ST STITHIANS GIRLS’ COLLEGE
GRADE 12
MATHEMATICS: PAPER 2
July 2014
TIME:
3 hours
MARKS:
150
EXAMINER:
Mr M Ancillotti
MODERATOR:
Mr P Statham
NAME:
___________________________________________________________
TEACHER:
Mr Statham
Mrs Germishuys
Mr Schaerer
Mrs van Rooy
Mr Ancillotti
PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY:
(1)
This question paper consists of 12 questions and 28 pages including the cover sheet
and is divided into two sections, Section A and Section B.
(2)
Read the questions carefully.
(3)
You may use an approved non-programmable and non-graphical calculator, unless
stated otherwise in any question.
(4)
Unless stated otherwise, round all answers to two decimal places where necessary.
(5)
Clearly show all details of your workings and calculations.
(6)
It is in your own interest to write legibly and to present your work neatly.
(7)
This question paper is also your answer booklet. Write all your answers in this booklet
in the spaces provided.
Grade 12 Preliminary Examination
St Stithians Girls’ College
Paper 2 - July 2014
SECTION A
QUESTION 1
In the figure, the capital A is symmetrical about the y-axis.
(a)
(b)
Given M (0 ; 6) and P (3 ; 0), show that the equation of the straight line SM is
given by 2 x  y  6. Show all your working.
(3)
If the coordinates of S are (a ; 5), calculate the value of a.
(2)
Page 2 of 28
Grade 12 Preliminary Examination
St Stithians Girls’ College
Paper 2 - July 2014
(c)
Write down the equation of BN if N is the point (0 ; 4).
(2)
(d)
Write down the coordinates of B, C, D and E.
(5)
[12]
Page 3 of 28
St Stithians Girls’ College
Grade 12 Preliminary Examination
Paper 2 - July 2014
QUESTION 2
The speeds, in kilometres per hour, of cyclists passing a point on
the route of the 94.7 Cycle Race were recorded and summarised
in the table below:
SPEED ( km/h)
Frequency (f)
0  x < 10
10
Cumulative
Frequency
10
10  x < 20
30
20  x < 30
45
30  x < 40
72
40  x < 50
170
(a)
Complete the table above.
(4)
(b)
Make use of the axes provided below to draw a cumulative frequency curve.
(3)
180
170
160
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
cf
x
10
(c)
20
30
40
50
Indicate clearly on your graph where the estimates of the lower quartile (LQ)
and median (M) speeds can be read, and write down these estimates.
Page 4 of 28
(2)
Grade 12 Preliminary Examination
(d)
St Stithians Girls’ College
Paper 2 - July 2014
Use your graph to estimate the number of cyclists that passed the point with
speeds greater than 35km/h.
(2)
[11]
QUESTION 3
(a)
Evaluate, without the use of a calculator, sin 400o.cos10o  sin 350o.cos140o
(b)
If sin 2A =

b
and A  45 , determine sin A in terms of b and c.
c
Page 5 of 28
(5)
(4)
Grade 12 Preliminary Examination
St Stithians Girls’ College
Paper 2 - July 2014
(c)
Determine the general solution of: sin  x  30   cos 2 x  0
(d)
i)
Show that sin( A  B)  sin( A  B)  2 cos A sin B
ii)
Hence, evaluate sin 80  sin 20 , without the use of a calculator, if it is
further given that cos 50  0, 64 .
Page 6 of 28
(6)
(2)
(5)
Grade 12 Preliminary Examination
St Stithians Girls’ College
Paper 2 - July 2014
[22]
QUESTION 4
Refer to the diagram.
RQ is a tangent to circle QTSUP with centre O. SOQ and PT are straight lines.


P T S  40 and SQ T  50
Find, with reasons, the size of the lower case letters marked (a) to (e).
Fill in your answers in the table below:
P
Q
d
U
b
c
O
R
a
e
S
Angle
T
Answer
Reason
a
b
c
d
e
[10]
Page 7 of 28
Grade 12 Preliminary Examination
St Stithians Girls’ College
Paper 2 - July 2014
QUESTION 5
In each case below, you are given a statement and a reason that are true for the
incomplete diagram. Complete the diagram, showing what was necessary so that the
statement and the reason are true.

(a)

Statement: A  B  180
Reason: Opp L’s of cyclic quad
O

(b)

Statement: T1  T2
Reason: Line from centre to midpoint
of chord
O
(2)
Page 8 of 28
St Stithians Girls’ College
Grade 12 Preliminary Examination
(c)
Paper 2 - July 2014
In the figure below, A, B, C and D are points on a circle with centre O.



It is further given that A  2 x  20 , reflex D O B  4x and C  2y .
A

O
D
B
C
Calculate the size of y, stating all reasons:
(5)
[7]
Page 9 of 28
St Stithians Girls’ College
Grade 12 Preliminary Examination
Paper 2 - July 2014
QUESTION 6
(a)
Name, stating a reason, two triangles below that are similar.
(2)
6
3
6
A
4
(b)
2
B
3
C
12
5
8
7
D
6
4
Two circles intersect at A and B. AC is a tangent to circle ABD at A and AD is a
tangent to the circle ACB at A. Straight line CEFD intersects the circles at E and
F. AE = AF.
A
1
C
2 3
2
1
1
E
2
F
B
Page 10 of 28
D
Grade 12 Preliminary Examination
St Stithians Girls’ College
Paper 2 - July 2014
i)
Prove: ΔACE /// ΔDAF
(3)
ii)
Show: AC.DF = AD.AF
(3)
Page 11 of 28
Grade 12 Preliminary Examination
(c)
St Stithians Girls’ College
Paper 2 - July 2014
In the figure below, ABC has D and E on BC. BD = 6cm and DC = 9cm.
AT : TC = 2 : 1 and AD // TE.
A
T
F
C
E
D
B
i)
Write down the numerical value of
ii)
Show that D is the midpoint of BE.
Page 12 of 28
CE
ED
(1)
(2)
Grade 12 Preliminary Examination
iii)
St Stithians Girls’ College
If FD = 2cm, calculate the length of TE.
Paper 2 - July 2014
(2)
[13]
END OF SECTION A
Page 13 of 28
Grade 12 Preliminary Examination
St Stithians Girls’ College
Paper 2 - July 2014
SECTION B
QUESTION 7
A circle, with centre O, is defined by the equation ( x  6)2  ( y  1)2  20 . From a point A
(2; 0) a tangent is drawn, touching the circle at B (a; b). The equation of this tangent is
2y  x  2  0 .
a)
What is the gradient of the tangent?
(1)
b)
What is the gradient of OB, in terms of a and b?
(2)
c)
Hence, or otherwise, determine the coordinates of B.
(5)
Page 14 of 28
Grade 12 Preliminary Examination
St Stithians Girls’ College
Paper 2 - July 2014
d)
Determine the length of AB.
(2)
e)
A second tangent is drawn to the circle, at point (4; 5). Find the point D, the
point of intersection of the two tangents.
(6)
[16]
Page 15 of 28
St Stithians Girls’ College
Grade 12 Preliminary Examination
Paper 2 - July 2014
QUESTION 8
a)
Three circles are sketched below, with centres A, B and C respectively.
The equation of the first, centred at A, is x 2  y 2  4 x  6 y  12  0 .
Note: The radius of the circle, centred at B, is 1 unit greater than the circle centred at A
and the radius of the circle, centred at C (p; q), is 1 unit greater than the circle centred at
B. Each circle centre is shifted 1 unit right and then 1 unit up to determine the next circle
centre.
y
y
xx
C
B
A
i)
Determine the radius and the coordinates of the centre of the circle
centred at A.
Page 16 of 28
(3)
Grade 12 Preliminary Examination
ii)
(b)
St Stithians Girls’ College
Paper 2 - July 2014
Determine the equation of the circle, centred at C, in the form:
( x  p) 2  ( y  q) 2  r 2
(3)
Refer to the diagram
alongside:
If the equation of PQ is given by 3y  x  k , find:
i)
the size of , the inclination of PQ, rounded to one decimal place.
(3)
ii)
the gradient of QS.
(3)
[12]
Page 17 of 28
St Stithians Girls’ College
Grade 12 Preliminary Examination
Paper 2 - July 2014
QUESTION 9
Bonuses were paid to each of the players in the German Soccer team who won the recent
World Cup in Brazil. 12 of the 15 bonuses paid to the final playing squad are given in the
table below. However, neither the median nor the upper or lower quartiles have been
given. The figures represent thousands of Euros.
100
140
150
170
180
275
450
600
650
950
1550
1550
It is now given that the median is 375, the Upper Quartile is 660 and the Inter-quartile
range is 500.
(a)
Complete the table below, ranking all 15 data items.
(4)
100
1550
SPACE FOR ROUGH WORKING
(b)
Find the standard deviation of the above data.
(2)
(d)
Determine what percentage of the bonuses lie within 2 standard deviations of the
mean.
(3)
Page 18 of 28
St Stithians Girls’ College
Grade 12 Preliminary Examination
Refer to the ogive sketch below, showing the speeds of motorists travelling
along a road where the permitted speed limit is 80 km/h.
Determine the modal class of the speeds travelled.
Cumulative Frequency
(e)
Paper 2 - July 2014
125
120
115
110
105
100
95
90
85
80
75
70
65
60
55
50
45
40
35
30
25
20
15
10
5
0
(2)
y
x
20
30
40
50
60
70
80
90
100
Speed in Km/h
[11]
Page 19 of 28
Grade 12 Preliminary Examination
St Stithians Girls’ College
Paper 2 - July 2014
QUESTION 10
(a)
i)
If the period of y  tan 2 x is halved, what is the new equation?
(1)
ii)
If the amplitude of y  cos 2 x  1 is doubled, what is the new equation?
(1)
iii)
If the graph of y  sin  2 x  30
 is translated left by 30 , what is the
new equation?
(b)
(1)
Given: f ( x)  1  sin x and g ( x)  cos 2 x
i)
Determine algebraically the values of x for which f ( x)  g ( x) if
x   180 ;180 
(7)
Page 20 of 28
St Stithians Girls’ College
Grade 12 Preliminary Examination
ii)
Paper 2 - July 2014
The graph of f ( x) is sketched below, for the interval x   180 ;180  .
Sketch the graph of g ( x ) on the same set of axes.
(3)
y
2
f
1
x
−180
−150
−120
−90
−60
−30
30
60
90
120
150
180
−1
−2
iii)
Use your graphs to write down the values of x for which f ( x).g ( x)  0
in the given domain.
(2)
[15]
Page 21 of 28
St Stithians Girls’ College
Grade 12 Preliminary Examination
Paper 2 - July 2014
QUESTION 11
In the sketch below, DA and DB are tangents to the circle at A and B. AF = FB.
AB produced cuts the line produced through DE, which is parallel to FB, at C.


AF produced meets DC at E and E A D  C  x .
D
2
E
1
1
2
3
x
F
1
2
3
2
1
C
B
x
1
2
A
(a)
Find, with reasons, 4 other angles each equal to x.
Page 22 of 28
(8)
St Stithians Girls’ College
Grade 12 Preliminary Examination
(b)
Prove that ABED is a cyclic quadrilateral.
(c)
Prove that A B E  3D A E .

Paper 2 - July 2014
(2)

(3)
[13]
Page 23 of 28
St Stithians Girls’ College
Grade 12 Preliminary Examination
Paper 2 - July 2014
QUESTION 12
(a)
AP  sin x , PB  2 , AQ  2 cos x and QC  3 .
Determine the value of x, x  0o ; 360o  , given PQ // BC.
A
sinx
2cosx
P
2
B
Q
3
C
Page 24 of 28
(4)
St Stithians Girls’ College
Grade 12 Preliminary Examination
(b)
Paper 2 - July 2014
While studying for this examination, Nicola came across the following statement:
A quadrilateral formed by the angle bisectors of any quadrilateral is always
cyclic.
With the aid of the diagram below, prove that this statement is true.
(i.e show that quadrilateral 𝐴𝐵𝐶𝐷 is cyclic). Do not provide reasons.
(4)
P
Q
w
w
z
𝐴 1
2
𝐵
z
1
2
1
2
2
y
𝐷
1
𝐶
y
S
x
x
R
[8]
END OF SECTION B
Page 25 of 28
Grade 12 Preliminary Examination
St Stithians Girls’ College
SPACE FOR ROUGH WORKING
Page 26 of 28
Paper 2 - July 2014
Grade 12 Preliminary Examination
St Stithians Girls’ College
Page 27 of 28
Paper 2 - July 2014
Grade 12 Preliminary Examination
St Stithians Girls’ College
Paper 2 - July 2014
ST STITHIANS GIRLS’ COLLEGE
GRADE 12
MATHEMATICS: PAPER 2
July 2014
MARK RECORD SHEET
FOR OFFICIAL USE ONLY
QUESTION
AIM 3
AIM 4
AIM 5
Trigonometry
Geometry
Statistics
1
/12
2
3
/11
/22
4
/10
5
6
/7
/13
7
/16
8
/12
9
10
/11
/15
11
/13
12
/8
TOTALS
/37
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