Download n west maths p2_sept 2014

NATIONAL
SENIOR CERTIFICATE
GRADE 12
MATHEMATICS P2
SEPTEMBER 2014
MARKS: 150
TIME: 3 hours
This question paper consists of 10 pages, 2 diagram sheets and 1 information sheet.
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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 the 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.
TWO diagram sheets for QUESTION 1.2, QUESTION 2.1, QUESTION 7.1 and
QUESTION 9.1 are attached at the end of this question paper. Write your name on
these diagram sheets in the spaces provided and insert the diagram sheets inside the
back cover of your ANSWER BOOK.
9.
An information sheet with formulae is included at the end of this question paper.
10.
Number the answers correctly according to the numbering system used in this
question paper.
11.
Write neatly and legibly.
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QUESTION 1
The distance travelled (in kilometres) by a group of 9 learners from Thohoyandou District to
Thengwe Secondary school every morning is given as follows:
24
34
50
66
76
82
88
94
1.1
Calculate (where necessary to ONE decimal digit):
100
1.1.1
the mean
(1)
1.1.2
the median
(1)
1.1.3
the interquartile range of the data
(3)
1.2
Draw a box and whisker diagram on DIAGRAM SHEET 1 to represent the data. (3)
1.3
Identify any outliers in the data set. Motivate your answer.
(2)
[10]
QUESTION 2
The table below compares the number of hours spent by 7 Mathematics learners and the
learners’ performance in the test.
Learners
Hours
Marks (%)
1
1
35
2
3
55
3
5
60
4
6
65
5
8
75
6
10
70
7
11
80
2.1
Represent the data in a scatter plot on DIAGRAM SHEET 1.
(3)
2.2
Determine the equation of the line of best fit for the data.
(3)
2.3
Hence, draw the line of best fit on the scatter plot.
(2)
2.4
Determine the correlation coefficient of the data. Hence, comment about the
strength of the relationship between the two variables.
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(3)
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QUESTION 3
In the accompanying figure, K(x; y), L(−2; −1) and M(4; 3) are the vertices of triangle KLM.
The equation of the side KL is y – 5x – 9 = 0 and that of KM is 5y + x – 19 = 0.
N is the midpoint of LM.
y
K(x ; y)
M(4 ; 3)

x

L(
3.1
Calculate the coordinates of N.
(2)
3.2
Show that the coordinates of K are (−1 ; 4).
(4)
3.3
Determine the equation of the line through K and N in the form y = mx + c
(3)
3.4
Determine the gradient of the line LM. Hence, prove that KN is the perpendicular
bisector of LM .
(3)
3.5
If L, M and the point J(7; a) are collinear, calculate the value of a.
3.6
Determine the size of the angle of inclination between KL and the positive x-axis. (2)
[17]
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(3)
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QUESTION 4
In the figure below, the origin O is the centre of the circle. P(x ; y) and Q(3 ; −4) are two
points on the circle and POQ is a straight line. R is the point (k ; 1) and RQ is a tangent to the
circle. T is an x-intercept of the circle.
y
P(x ; y)
R(k ; 1)
T
O
x
Q(3 ; −4)
Determine:
4.1
the equation of the circle.
(3)
4.2
the length of TQ. (Leave your answer in simplified surd form.)
(3)
4.3
the equation of OQ.
(3)
4.4
the coordinates of P.
(2)
4.5
the equation of the circle with centre P, that passes through (0; 0) in the
form x2 + y2 …
(3)
4.6
the equation of QR.
(4)
4.7
the value of k.
(3)
[21]
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QUESTION 5
5.1
5
and 0    180 , use a sketch to determine the value of the
12
following expression (WITHOUT USING A CALCULATOR): 3 sin   2 cos 
If tan   
(4)
5.2
Simplify the following expression to ONE trigonometric ratio of P:
sin (360  P)  cos 270
sin (90  P)  tan (180  P)  tan (360  P)
5.3
Calculate the value(s) of x , x [ 90 ; 270] if sin x = cos 2x – 1
(6)
(6)
[16]
QUESTION 6
The sketch represents the graphs of the following functions for x [180; 0] :
1
f ( x)  sin 2 x and g ( x )  tan x .
2
Line segment ABC is perpendicular to the x-axis at C(135 ; 0), with A on f and B on g.
D( 120;
3
)
2 is the point where f and g intersect.
6.1
Calculate the length of AB.
(3)
6.2
Write down the period of f.
(1)
6.3
Without calculations, use the graph to write down the values of x for which
f ( x)  g ( x) in the given interval.
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(5)
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QUESTION 7
7.1
In ∆ ABC, Â is obtuse.
C
A
B
Redraw the sketch in your ANSWER BOOK and prove that
7.2
a
b

sin A sin B
(5)
From A the angle of elevation to the top of a vertical tower CD is x and from a point
B, d metres closer to the tower, the angle of elevation is y  .
D
1
h
y
C
d
B
7.2.1 Show that the height of the tower is given by h 
x
A
d . sin x sin y
sin ( y  x)
7.2.2 Calculate the height of the tower if d = 85 m, x 10 and y  38 .
(5)
(2)
[12]
QUESTION 8
8.1
Prove that sin (x + y) + sin (x – y) = 2 sin x . cos y .
8.2
Hence, or otherwise, prove that:
sin 3 x  sin x
 2 sin x .
1  cos 2 x
(2)
(5)
[7]
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NOTE: GIVE REASONS FOR YOUR STATEMENTS AND CALCULATIONS FROM
QUESTIONS 9 TO QUESTION 11.
QUESTION 9
9.1
In the diagram below, O is the centre of the circle and OS is perpendicular to the
chord RT.
O
1 2
R
T
S
Prove, using Euclidian geometry methods, the theorem that states that RS = ST.
9.2
(5)
The line ABCD intersects two concentric circles with centre O as shown in the
diagram below. OA = 25 cm and OC = 17 cm.
O
A
1 2
B
X
C
D
9.2.1 Determine AC if OX = 15 cm.
(5)
9.2.2 Show that AB = CD.
(4)
[14]
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QUESTION 10
10.1
Complete the statements of the following theorems by writing down only the missing
word(s) in each case.
10.1.1
The opposite angles of a cyclic quadrilateral are . . .
(1)
10.1.2
If two triangles are equiangular, the corresponding sides are . . .
(1)

10.2

PQR is a tangent to circle QABCD. AB ║ QD. CB = CD. Let Q3  30 and D3  70 .
B
C
2
1
70º
1
2
A
1
2
3
1
2
3
D
30º
R
Q
P

10.2.1
Calculate Q1 .
10.2.2
Prove that C 110
10.2.3
Calculate B1
(2)

(4)

10.3
(4)
DOE is a diameter of a circle with centre O. DF and EF are chords of the circle.
GH  DE.
F
G.
12
D

1 2
H
.
O
E

Prove that G1  E .
(4)
[16]
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QUESTION 11
11.1
In the accompanying diagram, AD = 15 m, DB = 10 m and AF = 12 m.
A
15m
12m
D
F
10m
E
B
C
If DE ║ BC and DF ║ BE, calculate the length of AC.
11.2
(8)
PQRS is a rectangle. A is a point on QR such that PAˆ S  90 .
Q
P
1
2
A
3
1
2
S
R
Prove that:
11.2.1
PÂQ  AŜR
(3)
11.2.2
ΔAPQ /// ΔSAR
(3)
11.2.3
If RS = 8 units, QA = x units and RA = y units, express y in terms of x. (3)
[17]
TOTAL: 150
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DIAGRAM SHEET 1
NAME:
QUESTION 1.2
QUESTION 2.1
Scatter Plot
100
90
80
Marks (%)
70
60
50
40
30
20
10
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Hours
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DIAGRAM SHEET 2
NAME:
QUESTION 7.1
C
A
B
QUESTION 9.1
O
R
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S
T
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INFORMATION SHEET: MATHEMATICS
 b  b 2  4ac
x
2a
A  P(1  i) n
A  P(1  ni ) A  P(1  ni )
Tn  a  (n  1)d
Tn  ar n 1

Sn 

n
Sn 
n
2a  (n  1)d 
2


a r n 1
r 1
x 1  i   1
F
i
P
f ( x  h)  f ( x )
h
h 0
 x  x2 y1  y 2 
M  1

;

2
2

m
y 2  y1
a
b
c


sin A sin B sin C
a 2  b 2  c 2  2bc. cos A
cos 2   sin 2 

cos 2  1  2 sin 2 
2 cos 2   1

area ABC 
sin 2  2 sin . cos 
 fx
x
 
2
n
n( A)
nS 
2
 x  x 
i 1
i
n
P(A or B) = P(A) + P(B) – P(A and B)
b
1
ab. sin C
2
cos     cos . cos   sin .sin 
n
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m  tan 
sin      sin . cos   cos.sin 
cos     cos . cos   sin .sin 
yˆ  a  bx
x 2  x1
 r2
sin      sin . cos   cos.sin 
P( A) 
a
; 1  r  1;
1 r
f ' ( x)  lim
y  y1  m( x  x1 )
x  a 2   y  b 2
In ABC:
S 
r 1
;
x[1  (1  i)n ]
i
d  ( x 2  x1 ) 2  ( y 2  y1 ) 2
y  mx  c
A  P(1  i) n
 x  x ( y  y )
(x  x)
2