1
Grade 12 tasks:
Grade 12 learners must complete 7 formal assessment tasks.
Term 1: an assisgment; a project or investigation and a test = 3 SBA tasks
Term 2: a test and the midyear examination
= 2 SBA tasks
Term 3: a test and a midyear examination.
= 2 SBA tasks
These exemplars will not deal with the two examinations as we are still waiting for the
national examination guidelines. The test for each term may be compiled from the topical
assignments.
Term 1
Topical Assignments
Topic 1; Sequences and series
Question 1
1.1
Consider the sequence:
6 ; 10 ; 16 ; 24 ; 34 ; …
1.1.1 If the sequence behaves consistently, determine the next two terms of
the sequence.
1.1.2 Calculate a formula for the nth term of the sequence.
1.1.3 Use your formula to calculate n if the nth term in the sequence is 1264.
(2)
(5)
(3)
1.2
Consider the following sequence: 6; 18; 54; 162 ……..
1.2.1 If the formula for the general term of the sequence is Tn  ar n1 , give the
value of a and r.
1.2.2 Which term of the sequence is equal to 1458?
(2)
(4)
Given the geometric series: 5. (3) 4 + 5. (3) 3 + 5. (3) 2 + ……..
1.3.1 Explain why the series converges.
1.3.2 Calculate the sum to infinity of the series.
1.3.3 Calculate the sum of the first 9 terms of the series, correct to TWO
decimal places.
1.3.4
(2)
(3)
(4)
Use your answers to QUESTION 1.3.2 and QUESTION 1.3.3 to determine

 5.(3)
5 n
(correct to TWO decimal places)
n 10
1
(2)
2
Question 2
2.1
2.2
2.2.1
2.2.2
3x + 1 ; 2x ; 3x – 7 are the first three terms of an arithmetic sequence.
Calculate the value of x.
The first and second terms of an arithmetic sequence are 10 and 6 respectively.
Calculate the 11th term of the sequence.
The sum of the first n terms of this sequence is –560. Calculate n.
(2)
(2)
(6)
2.3
In the diagram below, the first (outer) triangle is equilateral triangle with sides of 8
units. Another equilateral triangle in drawn within this triangle, by joining the
midpoints of the sides of the outer triangle.
This process is continued without end.
2.3.1 Calculate the perimeter of the fourth triangle.
(2)
2.3.2 Show that the sum of the perimeters of all the
inner triangles will never exceed the perimeter
of the outer triangle.
(6)
Question 3
3.1
3.1.1
3.1.2
3.2
3.2.1
3.2.2
3.3.3
Given the sequence: 4 ; x ; 32
Determine the value(s) of x if the sequence is:
Arithmetic
Geometric
The following sequence is a combination of an arithmetic and a geometric sequence:
3 ; 3 ; 9 ; 6 ; 15 ; 12 ; …
Write down the next TWO terms.
Calculate T52  T51 .
Prove that ALL the terms of this infinite sequence will be divisible by 3.
QUESTION 4
4.1
Given the arithmetic series: – 7 – 3 + 1 + … + 173
4.1.1
How many terms are there in the series?
4.1.2
Calculate the sum of the series.
4.1.3
Write the series in sigma notation.
4.2
4.2.1
4.2.2
4.2.3
4.3
Consider the geometric series: 4 + – 2 + 1 + …
Write the series in sigma notation
1
Determine n if the nth term is
.
64
Calculate the sum to infinity of the series …
If x is a REAL number, show that the following sequence can NOT be geometric:
1;x+1;x–3…
2
(2)
(3)
(4)
(5)
(4)
(4)
(3)
(2)
(3)
(4)
(2)
(4)
3
Topic 2: Inverse functions
Question 1
1.1
1.1.1
1.1.2
1.1.3
1.1.4
1.1.5
Say whether the statements that follow are true or false. Give a reason for your
answer.
The inverse of 𝑓 = {(2; 3); (4; 7)} is {(3: 2); (7; 4)}
𝑓 = {(2; −3); (4; 6); (−2; −3); (6; 4)} is a many to one function
The inverse of 1.1.2 is a function
The domain of 1.1.2 is 𝐷 = {2; 4; 6}
The function 𝑓 and its inverse 𝑓 −1 are reflections in the line 𝑦 = −𝑥
(2)
(2)
(2)
(2)
(2)
1
1.2 Given 𝑓(𝑥) = − 2 𝑥 2 .
1.2.1 Determine 𝑡ℎ𝑒 𝑖𝑛𝑣𝑒𝑟𝑠𝑒 𝑜𝑓 𝑓(𝑥)
1.2.2 Is 𝑡ℎ𝑒 𝑖𝑛𝑣𝑒𝑟𝑠𝑒 𝑜𝑓 𝑓(𝑥) a function or not? Give a reason for your answer.
1.2.3 How will you restrict the domain of the original function to ensure
that 𝑓 −1 (𝑥) will be a function?
1.2.4 Draw 𝑓(𝑥)𝑎𝑛𝑑 𝑓 −1 (𝑥) on the same set of axes
1.2.5 Determine the point(s) of intersection of 𝑓(𝑥)𝑎𝑛𝑑 𝑓 −1 (𝑥)
(3)
(2)
(1)
(3)
(4)
Question 2
2.1 Determine the inverses of the following functions:
2
2.1.1 𝑓(𝑥) = 3 𝑥
2.1.2 𝑔(𝑥) = −3𝑥 − 9
1.4
Determine the point(s) at which 𝑓 −1 𝑎𝑛𝑑 𝑔−1 in question 1.3 will intersect.
1.5
Given the function f ( x)  x 2 .
5.1
Determine f 1 ( x) .
5.2
5.3
5.4
6.1
(3)
1
Sketch the graph of f ( x) .
(2)
1
Explain why f ( x) is not a function.
Explain how you would restrict the domain of f (x)
(1)
so that f 1 ( x) is a function.
(2)
The figure represents the graph of f ( x)  a x .
Calculate the value of a.
(2)
6.2.2 Draw a sketch graph of k (x) if k is the inverse
of f . Indicate the intercept(s), the coordinates
of one other point and the asymptote(s).
3
(0;0)
4
Question 2
y
2.1
The diagram alongside represents the functions:
.
𝑥
𝑓(𝑥) = 𝑘 ;
𝑝(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
𝑎𝑛𝑑 𝑔(𝑥) = 𝑙𝑜𝑔𝑚 𝑥
. Q(1 ; 3)
G.
.
.H . E
The minimum value of the function
𝑝(𝑥) equals 1 when 𝑥 = 3.
The turning point of the parabola is
at the point F.
EF is drawn parallel to the y-axis.
2.1.1
2.1.2
2.1.3
2.1.4
p(x)
f(x)
g(x)
x
F
Destermine the value of a, b, c, m and k.
(5)
Calculate the length of EF and GH correct to two decimal places
(3)
Determine the equations of 𝑓 −1 (𝑥)𝑎𝑛𝑑 𝑔−1 (𝑥)
(4)
Hence explain why of why not 𝑓(𝑥)𝑎𝑛𝑑 𝑔(𝑥) are symmetrical; with respect to the
line 𝑦 = 𝑥
(2)
y
2.2
The graph alongside shows the functions g, f and h.
f and g are symmetrical with respect to the y-axis
f and h are symmetrical with respect to the line
y = x. If 𝑓(𝑥) = 𝑎 𝑥 and the point (1 ; 4) lies on 𝑓(𝑥)
2.2.1
2.2.2
2.2.3
f
g
determine the value of a
(2)
write down the coordinates of P and Q (2)
write down the equations of 𝑔, ℎ 𝑎𝑛𝑑 𝑔−1 (6)
=…
(2)
in the
(3)
2.3.3 Write down the domain of
2.3.4 Draw graphs of both
same system of axes
x
Q
2.3
Given:
2.3.1 Explain why, unless the domain of this function is restricted, its inverse is
not a function.
2.3.2 Write down the equation of inverse,
of
for
form
h
P
(1)
for
and
on the
(4)
4
5
Topic 3: Present and Future value formulae
Question 1
1.
1.1
1.2
1.3
1.4
1.5
Felicity purchases a new house. The price of the house is R 825 000. She pays 20%
deposit and takes out a bond over 20 years to pay the balance.
Calculate her monthly instalments if the bank charges 11,5% p.a. compounded
monthly
(4)
Calculate the total amount she repays on the house.
(3)
If she increases his monthly repayments by R500, how long will he take to repay the
loan.
(4)
How much will he save by increasing his monthly instalment by R 500
(2)
If she wins the lotto and wishes to settle her bond after 12 years, calculate the
outstanding balance
(4)
Question 2
2
2.1
2.2
2.3
2.4
The Mathematics, Science and Technology Academy has just bought a luxury school
bus R 750 000. The intention is to replace this bus after 8 years. The bus depreciates
at a rate of 15% p.a. calculated on a reducing balance.
What is the trade-in value of the bus after 8 years.
(3)
Due to inflation the purchase price of a new bus in 8 years’ time will increase by 25%
p.a. simple interest. Calculate the purchase price of the new bus
(3)
The school wants to pay cash for the new bus after trading in the old bus.
Calculate the monthly deposits into a sinking fund order to pay cash pay for the new
bus after 8 years. This investment renders interest at 8,5% p.a. compounded monthly.
(4)
Suppose that 12 months after the purchase of the present bus and every 12 months
thereafter, the school withdraws R5 000 from his account, to pay for the maintenance
of the new bus. They makes 5 such withdrawals, what will the new monthly deposit
be?
(6)
Question 3
A construction worker wants to save a monthly amount towards his retirement. His bank
offers 11,5% p.a. compounded monthly.
3.1
If he pays R250 per month for the next 10 years, how much will he receive at the end
of this period.
(4)
3.2
If he wants to save R100 000 over this period, how much must he deposit monthly?
(4)
3.3
How long will it take him to save R100 000 if he saves R1000 monthly.
(4)
5
6
Topic 4 Compound Angles
Question 1
1.1
If 4 tanθ = 3 and 1800<θ<360determine using a diagram:
1.1.1 sinθ + cosθ
1.1.2 sin 2θ and cos 2θ
1.1.3 tan 2θ
3
8
and cos   
where  ,   [90 0 ;180 0 ] . Calculate the value
5
17
of the following without using a calculator:
1.2.1 sin    
(6)
1.2.2 tan    
(4)
1.2
Given: sin   
1.3
If sin 27 0  t , Express each of the following in terms of t :
1.3.1 cos 27 0
1.3.2 tan 153 0
1.3.3 cos 2430
cos 54 0
(10)
0
1.4
If sin 36 .cos 12= p and cos 36.sin 12= q, determine in terms of p and q the value of:
1.4.1 sin 48°
1.4.2 sin 24°
1.4.3 cos 24°
(8)
1.3.4
Question 2




2.1
Solve: cos 2 x  sin x  30 0 waar x   180 0 ;180 0
2.2
Determine the general solution:
2.2.1 tan 3x  750  1  0
2.2.2 1  sin 2  4 sin 2   0 , correct to 1 decimal place if necessary.





(6)
(6)

Solve the equation cos 2 x  sin x  30 0 for x   180 0 ;180 0
(6)
0
2.2.2 Sketch graphs of f x   cos 2 x and g x   sin x  30 on the same system of
axes for x   180 0 ;180 0 . Show the co-ordinates of all points of intersection with
the axes, all turning points and all points at which f x   g x 
(8)
2.2.1

2.3

Prove that :
2.3.1
sin 3x cos 3x

2
sin x
cos x
2.3.2
sin 4x  sin 2x
 tan x
cos 4x  cos 2x


(5)
(7)
6
7
Term 2
Topic 1: heights and distances
H
Question 1
1.1
A, B and L are points in the same horizontal plane.
HL is is a vertical pole with lenght 3 meter,
AL = 5,2 m, and AL̂B = 113°
The angle of elevation from B to H is 40°.
3m
L
5,2m
1.1.1
1.1.2
Calculate the length of LB.
Hence calculate the lenght of AB.
(2)
(4)
1.2
The diagram below represents the course of a swimming race in a bay on the coast.
 P is the starting and finishing point;
 A is a buoy, 300 metres from P on a bearing of 120 0
 B is a buoy, 500 metres from P on a bearing of 240 0 .
B
A
North
P 120
500 metres
240
300 metres
B
A
Calculate:
1.2.1 the bearing of the point P from the buoy, B and
(1)
1.2.2 the distance competitors must swim (from P to A, then to B and then back to P). (6)
1.3
In ABC , ADˆ C   , DA  DC  r , BD  2r ,
AC  k and BA  2k
A
k
2k
r

B
2r
D
1.3.1 In ADC , express cos in terms of r and k
1.3.2 In ABD , express cos in terms of r and k
1
1.3.3 Hence show that cos  
4
7
r
C
(2)
(3)
(3)
8
Question 2
2.1
AB .is a vertical tower in a horizontal plane BCD. The angle of elevation of A from C
is  , ACˆ D   , ADˆ C   and the distance CD  x metres.
A
B

C


D
x metres
x sin  sin 
(5)
sin    
2.1.2 Calculate the height of the tower if x = 40 metres ,   50 0 ,   70 0 and   15 0 .(2)
2.1.3 Calculate the area of ACD
(3)
2.1.1 Prove that the height of the tower AB =
2.2
AC represents a vertical tower, with A at
the top and C at the foot of the tower.
D is a point on the tower h metres below A.
On the same horizontal level as C is point
B, p metres from C. The angles of
depression of A and D to B are x and y
respectively.
2.2.1 Prove that p =
h cos x . cos y
.
sin (x  y )
(6)
2.2.2 Now calculate the value of p if h = 50 m; x = 32,3
Give the answer to one decimal place.
and y = 25,8.
(3)
8
9
Topic 2: Calculus
Question 1
1.1
1.2
2
Given f ( x )   , determine f ' ( x) from first principles.
x
Determine the derivative of:
1.2.1 f ( x)  7 x3  4 x  6
1
1.2.2 f ( x)  3 x 
3x
(5)
(3)
(4)
Question 2
Given:
f ( x)  2 x 3  x 2  4 x  3
2.1
Show that ( x  1 ) is a factor of f (x).
2.2
Hence factorise f (x) completely.
2.3
Determine the co-ordinates of the turning points of f.
2.4
Draw a neat sketch graph of f indicating the co-ordinates of the turning points as
well as the x-intercepts.
2.5
For which value of x will f have a point of inflection?
(2)
(2)
(4)
(4)
(4)
Question 3
A cubic function f has the following properties:
1

f    f (3)  f (1)  0
2
 1

f (2)  f     0
 3
 1 

f decreases for x   ; 2 only
 3 
Draw a possible sketch graph of f, clearly indicating the x-coordinates of the turning points
and ALL the x-intercepts.
(5)
Question 4
The graph of the function f ( x)   x 3  x 2  16 x  16 is sketched below.
y
f
x
0
4.1
4.2
Calculate the x-coordinates of the turning points of f.
Calculate the x-coordinate of the point at which f (x) is a maximum.
9
(4)
(3)
10
Question 5
The graph of y  f (x) , where f is a cubic function, is sketched below.
y

4

x
y = f /(x)
Use the graph to answer the following questions:
5.1
For which values of x is the graph of y  f (x) decreasing?
5.2
At which value of x does the graph of f have a local minimum? Give reasons for
your answer.
(1)
(3)
Question 6
6.1 A pasta company has packaged their spaghetti in a box that has the shape of a rectangular
prism as shown in the diagram below. The box has a volume of 540 cm3, a breadth of 4
cm and a length of x cm.
6.1.1 Express h in terms of x.
6.1.2 Hence show that the total surface area
of the box (in cm2 ) is given by:
A  8 x  1080 x 1  270
(3)
6.1.3 Determine the value of x for which the total
surface area is a minimum. Round the
answer off to the nearest cm.
(4)
(2)
h cm
x cm
4 cm
6.2 Water is flowing into a tank at a rate of 5 litres per minute. At the same time water flows
out of the tank at a rate of k litres per minute. The volume (in litres) of water in the tank at
time t (in minutes) is given by the formula V (t )  100  4t .
6.2.1
What is the initial volume of the water in the tank?
6.2.2
Write down TWO different expressions for the rate of change of the volume of water
in the tank.
6.2.3
Determine the value of k (that is, the rate at which water flows out of the tank).
(3)
(2)
6.3 A particle moves along a straight line. The distance, s, (in metres) of the particle from a
fixed point on the line at time t seconds ( t  0 ) is given by s(t )  2t 2  18t  45 .
6.3.1
Calculate the particle's initial velocity. (Velocity is the rate of change of distance.)
6.3.2
Determine the rate at which the velocity of the particle is changing at t seconds.
6.3.3
After how many seconds will the particle be closest to the fixed point?
(3)
(1)
(2)
10
(1)
11
Topic 3: Analytical geometry
Question 1
^y
A(0;3)
>x
C(5;0)
B( )
1.1
Show that ABC is a right angled isosceles triangle.
(6)
1.2
Determine the area of ABC
(3)
1.3
Given that BC is a diameter of the circumscribed circle of ABC , show that the
centre of
this circle is M, the point (1;1)
(2)
1.4
Calculate the equation of the circumscribed circle of ABC
(4)
1.5
Determine the equation of the tangent to the circle at C and show that this tangent is
parallel to MA.
(8)
Question 2
2.1
2.2
2.3
(5)
Show that if Q is the point Q( 0;7 ) the perpendicular bisector of PQ passes
Through the centre of the circle.
Does the point R (1; 2) lie on the circle, inside the circle or outside the circle?
Justify our answer
(6)
(3)
Question 3
3.1 Determine the equation of the tangent to the circle x2 + y2 = 10 at the point P(–1 ; 3).
(4)
3.2
Determine the equation of the tangent to the circle x2 + y2 + 2x – 4y = 20 at the
point (–4 ; –2).
(7)
3.3
Determine the coordinates of S, the fourth vertex of the parallelogram PQRS,
if P(–2 ; –1), Q(1 ; 6) and R(3 ; 4).
(4)
3.4
ABCD is a rhombus with diagonals meeting in K. A(0 ; 4), B(– 3 ; – 2)
and K(1½ ; 2½)
3.4.1 Determine the co-ordinates of C.
(2)
o
3.4.2 Show by analytical means that CK̂B  90 .
(4)
11
12
12
13
Question 4
4.1
The circle with centre O
and the straight line
intersect at the points
A(0 ; 5) and B(4 ; 3).
y
A(0 ; 5)
4.1.1 Determine the equation of
the circle.
(5)
4.1.2 Calculate the length of
chord AB.
(3)
B(4 ; 3)
x
O
y
4.2
C(1 ; 26)
In the figure B(1 ; 1) is the
centre of the circle. CA is
a tangent at A. C is the point
(1 ; 6). CB̂A  AR̂O  .
CA = 20 units.
Calculate:
4.2.1 the length of the radius of the
circle
(2)
4.2.2 the equation of the circle (3)
4.2.3 the equation of the tangent
CR
(4)
4.2.4 the equation of the radius AB
(4)
4.2.8 the co-ordinates of A
(4)
20
A

R

B(1 ; 1)
O
13
x
14
Term 3
Topic 1 Geometry
Question 1
1.1 In the diagram two circles intersect in A and C. BA is a tangent to the larger circle at
point A. Straight lines ATD and BCD intersect the circles in T and D, and C and D
respectively. The larger circle passes through centre O of the smaller circle.
Let B̂ = x.
1.1.1 Prove that D̂ = 180° – 2x.
1.1.2 Prove that AD = BD.
1.1.3Prove that TC  AB.
1.2
(4)
(5)
(2)
[17]
In the diagram below, two circles have a common tangent TAB. PT is a tangent to
the smaller circle. PAQ, QRT and NAR are straight lines.
Let Q̂  x.
T
1 2
1
R
2
P 1
3 2
3
4
5
2
A1
6
N
1.2.1
B
Name, with reasons, THREE other angles equal to x.
1.2.2
Prove that APTR is a cyclic quadrilateral.
14
x
Q
(5)
(5)
15
Question 2
2.1
In PQW, S is a point on PW and R is a point on QW such that SR || PQ.
T is a point on QW such that ST || PR. RT = 6 cm WS : SP = 3 : 2
P
S
Q
R
6 cm
W
T
Calculate:
2.1.1
WT
2.1.2
WQ
(3)
(4)
2.2 In the diagram PQRS is a parallelogram. Side RS is produced to W. WQ intersect PS in
X. M is a point on XQ so that MX = XW.
W
X
P
S
M
T
Q
R
Given MT // XS, PQ = 12 cm, WS = 4 cm
2.2.2 Determine the length of TR
XM
2.2.3 Determine the value of
XQ
15
(2)
(4)
16
Question 3
3.1
In the figure, AB is a tangent to the circle with centre O. AC = AO and BA // CE. DC
produced, cuts tangent BA at B.
1
E
1
2
O
A
234
2
1
34
B
F
2 3
1
C
21
ˆ
D
ˆ
3.1.1 Show that C 2  D1
3.1.2 Prove that  ACF ///  ADC
3.1.3 Prove that AD = 4 AF
3.2
(4)
(5)
(5)
In the diagram DA is a tangent to the circle ACBT at A. CT and AD are produced to
meet at P. BT is produced to cut PA at D. AC, CB, AB and AT are joined.
AC // BD. Let  1  X .
A
3 2X
1
C
2
1
1 2
2 3 4
1
T
1 D
2
P
B
3.2.1
3.2.2
3.2.3
3.2.4
Prove that  ABC ///  ADT.
Prove that PT is a tangent to the circle ADT at T
Prove that  APT ///  TPD.
2
If , 𝐴𝐷 = 3 𝐴𝑃show that 𝐴𝑃2 = 3𝑃𝑇 2
16
(5)
(3)
(5)
(4)
17
Topic 2: Statistics
QUESTION 4
As part of an environmental awareness initiative, learners of Greenside High School were
requested to collect newspapers for recycling. The cumulative frequency graph (ogive) below
shows the total weight of the newspapers (in kilograms) collected over a period of 6 months by
30 learners.
Weight of newspaper collected (in kilograms)
4.1
Determine the modal class of the weight of the newspapers collected.
(1)
4.2
Determine the median weight of the newspapers collected by this group of learners.
(1)
4.3
How many learners collected more than 60 kilograms of newspaper?
(2)
17
18
QUESTION 3
The length of time, in minutes, of a certain number of telephone calls was recorded. No call
lasted 25 minutes or longer. A cumulative frequency diagram of this data is shown below.
Cumulative frequency graph of duration of calls
45
40
Cumulative frequency
35
30
25
20
15
10
5
0
0
5
10
15
20
25
30
Time (in minutes)
3.1
3.2
3.3
3.4
Determine the total number of calls recorded.
Complete the frequency table for the data
Hence, draw a histogram for the data
Draw a box and whisker plot using the data on the ogive
18
(1)
(3)
(3)
(4)
19
QUESTION 4
In the grid below a, b, c, d, e, f and g represent values in a data set written in an increasing
order. No value in the data set is repeated.
a
b
c
d
e
f
g
Determine the value of a, b, c, d, e, f and g if:
 The maximum value is 42
 The range is 35
 The median is 23
 The difference between the median and the upper quartile is 14
 The interquartile range is 22
 e  2c
 The mean is 25
[7]
Topic 3: probability
QUESTION 1
1.1 Every client of CASHSAVE Bank has a personal identity number (PIN) which is made up
of 5 digits chosen from the digits 0 to 9.
1.1.1
How many personal identity numbers (PINs) can be made if:
(a)
Digits can be repeated
(b)
Digits cannot be repeated
1.1.2
Suppose that a PIN can be made up by selecting digits at random and that the digits
can be repeated. What is the probability that such a PIN will contain at least one 9?
1.2 Consider the digits 1, 2, 3, 4, 5, 6, 7 and 8 and answer the following questions:
1.2.1
How many 2-digit numbers can be formed if repetition is allowed?
1.2.2
How many 4-digit numbers can be formed if repetition is NOT allowed?
1.2.3
Determine the probability that numbers between 4 000 and 5 000 can be formed if
repetition is not allowed?
1.3
Consider the word PRODUCT
1.3.1 How many different letter arrangements are possible
(3)
1.3.2 How many different letter arrangements are possible if the first letter is T
(3)
and the fifth letter is C
1.3.3 What is the probability that different letter arrangements are formed with the letters R,
O and D following each other in any order.
(4)
19
(2)
(2)
(4)
[8]
(2)
(3)
(4)
[8]
20
Question 2
2.1
M1, M 2 and M 3 are 3 machines in a factory that manufactures nuts and bolts. They
2.1.1 Represents this data in a tree diagram.
2.1.2 Determine the probability that
(i).
The defective products are manufactures by machine M1
(ii)
The products are defective
2.2
(2)
(3)
[11]
A survey of 200 learners regarding their preferences for chicken, lamb or beef
yielded the following data:








90 prefers chicken as first choice.
64 prefers lamb as first choice
77 prefers beef as first choice
8 prefers all three types as first choice.
18 prefers beef and lamb.
27 prefers chicken and beef.
26 are vegetarians.
x number of learners prefers chicken and lamb above beef.
2.2.1 Represent this information ion a Venn diagram.
2.2.2 How many learners prefers chicken only
2.2.3 Determine the probability that a randomly chosen learner prefers chicken or beef
Question 3
The following contingency table shows the results of a survey amongs male and female
drivers stopped along a highway in Cape Town.
Speed fine
No speed fine
total
Male drivers
Female drivers
total
3.1
3.2
3.2.1
3.2.2
3.3
produce 2
(6)
45
35
80
25
45
70
70
80
150
How many persons participated in this study
(1)
Calculate the following probabilities
P(male drivers)
(2)
P(speed fines)
(2)
Are the events male driver and speed fine independent. Justify your answer through
calculations.
(5)
20
(6)
(4)
(3)
[13]
21
Topic 4: Regression
QUESTION 1
A group of students attended a course in Statistics on Saturdays over a period of 10 months.
The number of Saturdays on which a student was absent was recorded against the final mark the
student obtained. The information is shown in the table below and the scatter plot is drawn for
the data.
Number of Saturdays absent
Final mark (as %)
0
96
1
91
2
78
2
83
3
75
3
62
5
70
6
68
7
56
SCATTER PLOT SHOWING THE NUMBER OF SATURDAYS ABSENT
AND THE FINAL MARK ACHIEVED
100
Final Mark (as %)
90
80
70
60
50
0
1
2
3
4
5
6
7
8
Number of Saturdays absent
1.1
1.2
1.3
1.4
1.5
Calculate the equation of the least squares regression line.
Draw the least squares regression line on the grid provided on DIAGRAM SHEET 1.
Calculate the correlation coefficient.
Comment on the trend of the data.
Predict the final mark of a student who was absent for four Saturdays.
Question 2
A learner conducted an experiment to investigate the relationship between age and resting
heart rate (in beats per minute). He sought assistance of the local clinic. The information for
12 people is shon in the table below.
Age
Resting heart rate
(beats per minute)
2.1
59
88
32
74
42
74
50
93
22
85
Represent the data in a scatter plot
39
71
21
78
20
82
27
70
40
75
29
95
(3)
21
47
75
(4)
(2)
(2)
(2)
(2)
22
2.2
2.3
2.4
2.5
2.6
determine the equation of the least squares line
Draw the least squares line on the scatter plot
Calculate the correlation coefficient for the data
Use the correlation coefficient to comment on the relationship
between age and the resting heart rate
If a learner uses the least squares line to predict the resting
heart rate of a 45 year old person, will his answer be reliable?
Motivate your answer.
(4)
(2)
(2)
(2)
(2)
Question 3
The student enrolment at an FET college for the past 5 years is represented in the table
below: ( NB: we denote 2005 as year 1, 2006 as year 2, 2007 as year 3, 2008 as year 4 and
2009 as year 5, etc for convenience)
Year
Enrolment
1
60
2
65
3
145
4
220
5
312
3.1 Represent that data on a scatter plot
3.2 Determine the equation of the “least squares” ( regression) line for the data
3.3 Draw the least squares line for the data
3.4 Estimate the enrolment in 2012
3.5 Determine the correlation coefficient for the data. ( Explain what this value implies.)
(4)
(4)
(1)
(3)
(3)
Question 4
4.1
4.2
4.3
4.4
4.5
22