WME Mathematics Extension 1 HSC Exam 2021 PDF

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WME Mathematics Extension 1 HSC Exam 2021...

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Western Mathematics

2021

TRIAL HIGHER SCHOOL CERTIFICATE EXAMINATION

Mathematics Extension 1 General Instructions



Reading time – 10 minutes



Working time – 2 hours



Write using black pen



Calculators approved by NESA 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 – 6) 

Attempt Questions 1 – 10



Allow about 15 minutes for this section

Section II – 60 marks (pages 7 – 13) 

Attempt Questions 11 – 14



Allow about 1 hour and 45 minutes for this section

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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. A bag contains three green counters and twelve other counters, some of which are red, some yellow and some blue. Sixty trials of an experiment are run in which one counter is selected at random. The counter is replaced in the bag after each trial. Let X be the random variable representing the number of times that a green counter is selected. This random variable will have a binomial distribution. Given that

E ( X ) =12 , which of the following is closest to the standard deviation of X?

A. 3.1 B. 9.6 C. 3.5 2.

D. 4 Peter draws a vector from the origin to the point A (−2, 5 ) . H e then draws a vector from A, ending at point B. How far is point B from the origin? A.

√ 10 units

B.

√ 74 units

C.

√ 13 units

D.

√ 34 units

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3.

Consider the graph of

shown here:

Which one of the following would have 2 more roots than

?

A. B. C. D. 4. Which of the following is an anti-derivative for

A. B.

C. D.

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?

5.

, it is known that

For the polynomial

.

Which of the following statements is incorrect? A.

6.

.

B.

has one of its roots at

C.

is divisible by

D.

has a double root at

. . .

Billie was trying to prove that is divisible by 3 for all integers n, . She used mathematical induction and proved the statement to be true for . Billie wrote the assumption step as:

Billie then started Step 3 as shown:

Which one of the following lines could form part of Billie’s working following on from this?

A. B. C. D.

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7.

It is known that What is the value of

, where ?

.

A. B.

C.

8.

D. Which of the following best represents the direction field for the differential equation ? A.

B.

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C.

9.

D.

A company has a board of twelve directors. Six of these were selected at random to be candidates in the election for the positions of President, Vice-President and Treasurer. In how many ways can these three senior positions be filled? A.

B.

C.

D.

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10. A particle is initially positioned at . Its motion has velocity of Which of the following will be true when the particle reaches the origin?

A. B.

C. D.

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.

Section II 60 marks Attempt Questions 11-14 Allow about 1 hour and 45 minutes for this section Answer each question in the appropriate writing booklet. Extra writing booklets are available. For questions in Section II, your responses should include relevant mathematical reasoning and/or calculations.

Question 11 (15 marks) Use the Question 11 Writing Booklet (a) Let (i) Show that

.

1 2

.

(b) For what values of x is

3

?

2 (c) Find the angle between the vectors (d) The diagram shows the graph of

, correct to the nearest degree. .

(i) Sketch

ain of

.

(ii) Sketch 1 2

(e) By expressing .

, solve End of Question 11

Question 12 (16 marks) Use the Question 12 Writing Booklet -9-

, for

4

(a) Use the principle of mathematical induction to show that for all integers

,

3

. 3 (b) Use the substitution

to find

.

(c) A store has reduced the prices on two of their sale items as there have been a number of faults reported by customers. One of the items is a movie DVD. The store has 3 of these left in stock and it is known that there is a 2.5% probability that each is faulty. The other item is a music CD with each CD having a 10.8% probability of being faulty. The store has 5 of these left to sell. A customer chooses to purchase 2 of the movie DVDs and 3 of the music CDs as gifts. (i) What is the probability that none of these are faulty?

3

(ii) What is the probability that at least one of the items is faulty?

1

(d) Use the pigeonhole principle to determine how many integers must be selected from the numbers 5, 6, 7, 8, 9 and 10 so that at least 2 of the numbers will have a difference of 3?

Question 12 continues on page 9

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2

Question 12 (continued) (e) Two straight roads, Lagoon Road and Mitchell Street, intersect at an angle of . Audrey leaves the intersection at 9:00 am and travels on Lagoon Road at an constant speed of 50 kilometres per hour. Braxton leaves the intersection at 10:00 am, travelling along Mitchell Street at an constant speed of 60 kilometres per hour. The diagram below shows the direction of Audrey and Braxton’s travels.

Taking t = 0 at th (i) Show that the distance between Audrey and Braxton at time, t hours can be expressed as

2

. (ii) Find the rate of change in the distance between Audrey and Braxton at 12:00 noon, correct to the nearest km/h. End of Question 12

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2

Question 13 (15 marks) Use the Question 13 Writing Booklet (a) 1 (i) Show that

(ii) The region bounded by the curve below. If this region is rotated about the formed.

(b) Solve the equation (c) (i) Differentiate

3

. , the y -axis and the x -axis is shaded x -axis, find the exact volume of the solid

3 with respect to

x .

1

2 (ii) Hence find Question 13 continues on page 11

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Question 13 (continued) (c) Mick hits a tennis ball from a height 2.8 metres above the ground. The angle of elevation as the racket hits the ball is and the velocity is 202 km/h. The ball lands at a horizontal distance of d metres from Mick’s feet.

(i) Using tennis ball is:

3

as acceleration due to gravity, show that the vector displacement of the .

(ii) If d is greater than 26.2, the ball will be called “out”. Find the value of d and determine if the ball will be called “out”.

End of Question 13

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2

Question 14 (14 marks) Use the Question 14 Writing Booklet (a) (i) Show that . (ii) The graph of the polynomial y =

is shown below.

By letting decimal places.

rrect to 2

(b) Consider the differential equation:

(i)

2 3

dy 3 x 2+4 = 2y dx

.

Find a solution to the equation in the form y = f(x).

(ii) A direction field diagram for the differential equation is shown below. Use it to sketch two possible solution curves, one through (–1, 0) and the other through (2, 0).

Question 14 continues on page 13 Question 14 (continued)

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2 2

(c)

Consider the graphs of

The point A lies on A on (i)

and

where

shown below.

and has an x-coordinate of 3. The point A’ is the image of point

. 1 2

Find the exact coordinates of A’.

(ii) Show that the equation of the tangent to (iii) Find the coordinates of the point where the tangent to from part (ii). End of paper

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. meets the tangent

2

Blank Page

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2021 Trial HSC Examination

Mathematics Advanced Mathematics Extension 1 Mathematics Extension 2 REFERENCE SHEET

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Western Mathematics 2021 Trial Higher School Certificate Examination 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

(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

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C

D...