syllabus dot points ME

Question 1

understand the notion of a combination as a set of 𝑟 objects taken from a set of 𝑛 distinct objects

Answer 1

A

Question 2

use the notation (
𝑛
𝑟
) and the formula (
𝑛
𝑟
) =
𝑛!
𝑟!(𝑛−𝑟)!
for the number of combinations of 𝑟 objects
taken from a set of 𝑛 distinct objects

Answer 2

A

Question 3

investigate Pascal’s triangle and its properties to link (
𝑛
𝑟
) to the binomial coefficients of the
expansion of (𝑥 + 𝑦)
𝑛
for small positive integers

Answer 3

A

Question 4

review the concepts and language of outcomes, sample spaces, and events, as sets of outcomes

Answer 4

A

Question 5

use set language and notation for events, including:
a. 𝐴̅(or 𝐴
′
) for the complement of an event 𝐴
b. 𝐴 ∩ 𝐵 and 𝐴 ∪ 𝐵 for the intersection and union of events 𝐴 and 𝐵 respectively
c. 𝐴 ∩ 𝐵 ∩ 𝐶 and 𝐴 ∪ 𝐵 ∪ 𝐶 for the intersection and union of the three events 𝐴, 𝐵 and 𝐶
respectively
d. recognise mutually exclusive events.

Answer 5

A

Question 6

use everyday occurrences to illustrate set descriptions and representations of events and set
operations

Answer 6

A

Question 7

review probability as a measure of ‘the likelihood of occurrence’ of an event

Answer 7

A

Question 8

review the probability scale: 0 ≤ 𝑃(𝐴) ≤ 1 for each event 𝐴, with 𝑃(𝐴) = 0 if 𝐴 is an impossibility
and 𝑃(𝐴) = 1 if 𝐴 is a certainty

Answer 8

A

Question 9

review the rules: 𝑃(A̅) = 1 − 𝑃(𝐴) and 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 ∩ 𝐵)

Answer 9

use relative frequencies obtained from data as estimates of probabilities

Question 10

understand the notion of a conditional probability and recognise and use language that indicates
conditionality

Answer 10

1.1.12. use the notation 𝑃(𝐴|𝐵) and the formula 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴|𝐵)𝑃(𝐵)

1.1.13. understand the notion of independence of an event A from an event B, as defined by
𝑃(𝐴|𝐵) = 𝑃(𝐴)

Question 11

establish and use the formula 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴)𝑃(𝐵) for independent events 𝐴 and 𝐵, and
recognise the symmetry of independence

Answer 11

use relative frequencies obtained from data as estimates of conditional probabilities and as
indications of possible independence of events

Question 12

recognise features of the graph of 𝑦 = 𝑚𝑥 + 𝑐, including its linear nature, its intercepts and its slope
or gradient

Answer 12

determine the equation of a straight line given sufficient information; including parallel and
perpendicular lines

Question 13

examine examples of quadratically related variables

Answer 13

recognise features of the graphs of 𝑦 = 𝑥
2
, 𝑦 = 𝑎(𝑥 − 𝑏)
2 + 𝑐, and 𝑦 = 𝑎(𝑥 − 𝑏)(𝑥 − 𝑐), including
their parabolic nature, turning points, axes of symmetry and intercepts

Question 14

solve quadratic equations, including the use of quadratic formula and completing the square

Answer 14

determine the equation of a quadratic given sufficient information

Question 15

determine turning points and zeros of quadratics and understand the role of the discriminant

Answer 15

recognise features of the graph of the general quadratic 𝑦 = 𝑎𝑥
2 + 𝑏𝑥 + �

Question 16

examine examples of inverse proportion

Answer 16

recognise features and determine equations of the graphs of 𝑦 =
1
𝑥
and 𝑦 =
𝑎
𝑥−𝑏
, including their
hyperbolic shapes and their asymptotes.

Question 17

recognise features of the graphs of 𝑦 = 𝑥
𝑛
for 𝑛 ∈ 𝑵, 𝑛 = −1 and 𝑛 = ½, including shape, and
behaviour as 𝑥 → ∞ and 𝑥 → −∞

Answer 17

identify the coefficients and the degree of a polynomial

Question 18

expand quadratic and cubic polynomials from factors

Answer 18

recognise features and determine equations of the graphs of 𝑦 = 𝑥
3
, 𝑦 = 𝑎(𝑥 − 𝑏)
3 + 𝑐 and 𝑦 =
𝑘(𝑥 − 𝑎)(𝑥 − 𝑏)(𝑥 − 𝑐), including shape, intercepts and behaviour as 𝑥 → ∞ and 𝑥 → −∞

Question 19

factorise cubic polynomials in cases where all roots are given or easily obtained from the graph

Answer 19

solve cubic equations using technology, and algebraically in cases where all roots are given or easily
obtained from the graph

Question 20

recognise features and determine equations of the graphs of 𝑥
2 + 𝑦
2 = 𝑟
2
and
(𝑥 − 𝑎)
2 + (𝑦 − 𝑏)
2 = 𝑟
2
, including their circular shapes, their centres and their radii

Answer 20

recognise features of the graph of 𝑦
2 = 𝑥, including its parabolic shape and its axis of symmetry

Question 21

understand the concept of a function as a mapping between sets and as a rule or a formula that
defines one variable quantity in terms of another

Answer 21

use function notation; determine domain and range; recognise independent and dependent
variables

Question 22

understand the concept of the graph of a function

Answer 22

examine translations and the graphs of 𝑦 = 𝑓(𝑥) + 𝑎 and 𝑦 = 𝑓(𝑥 − 𝑏)

Question 23

examine dilations and the graphs of 𝑦 = 𝑐𝑓(𝑥) and 𝑦 = 𝑓(𝑑𝑥)

Answer 23

recognise the distinction between functions and relations and apply the vertical line test

Question 24

review sine, cosine and tangent as ratios of side lengths in right-angled triangles

Answer 24

understand the unit circle definition of cos 𝜃, sin 𝜃 and tan 𝜃 and periodicity using degrees

Question 25

examine the relationship between the angle of inclination of a line and the gradient of that line

Answer 25

establish and use the cosine and sine rules, including consideration of the ambiguous case and the formula 𝐴𝑟𝑒𝑎 = 1 2 𝑏𝑐 𝑠𝑖𝑛 𝐴 for the area of a triangle

Question 26

examine the relationship between the angle of inclination of a line and the gradient of that line

Answer 26

establish and use the cosine and sine rules, including consideration of the ambiguous case and the formula 𝐴𝑟𝑒𝑎 = 1 2 𝑏𝑐 𝑠𝑖𝑛 𝐴 for the area of a triangle

Question 27

understand the unit circle definition of sin 𝜃, cos 𝜃 and tan 𝜃 and periodicity using radians

Answer 27

recognise the exact values of sin 𝜃, cos 𝜃 and tan 𝜃 at integer multiples of 𝜋 6 and 𝜋 4

Question 28

recognise the graphs of 𝑦 = sin 𝑥, 𝑦 = cos 𝑥 , and 𝑦 = tan 𝑥 on extended domains

Answer 28

examine amplitude changes and the graphs of 𝑦 = 𝑎 sin 𝑥 and 𝑦 = 𝑎 cos 𝑥

Question 29

examine period changes and the graphs of 𝑦 = sin 𝑏𝑥, 𝑦 = cos 𝑏𝑥 and 𝑦 = tan 𝑏𝑥

Answer 29

examine phase changes and the graphs of 𝑦 = sin(𝑥 − 𝑐), 𝑦 = cos(𝑥 − 𝑐) and 𝑦 = tan (𝑥 − 𝑐)

Question 30

examine the relationships sin (𝑥 + 𝜋 2 ) = cos 𝑥 and cos (𝑥 − 𝜋 2 ) = sin 𝑥

Answer 30

prove and apply the angle sum and difference identities

Question 31

identify contexts suitable for modelling by trigonometric functions and use them to solve practical problems

Answer 31

solve equations involving trigonometric functions using technology, and algebraically in simple cases