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Mathematical Methods 2019 v1.
Units 1 and 2 sample assessment instrument
September 2018
Examination โ short response
This sample has been compiled by the QCAA to assist and support teachers in planning and
developing assessment instruments for individual school settings.
Schools develop internal assessments for each senior subject, based on the learning described
in Units 1 and 2 of the subject syllabus. Each unit objective must be assessed at least once.
Assessment objectives
This assessment instrument is used to determine student achievement in the following
objectives:
1. select, recall and use facts, rules, definitions and procedures drawn from Units 1 and 2
2. comprehend mathematical concepts and techniques drawn from Units 1 and 2
3. communicate using mathematical, statistical and everyday language and conventions
4. evaluate the reasonableness of solutions
5. justify procedures and decisions by explaining mathematical reasoning
6. solve problems by applying mathematical concepts and techniques drawn from Units 1
and 2.
Mathematical Methods 2019 v1.
Units 1 and 2 sample assessment instrument
Queensland Curriculum & Assessment Authority
September 2018
Subject Mathematical Methods
Technique Examination
Unit 1: Algebra, statistics and functions
2: Calculus and further functions
Topic Representatively sample subject matter from topics in Units 1 and 2
Conditions
Response
type
Short response
Time Paper 1 (technology-free): 50 minutes
Paper 2 (technology-active): 70 minutes
Perusal 5 minutes (Paper 2)
Other
- QCAA formula sheet must be provided
- Notes are not permitted
- Approved non-CAS graphics calculator
Instructions
- Show all working in the spaces provided.
- Write responses using black or blue pen.
- Unless otherwise instructed, give answers to two decimal places.
- Use of a non-CAS graphics calculator is permitted in Paper 2 (technology-active) only.
- A scientific calculator may also be used in Paper 2 (technology-active).
Feedback
Mathematical Methods 2019 v1.
Units 1 and 2 sample assessment instrument
Queensland Curriculum & Assessment Authority
September 2018
Paper 1 (technology-free) โ total marks: 55
Question 3 (11 marks)
Determine the derivatives of the following functions. Do not simplify.
a. 2 ๐ฅ
4
2
5
๐ฅ
b. (๐ฅ
3
5
c. ( 2 ๐ฅ
2
d.
1 โ๐ฅ
๐ฅ
3
โ 3
Mathematical Methods 2019 v1.
Units 1 and 2 sample assessment instrument
Queensland Curriculum & Assessment Authority
September 2018
Paper 1 (technology-free) โ total marks: 55
Question 4 (4 marks)
Part of the graph of ๐
3
2
is shown below.
The point ๐ lies on the graph of ๐
. At point ๐, ๐ฅ = 1.
The graph of ๐
has a gradient of 3 at point ๐. Determine the value of ๐.
Question 5 (5 marks)
The following table shows the probability distribution of a discrete random variable ๐.
a. Determine the value of ๐.
b. Determine ๐ธ(๐).
Mathematical Methods 2019 v1.
Units 1 and 2 sample assessment instrument
Queensland Curriculum & Assessment Authority
September 2018
Paper 1 (technology-free) โ total marks: 55
Question 7 (6 marks)
Determine the function for the graph below.
Use mathematical reasoning to justify your response.
Mathematical Methods 2019 v1.
Units 1 and 2 sample assessment instrument
Queensland Curriculum & Assessment Authority
September 2018
Paper 1 (technology-free) โ total marks: 55
Question 8 (7 marks)
Raoul and Alice play two games of tennis. The probability that Raoul wins the first game is
7
9
If Raoul wins the first game, the probability that he wins the second game is
5
7
If Raoul loses the first game, the probability that he wins the second game is
2
3
a. Complete the following tree diagram.
b. Calculate the probability that Raoul wins the first game and Alice wins the second game.
c. Given that Raoul wins at least one game, calculate the probability that he wins both games.
Raoul wins
Raoul wins
Alice wins
Alice wins
Raoul wins
Alice wins
First game
Second game
Mathematical Methods 2019 v1.
Units 1 and 2 sample assessment instrument
Queensland Curriculum & Assessment Authority
September 2018
Paper 2 (technology-active) โ total marks: 42
Question 1 (2 marks)
Let ๐ถ and ๐ท be independent events, with ๐
= 2 ๐ and ๐
2
, where 0 < ๐ < 0. 5.
Calculate ๐ if ๐(๐ถ โฉ ๐ท) = 0. 162.
Question 2 (9 marks)
The running cost of an electric bike is ( 2 +. 001 ๐ฃ
3
) dollars per hour when the speed is ๐ฃ kilometres
per hour.
a. Show that the cost ๐ถ of a trip of ๐ kilometres is given by ๐ถ = 2 ๐๐ฃ
โ 1
2
b. Verify that the speed that minimises the running cost is independent of ๐.
Use mathematical reasoning to justify your response.
Mathematical Methods 2019 v1.
Units 1 and 2 sample assessment instrument
Queensland Curriculum & Assessment Authority
September 2018
Paper 2 (technology-active) โ total marks: 42
Question 3 (4 marks)
Consider the following income statistics for businesses in a particular suburb.
Yearโs earnings Percentage
$40 000 loss 10%
$40 000 profit 45%
$200 000 profit 10%
Determine the expected value and the standard deviation.
Mathematical Methods 2019 v1.
Units 1 and 2 sample assessment instrument
Queensland Curriculum & Assessment Authority
September 2018
Paper 2 (technology-active) โ total marks: 42
Question 5 (6 marks)
The following diagram shows part of the graph of ๐(๐ฅ) = โ
10
27
3
5
9
2
20
9
19
27
The graph of ๐ is translated to the graph of ๐ by a movement of ๐ units vertically.
Determine all the values of ๐ so that ๐
= 0 has exactly one solution.
Mathematical Methods 2019 v1.
Units 1 and 2 sample assessment instrument
Queensland Curriculum & Assessment Authority
September 2018
Paper 2 (technology-active) โ total marks: 42
Question 6 (9 marks)
The function ๐(๐ฅ) has a stationary point at
Given that ๐(๐ฅ) = (๐ฅ + ๐)(๐ฅ
2
- ๐), where ๐ and ๐ are constant, determine the values of ๐ and ๐.
Evaluate the reasonableness of the solution and include a written record of the method that was used.
Mathematical Methods 2019 v1.
Units 1 and 2 sample assessment instrument
Queensland Curriculum & Assessment Authority
September 2018
Student results summary
Paper 1
(technology-free)
Simple familiar
(SF)
Complex
familiar (CF)
Complex unfamiliar (CU)
Totals 45 4 6
Paper 2
(technology-
active)
Simple familiar
(SF)
Complex familiar (CF) Complex unfamiliar (CU)
Totals 14 15 13
