Name:
......................................................................................................
Question Q1 Q2 Q3 Q4
Q5
Q6
Q7
Q8
Q9
Q10
Q11
Q12
Q13
Max Mark
14
12
6
6
7
23
16
9
13
8
13
8
15
Mark
TOTAL
DIAGRAM SHEET 1 for Question 2.2
Mass (in kg)
80 ≤ x < 90
90 ≤ x < 100
100 ≤ x < 110
110 ≤ x < 120
120 ≤ x < 130
Frequency
1
2
9
2
1
Cumulative Frequency
DIAGRAM SHEET 2 for Question 2.3
Mass in kg
DIAGRAM SHEET 3 for Question 3.1
English Marks
TV Hours
DIAGRAM SHEET 4 for Question 9.1
DIAGRAM SHEET 5 for Question 13.5
RONDEBOSCH BOYS’ HIGH SCHOOL
Mathematics
Grade 12
Tuesday 25th September 2012
Set by: R Harmuth
3 Hours ; 150 Marks
Moderated by: D Geldenhuys
PAPER TWO
INSTRUCTIONS
1. Calculators can be used, unless otherwise stated, with answers corrected to two
decimal places.
2. All necessary working MUST be shown.
3. When necessary, leave answers with positive exponents.
4. Number your answers as the questions are numbered.
5. Untidy work will be penalised.
6. Only blue and black pens may be used.
7. Sketches may be done in pencil.
8. This exam contains thirteen questions.
Question One
The table below gives a breakdown of the Western Union Currie Cup Rugby
log standings for the top 8 teams at the end of the 2001 season in Wales:
TEAM
Harlequins
Wasps
Sparrows
Robins
Wonders
Sharks
Dolphins
Golden Men
POINTS
56
52
50
50
44
40
32
24
1.1
Determine the median (number of points scored).
(2)
1.2
Determine the lower AND the upper quartile.
(2)
1.3
Draw a box and whisker to represent the points scored.
(3)
1.4
Use your box and whisker to comment on the spread of the
points scored by the teams.
(1)
[8]
Question Two
The masses (in kg) of the Springbok Rugby Team to play New Zealand in
September are given below:
95
105
100
118
109
126 107
82 108 116 109 98 100
105
2.1
Calculate the mean mass of the Springbok team.
2.2
Complete the following table on DIAGRAM SHEET 1.
Mass (in kg)
80 ≤ x < 90
90 ≤ x < 100
100 ≤ x < 110
110 ≤ x < 120
120 ≤ x < 130
Frequency
1
2
9
2
1
102
(2)
Cumulative Frequency
(2)
2.3
2.4
Draw an ogive (cumulative frequency curve) of the above
information on the grid provided on DIAGRAM SHEET 2.
(4)
Calculate the percentage of the team who have masses within
one standard deviation of the mean. Show ALL calculations.
(5)
[13]
Question Three
A group of 12 learners was randomly selected from a class. The marks
scored in a standardised English test (out of 100 marks) and the average
number of hours they spent playing TV games (on a computer) per week
was recorded.
English Marks 80 65 50 30 70 60 85 45 45 90 35 75
TV hours
10 20 35 45 15 25 10 37 23 5 40 18
3.1
Represent this data as a scatter plot on the grid provided on
DIAGRAM SHEET 3.
(4)
3.2
Draw a line of best fit for your scatter plot.
(1)
3.3
What conclusion can you make about the learners’ marks and
the average number of hours spent on TV games?
(1)
Another learner from the class spends 30 hours playing TV
games. Predict his English mark.
(2)
3.4
[8]
Question Four
In the diagram below, ∆ PQR with vertices P(3 ; 1), Q( 8 ; 2)
and R(2 ;  3) is given:
4.1
Calculate the length of QR, leaving your answer in surd form.
(2)
4.2
Determine the co-ordinates of M, the mid-point of QR.
(2)
4.3
Determine the equation of the line parallel to PR, passing
through M.
(4)
4.4
If PQTR is a parallelogram find the co-ordinates of T.
(2)
4.5
Calculate  PRQ.
(5)
[15]
Question Five
5.1
5.2
The equation of a circle is given as x 2  2 x  y 2  6 y  30  0
5.1.1
Prove that the point P( 1 ;  9) lies on the circumference
of the given circle.
(2)
5.1.2
Determine an equation of the tangent to the circle at the
point P( 1 ;  9).
(7)
Calculate the length of the tangent DE, drawn from the point
D(2 ; 6) to the circle with equation ( x  4 )2  ( y  1 )2  12 .
E is a point on the circumference of the circle.
(5)
[14]
Question Six
The circle with centre C and equation ( x  3 )2  ( y  4 )2  34 is drawn
below. A is a y-intercept of the circle.
6.1
Determine the value of y at A.
6.2
The circle is enlarged by a scale factor of 2½ about the origin.
Write down the equation of the new circle in the form
( x  a )2  ( y  b )2  r 2 .
6.3
(4)
(3)
In addition to the given circle with centre C and equation
( x  3 )2  ( y  4 )2  34 , another circle with centre P and
equation ( x  9 )2  ( y  10 )2  49 is now given.
6.3.1
Calculate the distance between C and P.
(2)
6.3.2
Do these two circles cut once, twice, or not at all?
Justify your answer.
(3)
[12]
Question Seven
The point A( 3 ; y ) is rotated about the origin through an angle of 120° in
1
an anti clockwise direction to give the point B( x ; ).
2
Calculate the values of x and y .
[6]
Question Eight
Consider the point P(14 ; 5). The point is reflected about the y -axis to PꞋ.
8.1
Write down the co-ordinates of PꞋ.
(1)
8.2
An alternative transformation from P to PꞋ is a rotation about the
origin through an angle of °, where  < 180°. Calculate .
(5)
[6]
Question Nine
In the diagram below, ∆ MNR is drawn with vertices M(6 ; 2), N( 6 ; 8) and
R(1 ;  6). ∆ MNR is now enlarged by a factor of 2 to ∆ MꞋNꞋRꞋ.
9.1
Draw ∆ MꞋNꞋRꞋ on the grid provided on DIAGRAM SHEET 4.
9.2
Write down the values of:
9.2.1
9.2.2
M N 
MN
Area of Δ M N R 
Area of Δ MNR
(3)
(2)
(2)
[7]
Question Ten
10.1
10.2
SHOW ALL NECESSARY STEPS IN THIS QUESTION
If sin 24   m , and cos 35  n , determine the following in terms of
m and/or n :
10.1.1
tan 66
(3)
10.1.2
sin 70 
(3)
10.1.3
cos 114 
(2)
10.1.4
sin 59 
(3)
Without using a calculator, find the value of A if:
10.2.1
A
cos 220   tan 140 
tan 30   sin 630   cos 50 
(8)
10.2.2
A  ( 2 cos 75  1)( 2 cos 75  1 )
(4)
[23]
Question Eleven
11.1
Given:
cos 2x  5 cos x  sin 2 x
Determine the general solution.
11.2
(8)
tan 2 x  1
Consider the expression
tan x
11.2.1
11.2.2
For which values of x , x  [0° ; 180], will this
expression be undefined?
Prove that
tan 2 x  1
2

tan x
sin 2 x
(3)
(5)
[16]
Question Twelve
A person standing at point P looks up to the top, Q, of a double- storey
house. P is 17m from the foot of the house. The angle of elevation of Q from
P is 33°. He turns around and walks in the opposite direction from the house
at an inclination of 8° for a distance of 12m to a point M. Let  PQM  .
Q

M
12m
33°
F
17m
8°
P
12.1
Calculate PQ.
(2)
12.2
Calculate the length of QM
(4)
12.3
Hence, calculate .
(3)
[9]
Question Thirteen
The graph of f ( x )  2 cos x for  180  x  180 is drawn below :
13.1
Write down the period of f .
(1)
13.2
For which values of x is f ( x )  0 ?
(2)
13.3
Write down the x - intercepts of g ( x ) if g( x )  f ( x  60 ) .
(2)
13.4
Write down the amplitude of h( x ) if h( x ) 
13.5
Draw k ( x )  sin( x  45 ) ON DIAGRAM SHEET 5
PROVIDED showing intercepts with axes and turning points. (4)
13.6
For which values of x is k (x)  0 and f ( x )  0 ?
 f (2x )
4
(2)
(2)
[13]