SECTION A
P(3; −1), Q(0; t) and 𝑅(4; 6) are the vertices of ∆𝑃𝑄𝑅.
1.
The equation of 𝑃𝑄 is 𝑦 + 𝑥 − 2 = 0 and the equation of 𝑃𝑅 is 𝑦 = 7𝑥 − 22.
𝑦
𝑅(4; 6)
𝜃
𝑇
Q(0; t)
𝑥
0
𝑃(3; −1)
a)
Show that 𝑡 = 2.
(2)
b)
Show that ∆𝑃𝑄𝑅 is right-angled at 𝑄.
(3)
c)
Calculate the area of ∆𝑃𝑄𝑅.
(4)
d)
Find the midpoint of 𝑃𝑅.
(2)
e)
Find the co-ordinates of point 𝑆 if 𝑃𝑄𝑅𝑆 is a parallelogram.
(2)
f)
Find the value of 𝜃 to 1 decimal place.
(4)
[17]
2a)
2b)
Use your calculator to find the following to 3 decimal places:
i)
(sin 72,3° – √𝑐𝑜𝑠66,6° )tan135°
(2)
ii)
∝ if 3cos2∝ = -0,369 and 0°≤ ∝≤ 90°
(3)
Study the diagrams below and determine the following without the use of a calculator.
(Leave answers in simplest surd form)
i)
OA
𝜃
∝
ii)
a
O
O
iii) the value of
10
1) sin𝜃
A(-2; -3)
B(6; a)
2) cos (-∝)
∝
3) 1 - 2cos2( 2 )
1
(2)
(2)
(1)
(2)
(3)
cos(90°+𝑥)
2c)
Simplify as far as possible:-
(4)
2d)
The figure shown below is a rectangular envelope with the flap open.
̂ = 130o PL = 15cm LV = VN = 9cm.
Find the area of the shape (ENVLP). V
( 1−𝑠𝑖𝑛2 𝑥).tan(𝑥−180°)
P
15
L
9
130° V
9
N
E
3.
(4)
[23]
The vertices of a polygon PQRS are shown in the grid below:
P ( -3 ; 2) , Q ( -2 ; 2) , R ( -1 ; 0) and S ( -4; 1) are given.
Each of the points of PQRS is rotated 90° about the origin in a clockwise direction.
a)
Write down the co ordinates of Qˈ the image of Q after the transformation.
(2)
b)
Sketch and label the vertices of the image PˈQˈRˈSˈ on the grid provided on the
diagram sheet.
(2)
PˈQˈRˈSˈ is then enlarged by a scale factor of 2 to give P ̎ Q ̎ R ̎ S ̎.
Write down the co-ordinates of P ̎.
(2)
State whether the transformation from PQRS to P ̎ Q ̎ R ̎ S ̎ is rigid or not, giving a
reason for your answer.
(2)
Write down the general transformation of a point (𝑥; 𝑦) in PQRS to a point in
P ̎ Q ̎ R ̎ S .̎
(2)
c)
d)
e)
[10]
2
4.
The following table shows the number of points the Sharks rugby team scored in the
Vodacom Super 12 Rugby Tournament during April, May and June this year.
Date
22/04 30/04 7/05 21/05 28/05 4/06 11/06 25/06
Sharks
40
12
34
22
26
23
30
8
Opponents
24
32
16
32
21
18
30
36
a)
How many games did the Sharks win?
(1)
b)
Determine the mean of the number of points the Sharks scored in these three
months.
(2)
c)
Determine the 5 number summary plot for the number of points the Sharks scored.
(5)
d)
Use your calculator to find the standard deviation of the Sharks scores.
(2)
Another rugby team’s scores are shown in the box and whisker plot drawn below.
a
15
28
b
40
e)
If the interquartile range is 20 and the range is 35, write down the values of a and b.
(2)
f)
Is the data that is shown in the box and whisker plot positively or negatively
skewed?
(1)
[13]
3
5.
The speed in kilometres per hour, of snow skiers passing a certain point on a ski slope
was recorded and summarized in the table below:-
ANSWER THIS QUESTION ON YOUR DIAGRAM SHEET.
SPEED ( km/hr)
Frequency ( f)
0 ≤ 𝑥 < 10
10 ≤ 𝑥 < 20
20 ≤ 𝑥 < 30
30 ≤ 𝑥 < 40
40 ≤ 𝑥 < 50
10
20
45
71
21
Cumulative
Frequency
a)
Complete the table on the diagram sheet.
(3)
b)
Make use of the axis provided on the diagram sheet to draw a cumulative
frequency curve.
(4)
Indicate clearly on your graph where the lower quartile (LQ) and median speeds (M)
can be read.
(2)
Use your graph to estimate the number of skiers that passed the point with speeds
greater than 15km/hr.
(2)
Indicate the 75th percentile on your graph. (P)
(1)
c)
d)
e)
[12]
TOTAL FOR SECTION A: 75 MARKS
4
SECTION B
𝑦
𝐵
. 𝐸(−4; 4)
𝐴
0
𝐶
𝑥
𝐷
6.
The line 𝐵𝐶 , with equation 𝑦 = −𝑥 + 2, is a tangent at 𝐵 to the circle with centre
𝐸(−4; 4).
𝐴𝐵 is a diameter of the circle. 𝐵𝐶 ∕∕ 𝐴𝐷, where 𝐶 lies on the 𝑥-axis and 𝐷 lies on the
𝑦-axis.
a)
Determine the equation of the diameter 𝐴𝐵.
(3)
b)
Calculate the co-ordinates of 𝐵.
(2)
c)
Determine the equation of the circle, centre 𝐸.
(3)
d)
Write down the co-ordinates of 𝐴.
(2)
e)
If the length of the diameter is doubled and the circle is translated horizontally 6
units to the right, write down the equation of the new circle.
(2)
[12]
5
The equation of a circle is 𝑥 2 + 𝑦 2 − 8𝑥 + 6𝑦 = 24 is given.
Use analytical methods to find the shortest distance between the circle and the line
𝑦 -10=0.
(4)
7.b) Calculate the length of the tangent AB drawn from the point A(6 ; 4) to the circle with
equation (𝑥 − 3)2 + (𝑦 + 1)2 = 10. (Leave answer in simplest surd form)
(5)
7a)
[9]
8.
ANSWER THIS QUESTION ON YOUR DIAGRAM SHEET.
A graph of h(x) is drawn below
[
T
y
p
e
[
T
y
p
e
a
a)
a equation for h.
Write down the
b)
q – 30°) on the same system of axes as h.
Draw f(x) = sin(x
c)
d)
e)
(2)
q
u
u
Show on your ograph where you would read off the solutions for h(x) o– f(x) = 0
t
t A , B , C , D … etc)
(Use the letters
e
e
(4)
Write down the new equation of f if the x axis is moved 2 units upwards.
(1)
f
f
r
r f was moved 60° to the right to form an equation
If the graph of
g, write the
o
o
equation of g using
the cosine trigonometric ratio.
m
m
t
h
e
t
h
e
d
o
c
u
m
e
n
t
d
o
c
u
m
e
n
t
6
(1)
(3)
[11]
9.
Given:
sin43°cos31° = a
sin31°cos43° = b
cos43°cos31° = c
sin43°sin31° = d
express the following in terms of a, b, c and d.
i) sin74°
(2)
ii) cos274°- 1
(3)
[5]
10.
Prove the identity
2+𝑐𝑜𝑠𝑥−𝑐𝑜𝑠2𝑥
3𝑠𝑖𝑛𝑥−𝑠𝑖𝑛2𝑥
=
1+𝑐𝑜𝑠𝑥
[5]
𝑠𝑖𝑛𝑥
11a) Solve for 𝜃
i) sin(𝜃 + 30°)=cos 2 𝜃
(4)
ii) 6sin𝜃cos𝜃 = 0,234 and 𝜃𝜖 [-180° ; 180°]
(5)
11b) If the graph of f(x)=6sin𝑥cos𝑥 is drawn give the range and the period of the graph.
(3)
[12]
12.
Given:
𝑠𝑖𝑛1°.𝑐𝑜𝑠1°
𝑐𝑜𝑠88°
=
1
2
;
𝑠𝑖𝑛2°.𝑐𝑜𝑠2°
𝑐𝑜𝑠86°
=
1
2
;
𝑠𝑖𝑛3°.𝑐𝑜𝑠3°
𝑐𝑜𝑠84°
=
1
2
; … … … ….
a)
State the next equation in the above sequence.
(1)
b)
State your generalisation (use 𝜃). (ie: your conjecture)
(2)
c)
Prove this resulting identity using your standard identities.
(3)
[6]
7
13.
You may recognize the given pattern from the cover of your ‘Grade 11 Classroom
Mathematics’ text book by P. Laridon et. al.
One of the hexagonal shapes that make up the pattern is drawn on a Cartesian plane below so
that its centre is at the origin.
𝑦
.
𝐴ˈ
𝑥
.
𝐴(1.5; −4)
Given that the co-ordinates at point A are (1.5; -4), calculate the co-ordinates of Aˈ.
14.
A triangle PQR is enlarged by factor k, through the origin to form PˈQˈRˈ. If the area of
triangle PQR is 18 and the area of triangle PˈQˈRˈ is 50 and P = (-27; 15), find the coordinates of Pˈ.
8
[4]
[4]
15.
Disney World decides to build a house for Mickey Mouse looking like a piece of cheese as
shown.
E
𝜽
60◦
∝
K
C
Y
M
̂ E=60◦
AC
I
𝒑
̂ C=∝
EK
̂I=𝜃
CE
MI = 𝒑 units
Show that the height of the house (CI) is : CI =
2p sin∝ tanθ
√3 cos∝ +sin∝
TOTAL FOR SECTION B: 75 MARKS
9
MICK is a rectangle.
[7]