HERZLIA SENIOR HIGH SCHOOL
“If you will it, it is no legend”
PRELIMINARY
EXAMINATION
GRADE 12
MATHEMATICS PAPER 2
MONDAY 1ST SEPTEMBER 2014
MARKS:
150
TIME:
3 HOURS
This question paper consists of 16 pages, 6 diagram sheets and an information sheet.
2
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.
3.
Clearly show ALL calculations, diagrams, graphs, et cetera that you have used in
determining your answers.
4.
Answers only will not necessarily be awarded full marks.
5.
You may use an approved scientific calculator (non-programmable and non-graphical),
unless stated otherwise.
6.
If necessary, round off answers to TWO decimal places, unless stated otherwise.
7.
Diagrams are NOT necessarily drawn to scale.
8.
6 diagram sheets for answering QUESTION 1.1, QUESTION 1.3, QUESTION 2.1,
QUESTION 2.2, QUESTION 8, QUESTION 9, QUESTION 10 and QUESTION 11 are
attached at the end of this question paper. Write your name in the space provided and
insert the sheets inside the back cover of your ANSWER BOOK.
9.
An information sheet, with formulae, is included at the end of the question paper.
10.
Number your answers correctly according to the numbering system used in this question
paper.
11.
Write neatly and legibly.
3
QUESTION 1
The data below shows the pulse rate of a sample of 12 people when they rest
and then again after 2 minutes of jogging.
1.1
Draw a scatter plot of the data given on the grid provided on DIAGRAM
SHEET 1.
(3)
1.2
Calculate the equation of the least squares line for this data.
(4)
1.3
Draw the least squares line on the scatter plot.
(2)
1.4
Calculate the correlation coefficient.
(1)
1.5
Comment on the correlation of the data.
(1)
1.6
If Joan’s heart rate after jogging is 86 beats per minute, what is her
resting heart rate, in beats per minute?
(1)
[12]
PLEASE TURN OVER
4
QUESTION 2
The time taken (to the nearest minute) for a certain task to be completed was
recorded on 48 occasions and the following data was obtained:
2.1
Complete the cumulative frequency table on DIAGRAM SHEET 2.
(2)
2.2
Draw an ogive (cumulative frequency curve) for the given data on
DIAGRAM SHEET 2.
(3)
Determine, from the ogive, the median, lower quartile and upper
quartile for the data.
(3)
Draw a box and whisker diagram of the data (assume minimum = 11
and maximum = 30).
(2)
2.3
2.4
[10]
5
QUESTION 3
ABCD is a quadrilateral with vertices A(1;6), B(3;0), C(6;1) and D(7;t) in a
Cartesian plane. AD || BC.
3.1
Calculate the gradient of BC.
(2)
3.2
Determine  , the angle of inclination of BC.
(2)
3.3
Show that AB is perpendicular to BC.
(2)
3.4
Find the value of t.
(3)
[9]
PLEASE TURN OVER
6
QUESTION 4
4.1
In the diagram below, the equation of the circle having centre M
is x 2  y 2  4 x  10 y  7 . Tangent PQ touches the circle at point P
and Q is the point (8;3).
y
Q(8;3)
P
O
x
M
M
4.1.1
Show that M has coordinates (2;–5).
(4)
4.1.2
Find the coordinates of the x-intercepts of the circle.
(4)
4.1.3
Give the radius of the circle.
(1)
4.1.4
Find the length of PQ.
(4)
7
4.2
In the diagram below, a circle has a diameter with equation y  2 x  3 .
The tangent at point E on the circle cuts the x-axis at F(12;0).
4.2.1
Determine the equation of the tangent at E.
(3)
4.2.2
Determine the coordinates of E.
(3)
[19]
PLEASE TURN OVER
8
QUESTION 5

5.1

Given: cos G  0,726 and 180   G  360 

5.1.1
5.1.2
5.2
Use a calculator to determine G , correct to one decimal place.
(2)
2 

Hence determine the value of tan  G  100   , correct to one
3

decimal place.
(1)
In the diagram below, T(8;k) is a point in the first quadrant.
If tan  
5.3
1
, determine without using a calculator:
4
5.2.1
The value of k
(2)
5.2.2
The value of sin  (leave your answer in surd form)
(3)
Simplify the following as far as possible without using a calculator:
5.3.1
sin( 180   x)
cos(90   x)  sin( 360   x)
(4)
5.3.2
cos( 45   )
 tan 
cos 45. cos 
(5)
9
5.4
5.5
By replacing 2A with 3A – A and 4A with 3A + A, prove the identity
cos 2A  cos 4A
 tan 3A .
sin 4A  sin 2A
Determine the general solution of 1  sin x  cos 2x .
(6)
(7)
[30]
QUESTION 6
In the diagram below, the graphs of y  sin a  b and y  c cos  and a third
trigonometric graph are sketched for the interval [0  ;360  ] .
6.1
Write down the values of a, b and c.
6.2
Use the graphs to solve the following for   [0  ;360  ] :
6.3
(3)
sin a  b  c cos
(2)
The equation of the third graph sketched is given by y  tan p  q .
Determine the values of p and q.
(3)
[8]
PLEASE TURN OVER
10
QUESTION 7
sin A sin B

.
a
b
7.1
Prove that in any acute-angled  ABC,
7.2
In the figure below, A, B and C are in the same horizontal plane. P is a
point vertically above A and B and C are 20 units apart. The angle of
elevation from B to P is  .
(3)
P
A
B



C
20
20 sin  tan 
.
sin(    )
7.2.1
Prove that AP 
7.2.2
Given that AB = AC, show that AP 
(4)
10 tan 
.
cos 
(2)
[9]
11
QUESTION 8
In the diagram below, O is the centre of the circle. Chord AB is perpendicular to
diameter DC. CM : MD = 4 : 9 and AB = 24 units.
8.1
Determine an expression for DC in terms of x if CM = 4x units.
(1)
8.2
Determine an expression for OM in terms of x.
(2)
8.3
Hence, or otherwise, calculate the length of the radius.
(4)
[7]
PLEASE TURN OVER
12
QUESTION 9
9.1
In the diagram below, O is the centre of the circle. PQ is a tangent to the
circle at A. B and C are points on the circumference of the circle. AB,
AC and BC are joined.


Prove the theorem that states CAP  ABC .
(6)
13
9.2
In the diagram below, two circles have a common tangent TAB. PT is
a tangent to the smaller circle. PAQ, QRT and NAR are straight lines.

Let Q  x .
9.2.1
Name, with reasons, THREE other angles equal to x.
(5)
9.2.2
Prove that APTR is a cyclic quadrilateral.
(5)
[16]
PLEASE TURN OVER
14
QUESTION 10
10.1
Use the diagram below to prove the theorem which states that if VW || YZ
XV XW

then
.
VY WZ
(6)
10.2
In  PQR below, B lies on PR such that 2PB = BR. A lies on PQ such that
PA : PQ = 3 : 8. BC is drawn parallel to AR.
10.2.1
Calculate (by showing all your working) the value of the ratio
10.2.2
Write down the value of
area of PRA
.
area of QRA
BD
.
BQ
(5)
(2)
[13]
15
QUESTION 11
11.1
Two circles touch each other at point A. The smaller circle passes through O,
the centre of the larger circle. Point E is on the circumference of the smaller
circle. A, D, B and C are points on the circumference of the larger circle.
OE || CA.
11.1.1
Prove, with reasons, that AE = BE.
(2)
11.1.2
Prove that  AED |||  CEB.
(3)
11.1.3
Hence, or otherwise, show that AE2 = DE.CE.
(2)
PLEASE TURN OVER
16
11.2
In the accompanying figure, AB is the diameter of circle ADCB. Chords
AC and BD intersect at E. EP is perpendicular to AB.
11.2.1
Prove that  BPE |||  BDA.
(3)
11.2.2
Hence show that
BP PE

.
BD AD
(2)
11.2.3
Prove that AB 2  BD 2 
BD 2 .PE 2
.
BP 2
(5)
[17]
TOTAL : 150
HERZLIA SENIOR HIGH SCHOOL
GRADE 12 MATHEMATICS PAPER 2
MONDAY 1ST SEPTEMBER 2014
DIAGRAM SHEET
(PLEASE PLACE IN YOUR ANSWER BOOK)
NAME:………………………………………………..
DIAGRAM SHEET 1
QUESTIONS 1.1 AND 1.3
DIAGRAM SHEET 2
QUESTION 2.1
QUESTION 2.2
DIAGRAM SHEET 3
QUESTION 8
QUESTION 9.1
DIAGRAM SHEET 4
QUESTION 9.2
QUESTION 10.1
DIAGRAM SHEET 5
QUESTION 10.2
QUESTION 11.1
DIAGRAM SHEET 6
QUESTION 11.2