THIS BOX IS FOR ILLUSTRATIVE PURPOSES ONLY

2017 Trial Examination

STUDENT NUMBER Letter

Figures
Words

MATHEMATICAL METHODS
Written examination 1

Reading time: 15 minutes

Writing time: 60 minutes

QUESTION & ANSWER BOOK

Structure of book
Number of Number of questions Number of
questions to be answered marks

8 8 40

 Students are permitted to bring into the examination room: pens, pencils, highlighters,
erasers, sharpeners, rulers.
 Students are NOT permitted to bring into the examination room: blank sheets of paper
and/or white out liquid/tape, notes of any kind, or a calculator of any kind.

Materials supplied
 Question and answer book of 12 pages.

Instructions
 Print your name in the space provided on the top of this page.
 All written responses must be in English.

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

 TSSM 2017 Page 1 of 12

2017 MATHEMATICAL METHODS EXAM 1

This page is blank

 TSSM 2017 Page 2 of 12

2017 MATHEMATICAL METHODS EXAM 1

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 (4 marks)

a. Let ( ) .
Find , expressing your answer in factorised form. 2 marks

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

( )
b. Let ( )

i. Find ( ) 1 mark

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

ii. Find ( ) 1 mark

__________________________________________________________________________

TURN OVER

 TSSM 2017 Page 3 of 12

2017 MATHEMATICAL METHODS EXAM 1

Question 2 (5 marks)

Let * + ( ) .
( )

a. Sketch the graph of ( ). Label the axis intercepts with their coordinates and label any
equation(s) of asymptotes. 3 marks

b. Find the area enclosed by the graph of ( ), the lines and , and the x-axis.
2 marks

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

 TSSM 2017 Page 4 of 12

2017 MATHEMATICAL METHODS EXAM 1

Question 3 (4 marks)

Let , - ( ) ( )

a. Find the equation of the tangent to the graph of ( ) at . 2 marks

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

b. Hence, find the angle θ from the positive direction of the x-axis to the tangent to the graph of
f(x) at , measured in the anticlockwise direction. 1 mark

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

c. Find the x values of the stationary points of the function f. 1 mark

__________________________________________________________________________

__________________________________________________________________________

TURN OVER

 TSSM 2017 Page 5 of 12

2017 MATHEMATICAL METHODS EXAM 1

Question 4 (4 marks)

A class contains 5 boys and 10 girls. The teacher runs a four question quiz at the end of each
lesson and selects a student at random to answer these questions. The teacher can select the same
student to answer any number of questions.

a. What is the probability that the number of boys selected in a given lesson is zero? 1 mark

__________________________________________________________________________

__________________________________________________________________________

b. What is the probability that at least one of the students selected in the lesson is a boy?
1 mark
__________________________________________________________________________

__________________________________________________________________________

c. What is the probability that no boy is selected in exactly three of the five consecutive
lessons? Give your answer in the form , where a, m and n are positive integers and p
and q are rational numbers. 2 marks

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

 TSSM 2017 Page 6 of 12

2017 MATHEMATICAL METHODS EXAM 1

Question 5 (10 marks)

a. Let ( ) ( )

i. Find the maximal domain, D, of f. 1 mark

__________________________________________________________________________

__________________________________________________________________________

Let ( - ( ) ( )

ii. Find the rule for . 2 marks

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

iii. Find the domain and range of . 2 marks

__________________________________________________________________________

__________________________________________________________________________

b. Let ( ) √ ( ) .
Given that the composite function ( ( )) is defined,

i. Find the rule for ( ( )) 1 mark

__________________________________________________________________________

__________________________________________________________________________

Question 5 - continued
TURN OVER

 TSSM 2017 Page 7 of 12

2017 MATHEMATICAL METHODS EXAM 1

ii. State the domain of ( ( )). 1 mark

__________________________________________________________________________

__________________________________________________________________________

iii. Find the stationary point(s) of ( ( )). 3 marks

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

 TSSM 2017 Page 8 of 12

2017 MATHEMATICAL METHODS EXAM 1

Question 6 (5 marks)

Let , - ( ) ( )

a. Find the x-intercepts of the graph of f. 2 marks

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

b. Calculate the average rate of change of f over the interval , -. 1 mark

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

c. Calculate the average value of f between the x-intercepts of f. 2 marks

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

TURN OVER

 TSSM 2017 Page 9 of 12

2017 MATHEMATICAL METHODS EXAM 1

Question 7 (3 marks)

A company produces batteries for calculators from two different machines A and B. 4% of the
batteries produced by Machine A are faulty and 5% of batteries produced by Machine B are
faulty. At the end of one day, Machine A produces 50 batteries and Machine B produces 80
batteries. The company owner selects one battery at random from all batteries produced in that
day.

a. What is the probability that the battery selected by the manager is faulty? 1 mark

__________________________________________________________________________

__________________________________________________________________________

b. A battery is selected and found to be faulty.

What is the probability that this faulty battery was produced by Machine B? 2 marks

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

 TSSM 2017 Page 10 of 12

2017 MATHEMATICAL METHODS EXAM 1

Question 8 (5 marks)

Let X be a continuous random variable with probability density function

( ) {

Part of the graph of f (x) is shown below.

a. Show that ( ( )) ( ) 1 mark

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

Question 8 – continued
TURN OVER

 TSSM 2017 Page 11 of 12

2017 MATHEMATICAL METHODS EXAM 1

b. Hence, find . / 2 marks

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

c. Show that the median , of X is a solution of the equation

2 marks

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

END OF QUESTION AND ANSWER BOOK

 TSSM 2017 Page 12 of 12

 THIS BOX IS FOR ILLUSTRATIVE PURPOSES ONLY

2017 Trial Examination

STUDENT NUMBER Letter

Figures
Words

MATHEMATICAL METHODS
Written Examination 2
Reading Time: 15 minutes
Writing Time: 2 hours

QUESTION AND ANSWER BOOK

Structure of book

Section Number of Number of questions to be Number of

questions answered marks
1 20 20 20
2 4 4 60
Total 80

 Students are permitted to bring into the examination room: pens, pencils, highlighters,
erasers, sharpeners, rulers, one approved graphics calculator or CAS (memory DOES
NOT have to be cleared) and, if desired, one scientific calculator, one bound reference
(may be annotated). The reference may be typed or handwritten (may be a textbook).
 Students are not permitted to bring into the examination room: blank sheets of paper
and/or white out liquid/tape.
Materials Supplied
 Question and answer book of 25 pages.
 Working space provided throughout the book.
Instructions
 Print your name in the space provided at the top of this page.
 All written responses must be in English.
Students are NOT permitted to bring mobile phones and/or any other electronic
communication devices into the examination room.

TSSM 2017 Page 1 of 25

2017 MATHEMATICAL METHODS EXAM 2

This page is blank

TSSM 2017 Page 2 of 25

2017 MATHEMATICAL METHODS EXAM 2

SECTION 1

Instructions for Section 1

Answer all questions in pencil on the answer sheet provided for multiple-choice questions.
Choose the response that is correct for the question.
A correct answer scores 1, an incorrect answer scores 0.
Marks will not be deducted for incorrect answers.
No marks will be given if more than one answer is completed for any question.

Question 1
The range of the function ( ) ( ) is

A. * +
B. * +
C. ( )
D. ( )
E. ( )

Question 2
The turning point of the function with the rule is

A. . /
B. . /
√ √
C. . /
D. ( )
E. ( )

Question 3
The number of x-intercepts of the function , - ( ) ( ) is

A.
B.
C.
D.
E.

SECTION 1 – continued
TURN OVER

TSSM 2017 Page 3 of 25

2017 MATHEMATICAL METHODS EXAM 2

Question 4
The period and range of the function ( ) . / are respectively

A. , -
B. , -
C.
D. , -
E. , -

Question 5
The average rate of change of the function f with rule ( ) √ , between x = 0 and
x = 5, is

A.
B.
C.
D.
E.

Question 6
Which one of the following is the inverse function of g ( - ( ) ?

A. , ) ( ) √
B. , ) ( ) √
C. ( ) √
D. ( - ( ) √
E. , ) ( ) √

SECTION 1 – continued

TSSM 2017 Page 4 of 25

2017 MATHEMATICAL METHODS EXAM 2

Question 7
The number of cars, X, owned by each employee in a particular department of a company is a
random variable with the following discrete probability distribution

x 0 1 2 3
Pr(X = x) 0.2 0.25 0.45 0.1

If two employees are selected at random, the probability that they do not own the same number
of cars is

A.
B.
C.
D.
E.

Question 8

The graph above is most likely to be the graph of

A. . /
B. . /
C. . /
D. . /
E. . /
SECTION 1 – continued
TURN OVER

TSSM 2017 Page 5 of 25

2017 MATHEMATICAL METHODS EXAM 2

Question 9
Given that ( ( )) ( ( )) then ∫ ( ) is equal to

A. ( )

B. ( ) ∫
C. ( ) ∫
D. ∫ ( ) ∫
E. ∫ . ( ) /

Question 10
The tangent to the curve at passes through the point (-4, -1).
The value of c is equal to

A.
B.
C.
D.
√
E.

Question 11

The graph of the function ( ) is obtained from the graph of a function f by a

reflection in the x-axis followed by a dilation by a factor of ½ from the y-axis.
The rule for the graph of f is

A. ( )
B. ( )
C. ( )
D. ( )
E. ( )

SECTION 1 – continued

TSSM 2017 Page 6 of 25

2017 MATHEMATICAL METHODS EXAM 2

Question 12
Consider the graphs of the functions f and g shown below.

The area of the shaded region above could be represented by

A. ∫ ( ( ))
B. ∫ ( )
C. ∫ ( ( ) )
D. ∫ ( )
E. ∫ ( )

Question 13
Let ( ) ( ) . / ( )
Which of the following is true for the graph of ?

A. ( )
B. ( ) ( ) ( )
C. ( )
D. ( ) ( )
E. ( ) ( )

SECTION 1 – continued
TURN OVER

TSSM 2017 Page 7 of 25

2017 MATHEMATICAL METHODS EXAM 2

Question 14
The random variable, X, has a normal distribution with mean 24 and standard deviation 0.35
If the random variable, Z, has the standard normal distribution, then the probability that X is
greater than 23.3 is equal to

A. ( )
B. ( )
C. ( )
D. ( )
E. ( )

Question 15

The continuous random variable, X, has a probability density function given by

( ) { √ ( )

√
The value of for which ( ) is

A.
B.
C.
D.
E.

SECTION 1 – continued

TSSM 2017 Page 8 of 25

2017 MATHEMATICAL METHODS EXAM 2

Question 16
Consider the transformation T, defined as

.0 1/ 0 10 1 0 1

The transformation T maps the graph of y = f (x) onto the graph of y = g (x).
If ( ) ( ), then the rule for g is

A. ( ) ( )
B. ( ) ( )
C. ( ) ( )
D. ( ) . /
E. ( ) ( )

Question 17
A machine produces 10 000 coloured counters in one day. It is known that 20% of the counters
are white. A sample of 32 counters is taken from these 10 000 counters. For samples of 32
counters, is the random variable of the distribution of sample proportions of white counters.
(Do not use a normal approximation).
.̂ / is closest to

A.
B.
C.
D.
E.

SECTION 1 – continued
TURN OVER

TSSM 2017 Page 9 of 25

2017 MATHEMATICAL METHODS EXAM 2

Question 18
Consider the discrete probability distribution with random variable X shown in the table below.

x 1 2 3 4 5
Pr(X = x) 0.1 0.15 a b 0.2

If ( ) , the value of a and b respectively are

A.
B.
C.
D.
E.

Question 19
The graph of intersects the graph of ( )( ) at two distinct points for

A.
B.
C.
D.
E.

SECTION 1 – continued

TSSM 2017 Page 10 of 25

2017 MATHEMATICAL METHODS EXAM 2

Question 20
The graph of is shown below.

The point (u, v) is any point on the graph.

The minimum distance from the point (0, 2) to any point on the curve is

A.
B.
√
C.
√
D.
E. √

END OF SECTION 1
TURN OVER

TSSM 2017 Page 11 of 25

2017 MATHEMATICAL METHODS EXAM 2

SECTION 2
Instructions for Section 2
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 (11 marks)

Let , - ( ) . /.

a. Find the period and range of . 2 marks

__________________________________________________________________________

__________________________________________________________________________

b. State the rule for the derivative function f ′. 1 mark

__________________________________________________________________________

__________________________________________________________________________

c. Find the equation of the tangent to the graph of at 1 mark

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

SECTION 2 – Question 1 - continued

TSSM 2017 Page 12 of 25

2017 MATHEMATICAL METHODS EXAM 2

d. Find the equations of tangents to the graph of , - ( ) . / that

have gradient . 2 marks

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

e. The rule of f can be obtained from the rule of under a transformation T, such that

.0 1/ 0 10 1 0 1

Find the value of a and value of b. 3 marks

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

SECTION 2 – Question 1 - continued

TURN OVER

TSSM 2017 Page 13 of 25

2017 MATHEMATICAL METHODS EXAM 2

f. Find the values of x, such that ( ) ( ) 2 marks

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

Question 2 (14 marks)

The diagram below shows part of the graphs .

a. Find the coordinates of A and B. 2 marks

__________________________________________________________________________

__________________________________________________________________________

SECTION 2 – Question 2 - continued

TSSM 2017 Page 14 of 25

2017 MATHEMATICAL METHODS EXAM 2

b. Show that the area bounded between the graphs of between

and is 2 marks

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

A tangent is drawn to the graph of at . This tangent cuts the y-axis at the point
C as shown below.

SECTION 2 – Question 2 - continued

TURN OVER

TSSM 2017 Page 15 of 25

2017 MATHEMATICAL METHODS EXAM 2

c. Find the coordinates of C. 2 marks

__________________________________________________________________________

__________________________________________________________________________

d. Find the angle, that the tangent line at makes with the positive direction of the
. Write your answer to the nearest degree.
2 marks

__________________________________________________________________________

__________________________________________________________________________

e. Find the area of the shaded region in the diagram above. 2 marks

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

SECTION 2 – Question 2 - continued

TSSM 2017 Page 16 of 25

2017 MATHEMATICAL METHODS EXAM 2

f. A line parallel to the tangent to at cuts the graph of at

and passes through the origin.
Find the value of a, correct to two decimal places. 3 marks

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

g. Find the exact length of . 1 mark

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

SECTION 2 – continued
TURN OVER

TSSM 2017 Page 17 of 25

2017 MATHEMATICAL METHODS EXAM 2

Question 3 (15 marks)

Let * + ( ) .

a. Express ( ) in the form , where a and b are non-zero integers. 2 marks

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

b. State the range of f. 1 mark

__________________________________________________________________________

c. Find the rule of , the inverse function of . 2 marks

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

SECTION 2 – Question 3 - continued

TSSM 2017 Page 18 of 25

2017 MATHEMATICAL METHODS EXAM 2

d. Find the point(s) of intersection of and . 2 marks

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

Part of the graph of f and the line is drawn below.

SECTION 2 – Question 3 - continued

TURN OVER

TSSM 2017 Page 19 of 25

2017 MATHEMATICAL METHODS EXAM 2

e. Write down an integral that will calculate the area of the shaded region. 2 marks

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

f. Let , ) ( ) . The graphs of f and g are drawn below.

i. Write down the domain of h where ( ) ( ) ( ). 1 mark

__________________________________________________________________________

__________________________________________________________________________

SECTION 2 – Question 3 - continued

TSSM 2017 Page 20 of 25

2017 MATHEMATICAL METHODS EXAM 2

ii. Find the stationary point(s) of , leaving the coordinates in exact form. 2 marks

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

iii. Find the coordinates of the point on , with coordinates correct to two decimal places,
which is at a minimum distance from the origin. Find this minimum distance correct to
two decimal places. 3 marks

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

SECTION 2 – continued
TURN OVER

TSSM 2017 Page 21 of 25

2017 MATHEMATICAL METHODS EXAM 2

Question 4 (20 marks)

A school has a policy of each student placing their chair on the table at the end of the day. Every
classroom in the school has 24 chairs in the room. On a particular day 24 students are present in
a class and each student is expected to place their chair on the table. The probability that a
student does not put their chair on the table at the end of the day is 10%. The expectation from
one student of placing the chair on the table at the end of the day, is completely independent
from another student.

a. Determine the probability that at least one of the chairs is not placed on the table at the end
of the lesson. Give your answer correct to four decimal places. 2 marks

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

b. A teacher observes that at least one of the chairs is not placed on the table at the end of the
lesson.
Given this, find the probability that fewer than three chairs are not placed on the table. Give
your answer correct to four decimal places. 2 marks

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

SECTION 2 – Question 4 - continued

TSSM 2017 Page 22 of 25

2017 MATHEMATICAL METHODS EXAM 2

The time it takes for students to place the chairs on the tables is approximately normally
distributed with a mean of one minute and standard deviation of 12 seconds.

c. Find the probability that a particular student takes at most 40 seconds to place the chair on
the table. Give your answer correct to four decimal places. 2 marks

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

d. The probability that a particular student takes at least m seconds to place the chair on the
table is 0.4062.
Find the value of m, correct to the nearest second. 2 marks

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

SECTION 2 – Question 4 - continued

TURN OVER

TSSM 2017 Page 23 of 25

2017 MATHEMATICAL METHODS EXAM 2

In another class of 24 students of the same school, the time it takes for students to place the
chairs on the tables is approximately normally distributed with a mean of one minute and
standard deviation of c seconds. In this class, the probability that a student takes fewer than 46
seconds is 28%.

e. Find the value of c correct to two decimal places. 3 marks

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

f. Find the probability that exactly three students of this class take fewer than 46 seconds to
place the chairs on tables. Give your answer correct to four decimal places. 1 mark

__________________________________________________________________________

__________________________________________________________________________

The principal of the school decides to take a sample of 50 students from a number of different
classes. For samples of size 50 from the population of students with a mean time of placing
chairs on tables of 1 minute and standard deviation of 12 seconds, is the random variable of the
distribution of sample proportions of students with a mean time of less than 40 seconds.

g. Find the probability that Pr( ≥ 0.06 | ≥ 0.04). Give your answer correct to three decimal
places. Do not use a normal approximation. 3 marks

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

SECTION 2 – Question 4 - continued

TSSM 2017 Page 24 of 25

2017 MATHEMATICAL METHODS EXAM 2

The principal finds that, in a particular sample of 50 students, five of them take less than 40
seconds to place chairs on tables.

h. Determine the 95% confidence interval for the principal’s estimate of the proportion of
interest. Give values correct to two decimal places. 2 marks

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

i. The probability density function for lateness to class, x minutes, is

( )
( ) {

i. Find the mean time for lateness to class. 2 marks

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

ii. Find the median time for lateness to class, correct to two decimal places. 1 mark

__________________________________________________________________________

__________________________________________________________________________

END OF QUESTION AND ANSWER BOOK

TSSM 2017 Page 25 of 25

 MATHEMATICAL METHODS
Written examination 1

2017 Trial Examination

SOLUTIONS

© TSSM 2017 Page 1 of 5

2017 MATHEMATICAL METHODS EXAM 1

Question 1

a. ( ) ( ) 1 mark
( )

b.
( )
i. ( ) 1 mark
( ( ))
1 mark

( )

ii. ( )
1 mark

Question 2

a.

1 mark for equations of asymptotes, 1 mark for y-intercept, 1 mark for shape

b. ∫ ( ( ) ) 1 mark
* + ( ) ( )
1 mark

© TSSM 2017 Page 2 of 5

2017 MATHEMATICAL METHODS EXAM 1

Question 3

a. ( ) 1 mark
√
( ) ( )
√
( ( )) 1 mark

b.

1 mark
c. 1 mark

Question 4

a. ( )
1 mark
b. ( ) ( )
1 mark

c. ( ) ( ) ( ) 1 mark

( ) ( )
1 mark

Question 5

a.
i.
Domain: ( )
1 mark

ii. ( )
√ 1 mark
( ) √ 1 mark

iii. (
(
1 mark each

© TSSM 2017 Page 3 of 5

2017 MATHEMATICAL METHODS EXAM 1

b.
i. ( ( )) √
1 mark

ii. ( )
1 mark

( ( ( ))
iii. ( ) 1 mark
√

For stationary point, numerator must equal zero

1 mark
( √ ) 1 mark

Question 6

a. ( ) 1 mark
( ) ( ) 1 mark

( ) ( )
b. 1 mark

c. ∫ ( ( ) ) 1 mark

( )
( ) 1 mark

( ) 1 mark

Question 7

a. ( )
1 mark

( )
b. ( | ) 1 mark
( )

1 mark

© TSSM 2017 Page 4 of 5

2017 MATHEMATICAL METHODS EXAM 1

Question 8

a. ( ( )) ( ) ( )
1 mark

b. ( ) ∫
Use
( ) ( ( )) ∫

( ) ( ( )) ( ) 1 mark

( ) ( )
1 mark

c. ( )
∫
( ( )) ( )
( ) ( ) 1 mark

1 mark

© TSSM 2017 Page 5 of 5

 MATHEMATICAL METHODS
Written Examination 2

2017 Trial Examination

SOLUTIONS

TSSM 2017 Page 1 of 10

2017 MATHEMATICAL METHODS EXAM 2

SECTION 1

Question 1

Answer: C

Explanation:

Question 2

Answer: B

Explanation:

. /

Question 3

Answer: E

Explanation:

Question 4

Answer: A

Explanation:

, -

Question 5

Answer: C

Explanation:
( ) ( )

TSSM 2017 Page 2 of 10

2017 MATHEMATICAL METHODS EXAM 2

Question 6

Answer: B

Explanation:

Find inverse rule on CAS. , )

Question 7

Answer: E

Explanation:

( ) ( )

Question 8

Answer: D

Explanation:

Question 9

Answer: B

Explanation:

. ( )/ ( )

( ) ∫ ∫ ( )

Question 10

Answer: A

Explanation:

( ) ( )

TSSM 2017 Page 3 of 10

2017 MATHEMATICAL METHODS EXAM 2

Question 11

Answer: B

Explanation:

. /

Question 12

Answer: A

Explanation:

∫ ( )

∫ ( ( ))

Question 13

Answer: B

Explanation:

Sketch on CAS

Note that option A is incorrect because is outside the domain of the function.

Question 14

Answer: D

Explanation:

( ) ( ) ( )

TSSM 2017 Page 4 of 10

2017 MATHEMATICAL METHODS EXAM 2

Question 15

Answer: E

Explanation:

√
∫ ( ) ∫ ( )

Question 16

Answer: C

Explanation:

and

and

Substituting into ( ) ( ) gives

( ) ( )

Question 17

Answer: D

Explanation:

.̂ / ( ) ( )

Question 18

Answer: A

Explanation:

Check for solution that satisfies both equations-

TSSM 2017 Page 5 of 10

2017 MATHEMATICAL METHODS EXAM 2

Question 19

Answer: B

Explanation:

( )( )

Question 20

Answer: D

Explanation:

.√ ( ) / √

√
When u = 0 the distance is 1, when u = 0.5√ the distance is that is <1.

√
Minimal distance occurs when √ , distance =

TSSM 2017 Page 6 of 10

2017 MATHEMATICAL METHODS EXAM 2

SECTION 2

Question 1

a. 1 mark
, -
1 mark

b. ( ) . /
1 mark

c. ( ) (on CAS)
1 mark

d. . / 1 mark
( ( ) )
( ( ) )
1 mark

e. 1 mark

. ( )/ 1 mark
. /
1 mark

f.
2 marks

Question 2

a. ( ) ( )
2 marks

b. ∫ ( ) 1 mark
. / 1 mark

TSSM 2017 Page 7 of 10

2017 MATHEMATICAL METHODS EXAM 2

c. 1 mark
( )
1 mark

d. ( )
2 marks

e. ∫ (( ) ( )) 1 mark

1 mark

f. 1 mark
1 mark
1 mark

g. Length = √ ( )
=√
1 mark

Question 3

a. ( ) (Use propfrac on CAS or long division)

1 mark for “2”
1 mark for “-7”
b. * +
1 mark

c. 1 mark
( )
1 mark

d. √
( √ √ ) ( √ √ )
2 marks

TSSM 2017 Page 8 of 10

2017 MATHEMATICAL METHODS EXAM 2

√
e. ∫ √
. /
1 mark for correct integral and 1 mark for correct terminal values

f.
i. , ) ( )
1 mark

ii. ( ( )) √ 1 mark
( √ √ ) ( √ √ )
1 mark

iii. √ . / 1 mark
( )
( ) 1 mark
Minimum distance = 0.22

1 mark

Question 4

a. ( )
2 marks

( )
b. ( ) 1 mark
( )

1 mark

c. ( )
2 marks

d. ( )
1 mark

1 mark

e. ( ) 1 mark
1 mark
1 mark

TSSM 2017 Page 9 of 10

2017 MATHEMATICAL METHODS EXAM 2

f. ( )
1 mark

g. (̂ ̂ ) 1 mark
( )
( ) ( )
2 marks

. / . /
h. ( √ √ ) ( )

2 marks

i.

i. ∫ ( ) 1 mark
∫ ( )
1 mark
ii. ∫ ( )

1 mark

TSSM 2017 Page 10 of 10