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YOUR NAME:
YOUR TEACHER:
PRELIMINARY EXAMINATION 2015
GRADE 12 MATHEMATICS PAPER 2
TIME: 3 HOURS
MARKS: 150
EXAMINER: M. Phungula
MODERATORS: H. Botes
S. Dzingwa
PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY
1. Write your name on top of this front page.
2. This question paper consists of 21 pages and a separate Information Sheet.
3. Answer ALL the questions on this question paper.
4. Diagrams are not drawn to scale.
5. You may use an approved non-programmable and non-graphical calculator, unless otherwise
stated.
6. Round off your answers to one decimal digit where necessary.
7. All the necessary working details must be clearly shown.
8. It is in your own interest to write legibly and to present your work neatly.
DO NOT WRITE IN THIS GRID
Q1 Q2 Q3
Q4
Q5 Q6 Q7 Q8
Q9
Q10 Q11 Q12 Q13
Q14
TOTAL
GRADE 12 PRELIMINARY EXAMINATION 2015
MATHEMATICS PAPER 2
Page 2 of 21
SECTION A
QUESTION 1
a)
In the diagram, A is the point (0;4) and B is the point (4;12). The straight line CAT
1
has a gradient of . KAB is a straight line.
3
Determine:
(1) 𝐶𝑇̂𝑋
(2)
ˆ 1
tan CTX
3
ˆ  18, 40
 CTX
(2)
𝐵𝐴̂𝐶
(4)
12  4
2
4
ˆ 2
tan BKO
ˆ  63, 40
BKO
mBAK 
ˆ  63, 4  18, 4  450
TAK
or
ˆ  BAC
ˆ  450
TAK
PRELIM 2015
ext angle
vert opp angles
GRADE 12 PRELIMINARY EXAMINATION 2015
MATHEMATICS PAPER 2
Page 3 of 21
b)
A wheel rolls down a hill represented by the straight line f.
y
x

A
A
f
The wheel starts in position A and has equation:
 x  0,5
2
B
  y  0,3  0, 25
2
The wheel rolls to a position B where its equation is then:
x 2  y 2  4 x  2 y  4,75  0
(1)
State the co-ordinates of the centre of the wheel when it is in position A.
(1)
(2)
Find the co-ordinates of the centre of the wheel when it is in position B.
(3)
 0,5
; 0,3
x 2  4 x   2  y 2  2 y  1  4, 75   2   12
2
 x  2    y  1
 C  2 ;  1
2
(3)
2
2
2
 0, 25
Find  , the angle of inclination of the hill.
A
1  0,3
mAB 
 0,86
2  0,5
tan   0,86
  139,10
PRELIM 2015
(4)
GRADE 12 PRELIMINARY EXAMINATION 2015
MATHEMATICS PAPER 2
Page 4 of 21
(4)
If the units are in metres, how far has the wheel travelled from A to B down the
hill?
d
(5)
(3)
 2  0,5   1  0,3
2
2
 1,98m
Through what angle has the wheel turned about its own axis?
(3)
circumference of wheel : 2  0,5  3,14
Angle :
1,98
 3,14  22, 60
3,14
[20]
QUESTION 2
a)
If sin  
2 6
and 90    270 , without the use of a calculator and with the use of a
5
diagram, calculate the value of :
5
2 6

(1)
tan θ
(3)
tan   2 6
(2)
cos 2θ
(3)
1  2sin 2 
2 6
1  2  

 5 
23
25
PRELIM 2015
2
GRADE 12 PRELIMINARY EXAMINATION 2015
MATHEMATICS PAPER 2
Page 5 of 21
b)
Simplify the following to a single trigonometric ratio of θ.
1  tan 2  . cos(  ). cos(180   ) if θ ∈ (0°; 90°)
1
(5)
sin 2 
 cos    cos  
cos 2 
1  sin 2 
cos 2 
cos 
c)
Evaluate


4sin15 cos15
without the use of a calculator. Show all working out.(4)
2 cos 405
4  2sin 30
2  cos 45
4  2  0.5
1
2
2
4
[15]
PRELIM 2015
GRADE 12 PRELIMINARY EXAMINATION 2015
MATHEMATICS PAPER 2
Page 6 of 21
QUESTION 3
A portable metal tubing support for a basketball net is to be constructed for an indoor court
with dimensions as shown in the diagram.
a)
Show that distance AC is 3, 26 m.
(2)
AC
1

3,1 sin 72
AC  3, 26
b)
If 𝐵𝐶̂ 𝐴 = 92° find the length of tubing needed for AB.
ˆ  60
BAC
(3)
 Bˆ  820
sin 82 sin 92

3, 26
AB
AB  3,3
[5]
PRELIM 2015
GRADE 12 PRELIMINARY EXAMINATION 2015
MATHEMATICS PAPER 2
Page 7 of 21
QUESTION 4
1) In the diagram, O is the centre of the circle. PWSR is a cyclic quadrilateral. PS,
WO and OS are drawn. PW||OS and 𝑃̂1 = 36°.
W
2 1
1
1
P
2
36
S
3
2
1
O
R
Calculate the sizes of the following angles with reasons:
a) 𝑂̂1
Oˆ1  720
(2)
Angle @ centre  2 
𝑏) 𝑆̂2
Sˆ2  360
(2)
alt  ' s , PW / / SO
c) 𝑂𝑆̂𝑊
(3)
ˆ isos OWS
Wˆ  OSW
Wˆ  540
d) 𝑅̂
Wˆ1  Wˆ2  72  54  126
 Rˆ  540 opp  ' s of cyclic quad sup
PRELIM 2015
(3)
GRADE 12 PRELIMINARY EXAMINATION 2015
MATHEMATICS PAPER 2
Page 8 of 21
2)
In the figure O is the centre of the circle. Use the sketch to prove that the angle
subtended by an arc at the centre of the circle is double the size of the angle
subtended by the same arc at any point on the circumference of the circle.
(6)
R
1
1
2
2
O
K
P
Q
Let Rˆ 2  x
Qˆ  x
and Rˆ1  y
Pˆ  y
Oˆ  2 y
radii
radii
3
Oˆ 4  2 x
ext  ' s
Oˆ  Oˆ  2  x  y 
3
4
[16]
PRELIM 2015
GRADE 12 PRELIMINARY EXAMINATION 2015
MATHEMATICS PAPER 2
Page 9 of 21
QUESTION 5
The data below shows the marks of the Grade 12 trial examination and the corresponding
final examination marks for 11 learners.
Trial examination
marks(x)
Final examination
marks(y)
80 68 94 72 74 83 58 68 65 75 88
72 71 96 77 82 72 58 83 78 80 92
a) Draw a scatter plot of the data above on the grid
(3)
b) Calculate the values of the point P ( x ; y )
(4)
x  75
y  78,3
c) Give the equation of line of best fit.
(2)
y  24, 6  0, 7 x
[9]
PRELIM 2015
GRADE 12 PRELIMINARY EXAMINATION 2015
MATHEMATICS PAPER 2
Page 10 of 21
QUESTION 6
The following data is collected in a Curro High School. It depicts the results of
Mathematics Portfolio marks of 100 learners.
(a)
Complete the Cumulative frequency column.
Mark Intervals
30 ≤ 𝑥 < 40
40 ≤ 𝑥 < 50
50 ≤ 𝑥 < 60
60 ≤ 𝑥 < 70
70 ≤ 𝑥 < 80
80 ≤ 𝑥 < 90
90 ≤ 𝑥 < 100
(b)
PRELIM 2015
Number of learners
3
15
18
19
28
15
2
(2)
Cumulative
Frequency
3
18
36
55
83
98
100
Draw an ogive representing the given data on the following graph.
(4)
GRADE 12 PRELIMINARY EXAMINATION 2015
MATHEMATICS PAPER 2
Page 11 of 21
Cumulative Frequency Graph
110
100
90
Final Examination Mark
80
70
60
50
40
30
20
10
0
0
10
20
30
40
50
60
70
80
90
100
Portfolio Marks
(c)
Show on the curve where you would find the median, lower quartile and
upper quartile. Give the estimated values on the graph.
(3)
Q1  54
Q2  68
Q3  77
 3 up or down 
(d)
Determine the estimated mean and standard deviation.
Mean  65%
(3)
ST    20%
[12]
TOTAL FOR SECTION A = 77 MARKS
PRELIM 2015
GRADE 12 PRELIMINARY EXAMINATION 2015
MATHEMATICS PAPER 2
Page 12 of 21
SECTION B
QUESTION 7
a)
Prove the identity:

sin 2 x  cos x
cos x

sin x  cos 2 x sin x  1
(5)
2sin x cos x  cos x
sin x  1  2sin 2 x 
cos x  2sin x  1
 2sin x  1 sin x  1
cos x
sin x  1
b)
Find the general solution for 𝜃 if 𝑡𝑎𝑛²𝜃 +
3
𝑐𝑜𝑠𝜃
+3=0
(6)
sin 2 
3

3 0
2
cos  cos 
sin 2   3cos   3cos 2   0
1  cos 2   3cos   3cos 2   0
2 cos 2   3cos   1  0
 2 cos   1 cos   1  0
cos  
1
2
or
  60
  120  k 360
or
  240  k 360
cos   1
  180  k 360
kZ
[11]
GRADE 12 PRELIMINARY EXAMINATION 2015
MATHEMATICS PAPER 2
Page 13 of 21
QUESTION 8
In the diagram below, A, B and P lie on a circle centred at O. The tangents to the
circle at A and B meet at the point T and 𝐴𝑇̂𝐵 = 𝜃. Express 𝐴𝑃̂𝐵 in terms of 𝜃. (5)
A
P
O
T
𝜃
B
Join AO and OB
ˆ  OBT
ˆ  90
OAT
Tand  rad 
AOBT Cyclic quad  opp  ' s sup 
ˆ  180  
AOB
1
Pˆ  180   
2
1
 Pˆ  90  
2
  @ centre 2 
[5]
PRELIM 2015
GRADE 12 PRELIMINARY EXAMINATION 2015
MATHEMATICS PAPER 2
Page 14 of 21
QUESTION 9
In the diagram O is the centre of circle CNEL. KE is a tangent at E.
N and T are points on KO. KO // EL.
Complete the following table:
Statement
̂3 = 90°
a) 𝐸
Reason
Subt by a Diameter LC
∴ 𝑇̂2 = 90°
Co-int angles, KO//EL
∴ 𝐶𝑇 = 𝑇𝐸
Rad perpendicular chord, bisect chord
(3)
Statement
̂1 + 𝐸
̂2 = 𝐿̂
b) 𝐸
̂3
but 𝐿̂ = 𝑂
Reason
Tan-Chord
Corresp angles, KO//EL
̂3 = 𝐸
̂1 + 𝐸
̂2
∴ 𝑂
∴ 𝐶𝑂𝐸𝐾 is a cyclic quadrilateral
Angles subt by same chord CK
(3)
PRELIM 2015
GRADE 12 PRELIMINARY EXAMINATION 2015
MATHEMATICS PAPER 2
Page 15 of 21
c)
Prove that  COT ///  KET.
(3)
Cˆ  Kˆ angles subt by segment EO
Oˆ 3  Eˆ1  Eˆ 2 angles subt by segment KC
Tˆ  Tˆ  vert opp L ' s 
3
1
 COT / / / KET
d)
 AAA
Determine the length of CO if OT = 1 unit and KT = 9 units.
(3)
CT OT

KT ET
CT OT

KT CT
 CT 2  9
CT  3
CO 2  CT 2  OT 2
Pythag
CO  32  12
 10  3, 2
e)
̂ correct to 2 decimal digits.
Determine the size of 𝐾
(2)
TE 3
tan Kˆ 

KT 9
Kˆ  18, 40
[14]
PRELIM 2015
GRADE 12 PRELIMINARY EXAMINATION 2015
MATHEMATICS PAPER 2
Page 16 of 21
QUESTION 10
2
NL = 6 units. RE = 𝑥 units; RD = 𝑥 units; EY = 9 units and DL = 𝑥 + 3 units.
3
S is a point on YL and ED // YL.
a) Show that L is the midpoint of DN by first solving for 𝑥.
2
x
x 3

9 x3
x 2  3x  6 x
(4)
 line / /, divide proportionally 
x 2  3x  0
x  0
x3
DL  6
LN  6
L is midpt
b) If SL = 1,4 units write down the length of DE.
1
DE Midpt
2
1, 4  0.5DE
DE  2,8
SL 
PRELIM 2015
(1)
GRADE 12 PRELIMINARY EXAMINATION 2015
MATHEMATICS PAPER 2
Page 17 of 21
c) If Area  RED = 2, 7 units² determine the area of  REN.
(3)
RED  0,5( RD)  h 
2 
2, 7  0,5   3  h
3 
h  2, 7
REN  0,5RN .h
 0,5  2  6  6   2, 7
 18,9
[8]
QUESTION 11
The median of an odd set of n numbers is p, and the mean is q. Give answers to the
following in terms of n, p and q.
(a)
If the largest number is increased by 10, write down the new mean and
the new median.
Median : p
(b)
nq  10
Mean :
n
A number, m, is inserted between the median and the next lower number.
Write down the new mean and the new median.
Median :
m p
2
Mean :
(3)
(4)
nq  m
n 1
[7]
PRELIM 2015
GRADE 12 PRELIMINARY EXAMINATION 2015
MATHEMATICS PAPER 2
Page 18 of 21
QUESTION 12
Two of the cogs that form part of the clock in the tower are represented in the
diagram below. The two cogs are represented by a smaller circle with centre O, the
origin, and larger circle with centre M. The point of contact of the two circles is at
point P (-3;2).
The radius of the larger circle is 2√13.
a) Determine the equation of the smaller circle centred at the origin.
(2)
OP  22   3  13
2
 x 2  y 2  13
b) Determine the equation of the line OM.
M OP 
y
PRELIM 2015
0  2 2

03 3
2
x
3
(2)
GRADE 12 PRELIMINARY EXAMINATION 2015
MATHEMATICS PAPER 2
Page 19 of 21
c) Determine the equation of the common tangent, 𝑡.
(4)
3
2
3
y  xc
2
3  3
2
c
2
13
c
2
3
13
y  x
2
2
m
d) If 𝑥 = 𝑎 at point M, write 𝑏 in terms of 𝑎.
y
(1)
2
ab
3
e) Determine the equation of the larger circle.
(8)
 2 
M  a; a 
 3 
2

2a
 x  a    y    2 13
3 

2

2
2
2a
 3  a    2    52
3 

8a 4 a 2
2
9  6a  a  4  
 52
3
9
13a 2  78a  351  0
a  3 or a  9
2
M  9;6 
 x  9   y  6
2
2
 52
[17]
PRELIM 2015
GRADE 12 PRELIMINARY EXAMINATION 2015
MATHEMATICS PAPER 2
Page 20 of 21
QUESTION 13
The sketch below represents the curves f ( x)  sin( x   ) and g ( x)  cos x
Y
C
f
g
A
B
60
O
(1) Using the graph, determine co-ordinates of:
(i)
A
30 0
(1)
(ii)
B
1500
(1)
(iii)
C
1200
(2) Determine the value of:
(1)
(i)
  30
(1)
(ii)
 3
(1)
[5]
PRELIM 2015
X
GRADE 12 PRELIMINARY EXAMINATION 2015
MATHEMATICS PAPER 2
Page 21 of 21
QUESTION 14
In the diagram, AB is a diameter of a circle with centre O . The point C is chosen such
that ABC is acute – angled. The circle intersects AC and BC at P and Q respectively.
C
1 Q
11
P
3
2
3
2
A
B
O
Let P̂2  x and Q̂1  y
ˆ?
a) Why is Qˆ1  A
(1)
Qˆ1 is an exterior angle of
quad ABQP.
√A
b) Prove that OP is a tangent to the circle passing through P , Q and C .
(5)
Aˆ  Qˆ1  y.............................proved in a)
√A
√A
 Pˆ  Aˆ  y...........................OP=OA
3
ˆ ....................Ext
But Cˆ  Qˆ1  APQ
 of CPQ
√A
 Cˆ  y  x  y
 Cˆ  x  Pˆ
√A
√A
Therefore, OP is a tangent to circle through P, Q and C.
[6]
TOTAL FOR SECTION B = 73 MARKS
PRELIM 2015