SECTION A QUESTION 1 E ;3) - St Stithians Collegemaths.stithian.com/New CAPS 2016 Prelim...
Transcript of SECTION A QUESTION 1 E ;3) - St Stithians Collegemaths.stithian.com/New CAPS 2016 Prelim...
Page 1 of 22
SECTION A
QUESTION 1
In the diagram below:
DC CB
A is the centre of the circle.
E is the midpoint of AB.
The equation of line BA is: 3 7y x
DF is a tangent to the circle at F.
(a) Find the co-ordinates of E. (2)
4 2 5 1;
2 2
(3;2)
E
E
(b) Determine the equation of the circle, centre A, passing through point E.
Give the equation on the form 2 2 2( ) ( )x a y b r (3)
2 2 2
2
2
2 2
(4 3) (5 2)
1 9
10
( 4) ( 5) 10
r
r
r
x y
B(2;-1)
E
A(4;5)
F
D
C(10;3)
y
x
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(c) Find the gradient of line BC. (2)
3 1 1
10 2 2BCm
(d) Hence, or otherwise, determine the equation of line DC. (3)
2
2
(10;3)
3 2(10)
23
2 23
DCm
y x c
sub
c
c
y x
(e) Show, by calculation, that the co-ordinates of D are (6 ;11) . (3)
3 7 2 23
5 30
6
3(6) 7 11
D(6;11)
x x
x
x
y
(f) Find the size of ˆEBC . (5)
3
tan 3
71,6
1
2
1tan
2
26,6
ˆ 71,6 26,6 45
EB
BC
m
m
EBC
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(g) Find the length of DF. (6)
2 2 2
2
2 2 2
2
ˆ 90 (tan )
10
(6 4) (11 5)
40
( )
40 10
30
DFA chord
FA
AD
AD
DF AD FA pythag
DF
DF
[24]
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QUESTION 2
(a) Simplify:
2
2
(sin cos ) 1
sin 1
x x
x
(4)
2 2
2
2
sin 2sin .cos cos 1
cos
2sin .cos
cos
2sin
cos
2 tan
x x x
x
x x
x
x
x
x
(b) Solve for x in the interval [ 180 ;180 ] :
cos( 30 ) sin3x x . (7)
cos( 30 ) cos(90 3 )
30 (90 3 ) 360
30 90 3
4 120
30 90
30 (90 3 )
2 60
30 180
150 ; 60 ; 30 ;120
x x
x x k
x x
x
x k
or
x x
x
x k
x
[11]
Page 5 of 22
G
1
1
1
1
21
E
D
C
B
A
QUESTION 3
(a) Refer to the figure:
Below are three statements that refer to the above figure. What additional
information would be required, if any, to make each individual statement
true?
(1) 1 1 1ˆ ˆ ˆA E D
D must be on the circumference
(2) 1 1ˆB C
GB = GC or G is the centre of the circle
(3) 1 2ˆ ˆ 90C C
BE is the diameter
(3)
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O
S
R
Q
P
(b) Use the figure below to prove that ˆ ˆ 180S Q .
O is the centre of the circle. (6)
Given: Cyclic quad PQRS, O centre of the circle
RTP: ˆ ˆ 180S Q
Const.: Join PO and OR
Proof:
ˆ
ˆ 2 (2 )
ˆ 360 2 ( ' int)
ˆ 180 ( )
ˆ ˆ 180
Let S x
POR x at circum
reflexPOR x s around a po
Q x half at centre
S Q
[9]
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32
1
21
2
1
D
FE
C
B
A
QUESTION 4
In the figure:
ABCD is a cyclic quadrilateral.
AB = AF
(a) Prove that DE = EF. (3)
1
3
ˆ
ˆ ( )
ˆ ( )
Let F x
B x AB AF
D x ext of cyclic quad
ED EF
(b) If it is further given that ED bisects ˆCDF , prove that FB bisects ˆABC .
(3)
2
2
1 2
ˆ ( sec )
ˆ ( )
ˆ ˆ
ˆsec
D x bi ted
B x in same seg
B x B
FB bi ts ABC
[6]
Page 8 of 22
BDA
O
C
QUESTION 5
In the figure:
O is the centre of the circle.
AC BC
OD CB
Prove that:
(a) / / /ODB ACB (3)
ˆˆ1) 90 ( )
ˆ ˆ2) ( )
ˆ ˆ3) (3 )
/ / / ( )
In ODB and ACB
OBD C given
B B common
DOB CAB rd of triangle
ODB ACB AAA
(b) 22 .AD OA BC (4)
2
(/ / / ' )
( )
( )
2
2 .
OD DB OBs
AC CB AB
AD DB OD AB
OB OA radii
AD OA
CB AD
AD OA BC
[7]
Page 9 of 22
QUESTION 6
A consumer testing company studied three different brands of washing machines
to see how much water was used during each wash. Each washing machine was
tested 15 times. The box and whisker plots below show the results of this study.
Washing machine A
Washing machine B
Washing machine C
Number of litres used by washing machine.
(a) Which brand of machine (A, B or C) used up most water on average? (1)
C
(b) Which brand of machine (A, B or C) is the most predictable? (1)
B
(c) The results of Washing Machine C are shown below:
81 85 85 88 89 90 92 101 104 105 106 106 108 112 112
(1) Determine the standard deviation of the litres of water used. (2)
10,4
(2) Based on this data, how many litres of water would be used in 67%
of the washing loads? (2)
[97,6 10,4 ; 97,6 10,4]
[87,2 ; 108]
[6]
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5 10 15 20 25 30
1
2
3
4
5
6
7
8
9
10
QUESTION 7
The distance (𝑥) in kilometres that the staff at a certain school in Durban travel to
work each day is summarised in the histogram below:
(a) Is the data positively or negatively skewed? (1)
Skewed to the right, positively skewed
(b) Which of the intervals is the modal interval? (1)
Mode 5 - 10
(c) Use the histogram to complete the given table. (3)
Intervals Frequency Cumulative Frequency
0 ≤ 𝑥 < 5 7 7
5 ≤ 𝑥 < 10 9 16
10 ≤ 𝑥
< 15
8 24
15 ≤ 𝑥
< 20
3 27
Kilometres
Fre
qu
ency
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20 ≤ 𝑥
< 25
2 29
25 ≤ 𝑥
< 30
1 30
(d) Use your frequency table to draw an ogive below. Label your axes. (3)
(e) Determine the following using your ogive:
(1) the interquartile range (2)
14,1- 5,6 = 8,5
(2) the percentage of these staff members that stayed between 4km and
14km from the school? (2)
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22 4
100 60%30
[12]
75 marks
Page 13 of 22
3
3
2
1
2
1
32
1
F
E
D
C
BA
SECTION B
QUESTION 8
(a) In the figure:
ABC is a tangent to the circle at B.
1 3ˆ ˆD D
Prove that:
(1) DCBF is a cyclic quadrilateral. (4)
1 2 3
2 1 3 1
ˆ ˆ (tan )
ˆ ˆ ˆ( )
( int )
B D AC chord EB
D D D
DCBF cyclic ext opp
(2) FC is a tangent to the circle FED. (4)
3
3 2
2
ˆ ˆ (tan )
ˆ ˆ ( )
ˆ ˆ
tan (tan )
E B AC chord BD
B F in same seg
F E
FC to circle FED chord converse
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xx
y
30°
CA
D
B
(b) In the figure:
ABCD is a cyclic quadrilateral.
AD = DC = x
AC = y
ˆ 30B
Show that 2 3y x (5)
2 2 2
2 2 2
2 2
ˆ 150 ( )
2 . cos150
2 3
(2 3)
2 3
D opp of cyclic quad
y x x x x
y x x
y x
y x
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Q
O
P
K
L
M
N
(c) In the figure (not drawn to scale):
OM // PL
KN : KO = 8 : 3
LM = 3KL
Calculate:
(1) Area of KMO
Area of NMO
(3)
1.
321 5
.2
KO hKO
ONON h
(2) LQ
QN (3)
( / / )
3 .3 9
4 4
( / / )
994
5 20
PO MLPL OM
OK MK
p kPO p
k
LQ POPL OQ
QN ON
pLQ
QN p
[19]
Page 16 of 22
QUESTION 9
The sketch below shows the graphs of 𝑓(𝑥) = − sin 𝑥 and 𝑔(𝑥) = cos𝑥
2 for
𝑥 ∈ [−180°; 180°]. The coordinates of 𝐴, the point of intersection of the two
graphs, are (𝑡;√3
2).
(a) Determine the value of 𝑡. (2)
3sin
2
60
t
t
(b) Use the sketch to solve the following equation: cos𝑥
2= −2 sin
𝑥
2cos
𝑥
2
for 𝑥 ∈ [−180°; 180°]. (3)
cos 2sin .cos sin
2 2 2
180 ; 60 ;180
x x xx
x
(c) If 𝑓 is translated 10° to the left and 2 units up, what will the new equation
of 𝑓 be? (2)
sin( 10 ) 2y x
[7]
(𝑡;√3
2)
𝑓
𝑔
𝐴
Page 17 of 22
QUESTION 10
(a) Simplify: sin170 .cos10
sin(20 ).cos cos(20 ).sinx x x x
(4)
sin10 .cos10
sin(20 )
1sin 20
2
sin 20
1
2
x x
(b) Prove the following identity:
cos cos2 1 1
sin sin 2 tan
x x
x x x
(5)
2
2
cos (2cos 1) 1
sin 2sin cos
cos 2cos
sin (1 2cos )
cos (1 2cos )
sin (1 2cos )
cos
sin
1
tan
x xLHS
x x x
x x
x x
x x
x x
x
x
x
RHS
Page 18 of 22
(c) Determine the general solution for :
27sin 2cos 5 0 (7)
2
2
2
7sin 2(1 sin ) 5 0
7sin 2 2sin 5 0
2sin 7sin 3 0
(2sin 1)(sin 3) 0
1sin sin 3
2
180 30 360
210 360
360 30 360
330 360
or
k
k
or
k
k
(d) If 31
sinsin
x ax and
31cos
cosx b
x ,
Prove that tan .b
xa
(5)
2 3 2 3
2 2
3 3
2
2 33
2 3 2
3
3
3 2
33
3
1 sin sin 1 cos cos
1 sin 1 cossin cos
1 sin
sin costan
1 coscos sin
tantan
tan
tan
x a x x b x
x xx x
a b
xx x bax
xx a x
b
bx
a x
bx
a
bx
a
[21]
Page 19 of 22
A
BO
Q(4;3)
P
y
x
QUESTION 11
The two circles in the diagram represent two interlocking gears, which touch at
point Q (4 ; 3).
The circles have the following equations:
x2 + y
2 – 25 = 0 and x
2 – 12x + y
2 – 9y + 50 = 0
(a) Show that the co-ordinates of P are (6; 4½). (3)
2 2
2 2
2
2
1 112 36 9 4 50 36 4
2 2
1 25( 6) 4
2 4
16;4
2
x x y y
x y
P
(b) Determine the equation of common tangent AB. (4)
3 4
4 3
4(4;3)
3
43 .4
3
25 4 25
3 3 3
OB ABm m
y x c sub
c
c y x
Page 20 of 22
(c) If the larger gear makes one full revolution, how many times will the
smaller gear turn completely? (4)
52 2 5
2
5 10
small bigC C
C C
twice
(d) Find the area of AOB . (3)
2
25 250; ;0
3 4
1 25 25 625
2 4 3 24
A B
Area units
(e) Another tangent to circle O, drawn from A touches the circle at C.
Determine the length of CQ. (2)
(4;3) ( 4;3)
8
Q C
CQ units
[16]
Page 21 of 22
QUESTION 12
The diagram shows a pyramid
shaped ‘cone’.
Each face is an isosceles triangle
with base angles of 640. The base
is a square of side 6cm.
EG is the slant height of the
pyramid.
EF is the perpendicular height of
the pyramid.
𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑎 𝑝𝑦𝑟𝑎𝑚𝑖𝑑 = 1
3× 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑏𝑎𝑠𝑒 × 𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 ℎ𝑒𝑖𝑔ℎ𝑡
(a) Determine the length of edge AE. (3)
3 ( )
3cos64
6,8
BG GA cm EG BA in isos
AE
AE cm
(b) Calculate the height EF. (4)
2 2 2
2 2 2
tan 643
6,2
( )
6,2 3
5,4
EG
EG cm
EF EG FG pythag
EF
EF cm
6 cm
64F
G
D
E
C
B
A
Page 22 of 22
(c) Determine the volume of the pyramid. (2)
3
1.(6 6).
3
136 5,4
3
64,8
V EF
V
V cm
(d) The pyramid is to be wrapped in a single layer of gold foil, with no
overlaps.
Calculate the total area of foil that would be needed. (3)
2
14 6 6 6
2
14 6 6,2 6 6
2
110,4
TSA EG
TSA
TSA cm
[12]
75 marks
Total: 150 marks