Mathematics Advanced

(1)

Western

Mathematics

2020

TRIAL HIGHER SCHOOL CERTIFICATE EXAMINATION

Mathematics Advanced

General

Instructions  Reading time – 10 minutes _{ Working time – 3 hours}  Write using black pen

 Approved calculators may be used

 A reference sheet is provided at the back of this paper

 In Questions in Section II, show relevant mathematical reasoning and/or calculations

Total marks : 100

Section I – 10 marks (pages 2 – 5)  Attempt Questions 1 – 10

 Allow about 15 minutes for this section Section II – 90 marks (pages 6 – 27)

 Attempt Questions 11 – 27

(2)

- 2 -

Section I

10 marks

Attempt Questions 1–10

Allow about 15 minutes for this section

Use the multiple-choice answer sheet for Questions 1 – 10

1. What amount does an investment of $20 000 grow to after 3 years at 5% p.a. compounded

quarterly? A. $20 759.41 B. $23 152.50 C. $23 215.09 D. $23 223.68

2. The weekly pay for workers at the Prosper Factory is normally distributed, with a mean of

$750 and a standard deviation of $35.

What percentage of workers earn below $680 a week? A. 0.15%

B. 2.5% C. 5% D. 47.5%

3. The function is transformed by first being dilated vertically by a scale factor of 3

and then translated horizontally 4 units to the left. Find the equation of the transformed function.

A.

B.

C.

(3)

4. _{For the series} _{, calculate the exact value of the sum of the first 6 terms.}

A.

B.

C.

D.

5. The 7th_{term of an arithmetic sequence is 45 and the 11}th_{term is 77.}

Find the first term (a) and the common difference (d).

A.

B. C. D.

6. Twenty students sit a Chemistry test and the mean of their scores is 78.

Two students sit the test late and their scores are 95 and 83. What is the new mean for the Chemistry test?

A. 79 B. 80 C. 83 D. 89

(4)

- 4 -

7. What is the equation of the axis of symmetry of the quadratic function

? A. B. C. D. 8. A function is given by

If this function is a continuous probability distribution, what is the area under the curve? A. –1

B. 0.5 C. 1 D. 2

9. Find the derivative of .

A.

B.

C. D.

(5)

10. Holly drew a scatter-plot of a binomial data set which compared the construction time of houses with their cost.

The construction times ranged from 6 weeks to 6 months.

She found the equation of the line of best fit and used it to estimate the cost of a house which took 10 months to build.

What term would describe this process? A. Causality

B. Correlation C. Extrapolation D. Interpolation

(6)

Western

Mathematics

2020 TRIAL HIGHER SCHOOL CERTIFICATE

EXAMINATION

Mathematics Advanced

Section II Answer Booklet

Class and Teacher

Student Number

Student Name 90 marks

Attempt Questions 11 – 27

Allow about 2 hours and 45 minutes for this section

Instructions _{ Answer the questions in the spaces provided. Sufficient spaces are} provided for typical responses.

 Your responses should include relevant mathematical reasoning and/or calculations.

 Extra writing space is provided at the back of the booklet.

(7)

Question 11 (4 marks)

(a) _{Show that the derivative of} _.

………. ………. ………. ……….

2

(b) Hence or otherwise find

………. ………. ………. ……….

(8)

- 8 - Question 12 (5 marks)

Describe the features of the periodic function .

In your answer include the amplitude of the function, its period, the centre and the upper and lower endpoints of the vertical oscillation and the phase shift of the curve. You do not need to find x or y intercepts.

You may use a sketch to illustrate you answer if you wish, but it is not required.

5 ………. ………. ………. ………. ………. ………. ………. ………. ………. ………. ………. ………. ………. ……….

(9)

(a) Show that

………. ………. ………. ………

1

(b) Hence or otherwise, sketch the graph of showing any asymptotes and the

x-intercept.

2

(10)

- 10 -

Fred sits his Trial exams in Modern History and Ancient History.

The marks for the Modern History class have a mean of 54 and a standard deviation of 5.6. The marks for the Ancient History class have a mean of 76 and a standard deviation of 2.1.

(a) Compare and contrast the distribution of marks for the two classes.

………. ………. ……….

2

(b) Fred scored 65 for Modern History and 80 for Ancient History.

Using z-score calculations, explain which subject he performed better in and why. ………. ………. ………. ………. ………. ………. ………. 3

(11)

(a) Solve in the domain .

………. ………. ………. ………. ………. ………. ………. ………. ………. ………. 3

(b) Find the median of the continuous probability distribution defined as in the

domain [ 0, 4 ]. ………. ………. ………. ………. ………. ………. ………. ………. 3

(12)

(a) Solve the equations simultaneously and show that there is

only point of intersection. Give its coordinates.

………. ………. ………. ………. ………. ………. ………. ………. ………. 3

(b) Sketch on the axes below. 2

Question 16 continues on page 13

(13)

Question 16 continued

(c) Calculate the area bounded by the curves and the x-axis.

………. ………. ………. ………. ………. ………. ………. ………. ………. ………. 2

(14)

It is known at the beginning of winter in a large population, 15% of people will be infected with a particular virus.

(a) Four people are selected at random, find the probability that at least one of them has the

virus. ………. ………. ………. ………. ………. 2

(b) What is the smallest number of people a drug company would need to test to have a

greater than 95% chance that at least one of the tested people had the virus.

………. ………. ………. ………. ………. ………. ………. ………. ………. ………. ………. ………. 3

(15)

Question 17 continued

(c) As winter progresses the virus spreads much more and the health authorities decide they

want to stop the virus and have been given a new medication to trial. The two-way table shows the number of people in a trial.

Taking Medication Control Group

Virus 204 205

No Virus 212 209

(i) What percentage of people in the trial had the virus?

………. ………. ………. ……….

1

(ii) What percentage of people in the control group had the virus?

………. ………. ………. ……….

1

(iii) Determine if it is worth the heath authorities using this new medication.

………. ………. ………. ………. 1

(16)

- 16 - Question 18 (4 marks) (a) Evaluate . ………. ………. ………. ………. 2

(b) Henderson’s harvests oranges to sell to Cottonworths, and the weights of the oranges

they sell are normally distributed.

Oranges that weigh less than 100 grams are rejected, and this harvest season 97.5% of their oranges are accepted to sell.

Cottonworths also offers a bonus for premium oranges that are greater than 130 grams and 16% of this seasons harvest are classed as premium.

Find the mean and standard deviation of the weights of the oranges.

………. ………. ………. ………. ………. ………. ………. ………. ………. ………. 2

(17)

Max did a survey of a group of people he knew about their age and how much they earn each week. The results are shown in the table below.

Age (years) (x) 18 45 28 15 32 68

Wage ($/week) (W) 715 2350 1530 438 1690 1320

(a) Using your calculator find (r) the correlation coefficient and explain what type and

strength of correlation this data gives.

………. ………. ………. ……….

2

(b) Using your calculator find the equation of the least-squares regression line in the form

………. ………. ………. ………. 1

(c) Use your equation to estimate the earnings of a 50 year-old worker.

………. ………. ………. ……….

1

(18)

- 18 - Question 19 continued

(d) Could your equation from part(b) be used to make valid estimates for ages greater than

68 and less than 15 years?

Validate your response with calculations and or reasons.

………. ………. ………. ……….

(19)

A swimming pool is to be emptied for maintenance. The quantity of water, Q litres, remaining

in the pool at a time, t minutes after it starts to drain, is given by:

2

( ) 2000(25 ) , 0

Q t  t t .

(a) At what rate (in litres/min) is the water being removed at any time (t)?

………. ………. ……….

1

(b) How long will it take to remove half of the water from the pool to the nearest minute?

………. ………. ………. ………. ………. ………. ………. 2

(c) At what time does the rate of flow of water from the pool reach 20 kL/minute?

………. ………. ………. ……….

2

(d) Describe how the amount of water remaining in the pool changes as the pool empties.

Include mention of how the rate itself changes.

………. ………. ………. ……….

(20)

Three towns, A, B and C form a triangle.

Town A is 80 km from Town B and Town C is 40 km from Town A as shown below:

The bearing of Town B from Town A is 130. The bearing of Town C from Town A is 240.

(a) Find the area of the triangle formed by the three towns, to the nearest square kilometre.

………. ………. ………. ………. ………. 2

(b) Using the cosine rule, find the distance between Town B and Town C, to the nearest

kilometre. ………. ………. ………. ………. ………. ………. ………. 2

(21)

(a) Given _{f x}_{( )}_ ₄__x2 _{complete this table of values, correct to 3 decimal places.}

x 0 0.5 1 1.5 2

f(x)

1

(b) Use the Trapezoidal rule, with four sub-intervals, to estimate the value of

2 2 0 4x dx



. ………. ………. ………. ………. ………. ………. 2

(22)

For the curve _y__x3_3_x2_9_x_4_:

(a) Find any stationary points and classify them.

………. ………. ………. ………. ………. ………. ………. ………. ………. ………. 3

(b) Find the point of inflexion.

………. ………. ………. ……….

1

(c) Sketch the curve, showing all main features. 2

(23)

A particle is moving in a straight line with velocity

v



3 e

t



6 e

t with t measured in minutes

and v in ms-1_.

The particle begins its motion at origin.

(a) What is the initial velocity?

………. ………. ……….

1

(b) Find an equation for x, the displacement of the particle.

………. ………. ………. ……….

2

(c) Show that when x = 10, 3_e2t_7_et_6_0

. ………. ………. ………. ………. 2

(d) Hence, find the value of t when x = 10.

………. ………. ………. ………. ………. ………. 2

(24)

Kate and Dave are buying a house for $1 700 000, they have a $200 000 deposit and will need to

borrow the remaining balance.

An interest rate of 3.6% p.a. compounded monthly is charged on the outstanding balance. The

loan is to be repaid in equal monthly payments (M) over a 30 year period.

How much should Kate and Dave be paying each month to fully pay off the house in the 30 year time period and how much interest do they pay over the life of the loan?

………. ………. ………. ………. ………. ………. ………. ………. ………. ………. ………. ………. ………. ………. ………. ………. ………. ………. ………. ………. ………. ………. 5

(25)

A leak from a tanker has accidently contaminated a farmer’s paddock with a toxic chemical. The chemical concentration in the soil was 6kL/ha immediately after the accident.

One year later the concentration was measured to be 2.4kL/ha. It is known that the concentration, C, is given by:

Where C0 and k are constants and t is measured in years.

(a) Evaluate C0 and k.

………. ………. ………. ………. ………. ………. ………. ………. ………. 3

(b) It will not be safe for the farmer to plant a new crop until the concentration falls below

0.2kL/ha.

How long, to the nearest month, after the spill does the farmer need to wait for the paddock to be safe to use?

………. ………. ………. ………. ………. ………. ………. ………. 2

(26)

Nathan is on a paddle board in the ocean 3 kilometres from the nearest point O on a straight beach.

Nathan needs to meet his friend Jarrod who is located 6 kilometres along the beach from the point O.

Nathan paddles at a rate of 4 km/h to a point C on the beach and then walks at a rate of 5km/h along the beach to Jarrod.

Show that the total time it takes Nathan to reach Jarrod is given by:

2 _{9 6}

( )

4 5

x x

T x     .

Hence, find the minimum time it will take Nathan to reach Jarrod.

………. ………. ………. ………. ………. ………. ………. ………. ………. ………. Answer Space for Question 27 continues on page 27

(27)

Question 27 continued ………. ………. ………. ………. ………. ………. ………. ………. ………. ………. ………. ………. ………. ………. ……….

End of Paper

(28)

- 28 -

(29)

Section II Extra writing space

If you use this space, clearly indicate which question you are answering.

(30)

- 30 -

Section II Extra writing space

If you use this space, clearly indicate which question you are answering.

(31)

2020 Trial HSC Examination

Mathematics Advanced

Mathematics Extension 1

Mathematics Extension 2

(32)

(33)

(34)

(35)

Western

Mathematics

2020 Trial Higher School Certificate Examination

Mathematics

Advanced

Name ________________________________ Teacher ________________________

Section

I

–

Multiple

Choice

Answer

Sheet

Allow about 15 minutes for this section

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 the

correct answer by writing the word correct and drawing an arrow as follows.

_A _B _C _D 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

(36)

Western Mathematics

2020

TRIAL HSC EXAMINATION

Mathematics Advanced

Solutions

Section

I

No Working Answer 1. C

(37)

No Working Answer

2.

< $680 means a z-score lower than -2

Therefore 100 – 50 – 47.5 = 2.5% so 2.5% of people will earn below $680. B 3. D 4.

This is a geometric series with common ratio and first

term 𝑎 2π.

(38)

No Working Answer 5. A 6. A

7. Since (x + b) parabola is translated b units to the left, so axis is

x = –b

OR Axis of symmetry

(39)

No Working Answer

8. Area under the curve must equal 1.

C 9. B

10. Predicting values outside the original data set is called extrapolation.

C

(40)

Western

Mathematics

2020 Trial Higher School Certificate Examination

Mathematics

Advanced

Name ________________________________ Teacher

________________________

Section

I

–

Multiple

Choice

Answer

Sheet

Allow about 25 minutes for this section

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 the correct answer by writing the word correct and drawing an arrow as

follows. _A _B _C _D 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

(41)

Western Mathematics

Mathematics Advanced Trial HSC

Solutions

2020

Section

II

Question Working and answer Marks Mark

Allocation 11. a) 2 2 marks for correct solution 1 mark for correct use of log laws or correct steps in differentiation or equivalent merit b) 2 2 marks for correct solution

1 for taking out

the common factor to get part (a) answer or equal merit

(42)

(43)

Allocation

14. a) The Ancient History class has a higher mean and a smaller standard

deviation, meaning the scores are grouped more closely around the mean which indicates more consistent marks. While the Modern History has a lower mean and a larger standard deviation, meaning the scores are more spread out around the lower mean, so are much less consistent marks.

2 2 if correctly

commented on

both mean and

standard deviation 1 if only comment on either mean or standard deviation b)

Fred performed better in MH as he had a higher positive z -score meaning he performed further above the mean, compared to AH, even though he had a lower raw mark for MH.

3 3 marks for two correct z–score calculations and analysis 2 marks for correct z scores but incorrect analysis or equivalent merit 1 for a correct z – score calculation or equivalent merit

(44)

Question Working and answer Marks Mark Allocation 15. a)

3 3 marks for all

correct solutions with no extras 2 marks for most correct solutions with some missing or extras or equivalent correct working 1 mark for some relevant and correct wotking

(45)

Question Working and answer Marks Mark Allocation b) 3 3 marks for correct solution 2 marks for setting up integral correctly with error in calculation or integration or similar merit 1 for some attempt to set integral and evaluate

(46)

Allocation

16. a)

$/week

(52)

d) The equation is not a valid way to extrapolate for younger and older workers, as it has only a moderate correlation coefficient. There is one data point (for a 68-year old) which affects the results causing this. People below 15 would not be working because they are too young, and at the older end, people have retired or are working less, and this equation does not take this into account.

The equation is only valid for interpolation between 15 and 45 and, even then, has been affected by the last data point, a possible outlier. Max would need more data for ages outside of this if he is to make a more accurate prediction. Had he left the last data point out the

correlation would have been much stronger and more accurate in the 15 – 45yr range.

2 2 marks for any

valid response which shows understanding of correlation, linear regression and extrapolation

(53)

20. a) 𝑄 𝑡 2000 25 𝑡 ,𝑡 0 𝑄′ _𝑡 _{4000 25} _𝑡 ∴ it is emptying at a rate of 4000 25 𝑡 litres/minute 1 1 mark for correct answer b) Pool full at 𝑡 0 𝑄 𝑡 2 000 25 0 1 250 000𝑙𝑖𝑡𝑟𝑒𝑠

∴ half full 625 000 litres

625 000 2 000 25 𝑡 312.5 625 50𝑡 𝑡 𝑡 50𝑡 312.5 0 2𝑡 100𝑡 625 0 𝑡 100 √100 4 2 625 2 2 𝑡 100 √5000 4 𝑡 7.322 or 42.68 ∴ 𝑡 7 minutes

∴ it will take 7 minutes to half empty the pool

2 2 marks for correct answer 1 mark for finding a quadratic equation and attempting to solve or equivalent merit c) 20𝑘𝐿 20 000 L/min 20 000 4 000 25 𝑡 20 000 100 000 4 000𝑡 4 000𝑡 80 000 𝑡 20 𝑚𝑖𝑛

∴the flow rate will be 20kL after 20 minutes

2 2 marks for correct answer 1 for correct conversion from kL or correct calculation without conversion or similar merit

d) Pool is empty when 2000 25 𝑡 0

ie when 𝑡 25

Rate 𝑄 𝑡 4000 25 𝑡

When 𝑡 0, Rate 4000 25 100 000

When 𝑡 25, Rate 4000 0 0

As the pool empties, the rate of flow remains negative, but its value is decreasing meaning that the rate of flow of water slows, becoming zero when the pool is empty

2 2 marks for any

explanation

that includes

that the rate of

flow is

decreasing.

(54)

21. a) 2 2 marks for finding correct area 1 mark for finding the

angle CAB from

the bearings or equivalent merit b) 2 2 marks for correct answer

1 mark for the

correct substitution or equivalent merit 22. a) x 0 0.5 1 1.5 2 f(x) 2 1.936 1.732 1.323 0 1 1 mark for correct answer b) 𝑓 𝑥 𝑑𝑥 𝑏 𝑎 2𝑛 𝑦 𝑦 2 𝑦 𝑦 ⋯ 𝑦 2 8 2 0 2 1.936 1.732 1.323 2.9955 2 2 marks for correct estimate using formula or 4 trapezia 1 mark for some progress by any method

(55)

23. a) 3 3 marks for correct points and (checked) nature 2 marks for

finding y’ and

y” and turning

points but not

classifying or

equivalent

merit

1 marks for

finding y’ and

y” or

equivalent

merit b)

Concavity changes so (1, -7) is an inflexion

x 0 1 2

y’’ -6 0 6

1 1 mark for

(56)

c) 2 2 marks for correct shape of the curve and the 4 points shown. 1 mark for an incorrect graph with some points correct 24. a) 1 1 mark for correct answer b) 2 2 marks for giving the correct equation 1 mark for correct integration but not finding C or equivalent merit

(57)

c) 2 2 marks for correct substitution and manipulation to show required equation 1 mark correct substitution and some manipulation d) 2 2 marks for substitution and solving a quadratic and finding 2 solutions 1 mark for substitution and attempt at solving the quadratic

(58)

25. 5 5 marks for correct repayment and amount of interest (NB allow for differences in calculations with rounding) 4 marks for correct repayment but not interest or equiv merit 3 marks for significant progress on series and equations or equiv merit 2 marks for setting up first 2 months or equiv merit 1 mark for some relevant calculations or equiv merit

(59)

26. a) 3 3 marks for finding C0 and writing a new equation for C and finding k 2 marks for finding C0 writing a new equation for C and attempting to solve 1 mark for some relevant calculations and exponential manipulation b) 2 2 marks for finding t and stating how

long the farmer

should leave the field (Numbers could be different depending on rounding in working) 1 mark for showing some progress to being able to find t

(60)

27. 5 5 marks for showing required expression and finding minimum 4 marks for showing required expression and minor error in finding minimum or equiv merit 3 marks for showing required expression some progress in finding minimum or equiv merit 2 marks for error in showing required expression and some progress in finding minimum or equiv merit 1 mark for some relevant calculations

(61)