AOS GENERAL exam - Art of Smart Mock Exams/AOS_GENERAL_exam.pdfmathematical reasoning and/or calculations Total marks – 100 Section I Pages 1–8 25 Marks ... Sabrina plays a game

General Mathematics

General Instructions

• Reading time – 5 minutes

• Working time – 2.5 hours

• Write using black pen

• Board-approved calculators maybe used

• A reference sheet is provided atthe back of this paper

• In Questions 26–30, show relevantmathematical reasoning and/orcalculations

Total marks – 100

Section I Pages 1–8

25 Marks

• Attempt questions 1–25

• Allow about 30 minutes for thissection

Section II Pages 9–16

75 Marks

• Attempt questions 26–30

• Allow about 2 hours for thissection

Section I

25 marks

Attempt Questions 1–25

Allow about 30 minutes for this section

1. What is 1 780 900 km written in scientific notation correct to two significantfigures?

(A) 1.78⇥ 104 km

(B) 1.8⇥ 105 km

(C) 1.78⇥ 106 km

(D) 1.8⇥ 106 km

2. Which of the following is 4x+ 3y � 2x� 5y in its simplest form?

(A) 3x+ 2y

(B) 2x+ 8y

(C) 5x� 2y

(D) 2x� 2y

3. Anurag’s gross pay each week is $752.58. The following deductions are takenfrom his gross pay each week:

• tax $70.93

• superannuation $45.70

• health fund $38.15.

What is Anurag’s net pay each week?

(A) $326.18

(B) $597.80

(C) $771.32

(D) $952.25

1

4. The diagram below shows the graph of an equation.

1

Which of the following equations does the graph best represent?

(A) y = 1� 3x

(B) y = 1� 3x

(C) y = 1� 3x2

(D) y = 1� 3x3

5. Which of the following is 3x0 + (5x)0 in its simplest form?

(A) 6x

(B) 8x

(C) 1 + 5x

(D) 4

6. The length of a wrench was measured to be 38 cm, correct to the nearest cm.

What is the percentage error in this measurement, correct to one significantfigure?

(A) ±0.01%

(B) ±0.5%

(C) ±1%

(D) ±2%

2

7. A watch costs $449, including 10% GST. What is the price of the camera withoutGST, correct to the nearest dollar?

(A) $395

(B) $401

(C) $408

(D) $503

8. The diagram below shows the graph of an equation.

2

1O

Which of the following equations does the graph best represent?

(A) y = 2x+ 2

(B) y = 2x+ 1

(C) y = 2� 2x

(D) y = 1� 2x

9. The probability of winning a game is3

7. Which expression represents the prob-

ability of winning two consecutive games?

(A)3

7⇥ 1

3

(B)3

7⇥ 2

7

(C)3

7⇥ 2

(D)3

7⇥ 3

7

10. What amount must be invested now at 4% per annum, compounded quarterly,so that in six years it will have grown to $50 000?

(A) $8919

(B) $11 156

(C) $39 378

(D) $49 316

3

11. The graph shows a line which has an equation of the form y = mx+ b.

x

y

O

Which of the following statements is true?

(A) m is positive and b is negative

(B) m is negative and b is positive

(C) m and b are both positive

(D) m and b are both negative

12. A swim team consists of eight members. Three of the members are being selectedto attend nationals.

In how many ways can the three members be selected?

(A) 10

(B) 24

(C) 56

(D) 70

13. A machine produces drain pipes. The mean diameter of the pipes is 8cm andthe standard deviation is 0.04cm.

Safety guidelines insist drainpipes can have a diameter no smaller than 7.96 cm.Assuming a normal distribution, what percentage of pipes produced will violatethe safety guidelines?

(A) 16%

(B) 32%

(C) 34%

(D) 68%

4

14. Young’s formula below is used to calculate the required dosages of medicine forchildren aged 1-12 years.

Dosage =age of child (in years)⇥ adult doage

age of child (in years) + 12

How much of the medicine should be given to a 30-month old child in a 24-hourperiod if each adult dosage is 30 mL? The medicine is to be taken every 8 hoursby both adults and children.

(A) 5.5 mL

(B) 15.5 mL

(C) 27.5 mL

(D) 30.5 mL

15. How many kilobytes are there in 2 gigabytes?

(A) 220

(B) 221

(C) 230

(D) 231

16. In a household, a heater is run for 50 minutes each night, in four separatebedrooms. The household uses a 9 kW heater unit. Electricity is charged at12.01c/kWh when the heater is being used. What is the electricity cost due tothe heater in this household in one week?

(A) $1.63

(B) $6.54

(C) $25.22

(D) $65.56

5

17. A is 38 km due north of B. C is on a bearing of 145� from A and 60� from B.Which diagram represents this?

(A) A

38 km

B

C

60�

55�

(B)

A

38 km

B

C

60�

145�

(C) A

38 km

B

C60�

35�

(D) A

38 km

B

C

35�

60�

18. A table of future value interest factors is shown below.

A certain annuity involves making equal contributions of $35 000 into an accountevery 6 months for 1.5 years at an interest rate of 3% per annum.

Based on the information provided, what is the future value of this annuity?

(A) $50 500

(B) $51 000

(C) $92 727

(D) $106 162

6

19. Zhiyu works a 40-hour week and is paid at an hourly rate of $22.50. Any overtimehours worked are paid at time-and-a-half. In a particular week, he earned $1440.

How many hours in total did Isabella work in this week to earn this amount?

(A) 42

(B) 49

(C) 56

(D) 67.3

20. The value of E varies directly with the square of S. It is known that E = 10when S = 10. What is the value of E when S = 20?

(A) 40

(B) 80

(C) 320

(D) 400

21. The shoe colours of a sample of children were recorded. When analysing thisdata, which of the following could be found?

(A) Mean

(B) Median

(C) Mode

(D) Range

22. How many square millimetres are in 0.0075 square centimetres?

(A) 0.75

(B) 7.5

(C) 75

(D) 750

7

23. Jennifer used the ‘capture-recapture’ technique to estimate the number of carpliving in a dam.

• She caught, tagged and released 20 carp.

• Later she caught 36 carp at random from the same dam.

• She found that 8 of these 36 carp had been tagged.

What estimate should Jennifer give for the total number of carp living in thisdam, based on her use of the ‘capture-recapture’ technique?

(A) 56

(B) 90

(C) 160

(D) 162

24. The mean of a set of 10 scores is 10. Another two scores are included and thenew mean is 12. What is the mean of the two additional scores?

(A) 18

(B) 20

(C) 22

(D) 26

25. Which of the equations below is the equation 4z = 3t+2r2 rearranged such thatr is the subject?

(A) r = ±p3t� 4z

2

(B) r = ±p4z � 3t

2

(C) r = ±r

3t� 4z

2

(D) r = ±r

4z � 3t

2

8

Section II

75 marks

Attempt Questions 26–30

Allow about 2 hours for this section

Question 26 (15 marks)

(a) 1Expand f(5f + 5) + 5(f + 5).

(b) 2Calculate the value of d correct to two decimal places.

4

d20�

(c) 3Solve the equation4x+ 2

3� 7 = 5� 9x.

(d) 3Solve these simultaneous equations to find the value of x and y.

y = 2x+ 3

x� 2y � 7 = 0

(e) 2The mass of concrete varies directly with the mass of sand in the mixture. Adeck has 80 kg of concrete. The concrete had 15 kg of sand in it.

Calculate how much concrete could be made with 35 kg of sand.

(f) 2Emma lives in Sydney which is located at 34�S 151�E.

Elise lives in Ashburn. Elise knows she is on the same line of latitute as herfriend Emma, she also knows she is 4 hours ahead of Sydney time. DetermineEmma’s longitude.

(g) 2Find the area of the triangle below

A

4 m

B

3 m

C

120�

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Question 27 (15 marks)

(a) Yubert is buying a used car which has a sale price of $22 149. In addition to thesale price there are the following costs: Transfer of registration $50, and stampduty.

(i) 1Stamp Duty for this car is calculated at $2 for every $100, or part thereof,of the sale price. Calculate the Stamp Duty payable.

(ii) 4Yubert borrows the total amount to be paid for the car including StampDuty and transfer of registration. Interest on the loan is charged at a flat rateof 6.8% per annum. The loan is to be repaid in equal monthly instalmentsover 3 years. Calculate Yubert’s monthly repayments.

(iii) 3Yubert wishes to buy some insurance for the car for 24 months. The cost ofinsurance is calculated using the following:• Base Rate $1500.

• Luxury Car Tax (LCT) 3% of base rate.

• Stamp Duty 5.5% of the total of base rate and LCT.

• GST 10% of the total of base rate and LCT.

If Yubert pays the insurance in quarterly installments over two years. Howmuch must he pay in each installment?

(b) 3Sarah wants to ensure her commute to University is as environmentally friendlyas possible. She wants to compare the amount of fuel used via two di↵erentmethods of transport.

Sarah’s car uses 9 litres of fuel for every 100km travelled. The distance from herhome to the university car park is 34 km.

She could instead travel to her friend Kirsten’s house 6 km away. Kirsten’scar only uses 6 litres of fuel for every 100km travelled. They could then drivetogether in Kirsten’s car to uni. The distance from Kirsten’s house to the Uni-versity is 40 km.

Which way of travelling uses less fuel?

(c) 4Jaeger wanted to find out the volume of the shed in his backyard. He asked hisfriends to help measure, unfortunately they all used di↵erent units! Jaeger wastold his shed is 2 m wide, 400 cm long, and 10 000 mm high.

What is the volume of the shed in metres cubed?

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Question 28 (15 marks)

(a) 2Sabrina plays a game where she rolls a die and wins the amount shown on thedie. For example, if she rolls a 5, she would win 5 dollars.

How much should Sabrina pay to play this game?

(b) An o↵set survey of a 5 sided room is shown below. (NOT TO SCALE)

W

V

U

S

R

Q

P

T

21

6 36

27

26

3012

(i) 1Show that the length of TQ is 62.29 m.

(ii) 2Calculate the length of PT and PQ to two decimal places.

(iii) 2Hence find the angle TPQ.

(c) 3A fair coin is tossed four times. Using a tree diagram, or otherwise, calculatethe probability of obtaining three tails and one head in any order.

Question 28 continues on next page.

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(d) An aerial diagram of a pond is shown.

4

1512

188

(i) 3The pond is 20m long. A surveyor measures the length across the pondin 5 metre intervals. The pond appears to have a uniform depth of 2.4 m.Find the volume of the pond in cubic metres. In your calculations, use twoapplications of Simpson’s rule.

(ii) 25% of the water needs to be sampled by the environmental protection agencyfor water pollution testing. How many litres of water need to be extracted?

End of Question 28

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Question 29 (15 marks)

(a) The cost of purchasing new computers for a school is $95 000. The cost will beshared equally by the students attending the school, so that C (in dollars) is thecost per person when n students attend the school.

(i) 1Draw a table of values to determine C for 500, 1000, 1500, 2000 and 2500students.

(ii) 2Using the values from the table, draw a graph showing the relationship be-tween C and n. Make sure to label your axis appropriately.

n

C

(iii) 1What equation represents the relationship between n and C.

(iv) 1Give one limitation of this equation in relation to this context.

(v) 1Is it possible for the cost per student to be $104? Support your answer withappropriate calculations.

(b) 3What is the maximum number of standard drinks that a female weighing 70 kgcan consume over 4 hours in order to maintain a blood alcohol content (BAC)of less than 0.05?

Question 29 continues on next page.

13

(c) John and Eric record their 400m times over a month. They each run a total of30 times. A five number summary of their times is listed below. (Times are inseconds)

• John’s 5-number summary for the runs is 67, 68, 69, 74, 76.

• Eric’s 5-number summary for the runs is 67, 69, 71, 73, 75.

(i) 2Draw two box-and-whisker plots to display their results.

(ii) 2What percentage of John’s times were under 69 seconds?

(iii) 2Eric claims that he is the superior runner to the coach, and that he shouldbe put on the track team. Is he correct? Justify your answer by referring tothe summary statistics and the skewness of the distributions.

End of Question 29

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Question 30 (15 marks)

(a) 2Nitij and Claudia plan to have $50 000 in an investment account in 15 yearstime for their son’s university fees.

The interest rate for the investment account will be fixed at 5% per annumcompounded monthly. Calculate the amount that they will need to depositinto the account now in order to reach their goal.

(b) The scatterplot shows the relationship between Ice Cream sales and Tem-perature for the last 12 days:

(i) 1For the given data, the correlation coe�cient, r, is 0.81. What doesthis indicate about the relationship between Ice Cream sales and tem-perature?

Question 30 continues on next page.

15

(ii) 4The Ice Cream sales and Temperatures used to produce the scatterploton the previous page are shown in the table below.

Temperature Ice Cream Sales14.2 21516.4 32511.9 18515.2 33218.5 40622.1 52219.4 41225.1 61423.4 54418.1 42122.6 44517.2 408

Calculate the mean Temperature, T , and the standard deviation, �T

,as well as the mean Ice Cream Sales I and standard deviation of theIce Cream sales �

I

. Calculate all values to two decimal places.

(iii) 2Using the values from part (ii), find the equation of the least-squaresline of best fit.

(iv) 2Draw the least squares line of best fit.

T

I

(v) 1Using the line, or otherwise, estimate the number of Ice Creams soldon a 20� day.

(vi) 2What is the gradient and what does it represent?

(vii) 1Why is this line NOT useful for predicting sales in a much warmer placesay with an average temperature of 35�?

End of Exam Paper

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2016 HIGHER SCHOOL CERTIFICATE EXAMINATION

Mathematics General 2

FORMULAE AND DATA SHEET

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