western region€¦ · web viewhsc preliminary course . final examination. general mathematics...

2006HSC Preliminary

Course FINAL EXAMINATION

General Mathematics

General Instructions

o Reading Time- 5 minutes

o Working Time - 2 hours

o Write using a blue or black pen

o Calculators may be used

o A Formulae Sheet is provided at the back of this paper which may be detached and used throughout the paper.

Section I pages 3 – 7

Total marks (20)

o Attempt Questions 1-20

o Answer on the Multiple Choice answer sheet provided.

o Allow about 30 minutes for this section

Section II pages 8 - 14

Total marks (60)

o Attempt questions 21 – 25

o Answer on the blank paper provided, unless otherwise instructed. Start a new sheet for each question.

o Allow about 1½ hours for this section

Preliminary Course General Mathematics 2006 Section I

Total marks (20)

Attempt Questions 1-20

Allow about 30 minutes for this section

Use the multiple choice answer sheet.Select the alternative A, B, C or D that best answers the question. Fill in the response oval completely.

Sample

2 + 4 = ? (A) 2 (B) 6 (C) 8 (D) 9

A B C D

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

A B C D

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

correct

A B C D

2

Preliminary Course General Mathematics 2006

1. Use the formula to find the value of v when u = 5, t = 2 and

a = 10

(A) v = 210 (B) v = 30 (C) v = 420 (D) v = 44100 2.

(A) (B) (C) (D)

3. The value of given that and that , and is:

(A) 9 (B) 3 (C) 1 (D) 6

4. Simplify

(A) (B) (C) (D)

5. Angela invested $25 000 at 12% p.a. interest, compounding monthly. The value of the investment after 3 years would be given by the calculation:(A) (B)

(C) (D)

6. Joseph works as a carpenter and is paid an hourly rate of $12.40 for a 40 hour week and time and a half for all overtime. What is he paid for a week when he works for 45 hours?

(A) $496.00 (B) $558.00 (C) $589.00 (D) $837.00

7. Shaylee borrows $12 000 to buy a motorbike. She is to pay the money back over three years in equal monthly repayments and is charged simple interest at 8% p.a. How much is each instalment?

(A) $286.15 (B) $413.33 (C) $1240.00 (D) $572.30

8. Aaron has a salary of $41 600 which is paid fortnightly. What is his fortnightly net pay if he has fortnightly deductions of $345.50 in taxation, $65.00 for medical insurance and $35.00 for union fees?

(A) $1582.87 (B) $791.43 (C) $1154.50 (D) $354.50

3


9. A die is biased so that the probability of throwing a 6 is twice that of any other single

number. The probability of throwing a 5 on a single roll of the dice would be:

(A) (B) (C) (D)

10. A restaurant offers the choice of 5 entrees, 10 main courses and 8 desserts. Mark eats a three course meal at the restaurant every night. If he tries a different combination of entrée, main and dessert every night of the week, including weekends, how long would it take him to try all the possible combinations?

(A) Less than 6 months (B) Between 6 and 9 months (C) Between 9 and 12 months (D) More than 12 months

11. Tran and Jan are drawing cards from a normal pack of 52 playing cards. They have drawn 20 cards without replacing any and no Aces have been drawn. Tran says: “The probability that we will draw an Ace is greater now than it was when

we started.”Jan says: “There is a better chance of drawing a black Ace than a red Ace now.”

Who is correct?

(A) Both are correct. (B) Neither are correct?. (C) Only Tran is correct. (D) Only Jan is correct.

12. The weather bureau says that “It is highly likely that there will be storms in the region this afternoon.” Based on their prediction, the probability that it doesn’t storm in the region could be:

(A) 0.05 (B) 0.5 (C) 0.95 (D) 0.9

13. The exact area of the circle shown is:

(A) m2 (B) m2 (C) m2 (D) m2

4

8 m


14. In the diagram below, the value of = ?

(A) (B)

(C) (D)

15. Find the volume of the rectangular pyramid shown below.

(A) 3840 cm2 (B) 468 cm2 (C) 960 cm2 (D) 2880 cm2

16. Mitchell measures the length of a piece of pipe using a tape measure marked in cm. He records the length as 2.00 metres. The percentage error in his measurement is

(A) 2.5% (B) 5%(C) 0.0025% (D) 0.25%

5


17. The temperature on the same day in two towns called Summerville and Springfield are shown on the radar chart below.

Temperature in Two Cities

02468

1012141618202224262830

1 a.m.

2 a.m.

3 a.m.

4 a.m.

5 a.m.

6 a.m.

7 a.m.

8 a.m.

9 a.m.

10 a.m.

11 a.m.

12 noon1 p.m.

2 p.m.

3 p.m.

4 p.m.

5 p.m.

6 p.m.

7 p.m.

8 p.m.

9 p.m.

10 p.m.

11 p.m.

12 p.m.

SummervilleSpringfield

What is the difference in temperatures between the two towns at 1p.m.?

(A) 10 o C (B) 5 o C (C) 15o C (D) 30 o C

18. The graph show the ages of people at a party. Which age range below comprised about a quarter of those present?

6

(o C)

Preliminary Course General Mathematics 2006 Ages of People at a Party

0 to 5

6 to 12

13 to 18

19 to 2526 to 35

36 to 45

over 45

(A) 0 to 12 (B) 6 to 18 (C) 13 to 25 (D) 26 to 45

19. The scores below are Karina’s results on ten quizzes, each out of a possible 12.

4, 5, 6, 7, 7, 8, 9, 10, 11, 12

The median and interquartile range respectively for the scores are:

(A) 7. 9 and 4 (B) 7.5 and 5 (C) 7.9 and 8 (D) 7.5 and 4

20. Mark records the maximum temperature each day for the month of September. He wishes to represent the data on a graph. Which type of graph would be suitable to show all the maximum temperatures and how they changed during the month?

(A) A Line Graph (B) A Sector Graph (C) A Divided Bar Graph (D) A Box and Whisker Plot.

7

Preliminary Course General Mathematics 2006 Section II

Total Marks (60)Attempt Questions 21 – 25Allow about 1½ hours for this section.

Answer all questions, starting each question on a new sheet of paper with your name and question number at the top of the page. Do not write on the back of sheets.___________________________________________________________

Question 21 (12 marks) Start a new sheet of paper.

(a) (i) Expand and simplify

(ii) Solve for x

(b) The formula below gives the infant dosage of a drug “Relaxamax” in comparison with the adult dosage:

where D = infant dosage in mLand m = age of infant in months

(i) Find the Infant dosage of “Relaxamax” (to the nearest tenth of a ml) for a child who is 9 months old.

(ii) A child is given a dosage of 3.8 ml of “Relaxamax”. How old is the child to the nearest month?

8

Marks

1

2

1

2

Question 21 (continued)

(c) The cost of hiring a venue called “The Palace” for a fully catered party is given by the formula:

where C = cost to hire the venue and G = the number of guests

(i) Complete the table for on page 18.

(ii) Draw the line representing on the graph provided on page 18 and label it “The Palace”.

(iii) A second venue, “The Dive” also charges on a linear scale. The cost for 10 guests at “The Dive” is $400 and for 50 guests is $1600. Draw and label the line representing the cost if hiring “The Dive” on the same graph on page 18.

(iv) Give the gradient of the line for “The Dive”.

(v) Write the formula for the cost of hiring “The Dive” in the form

(vi) From the graph, for how many guests is the cost of hiring the two venues the same?

Marks

1

1

1

1

1

1



(a) Angelina is employed by a Real Estate Agent as a salesperson and is paid by commission. The commission is 3% on the first $2 000 and 0.8 % on the remainder of the price of each property that she sells.

(i) In September she sold two properties valued at $165 000 and $550 000 respectively. How much was she paid for the month of September?

(ii) She is offered an alternative pay contract where she is paid a retainer of $40 per day plus commission at 0.9% of her monthly sales. How much more or less would she have earned under the new contract for the month of September if she worked on 25 days?

(b) Two siblings, Katrina and Evan each receive a gift from their grandparents of $4 000.

(i) Katrina invests the money in a 2 year fixed term investment account paying 6% interest compounding six monthly. If she withdraws the money after the two years, how much interest will she have earned? (Answer to nearest dollar.)

(ii) Evan purchases 1000 shares in a company at a price of $4.00 per share. The shares have an annual dividend yield of 5%. Brokerage costs are a one off payment of 1% of the purchase price. After two years, the shares are sold at $4.90. Calculate the total earnings over the two years, after costs.( To nearest dollar.)

(iii) What rate of compound interest on a 2 year fixed term investment account, compounding six monthly would have paid the same earnings as Evan received from the shares?

10

Marks

2

2

1

2

1


(c) Jim has a gross weekly pay of $824 and a net weekly pay of $600 after tax.

(i) He has allowable tax deductions of $450 for medical fund and $250 for union fees, what is annual taxable income? (Assume 52 weeks in a year.)

(ii) Use the tax table below to calculate the income tax payable on Jim’s taxable income.

(iii) With his tax, Jim also has to pay a Medicare levy of 1.5% of his taxable income. What is his total tax for the year?

(iv) Calculate Jim’s tax refund or bill for the year.

Marks

1

1

1

1



(a) A class observes an intersection for 5 hours over a week and records the traffic data in the table below.

Type of vehicle FrequencyPrivate vehicles Large Passenger Sedan 300

Small passenger Sedan/Coupe

210

Station wagon 100Large passenger 4WD 90Small passenger 4WD 80Utility 30People Mover 40Motor cycle 50

Commercial Industrial vehicles

Utility 70Van 30Other small Commercial vehicle

20

Truck 80Bus 60Other large Commercial Vehicle

40

Based on the results of this survey, complete the following :

(i) If a Private vehicle were chosen at random, what is the probability that it would be a station wagon?

(ii) If a Commercial Vehicle were chosen at random, what is the probability that it would be a bus?

(iii) If any type of vehicle were chosen at random, what is the probability that it would be a truck?

(iv) If any type of vehicle were chosen at random, what is the probability that it would be a Utility?

(b) There are a number of different styles of numberplates available in NSW. How many distinct arrangements are possible for the following styles of number plate:

(i) Three letters followed by three numbers

(ii) Three letters, then two numbers then another letter.

(iii) A six character numberplate where each character can be a number or a letter.

12

Marks

1

1

1

1

1

1

2



(c) A TV game show, “Contract or No Contract” has 26 cases which each contain a label representing an amount of money. The amounts are as follows:

$1 $1 000$2 $1 500$5 $2 000$10 $2 500$20 $3 000$25 $4 000$50 $5 000$75 $7 500$100 $10 000$200 $12 000$250 $15 000$500 $20 000$750 $25 000

The contestant chooses a single case from the 26 cases available. What is the probability that it contains:

(i) An amount of $1 000?

(ii) An amount less than $1 000?

(iii) An Amount between $75 and $750 inclusive?

(iv) An amount which is not between $75 and $750 inclusive?

13

Marks

1

1

1

1


(a) Harriet drove from Sydney to Finley, a distance of 712 km in 8 hrs and 30 minutes. Josie drove from Narranderra to Dubbo a distance of 394 km at an average speed of 95 km /h.

(i) Find Harriet’s average speed in km/h (to the nearest unit).

(ii) How long did it take Josie to complete her journey (to the nearest 5 minutes)?

(iii) What was Josie’s equivalent speed in m/s (to nearest 10th m/s)?

(b) A room is in the shape of a trapezoidal prism and has floor dimensions 8 metres by 6 metres as shown below. The height increases from 2 metres to 3 metres.

(i) Find the number of litres of air that the room would hold. ( 1 m3 = 1000 litres)

(ii) Find the total area of the 4 walls.

(iii) Windows and doors which have a total area of 20 m2 are to be placed in the walls. The remaining wall area is to be painted with a paint which has a coverage rate of 15 m2 per litre. How many litres of paint would be needed for one coat? (Answer correct to one decimal place.)

Marks

1

1

1

1

1

1


(c) While on holidays, Andrea takes a photograph of her two brothers standing by the bus. Some of the dimensions on the photograph are shown.

(i) If the taller brother’s height is really 150 cm, find the scale of the photograph.

(ii) Find the shorter brother’s real height.

(iii) If the tree had a real height of 4.5 metres, what would be it’s height in the photograph?

(d) A wall is constructed in a garden. Two girls Angela and Cleo are

on opposite sides of the wall.

(i) Angela stands a 4 metre ladder against the wall. When it touches the top of the wall it’s foot is 1.2 m from the base of the wall. Show that the height of the wall is 3.8 metres, correct to 2 significant figures.

(ii) Cleo stands so that the angle of elevation from her feet to the top of the wall is 32o. How far are her feet from the base of the wall? (correct to 2 significant figures.)

Marks

1

1

1

1

2



(a) The points A to F below are stages in a statistical investigation. Re-write the letters A to F in the correct chronological order.

A. Analysing the data and drawing conclusions.B. Writing a reportC. Summarising and displaying the data. D. Posing the questions E. Organising the data.F. Collecting the data.

(b) In a survey of birdlife in a woodland, 200 birds are netted, tagged and released. A week later another 200 birds are netted and 5 are found to be tagged. Calculate an estimate of the number of birds in the woodland.

(c) The rainfall in Hardies Plains for 2005 is recorded in the table below.

Month J F M A M J J A S O N DRainfall (mm)

60 40 55 30 60 70 30 25 30 50 25 85

Use the grid provided on page 19 to draw and label a radar chart for the rainfall data.

16

Marks

2

3

3


(d) The length of the right foot of a sample of 12 students at Riverdale High is recorded below.

24cm 21cm 23cm 20cm 23cm 26cm25cm 23cm 20cm 29cm 29cm 30cm

The length of the right foot of another sample of 32 students at Sunnydale High is recorded in the table below.

Length of Right Foot

Frequency

20 221 322 123 524 625 726 427 228 129 1

(i) Find the mean and sample standard deviation of the Riverdale High data correct to 2 decimal places.

(ii) Find the mean and sample standard deviation of the Sunnydale High data correct to 2 decimal places.

End of Exam

.

Marks

2

2


Table and graph for Question 21 (c)(i) and (ii)

Number of Guests (G) 10 40 70 100

Cost of hiring venue (V)

18

Number of guests

Cos

t of h

iring

Ven

ue ($

)


Graph for Question 25 (b)

0

20

40

60

80

100J

F

M

A

M

J

J

A

S

O

N

D

19


20


21

General Mathematics Preliminary Examination 2006

Multiple Choice Answer Sheet

Name ______________________________

Completely fill the response oval representing the most correct answer.

1. A B C D

2. A B C D

3. A B C D

4. A B C D

5. A B C D

6. A B C D

7. A B C D

8. A B C D

9. A B C D

10. A B C D

11. A B C D

12. A B C D

13. A B C D

14. A B C D

15. A B C D

16. A B C D

17. A B C D

18. A B C D

19. A B C D

20. A B C D

23

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