MATHEMATICS
PAPER 2
GRADE 12
JUNE 2015
TIME: 3 HOURS
MARKS: 150
RONDEBOSCH BOYS’ HIGH SCHOOL
______________________________
Examiner: P Ghignone
Moderator: T Edwards
______________________________
This question paper consists of 14 pages (including an
information sheet.
______________________________
Page 1 of 14
INSTRUCTIONS AND INFORMATION
Read the following instructions carefully before answering the questions.
1.
This question paper consists of 11 questions.
2.
Answer ALL the questions in the ANSWER BOOK provided. Additional space is
available at the back of the answer book. Number your work carefully.
3.
Use the DIAGRAMS in the ANSWER BOOK for any markings you make.
4.
Clearly show ALL calculations, diagrams, graphs, et cetera that you have used in
determining your answers.
5.
Answers only will not necessarily be awarded full marks.
6.
You may use an approved scientific calculator (non-programmable and nongraphical), unless stated otherwise.
7.
If necessary, round off answers to TWO decimal places, unless stated otherwise.
8.
Diagrams are NOT necessarily drawn to scale.
9.
An information sheet with formulae is included at the end of the question paper.
10.
Number the answers correctly according to the numbering system used in this
question paper.
11.
Write neatly and legibly.
Page 2 of 14
QUESTION 1
Mr Jacobs owns a laundromat and charges his clients by the weight of their
laundry. He kept a record of the weights (in kg) brought in by 22 clients.
0,5
4,0
1.1
1.2
1.3
1.4
0,8
4,5
1,5
5,4
1,9
6,2
2,1
6,4
2,8
6,5
2,8
7,7
3,3
7,8
3,4
9,2
3,4
9,8
3,6
13,0
Determine the mean weight of the laundry.
(2)
Calculate the standard deviation of the data.
(2)
How many clients have laundry weights that are within one standard
deviation of the mean?
(2)
He calibrates his scale and discovers that the scale was recording a weight
0,5 kg greater than the actual weight. Determine the actual mean and
standard deviation.
(2)
[8]
Page 3 of 14
QUESTION 2
Cumulative Frequency
Further to his investigations, Mr Jacobs collects data for the whole week, and
prepares an ogive (cumulative frequency curve) to summarise his data.
Amount of Laundry (kg)
2.1
Use the table in the ANSWER BOOK and complete the table below. (3)
Weight (in kg)
0≤𝑥<2
2≤𝑥<4
4≤𝑥<6
6≤𝑥<8
8 ≤ 𝑥 < 10
10 ≤ 𝑥 < 12
12 ≤ 𝑥 < 14
2.2
2.3
2.4
2.5
Cumulative Frequency
Frequency
Use the frequency data to determine an approximate mean weight of
laundry for the week.
(3)
The data set has a median laundry weight of 5,6 kg. Use the ogive to
estimate the lower quartile and upper quartile. Show evidence on the ogive.
(2)
A client brings in a load of 13,6 kg. Determine (by calculation) whether
this weight is an outlier.
(3)
Mr Jacobs uses a heavy duty machine to wash loads greater than 11 kg.
Determine the number of clients who will have their laundry washed in
the heavy duty machine.
(2)
[13]
Page 4 of 14
QUESTION 3
3.1
Given the points 𝐴(−2; −5), 𝐵(3; −1) and 𝐶(9; −3).
y
x
B(3;-1)

C(9;-3)
A(-2;-5)
3.1.1
3.1.2
3.1.3
3.1.4
3.2
Determine the length of BC. Leave your answer in surd form.
Determine the coordinates of M, the midpoint of AB.
Find the equation of the line parallel to AB, passing through C.
Calculate the size of 𝜃.
(2)
(2)
(4)
(5)
PQRS is a rectangle with 𝑄(3; 5), 𝑆(−5; 3) and SR parallel to the 𝑥-axis.
y
P
Q (3 ; 5)
R
S (-5 ; 3)
O
x
3.2.1 Write down the coordinates of R.
(2)
3.2.2 Show that OQ is perpendicular to OS.
(3)
3.2.3 Explain why Q, R, O, and S lie on the same circle and give the
coordinates of the centre of this circle.
(5)
[23]
Page 5 of 14
QUESTION 4
4.1
The equation of a circle is 𝑥 2 + 𝑦 2 − 2𝑥 + 4𝑦 − 4 = 0.
y
x
A (x ; y )
4.1.1 Determine the coordinates of A, the centre of the circle and the
length of the radius, r.
(5)
4.1.2 Calculate the value of p if N (1; 𝑝) with 𝑝 > 0 is a point on the
circle.
(1)
4.1.3 Determine the equation of the tangent to the circle at N.
(2)
4.2
A second circle, centre B, with equation (𝑥 − 4)2 + 𝑦 2 = 𝑘 2 cuts the
circle given in (4.1) twice. Determine the values of 𝑘 for which point A
will be inside the circle B.
(4)
[12]
Page 6 of 14
QUESTION 5
5.1
Given: sin 2𝐴 =
√15
8
for 0° ≤ 2𝐴 ≤ 90°
5.1.1 Determine the value of cos 2𝐴 without the use of a calculator. (3)
5.1.2 Hence, find the value of cos 𝐴 without the use of a calculator.
Leave your answer in surd form.
(4)
5.2
Given
cos(𝐴 − 𝐵) = cos 𝐴. cos 𝐵 + sin 𝐴. sin 𝐵
5.2.1 Derive the formula for cos(𝐴 + 𝐵).
5.2.2 If 𝐴 + 𝐵 = 90°, show that:
cos 2 (𝐴 − 𝐵) = 4 cos 2 𝐵 sin2 𝐵
(3)
(4)
[14]
QUESTION 6
6.1
Simplify, without the use of a calculator:
sin 140°.sin 120°
sin 110°.sin 340°
6.2
Given:
sin 𝑥+sin 2𝑥
1+cos 𝑥+cos 2𝑥
(6)
= tan 𝑥
6.2.1 Prove the identity.
(4)
6.2.2 Determine for which value(s) of 𝑥 the identity is invalid if
𝑥 ∈ [0°; 360°].
Page 7 of 14
(3)
[13]
QUESTION 7
7.1
Given 𝑔(𝑥) = 2 cos(𝑥 − 30°)
7.1.1 Make a neat sketch of 𝑔(𝑥) for the interval [−90° ; 270°] on the
grid provided.
(3)
7.1.2 A straight line is now drawn through the turning points of the
graph. Determine the equation of the straight line.
(4)
7.2
Determine the general solution for:
32 sin 𝜃 − 10.3sin 𝜃 + 9 = 0
(6)
[13]
QUESTION 8
A vertical post, PT, has its foot in the same horizontal plane as points Q and R.
𝑇𝑄 = 𝑇𝑅 = 𝑦 and 𝑃𝑄 = 𝑃𝑅. The angle of elevation from Q to P is 𝜃 and
𝑃𝑅̂𝑄 = 𝛽.
P
Q

y
T

y
R
8.1
Express PQ in terms of 𝑦 and 𝜃.
8.2
Prove that 𝑄𝑅 =
2𝑦.cos 𝛽
(2)
(4)
cos 𝜃
[6]
Page 8 of 14
QUESTION 9
9.1
In the diagram below, P, Q, R and S are points that lie on the
circumference of the circle with centre O.
Given below is the partially completed proof of the theorem that states
that 𝑃̂ + 𝑅̂ = 180°. Complete the proof by filling in the missing sections.
P
2
O
1
Q
S
R
Construction:
Join SO and QO.
Statement
𝑂̂1 = ⋯
𝑂̂2 = 2𝑅̂
𝑂̂1 + 𝑂̂2 = 2𝑃̂ + 2𝑅̂
And 𝑂̂1 + 𝑂̂2 = ⋯
∴ 2𝑃̂ + 2𝑅̂ = 360°
∴ 𝑃̂ + 𝑅̂ = 180°
Reason
…
….
…
(5)
Page 9 of 14
9.2
In the figure below, UVWX is a cyclic quadrilateral, with 𝑈𝑊||𝑌𝑍 and
̂ 𝑋 = 58°.
tangent YXZ touching the circle at X. 𝑈𝑊
V
1
2
58
U
3
Y
2
W
1
X
Z
Determine the values of the following angles, showing all steps and
reasons:
9.2.1
𝑋̂1
(2)
9.2.2
𝑋̂3
(2)
9.2.3
𝑋̂2
(2)
9.2.4
𝑉̂1
(2)
Page 10 of 14
9.3
In the diagram below, AC is a chord of circle ABCDE.
𝐴𝐹𝐷 ⊥ 𝐸𝐹𝐵, ∠𝐷1 = 40° and ∠𝐸 = ∠𝐵1 .
A
2
E
1
F4
2
1
32
1
B
40
1
2
1
2
C
D
In the following questions, give a reason for each statement:
9.3.1
Name THREE angles each equal to 50°.
(4)
9.3.2
Calculate the size of ∠𝐷𝐶𝐵.
(2)
9.3.3
Prove that EB||DC.
(2)
9.3.4
Prove that AC is a diameter of the circle.
(3)
[24]
Page 11 of 14
QUESTION 10
M
In the diagram alongside, Q is a
point on MP and T is a point on PN
of ∆𝑀𝑁𝑃, such that QT||MN.
S is the midpoint of MN. R is
another point on MN such that
QR||PS.
MQ:QP = 5:2
R
S
Q
Calculate, with reasons, the
numerical value of the following:
10.1
10.2
P
𝑃𝑇
N
T
(2)
𝑇𝑁
𝑅𝑆
(5)
𝑀𝑁
[7]
Page 12 of 14
QUESTION 11
In the diagram, WPTU is a cyclic quadrilateral with 𝑈𝑊 = 𝑈𝑇. Chords WT and
PU intersect at Q. PW extends to S such that 𝑆𝑈||𝑊𝑇.
S
W
3
2 1
1
U
2
3
1 2
4 3
4
2
1
Q
P
1 2
T
11.1 Complete the statement: The angle between the tangent to a circle and
the chord drawn from the point of contact is …
(2)
11.2 Prove that:
11.2.1
US is a tangent to circle PWUT at U.
(5)
11.2.2
∆𝑆𝑃𝑈|||∆𝑆𝑈𝑊
(4)
11.2.3
𝑆𝑈 2 = 𝑆𝑃. 𝑆𝑊
(2)
11.2.4
𝑆𝑈 2 . 𝑄𝑈 = 𝑃𝑈. 𝑆𝑊 2
(4)
[17]
Page 13 of 14
INFORMATION SHEET: MATHEMATICS
𝑥=
−𝑏 ± √𝑏 2 − 4𝑎𝑐
2𝑎
𝐴 = 𝑃(1 + 𝑛𝑖)
𝑛
𝑛
∑1 = 𝑛
∑𝑖 =
𝑖=1
𝑖=1
𝑇𝑛 = 𝑎𝑟
𝐹=
𝐴 = 𝑃(1 − 𝑖)𝑛
𝐴 = 𝑃(1 − 𝑛𝑖)
𝑛−1
𝑛(𝑛 + 1)
2
𝑆𝑛 =
𝑎(𝑟 𝑛 − 1)
𝑟−1
𝑇𝑛 = 𝑎 + (𝑛 − 1)𝑑
ℎ→0
𝑆∞
; 𝑟≠1
𝑥[(1 + 𝑖)𝑛 − 1]
𝑖
𝑓 ′ (𝑥) = lim
𝐴 = 𝑃(1 + 𝑖)𝑛
=
𝑃=
𝑎
1−𝑟
𝑆𝑛 =
𝑛
(2𝑎 + (𝑛 − 1)𝑑)
2
; −1 < 𝑟 < 1
𝑥[1 − (1 + 𝑖)−𝑛 ]
𝑖
𝑓(𝑥 + ℎ) − 𝑓(𝑥)
ℎ
𝑥1 +𝑥2 𝑦1 +𝑦2
𝑑 = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2
𝑦 = 𝑚𝑥 + 𝑐
M(
2
𝑚=
𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 )
;
2
)
𝑦2 − 𝑦1
𝑥2 − 𝑥1
𝑚 = tan 𝜃
(𝑥 − 𝑎)2 + (𝑦 − 𝑏)2 = 𝑟 2
In ΔABC:
𝑎
𝑏
𝑐
=
=
sin 𝐴 sin 𝐵 sin 𝐶
𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐 . cos 𝐴
1
𝑎𝑟𝑒𝑎∆𝐴𝐵𝐶 = 𝑎𝑏. sin 𝐶
2
sin(𝛼 + 𝛽) = sin 𝛼 . cos 𝛽 + cos 𝛼 . sin 𝛽
sin(𝛼 − 𝛽) = sin 𝛼 . cos 𝛽 − cos 𝛼 . sin 𝛽
cos(𝛼 + 𝛽) = cos 𝛼 . cos 𝛽 − sin 𝛼 . sin 𝛽
cos(𝛼 − 𝛽) = cos 𝛼 . cos 𝛽 + sin 𝛼 . sin 𝛽
cos2 𝛼 − sin2 𝛼
cos 2𝛼 = {1 − 2 sin2 𝛼
2 cos 2 𝛼 − 1
sin 2𝛼 = 2 sin 𝛼 . cos 𝛼
(𝑥; 𝑦) → (𝑥 cos 𝜃 − 𝑦 sin 𝜃 ; 𝑦 cos 𝜃 + 𝑥 sin 𝜃)
𝑛
𝑥̅ =
∑ 𝑓𝑥
𝑛
𝑃(𝐴) =
𝑛(𝐴)
𝑛(𝑆)
𝑦̂ = 𝑎 + 𝑏𝑥
)2
𝜎 2 = ∑(𝑥𝑖 − 𝑥̅
𝑖=1
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
𝑛
𝑃(𝐴 𝑜𝑟 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 𝑎𝑛𝑑 𝐵)
𝑏=
∑(𝑥 − 𝑥̅ )(𝑦 − 𝑦̅)
∑(𝑥 − 𝑥̅ )2
Page 14 of 14