GRADE 10
MATHEMATICS
ASSESSMENT EXEMPLARS
Dear Teacher
1. This resource is a compilation of 14 tests that cover all the grade 10 topics.
2. Each test is out of 50 marks and is 1 hour long.
3. These tests are exemplars which may be used as they are or adapted to suit your context.
4. They may also be used to compile your midyear or end-of-year examination.
5. When you do that you must ensure that you adhere to the topics weighting as specified in the
CAPS documents
Summary of Schools Based Assessment Tasks for Grades 10
Total marks
50
50
50
100
50
50
50
Convert to a
mark out of:
10
20
10
30
10
10
10
Grade 10
-2-
100
TOTAL
End-of-year
examination
PROMOTION MARK
Year-mark
Term 4
Test
Test
Test
Term 3
Examination
Test
tasks
Term 2
Assignment/T
est
Assessment
Project/Invest
igation
Term 1
200
300
200
300
Exemplar Assessments 2012
Test 1: Algebra 1
Time: 1 hour
Marks: 50
Question 1
1.1
9
11  x
Consider: A=
If x   14 ;
A:
 11 ;  5 ; 0 ; 5 ; 11 ; 14, which value(s) of x will make
Rational
1.1.1
(1)
Irrational
1.1.2
(1)
Undefined
1.1.3
1.1.4
(1)
Non-real
(1)
[4]
Question 2
2.1
Calculate the following products:
2.1.1
 5y
2  b3  a
 p  2 p 2  2 p  4
2.1.2
2.1.3
2.2
2.2.1
2.2.2
2.2.3
3x
2

2
(4)
(4)
(4)
Factorise fully:
2 x 4  32
2m 2  5m  3
x 3  y 3  2x  2 y
(4)
(4)
(4)
[24]
Question 3
3.1
3.2
What must be added to
With what expression must
3.3
Evaluate
3.4
With what expression must
to make it equal to
be divided to get a quotient of
if x =7,85 without using a calculator. Show all your work.
be multiplied to get a product of
(3)
(3)
(4)
(3)
[13]
Question 4
Simplify the following algebraic fractions
4.1
(4)
4.2
(5)
[9]
Grade 10
-3-
Exemplar Assessments 2012
Test 2: Algebra 2
Time: 1 hour
Marks: 50
Question 1
1.1
Show that
can be simplified to
(6)
1.2
Solve for x :
1.2.1 2 x 2  x  3  0
1.2.2 x(2x+3) = 2
(3)
(4)
1.3
Represent graphically:
(5)
1.4
Solve the following system of equations:
3x –2y + 8 = 0 and 4y–6 = 2x
(6)
[18]
Question 2
One side of a rectangular field is x  y  metres long. Given that the area is
( x 2  x  y  y 2 ) square metres, calculate the perimeter in terms of x and y
[8]
Question 3
3.1
Simplify the following expressions(Give your answer with positive exponents)
 9x 2
 4
 y
3.1.1




1
2
(3)
(3)
3.1.2
3.1.3
(4)
3.1.4
(4)
3.2
Solve for
if
3.2.1
(3)
3.2.2
(3)
3.2.3
(4)
[24]
Grade 10
-4-
Exemplar Assessments 2012
Test 3: Number Patterns
Time: 1 hour
Marks: 50
Question 1
1.1
Write down the next three terms and the general (or nth term) of each pattern:
1.1.1
2; 4; 6; 8; …
1.1.2
1; 7; 13; 19; …
1.1.3
1; 4; 9; 16; …
1.1.4
25; 21; 17; 13; …
1.1.5
x  1; 2 x  2; 3x  3; 4 x  4;...
1.1.6
1.1.7
1
3
; 1; ; 2;...
2
2
1.1.8
1 1 1 1
; ; ; ;...
2 3 4 5
3 1 1
3 ; 3 ; 3 ; 3;...
4 2 4
(24)
Question 2
2.1
Consider the following pattern.
Arrangement 1 Arrangement 2 Arrangement 3






th
2.1.1
2.1.2
2.1.3
How many flowers will be used in the 4 arrangement?
th
How many flowers will be used in the n arrangement
Which arrangement will have 99 flowers
2.2
The height of water in a tank is recorded whilst the tank is being filled. The results have been
recorded at five minute intervals:
First reading
Level in cm
2.2.1
2.2.2
2.2.3
2.2.4
2.2.5
3
After 5
minutes
11
After 10
minutes
19
What will be the height of the water after 25 minutes?
What will be the height after an hour?
At what rate is the level rising? Give your answer in cm / minute.
What will be the water level after 5n minutes?
After how many minutes will the water level be 403 cm ?
Grade 10
-5-
(1)
(2)
(3)
After 15
minutes
27
After 20
minutes
35
(1)
(2)
(2)
(3)
(3)
Exemplar Assessments 2012
2.3
Consider the following sequence of Es:
2.3.1
2.3.2
2.3.3
2.3.4
How many blocks will be needed to build the 10th E?
How many blocks will be needed for the nth E?
116 blocks are needed for the kth E. Calculate the value of k.
Can the total number of blocks ever be a multiple of 10? Explain.
(3)
(3)
(3)
(3)
[26]
Test 4 Finance
Time: 1 hours
Marks: 50
Question 1
1.1
Simphiwe plans to travel to New Zealand. Her flight is paid for. However,
she will need spending money of about NZ$ 300.
If the exchange rate is currently 5,1265 Rand to the NZ$ how much will
she need in South African Rand ?
1.2
Simphiwe has just received a gift of R 500 from her grandmother and she promises
another R 500 in a year’s time. She decides to invest this money in an account which
pays 8,5% p.a. compounded annually. How much additional money will she have to
save to add to her investment after two years if she withdraws all the money for
her trip.
(2)
(8)
[10]
Question 2
2.1
Determine through calculation which of the following investments will be more profitable:
(a)
R7 000 at 10% p.a compound interest for 5 years.
(4)
(b)
R7 000 at 12% p.a simple interest for 5 years.
(3)
2.2
Inflation is set at 5,5% for the next three years. A small scooter currently
costs R7999,00. I want to buy this scooter on my birthday in 3 year’s time.
2.2.1
2.2.2
What will it cost in three year’s time?
(3)
The dealer offers me a hire purchase on the scooter on the following
terms: 15% deposit and the balance payable over 36 months. The current
interest rate on a hire purchase deal is 23% per annum. Calculate my monthly
payments.
(6)
What is the total amount that I pay to the the dealer
(2)
Every year on my birthday I receive R3000 from my grandfather.
If I save this money in an account that pays 11% per annum compound interest,
will I be able to pay in cash for the scooter after 3 years? Show all calculations to
justify your answer
(8)
2.2.3
2.2.4
Grade 10
-6-
Exemplar Assessments 2012
Question 3
3.1
Below is a table with the buying and selling prices of different currencies: (13 /09/ 06)
Country
Currency
Symbol
Switzerland
New Zealand
Japan
Franc
Dollar
Yen
Swiss Franc
NZD
Y
3.1.1
3.1.2
3.1.3
Exchange Rate
(units per R1)
0.1697
0.21
16.03
You have R5000 to spend in Switzerland. How much Francs can
you buy?
What will it cost you in Rands to purchase 5000 Yen?
If you exchange 1000 NZD how much Rands will you get?
(2)
(2)
(2)
3.1.4
3.2
You want to import a personal computer from Japan at a total cost
of 96180 Yen. The equivalent computer cost R8 500 in South Africa.
Will you import or buy locally? Show all calculations to justify
your answer.
(3)
Draw two graphs on the same set of axes to illustrate the difference between
compound interest and simple interest graphically.
(5)
[14]
Test 5: FUNCTIONS 1
Time: 1 hours
Marks: 50
Question 1
If and g ( x ) 
1.1
1
and h( x)   x 2 , answer the following questions;
x
Determine the values of the following;
f (1)
1.1.1
1.1.3 x if f ( x)  0
1.1.5 g ( 2)
1.1.7 h(2)
1.1.2
1.1.4
1.1.6
1.1.8
f ( 2)
g (1)
x if g ( x)  2
h ( 2)
(16)
1.2
Name the type of function that is defined in each case.
(3)
1.3
Draw a sketch graph of each of the functions showing all critical points,
asymptotes, axes of symmetry and intercepts with the axes. You can use the
values in question 1.1 to assist you if necessary. Each function must be
sketched on a separate set of axes.
(6)
Determine the domain and range of the functions f , g and h .
(6)
1.4
[31]
Grade 10
-7-
Exemplar Assessments 2012
Question 2
Consider the functions s( x)  x 2  9 and t ( x)  2 x  6
Sketch the graphs of s and t on the same system of axes, showing ALL
intercepts with the axes and relevant turning points
2.1
(7)
Use your sketch to find the values of x if;
2.2
2.2.1
2.2.2
2.2.3
2.2.4
s ( x)  t ( x)
(3)
(2)
(3)
(2)
s(x) – t(x) = -3
s ( x)  0
s(x) < t(x )
Write down the equation of q if q(x) results from shifting s (x ) 2 units up.
2.3
Test 6: FUNCTIONS 2
Time: 1 hours
(2)
[19]
Marks: 50
Question 1
Sketched below are the functions g ( x)  b x  c and h( x) 
k
and A, the
x
point of intersection, is ( 1 ; 1 )
1x
h x  =
2
1
+
g
2
1.1
Find the values of k , c and b
(3)
1.2
What is the equation of the
asymptote of g
(2)
1.3
What is the range of g
(2)
1.4
What is the equation of f if f (x )
is the reflection of g (x ) in the y-axis
2
h
1
A(1;1)
-2
2
-1
(2)
[6]
-2
Question 2
Given the two functions f x    x 2  9 and g x   x  3
2.2
Sketch the graphs of f and g on the same system of axes, showing the co-ordinates of all
intercepts with the axes.
(6)
Calculate the co-ordinates of the points at which f x   g x 
(4)
2.3
Read from your graph, the values of x for which f x   0 .
(2)
2.4
Draw a dotted line on your sketch, showing graph of y 
2.1
on the axes must be shown.
1
f x  . The intercepts
2
(3)
[15]
Question 3
Grade 10
-8-
Exemplar Assessments 2012
3.1
3.2
3.3
Determine the equation of a linear function f (x) = mx + c, if f (0)  7
and f (2)  0
(2)
4
 2 . Clearly indicate the asymptotes.
x
Y
A sketch graph of f ( x)   x 2  9 is shown.
P
Draw a sketch graph of y 
(3)
3.3.1
What is the length of line segment PR?
(4)
3.3.2
For which values of x is
3.3.3
Compare the shapes of
f ( x)  0 ?
(2)
-3
y  f ( x)   x  9
and y  3  f ( x)
2
O
X
R
(2)
[13]
Question 4
4.1
Sketched below are graphs of f x   a.b x and g  x  
p
 q for x  0
x
^y
C(1;6)
A(0;2)
B(4;0)
O
4.1.1
4.1.2
4.1.3
Given that f x  cuts the y -axis at A(0;2) and that C (1;6) lies on both graphs,
calculate the values of a and b
(4)
It is further given that g x cuts the x -axis at the point B4;0 . Calculate the
values of p and q .
(4)
Write down the equation of the horizontal asymptote of g x
(2)
4.2 In the sketch f is the graph of the function
4.2.1
>x
y  ax + q
Calculate the value of a and q if the
Grade 10
y
-9-
f
Exemplar
Assessments
2012
 P (3
; 9)
4.2.2
point P(3 ; 9) lies on the graph.
(4)
Write down the equation of the
asymptote of f.
(2)
[16]
Test 7 : Analytical Geometry
Time: 1 hours
Marks: 50
Question 1
P( -5 ; 9 )
Q( 4 ; 6 )
S(x ; 3 )
R( -1 ; 1 )
1.1
1.2
1.3
1.4
Find the gradient of line PQ
If it is given that PQ  PS , find the value of x in the point S
1.3.1 Assuming that x  7 , find the gradient of line SR
1.3.2 What does this tell us about SR? Explain.
Join PR
1.4.1 What type of triangle is PRS ? Show all working.
1.4.2 Find the area of PRS
(3)
(5)
(2)
(2)
(5)
(3)
[20]
Question 2
Grade 10
- 10 -
Exemplar Assessments 2012
y
A ( -3 ; 7 )
C(4;6)
D ( -2 ; k )
B ( -8 ; 2 )
x
2.1
2.2
2.3
2.4
Calculate the gradient of BC
D(-2; k) lies on BC. DA  AC. Find the value of k.
Find the length of BC.
Assuming the value of k is 4, calculate the area of  ABC.
(3)
(4)
(3)
(5)
[15]
Question 3
Sketched below is
3.1
3.2
3.3
3.4
. The co-ordinates of the vertices are as indicated on the sketch.
Calculate the co-ordinates of the mid-points D and E of AB and AC respectively.
Show that DE // DC
Given the points
and
, show that
PARM is a parallelogram by proving both pairs of opposite sides parallel.
The vertices of a rhombus
,
. Prove that:
the diagonals RO and HM bisect each other.
(4)
(3)
(4)
(4)
[15]
Test 8: Trigonometry 1
Time: 30 minutes
Grade 10
- 11 -
Exemplar Assessments 2012
Marks: 25
Question 1
In each of the following right-angles triangles, write down the value of the required trigonometric ratio (leave your
answers as ratios) and calculate the size of the angle marked  :
8
1.1
Find the value of sin  and the size of angle 
7
(3)

29
1.2

Find the value of cos and the size of angle 
7
5
(3)
Find the value of tan  and the size of angle 
1.3
4

(3)
[9]
3
Question 2
2.1
If sin  =
5
, determine each of the following without the use of a calculator:
13
(Hint: Use a sketch) (  < 90o)
2.2
2.1.1
cos 
2.1.2
tan 
2.1.3
2.1.4
sin 2 
2.1.5
cos 2 
2.1.6
sin 2  + cos 2  (8)
Make a conjecture about
a)
b)
2.3
sin 
cos 
sin 
cos 
(2)
sin 2  + cos 2 
(2)
Simplify the following expressions without the use of a calculator and show ALL the workings:
sin 45 sin 90
cos 0 cos 60

(5)
[17]
Test 9 : Trigonometry 2
Time: 1 hour
Grade 10
- 12 -
Exemplar Assessments 2012
Marks: 50
Question 1
Use a calculator to determine  (correct to ONE decimal place), (  < 90o) in each of the following:
1.1
1.2
1.3
3 cos  = 5
3 tan  = 5
5 sin (2  + 10o) – 4 = 0
(10)
Question 2
ˆ C  300
In the sketch below, BCD is right angled at C, BD = 3 units, BD
and ABˆ E  20 0 . Also, BCDE is a rectangle.
calculate the lengths of
A
2.1
BC
(4)
2.2
CD
(4)
2.3
AD
2.4
Angle DBA
B
20
E
(6)
(3)
[17]
30
Question 3
C
D
A
With reference to the figure alongside:
D
3.1
Write down two ratios for cos 34.
(4)
3.2
If CD = 8,3 cm, calculate the value of BD
(4)
3.3
AC
Write down a trigonometric definition for
.(2)
BC
34
B
C
[11]
Question 4
In the figure alongside MN  NR, MRN = 42o, MN = 8 units, PR = 5 units and PR  NR.
M
4.1
Calculate NR.
(4)
4.2
Calculate MR
(4)
4.3
Calculate PN
(4)
P
[12]
42o
R
N
Test 10: Trigonometry 3
Time: 1 hour
Grade 10
- 13 -
Exemplar Assessments 2012
Marks: 50
Question 1
A shark spotter is standing at lookout point 25m above the waters edge. He spots a shark in the
water at an angle of depression of 50  . If the swimmer that is also in the water is 5m from the foot
of the lookout spot, how far is the shark from the swimmer?
50
25m
5m
[6]
Question 2
A laser speed trapping device is mounted on a pole that is 3m high. The device measures the initial angle
of depression of a car (x) and then measures the angle of depression again 1 second later (y).
These measurements are used to work out how much distance the car has covered in 1 second and then to
determine whether or not the driver is breaking the speed limit.
2.1
Show that the initial distance from the camera is given by: D1 
from the camera, one second later, is given by D2 
Grade 10
- 14 -
3
tan y
3
and the distance
tan x
(3)
Exemplar Assessments 2012
2.2
2.3
If x  3,5 and y  6 , calculate the distance (in metres) covered by the car
in 1 second.
(3)
If the speed limit is 60 km/hour, determine whether or not the motorist is exceeding
the speed limit. Show all calculations. (If you were unable to do 6.1, assume
that the car covered 20,51 m in 1 second.)
(2)
[8]
Question 3
In the diagram, AC=13 units, AB = 12 units and BD = 4 units. CBˆ A and BD Cˆ are right-angles.
3.1
C
3
13
3.2
Calculate the measurement of CB
and BD.
Now determine the value of:
3.2.1
D
3.2.2
3.2.3
A
12
B
3.3
(4)
tan A
sin CBˆ D
tan ACˆB  cos DBˆ C
Use you calculator to determine the
size of angle A, correct to one
decimal place.
(1)
(1)
(3)
(2)
[11]
Question 4
Sketched are graphs of y  f x   a tan x and y  g x  b sin x  c
Write down the:
4.1
value of a ;
Grade 10
(1)
- 15 -
Exemplar Assessments 2012
4.2
4.3
4.4
4.5
period of f ;
values of b and c
range of g
(1)
(2)
(1)
values of x  [180 0 ;180 0 ] for which f x   g x  .
(3)
[8]
Question 5
The following graphs have been drawn below: f ( x)  tan x and g ( x)  a sin x + q
y
4
g
2
x
50
100
150
-2
5.1
5.2
5.3
5.4
5.5
200
250
300
350
f
Determine the value of a and q.
Write down the values of x for which g ( x)  f ( x)  1 .
Write down the period of f.
Write down the equations of the asymptotes of f
For which values of x is f(x) = 0
(2)
(2)
(1)
(2)
(3)
[10]
Question 6
Below is a sketch of f ( x)  cos x  q and g ( x)  a sin x
6.1
6.2
6.3
6.4
Write down the amplitude of f and g
What is the range of f
What is the period of f
Determine the values of a and q
2
(2)
(2)
(1)
(2)
[7]
g
y
1
x
-90
-60
Grade 10
-30
0
30
60
- 16 -
90
120
150
180
Exemplar Assessments 2012
Test11: Statistics
Time: 1 hour
Marks: 50
Question 1
1.1
1.1.1
1.1.2
1.1.3
1.2
1.2.1
1.2.2
1.2.3
1.2.4
1.2.5
At a recent interschool athletics event, the following distances were recorded for all the
participants in the girls long jump: 3,1m; 3,2m; 5,0m; 3,6m; 4,1m; 2,9m; 3,2m; 4,3m; 4,9m; 3,9m;
2,8m; 4,6m.
Calculate the
mean,
(3)
median and
(2)
mode
(1)
The amount of time, in minutes, that a group of Grade 10 learners spent on their cellular phone
in a week was measured and recorded below. Use classwidths of 20 minutes
82
102
108
142
150
170
145
115
154
121
128
126
147
92
137
158
81
146
177
152
152
130
88
132
Construct a frequency table for this data set
Calculate the estimated mean time
What is the modal class?
What percentage of learners used their cellular phones for 2 hours or more
in the week?
What comment can you make regarding the use of cellular phones amongst
this group of learners?
(5)
(5)
(1)
(2)
(2)
[20]
Question 2
2.1
2.1.1
The marks in a class test of 15 girls in Mrs Mbusi’s Science class s given below. The test is out of
50 marks:
43
42
31
32
22
13
44
38
25
50
9
15
25
35
41
Determine the mean, median and mode of the marks.
(3)
Grade 10
- 17 -
Exemplar Assessments 2012
2.1.2
2.1.3
Calculate interquartile range for the class
Draw a box and whisker plot to represent the marks
(3)
(2)
2.2
Consider the box-and-whisker diagrams below representing the marks of two Grade 10 classes in
a test out of 50
10A
10 B
10
2.2.1
2.2.2
2.2.3
2.2.4
2.2.5
40
30
20
50
Write down the five number summary for 10B
What is the inter quartile rage for 10A
What is the median mark for 10B
What percentage of marks in 10A lies between the highest mark and the median
Which class in your opinion did the best? Give a reason
(3)
(1)
(1)
(1)
(2)
[16]
Question 3
The pulse rate of patients at a clinic was measured and the results tabulated in a frequency table shown
below
3.1
Complete the frequency table
(4)
Pulse rate
150≤x< 160.
160≤x< 170.
170≤x< 180.
180≤x< 190.
Tally marks
Frequency
3
//// ///
13
////
Use the frequency table to determine:
3.1.1
3.1.2
3.1.3
3.2
3.2.1
How many patients were measured on that day?
the estimated mean
the modal class
(1)
(2)
(1)
A music shop records the sales of CDs over a two year period. The monthly sales figures are
given below:
204
255
310
283
288
393
282
359
364
172
158
407
458
299
109
307
272
283
285
367
479
280
382
258
111
Complete the frequency table below
(4)
Class interval
Tally marks
Frequency
Class midpoint
100
////
4
150
Grade 10
- 18 -
Frequency x
class midpoint
600
Exemplar Assessments 2012
2450
3150
Total
3.2.2
Total
Determine the mean monthly sales to the nearest whole number
(2)
[14]
Test 12: Probability
Time: 1 hour
Marks: 50
Question 1
1.1
1.1.1
A fair die is rolled once.
Write down the sample space
1.1.2
what is the probability of getting the following:
(a)
(b)
(c)
(d)
(2)
The number 6
A number less than 3
the number 3 or 6
the number 3 and 6
(1)
(2)
(2)
(1)
1.2 . A bag contains 7 red marbles and 5 green marbles. One marble is drawn out of the bag.
Determine the probability that it is:
1.2.1
a red marble
(1)
1.2.2
a green marble
(1)
1.2.3
a yellow marble
(1)
1.2.4
a red or green marble
(2)
1.2.5
not a red marble
(2)
[15]
Question 2
Two dice are rolled simultaneously.
2.1
Write down the sample space
2.2
Determine the probability that :
(a)
the number 2 is obtained
(b)
the sum of the numbers equals 8
(c )
one of the numbers is an odd number
(d)
both of the numbers are factors of 6
1
2
(3)
(2)
(2)
(2)
1
4
5
3
3
Figure 1
4
2
3
2
Figure 2
2.3
Mandy spins the spinner in figure 1 and Louisa spins the spinner in figure 2.
2.3.1
2.3.2
Calculate the probability for each one to get a 3
Who has the greater chance of getting a 2? Give a reason for your answer
Grade 10
- 19 -
(2)
(2)
Exemplar Assessments 2012
2.3.3
Who has the greater chance of getting a 1? Give a reason for your answer.
(2)
[15]
Question 3
There are 120 grade 10 learners at a school. 55 learners offer Mathematics and 80 offer Life Sciences.
There are 25 learners who does not offer Mathematics or Life Sciences
3.1
Represent the information in a Venn diagram
(4)
Use the Venn Diagram to calculate the probability that a randomly chosen learner:
3.2
offer Mathematics only
(2)
3.3
offer Life Science only
(2)
3.4
offer Mathematics and Life Sciences
(2)
[10]
Question 4
In a staff of 20 teachers a survey was conducted to establish how many drink coffee and how many drink
tea.. The following was found: 3 staff members did not drink either coffee or tea; 11 drank coffee and 8
drank tea
4.1
Represent this information in a Venn diagram. Let C={staff members who drink coffee} and
T={ staff members who drink tea}. Let the number of staff members who drink both coffee and
tea = x.
(4)
4.2
Calculate the value of x
(2)
4.3
If a staff member is chosen randomly, calculate the probability that s/he drinks
a. Only coffee
(1)
b. Only tea
(1)
c. Coffee and tea
(1)
d. Coffee or tea
(1)
[10]
Grade 10
- 20 -
Exemplar Assessments 2012
Test 13 : Measurement
Time: 1 hour
Marks: 5
Question 1
1.1
In the figure, the cube and the cylinder have equal volume. If two sides of the cube are
doubled, and the diameter of the cylinder is doubled, will the volumes of the resulting
prisms still be equal? Show all calculations.
(5)
y
y
x
x
x
1.2
Consider the figures below and in each case determine:
1.2.1
the surface area and the volume
of the prism
1.2.2
the surface area and the volume
of the cylinder
15 cm
30 cm
24 cm
18 cm
18 cm
1.2.3
Determine the surface area if the edge of the base of the prism is doubled.
1.2.4
Determine the volume of the cylinder if the radius is doubled.
(10)
[15]
Grade 10
- 21 -
Exemplar Assessments 2012
Question 2
Dylan made a right square pyramid out of
plaster for an art project as shown alongside.
Each side of his pyramid's base measures 8 cm.
The height of the slant triangle of this
pyramid measures 5 cm.
5cm
h
8cm
2.1
What is the area, in square cm’s, of the base of Dylan’s pyramid?
2.2
What is the total surface area, in square cm’s, of Dylan’s pyramid? Show
your working.
(5)
What is h, the height, in cm’s, of Dylan’s pyramid? Show or explain how
you got your answer.
(4)
2.3
2.4
(2)
Using the height you determined in part 2.3, what is the volume, in cubic
cm’s, of Dylan’s pyramid? Show your working.
(3)
[14]
P
3.1
12
12
Question 3
h
 PQR is an equilateral triangle, with the
measurement of each side equal to 12cm.
R
3.2
12
Q
3.1.1
Use any appropriate method to show that the perpendicular height of the
triangle is 10,39cm.
(3)
3.1.2
Hence, calculate the area of the triangle.
(2)
A triangular pyramid is constructed using four
triangles that are the same as the triangle
in 3.1 (i.e. an equilateral triangle with sides
measuring 12cm).
3.2.1
3.2.2
3.2.3
Calculate the perimeter of the base of
the pyramid.
(1)
Explain how you know that the height
of the slanted triangles is 10,39cm.
(1)
Calculate the total surface area of the
pyramid.
(3)
[10]
Grade 10
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Exemplar Assessments 2012
Question 4
A product designer is designing a set of three open baskets according to the following
specifications:

The capacity of the baskets:
4.1
Has the same scale factor been applied when reducing the basket capacity
from large to medium and from medium to small? Substantiate your answer
with calculations.
(3)
4.2
Convert the capacity of each basket to cm 3 ( 1ml  1cm3 ).
(3)
4.3
The designer decides that the large basket should be 15cm long and
12cm wide. Show that the height of the basket will be 8.3cm.
(3)
large basket = 1,5l
medium basket = 1,25l
small basket = 1,0l
4.4
The baskets fit inside each other with a 0,5cm gap all around. What is the length
and breadth of the medium basket?
(2)
4..5
The small basket will be covered with decorative paper. Calculate the area of
paper required to cover the bottom and sides of the small basket. (The
dimensions of the small basket are length = 13cm, breadth = 10cm and
height = 7,7cm.)
Grade 10
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(3)
[14]
Exemplar Assessments 2012
Test14 : Geometry
Time: 1 hour
Marks: 50
Question 1
1.1
In the diagram below, AB and DC are two parallel lines cut by two transversal lines at X, Y and Z
respectively.
A
X
Z
x
o
60
B
D
Y
x - 20o
C
1.1.1
1.1.2
1.1.3
1.1.4
1.1.5
Determine giving reasons, the value of x in the diagram:
(6)
Name one pair of co-interior angles
(1)
Name one pair of alternate angles
(1)
Complete: If two parallel lines are cut by a transversal, then the co-interior angles are .......(1)
Complete: The size of angle XYD =
, Reason .....................
(2)
1.2
Determine with reasons, the value of y ( XY) in the diagram below. Given that AB = 4 units
and BC = 3 units. X and Y are the midpoints of AB and BC respectively.
(5)
A
=
4
X
= y
B
Y
C
3
[16]
Grade 10
- 24 -
Exemplar Assessments 2012
Question 2
2.1
6
2.2
6
12
Question 3
3.1
10
Grade 10
- 25 -
Exemplar Assessments 2012
3.2
12
Question 4
4.1
8
4.2
8
Grade 10
- 26 -
Exemplar Assessments 2012
PROJECTS AND INVESTIGATIONS
INVESTIGATION: Functions
1
Draw the graphs of each of the below by means of point-by-point plotting .
1
y  2 
 x
y
1
2
x
1.14
y
y  3 x  4
1.15
y  x2
1.7
1.2
y  2x
1.8
y
1.3
y  2 x
1.9
y
1
x
1.10
1.5
y  x2  2
1.11
1.6
y  2x 2
1.12
x
3
1.13
1.1
1.4
2
Marks: 50
y
1
2
x 1
y  5
y
1
2
x3
1.16
2
x
y   x  1
2
y  x 2  2x  8
1.17
y  2 x  3
1.18
x 2  y 2  25
(18)
Now indicate whether it is a straight line, parabola, hyperbola or any other function by filling the
algebraic equations in the correct column in the table below:
Straight line
Parabola
Hyperbola
Other
3
What do the equations representing straight line graphs have in common?
(4)
(3)
4
What do the equations representing parabolas have in common?
(3)
5
What do the equations representing hyperbolas have in common?
(3)
6
Write down 4 other formulas that each make : straight lines, parabolas or hyperbolas .
(4)
7
Is the graph of y  3x 2  2 x  1 a line, a parabola or some other shape? Explain.
(3)
8
Is the graph of y  3x 3  2 x  1 a line, a parabola or some other shape? Explain
(3)
9
What do you notice about the graphs of the following equations:
y
1
1
, y   2 , y  x2 , y  x2  2
x
x
Grade 10
- 27 -
(4)
Exemplar Assessments 2012
10.
Make a conjecture about the effect of the ‘+2’.
Project: Finance
(3)
Marks: 50
Section A – Exchange Rates ( 2 marks for each correct answer =30)
The table below shows the average Rand (R)/US dollar ($) exchange rate from 2000 to 2007. The figure
given under the column “Exchange Rate” is how many Rands were required to get one US Dollar.
YEAR
2000
EXCHANGE RATE
6.94
2001
8.58
2002
10.52
2003
7.57
2004
6.45
2005
6.37
2006
6.78
2007
7.06
1
In 2006, a book published in America cost $15.00. How much would you expect to pay for
the book
in South Africa?
2
You go to a bookshop and see the book on the shelf. It costs R136.00. Is this more or less than
you expected? Suggest some reasons for the difference between the price you expected to pay
and the marked price of the book.
3
In which year would you expect the book to cost the least? Substantiate your answer.
4
much
In 2007, A T-shirt made in South Africa costs R95.00. The shirt is exported to America. How
would it cost in $?
5
If you were running a clothing factory where the clothes were made from imported
American cotton,
in which year would the exchange rate have been best for you and in
which year would it have been worst? Explain.
6
If you were running a business that exported South African chocolates to America, in which year
would the exchange rate have been the best for you and in which year would have been the worst?
Explain.
7
The banks only give a few exchange rates in the same format as the table above, i.e. how many
Rand you need to buy one unit of foreign currency ($, €uros (€) and British Pounds (£)). All other
exchange rates are quoted in the amount of foreign currency that can be purchased with R1.00. In
2007, the Rand/Australian dollar (A$) exchange rate was R1.00 to A$ 0.17. If you went to the
with R100, how many Australian
dollars would you be able to get?
bank
8
How much would you need in R to get A$ 1? Is this a better or worse exchange rate than the
Rand/Dollar exchange rate in 2007? Explain your answer.
9
Work out the exchange rate between Rands and US dollars in the same format as the exchange
rate for Australian dollars, i.e. work out how many US dollars you can get for R1.00.
Grade 10
- 28 -
Exemplar Assessments 2012
10
Using the 2007 exchange rates for Australian dollars and US dollars, work out how many
Australian
dollars you would need to get 1 US dollar.
11
In December 2007, the price of crude oil, which is used to make petrol, was $92.00 per barrel.
How much did South Africa pay for a barrel of oil and how much did Australia
pay for a barrel
of oil?
12
In 2007, the exchange rate between the Rand and the British pound was R14.13 to 1£. The
Rand/US dollar exchange rate was R7.06 to 1$. Estimate the exchange rate between the US dollar
and the British pound.
13
the
You are visiting America and decide to buy a Big Mac burger for lunch. It costs $3.20. What is
Rand equivalent of the Big Mac Burger. (Use the 2007 exchange rate given in the table above.)
14
While you are eating your Big Mac, you compare what you have just paid in Rands for your
burger with what you would have paid in South Africa (R15.50). Is the Big Mac more or less expensive
in
America?
15
Experiment with exchange rates, using any method you think suitable, to work out a
Rand/Dollar exchange rate that would result in the price of a Big Mac in America being
equivalent to the price that you pay for a Big Mac in South Africa. Does the official
Rand/Dollar exchange rate undervalue or overvalue that Rand?
Section B – Inflation and Interest (20)
1
In 2000, a pair of track shoes costs R450. The inflation rate is 5,4%. What would you expect the
shoes to cost in 2007?
(3)
2
The table below gives that actual annual inflation rates from 2000 to 2007. Use the table to work
out the 2007 price of the track shoes. How does this compare with your answer in 1. Explain why
your answers are different.
(4)
YEAR
2000
2001
2002
2003
2004
2005
2006
2007
INLFATION
RATE (%)
5,4
5,7
9,2
5,8
1,4
3,4
4,7
7,7
3
You decide to buy a cellphone costing R890. You have saved R320 towards the cost of the
cellphone and your parents have agreed to lend you the balance at 8,5% per year simple interest.
You will pay your parents back in equal monthly payments for a period of 2 years. Work out your
monthly repayments.
(4)
4
your
Instead of borrowing money, you decide that you are going to save for two years and then buy
new cellphone.
4.1
You invest the R320 that you have in a special savings account which pays 8% per
annum interest, compounded monthly. Calculate the balance in this account after 2 years.
Grade 10
- 29 -
Exemplar Assessments 2012
4.2
4.3
4.4
After six months, you deposit R170 into an ordinary savings account, at 7.5% interest per
annum, compounded six monthly. Six months later, you deposit R160 and six months later you
deposit R170 into the same account. Calculate the balance in this account at the end of two years
(3)
During the two-year period that you are saving, the rate of inflation is 6,7% per annum.
What will the cost of the cellphone be at the end of the two years?
(3)
Using your results from questions 4.1 – 4.3, determine whether or not you will have
enough money to pay cash for the cellphone.
Grade 10 Project: Shape, Space and Measurement
1
Marks: 75
On a sheet of unlined paper, construct line AB = 9cm.
Now construct line DC so that DC is parallel to AB and equal
to AB. Join B to C and A to D.
(2)
1.1
What type of quadrilateral is ABCD?
(1)
1.2
Write down four conjectures about ABCD that involve either equal lines
or equal angles.
(4)
1.3
2
(3)
Confirm each conjecture by measuring the lines or angles and write
down the measurements.
Using a compass, construct a circle with a radius of 5cm. Label the centre of
the circle K.
2.1
2.2
2.3
2.4
Draw any two diameters of the circle
and label them PQ and RS as
shown in the diagram.
P
(1)
(2)
R
How long is PQ? Explain how
you know this without
measuring PQ.
(2)
K
Which other lines equal PK?
Give a reason for this.
Join points P, R, Q and S.
What type of triangle is
 PKR? Give a reason.
(4)
[11]
(2)
S
Q
(2)
2.5
What type of triangle is  PKS?
2.6
Prove that SP̂R  90 .
(4)
2.7
Are there any other angles that are 90 ? If so, name them.
(2)
2.8
Do you think that you have proved that quadrilateral PRQS is a
rectangle. Explain.
Grade 10
Give a reason.
- 30 -
(2)
(2)
[19]
Exemplar Assessments 2012
3
On the grid provided, plot points F(1;1) and G(6;1).
(1)
(0;0)
4
3.1
What is the length of FG?
(1)
3.2
Plot point E so that FE is the same length as FG and the coordinates of
E are integers, but FE is not parallel to the y-axis. What are the
coordinates of E? Explain the method you used to plot E.
(3)
3.3
Plot point H so that EFGH is a rhombus. What are the coordinates of H. (1)
3.4
Which property of a rhombus did you use to draw EFGH?
3.5
Draw the diagonals FH and DE. Using coordinate geometry, prove that
the diagonals bisect each other.
(1)
(3)
[10]
What is the definition of a regular polygon? Using your definition of a regular polygon, decide
which of the following are regular polygons. In each case, you must state which requirements
of your definition are true (if any) and which are not true (if any). Make use of diagrams
to illustrate your answers.
4.1
Scalene triangle
(2)
4.2
Rhombus
(2)
4.3
Isosceles trapezium
(2)
4.4
Isosceles triangle
(2)
4.5
Square
(2)
4.6
Kite
(2)
4.7
Parallelogram
(2)
4.8
Equilateral triangle
(2)
[16]
Grade 10
- 31 -
Exemplar Assessments 2012
5
On the grid paper, plot the points A(-1;2) B(0;-5) and (4;7). Join the points.
(4)
5.1
What type of triangle is  ABC?
(1)
5.2
Prove the conjecture you have made in 5.1.
(5)
Note: The coordinates of A, B and C are all integers and no integer is used more
than once. Use this rule to answer the next question.
5.3
5.4
On the grid, plot points J, K, L and M so that JKLM is a kite. Give the
coordinates of the four points you have plotted.
(4)
Prove that JKLM is a kite.
(5)
[19]
Grade 10 Project: Trigonometry
Marks: 85
You are reminded of the definitions of sine, cosine and tangent, abbreviated as sin, cos and tan:
In the right angled triangle below:
side opposite  QR

hypotenuse
PR
side adjacent to  PQ
cos  

hypotenuse
PR
side opposite 
QR
tan  

side adjacent to  PQ
side opposite  PQ
,

hypotenuse
PR
side adjacent to  QR
,
cos  

hypotenuse
PR
side opposite 
PQ
tan  

,
side adjacent to  QR
sin  
sin  
P

hypotenuse
side opposite 
 side adjacent to 

side adjacent to 
side opposite
Q
R
A
Task 1
E
B

C
D
1.1
Name all the similar triangles in the sketch above.
1.2
Given that BD = 8 units, DC = 4 units and AD  4 2 units, calculate the lengths
Grade 10
- 32 -
(3)
Exemplar Assessments 2012
of all the other line segments in the sketch. Leave your answers in surd form
(5)
Express sin  , cos  and tan  in as many different ways as possible.
1.3
For example sin  
BA AD

 ...
BC AC
(15)
Given that ABˆ C   , express sin  , cos  and tan  in as many different
ways as possible.
1.4
(15)
[38]
Task 2
The three circles have radii 2, 3 and 5 units.
4
B3
A3
B2
A2
2
A1
B1
D
-5
5
C1
C2
-2
C3
-4
2.1
Read, as accurately as possible, the co-ordinates of the points marked:
A1 , A2 , A3 ,...C3 and hence complete the following table (work correct to 1 decimal place).:
x coordinate
y coordinate
x
r
y
r
y
x
A1
A2
A3
B1
B2
B3
C1
C2
C3
Grade 10
- 33 -
Exemplar Assessments 2012
(10)
2.2
Write any observations about what you have read and calculated using the
co-ordinates of the nine points.
(9)
[19]
Task 3
3.1
3.2
3.3
ˆ D , BOˆ D and COˆ D and use your calculator to determine the sine,
Measure AO
cosine and tangent of each of these three angles.
(6)
How, if at all, are the ratios determined in the previous question related to the
values in the table completed in task 2?
Use your calculator to investigate
3.3.1.
the maximum and minimum values (if they exist) of sin  , cos
and tan  for any values of  (try multiples of 10 0 )
(10)
3.3.2
the value/s of sin 2   cos 2  for at least 5 of values of  (use   0 0 ,
some positive values and some negative values).
(2)
3.3.3
the values of sin  , cos  ,
values of 
3.4
(2)
sin 
and tan  for at least 5
cos 
(4)
Write any observations about the results you obtained through your calculator
work.
(4)
[28]
Grade 10
- 34 -
Exemplar Assessments 2012