cost 99 cents per minute, or part thereof. SMS messages cost 25.3 cents. How much would Riley be charged for a phone call lasting 3 minutes and 5 seconds? A $3.37 c $3.87

B $3.45 D $4.36

1 Stuart buys a ticket in a raffle in which first prize - ~ is $700 and second prize is $200. If 1000 tickets

are sold for a dollar each, what is Stuart's financial

- --- l i

' I

expectation? l-i -$0.90 c $0.10

B -$0.10 D $0.90

A motorcycle was purchased for $9350 six years ago. If its current value is $2900 and it is being depreciated by the straight-line method, how much depreciation is apportioned every year? A $1105 B $1090 C $1290 D $1075

Consider the scores 8 2 7 9 5 7 4. Which is NOT equal to 7? A mean C median

What is the area of the

B mode D range

triangle (to one decimal place)? A 21.1 m 2

5m~Jm 6 --~ B 7.7 m 2 9 m

C 20.4m2 D 36.8 m2

12 Which could be the graph of y =- (x > 0)?

X

D y

121/-\ ~

Sample HSC Examination 1 303

~ I

I ! I

Will borrows $24 000 and agrees to repay it over five years with equal monthly instalments. The flat rate of interest charged on the loan is 6% p.a. How much is each instalment? A $520 c $464

B $475 D $424

Which scatterplot shows weak negative correlation? AL BL 0 0 0 0 ' f> '1) ~ t:" G "' "' " " " "' "' "

" ., "' "

The diagram shows a washer that is 5 mm thick. - Find its volume to one decimal place.

A 3.1 cm3 C 31.4 cm3

B 62.8 cm3 D 12.6 cm3

Expand and simplify 4(x- 1) - 3(x- 2). Ax+2 Bx+5 C x-10 ux-7

A patient is to receive 750 mg of a drug at 15 mg - per hour. The drug is available as an infusion vvith

500 mg in 250 mL. What flow rate should be set? J'- 2.5 mL/h B 7.5 mL/h C 25 mL/h D 75 mL/h

l~ -- Consider this set of scores: ,-

l ~ 13, 15, 17,18,20,2~2~27,30,44. Vvhich of the scores are outliers? l'- none B 4 only C 44 only D 4 and 44

} . A two-way table has been drawn up showing the results of a lie-detector test.

j Detected true 'fn:1e stat~I.TlLeEt I 84 Fclse staten:-~ent 36

Detected. f2clce 16

124

What is the probability that a statement was correctly detected?

/. [ __ 21 65

1-::j, _j

31 4 -

(c 65 5

304 ExceJJ HSC i'~athematics General 2

6 13

Olivia paid her credit card bill 25 days late. If the amount due was $7600 and the simple interest rate was0.048% per day, how much interest will Olivia have to pay? A $0.91 B $9.12 C $91.20 D $912.00

A plane left X and flew 32Q km toY on a bearing of 125. From Y it flew due east 450 km to Z. Which diagram shows the path of the plane?

A N B N ~ I

125 X

____;2 320 km l~m y 450 km z

c D

z 450 km

N ! I

y

125 X~ z 450 km Y

l Given that V = -1rr 2h, which expression will give a

3 correct value of r?

l ~ B J3V . h nh & nh c D J3V

In a normal distribution, the mean is 64 and the standard deviation 12. Appro:ximately what percentage of scores lie between 40 and 76? A 68% B 95% c 81.5% D 99.7%

Some number plates have two letters, two numbers and then two more letters. How many of these number plates will have the same letter four times?

"\ A 456976 B 14950 C 10400 D 2600

. It is known that D varies inversely with V. If D = 8 when V = 315, what is the value of D when V = 40? i_ 1575 B 5 C 283 D 63

A cylindrical tank holds 20 000 litres of water. The -- - tank is 2. 5 m high. What is the diameter of the tank

closest to? A 6.5m B 3.2m C 2.4m D 1.6 m.

~ j Suggest,ed time: l hour 5.5 mim.lt,es Total: 75 marks Attempt Questions 26-30. Show all working.

(15 marks) a A financial institution published the following

table ofhome-loan repayments per $1000 borrowed at various interest rates over various periods.

Montl:ily repayments per $1000 borrmved Period Interest rate (years) 6% 7.~%

7 $14.61 $15.34 12 $9.76 $10.55 15 $8.44 $9.27 25 $6.44 $7.39

i How much would the monthly repayments be on a $70 000 loan

9% $16.09 $11.38 $10.14

$8.39

over 12 years at 6% p.a. interest? 1 mark ii How much interest would be paid on a

$110 000 loan over 25 years at 9% p.a. interest? 2 marks

ill Brock pays $613.60 per month on his loan. If the loan is over 7 years at 7.5% p.a., how much did Brock borrow? 1 mark

iv Rebel was originally offered a loan of $87 000 over 15 years at 7.5% p.a. Before she accepted the loan, interest rates rose. She finally took the loan over 15 years at 9% p.a. interest. How much extra were Rebel's monthly repayments? 1 mark

b The diagram is a plan of a small farm, which is bounded on one side by a straight road and on the opposite side by a river.

515 m 656 m 597 m 684 m

736 m

river

i Use Simpson's rule twice to approximate the area of the farm in hectares. 2 marks

ii The measurement along the road w.as not accurate. The actual measurement is 1206 m. How many square metres difference does this make to the area? 2 marks

c A back-to-back stem-and-leaf plot has been dra,Nn showing the scores by students in two different classes, X and Y.

Class scores X y

9 3 4 6 7 7 2. 0 5 1 4 8

9 8 5 4 1 6 0 3 5 6 8 6 5 3 2 7 1 2 4 7 8

9 6 5 3 8 2 4 4 7 9 9 4 1 9 0 3 5

i. What is the range of scores in class X? 1 mark ii What is the median score in classY? 1 mark iii Compare the results, referring to the

shape and patterns of the display. 2 marks

d Solve the equation .J3x + 4 = 5 . 2 marks

(15 marks) a A box holds 5 black and 4 white buttons. Two

buttons are selected at random, one after the other without replacement.

b

2nd 1st button

bu~

c This is a table of present-value interest factors.

~~o I i :J:-enod: :-----l 1 120 1 I 16o I I 200 I i 240

0.6% 0]2% 85.367 '

1

1 80.170 102.67 94.818 116.29 I 105.81

I

----~---~

0.8% i l.2% I ' l

76.955 i 63.419 ! 70.976 I 75.665 1,1 78.575

90.068 99.602

106.53 127.01 _j__ 114.06 L-~--~------ -~------~------~

Craig wants to borrow $160 000. The reducible-interest rate is 1.2% p.a., charged monthly. The term of the loan is 20 years.

Find the amount of each monthly instalment. 1 mark

ii If the lending institution also charges a fee of $15 per month, find the total amount that Craig pays in fees and charges (including interest) over the term of the loan. 2 marks

d Yvonne makes pies to sell at a stall each weekend. She has fixed costs of $150 and there is an additional cost of $3 for every pie. Yvonne sells her pies for $5 each.

How much profit will Yvonne make in a weekend when she sells 200 pies? 1inark

n How many pies must Yvonne sell to break even? l mark

(15 marks) a In a test, the results are given as z-scores.

Alison did the test and her z-score was 1.5. Explain the meaning of Alison's z-score. l m.ark

ii If Alison's raw score was 156 and the mean was 142, find the standard deviation. l mark

b In the diagram, 75 km ~--------..,_Q Pis 75 km due west ofQ. 125 km Ris 125 km 175 km from P and 175 km from Q. R i Use the cosine rule to find the size of

angle QPR. 2 marks ii Find the bearing of R from P. l mark

c Tim made a study of the arm length (from elbow to wrist) and leg length (from knee to ankle) of some students. He found that the arm lengths had mean x = 26 em and standard deviation 4.25 em. The leg lengths had mean y = 45 em and standard deviation 5 em and the correlation coefficient was 0.935.

Briefly describe the correlation between the measurements. l mark ~I Find the equation of the least squares

line of best fit. 2 nurlcs iii Tim's leg length was 56 em. What would

you predict his arm length to be? l mark

BxceJJ HSC /V\athematics General 2

A and B lie on the equator. A and Q lie on the 20oW meridian. P and Q lie on the same parallel oflatitude. LAOB = 90. LAOQ=40. i. What is the longitude ofB? ii What are the position coordinates

1 mark

ofP? 2 marks iii Given that the radius of the Earth is

approximately 6400 lan, what is the distance in kilometres between A and B? 1 mark

e The capture-recapture technique was used in a study of bandicoots on an island. In the first stage of the study 48 bandicoots were captured, tagged, and released. Later another 45 bandicoots were captured and released. 16 of these were found to be tagged.

What estimate does the study give for the number of bandicoots on the island? l mark

ii Later again, 27 bandicoots were captured. If the number of bandicoots was estimated to be 20% less than the previous estimate, how many were tagged?

(15 marks)

1 mark

a The number of beads (B) sewn on a wedding dress varies with the square of the length of the skirt (l em). A skirt oflength 30 em has 1000 beads.

How many beads will be on a skirt oflength 60 em? 2 nlarlcs

E How long is the skirt of a dress with 6250 beads? 2 m.arks

b The following graph shows Lhe life expectancy for females and males according to the area ofNSW where they live.

Life expectancy at birth by local health district and se)(, NSW, 2003-2007 Years Males Local Health District Females Years 78.5 Sydney 84.3 79.2 South Western Sydney 83.9 80.9 South Eastern Sydney 85.3 79.1 Illawarra Shoalhaven 83.8 79.5 78.7 81.9 78.1 77.8 78.6 78.1 78.4 77.7 76.5

'Western Sydney 1~~~ Nepean Blue Mountains ~~~~ Northern Sydney

Central Coast Hunter New England

Northern NSW c=:= Mid North Coast

Southern NSW Murrumbidgee Western NSW

84.0 83.2 85.5 83.2 82.9 83.9 83.4 82.9 83.0 81.9

7 9. 2 1-------,---F===T==i NSW \==r="==r='-----,---1 84.0 100 90 80 70 60 60 70 80 90 100

Life expectancy at birth Life expectancy at birth

Source: ABS mortality data population estimates (SAPHaRI). Centre for Epidemiology and Evidence, NSW Ministry of Health.

i What is the life expectancy of a male in Northern NSW? 1 mark

ii How much greater is the life expectancy of a female in Northern Sydney than that of a male in Western NSW? l mark

iii Briefly comment on any patterns or trends that can be seen. 2 marks

I th ul F kQq . al . . c n e orm a == -,- , r 1s ways pos1t1ve r-

and k is a constant. Make r the subject of the formula. l mark

ii. If k = 9 X 109, Q = 4 X 10-12, q = 1.2 X 10-10 and F == 2.7 x 10-11, find r. 2 marks

d The scatterplot shows th.e number of car accidents over a certain period of time in particular regions and the number of bottle shops in those regions.

4-~ lQI~~~-4~"~+-~-~---~4~

instructions: Reading time- 5 minutes Working time- 21fz hours

'' Write using black or blue pen. :: Calculators may be used. A formulae and data sheet is provided on pp. 400-402.

Choose the correct answer (A, B, Cor D) for each question. Suggested time: 35 minutes Attempt Questions 1-25. (Each question is worth one mark.)

In eight weekly tests, Maria scored the following marks: 8 9 6 8 5 7 8 7 What is the median? A 6.5 B 7 c 7.5 D8

Which expression will give the value of x?

c

12 cos 69 12

sin 69

Simplify 8x- 3(5- x). A Sx- 15 C 9x- 15

B 12 sin 69 12

n--cos 69

B 7x- 15 D llx- 15

. How much compound interest, to the nearest dollar, is earned when $30 000 is invested at 9% p.a. interest, compounded monthly, for seven years?

$18 900 B $33 677 C $24 841 D $26 196

. A box holds one blue, one yellow and two red ribbons. If two ribbons are chosen at random, what is

the probability that both are red? 1 1

A B 3 2 1 1

c - D -4 6

Given that E = mr?, find E if m = 0.8 and c = 3 x 105. .A 7.2 X 1010 B 5.76 X 1010 c 7.2 X 105 D 5.76 X 105

Bxcel! HSC 11!\athematics General 2

. Cameron's pay-rate is $21.60 per hour. What does Cameron earn in a week where he works 37} hours at the normal rate and 6 t hours at time-and-a-half? A $835.92 B $950.40 C $1009.80 D $1020.60

In a state-wide examination, the results are normally distributed with mean mark 62% and standard deviation 12%. What is the best approximation for the percentage of candidates who scored more than 50%? A 34% B 68% c 84% D 95%

Joanne borrows $16 000 and repays it with 60 monthly repayments of $332. What annual flat rate of interest is charged? A 4.9% B 5.4% C 24.5% D 24.9%

Use Simpson's rule to find the approximate area of the land shown in the diagram.

225m

A 11.4 ha C 22.8 ha

600 m

175m

B 12.3 ha D 68.4 ha

A survey was taken of the number of cars in households. The results are shown in the table. Number of cars 0 Frequency 15

What is the median? A 1 c 1.5

l 23

2 30

1.38 D2

3 4 5 l

Simplify ---o-ab 2ab2

l a 2 A 2b2 B 2

b2 D 2b2 c

2

- -- j Which diagram could shoV:, the graph of y = 3(2x)? - .A y.\ D y

X X

c y D Y'

--3 --!1-------~ 0 X 0 X

Over 2 hours Harry, who has a mass of 88 kg, drinks three cans of rum and cola. Each can is equivalent to 1.8 standard drinks. Which is closest to what his blood alcohol content would be when he finishes the third can? A 0.03 c 0.07

B 0.05 D 0.09

A car was bought for $24 000. It is being depreciated using the straight-line method, with the amount of depreciation apportioned per period being 12.5% of the original value. What is the salvage value of the car after six years? A $6000 c $13 000

B $11000 D $18 000

The correlation between two sets of data has coefficient -0.6. Describe the correlation. A weak negative B moderatively-weak negative C moderately-strong negative D strong negative

Notebook entries have been made of a survey of a block ofland. How long is the boundary from A to C? A 30m B 40m C 50m D 70 m

B 100

D 50 70 30 40 c 0 A

Annabel has 3000 shares in a company and received a total of $630 when dividends were paid. At the time the shares were vaiued at $5.60 each. What was the dividend yield? A 3.75% B 8.8% C11.25% D21%

1 What is the equation of the line? I -

~

1 2 3 X

3 A. y=-x+2 2

3 B y:=2x+-2

2 c y=-x+2 3

2 D y=2x+-

3

A committee of three people is to be chosen from a _ group of 12 people. How many different committees

are possible? 1\ 1320 B 479 c 220 D 36

which expression will give the value of x?

~ X

8 sin 30 sin 70 A

sin 70 B 8 sin 30o

8 sin 70 8 sin 80 c

sin 30 D sin 70

Which is closest to the length of time it would take to download a 28.2MB file at 32 kB per second? A 90s B 15 min C 25 min D 2lh h

Dave plays a game in which he has one chance in five of winning $200. If he doesn't win, Dave will lose $50. What is Dave's financial expectation from playing the game? A -$10 c $10

B $0 D $40

What is the solution to the equation

8x + 1 _ x + 3 = 17? 5 4

A X=7 C X= 27

B X= 13 D x = 239

The average (mean) score of 14 students in a test is 85%. One student was absent and did the test later. He scored 70%. What was the new average? A 90% B 84% c 81% D 78%

Sample HSC Examination 2 309

55 rminul:es

26-30. Show ail working.

(15 marks) 360

a The formula n = --- is used to find the 180-a

number (n) of sides of a regular polygon where all the angles measure a0 . (A regular polygon has all its sides equal arid all its angl,es equal.)

How many sides must a polygon have if each of its angles measure 135? l mark

ii What type of polygon has all its angles 108? 2 marks

iii Rearrange the formula so that a is the subject. 2 marks

iv What is the size of each angle of a regular hexagon? 2 marks

b The diagram shows the graph of y = lOOO(aY

y 1000k-~r-~,~~--;-,-------c--~--~-~--~,-,-,-~~-~

9001-~~--;-----;-~----;-~--T--~-~-;---~-~----~----,--;-----

8001-c--;~+-----~~--+--'-----C--~--4-----c--~--~-i---C-

700~--; ____ .J:~~~-c+~-:-_c--~--+--~--~-T-~~~~~~-'-

6 o o 1--,___,.---;----'---"-..c-:--'c----.--- ,--- ~--o---1---c--',---:"~'--'- --'=

5001-----~-~--'---+~-~,-~--,~~-~-r-7-~-r-,-~-~

4001----C-'---,---;-r--+-~~--~~:-+-+---~~-~-~-~---:---+-

3001~---'------C--+--C-4--:---,~-~~~,--"--~--+--c--~-

2001-----:----~----:-+-~---'----,-~-~--~~~~---;-

1001~-:_--,--,--,~- '-L-~-------'--c-~-'---~--~

oL_~~~---~~~~~~-~~-~ 1 2 3 4 56 7 91011121314151 1718 X

What is the value of y when x = 2? l m.ark H For what value of xis y approximately

350? l mark The expression y = lOOO(aY can be used to model the depreciation of an asset, originally valued at $1000, under the declining-balance method.

Hi How much vvill an asset originally valued at $1000 depreciate during the fourth year? l mark

iv What is the rate of depreciation? 1 mark v If the asset was being depreciated by the

straight-line method, and the value was the same as the declining-balance method after ten years, show the depreciation on the graph. l mark

vi How much depreciation is allocated per period under the straight-line method? l mark

vii. For how many years could the straight-line method be used? Justify your answer. 2 marks

31 0 JExceJI HSC Mathematics General 2

(15 marks) a Ava borrows $12 000 over three years, The flat rate

of interest charged on the loan is 7% p.a. i Find the total amount of interest Ava

will pay. l mark ii If Ava repays the loan with three equal

annual instalments, find the amount of each instalment.

iii If the interest rate was reducible instead of flat, Ava's annual repayments would be $4572.62. How much would Ava save altogether with the reducible rate of interest compared to the flat rate?

2 n1arks

l mark

b A helicopter leaves P and flies on a bearing of 070 to Q, a distance of375 km. From Q it flies 515 km on a bearing of 130 to R.

' 170 ' 375 km 515 km

Explain why the size of LPQR is 120. l mark ii Find the distance from R to P to the

nearest kilometre. 2 marks

c The diagram shows the plan of the verandah extension that is being added to a house. The verandah is made up of a right-angled triangle and a sector.

d

6.8 m

6 m existing house

Find the area of the verandah. 3 m.arks ii Find the length of railing required if it is to

be placed on ~p sides of the verandah except where it adjoins the house. 2 m.arks A patient has been ordered 75 mg of medication. The medication is available as 5 mg/100 mL. How many millilitres must the patient receive? l mark

i.i. The patient is to receive the medication through an IV set at 15 drops/mL. If the IV runs at 30 drops per minute, how long, in hours and minutes, will it take for the patient to receive the medication? 2 marks

-1 (15 a A teacher gave a test to her class where all the

questions had to be answered as either true or false. She checked the answers and dww up the following table.

,Question. true Question fabe

104 75

78 193

If 18 students did the test and each one answered every question, how many questions were on the test?

ii What percentage of questions were answered correctly? "

iii What is the probability that a true

2 lTLarks

2 rnar~(S

question was answered true? l m::crlc iY What is the probability of having a correct

answer when the question was answered as true? l mark

ib i A building has a rectangular roof 5 m long and 3m wide. If 15 mm of rai.n falls, how much water should flow from the roof?

2 rnarks ii The water flows into a cylindrical tank.

If the radius of the tank is 1.2 m, by how many centimetres will the water level rise in the tank? 3 marks

c A plane flies from Sydney (34S, 151 aE) across the Pacific Ocean to Mexico City (19N, 99W). The flight takes 16'li hours. If the plane left Sydney at 4:50pm on Thursday Sydney time, what is the local time and day when the plane arrives in Mexico? (Ignore time zones.) 4 marks

(l5 marks) a The masses of certain packets of chips are

normally distributed, with mean mass 210 g and standard deviation 5 g.

Explain what it means to have a z-score of -1. l mark

ii What mass would a packet have if it has a z-score of 3. 1 mark

iii The packets are labelled as having a weight of 200 g. Approximately what percentage of packets would have less than 200 g? 2 marks

b A survey was taken on different days of the number of vehicles passing through a particular intersection in particular time periods during each day. The results for Friday and Saturday were analysed and two box-and-whisker plots were drawn on the same scale.

0

Number of vehicles on Friday ancl Saturday

Friday~ Saturday~

5 10 15 20 25

'What was the highest number of vehicles and on what day did that occur? l mark On Saturday, approximately what percentage of the time were there fe"vver than 8 vehicles? l mark Compare and contrast the patterns for tl1e two days. l marl

(15 rnarks) Express 3. 7 x 1010 litres in gigalitres. 1 mark

b Expand and simplify 2(.!? + 5x + 1) + 3x(x- 1) 2 marks

c The 75 students in Year 4 at a primary school were asked the number of pets in their household. The reshlts are found in the table.

Number of pets (} l 2 3 4 5 6 Frequency 18 22 ll 9 8 4 3

What is the mean? 1 mark H What is the standard deviation?

(Give the answer correct to one decimal place.) l mark

iii Is the data normally distributed? Justify your answer. 1 mark

i.v What percentage ofscores are within one standard deviation of the mean? 2 marks

100 Complete the table of values for T =- . s

l mark

Sketch the graph ofT= 100 (S>O). 2 marks s

100 iili The rule T = - can be used to give the

s time (T) in seconds to travellOO metres at a given speed (S) in metres per second. Use the graph to find the approximate time to travellOO mat 15 m/s. 1 mark

e i On 19 March Billy uses his credit card to get a cash advance of $2400. A fee is charged. The fee is the greater of $5 or 2.5% of the advance. How much is the fee that Billy must pay? l mark

ii. Cash advances attract immediate simple interest at 18.6% p.a. charged from, and including, the date taken and to, and including, the date repaid. If Billy repaysthe advance (plus fee) on 5 May, how much interest will he pay? 2 m2crks

312 Excel! HSC Mathematics General 2

or to pp. 397-399 for

D diameter = 2.4 m radius= 1.2 m; height= 2.1 m v= 1Trh

= n X 1.22 X 2.1 = 9.500 175 ... = 9.5 (1 d.p.)

The volume is 9.5 m3 to 1 decimal place. Now 1 m3 holds 1000 L. So the capacity of the tank is 9 500 litres. B

m Electricity used= (2500 X 6) Wh = 15000 Wh = 15kWh

Cost= 15 x $0.30 = $4.50 B

a 35mm=0.035m Volume= (18 x 12 x 0.035) m3

= 7.56m3 Now 1 m3 = 1000 L. So 7560 L would be caught. D

h I] A=2(a+b) 390m

135 = -(318 + 390)

2

318m

= 47 790 m2 135 m = 4.779 ha

The closest area is 4.8 ha. A

m Number of days= 25 000 + 320 = 78.125

Now 78.125 + 7 = 11.1607 ... So the water will last 11 weeks. B

II!J Scale = 1 : 500

m

So 5 mm represents (5 x 500) mm. This is 2500 mm or 2.5 m. Now 2.5 x 2.5 = 6.25. So each square represents an area of6.25m2. Estimated area= 137 x 6.25 m2

= 856.25 m2 So the area is approximately 856m2. C

Year Usage (kWh) 1990 1.04 X 1014 2000 1.19 X 1014 2010 l.S4 X 1014

a 1.54 x 1014 kWh= 1.54 x lOn MWh = 1.54 X 108 GWh

In 2010 the world's electricity usage was 1.54 x 108 GWh. -./

b From 1990 to 2000 difference = 1.19 X 1014 - 1.04 X 1014 = 1.5 X 1013

From 2000 to 2010 difference = 1.54 X 1014 - 1.19 X 1014 = 3.5 X 1013

Number of times greater 3.5 X 1013

= 1.5 X 1013 1

=2-3

c Without knowing the source of the data no assumptions should be made about its accuracy. -./

m a 20mm=0.02m Volume= (24 x 11 x 0.02) m 3

= 5.28m3 = 5280 L -./

b 15% is not caught so 85% is caught. 85% of 5280 L = 0.85 x 5280 L

=44881 Flow into tank = ( 4488 - 60) L

=4428 L -./ c radius= 1.6 m

4428 L = 4.428 m3 V=nrh

nx 1.62 x h = 4.428 h- 4.428

- 1T X L62

= 0.550 57 ... = 0.55 (2 d.p.)

The height of water in the tank will increase by about 0.55 m or 55 em. -./

d Number of days = 4428 + 220 = 20.127 ... days

The water will last 20 days.

m a Usage cost= 27 X $2.34 = $63.18

b Last reading= 783 - 27 = 756

c Consumption = 27 kL = 27000 L

Average use = 27 000 + 90

-./

= 300 litres per day -./ d Total cost

= $63.18 + $36.94 + $157.24 = $257.36 Cost per day= $257.36 + 90

= $2.859 555 ... = $2.86 (nearest cent) -./

IIJ a Cost= 341 x $0.274 = $93.434 = $93.43 (nearest cent) /

b Cost for 10 years for first fridge = $93.434 X 10

89 m

= $934.34 Cost for second fridge for 10 years = 586 X $0.274 X 10 = $1605.64 Difference= $1605.64- $934.34

= $671.30 It would cost $671.30 more over 10 years to run the second fridge. -./

h a A""-(d1+4d +d1) 3 . m

38 = -(0 + 4 X 32 + 49)

3 38 .

+ -(49 + 4 X 40 + 0) 3

= 4889.3333 .... The surface area of the dam is approximately 4889 m2 .

b Volume= (4889 x 1.8) m3 = 8800.2 m3

The volume is approximately 8800 m3 Now 1 m3 = 1 kL. Capacity= 8800 kL

=8.8ML The capacity of the dam is about 8.8ML. /

SAMPLE HSC EXAMINATIONS Sample HSC Examination 1 ....... pp. 303-307 IJ Ordered selections of 3 from 5

=5x4x3 = 60

If x = 4 andy = 5, 1 2 1 2 -xy = -x4x5 2 2

=50

IJ z = 1.4, :X= 69, s = 5 X-X

z=--s

x-69 1.4= --

5 7=x -69 X =76

Mary's score is 76%.

IJ Narrabri (30S, 150E), St Petersburg (60N, 30E) Degrees difference= 150- 30

= 120 Time difference= (120 + 15) hours

= 8 hours St Petersburg is 8.hours behind

D

c

Narrabri. It is 6 am. B

11 Energy used= 1800 W x 9 h x 30 =486000Wh =486kWh

Cost= 486 x $0.286 = $138.996 ""$139 c

IJ The call lasts more than 3 minutes but less than 4 minutes. Charge will be for 4 minutes. Cost= $0.40 + 4 x $0.99

= $4.36 D

Worked Solutions Ch. 14: Mathematics and Resources Focus Study 2 Test Q5-15 and Sample HSC Examination 1 Ql-6 393

:I

Financial expectation l l

= --x$700+ --x$200-$1 1000 1000

=-$0.10 B

Loss in value = $9350 - $2900 = $6450

Yearly loss = $6450 7 6 '~ $1075 D

8 2 7 9 5 7 4 Mean= 42 77

=6 The mean is not equal to 7. A

9 m

A=~absinC 2 1

= -X5X9Xsin 70o 2

= 21.143 083 97 ... = 21.1 (1 d.p.)

The area is 21.1 m 2 .

12 y =- is a hyperbola. X

x cannot be equal to 0.

I=Prn = $24 000 X 0.06 X 5 = $7200

Total to repay = $24 000 + $7200 = $31 200

Each instalment = $31 200 7 60

A

B

= $520 A

I ...

~ C shows weak negative correlation. C

em

A= 1r(R2 - r) = 7r X (1.52 - 0.52) = 6.283 185 ...

V=Ah = 6.283 185 ... X 0.5 = 3.14159 .. . =3.1 (ld.p.)

The volume is 3.1 cm3 A

4(x- 1) - 3(x- 2) = 4x- 4- 3x + 6 =x+2

500 mg in 250 mL = 5 mg in 2.5 mL = 15 mg in 7.5 mL

So a flow rate of 7.5 mL/h is required.

4, 13, 15; 17, 18,20,24,24,27,30,44 Median= 20 QL = 15 and Qu= 27 IQR = 27-15

= 12

So QL- 1.5 x IQR = 15- 1.5 x 12 = -3

Qu + 1.5 x IQR = 27 + 1.5 x 12 =45

A

B

There are no scores less than -3 or greater than 45, so there are no outliers.

Number correctly detected= 84 + 124 =208

Total statements= 208 + 16 + 36 =260

208 P( correctly detected) = -

260 4 5

I=Prn = $7600 X 0.00048 X 25 = $91.20

The correct diagram is C.

V=I_7rr 2h 3

3V=7rr2h 2 3V

r =-7rh

r = {3V (r > O) ~;];

A

c

c

c

A

40 is two standard deviations below the mean.

76is one standard deviation above the mean.

68% of scores lie within one standard deviation of the mean. 95% of scores lie within two standard deviations. 95% - 68% = 27% lie between 1 and 2 standard deviations. Percentage= 68% + t of27%

= 81.5% c

Worked Solutions Sample HSC Examination1 Q7-26

There are 26 possibilities for the repeated letter. For each of these, there are 100 possibilities for the two numbers (10 X 10). Number of arrangements= 26 x 100

= 2600 D k D=-V

When V=315, D=8 k

8=-315

k=8X315 =2520

2520 .. D=--

V When V=40,

2520 D=--40

=63 D

1000L=lm3

so 20000 L =20m' Now V=nr'h

20=7rXr2 x2.5 20 2

---=r nx2.5

r 2 =2.54679 ... r = 1.595 769122 ... (r > 0) r = 1.6 [l d.p.]

The radius is 1.6 m. The diameter= 2 X 1.6 m

=3.2m

Monthly repayments per $1000 borrowed Period Interest rate (years) 6% 7.5%

7 $14.61 $15.34 12 $9.76 $10.55 15 $8.44 $9.27 25 $6.44 $7.39

12 years at 6% Repayments= 70 x $9.76

= $683.20

U 25 years at 9% Repayments= llO x $8.39

9% $16.09 $11.38 $10.14

$8.39

B

= $922.90 / Total paid= $922.90 x 25 x 12

= $276 870 Interest= $276 870- $ll0 000

= $166 870 /

iii 7 years at 7.5% ~. Repayments= $15.34 per $1000

$613.60 7 $15.34 = 40 Brock borrowed $40 000 /

ivAt 7.5%: Repayments = 87 x $9.27

= $806.49 At9%: Repayments = 87 x $10.14

= $882.18 Extra repayments

= $882.28 - $806.49 = $75.69

1200 m

515 m 656m 597m

h = 1200 ~4 = 300

river

h A""-(d1 +4d +d1) 3 m

736 m

300 = -(515 + 4x656 + 597)

3 300

+ -(597 + 4x684 + 736) 3

= 780 500 Area ~ 780 500 m 2

= 78.05 ha

ii h = 1206 ..,_ 4 = 301.5

/

Actual area= (780 500 ..,_ 300) x 301.5 = 784 402.5 /

Difference = 784 402.5 - 780 500 = 3902.5

The difference is 3902.5 m2 . [Or: Use Simpson's rule again with the different measurement.]

X y 9 3 4 6

7 7 2 0 5 1 4 8 9 8 5 4 1 6 0 3 5 6 8 6 5 3 2 7 1 2 4 7 8

9 6 5 3 8 244799 4 1 9 0 3 5

X: range = 94 - 43 =51

ii Y: median= 74 +77 2

= 75.5

iii ClassY did better than class X with more higher scores and a higher median (75.5 compared to 70.5). / The results in class X are symmetrical but those in Y are negatively skewed.

/

d .)3x+4=5 3x +4=25

3x =21 X =7

a i 2nd 1st 4 button

butt~B

~" :;: g w~

i w 8 //

ii P(same colour)= P(BB) + P(WW) 5 4 4 3

= -x-+-x-9 8 9 8 4 9

ii

,;,:

d

y 60 55 50 45 40 35 30 25 20 15 10

y l~x ~~2 I \

!7 i\ "\

7 I l\

\ I I

7 -~ L

1/ i\ 0

! I // 2 4 6 8 10 12 14 16 X

is a maximum when x = 7.5. The graph is symmetrical. The maximum will be in the middle of X = 7 and X = 8. / Whenx= 7.5, y = 15 X 7.5- 7.52

= 56.25 The ma.,"'Ci.mum value of y is 56.25. Period = 240; rate = 0.6% From the table the interest factor is 127.01 Instalment= $160 000 ..,_ 127.01

= $1259.743 32 ... = $1259.74 (nearest cent)

ii Total paid each month = $1259.74 + $15 = $1274.74 Total repaid= $1274.74 x 240

= $305 937.60 Total charges = $305 937.60- $160 000 = $145 937.60 200 pies: Income = 200 x $5

= $1000 Cost= $150 + 200 x $3

= $750 Now $1000- $750 = $250 So Yvonne will make a profit of$250.

i i If x pies are sold: I= 5x C = 150 + 3x Break-even when I= C i.e. 5x = 150 + 3x

2x = 150 x= 75

So, Yvonne must sell 75 pies to break even.

a i Alison's raw score is 1 Vz times standard deviation above the mean.

/

Difference in scores= 156- 142 = 14

14 = 1.5s s=14~1.5

= 9.3333 ... The standard deviation is 9t .

75 km Q

125 km 175 km

q' +r'- p' cos p = ~------=--

2qr 1252 + 752 -1752

2X125X75 = ,-0.5

p = 120 N 1 >x

p 1 75 km Q 120

125 km 175 km

R

Let X be a point due north ofP. P is due west of Q. LXPQ = 90 Reflex angle XPR= 90 + 120

= 210 The bearing ofRfrom Pis 210. /

.c i

ii

There is a strong positive correlation between the measurements.

5 gradient= 0.935 x --

= 1.1 4.25

y-intercept = 45 - 1.1 x 26 = 16.4

/

/

So the equation is y = 1.1x + 16.4 / iii When y =56,

l.lx + 16.4 =56 l.lx = 39.6

X= 36 So, a prediction for Ti.m's arm length is 36 em.

Longitude of A is 20W. LAOB=90 Longitude ofB is 70E.

ii P(40N, 70E)

iii I= _!_27fT 360 90

= -X2X1TX6400 360

= 10 053.096 49 ... ' The distance from A to B is

approximately 10 050 krn.

//

IJVorked Solutions Sample HSC Examination 1 Q27-28

i:

! i I I I I

e i Let x be the number of bandicoots. X 45

Now-=-48 16

45 x=-x48

16 = 135

Th.e,.study suggests there are 135,bandicoots on the island. ./

ii 20% of 135 = 27 Now 135 - 27 = 108 So the new estimate is 108 bandicoots. Let n be the number that were, tagged.

n 48 So-=--

27 108 48

n=-x27 108

= 12 So 12 of the bandicoots were tagged. ./

z: i B=kl2 When l = 30, B = 1000 1000=/cx302

=900/c lc = 1000

900 10 9

B=10f 9

When/ =60,

B= 10 x602 9

=4000 There will be 4000 beads.

B= 10/' 9

WhenB =6250

6250 = 10 f 9

12 = 6250 7 10 9

= 5625 l = .J5625 (l > o) = 75

The skirt is 75 em long.

b i The life expectancy of a male in Northern NSW is 78.6 years.

ii The life expectancy of a female in Northern Sydney is 85.5 years and of a male in Western NSW is 76.5 years. Difference= 85.5 - 76.5

=9 So the life expectancy of a female in Northern Sydney is 9 years longer than that of a male in WesternNSW

m [Some possible answers] In every district the life expectancy of a female is greater than that of a male. The pattern for males is very similar to that for females. In both cases, for example the greatest life expectancy is in Northern Sydney and the next greatest is in South-Eastern Sydney. Life expectancy appears to decrease the further the distance from Sydney. Life expectancy appears to be greater in the more affluent areas.

././ Life expectancy at birth by local health district and sex, NSW, 2003-2007

Years Males Local Health District Years 78.5 Sydney 84.3 79.2 South Western Sydney 83.9 80.9 South Eastern Sydney 85.3 79.1 IllawarraShoalhaven 83.8 79.5 WestemSydney 84.0 78.7 Nepean Blue Mountains 83.2 81.9 NorthemSydney 85.5 78.1 Ce11tral Coast 83.2 77.8 Hunter New England 82.9 78.6 Northern NSW 83.9 78.1 Mid North Coast 83.4 78.4 SouthemNSW 82.9 77.7 Murrumbidgee 83.0 76.5 Western NSW 81.9 79.2 NSW 84.0

100 90 80 70 50

Life expectancy at birth

60 70 80 90 100

Life expectancy at birth

Source: ABS mortality data population estimates (SAPHaRI). Centre for Epidemiology and Evidence, NSW Ministry of Health.

c i F= kQq r'

d

Fr 2 = lcQq 2 lcQq

r =--F

r=p;q (r>o)

r= =0.4

31 accidents occurred in the region that has 20 bottle shops . ./

fi There is moderately strong positive correlation.

iii The bottle shops don't cause the accidents. It might be, for example, that there are more bottle shops and

Solu::icns Sample HSC Examination 1 029-30

more accidents in regions with higher populations.

a i

Q

Q is due south of 0. The bearing of Q from 0 is 180 . ./

ii LROP = (360- 300)0 +soc = 140

iii Let x m be the length ofRP. By the cosine rule: :x? = 1752 + 1522 -2 X 175 X 152

X COS 140 ./ = 94 482.564 37 .. .

x = 307.380 1626 ... (x > 0) The length of the boundary RP is 307 m, to the nearest metre. ./

b i Types = 3 x 2 x 2 = 12 ./

ii Three different tickets without black 3

P(ticket without black)= -12 1 4

1 Expected number = - x 60

4 = 15

c i Value after 6 years= $15 211 Amount invested= $500 x 4 x 6

=$12000 Difference= $15 211- $12 000

= $3211 So Fiona will earn $3211 in interest

ii PV=~ (1+r)"

./

$15 211 ./

(1.02f = $9457 (nearest dollar) ./

iii FV = $15 211 when annuity is $500 Factor= $15 211 7 $500

= 30.422 When annuity is $600 FV = 30.422 X $600

= $18 253.20 Difference= $18 253.20- $15 211

= $3042.20 Fiona would have $3042.20 more at the end of 6 years if she invested $600 per quarter instead of $500 per quarter. ./

d i x = 152 [by calculator] ./

ii s=L9966638 ... = 2.00 (2 d.p.)

iii Number of packets= 4 + 1 =5

5 Percentage of packets= -xlOO%

150 1

=3-% 3

iv 150 is one standard deviation below the mean. / 68% are within 1 standard deviation of the mean. Percentage Dutside of 1 standard deviation = 100%- 68%

= 32% \ Half of these are below 150. 16% of packets will have less than 150 toothpicks. /

8 9 6 8 5 7 8 7 In order: 5 6 7 7 8 8 8 9

7+8 median=--

2 =7.5

cos 69 = _.:._ 12

X= 12 COS 69

8x- 3(5- x) = 8x- 15 + 3x =11x-15

A=P(1+r)" = $30 000(1.0075)84 =$56 196 (nearest dollar)

I= $56 196 - $30 000 = $26 196

2 P(fust is red) = -

4 1 2

. 1 P(second 1s red) = -

3 1 1

P(both red) = -X-2 3 1 6

E= mr? Ifm = 0.8 and c= 3 x 105, E = 0.8 X (3 X 105?

= 7.2 X 1010

Pay= 37.5 x $21.60 + 6.5 x L5 x = $1020.60

c

50% is one standard deviation below mean.

Percentage = 34% + SO% =84%

Total repaid= 60 x $332 = $19 920

Interest= $19 920-$16 000 = $3920

Annual interest= $3920 + 5 =$784

Annual interest rate $784

=---xlOO% $16 000

=4.9%

h A~ -(d1+ 4dm+dl) 3 300

= -(225 + 4xl75 + 215) 3

= 114000 Area~ 114 000 m 2

=1L4ha

A

A

Total frequency = 15 + 23 + 30 + 5 + 1 = 74

The middle scores are the 37th and 38th scores.

Median~ 1 [15 + 23 = 38. The 16th to 38th scores are all L]

a2 b a 1 ---':-ab=-X-2ab2 2b ab

y =3(2x) Whenx =0, y = 3(2)

=3Xl =3

1 2b2

Cis the correct graph.

Harry is male. N= 3 X L8

= 5.4 H=2 M=88

10N-7.5H BAC=----

6.8M 10 X 5.4- 7.5 X 2

6.8 X 88 = 0.065 1737 ... = 0.07 (2 d.p.)

D = 12.5% of $24 000 = 0.125 X $24 000 = $3000

S = V 0 -Dn = $24 000 - $3000 X 6 = $6000

The correlation is moderately-strong negative. C

B B

100 "~, D 50 70 30 40 c 0 A 30m

By Pythagoras' t_heorem, AC2 = 302 + 402

= 2500 AC =50

A

The length of AC is 50 m. Dividend per share= $630 + 3000

= $0.21 $0.21

Dividend yield= --x 100% $5.60 = 3.75%

y

7 5

!/ v

3 1/

!-

1

123456X

2 Gradient=-

3 Vertical intercept = 2

2 Equation is y = -x + 2

3

Number of committees 12X11Xl0

c

A

c

3x2xl =220

c

~ X

Third angle = 180- (70 + 30t = 80

By the sine rule: a b

sinA sinB X 8

sin 80 sin 70 8 sin 80

x=

28.2MB = 2S.2 X 1024 kB = 28 876.8 kB

Time taken= (28 876.8 + 32) seconds = 902.4 seconds

D

= 15.04 minutes B 1

P(winning) = -5

4 P(not winning) = -

5 Financial expectation

1 4 = -x$200- -x$50

5 5 =$0 B

"!NO'ik2d Solutions Sample HSC Examination 2 01-23

8x+1 x+3 -----=17

5 4

(8x + 1) (x + 3) 20 X - 5-L - 20 X - 4 - = 20 X 17 4(8x + 1)- 5(x + 3) = 340

32x + 4- 5x- 15 = 340 27x -11 = 340

/ 27x= 351 x=13 B

EEJ 14 students average 85% Total marks =14 x 85

= i190 New total =1190 + 70

=1260 New number of students = 14 + 1

= 15 New average = 1260 + 15

= 84 The new average is 84%.

360 n=--

180-a If a= 135,

360 n=---

180-135 =8

The polygon has 8 sides.

360 ii n=--

180-a

m

If a= 108, 360

n= 180-108

=5 A regular pentagon has all its angles 108.

360 n=--

180-a n(180-a) = 360

360 180-a=-n

360 -a=- -180

n

a= 180- 360 n

[or: a= 180n: 360 l iv A hexagon has 6 sides.

Whenn =6, . 360

a=180--6

= 120 Each angle of a regular hexagon measures 120.

b i y = 1000(a)"' Whenx= 2, Y"' 810 ~i When y = 350,

X"' 10 iiiWhenx= 3, Y"' 730

Whenx = 4, y"' 660 Depreciation = $730 - $660

= $70

B

y 000

900 800

700

600

400

300

200

The asset would depreciate by approx. $70 in the fourth year.

iv In the first year the asset depreciates by$100. Rate of depreciation

$100 X 100%

$1000 =10%

The rate of depreciation is 10%. / v

I~ \. \c'c l.j. i

ii Questions answered correctly = 104 + 193 = 297 /

297 %answered correctly= - x 100% 450

=66%

m True questions= 104 + 78 = 182

104 P(answered true)= -

182 4 7

True answers= 104 + 75 = 179

104 P(true question)=-- /

179

b v = lbh

ii

= 5 X 3 X 0.015 = 0.225

Volume= 0.225 m 3 / Amount of water= 0.225 x 1000 L

= 225litres

V = nr2h 0.225 = 1r X 1.22 X h

h 0.225

= 0.049 735 919 ... = 0.05 (2 d.p.)

The level of water in the tank will rise by approximately 5 em.

Angular difference= 151 + 99o = 250

Time difference = (250 + 15) hours = 16 hours 40 min

Sydney is 16 hours and 40 minutes ahead of Mexico City. The plane leaves at 12.10 am Thursday, Mexico time. It arrives 16'h hours later. It arrives at 4:40 pm, Thursday.

a i A z-score of -1 means that the score is one standard deviation below the mean.

ii Mass~ 210 g + 3 x 5 g = 225 g

iii 200 g is two standard deviations below the mean. 95% of packets will be within standard deviations of the mean. 5% will be outside two standard deviations. 2.5% will be less than 200 g.

Number of vehicles on Friday and Saturday

Friday~ Saturday~

10 15 20

The highest number was 23 on

25

Friday. ./

8 is the lower quartile. Approximately 25% of the time there were less than 8 vehicles. ./

m Both displays are fairly symmetrical, showing similar patterns. The data for Friday was consistently higher than that for Saturday. ./

2nd 1st 5 battery

batt~C yc t 7 F ~ 9 c

lO F~ 3_ F 9

7 6 ii P(both charged) = - x-

10 9 7 15

.... P( . fl ) 7 3 3 7 m one 1s at =-X-+-x-10 9 10 9 7 15

Yes, Natasha is correct.

Energy used= 1200 W x 6 h = 7200Wh = 7.2 kWh

Cost= 7.2 x $0.284 = $2.0448 = $2.04 (nearest cent)

ii Cost before GST = $42.70 + 836 X $0.328 = $316.908 = $316.91 (nearest cent) GST = 10% of$316.91

././

= $31.69 (nearest cent) Total Cost= $316.91 + $31.69

= $348.60

a 1 GL = 1 x 109 L 3.7 X 1010 L = 3.7 X 10 GL

=37GL b 2(? + Sx + 1) + 3x(x- 1)

= 2? +lOx+ 2 + 3?- 3x =5.l?+7x+.2

mean= 1.88 [by calculator]

ii 5 = 1.704 582 06 ... = 1.7 (1 d.p.)

iii No, the data is not normally distributed. It is skewed with more

./ /

lower than higher scores. ./

i'; x+s=l.88+l.7 =3.58

x- s = 1.88- 1.7 =0.18

Scores between 0.18 and 3.58 are l, 2 and 3. Number of scores = 22 + 11 + 9

= 42 Percentage within 1 standard

42 deviation = - x 100%

75 =56%

T= 100 s

ii T

m Approximately 7 seconds.

2.5% of $2400 = 0.025 x $2400 = $60

Billy must pay a fee of $60. ./ "ii From 19 March untilS May

inclusive is 48 days. [13 + 30 + 5] / Total amount = $2400 + $60

= $2460 I=Prn

0.186 = $2460 X -- X 48

365 = $60.172 273 ... = $60.17 (nearest cent)

Billy must pay $60.17 in interest. /

Worked Solutions Sample HSC Examination 2 029-30

practice hsc papers general 2

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