Mathematics/P2
2
DoE/Feb. – March 2011
NSC
PAGE
MARKS: 150
TIME: 3 hours
This question paper consists of 9 pages, 5 diagram sheets and 1 information sheet.
INSTRUCTIONS AND INFORMATION
Read the following instructions carefully before answering the questions.
1.
2.
3.
4.
This question paper consists of 12 questions.
Answer ALL the questions.
Clearly show ALL calculations, diagrams, graphs, et cetera that you have used in determining the answers.
Answers only will not necessarily be awarded full marks.
5.
6.
7.
An approved scientific calculator (non-programmable and non-graphical) may be used, unless stated otherwise.
Round your answers off to TWO decimal places if necessary, unless stated otherwise.
Diagrams are NOT necessarily drawn to scale.
8.
FIVE diagram sheets for QUESTION 1.2, QUESTION 2.1, QUESTION 2.2, QUESTION 3.1, QUESTION 8.1 and QUESTION 12.3 are attached at the end of this question paper. Write your centre number and examination number on these sheets in the spaces provided and insert them inside the back cover of your ANSWER BOOK.
9.
10.
11.
An information sheet, with formulae, is included at the end of this question paper.
Number the answers correctly according to the numbering system used in this question paper.
Write legibly and present your work neatly.
QUESTION 1
The table below gives a breakdown of the PSL log standings for the 8 top teams at the end of 2008/2009.
POSITION
TEAM
POINTS
1
SuperSport
55
2
Orlando Pirates
55
3
Kaizer Chiefs
50
4
Free State Stars
47
5
Golden Arrows
x
6
Bidvits Wits
x
7
Ajax Cape Town
x
8
Amazulu
42
[Source: http://www.safa-_psl log]
1.1
If the average points for these 8 teams is 48,375, show that
.
(2)
1.2
Draw a box and whisker diagram of the information given on DIAGRAM SHEET 1.
(4)
[6]
QUESTION 2
The individual masses (in kg) of 25 rugby players are given below:
78 102 88 93 81 90 75 60 76 75
68 90 80 77 81 69 60 83 91 100
80 70 81 64 70
2.1
Complete the following table on DIAGRAM SHEET 1
MASS (kg)
FREQUENCY
CUMULATIVE FREQUENCY
60
70
80
90
100
(4)
2.2
Draw an ogive (cumulative frequency curve) of the above information on the grid provided on diagram sheet 2.
(3)
2.3
Calculate the mean mass of the rugby players.
(2)
2.4
How many rugby players have masses within one standard deviation of the mean? From your calculations, calculate the percentage of the rugby players who have masses within one standard deviation of the mean.
(5)
[14]
QUESTION 3
A group of 12 learners was asked to measure their arm span (from fingertip to fingertip) and their height. The data below was gathered.
Arm span (cm)
156
157
160
161
162
165
170
177
184
188
188
194
Height (cm)
162
160
155
160
170
166
170
176
180
187
192
193
3.1
Represent the data as a scatter plot on the grid provided on DIAGRAM SHEET 3.
(4)
3.2
Draw a line of best fit for this scatter plot.
(2)
3.3
Would you expect a person with below average arm span to be below average in height? Give a reason for your answer.
(2)
[8]
QUESTION 4
In the diagram below
PQR with vertices P(– 1 ; 2), Q(– 2 ; – 2) and R(3 ; 0) is given.
4.1
Calculate the angle that PQ makes with the positive x-axis.
(3)
4.2
Determine the coordinates of M, the midpoint of PR.
(2)
4.3
Determine the perimeter of
PQR to the nearest whole number.
(5)
4.4
Determine an equation of the line parallel to PQ that passes through M.
(3)
[13]
QUESTION 5
5.1
The equation of a circle is
.
5.1.1
Prove that the point (2 ; – 9) is on the circumference of the circle.
(2)
5.1.2
Determine an equation of the tangent to the circle at the point (2 ; – 9).
(7)
5.2
Calculate the length of the tangent AB drawn from the point A(6 ; 4) to the circle with equation
.
(5)
[14]
QUESTION 6
The circle, with centre A and equation
is given in the following diagram. B is a y-intercept of the circle.
6.1
Determine the coordinates of B.
(4)
6.2
Write down the coordinates of C, if C is the reflection of B in the line x = 3.
(2)
6.3
The circle is enlarged through the origin by a factor of
.
Write down the equation of the new circle, centre A/, in the form
.
(2)
6.4
In addition to the circle with centre A and equation
, you are given the circle
with centre B.
6.4.1
Calculate the distance between the centres A and B.
(2)
6.4.2
In how many points do these two circles intersect? Justify your answer.
(2)
[12]
QUESTION 7
The point (x ; 2) is rotated about the origin through an angle of 150( in an anticlockwise direction to give the point (– 3 ; y). Calculate the values of x and y.
[5]
QUESTION 8
In the diagram below
MNP is given with vertices M(– 5 ; 2), N (6 ; 4) and P(2 ; – 4).
MNP is enlarged by a factor of 1,5 to
.
8.1
Draw
on the grid provided on DIAGRAM SHEET 4.
(3)
8.2
Write down the values of:
8.2.1
(2)
8.2.2
(2)
8.3
If the above transformation is applied to
n more times to get the image
, write down the value of
.
(2)
[9]
QUESTION 9
Consider the point A (– 12 ; 6). The point is reflected about the x-axis to A/.
9.1
Write down the coordinates of A/.
(1)
9.2
An alternative transformation from A to A/ is a rotation about the origin through α°, where
. Calculate α.
(6)
[7]
QUESTION 10
10.1
If sin 28° = a and cos 32°= b, determine the following in terms of a and/or b :
10.1.1
(2)
10.1.2
(3)
10.1.3
sin 4°
(4)
10.2
Prove without the use of a calculator, that if sin 28° = a and cos 32° = b, then
.
(4)
10.3
Evaluate each of the following without using a calculator. Show ALL working.
10.3.1
(7)
10.3.2
(4)
10.4
Determine the general solution of:
(7)
10.5
Consider:
10.5.1
For which values of x,
, will this expression be undefined?
(3)
10.5.2
Prove that
for all other values of x.
(5)
[39]
QUESTION 11
The sketch below shows one side of the elevation of a house. Some dimensions (in metres) are indicated on the figure.
Calculate, rounded off to ONE decimal place:
11.1
EC
(3)
11.2
(3)
11.3
Area of
DEC
(2)
11.4
The height EF
(3)
[11]
QUESTION 12
The graph of
is drawn below.
12.1
Write down the period of f.
(1)
12.2
Write down the amplitude of h if
.
(2)
12.3
Draw the graph of
for
on the grid provided on DIAGRAM SHEET 5.
(3)
12.4
Use the graph to determine the number of solutions for
,
.
(1)
12.5
For which values of x is g(x) ( 0?
(2)
12.6
For which values of x is
and
?
(3)
[12]
TOTAL:
150
CENTRE NUMBER:
EXAMINATION NUMBER:
DIAGRAM SHEET 1
QUESTION 1.2
QUESTION 2.1
MASS
(kg)
FREQUENCY
CUMULATIVE FREQUENCY
60
70
80
90
100
CENTRE NUMBER:
EXAMINATION NUMBER:
DIAGRAM SHEET 2
QUESTION 2.2
CENTRE NUMBER:
EXAMINATION NUMBER:
DIAGRAM SHEET 3
QUESTION 3.1
CENTRE NUMBER:
EXAMINATION NUMBER:
DIAGRAM SHEET 4
QUESTION 8.1
CENTRE NUMBER:
EXAMINATION NUMBER:
DIAGRAM SHEET 5
QUESTION 12.3
INFORMATION SHEET: MATHEMATICS
;
;
M
In (ABC:
P(A or B) = P(A) + P(B) – P(A and B)
MATHEMATICS P2
FEBRUARY/MARCH 2011
B
C
A
M(– 5 ; 2)
GRADE 12
NATIONAL
SENIOR CERTIFICATE
A
B
C
E
D
F
G
7,5
3,5
9,4
� EMBED Equation.3 ���
f
P(2 ; – 4)
N(6 ; 4)
N(6 ; 4)
P(2 ; – 4)
M(– 5 ; 2)
Copyright Reserved
Please turn over
Copyright Reserved
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Chart1
0
5
12
19
23
25
Mass (kg)
Cumulative frequency
CUMULATIVE FREQUENCY CURVE
Sheet1
60 0
70 5
80 12
90 19
100 23
110 25
Sheet1
0
0
0
0
0
0
Mass (kg)
Frequency
Cumulative Frequency Curve
Sheet2
Sheet3
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_1352717867.xls
Chart1
162
160
155
160
170
166
170
176
180
187
192
193
Arm span (cm)
Height (cm)
Scatter plot showing arm span vs height
Sheet1
156 162
157 160
160 155
161 160
162 170
165 166
170 170
177 176
184 180
188 187
188 192
194 193
Sheet1
0
0
0
0
0
0
0
0
0
0
0
0
Arm Span (cms)
Height (cms)
Scatter Plot showing Arm Span vs Height
Sheet2
Sheet3
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