3U WME 2020 with Sols - Extension 1 Mathematics Paper PDF

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Extension 1 Mathematics Paper ...

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Western Mathematics

2020

TRIAL HIGHER SCHOOL CERTIFICATE EXAMINATION

Mathematics - Extension 1 General Instructions



Reading time – 10 minutes



Working time – 2 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: 70

Section I – 10 marks (pages 2 – 7)  

Attempt Questions 1 – 10 Allow about 15 minutes for this section

Section II – 60 marks (pages 8 – 14)  

Attempt Questions 11 – 14 Allow about 1 hour and 45 minutes for this section

‐1‐ 

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.

There are eight questions in a multiple-choice test. Each question has four possible answers, only one of which is correct. What is the probability of answering exactly five questions correctly by chance alone, correct to 3 significant figures?

2.

A.

0.0000153

B.

0.000412

C.

0.000977

D.

0.0231

1 4 Find the vector projection of 𝑝  󰇡 󰇢 onto 𝑞  󰇡 󰇢. 2 3 2 A. 󰇡 󰇢 4 2 B. 󰇡 󰇢 4 C. D.



2 5  √  4√ 5

2 5  √  4√5 

‐2‐ 

3.

Write the expression

in the form

.

A.

B.

C.

D. 4.

Which of the following differential equations could be represented by the slope field diagram below?

A. B. C. D.

‐3‐ 

5.

Given that

and

, find an expression for

.

A.

B. C. D. 6.

The polynomial

has roots -1 and 2, one of which is a triple root.

Find the values of a and b.

A. B. C. D.

‐4‐ 

7.

The function

is shown in the diagram.

If this function is transformed using steps I, II and III as below: I. II. III.

Reflected about the Vertically translated 1 unit down Dilated horizontally by a scale factor of 2.

Which equation would represent the transformed function? A. B. C. D.

‐5‐ 

8.

The area enclosed by the curves

is shaded in the diagram below.

Which expression could be used to calculate this area? A.

B.

C.

D.

‐6‐ 

9.

A body of still water has suffered an oil spill and a circular oil slick is floating on the surface of the water. The area of the oil slick is increasing by At what rate is the radius increasing when the area is

?

A. B. C. D.

10.

A four-digit security code is to be created for a building alarm, using any selection of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The code must be entered in the correct order to disarm the alarm when entering the building. Digits may be repeated. It has been decided that the code will contain exactly two different digits, for example 4224 or 7177. If an intruder, who knew about this restriction, tried to guess the alarm code, what is the probability they would get it correct?     B.  

A.

C. D.

  

 



‐7‐ 

 Section II

60 marks Attempt Questions 11 – 14. Allow about 1 hour and 45 minutes for this section.

Answer each question in a SEPARATE writing booklet. Extra writing booklets are available. In Questions 11 – 14, your responses should include relevant mathematical reasoning and/or calculations.

Question 11 (15 marks) Use the Question 11 writing booklet. (a)

i)

Write an expression for

ii)

Hence, find

in terms of

2

2

.

(b)

Evaluate

(c)

The continuous random variable, X, has the following probability density function:

using the substitution

i)

Find the value of a.

ii)

Write an expression that could be used to correctly calculate

2

(Do not evaluate your expression) Question 11 (c) continues on page 9 



-8

3

.

.

1



Question 11 (c) continued iii)

The graph of the probability density function for f(x) is shown below.

1

The line x = 2.4 creates an area of 0.5 square units to the right of the line and under the curve, as shown.

Explain what measure the value of x = 2.4 represents in relation to f(x). (d)

A particle is moving such that, at time, t seconds, its displacement, x metres, satisfies the equation

i)

.

Show that the velocity of the particle can be represented by 2 .

ii)

Using the formula

, or otherwise, find the acceleration of the particle after

1 second.

End of Question 11





-9

2



Question 12 (15 marks) Use the Question 12 writing booklet. (a)

Consider vectors

i)

Find the magnitude and direction of

ii)

Find the resultant vector of

iii)

Calculate the dot product

iv)

Find the angle between the vectors

(b)

2

.

2

.

1 2

.

A game is played in which players throw netballs, aiming to get them through the hoop. The players are given two options in how they play the game. Option 1: Players take 2 attempts and win a prize if at least one of the balls goes through. Option 2: Players take 3 attempts and win a prize if at least two of the balls go through. Karen decides to play. The probability that she will get a ball through on any attempt is p, where p > 0. 1

i)

Show that the probability that Karen wins if she takes Option 1 is

ii)

Show that the probability that Karen wins if she takes Option 2 is

iii)

Karen has calculated that she is 3 times as likely to win by choosing Option 1 over Option 2.

. .

1 2

Find the value of p.

(c)

i)

Show that

ii)

Hence, find

1

where m and n are both positive, even integers and

.

End of Question 12



 - 10 -



3



Question 13 continued (c)

Sienna intends to row her boat from the south bank of a river to meet with her friends on the north bank. The river is 100 metres wide. Sienna’s rowing speed is 5 metres per second when the water is still. The river is flowing east at a rate of 4 metres per second. Sienna’s boat is also being impacted by a wind blowing from the south-west, which pushed the boat at 8 metres per second. She starts rowing across the river by steering the boat such that it is perpendicular to the south bank. i)

Show that the velocity of Sienna’s boat can be expressed as the component vector:

2

ii)

Calculate the speed of the boat, correct to 2 decimal places.

1

iii)

Determine the distance rowed from Sienna’s starting point to her landing point and how long it will take her to reach the north bank.

3

End of Question 13

 



- 12 



Question 14 (15 marks) Use the Question 14 writing booklet. A golf ball is hit at a velocity of

(a)

at an angle

to the horizontal.

The position vector s(t), from the starting point, of the ball after t seconds is given by

i)

Using gravity of

show that the maximum horizontal range of the ball is

2

metres.

ii)

To ensure that the ball lands on the green, it must travel between 400 and 450 metres. What values of

iii)

1

, correct to the nearest minute, would allow this to happen?

The golfer aims accurately and hits the ball directly towards the green.

3

After 3.4 seconds of flight, at a point 8 metres above the ground, the ball hits a low flying TV drone. If it had not hit the drone or any other obstacles, would the ball have landed on the green?

Use mathematical induction to prove that

(b)

for all integers

.

For the differential equation

(c)

3

:

i)

Show that

.

ii)

Find the solution to the differential equation; given that when

1

Question 14 continues on page 14 



- 13 

2



Question 14 continued (d)

Chris is planning to survey 36 people. He will capture the birth month of each person in the data. i)

What number of people born in January would Chris expect in this sample? (Assume each month has an equal probability of being selected.)

1

ii)

Calculate the probability that the number of people in Chris’ sample born in January is within one standard deviation of the mean.

2

End of Paper

- 14 



2020 Trial HSC Examination

Mathematics Advanced Mathematics Extension 1 Mathematics Extension 2 REFERENCE SHEET

15 



16 

- 17 





- 18 





WesternMathematics 2020 Trial Higher School Certificate Examination MathematicsExtension1

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

 

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



- 19 

C

D

Western Mathematics

TRIAL HSC EXAMINATION

2020

Mathematics - Extension 1 Solutions Section I No

Working

Answer

1.

D

2.

Proj 𝑝  

.

 

. 𝑞



4 1 𝑝  󰇡 󰇢 onto 𝑞  󰇡 󰇢 . 2 3 𝑝. 𝑞  4  1   3  2  10 

 󰇛1󰇜  2  5 10 1 . 󰇡 󰇢 Proj 𝑝  2 5 1  2 󰇡 󰇢 2 2 󰇡 󰇢 4  𝑞

B

1

No

Answer

Working 6.

For the triple root cannot be 2 since the constant term is -2. A triple root of 2 would lead to a constant of -8. So the triple root is . Substituting -1 into the polynomial gives:

First derivative of the polynomial is The second derivative is Substituting

Substituting

7.

.

into second derivative gives:

into

D

gives:

OR substituting x = -1 and 2 into f(x) and solving simultaneously works. Step I has the function reflecting about the x-axis, so it is the negative of the original function, ie. . Step II has us moving the function moving down by 1 unit vertically, which means we must subtract 1 from the function. ie. Step III has us stretching the graph horizontally by a factor of 2, so it’s frequency will be halved. ie.

3

A

No

Answer

Working 8.

C

The area can be found by doubling the integral between zero and of the difference between the two curves, since the points of intersection are at

.

4

No

Answer

Working 9.

C

10.

We have

ways of choosing the pair of numbers to use in

the code. With the restriction, we have 14 ways of arranging any pair of digits. Eg. If we used 0 and 1 We could have 2 of each digit: 0011 0110 0101 1100 1001 1010 (6 arrangements) We could have 3 x 0 and 1 x 1 0001 0010 0100 1000 (4 arrangements) Or we could have 3 x 1 and 1 x 0 1110 1101 1011 0111 (4 arrangements) So we have

D

arrangements with exactly 2

digits. Probability of guessing the correct code is

.



5



TrialHSCExamination2020 MathematicsExtension1

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 A



(B) 6 B

(C) 8 C

(D) 9 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

 

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

6

C

D

Solution

Marks Allocation of marks 3

(b)

3 marks for correct answer obtained from correct use of given substitution 2 marks for obtaining correct integral in terms of u, including upper and lower bounds or equivalent merit 1 mark for some relevant work towards using the given substitution

(c)

i)

2

2 marks for correct answer with valid working 1 mark for using total probability equals 1 to attempt to find a.

8

Solution

Marks Allocation of marks

ii)

1

(ii) Also accept 𝑃󰇛3  𝑋  4󰇜 

    𝑥 󰇛4  

 𝑥󰇜𝑑𝑥

iii) Since it is a probability density function, the area below the 1 curve is 1 square unit. The area to the right of equals 0.5. So divided the total area in half, so it is the median. (d)

i)

Any correct expression acknowledging the difference

2

OR

Correct answer and valid explanation of finding half of the area.

2 marks- correctly finds or rearranges to make t the subject and finds

1 mark - attempts to differentiate t in terms of x or correctly expresses x in terms of t

9

Solution ii)

Marks Allocation of marks 2

OR

2 marks - correct answer after obtaining correct expression for acceleration in terms of t or x, or equivalent 1 mark - attempt to use acceleration formula or to find acceleration in terms of t.

10

Question 12

(a)

2020

Solution

Marks Allocation of marks

i)

2

2 marks for both direction and magnitude correct 1 mark for one answer correct

ii) Graphically

2

2 marks for correct result using a diagram or algebraically 1 mark for attempt with a minor error

Algebraically 2𝐛  𝐚  2  2  4  4  5  4  8  4  5  8  13 iii) 𝐚 . 𝐛  𝑥 𝑥  𝑦 𝑦  4  2   5  4  8  20  28

1

11

1 mark for correct answer

Solution

Marks Allocation of marks

iv) 𝐚 . 𝐛  |𝐚||𝐛 | cos θ where θ is the angle between the vectors. | 𝐚 |  4  󰇛5󰇜  √41

2

2 marks for correct solution

1 mark for equating the expressions for scalar product of vectors but with an error in algebra or calculation

|𝐛|  󰇛2󰇜  4  √20 𝐚 . 𝐛  √41  √20 cos θ 𝐚 . 𝐛   28 󰇛 from iii󰇜 󰇜 ∴ √820 cos θ  28 28 cos θ   √820 28  θ  cos   √820  󰇛0.9778󰇜 θ  cos  167.905  167° 54′ 󰇛 nearest minute 󰇜

12

(b)

1

Correct answer

ii) Probability of at least 2 ...