Grade 10 Mathematics Platinum Navigation Pack
Grade 10 Mathematics Platinum Navigation Pack
Grade 10 Mathematics Platinum Navigation Pack
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Acknowldgements
MyPedia Assessments Maths Grade 10, Calvin Dube, (Ed) 2019. Reprinted by permission of Pearson South Africa
Pty (Ltd)
At Pearson South Africa, we believe that education is the key to every individuals’ success.
To ensure that despite the challenges, teachers and learners can meet all the necessary
learning outcomes for the year, we have created the Navigation Guide, a free resource to
support teachers and learners during this challenging time.
The Navigation Pack aims to summarise and highlight the changes in the 2021 DBE ATP
and provide teachers and learners with worksheets that focus on impacted topics in the
curriculum.
Due to resequencing of topics, the order of topics in the textbook that is currently used
in the classroom may not be aligned to the new sequence of topics in the ATP. Pearson
has included page numbers from one of our tried and tested series, Platinum, to guide
the teacher and learners as they navigate through the textbook, with the 2021 ATP. The
Navigation Pack has a set of assessments based on the Section 4 changes and the revised
assessment guidelines.
Introduction 5
COVID-19 safety guidelines for teachers and learners
Gatherings at school
Where schools are open for learning, it is up to management to take decisive action to
ensure sites are not simultaneously used for other functions such as shelters or treatment
units in order to reduce the risk.
Implement social distancing practices that may include:
• A staggered timetable, where teachers and learners do not arrive/leave at the same
time for the beginning and end of the school day.
• Cancelling any community meetings/events such as assemblies, cake sales, market d y,
tuckshop, after-care classes, matric dance, Eisteddfod and other events.
• Cancelling any extra-mural activities such as ballet classes, swimming lessons, sport
games, music class and other events that create a crowd gathering.
• Teaching and modeling creating space and avoiding unnecessary touching.
• Limiting movement and interaction between classes.
• Schools with an established feeding scheme plan are to ensure that hygiene and social
distancing is always implemented. Teachers and staff members assisting with food
• distribution are to wear masks, sanitise prior to issuing food items and learners are to
stand 1,5m apart in the queue.
1. Restrooms/toilets
Hand washing
• eating
• entering the classroom
• using the toilet
• blowing your nose or coughing
• touching tears, mucous, saliva, blood or sweat.
When schools open, classroom settings should be altered in order to promote hygiene,
safety and social distancing.
Social distancing
Introduction 7
3. Social behaviour
It is extremely vital during a pandemic that focus is not only directed towards optimal
physical health and hygiene but fi nding ways to facilitate mental health support.
*11
MEASUREMENT Unit 1: Volume and Revise the volume and surface areas of 2 weeks Platinum LB Page 268 – 275
surface area of right-prisms and cylinders. Platinum TG Page 195 – 196
shapes Study the effect on volume and surface
area when multiplying any dimension by a
constant factor k .
Unit 2: Volume and Calculate the volume and surface areas of Platinum LB Page 276 – 279
surface area of spheres, right pyramids, right-cones and a Platinum TG Page 202 – 205
combined shapes combination of those objects (figures).
TOTAL WEEKS = 10
*11
Measurement has moved from Term 3 to Term 2.
Mathematics**1
Euclidean Geometry 3
TERM 2 Trigonometry 2
Number Patterns 1
Measurement 2
Statistics 2
Probability 2
Analytical geometry 2
Programme of Assessment**2
TERM 1 TERM 2 TERM 3 TERM 4
**1
No important aspect in Mathematics curriculum is compromised.
**2
The amended School Based Assessment (SBA) is aligned to the content and time available. Informal tasks
and activities should be used as assessment for learning, to prepare for formal assessment.
*1
LB is Learner’s Book
*2
TG is Teacher’s Guide
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TOPIC UNIT CONTENT SPECIFIC CONCEPTS TIME LINKS TOPLATINUM SERIES AND PAGE REFERENCE
PEARSON NAVIGATION PACK
Term 1
EXPONENTS, Unit 8: Revise the Revise laws of exponents learnt in 3 weeks Platinum LB Page 32 – 40
EQUATIONS, laws of exponents Grade 9 where: Platinum TG Page 23 – 26
AND x, y > 0; m, n ∈ ℤ
INEQUALITIES*3 x m × x n= x m+n
x m ÷ x n= x m−n
(x m) n= x mn
x m × y m= (xy) m
Also, by definition: x −m= _ 1 ; x ≠ 0and
x m
0
x = 1; x ≠ 0.
*3
Exponents, equations and inequalities are covered as one topic in Term 1.
REVISED DBE ANNUAL TEACHING PLAN NAVIGATION PLAN
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REVISION Platinum TG: Worksheet A (solutions) Page 247 (279)
Platinum TG: Worksheet B (solutions) Page 248 (280)
EUCLIDEAN Unit 1: Geometry: Investigate and form conjectures about 3 weeks Platinum LB Page 162 – 169
GEOMETRY *4 Revision from earlier the properties of special triangles, Platinum TG Page 119 – 120
grades quadrilaterals and other polygons. Try
to validate or prove conjectures using
any logical method (Euclidean, co-
ordinate or transformation geometry
from Grade 9).
Unit 2: Conjectures*5 Disprove false conjectures by producing Platinum LB Page 170 – 177
counter-examples. Platinum TG Page 122 – 128
Unit 3: Investigate Investigate alternative definitions Platinum LB Page 178 – 179
special quadrilaterals of various polygons (including the Platinum TG Page 129
isosceles, equilateral and right-angled
triangle, the kite, parallelogram,
rectangle, rhombus, square and
trapezium).
REVISION Platinum LB: Topic Revision Page 180 – 181
Platinum TG: Topic Revision Memorandum Page 129 – 133
Platinum TG: Worksheet A (solutions) Page 253 (288 – 289)
Platinum TG: Worksheet B (solutions) Page 254 (290 – 293)
ASSESSMENTS Task 1: Investigation or project
Task 2: Test Navigation Pack: Test Exemplar Page 22 – 24
TOTAL WEEKS = 10
*4
Euclidean Geometry has been moved from Term 2 to Term 1. The other section of this topic will be covered in Term 4.
*5
Learners need to be guided on explaining their mathematics reasoning when working with conjectures.
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Term 2
TRIGONOMETRY*6 Trigonometric Extend the definitions of s in θ, cos θ and 2 weeks Platinum LB Page 84 – 106
definitions tan θfor 0
° < θ < 360°. Platinum TG Page 60 – 73
Effect of parameters Study the effect of a and q
on the Platinum LB Page 137 – 143
aand q on graphs defined by: Platinum TG Page 100 – 104
trigonometric graphs y = a sin θ + q,
y = a cos θ + q,
y = a tan θ + q, where aand q ∈ ℚ and
θ ∈ [0°; 360° ].
Sketch, use and Sketch graphs, find the equations of Platinum LB Page 144 – 157
interpret graphs*7 given graphs and interpret graphs. Platinum TG Page 105 – 112
Note: Sketching of graphs must be
based on the observation of the effect
of aand q
.
ASSESSMENTS Navigation Pack: Targeted worksheet 1 Page 22 – 24
Page 34 – 35
NUMBER Investigate number Patterns: Investigate number patterns 1 week Platinum LB Page 48 – 51
PATTERNS*8 patterns leading to those where there is Platinum TG Page 32 – 33
a constant difference between
consecutive terms, and the general
term (without using a formula – see
content overview) is therefore linear.
REVISION Platinum LB: Topic Revision Page 52 – 54
Platinum TG: Topic Revision Memorandum Page 33 – 35
Platinum TG: Worksheet A (solutions) Page 245 (275 – 276)
Platinum TG: Worksheet B (solutions) Page 246 (277 – 278)
*6
Trigonometry has been moved from Term 1 to Term 2.
*7
Use the knowledge of functions to help learners understand this unit.
*8
Number patterns has moved from Term 1 to Term 2.
REVISED DBE ANNUAL TEACHING PLAN NAVIGATION PLAN
TOPIC UNIT CONTENT – SPECIFIC CONCEPTS TIME LINKS TOPLATINUM SERIES AND PAGE REFERENCE
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FUNCTIONS *9 Unit 1: The concept The concept of a function, where a 5 weeks Platinum LB Page 116 – 120
(INCLUDING of a function certain quantity (output value) uniquely Platinum TG Page 81 – 84
TRIGONOMETRIC depends on another quantity (input
FUNCTIONS value). Work with relationships between
variables using tables, graphs, words
and formulae. Convert flexibly between
these representations.
Unit 2: Plot basic Point by point plotting of basic graphs Platinum LB Page 121 – 124
graphs and defined by: Platinum TG Page 81 – 84
investigate the effect x(where b
y = x 2; y = _1x and y = b > 0
of aand q
on the and b ≠ 1)
graphs to discover shape, domain (input
values), range (output values),
asymptotes, axes of symmetry, turning
points and intercepts on the axes
(where applicable). Sketch graphs, find
the equations of given graphs and
interpret graphs.
Investigate the effect of a and qon the Platinum LB Page 125 – 136
graphs defined by: Platinum TG Page 84 – 100
y = af (x) + q, where f (x) = x, f (x) = x2,
f (x) = bx, b > 0, b ≠ 1.
Sketch graphs, find the equations of
given graphs and interpret graphs.*10
ASSESSMENTS Navigation Pack: Targeted worksheet 2 Page 36 – 37
*9
Assess and use learners’ prior knowledge to cover the topic. Use assessment for learning activities to provide learners with opportunities to practise.
*10
Note: Sketching of the graphs must be based on the observation of the effect of a and q .
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Term 2
MEASUREMENT*11 Unit 1: Volume and Revise the volume and surface areas of 2 weeks Platinum LB Page 268 – 275
surface area of right-prisms and cylinders. Platinum TG Page 195 – 196
shapes Study the effect on volume and surface
area when multiplying any dimension by
a constant factor k .
Unit 2: Volume and Calculate the volume and surface areas Platinum LB Page 276 – 279
surface area of of spheres, right pyramids, right-cones Platinum TG Page 202 – 205
combined shapes and a combination of those objects
(figures).
TOTAL WEEKS = 10
*11
Measurement has moved from Term 3 to Term 2.
REVISED DBE ANNUAL TEACHING PLAN NAVIGATION PLAN
TOPIC UNIT CONTENT – SPECIFIC CONCEPTS TIME LINKS TOPLATINUM SERIES AND PAGE REFERENCE
PEARSON NAVIGATION PACK
TRIGONOMETRY Unit 1: Trigonometric Revision: Trigonometric problems with Self-study Platinum LB Page 254 – 258
(2D) problems with geometric properties. 2 weeks Platinum TG Page 185 – 188
geometric properties
Unit 2: Problems in Solve two dimensional Problems Platinum LB Page 259 – 264
two dimensions involving right-angled triangles. Platinum TG Page 188 – 191
REVISION Platinum LB: Topic Revision Page 265 – 266
Platinum TG: Topic Revision Memorandum Page 191 – 194
Platinum TG: Worksheet A (solutions) Page 263 (310 – 311)
Platinum TG: Worksheet B (solutions) Page 264 (312 – 313)
STATISTICS Unit 1: Measures of Revise measures of central tendency in 2 weeks Platinum LB Page 224 – 226
central tendency: ungrouped data. Platinum TG Page 163
mean, median and Measures of central tendency in Platinum LB Page 226 – 230
mode grouped data: calculation of mean Platinum TG Page 163 – 165
estimate of grouped and ungrouped
data, and identification of modal interval
and interval in which the median lies.
Unit 2: Measures of Revision of range as a measure of Platinum LB Page231 – 235
dispersion: range, dispersion and extension to Platinum TG Page 165 – 167
percentiles, quartiles include percentiles, quartiles,
and interquartile and interquartile and semi-interquartile
semi-interquartile range.
range
Unit 3: Five-number Five-number summary (maximum, Platinum LB Page 236 – 237
summaries and box- minimum and quartiles) and box-and- Platinum TG Page 167 – 168
and-whisker plots whisker diagram.
Unit 4: Analysing Use the statistical summaries Platinum LB Page 238 – 241
and interpreting (measures of central tendency and Platinum TG Page 168 – 170
statistical summaries dispersion), and graphs to analyse and
of data make meaningful comments on the
context associated with the given data.
REVISION Platinum LB: Topic Revision Page 242 – 244
Platinum TG: Topic Revision Memorandum Page 170 – 172
Platinum TG: Worksheet A (solutions) Page 261 (305 – 306)
Platinum TG: Worksheet B (solutions) Page 262 (307 – 308)
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Term 3
PROBABILITY*12 Unit 1: Probability The use of probability models to 2 weeks Platinum LB Page 288 – 293
models compare the relative frequency of Platinum TG Page 216 – 218
events with the theoretical probability.
Unit 2: Venn The use of Venn diagrams to solve Platinum LB Page 293 – 299
diagrams and probability problems, deriving and Platinum TG Page 218 – 219
symbols applying the following for any two
events in a sample space
S: P(Aor B)= P(A) + P(B) − P(Aand B);
Aand B are mutually exclusive if
*12
Probability has moved from Term 4 to Term 3. Give learners numerous activities to build their terminology.
REVISED DBE ANNUAL TEACHING PLAN NAVIGATION PLAN
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ANALYTICAL Represent geometric Derive and apply for any two points 2 weeks
GEOMETRY figures on a (x1; y1) and (x2; y2) and the formulae for
Cartesian coordinate calculating the:
system.
Unit 1: The distance Distance between the two points. Platinum LB Page 186 – 190
between two points Platinum TG Page 139 – 142
Unit 2: The gradient Gradient of the line segment connecting Platinum LB Page 191 – 195
of a line segment the two points (and from that identify Platinum TG Page 142 – 144
parallel and perpendicular lines).
Unit 3: The midpoint Coordinates of the midpoint of the line Platinum LB Page 196 – 202
of a line segment segment joining the two points. Platinum TG Page 145 – 149
REVISION Platinum LB: Topic Revision Page 203 – 204
Platinum TG: Topic Revision Memorandum Page 149 – 151
Platinum TG: Worksheet A (solutions) Page 255 (294 – 295)
Platinum TG: Worksheet B (solutions) Page 256 (296 – 297)
Navigation Pack: Targeted worksheet 3 Page 30 – 33
Page 38 – 40
ASSESSMENTS Task 3: Test 1 Navigation Pack: Term 3 Control Test 1 Page 45 – 47
Exemplar Page 66 – 67
Task 4: Test 2 Navigation Pack: Term 3 Control Test 2 Page 47 – 50
Exemplar Page 68 – 69
TOTAL WEEKS = 10
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Term 4
EUCLIDEAN Unit 1: Problems and Solve problems and prove riders using 2 weeks Platinum LB Page 251 – 252
GEOMETRY*13 proofs using the properties of parallel lines, triangles Platinum TG Page 174 – 181
geometric properties and quadrilaterals.
REVISION Platinum LB: Topic Revision Page 203 – 204
Platinum TG: Topic Revision Memorandum Page 181 – 183
Platinum TG: Worksheet A (solutions) Page 261 (305 – 306)
Platinum TG: Worksheet B (solutions) Page 262 (307 – 309)
ASSESSMENTS Task 5: Test Navigation Pack: Term 4 Control Test Page 51–53
Exemplar Page 70 – 71
TOTAL WEEKS = 2
*13
Topic has been moved from Term 3 to Term 4. No content trimmed.
Navigation
Targeted
Worksheets
Pack
Mathematics Grade 10
1 Trigonometry
2 Functions
3 Analytical Geometry
Targeted Worksheet 1
Topic: Trigonometry
Content summary
This worksheet focuses on:
Targeted Worksheet 1
Time: 60 minutes
Name: Surname:
Topic: Trigonometry
This worksheet consists of 3 questions.
Instructions
Read the following instructions carefully before answering the questions
1. Answer ALL the questions.
2. Clearly show ALL calculations.
3. You may use a non-programmable scientific calculator.
4. Write neatly and legibly.
θ
x
0
13
P(x; –12)
With the aid of the above sketch, determine the value of:
1.2.1 tan θ (3)
1.2.2 2 sin θ (2)
1.2.3 cos 2θ (2)
Targeted Worksheet 1
1.3 If 4 cos B − 5 = −2 and 0° < B < 90°, without the use of a calculator:
1.3.1 write down cos B = … (1)
Make use of a diagram and calculate:
1.3.2 tan B (1)
1.3.3 sin(90° − B) (2)
tan B
1.3.4 _
sin B
(4)
[23]
1_ cos 30° + cos 0° + sin 90°
2.2.2 _ (5)
√ 3
[13]
Targeted Worksheet 2
Content summary
The basic functions learners should know include:
Question 1
• Tests the learners’ ability to analyse different functions and determine the domain and range
of that function, when given the equation and the graph.
Question 2
• Tests the learners’ ability to apply input values, understanding of function notation, when
given the equation of certain graphs.
Question 3
• Tests the learners’ ability to answer functional questions based on observations of a given
graph. This question shows two intersecting graphs (Parabola and line), learners are expected
to analyse and interpret the graphs. In addition, learners should know what the x -values of
the graph are when the two graphs are equal (x-coordinates of the points of intersection)
Targeted Worksheet 2
Time: 60 minutes
Name: Surname:
Instructions
Read the following instructions carefully before answering the questions
1. Answer ALL the questions.
2. Clearly show ALL calculations.
3. You may use a non-programmable scientific calculator.
4. Write neatly and legibly.
Question 1
1.1 Consider the function: f (x) = 2x2 − 4.
y
x
– √2 0 √2
–4
Targeted Worksheet 2
3
1.2 The diagram shows the graph of the function: f (x) = −_x − 1.
y
x
0
–1
x
0 2
–6
–8
Question 2
2
(x) = _
2.1 Given: f (x) = 2x − 7, g and k (x) = x 2 − 2x + 7. Calculate the following:
x + 5
2.1.1 f (10) (2)
2.1.2 k(−2) ⋅ g(1) (4)
[6]
Targeted Worksheet 2
Question 3
3.1 Study the diagram below and answer the questions that follow.
y
g
(–1; 8)
f
x
–3 0 3
Question 4
4.1 Given: h(x) = 2x − 12.
4.1.1 Write down the equation of the asymptote of h. (1)
4.1.2 Sketch the graph of h
. (3)
4.1.3 If m
(x) = h(−x) − 3, describe the transformation to obtain the graph m
. (2)
4.1.4 Give the range of m(x). (2)
[8]
Targeted Worksheet 2
Question 5
x 2 + q. Study the graphs and
The sketch below shows the graphs of f (x) = −2x + 6 and g(x) = a
answer the questions that follow.
y
9
P
g
f
x
0 3
Question 6
k
Given: p(x) = _x + q
y
(–1; 2)
x
0
–2
Targeted Worksheet 3
Content summary
Questions 1 – 3 test learners’ ability to apply formulae for calculating the:
• Distance formulae
• Gradient of line between two points (and from that identify parallel and perpendicular lines)
• Coordinates of mid-point
Question 1
• Tests the learners’ ability to apply the distance formula. Finding the length of a line given two
points, as well as finding coordinate points when given a length between two points.
Question 2
• Tests the learners’ ability to find the midpoint of a line segment, given two points on the line.
Additionally, learners are expected to be able to calculate the endpoint of a line when given
the midpoint.
Question 3
• Tests the learners’ ability to calculate parallel and perpendicular lines, prove parallel and
perpendicular lines and determine coordinate points when given the gradient of collinear
points.
Questions 4 and 5
• Examination type questions - are integrated application questions involving distance, gradient
and midpoint of a line segment. Learners are expected to know, and are tested on the
properties of quadrilaterals and triangles.
Targeted Worksheet 3
Time: 60 minutes
Name: Surname:
Instructions
Read the following instructions carefully before answering the questions
1. Answer ALL the questions.
2. Clearly show ALL calculations.
3. You may use a non-programmable scientific calculator.
4. Write neatly and legibly.
Question 1
Lis drawn with K(−3; 2)and L(5; − 1).
1.1 In the diagram below, the line segment K
y
K(–3; 2)
x
0
L(5; –1)
Targeted Worksheet 3
Question 2
2.1 Given M(−2; −1)and L
(4; 5). Determine the coordinates of J, the midpoint of M
L. (3)
2.2 In the diagram below, K(4; 2)is the midpoint of F
P. Determine the coordinates of P
.
y (4)
P
K(4; 2)
x
0
F(–3; –4)
[7]
Question 3
3.1 If A(−4; 8), B(−10; 0), C(6; −12) and D(12; −4)prove that:
3.1.1 AD ∙ BC (5)
3.1.2 AB ⊥ BC (4)
3.2 In the diagram below, the points A , B, Cand D (−1; 3), B(2; 4), C(6; 0),
are given by A
D 1(a; 4) and D 2(b; 4).
y
B(2; 4)
D1
D2
A(–1; 3)
x
0 C(6; 0)
Targeted Worksheet 3
Question 4
A(1; 3), B(4; 1)and C
(6; 4)are three points in the Cartesian plane.
y
C(6, 4)
A(1, 3)
B(4, 1)
x
0
Question 5
5.1 C(−3; 3), D(−1; 2)and E
(x; −4)are vertices of a triangle shown below. C
D ⊥ DE.
y
C(–3, 3) D(–1, 2)
x
0
E(x, –4)
5 ✓
1.1.1 tan θ = _
12
Hypotenuse = 13 ✓
5 ✓
_
sin θ = 13
(3)
12
1.1.2 cos θ = _
✓✓
13
(2)
_ 12 5 17
1.1.3 sin θ ✓ + _
+ cos θ = 13 _
✓ = 13
13
✓ (3)
1.2.1 (13) − (−12) = (x)
2 2 2
(Theorem of Pythagoras) ✓
169 − 144 = (x)2
25 =(x)2
x = ± 5 units ✓
Choose x = − 5
tan θ = (
− 152 )
−
_ − 12
− 5 = 12
= _ _
5 ✓ (3)
√ ( 2 )
√ 3
1_ cos 30° + cos 0° + sin 90° = ____________
____________
2.2.3 √ 1_ ____________
✓+ 1 ✓+ 1 ✓ = _12 + 2 ✓= 2_21 ✓ (5)
3 3
[13]
_2
x =
3.1.1 tan 2,5
tan x = 5 ✓
x = 78,69° ✓
(2)
θ = sin −1(_35 )✓
θ = 36,87° ✓
(3)
3.2.2 3θ − 18° = tan −1(5)
3θ − 18° = 78,69° ✓
3θ = 96,69°
θ = 32,23° ✓✓
(3)
3.2.3 cos θ = 0,66 ✓
θ = cos −1 0,66 ✓
θ = 48,70° ✓
(3)
[14]
Total: [50]
x
0 2
–9
–12
x-intercepts ✓
asymptote ✓
y-intercept ✓
(3)
4.1.3 The graph is reflected through the y -axis ✓ and translated 3 units down ✓ (2)
4.1.4 y ✓ > −15 ✓ (2)
[8]
2.1 J(_ 2 )
x + x y + y
; _
1 2 2
1
2
J(_ )
(4) + (−2) (5) + (−1)
; _
2
2
✓
J(1; 2) ✓
(2)
2.2 M(_ 2 )
x + x y + y
; _
1 2 2
1
= (x; y)
2
(x; y)is the midpoint
_2 − 3 + x − 4 + y
= 4 ✓ and _
2
2
2
= 2 ✓
− 3 + x 2= 8and −
4 + y 2= 4
x 2= 11and y 2= 8
(11; 8) ✓(4)
P
[5]
3.1.1 Two lines are parallel if their gradients are equal, m 1= m
2.
y − y
m=_
2 1
x 2 − x 1
;
(− 4) − (8)
AD= (_
m
12) − (− 4 ✓ = − _34 ✓
)
(− 12) − (0)
BC= _
m
✓ = − 3_4 ✓
(6) − (− 10)
m
AD= m
BC
AD ∥ BC✓ (5)
3.1.2 Two lines are perpendicular if the product of their gradients is equal to –1.
(0) − (8)
AB= ___________
m
(− 10) − (− 4)
✓ = _43 ✓
BC= − _34
m
4 − 3
_
2 + 1
0 − 4
= _
6 − x
✓✓
D
1 × (6 − x D)= − 4 × 3 ✓
x D= 18 = a
D 1is the point (18; 4) ✓(4)
3.2.2 m
AB × m CD= − 1
y − y y − y
_
B A
× _
x B − x A
C D
x C − x D
= − 1
(_ ) × ( 6 − x D
)
4 − 3 0 − 4
_
2 + 1 ✓ = − 1 ✓
1
_
× − 4 = − 6 + x D ✓
3
x D= 4_23 = b
(1) − (− 1)
BD= _
4.4 m ✓ = − _23 ✓
(4) − (7)
AB= − _23
m
(proven in 4.3) ✓
m
AB= m
BD ✓, therefore A, Band D
are collinear. (4)
(3✓; 6 ✓)
4.5 G (2)
[12]
(3) − (2)
CD= (___________
5.1.1 m − 2) − (− ✓ = − _12 ✓
1)
(2)
5.1.2 m
CD × m DE= − 1, CD ⊥ DE
− _12 × (_
− 4 − 2
))
x − ( − 1 ✓ = −1 ✓
5.2 M(_ )
(k) + (5) (− 4) + y
; _
2
2
= (
2
− _32; y)✓
k + 5
_
= − _32 and _
2
− 4 + 10
2 = y ✓
k + 5 = − 3and 6
y = 2
k = − 8 ✓ and y = 3 ✓(4)
[10]
Total: [60]
Exemplar Assessments
Time: 1 hour
Name: Surname:
Instructions
Read the following instructions carefully before answering the questions.
1. This question paper consists of 4 questions.
2. Answer ALL the questions.
3. Clearly show ALL calculations, diagrams, graphs, etc. which you have used in
determining the answers.
4. Answers only will NOT necessarily be awarded full marks.
5. You may use an approved scientific calculator (non-programmable and non-graphical),
unless stated otherwise.
6. If necessary, round off answers to TWO decimal places, unless stated otherwise.
7. Diagrams are NOT necessarily drawn to scale.
8. Write neatly and legibly.
Question 1
1.1 A marquee has a rectangular base of 10 m by 4,3 m with an equilateral triangular cross-
section and rectangular sides as shown below.
4,3 m
h
10 m
Calculate the area of the canvas needed to make the tent excluding the floor. (5)
1.2 The floor of the tent is to be covered with a green carpet. If the cost of the carpet
is R25 per square metre, how much is required to do so? (4)
Exemplar Assessments
1.3 A bus stop shelter is built in the shape shown below. The shelter has a circular
cross-section of radius 200 cm and length of 450 cm.
200 cm
450 cm
200 cm
Calculate the area of material needed to construct the shelter in square metre. (6)
[15]
Question 2
The diagram below shows a sketch of trash can made up of a hemisphere placed on top of a right
cylinder with radius, r , and height 6,5 m. The volume of the cylinder is 250,15 m.
r
Formulae:
Surface area of sphere = 4πr 2
Question 3
Given: f (x)= ax 2 + qand f cuts the x -axis at points (p − 5)and (p + 1),where p
∈ ℝ.Determine:
3.1 the value of p
. (2)
3.2 the values of a
and q
if it is also given that f (−1)= 4. (4)
3.3 Hence, sketch the graph of f (x). (2)
[8]
Exemplar Assessments
Question 4
4.1 Consider the functions: g(x) = 4 tan x and f (x)= 4 sin x.
4.1.1 Draw sketches of these functions for 0° ≤ x ≤ 360° clearly indicating: (5)
• the asymptotes if any
• all intercepts with the axes
• turning points if any
• the endpoints of both functions.
4.1.2 What are the values of x for which 4 tan x = 4 sin x? (3)
4.2 The diagram below shows two graphs of f (x)= a cos x + qand g
(x)= 2 intersecting
at Aand B
.
y
3 f (x)
g(x)
2
A B
x
0 90° 180° 270° 360° 450°
–1
C(180°; –1)
–2
Exemplar Assessments
Time: 1 hour
Name: Surname:
Instructions
Read the following instructions carefully before answering the questions.
1. This question paper consists of 5 questions.
2. Answer ALL the questions.
3. Clearly show ALL calculations, diagrams, graphs, etc. which you have used in
determining the answers.
4. Answers only will NOT necessarily be awarded full marks.
5. You may use an approved scientific calculator (non-programmable and non-graphical),
unless stated otherwise.
6. If necessary, round off answers to TWO decimal places, unless stated otherwise.
7. Diagrams are NOT necessarily drawn to scale.
8. Write neatly and legibly.
Question 1
1.1 In the diagram below, A C = 15cm; Bˆ
D = 7cm; B AC = 52,5°and B
D ⊥ AC.Use the diagram
to calculate:
A
7 cm
52,5°
D
B 15 cm C
Exemplar Assessments
20 m
R Q P
Calculate the distance between the speed boats at the time. (5)
[11]
Question 2
An envelope is in the shape of a rectangle formed by two triangles A BCand A
DC. The length of
the rectangle A B = 25cm and the diagonal A C = 30,52cm. Two perpendiculars B Fand D
Eto the
line ACare dropped from the opposite vertices B and D
respectively as shown below:
A 25 cm B
D C
Calculate:
2.1 The length of B
C. (3)
2.2 The size of angle B
AC. (3)
2.3 The length of D
E. (3)
2.4 The area of △ADC. (3)
[12]
Question 3
The data below represents the marks obtained by 15 Grade 10 learners in a Mathematics test.
34; 54; 48; 62; 43; 52; 78; 68; 46; 90; 58; 84; 48; 73; 37
3.1 Calculate the mean mark. (2)
3.2 Determine the median mark. (2)
3.3 Determine the interquartile range. (3)
3.4 Draw a box-and-whisker diagram for the data. (3)
[10]
Exemplar Assessments
Question 4
The table below shows the distance in metres travelled by all the learners in Grade 10 at a certain
school from their homes to school.
Distance Frequency
1 300 ≤ d < 1 400 2
Question 5
A survey was conducted among 45 Grade 10 learners at a certain school to establish how many
learners had Smart phones (S) or Tablets (T) at school.
Results showed that:
Exemplar Assessments
Time: 1 hour
Name: Surname:
Instructions
Read the following instructions carefully before answering the questions.
1. This question paper consists of 5 questions.
2. Answer ALL the questions.
3. Clearly show ALL calculations, diagrams, graphs, etc. which you have used in
determining the answers.
4. Answers only will NOT necessarily be awarded full marks.
5. You may use an approved scientific calculator (non-programmable and non-graphical),
unless stated otherwise.
6. If necessary, round off answers to TWO decimal places, unless stated otherwise.
7. Diagrams are NOT necessarily drawn to scale.
8. Write neatly and legibly.
Question 1
(a; b)are the vertices of a
(0; 6), B(12; 12), C(14; 8)and D
In the diagram below, it is given that A
quadrilateral ABCD.
y B(12; 12)
A(0; 6) C(14; 8)
D(a; b)
x
0
Exemplar Assessments
Question 2
(x; 1), B(−1; 4), Cand D
The points A are shown on the Cartesian plane below:
y
B(–1; 4)
A(x; 1)
x
0
Question 3
3.1 A hardware store offers a discount of 10% on the first 20 bags of cement bought
and a further 12% discount for every additional bag bought. How much did a
customer pay for 50 bags of cement if the cost of each bag is R50? (3)
[3]
Question 4
Karabo likes travelling. She has saved R25 000 for an air ticket and other expenses on her
trip to Britain for a vacation at the end of 2021.
4.1 Use the table below of exchange rates to determine how many British pounds she
can buy for R25 000. (2)
SOUTH AFRICAN RAND RATES TABLE
Equivalent Value of Rand Equivalent of 1 Unit of
Foreign Currency
R1 Currency
US dollar 0,066537 15,029313
Euro 0,057370 17,430570
British pound 0,046380 21,560961
Australian dollar 0,084634 11,815569
Exemplar Assessments
4.2. Karabo plans to make another trip to Germany at the end 2024.
4.2.1 Assume that the annual rate of inflation in South Africa will be 7,1% over the
next 3 years. In 2024, what amount of money will be equivalent to the value of
R25 000 now? (3)
4.2.2 Karabo plans to invest equal amounts of R14 350 into a savings account on
1 December 2022 and on 1 December 2023. If the account earns interest at
12% p.a. compounded annually, how much will she have in the account on
the 1 December 2024 for her trip? (3)
[8]
Question 5
A grandfather sold 10 tables for R15 000 which he invested in a savings bank to take his
granddaughter to university. The money was invested on her second birthday and will only
be available to his granddaughter on her 19th birthday. If the money is invested in a fixed
deposit at a simple interest rate of 4% per annum:
5.1 find how much she will get when she turns 19 years. (3)
5.2 find the interest generated by the investment over the years. (2)
5.3 calculate the interest on the investment if it was made on compound interest terms. (5)
[10]
Total: [50]
Exemplar Assessments
Time: 1 hour
Name: Surname:
Instructions
Read the following instructions carefully before answering the questions.
1. This question paper consists of 4 questions.
2. Answer ALL the questions.
3. Clearly show ALL calculations, diagrams, graphs, etc. which you have used in
determining the answers.
4. Answers only will NOT necessarily be awarded full marks.
5. You may use an approved scientific calculator (non-programmable and non-graphical),
unless stated otherwise.
6. If necessary, round off answers to TWO decimal places, unless stated otherwise.
7. Diagrams are NOT necessarily drawn to scale.
8. Write neatly and legibly.
Question 1
In the diagram below, △HIJ has K and L
on H
Iand JIsuch that K
L ∙ HJ. Gis the intersection of H
L
and KJ. KI = 6cm, K
L = 4,2cm, G
L = 2,5cm, K G = 2cm and G J = 6cm.
H
K
6c
m
m
2c
G 4,2 cm I
2,5
m
cm
6c
Exemplar Assessments
Question 2
2.1 Use the sketch below to prove that:
P S
2 1 1
2
M
4
1 3
2
2 2
1 1
Q R
O
1
2 4
3
1 1
2 2
N M
2.2.1 ˆ
Write down the size of O
1. (1)
2.2.2 Calculate giving reasons:
2.2.2.1. The size ofˆ 1 .
L (2)
ˆ
2.2.2.2. The size of K M
N . (3)
[17]
Question 3
In the sketch below, K
PMNis a parallelogram. O ˆ
Nbisects K NM
and O ˆ
Mbisects N M P.
K O P
1 2 3
2
1
2 1
N M
Exemplar Assessments
Time: 2 hours
3.3 In the diagram below, D is the midpoint of △ABC. Eis the midpoint of A
C. DEis produced to
Fsuch that D
E = EF. CF ∙ BA.
A
E F
D
C
B
Question 4
ABCDis a parallelogram, D
E = BGand A
F = CH.
F
E
A
2
D
1
2
C
G
H
Prove that:
ˆ
4.1 2 =ˆ
A C
2 . (3)
4.2 EC = AG. (2)
4.3 FG = EH. (5)
4.4 EFGHis a parallelogram. (3)
[13]
Total: [50]
Exemplar Assessments
Time: 2 hours
Name: Surname:
Instructions
Read the following instructions carefully before answering the questions.
1. This question paper consists of 7 questions.
2. Answer ALL the questions.
3. Clearly show ALL calculations, diagrams, graphs, etc. which you have used in
determining the answers.
4. Answers only will NOT necessarily be awarded full marks.
5. You may use an approved scientific calculator (non-programmable and non-graphical),
unless stated otherwise.
6. If necessary, round off answers to TWO decimal places, unless stated otherwise.
7. Diagrams are NOT necessarily drawn to scale.
8. Write neatly and legibly.
Question 1
1.1 Factorise the following expressions fully:
1.1.1 12 x 3 − 3x (3)
2
1.1.2 x + 11x − 42 (2)
1.1.3 x − xy + y − 1 (3)
3
2x − 1
1.1.4 _ (2)
x + x + 1
1.2 Simplify the following:
x+1 x−1
3 − 3
1.2.1 ___________
3 x
(3)
2
12 x − 3 6x + 3
1.2.2 ___________
2 ÷ ___________
2 (6)
2x + x − 1 x + 2x + 1
[19]
Question 2
2.1 Solve for x :
2.1.1 x(x − 1)= 20 (4)
1
2.1.2 4 x+1 − _ = 0
64
(4)
2.2 Solve simultaneously for x and y :
3x + 4
y + 2x + 10 = 0and y = _
4 (4)
[12]
Exemplar Assessments
Question 3
3.1 Given the linear number pattern: 8
; 2; − 4; …
3.1.1 Write down the next two terms of the sequence. (2)
3.1.2 Determine the n
th term of the sequence. (2)
3.1.3 Determine the 25th term of the sequence. (2)
3.1.4 Which term of the sequence will equal to −
76? (2)
3.2 Thabo is investigating the number of white and dark tiles used to build the floor space of his
room as represented by the patterns below.
Exemplar Assessments
Question 4
x
(x)= _32 (_12 ) − 6. Aand B
Sketched below are the graphs of f (x)= ax 2 + qand g are the x
and point E(1; 3)lies on f. Cis the turning point of f and
-intercepts of f. The graphs intersect at A
-intercept of g
Dis the y .
C E(1; 3)
f
g
B
x
A 0
Exemplar Assessments
Time: 2 hours
Question 5
The sketch below shows f and g ,the graphs of f (x)= −_1x + 1and g
(x)= ax + qrespectively. Points
A(−3; 4)and B(5; −4)lie on the graph of g .The two graphs intersect at points C and D
. Line B
E is
drawn parallel to the y -axis, with E on f .
A(–3; 4)
f C
E
1
x
0 D
g
B(5; –4)
Question 6
6.1 Calculate the interest rate on the amount R8 000 if it triples after 3 years at a compound
interest. (3)
6.2 The cash price of an item is R5 000. Sibusiso wants to buy the item using a hire purchase
agreement. He has to pay a deposit of 10% and pay the balance in 36 monthly instalments
at a simple interest of 8% p.a. Calculate:
6.2.1 The monthly instalments he pays. (5)
6.2.2 The total interest he pays. (2)
6.3 The cost of a shirt at a certain retail outlet is about R950. What would be the price of
4 such shirts in USD if 1USD = R13,44? (3)
[13]
Exemplar Assessments
Question 7
7.1 Mr Smith lives in a block of 90 apartments labelled from 01 up to 90. You are not sure
what number his flat is, so you do not know which number to ring at the front gate.
7.1.1 What is the probability of getting either Mr Smith’s flat or one of his immediate
neighbours, if you ring a number at random? (2)
7.1.2 You have remembered that his flat number has a 9 in it. What is the probability
that you will be able to guess his flat number by considering this information? (2)
7.2 On a particular day, 150 learners visited the school tuck shop. These learners were
interviewed to find out what they bought.
• 125 learners had bought pies.
• 85 learners had bought muffins.
• 70 learners had bought both pies and muffins.
7.2.1 Draw a Venn diagram to represent the above information. (4)
7.2.2 What is the probability that a learner chosen at random had bought the
following:
7.2.2.1 pies but not muffins? (1)
7.2.2.2 neither pies nor muffins? (1)
[10]
[100]
Exemplar Assessments
Time: 2 hours
Name: Surname:
Instructions
Read the following instructions carefully before answering the questions.
1. This question paper consists of 8 questions.
2. Answer ALL the questions.
3. Clearly show ALL calculations, diagrams, graphs, etc. which you have used in
determining the answers.
4. Answers only will NOT necessarily be awarded full marks.
5. You may use an approved scientific calculator (non-programmable and non-graphical),
unless stated otherwise.
6. If necessary, round off answers to TWO decimal places, unless stated otherwise.
7. Diagrams are NOT necessarily drawn to scale.
8. Write neatly and legibly.
Question 1
The time taken, in minutes, to complete a 5 km race by a group of 13 learners is given below:
19; 21; 16; 22; 27; 20; 25; 34; 19; 22; 13; 30; 22
1.1 Identify the median time taken by the runners to complete the 5 km race. (2)
1.2 Determine:
1.2.1 the mean time. (3)
1.2.2 the range. (1)
1.2.3 the interquartile range. (3)
1.3 Draw a box-and-whisker plot to represent the data. (3)
[12]
Exemplar Assessments
Question 2
The weights of 45 boys are distributed as shown in the table below.
Weight w
(kg) Number of boys
Question 3
3.1 Show that the quadrilateral bounded by the lines:
y = 2x + 1; y − 3x = 6; y + 3x = 2and y = 3x + 1is a trapezium.
(4)
3.2 In the diagram below, A BCis a triangle in the Cartesian plane with vertices A(−5; −3),
B(7; 2)and C(x; y).BN ⊥ CAand M(−1; 3)is the midpoint of A
C.
y
C(x; y)
M
B(7; 2)
x
0
θ
A(–5; –3)
Exemplar Assessments
Question 4
4.1 y
(x; 5)
13
θ
x
0
Question 5
5.1 In the diagram below, △PQR is right-angled. Q ˆ
T ⊥ PRand T PQ = α.
P
Q R
5.1.1 Use PQ, QR, QT, TR, PTor PRto write down the following:
5.1.1.1 any two possible ratios of c os α. (2)
5.1.1.2 all the possible ratios of t an α. (2)
5.1.2 P = 5and α
If it is given that T = 40° ,calculate the numerical value of Q T. (3)
5.2 In triangle P QT,ˆ 30°and ˆ
=
P Q = 60° . Calculate the value of 2 sin P + cos 3Q. (3)
[10]
Exemplar Assessments
Question 6
D ⊥ BC, ACis 18 units,ˆ
6.1 In the figure below, A C
=
48° and BD = 12units.
A
18
48°
B C
12 D
2m
10 m
B
65°
C E
D
6.2.1 At what height is the top of the ladder above the ground? (2)
6.2.2 If the painter lowers the ladder by 2 m, what will be the size of the angle
between the ladder and the ground? (3)
[11]
Exemplar Assessments
Question 7
are shown for x ∈ [ 0°; 360°].
The graph of f (x) = 2 cos x and g
2
f
1
g
–2
Question 8
8.1 In the diagram, A
BCis a right-angled triangle. P
Q ∙ CBand P
is the midpoint of A
C.
A
P Q
C B
2p 2
1 3
4
1 2
2 60° 1
N
M
2.2 = 1_2 (
Volume of hemisphere: V 4_3 πr 3)✓
q = _92 ✓(4)
3.3 y
4,5
x
–3 0 3
✓ correct shape
✓ correct intercepts (2)
[8]
4.1.1 y
✓ Intercepts of f (x)
5 ✓ Turning point of f (x)
4
✓ Shape of f (x)
3
2 ✓ Asymptotes of g
(x)
1
✓ Shape of g
(x)
x
90° 180° 270° 360°
–1
–2
–3
–4
–5
(5)
4.1.2 x = 0° ✓x = 180° ✓x = 360° ✓ (3)
4.2 f (x) = a cos x + q and g (x) = 2
4.2.1 3 = a cos 0° + q ✓
3 = a + q ①
− 1 = acos180° + q ✓
− 1 = −a + q ②
2 = 2q
q = 1; ✓ a = 2 ✓
f (x)= 2 cos x + 1 ✓ (5)
4.2.2 2 = 2 cos x + 1 ✓
cos x = _12 ✓
x = 60°; x = 300° ✓
A(60°; 2)✓ and B
(300°; 2)✓(4)
[17]
Total: [50]
35 40 45 50 55 60 65 70 75 80 85 90
[10]
4.1.1 29 ✓ (1)
4.1.2 1 950 m ✓ (1)
_ ∑ (f. x)
= _
4.1.3 x n
_ _
= 49 850
x 29
= 1 718,97 m ✓ (3)
5 × 100 ✓= 17,24% ✓
_
4.2.1 29 (2)
12
_
4.2.2 29
× 100 ✓= 41,38% ✓ (2)
[9]
5.1 (4)
S T
✓ ✓ ✓
15 18 7
✓
5
5.2.2 7 ✓
Had a Tablet only: _
45 (1)
5.2.3 5 + 18 + 7 = 40 ✓
Had a Smart phone or a Tablet: 1
40
= _89 ✓
_
45 (2)
[8]
Total: [50]
12 − 8
BC= _
m 12 − 14
= −2 ✓
ˆ
1= ˆ
3.2 O N
2 (alt. ∠
s; KP ∙ NM) ✓
ˆ
1= ˆ
O
N
1 (given)
ˆ 3= M
O
ˆ
1 (alt. ∠
s; KP ∙ NM) ✓
ˆ
O
3= M
ˆ
2 (given)
KO = KN (sides opp.=∠s) ✓
KO = PM (opp. sides of parm.) ✓
KO = OP
(sides opp.=∠s) ✓ (5)
3.3.1 SAS ✓ (1)
3.3.2 CF = DA (≡ △ s) ✓
DBCFis a parallelogram (pair of opp. sides =
and ∙ ) ✓ (2)
3.3.3 2DE = BC ✓ (opp. sides of parm) ✓
DE = 1_2 BC
(2)
[13]
4.1 2 = ˆ
ˆ
A D
1 ✓ (corr. ∠s; AD ∙ BC) ✓
ˆ
= C 2 (alt. ∠
s; AD ∙ BC) ✓ (3)
4.2 BG = DE (given)
AB = DC (opp. sides of parm) ✓
AB + BG = DC + DE ✓
AG = EC (2)
4.3 AF = CH (given) ✓
AG = EC (proved) ✓
A 2 =ˆ
ˆ
C
2 (proved) ✓
△FAG ≡ △HCE (SAS) ✓
FG = EH (≡ △s) ✓ (5)
4.4 Similarly; △ FDE ≡ △ HBG (SAS) ✓
EF = HG (≡ △s) ✓
EFGHis a parallelogram (2 pairs of opp. sides =) ✓ (3)
[13]
Total: [50]
y = −9_2 ✓
D(
0; − _92 )
(2)
4.1.2 y = −6 ✓ (1)
4.1.3 y > − 6 ✓ (1)
x
4.2.1 0 = _32 (_12)
− 6 ✓
2= 2 −x
4=2
x = −2 ✓
(− 2; 0)
A (2)
4.2.2 At A(−2; 0): f (−2)= a (−2) 2 + q
0 = 4a + q
①✓
v At E(1; 3): f (1)= a (1) 2 + q
3 = a + q
②✓
① − ②: − 3 = 3a`
a = −1 ✓ and q = 4 ✓(4)
4.3.1 CD = 4 − (− 4,5)✓ = 8,5 ✓ (2)
0 − (− 4,5)
AD= _
4.3.2 m = −_94 ✓✓
− 2 − 0
y = −_94 x − _92 ✓
(3)
4.4.1 −2 < x < 2 ✓✓ (2)
4.4.2 x < 0 ✓ (1)
[18]
(− 4)
4 −
_
5.1 m = − 3 − 5 = −1 ✓
y − 4 = −1(x + 3)
y = −x + 1 ✓
a = −1; q = 1
(2)
Exemplar Assessments
√
3
24 000
_
8 000 = 1 + i
i = 44,22% ✓
(3)
6.2.1 10 × 5 000 = 500 ✓
Deposit = _
100
Balance = 4 500 ✓
A = 4 500(1 + 3 × 0,08)✓
A = 5 580 ✓
5 580
Instalments = _
36
= R155 ✓ (5)
6.2.2 Interest =
5 580 − 4 500 ✓ = 1 080 ✓ (2)
6.3 4 × 950 = 3 800 ✓
3 800
_
Cost in USD = 13,44
✓ = 282,74 ✓ (3)
[13]
3 = _
7.1.1 _
90
1 ✓✓
30
(2)
7.1.2 Event =09; 19; 29; 39; 49; 59; 69; 79; 89; 90 ✓
10
Prob = _
90
= _19 ✓ (2)
7.2.1 (4)
P M
✓ ✓ ✓
55 70 15
✓
10
55 = _
7.2.2.1 _
150
11
✓
30
(1)
10 = _
7.2.2.2 _
150
1 ✓
15 (1)
[10]
Total: [100]
Exemplar Assessments
[12]
2.1 x = 11 ✓ (1)
2.2 60 < w ≤ 70 ✓ (1)
2.3 ∑ x = 2 875 ✓
_ 2 875
= _
x 45
✓ = 63,89 ✓ (3)
[5]
3.1 A trapezium has one pair of sides parallel. ✓
y = 2x + 1
y = 3x + 6 ✓ [arranging in the form y = mx + c]
y = −3x + 2and y = 3x + 1
The lines y = 3x + 6and y = 3x + 1have the same gradient. ✓
Therefore, they are parallel. ✓(4)
x − 5 y − 3
3.2.1 _ = −1 ✓and _
2 2
= 3 ✓
x = 3and y = 9
(2)
9 − (− 3)
AC= _
3.2.2 m ✓ = 3_2 ✓
3 − (− 5)
(2)
BN= −_23 ✓
3.2.3 m
y − 2 = −_23( x − 7)✓
20
y = −_23 x + _
3
✓ (3)
_________________ _
3.2.4 AC = √ 3 + 5) 2 + (9 + 3) 2 ✓ = 4 √ 13 ✓
( (2)
_ _
3.2.5 Area = 1_2 × 4 √ 13 × 2 √ 13 ✓✓ = 52 ✓ (3)
[16]
2 2 2
4.1.1 x + 5 = 13 ✓✓
____________
x=√
169 − 25
✓ = 12 ✓(4)
5 ✓✓
4.1.2 tan θ = _
12
(2)
Exemplar Assessments
2 2
4.1.3 cos 2 θ − 2 sin 2 θ = ( )
12
_
13 − 2(_ 5
) ✓✓
13
50
144
= _
169
− _ 94 ✓(4)
✓ = _
169 169
1 ✓✓✓
1 × 1 × __
4.2 sin
90° ⋅ sin 90° + tan 45° ⋅ cos 60°
______________________________
tan 45° ⋅ sin 90°
= __________
1 × 1 ✓✓
2
1 + _1
= _ 2
1
✓= _32 ✓ (7)
[17]
PQ QT
PT ✓ and c os α = _
_
5.1.1 cos α = PQ PR ✓ or cos α = _
QR
QR QT
tan α = _
PQ ; ✓ tan α = _ TR ✓
PT ; ✓ and t an α = _
QT
(5)
QT
5.1.2 tan 40° = _
5 ✓
QT = 4,20 ✓ (2)
5.2 2 sin P + cos 3Q ✓ = 2 sin 30° + cos 180° ✓ = 0✓ (3)
[10]
AD
6.1.1 sin 48° = _
18
✓
AD = 18 sin 48° ✓ = 13,38 ✓ (3)
13,38
6.1.2 tan B = _
12
✓
tan −1(_ )
13,38
ˆ
=
B 12
✓ = 48,1° ✓ (3)
AC
6.2.1 sin 65° = _
10
✓
AC = 9,06 ✓ (2)
6.2.2 BC = 9,06 − 2 = 7,06 ✓
7,06
sin E = _
10 ✓
ˆ
E=
44,9° ✓ (3)
[11]
7.1 y = sin x ✓ (1)
7.2.1 360° ✓ (1)
7.2.2 1 ✓ (1)
7.3 − 3 (1)
7.4 0° < x < 90° ✓✓ (2)
7.5.1 h(x)= f (x) − 3 ✓ = 2 cos x − 3 ✓ (2)
7.5.2 − 5 ≤ y ≤ − 1 ✓✓ (2)
[10]
8.1.1 PQ ∙ CB ✓ (given)
Pˆ
Q
A ˆ
= C
B
A = 90° ✓ (corr. ∠s; PQ ∙ CB) ✓ (3)
PQ ⊥ AB (proven)
Exemplar Assessments