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GRADE 12 - MATHEMATICS PAPER 2
EXAMINER:
Mr I.L. Atteridge
MODERATOR: Mrs K. Hulme
DATE: 20 July 2009
TOTAL: 150 marks
TIME: 3 hours
CANDIDATE’S NAME:..................................................................................
CANDIDATE’S MATHEMATICS TEACHER:......................................................
INSTRUCTIONS TO CANDIDATES:
1. Answer all questions in the answer booklet provided.
2. All written work must be done using blue or black ink. Diagrams and graphs must be
drawn neatly using pencil.
3. No correction fluids or tape may be used
4. Scientific, non-programmable calculators may be used unless otherwise stated.
5. Round off to TWO decimal places unless otherwise stated.
6. It is in your own interests to work neatly and to show all necessary steps in calculations.
7. Insert your question paper into the back of your answer booklet when handing in.
THIS EXAMINATION CONSISTS OF 10 PAGES
Mathematics
Grade 12 Paper2
20 July 2009
Page 1 of 10
SECTION A
QUESTION 1
In the diagram below, A, B (-5; 2) and C (-3; 8) are vertices of ∆ABC.
E (1; 0) is the midpoint of AB. DB is a line perpendicular to the 𝑥-axis.
C (-3; 8)
y
B (-5; 2)
D
x
E (1; 0)
A
a) Find M, the midpoint of BC.
(2)
b) Give the coordinates of point D.
(1)
c) Show that the coordinates of A are (7; -2)
(2)
d) Determine the length of line AB. (leave answer in simplest surd form)
(2)
e) Determine the coordinates of F, if F lies in the first quadrant and ABCF is a parallelogram.
(2)
f)
(4)
Determine the equation of the perpendicular bisector of BC
g) Determine the angle of inclination of line EB to the nearest degree.
(3)
h) Determine the equation of the line parallel to BC passing through E.
(2)
i)
Calculate the size of angle 𝐸𝐵̂ 𝐶.
(2)
j)
Determine the area of ∆𝐴𝐵𝐶
(4)
[24]
Mathematics
Grade 12 Paper2
20 July 2009
Page 2 of 10
QUESTION 2
cos(360𝑜 +𝑥) cos(90𝑜 +𝑥) tan2 (𝑥−180𝑜 )
a)
Simplify to a single trigonometric ratio:
b)
Simplify the following without a calculator and showing all your working :
sin(180𝑜 −𝑥) sin(𝑥−360𝑜 ) tan(−𝑥)
sin(−120𝑜 ) cos660𝑜 tan315𝑜
cos310𝑜 sin140𝑜 +sin2 410𝑜
c)
Use your calculator to simplify
cos2𝜃
2
+ tan2 𝛼
(6)
correct to four decimal places
if 𝜃 = 45𝑜 and 𝛼 = −116𝑜
d)
(6)
(2)
Solve the following equations:
(1)
sin 2𝑥 cos 𝑥 − sin 𝑥 cos 2𝑥 = 0,500 for 0𝑜 ≤ 𝑥 ≤ 90𝑜
(3)
(2)
sin 2∅ = −2 cos ∅ for ∅ 𝜖 [−180𝑜 ; 360𝑜 ]
(6)
[23]
QUESTION 3
Consider the functions 𝑓(𝑥) = sin(𝑥 − 450 ) , and 𝑔(𝑥) = cos 2𝑥, for 𝑥 ∈ [0°; 180°].
a)
Sketch the graphs of f(x) and g(x) on the same system of axes, clearly indicating the coordinates of all
turning points, end points and intercepts with the axes.
(8)
b)
What is the period of f ?
(1)
c)
What is the period of g ?
(1)
d)
Use your graph to solve for 𝑥 if sin(𝑥 − 45°) = cos 2𝑥 , where 𝑥 ∈ (0°; 180°).
(2)
[12]
Mathematics
Grade 12 Paper2
20 July 2009
Page 3 of 10
QUESTION 4
Write down a possible data set of 9 values that would be represented by the above box and whisker plot.
[5]
QUESTION 5
a)
State the following transformation of 𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻 to 𝐴′ 𝐵′ 𝐶 ′ 𝐷′ 𝐸 ′ 𝐹 ′ 𝐺 ′ 𝐻 ′
in notational form : ( 𝑥 ; 𝑦 ) → (
;
) → (
5
;
(4)
)
G
H
4
E
F
D
C
3
2
1
A
-4
B'
C'
F'
G'
A'
-2
B
2
4
6
-1
D'
E'
-2
H'
-3
Mathematics
Grade 12 Paper2
20 July 2009
Page 4 of 10
b)
A square with coordinates 𝑃( 0 ; −5 ), 𝑄( 5 ; 0 ), 𝑅 ( 0 ; 5 ), and 𝑆 (−5 ; 0 ) is given :
(1)
If 𝑃′ 𝑄 ′ 𝑅′ 𝑆 ′ is the reflection of 𝑃𝑄𝑅𝑆 about 𝑦 = −𝑥 , give the coordinates
of 𝑃′ and 𝑅′ .
(2)
Calculate, correct to 2 decimal places, the coordinates of 𝑃′′ and 𝑅′′ if 𝑃′ 𝑄 ′ 𝑅′ 𝑆 ′ is
rotated about the origin through 450 anticlockwise to 𝑃′′ 𝑄 ′′ 𝑅′′ 𝑆 ′′.
(3)
(2)
(4)
If 𝑆 is rotated from its original position counter-clockwise about the origin through
an angle of 60𝑜 , to a new position 𝑆 ′′′ find the coordinates of 𝑆 ′′′ without a calculator,
leaving your answer in surd form
(4)
[14]
Mathematics
Grade 12 Paper2
20 July 2009
Page 5 of 10
SECTION B
QUESTION 6
The coordinates of trapezoid 𝐴𝐵𝐶𝐷 are as follows: 𝐴( 1 ; 4,5 ), 𝐵( 3 ; 5 ), 𝐶( 3 ; 2 ), 𝐷( 1 ; 2,5 ) .
𝐴′ 𝐵′ 𝐶 ′ 𝐷 ′ is the reflection of 𝐴𝐵𝐶𝐷 in the line 𝑥 = 4, while 𝐴′′ 𝐵′′ 𝐶 ′′ 𝐷 ′′ is the reflection of 𝐴′ 𝐵′ 𝐶 ′ 𝐷 ′
in the line 𝑥 = 9 .
Y
x4
x 9
B
A
D
C
X
a)
Write down the coordinates of:
(1)
𝐶′
(2)
𝐷′
(3)
𝐵′′
b)
Write down a single rule that would transform 𝐴𝐵𝐶𝐷 to 𝐴′′ 𝐵′′ 𝐶 ′′ 𝐷′′
c)
What would the coordinates of 𝐵′′ be if the order of the reflection was changed,
i.e. reflected in the line 𝑥 = 9 before 𝑥 = 4
(3)
(3)
(3)
[9]
Mathematics
Grade 12 Paper2
20 July 2009
Page 6 of 10
QUESTION 7
The hexagonal shaped nut or bolt we often use is cut from a cylindrical rod of iron. Usually the machinist
knows the “flat distance” (x) from one side of the nut to the opposite side.
Each nut is classified according to the flat distance (x)(eg 10mm or 3 inch, etc), which determines the size
4
spanner we use. By convention, the length of one side (s) of the hexagonal nut is 2
3
of the “flat distance” (x)
s
x
d
a)
b)
Use the Theorem of Pythagoras and the diagram to find a formula for the diameter of the
cylindrical rod (d) in terms of the flat distance (x).
(4)
Use your formula to determine the size of the distance (d) for a 22mm nut.
(3)
[7]
QUESTION 8
a)
Find the equation of the resultant line if the line 𝑦 = 2𝑥 + 4 is rotated through the origin
90𝑜 in a clockwise direction.
b)
(2)
If this resultant line is to be further rotated, but this time through its y-intercept, through
how many degrees must it be rotated if its defining function is to become 𝑦 = 4 ?
(3)
[5]
Mathematics
Grade 12 Paper2
20 July 2009
Page 7 of 10
QUESTION 9
Given:
2 tan 𝑥 − sin 2𝑥
2 sin2 𝑥
= tan 𝑥
a) Prove the above identity.
(6)
b) For what value(s) of 𝑥 is the identity not valid in the interval 00 ≤ 𝑥 ≤ 1800
(5)
c) Solve for 𝑥 if sin 2𝑥 = −0,582 for 𝑥𝜖𝑥[−180𝑜 ; 180𝑜 ] to the nearest degree.
(4)
d) Determine the general solution sin 2𝑥 = cos(𝑥 + 30𝑜 )
(6)
[21]
QUESTION 10
̂ = 𝟏𝟓𝟎𝒐 , 𝑴𝑷 = 𝒅 , 𝑷𝑹 = 𝒆 and 𝑴𝑹 = 𝒇.
a) In MPR, 𝑷
Show, without using a calculator, that 𝐬𝐢𝐧 𝑴 = 𝟐𝒇.
𝒆
M
f
d
P
e
R
(3)
Mathematics
Grade 12 Paper2
20 July 2009
Page 8 of 10
b)
In the figure below Hector is standing at point A on top of building AB that is 50 m high. He observes
two cars, C and D, that are in a triangular parking lot. They are in the same horizontal plane as B.
The angle of elevation from C to A is 55° and the angle of elevation from D to A is 48°. 𝐶𝐴̂𝐷 = 71𝑜 .
A
71°
50 m
48°
B
D
55°
C
(1)
Calculate the lengths AC and AD, correct to 1 decimal place.
(4)
(2)
Calculate the distance between the two cars i.e. the length of CD.
(3)
(3)
Calculate the area of the parking lot ( ΔBCD )
(6)
[16]
Mathematics
Grade 12 Paper2
20 July 2009
Page 9 of 10
QUESTION 11
Consider the data set below :
2; 5; 7; 11; 23; 46; 61
a) Calculate the mean
(1)
b) Calculate the variance
(3)
c) If k (a constant value) is added to each element of the data set, write conjectures about the effect of k on
the mean and variance (show your working to prove your conjecture)
(4)
d) If each element of the data set is multiplied by m (a constant value), write conjectures about the effect of
m on the mean and variance (show your working to prove your conjecture)
(6)
[14]
Mathematics
Grade 12 Paper2
20 July 2009
Page 10 of 10