Year 11 & 12 Functions & Relations Tests

Please note, these exam style questions are the same questions that appear under the topics but put into a random order.

This is perfect to prepare you for a test or exam and will help you revise the whole of Functions & Relations (NOT including Logs & Exponentials and Circular Functions). If you would like to do exam questions on a specific topic (such as "Linear Equations"), select it on the left.

5 MINUTE FUNCTIONS TESTS (7 Tests)

10 MINUTE FUNCTIONS TESTS (3 Tests)

15 MINUTE FUNCTIONS TESTS (1 Test)

Click the question to reveal the solution

5 Minute Functions Tests

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Question 1 [2024 Exam 2 Section A Q1]

The asymptote(s) of the graph of $ y = \log_e(x + 1) - 3 $ are

a. $ x = -1 $ only
b. $ x = 1 $ only
c. $ y = -3 $ only
d. $ x = -1 $ and $ y = -3 $

2024 VCAA Maths Methods Exam 2: Test 1

Marks: 16 Reading: 1 minute Writing: 24 minutes

Section A – Multiple Choice (4 marks)

Question 1 [Exam 2 Section A Q1]

The asymptote(s) of the graph of $y = \log_{e} (x + 1) - 3$ are

A. $x = - 1$ only
B. $x = 1$ only
C. $y = - 3$ only
D. $x = - 1$ and $y = - 3$

Question 2 [2024 Exam 2 Section A Q2]

A function $g : R \to R$ has the derivative $g^{'} (x) = x^{3} - x$ . Given that $g (0) = 5$ , the value of $g (2)$ is

A. 2
B. 3
C. 5
D. 7

Question 3 [Exam 2 Section A Q3]

A discrete random variable $X$ is defined using the probability distribution below, where $k$ is a positive real number.

x	0	1	2	3	4
$P (X = x)$	2k	3k	5k	3k	2k

Find $Pr (X < 4 ∣ X > 1)$ :

A. $\frac{13}{15}$
B. $\frac{11}{13}$
C. $\frac{4}{5}$
D. $\frac{8}{15}$

Question 4 [Exam 2 Section A Q4]

If $\int_{a}^{b} f (x) d x = - 5$ and $\int_{b}^{c} f (x) d x = 3$ , where $a < b < c$ , then $\int_{a}^{c} 2 f (x) d x$ is equal to

A. –16
B. 16
C. –4
D. 4

Section B – Short Answer / Extended Response (12 marks)

Question 1 [Exam 2 Section B Q1]

Let $f (x) = (x + 1) (x + a) (x - 2) (x - 2 a)$ , where $a \in R$

a. State, in terms of $a$ , the values of $x$ for which $f (x) = 0$ . (1 mark)

b. Find the values of $a$ for which the graph of $y = f (x)$ has:

i. exactly three x-intercepts (2 marks)

ii. exactly four x-intercepts (1 mark)

c. Let $g (x) = (x + 1)^{2} (x - 2)^{2}$ , which is the function $f$ where $a = 1$

i. Find $g^{'} (x)$ (1 mark)

ii. Find the coordinates of the local maximum of $g$ (1 mark)

iii. Find the values of $x$ for which $g^{'} (x) > 0$ (1 mark)

iv. Consider the two tangent lines to the graph of $y = g (x)$ at the points where $x = \frac{- \sqrt{3} + 1}{2}$ and $x = \frac{\sqrt{3} + 1}{2}$ . Determine the coordinates of the point of intersection of these two tangent lines. (2 marks)

d. Let $g (x) = (x + 1)^{2} (x - 2)^{2}$ and define a second function $h (x) = (x + 1) (x - 1) (x + 2) (x - 2)$ , which is the function $f$ where $a = - 1$

i. Using translations only, describe a sequence of transformations of $h$ , for which its image would have a local maximum at the same coordinates as that of $g$ . (1 mark)

ii. Using a dilation and translations, describe a different sequence of transformations of $h$ , for which its image would have both local minimums at the same coordinates as that of $g$ . (2 marks)

END OF QUESTIONS

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2024 VCAA Maths Methods Exam 2: Test 2

Marks: 16 Reading: 1 minute Writing: 24 minutes

Section A – Multiple Choice (4 marks)

Question 1 [Exam 2 Section A Q5]

Consider the functions $f : (1, \infty) \to R, f (x) = x^{2} - 4 x$ and $g : R \to R, g (x) = e^{- x}$
The range of the composite function $g (f (x))$ is

A. $(0, e^{3})$
B. $(0, e^{3}]$
C. $(0, e^{4})$
D. $(0, e^{4}]$

Question 2 [Exam 2 Section A Q6]

Consider the function $f (x) = \frac{2 x + 1}{3 - x}$ with domain $x \in R ∖ {3}$ .
The inverse of $f$ is

A. $f^{- 1} (x) = \frac{3 x - 1}{x + 2}$ with domain $x \in R ∖ {3}$
B. $f^{- 1} (x) = \frac{3 - x}{x + 2}$ with domain $x \in R ∖ {- 2}$
C. $f^{- 1} (x) = \frac{3 + x}{x + 2}$ with domain $x \in R ∖ {- 2}$
D. $f^{- 1} (x) = \frac{1 - 3 x}{x + 2}$ with domain $x \in R ∖ {- 2}$

Question 3 [2024 Exam 2 Section A Q7]

A fair six-sided die is repeatedly rolled. What is the minimum number of rolls required so that the probability of rolling a six at least once is greater than 0.95?

A. 15
B. 16
C. 17
D. 18

Question 4 [Exam 2 Section A Q8]

Some values of the functions $f : R \to R$ and $g : R \to R$ are shown below.

$x : 1 2 3$
$f (x) : 0 4 5$
$g (x) : 3 4 - 5$

The graph of the function $h (x) = f (x) - g (x)$ must have an x-intercept at

A. (2, 0)
B. (3, 0)
C. (4, 0)
D. (5, 0)

Section B – Short Answer / Extended Response (12 marks)

Question 1 [Exam 2 Section B Q2]

A model for the temperature in a room, in degrees Celsius, is given by

$f (t) = \frac{12 + 30 t}{22}$ , where $t > \frac{1}{3}$

a. Express the derivative $f^{'} (t)$ as a hybrid function.

b. Find the average rate of change in temperature predicted by the model between $t = 0$ and $t = \frac{1}{2}$ .
Give your answer in degrees Celsius per hour.

c. Another model for the temperature in the room is given by $g (t) = 22 - 10 e^{- 6 t}$ , $t \geq 0$ .

i. Find the derivative $g^{'} (t)$ .
ii. Find the value of $t$ for which $g^{'} (t) = 10$ , correct to three decimal places.

d. Find the time $t \in (0, 1)$ when the temperatures predicted by the models $f$ and $g$ are equal.
Give your answer correct to two decimal places.

e. Find the time $t \in (0, 1)$ when the difference between the temperatures predicted by the two models is the greatest.
Give your answer correct to two decimal places.

f. The amount of power, in kilowatts, used by the heater $t$ hours after it is switched on, can be modelled by the continuous function $p$ , given by

$p (t) = {\begin{cases} 0.3 + A e^{- 10 t} & 0 \leq t \leq 0.4 \\ 1.5 & t > 0.4 \end{cases}$

i. Given that $p (t)$ is continuous for $t \geq 0$ , show that $A = 1.2 e^{4}$ .
ii. Find how long it takes until the heater has used 0.5 kilowatt hours of energy. Give your answer in hours.
iii. Find how long it takes until the heater has used 1 kilowatt hour of energy. Give your answer in hours, correct to two decimal places.

END OF QUESTIONS

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MathsMethods.com.au
FUNCTIONS TEST 3

Number of marks: 4

Reading time: 45 seconds

Writing time: 6 minutes

Read this carefully: Do the following questions under timed conditions.

You may not be able to answer any of the questions, that's totally fine. These exam questions are really difficult and most students struggle with them.

Once you've finished, look at the answers and then mark yourself. After that, work out how to do the questions that you couldn't answer.

Now complete these questions again (under timed conditions and without looking at the answers). Then mark yourself again. Keep doing the questions over and over until you feel happy with the mark you got.

SECTION A – Calculator Allowed

Maths-Methods-Detailed-Exam-Questions-Martix-Transform-1-1

Maths-Methods-Detailed-Exam-Questions-Intercept-1-2

Maths-Methods-Detailed-Exam-Questions-Martix-Transform-1-4

END OF QUESTIONS

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MathsMethods.com.au
FUNCTIONS TEST 4

Number of marks: 4

Reading time: 45 seconds

Writing time: 6 minutes

Read this carefully: Do the following questions under timed conditions.

You may not be able to answer any of the questions, that's totally fine. These exam questions are really difficult and most students struggle with them.

Once you've finished, look at the answers and then mark yourself. After that, work out how to do the questions that you couldn't answer.

SECTION A – Calculator Allowed

Maths-Methods-Detailed-Exam-Questions-Inverse-1-1

Maths-Methods-Exam-Questions-SketchingTP-1-2

3 marks

END OF QUESTIONS

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MathsMethods.com.au
FUNCTIONS TEST 5

Number of marks: 4

Reading time: 45 seconds

Writing time: 6 minutes

Read this carefully: Do the following questions under timed conditions.

You may not be able to answer any of the questions, that's totally fine. These exam questions are really difficult and most students struggle with them.

Once you've finished, look at the answers and then mark yourself. After that, work out how to do the questions that you couldn't answer.

SECTION A – Calculator Allowed

Maths-Methods-Detailed-Exam-Questions-Quadratic-1-1

SECTION B – No Calculator

Maths-Methods-Exam-Questions-SketchingTP-1-1

3 marks

END OF QUESTIONS

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MathsMethods.com.au
FUNCTIONS TEST 6

Number of marks: 5

Reading time: 1 minute

Writing time: 7.5 minutes

Read this carefully: Do the following questions under timed conditions.

You may not be able to answer any of the questions, that's totally fine. These exam questions are really difficult and most students struggle with them.

Once you've finished, look at the answers and then mark yourself. After that, work out how to do the questions that you couldn't answer.

SECTION A – Calculator Allowed

Maths-Methods-Detailed-Exam-Questions-Domain-1-2

Maths-Methods-Detailed-Exam-Questions-Inverse-1-3

Maths-Methods-Detailed-Exam-Questions-Domain-1-4

Maths-Methods-Detailed-Exam-Questions-Inverse-1-5

END OF QUESTIONS

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MathsMethods.com.au
FUNCTIONS TEST 7

Number of marks: 5

Reading time: 1 minute

Writing time: 7.5 minutes

Read this carefully: Do the following questions under timed conditions.

You may not be able to answer any of the questions, that's totally fine. These exam questions are really difficult and most students struggle with them.

Once you've finished, look at the answers and then mark yourself. After that, work out how to do the questions that you couldn't answer.

SECTION A – Calculator Allowed

Maths-Methods-Detailed-Exam-Questions-Polynomials-1-1

Maths-Methods-Detailed-Exam-Questions-Transform-1-3

Maths-Methods-Detailed-Exam-Questions-Inverse-1-4

Maths-Methods-Detailed-Exam-Questions-Domain-1-5

END OF QUESTIONS

10 Minute Functions Tests

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MathsMethods.com.au
FUNCTIONS TEST 8

Number of marks: 6 (2 Year 11)

Reading time: 70 seconds (20s Year 11)

Writing time: 9 minutes (3m Year 11)

Read this carefully: Do the following questions under timed conditions.

You may not be able to answer any of the questions, that's totally fine. These exam questions are really difficult and most students struggle with them.

Once you've finished, look at the answers and then mark yourself. After that, work out how to do the questions that you couldn't answer.

SECTION A – Calculator Allowed

Maths-Methods-Detailed-Exam-Questions-Martix-Transform-1-2

SECTION B – No Calculator (YEAR 12 ONLY)

Maths-Methods-Detailed-Exam-Questions-Composite-2-1

Maths-Methods-Detailed-Exam-Questions-Composite-2-2

END OF QUESTIONS

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MathsMethods.com.au
FUNCTIONS TEST 9

Number of marks: 6

Reading time: 1 minute

Writing time: 9 minutes

Read this carefully: Do the following questions under timed conditions.

You may not be able to answer any of the questions, that's totally fine. These exam questions are really difficult and most students struggle with them.

Once you've finished, look at the answers and then mark yourself. After that, work out how to do the questions that you couldn't answer.

SECTION A – Calculator Allowed

Maths-Methods-Exam-Questions-Functions-1-1

Maths-Methods-Detailed-Exam-Questions-Polynomials-2-1

Maths-Methods-Detailed-Exam-Questions-Martix-Transform-1-3

SECTION B – No Calculator

Maths-Methods-Detailed-Exam-Questions-Intercept-1-3

END OF QUESTIONS

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MathsMethods.com.au
FUNCTIONS TEST 10

Number of marks: 8 (3 Year 11)

Reading time: 90 seconds (30s Year 11)

Writing time: 12 minutes (4.5m Year 11)

Read this carefully: Do the following questions under timed conditions.

You may not be able to answer any of the questions, that's totally fine. These exam questions are really difficult and most students struggle with them.

Once you've finished, look at the answers and then mark yourself. After that, work out how to do the questions that you couldn't answer.

SECTION A – Calculator Allowed

Maths-Methods-Detailed-Exam-Questions-Transform-1-1

Maths-Methods-Detailed-Exam-Questions-Inverse-1-2

SECTION B – No Calculator (YEAR 12 ONLY)

Maths-Methods-Detailed-Exam-Questions-Composite-3-1

Maths-Methods-Detailed-Exam-Questions-Composite-3-2

Maths-Methods-Detailed-Exam-Questions-Composite-3-3

Maths-Methods-Detailed-Exam-Questions-Composite-3-4

END OF QUESTIONS

15 Minute Functions Test

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MathsMethods.com.au
FUNCTIONS TEST 11

Number of marks: 10

Reading time: 110 seconds

Writing time: 15 minutes

Read this carefully: Do the following questions under timed conditions.

You may not be able to answer any of the questions, that's totally fine. These exam questions are really difficult and most students struggle with them.

Once you've finished, look at the answers and then mark yourself. After that, work out how to do the questions that you couldn't answer.