WME Adv Trial 2020 with Solutions PDF

Preview

Summary

Due to the implementation of the new syllabus, past paper resources that cover the new content is almost impossible to find. HOWEVER, I have collated 35+ past papers from different schools for their 2020 trials :)...

Description

Western Mathematics

2020

TRIAL HIGHER SCHOOL CERTIFICATE EXAMINATION

Mathematics Advanced General Instructions

Total marks : 100



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

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



Allow about 2 hours and 45 minutes for this section



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%

3.

B.

2.5%

C.

5%

D.

47.5%

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. D.

-2

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 11th 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

-3

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?

9.

A.

–1

B.

0.5

C.

1

D.

2

Find the derivative of

.

A. B. C. D.

-4

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

-5

Western Mathematics 2020 TRIAL HIGHER SCHOOL CERTIFICATE EXAMINATION Class and Teacher

Mathematics Advanced Student Number

Section II Answer Booklet 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. If you use this space, clearly indicate which question you are answering.

Question 11 (4 marks) (a)

Show that the derivative of

. 2

……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. …………………………………………………………………………………………….

(b)

2

Hence or otherwise find

……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. …………………………………………………………………………………………….

-7

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. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. …………………………………………………………………………………………….

-8

5

Question 13 (3 marks) (a)

1

Show that

……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. …………………………………………………………………………………………… (b)

Hence or otherwise, sketch the graph of

showing any asymptotes and the

x-intercept.



-9

2



Question 14 (5 marks) 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. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. …………………………………………………………………………………………….

          



- 10 

3

Question 15 (6 marks)

(a)

Solve

in the domain

3

.

……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. (b)

Find the median of the continuous probability distribution defined as

in the

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





- 11 

3

Question 16 (7 marks) (a)

Solve the equations

simultaneously and show that there is

3

only point of intersection. Give its coordinates. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. (b)

Sketch

on the axes below.

Question 16 continues on page 13 



- 12 

2

Question 16 continued (c)

Calculate the area bounded by the curves

and the x-axis.

……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. …………………………………………………………………………………………….

- 13 

2

Question 17 (8 marks) 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. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. Question 17 continues on page 15





- 14 

3

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 1

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





- 15 

1

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.

……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. …………………………………………………………………………………………….  

- 16 

2

Question 19 (6 marks) 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.

(a)

Age (years) (x)

18

45

28

15

32

68

Wage ($/week) (W)

715

2350

1530

438

1690

1320

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.

……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. …………………………………………………………………………………………….

Question 19 continues on page 18 



- 17 

1

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.

……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. …………………………………………………………………………………………….

- 18 

2

Question 20 (7 marks) 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:

Q(t )  2000(25  t )2 , t  0 .  (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. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. …………………………………………………………………………………………….

- 19 

2

Question 21 (4 marks)

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. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. …………………………………………………………………………………………….

- 20 

2

Question 22 (3 marks)

(a)

Given f ( x)  4  x 2 complete this table of values, correct to 3 decimal places.

x

0

0.5

1

1.5

1

2

f(x)

2

(b)

Use the Trapezoidal rule, with four sub-intervals, to estimate the value of  4  x 2 dx . 0

……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. …………………………………………………………………………………………….

- 21 

2

Question 23 (6 marks) For the curve y  x 3  3 x 2  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.



- 22 

2

Question 24 (7 marks) t t A particle is moving in a straight line with velocity v  3 e  6 e 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, 3e 2t  7et  6  0 .

2

……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. (d)

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

2

……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. ……………………………………………………………………………………………. 

- 23 

Question 25 (5 marks) 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 peri...