VCE Maths Methods Practice EXAM 2 PDF

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Methods practice exam for units 3 and 4....

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2019 PRACTICE EXAM

MATHS METHODS UNITS 3&4 Exam 1 Question Booklet Exam 2 Question Booklet Worked Solution Booklet

A co ll ab or at io n by : L UKE SANT OMARTINO MICHAEL REHFISCH R EBECCA W IL LIAM S SEPEHR RASEKHI

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Letter

STUDENT NUMBER

MATHEMATICAL METHODS Written examination 2 2019 Reading time: 9:00 a.m. to 9:15 a.m. (15 minutes) Writing time: 9:15 a.m. to 11:15 a.m. (2 hour)

QUESTION AND ANSWER BOOK Structure of book Section

Number of questions

Number of questions to be answered

A B

20 4

20 4

Number of marks

20 60 Total 80

.

• Students are permitted to bring into the examination room: pens, pencils, highlighters, erasers, sharpeners, rulers, a protractor, set squares, aids for curve sketching, one bound reference, one approved technology (calculator or software) and, if desired, one scientific calculator. Calculator memory DOES NOT need to be cleared. For approved computer-based CAS, full functionality may be used. • Students are NOT permitted to bring into the examination room: any technology (calculators or software), notes of any kind, blank sheets of paper and/or correction fluid/tape. Materials supplied • Question and answer book of 19 pages • Formula sheet • Answer sheet for multiple-choice questions Instructions • Write your student number in the space provided above on this page. • Check that your name and student number as printed on your answer sheet for multiple-choice questions are correct, and sign your name in the space provided to verify this. • Unless otherwise indicated, the diagrams in this book are not drawn to scale. • All written responses must be in English. At the end of the examination • Place the answer sheet for multiple-choice questions inside the front cover of this book. • You may keep the formula sheet.

Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the examination room.

© ATAR Notes 2019

2019 MATHMETH EXAM 2

2

Instructions Answer all questions in the spaces provided. In all questions where a numerical answer is required, an exact value must be given, unless otherwise specified. In questions where more than one mark is available, appropriate working must be shown. Unless otherwise indicated, the diagrams in this book are not drawn to scale.

Question 1 Let ฀฀: ฀฀ → ฀฀, ฀฀(฀฀) = −3 sin(2฀฀) + 2. The period and range of this function are respectively A. B. C. D. E.

฀฀ and [−1,5] ฀฀ and [−3,2] 2฀฀ and [−1,5] 2฀฀ and [−3,2] 4฀฀ and [1,5]

Question 2 Let ฀฀: ฀฀ → ฀฀, ฀฀(฀฀) = 2฀฀ 3 − 3฀฀ 2 − ฀฀฀฀ + ฀฀. When ฀฀(฀฀) is divided by (฀ ฀ + 1), the remainder is 5. The sum of the values of ฀฀ and ฀฀ is A. B. C. D. E.

−5 −1 1 5 10

Question 3 The midpoint of the points (4,5) and ฀฀(฀฀, ฀฀) is (1,1).

The coordinates of the point ฀฀ are A. (7,9) B.

3

� , 3� 2

C. (5,6) D. (−2,3) E. (−2, −3)

SECTION A – continued

3

2019 MATHMETH EXAM 2

Question 4

The function ฀฀: ฀฀ → ฀฀, ฀฀(฀฀) = ฀฀ 2 + ฀฀฀฀ + 2, where ฀ ฀ > 0, has a range of [−14, ∞). The value of ฀฀ is A. B. C. D. E.

2 3 11/2 6 8

Question 5

The function ฀฀ is strictly decreasing over the interval (฀฀, ฀฀).

If ฀฀, ฀ ฀ > 0, then which of the following must be true? A. B. C. D. E.

฀฀(฀฀ − ฀฀) > 0 ฀฀ ′ (฀฀) > 0 ฀฀ ′ (฀฀) < 0 ฀฀ ′ (฀฀ − ฀฀) = 0 ฀฀(฀฀) − ฀฀(฀฀) > 0

Question 6 A large random sample of visitors to the Wonder Wharf amusement park were surveyed and asked if they enjoyed their visit to the park. The survey results state, with a 95% confidence interval, that between 85% and 93% of visitors enjoyed their visit to the park. The number of surveyed visitors who enjoyed their visit to the Wonder Wharf amusement park is closest to A. B. C. D. E.

89 148 167 209 235

Question 7 The function ฀฀ has an average value of ฀฀ over the interval [฀฀, ฀฀]. ฀฀

The value of ∫฀ �2 − ฀฀(฀฀)�฀฀฀฀ ฀ A. B. C. D. E.

is equal to

2฀฀ − 2฀ ฀ + ฀฀ 2฀฀ − 2฀฀ − ฀฀ ฀ ฀ + 2฀฀ − 2฀฀ (฀฀ − ฀฀)(฀฀ − 2) (฀฀ − ฀฀ )(฀ ฀ + 2)

SECTION B – continued TURN OVER

2019 MATHMETH EXAM 2

4

Question 8

The average rate of change of the function with the rule ฀฀(฀฀) = (฀฀ 2 − 1)2 over the interval [−2, ฀฀], where ฀ ฀ > −2, is −3. The value of ฀฀ is A. 1 3 B. 4

C. −1 1 D. E.

2

1

−2

Question 9 The function ฀฀ has rule ฀฀(฀฀) = �log฀฀ (฀฀฀฀), where ฀฀ is a real positive constant.

The maximal domain of ฀฀ is 1

A. � , ∞� ฀฀ 1

B. �

฀฀

, ∞�

1

1

C. �−∞, − ฀฀ � ∪ � 1

D. �−∞, − � ฀฀

฀฀

, ∞�

E. (−∞, ฀฀]

Question 10

Consider the transformation ฀฀1 , defined as

1 ฀฀ ฀฀ ฀฀1 : ฀฀ 2 → ฀฀ 2 , ฀฀1 ��฀฀�� = �2 0� �฀฀� 0 1

The transformation ฀฀1 maps the graph of ฀ ฀ = ฀฀(฀฀) onto the graph of ฀ ฀ = ฀฀(฀฀), where ฀฀(฀฀) = √฀฀3 . The transformation ฀฀2 : ฀฀ 2 → ฀฀2 also maps the graph of ฀ ฀ = ฀฀(฀฀ ) onto the graph of ฀ ฀ = ฀฀(฀฀). The transformation ฀฀2 could be given by

฀฀ 1 A. ฀฀2 �� ฀฀�� = � 0 ฀฀ 1 B. ฀฀2 �� ฀฀�� = � 0 1 ฀฀ C. ฀฀2 �� ฀฀�� = �0 ฀฀ 1 D. ฀฀2 �� ฀฀�� = � 0 1 ฀฀ E. ฀฀2 �� ฀฀�� = �0

0 ฀฀ �� � 2 ฀฀ 0 ฀฀ �� � 8 ฀฀ 0 ฀฀ 1�� � ฀฀ 8 0 ฀฀ �� � 2√2 ฀฀ 0 ฀฀ 1 − � �฀฀� 2

SECTION A – continued

5

2019 MATHMETH EXAM 2

Question 11

The continuous random variable ฀฀ has a normal distribution with mean 1 and standard deviation 2. The continuous random variable ฀฀ has the standard normal distribution. The probability that ฀฀ is between −1 and 2 is equal to A. B. C. D. E.

Pr(0 < ฀ ฀ < 2) Pr (0 < ฀ ฀ < 3) Pr (−1 < ฀ ฀ < 3) Pr(X < −1) + Pr (฀ ฀ > 3) Pr (−3 < ฀ ฀ < 3)

Question 12 The number of pets, ฀฀, owned by a random individual is a discrete random variable with the following probability distribution. ฀฀

Pr(฀ ฀ = ฀฀)

0

1

2

3

4

0.2

0.34

0.24

0.12

0.1

If two random individuals are chosen, the probability that they own a total of 2 pets is A. B. C. D. E.

0.0576 0.2116 0.096 0.2376 0.1636

Question 13 The continuous random variable ฀฀ has the standard normal distribution. If Pr(฀ ฀ > ฀฀) = ฀฀, where ฀ ฀ > 0, then Pr(−฀฀ < ฀ ฀ < 0|฀ ฀ < ฀฀) is

A. B. C. D. E.

1−฀฀ 2 ฀฀

1−฀฀ 1

2฀฀ 2฀฀−1 ฀฀−1 2฀฀−1 2฀฀−2

SECTION B – continued TURN OVER

2019 MATHMETH EXAM 2

6

Question 14 In a particular scoring game, players attempt to throw a small hoop onto a pole. Players score a point each time they throw the hoop onto the pole. For each throw, the probability of throwing the hoop onto the pole is ฀฀. If a player takes two throws, the expected number of points that this player will score is A. B. C. D. E.

2฀฀ ฀฀2 2฀฀2 2฀฀(1 − ฀฀) ฀฀(2 − ฀฀)

Question 15 Consider the functions ฀฀: [฀฀, ฀฀] and ฀฀: [฀฀, ฀฀].

If ฀฀′ (฀฀) > 0 over the interval [฀฀, ฀฀], then ฀฀�฀฀(฀฀)� exists if A. B. C. D. E.

฀ ฀ > ฀ ฀ > ฀ ฀ > ฀฀ ฀ ฀ > ฀ ฀ > ฀ ฀ > ฀฀ ฀ ฀ < ฀ ฀ < ฀ ฀ < ฀฀ ฀ ฀ > ฀฀(฀฀) > ฀฀(฀฀) > ฀฀ ฀ ฀ < ฀฀(฀฀) < ฀฀(฀฀) < ฀฀

Question 16 A box contains three red balls and four green balls. Two balls are drawn at random from the box without replacement. The probability that the marbles are of the same colour is A. B. C. D. E.

1

7 25

42 25

49 3 7 4 7

Question 17

The graphs of ฀ ฀ = 3฀฀฀฀ 2 and ฀ ฀ = ฀฀฀฀ + ฀฀, where ฀฀ ≠ 0, will have two points of intersections if A. B. C. D. E.

฀฀>0 ฀฀ 0 and ฀ ฀ < 0 ฀ ฀ < 0 and ฀ ฀ < 0 ฀ ฀ > 2√3฀฀฀฀

SECTION A – continued

7

2019 MATHMETH EXAM 2

Question 18

Let ฀฀(฀฀) = ฀฀฀฀ ฀฀฀฀ + ฀฀, where ฀฀ ≥ 0.

The stationary point of ฀฀ is closest to the origin when ฀฀ is closest to A. B. C. D. E.

0 0.69 1 1.03 1.20

Question 19

A probability density function ฀฀ is given by

where ฀ ฀ > 0.

1 ฀฀ ฀฀(฀฀) = �฀ ฀ sin � ฀฀� 0

0 < ฀ ฀ < ฀฀ elsewhere

The value of ฀฀ is A. ฀฀ ฀฀ B. 2 C. 2฀฀฀฀ ฀฀฀฀ D. E.

2

4฀฀฀฀

Question 20 In a large flock of birds, it is known that 60% of the birds are magpies. A scientist takes a sample of 10 birds from the flock. For samples of 10 birds, ฀฀� is the random variable of the distribution of sample proportions of magpies. (Do not use a normal approximation.) Pr�฀฀� > 0.8 | ฀฀� > 0.5� is closest to A. B. C. D. E.

0.05 0.07 0.17 0.20 0.63

END OF SECTION A

2019 MATHMETH EXAM 2

8

Instructions Answer all questions in the spaces provided. In all questions where a numerical answer is required, an exact value must be given, unless otherwise specified. In questions where more than one mark is available, appropriate working must be shown. Unless otherwise indicated, the diagrams in this book are not drawn to scale. Question 1 (13 marks)

Let ฀฀: [−฀฀, 2฀฀] → ฀฀, ฀฀(฀฀) = 2฀ ฀ sin(฀฀).

a.

State the rule for the derivative function ฀฀′.

1 mark

_____________________________________________________________________ _____________________________________________________________________ b.

Find the range of ฀฀. Give values correct to two decimal places.

1 mark

_____________________________________________________________________ _____________________________________________________________________ c.

Find the total area of the regions bounded the ฀ ฀ = ฀฀(฀฀ ) and the ฀฀ -axis. Give your answer in exact form. 1 mark _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________

d.

฀฀ ฀฀ ฀฀ The transformation ฀฀: ฀฀ 2 → ฀฀2 , ฀฀ �฀฀ � �� = �฀฀� + � � maps the graph of ฀ ฀ = ฀฀(฀฀) onto the ฀฀ graph of ฀ ฀ = (2฀ ฀ + ฀฀) cos(฀฀) + 2. State the values of ฀฀ and ฀฀. 2 marks _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________

SECTION B – continued

9

2019 MATHMETH EXAM 2

Let ฀฀: [฀฀, ฀฀] → ฀฀, ฀฀(฀฀) = 2฀ ฀ sin(฀฀ ), where −฀฀ ≤ ฀฀ < ฀฀ ≤ 2฀฀. e.

State the largest difference between ฀฀ and ฀฀ for which ฀฀−1 , the inverse function of ฀฀, exists. Give the value correct to two decimal places. 1 mark _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________

Let ℎ: [−฀฀, 2฀฀] → ฀฀, ℎ(฀฀) = 2฀ ฀ sin(฀฀) + ฀฀, where ฀฀ ∈ ฀฀−. f.

8

Find the value of ฀฀ for which the average value of ℎ is −3.

2 marks

_____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ g.

฀฀

Find the value of ฀฀ for which the tangent to the graph of ℎ at ฀ ฀ = − passes through the point (4, −21).

2

2 marks

_____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________

SECTION B – continued TURN OVER

2019 MATHMETH EXAM 2

h.

10

Find the value of ฀฀ for which the area bounded by the graph of ℎ, the ฀฀ -axis and the lines ฀ ฀ = −฀฀ and ฀ ฀ = 2฀฀ is 200฀฀.

2 marks

_____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ i.

Find the value of ฀฀ for which ℎ(฀฀) has only two solutions, correct to two decimal places. 1 mark _____________________________________________________________________ _____________________________________________________________________

SECTION B – continued

11

2019 MATHMETH EXAM 2

Question 2 (10 marks) Dan runs a company which produces bubble bath mixture. If one measure of bubble bath mixture is added to a bathtub of water, the volume, ฀฀ litres, of bubbles in the bath after ฀฀ minutes of adding the mixture is given by rule ฀฀ ฀฀

− 100 ฀฀(฀฀) = ฀฀ 2 ฀฀ ฀฀

where ฀฀ is the percentage (%) of sodium laureth sulphate in the mixture. a.

Find the time when the volume of bubbles is a maximum. Express your answer in terms of ฀฀.

1 mark

_____________________________________________________________________ _____________________________________________________________________ b.

If Dan decides to use 5% sodium laureth sulphate in his bubble bath mixture, what is the maximum amount of bubbles that will form? State your answer in exact form. 1 mark _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________

c.

After 30 minutes, Dan would like there to be at least 100 litres of bubbles. Which values for ฀฀ give this result? Give values correct to two decimal places.

1 mark

_____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________

SECTION B – continued TURN OVER

2019 MATHMETH EXAM 2

12

Dan’s company also manufactures detergent. If one measure of detergent is added to a sink of water, the volume, ฀฀ litres, of foam in the sink after ฀฀ minutes of adding the detergent is given by ฀฀(฀฀) = ฀฀฀฀ −฀฀฀฀

where ฀฀ and ฀฀ are positive real numbers and ฀ ฀ > 0. d.

Find the time, in terms of ฀฀, when only half of the initial volume of foam will remain.

2 marks

_____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ e.

Find the rule for ฀฀ −1, in terms of ฀฀ and ฀฀.

2 marks

_____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________

SECTION B – continued

13

2019 MATHMETH EXAM 2

For the foam produced by the detergent from Dan’s company, ฀ ฀ = 1. For the foam produced by the detergent from a rival company, ฀ ฀ = 2. Dan runs an experiment. In Sink 1, he puts one measure of the detergent from his company. At the same time, he puts one measure of the detergent from the rival company in Sink 2. Initially, the volume of foam in each sink is the same. f.

At ฀ ฀ = 2, find the exact ratio of the volume of foam in Sink 1 to the volume of the foam in Sink 2. Express your answer in the form 1 ∶ ฀฀, where ฀฀ is a real number in exact form.

1 mark _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ g.

At a certain time, the volume of foam in Sink 1 is ฀฀. Find the volume of foam in Sink 2 at this time. Express your answer as a fraction, in terms of ฀฀ and ฀฀.

1 mark

_____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ h.

Find the value of time such that rate at which the volume of foam in Sink 2 is changing with respect to time is double the rate at which the volume of foam in Sink 1 is changing with respect to time. Express your answer in the form log฀฀ (฀฀), where ฀฀ is a positive integer. 1 mark _____________________________________________________________________ _____________________________________________________________________ ______________________________________________...