2011
Secondary School
Examination
Papers
Secondary Two Express
Mathematics
Paper 1 & 2
1
Admiralty Secondary School
SA2
2
Ang Mo Kio Secondary School
SA2
3
Beatty Secondary School
SA2
4
Bedok Town Secondary School
SA2
5
Bowen Secondary School
SA2
6
Bukit Batok Secondary School
SA2
7
Bukit Merah Secondary School
SA2
8
Chung Cheng High School
SA2
9
Clementi Town Secondary School
SA2
10
Geylang Methodist Secondary School
SA2
11
Jurong West Secondary School
SA2
12
Northland Secondary School
SA2
13
Ping Yi Secondary School
SA2
14
Shuqun Secondary School
SA2
15
Tanjong Katong Secondary School
SA2
16
Yishun Secondary School
SA2
17
Yuhua Secondary School
SA2
I
NO:
N AME:
CLASS:
ADMIRALTY SECONDARY SCHOOL
4
"
~.!'!..~
II
END OF YEAR EXAMINATION 2011
SUBJECT
: Mathematics
PAPER
:1
LEVEUSTREAM
: Sec 2 Express
DATE
: 10 October 2011 (Monday)
TIME
: 1000 - 11 00
DURATION
: 1 hour
II
Instructions lo candidates:
1. Write your name, class and index number.
2 . Answer ALL questions.
3. Use an electronic calculator lo evalu ate explicit numerical expressions.
If the degree of accuracy is not specified in the question. and if the answer
is not exact, give the answer lo three significant figures.
Give answers in degrees to one decimal place. For rr. use either your
calculator value or 3.142, unless the question requ ires the answer in terms
of rr.
4. EssenLial workings must be shown . Loss of essential workings and illegible
handwriting will lead to loss of marks.
DO NOT TURN OVER THIS PAGE UNTIL YOU ARE TOLD TO DO SO.
This question paper consists of 11 printed pages ineluding this cover page.
Mathematical Formulae
Mensuraiion
Curved Surface Area of a Cone = 7frl
Surface Arca of a Sphere
=4 7fr 1
.. I
2
Volume of a Cone = - Jff
3
4
'1
Volume of a Sphere = :;- Jrr
.>
S1atistics
"[, j-.;
Mean=
2
J
l
1. Express eacii map scale in the fomi I : r .
(a) 3 cm : 24 m
(b) 7 cm : 42 km
Answer.
2.
A 1s inversely proportional to 8 . Given lhat A = 8 and 8
(a)
12)
(b)
[21
= 36, find an equation connecting A
and B.
Answer:
3
...................... ...
(3)
3.
Polygon ABCDEFGH is similar to polygon PQRSTUVW. find PQ. The two polygons are not
drawn to
~l e.
1.5
T
s ~m
A
Q, _ - - - - - .R
Clll
~--' s
--
u
f'
n
F
--
t;
6 Clll
v
11
w
x .Y =.!., find the val1Je
4· If 2y+3x
4
II
II
G'
D
11
II
2 Clll
(.
Answer:
(2)
Answer:
.... ............... ····· (2]
of~ .
y
5. Simplify the following expression:
(a) (2A ; Sy)-3(5., -2y)
5a + 4h
(b)
2
cd'
(c) -
'
+
c~
2h-3o
3
cdl!
>< - + 4
2
Answer:
s
(a)
........................
[2]
(b)
........................
[2]
(c)
........................
[2]
6. Expand (9a+4b)(1a-5b) .
7. Factorise 2x'
8.
1 J.t -
Answer:
.. ...................... [2)
Answer:
........................
27.
[1)
The diagonal of a square is 9 cm. Find lhe perimeter of the square.
Answer:
6
..................cm
(4)
9.
A pottion of a pyramid with a square base ls removed, resulting in the following sohd. 11J and
SQ are the diagonals or the base or lho original pyramid and Rw is the perpend1culnr height.
Grvon that TS = 29 cm and RW = 57 cm. find
(a) lhe base area or the solid.
(b) lho volume or the solid.
w
T
Q
~._ ~7
cm
s
s
LI
Answer:
7
(a)
...cm'
(1]
(b)
.. cm3
(2)
10. A solid 1s made up ol a cylinder and a cone with dimensions as shown below. Taking
If -
l.142 . find
(a) lhe slant height of the cone.
(b) (I) the ouNed surface area of cone.
L------.i.
(II) the to tal surface areil or solid.
I
Ycm
--...lll cm
(c) (t) the volume of cone,
(ii) the vofume of the solid.
Give your answer in 3 significant figures
8cm
..
Answer:
[3]
............... ...cm
(a)
2
(1I
(ii)
cm2
(3)
(c) (i)
... .............. cml
(1 J
(b) (i) .................. cm
(ii)
............
cm3
(2)
11. The following numbers are the number of words in each line from a page or a novel.
(a)
Fill in the given frequency table.
(b)
Calculate the
(i) mean,
(ii) median .
(iii) mode.
8
8
11
14
9
11
7
8
13
12
2
13
14
3
8
12
11
11
14
10
13
2
11
9
13
13
13
13
4
8
11
.13
10
11
13
12
13
10
3
13
9
14
Solution:
(a)
x
f
2
f.~
-
3
4
7
-
8
g
10
11
12
- -
13
14
Total
[3]
Answer:
9
(b) (i) Mean•............. ..
[1]
(ii) Median• ...... .. .....
(1)
(iii) Mode • .. ......... ...
[ 1]
12. It is given that
&
= (x: x is an integer. I < x < 50) and sets A, 8 and Care defined as follows:
A - {x: x is divisible by 7)
H = { ..: x is a multiple of 14)
c;
{x: x
is multiple or 3)
(a) List the elements of A n C .
(b) Find
n(<A n C )').
(c) Stale 11(8 n c}.
(d) Draw a Venn Diagram to represent the above information. Shade ,1 ,.., B•,-..,
c.
[2]
(d) Solution
Answer:
10
(a)
[2]
(b)
(1]
(c)
(1 I
13. A bag contained 6 blue balls and 7 red balls. If a ball is picked at random from tho
bag, find the p robability that a red ball is picked.
Answer:
End of Paper
11
[1)
1. Express each map scale in the rorm l : r .
(a) 3 cm: 24 m
(b) 7 cm: 42 km
Solulion:
(a) 3cm : 24 m
1 cm:8m . ......... Ml
1 : 800 ............... A1
(b) 7 cm : 42 km
1 cm:6 km ....... .. M1
1 : 600000 .......... A1
Answer:
(a)
(21
(b)
[2)
2. A is inversely proportional to B. Given that A; 8 and B; 36, find an equation connecting A
and B.
Solution·
k
A -JJ
8-~
16
K Ml
k =8x36
-288 K Ml
288
II : - K Ill
R
Answer:
3
............... ......... . . [3]
3.
Polygon ABCDEFGH is similar LO polygon PQRSTUVW. find PO. The two polygons are not
drawn to scale.
Q
R
8cm
A
1.5 cm
T
s
p
u
,..
6 (an
,...---------. £
v
w
G
Solution:
D
2cm
c
Sine<> polygon PQRS"J"UVWis similar to polygon ABCDEFCH.
l'Q RS
-,=AB CD
PQ
= l.S
8
K K Ml
2
f'Q =~x8
2
= 6cm K K Al
4
·
If
Answer:
.................. cm
(2]
Answer:
•..•. .•. . ..... ... .. .....
[2)
x- y
I
r
- ·- =-, find the value of:.. .
2y+3.r 4
y
Solution:
x-y
2y +3x 4
4(x y) = 2y+3xKKM I
~
4x-4y=2y+3x
x=6y
~=6 KKA I
y
4
5. Simplify the following expression.
(a) (h+Sy)-3(5r-2y)
(b)
(c)
5,,. 411
2
+
2b - lo
3
cd' Ct' cdt'
x +.1
4
2
$olut10n
(a) (2n 5y) - 3(Sr - 2y)
·2•~
Sy- ISx+6yKKM I
• - 13.\ • 11.v KKK Al
(b)
2b
'\11 I 4/1
t
2
1(S"
)(I
3
I 4b)
6
1
2(2/i 3n2 K K Ml
1511 I 12h I 4/,
6
9u I l(Jb
KK
6
(c)
c·dJ
~
•
"
T
2
.. "' x 2
..,,'
K K Al
,/
K K Al
e1.lc:
C('
4
1
6
(111
x
-I
,.,,~
KKMI
6
Answer:
(a)
.......................
(2)
(b)
........................
l:?J
(c)
························
[2]
6. Expand (9C1 ~ 4b}(7C1 - 5b}.
Solution:
(9CI ·I 4b)(7a - 5b)
= 9a(?a-5b) +4h(?a -Sb)
- 6JC1' - 45ab+ 28a/1- 20b'
=63a ' - 17ab-20b
2
KK
K K Ml
KKAI
Answer:
7.
........................ [2]
Factorise 2x 2 +3x-27.
Solution:
2x ••
2x 1 +3x - 21
.< /
·,'
,,
.., +9
' • -3
= (2x+9)(x -3) KAI
-27
Answer:
+9x
-6x
+)x
.............. .......... (1 ]
8. The diagonal of a square is 9 cm. Find the perimeter of the square.
Solution:
Let the breadth of the square = b cm
By Pythagoras' Theorem,
b 1 +b 1 - 9 2
K K Ml
2b' = 31
h ' - 40.5
h
.J40.5 K Ml
=6.36396103067
Perimeter of the Square
= 4 x J40.5 OR = 4 >< 6.364 (at least 4 s.j .) K Ml
= 25.5 (3 s.J.)
= 25.456
K K Al
Answer:
6
.............. . ... cm
[4)
9. A portion ol a pyramid with a square base is removed. resulting on the follOWJng solid.
ru and
SQ are the diagonals or tho base or the O<igonal pyramid and RW is the perpendicular height
Given that TS• 29 cm and RW • 57 cm. nnd
(a) the base area of the solid.
(b) the volume of the solid.
w
r
29c:m
s
s
Soluhon
(a) Base area or the solid
3 x29x29
.j
• 630.75 cm 1
K Al
(b) Volume ol lhe solid
~
I
x 630.75x57 K Ml
3
= It 984.25 cm'
K Al
Answer:
7
(a)
.................cm2 (1)
(b)
............ ...
cm3
(2]
10. A solid is made up or a cylinder and a cone with dimensions as shown below. Taking
n
9
3.142, find
,_____, 19
(a) the slant height of the cone,
____,Ill
(b) (i) the curved surface area of cone.
(II) !he total surface area of solid,
(c) (i)
cm
the volume of cone,
cm
(ii) the volume or the solid.
Give your answer in 3 significant figures.
Scm
Solution:
(a) Let the slant height or the cone be I cm. }
Sy Pythagoras' Theorem.
Ml
I= J97
: 9.849957802
11 :91 + 41
"9.85 cm (3 s.f.) K Al
- 81+16
= 97KK Ml
(b) (i) Curved surface area or cone
- 3.1 42
'I(
(c) (i) Volume of cone
4 )( ~) . 84?
I
~-xl. 1 42
- 123 .782232
= IZ·lcm'
3
~ 150.816
(3 s.f.)K Al
= 151 <m
(Ii) Curved surroce area of cylinder
1
x 4 x4x9
CIU l
(3<ig fig)K K Al
"'3. 142 xSx I I
~ 276 .496
(ii) Vol\•me of cylinder
cm' K MI
-3.142x4x4x ll
Base area of cylinder
~3. t 42
= 552.992 cm' K K Ml
x4x4
= 50 .272 cm' K M I
Volume or the solid
Total surface area of solid
= 552.992 + 150.816
= 703 .808 cm '
= 704 cm' (3 sig jig )K K 8 1
= 123 .78+276.496 +50.272
=•150.548
- 451 cm ' (Js.f.)K 81
Answer:
8
(a)
...................cm
(3)
(b) (i) .................. cm 2
11 I
(ii) .................. cm2
(3]
(c) (i) .................. cm3
[1]
3
(ii) .................. cm
[2]
11. The following numbers are the number of words in each line from a page of a novel.
(a)
Fill in the given frequency table.
(b)
Calculate the
1nean,
(i)
(ii) median.
(iii) mode.
8
8
8
11
14
9
11
7
3
4
12
11
14
10
8
11
11
13
10
11
13
13
2
12
13
12
2
13
14
9
13
13
13
13
10
3
13
13
9
14
Solution:
(a)
I
8
11
.<
I
f<
2
2
4
3
2
6
4
- 7-
1
4
1
7
8
5
9
3
27
10
3
30
11
7
77
12
3
36
13
11
143
14
4
Total
-
I
42
-
---40
-
56
-
~
(3)
430
-
l"requency Table - A3, 1 mark per column+ 1 mark for totals
(b) (i) Mean= 430 + 42 =1 0.2 (3 s.I.) ....... ........ 81
(Ii)
42
+I = 2 1.5
2
Middle Values are 21" value and 22"" value
Median = 11 ..••.•...... 111
(iii) Mode = 13 ............ 11 1
Answer:
9
(b) (I) Mean= . ........ . ... . .
(1 )
(ii) Median = .. ........ ...
(1)
(Iii) Mode= . .............
[1)
12. ll is given that c = {x: x is an integer. I < x < 50) and sels A, Band Care defined as follows:
A - {x: x is divisible by 7)
8 : {.<:" is a multiple of 14}
C ~ {x: x is multiple of 3}
(a) List the elements of A" C.
n((A I'\ C)').
{b) Find
C).
(c) Slate 11(8 n
(d) Draw a Venn Diagram to represent lhe above infonnalion. Shade A I'\ 8'1'\ C .
(2)
. ·Solution:
(a)
A=(7.14,21.28.35.11.49)
c=
(3. 6. 9 . 12, 15. 18, ll. 24, 27. 30, 33, 36, 39@ 45. 4$}
A" C - (2 l, 42 J
K AI
(b) n((A I'\ CJ')= 48 - 2
(c)
= 46
K 1/1
B ~ {t4.28@
11(/J" C) c
I
K Al
Solution
E
2
47
46
7
.A
4
5
35
8
28
ll
16
17
20
19
23
25
33
"
26
18
15
36
32
22
1
Shading of
30
•12
31
29
An B'r-, C - HI
24
/.:
43
0
37
34
38 41
Answer:
10
Draw Venn
Diag.ra"' - Ml
15
27 29
14
49
II
44
9
(>
12
8
10
c
l
(a)
........................
[2]
(b)
........................
[1]
(c)
························
(1 I
13. A bag contained 6 blue balls and 7 rod balls. If a ball is picked at random from the
bag. find the probability that a red ball is picked.
Solution:
/~Red
ball is picked)
7
6
7
13
I
7
K K
A I
Answer:
End of Paper
11
. . . . . . . ¥ . •• • • • • • • • •
11r
I
NO:
NAME:
CLASS:
ADMIRAL TY SECONDARY SCHOOL
II
END OF YEAR EXAMINATION 2011
II
SUBJECT
: Mathematics
PAPER
:2
LEVEUSTREAM
: Sec 2 Express
DATE
: 12 October 201 1 (Wednesday)
TIME
: 07 50 - 09 20
DURATION
: 1hour 30 min
Instructions to candidates:
1. Write your name, class and index number.
2. Answer ALL questions.
3. Use an electronic calculator to evaluate explicit numerical expressions.
If the degree of accuracy is not specified in the question, and ii the
answer is not exact, give the answer to three significant figures.
Give answers in degrees to one decimal place. For;r , use either your
calculator value or 3.142, unless the question requires the answer in
terms Of IT ,
4. Essential workings must be shown. Loss of essential workings and
illegible handwriting will lead to loss of marks.
DO NOT TURN OVER THIS PAG E UNTIL YOU ARE TOLD TO DO SO.
This question paper consists of!! printed pages including this cover page.
Mathematical Formulae
Mensuraiion
Curved Surface Area of a Cone = 7frl
Surface Arca of a Sphere
=4 7fr 1
.. I
2
Volume of a Cone = - Jff
3
4
'1
Volume of a Sphere = :;- Jrr
.>
S1atistics
"[, j-.;
Mean=
2
J
l
1. A map is drawn to a scale of 1 • 200 000
(a) The MRT station and the
s~
1s O.7 cm apart. Find the actual distance, 1n km
(b) The school field has an area of 2.6 km2 Find its area on the map.
Answer:
{a)
(b) _ _ __
km
cm'
2. Two pipes, A and B, take 7 hours and 11 hours respectively to fill up a swimming pool. After
two pipes are turned 011 for an hour. Pipo A is turned off. How long does ii take for Pipe B to fill
the remainder'?
hrs
Answer:
.1
r~
3.
Answer this whole question on a piece of graph paper.
The following table gives corresponding values of x and y which are connected by the equation
y ~ l 9 - 4x - 3x 1 .
3
1
~j~;=:~,_,__·~-3~'-,--~-. .,- ._,_~~:~.. .,. -~_~'~_:_9~~-:_2~·~-~~~~:~1 -~
(a) Calculate the value of p and the value of q.
(b) Using a scale of 2 cm lo represenl 1 unit on the x-axis and 2 cm for 1O unils on lhe y-axis.
draw the graph of y = 19 - 4x - Jx 2 for- 4 5. x 5. 4.
(c) Draw and label the line of symmetry for the curve.
(d) Draw y
~
15- 2x on the same graph.
(e) Solv'e y = 19 - 4x - 3x 2 and y = 15- 2x graphically.
4. There are" benches and y students in an auditorium . When a bench is seated with 3
students. 18 are left standing. If a bench is sealed wilh 4 students. 3 benches would be
empty.
(a) Write 2 equations showing the relationship between x benches and y studenls.
(b) Find the number or benches and students.
Answer:
(a)
(b)
benches
students
(;
5. The heights of 20 students are shown on a stem and lear diagram below.
Steam
Leaf
12
1
3
3
3
4
5
13
3
4
4
4
x
5
14
1
2
,,
5
6
Key:
12 11
means 1?1
6
9
8
cm
(a) 1r the modal height is 13~ cm and the mean height or the students is
133.65 cm, !ind the
values or x and y.
(b) Betty is the shortest in the group of studonts. How tall is she?
(c) tr a student is pieked at random What Is the probability of choosing a student with hoi9ht
123 cm?
Answer:
(a)
=
)' =
'
cm
(b)
(c)
I
6 The diameters of 50 ball .bearings produced by a factory measured in mm (correct to 2
significant figures) are given in the table below.
Diameter (mm)
Frequency
5.0 - 5.2
5.3 -5.5
5.6 -5.8
5.9 - 6.1
6.2-6.4
6.5- 6.7
6
8
12
11
7
6
(a) State the median class.
(b) State the modal class.
(c) Fill in the frequency table below.
(d) Calculate the mean.
(c)
Solution:
Diameter
Mid-Value (x)
f
J<
5.0- 5.2
5.3 - 5.5
5.6 -5.8
.
5.9-6.1
6.2 - 6.4
6 .5 - 6.7
[•
Total
Answer:
r
(a)
~~~~~~~~
['
(b)
~~~~~~~~
(d)
6
1·
7
A hollow hemispheric container has an external radius of 15 cm and an internal radius 12 cm.
Given that ,,. = 3.142,
(a) find the total surface area of the container.
{b) find the volume of the container.
(c) If the hemispheric container is melted and recast into small cubes of length 2 cm
side. Find the maximum number of cubes which can be made.
on each
t5 cm
· - --- -- --- ····~
<
Answer:
12cm
(a)
cm 2
[:
(b)
cml
[:
1:
(c)
7
8 A square pyramid has a base renglh of 10 cm The height of one of its triangular f<iocs os h cm.
Goven Ural h is 8.5.
(a) find the are<i of one lriangular face
(b) find the tolal surface area.
IOcm
Answer:
END OF PAPER
(a)
cm1
[
(b)
cm 1
[:
(a) The MRT station and the school is 0.7 cm apart. Find the actual distance, in km.
(b) The school field has an area or 2.6 km'. Find its area on the map.
Solution:
(a)
1 : 200 000
0.7: 0.7 x 200000 ...... M1
0.7 : 140 000
Actual distance = 140 000 + 100 000 ..•• M 1
= 1.4 km .................. A1
1 cm : 200 000 cm
1cm:2km
1 cm 2 : 4 km2 .• : • . . .•.•....•.•• . ••••• M1
Area on the map= 2.6 + 4 ..••. ... M1
= 0.65 cm2 .•.. •. A1
(b)
2. Two pipes, A and B, lake 7 hours and 11 hours respectively to fill up a swimming pool. After
two pipes are turned on ror an hour, Pipe A is turned ofl. How long does it take for Pipe B lo fill
the remainder?
Solution:
In 1 hour,
Pipe A can fill
1
or the swimming pool.
7
Pipe B can fill
~or lhe swimming pool.
11
Arter 1 hour, Pipe A and Pipe B filled ( J
7
18
(1 - - )
: I -
77 I I
59
11
59
x II
K
K
K
K Ml
77
7
3
7
- l) -
!.! of the swimming pool. . ..... M 1
77
. . is
. not 1·1
ct
-59 or the swimming
1 le ....... M1
77
77
59
+ ~) :
OR ""8.43(3s. j.) K K Al
3
3.
Answer this whole question on a piece of graph paper.
The following table gives corresponding values of x and y which are connected by the equation
y=l9 - 4x - 3x' .
(a) Calculate the value of p and the value of
(b) Using a scale of 2
0
1
2
3
4
19
12
-1
q
-45
'I·
cm to represent 1 unit on the x-axis and 2 cm for 10 units on the y-axis,
draw lhe graph of y = 19 - 4x - 3x 2 for - 4 :S x :S 4 .
(c) Draw and label the line of symmetry for the curve.
( d) Draw y = I 5 - 2x on lhe same graph.
(e) Solvey = 19 - 4x - 3x 2 andy = 15-2x graphically.
Solution:
(a)
p = l9 - 4(- 1)
= 20
3(- 1)'
KKK Al
q = l 9 - 4(3) - 3(3)'
= -20 K K K ; fl
(b) Scale [.41]
Plot points for y = 19 - 4x - 3x ' [M1J
Draw and label curve y = 19 - 4x - 3x' (A 1]
(c) From lhe graph, line of symmetry: .r " - 0.65 (,t1]
(d) Plot points for y = 15 - 2,,· [M1J
x
-3
y
21
1
0
3
15
9
Draw and label line y ~ 15 - 2x (111 )
(e) From lhe graph,
x = - 1.5 ± 0. 1
y = l8 ± 1
KK Al
x = 0.85 1: 0. 1
y = 13.5 ± I KK Al
4.
There are •benches and y students 1n an auditorium. When a bench 1s seated with 3 students,
18 are left standing. If a bench is seated with 4 students. 3 benches would be empty.
(a) Write 2 equations showing the relationship berween x benches and y students.
(b) Find the number or benches and students
Solution
(a) 3.r+ 18
}'/\ /\ /\ Al
4x - 12 - y/\/\/\ Al
(b) 3x t 18 • y /I /I (I)
4A
12
)'
/I /I (2)
Substituting ( 1) 1rito (2).
t2=3x+l8
4x
30 KKBI
.>
Subs111u11ng this into (1 ),
3(30)
t
t 8 • )'
y
108 K K Bl
There arc 30 benches and 108 students
•.. .A I
5. The heights of 20 students are shown in o stem and leaf diagram below.
Leaf
Steam
12
·1
3
3
3
4
5
13
3
4
4
4
x
5
14
1
2
y
5
G
Key
121 1
8
G
9
means 121 cm
(a) If the modal height 1s 134 cm and the mean height of the students is 133.65 cm, find the
values of 1 and y.
(b) Betty 1s the shortest in the group or students How tall is she?
(c) If a student is picked et random Whal is the probability of choosing a student with height
123 cm?
Solution:
(a) x - 4
/\ /\ Al
Total height - 20 >< 133.65
2673 A/\ Ml
Remaining l teight
-2673 - 121
1(123) - 124 - 125
IJJ
- 14}
y = 3 A A II
(b) Betty's height 1s 121 cm.
• .....A1
4(134) - IJS - 136 - IJS - 119
141
142
l4S
116
(c) ?{choosing a student .with height 123 cm) =
2.
K Al
20
6 The diameters or 50 ball bearings produced by a factory measured in mm (correct lo 2
significant figures) are given in the table below.
Diameter (mm)
Frequency
5.0 - 5.2
5.3 - 5.5
5.6 -5.8
5.9 - 6.1
6.2 - 6.4
6.5 - 6.7
6
8
12
11
7
6
(a) State the median class.
(b) State the modal class.
(c) Fill in !he frequency fable below.
{d) Calculate the mean.
Solution:
(a)
50 1
+ = 25. 5
2
Middle values are 25°' value and 26"' value
Median Class = 5.6 - 5.8 K Al
(b)
ModalClass =S .6 -5.8
K Al
(c)
Diameter
I
Mid-Value (x)
/>:
5.0-5.2
6
5.1
30.6
5.3 - 5.5
8
5.4
43.2
5.6 -5.8
12
5.7
68.4
5.9-6.1
11
6
66
6.2-6.4
7
6 .3
44.1
6.5 - 6.7
6
6.6
Total
50
39.6
-
291 .9
1 mark per column A3, I mark 2 lol3ls 1/ I
(d)
Mean = 291 .9 • 50 = 5.838 •... .. A1
6
7
A hollow hemispheric container has an external radius of 15 cm and an internal radius of 12 cm.
Given that ;r = 3.1 42,
(a) find the total surface area of the container.
(b) find the volume or the container.
(c) If the hemispheric container is melted and recast into small cubes of length 2 cm on each
side. Find lhe maximum number of cubes which can be made.
t 5 cm
· -·---------·
Solution:
(a) Surface Area of brim or hemisphere = 3.142(15)' - 3.142(12)'
= 254.502K M l
Total Surface Area or the hemisphere
= lx4x3.142x(15}' +lx 4 x3. t42x(12)'
2
2
~254.502 K
1413.9 ~904.896+254.502
-2573 .298 cm '
K Al
(b) Volume of lhe outer hemisphere =
lx~x3.142x(l 5)'
2
.l
=7069.Scm '
Volume or the inner hemisphere =
K Ml
.2!. x~.) x3.142 x (12)'
= 36 I 9.584 cm'
K MI
Volume of the hemisphere = 7069.5-3619.584
-3449.916 cm '
(c) Volume of each cube= 2x 2x 2
-8cm' K Ml
Number of cubes= 3449.9 16 + 8
K MI
= 43 1.2395
.:431
K Al
1
K Al
Ml
c
12 cm
6 A sQuare pyramid has a 'base length of 10 cm. The height of one of its triangular faces is h cm.
Given that h is 8.5,
(a) find the area of one triangular lace.
(b) find the total surface area.
JO cm
Solution :
(a) Area of one triangular
face =~ x I0 x 8.5
2
= 42.5 cm' K Al
(b) Base area = IO x l O
- IOOcm '
To1al surface area
K Ml
= 42.5 x <l +I 0-0
c
270 cm'
K 111
Answer:
(a)
(b)
END OF PAPER
------
cm 2
cm'
Class
J
Index Number rmc
ANG MO KIO SECONDARY SCHOOL
FINAL EXAMINATION 2011
SECONDARY TWO EXPRESS
Mathematics Part One
Seiter: Mr Tan Wee Hong
..
M onday
10 OCTOBER 2011
1 hour
candidates answer on !he OUeshon Paper
R EAD THESE INSTR UCTIONS FIRST
Write your name, index number and class on all lhe work you hand in.
Write in dark blue or black pen on both sides of lhe paper.
You may use a pencil for any diagrams or graphs.
Do not use highlighters. glue or correclion Ould .
/\nswor alt q uestions.
If working is needed for any queslion it must be shown with lhe answo1.
Omission of essential working will reStJlt in loss of marks.
Calculators should be used where appropriflte.
If the degree of accuracy is not spcohocf in tho queslion. and if the answer is not exact,
give lhe <1nswer to three significant figures. Give answers in degrees to one decimal
place.
For"· use either your calculator value or 3.142. unless the queslion requires lhe
answer in terms of n .
At lhe end of the examinallOn, fasten ;ill your work securely together.
The number of marks is given in brackets [ ) at tho end of each question or part
question.
The total or U1c marks for U1is paper Is 50.
EXamlner's Use
Far
This document consists of 12 printed pages.
[Turnover
Part I (50 Marks)
Answer AL L the quest ions.
In the figure bcl(>W, A8CD is a trnpezium
s~ich
that LDAB = LADC .._90• and
CD - 21 cm. E is a point on AC such 1ha1 A£ = 24cm, EB ~ 18 cm and .t.AEB = 90°.
D
21 cm
24cm
18 cm
8
A
(i)
(ii)
(iii)
F'ind the leng1h of A8.
Given rhat !he area of rrapeziurn ABCD is 510 c·m 2, !ind !be length of AD.
Hence, calculate the lenglh of EC.
Ans:
(i) _ _ _ _ _ _ _ _ _c;::.m"- [I]
(ii)
---------"""--
[2]
(iii)
- - - - - - - - - - C " -11"1-
f2)
cm
2
(.1)
·1 he distance be1ween Ang Mo Kio Secondary School and Tan Tock Seng
llospital on a map ts .17 cm.
( i)
Given that lhe actual distance bc1ween the two locations is 6.8 km. find the
scale of the map rn 1hc fonn I · n .
(ii)
Calculate the actual area of Oishan Park, in km2 , if its area on 1hc map
is 3.25 cm2 •
(b)
h takes 5 air pumps working 1ogcthcr at the same ra1c to inflate a bouncy castle
completely in 36 minutes. If one of the nir pumps is out o f o rder, how much
longer do the remaining air pumps take to inflate the bouncy c:~~t lc?
[I I
(ii)
(b) _ _ _ _ _ __
km 1 [2)
minut~
[2)
(Tum ovct
3
Given that f) and q are integers such that - 2 5 p 5 5 and - I 5 q 5 6, fi nd
(a)
the least possible value of pq,
(b)
the greatest possible value of p-3q ,
(c) the least possible value of q; .
p
Ans:
.
Solve the equation
x+ I
--+
I
(a)
---------~
(I]
(b) - - -- - - - - - --
[I J
(c) - - -- - - - - --
(I I
I
=- .
5x - 1 2(1 - 5x) 4
Ans: _::x_=_ _ _ _ _ _ _ _ _ _ _ _ 13]
5
l11s given that 2p = Jq' -3p .
{i)
(ii)
Express q in terms cif p,
Hence, or olherwise, find lhe value of q when p = -3 .
Ans:
l 2J
(ii)
6
q=
[ IJ
Factorise 1he following expressions complclcly.
(a)
36x' y-49y',
{b)
4a' + 2ab-10a -5b .
.
[2 J
(b) _ __
_
[2)
[Turn over
7
Simpli fy the following expressions.
(a)
7a(a+5)-(a-2)(a+l),
(b)
x-y . x 2 -y '
2a -3b 4a -Gl>
y
A ns:
(a) - -- - -- - - - 13 1
(b) - - - - - --
-
[2]
8
(ii) The following shows 1hc number of books read by a group of chi ldren dunns
1hcir December hohday:;.
s,
7, 5, 3, 4, 7, 2, 6, x, 5.
(i)
Jf tl1ere are 2 values for mode, wrilc down 1he v;ilue ofx.
(ii)
Hence. calculate the mean.
(b) During a health check-up, lhc weights uf20 Secondary One s11Jde111s of1hc same
hcighl were recorded . The s1crn-und-leafdingram below illustrate:> their results.
3
5
6
7
4
0
3
4
6
4
...
7
8
0
2
3
9
5
8
6
0
1
4
8
0
(i)
If the median weight of the s1111fo111s
(ii)
A student is considered undcrwciglu if his/her weight is less than 4S kg.
1s
56 kg. writ<' down the value of w.
Find the percentage ot s tudents who :ire nnderweigh1.
AJ1s:
(:i)
( i)
.( =
11 J
(ii)
·.
(h)
(i)
(ii)
fll
w=
[I]
%
111
[Turn over
9
(a)
Given c = {all the households in a HOB block},
A ~
{households who subscribe to broadband intcmet} and
fl - {households who subscribe to cable TV}.
(i)
(ii)
Express An B 'f. ¢ in words.
F.xpress, in set notation, the statement that all households who subscribe to
c.a ble TV r1lso subscribe 10 broadband internet.
{b) Shade the region r~-presenting the sci
(CvD)' v(C n D) in the Venn diagram
below.
c
c
D
[,
- -- - - - - - -- -- - - --
,
- - - [I]
(ii) - - - - - - - -- -
Ill
10
The diagram below <how< the graph of y - 2 (' + 4x - 6. The graph cuts lhc y-axis at
pomt A. Fi11d
(a)
1he coordmates of poinl A.
(b)
lhe values of k and Ii,
(c)
the equation of 1hc line of syn11nc1ry.
0
A
Ans:
(
)
[I I
(a)
A•
(b)
k=
(I I
h=
[I I
-.
(c)
[I I
[Tum over
11
The diagram below shows a sequence of figures made up of sin~ll s haded ;ind
unshaded triangles.
h.
Fig. I
A
Fig. 2
Fig. 4
Fig.3
The !able below shows the number of each type of small triangles in each figure.
Figure
I
2
3
4
5
...
Total number of triangles
4
9
16
25
p
...
Number of shaded triangles
3
6
9
12
q
...
Number of unshaded triangles
I
3
7
13
21
...
~-
(a)
(b)
n
r
Stale lhc v alues of p and '/·
(i)
( ii)
In figure
11 ,
the number of unshaded triangles is r, express r in tem1s of11.
Figure k has 133 unshaded triangles, find the value of k .
Ans:
(a)
( IJ
p=
_.:_
q _= _ _ _ _ _ __
(b)
_
_
[I]
(i) - - - - - - - - - [ q
(ii)
k =
(I I
12
11ic dingram below shows a mclallic cube. A pyramid WXYZ is cur our from 1he cube
such 1ha1 XY =
rz = WY - 6 cm.
:w
F
r.
,
,
,·
(a) Calculate 1he volume oft he pyramid ll'Xl'Z.
(h) 111c pyramid WXYZ is mcllcd and used
10
make spherical mcrallic bc.1ds of
radius 0.3 cm. Find the number ofbc.1ds 1ha1 can be made from 1he pynunid
WXYZ. (Take
tr -
3.142)
Ans:
(a) _ _ _ _ __
(b)
cml (2)
beads (2)
(Turn over
1l
The diagram below shows 1hc graph of x -
>
,.
•
3
7
:r:
J
:r: - y ~
3
"
I
.7
(a) The 1ablc <hows lhc com:spondmgx unJ y volue< for the <-'<JUilllOn 2.t t Jy ., 7.
(b)
(i)
l'ind the value of q.
(ii)
Draw 1hc grnph of 2x + 3y = 7 for - 4 < x S 5 on 1hc same axes nhovc.
u~c
[ I]
your gmph to solve lhe simultaneous cquniions
.t-y=I,
2x+3y - 7 .
Ans:
(ll)
(b)
(i).,..:L..~--------- (I J
x-
t; nd or Pan I
[I]
Secondary Two Express Fin:il Examination 2U 11 Parer One
Marking Scheme
Qn
l(i)
l(ii)
-I(iii)
Ans,vcr
M~rk.ing
1
Scheme
Bl
1
AB =J24 + 18 =30cm .
.!_ (21+30XAD) •5 10
2
Ml
AD= 20cm.
Al
AC = J20' ~2 1 1 w29cm.
EC = 29- 24 5 cm.
Ml
Al
(Ambiguous case)
A1ter-na1ive :inS\\'Cr I :
AreaofBEC
I
I
- SJO - - x 2 1x 20- x24x l 8 = 84cm2.
2
2
1
1
Hence, EC • 84 ~ ~ 18 w ') cm.
2
3
or
Allernative llOS\Yt r 2:
BC' 20 2 +9 1 = 481
l!.'C' + 18' BC'
EC' = 48 1 324 157
F.C = 12. 5 cm (3sf)
2(:t)( i)
--
17 r,m : 6.8 km
17; 680000
I : 40000.
--
.
01
~
!(aXnl
!(b)
lcm 0 4km
I cm2 : 0. 16 knl
Actual Area - 3.25x 0. 16
0.52 km'.
~--
Ml
Al
·-
5 pump~ 4 36 nun
l pump -+ 180 min
4 pumps-+ 45 min
Additional tim~ laken - 45 - 36 = 9 min .
Ml
Al
l(a)
(- 2X6) =-12.
Bl
l(b)
(5) - 3(-1) =8
Bl
-
-
6
l(c}
4
fl I
6.
( I)'
.\+ I
+
=5, I 2(1- 5x) 4
2(x I 1)
I
I
2(5x - 1)- 2{5x I) • 4
21
I
~2 - 1
2(5x 1)
2x t I
Ml
4
I
2(5x -1)"4
Ml
4(2.i + 1) = 2(5x - 1)
BA
~ 4 = 10x - 2
6
2x
Al
x 3
Altcmnhvely,
2(\ 1 1X1 - 5x)+(SA- 1) I
J(~ r - IX1 - s.•)
"'4
!Ot' - 8x+2+Sx - 1 I
2( 2sl +10;-if - 4
10t'-3x+ I
--=-I
20x-2 11
·10t 1 - 12.-+4
SOr' +20.r 2
10• 1 -32x+6=0
s.' 16xi3-0
Ml
~0.\ 1 ~
Ml
5.• 1Xx-3) =o
I
s
r~1
or
Al
rcJcctc<I)
5(1)
r,
2p -..Jq - "ip
1
lp - q '
Ml
3p
q' =4p 1 +3p
q
\(ii)
)(n)
Al
'.}4p'13 p.
01
q - l}4(-3)' +3(-3) - 3.
-
36x'y - 49y 1
,{36x' 49y') y(6x+ 7yX6x -7y).
_________
Ml
Al
_.__
-
(,(hi
4a 1 +2ab - 1011 5b
211(211 t b)-5(2a +I>)
..
Ml
Al
(2a 1-bX2a -S} .
7(a)
1a(a t 5) - (o - 2~a+I)
1a' 1-350 - (11' - a-2)
7a 2 +35a-a' H1+2
6a' t36a+2.
"
7(b)
~y
Ml
Ml
Al
. ..,1 -y'
2a - 3b 4a-6b
l - y
- )( 2(2"-3b)
2o - 3h (x I yXx - y)
2
A+y
-
Ml
Al
Bl
7
8(a)(i)
x
8(•)(ii)
Menn - 5.1.
Bl
8(b)(i)_ •
w
5.
Bl
xl00=30%.
01
S(b)(ii)
-
6
20
9(n){i)
\l(a)(1i)
Some household~ subscnbe to both broadband
1111cmc1 and cable TV.
II ~
IO(b)
Bl
A.
9(b)
131
t
Bl
2x' +4x 6•0
(2x-t 6Xx- 1)=0
k
- ), /r s l
Bl. 131
IO(c)
---
I orIC 0 f Symmetry·
.l +I
2
I
x
ll(a)
p
"
ll(b)(i}
Bl
(5 I I)' =36 .
3(5)- 15.
Bl
Bl
(11 I t)' - 311
81-
Alt cn1ntivc anS\YCrs:
r (11 - 1)1 +11
r 11 1 - 11 .. 1
II (bXii}
Jr '
i+l=l33
k' k - 132 =0
(k 12Xk+11) ~ 0
k s 12ork=- ll.
HI
(rejected)
Allow tnal-and-error.
12(a)
-
-
Volume of ryrarn1d
1
3
xGx6x6)x6
Ml
36 cm 1•
12(h)
.
Nu mhcr of beads
3" :
~ (3.1 42Xo.J')
Al
-
Ml
3 18.268 187
3 18
13(a)(i)
--
q 3.
Al
111
17
I 3(a)(1i)
Graph of
Bl
2x+3y = 7
'
•
.
I3(b)
'
<-2, Y "' I.
•
Bl
ANG MO KJO SECONDARY SCHOOL
FINAL EXAMINATION 2011
SECONDARY TWO EXPRESS
MATHEMATIC S PART TWO
Setter: Mdm Leong Cbuin Sia
Frid•y
7 October 20 II
I llOUR IS M INUTES
AddutoN1 matC"fl.als.
Ans•'Cf Paper
flectron1c Caleula1ur
Gl'\l hPa r I ;,.hc...-ct)
----~---------------
R EAD T H ESF. INSTIUJCT IO NS fl!RST
Write your na1nc, clnss and index 11u1n1Jcr in lhc spaces provided o n the answer ptlpcr
Write your answers and working on 1hu separate answer paper provided.
Write in dark blue or black pen o n holh side~ of the paper.
You may use a pencil for any diagram~ or graphs.
Do not use staph~. paper clips. highlighters. glue or correction fluid.
Answer all questions.
Show all your working on the same page ns the rest of the an.-.vcr.
Orrnss1on of essential working wrll result rn loss of marks.
Cnlculnt<1rs should be used where appropnalc.
If the degree of accuracy is no t speerlied m 1he qucsuon, and if the answer is not ex net. give lhc
answer lo three sign ificant figures. Give 1111swcrs in degrees to one decimal place.
ror n, use either your calculator vnluc or 3.142, unless the question rcqui1cs the nnswcr in tcnns
of n.
At the end of the examination, fostcn all your work securely together.
The numhcr of marks is given in brackets ( I at the end of each question or part qu.,st1on.
The total of the marks for this p;ipcr is SO
This question paper consists of 6 printed p age..
(Tum Over
2
Scclion A (22 ma.-ks)
Answer ALL the qucsrions
Make m the subject of the following fonnula
2- m
n= - - .
(3]
2111 ·13
2
3
(•)
Solve rhc inequal ity 3.(-5 < x+ I S 2(x+.!..).
2
(2)
(b)
Hence, state the small cs1 value of x'.
!I J
( a)
Expr~s
(a)
x -+
1
,1y
y -- - ·
Y - a.."i ;1
X)'
- y'
x' - y'
c
•
1rtt\;t1on
•
•
Ie
"'1th<•
Stnt;
denominator.
[3 J
If e' + J' = 57 and ef = 3. find the value of 2(e + [)1 •
[3]
(b)
4
x'
Express in set notation. the region shaded in rhe Venn diagram.
fl I
p
·o
(b)
E: = { x: > i~ ~n intcgc.r, I$ x $ 30},
A = { x: x is a m ultiple nf3 an<l x ;;, 6 },
fl
=(x: 3x + 2 = I4) imd
C = { x: xis ll positive integer such Umt
2.~ + I<
12 I.
Fine!
(i)
fl ('. C,
[ I)
(ii)
A('. fl,
111
(iii)
n(A v B)'.
[I I
AMK~~- 2011 FY 21:_MATllS_l'2
3
5
(a)
Given that x
=- I is a solution of the equation mx' + (m 2 + l)x + 3 =O, tind
[31
the possible values of m.
(b)
II is given lhal A is directly proportional to r' . When r = 5, A= 500 .
(i)
fonn an equation conne<:ting A and r.
[2)
(ii)
f ind ihe value of r when A = I08.
[I ]
Section B {28 marks)
Answer ALL the questions
<>
f\ survey was conducted among 50 fomi lies to find out the number of children
they have. The data collected is rc1>rcscntcd in the table below.
Nun•berof chi ldren
I
2
3
4
Number of fa milics
x
23
y
5
(a)
Show thal x
(b)
If the mean number of children per family is 1.88, show that x+ 3y = 28 .
(c)
Solve lhc s imultaneous equat ions from (a) and (b) 10 find the values of.•
-i
y
~
22.
(I l
(21
::incl ,v.
f21
(d}
Find the percentage offomilies with J or more children.
[I)
(e)
Using your answer fro m (c), state the median.
(I]
AMK~S _201 l
rv
2E MATHS P2
(Tum Over
4
7
Jn lhe figure, WXYZ is a rectangle in which XY = (3.<+9) cm, YZ = 3x cm ,
I
WA = AB = BZ and XC ,,- XW .
3
3x + 9
y
3x
7.
(a)
Express and simplify lhc area of triangle ABC in tcm1s or x.
(b)
Given that the area of the sltaded region is 80 cni', fonn an equation in x
[2]
and show that it reduces to x' + 3.r - 10 = 0 .
[2]
(c)
Solve the equation x' + 3x - I 0 = 0 .
f21
(d)
I lcnce, lind the perimeter of reclanglc WXY2.
(I]
,
AMKSS 2011 _ FY 2( MA"n1S_P2
s
8
Figur" l shows an invened right circular cone with IOp radius 6 cm and height
14.4 cm.
14.4 cm
Figure I
(>i)
Figure II
Calculate
1IJ
(i)
lhe slant height of the cone,
(ii)
the curved surface area of the cone, leaving your answer in 1erms of
[2J
Jr.
(b)
Water is poured into tltc cone to till up
-~of its capacity. Find the volume
121
of water in the cone::, le.aviug your ans,vcr in terms of ff.
(c)
Figure II s hows a container formed by atlaching a hollow cylinder IQ a11
open hcn1isphcrc. A ll the water in the cone is 00\V p0urcd intv the container
10
figure II. 111c container is tille<l to lhe hrim.
Given lhat the height of the cylindric,il p:1r1 rs equal to the radius of the
hemispherical part, find the radius of the hemisphere.
AMKSS_20 1t FY_2E MA11-IS_P2
[3)
[Tum Over
6
9
Answer the whole of this question on a sheet of graph paper.
A ball is thrown upwards fro m the edge of the top of a ven ical building. Its
[Tum Over
JlOsition during its flight is represented by the equation h; -x' +9,r ·~ 12, wnere
It metres is the height of the ball above the ground and x metres is its horizontal
distance from the foot o f the building.
(a)
Some corresponding values of .r and h arc given in the following table.
x
0
I
2
3
4
5
6
7
8
9
h
1'2
20
26
30
32
32
30
26
20
12
Using a scale of2 cm to I unit, draw a horizontal x- ax is fo r 0 :S x :S 9.
Using a scale of 4 cm to 5 units, draw a venical h-axis for I 0 5 It 5 35.
On your axes, plot the points given in the table and join them w ith a
131
~111ooth curve.
( h)
Use your graph to find
(i)
the greatest height reached by the ball,
(ii)
how fa r the ball has travelled horizontally when it first reaches the
height of 25 metres.
(c)
fI]
(I]
A tree of height k metres is located 8.4 metres from the foot of the
building. Given that the ball pt!S$CS 3 metres above the top of the tree, use
yot1r graph to estimate the value of k.
l".nd of Part 2
AM KSS_20 1I FY_2E_MA111S_P2
rIJ
7
Secondary Two Express Final Examination 2011 P•1>cr Two
Marking Scheme
_Qn
I
Solutions
- - -
2m +3
n(2m + 3) = 2 - m
21nn+Jn-... 2-m
2m11+m=2-311
m(2n + 1)=2 - 311
Marks
2- m
11 - - -
-
Ml
Ml
2 - 311
Al
111= - -
2n+I
311 - 2
.Or Ill=
-2n-I
a
2
-
I
3x-5<.r+IS2(x+ ·)
2
3x 5<x+ I and x+ I s2(x •
,_
1x-x< 6
.,+1 S2 t + I
x <3
OS.>.
I
2
Ml
)
-
-
-
x2:0
I•
,.
-
0Sx<3
-
Al
:
I"
I
3
-
0
Bl
x
)'
)'
- +
x
z
.l' '
...i +.l)'~-y'
- x + y
x(x+ Y)_li<- Jj (x + y)(x-y)
I
I
l
- -+
(x+ y} (x -y} <.r + 1·15.·' -x~
-
-
-
x
y•x~y-y
01
(x+ y)(x- y)
....__
2x'y ~!"
ry( ... y)(< y)
2x - y
=
(x + y)(x - y)
1-
'""
,_3
b
+ f' + 2e/)
- 2(57 + 2(3})
= 2(63)
- 126
= 2(c
-·-
Ml
Ml
-
Al
-
2(e + [)'
Ml
= 2(e' + 2ef + /'}
--
,.
-
2
Ml
--
AMKSS 20 1l fY _2E_MA111S r 2
Al
-
·-~---
8
~
4
5
u
P/"\R 1or(P 1 V R) 1 or (P/"\R)'riP
BI
b
bi
bii
bii i
A = 6, 9, '~-· 15. 18, 2 1. 24. 27. 30}
B; 41
Ca 1.2.3.4.51
BriC =l4l
A r'I B = por{J
nlAuB) ' ~ 20
Bl
BI
BI
a
When x = - 1,
m(-1)2 +(m 2 +1)(- 1) +3= 0
I
m - 111 1 - I ~· 3 = 0
2
- 111-2 ;;Q
(m + l)(m - 2) = 0
1n e - l or 1n e 2
111
bi
- - -6
I
Al
-
Ml
Al
1
hii
Ml
A = kr'
When r = 5 , 11= 500
500 = k(S)'
k =4
'---
MI
A = 4r
When A = 180,
I 08 = 4r'
r =3
fl I
a
x + y + 23 + .s = 50
x + y = 22 (shown)
Bl
b
x + 2(23) + 3y + '1(5) = 1.88(50)
x+ 46+3y+ 20=94
Ml
x+3y=28 (shown)
Al
c
..:+ y= 22 --------{I)
x+3y = 28 -------- (2)
f-
Eon 12\
2y = 6
I I ).
Ml
.v=3
Subs1itutc
d
y
~3
into ( l ),
x+3 = 22
x= 19
TI1crcforc, x = 19 and y = 3.
/\ l
16%
Bl
AMKSS_20 1 I J
Y_2E_MATllS_112
·-
9
2
e
7
Bl
I 2
2 3
a
I
J
Arca of /\A/JC --(- X W)( X l')
I 2
I
2 3
3
I
(2x)(x+3)
2
= (x)(x-+ J) or x 1 +3x
Ml
= {(-(3.r)](-(3r+9)J
b
Arcn ofrcc1ongle
Al
=(3x + 9)(3.r)
Shud~d region = (3.r + 9)(3x) - (x 1 1 3x)
Ml
1
80 9x +27x-x' - Jx
l!O Sx' 1 24.r
-I
-~
o-sx' +24x
-
80
-1''"'"'' "
Al
(ShO\\n)
- I
..=_j..:
o - ~'•3x 10
0 ( ' - 2)(x + 5)
\' .. 2 or x=-5 (NA)
•
~
t ti
'=2.
Al
When
l'cnmc1cr = 2{J(2) + 9 + J(2)j
-1
- f,
~i
-
2
-
I · 15.6cm
Curvctl surface area
1rrl
IT X 6 x 15.6
9J.6JT
I
~
b
.
Bl
. ,, - 6' + 14.4:
=--~ ""
- -
81
Ml
Al
Volume of\\alcr
5 (I nr' lz)
6 1
5 I
2
= )( x n x6 x14.4
6 1
~ 144JT c1n"
>---
Ml
c
-~
Ml
-
Lei lhc radius of1he hemisehere be r cm.
Al
-
Radius ofcvlindcr- rem
Hcioh1 of cvlonder r cm
0=j
=
Volume of waler in conl.Uner = 144.T ems
AMKSS_ZUl l FY _2r MAlHS
~2
-
10
Vol of hcmisnhcrc +Vol of cvlinder = 14411'
2 .
• m·' + m·' (1') = I44tr
3
-·
Ml
2 1
r ~ r'= \ 44
3
' l44
l-2 r·=
~
9
a
bi
I
bii
~
-
-
3
r' =86.4
r = 4.42 cm 13sl)
Ml
Al
Please refer-to l!raoh in a seoarate file.
-- correct scale and labellino of axis
-- corrects •JOints
-- smooth cu1ve
Bl
Bl
Bl
Max h = 32.25in
Bl
lacccntable ran"C J2 < It$ 33)
x~ I.Sm(± O. l l
Manv do not understand Quc.~tion.
k = 14m (acceptable range
13.75 ~ k ~ 14.25)
_,
/:,i1d ofpoper
AMKSS 2011 _FY 2E_MATHS 1'2
Bl
01
)v
(,]11)
(P\)_._(.)!iYtcl f (A,ll.
.
- ~/}j15 1 llAhtll1j1-~ I lb11)
~
·- ...
·
~ttt·
,m.m - W1 I
!VI AX
'f./-0
(.1.,) ~
n_"'
_.....B I
3.J.. JS '(YI_,f' ( :!:. C» 5)
l·N-(11\.~
'b"/,\r ~;
t-J:r ~ \vtt
.
- ...; - _]b.1)
'
..-·
.-
. -. ....... !-· ...t
~
(t.0·1)- BL ..
J.:' K-4- J I,;:. ! i-"/-
~
\l
Ii - 3
~ /lt M~--
....
'
h = - -~;1-i«h. t I).
Jo
-. -- -
. )5-
\\
'\
\
\
\
\
.Bl
BEATIY SECONDA RY SCHOOL
END-OF-YEAR EXAMINATION 2011
SUBJECT : Mathematics
LEVEL
PAPER
DURATION : 1 hour 15 minutes
:1
SETTER : MdmToh. PL
I
CLASS:
DATE
: Sec 2 Express
: 7 October 2011
! NAME :
I
REG NO ;'
~
.....................•.••..•.•.•. ..•.•...................•.........................
READ THESE INSTRUCTIONS FIRST
\\' rite y<>ur name, closs and index numher 1n the sprtces on 1bc cop oflhis page.
\\'rite in dark blue or h1ack r>c'"·
You n1ay use a pencil for any d1ag.ru1ns 01 graphs.
Oo 001
u~ ~1apl c~.
p.ipcf c liJ)S. highlighters. gh1e or corT~tivn OuiJ.
Ans"'cr all qucstio11s.
(f \VOrking is needed fOr any question, ii n1usc be ~ho\vn \Vi1h the l'.lnswcr.
On1ission of essential \vorking 'viii result in loss: of n1arks.
You are expected to use a scientific calculator to evaluate cx1>ficiLriuo1cnc.:al cxpr~s.~1011!'.i.
If chc degree of accorncy is nol specified in the question. and if the ails,vcr is 001 exact. give the ans\''er IQ
three significant figures. d ive an.s,vers in degrees 10 one dccinH1l place.
F<.11 11. use cilhl:r your calcul:uor value Qr 3. 142. unlcs.'i the qucsilon requires 1hc ans,vcr in tenns of n .
'J'h.; nun1bcr of rnarks is givcli in bracke1s [ ] a1 1hc cntl of each <1ucs1ion or f1'1rl
fho total numbet of marks for this P"I"'' is 50.
qu~tion.
For Examiner's Use
50
This paper consists of!l p1intcd pages (including 1his cover page)
[Tuni over
3
Answer aU the questions.
I
Use
(n)
Solve the inequ~1li1y 2(1 - x) < - 6.
(h)
Hence write down 1hc smallcsr square number which satisfies 2( I - x) < - 6.
A 11.ttver:
121
(a)
(b) _ _ _ _ __
l
F"r
Eram111~
_
Ill
x1 - a
Givenlha1 P; - - ,
a
(a)
find, in tenns of"• the v~luc of P when x- 5a,
(b} expressx in tcnns of Panda.
Answer: (a)
P*
(b) - - - - - - - -
121
(21
4
(n)
(b)
i!xomfnt:t
u,,.
Hence, find the least integer value of q given that 540q is a pe1fcct square.
A11s11rer:
(ll)
121
540 =
(b)
4
for
Express 540 as a product of its prime factors, giving your answer in index
no1ation.
q ~
_ _ _ _ _ __ Ill
A bag contains>' red sweets, 3 yellow sweets and I green sweet.
3
.
5
(•)
Find the value ofy given that tbe probability of drawing a red sweet is
(b)
Two red sweets arc drawn out of the bag and are not replaced. Find the probability
that the nexi •weet clr.1wn will be yellow.
(")
y=
_ _ _ _ _ _ _ 121
(h) - - - - - - - - - (21
5
5
(n)
(b)
f'or
Factorise Sxy- IOx - y + 2.
.
S11npl1fy
x2
25
I· uunllf(
u,,
_
.r1 -2.r-l)
121
(b) _ _ _ __
6
131
A wall has an area of 40 m2. Eight tms of paint
:ire
the minunum numhc1 of tins of paint needed
coat a total of 4 times over another wall
o f ao e(l 70
10
needed to coat the wall S lime<. Find
1111•
AI LfW~r:
llllS
131
6
7
The diagram shows an cquil>1cral triangle f'QR wnh PQ ~ (2x- I) em, QR~ (2y + 3) em
and PR = (r + y + 2) cm.
p
x+y+2
Q'- -- l -- - l R
x + 2y - I
Usmg sm1ultancou.< equation,, find the value of x und ofy.
A11s wer:
\' •
----
(41
7
8
,..
(},.
Lei c {I, 2, 3, 4, 5, 6, 7, 8, 9 }, A= (x: x is n prime number) and
£tu1nl11t
R
(a)
/> : ' is an odd number}
lllu~lralc
'''"
lhc infonnation in the Venn diagram below.
121
(b)
L1s11hc clcmcnt(s) of ser (A v B)'
(c)
Find 11(111'\ B)
A 11swcr :
(b)
III
(c) - - -
Ill
8
9
In the diagram. MBC is similar 10 AEBD.
AD-7 cm,/)/)' 2 cm, BE ~ t:C • 3 cm and AC~ 12 cm.
/.BDE - 47° and L.BAC = 29".
A
7cm
12cm
B
£
3cm
(a)
Find /A/IC.
(b)
Calcululc 1he length of /)Ji.
c
•
Answer:
111
(11)
(b)
/)~;-
- - - - - - ,.,,, 121
9
10
24
24
Consider two numbers - and - - , where x is a positive integer. If the difference
x
x+ I
. I
bcrwccn the numbers is
3,
(•)
show that x' + x - 72 = 0 .
(b)
Hence, find the value of x, where x ¥ 0 and xis a positive integer.
121
(b)
X
G
---------
121
FOr
E'<(lf/llllf!I
Use
10
11
(a)
Expand and simplify (a+ 3)' +(a+ 7)(a - 4).
(b)
Express os u single fraction
Fer
E;an1111r1
Use
I-~.
<+l
(;i)
(31
(h)
121
11
12
The graph below shows a line/, which crosses the y-axis and x-axis at A and B(- 5, 0)
For
lixo111i11c.
respectively.
Utt
y
I
B (-5, 0)
0
(a)
Given the gradient of line I is 1.5, write down the coordinates ofA.
(b)
Find the cquution of the line I.
Ans•ver:
(a) - - - -- - - - - [21
(b) - - - - - - - - - [ l J
12
13
Tho diagram shows lhe unifom1 cross-seclion ABCD
or a tank. ABCD is a trapezium
such that AL) = 35m. BC - ISm and CD - 29m.111e iank is completely filled with water.
A
lS "'
B
11fjir,, ~"'
lllili llii
1
iii!!i!/~:;::=::·:··
29
Ill
D
(a)
Find the area A/JCD.
(b) Water is discharged through an outlet at D al a constant rate. It takes 7 hours to
empty the tank. By letting the length o r the tank be Im or otherwise. !ind lhe time
taken fo r the water level to fall 5 m below AB. Oiv,, your answer in hours and
n1inutes
121
(a)
(b)
- - - hours
End of Paper
J)o nt>t fin-get 10 clu~ck ; 1011r H1ork! :)
1nint1tes
[31
Far
13
Answer Key
2
3
(a)
x <2
(b)
Smallest square number = O
(n)
25n- I
(b)
x = ±JPa+a
(a)
540 = 21 )( 3} x 5
q = 15
(h)
4
5
(n)
y
(b)
3
P(yellow) = 8
(a)
(y - 2)(5x - 1)
(h)
(x1 5)
(x +J)
6
6
12 tins
7
.r -5,y=3
8
(a)
c
4
2
9
6
8
(b)
(AuB)' = {4, 6j
(c)
11(A l'"\ 8)
3
14
9
(a}
LABC= 104°
(b)
D£=4cm
JO
(b)
.r-8 or
11
(a)
2a' +9u- 19
(b)
5
x+3
(•)
p=7.5
J2
13
x =-9 (NA)
(b)
y = l.5x+1.5
(a}
AB = 21
(b}
I hour 24 minutes
BEATTY SECONDARY SCHOOL
END-OF-YEAR EXAMINATION 2011
..
SUBJECT : Mathematics
LEVEL
PAPER
DURATION : 1 hour 30 minutes
:2
SEITER : Mr Bernard Lee
l
CLASS :
DATE
: Sec 2 Express
: 13 Oct 2011
I
l NAME:
REG NO:
•••..................................................................... .....
READ THESE INSTRUCTIONS FIRST
\\'rite:. your n(lmc;, class and index nun1bcr in Lhc spaces 011 the top of thii:: pttge.
\Vrite in dal'k blue or blru;k pen,
You Jnay use a pencil for anydiagnuns or gr(lf>hs.
Do not use staples, paper clips. higl11igl11cr.;, gluo or COITCCtion Ouid.
Answer nll questions.
rf \\'Ofk.iug is needed for any qucs11011, ii n1ust he shown \vilh !he 30S\VC(
Omission of cssc-ntial worki1lg v.•ill rcsuh in loss or n1ttrks.
Calculnh)rs: should be used u•hcrc apprQpria1c.
If ihc degree of accuracy is not specified in 1he question, .a11ct if die aus,vcr is not c.'act. give th(!
an~v.icr
10
thri'..e significant figures. Give a1'1S\vers 111 degfecs 10 one decnnal ploce
For tr, use either your ca1culator valut' or 3.142> unit!SS lh~ qtrcstion requires 1he ans,i,.•er in tern1s of
'"
Af the c11d of lhe examination, fasten all your \11ork securely to~cthcr.
The 11u1nbcr of m3rks is given in bracke1s [ 1at the end of eaeh question or 1xu1 question.
111e 101(11 number of n1arks for this p3per is SO.
This paper consists of
2.. printed pages (including this cover page)
!'furn over
2
A road of20 km is represented by a 10 cm line on Map A.
(n)
Express the scale of Map A in the form I : 11.
(2]
The scale of Map B is I : 50 000.
121
(h)
What is the length, in cm, of the same road in Map B?
(c)
A lield is represented by an area or 3 cm' on Map A. What is the area of the
same lield on Map B'?
2
(3]
A cone has a volume o1'2592n cm' and a radius of 18 cm.
(:i)
Calculate the vertical height of the cone.
121
(b) Show that the slam height of the cone is 30 c m.
+
0
-
Cylinder D
111
'
..'
:
0
'
Rocket Toy
The cone is cur horiwntally to form Cone /J and f'rustrum C. Cyli nder D of radius
9 cm 1s then placed between Cone /J and Frustrum C to produce a rocket toy. as
shO\vn 1n the diagran1.
(c) The rocket toy has twice the volume of the original cone. Calculate the height
121
of Cylinder D.
(d) C"lculate the total surface area of the rocket toy.
lVolume of cone = ~ nr' h, Curved surf.lee area of cone = rrrl 1
'
Ill
3
3
A photo fr.une consists of a centre measuring I 0 cm by 15 cm, and a wooden
hordcr of thickness .r cm surrounding H. as shown in the figure.
)( cfu
(•)
'.
i xcm
15cm
JO
cm
Show that the area of the border 1s 4.r' + 50x.
(b) Given that the area of the honlcrcquals to the area of the centre, find the
121
141
valueofA.
4
2 laddc'fs. PQ and A 8 are resting ugnins1opr>Osllc walls of an alIcy. PQ and Afl
are 8 metres and 4 metres above lhc ground rcspcctivdy. Tis 1hc point where the
2 luddc-n; mccl. The diagram b<:low shows lhc side view of the ladders.
(•)
<.liven that TS ~ v metres. TV - • metre:> and that IYf'BP is similar to /J.TAQ.
121
cxpn:ss x in tcnns ofy.
tcnn> ofy .
(h )
Express the length of PA
(c)
Given also that /\PT/\/ 1s s11mlar to APQA, lind the length of1M.
in
111
121
4
5
The do1 diagram below shows the number of people living in each house on a
street.
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
-1-f- I 1-J-J
0
1
2
3
4
5
•
•
I
6
No . of people living in each house
{a)
6
Sl;ile
(i)
the median.
111
(ii)
the mode.
111
121
121
(b)
Calculate lhc rncM 1lUmbcrofpcoplc living in each house
{c)
Calculn1e Ute percentage of houses that are not occupied.
(a)
Show lhal (2x 1 3) 2 - {2x + y)(2x- y)- y(y + 3) can be reduced LO
3(4.1· - y + J).
(h) Hence. solve 1hc following sinu~taneous equations.
(2.i +3)' - (2x+ y)(2> - y) - y(y+ 3) : 0
4y - 3x=l2
141
5
7
The figure below shows o circle of centre 0, with mdius r metres. A and B an:
points on the circle such that AH
16 m. M 1s the midpoint of All and Mis on
ON such that MN = x meircs.
o,
'' ' ' '
'
A
r
''
''
8
M, x
~
N
-16m
~
(a)
B)«onsidcriug 60MH. show that " ' - 2r.<+ 64 - 0 .
121
(h)
1:;xprcss r in terms of.<
(e)
Hence. given that MN
121
Ill
4 m, calculate !he radius of the sphere.
6
Answer the whole of this question on the graph paper provided.
8
Mr Ong stood on the roof of a building ond threw a ball upwards. Al time 1 se<:onds,
the height, '1 metres, of the ball rrom the roof is given by h
~
I St - Sf'.
The following table gives some corresponding values of 1 and It.
Is
0
0.5
I
1.5
2
2.5
3
3.5
hi m
0
6.25
10
p
10
6.25
q
-8.75
I
(n)
Calculate the values ofp and of q.
(h)
Using 4 cm to represent I unit on the horizontal 1-axis and I cm to
121
131
rcpresem I Wlit on the vc11ical h-axis, draw the graph of It = 151 - 51' for
OS 1S 3.5 .
(c)
Use the gr:1ph to detem1ine the maximum height of the ball from the roof.
111
(d)
Whal does the negative value of It when t ~ 3.5 s mean?
fl I
- - Emf nf Paper - ()II
l(a)
I(hl
!(c l
2('.!)
2fh\
2fc)
2{d)
<n
Answer
I : 200 000
40cm
48 cm'
24 cm
30cm
--
32cm
4520 cm'
A .JlSl\'Cr
5 a)lil 4
5 a)(ii) 4
5 bl
2.88
5{c)
12%
-
6(b)
x = Oand y • 3
7(b)
.r' + 64
r = --
7(c)
I0 1n
2x
j ib\
4;;-,,
4(b)
.<= 2.5
8
x = 2>
PA = 3y
2
2
3
a)
.!~)
8(cl)
-
D
11.25, (I = 0
11.25 m
It means that the ball is 8. 75 m
below the roof f below Mr Ong.
p
--
1--
4(c)
--
Ill
Name:
Class: Sec
- -- - - -
Expected Grade:
Value-Added Grade:
Review:
Bedok Town Secondary School
Challenged to Excel
Success Through Perseverance
2nd Semester Examination 2011
Mathematics [4016/1]
Paper1
Secondary 2 Express
5 October 2011
Wednesday
1040 -- 1155hrs
1 hr 15 min
READ THESE INSTRUCTIONS FIRST
Write your name, class and register number on all the work you hand in.
Write in dark blue or black pen.
You may use a pencil for any diagrams or graphs.
Do not use staples, paper clips, highlighters, glue or correction fluid.
Answer aU questions.
If working is needed for any question it rnust be shown with the answer.
Omission of essential working will result in loss of marks.
You a.-e expected to use a scientific calculator to evaluate explicit numerical expressions.
If the degree ofaeeuracy is not specified in the question, and it: the answer is not c;xact, give the answer ·
three significant figures. Give answers in degrees to one decimal place.
'for n, use either your calculator value or 3.142, unless the question requires the answer in tcnns of n.
At the end of the examination, fasten all work securely together.
'Ilic number of marks is given in brackets [] at the end of each question or part question.
The total of the marks for this paper is 50.
This question paper consists of
· Setter's Name: Mdm Yeo Liew ctieng
t)
printed pages.
2
il1athematical Formulae
Compound interesr
/1tfens11ra1ion
Curved surface area of a cone ; wl
Surface area of a sphere = 4m· 2
1
Volume ofa cone ~ - nr 2h
3
. I
1· asp h ere = 4 1rr I
Voumeo
3
Arca ofa triangle ABC - .!.ahsin C
2
Arc length = rB , whereO is Ill radians
Sector area
~ .!.,. 29, whereB
2
is in radians
Trigo110111et1J'
a
b
- - =sin A sin R
= -csin C
a' - b1 +c' - 2hccos A
Sratistics
Mean = _
,~'i·
_,,_..
'£.f
r-;- -( "i,fx
.
Standard deviation = / 'f.Jx
~
4(
4!
)2
3
Fur
£xcuni
(h;
Answer all the questions.
I.
Evaluate
0.325
~
-.V20.5 - 2.74
,
(a)
showing all the figures on your calculator display,
(b)
giving your answer correct to 2 significant figures .
..
Answer:
(a) _ _ __ _ __
(b) ··2.
(a)
- - - - - - --
Write lhe following numbers in descending order
2.35, 2.335 '2.:l5' 2.J.5' 2;')
(b)
State the largest integer which satisfy
2
47
x 5 3_.
3
Answer:
(a) _
_ , _ , __ , __ , _
(b) _ _ _ __ _ _ _ __
20l l -SA2-2E· F.M-l'I .doc
_
(2]
(2)
Fo
5
Exa11
5.
(a)
Sublracl lhe sum of 3x3
(b)
Factorise completely
(i)
8a 3 - 72ab2,
(ii)
;~.x - 1 2 .
t-
2
7 x - 8 and 9.r2 - 6x + 12 from 8x3 - 5x + 17 x - 21 .
Answer:
(a) _
_
_ _ _ _ __
(bi) _ _
(bii) _
201 1-S/\2-2E-EM-1'1.doc
_
(3)
(2)
_
(I)
6
£ rain.
u.,
6.
In the diagram, find angles x and y. State your reasons clearly for your working.
Answer:
(a) x
~
·- - -- --
-
0
0(2]
(b)y -
7.
(a)
(2]
London time is 7 hours behind Singapore time. A non-stop flight from London to
Singapore is scheduled to take 13 hours and 29 minutes. If the flight is to arrive in
Singapore on Saturday at 17 18 Singapore time, find the departure day and time
of the flight in London time.
(b)
Given y is inversely proportional to (x+ I). lfy ~ 7 when x = 4, find
(i)
the equation relating x and y,
(ii)
the value ofy when x = 2.5.
Answer:
(a) _ _ _ _ _ _ __
(bi) - -
-
_
(2}
- - --- -·- - - l2]
(bii) - - - - - - - - - - (1}
?.OI l -SA2-2E-J::M-Pl.do<:
7
Fo.
£<om
u
8_
Solve the following pair of simultaneous equations.
5x - Gy " 27
2y = 3x- IJ
Answer:
x - - - - - - -- - (3)
y= _
9.
A right circular cone has a curved surface area of 136 cm2 . Given the radius
of cone is 5 cm, find
(a)
the slant length, /,and
(b)
the height of the cone, h.
Q _____ ___ _
5
Answer:
201 l-SA2-2£-LM -Pl.doc
(a} _ __ _ _ __ _ cm
(2)
(b)_ __ _ _ __ _ cm
[2)
8
FfJr
E\"Ullll
Us
I 0.
ln the year 2009, Sandra earned 20% more than what she earned in 2008.
In 20 I 0, she earned 15% more than what she earned in 2009.
(a)
By letting $x to he the amount Sandra earned in 2008, express in terms of x, the
amount Sandra earned in 2010.
(h)
Calculate the total percentage increase in her earnings from 2008 to 2010.
Answer:
(a)$ _ _ _ __
- - - [2)
(b) _ _ _ _ _ _ _ _ _ % [2]
11.
In the diagrams shown below PQRS
= J"UVS.
L.STU = 113°, RS = l 8 cm, QR= 12 cm,
PS - 9 cm, / PSN= 62° and L.QRS= 74°. Find
(a)
the length of SV,
(b)
the length of l'V,
(c)
L.PQR,
(d)
L.QP/I.
R
12cm
Q
T
Ans,ver:
20 I 1-SA2-2E-EM-P I.doc
(a)
cm
[ 1J
(b)
cm
[I ]
(c)
o
( d)
___ 0
[I ]
11 1
9
Fo1
E.ta111.
U:
12.
The following data shows the number of hours 24 students of a class spent studying per
day.
4
3
~
.>
0
(a)
3
4
6
4
4
4
0
I
4
4
0
5
4
5
6
0
3
3
Using the grid below, draw a dot diagram to represent the above data.
{2]
I-
(b)
What is the modal number of hours the students spent srudying"
(c)
Calculate the mean number of hours the students spent studying.
(d)
If the above data was presented on a pie chart, what is the size of the angle that
represents the number of students who study at least 3 hours a day.
(c)
What is the probability of a student studying 4 hours a day')
Answer:
(b),_ __:__ _ __
hours [l J
(c)
hours (I)
(<!)_
_
_
_ _ _ __
(c), _ _ _ _ _ _ _ __
ENO OF PAPER
20 t J-SA2-2E-EM-P I.doc
0
[I l
[I}
Class: Sec_ __ _ __
Expecled Grade:
- - -- -
Value-Added Grade:
Re1tlew:
Bedok Town Secondary School
Challenged to Excel
Success Through Perseverance
2nd Semester Examination 2011
Mathematics [4016/2]
Paper2
Secondary 2 Express
0930 - 1045 hrs
7 October 2011
Frida y
1hr15 mins
Ri,;A 0 THESE INSTRUCTIONS Fl RST
Write your name, class and register number on all the work you hand in.
Write in dark blue or black pen.
You may use a pencil for any diagrams or graphs.
Do 1101 use staples, paper dips, highlighters. glue or correction fluid .
Answer all questions.
I f working is needed for any question it must be shown with the answer.
Omission of essential working will result in loss of marks.
Ca cul ator.. should be used where appropriate.
If the degree of accuracy is not specified in the question, and if the answer 1s nol exact, give the answer
three significant figures. (jive an~wcrs in degrees to one decimal place.
For n, use either your calculutor va lue or 3. 142, unless the question requires the answer in tenns of n.
At the end of the examination, fasten ull work securely together.
The number of marks is given in brackets r J al the end of each question or part question.
The total of the marks for this paper is 50.
This queslion paper consists of 6 printed pages.
Setter's Name: Mdm Yeo Liew Cheng
2
il1athematical Formulae
Compound interesr
/1tfens11ra1ion
Curved surface area of a cone ; wl
Surface area of a sphere = 4m· 2
1
Volume ofa cone ~ - nr 2h
3
. I
1· asp h ere = 4 1rr I
Voumeo
3
Arca ofa triangle ABC - .!.ahsin C
2
Arc length = rB , whereO is Ill radians
Sector area
~ .!.,. 29, whereB
2
is in radians
Trigo110111et1J'
a
b
- - =sin A sin R
= -csin C
a' - b1 +c' - 2hccos A
Sratistics
Mean = _
,~'i·
_,,_..
'£.f
r-;- -( "i,fx
.
Standard deviation = / 'f.Jx
~
4(
4!
)2
J
Answer all the questions.
l.
(a)
Factorise completely
(i)
(ii)
2x 2 + 6x,
[I J
2
2x -18.
(2)
2
(b)
s·
(c)
(i)
Make x the subject of y
(ii)
Hence, find the value of x when y
(d)
rf 2x +6x
imp i Y 2x2 -18 .
(l J
·
=
2x 2 .l.6x
•
2x 2 - 18
=
•
[3]
5.
[l]
The masses of 12 bags of potatoes arc shown in the stem-and-leaf diagram below.
Stem · Leaf
0
7
9
3
5
2
0
3
4
6
6
8
5
7
Key: 0 17 means 0.7 kg.
2.
or apples.
(i)
Calculate the mean mass of the bags
(ii)
Find the median mass of the bags of apples.
(2)
[l]
It is given that. f>- { lett«rs in the word 'examinations' }, A= {letters in the word 'taxation' }
and B - {letters in the word 'extension'}.
(a)
L ist the elt::ments of F:, A and B.
(3]
(b)
List the clements of AuB and Ar>B.
f2]
(e)
Represent the given set using a Vem1 diagram.
[21
(d)
lf a letter is picked at random, what is the probability that the letter
(i)
is found in both set A and B,
[I]
(ii)
is found in A hut not in B?
[I]
201 I ·SA2-2E-EM-l'2.doc
3.
The diagram below shows a trapezoidal plot of land such that AB = (x + 4) m,
!JC ~
( 11 - 2x) m, CD = (3x + l) m and AD = (3x - 5) m.
(x+4) m
{3x- 5) m
D
(a)
c
(3x +I) m
Write, without expanding, an expression, in tenns ofx, for the area ofirapezimn
[I]
ABCD.
(b)
Given that the area of the trapezium is
reduces to 12x
1
-
5x - 93
34 m2,
fonn an equation and show that it
= 0.
121
(c)
Solve 12x 2 - 5x - 93 = 0, and hence state the leni,>th of BC.
( d)
The owner of the land wants to fence up the area. The fencing costs$ 89.00 perm
run. Calculate the total cost the owner has to pay for the installation. ·
(e)
(3]
[2]
This area of land is plotted onto a map with a scale of l: 200. What would be the
2
area of this land on the map in crn ?
2011 -SA2-2E-EM-P2.doc
[2]
5
4.
(a)
The diagram below shows the base of a righl regular hexagonal pyramid such
CD = 6 cm and GJ-1 = 5.2 cm.
B
A
''
F
'\G.'
'
--------:t:---.
. ·---'. ,
'
,
,
.:s:2 c
:' .
c:·
'
'
'
11
f IJ
(i)
Cal~ulatc
(ii)
Given that the height of the pyramid is 7 cm, find the volume oflhe PJ~·amid.
the base area of the. pyrnmid.
f2]
(iii) This solid pyramid is melted down and moulded into small spherical halls of
diameter 4 cm. Show that the maximum number of whole balls that can be
made is6.
(b)
(3]
/\.sheet of coloured paper has length 47 cm and width 35 cm. What is the maximum
number of 3 cm hy 3 cm whole squares that can be cut from this sheet of paper?
[2)
-~--------------
201 l -SA2 -2F.- F.M-P2.doc
6
5.
Answer the whole of this question on a piece of graph paper.
The variables x and y are connected by the equation y "' -x1
-
x + 6.
Some corresponding values ofx andy arc shown in the table below:
I : I ~ I ~ I ~ 1 -~~-1 : 1->I :] ~. I
121
(a)
Calculate the value of a and of b.
(h)
Using a scale ot2 cm to I unit on the x-axis and I cm to I unit on they-axis,
draw the graph of y
(c)
= ~x
2
-
x -1· 6 for values of x in the range - 4 $ x ~ 3 .
(31
On your graph, draw and state the equation of the line of symmetry for this
curve.
[2]
(d)
Use your graph to find the values ofx wheny = -2.
(2)
(c)
On the same graph, draw the line y = I - 2x .
(2)
(I)
Using your graph drawn in pa1t (e), state the solutions of - x 2
End of Paper
20 I l -SA2-2F.-F.M-P2.doc
-
x + 6 = I - 2x.
(11
Answers
x
x- 3
lai.
2x(x + 3)
ci.
ii.
3 .75
di.
I. 925 kg
11.
y-l
b. = {e,x,a,111,i,n,t,o,s} , .11 = {r,a,x,i,0,11} , B = {e,.x,t,n,s, i,o}
Au 8 = {t,a,x,i,o,n,e,s}, AnB = {t,x,i,o,n}
5
l
2a.
b.
di.
3a.
d.
4ai.
II.
2(x+3Xx - 3)
b.
3y
x= --
II.
9
I
(3x - 5X4x+5)
2
$2314
93.6 cm
11.
2
11.
9
c.
8.5 cm 2
218.4 cm 3
31 3 •
x= - -·
12 ' . ' .~cm
b.
165
l.7kg
c:a_s_s_ _
~l_F_u_n_N_a_m_e_ _ _ __ _ __ ~·-----~'-ln_d_e_x_N_u_m_b_e_r_
>·- ,
-, ""
END OF YEAR EXAMINATION
l:>o\,ven
_ __
0
2011
I l>elieve, therefore I arn
4016~01 j·.
MATHEMATICS PAPER 1
Secon dary 2 Express
11 October 2011
1 hou r 15 min
-----
l
INSTRUCTIONS TO CANDIDATES
Write your name, class and index number on all the work you hand in.
Write in dark blue or black pen on the question paper.
Answer all the questions.
Write your answers and working on the space provided.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the
case of angles in degrees, unless a different level of accuracy is specified in the question. f
At the end of the test, fasten all your work securely together.
INFORMATION FOR CANDIDATES
The number of marks is given in brackets [ ) at the end of each question or part question.
The total number of marks for this paper is 50.
The use of electronic calculators is allowed in this paper.
You are reminded of the need for clear presentation in your answers.
DO NOT OPEN THIS PAPE R UNTIL YOU ARE TOLD TO DO $0
For Examiner's Use
This document consists of.!!. printed pages, including this cover page.
Setter: Miss Melissa Chong
ELECTRONIC CALCU LATORS ARE ALLOWED TO BE USED IN
THIS PAPRR.
l
In the figure, if All is para! lei to CD,
~11_ = ~
CD 3
Nothing i~
to be
wrillcn oil
and AE = 9 cm, calculate CE.
tlus
margin.
(The diagram is not drawn to scale.)
Ans: CE -=
_ _ _ _ __ cm f2)
Solve the simultaneous equation using the elimination method only.
5y - 7.\=-38
5x + 2 ~-Sy
I.
Ans: x
2
=
.Y " - -
[3 J
3
Notl!iff8 is to Ix.
written Ol'l ll1 tJ
margin,
p
x-2
R~---------__LJ Q
x+ S
The diagram shows t\J>QR in which L.PQR = 90•, PQ - (x
2,) cm,
QR = (x + 5) cm and PR-= (2x - I) cm.
Using Pythagoras' Theorem, fonn an equation in x and show Lhat it
reduces to x 2 - Sx - 14=0.
(h) IJenee, ealculnte the perimeter and Area oft!J'QR .
(a)
Ans: (b) Perimeter =
_____ cm Arca = _ __ _ cm'
3
f2]
(41
4
(a) Ex.press in set notation, the set represented by the shaded area in terms of
A and B.
Norki.rig u to IN
'-'t'iuen C)n rlru
marg1'n.
I I]
Ans: (a)
(b) On the Venn diagram shown below , shade the set {Au B)'v (A !l B).
[I j
f;
5
Al a strawberry farm, the number of strawberries Sus ie gathered over the past
I 0 days are as follows 39, 37, 35, 48, 47, 44, 39, 44, 35, 44.
(a) Construct a stem and leaf diagram to represent the above data.
(h) Fincl the median of this distribution
Ans: (b)
4
M edian ~
_ _ _ __ _ _
f2]
t21 I
N<JJlnng 11 IO +
6.
Ml'frten on th/:.
/
/
/
margir..
\,
.
·'.-·
..,.··
'
'\
-·
\'•·
·.
.....
The diagram shows a solid made up of a hemisphere, a cone and a right
cylinder. The radii
the hemisphere and the cone are 12 cm each and the
cylimler has a radius of 4 cm. The heights of both the cylinder and the cone
are 16 cm.
or
(a) Find the volume of the solid.
(b) Find the total surface area of the solid.
Round both your ans wers off to 2 decimal places.
Ans: (a) _ _ _ __ ___ cm 3 [4)
(b) _
5
_
~
_ __ cm2 LSJ
7.
It is given thaty is inversely proportional toV~. When x = 27,y - 12.
1Wwhing i> tu bt
l'.d11e" ou rl1tJ
margm.
(a) Find the equation eonnectingx and y.
(b) Find the value of x when y ~ 7.2.
(a) _
_
_
_
_
_ __
_
(b) _ _ __
8.
[2)
L2J
6 cards numbered 1, 3, 4, 5. 7 and 9 are placed into a bag. A card is then
drawn at random from the bag. Find the probability that the number on the
card dt :iwn is
(a) a 4,
(b) an even number,
(e) a prime number,
(d) greater than 9.
Ans: (a) _ _ ___ _-_. - (1]
(b) _ _ __
_
(1]
(c)
(d) _
6
llJ
_
_
_
_
II J
9.
The graph of y
= (x + h)(2- x), where Ir is a constant, cuts the x-axis at B
Nothing H. to b.
wnl/~u on
and C. Tt cuts the y-axis at A(O, l 0).
mU~ffl
(a) Show that the value of Ji is S.
(b) Write down the CoQrdinal~ of 8.
(c) J\ line parallel to AB passes through C. Find the equation of this line.
(b) ' -- - - ' _ _ ___,
(c) _ _ __ __ _
7
[ l]
(2]
131
tl1l'S
10.
l
3x - 2
Express - - + 1
xd 3x 47x - 6
3
. gl fi
.
as a s m e ract1on.
9-x
NtJllr"'X 11 to bt
KTIU('ff OfJ t#iiS
- -1
'""'~"'
Ans:-- - - -- - -- - - [41
11.
2
(a) Solve-x +2
x_ = I.
x 13
(b)(i) Factorise 2x1 + 13x + 15.
(ii) Hence, write down two factors of2 1 315.
[J I
Ans: (a) __
(b)(i),_ __
(b)(ii)
END OF PAPER
8
_
_
_
[ 1]
12]
c_i_a_ss_ _
._j
IIndex Number
_.I Full Name
I
- ---'
-.,' .I"
_> ·-· '-..
~
b (. \). ! ( '"
)
/V
-·
END OF YEAR EXAMINATION
0
2011
')
JI.
I bcltcv<', :fH.•rcfor<;o I cur1
__ _J
ATHEMATICS
condary 2 Express
. 4 016101
Oc to ber 2011
I
- -- 1 hour r
INSTRUCTION S TO CANDIDATES
Write your name, class and index number on all the work you hand in.
Write in dark bfue or black pen on the question paper.
Answer all the questions.
Write your answers and working on the space provided.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the
case of angles in degrees, unless a different level of accuracy is specified in the question.
At the end of the test. fasten all your work securely together.
INFORMATION FOR C ANDIDATES
The number of marks is given in brackets ( ] at the end of each question or part question.
The tota l number of marks for this paper is 50.
The use of electronic calculators is allowed in this paper.
You are reminded of the need for clear presentation in your answers.
DO NOT OPEN THIS PAPER UNTIL YOU ARE TOLD TO DO SO
For Examiner's Use
50
--
,,_
.. -
--- - - - - - - ---- ... --------
This document consists of
I..
Setter: Miss Melissa Chong
~
·------~
printed pages, including this cover page.
RLECTRONIC CALCULATORS ARE ALLOWED TO 8 £ USE D IN
THIS PAPER.
=~
and AE = 9 cm, calcu late CE.
Ans:
CE ~
In the figure, if AB is parallel to CD, AB
CD
(The diagram is not drawn to scale.)
3
Nothing i.
to be
written 01
this
margin.
Triangle ABE is similar to Triangle CDE.
AB 4
.
A
-
CD
=-
3
<£ =~ rMil
~
Thus, CE
9
CR
3
4
3
4CE = 27
CE = 27/4 = 6.75
El
_ _ _ __ _
cm
(2)
2
Solve the simultaneous equa tion using the elimination method only.
Sy - 7x = -38 - - - -(1)
Sx ~ 2 = - 5y
(2)
From (2), 5y + Sx =-2
(3)
(1)-(3): -· 7x - 5x= - 38 - (- 2 ) 6
- 12x = - 36
x=3
El
Subst. x - 3 into (3).
Sy ~ 5(3) = -2
Sy = - 2 - 15
Sy = - 17
y=-3.4
El
Ans: x = _ __ _ y = _ _ _
2
(3]
3
,\'olhmx is to f
wrifu:n.on thi,f
p
lttOl"gi ll
.x - 2
R
..::;__~~~~~~~~~-LIQ
i\'
+5
The diagram shows t;.PQR in which L PQR
= 90•, PQ =(x -
2) cm,
QR = (x t 5)cmandPR= (2x - l)cm.
(a) Using Pythagoras' Theorem. fonn an equation in x and show lhat it
reduce..~ to x 2 - 5x - 14 =0 .
(b) 1lcnce, calculate lhc perimeter nnd area ofM'QN .
2
(t+S)'+(x - 2) - (2x-l)
x ' + I Ox 1 25 1 x
2
IMiJ
4x 1- 4 =4x1
2
· [2 ]
4x 1 I
2
(a)
2x l· l0x +28 '-' 0
2x 2 -lOx - 28-0
' ' - 5x-14 = O(show11)EJ
(2.\ - 14)(x I 2)= 0
2x - 14 =0
(b) 2 .. - 14
x =7
OR
x~
.\.11
2 0
x - 2(rejected)
~
Perimeter - 2(7)- 1 1(715) 1(7-2) =30cm L J
Arca
Y, x (7+5) x (7-2) = Yi x 12 x 5 - 30cm
Ans: (h) Peri meter
2
~-]
_ _ __ _ cm Area = _ __ _ cm'
3
[41
l
4
(a) Express in set notation, the set represented by the shaded area in terms of
A and B.
/\ns: (a) _ _ _ AnB' _ _ _ _ _ _
(b) 011 the Venn diagram shown below, shade the set (A v B)V(/I n 8).
Nothmg is tu Ix·
wnlft•n on ll1b'
rnarxin.
[ I)
[ I)
t;' __- - - - - : :______
:
A
5
Al a strawberry fann , the number of strawberries Susie gathered over the past
JO days arc as follows 39, 37. 35, 48, 47, 44, 39, 44. 35, 44.
(a) Construct a stem and leaf diagram to represent the above data.
(b) Find the median or this distribution
Stern
Leaf
(a}
55 7 9 9
3
4 4 4 7 8
4
Key: 3j5 represents 35
~
(b) Median = (39+44) / 2 - 83 I {
- 41.5
I
Al
)~
I
MI
l
4
[2]
Ans: (b) Median ~ ------
6.
,; ·.....
/
'
No11tmx if rob.
•••nll~lt Oil lhis
'
l
\,\.
·~
... ..
r-·-
'----...
"""R'"
\
///
(
(2J
)
--/
!
T he diagram shows a solid made up of a hemisphere, a cone and a right
cylinder. The radii of the hemisphere and the cone are 12 cm ench and the
cylinder has a radius of 4 cm. The heights of both the cylinder and the cone
arc 16 cm.
(a) Find the volume of the solid.
(b) Find the total surface area of the solid.
Round both your answers off to 2 decimal places.
(a) Volume= -I tr{l2) 1 (16)+tr(4) 2 {16)+ -I x -4 tr(l2) J =6836. 1Jcm'• (2dp)
~ '-;==-, .J ~
~
~
R
Al
-
~J
lb) Let the s lanted height of the cone be I cm.
•
127 + 162
,,, .. 400
f
I
20
G
~'Artr(12)(20)+2n(4){16)+
Total surface area =
1
2
= 2865. 13cm'(2dp)
'
~
7
1
x4tr( l2) +2;r(1 2 - 42)
~
~
Ans: (a) - - -- - -
cm 3 (4)
(b) _ _ __ __ _ cm 2 l5J
5
II is given thal y is inversely proportional to !Jx. When x = 27, y - I 2.
Noll1mx u· to b.
Wl'illt.'H
nr.;rxtn
(a) Find the equation conncctingx andy.
(b) Find the value ofx wheny = 7.2.
(a)y(Vx) - k
G
12(V27>= k
Therefore k - 12 x 3 = 36
v('Jx )-
360
(b) 7.2(Vx )= 36
v;_5G
x
· 8.
-1250
(a)
l2]
(b) _ __
[2)
cards numbered I, 3, 4, 5, 7 and 9 are placed into a bag. A car<l is then
drawn at random from the bag. Find the probability that the number on the
card drawn is
(a) a 4 ,
(b) an even number,
(c) a prime number,
(d) greater than 9.
(l
(a) I'(drawing a card with number 4) =
(h) I'( drawing an even number) =
.!.
(c) P(drawing a prime numher) =
~=
6
6
~ ~
0
1
0
2 -
(d) P{drawing a number greater than 9) ~ 0
0
[I)
Ans: (a)
(h) _ _
_
_
_
[I}
(c)
{d) _ __
6
[I)
_
_
[I)
on tllis
9.
The graph of y =(~ • h )(2 - x) , where h is a constant, cuts the x-axis at B
and C. It cuts the y-axis at A(O, I0).
Notlung is 10 b.
1t-n11enontl1lr
margi11.
y
s:
(a) Show that the value of h is
(b) Write down the coordinates of B.
(c) A line parallel to AR passes through C. Find the equation of this line.
[1I
(a) Subst. (0, I 0) into the equation.
o- co +
1
h)(2 l0 - 2h
h =5 (shown)
oG
(b) y = (x ~ 5)(2 - x)
Subst. y - 0 into the equation.
O=<~ + 5)(2 - x)
x + 5 - 0 or 2 x ~ 0
x ~ -5
or x = 2
B(-5,0)0
.
1· . .
(c) Gra d1cnt o this hne
y~
G
- 0 =2 ~
= -100-+ :i
-Ml
2x + c
Subst. (2,0) into the equation.1~
0 - 2(2) IC
~
c =_,,
Equation is y ~ 2x - 4.
0
121
(c) _ _ __
7
[3)
10.
No11J1111,t11ol
3
. I r
·
Express -I- -1 13x - 2
- - - as a sing c iracllon.
x+3 3x +1x - 6 9 x 1
I
3x - 2
3
-- + 2
- x+3 3x ~7x - 6 9 - x 2
I
3x - 2
3
"'-l'fllen Oft Jlus
mcl~'"
- --+
x+J
=
I
x+3
Q,£-l)JJJ},
GI
+
I
3
x+3
(3+x)(3-x)
_ (3-x)+(3 - x) - 3 [
(3 + x)(3 - x).
-
3 -2x
= (3 -tx)(3 - x)
Ml
I
0
An~: ___ __
11.
(a) Solve
2
X+2
-
x
A+)
(b)(i) Factorise 2x2
[4}
- - --
=1.
-1
I 3x -t 15.
(ii) Hence, write down two factors of 21 3 15.
2
x
x+2 x +J
2(x 1 3) - x(x-t 2) =
1
(x+2)(x+3)
- x2 + 6
•
=I
x· +5x+6
(a) x 2 ~6 = A 2 +5x+6
-- - - - = I
-2x 2 -5x O
x(- 2x - 5) - 0
l?iJ
G
x =O
OR
-2x- 5 = 0
-2x = 5
x: -2.5
(bXi) 2x7 + 13x + 15 = (2x+ 3)(x + 5)
0
(ii) Let x be 100.
2(100)2 + 13(100) + 15 = 21 315 = (200 ;- 3)(100 + 5)
The two factors are 203 and l 05.
0
ENO OF PAPER
8
0
-
--
Paper 2
.·
for
fo•
3.
F..xand,,er's
""
The number of hours per week spent in doing Mathematics homework by some pupils is
shown in the table below
7
8
No of hours per week
5
6
6
Frequency
10
II
x
(a) Write down the largest possible value of x if the mode is 7.
( b) Write down the value ofx if the median is 6.5.
( c) Calculate the value ofx if the mean is 7.
Answer ( a) _
_I I I
(b) _ __ [ I)
(c) _ __
4.
(a) Factori7.e completely 471rl/r + 2m· 2 ,
( b) Solve the simultaneous equations 5x - 6 y
method.
= 27
[2)
and 4x - 2y = 16 by substitution
I
'l
Answer (a) _ _ _ _ __ _ _ __
(b)x -
3
___
[11
,y- _ _ _ [3]
Given that a=
J§,
(a) find the value of a 2 if b =I , c =
!4
and d = 5,
( b ) express b in terms of a, c and d.
Answer (a) - -- - - - - [I
(b) _
, 6.
I
J
_ _ __ [3J
/\ 13 m wire connects ·two poles of different heights. The height of the taller pole is
equal to the perpendicular distance between the two poles: If the height of the shorter
pole is 7 m, find the height of the taller pole.
I
!
i
?m
Answer _ __ _ m [
4
4]
7.
Diagram I shows a pencil. It is made up of a cylinder and a cone. The cylinder has a
diameterof0.7 cm and height of 15 cm. The cone has a diameter of.0.7 cm and height
of2 cm. Take tr = 3. 142.
( a } Calculate the volume of the pencil.
Diagram I
Answer ( a ) -·- __
cm.Ir 3 J
Diagram II ~hows twelve of these pencils, which just fit into a box.
Diagram
n
( b) Show that the volumeofthc box is 99.96 cm3•
(2]
( c ) Calculate the percentage of the volume of the box that is not occupi ed by the
pencils.
Answer ( c) _ _ __
5
%(2]
F«
F:,om1r..er"s
For
8.
""
3
x
Express 2
x
3
£1tumnrr'
as a single fraction in its simplest form.
•Jt
Answer
9.
_ _ f31
Solve the followir1g by factorization.
(a) x'
2x
Answer (a) x = _ _ __ _ [ 2)
( b) a(2a - 7) = 30
An.\wer ( b) a - - -- - -- - [ 4
10.
l
Answer the whole of this question on a sheet of g.r>lph paper.
( a) Given that y " 4 - 3x - 2x2 , copy and complete the following table.
[2)
B I :~ j1--·2=.2---1--·.:..l---1_..:~'--+--_.:.1=====~2=~
( b) Using a scale of2 cm t-0 l unit on the x-axis and l cm to l unit on they-axis, draw
the graph of y = 4 - 3x - 2x' for - 3 !> x :> 2.
·
[3)
( c ) Write down the value(s) of x when
[ I )
( i) y 0 and
(
I )
(ii) y = 4 respectively.
( d) Find the equation of the line of symmetry of the curve.
fl l
[ 1)
( c) On the same axes, draw the graph of y = x ~ I .
=
( f) Hence, tin<l thc solution of 4 - 3x - 2x'
6
=x +I.
[2)
Fw
F.xaminer ':;
I.
A container has 80 markers, some of which are pink, some are yellow and the rest are
blue. A marker is drawn at random from the container. "Jbe probability of drawing a
I
1
f.:Sautiner
115('
pink marker is JO and the probability of drawing a blue marker is 8. Find the
(a) number of yellow markers in·tbe.container,
( b) the munbcr of yellow markers that needs to be removed from the container so that
the probability of drawing a yellow marker from the remaining markers will be
0.75.
I
I'
(a) No. of yellow markers= 80x ( 1- - -- )
.
10 8,
= 80x -3
4
=62
[ A1 )
( b ) Let x be the no. of yellow markers being removed.
62 - x
3
80 - x
4
Answer (a ) _ 6_2_ _ ( I
240 - 3x = 248 - 4x
(b)~
[ I
l
J
x=S [Al]
2.
(a) In the space provided, draw the Venn diagram to represent two sets A and B such
[ IJ
that An B = A .
((--)-A~
/
~~-(b)
i;
= {x : x is an integer , 4 $ x $
-~--~-
14 }
T = ( x : xis divisible by 2}
F = { x: xis divisible by 5}
( i) Draw a Venn diagram to illustrate this information.
t
I t5
I \____
contain the correct elements
10
s
t4
12
)
/
l - . --79'"11
( ii ) List the clements of the set
(iii) Write down
J
y~-----46~
El
1 mark if one of the sets
[2
13.
(F v T)' .
n(T n F'):
Answer ( b ) ( ii ) (Fu Tf
= { _2, ~-'-"-'11=-=-1-=-3_ _ __
(iii) _5 __
2
)[ I J
[IJ
For·
€l'w :1i1:er '.f
3.
/ll('
The number of hours per week spent in doing Mathematics homework by some pupils is
shown in the table below.
No of hours per week
5
6
7
8
Frequency
6
10
II
x
(a) Write dovn1 the largest possible value ofx if the mode is 7.
( b) Write down the value of x if the median is 6.5.
( c) Calculate the value of x if the mean is 7.
( b ) 10 + x = 16 - 1
= 15
x = 5 (Al ]
(c)
5x6+6x l0+7"xll+8x = 7 [Ml]
6 + JO+ll+x
30 + 60 + 77 + 8x =
7
27+ x
167 ~~x= 7
27+ x
167 +Bx= 189 + 7x
x = 22 [Al )
Answer ( a) _1_0_ _ [ I J
(b) _ 5_
(c)
4.
22
_
[ I)
[2)
(a) Factorize completely 4m·'h + 2m·2 ,
( b) Solve the simultaneot1s equations 5x - 6y = 27 and 4x - 2y =16 by substitution
method.
= 27 . . . (!)
4x - 2y = 16 . . . @
fr. ® : 2x - y = 8
( b ) 5x - 6y
2 x -8 ··· 17'>
"-"?/
subst.@ into (!) : 5x - 6 (2x - 8) = 27
5x - 12x + 48 = 27
- 7x = - 21
[Ml ]
y ~
x=3 ... (4)
subst.@ into @ : y
[Al)
=2 x 3 - 8
=6 - 8
=- 2
[Al]
Answer (a)
~nr (_~h
2
+
!)
(b)x = _ 3_ _ _ ,y ~
3
[ I ]
-2
[ 3]
For
E:wminc•r's
Fo,
I
F..'ff'"'"U'I'',(
t
E.1.urmner·~
5.
11;.lf
·I
(a ) find the value of a 2 if b = I , c = - and d
4
( b ) express b in tenns of a. c and d.
(a)
02
= 5,
=b-c
b+d
1- 1
= 4
1+5
3
=~
6
= ~ [Al]
2
b-c
( b) a = --
b+d
a 2b + a 2 d =b - c
a 2b - b = - a 2d - c
b ( a 2 - 1 } =- a 2 d - c
[Ml]
[Ml]
I
Answer (a) _::,.
8 _~---
- a d- c
2
-a d-c
b= - - a' - I
6.
[ 1J
b= - -(b) _ ___..a~2~-~1_ _ [3}
(Al]
A 13 m w;re connects two poles of different heights. The height of the taller pole is
cyual to the perpendicular dis tance between the two poles. If the height of the shorter
role is 7 m, find the height o f the taller pole.
7m
Let the ht. of the taller pole
be x metres.
( x - 7 )2 + x z = 13 7 ( Pythagoras' thm ) [ Ml]
x 2 - 14x + 49 + x 2 : 169
2x 2 - 14x - 120 =0
x2 - 7x- 60 =0
[Ml ]
( x + 5 ) ( x - 12 ) =0
x =- 5 ( rejected ) or 12
[Ml]
[Al)
4
An~wer
12
__ m [ 4]
Fr~
'"''''"'"''/''j 7.
U,\('
For
l>iagram I shows a pencil. It is made up of a cylinder and a cone. The cylinder has a
diameter of0.7 cm and height of 15 cm. The cone has a diameter of.0.7 cm and height
of2 cm. Take tr= 3.142.
(a) Calculate lhe volume of the pencil.
(07)' xl5+ JI;rrx (0 7) x2
1
Volume = tr~
~
= l.8375tr + 0.081611"
= 1.9t9t6tr
= 6.0300216
Diagram I
"' 6.03 cm 3 ( 3 s. f.)
Answer(a)
6.03
Diagram II shows twelve of these pencils, which just lit into a box.
Diagram II
r 21
( h) Show that the Vlllume of the box is 99.96 cm3 .
Volume= ( 6 x 0 .7) x ( 2 x 0.7)
= 4.2 x 1.4 x 17
=99.96 cm 1 (Shown)
x ( 15 + 2)
( c ) Calculate the percentage of the volume of the box that is not occupied by the
pencils.
'Yo required= 99.96 - 12x6.03 x!OO"/o
99.96
: 99.96 72.36 x I OO%
99.96
27 6
= · x 100%
99.96
=27.611%
:.: 27.6 'Yo ( 3
5
S.
f. )
Answer ( c) 27.6
% [2]
fuomincr '.
''"
I or
1:~cm1'll"' 'r
,,,,,
.
8.
Express 2 -
3
x
F.XOJtl(ltf'T '.
x
3
as a single fraction in its simplest fonn.
2(x - 3) - 3x = 2x - 6-3x
x
3
[ M2
x- 3
- x- 6
=- x- 3
J
(Al ]
-x-6
A11swer
9.
x- 3
{3 J
Solve the following by factorization.
(a) x 1 = 2x
(a ) x 2 - 2x = 0 · ( Ml ]
x(x-2) = 0
x = 0 or 2
(Al ]
Answer (a) x • _ 0 o_: 2 __ [ 2 )
(b) a{2a -7) = 30
( b)
2oz - 7a = 30
2a - 7a - 30 = 0
( a - 6 )( 2o + 5 ) =0
o = 6 or - 2.5
7
[Ml )
[Ml]
(Ml]
(Al]
Amwer ( b) a -
I0.
Answer the whole of this question
(a) Given that y
=4 -
011
6 or - 2.5
[4l
a sheet of graph paper.
3x - 2x 1 , copy and complete the followi ng table.
I
- I
[2 J
2
( b) Using a scale of2 cm lo I unit on thcx-axis and I cm to I unit on they-axis, d raw
the graph of y = 4 - 3x - 2x 1 for - 3 ::; x :>: 2.
( c) W1ite down the value(s) of x when
( i) y - O and
( ii ) y = 4 respectively.
( d ) Find the equation of the line of symmetry of the curve.
( c) On the same axes, draw the graph of y = x ~ l .
2
( f) Hence, find the solution of 4 - 3x- 2x = x +I.
6
[3 ]
{I
J
( JJ
[ I
l
J
[2J
( I
- - ..---,
·! -r.
L
'
I
1-3
- r ---r
j-·
'
'
1~
' II - ·1 .
' ;i >
j ·-/
() i t' ; •
! ...!<CL._L.!._!, ,,._
.
'
,.
L'L-12~-="--
i
I
4 i --; !:!t2
.7~,.H;;
<.:
'.
\·
i,
i
'
~·
7
I
b:____ - 1
IRogNo
~fl
.....
~~
-·
Clm 2E_
BUKIT BATOK SECONDARY SCHOOL
SECOND SEMESTRAl EXAMINATION 2011
11111 IAIK
Ut• Ut llllll
Seoondary Two Express
MATHEMATICS Part I
11 Oct 2011
101 5-11 15
Candidates answer on the Question Paper
1 hour
READ THESE INSTRUCTIONS FIRST
Write your name, register number and class on this cover page.
Write in dark blue or black pen in the spaces provided on the Question Paper.
You may use a pencil for any diagrams or graphs.
Do not use staples, paper cJips, highlighters, glue or correction fluid.
Answer all questions.
The number of marks is given in brackets f ) at the end of each question or p.art
question.
The total number of marks for this paper is 40.
If working is needed for any question, 1t must be shown in the space" hclow ll1<JL
question.
Omission of e:;scntial working will result in loss of marks.
Calculators are NOT allowed to be used in this paper.
You should not spend too much time on any one question.
-For Examiner's Use-
This paper consists of 9 printed pages.
S.i Z ZUIJ 11
t:xpress :lla1lislf'or; J
ANSW ER Al .L Tll E QUESTIONS ( 40 marks )
I. Oiven thatx 2
+ y 2 = 25 andxy = 12, lind the v:tlueof(x+ y)z.
Ans: ________ _ [2]
2. Factorise completely x 3
-
2x 2
+ 2x - 4.
Ans: _ _ _ _ _ _ _ __ (2]
3. If y is directly proportional to .JX and if y
= 6 wh<:n x = 16, express y
in tcnns of x.
Ans: - - - - - - - --
[2]
2
llabil5 nfMind: St1iving_{or accurc":y an(/ precision
,~A .l 11111 t l bµrt•3l /./~JJlt5 /Part
I
4. Simplify
3b 1 ac'
(a) _ 2_c_
+
18
a 2b
X b 4 a3'
c2
y z _ ,,
(b) y2+1y l 3
A ns: (a)
_
(b) _
5. If p =
J
s
y- 2S
[2]
- [2]
-
,
(a) cxprC$S sin tem1s o f p a nd y ,
(h) find the value of s when p =
J3 a nd y
= -1 .
Ans: (a) _ _
(b) _
Hab1Lt ofMind: Strivi11gfor accnrac,1 and pri'rision
_
_
_
_
_
_
_
_
[3]
_ [1)
6. In 1he figure below, liAFC is similar to liBDC. AC
FC
= 20 cm, BC = 8 3I cm mid
= 3 cm. Find the length of AD .
Ans:
HahiL~ qrMlnd:
Strii•ingfOr a ccurncy a1td pre<:ision
_ _ __ __ [1J
SAl 101112 F...xpre'O !tfotlt.s I Por1 I
7. Solve 1hc following cqua1ions:
a) 3x l - 14x = -8
b) 1
+ _s_ = _ B_
X~ I
zl-1
Ans: a) _ _ _ _ _ _
b) _
_
_ [2)
_ _ __ [3)
5
Hubits of Mimi. Sit fri11gfnr accuracy and precision
SAl 10! I ll E.r.pre..._., .A1atliJ I l'nrr I
8. A stmight line l has a gradient of2 and passes through the point (3, 7).
a) f ind the equation of line / ..
[2)
b) Complete the following table o f values for line L.
x
F
- 1
[I)
2
0
s
y
=
c) In the axes below, the line y
x + 2 is clrnwn. By drawing line l on the same
axes, find the coordinates of the point of intersection o f y
x + 2 and line/,.
=
ljl+r-
~·++ -r
;_ ~ : ; ~""
l - -- ...H
. ~ !tj-_ ==. _:_r_;' ;i ;I -r
r·~· • !/ ; . I
-H11 ! - l ; : • l
- ~·1- .;~; ~ ~ ~
.. .,
H+-+'+'i__j~~+......·,_+·;-'-;'-t....,+H-t-'/·~
·~-1-~'-:H--!1.. ,.'-;
H-+-'-+-H_._. .;,-++1 -+, 1
,_
i i •
....--
!
I
.
'
1
;'-t'_,'-t
' + +H-t++'-:;:...;;...;~..
1 1
-iH + +H-+++;..,,.;,-"-,-I
, 1-'-,-l
1 ; : f H -+-t--!"'1H-H
I
! f
I;:
H-r..--+-'-.;...;~+-': +~ ~
.
H,-+:·~'1-+-1-~;...;:~
_·- "'"~''.'~'-t~-t+"-l~.+-·1-'·-1~~ ~
-t++· ~
l
-lJT-F i-+'
i
'I
1
H-t- ._. j i ~-~·
~~
-1-..l..J·.'·-.~·......_,··:-. .
: i I ·H-++H
I I
l
l
- 1·+.-f-1·- t++-H-+-+',-+,:-+-+-1
!--
I!;!
I !
t-f"J.' -f+H-H-W-t.. ! j ! : ++-,H-++ +-1-+-+-+.1-;·.;:...;....q
,_,_+-i
''>-+''-+--+<r/~if
,.·..._'t-+'-+-'!;'+.:;.-·:·+-~H-'-l,-f.~ili r-H··+
,' -,,1-_.J,; -~.-:.:.+_:"'..,+~~1.:t•-_.;.,.:;;.::
:l ;
..-1
1i i
. LL_LJ_
1:~
l-1-+1.'i,M,+
,,'11,'-t-°l;~
... +
+,-+-
1 J
1
·-·+
l-!-·:H~n!',~1"f--t'-+-t1""'-:~-+,1
,
1
i/·
- ;or L1
1
~
i
i
! :. :
I
~···~
H-,.
'rTTt-
L
!1 !1 -f, 1L
·~+;-ir+H-++-+-H-l·~-~··..,'~~··_,•-+-t--,-t-i-+++-o...;-+-+-+...-+·~
!,- =.~·~-r;-11-+H-++-+-H-+++'
-;'-+'+ +-r-t"'-t++o...;-+-''-+'-';,.;.-1
i
t
i
i .:.
I ""'
1
1 I I
I
I
l .:...
I J l
1 i
i
j
:
!
!
Ii
i"'l
H...,--,'-+"
a H..;...;,-t.
, ~;-..,-.;, ~; •.._._...._++-'
''-+'-~
-+-'H'-+4
:
~w..:
..
i
t
- i
I
'
f _.;,.__
i
Jl.·-1....._..c_;.+.._-'-'-i-l
!
'I
;
!. ; ;- !
'
I
!
l
t
;
j
j
HH-~+f-'µ-1·-''-;..'·+.<-1:-.;.'··'H-+++-Jl-.f..;..+.1'· -<1-+-+-I
_.L
; ; ' - ']! ...!
;- ~ ! !
!
'..;..-1.J,- ',-:,. Tl .
r l-"1-+-+-+-1-!'-;..;'
l
!
t
I
'
I
Ans: point of in1ersec1ion: (
!
II l
)[!)
6
Habits (Jf M1.nd. Strh·ingfor accuracy and prtti.fio11
S.tl JIJ/ f ll F.l:pn.•nc/i/uths/Pan l
9. Given tha t £ = (x: x is an in t eger, 1 $ x $ 10}. A
B = {x: x is a prime number}.
={x: xis a factor of 6},
a) List 1hc elemenls of 1hc set (A fl 8).
h) Fiacl n(A V 8)'.
_
Ans: a) _ _ __
b) _ __
_
_
_
_
_
[I]
_
. 12J
10. a) In the Venn Diagram helow, shade AV B'.
..--------~
E
[IJ
h) Express in set notalio n, as s imply as possible, the scl shaded in the Venn Diagram .
-----~t:
Ans: b)
---
_
_
_ [I]
7
Hnbit.f ofMmd: Stm'lngfor accurncy and precisf(ln
SAl 1011 /l
t:xprt•ss .Hatlts /Pan I
11. The diagram shows the graph of a quadmtic equation. The curve cuts the x-axis al
P( I, 0) and Q(?, 0).
y
a) Find the equation of the line of S)mmetry.
b) Given that the equation of the curve is y
-x 2 +ax - 7, find
=
(i)
(ii)
the value of a,
the !,'featest vah1e ofy.
Ans: a)
_
b) (i)._ __
_
_ _ [I]
_
(ii)_ _ __
_
12)
[I)
8
Hubits ql:\1i11d: Striving.for C1cc,·urac,:y and precision
S.411011 il /:Jpretr ,tfalhs i l'a11 l
12. John h,~s x rwo-dollar notes and y live-do llar notes. 'f he total va lue of the notes is S48.
a) Form an equation connecting x :m<l y.
b) If the number of two-dollar notes is 4 less than the number of five-dollar notes, by
fonning another equation in x and y, find the number of two-dollar notes,
Ans: a) _ _ __ _ _
b) two-dollar 11otes:
•
0
_
_
_
[I]
[3]
£11d "/Part/***
9
f /(l//its ofMind: :Striving/or acc.umcy <md precisio11
Ma r king Scheme (Pa rl ll
I. (x+y) 2 =x 2 + 2xy+y 2
Ml
= 25
= 49
Al
+ 2(1 2)
2. x 3 - 2x 2 + 2x-4 = x 2 (x-2) +2(x- 2)
(x 2 t- 2)(x - 2)
=
Ml
Al
k./X, k is a constanl
3. y =
Substitute y = 6, x
=16, k =i
:. y = ~./X
MI
Al
2
Ml
Al
12
y 2 -9
(b) --
y'+4y+J
(y-3J(y+3)
=
Ml
(y.i)(y+J)
y-3
Al
=y+J
Js
5. (a) p =
2
p =
y-2<
s
Ml
'Y-2s
= p1 y- Zs112
s(l + lp 2 ) = p 2 y
s
~ ~
.
Al
l+lp2
2
(b) s
Ml
,,2.v
(J3} (-t)
!+2(,/i)2
l
= -176 = -
)
,
ll l
6. ·.- t.AFC is similar i\IJDC.
AC
BC
FC
DC
20
-
8~
= ...1.
3
DC
Ml
DC
Ml
= :'4. cm
AO = AC - DC
AO = L.0- ~= 18 ~cm
•
•
Al
Hx = -8
l4x + IJ = 0
Ox - Z)(x - 4) = 0
7. (a) 3x 2
-
3x 2 -
2
x= -3 orx=1·
(b) 1 + -2._
x+t
x+6
-
x+1
=
Ml
1\ I
=- 8-
xZ-1
8
Ml
(xH)(x- 1)
(x - l)(x + 6)
= !J
Ml
x 2 + Sx -14 = 0
(x + 7)(x - 2)
=O
x = -7 orx = 2
8. (a) grndient
y
=
Al
2.
= 2x + c
Since (3. 7) is a point on the line, 7
=2(3) + c,
c=l
Ml
Equation of line : y
=2x + 1
AI
111 ft
(c) Correctly drown line with label
Point of intcrScction = ( 1. 3)
9. (a)
BI ft
BI
A= {l,2,3,6}
=
B (2.3.5, 7}
A n B = (2.3}
(b) AU 8
BI
={l.2.3,5.6,7}
(AU B)'
= (4.8,9,10)
n(AU8)' = 4
Ml
Al
10. n) In the Venn Diagram below, shade AU 8'.
E
'.
Bl
b)A n 8'
Bl
Il.a)x = 4
Bl
b)(i) y = - x 2 +ax - 7 and (1, O)is point on the curve,
0 =-l+a -7
a=8
b)(ii)gre;itcsty-- 12 I 8(4) - 7 =9
12.a)2x+Sy=48
b) x+ 4 =y
2x ~ S(x + 4) = 18
x =4
Ml
Al
BI
Bl
131
MI
Al
I
I
"'•No -.
Cl"" 2E -
.
BUKIT BATOK SECONDARY SCHOOL
SECOND SEMESTRAL EXAMINATION 2011
lllrf .....
Secondary Two Express
HCllHU ltlltl
MATHEMATICS Part II
11Oct2011
1125 - 1255
1 hour 30 minutes
Write all your answers on the writing papers provided.
READ THESE INSTRUCTIONS FIRST
i
Wlite your name. register numbc1 and class on all the work you hand in.
Write in dark blue or black pen.
You may use a pencil for any diag1ams or graphs.
Do n<>l use staples, paper clips, highlighters. glue or correction fluid.
I
Answer all the questions.
If working is needed for any question it must oo shown with the answer.
I Omission or essential working will result in loss ol marks.
Calculators should be used where appropriate.
If the degree of accuracy is not specified in the Question, and if the answer is not exact,
give the answer to three significant figures. Give answers in degrees to one decimal place.
The number or marks is given in brackets [ ] at the end of each question or part question.
The total number of marks ror this paper i s 60.
For Examiner's Use
You should not spend too much time on any one question.
This paper consists ofS printed pages.
Answer all qu estions. 160 mar ksl
I. (a) Simplify n.2
-
(11
+ a.)(n -
a)
(b) Hente cv~luatc 326541829 2
-
[I]
326541833 x 326541825.
121
(c)
x+Z
5x - 1
The lengths of the three sides of a right-angled triangle are (x + 2) cm,
(Sx - 1) cm and 5x cm respectively.
(i) Use Pythagoras' Theorem to form an equation in terms of x and show that
reduces to x 2 - 6x + 5 = 0.
(ii) Solve this equation to find the two possible values of x.
\I
i2]
12]
2. In the diagram, AD and BC are horizontal and AC is vertical. CE is perpendicular to
AB. It is given that AC = 4.2 m, CD = 5.6 m and LBAC
(a) the length of AD,
(b) LACD,
(c) the length of CE,
(d) the length of AR .
= 55°. Calculate
[2]
(2]
[2)
[2]
A
D
Page 2 of S
H'abits qfMind: Stri11ing.for accuracy and pret:ision
SAJ 101 J J 2 £.-/>' ~·.u .~101h:-. / l'r111 II
3. The scale of map Xis 1 cm : 4 km. Suppose 1ha1 lhe actual area of a park is 64
(a) Find, in cm 2, the area of lhe park on map X.
2
kn1 •
[21
The ar~-a o f 1he sam e park when drawn on map Y is 25 cm 2 •
(h) Find the scale of map Yin lhc form 1 :
(21
11.
(c) A road is repn:scnled by a ll:ngth of 10 crn on map Y. Find its acrual length in
I 1l
kilometres.
4.
Rec1angle B
Rectang le A
x
x ·I 4
D
i
Two rectangles, A and R, each have an ar~-a of 58 cm2• The le nglh of rectangle A is x
cm. The leoglh of rectangle Bis (x ~- 4) cm.
(a) Find, in tem1s of x, an expression for !he wid1h of
(i) rectangl<: A,
(I)
(ii) reclang le R.
[ IJ
(b) Given that the width of rectangle A is 2 cm more than 1he width of rectangle 13,
fom1 an equation in x a nd show that it simplifies to x 2 + 4x - 117 - 0.
(21
(c) Solve the equation and find the value of x.
[21
y•
x'
5. (a} Simplify - - + - - and show thai it reduces lo - (x + y) .
x- y
y-x
(h} Hence, simplify l
+
:t' Y
.,i.
(3)
[2]
-~­
~-,,
y-x
Page 3 of S
Habits ofMi1rd· Strfrillg for accuracy and precision
S:':IZ 2011 I l £:q>ress l1.f111hs,/ Pan JI
6. Crime has struck Gotham City again and Inspector Gorrlon is shining the Bar-signal
into the clouds to inform Batman. The searchlight iir on the top of a building 400 rn
high. It is given that the angle of elevation of the cloud from searchlight is 60°, and
the vertical distance of the clouds from the ground is 2000 m.
(a) Find the length of the.light beam.
[2]
(b) Batman is at point X as shown in the diagram when he saw the Bat-signal in the
clouds. Given that point Xis 3 km away from the building, find the angle of
elevation of the Bat-signal from the point X.
(4 J
Page 4 of S
Habirs ofMind: Striving for accuracy and precision
SA}]()// I) £xpr(•.u ,\/a01s I Pm1 JI
7. A cylindrical so lid piece of wax (Figure I) is me lted down to form the so hcl shown in
Figure 2, which consists ofa hemisphere joined to a·cone. The radius of1he
hemisphere is 22 cm and the height of the cone is 28 cm.
l
3r
!
- . ---.-..
~
Figure 2
Figure l
(a) Find the volume of the piece of wax in Figure 2.
Leave your answer in tcnns of rr.
(h) Given that the height of the cylinder i5 thrice it~ radius, find the radius
cylinder.
(c) find the toll\ I surface ar ea of 1hc cylrnder.
[4)
or01e
[4 J
PI
/J. Amwer the whofo ofthis q1111slion 011 u piect' 1ifgra{lll paper.
The variables x and y are conncc1ed by the equation y == 30x - 5x 2 . Some
corresponding values of x and y arc given in !he table below:
F-=;--+1-~
(a) ('alculatc the values of a and b.
(b) Draw !lie graph o fy = 30x - Sx 2 for 0 ~ x S 6, using a scale o f 2 cm 10
represcn1 I unit on the x ·axis and 2 cm 10 represent 5 units on 1hc y-axis.
(C) Write down the equation of the line o f S)'IDOletry of the graph.
(d) Use your graph to find
(i)
the value of y when x = 2.5.
(ii)
the values of x when y = 20,
(iii)
the maximum valucofy.
(e) Ry adding a su itable line to the groph, solve the equation Sx 2 - 30x + 23 =
[2)
[J]
[ l)
(I]
[21
[l)
0.
(2]
*"*End of Part II***
Pages of 5
ffubiis o/Mmd: Strfringfor accuracy and precl<ion
Mad d ng Scheme (Part II)
b) 326541829 2
=
a.) = 112 - (n2 - 11 2)
+ a)(11 -
I. a) nZ - (n
=a2
.
Bl
326541833 x 326541825
-
326541829 2 -
(326541829
:.
+ 4)(326541829 -
= 4 2 = 16
MI
Al
=
=0
c) (ii) x 2 - 6x + 5 = 0
(x-S)(x-1)=0
Ml
X"' Sorx = l
2. a) AD = ../S.6 2
Al
'1-.2 2
-
MI
1.f7
=-Ill
s
=3.70m
b) LACD
Ml
Al
c) (i) (x + 2) 2 + (Sx - 1) 2 = (Sx) 2
x 2 + 4x + 4· - lOx + 1 0
x 2 -6x + 5
4)
I
Al
42
= cos- 1 ( S.6)
Ml
= 41.4096°
::: 41.4°
Al
c) CE = 4.2 sin 55°
Ml
=3.44 rn
Al
d)All = ~
Ml
cos SS"
= 7.32 m
Al
3. a) J cm : 4 km, 1cm 2 :16 km2
Are.1of park on ma1>X=
~
= 4 cm
16
Ml
2
b)25cm 2 :64km2 ,Scm : 8km
Scale of map Y is 1 : 160 000
c) aclu<ll lcngth
= 10 x
160 000 = 16 km
Al
Ml
Al
Bl
o
4. (a) (i) wid1h of n.-ctangle A
=-so!x ' = -2r
111
(a) (ii) width of rectangle n =
(b).!..'2
-~
2r
2(x+4)
=2
+ 4x -
+ 4)
=0
Ml
Ml
x = -13 (rejecled) or x == 9
y2
81
117 = 0 (shown)
(c) (x + 13)(x-9)
xi
y-x
yz
x-y
¥ 2
5. a ) - + - = - - x- y
)
Ml
117(x + 4) - 117x = 4x(x
x2
117
,
2 ,x+•
Il I
x-y
yZ-x2
= r-- y-
Ml
(x+y){x- y )
x-y
Ml
=-(x+y)
Al
b)l + ~ "' 1 -2....
1 -~
x-y y-r
x+y
Ml
--x+y Y
x+y
:~
x•y
Al
Al
. .
6. a) length of light beam
=-.1600
-.
MI
= 1847.52"' 1850 m (3 s.f)
AI
S1n6Q
h) Distance from X to bottom of bat-signal, Xl'
= XZ-YZ
= 3000 -
= 3000 -
1600
t;in60..
1600
3
Jv11
J3
= 2076.239569
Ml
111
Let B be the angle of elevation.
tanO
=
2000 •
,..,0
Ml
3tl00- 7 J 3
/\nglc of elevation = tan- 1
2
3000
~ = 43.9° (I decimal place)
.1
>
I\ I
z
1
3
3
7. a)Volurneofwax = - rrr 3 + - rrr 2 h
b)h
2
Ml
3 + .!.rr(22) 2 (28)
=~rr(22)
3
3
Ml + Ml
= 11616rr
/\1
= 31·
rrr h
= ll6161r
r 2 (31·)
Ml
MI
= 11616
r = ?./3U72
r "' 15.7 cm (3 s.f.)
Ml
J\ 1
c) Total surface area of cylinder= 2-rrr 2
= 2rr(V3B72) + 2rr x 3(~3072)
2
= 6197.2178 crn~"' 6200cm
2
2
+ 2nrlt.
Ml, Ml
Al
I
i'
r
J
I
i'
r
J
2
Paper 1
2
3
l.'actorise each of the following completely.
(a) 3ac - 6ad - 5bc + I Obd
(h)
27m' - 3m
Answers
(a) -
---··-----
(b)
4
[2J
121
A stack of 4 cards have the numbers '2', '3', '4', and '5 '. Two cards arc drnwn
one aflcr another, at random, with out replacement, to fonn a 2 digit number.
(a) List the sample ~pace.
Find the probability that
(b) the two-digit number is divisible by 3.
(c) the two-digit number is prime.
Answers (a) _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _ _ _ _ __
(b) - - - - - · · - - -
(c)
[IJ
(I]
(lj
3
5
It is given tlml p is dircclly proportional lo the ·square root of q and that p
=6
when q =4 .
(a) Find the equation conncctingp and q.
(b) Find the value of q wh<.:n the value ofpis doubled.
Answers
6
(a)
[2)
(b)
LI J
The swimming pool al De lta Sports Complex is 1111dergoing maintenance. It takes
4 pipes to drain the water away in 50 minutes. How long will ii take to drain aw~y
~11
the water if 6 more pipes were to be u.~ed?
Ansl11ers
(2]
4
7
Fin<l lhe value of p'
+ q 1 , given that pq -"' 13 an<l p + q
[3 J
Answers
8
-
r - .v
- +
(a) Simplify 4s -t 36r
=8.
r s
I 8r + 2s.
3x
2
- - +(b) Express 2x' - 9 x - 5 5 - x as u single fraction.
A11swers
(a)
[2)
(b)
(3]
5
9
5d > .!.Q
(a) Solve the inequality 9 - 3
(h) Represent the solution on the number line below.
Answers
(h)
10
(a)
.. _ __ _____ __
[2)
.,,.
[I]
V - 2gli
,
I
3
Given that v = , 1
(a) make J: tbe subject fonnula.
{b) find the value of g when u = 6, h - 3 and •·=2.
Answers
(a)
Pl
(b)
[I}
6
11
Tn the diagram below, BC is parallel to EF. Given that the length of AF = 3.6 cm,
AE = 6 cm, EF - 4.8 cm and B.C is 12 cm,
MEF is a right-angled triangle.
(b) Given that 6AEF and MBC are similar, find the length of CF.
(a) Show that
A
Answers
(a) Shown in question paper
[I)
(b) CF= _ _ _ _ _ _ cm
(2)
7
I
12
.
The diagram he low shows the graph of -
2 x + y = - I am.I 0.25x - 3.v = 17.
0 .25 x - 3 y = 17
x
(a) !'ind the coordinates of Point A and l:l.
(b) What is the coordinates of Point C?
(a) Coordinates of Point A -
fl)
Coordinates of Point B =
(I]
=
[3]
(c) Coordinates o f Point C
.·
13
A c ube of length 12 cm is melted and recastcd in1o the shape of a cone. Given that
the cone has a diameter of 20 cm. find the height of the cone.
12 cm
Answers
(a)
9
14
(a) Shade 1hc region representing the set P n Q' in the Venn diagram below.
(h) c - { x: x is an integer ,
l <x
~
16},
A = { x: x is a perfect square},
B- { J: x is a faclorof32 }.
Wn lc down the valut! of
(i)
n( A),
(ii) n(AnB).
A11si.-e~
(a)
Shown in Venn Diagram
[I J
(b) (i)
(11
(ii)
[I)
10
IS
The heigh! of 12 employees in an office wa~ measured. The information is
displayed in 1he s1em and leaf diagmm below.
S tem
15
Leaf
5
16
17
0
' 18
3
6
7
4
5
7
8
9
9
Key: 1515 means 155 cm
Find
(a) the mode,
(b) the median, and
(c) the mc:m of 1he heights.
Ans1vers
cm
[ lj
(b) Median ~ ------- cm
( 1)
(a) Mode
(c)
Mean
END OF PAPER
=- - -- - --
-
- - - - CITI
[21
Seconda ry 2 Final Ycar Examination
Mathematics Paper I Answer Schem e
}(.~ 7. 12
Bl
Bl
7. 118
"<!!)_
2(a)
-
I cm: 250000cm = I cm: 2.5 km
Distance between the two towns on the map
= 32 + 2.5
(b)
~·- ~
Al
= 12.8cm
I cm: 250000 cm
=lcm:2.S km
:
Area scale of first map
} cm 2 : 6.25km 1
•
Actual area of plantation
l.6x 6.25
,
c
= I Okm
Jae - 6ad - 5hc + I Ob<J
- 3a(c - 2d) - 5b(c - 2d) ·
=(c - 2d)(3a - 5b)
-
' (h)
t_4(<1J
i (b)
r-·- ---
,I
~
Ml
- ~ --
·I
-~-·.
L
I(c)
IS(aJ
'
2
I
I
I
27m 1 -3m
=3m(9m1 I)
= 3m(3m + 1)(3m -1)
s ~ {23, 24, 25. 32, 34, :J:i, 42, 43. 45, 52, 53, 54)
I
- -
-
3
l
- -
-
Ml
Al
Al
t - - -· Al/Bl
-
-
-AliHi
-
4
p = kjq where k is a constant
lfp - 6and4~4,
6=k-f4
Al
6-2k
:. k ,. 3
p = 3h
The eouation is
-
Ml
-
-
-
I
(b)
lfp= 12,
12 = 3/q
4 = jq
Al
q=2
Let the time be x minutes and the number of pipes be p.
-7 inverse p ro(l<)rtion
6
xrr= k, where k is a constant
lfp=4, x= 50,
k = 200
Ml
:.xp =200
lfp = lO,
!
!
200
x=-
10
:.x=20
Al
Or
Ml for
4
: - x50 = 20
10
'
\VOi king
Al for
final
I
I
:tllS\Ver
!
--7
.
-
-
ptq=8
(p+q}l ,,, gl
p 2 +2pq+q 2 =64
p ' +2(13)+q ' - 64
1
8(:.1)
I
Al
r- s
2(9r t· s)
x
4(s + 9r)
r- s
Ml
I
Al
= -2_
__
I
I
Ml
Ml
r-s
r- s
4s+36r 18r +2s
r- s
r- s
-:: - - - +
4(s + 9r) 2(9r+s)
=
'·
-
2
:. p + q = 64 - 2(13) = 38
I
I
:
--
I
(b)
3x
2
- +2.' - 9x - 5 5 x
3x
2
= - - - + -(2x + l}(x - 5) 5 - x
2
3x
= - - -- (2x+l)(x - 5) x-5
Ml
3x - 2(2x+ I)
Ml
= (2x+ J)(x - 5)
=
3x- 4x-2
(2x + l)(.r - 5)
-x- 2
= (2..r + l)(x- 5)
=
<l(a)
x+ 2
(2.r + l )(x - 5)
- - - - -- -- -- - -
. 5d 10
- 29
3
15d 2 90
I
Al
- - - - - - - --tMl
J\l
d26
=.e--- -- - - ·
. _ _ --+--
5
--+--
6
_,_
___,_
7
8
-
-
-
Al
- ·
- -- - -
- - --
-- - -- -+-
-
MI
Ml
Al
J\l
----·--- -~ --
· ~ - ·--.------------~-------~-~-------.~~~--,
AE'
=6'
= 36
Since AF'+ FE 1 = A£1 , 6AEFis a right-angled triangle.
11 (b)
CB
FE
AC= AFxCB ,,. 3.6x 12
FE
4.8
Bl
AC
AF
%
Ml
9 cm
Al
CF=AC-AF=9-3.6 =S.4 cm
Coordinates Point A = (0, - 1)
.._
of
12(a)
Bl
Wheny =O,
!x+(0)= - 1
2
x= -2
Coordinalcs of Poinl B • ( 2, 0)
12(b)
131
I
, - x+ y = -I ------- (1)
I 2
0.25x - 3y - 17 - - (2)
Rearrange eqoacion (I)
I
y = - 1- .t
2
y- - I - 05x ------- (3)
, Substitute y = I 0.Sx into equacion (2)
0.25x - 3( I - 0.5x)- 17
0.25x + 3 + l.5x = 17
f 1.75.x ~ 17 - 3
i l.75x - 14
Ml for
I
sub.
I
x=S
Ml for
solution
of..> c.- y
Substilulex = 8 into equation (3)
y= I - 0.5( 8)
y = -1-4
J' = - 5
-·--_ Coordinates of Point C = (8, - 5)
,
13
- · - -- --i
Al _ _
Volume of cone
= Volume or cube
1
!
1
I= 12 x 12 x 12 - 1728 cm'
L_J. ,;, , ""'. ,.
= 20 < , = •• , ...
Ml
-1
Volume of cone =
1728 =
!
x
JC
1
3
x Dase Arca" lleight of cone
x 10' x Height of cone
3
Height of cone
Ml
3)(1728
2
")( 10
- 16.5 cm
Al
14(a)
Al
14(h)(i)
PnQ'
A = {4, 9, 16}
14(b){ii)
n(A ; 3'--- A A H ={4, 16)
Al
-·
n (A A 8) _=.o..2_ _ _
15~
(b)
(.;J
Mode- 169cm
1 Mcdian=~65 +J 69
;
IMod~
2004
2
= -p
=167cm
167cm
---------
-
-
Al
Al
Al
IMI
__lA1__
-l
2
Paper 2
2
3
The diagram below shows a sequence of tiling patterns fom1ed by using small square tiles
of length I cm.
(a)
11 =
I
II =
2
II -
Draw the figure
in the answer
paper
3
n=4
The table below shows some of the observations made.
-
fig.
No. of squares (S)
Perimeter of figure (/')
I
3
8
2
5
12
3
7
16
4
x
20
II
y
I
(a) Find the value ofx and draw the4'" figure in the answer paper.
121
(b) Express y in tem1s of n.
[I]
(c) Derive a fo1mula that shows the relat.ionship between P and 5. Hence, use it to
find P, the perimeter of the figure when Sis 43.
(2]
3
4
A group of n sn1dents are going for an excursion to the Science Centre. The total fare fur
booking a bus 10 Science Centre is $ 120. This amount is to be shared equally among the
studenL,.
(a) Writi.: an expression, in tcnns inn, for the amount paid by each student.
l lJ
(b) On the day of the excursion, 2 more students joined the group.
ronn an expression, in tem1s of 11, for the new amount that each student has to pay.
f 11
(c) If each student pays $2 less now, fonn an equation inn and show that it reduces to
11
+2n - J20 =0.
1
f3l
fl. i
(d) Solve the equation and hence, lind the <>riginal number of students.
S
Bukit Merah Shopping Centre is holding a lucky draw.
There arc 500 chips in the box of which there arc w red chips, x blue chips and y white
chips.
A reel chip means the customer wins $30 shopping voucher.
A blue chip means the customer wins digital camera woth $105
A white chips means the customer wins a an umbrella.
(a) If the probability of :;electing a red chip and a blue chip is -
6
25
and
I
10
respectively.
find the value or 111, x and y .
f3 J
(b) On the eve of Chnstmas Day, the manageme111 of the shopping mall would like to
increase the chance of winning a digital camera to celebrate the joy of christma~.
I low 111any blue chips must be added such that the probability of selecting a blue chip
.
IS -
I
4
?
[2]
4
6
In the recent Job Week carried out by a group of 36 scouts, the mnount of money
collcctc<l are shown in the ta ble below.
Amount
collected ($ w)
Number of
scouts
0 < "' s 20
-
5
--
20 < w$ 40
40 < w .S 60
12
4
-
60 < w .S 80
-
6
80 < w S IOO
3
(a) Copy and complete the following table.
Amount collected (S w)
Mid-value, x
[3)
fx
Frequency,/
0 < w S 20
·-
20 < w S 40
40 < w .S 50
50 < w .S RO
-
--
.... -
- ---·- -·
- - --
80 < w .S 100
'I. J =
"i.Jx=
(b) Estim ate the mean amount o f mo ney coll ected by each o f the 30 scouls.
f2)
(c) State the class interval wl1ich contains the median.
(I )
5
7
The diagram below shows a water- fi ll ing container at Wi ld Wild We1., a water theme park.
Tap
revolwing rod
The container is made up of a hollow circular cylinder of height 8 cm and internal radius
of 42 cm and a hemisphere of the same radius. The container is suspended in the air hy a
revolving rod. \Vhen the water in the container reaches a particular level , all the waler is
then discharged out from the container.
(a) A layer of paint was coaled on the internal surface of the container to prevent rusting.
Calculate the tot<il internal surface area of the open container.
f3J
(b) When the water in the container is 28.2 litres, the water is completely discharged
from the container. (Given that I litres - 1000 cm 3).
Calculate the overa ll vertical hei ght in the container before all the water is
discharged.
(4]
6
8
Auswer the whole of this question on a sheet of graph ,,aper.
The number of bacteria 11
i1\ 11 colOl\y afkr lime I hours
is given by
11
= 1501 2 +I 00.
The colony sla1ts with I 00 bacteria.
0
I
2
3
4
5
6
7
100
250
700
1450
2500
3850
5500
b
Ti me (t hours)
No. of Bacteda (11)
(a) Find the value of b.
[I J
(h) Using a horizontal scale of 2 cm lo represent I hour and a vertical scale of 2 cm to
represent I 000 bacteria. draw the graph of 11 against I .
(3}
(c) Use your graph lo !ind
(i) the value of 11 wheu
(ii) the value oft when
I
11
[I I
4. 3,
=6200.
(d) (i) The numhcr ofhactcii~ in another colony is represented as n =
(1 }
5500 - 500t.
On the same axes, draw a graph to represent the number ofhacteria in this colony.
[I J
(ii) Find the value of / wben the numbers of bacteria in both colonies are equal.
END OF PAPER
[I]
Secondary 2 Express Mathematics
End of Year Exnminations Paper 2
l(a)
Distance lm veiled by Alpha
J2x 2
- 24km
Ml
Distance lra veiled by Bela
= 16x 3
= 48km
I
(h) (i) -
Using pytha goras' theorem,
Let the dist,ance between Port A and B be D.
D 2 =24 2 + 48 1
Al
D=M+ 48'
:. D 53.66km = 53.6km
Ml
t
Strawberry
Blueberry
kiwi
l
orange
I
I
I \•i>
I
2 (a)
Thcr are dis~oint sclli.
27 +x+ 25 + y = JOO
~+y - 100 -27 -
25
:. x .. y"' 48
(h)
Mean = 3.5
£><27)+( 3x) I (4x25)+5y = J.
5
100
54+3x+ 10 015y =3SO
(c)
3x+5y= l 96
x+ y = 48
x= 48- y
J
~
·
Subx=48 - y intoJx+5y=l96,
3(48- y)+5y = 196
Ml
144 - 3y + Sy = 196
2y = 52
Al
y = 26
:. x = 48- 26
Al
x =22
Or any other valid method {s uch as eliminatio1:1_!11Cth~)_ _
(d)
Modal= 2
3 (a)
x=9
- · - - ·Bl
---·Bl
4th figure:
<---
,_
f---
I I I
~·-·---
{~
(c)
~.
4 (a)
(b}
(c)
I
I
v = 2n ·I I
P = 2(S + l)
Bl
Bl
IrS ;_ 43,
P = 88
IN
120
ew amount pa1'd = $ II+ 2
120 120
=2
II
n ~2
l 20(n + 2) - 12011
=2
11(11 + 2)
-
12011 + 240 - 12011 = 2n 2 + 411
2n1 + 411 - 240 = 0
n 1 +2n - 120 = 0 <shownl
Al
t-- -·-
~ --·-
Amount paid by each student = $
'----
120
n
-
Al!l31
- - ----.l\ 1 Bl
-··
1
-
Al/131
.
Ml
Ml
Ml
·-
(cl)
,,i + 211 120 0
=0
=- 12
(n - 1O)(n+1 2)
Ml
(rcjec1ed)
11=I0 or /1
Thc original number of stud ents is J 0.
5 (a)
6
P(R
Al
IV
) = 25 = 500
w -. 120
Ml
x
I
P(B
Ml
=ro=soo
>
x- so
y == 500 - 120 - 50
y =330
(b ) (i)
Al
Let the number of blue chips to be added be a.
5O+ x I
-= 50O+x 4
4( 50
f· x)
Ml
=500 + x
200 + 4 x - 500
t-
x
Al
3x = 300
x = 100
Th erefor<:, l 00 blue chips have to be added.
6 (a)
-
A 111ount
Mid-value, x
Frequency, f
0 < w S 20
10
5
50
20 < wS 40
30
12
360
40<w S 60
50
4
200
60 < wS 80
70
6
420
80<w S 100
90
3
270 ·-
collected ($ w)
i
l
--
Minus
-
-
I
fx
2
L./ = 30
.
mark
per error
-
-
I fx = 1300
(b)
(c)
M ean amount o f money collected
1300
-30
$43.33
Ml
Al
Since it is an even set of data, the middle position is 15th and
posit ion.
Th erefore, the median class interval is 20<w S 40 .
1 - - -- - - il 7( a)
surface area of hemisphere
I
l 6th
Al
..!. x(4nr 2 )
2
.!.x (4x1Tx21 2 )
2
2770.9c:m 2
Ml
surface area of cylinder
2nrh
2xnx21x8
Ml
1055.6cm2
Total surface area of container
- 27i0.9 +I 055.6
3826.5cm 2
Al
3830c11l
28.2 litres =28200cm3
v olurne of hemisphere part of the container
~ x(~m.>)
~x(~ xnx2 l})
Ml
19396cm'
9396 + Volume of waler in cylinder
l
h =28200 - 19396
3. 14 x 21' x h
.h
8804
;r 21 1
=- -
x
=8804
= 6.354cm
= 28200
'
Ml
Ml
Overall vertical height of container
=21+6.354
= 27.354
=27.4cm
Sa
(b)
Al
b ~ 7450
M2 for correct scale used
MI for correct plotting of points and joining _ll_.
1e_.p'"'oi1_1t-'
'- -s_ _ __ __
2900 ace t answer within ran e from 2850 to 2950)
6.4 (accept answer within range from 6.3 to 6.5
M I for corrcc! plotting of linear graph
4.6 acce t answer witJ1in ran c from 4.5 to 4. 7
--~-~-~-~-~-~~~-~
Al/B l
1 - --
Al
Al
~~--~~-~~-t--~~--r
Al
[ Name: _ _ _
:_ _ _
_ _ _ __ _ _ _ _ ....l_c1a_s_s_
IClass Register
No : .
..
,-
0
CHUNG CHENG HIGH SCHOOL
(MAJN}
Parent's
Signature
ScbpQI Ciiuna,CJll>~ . . ··· School.Churjj,£hang High Scbool Chuna.Cbeng \1igh School
1<&;il1foi :clfun~C.hr.r:..,,
hciOl~ti'uFijLcf«i~g"Hi.QnS.cl)b01 ChuH{l;c'!i'e'rlg .H• ti&hbol ·
·~Ooi'6hui>9'¢~- :SchoOt cilungtfieng Hi91i' scooo1 Chuaj:ciierig H~fi'.schoo1 ·
ti SchOol chung Cheng High School Chung' cheng.High School Chung'Ctieng Hlgh SchOO
Mathematics
Paper1
END-OF-YEAR EXAMINATION 2011
SECONDARY2
Tuesday 11 October 2011
1 hour 15 minutes
. ·1
Instructions to Candidates:
j
I
1. Write your name, class and register number in the space provided o n the cove r pagP i
or this booklet.
'
2. Write only in dark blue or black pen.
3. You may use a pencil for any diagrams or graphs.
4. Do not use staples. paper clips, highlighters. glue, correction fluid or tape.
5. If working is needed for any question il must be shown with the answer.
6. If additional papers are used, fasten the sheets together with the question paper.
7. If the degree of accuracy is not specified in the question. and if the answer is no t
exact, give the answer to three significant figures. Give answers in degrees lo one
decimal place.
Information For Candidates:
CALCULATORS ARE ALLOWED FOR THIS PAPER.
OMISSION OF ESSENTIAL WORKING WILL RESULT IN LOSS OF MARKS.
The intended marks for the question or parts of the question are given in brackets [ ).
The total number of marks for this paper is 50.
You are reminded of the need for clear presentation in your answers.
This question paper consists or 9 printed pages (including this cover page).
fm
F'u•
2
~·l"t1Mi1u•r't 1.1,.
" ' " " ''.,,,. .. ·, H,'(.
Answer all the questions.
Given that y is inversely proportional to (x + l), and y
(a) express yin tem1s of x ,
(b) find the value ofy when x = - 17.
I when x
~
3.
Answer (a) y=........ ... .. ..... .......... [2 1
I
(b)y- ............................
2
l lJ
A moving object travels s metres int seconds. It is given thats is directly
proportional to 12 • Express sin tenns of I and k, where k is n non zero constant.
If the object travels 75 metres when 1 - Q seconds, find the new value of s when
1 = 4Q seconds.
A11swer (a} s - ............................ [I J
(b) s= ............ :............... [2J
Chung Cheng High School (Mnin)
End n/Year Exum 1011
Maths Paper I
J
S11nplify
(a)
(a+ 2b)(a - 2b) - 4hl + <lab,
(h)
(p+q)(p - q ) .
- (p - q)'
~(q-p)
Answer (a) . ...................................... f2 I
(b) ······································· [2)
4
I
I
Factorise completely
(a) 2sx2 - 1,
(h} 3x+6y - x 2 - 2.ry.
(c)
I
6 - l5x - 36x' .
Answer (a) ............. ....................................... [ 1J
{b) .................................................... [2]
(c) .................................,..................... [2)
2•5
Chung Cheng Hill.Ir Sclwol (Mnin)
F.nd o/Year Exam 2011
Math.• l'oper I
,·;()(
F(J•
•"'°m1wu r '.t t~\·~
5
4
ABCDVis a pyTamid with a square base of sides 5 cm.
VO is vertical and VC = 13 cm.
Find the volume of the pyramid, con'ccl yom answer to 2 decimal places.
C'~Uffl/O:C'fl- JU'•
c
A
5cm
Answer .. ................................. cm:I [4]
6
Solve lhc s imultaneous equations x + .!.. y
2
=9 and
3x - 2y = 13.
Answer x = .................... y = ................... (3]
Chung Cheng lfigh School (Main)
End of Year Exam 2011
Math< Paper I
f'o•
s
F.,
('Yfllllinr.t~liJ('
7
trr11,,fort'1 11..11t
Two cars leave a point at the same time and travel in opposite directions in a
straight route. The speed of the faster car is 12 kmil1 more than the slower car.
At the end of 1.!.days. the two cars are 5616 km apart.
2
If the speeds of the faster and the slower cars arc x kin/h and y km/h respcclivcly,
form two equations involving x and y. Hence, solve the pair of simultaneous
equation and find the speeds of the 2 cars.
Answer Equation I: .......................................... [I]
Equation } ........................................... [I J
Speed o.ffaster car: .....................km/h Speed ofslower car: ..................... km/h [3]
C/1;mi: Clieng Nfgh Sclwol (Main)
End <ifYear Exam WI I
Ma1l1s Paper I
,...
,.~.:."'"
6
8
Express 84 and 280 as the product of its pri me factors in index notation.
If 84k is a multiple o f 280, fi nd the smallest possible integer, k.
Answer 84 = ......................................... (I]
:uw -......................................... (1]
k ~ ........................................
9
LI]
In 20 I I, there is a 15% decrease in the enrolment of a schoo l as compared to
in 2010. Gi ven that the enrolment of the school in 2011 is 1105.
Calculate
(a) the enrolment in 2010,
(h) the percentage increase in enrolment from 2011 lo 2012 if t.hc school forecasts
that the enrolm.:nt in 201 2 is 1326.
Answer.{a) ..................................................... [ 1]
(b) ..............-....................................% (1]
Chung Cheng lligh School (Main)
End of Year &am 1011
Maths Paper I
,..,
,.,.
7
r m1>1,.,-tr's 1aor
JO Makcx the subject of the fomwla y ~
...,, .:tr'J
Jx'- y
-Tx
llencc find the value(~) ofx when y = 6 and z = 4 .
Answer x = ................................................... LJ ]
x
11
= ... ...... ................ .... .... ......... .. ... ....
12J
Find lht: value o f an interior angle of a regular octagon .
A11.rwer: interior angle =
Chung Cheng Iligh Schuul (Main)
.................... ......... ....... ...... 0
End of Year £wm 2011
(2)
Maths Paper I
use
FN
tr.,mi11~r\
•
8
1de
J2
(a) Shade the following sets in the respective Venn <lia!,'Tams below.
(i) ArlB'
c.--~~~~~~~~~~~~
An.S\\ rr:
1
(11
(ii)
(A" B')u(A 'fl 8)
&
I IJ
(b) Write down the set Lhal represent the following Venn diagram.
(i)
&
(ii)
Answer {b){i) .................................................. (I J
{b){ii) ....................... ........................... [ I I
Chung Ch,.ng lligh s,hool (Main)
/•(H
' l'Jftl!l'mir-r:, 1.'je'
Endu/Year Exam 2011
9
The table below shows the numhcr of cars per household.
13
[No. nf cars
No. of household
J
I0
6
- 18
_±±-+I~
(a) If the median number of cars is I, find the largest possible value of x.
(b) Ifx w 5.
(i) state the modal number of cars,
(ii) find the prohability of choosing.a household with at least 2 ~rs.
Answer {a) x ""' .......................................,....... [ 11
(b)(i) .................................................. LI J
(b)(ii) ·················································· f2 J
14
The grouped frequency table hclow shows the height of 15 plants in an
experiment.
Complete the table hclow.
~ght ( H cm)
-1--- -M_i_
d ·_,_
,a_lu_e......_x..__-+- - -'--- '-"-'---t---
h~ ~ z ~;~
-
-
W<H S 80
__..f!.._x_
~
50<H -S 60
- - -1--
~cll S ~
(2)
--lf --
-
-
6
3
2
Find the mean height of the 15 plants.
Answer .................................................cm [I J
E11d of Paper
Clrung Cheng lligh School (Mam)
Em! of Year Exam }OJ/
,'i,farhs Paper I
Class:
I
Class Register N- ; -
50
CHUNG CHENG HIGH SCHOOL
(MAJN)
Parent's
Signature
Ct>ung Chong .High School Chang. Ctierm Hig!\~Ch091 C.flu'.'9 Ch<ino i:ifgh
.•Chung Cheng High SctliXil Ctf~ng'Cl~~ng Hig~ S?,lfo1:qH,Ung:Cti~'11if!aifi'
Chung Cheng High School Cl)l!!l!J C~ong High~ ChUhg Chong 1119~:
Chung Cheng High School Chung Cheng HigH School Chung Cheng Hlgh s.
Mathematics
Paper1
END-OF-YEAR EXAMINATION 2011
SECONDARY 2
Tuesday 11 October 2011
1 hour 15 minutes
Instructions to Candidates:
1. Write your name. class and register number in the space provided on the cover page
of this booklet.
2. Write only in dark blue or black pen.
3. You may use a pencil for any diagrams or graphs.
4. Do not use staples, paper clips, highlighters, glue. correction fluid or tape.
5. If working is needed for any question it must be shown with the answer.
6. If additional papers are used, fasten the sheets together with the question paper.
7. If the deg ree of accuracy is not specified in the q uestion, and if the answer is not
exact. give the answer lo three significant figures. Give answers in degrees to one
decimal place.
Information For Candidates :
CALCULATORS ARE ALLOWED FOR THIS PAPER.
OMISSION OF ESSENTIAL WORKIN G Will RESULT IN LOSS OF MARKS.
The intended marks for the question or parts of the question are given in brackets ( ).
The tolal number of marks for this paper is 50.
You are reminded of the need for clear presentati on in your answers.
This question paper consists of 9 printed pages (including this cover page).
F<w
/'():·
('!fffllli11.~1· \· l\f
t'.~OIJll!otf'$ JoSt
Answe1· a ll the questions.
1
Given that y is inversely proportional to (x+ I), and y = I when x = 3,
(a) cxprcssy in tcnns ofx,
(b) findthevalueofywhenx - -17.
k
(a) y = - - - , where k is a non zero constant
(x + I)
Givcny = I when x = 3,
l= - k -
(3+1)
k=4
4
. y= - -
.·
(x+ 1)
(b) whenx = -17,
4
y= -
(- 17+1)
l
y = --
4
4
Answer {a) y-.... .... y = - - .................... [2j :
(x + t)
I
I
{b)y= .. ......... y= - - ................. (I)
4
A moving object travels s metres in t seconds. It is given thats is directly
proportional to 12 . Express s in tenns oft and k, where k is a non zero constant.
If the object travels 75 metres when t ~ Q seconds, find the new value ofs when
1 = 4Q seconds.
s = kr', where k is a non zero constant
I
If the object travels 75 metres when t = Q seconds
75= kQ'
! new value ofs when t = 4Qseconds = k(4Q) 2
: l6kQ 2
- 16(75)
= 1200
Answer {a) s= .......... kt 2 .......... .. ...... [I)
{b)s""........ 1200.. :................. (2)
Clr11ng Cheng High School (Main)
Eml of Yet1r E:wm 1011
Maths Paper I
r.,
J
t'Yf1111ii-u:r's 11Se
3
Simplify
(a)
(a ~ 2b)(a - 2b) 4b 1 +4ab ,
(h)
(p + q)(p ~ q)
(p-q)-
+- '
(q-p)
(a+2b)(a-2b) - 4b2 ~ 4ab
(a)
= a 1 - (2b)' - 4b2 + 4ab
=a 2 - 4b 1 - 4b 2 +4ab
= a 1 - 8b 2 +4ab
(b) (p~ q)(p-q)
+- '-
(q-p)
(p-q)l
=(p+q><e-q2x(q -
p)
(p-q)I
= (p+q)(p-q)x [- (p - q)]
(p - q)I
=-(p ~q)
ur - p - q
Answer (a) .......... a' - 8h 1 + 4ab ......... ................ (2)
p - q .-................
12f
Answer (a) .................... (5x - J)(Sx + I) ................................
l l)
{b) ......... - (p I q)
4
0/
-
factorise com pletely
(a) 25x1 - 1,
(b) 3x+6y - x 2 - 2xy,
(c)
6 - 15x - 36x2 •
: (a) 25.l' - I - (5x)' - I
= (5x - l)(5x4 I)
(b) 3_, +6y x 1 -2Ay = J(x+2y)- <{t+2y)
=(J-:r)(x+2y)
2
1
(c) 6 15x - 36x =3(2 - 5x - 12x )
d(3:r 1- 2)(1 -4x)
(b) ................... (3 - x)(x + 2y) ........-..... _.................. (2)
(c) .................... (3x
Chung Chen/( Higlr & hoof (Main)
+ 2)(1 - 4xl ................................. 121
F.1ulo/Year Exam 101 I
1l-fa1Ja l'aprr I
I
f'c:f
t'UWJ1•1tf'f 'J
use
4
ABCDV is a pyramid with a square base of sides 5 cm.
VO is vertical and VC: = 13 cm.
Find the volume of the pyramid, correct your answer to 2 decimal
5
AC -
Js
AC =
J50 cm
1
place~.
+5 2
132 -[(~)JSO
VO =
/'(J•
t'.1.1m1••t'• :S- 11fr
r
c
VO - J156.5 cm
A
5cm
Therefore,
Volume of the pyramid
~
.!.x(5x5) x Jt56.5
=
104.25 cm3 (corr.to 2dp)
3
Answer ......... 104.25 .......................... cm 3 f4l
Solve the simultancot1s equations x
6
XI
I
2
y=9
1
x = 9 - 2 y··········· ( 1)
3x- 2y = 13 ........... (?.)
Sub ( I) into (2),
3(9 -
I
y) - 2y =d3
2
3
27 - - y - 2j>=l3
2
3
I
2
I
2
y = 9 and 3x 2y = 13 .
Method2:
1
x+ - y = 9
- (1)
2
3x 2y = l 3 - - - -(2)
Multiply (I) throughout by 4
4x+2y = 36 - - - -(3)
(I) i (2)
7x = 49
x=7
y = l4
y=4
Suby "' 4 into(!),
x=9
t
~ (4)
2
sub x = ?into (I)
4(7)12y =36
2y=8
y = II
:.x 7.y = 4
x=?
:.x = 7,y=4
Answer x = ...........? ......... y = ............4....... [3]
Clumg Cheng Higlr Sclrool (Mllin)
f:.'111/ tifYeor F.tam 2011
;Wat/rs l't1per I
II
. ......·~
,_,...·, .._,.,.
s
r..
Pf«lfll!tj(T'S l.:,f~
7
Two cars leave a point at the same tim e and trave l in opposite directions in a
str111~ht route. The speed of the faster car is 12 km/h more than the slower car.
I
At the end o f 1 days, the two cars are 56 16 km apart.
2
If the s1>eeds of the laster and the slower cars are x km/h and y km/h respectively,
fonn two equations involving x and y. Hence, solve the pair of simullanco\ls ·
equation and find the speeds of the 2 cars.
x - y = 12 - - - -(!)
l
. l
Numbers o f hours m I- days = I- x 24
2
2
=36
36x~ 36y=5616
x+ y = 156 - - - -(2)
From (1),
x = l 2+ y -- - - (3)
sub (3) into (2)
12+ y
I y~
156
2y d44
y =72
sub y = 72 into (3),
x
12 -~
72
.r 84
T herefore the speed of the faster car is 84km/h and the s lower car is 72km/h.
Answer equation/: ......... x- y
.,
=12 ................................. (I]
equation 2: ........... x+ y = 156 ............................... (I]
Speed n/faster car-........84 ............ kmlh Speed ofslower car: ..........72........... kmlh [3]
Chung Cheflg High Schon/ (Muin)
Emlo/YcarExmn 2011
Mmhs Paper I
8
6
Express 84 and 280 as the product of its prime factors in index notation.
lf84k is a multiple of280. find the smallest possible integer, k.
Least possible value o f k = 2 x 5
= 10
Answer 84 = ........... 2zx3x7 .............................. [l l
280 - ....... 2 3 x5x7 ................................. [I J
k= ................ 10.,...........................
9
[ll l
In 2011, there is a I 5% decrease in the enrolment of a school as compared to
in 2010. Given that the enrolment of the schoo l in 20 1lis 11 05.
Calculate
(a) the enrolment in 2010,
(b) the percentage increase in enrolment from 2011 to 2012 if the school forecasts
that the enrolment in 2012 is 1326.
(a)
.
Enrolment 111 20 I 0 -
1105
85
x IOO
- 1300
(b)
Percentage increase from 20 11 to 2012 =
1326 1
- l OS x 100%
1105
=20%
Answer.(a) .................... 1300................................. [ l]
{bJ ..................... 20..............................% [I I
Chung Cheng lligh School (Ma111)
End of Year Exam 2011
•
Maths Paper I
I
F•'
Pr.-r
7
rt:tt.milla 'J ,.,,.
10
l' ..Olt.';.,,.,.\-1(\.o'
3.t'z - y
Make x lhe subjecl ofche fonnula y
x~
Hence find the value(s) of x when y = Ii and z = 4.
3x'z-y
y ~ -
-
xl
x'y=3x 2 z-y
3x1 z-x2y = y
x 2 (3z-y) = y
·±rY
x
v~
Wheny =
6,z - 4,
x=±5t~~-6
x=
L~
x=±I
Answer x
~ ...................... ±~ )z~ y .............................
(3]
x = ....................... ±1............................ [21
ll
Find the value or an interior angle of a regular occagon.
(8 - :>)I X0°
interior angle of a regular octagon
S
= 135°
Answer: Interior angle= ................ 135..........................
Chung lheng ll1xh School (Main)
End n/ l'e<ir EJ:am 101 I
0
[2]
Matl1s />aper I
..
,
; '"''".. ·r·r ;t~t'
12
FM
8
(a) Shade the following sets in the respective Venn diat,'l'ams helow.
(i) Af\B'
rt.1m~""r'l 1:.w
& ~~~~~~~~~~~-
Ans•wtr:
( I]
(i i) (Af\B)u(A'nB)
£
II I
(b) Write down the set that represent the following Verm diagram.
(i)
£
(ii)
Answer{b){i) ........... BnA ' OR (An B)'nB ........(I]
(b)(ii) ................. Bv A' ................................. 11]
Chung Cheng Hlgh School (Main)
End of )'ear Exam 2011
Maths Poper I
/>'()..
r~arntittr'1 rlfl"
lJ
r.,.
9
T he table helow shows the number o f cars per household.
I
~fears
, ,,~
3
11
0
~ f household ...,c..6_-_-_-_-_-- -;-'-_o..5'-----_-_-_-..J+.-'='--
-
I?
"-~-
-'-·
,. ........ ' '; toJt
14
- -
_J
--'·--'--1
(a) Jf the median number of cars is I , find the largest possible value o f x.
(b) l f r = 5,
(i) state the moda l number of cars,
(ii) find the probability of choosing a household with al least 2 car.<.
(a}
6 + 4 = 3 +x+I
x
6
(b) Mode - 0 numbers of ca rs
(c) P(choosing a household with at least 2 cars)
3+5+ 1
20
9
=-
20
I
Amwer (aj x - ........6 ...................................... . [ I i
I
I
(b)(i) ...................0..............................
(b}(ii) ...................
14
The grouped frequency tab le below shows the height o f 15 plants
experiment.
Complete the table below.
111
Height ( H_ c_m_,_)_ -+-- Mid-value (x_,_
) -~-Frequency(/)
35
- -+--- - __
3_ _ _ .
30 < fl ~ -'-40
-+40 < II < 50
45
I
50
<
li
$ 60
55
6
- 65
3
60 < II $ 70
-'--- -1-70 < }{ $ 80
75
_ _ _ 2
15
'I
1·1. 1 I
_2_ ............................
20
F
-
f!
an
Pl
(x -
-
io5 _ _ _
45
330
195
- - -·
- -
_.J2Q_
825
- -
l
·-
Find the mean height o f the 15 plants.
.
825
mean height of the 15 plants = -
15
= 55 cm
Answer .................. 55...............................cm (I)
End of Pa per
Clrung ChPng lligh School (Ma111)
£ nd of Year Exam 2011
Maths Parwr I
I Class: _ _ _ _._
IC
_ lass Register No: -
-
---l
50
CHUNG CHENG HIGH SCHOOL
(MAIN)
Parent's
Signature
OlUng ~High Schoo(Cllung ~High SchOol Cl1Uriii Cheflg High sch:io.I Choog Chlinn.H[gh School"·'' '"
Chung Chen!JHigh Scllool€hooQ clieng HiQh School ~C~.High'. SCtlOOlC!ll.ng~l;j iiigh ~:
Ch)!ng ~ Hjgh School C:hurl!J.Chjirig High SchOOI y!!uii!i.\:~ l;!igh
Chung~
SdlOoi ;;; •
Chung'clieni.Bigh School Chung Che/lg High SchoOI ChiJ!lli C~ 11i9fi"s¢ti00r Chung~Cti'eng 'ttigli SCtfOOf;;:<!,-:
SctO:.f
Mathematics
Paper 2
8'9h
:=-;
END OF YEAR EXAMINATION 2011
SECONDARY2
Thursday 6 October 2011
1 hour 15 minutes
f/,iStructions to Candidates:
1. Write your name. class and register number in the spaces at the top of the page and on all
the work you hand in.
2. Write only in dark blue or black pen on both sides of the paper.
3. Start each question on a new page.
'1. You may use a pencil for any diagrams or graphs.
5. Do not use staples, paper dips, highlighters. glue, correction fluid or tape.
6. If working is needed for any question it must be shown with the answer.
7. Fa st en all the answer sheets together with the question paper.
8 . If the degree of accuracy is not specified in thP. 'JllP.stion, and if the answer is not exact.
give the answer to three significant figures. Give answers m degrees to one decim:il
place.
9. For ;r. use either your calculator value or 3.142, unless the question requires the answer
in terms of 1f .
I
i
Information For Candidates:
CALCULATORS ARE ALLOWED FOR THIS PAPER.
OMISSION OF ESSENTIAL WORKING WILL RESULT IN LOSS OF MARKS.
The intended marks for the question or parts of the question are given in brackets [ ].
The total number of marks for this paper is 50.
You are reminded of the need for clear presentation in your answers.
is_question papet.oonsists of 4 printed pages (including this cover page).
_ _ _ T'"'h""
Answer ALL the questions.
In the figure, IMPQ is s im ilar to.MCB.
AP ~
5 cm. BC ~ 4 cm, AQ- 3.5 cm, QC= 6.5 cm aml OC = 3 cm.
c
p
:\ crn
2
.d
()
B
(a)
f ind the lengths of /'Q and AO.
I11 J
(h)
Find the area of &!Cll .
11 !
A class of JO children took a Mathematics TesL
Their m arks are given below:
7
37
14
25
21
45
9
49
(a}
47
16
11
18
23
38
1.51
20 35 42
14
30-----~
45 42 3:5
39 27 40
2o..;8;.___4-'l - '->
-'5;.__~19
Construct a single ordered stem and leaf diagram to represent the marks of all
30 children. Part oftht: diagram has l:H..>cn drawn for you. Copy and compl ete
the diagram.
Stem
Pl
Leaf
0
7
9
I
2
3
4
Key:
314 means 34
[21
(b)
Find the modal mark and the median mark of the c la:>s.
(c)
The pass mark for the paper was 25 marks. Expressing the answer as a
fraction in its simplest form, calculate the probability that a child chosen at
121
random from the class passed the test.
CC/IMS Sec 1 Exprr"'.t F.nd-OfYear Examination 1011
2
3
The price of rice in Singapore was x cents p.,;r kg in January 2011. In September
20 I I the price has increased by 20 cents per kg.
·
(a)
Jack Catering bought $102 of rice in January 2011. Express the amount of rice
bought in kg in tenns of.r.
(h}
How many kg of rice could be bought by Jack Catering with $102 in
September 20 I I? Express your answer in terms of x.
(c}
4
[11
If the difference in 1.hc number of kg of rice oought in January and September
is 8, form
(d)
(I]
an equation in x and show that it reduces to x 2 + 20x - 25500 = O.
Solve the equation in (c) and use it to find the number of kg of rice that could
he bought in September 20 I I.
[3]
[31
Answer the whole of this question on " sheet of graph paper.
The variables x and y are connected by the equation
y =4- 2x I x 2 •
Svme corresponding values of.< and y arc given in the following table.
(11
(a)
Find the val ue u[p.
(h)
Using 2 cm to represent I unit on the x-axis and I cm to represent l unit
0 11
the
1
y-aitis, draw the graph of y = 4 - 2x + x for - 3 S: x $ 4. On your axes, plot the
points given in the tab le and join them with a smooth curve.
(c)
From I.he graph,
(i) find the minimum value of the curve.
(ii) write down the equation of the line of symmetry.
(iii} tind the two solutions
(d)
(i)
or 4 - 2x+x
2
=8 .
On the same axes, draw the graph of y = x + 4 .
[31
ri J
(I]
(2]
[I]
(ii} Hence, write down the coordinates of the points when y = x + 4
intersects y = 4 -2x+ x'.
CCHMS Serl Express l:.rul-Of-Year Exuminmion 101 I
(2]
3
5
The object below consists of a solid cylinder with a solid conical top and a hollow
hemispherical base. The conical top has a ve1tical height of 8 cm and a slant height
of 10 cm. The cylinder has a height of 12 cm and ·the hemispherical base !tas a
radius of r cm.
(a)
Show that the value ofr is 6.
[2]
(b)
Find the volume oftbe object in terms offf .
[4]
(c)
Find the total surface area of the object in
t.erms of ;r.
[4]
-6
(a)
The description of set A and set Bare given below:
A = {x: xis art integer s uch that x 2 <IO}
B = {x: - 5 < x < 4 and xis a natural number)
List the clements contained in each of the sets above.
(b)
[2]
It is given that:
c = {l,2,3,4,5,6, 7,8,9,10}
E = {2,4,6,8}
S = {I. 4, 9}
(i)
(c)
Draw a clearly labelled Venn diagram to illustrnte the relationship
between 1:, £and S.
(ii)
List the elements contained in Eu S.
(iii)
Find n(E' n S).
[2 J
l (J
111
In a survey, JOO children were asked to name their colour preferences.
The following results were obtained :
61 prefeffed Blue, 55 prefeffed Red and 7 did not prefer both Blue and Red.
(i)
(ii)
Let R be the set of children who prefer Red and B be the set of children
who prefer Blue, draw a clearly labelled Venn diagram to represent the
above infom1ation.
If a child is selected randomly from the above, what is the probability
of selectu1g one who likes Red only.
[2)
[I]
- - End of Paper- CCI/MS Sec 2 E."f>rc.<s End-Of-Yeor £xami1101io11 1011
4
IClass:
j Name:_ _ _
IClass Register No:
lV ~ "' ii----'--
~
-50
CHUNG CHENG HIGH SCHOOL
{MAIN)
Parent'-s Signature
.
Chung Cheog H"igh School Chung Chong High Sc:l\ool Chung Cheng High SchOOI Chung Cheng High Sd'lool
Chung Cheog Higll School Chung Cheng High Sc:l\ool Chung Cheng High School Chung ~ Hig11 SchOol
.. Chung Cheng High $diool Chung Chong High School Chung Cheng High School Chul)lj Ch~ !i'9h Sctioo'
·Chung Cheng High School Chung Cheng High School Chung Cheng High SchOol C~ung Cheng H"igh Schoof
Math ematics
Paper 2
END OF YEAR EXAMINATION 2011
SECONDARY 2
Thurs d ay 6 October 2011
1 hour 15 minutes
/nstrUCtions to Candidates:
1. Write your name, class and register number in the spaces at the top of the page and on
the work you hand in.
0 11
2. Write only in dark blue or black pen on both sides of the paper.
3. Start each question on a new page.
4. You may use a pencil for any diagrams or graphs.
5. Do not use staples, paper dips, highlighters, glue, correction fluid or tape.
6 . If working is needed for any question it must be shown wrth the answer.
7. Fasten all the answer sheets together with the question paper.
6. If the degree of occuracy is not specified in the question, :;in<I if lhe answer is not exact,
give the answer to three significant figures. Give answers in degrees to one decimal
place. ·
9. Four. use either your calculator value or 3.142, unless the question requires the answer
in terms of 1t .
Information For Candidates:
CALCULATORS A RE ALLOWED FOR THIS PA PER.
OMISSION OF ESSENTIAL WORKING WILL RESULT IN LOSS OF MARKS.
The intended marks for the question or parts of the question are given in brackets [ ).
The total number of marks for this paper is 50.
You are reminded of the need for clear presentation in your answers.
. __
This question paper consists of 4 printed pages (including this cover page~)_
Answer A l ,L the questions.
In rhe figure, AAPQ is similar tot.ACS .
AP = 5 cm, BC = 4 cm. AQ = 3.5 cm, QC = 6.5 cm and OC - 3 cm .
(a)
(4)
Find the lengths of PQ and AB.
Length of PQ
.
PQ AP
.
.
- ···- = -- (Rauos of corr. sides are equal)
CB AC
PQ 5
-
=10
4
5x4
PQ = .
10
PQ=2crn
Length of AB
. o,r corr. sic· 1es are equaI )
AB = AC(R
-- auos
1
AQ
AB
-
3.5
AP
10
=-
5
AB = IOx3.5
5
AB =7cm
(b)
Find the area of MCB.
(I )
1
Area oft.ABC = - x7 x 3
.
2
= 10.5cm1
2 A class of30 children took a Mathematics Test.
Their marks are given below:
1;,
37
45
(a)
Construe! a single ordered stem and leaf diagram to represent the marks of all
30 children. Part of rhe diagram has been drawn for you. Copy and complete the diagram.
25
9_ '!_9
14
47
16
11
23
18
38
20
35
JO
45
CCHMS Sec l Express End-OJ: Ycar F.xllminulion 201 J
42
42
14
35
39
28
27
41
~~-~_;;_I
[3)
2
Stem Leaf
0
7
I
I
2
0
3
0
4
0
(b)
9
4
4
5
6
I
3
5
7
5
5
2
5
7
5
2
8
8
9
8
9
9
5
7
Key: 314 means 34
Find the modal mark and the median mark of the class
(b) Modal score =35 marks
[2]
.
28+30
Median score = ·
2
• 29 marks
(c)
The pass mark for the paper was 25 marks. Expressing the answer as a
fraction in its si mplest fonn, calculate the probability that a child d1osen at
rJndom from the class passed the Lest.
(c) P(choosing a child who pass)=
(2]
~~
3
5
J
The price of rice in Singapore wa.~ x cents per kg in January 2011. In September 2011 the price has
increased by 20 cents per kg.
(a)
J:lck Catering bought $102 of rice in January 20 I I. Express the amount of rice
bought in kg in tcnns of.;,
[lj
. bought .m January 20 11 = -I 0200
Amowit. o f nee
- kg.
x
(h)
How many kg of 1ice could be b<Jught by Jack Catering with SI 02 in
September 2011? Expres.~ your answer in teons of x.
10200
Amount of rice bought in September 2011 kg.
x+20
(c)
If the difference in the number of kg of rice bought in January and September
[1)
(3)
is 8, fonn an equation in x and show that it reduces to x' + 20x - 25500 = 0.
CCHMS Sec 2 Express Et1d-Of-Year Elamlnarion 201 I
3
~ - 10200 = 8
x
x+20
(x+20)l0200 - l0200x =8x(x + 20)
I 0200x + 204000 - I 0200.r = Kx1 + I60x
8x 2 + I60x - 204000 = 0
x' + 20x - 25500 =0
(d)
Solve the equation in (c) and use it to find the number of kg of rice that could
be bought in September 20 I I.
(3)
x' + 20x - 25500='0
(x+ 170)(x-150) = O
x= - 110(N.A) or x=l50
The amount of rice bought in Sept 2011 =
12
o oo_
150+20
=60 kg.
4
Answer the whole of tf1is questiort on a sheet of grap h paper.
The variables x and y are connected by the equation
y = 4 - 2x+x2 .
Some corresponding values of x and y are given in the following table.
(a)
(b)
Find the value ofp.
(I)
Using 2. cm to represent I unit on the x-axis and I cm to represent l unit on the
y-axis, dmw the graph of y = 4- 2x + x' for -3 $ x $ 4. On your axes, plot the
points given in the table and join them with a smooth curve.
· (c)
From the graph,
(i)
(ii)
find the minimum value of the curve.
write down the equation of the line of symmetry.
(iii) find the two solutions of 4 - 2x + x 1 = 8 .
(d)
(i)
On the same axes, draw the graph of y = x + 4.
(3)
[IJ
(l l
[2]
(I)
(ii) Hence, write down the coordinates of the points when y = x + 4
intersects y = 4 - 2x + x'.
CCf/MS Sec 2 £xp1·ess End·Of: Year faamination 2011
(2)
4
lndex No
.
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£'"'"·" J::mf·Of-YPOr Examination 1011
.
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5
5 The obj(x;t below consists of a solid cylinder with a solid conical top and a hollow hemispherical
base. The conical top has a vertical height of 8 cm and a slant height of I 0 cm. The cylinder has a
height of 12 cm and the hemispherical base has a radios or r cm.
(a)
Show that the value of r is 6.
(2)
r = JI0 1 - 81
r=J36
r - 6 (SJrow11)
(b)
Find the volume of the object in terms of,,..
Volume of object s Vol. of cone+ Vol. of cylinder- Vol. of hemisphere
I
,
= - x.irx62 x8+.irx6-xl2 -
3
= 9611" cm 1 +432Jr cm 3
-
I
4
- x - x.irx6
f4)
i
2 3
144·.ir· cm1
= 1f(96 + 432-144)
=384'1'Clll
1
Find the total surface area of the object in tenns of .ir.
Surface area of object '- S urface area of cone+ Surface area of cylinder
Surface area of hemisphere
(c)
f4]
I
= ,T x 6x 10+ 2.irx6xl2 + - x4.ir x6 1
2
= 60.ir + I 441f + 72.ir
276ncm'
6 (a)
The description or sci A w1d set B are given below:
A = {x: xis an int eger suc h that x' < 10)
8 = {x: - 5 < x < 4 and xis a natural number}
List the clements contained in each of the sets above.
A= {-3,-2, l.O,l,2,3)
B = {l,2,3}
(b)
l2]
It is given that:
= {1,2,3,4,5,6, 7,8,9,10)
c
E = {2,4,6,8)
s = {1,4,9}
CCHMS.Sec l Express F.nd-0/-Year Etamination 1011
6
(c)
[2]
(i)
Draw a clearly lahellcd Ve nn diag ram to illustrate the relationship
hctwecn c , J:: and S .
(ii)
List the e lements contained in Ev S.
EvS = {1, 2,4,6,8,9}
fl]
(ii i)
Find 11(t:' n S).
11(E' n S) = 2
[l )
ln a survey, 100 children were as ked to name their colour preferences.
The following results were obtained:
6 1 prefe recl Blue, 55 prcfe red Red and 7 did not p refe r hoth Blue and Red.
(i)
Let R be the set o f c hildren who like Red and B be the set o f children
who like Blue, draw a clearly labelled Veru1 d iagra m to re present the
nhovc information.
(2]
n(R r~ ll)
= 11(R) + 11(B)-11(H v B)
= 55 161 - (100-7)
55+6 1- 93
= 23
(ii)
If a c hild is selected rando mly from the above, what is the probability
o f selecting one who likes Red only.
P(sclccting one who lik es Red only)
[I]
=E
100
8
25
- - End of Paper·- -
CCHMS
s..c 1 t:xpr<'lJ
F.nd-Of-l'et1r Ext1minotitm 1011
7
Index Number:
Class: _ __
Clementi Town Secondary School
End-of-Year Examination 2011
Secondary 2 Express
Mathematics
Paper1
1 hour
Additional Materials provided : Writing Paper for rough working
CUMfMll TOWN~ SCHOOl CltMENll TOWN SE~ SCHOOt. Cltf.IE:tm T(MH S«:.CJINDif,HY$c>IOOlClfMENTI T~ $CCOfC)f.RY SOC'W'll
Q.EMC.NTI f0W... SECONDARY SCHOOL ClEMEHTI TOWN SECONOAAY SCHOO&. CLEMCNTI TOWN &f(;.OMOARY SCH)()l ClFMENTt TOWN SE.CONOAA't'SC>!Of'M
CUMENTI TOWN !KC,()NOARY $CU00l CLE.a.tfNll TOWN SECONOAAY $CU00l CLEMENTI J()WN $C~Y SCHOOl. tlEMENTI TOWN SECONDARY ~C:..11'104
ClEMENTI TOWN SECONDARY SCH00l, Ct.EMt.NTt TOWN SECONOAAY SCllOOl Ct EMGNTI TOWN &t(.0NDARY SCtlOOI. Clt:Mi=~ll TOWN SCCONUAHY $f :1 !tiCI.
CLEME.Nl 1l0WN SECONOJ\lh' SCllOOL ClE.Ut:N1 I f()\VN SCCONOAR.Y $Ctl()()t. CU,.1t;f" tl 1()1//N S.CCONOAAY SCHOOL ClCa.iCNltTOWN S..('.OND.AAY SCHU(.)1
READ THESE INSTRUCTIONS FIRST
Do not open the booklets until you are told lo do so.
Write your name. register number and dass on all the work you hand in.
Write in dark blue or black pen on both sides of the answer paper.
Do not use staples, paper clips, highlighters, glue or correction fluid.
Answer all the questions.
Write your answers in the spaces provided on the question paper.
If working is needed for any question it must be shown with the answer.
Omission of essential working will result in loss of marks.
CALCULATORS ARE NOT ALLOWED TO BE USED IN THIS PAPER.
The number of marks is given in brackets [ ) at the end of each question or part question.
The total number of marks for this paper is 50.
FOR EXAMINER'S
USE
This question paper consists of 11 printed pages. including this cover page.
(Turn over]
2
C"lcmcft•• ·rown Scconcbry '-'""''
f'.nd-of~Ycar Ex<1m1natW'ln ?01 1
AI1swcr all the questi ons in the spaces provided .
Factorise the fo llowing expressions completely
(a)
(b )
2
3x1 -2x - 21,
Answer
(1]
."1ns•1'£1r
(2)
4x'+2xz-2xy-yz.
Express the following as a single fraction in its lowest terms
p(p - q) . q - p
- 4p
~ (2p) 2
•
Answer
Pl
3
C'lcmen1i To1,1,11 Secondary S.:-hool
l:nd~f Vt:¥ J.:umina1ioo 2011
3
.t
Solve
2
3
\-fatbrtM<ics; Pat'ltt I
x - 1\
5x
I.
Answer
x - .. . . . . . . . . . . . . . ...........
Anslver
1n
[3]
Solve the simul!aneous equations.
3m i
11
+ I = 9,
II
m + - - 1= 9 .
2
= ..........., 11
{3]
4
(' le111ot11f 1 T,)Wn
t.·bthnna11cs I
S
Sooonduy School
F,nd-6f·YcM FJtaminstion 2011
r~-ct I
Make c the su~jcct of the fonnula
IOc
c= - 5.
k
c - ... .. .. ... . . . . . . .. ...... ....
Answer
6
[3]
In a class of 35 student~. 18 play ba.~ketball only, l 0 play rugby only and 7 play both
basketball and rugby.
When a student is selected at random, find the probability that
(a)
the student plays both basketball and rugby,
(b)
the student plays only rugby,
(c)
the student plays haskctball.
Ans>n<r
(a)
.... ....... ................ [I]
(b)
.......... ... .... .......... [lj
(c)
............... ............ (I)
5
ClcmC'nC1 Town Sccond:..r)' ~ hnol
_F._
nd_·•_r._v_
.,_,_F.x_•_m_;._,,_;o_n_2•_
ll_I_ __ __ _ __ _
s_cc_·•-""-".;.
Y_
2_E~
•p<~c-ss___ ----------~1a1hern.-c1cs / rap<r 1
On a map, 3 cm represents 6. 9 km on actual ground . ·
7
(a)
Express the scale of tJ1e map in the form I : 11.
(b)
Calculate the actual distance, in km, represented by 5 cm on the map.
Answer
(bj
8
l2J
(a)
.................... . km
[2]
It is given that y is directly proportio nal to x' and that when x = 2, y - 4.
Find
(a)
the e4uation conn.:ctingx and y,
(h)
the valucofy when x - - l,
(c)
the value ofx wheny - 32.
Answer
{a)
........................ [21
(bj
y = .........................
fl J
{c)
x ~
.........................
fl J
6
C'km<nll
.;_l>_;l••::..""':...""cc'...;."";...'_P•-'--'' _ __
9
_ __
Thegraphof y = (<
_ _ _ _.;_S-.....;.;...;...IM)..:.·.;;.2--£...;:.P'=..;.;;;._ _ _ __
t
_ __
rC)....'ll Sccoodary Schonl
_;l:;;;
Pd>G(•Vnt Eumicui1ioo 2011
2)(4 - x)cutsthex-axisaLAandB.
It cuts the y-axis at C .
Write down
(a)
the coordinates of A and of B,
(b}
the coordinates of C,
(c)
the equation of the line of symmetry.
- . i . - - - - , - + -- -- - - - - '.......... .t
A
Answer
0
B
A(. . ....... . ' ..........)
[I)
B( ......... . , ..........)
[I)
(b)
c (. ·······.. ' ..........)
[I l
(c)
1•11o• • • t• •••• ••• ••• • H• • ·•
[I]
(a)
7
('lcm1..~n1i
T<>w•l Sccnnd11ry c;('hfl1ll
End-0f· "f'e;a, Examinn1i-0n 11.::11.;_1_ _ _ __
10
(a)
'----- - -- -- - - - -•..::1>..c'''..c'"..c".::'"::.'...:''.;_':1::;p~~.::.::
'I
_ __ _·.::
'•.::'".:.."<i;;;.
" -'"
'Y.:.
? .:..
F.C!...:p.::""·'.::
In the following figure, Mil<.: is congruent 1o·AXYz.
AB - 8.1 cm, BC= 6.5 cm, CZ= 8.2 cm andAY = 2 cm.
Find the length of
(i)
c
AZ,
(ii) Xl.
6.5cm
B
~2
x
A11s1Ver
(h)
(a}(i ) ........... ............... cm
[I]
(a)(i i ) ....................... ... cm
[I]
In the figure, BC = 15 cm, PQ ~ 6 cm, AP ~ 7 cm and AQ - 6.5 cm.
Ci iven that 6 ABC is similar to/\ Al'Q, calculate lhc Ieng.th of Bl'.
B
---
15 cm
R
A
A11swer
(b)
. ...... ................ cm f2]
8
M;1lh(;ma1iq;:
U
P..tv~:•
1 _ _ _ _ __
(a)
_
Clem('nli l uwn Se"._,,,,1,.1y Schol•I
_:S<:::":.;;o;;;,
•d::.
''Y
=l:...F""f"....:"'
:::__ _ _ _ __ _ __:F.:::'":..
h::.:>f...:~....:'ea::.'.::C': ::':::
"';;;,i
n::::
•li::.'
";..:
' l'-"0'11
_ __
The fol lowing set of data shows the average number of hours spent on Facebook
by a group of20 Secondary 2 students in a week during the holidays.
(i)
Ans~vcr
5
2
9
10
14
s
6
23
21
12
7
12
30
15
8
13
19
12
5
18
Complete the following stern-and-leaf diagram to represent the above data.
(a)(i)
Stem
0
Le:if
5
2
5
5
6
7
8
9
.r
2
3
Key:
(ii)
L2J
F111d the median number o f hours spent on f acebook.
Answer
(b)
(a)( ii) ...................... hours
[I J
l)ur ing n period of 60 da:r.; a weather station recorded number of hours of
sunshine each day. The results arc given in the table below.
Tim e
_ (x hou rs)
Freqm:ncy
0 ~ x $2
10
2 <x!>4
15
4 <x '.5' 6
6 <x!> 8
8 <xS l0
12
10
8
10 <.t ~
5
S tate the modal class interval.
Answer
(b)
(I]
°"" :)a;unc.bry Scb.."\OI
9
Cbtcn11 ·1
End-of Ytar L'-amfn..t1_000_20
_1_1 -
-
4
12
(a)
- -- --
-
- "'-·<on<_Luy 2 F ,~
In U1c follow111g Vmn diagram , shade the region (' fl D'.
Ans wer
{a)
D
[ I]
(b)
t
= { x: x is an integer, I ~ x S I 0 ) ,
A = { x: x is a multiple of 3 ) .
B - { ..>: .< is a perfect squa re}.
(i)
Answer
Complete.: the following Venn di agr.un to rqm:sent the above informat1on,
hy listing all clements in the sets t, A and B.
(b)(i)
B
.i
fl ]
(ii)
Write down 11(A n B) .
(iii)
List the elements of the set (/I v B)'.
Answer (b)(ii)
fl)
(b)(iii)
( I)
10
13
Clcrn-em1 To..••n Secondary School
End·of-Year- Exarnina1ion :?011
~condmy 2 ()(pre).."l
Malhl'maocs : Paper I
l.n the diagram, LBAC = 90°. AB= 15 cm, BC - 17 cm. AD = I 0 cm and CD= 6 cm.
(a)
Find the length of AC.
A
Answer
(b)
(a)
15cm
.. ... .. ..... ....... .... cm
B
[21
Show that /.\.ACD is a right-angled triangle.
Answer
(b) ................. ......... ................... .......................... . .. ...... . .... .
!21
11
Ckmmti Tn• 1' S«Md.ary Sdt00I
~f.Yr.u f,~mJN0011201 1
14
Th~
S.,...·11.-bry 2 L1;PJQ..l>
_
_ _I
_ __ __ _ -----''"_
11~
_
,_
••_
•~_
·s_
l_
P
so lid in the figure belo w is made up of a cone and a c ylinder.
T he cone lrns a s la nt height of 5 cm and height 4 cm.
The cylinder has a hciW-.t
or7 cm.
Calc ulate
(a)
the radius, r, of the cone,
(b) the volume o f' the figure, leaving your answer in te nns o f IC.
7cm
',
At1swer
END OF PAPER
(11)
.. ... ... ............. cm
(2)
(b)
............... ...... cm 3
[2}
Ql l
(a)
8crn
(b)
II(' ' ~ CD 2 = 8' + 6 ' = 64 ~ 36 I 00 I 0 ' = AD'
:. By the converse of P)1hagoras' Theorem.
MCD is a right-angled triangle.
= =
Q14 .
(a)
(b)
3 cm
15n cm 3
Name: --------------~--
Register Number : _ _ _
Class: _ _ _
Clementi Town Secondary School
End-of-Year Examination 2011
Secondary 2 Express
Mathematics
1 hour 30 m i nutes
Paper 2
Additional Materials provided : Writing Paper and Graph Paper
ca.0EH11 l()INN$C~ SCHOOl ClCKHTl TOWVN SE<XltCWn'so-t:X>l CLCMCNfl f O'fN S((()tO.t.RVSCHO()l Ca-tMOffi TO'WNSf.C<>NOARY SCHOOl
Q.EMENTI T~ St~ SO«X)L Q_t;Mf:_Nll TOWH UC(»IO,fJtYSCHOOl. CLUolfNTI ·~ s~vSCHOOl CU:MEHfl ·~sceotc:>Nn SQt(}()t_
tll::~NTI TOWN &fCOMJARY SCHOOi. CU.MENt 1 l()WN $1:CCINOARY SCHOOl CLtM(Nll TOWM &C~Y SCHOOL GLCMEHJJ TOWN $EGOHOARY SCHC)Ol
ClCMetfll TOWN S£CONOMYSC!i00l CU :t.!ENTI TO'Wr.t SEC0t40ARV SCHOOL Cl EMICNTI TOWN Sf:COl\OARY 50100\. Cl f:..t[Nll TOWN SECONDARY sc• tOOI.
Ct CMGHTI 1 OWN SE'<:ONOAAYSCI OOL Cl ..WFNJI TOWN S(CQllOAR'f SOIOOL CLf MfJITI TOWN SECONOARV sc.-ioOl CLf ~E Nfl lO'WN StCONlMl'h' SC. IOOL
READ THESE INSTRUCTIONS FIRST
Do not open the bookl ets until you are told to do so.
Write your name, register number and class on all the work you hand in.
Write in dark blue or black pen on both sides of the answer paper.
Do not use staples, paper clips. highlighters. glue or correction fluid .
Answer all the questions.
If working is needed for any question it must be shown with the answer
Omission of essential working will result in loss of marks.
Calculators should be used where appropriate.
If the degree of accuracy is not specified in the Ql•P.stion, and if the answer is not exact.
give tlB answer to three significant figures.
Give answers in degrees to one decimal place.
For n, use either your ca lculator value or 3.142. unless the question requires the answer
in terms of n.
At the end of the examination, fasten all your work securely together.
The number of marks is given in brackets [ ) at the end of each question or part question.
The total number of marks for this paper is 5 0.
FOR EXAM INER'S
USE
50
This question paper consists of 4 printed pages, including this cover page.
(Turn o ver)
Clen1enti Town Secondary Sch<>ol
Cnd-of-Yca1 Cxarnination 2011
2
Seconclary 2 express
r-.bthem3tics I P;:,per 2
Auswer all <1uestio11s. ·
Given thatp varies inversely a~ the cube root of q and p = 25 when q = 64, find
2
(a) pin terms of q.
[2j
(b) the value ofq when p = 200.
(2}
The scale of a map is I : 50000.
(a) The perimeter of the reservoir on this map is 26 cm. What is the actual perimeter of the
reservoir'!
[2]
(b) A lake is represented on this map by an area of20 cm2 • Find the actual are-.a of the lake in
hectares. [I hectare= I 0000 m2 )
[31
E
3
In the diagram, CDE. BFE and AFO are straight
line.~.
AD II BC and BA II CD.
(a) Name a triangle which is similar to triangle EBC.
(I)
(b) Write down another pair of similar triangles.
[lj
(c) Given that ED=9cm. AB = 6cm and BC=l2.5cm,
(i) state, with reason, the length of DC,
ri l
(ii) find the length of FD.
[2]
3
Clementi Town Scoof'1dary School
End·of·Year Examination 2_0_11_ _
4
(a) Express -
I
-
hi
_
_
1
x f3
(b) Expand and simplify
_
_Scconc:1>ry 2 Expres.s
_M_••_
M_m_a6cs I Paper 2
as a single fracuon in its simplest fonn.
(2 - 3x)' -
{c) Solve the c<1uutio11 (x-4)(2.<
S
_ __ _ __
(3x - 4){2x - l).
I) = 4.
[2]
[21
[2]
It is given that P - {3, 7. 8, 11, 12, 14}, Q: (2. 4, 7, 8, 10, 12, 13} and the universal set.
{=PvQ.
(a) Draw a clearly labelled Venn diagrom to ill ustrate the sets P. Q and~·
121
(b) List the elements of the set P n Q.
I 11
(c) Lisi thcclcmentsofthesel /"v Q.
[I]
(d) Write down n(P' vQ).
I 11
(e) If an element or; is chosen at rnndom. find the probability that it is a member of P.
121
6
In the diagram, PQRS is a square of side 17 cm. A. B. C and D arc points on the sides PQ, QR,
RS and SP respectively such that AQ - BR - CS = OP. ABCD is a square.
(a) Given that AP - x cm, write down an expression, in tenns of x, for the area of triangle APD.
II J
2
(b) If the area of ABCD is 205 crn , write down the equation in x and show that it reduces to
.r 2 -17.tt42 = 0.
(c) Solve this equation and write down the length of AP if A is nearer lo Q than to P.
131
131
Clern~nti Town Second~ry School
EJld-of-Year f;xami~tion 2_0_11_ _
4
_
_
Secondary 2 Express
Malhcrnatk:s 'raper 2
v
7
The diagram shows a toy consists of a coloured hard paper cone and a hollow plastic
hemispherical base. The bottom of the base is filled with metal which occupies I 0% o f the
volumt: of the hemisphere as shown in the diagram. 111c common radius OA is 5 cm :ind the
slant height of the paper cone is 13 cm. The vertex Vis directly above the centre 0 .
(a) Find the height of the toy.
121
(b) If the coloured hard paper costs S7.80 per m
2
,
find the cost , giving your answer in the \\ hole
[31
number of oents, of making this cone.
(c) The d<.:nsity of the metal attached to the hemisphere is 8.3 g/cm) Calculate the rna~:< p f the
l JJ
metal used.
8
An swer the whole of this <1uestion on the sa me side of a sheet of g raph paper.
·lbe variables x :inti y are connected by the equation y
values of x and y arc given in the tab I" below.
9 +Sir
2.\ '. Some corrc«ponchng
ft I
(a) Calculate the value ofp.
(b) Using a scale of 2 cm to represent I unit, draw a horizontal x-axis for 0 S x S 4
Using a scale of 2 cm to represent 2 units, draw a vertical y-axis for - 4 S y S t 4.
On your axes, plot the points given in the table and join them with a smooth curve.
f31
(c) Use your grnph to find
(i)
the largest value of y and the value of x for which this occurs,
(ii) the values of x when y = I 0.
[2)
12]
- End ofPaper -
Cll'ntt.•1Hi 'J'9wn Sec School End of rl':.ir [~am 2011 ~·1A f11£1\·1.·\ TICS Paper 2 1\·l:.rk Schemt
ft
(a)
-·-- - - - -
k
p = :jq
-
, where A1s <• cons.1a"1
When p
~
4
(>t)
- --I
x
3x - I x+3
·: c(x+ 3) -
25, q - 64
Ox-
I)
Ml
(3x- l)(x+ J)
Ml
..·25= _!_
\;'64
1
x 1 +1
(3x-tXx+ 3)
10-0
fq
.. p =
(b)
x + 3x - 3x + I
(Jx - 1)(.n 3)
.,-,k = 100
When p = 200,
100
(b)
(2x-3)' -(3x - 4}(2x - I)
(c)
= 4 - 12.r +9.< ' - 6x ' + 11.x - 4
= 3x - x
(x-4X2x- I) = 4
- 4 - 12.< + 9x' - (6x' -I Ix+ 4)
Ml
:. 200 = , /
'..Jq
=>
w= ]_
2
I
=> q =
8
Al
Al
Ml
Al
=> 2x' - 9.x + 4 = 4
141
=> 2.:c' - 9.:c =0
Ml
=> x(2x-9) = 0
2 (a) Actual perimeter = 26x500 m
13000 m = I Jkm
(b} I : 5()000
- I cm : 50()00 cm
= I cm : 500m
(I cm)' : (500 m)'
=
Ml
I
=> x = O or x -= 4 2
Al
~-3
(_a_
) _ A_
F:-F D
_ __
(b)
l
i
!
!<c)
·-·-- - - - - - -
~---------~~
131 : .;Ofl'CCi
diag nun
RI
B l: all
Ml
:\I
nurnhc r~
[51
cntl'rt'd
COfl'<.'Clly
HI
MJ·J) and D..DFE arc similar
RI
or 6.AFB and a CBE are sirnilar
DC ~ 6 cm
(opp. <ide< of a !!gm) Bl >vith
(i)
t('a...,on
FD
9
(ii)
~II
12.5
=> F{) : l2.5 x_'!_
IS
=> H> = 7.5cm
16]
5 (a)
l
lcm : 250000m '
. . the actual area : 20 x. 250000 n\ !
- 5000000 m ' = 500 ha
t\I
Al
DI
(b)
!51
~-'------------· ------'----·-
(c )
PuQ = !2. 4. 7. 8. 10, 12, 13 f
(d)
n(P\.1Q) = 7
(e)
n(~) = IO and n(P) = 6
Prob(a number choS<'n b<Olongs to /')
n(P)
= ~- =
n(;)
= 0.6
6
Bl
131
Mi
10
Pl
2
PD - AQ - 17 - x
6 (a)
8 (a)
.
I
: . an:ao f MPD= - x(J1 - x)
2
Bl
p : 9 I 5(4) ?(4 1 ) : -)
(b)
Bl
~ . -=,,
f
\2
- 289 - 34.< + 2x'
'!
II· : ..
: •: -H-fE
area of ADCD = J1 '-4{.!..:\t17 _ >))
(b)
r.-: 11 1:_
JJ I
OR
!By Pythagoras' theorem.
AD' - (1 7 - x)' +.r'
RI:
COfTeci
a~cs
scale
->AD' =289-34.r+.r · ._,-
(c)
=> AD' =2r' -34.r+ 289
·. area of ABCD = 2x ' - 34.e< 289 J
. . 289 - 34.r+ 2.r ' = 20;
=> 2x' - 34x+84 =0
::::> r ' - 17.r+42=0
Ml
::::> (r-3)(.r-14)=0
Ml :
Rl
.,
••
. I
Al
a)
>r=3orx = l4
AP - 14cm
Al
Al
VO - Ji3• - 5' = 12cm
heigh• oft0y = 12+ 5 ~ 17 cm
131
JI I
(b) curved surface of cone
m RI :
i.:
(7)
~u!f:.
t:
..
lt-::t
(c)
(i)
~.
l
tt;t
t: :·~:: c~
.. - r
:::.~
: _~g
Ille Jar11e'1valueof ,.1s12.1 and
Ml
- 65JtCl1l
:. cosr of m•king 1his cone
Ml
16 cents (correct to the w::an:sl c;cms)
Al
(ii)
Volume of hc1ni.sphere -
(c)
1he c;orrcspond1ng value of x is 1.25
65n(7.8)(100)
- 100·
3. n(5 ' )
250ir
-
Clll
x - 0 22 e>r
~.28
B.?: I nrnrk
1
1..:ach
for
lanS\Vl' r
&
Ml
I
.
250!7 10
Volume ofmatcnal used =
J<. l
100
25;r
'
= - - c.n1
3
- 2 I7 g
~ IO.
Bl
0 2 100 J
3
m:i.« of ma1erial used = (
When y
Bl ""\)
ac;cept l 'l. "'
bu1 nor J J_(}
\\·1thi11
3
= -
smoothncs-.
... ·~f I.~~ °ffu
:, t
=•(5)(13)
t•
· .I· ,. · -i:.i
must
Sh O\\
7
RI : all fi rst
8 point"
CC'.\uec1ly
plotk-d
2
~ .2 10 ~ . }
[S J
Al
~ir }s.3)g
(corrccllo 3 .<ig. fig.)
Al
[8J
No1c:
M
Mtll\Co-0 mark, awarded for a valid me1hod applied 10 1he problem.
Accuracy mark,. awarded fora corrccl answer or in1ennediate step correctly ob1a1ned.
Mar~ for a correct resull or s1atcmcn11ndcpenden1 of Method marks.
J)M or DB (or dc-p•) may be used 10 indicate lh31 a panicular M or B mark is dcpendcnl on an earlier Mor B in 1he scheme.
J\
U
Geylang Methodist School (Secondary)
End - of - Year Examination 2011
Candidate
Name
Class
F===:;---1n-d~xN-umb.er
----;::::=
["
I
j
MATHEMATICS
Paper 1
2 Express
Candidates answer on the Question Paper.
Setter :
1 hour
Mdrn Rachel Cheng
7 October 2011
READ THESE INSTRUCTIONS FIRST
Write your name, index number and class on all the work you hand 1n.
Write in dark blue or black pen in both sides of the paper.
You may use a penoil fo< any diagrams or graphs.
Do not use staples, paper clipS, highlighters. glue or correctt0n fluid.
Answer all questions
If working is needed for any question, it must be shown with the answer.
Omission of essential working will result in the loss of marks.
Calcul'1tnrs should be used where appropriate.
If the degree or accuracy Is not specified in the question, and if I/le answer is not exact. givt: the
answer to 3 significant figures. Give answers in degrees to one decimal place.
At the end of the examination. fasten all yaur work securely together.
The number of marks is given in brackets [ ] at the end of each question or part question.
The total number of matks for this paper is 50.
-- - -
- - - - - -,
For Examiner's Use
This document consists of 1O printed pages.
[Turn over
GMS(S)/Mathi P Ii EOY2011/2E
Total Score: 150 marks)
Answer ;\LI . questions in this paper.
Expand each of the following
(a)
..
- 7a(- 2a + 3b),
Ans:
2
{o)
fl J
{b)
[2]
The area of a school. 25 km 1 , is represented on a map hy I cm2•
(a)
Jf the map has a ~ale of I : r, find the value of r .
(b)
If the perimeter of the school on the map is 12 c rn , find the actual
perimeter of the school.
Ans:
(a)
{b)
rs
f2]
km (I]
2
GMS(S)//l.fath!l't / E OY2011 /2E
3
The shoe sizes of 12 baskc1ball players arc listed below.
8, 8, 10, 8. 10, 9, 12, 10, 8, 8, 12. 11
r:ind
(a)
the mode,
(b)
the m~-dian.
Ans:
(a)
_ _
_ ___ [I]
{h)
4
[2)
In the figure below, tvl/I(' is congruent lo /11:,/)F .
E
A
!'ind
(a)
/JJAC,
(b)
the lenglh of AD.
Ans:
(a) _
(b)
_
_
__
fl J.
-- ~ [2]
3
GMS(S)1M ath/P l/ t:OY20 l 1!2E
5
(a)
State the equation o f the linear graph shown below.
..
5 x
·- 5
- 6
(I>)
On the same axes, draw the graph of y = x + I for - 3 S x $ 3.
(c)
Hence, solve the simultaneous equations graphically.
Ans:
(a)
(c)
[2)
-
- [2]
X
•.:_"'_ __
_,___ __
-
[2)
4
GMS(S)/Math/I' 1/EOY201 !i2E
6
Factorise completely
(a)
x'y-3.xy,
(b)
2p 2 + 4pq - 3qp - 6q 2 •
Ans:
((/) _ _
(b) 7
-
_
_ _ __ [2]
-
·-
[2]
1: - {2,3, 4,5,6. 7. 8.9.10, ll , 12}
X - {x : x
~ multiples
Y ~ {y: y
f'
(a)
of 31
prime numbers}
[3)
Represent the sets using a Venn diagram .
Answer ((I)
(b)
Find n(X 11 Y°).
Ans:
(b) _ _
_ _ __
[11
5
GMS(S)/Math/ Pl/EOY2011 /2E
8
An outdoor camp wa.~ planned for a group of 20 Nl'CC cadets and enoug h
food was packed to last for 6 days. If I0 more cadets d<.'(;ided to tum up for
the camp, how long would the same runount of food last?
_ _
Ans:
Given thatp =
[21
2q +r1
7 '
(a)
find the valw ofp when q - 2 and r - - 5,
(b)
maker the suhjoct of the fonnula.
Ans:
(a)
[ I)
(b)
[JJ
6
GMS(S)iMa1h/PJ / EOY2011!2E
10
or people consis1ing of 9 men, 6 women, 12 boys and 3 girls went
a company rc1reat. A person is chosen al random from lhc group. Find
lhe probability that
/\ group
011
(a)
the person is a man.
(h)
the person is either a woman, a boy or a girl.
'.
Ans:
lJ
(a)
(I)
(b)
(2]
The table below shows the number of goals scored by soccer teams in the
Harclays Premier Le.ague last year.
(3)
Fmd the value of x if 40 teams took part in the competi1ion.
{b)
State the biggest possible value of x if the mode is 0 goal.
(c)
Write down the smallest possible value of x if the median is I gmol.
Ans:
(a)
-
- -- _
I I]
(b)
fl I
(c)
_ _ _ _ f2J
7
GMS(S)/l'v1ath/P I /EOY20 I li2E
12
For each of the diagrams, shade the region representing each of the
following sets.
(a)
[ I]
AriB
c
(b )
[I)
AvB
c
00
[2J
(c)
B
8
GMS(S)/Math/ P l/EOY2011!2E
13
In the figure below, 1\Xfz· is enlarged by a scale factor. k to form AXYZ.
I0.5 cm
z
GivcothatXY= I0.5cm, Y'Z' = I0.2 cm and X'Y'=7 cm, find
(11)
the scale factor, k of the enlargement.
(h)
tl11.: area of the shaded region to the nearest whole number.
Ans:
{a)
(b)
-
_ fl]
crn 2
[3]
9
GMS(S)!Math/P l/EOY201 l/2E
14
14
(11)
An cquila1eral triangle DEF has sides of 14 cm. Find lhc height, h
of 1his 1riangle.
(h)
I lence, find the area of MJEF .
Ans:
(a)
{b)
cm [21
cm
2
[I]
End of Paper
10
GMS(S~'Math1Pl/EOY20 11 /2E
Paper I Answer Scheme
Remarks
Nowori::inf.
re 11ired
('o.n <i'l()n'C
muse any
method
fKCOrdin~ 10
1 c7:
2(a)
2s 1uT
Al
- -
: 5 km
1 cm
_I _ : _1.00 00-0 _
ac1uul perimeter - 12 x 500000 = 60-00000
2(h)
3(a)
lheir
p1c1Crtncc.
2x x'
- I- - + J 9
8- - - - - - - - l-
13(h)
- - - -
~ fill km
_
·-
rcqutrcd
-
8, 8, 8. 8. 8. 9, 10. 10, 10 , 11 , 12. 12
Ml
Median
L4(a)
- - --180·
-- 100- · -38- -- 4=2.__
- - -- - - - - -- - /.. H1IC
_ __ _ _
4(b)
Ml
_A_I
A1
Al- - ~ki•a
- -Rearranging in ascending order,
-
I
I
- -
~ 1 ~ = 9.5
2
Al
BC - DC +DF 19 DC I 17
DC = 22
AD = 58~~- 4- 0
gradient - · - = -2
0-2
equation: y = -2.x ~ 4
- -
- - --
Ml
- - - --
- -- - - - A l
Ml
- - - - --
- - -- - -
--
Al
--
--
GMS(S)/Math/ PllMYE201 J!2E
S(h)
I ma'k for
plotting lh~
!,'Taph
3i;Cu.ratdy
i\2
-7
-~
= x
/ / >'
'----.;.-----'---4-+--'--~~L.._· -
4
I
I m:·1 rk fo1
+I
labeling th..:
8-~•ph
5
x
\
II
l
>
.'
~--~-- -~
~~----- ~
I
·s· /·~-\-+-1 --d-2
'
-A-2 --t--i-;~kc;;;·; -x ~ an_y~=
~~-~--~~~~~~~~~~~~~-+~-~1-' ~~-1
~i.S!
' 6(a)
6(h)
7(a)
x 2 y - 3xy = ~)·(x - 3)
2p 2 +4pq - 3qp - 6q 2
A2
2p(p 12q) - 3q(p ·1 2~)- = (p+2qX2p - 3q)
---------~-~~-~~------·-
l'
4
l
-·--Ml
-
Al
---- --
_,____,
I
I m.uk lflr
1\3
x
lal>t-ling
Arldi1io ual
2 m:irk!>< fur
pladn p,all
no:-
accvr;ltt l!'
8
I m;uk for
plac ing. at
k ~l~l
7(h)
·8
n(Xn Y') = J
Let NPCC cadets be m and days bed
Al
' Ml
k
20 = 6
k "' 120
120
:.m =-
6 IUJ'S
accurntcly
10
d
2
I
I
GMS(S)iMath/l'l/MYE201 li2P.
:. 30
Al
=4
2q + r i
p=
7-
Al
120
d
30d = 120
9(a)
2(2)+ (-5}2
= ~
1
9(b)
I:-
-
-
I
requ1r~tJ
.
- 4 - or4.14 (3 s 1g. fig)
7
_ 7;___ _
2,,;r'
p=2q+r 1
- 2q =r'
£
l()(a)
N nwnrl<ln~
Ml
Ml
1\1
v1p-2q
l'(man) -
Al
.2_ = 2_ or 0.3
_ _ _ ;'. ,:30::_
__
_l~O
I- i(l(b) P(woman. boy, girl) = I - P(men}
~
3
10
1- -
7
- - or0.7
10
Ml
An)' other
1nc1hod i$
:t~cepuablc
Al
,,,-
Jl(a) x= 40 - 20 - 6 - 5 - 5 - 0 = 4
x~ 19
·
-l l(c) r~- 5
r.
Tt(b)
12(b)
I
_J
.l
GMS{S)!Mathll'llMYE201 l i2E
82
I 2(c)
~13(a)
scale
factor ~
image
- - 01igi11al
= 10.5=1.5
7
IJ{b) Y.7. = 10.2 x 1.5 = 15.3 cm
Arca of shaded area s
Ml
(~x 15.3xl0.5)- (ix7x 102)
= 80.325 • 35.7 = il_(ncarest whole number)
14{a) " - ) 142
-CiJ
Ml
Al
Ml
- J 141
~{h)
Al
- i l l (3 sig. fig)
Area~ ~
x 14 x
.Jl47
l~ '-~~~~~-=-8_4.9_{_3_s-ig_._fi-g.~) ~~~~~~~~~A~I ~~
4
Geylang Methodist School (Secondary)
End - of - Year Examination 2011
.'
MATHEMATICS
Paper2
2 Express
Additional materials : Writing Paper
Graph Paper
Setter :
Mdm Rachel Cheng
1 hour 30 minutes
4 October 2011
READ THESE INSTRUCTIO NS FIRST
Wrile your name, index number and class on all the work you hand in.
Write in dark blue or black pen in both sides of the paper.
You may use a pencil for any diagrams or graphs.
Do not use staples. paper clips, highlighters, glue or correction fluid.
Answer all q uestions
If working is needed for any question, ii must be shown with the answer.
Omission of essential workir'1g will result in the loss of marks .
Calculalors should be used where appropriate.
tr the degree of accuracy is not specified in the question, and .t the answer is nol exacl, give the
answer to 3 significant figures. Give answers in degrees lo one decimal place.
At the end of the examinalion, fasten all your work securely together.
The number of marks is given in brackets ( I al the end of each question or part queslion.
The total number of marks for this paper is 50.
This document consists of 5 printed pages and 1 blank page.
[Tum ov er
(iMS(S)/MathiP2iEOY201 l/2E
Total Score: ISO marksf
Ans wer ALL questions in this paper.
With the help of a calculator, evaluate
(a)
{g] l x(29.75)'_
v 47.89 ,
(b)
(2~)1 +(- 1. 1)'.
.''
11]
[I I
Give your answers correct to 3 significan t figures.
2
The value. Vin dollars, of a silver plate is directly proportional to the square
of its mass, min grams. If a silver plate weighing I0 g is worth 5675, find
(a)
an equation connecting Vand m ,
[2]
(h)
the mass, in grams, ofa silver plate that is worth $16 875.
( 1J
Twice of Jenny's present age is 14 years less than her father's present age.
(41
7 years ago, her father is 5 times as old as she_ Find the present age o f Jenny
and her father.
4
Simplify the following expressions
(11)
(b)
5a ac' c'
x+ 9h 2b 8b''
-
3x2 +2x-8
,\ l
- 4
[2)
[2)
GMS(S)!Math/P2/EOY20l l/2E
S
A car traveled from town A to town C, passing hy town B along the way.
r:rom town A to 8, the car traveled for 50 km at a constant speed ofx km/h.
However. it broke down when it reached town B. The car took an hour to
repair and completed the remaining journey of 6 km to to,vrl ~· al a· constant
speed of (x-16) kmlh.
(a)
Write down the time spent to travel from town A to B, in terms ofx.
[11
(b)
Write down the time spent to travel from town D to C, in terms of x.
[ l)
((')
Given that the total time the car spent traveling on the road is 4 hours,
write down an equation in tcm1s of x. and show that it reduces to
x' - 30x+200
0.
13)
(ti)
Solve the equation and find the value of x.
[2)
(t>)
Hence, or otherwise, find the average speed of the car for the entire
(I]
journey.
<•
thin!\ algebraic rules, evaluate
(a)
68 1
{h)
2 .01 ?.
-
32 1 ,
[2]
[2J
- press - 2x- t- - 3x
. Ic firacllon
. 111
. its
. s1mp
. Iest Iiorm.
he
- - as a smg
I x x 2 -I
f3]
llie ratio of the interior angles of a pentagon is 3 : 4 ; 5 : 5 : 7, find the
(31
iarges1 interior and exterior angle.
3
GM S(::i)/Math/ P2/EOY20 11 i2E
9
1- - -
tOcm
' '
/\cone is cut from a solid hemisphere of diameter 60 cm. The base diameter
of the cone is found to be half that of the hemisphere's.
'I"ak e
;r
= -22
7 .
Curved Surface Area of a cone • m-1
Surface l\rea of a sphe1·e = 4711' 1
J
,
Volume of a cone = - ;rr· Ii
3
Volume o f a sphere =
(a)
(b)
4
w
•
3
Find
(i)
lhe radius of the cone,
11 l
(ii)
the volume of the cone,
[2J
(iii)
the volume of the remaining solid.
(2)
If the cul hemisphere
i~
fil led with water to the hrim, rind th e surfncc
[2]
area in contact with water.
I0
Mrs Lee baked 20 chocolate and cranberry muffins and a of them were
chocotate muffins.
(a)
Write in tenns of a. the probability that a cranberry muffin was chosen
[I)
at random.
(h)
lvlrs Lee baked 12 more c hocolate and 18 more crw1hen-y muffins.
{3 j
The probability that a cranberry muffin was clm~en at random from
the muffins became 0 .4 . Find the value of a.
11
Answer the whole of this <1uestion on a sh eet of gr aph pap~ r.
4
GMS(S)1Math/P2/EOY201 li21"
The variables x and y are related by the equation y
(a)
=- x 1 -
2x - I .
Copy and find the values of c and d for the table below.
(11
8=t41-J 1 -ili-~o~2~
y
(h)
- 9
-4
-I
--
c
- I
- 4
d
--
- -
[1J
Using a scale of 2 cm to represent I unit on the x-axis and I cm to
represent I unit on they-axis, draw the graph o f y
=- x 1 - 2x - I
for
- 45x52.
(c)
From your graph, find
(i)
the value(s) of x when y = - 5.5 ,
f2j
(ii)
the equation of the line of symmetry, and
[lj
(iii)
the coordinates of the maximum point.
[lj
End of Paper
5
GMS(S)iMath/ P2iEOY201 li2P.
Paper 2 Answer Scheme
l (a)
''
l(b)
- 24.2
2(a)
V = km '
675 = k(10')
k =6.75
- -- - - - -
,.
-·--
Noworti.io¥
·~!.._
Nowor1dup,
rcouired
Al
Ml
Al
:. V '- 6.75m 2
I•
16875 = 6.15m'
2(b)
m 1 =2500
m = 50g
3
-
Remarks
Marks
Al
Al
Let Jenny's ugc be x and her father's age by y.
2x+ 14 "' y
------(I)
I m3rk c.a;.h
forwnlm$
A2
chctqu:)U~
y-7=5(x -7) --··--- (2)
Sub (I) in (2)
2x+ l4 - 7 = 5x - 35
42 = 3x
:
x= l4
I
At1)' mt'lhod
1s accept<if1k
Al
Subx = 14 into (I)
y=42
Al
1~
! 4(a)
Sa uc 1 c'
Sa ac 1 8b1
)( x - +-= - x 9b 2b 8b}
9b 2b c'
Ml
40a 1 b 1c 1
I 8b 2c 1
20a 1b
= -9c
4(b)
3x1 +2x-8 _ (3x-4Xx+ 2)
,t 2 - 4
- (x+2Xx-2)
3x - 4
x- 2
Al
Ml
Al
-
_J
GM S(S)/Math/P2/EOY2011 i2E
5(a)
50
- hours
x
6
- - hours
x - 16
50
6
- + - - =4
x x-16
50(.r - 16)+
6x
_
4
x(x - 16) x(x 16) 50x -800+6x
4
x 2 - 16x
50x-800+6x = 4(r 2 - 16x)
--S(h)
S(c)
i
;
I
(
~
-
I
l
' .
Ml
Ml
4x - 64x -50x - 6x -t 800 =0
4x1 -120x+800 • 0
x2 - 30.r + 200 =0 (shown)
2
~
S(c)
Al
x - 30x + 200 =0
(x - 20Xx- l 0) = 0
x = 20 or x = 10 (tejcctcd)
S(d)
--
·-.
6(a)
k --··
I 6(b)
-7
-
2
Ml
Al
Average Speed = Total Distance I Total Time
~ 5615
= llZlun/h
I
!
6R' - 32 2 = (6s +nX6s-32)
~ ( I 00)(36) • 3600
2.01 2 = (2+0.01)'
=2 2 +2(2)(0.01)+0.01 2
= 4 + 0.04 ~ 0.0001
=4.0401
3x
2x
3x
2x
+- - - + -1- x x 2 - I
1-x (x+lXx - 1)
-2x
3x
• - -t
x- 1 (x+IXx - 1)
= -2xlx + 1)+3x
(x+ 1Xx- I)
2x2 -2x 1-3x
m
Tx'+ 1Xx - 1)
Al
-
Ml
Al
Ml
Al
Ml
Ml
2
+x
= (x-2x
+ IXx_:- 1}
Al
2
GMS(S}tMathi1'2/EOY201112E
8
- - Su~ofint~angle~ofpentagon= (5 - 2)x1W - - - - - - i ~ 540°
Ml
24 units represent 540°
I unit represent 22.5°
•
Thus1hc angles arc 67.5°, 90°, 112.5° , 11 2.5° and
J
57.5°
1The larges! inteiior angle 1s J 57.5°
9(ai)
9(aii)
0
The largest exterior angle is 180_£7 .5 = 112.5_
- -Radius - (60 + 2) + 2 = .Li£!ll
Volume of' cone ~
c::
9(aiii)
- -
~ x 22 x J 52 x 30
3
7071
7
3
M
cm3 or 7070 cm.1
7
-V-o-lum
- e-of remaining solid-
9(b)
:3
• 49500cm·'
Slant height . I= J 15' + 30; =
_J
Al
------ - -- ·- - - - --r--- -1---- -I
= Vol hemisphere - Vol cone
- .!_ x ~ x 2 x 30 2 - 7071
2 3 7
7
3
Al
--+-A_l_ _ _ __
/\I
1
Ml
/j I 25
·-
- - - - - -- - -
Al
Ml
I
L
l -
~ x I S x Jiii5 ~ 1580 cm (3 s ig. fig)
AI
j
7
--.,0- a - - -- - - -----t§I ~.-;,;;as
Surface area = wl =
2
2
- - - P (cranberry) = - -
_ _ _2.Q__ _
0.4 = 20 - 11+1s
50
20 = 38-a
a = l8
c= Oandd - -9
I ...,.....i
M-1
Ml
Al
-1-I
(iMS(S)!MathfP2iEOY2011!2E
II (b)
- - ···--· -
-y/
-4
~
,
i
- ll(ci)
·-
I m3rk for i
pJouing &he
axes
:t<:tOrtling t()
A3
I
-1
I
--
2.'.
"'
-
thescak
: - x
..
~2
.
-
v
x = -3.35 and 1.35
·--·~-
I mark tor
'.
ptou.ing
poinl'> and
·'\.
'\;=
d111wing;J
smooth
cur.,e
through lhe
1
- x - Zx - I
p<1i11~
\\
I m.irk fen
bbcling the
\
{:,l';Jph ;Jnd
axes c learly
\
,
A llow:m
82
i11u:r\«1IQf
±0.15
1----1-------------------------·--------1-c~--1-----1
lllciil
J.l.f.,iii)
-
x = -1
•.
Maximum ~>int (-1, 0)
-~-
Bl
Bl
4
Jurong West Secondary School
End-of-Year Examination 2011
Subject
Level/Stream
Date
Time
Duration
Mathematics
Secondary Two
5 October 2011
0800- 0915
1 hour 15 minutes
4016/1
(Part 1)
(Express)
(Wednesday)
Instructions to candidates
Do not turn over the Question Paper unless you are asked to do so.
2
Write your name, class and index number in the s pace provided at the
top of this page.
3.
Answer ALL questions.
4.
Write your answers in the spaces provided on the question paper.
ALL essential working must be shown in the spaces provided.
Omission of working w ill result in loss of marks.
5.
The number of marks is given in brackets l j at the end of each question
or part question .
6.
NO C ALCULATOR may be used in this Paper.
If the degree of accuracy is not spcci lied
in the question and if the answer is not exact,
the answer should be given to
three significant figures .
Answers in degrees should be given 10
one decimal place.
of an~r scri t
Signature Date
Student
1---
-
-
-
--l-
-
-
- -
-
Parent
- - - - - - -- - - - - --
- -
-- -
This question paper consis tli of9 printed pages, including this page.
- ---
J\""'
._
VS0<>S.,F...,..
11.._
d--0..,f~
· Y~•=ar~E=•=
• m=j=n•~H=o'~'2=0~
11....__
_
=M=•
••h,•matics <Pan
Stcond an.:._ 2~
I)
Answer ALL the questions. ISO marks!
Factorise each of the following
(a)
4 p 2 ~ 2p
(b)
3x1 + x - 2
(c)
18mr~
..
9ms-2nr - ns
Ans: (a)
[I)
(b)
- - - ·- -- - __ I! I
(c)
_ __ [2J
Simplify each of the following
(a)
(6wy + y){3w2
(b)
5t(t + 3)- J(St - 2)
(c)
-
3
l-x
1-
5
-
2y)
-
l - x2
Ans : (a)
(b)
(c)
- - - -
- -
fll
- - - - - - - _121
-
-
(21
Page 2 of9
3
Data on the size o f classes for 1wo university courses. Psycholo gy and Philosophy,
arc s hown below:
~- ·~- Psychology
'j
16
29
20
Philosophy
I --i-9
17
14
_ _ !l___
25
, --,6-
- - -- - - 31
--·-a=-rn
21
- ·
~2 1
24
-
31--
22
2_5 _
32
IS
33
18
18
20
--~·
14
-
20
-
(a)
Represent the data of these two courses in a back-to-back stem -and-lea f
diagram.
(b)
The maximum capacity ofa classroom is only 20. Classes which exceed this
capacity have to be split. Calculate the percentage of classes for both
Psychology and Philosophy that will be split.
I
(2]
Ans: (a)
(h)
4
-
- % [2]
Solve the following equations.
(a) 5x 2 = 12 x
(b) y -
12
=4
y
Ans: (a)
(b)
[2 ]
-
- - f3)
Page 3 of9
J~\~VS=S~£=
n~
d ..=
o(~·y..,.....
,, r_.[.._,,,,..
am
..,.j=
n•='i=on~2=0~11~--""l\l..,•,_,,
!h"'"e"'=a t_ig__(Pa rt Il
5
Let 1; = {O, J, 2, 7, 8. 12,21. 24, 36},A = {2, 7. 24, J6} and
Tl = {2, 8, 12, 21, 36}.
(a)
Represent the setsc, A and Bin the box given b..:low.
[ 1J
6
(b)
Find n(A v B)
(e)
List all the subsets of (Au B)'
Ans: (b)
[ 1)
(c)
(I J
In the Venn diagrams, shade
(a)
M'vN
c
l JJ
(b)
(PnQ)'
&
11 J
Page 4 of9
._l\VS5 Fnli-·nf~Vt , r (:\amination 2011
7
(a)
(h)
Sttonrht r)' l
Find the median of the following 8 values:
24
17
12
8
I0
6
15
9
Find the mode of the following 8 values:
4
1
4
2
16
2
0
7
F.x1>rc~s
.'
(e)
The daily mean number of DVDs rented from a shop is 80 for
weekdays and 150 for weekends. Find the daily mean number of DVDs
rented from the shop for the whole w~ek .
Median = _ _ __ _
[IJ
{b)
Mode = -
(I]
(c)
Mean=
Ans : (a)
- --- - -
---
DVDs [21
Page 5 of9
l\1arhrmsric::t CPart ll
8
Sscondary 2 £xprtsJ
In the figure below not drawn to scale. triangle ABC is similar to triangle ADE. BC
is parallel to DE. Given tha1 AB - 2 cm, BC = 3 cm, DE ~ 9 cm and /ABC~ 40",
find
(a)
LADE. stating Lhe reason clearly
(b)
the length of BO
..
D
E
Ans: (a)
(b)
RD ~
0
[I I
W I
[J)
J'agc 6 of9
.J \>VSS EncJ.o(. Ytn r F: ·uu11.~lt.,_,1a.,1,,io,.,n_.Z"'0.._11.__ _ __.Q'l"'
., o"'
t h,,e,.11...
ia.,_,1i,,_cs'-'(..,_P_.a,_,11wl"'l_ _ __ _ _ __,s"'
· e"'c,,,01.,11.,1a..,.r.._v_.2_,t"'
:x,..p._re...,s•
9
Solve 1hc simu ltaneous equations
5x +7 = - 3y
x1-2y= O
Ans :
x•
>' "'--
- --
- [t1J
Page 7 of9
J\\IS.S End-of-Year [x ami na.,.H,..nn~l..
0....
1._1---~~...,ta...,1h,.r=
·ma1ics <Pan 1t
10
(a)
Simplify (x - y) 2 + 2xy .
(h)
Uence, evaluate 9982 + 4000.
.
.
_
Ans: (a)
_ [2)
(h)
[2)
,
II
2
U1vcn that (w - 1)' = 4 and Jw1 = 8, find th e value of~
2
Ans :
.
-
-
_ [41
Page 8 of9
J\\'SS F.nrl-nf-\..eat £xan1ination 2011 _ _
12
It is given 1hat x
_
-";\'"'
lau.tlw!f"'""''""''i"-'9"'' -'ra,.rtl...!.lll_ _ __ _ _ _-"'Se"'c""on<l!lry 2 [\'press
= \;; ' - y 2
(a)
Find the value ol"x when z
(b)
Express= in tcm1s ofx and y .
5 andy
=
3
Ans : (a)
(b)
13
x=
(2)
_
_
_ _ [2)
A fair die is thrown. C'alculate the probability thnt
(a)
a fraction is obtaine<l,
(b)
a numho.:r 4 or more is obtained,
(c)
'6' is not ob1amw.
(I)
Ans: (a)
(b)
(c)
- - --- - --
[i J
(11
End of Part I
Page 9 of')
JURONG WEST SECONDARY SCHOOL
End-of-Year Examination 2011
Mathematics Part 1
Secondary 2 Express
Answer Kfil(
I
Answer
2p(2p ~ I)
(3x - 2)(x + I_) _
_ _
(9m - n)(2r + s)
- - - - - - - - - - - - - - - - - -- - - - -1
3
c
3x+8
- -(I t· x)(l - x)
a
Leaves for l'n•c hOJo"V Stem Leaves for Philosophy
87
8
6
4
I
3
6 s 4
4
9
0
0
2
I
0
5 2
9
5
,)
3
I
I
2
.- - - - - - - - - _;:_:
KcL!_l4 n~ l4
50% - - - - - - - ·- - - -12
x O or r = 5
-- --- ~
~
b
a
b
a
l
- - - --
---
-y=-6
- - = - 2-
or y
- - -- - - -
B
8
7
24
21
0 1
c
. ·. N
IT
&
p
'
.. "
7
'8
a
b
c
a
b
9
10 a
b
:
x2
+ y1
1 000004
-
66
11
12
-
Median = 11
Mode= 2 and 4
Mean= 100 DVDs
40°
BO = 4 cm
X • - 2; v=I
a
b
- 13
- a
b
c
x =4
z ± \/ .l. l
0
I
y'
--
1
-2
-s6
I
,
r
.
Cl ass Index no.
-1
[Nam e :
I
I
Jurong West Secondary School
End-of-Year Examination 2011
Subject
Level /Stream
Date
Time
Duration
Mathem atics
Secondary Two
5 October 2011
1030 - 1145
1hour1 5 minutes
(Part 2)
(Express)
(\Vedncsday)
/50
4016/2
Instru ctions to ca ndidates
Do not turn over the Question Paper unless you are asked to do so.
2
Write your name, class and index number in the space provided at the
to1> of this page.
3.
AJ1swer ALL questions.
4.
Write your answers in the spaces provided on the question paper.
Al,L essential working must be shown in the spaces provided.
Omission of working will result in loss of marks.
5.
The number or marks 1s given in brackets [ ) at the end of each <1ues1ion
or part question.
6.
SILENT, ELECTRONI C CALCULATOR may be used in this Paper.
If the degree of accuracy is not specified
in the question and if the answer is not exact,
the answer should be given to
three s ign iIicant figures.
Answ~rs in degrees should be given to
om: decimal place.
After c heckina of answer scriot
Checked bv Sia111ature Date. .
Student
..
Parent
i
I
___;
-~------- -
This question paper consists of9 printed pages, including this page.
J\\·SS End ..n~x11mi11ation 20l I
M:.1lht1n:Hks <P:art 21
Suondary 2
f~
Answer ALL t he questions. f 50 marks)
I.
(a)
The fol lowing dot diagram shows the amount of poc ket money
received by a class e>f students in a week.
•
• •
• •
• •
•
•
•
$20
(b)
$21
' '
• •
S22
•
• •
• •
• •
$23
$24
•
$25
t 26
(i)
What is the total number of students represented in this dot
diagram?
(ii)
How many students received at most $23 in a week?
The number of projecL~ completed by a company is tabulated
below.
Number of _P.rojocts completed 0 I 2 3
Number of months
2 7 6 x
(i)
4
4
5 6
2
1
Find the minimum val ue ofx if the mode is 3.
Using the value ofx in (bXi), find
(ii)
the mean number of projects completed.
(iii )
the median number of projects completed.
Ans: (a) (i)
students [I ]
(ii)
students (I]
(b)(i)
(ii)
Minx -
Mean =
(iii) Median =
(I]
projects (21
projects [I ]
Page 2 of9
:U_VSS F.nd::J•frYtar f xarnination 2011
2.
(a)
:\lathe1naHrs CP:u1 2)
A letter is chosen at random from the word 'SHENANIGAN'. Find
the prohabil ity that the letter chosen is
"N"
(i)
an
(ii)
a vowel
'.
(b)
The diagram shows a wheel. Joel spins the wheel once.
What is the probability that Joel obtains
(i)
a score of I
(ii)
a score of' an even number
(iii)
a score that is not a S
_
[11
[11
(I l
_
(Jl
Ans: (a) (i)
(ii)
(b) (i)
(ii)
(iii)
_
(I]
PJge 3 ol'9
,J\\'SS F:nd-of- Ytar Exantinalion 2011
3.
l\.1athtrHatil'S f Part 2)
2 cm on map A represen ts I km.
(a)
Express the scale of map A in the fnm1 I : n.
(b)
If the length ofa river on the map is 9.8 cm, !ind thei)ctual distance of the
river in km.
(c)
The actual area o f a forest on the map is 4.5 km1 . Find the map area o f the
forest in cm 2•
L2J
(a) _
(b) _
(c)
_
_
_
_ kmf2J
·-- - - - cni2 (2]
l'agc 4 nf9
J_\Y!"S End·-Of· Vtllr F. uun ina I ion 2"'0l..,l_ __ -'"'M"'a"'1l,,1t..,_m.,,a,_.;lk.,•.wfP'-'>"'r1...,2"'-l- - - - - - - "S"'""'"""'"du•OL
r>L
' 2,_F."'
,X°'P"''=•"
4.
The diagram shows 12 solid spheres packed tightly into a c losed cuboid such that
all the spheres iln: touching each other and at least one face of the cuboid.
(a)
Jfthe
radiu~
of each sphere is 5 cm, fi nd the dimensions (length, breadth,
height) of the cuboid.
(h)
(c)
Find the volume of one solid sphere. (Volume of a
sphere~
4
3
nr J)
The 12 sohd spheres are melted into liquid state and poured into the same
cuboid. Calculate the level of the liquid in tile cuboid now.
(a) _
cm by_
cm by _
_cm f3J
3
(b) _ _ _ _ _ cm [21
(c) _
cm (3)
:!_\\SS Enft.of·Yt"3t El::.mjnation 2011
6.
f\ J ath~m•1ig
Stto-ndJir>· 2 Expl"Mt
<Part 21
The diagram below shows a solid right circular cone with a curved surface
arcu of I44n c m 2 and base radius 9 tm.
(a)
Find 1hc length of OP.
(b)
Find 1he volume of the solid cone, correct to J d ecimal place.
.
. .
I
,
(Curved surface area ol a cone ~m-1; Volume ol a cone = - trr · fi)
3
(a) _
_ _ _ cml21
(b) _ _ __
cm·' [3j
Page 7 <)f9
,J\.VSS End-of-Yt":.r £~:.min:lfion 20_1I
7.
(a)
(b)
Sctondar\>' 2 Expr.:ss
y is inversely proportional to (4x + 5).
(i)
Given lhat y ~ l when x = 2.
find an equation relating x and y.
(ii)
Find the value ol v when x
.
.
-
=-5 .
4
A rope is cut into 3 pieces, A13, BC and CD respectively. The length
of CD isx cm and AB isy cm.
If AC : CD = 2 : l,
AB : BD = l : 3 and BC= 5 cm,
find the length of CD and AH.
(a) (i) _ _ __ _ _ _ __
_
_ [2]
(ii)y ~ _ _ _ _ _ __ __ (21
(b) CD = ·- - - cm; AB = _ _ _ cm [31
Page 8 of9
~1athcruati('\
J\\'SS End..of.. Yt-ar fu:aminalion 2011
8.
(Pllrt 2)
Srcondar)' 2
Exprc~,
A nswer the w hole of t his questio n on a piece of gra ph paper.
The following shows a table o r values for y = 5- 2x - x 1 •
-IE_R
01 'f2 j
~
x
~ffi2
-4
y
(a)
-3
~
- -
5
p
---
)
2
·- -
~
--
Calculate the value of p.
[I J
(b)
Using a scale of 2 cm to represent I unit on both the x - and y-axes, <lraw
the graph of y = 5 - lx x 2 for - 4 :> x ~ 2 .
[31
(c)
Draw an<l write down the equation of the line of symmetry.
r11
(d)
By drawing an appropriate line, use your graph to solve
5 - 2x - x ' =I.
(21
End of Part 2
P~gc
9 of 9
JURONG WEST SECONDARY SCHOOL
End-of-Y ear Examination 2011
Mathem atics Part 2
Secondary 2 Express
Answer Key
~
Answer
20
16 -- - - ·- ii - bi Minx = 8
.__ - 1ii- - Mean= 2.5
Median = 2 .5
iii
ai
3
2
On
'=1Tai
------
~
-
--
----
-
-ii -
-
10
2
._j_ _ _ __
bi
3
.
- -
L- -
-
-
~
ii
iii
-- - - - - --
-
8
- - 000
.2- 1·- : 50
- -- ~
-~
a
L4
b
1---
.£..
2..
~
6
a
c
b
7
-
~~
'-
--
3
8
7
3
,__
- - -- -- -
~---
- -----
- --
-
4.9km
- - 18 crTT""
30 cm b')'. 20 cm by 20 cm
524 crn 3 _ __ - -
--
- ·
- -- ---
- -
10.5 cm
x = 7.48, y_ = 16.3
h = 8.78
16 cm
1122.1 cm~
ai
13
y= - 4x+5
ii
b
a
c
d
v:: 1.3
CD = 4cm, AB = 3cm_
missina value =_§__ _ _
x =- 1
x = - 3.2±0.1, 1.2±0.1
----
---
-
-
Index No :
Name :
-.-
I
Calculator Model
C lass: 2E
- -
-- - - - -
Parent's Signature:
Actual Grade :
NORTHLAND SECONDARY SCHOOL
Motivated Learners, Assets to C"mmunity
N urt11rit1f; Mi11ds, Shaping Character, Stre11gtlte11illg Vix our
- - - - - -- --
-- - - - - - - - - - -
FINAL EXAMINATION 2011
Subject: MATH EMATICS
---Level : Seco ndary 2 Express
- - ·- - - -
-
- --- - - --- -
Durat ion : I hour I 5 minutes
- --- - -- - - - - Setter : Ms C inuy Oh
Vetter: Mr Low KM
Paper: I
-- - --
- - -- - -- - -
Date: I O'h Oct 2011
Time : 7.40 arn to 8.55am
- - - - - -
-READTHESF.INSTRUCTIONS
- - - -- -- - - - - FIRST
-- -
--
- - ---
- - -- - -
~-·
-
Write yt•ur index number and uamc at the top of this cover page.
Write in dark blue or black pen.
You may use a pencil for any di agrams or graphs.
Do not use staples, paper clips, highl ighters, glue or correction fluiu.
Answ1.:r all questions.
If working is needed for any questio n it must be shown in the sp:ice provid\.xL
Omissio n of essential working will ro;sult in loss of marks.
Calculators :;ho11ld be used where appropriate.
If the degree of accuracy is not specified in the question, and if the answer is not exact, give
the answer to three significant figures. Give answe rs in degrees to one decima l pl~cc.
For :r, use either your calculator value or 3. 142, unless the que~tion requi res the answer in
terms of ;r .
At the end of lhe examination, fasten all your work securely together.
The number of marks is given in brackets [ l at the e nd of each question or pan question.
The total of the marks for this paper is 50.
____
._
This paper cOi1sists o f 11 printed pages including the
.:;we;. pa~e
--
_J
Answer all questions
1.
(8.5) 1 +60 - \/2345
giving your answer to 3 decimal places.
Evaluate
JJ36xS -
Answer:
2.
(a)
(b)
_
_ __
[I]
_
Express 0.75% as a traction in Its simplest fonn.
The usual pri ce of on airline ticket is $420, inclu sive of7% GST.
(i) Find the price of the airline ticket before CiST.
(ii)
Find th<! percentage decrease in its usual price if the price drops to S294
during a sales promotion.
Answer:
(a)
-
(b)(i) $
(b)(ii)
-
-
-
-
--
[I I
- - -·- - - I l J
- - --
-- -- -
%
[21
---3
3_
(a)
(b)
Expand and simplify (3y - 2) 2 - 9y(y - 5).
Given that 5x
7y
=3(x -
8x .
4y), find the value of -
y
..
Answer:
4.
(a)
Factorise Sx' - '.!O completely.
(b)
2
Solve 6x - x
(a)
[2]
(b)
121
(a)
(2)
(b)
[J l
= I.
Answer:
5.
Written as the product of its prime factors,
160 .= 2} x 3' x 5
(a)
'Write I 08 as the product of its prime factors.
(b)
Find the smallest positive i nteger k such that 360k is a cube number.
(c)
Two lights 11ash every I 08 seconds and 360 seconds respectively.
Ir the
..
two lights Oash at 11 nm. \>then will they next flash at the same time?
Answer:
6
'tl .
I
.
Ma kc h the subject
ol 1e tommla -
!>'
/Ir
= "y
- ---
(b)
-- -
(c)
_
_ __
(I]
_
[2]
a
2h
Answer:
---------- - -~
(1 J
(a)
Pl
7.
A
J
'
\
•
c
l\}'>')o
D
/\
2' ,J \
11
a
P
,
.
E
From the diagram above, find
(a)
L CBK
the reflex angle
LBKH.
(b)
of
i
Answer: (a)
(b)
L CBK =
(I
[I)
reflex angle of L BKH [2)
8.
The two circles shown have radii x and 3x. A point is chosen, at random, inside the
larger circle. Find, in its simplest fonn, the probability that this point is on the shaded
area.
Answer:
[2 )
6
9.
An actual ground distance of5/m1 is represented by a distance of2cm on a city map.
(a)
Express the scale of the map in the fonn I: 11 .
(b)
If the distance between two buildings on the map is 15 cm find their actual
distance on the ground in km.
(c)
If the area ofa stad ium is 0 .3 cm' on the map, !ind its actual area in km'.
An~wcr:
111
(a)
(b)
(c)
- -- - - km
[I)
km : [2)
I
7
10.
(a)
SMRT employed l 15 men to construct an underground tunnel and completed the
construction in 285 days. Find the number of days required by 95 men to
complete the same project.
(b)
There arc two numbers. When the second number is subtracted from twice the
first number, the result is 18. When 9 is added to the first number, the result is
equal to twice the second number.
(i)
Usingx as the first number and y as the second number, write down a pair
of simultaneous equations.
(ii)
Hence find the numbers x and y, .
Answer: (a)
(b)(i)
(b)(ii)
12J
days
Equation I:
(1 J
Equation 2:._ ___
[ 11
x=
- -· y
"'-
-
'
[3]
8
I I.
l'QRS is a rectungk. Given 1lrnt PT = 4 cm, TQ = S cm
~nd
QR = I 2 cm,
s
Q
12cm
Find
(a)
the area of 6.PRT,
(b)
the perimeter of. 6.PRT .
Answer:
12 - (a)
Shade the region representing
cm '
111
(b)
l'/11
(2)
fI]
(A n B)'
_ __
(b)
(a)
___J
Write in set notation the ~haded region as shown in the Venn diagram.
Answer:
(b)
lJl
9
13.
(a)
The weekly pocket mom:y of 300 children are shown in the table below.
Weekly J)ocket
money($)
Number of
children
40<xS45
45<x.S50
50 < x s 55
75
<)0
13 5
Calculate an estimate of the m ean week ly pocket money offhc children.
(b)
The table below shows th e marks of a science
Marks
Nurnhcr of
students
23
0
24
14
25
26
4
5
te~t
scored by a group of students.
27
x
28
3
Find
(i)
the smallest possible value ofx if the modal mark is 27.
(ii)
the largest possible value ofx if the median is 25.
Answer:
(a)
$ _ _ __
(21
(b)(i)
x=-
(I I
(b)(ii)
- - - - x= - --
[I )
10
14.
Consider the pattern.
1 1 3
s, :-+
- =1 2 2
1 I 5
s, . -+-=-
(a)
(b)
(c)
2
3
6
1
I
7
' '
S3 : -+- = 3 4 12
Write down the 11"' line in the pattern.
1-'incl and simplify an expression, in terms of"· for the ,,•h line in the pattern.
Pind the value of S, -S6 ~ S , -S, ~s. -S, 0 •
[I)
Answer: (a)
(h)
-
(c)
-
f2)
fl l
- - - End o.fJ>uper - - -II
Answer all <1uestions
(8.5) 3 + 60 -V2345
1.
F.valuale
r.-.
'1l36x8
giving )i()tlr ar\SWCt !() 3 decimal places.
..
Solution:
20.035 (Bl)
Answer:
2.
[II
(a)
Express 0.75% as a fraction in its simplest fonn.
(b)
The usual price of an airline ticket is $420, inclusive of7% GST.
(i) Find the price of the airline ticket before GST.
(ii)
Find the percentage decrease in its usual price if the price drops 10 $294
during a sales promotion.
(a)
Solution:
0.75%
0.75
= -1()()
3
400
(b)
(i)
IBlj
Solution:
107%--) $420
100%
(ii)
420
> - xlOO
107
- $392.52 [D 1]
Solution:
420 - 294 xJOO (Bl)
420
= 30%
(Blj
Answer:
l I]
(a)
(h)(i) $ _
(b)(ii)
_
_
- - - - -%
{1)
[2)
3
3.
(a)
(b)
(a)
F,xpondand s implify (3y-?.) ; - 9y(y- 5) .
8x
.
y
Given that Sx - 7 y - 3(x - 4 y). find the value of -
Solution:
(3y- 2) 2 - 9y (y - 5)
= (9y1- 12y+4)181j -9y' +4 5y
= 33y +4 [Bl]
(b)
Solution:
5x - 7y = 3(x - 4y)
5x- 7y -dx - 12y
5x -3x = - 12y+ 7y
2x - - Sy [Bl )
Rx ' - 20y
Bx = - 20
[Bl}
y
Answer:
(b)
- - -4.
(a)
(b)
(a)
{2)
(a)
- - - -
- - --- -
(2 ]
Factorise Sx' -20 completcly.
Solve 6x -
2
x
= I.
Soluti on:
S.x' -· 20
~ 5(x'
- 4)( Bl]
-= 5(x - 2)(x + 2)( RI]
(b)
Solution :
2
6x -- = I
x
2
6x - 2
·-"' I
x
6xi - x - 2 = 0 (Bl]
-
(3x - 2)(2x + l) =0 (81)
2
x=-
3
or
I
x =- -
2
(Bl for both correct answers]
Answer:
(a)
[2)
(b)
[JI
4
5.
Wrillcn as the product of its prime factors.
360
2 1 x3 2 x5
(n)
Write I 08 as the product of its prim<.:: factors.
(b)
Find the smallest positive integer k s uch that 360k is a c ube number.
(c)
Two lights nash every IOS seconds and 360 seconds respectively. If the
two lights flash at I I am. when will they next Oash at
th~ sam~ time?
Solution:
(b)
k = 3x5
(c)
360 = 2'x3:x5
75 (Bl)
108 = 2 2 x~J
LCM =2 3 x3Jx5= 1080s [Bl]
1080s = 18 min
:. 11 18am
[Bl)
/\nswer:
6
(:i)
(I)
(b)
(I]
(c)
[2)
I _ fh- a
Make: h the subject ol' thc fonnuln s ~ ~ :;.. h
2
Solution:
/h-a
s=v2iiI
(
~)
2
•
,s
I
s2
2/r
h- a
2h
(Bl]
2h
s 1 1r - s 2 a
h-a
.~ 2 a= s 2 h - 2h
s;a = h(s 2 - 2) [81]
s 2a
Ir= - 2
(s
-
2)
[Bl)
Answer:
[ 3)
5
-j\5- f)
I \
7.
\
\
J-
From
(a)
(b)
(a)
the diagram above, find
(b)
Solution:
L CBK
the rencx angle of L BKH.
Solution:
L.CBK 55° (Blj
reflex LBKJI =(180" -45°)+(180° - 55°) [Ml]
reflex /RKJI
= 2609
[Al)
Answer: (a)
(b)
0
LCBK=
[ IJ
reflex angtc of LRKH l2)
"
8.
The 1wo circles shown have radii x and 3x. A point is chosen, al random , inside the
larger circle. Find, in its simplest form, the probability that this point is on the shaded
area.
Solution:
;r(3x) 2 - :r(x) 2 = 8;ix 2
[MI]
Answer:
~~~~~~~~~
·~~-====~=
[2)
6
9.
An actual ground distance of 5 km is represented by a distance of2 cm on a city map.
(a)
Express tht: scale ot'thc map in the fonn I: 11.
(b) If the distance between two buildings on the map is 15 cm lind their actual
distance on the ground in km .
(c)
(a)
If the area of a stadi um is 0.3 cm 2 on the map, find its a~tual a~ea in kn/ .
Solution:
2cm :Skm
2cm:5000m
2cm : 500000cm
I : 250 000 [RI]
(h)
Solution:
2cm -7 5km
15cm -+
( c)
5
x!5 =37.5km [Bl]
2
Solution:
2cm :5km
:
1
4cm :25km
2
1
4cm --+ 25km
0.3cm 1
[HI]
1
25
> - x0.3
4
J .875km
2
[Bl]
Answer:
(b)
10.
(a)
(J]
km
2
----- km [2]
- ---- -- - - - - - ·- ---- --- --- - - - - - (c)
·- -
[I)
(a)
SMRT employed I l 5 men to construct an underground tunnel and completed the
construction in 285 days. Find the number of days required by 95 men to
complete the same project.
Solution:
J l 5 men --> 285 days
-
285x115
- 1Ml] = 345days[Al]
95
9:>me11-+ -
7
(b)
There are two numbers. When the second number is subtracted from 1wice the
first number, the result isl8. When 9 is added 10 the first number, the result is
equal to twice the second number.
(i)
Using x as the first number and y as the second numh1:r, write down a pair
' '
.
of simultaneous equations.
(ii)
(b)
(i)
Hence find the numbers x and y, .
Solution:
2x-y=18 (81]
x+9=2y (Rt}
(ii)
Solution:
2x - y=18
2y - x=9
- -- -
(I)
- - - ---(2)
(1)x2
4x - 2y=36 - - - (3)
(2) + (3)
(2y - x)+(4x - 2y)
9+36 [Ml}
3x=45
x= l5 [Al]
2y-15=9
2y=24
y = l2 LAI]
Answer: (a)
(b)(i)
_ _ _ _days
Equation I:_ _ ___ [l)
Equation 2: _ _
(b)(ii)
L2J
_ _ [I]
x= _ _,, y = _ _ _
13J
8
9.
An actual ground distance of 5 1011 is represented by a distance o f 2 cm on a city map.
(b)
Express the scale of the map in the form 1: n.
If the distance between two bui ldings o n the map is I Scm find their actual
distance on the g round in km .
(c)
I f the area of a stadium is 0.3 cm 2 on the m ap, find its a<;tual area in km 1 •
(a)
Solution:
2cm : 5km
(a)
2cm :5000111
2cm : 500000cm
I : 250 000 (RI]
(b)
Solution:
2rm > Skm
•
bcm
-..+ -5 x l 5
37.Skm (Bl]
2
(c)
Solution:
2cm: 5km
4cm 1 :25/m1 2
[lll)
2
4cm --+ 25km '
O.kmi --)
~5 x 0.3 =
4
l.815km 1
[HJ]
Answer:
(b)
(c)
JO.
(a)
[I J
(a)
km
[I I
kn1' [2]
SM RT employed 115 men to construct an underground tunnel and completed the
construction in 285 days. Find the number of days required by 95 men to
complete the same project.
Solution:
I 15 men --+ 285 days
95 men --t
285x1 15
95
[Ml)= 345 days [Al ]
7
11 .
PQRS 1s a rectangle. Given that PT= 4 cm, TQ = 5 cm and QR = 12 cm,
s
p
'.
12cm
Find
(a)
the area of t!J'RT.,
the perimeter of !!.PRT .
Solution:
(b)
1
•
llPf?T = - x 4x12 = 24r m '
2
Solution:
(a)
(b)
[Bl)
9
PR= J9 + ll? =15}
TR = N 1121 =13 [Ml)
Perimeter
=15+Llt4
=32cm [Al]
Answer:
(a)
(b)
12.
_ _ _ _ cm 1
cm
(a)
Shade the region representing
(b)
Write in set notation the shaded region as shown in the Venn diagram.
(A 11 B)'
[1)
(2]
[lj
l I)
Solution:
P'uQ
9
13.
(a)
The weekly pocket money of 300 children are shown in the tilble below.
Weekly pocket
money($)
Numherof
children
40
<:x~45
45<x550
50 < x s 55
90
135
75
..
Calculate an estimate of the mean weekly pocket money of the children.
Solution:
1>1ec111
= (75x42.5) r(90x47.5) +(135x52.5)
- - r·1]
,. 1
1l.1ean
=- -
Mean
=$48.50
(b)
The table below shows the marks of a science test scored by :a group of students
..,
300
14550
300
[Al)
Marks
Number of
students
23
0
24
14
25
4
26
5
27
x
28
3
Find
(i)
the smallest possible value of x i f lhe modal mark is 27.
(ii)
the largest possible value of x if the median is 25.
b(i) Solution:
15 [J31J
b(ii)
Solution
17 =5+x+3
x=9 (RI)
Answer:
(a)
$
(b)(i)
x=
-----
[ 1)
(bXii)
x•
---
[I J
[2)
10
14.
Consider the pattern.
1 1
3
I
7
Si : - + = 1 2 2
1 I 5
s, :-~
-= 2 3 6
1
S3 :-+-=-
(a)
(b)
3 4 12
Write down the 11"' lin e in the pattern .
Find and simplify an expression, in lcnns of 11, for the nlh line in the pattern.
(c)
Find the value of Ss - S, + S, - S, +
(a)
Solution:
1
II
II
12
132
Solution:
=I
S
"
= ~+I)+ 11
+ _ 1_
11
11
+I
n(n + 1)
s. = -1 + -1- (HI)= ~ 1
(c)
0•
=_!_+ - =~ [Ill)
s
(b)
s. - S,
n n+I
Sol ution:
11(1111)
[Bl]
(~+*)- (~+~)+(~I ~)-(i+ ~)+(i+~) c~ +fi)
=
I
5
I
6
I
6
I
7
I
7
I
8
+ - - -- + - + - -
I
I I
I
I
I
- - +-+ - - -- - 8 9 9 10 10 11
=- - S
=-
6
55
11
[Bl]
Answer: (a)
[I)
(b)
f2]
(c)
- --
-
- -- - --- - -
( 1J
-End ofPoptr - - - -
II
NORTHLAND SECONDARY SCIIOOL
11fotivMed Lea,rners, Assets to Com111u11ity
l\'11rwri11g Jlt/inds, Shapi11g Character, Strengtheni111: Vigour
--
-=--
~-
FINALEX.AMINATION2011
ubjcc1 : MATHEMATICS
- - - -- - - - - -- -
~Lev_~~ : _Secondary2 Express _ _ _ _ - - - - Duration: I hour IS minutes -- -
- - -- -
tter : Ms Cindy Oh
etter: Mr Low KM
REAU
TH~R
- -
Paper: 2
-- - -- - -- -
- - -- -
Date: IO'"Oet201 I _ _
_ _
Time : 9.00 am to 10.ISarn _ _
- - -- - --- - -- - - - - -
- - - -- - -·- - - - - - -
INSTRUCTIONS FIRST
Write your index number and name on all the work you hand in.
Write ir 1 dark blue or black p.:n on both sides of the paper.
You may use a pencil for any diagrams or graphs.
Oo not usc staples, paper clips, highlighters, glue or correction lluid.
Answer all questions.
If working is needed for any question it mus l be s hown with the answer.
Omission of essential working will re.~ult in loss o f marks.
Calculators should be used where appropriate.
lfthe degree ofaecuracy is not specified in the question, and if the answer is not exact, give
the answer to three signi ficanl figures. Give answers in degrees to one decimal place.
For 7t , use either your calculator value or 3. 142, unless 1he question requires the answer in
tenns of ;r.
Al the l!nd of the examination, fasten all your work securely together.
The number of marks is given in brackets ( ) at the end of each question or part question.
The ·total of the marks for this paper is 50.
This paper consists of 6- pri nted pages including the cover page
Write your
1.
answer~
Answer all questions.
on separate wr it ing paper p1·0,·ided . (50 marks)
A classroom has 40 chairs. During a Music lesson, 25% of the chairs arc vacan1
while all the remaining chairs are occupied. l flhc boys sil on
chairs, calculate:
2
3.
2
5
of the occupie<l
' '
(a)
the number of cha irs which are occupu:d by the boys.
l2]
(b)
the ratio of number of girls to the total number of chairs in the c lass.
[2]
Givenrhal x+y = 7and xy= - 18 ,
(a)
Show that x 2 + y 2 = 85.
[2]
(b)
l lence or otherwise, evaluate (x- y)2.
[2]
(a)
<f + 3
(b)
(3)
(Zq)~ ~_!!_
1
1
Simpli fy
"..,..press - 43x -2
=!!__
q + 2q - 3
3- x + :>
-
a5
' .
. . I ,.
a s '111gl"e 11act1on
111 its sanp est 1orm .
l3]
--- - -- - - -- - -
4.
Given the above regular polygons, find the va lues of x , y and =·
5.
(a)
(b)
[3j
Given that a is inversely proportional to lie+ I and U1at a ~ 7 w hen c - 63.
(i) Express a in terms of c.
[2)
(ii) Find the value o f c when a - 14.
[1)
y is directly proportional to the square of x . It is known that y = I I for a
[21
particular value of x. Find the value o fy when thi s value of xis doubled.
3
6.
/\universal sel E and its subsets A. H :md Care given by
= {x: xis an integer for
e
A~
2 ~ x S"B },
{x: xis a factor of30},
B - {x : x is a prime number} ,
' '
C
= {x: xis a positive integer and
2x +I~ 7 }.
(a)
State the value of n(A).
[ I]
(b)
List the clements in each of the following sets,
(i)
(i i)
AuB
c
fl]
[ I)
C n B'
{IJ
- - - - -
(iii)
7.
p
\.
\.
\.3i: cm
'\
z +4)
A
cm', h
\
\.(3;r - 5l c:n
"
Q '-'---(..:J-..---~-L-lf-1- - - " J<
PQR is a right-angled triangle. A and B arc points on PQ and PR respectively
such that AB II QR. BR is (3:c - S)cm, AB is (:c + 4)cm , PB is 3x cm and QU is
(4:c - S)cm.
(a)
Using similar tliangles or olhcrwisc, show that 3x 2 -17 x + I 0 = 0
1
(h)
Solvetbcetiuation 3x
(c)
Hence, find the length of PQ.
17x+ l0 = 0
[2]
[2]
[ I]
8.
26 rrt1
.... .. -.. .
Di"lrun I
(a)
A model consists o f a solid hemi sphere altached to a solid cylinder
is shown
in Diagram I. T he height of the cylind er is 26 cm and the base area is 198
cm 2.
(i)
Find the radi us of the base.
(11
(ii)
The model in diagram I is full o f water. Ca lculate the amount of water
[31
in the model in litTes, giving your answer to 3 signi fican t figures.
(iii) lftbe water drains from tht: model at a constant rate of0.05 litres per
[I)
second , calculate the time taken to empty the water in th e model. in
minn1cs and sec{mds.
"~ -.
removed
Oi.tgre.sn 2
(b)
Part of the cylinder in the shape of a right circular cone is removed as show11
in Diagram 2.
(i)
Given that the volume of the cone removed from the model is
(2 1
630cmJ, calculate the height o f the cone.
(i1)
Calculate the total ~urfoce area o f the remaining model as shown m
(31
Diagram 2.
- - - - - - - -- - --
5
<) _
A11swer tire whole ofthis q11estio11
011
a sheet of graph paper.
A hall is thrown upwards from the top of a building where It is the height of the
ball above U1e ground in metres and
t
is the time of the flight in seconds. TI1e
motion is given by h = -1' + 41+8.
Some corresponding values oft and It are given in the following table.
(a)
Calculate the value ofp.
[I]
(b)
Using a scale of2 cm to represent I 1111it, draw a horizontal 1-axis and a
[3)
scale of I cm to represent I unit, draw a vertical Ii-axis, draw the graph
ofh
= - 1 2 +4t +8.
(c)
Find the maximum height of the ball.
[I)
(d)
What am the two possible times where the basketball is at I 0 m?
(2)
(e)
At the same instant. a second ball is being projected into the air with its
motion represented by h
ground in metres and
(i) Draw tl1e graph
I
=t + 8
where It is the height of the hall above the
i,,; the time of the flight in seconds
reprc.~enting
the motion of the second ball on the
r11
same axes.
(ii) Hence, find the time when both balls collide.
- ------- End of Paper -------
[!]
Answer all questions.
Write yo ur answers on separate writing paper p r ovided. (50 ma1·ks)
I.
A cla-;sroom has 40 chairs. During a Music lesson, 25% of the chain;. are vacant
while all the remaining chairs are occupied. If the boys sit on
~of the occupied
)
chairs, calculate:
(a) the number of chairs which are occupied by the boys.
(21
Solution:
75
.•
.
x 40 =30chmrsoccup1ed [Bl]
100
.,
~: x 30 = 12 chairs occupied by boys [BI)
5
(h)
the ratio of number of girls to the total number of chairs in the dass.
[2)
Solution:
30 - 12 = 18 chairs occupied by girls [B 1)
18 : 40
9: 20 [Bl)
- - - 2
Given that x + y
(a)
= 7 and
-·- - ---- - - - - - - - - - -
~y = - 18 ,
(2)
Show that x' + y2 = 85 .
Solution:
= x 2 + 2,ry + y2
(x.+ y)2
2
(7) "' x
2
+ y' + 2(- 18)
-l
x 2 + y 1 =49+36
1
x +/
(b)
(Bl]
=85 (shown)
J [B 11
Hence or otherwise, evaluate (x - y}'.
Solution:
(x - y) 2 = X 2 - 2.xy+ y·, (Bl l
~
1
l
(x. - y)"=x +y - 2~y
(x - 1>) 2 = 85 -2(-18)
(x- y/ .,, 121 (Bll
- -·
- -
- - - - --
[2)
3.
(a)
1
2
Simplify <2q> + q - q
q+3 q 1 +2q - 3
Solution:
(2q)2 . ql - q
[3]
q+3 .,. q;-;2q - 3
1
q(q - 1}
4q
=- .,.•
(q + 3)
(q + 3)(q - 1)(81)
=--2L_x(q i 3)(q-!} (Bl)
(q +3)
q(q-1)
=4q [Bl )
(b)
4
3
Express - - - - - as a single fraction in its simplest fonn.
3x - 2 x+5
[3]
Solution:
4
3
3x - 2
x +5
[ R1 for (3x - 2)(x + 5)J
4(x + 5) - 3(3x 2)
(3x - 2)(x + 5)
= 4x+ 20 - 9~ +6
[Bl for -9x+6]
(3x 2)(x+ 5)
26-5x
[Bl]
(3x - 2)(x t 5)
; --
4.
/
\
Given the above regular polygons, find lhc values of x, y and z.
Solution:
(6 - 2) x I 80~
= 720°
L ' ~ no= 120° [Bl]
6
180° - 120_'.'_ = 300 [Rl]
2
L.z - 360" - 2(120°) = 120" [Bl]
Ly
[3]
5.
(a)
Given that a is inversely proportional to ~./~:-;] and that a - 7 when c
(i)
Express a in tcnns
= 63.
or c.
[21
Solution:
k
a =
' ·=
l.ic + l
k
7 = - -V63+ I
k = 28[BIJ
28
:. a = - ,=
~c+I
(ii)
[Bl)
Find the value of c when a = 14.
[I]
Solution:
28
:.14 = - =
'Jc+ I
Vc+1 -= 2
c=7
(b)
[Bl]
y is directly proportional to the square of x. It is known that y = 11 for a
particular value of x. Find the value ofy when this value of x is doubled.
Solution:
y=lcr 1
11 =kx~
11
k =-
xi
[Ml]
\Vhen x = 2x
y
=k(2x}2
11
,
2x x 4x"
y=44 (Al]
y=
[2)
6.
A uni versal set
E
f.
and its subsets A, Band Care given hy
= {., :xis an integer for
A~
2 $ x SS),
{x: x 1s a factorof30},
..
fl = {x: x is a prime number},
C
= {x : xis a positive integer and
(a)
2x + I !> 7 t.
State the value of n(A).
[ I]
Solution:
={2,3,4,5,6,7,8}
A ={2,3,5,6}
H = 12,3,5, 7}
F:
C = {2,3}
11(A) - 4 [BJ)
(b)
List the elements in each of the following sets
( i)
Solution:
c
[I)
C :. {2,3} (81 1
(i i)
AvB
[11
Solution:
Av B = {2,3,5,6,7} [Bl J
(ii i)
C riB'
[IJ
Solution:
C nB' =
{¢}
[BI]
- - - - -- - - - -
7
7.
E'
'\
'\
\ JT. cm
\
A
\
(z + 4) cm\
h
~,(3x-5)c:n
"\
'
Q '-'-- - -- -----").
(4x- :) cm
PQR is a right-angled triangle. A and 8 are points on PQ and PR res pecti vely
such that AR II QR. BR is (3.t
S)cm, AR is (x + 4)cm, PB is 3x cm and QR is
{4.x - S)cm .
(a)
1
Using similar triangles or o therwise, show that 3x -17x+ I0 = 0
f2]
Solution:
x+4
3x
4x - S
3x+ (3x - 5)
x~
= 3x - rs11
4 ' - 5 6x 5
(x + 4)(6x-5) = 3x(4x - 5)
4
6x' - 5x+24x-2 0 = 12x 1 - 15x
6x 2 +19x-20 = 12x 2 - 15.x
(j< - 34x 120 = 0}[1/lj
(b)
3x -I 7x + 10 = 0
Solve the equation 3x 1 - 17.x+ 10
=0
(2)
Solution:
3x 2 - 17x+ l 0
0
(3x - 2)(x - 5) - 0 (RI )
2
x =-
(c)
or x = 5 [Bl]
3
Hence, find the lcng1h o f J>Q.
[ 1J
Solution:
Using x
=5
PQ = .J25' - 15 1
PQ
= 20
[Bi i
s
8.
26cm
.. .. ...........
---
Diagiem I
(a)
A model consists of a solid hemisphere attached to a solid cylinder is shown
in Diagram I. The hei~I of the cylinder is 26 cm and the ba~e area is 198
cmz.
(i)
Find the radius of the base.
lI ]
Solution:
711"
r
(ii)
2
=198
7.938851143
r = 7.94 cm [Bl]
The model in diagram l is full of water. Calculute the amount of water
[3)
in the model in litres, giving your answer to 3 significant figures.
Solution:
3.711"~ +(198)(26)
3
= 3_n(7.938851143f +(198)(26) (M l, Ml]
3
- 6195.92835 lcm 1
= 6.195928351 f
6.20f [Al]
(iii) If the water drains from the model at a constant rate of0.05 litn.-s per
111
second, calculate the time taken to empty the water in the model, in
mintues and seconds.
Solvtion:
0.05f >Is
6.20f - > 124s =2min4s fBI]
9
(
-~
'
20an
;,:::: ~ r. . ~_~:_.
>
•
-
rc1novecl
;_-i_ __ ___/
....____..
Dic..tp111112
(b)
Part of the cyli nder in the shape of a-right circular cone is removed as shown
in Diagram 2.
(i)
Given that the volume of the cone removed from t he model is 630
cm3,
(2)
calculate the height o f the cone.
Solution:
.!.m- h = 630
2
3
.!.;r(7.938851143)' It = 630 [Bl]
3
h
~
9.545454 544 cm
Ii "' 9.55cm [BI)
(ii)
Calculate the total surface area o f the remaining model as shown in
[3]
Diagram 2.
Solution:
Surface area of hemisphere
= 2m'
Curved surface of cylinder
m
2mh
Curved surface of cone
= ml
I -/ h>.;;'
I = J(9.545454544)' +(7.938851143) 2
[Ml]
{ = 12.4 153558 1
Total surface area
= 21lr 2 + 2mh +ml
[M 11
- m-(2r + 2/i r /)
"' n(7.93885 I 143)(2 x 7.9388511 43+2x26+12.41535581)
= 2002.80038
"'2000cm' IA I J
10
P-
~
..
.
. _).__ _:____ ~-_,
'·· ~
;.
-.
0
------ - End "f Paper -
--
12
Mathematics
Paper 1
3
4.
Ex1)ress the following as a s ingle fraction in its simplest form
(a)
4x 2 - 25
6/-11.x - 10 '
2
x- 2
x+5
x -4
(b)
- 2- - - - .
(c)
- - +- .
3
4
x+2
x
I
-
- - - - - -- - -
Ans : (a) . . . . . . . .. . . .. • .. . .... ... .. . . .. .
12)
(b) ...... ........... ····· ······ ··
[3]
(c) .. ....... •. ..... ...............
[3]
4
- - - -- 5.
Factorise completely
(a)
2x 1 -9x 5 ,
(b)
(i)
y 2 - 81.
(ii)
Hence, calculate 87.655 2 -1 2.3452 • Show your workings clearly.
'.
Ans : (a) . ... . .. . ... ... .. . .... .. ... .. .. .
[2j
(b) (i) ....... .. ...... . .. .....
[I ]
f2J
(ii)
6.
L
(i)
Given the fonnula 3s _ !_ '- p . Make s the ~ubject of the fonnula.
(ii)
Hence, find the value of s if r
2r
~------
=2
and p = - I .
--
Ans : (i) ... ........... ............ ... ..
(31
(ii) . . . . . .. . .. . . .. .. ..... .. . . . . . . . .
[21
- -
-
--..,..,----
"Hht oa1n1~"'"PIPfJ •tlcP.11~) or P'll\o.: 'I• Se-c.<1n.!i:ir) S<liool, 11 muu llOC kdwil..::11re.:t "' p.t vo .....,~ •
-
·-- - -'
5
Expand and simplify
8.
(a)
(2t 3)(x + 4),
(b)
(x - 1) 2 - 3(x -~ 2)(x- 2).
Ans : (a) . . .. . ... . .. ... .. .... ... . . . . . . ..
[2)
(b) .... · ········ .......... . ... ... .
(3)
On a certain map, an area o f 100 crn 2 represent an actual area of 4 km2•
(i)
If the map has a scale of I : n, find the value of n.
(ii)
If the distance between point A and point Bon the map is 5 cm, find the
actual distance, in km, between the two points.
Ans : (i) n = . . .. .. .. . .. ......... ... .....
Pl
(ii) .......................... .. km [2)
1
I I. Solve the simultaneous equations
3y+ 2x = l4 ,
4x= l3 - y.
Ans:
x =..........,y = . ........ . (3)
12. Solve
(a)
- 18x2 + 9x+ 2= 0 ,
(b)
x+(2x - 1)2 = (2x+3)(x - l) -7x t 9 .
Ans : (a) . . . . . .. . .. .. . . .. . ... . . . . . . . . . . .
(b) ........ .
End of Paper
•rllis u ..._,..,. ..,. •
Ille- rop:rry tliPillc Yf S.9flJM) ~ ti e ...'ll -
btt:lui'flC•~ • Plftqf ..,..,_. p
[2]
~I
S1.;c 2 E End Year Examination Maths Paper J Answers Year 2011
Qn
Part
l.
(i) (ii)
2. 1 (a)
Mi1rks
840
839.3
BI _
-
'~: -:;oy
·1 \
t.
··B l ·
-
Comments
-
Bl
._,,
\
" )
J
.-:' - '·' _,.,.'
(b)
-
•
.
.I
'(
•<)
: ' •/
·, ;_,
-;,. ,
'·
.. ..>:....,~(
-.-3.
x ·9
4.
I
(a)
(I.)
!
I
I
I
I
I
(c)
01
i ..~ - ·--. - - ; ,......'-'
iI/
..
.
22
(2x - 5)(2x+
(2x - 5)(3x + 2)
2x+5
=-3x+2
2(x ·~ 2)
x +5
(x-2)(x 1 2) (x - 2)(x + 2)
x+ 5 - ? (x+ 2)
= (x - 2)(x+ 2)
r+ 5 - 2.r - 4
=(x - 2)(x + 2)
- x+ I
=(x - 2)(x+ 2)
-
--
-
I
02
I
-
Ml
Al
Ml
--
Ml
-
Al
4(x+ 2)
+ - x + 2(x) x(x+ 2)
3x+ 4(x+2)
=x(x+ 2)
3x+4x+8
= x(x + 2)
7x+8
=x(x+ 2)
3(x)
-
-
-i
i1
I
Ml
Ml
Al
-~
-
5.
6.
(a)
(b)(i)
(b)(ii)
(i)
I
(ii)
(a)
(b)
8.
'·-
9.
(i)
(ii)
(j)
(ii)
(iii)
Ml
"A l
3s(2r) s p(2r)
=-2,.
2r
2r
6sr - s = 2rp
s(6r - 1) = 2rp
2rp_
s=
6r-l
Ml
Ml
Al
2(2)(- 1)
s =-·.
6(2) - 1
Ml
-4
Al
s=
7.
132
01
(2xi· l)(x-5)
(y - 9)(y1 9)
(87.655-12.345)(87.655 + 12.345)
= 75.3lx l00
=7531
11
-
2x 2 +8x-3x - 12
= 2x' +5x 12
x ' - x xtl -3(x 2 - 4)
=x 1 -2x+l -3x' t12
= - 2x 2 - 2x+l3
Ml
-
lO crn: 2 km
10 cm: 200 000 cm
I cm : 20 000 cm
J : 20 000
I: 20000
5cm : 100 OOOcm
Actual distance " I km
RD= 9cm
LBED =LACB
= 180°- 90°-38°
= 52°
I
Area = - x9x6
2
= 27cm1
Al
Ml
Ml
Al
-
Ml.
Ml
Al
Ml
__.__ ~A~I---''---~~~~·-'
Bl
Ml
Al
Ml
Al
I 0.
k.~J
5
!
When 5 = /aJ is doubled.
y = k(2x) 1
y=Rla'
l y 8(5) = 40
I J.
Met hod I ; Substitution
Suh y = 13-4x into Eqn (I)
3(13 - 4x) + 2x = 14
39- 12x + 2x =14
x 2.5
Sub x 2 .5 into Eqn (2)
y 13 - 4(2.5)
y=3
: . x =2.5, y =3
1 - -- f -Method 2: Elimioation
Ml
' '
l
Ml
Ml
Al
Ml
6y+4x = 28 - >Eqn(3)
Eqn (3) - 2 : 5y =15
y=3
Sub y =3 into Eqn {2)
i-.
112.
I
13 - 3
Ml
:.x =2.5,y 3
Al
x
.
Al
I
l Eqn (l)x 2:
4x ~
=2.5
- -- - - -
(a)
( -3x+2X6x+l) ~ O
Ml
Al
(b)
(· Jx-12)=0 or (6T+ 1) = 0
- I
2
x=or x= 3
6
x+(4x' - 2x - 2x+I) = {2x2 - 2x+3x - 3)-7 x+9
.
!
Ml
4x2 -3x+l=2x 2 6x +6
2x 2 +3.x - 5=0
(2x ~ 5)(x- I) = 0
(2x + 5)=0 or(x-1) = 0
-5
x =or x=I
2
- ---"--- --
Ml
--
Ml
I
Al
·-
-
-
Mathematics
Paper 2
3
~(a)
(b)
4.
If x ~ y = - 4 and xy =
6, find lhe value of (x+ 2)(y I 2).
Factorise R.ry - 98:ry 1 completely.
Ans:(a) .............. ... .... .. ........
[2]
(b) .. .. . . .. .. . . . . .. . ... . . .. . . ... . .
[21
Mr Lee and his family visited Ei ffel Tower in Paris. The tower wa5 320 m high.
From the lop of the tower, they saw the playground at 8 and the shop house at A,
650 rn and 690 rn away respectively as shown in the diagram hclow. Find the
distance between 1he playground and the shop house.
320 l.'ll
L_________________
A_n_s_:_ ._·_·._.._. ._._.._· ._.._.._.._.._........ m
Pl
J
4
5.
The cost of a cup of coffee and a bowl of chicken noodle is Sx and Sy
respectively. Tessa bought two cups of coffee and three bowls of chicken noodle
at a total <:Ost ofS 14. Jordan bought four cups of coffee and five bowls of chicken
noodle al a total cost of$24.20.
(i)
Write down 2 equations in terms of x and y.
(ii)
Hence, find the cost ofa cup of coffee and a ,bowl of chicken noodle.
Ans : (i) . . .... .. ........... . . ... ...... .
fl]
(ii) x = ........... , y= ........ ...
6.
(3]
It is given that ( 2, 3) is a solution of the equation 5 y - kx = 17 .
(i)
Find the value of k.
(ii)
Detenninc whether ( - 2, 5) is a solution of 5 y -kx =- 17 . Show all of your
workings clearly.
Ans : (i) k = .. .. ... .. . .. .. .. . .. .... ..
(ii) .. .. .. .. .. ..... .... . .. .. .....
[2]
[2J
5
7.
1: = {x : xis an integer, 2 :5 x :5 12 }
A - {x:xisa fac1orofl8}
JI = {x : 3x - 8 < 13 }
(a)
(b)
(i)
List the clements in set B.
(ii)
List the clements of 1he set Ar\ B.
A youth club has facililics for members to play soccer and floorball. The
club has I 00 members and each member plays at least one game. 50
members play soccer and 80 members play floorba ll. Sand F represent 1hc
sets of members who played soccer and floorball respectively. Find lhc
number of members who play bolh soccer and floorball.
I
Ans : (a) (i) . . .. . . . . . . . . . .. . .. . . . . . ........... .. . . .
(I)
(ii)... .. ......... ............ .. . .......... [21
(b). ...... ... .. .. ... ..... .. ..... ... . ...... .
[2)
6
The hcigh1s of20 students in C lass 2A arc given below.
157
183
(a)
171
149
194
169
166
160
183
194
149
174
166
160
177
157
180
174
160
169
R epresent the dala above in a Siem-and-leaf diagram.
'.
Stem
Leaf
(2]
(b)
(c)
For this <lis1ribution, find the
(i)
median,
(ii)
mode.
(iii)
mean.
Suppose one new stuclen1entered1he class. If the new mean heigh! of lhe
class is 171, find the height of the new student.
Ans: (b) (i) .... .... .. . ... .. ....... ..
fl I
(ii).... ....... .... ...... .. ..
[lI
(iii) ....... --- .... ... ·- ·- ·-·
(2)
(c).... .. ...... .. ...... . ... .....
(21
J
7
9.
The diagram below shows two paper cups without cover. Cup A is in the shape of
a sqLtarc pyramid with a square base of side 7 cm and its slanted height is 8.3cm.
Cup B is in the shape of a cone w ith a base radius of 4 cm and its verti cal height
and slanted heights arc 7.Scm and 8.5cm respectively.
(i)
Find the height of cup A.
(i i)
Cup A is filled with water to the brim. Find the volume of water in Cup A.
(i ii)
Whi ch cup will be able to hold more water? Show your workings clearly.
(iv)
Calcul ate the area of pape r needed to make cup B.
7.5 cm
C up A
CupR
(D iagram s arc not drawn to scale)
I I)
Ans: (i) . . .... .................... .cm
(ii) . .......... .............. . cm 3
{21
(iii) Paper Cup..... ... . ...... .. .
(3)
2
_ _ __ _ _<i_v_>_··_··_·· ·· ... .... .. .... .....cm _
~
8
~
Answer lhc whole of this qu estion o n a
shc~I of graph 1>ap cr.
(a) Copy and complete the table using the equation y = x' + 2x - 3 .
L
-= x
I -~
~ + 2., - 3 I
I -3-1
I I- I
o
- 2
- 1
0
3.
2
(b) Using a scale of 2 cm to I uni t on the x-axis and I cm to I un it on they-axis,
draw the graph of y = x 7 + 2x - 3 .
fl J
[3]
(c) Using your graph,
(i) stale the minimum value ofy,
(ii) find the values of x when y
= 5,
(iii) solve the equation x' + 3x = 4 .
End of Pap~r
l lJ
fl l
131
Sec 2 E £ n<l Ycar l~xamination Maths Pape r 2 A nswers Yea r 20 11
I
Q~
Part
I.
(3p - l)(p+ 2) 2(p - 2)
- x- - (p 2)(p+ 2)
3p+ J
Marks
2.
' '
3p+I
Al
-
(a...____,-'x = 6,_
5°_ _ _
3cm yon
=- ?cm 15cm
131
(h)
Ml
7y = 45
3
3.
Al
y=6 - c:m or y =:: 6 .43cm
_J_
:cy+ 2x+ 2y + 4
(a)
Ml
- - -
I
4.
(a)
~
(a)
-
Ml
Al
··-
-
6 11.3111
Ml
BO = / 6502 3201
.
565.Rm
Ml
Al
6 11.3 - 565 .8 =45.5 m
- -
Eqn I : 2x + 3y = 14
Eqn 2: 4T+ Sy = 24.20
Bl
--4__:_
(b)
Al
=Jwo' - 3201
AO
Distance
I-·
I mark is
deducted from
overall score if
answers arc not
rounded off to 3
s. f.
7 - - -
--'L....--
=6 +2( 4)+4
=2
c1i)
2..y(4 - 49yl>
__j_:_2.ry(2 - 7 y )(2 + 7y)
1
-
Ml
= 3p -_!
J
Comments
Method I : Subst itutio n
14 - 3y .
. (~)
S ub x=-- mto h1n L
2
4(
14
Ml
- ]y) +5y = 24.2:0
2
y = 3.80
Suh y = 3.80 into C:qn ( I)
2x+ 3(380) = 14
Ml
x = l.3
:. x - $ 1.30,y = $3.80
-----~­Method 2: Eliminati o n
Eqn ( l)x2 : 4x + 6y = 28 ~ Eqn(3)
Eqn (3) - (2): y 3 80
Sub y 3.80 into C:qn (2)
2x + 3(3.80) 14
- - --'--
=
- --
- --
-
Ml
MI
- - - - L-'-:.:..:...
-
Both F.qns must
be provided.
----
x = 1.30
-6_
(a)
-
(b)
7_
(a)(i)
--
8.
Al
5(3) - k(2) = 17
k = -I
Working:
5(5) - (- 1)(- 2) = 23 ;e l 7
Answer: No
Ml
Al
..
Mt
Al
3x-8<13 ·
x<7
8 = {2,3,4,5,6}
Bl
Ml
Al
A = {2,3,6,9}
(b)
A n8 = {2,3,6}
Let x be those who played both games
(50 - x)+x+(80-x) = 100
Ml
x=30
Al
(a)
-- Stem
14
15
16
l7
18
19
>---
- - - lb)(i)
(ii)
(iii)
9·
.-. x "' $ l JO,y = $3-80
(ii)
I
'----·
-
--
---
Leaf
9 9
7 i
0006699
I 4 4 7
0 3 3
4 4
B2
I mark is
deducted if
students write
comn1as 111
between
numhers_
169 cm
160 cm
swnof20d_<_!IO
Bl
Bl
20
= l69.6cm
"'l70cm
··(c) Sumofheightof2l students = 171 x 21 =: 359l
Height of new student = 3591 - 3392
=199cm
"8.3 ::.3.S
1
1 "'7.53cm
(i) - ·Hclght of Cup A =
(ii)
Volume of Cup A = _!_x (7x7)x7.53
3
= 122.99cm1
"'12Jcm 3
(iii)
1
VolumeofCupB = x(;rx4 1 ) x 7.5
3
=125 .66cm'
- --
Ml
-
Al
Ml
Al
Bl
Ml
Al
Ml
Ml
·-
t - -·
---
I
3
""126cm
1-
-
(iv)
-1
/\nswer: Cup B.
- - - Area of paper = tr x 4 x 8.5
---10.
Al
Ml
=106.&m'
"' 107cm 2
'Al
r----lx
! _>"_..._.
0
-3
2.:c 3
-I
0
01
(b)
..
~·
4
=±ft=
I
(c)(i)
(ii)
(iii)
L__
Im each
awarded for
• correcl scale
• correcl
points
plolted
smooth
line
•
joining all
points
l
..
4
B3
Bl
v = -4
x =-4, x
=-x+I
2
01
x' +3x =4
x 2 +2x - 3= x+I
Ml
Plot the line y = - x 1 I
x= - 4,x = I
Ml
Al
I
Class_: _ _ ...__c_a_n_d-id- al_e_N
·_a_m
_e
_:_ _ _ _ _ _ __ _.__
c _a_ndidate Index Number:
J
SHUQUNSECONDARYSCHOOL
2011 End-of-Year Examination
Secondary 2 Express
MATHEMATICS
10 O ctober 201 1
Paper 1
Candidates answer on the Question Paper
1 h our 30 minutes
[_
I
INSTRUCTIONS TO CANDIDATES
I
W 11te your name, class and class index number in the spaces at the top of this page
- 1
;ind all the work you hand in.
Writ~
in blue or black pen.
Yo u may use a pencil for any diagrams or graphs.
Do not use staples, paper clips highlighters, glue or correction fluid.
Answer all questions.
If working is needed for any question. it must be shown with the answer.
Omission of essential working will result in loss of marks.
You are expected to use a scientific calculator to evaluate explicit numerical
expressions.
If the degree of accuracy is not specified in the question, and if the answer is not
exact . give the answer to three significant figures. Give answers in degrees to one
decimal place.
For
TT.
use either your calculator value or 3.142, unless the question requires the
answer in terms of
;r .
At the end of the examination. fasten all your work securely together.
The number of marks 1s given in brackets ( ] at the end of each question or part
question.
The total of marks for this paper is 50.
'---===-=============-=======================~
This question paper consists of 11 printed pages.
l
[Turn over
2
Express I .03528
(a)
correct to I decimal place.
(b)
correct to 3 signilicanl ligures.
Ans: (a} _ __ __ _
(b) _
2
_
_
[lJ
fl]
Three bells are set to ring al intervals of 4 minutes, S minutes and 6 minutes
respectively. Jf all the bells ring together at 0730, at what time will they next ring
toge1her again?
Ans: _ _ __ _ _ [2)
:>
Observe the number pattern below.
Line 1: 9 x 3 "' 27
Line 2: 99 x 93 '" 9207
Line 3: 9911 x 993 - 992 007
(a)
Write down the 4•h line in this pattern.
(b)
From the pattern, find the value of9 999 200 007 + 99 999.
Ans: (a) _ _ _ _ _ _ _ __ _ (I l
(b)
sass Eov 2011
Mathematics
___ _Ill
3
4
For each of the following Venn diagrams. shade lhe region rcpresenling:
(a)
[I)
(Av B)',
R
(b)
fl I
A'nB.
13
A
5
Given lhat A is dircclly rroponional to ,J, and when r = 2, A= 14,
(i)
exrress A in lerrns of r,
(ii)
lind the value of A when r - 6.
Ans: (i)
(2)
(ii) _ __ _ __ [I)
sass Eov 2011
Malhematic.s
4
6
In the diagram below, a;1gle ABD = 90° and lJCD is a straight line. Calculate the value
of
(a)
x,
(b)
y.
A
-.
ycm
4cm
Ans: (a)
(b) _ ._
sass EOY 20 11
Mathematics
- [I)
_
_
_
_
121
5
7
In the diagram below, triangles ABC and DEF are si milar.
CaJ..:ulate the value of x.
A
()
2 cn~
c
IJ
E
A ns:
sass FOY 2011
Mathemalies
F
_ [31
6
8
If 4 water pipes can fill
atank in 42 minutes,
(a)
how many minutes would 3 similar pipes take to fill the tank?
(b)
how many pipes would be needed lo fill the same tank in 24 minutes'!
Ans: (a) _ _ _ __ _ minutes [2]
(b) _ _ ____ pipes
9
[2]
At an electronics store, a thumb drive costs $x and a pair of earphones costs $y. It is
given that x ~ 2y = 39.
(a)
Given also that 2 thumb drives and 3 pairs of earphones cost a total ofS62, write
down a second equation in x and y .
(b)
Solve the simultaneous equations to find the value of x and ofy.
[I]
Ans: (a)
(h)x - _
SQSS EOY 2011
Mathematics
_ , y - __
[3)
7
I 0 The stem-and-leaf diagram below shows the pulse rates <lf 20 ~1uden1s measured after a
PE lesson.
Stem
Leaf
8
6
7
9
9
0
2
4
5
8
10
2
J
5
5
6
II
0
6
6
8
9
2
LKey: 8 I 6 means 86 beats per minute ]
(a)
Stale the median and mode of the distribution.
(h)
What percentage of the students has pulse rates greater than I 07 beats per
rnin.ie'!
Ans: (a) Median
~_
_ _ _ beats
Mode - _ __
Mathematics
minute,
beats per minute 12]
(b) _ _ _ __ _ _ _ _
sass Eov 2011
~ler
_ _ _ % 121
8
II
Box P contains IS blue marbles and I 0 yellow marbles. Box Q contains I 8 white
marbles and 7 blue marbles.
(a)
If a marble is chosen randomly from box P, calculate the probabi lity that it is a
blue marble.
All the marbles from box P and box Qare put into a bag. A marb le is chosen randomly
from the bag.
(b)
Calculate the probability that the marble is
(i)
blue,
(ii)
not yellow.
- __
Ans: (a) _ _
(b)(i) _
_
[I]
_ (I)
(ii) - - - -- -
sass EOY 2011
Mathematics
(2)
9
12 The graph below shows the CQSI of taxi fares for journeys up to 12 kilometres.
:
.1
"
!
I
~ ·'
;
' rt:
!
;
2
..
..
I -
: ' ..
I
-
I
' !
f
I
0
I-
'
'
2
4
I
I
I
6
12
10
8
Distance (Im•)
(a)
Find the cost shown on the taxi meter whom :i passenger first boards a tnxi.
(b)
The cost for a particular journey is $4.60. I low far was the journey'!
(c)
Find the gradient of the graph.
(<l)
State the equation of the graph.
(I J
Ans: (a)$
(b) -
-
- - - - km
[I]
(c) _
(d)
SOSS FOY 2011
Mathem:ihcs
fl 1
_ _
Ill
10
13
Expand and simplify the following:
(a)
(1 - 3nr)(5nr + 2),
(b)
- 5 (4x- J)i.
Ans: (a)
(2]
(b) _
14
___ -- .
(a)
Using factorisation, evaluate 96?2 - 33i without the use of a calculator.
(h)
.fy 4m 2 11'' 12mn 2
.
1 -+-.
S1mp1
4
3p 3
2lp
Ans: (a)
- -
(b) _ __
sass EOY 2011
Mathematics
.. _
_
(2)
[21
(2]
11
15
Factorise the following expressions t>Ompletcly.
4
(a)
3a7 bl
(b)
2x-' + 1 lx - 21,
(c)
'
3p - q - 6pq + 2q-.
l 5ab
,
Ans: (a) _ _ _ _
_
(b)
(c) _
End of Paper
sass eov 2011
Mathematics
[I]
_
_ __
(21
_
r21
Answers - Sec 2E EOY 2011 Paper 1
_ _ __ _
1.04
LCM 4, 5 and 6 "' 60
'-T:.;;im
""'c 0:..::8..:c
30"--- 9999 x 9993 = 99920007
99993
.:::.a,__~
1 .0
..:..i1
lt~
of
2
·-
-~~
3(b}
=
4(a)
81
81
81
81
81
4(b)
5(i)
5 ii
6(a)
I
i\ = k?
k = 14/4 = 3.5
So A = 3.5 r2
i\ 3.5 (62) = 126
=
81
81
81
x2 = = 52 + 42
x = J4i = 6.4031
::; 6.40
6(b)
2
BD
=21
2
-
81
M1
2
5
BD =J11 6 = 20.396
=20.396 7
8(a)
8(b)
4
= 16.4
A1
x
2
-= 6 x+ 9
x + 9 = 3x
x = 4.5
4(42) = 31
Ml
t 56 minutes
4(42) 24p
= 7 pii:ies
A1
M1
A1
=
=
M1
M1
A1
9(a)
-9(b) --
_lx_ + 3y = 62
-
x + 2y = 39 ----(1)
-
81
-
-
- -
2x + 3y = 62 ----(2)
(1} x 2: 2x + 4y = 78
10{a)
10(b)
1Hal
11(b)(i)
(3) - (2):
y = 16
Sub into (1):
x=7
-
· -(3)
M1 (or correcl substitution into an eqn)
A1
A1
-
-
- ~1
81
81
-
11(b)(ii) P(yellow) = 10/50
P(not yellow) = 1 - 1/5 = 415
Or P(yellow)
= 40/50
-
81
81
M1 (for 512 0)
Median= ( 103 + 105)/2 = 104
Mode= 106
5/20x 100%
= 25%
15/25 = 3/5
22/50 = 11/25
Ml
A1
-......
Minus max 1 mark
for non-simplified
fraction
-
M1
= 4/5 A1
$2.80
5.4 km
Grad = rise/run
= 4 / 12 113
..1£@_ _ y = 1/3x + 2.80
13(a)
5m + 2 - 15m 2 - 6m
2 -m - 15m2
13(b)
-5 (16x2 - Sx + 1)
= - 80x2 + 40x - 5
: 14(a)
(967 + 33X967 - 33)
= (1000)(934)
= 934 000
4nl'' rr'
I 21nn ..
14(b)
• 12( al
12(b)
12(c)
- - - -- ·- - -···
81
81
I
=
=
--
-
-
81
81
M1 (correct expansion)
Al
M1 (correct expansion)
A1
81
61
-· --
3p; . -;. 21p'
15Ca\
15Cb\
15(c)
..
~
4m 2 n 1 21p'
= -- 3 x - - 2
3p
12m11
M1
7nmp
= -3
A1
3ab'Ca - 5bl
f2x - 3Xx + 7)
3p - q - 2q(3p - q)
= {3~ - g)(1 - 2g)_
-----
-
- --~
61
82
M1 (any correct grouping)
A1
----· -
---
I
- 1
Candidate Name:
__j___ ~~
~~-
SHUQUN SECONDARY SCHOOL
2011 End-of-Year Examination
Secondary 2 Express
MATHEMATICS
Paper 2
11 October 2011
Additional Materials: Answer Paper
Graph paper ( 1 sheet)
1 hour 30 minutes
INSTRUCTIONS TO CANDIDATES
Write your name, class and class index number in the spaces at the top of this page
and all the work you hand in.
Write in blue or black pen.
You may use a pencil for any diagrams or graphs.
Do not use staples, paper clips highlighters. glue or correction fluid.
Answer a ll questions.
II working is needed for any question. it must be shown with the answer.
Omission of essential working will result in Joss or marks.
You are expected to use a scientific calcula tor to evaluate explicit numerical
axpressionc
If the degree of accuracy 1s not specified in the question, and ii the answer is not
exact, give the answer to three sign ifica nt figures. Give answers in degrees to one
decimal place.
For
;r,
use e11)'ler your calculator value or 3.142, unless the question ~equires the
answer in terms or
;r .
At the end of the examination, fasten all your work securely together.
The number of marks is given in brackets [ ) at the end of each question or part
question.
The total or marks for this paper is 50.
This question paper consists of 4 printed pages.
[Tu r~ ove r
2
Answer all the que~tions.
1.
2.
A map of a reservoir is drawn on a scale of 1: 25 000.
(a)
Fin<l the ratio of the area of the reservoir on the map to lhc acrual
area .
(I}
(h)
TI1e length of a foot trail along the r~ervoir on the. map is 2.4 cm.
Find the actual length of the foot trail, in km.
(I]
(c)
·111e reservoir has an area of 1.2 km . Find the area of the reservc.>ir
. .
•
?
on thl' map, giving your answer 111 cm· .
¢
2
= {x: x is an integer
[21
an<l - 4 $ x $ 4}
A ~ (x: -3 <x< 2}
R = {x: 5x - 3 <! 2)
(a)
J.
List the elements of
(i)
B,
[I )
(ii)
Av JI
i 1]
(b)
Find n(A !l B).
[I]
(c)
Lis.I all the subsets for ii" B.
[I ]
(d)
Describe R' in set-builder notation.
lIJ
The table shows the number of story books read by a group of students during
the school holidays .
N
-' u_m_b_e_
r o-f-st_o_ry_b_o-ok-s-rc_a._d _,_,_ o_ I
Nu mber of students
!.....
.
4
I.
2
4
I.
3
5
I.
4.>'
I.
x5
l
]
(a)
Write down the largest possible value ofx given that the modal
number of story books read is 3.
·
[l]
(b)
Calculate the value ofx given that the mean is 3.
f2l
(c)
Write down the largest possible value of x given that the median is -3 .
(1]
(d)
Given that the probability of one student chosen at random had read
2
more than 2 story books is
, find the value of x.
3
[2]
sass EoY 2011
Mathenlalics
.I
4.
(a)
Simpl ify
(i)
18xy 2
(2]
-JOx' .v
9/ - l5xy'
(ii)
3x
(2x - 5)
(b)
5.
2
I
131
I
.
15 - (>x
. .
x 2(1 Jb
fl 1s given that - = , express x .m tenns o r 11 tln ct h .
4
5
121
J'he diagram shows a cross-s1.-ctional area of a toy where AB - (x + 4) cm,
!1C' - J5(x - 3) cm, Cf)
/ MD
A
~ -3
(3 t
2) cm, Af) ~ (x - 3) cm and
~0C=9U'' .
r-
'
JJ
x+4
\.
'-. Js(x - 3)
6_
I)
-
3x - 2
',
·--- - '~c
(a)
f.ind an expression for the cross-sectional area.
l2J
(h)
Given that the cros~·sectional area is 114 cm2 , form an equation in x
and show that it reduces to 2x 2 - 5x - 117 = 0 .
[21
( c)
Solve the ctJUalion 2x'
(d)
Explain why one of the values of x has to be rejected.
SQSS £-OY 2011
5x -11 7
= 0.
f\1athematics
P)
[I)
6.
Thc cliaw-am shows a solid obj eel made from a cone and a hemisphere. Th.:
cone has a height of 12 cm. Tiic cone and the hemisphere both have a radiu' .,r
4 cm. raking 17 as 3.142. find
(a)
Liu: lolal volume of the ohjecl,
13]
(h)
the length of the slan1 heighl l,
(2]
(c)
lhc total surface area oflhl! object.
[J)
(Curved su1face area nfa sphere • 4nr 1 ; Curved surlacc area of cone= :rr/
4
I
Volumeofsphere - - nr1 ; Volume of cone= nr 2h).
3
3
\
/~·-
--
' •
4 ~~i .
·, .
\\, -- --- -·1
/
/j
\.,
12 rm
\
\
l
'
f
\/ ....
7
Answer the whole of this qucs lion on a sheet of graph p aper.
The variables x and y are cnnneclctl by the equation y = x 2
shows some values ofx and the corresponding values ofy.
I ~ I =~ I
4
f ;
-
2.x +I. The lahlc
-2
I
(a)
Find Che value of a.
[I)
(b)
Using a scale of2 cm to represent I unit on both axes, plot the graph
of y -x 2 - 2x +I for - 4 S x::; 2 on the graph paper provided.
13 l
(c)
From your graph, find the
=
(i)
values of x when y - - 1,
[2]
(ii)
coordinates of the maximum point,
[ 1J
(iii) equation for the line or symmetry.
~ame
= -2x -
(d)
Using lhe
(c)
From the graphs, find the coordinates of the points where
v = - .x
2
-
axes. draw the graph of y
I II
2.x +I and y
=
2x - 3 intersect.
F,nd of Paper
sass Eov 201 1
Mathematics
3.
121
12)
Sltu~1un S.:condmy School
S.:.:2 Express F.OY 20 I I Math P2 Marking Scheme
,-~~·1 ~~J111i,ns
Ii?
Maik
Map Area = (- '- )'
Acwal Area
25000
Map Aren
I
Actual Area
131
625000000
Altematjvc
Map A!·ea
Actual Arl'fJ
=_ _I _c_·m_2_
Bl
0 .0625 km '
L
(b)
~
Map Length_ =- (- l- )
Actual frng tlt
25.000
2.4
x
(- 1 )
25,000
x = 60,000 cm
.>.
= 0.6 km
Bl
Alternative
I
~
I : 25.000
2.4 : 25,000 x 2.4
2.4: 60,000
2.4 cm : 0.6 km
Bl
(<)
Map Area
I
=
Area 625,000,000
Actual
y
12,000,000,000
yy
Ml
625,000,000
12,000,000,000
625,000,000-
=19.2
cm'
Alternative
Al
Shuqun Secondary School
Scc2 Express EOY 2011 Math P2 \farking Scheme
! _ I__ _
I
l
l
I
- --
Map Area
·=
--~-
Acwal Area
-
2/7
- -- · - -- - -
I cml
0.0625 km'
Ml
1 cm 2
y
J2 km
1
= 0.0625 km'
l.2
l
-- - cm
0.0625
y = 19.2 cml
y=
l
AI
i
li 2---+---1- - - - -- -- -! (ai)
I
I
l
li
(aii)
5x - 3 <: 2
5x ~ 5
x~
I
I
I
I Bl
H = { I, 2, J , 4)
-+- - -- -- - -- ------ - - - - - - --1-A - (- 2, - 1, 0, I }
1--1
-1- -
--
- - -
·----~- ·---
(b)
AnB={I}
11(An B) = I
(c)
All subsets of An JJ are ¢, {I}.
(d)
___,__II ___ _
Bl
Avfi = {- 2, - 1,0,1,2,3,4)
-
..
~
[31
Bl
- - -1---·---- ·--
H' ,:: /-4, -\-2, -1, O}
fl' = { x: xis an integer and - 4 s x $ OJ o r
B' ~ { x : xis an integer and x $ O}
J
Bl
(a)
- - -· - -- - - -- -- - - -- - - - - +-Bl
for mode= 3, largest possible value of x is 4.
(b)
- - ·- -Mean = 1(0) + 4(1) + 4(2) + 5(3) + 3(4) + x(5)
.
- - --
-
1+4+4+5+3+x
3
= 0+4+8+15+12+5x
17+x
5 1+3x = 39+ 5x
12=2x
x=6
Ml
I;
i
, __ _J
Shuqun Sc..:ondary School
Sec! fa pr~ss EOY 2011 Malh P2 Marking Scheme
317
r(0f:3+~1 3 1 +t1\1 4 - 3 + x
~·
I;·_
10
(d)
Total number of s tudents = I + 4 + 4 + 5 + 3 1 x .= l 7 ~ x
Number of students reaciing more than 2 story books = 5 + 3 +
2
P(studenlS reading more than 2 story books) - -
x
I
•
3
8+ x
2
-- =
l 7+x. 3
3(8 + x) - 2( I 7 + x)
x - 34 - 24
<= 10
Ml
. Bl
i\ ltemmivc
9
11+x
3(9) = 1(17 + x)
x = 2 7 - 17
x= 10
I
1-·
l
=3
Ml
Bl
18.ry' - 30x 1 y
i (ai)
9y 3 15xy 1
6xy(3y - 5x)
I
I
Ml
3y 2 (3y - 5x)
2x
y
Al
Alternative
18xy' - 30x' y
9y 3 - l 5.\y 2
i_J_I
6x(3y 2 - Sxy)
3y(3y1 - 5xy)
Ml
2x
Al
y
Shuqun Secondary School
Sec2 Express EOY 20 l l lvlath P2 Marking Schen1e
(aii)
i
I II
I
15 -
3x
(2x - 5)
1
1- (2x3x- 5)
1
1-
I·
2
+
6x
I
3(5 - 2x)
:Ix
I - (2x -
4.n'
1
5) 1
Ml
3(2x - 5)
ljMI
3x(J) - 1(2x - 5)
3(2x - 5) 1
9x - 2x+5
·3(2x - 5)'
I
7x+ ~--
l
~-
=
.- {b)
- - lAI
3(2x - 5)'
- - + - - ' - - - - -- - -- - -- - - x - 2a 3h
4
5
5(x - 2a) = 4(3b)
5x - 10a ~ 12/J
I
5
~
1x(4x+2)(x-3)
J=
~x(2x+ l)(x - 3)
=2~'
1 (b)
I
1
- - - -3)
1
2x -Sx-3= 11'1
2x 2 - Sx - 11 7 = O(shown)
II
I.;...,-
Al -- -···-· -
f
i
1
Ml
_t
1
- 6.x+x - 3
! __..__=_2_x' - 5x - J
1-
~ x (x + 4 + 3.x - 2)(x -
Al-
- - --- - -- --
Ml
Al
(c)
2x' - 5x - 117=0
(2x +I 3)(x - 9) = 0
I
j
x= -- --·
Area of trapezium ABC/) =
-- j
! Ml
5.x =12.b + 1Oa
2(6/J +Sa)
(a)
I
Ml
-- 1
Shuqun Scconc1ary Scl111ol
Sec? Express EOY 20 I I M:ith P2 Mark111g Scheme
2x•I)
0
01
l
9=0
x 9
517
Al mark
for each
correct x
,,,,~
(d)
l
I
l
I
t
AB - x + 4
All ~
.
2
I
2
cm
z
.
.
Reject ' -' - 6 I smcc
the le ngth o f AB cannot he negative.
l't
.
'
Volume of hemisphere -
131
I 4 ,
(- m· )
2 1
'3
- - ;r(4)'
Ml
~ 134.058 cm 1
Volume of cone ~
I
nr' h
3
~i. ;r(4) 2 ( 12)
3
~ 201.088 cm'
Ml
Total volume of solid ~ 134.058 + 201.088
= 335. 146 cml
,.335 crni (con=t to 3 sig. figures)
Al
(h)
Slant
heighrofcone,/ ~ J 12 2 + 42
/= /144+16
I= 12.649cm
I "' 12.6 cm (correct to 3 s1g.figurcs.)
Ml
J\ I
Shuqun Secondary School
S.:..:2 Exprc~s EOY 20 I l Math P2 Marking Scheme
. (c)
1
Curved surface area of hemisphere= - (4;rr 1 )
2
= 2,-rrl
~ I 00.544 cm2
i MI
Curved surface area of cone - 711'1
- ,,(4)(12.649}
- 158.972 cm 2
Total surface area of solid - I00.544 + 158.972 t:m 2
"' 259.5 16 cm1
"'260 cm 2 (correct to 3 sig. ligures).
I'
(a)
a- - 7
[
, Bl
~
. (h)
11 n1ark
1r \:On l"{ t
'
~"'· .1ml
: scah.:s .
...
MI mar"for all
com..>ct
points
plotted.
Ml mark
for Sllll)(lth
CUJ'\C
COH!lCClit1g
all poinls.
I
(ci)
From the graph,x= - 2.75 or 0 .70.
Acceptable answers: x ~ 0.70 or -2.75 (± 0.05)
From the graph, the ma"irnum point is ( 1,2)
- ---1
Bl.Bl_
Bl
.,
Shuqu11 S..x:ondary School
Sec2 Expr <:ss F.OY 20 11 Math P2 Marking Scheme
r- ---1{.: iii)
I Frorn the gr:;ph, the equation for-line of symmetry is x -
(d)
717
I.
Bl
--;
- - Ml
i"""'
~
for correct
grad ient.
M l mark
for correct
y-
intcrcept.
t
(c)
f rom the graph, the coordi nates are ( 2, I) and (2,- 7).
rAN..IONG KATONG SECON DARY SCHOOL
2011 YLJIR l:NO EXAMINATION
SC CON DARY 0Nf;
Mathematics
Paper 1
··"'m'"'"'
u.\e
r"'
F~tmtlfl('r'.i
( ill'
AnS\\Cr A LL q uestions. A"'L diagra ms in rh is paper arc nol d raw n to scale.
(a) Express
6(1
<llld 126 as a pro.Juel of its prime
foc101s.
(b) Find the HCF und LCM of 66ahr2 and 126n2bc
Answer (a) 66 =
126 (b) l!CF -
LCM-
2
[ I
I
[ I ]
( I J
[ I
l
Fact(lrise each of the following completely.
{b) x 2 - (y - 2)1
Answer (a)
(2 J
( fl)
[2l
~:n
For
f.-. Jr
G1vcn lhal
J. ,):•."lttwt'
2v+ I
!)rt·
~v =-- ,
3v- I
(a ) ex pres' ' in terms of w.
I
(b) cakulatc the value of•' when w = 4
.
·1
For
F..sam;"'frS
(i..-
4
·1 he table below shows som e corresponding val ues of z and x.
I- J__ sI _ .c.=8
[ _=
2
I ..
~
3
II
(ii) Is (.r - 2) dircc1ly proportional to z" Show your workings.
(b) Hence, express (x - 2) in tem1s of z.
,
_j
1·~·1
::miner's ,
5
The picture shows the lloor plan of a hoosc wi th the scale indicated.
{;<>i,
(a) A cab inet will be placed a long line AB. Calculate the actual length of line AB in
metres.
(b) Find the area of the study room on the map in c;r1
2
.
(c) Find the actual area of the study room in m 2.
A
Study Roo111
1-·-····---·
B
Living Room
·-
-
-
- - - --
S..:ak Hi
- - -
· - ••• • ...J
for
F.,;r
£.r.amirwr">:
6
Solve the simultaneous equations 3x - 5y - 6 = 0 and 2x - 3 y - 5 = 0.
u .....
l·m
Th..: c4uatio11 of a llllcar graph 1s 2x + 3y = 5.
(a} Write down the gradient and y·intcrcept of the equation.
(h} lf(q, q - 6) is a solutton of the equation, find the value of q.
(c} Given that there arc no int ersections between the equations 2x + 3y
2.i I 3y = p. State a suitable value for p.
Ans wer (a)
5 :111d
Gradient ~
(I ]
y-intcrccpt =
II I
(b) fl =
I2J
(c) p
(I )
~
f n,
1-;xaminer·:;
Us'-'
,.,,,
0
8.
/~\'1Jml11t'f' .'.l
The diagram shows the graph of y =2x - 4x +c . The curve meers 1l1e x·nxis at R
and the origin. !Graph is not drnwn to sca le.)
7
{h1•
y
(a) State the value of c.
( h) Find the coordinates of fl.
(c) Find the coordinates of the minimum point.
x
Answer (a) c -
{I
_)
l2J
~'- )
f2 1
(b) ( _,_
(c) (
J
1 ·,"jr
11 and Han· sub~c1s pflhc universal set€. The number of clements in c;ich ~uh~cl"
as shown in the Venn Diagram.
Given that n(A) 2.r. n(B) ~ 5x - 7 and n(A n B) ~ n{A v B}', find the value of'
(:i) '·
(h)
n(A 'r. B).
(c) n{e) .
JO
Answer (a)
[ 1J
(b)
l 2]
(c)
[ I J
Shade the requi red regions in the Venn Diagrams.
(a) A' r. IJ'.
rI I
(b) (A'r.B)' ~,) (A n B).
(~!\--"\
~, _)
{ 1l
,. ,,..,,.,,, , .
10
Fo··
:x:m1;:;.,,.'s
(,:f;'
/'fll
11
/ ,\(Jmmn
ll1e scores of a basketball team in a series of matches are given below.
63
70
63
52
75
50
67
63
69
56
{ /,,.
52
51
(a) Find the mean, median and mode of the scores.
(h) Find lhe probabil ity thal the team scored more than 60 puinls.
(I J
Answer (") Mean=
Median
Mocle=
(IJ)
~
( I
I
IIl
( I J
12
lllUh'I' '.)
The lirsl term of a sequence of numbers is 56. The following le1ms arc given hy
subtracting five from the previ ous 1enn.
(a) Write: down the next two terms in 1hc sc4uencc.
1
(b) Write down an ell:pression , 1n 1em1s of 11, for the 11 h tenn .
12
fo•
f't.T
'!.¥atr.:rrt'r"'i
I.At'
'l11c: diagram
show~
the siue view of a wheel s1op.
fjQnUf1C'I'
U?J'o.'
Find
4
(a) the angle x,
(b) the value ofy, 1l1c height of the wheel stop.
y
7
l)imensions are no\ 10 scale.
$
TKSS 201 l Sec 2 Mathematics Mark ing Scheme Paper I
End of Year Ex ominations
Qns ~ ! Details Answers
la)
66 = 2x3x I I
126 = 2 x 32 x 7
- ·~
I
HCF ~ 2 x 3 x a x b x c = 6abc
LCM ~ 2 x ) 2 x 7 x 11 x a2 x b x c2
lb)
= 1386a2 bc1
n 2(n + l)- 3(n+ I)
(n 2 - 3)(n+I)
I
2b)
x1 - (F - 2f
[x - (y - 2Jix~· (y - 2)j
, [x
y+2][x +y - 2]
'
-1
'3ai- --1W"' 2;,+J3v - J
!
i
I ;~~~~~:~~?:1+
1
)
3vw-2v = w + l
v(3w - 2) = w+ I
I
w+I
v= - 3w - 2
3b)
I
I +l
v = - 4-
3 -2
4
v
=- 1
4a)
I
z
x
<x-2)
x- 2
- -
l
2
5
3
3
8
6
3
~'
Conclusion:
~---'-=
(x-=-2) = 3z
3
11
I
9
3
I
=~-=1
TKSS 20 I I Sec 2 Ma1hematics l\farking Scheme Paper I
End ol'Ycar Exam inations
! 5.1)
l-5b)_ _..
sc)
1
I
6a)
3.5 x 2 =7 m
-- - l
Area of study room on map = 12 -- 12.?c...:l....:c::.n:.:_
1 _ _ ____ _
Linear scale lcm: 2m
Area Scale Icni2: 4m2
Actual area ofstudv room = 48 - 50.84·-'-'-m2 -·
I 3x - 5 y - 6 = o-------·-···-(l )
I
' 2x 3y 5 = 0 ---··---·----(2)
( l)x2 : 6x- 10y- 12 = 0----------(la)
(2)x3: 6x - 9y - l 5 = 0 -----··---·-(2a)
( l a)-(2a):
- IOy - 12 + 9y -~
15 =0
- y+3 =- 0
y =3
=3 into (I):
6x - 10(3) - l2 = 0
subt. y
7a)
x = ?:___ _ __ _ __ __ _ __ _ __ _
2x+ 3y = 5
2
5
Y"'- 3x+:i
.
2
.
·5
gradient = y-mterccpt =
3
3
7~
,
2
5
Subst (q, q + 6) into y = - - x+ 3
3
c1+6 = - Zq + ~
3 3
2q
5
3
3
q+ - ~ + - - 6
l/
7c)
~-8b)
8c )
9a)
13
=-
5
Any value cxccp1 5
1
1 2x - 4x = 0
1 x(x - 2) = 0
x = Oor x = 2
B (2, ())
Equation of line of symmetry: x = I
Subt. x = I into y= 2x 2 - 4x
y =- 2
n(AnB) ~ n(AvB)'
2x = 8
X=4
2
TKSS 2011 Sec2 Mathcma11c~ Marking Scheme Paper I
I:nd of Ycar E.\amin.itiuus
- --r
i %)
I n(A'nB)-3x - 7
I n(A'n8)=3(4}
I n(A 'n B) = 5
9c)
i
7
n(s ) -= 5.x - 7 +8
'11(1;)=21
IOa)
-
I Ob)
11 a)
63+63+ 75+67+69 • )2 I 70 t 52 +50+63+ 56• 51
12
niean- - - - mean - 60.9 (3 s.f.)
50,5 1,52,52,56,63,63, 61, 67,69, 70, 75
median - 63, m()dc =63
I lb)
7
P(rnorc than 60 points) = 12
12a)
Next 2 tcnns - 51 , 46
12b)
T1 =56
•
T i - 56 - 5
T i - 56 - 5 - 5
T. - 56 - 5(11 - I)
T,, = 56 - 511 + 5
T. = 61 - 511
I3a}
~ 3
cosx-=-
7
3
y
TKSS 2011 St.><: 2 :-.1athcn1a1ic.-; Mai king S~hcmc Pap.:1
End of Year Exa111111ations
1 .-" - M .6(1 d.p)
4
Class
Rc9 Number
c -r
Candio:itc Name
TANJONG KATONG SECONDARY SC HOOL
End of Year Examination
Seco nd ary Two
MATHEMATICS
PAPER2
0750 - 0905
Thui·sday
6 October 2011
-
- I hour 15 minutes
READ THESE INSTRUCTIONS FIRST
Write your name, class and register number on all the work you hand in.
Write in dark blue or black pen.
You may use a pencil for any diagrams or graphs.
Do not use staples, paper clips, highlighters, glue or correction lluid.
Answer all questions m this paper.
Write your answers on the writing papers provided.
If working is needed for any question it must be shown with the answer.
Omission of essential working will result in loss of marks.
Calculators should be used where appropriate.
If the degree of accuracy 1s not specified in the question, and if the answer is not
exact, give the answer to three significant figures. Give answers in degrees to one
decimal place.
For rr, use either your calculator value or 3.142, unless the question requirc:s the
answer in terms of rr.
At the end of the examination, fasten all your work securely to gel her.
The number of marks is given in bracktls [ ] at the end of eiich question or part
question.
The total of the marks for this paper is 50.
You arc reminded of the need for clear presentation of your answers.
This question pnpcr consists of 6 printed pages.
rTurn ove r
J
3
Answer all the questions
(:.) Expand and simplify the following.
in
this paper
(i) (2a - S)(3a + 2)
(i i) (y
2
-
II I
yJs)(/ + y/5)
(21
(b) Simplify the followint: expressions.
- - +-3m'
. (-411111) l
( 1)
12
2
1
(2]
II
(ii) ~4
x·- a_+6 _ _,__:i_+4
2a'+6a a 2 +5a - 6 4(a~3)
131
2 .,.:fhc: ;,tern ancl leaf diagram bdow shows the marks in a Mathematics test of a group of
~tudcnls.
Stem
Leaf
4
2
0
5
6
3
0
4
0
3
7
I
3
2
8
4
4
6
7
3
8
5
(a) Find
( i) the mode,
(ii) the mean,
(I J
(2j
(iii) the median.
111
(h) The passing mark for the test was 40 marks.
Calculate the percentage of students who passed the test.
(c) S tudents who scored more 60 and above are awarded a distinction.
One student is chosen from the class. Calculate the probability that the student scored
a distinction.
(2)
[I 1
3
4
A motorist rode 90 km from P to {J at a n average speed of x km/ h o n hi> ti rst joun"'Y
l le returned from Q to I' by the s ame route. On this return journey. hi~ average s p.;c.t fot
the return journey was 10 krn/h less than the first journey.
(a) Write down an expression , i11 terms of x, fo r the number of hours taken for his first
(I I
journey.
(h) Write down an expression. iri terms of x, for the number of hours taken for his return
[I)
journey.
(c) He took 1.5 hours more on the rel um JOUmey than the firs1 journey.
Write down an equation in x and sho w that it reduces to x ' - I Ox
600 = 0 .
(d) Solve the equation in (c) .
(31
[JJ
[I ]
(c) Calculate his average speed for the return journey.
'-
4
Given that & = {x: - 5 !': x S 5, x F £}, A= {x: 2x + 7 <:: - 3(1
(a) Solve the inequality 2.1 4 7 > 3(x ~ I).
Hence, list the elements in set A.
t
I)} aml B {x: - 5 < x < 6 ~.
13 l
(b) Represent & , sci A and set Busing a Venn diagram .
List the members of each sec in your diagram.
[31
(c) An integer is chosen from Ii. Find the probability chat the integ er is in the region
(i) Ar. B
(ii) Av B
11 J
11I
5
A nswer th e w hole of this
t JU es t ion
5
on a s heet of g.-ap h paper.
The tahlc helow shows the corresponding values of x and 1• for the function
y
2~ 1 - x +I.
I -I _.:--3"x
y
_ 22
-+----=2_
_
11
-a1,--_ln
--'01_~==21 = =:== 7
2
l__I_
I ii3
[I J
(a) Find the value of a.
( h) Using a scal e of2 cm to represent I unit on the x-ax is and 2 cm to represent 2 units o n
they-axis. draw the graph of y = 2x' - x +I for the va lues o f x in the range
[3)
J<:x<;J.
(c) Using your graph, find the values of x when y = IO.
(2)
(d) Writ e down the line of symmetry of the graph.
I. I j
1
(c) Us ing your graph, solve 2x - x - 7
= 0.
(3]
6
6
/\ water t;mk is nwdc with a cylinder and a cone as shown in lhc following diagram . fl 1<!
cylinder has a diamch:r of 4 m and a height of 3 m. The cone has a slant height of 6 m.
3m
--------·
··- -- - - -·_.,./
(Use n as 3.142)
(a) Calculate the total volume of the water tank.
(b) l tin of paint is needed to pain! 12 m 2 of surface. Calculate the minimum number of
tins of paint needed to paint the total outer surface area of the water tank.
(c) 5 workers needed 6 <lays to paint the whole water tank. Calculate the number of
workers needed if the painting is to be done in I 0 days.
End of Paper
(3)
[4]
[I I
-
Qn
la(i)
Sululion
(2a - 5) (3a I 2)
-
-
·-
- 611 1 + 4a I 511 - 10
- 6a 1 - Ila 10
la(i1)
I-
lb(i)
,
<.v' - )'"S)(y 2
y' 5 v'
I
yJ5)
,
(-411111)
3n1
-+-
2' fl
2' n
12
( 411111)J
)(
12
3. n1 '
2 '11
-- 641112
Jm'
121/' fl
=
9
Jh(ii)
ti
.....
-
{/ ~ 6
4
{/ +4
4((1 + J)
4(a + 3)
x- x- 1
// l +Sa 6
2Cl
I 6(l
a+4
Cl I 4
{/ i 6
4(a i 3)
)(
- x- 2a(a +3) (a+6)(a - I)
a+4
I
2Cl 2 I (1 (1
(J +4
x-o' +Sa
(I +6
-,___
2a(i)
2a(i i)
=
(I
- --
- --
2
a(t1
I)
Mode - 53
32 +3 4 + 37 ; 38+ 40+40 ~ 41 + 44 +44+ 45 ...68
Mean - -
-
20
- 4 9.05
2a(iii)
M<Xlian - 49
2(h)
16
rercentagc ra:s.~ = - xlOO
20
2(c)
J(a)
= 80%
5
1
P(distinc lion)- - =20 4
90
90
-x - 10
..\
3(b)
-
- ·
-
j
l(c)
;
90
90
~-~ - -:;:-
i
l.)
90x - 90(x - IO) = U
I, (x - IO)(x)
90x - 90x_~ 900 = LS
(x - I O)(x)
900 = 1.S( x I O)x
]~00 =
!
I I
.5x'
0 = 1_sx 1
I.
O=x'
r:•N- - 0
i 3(d)
-
Sx
I 5x - 900
IOx -· 600
1
x - 10x - 600 = 0
1·
(x - 30)(x + 20) = 0
f
l- --l x ~ 30 _
J(el
I_
4(a)
!
~
x = - 20,_ _ __ __
Average speccI ::: - 90
== 20 k 1111'Il
I
_
- --
-1
3Q_:- IO
2x+7~
- J(x + I)
2x+7 2: - Jx - :l
.5x 2: - 10
x ;::: - 2
I
2, - l,0,l,2,3,4,5
I
4(b)
l
-5
!
4(c)Q)
P(Ar.B) ~ 8
Il
4(c)(ii)
P(A u B) = .!_0
- - · - · - - -- - ' I I'-- - - - -- -- - - - - - - - 1
5(a)
a = 2(-l) 1 -(-l)+l=4
L2 (b) ---~--
25 - ·
• ...; _ _ _ .:.__.!
~\
.
10 :·.
'\
: ...,
·~
:
.
! -
1
. is• ,-:;
I
I
. '.
5(c)
x
+-"x_=
S(c)
'
. ·--·· ··- ... .
:
·2
s,~
~
,, .--r
0.
·1
'
·
I
y F 2x1 ~ x t 1/ .
10
: ··· \,
1-.'i_,,(d
__,)_
'
• !
\ · :: .
>-----
' i
. ':...t:
!
. , \. .
.3
.. I
:
/
/'
..
.
.. -·
0
2.3 to 2.5 and x = I .75 to - 1.95
0. 2 - 0. 4 _
2x1 x 7 =0
, 2x'-r+1"'8
I
x = 2 05 to 2.25 and x =_
- 1.5_l.0'-- .:..
1:..:
·7_ __ __ __ _-1
6(a)
Volumcofcyl indcr ~ ar 2 /r
= 3.142><2 1 ><3
= 37.704 m3
volume of cone
~
itr' Jr
3
= 12.568 m 3
_!_
Iota! volume~ 50.3 m3
Surface area of cone = itr
= 3.142x2x6 =37. 704m2
f -- --+--'-
6(b)
surface nrca of cylinder - 211 rh + itr2
= 2x3. 142x2x3+3. 142 x2 2
2
= 50.272 111
total
area ~
87.976m'·
.
num her o f ttns
6(c)
87.976 = 7 .3'.>
12
= -·---
>-- - -- -- - - - - - - - - ---- - -- - - -
Numhcrof'workcrs ~ ~ 6 = 3
IO
- - -- - - --"'"------- - - - ·-
YI SHUN SECONDARY SCI IOOL
We Seek, We Strive, We Soar
END-0.F -YEAR EXAMINArflON 201.l
Name: - - -- - - -- -- - -- -- -
Reg. No: _ _ ___ Class:
Sec 2 Express
Date: 7ttt October 2011
MATHEMATICS
PART 1
Duration: 1 hour 15mins
MAX MARKS: 50
READ THESE INSTRUCTIONS FJRST:
Write your name, register number and class in the spaces provided at the cop or this page
Write in t!ark blue or black pen.
You may use a pencil for any diagrams or graphs.
A.n swer all questions .
If working is needed for any question it must be shown with the answer.
Omission of essential working will result in loss of marks.
You are expected lo use a scientific calculator to evaluate explicit numerical expressions.
lfthc degree of accuracy is not specified in the question, and if the answer is not cx.11ct, give the answer to
tlm:c significant figures. Give answers in degrees to one decimal place.
use either your calculator value or 3. 142, unless the question requires the answer in terms of ;r.
For
7f,
The
num~cr or m<orks
is given in brackets [ ) at the end of each question or pai1 question.
The total number or marks for this paper is 50.
This document consists of2 printed' pages including the oovcr page
(Tum Over)
Answer aU the questions
I.
The temperature, in ·c, at night, on three successive days was 6. -9 and I .
Find an expre~sion, in terms of I , for
(a)
the difference in temperature between the sccoud and the third day,
(b)
the mean temperature for the three days.
____ [ II
(h)
).
-··-- -- [1)
Given that WO, PQ, and RS are parallel, L. WOf' = 7k and
/.SHY = 23k, find the value of k .
L2J
Anslver
3.
Solve the inequality7 - 3x > 3x- 7 .
Show your solution on the number line below.
12]
Answer:
-5
-4
-3
-2
_,
0
2
2
3
4
5
4.
The diagram shows a
Find the shaded area.
liqU.1r c
111seribctl in a circle. The radius of lhc circle 1s 9 cm.
2
Answer
5.
cm
(2)
In~ triangle XYZ, given that XY
12 cm, YZ = 5 cm and ZX- 14 cm.
lh->.-rm1nc whelher the triangle XYZ is a righl-angled triangle. Show your working clearly.
/111.\'tV('r
6.
,\ b0y scout is planning to build a lent to camp overnight but unfortunutely he lost a wooden
-urport pole RZ and he woLJld be looking for a replacement pole. He measured the groundsheet
;ind cover sheets and fount! that PQ "'4 metres and PR = RQ 2.5 metres.
Wh.11 should be the ideal height of the replacement pole?
R
p
z
Q
4"1 - - -- - -
Answer
3
_ _ __ _ _ _ _
m_ (2)
7.
One solution of 2.r1 ~ h IS = 0 is .t = 3. Find
(a)
the value of k .
(h)
the <•ther solution oft lw equation.
Answer (a)
k- _ _
ii l
Pl
(b) ~
R.
Evaluate the expression
29 7
·
and correct to 2 decnnal places.
- 3.04(9.7 1- 17.26)
(h)
19
Express - as a percen tage.
37
(c)
Express 23 ~ % as a fraction in its simples t fonn.
4
ll J
Answer (a) --~
4
[11
(b)
%
(c)
- - .. [I I
9.
A map 1s drawn to scale oft· 200 000.
Find the actual distance, in kilometers, represented hy IS S cm on the map.
(h)
A nature park covers an area ol 150 s4uare kilo111etc1s.
Find, in s4uurc cen11111etcrs, the area r.:prescnt111g the ~i ty on 1hc map.
(a)
Answer (a)
_ _ _ __ krn
(2 j
2
Lil
(l>)
Id
On lil<: ..inswcr space hclow, ltnc PQ has been drawn for you. On the same drawing.
(a)
construct a triangle PRQ with QR = 8 cm and L PQR - 1204 •
(h)
construct the perpendicular bisector of PQ.
(c}
construct the angle bisector of L PQR.
Answers (a), (b) and (c)
p
Q
5
cm
[2)
[I)
111
I I.
In 1he d1agrom. !he conrdinalcs for 1hc puinls 1\, Band Care ( 5, - 2). ( l, -2) and
(~. ·I I
respectively.
(a)
(b)
(c)
Find the gradient of UC.
Ca lculate the area oflrianglc AUC'.
Draw !he line y = - 3 in the diagram and label your equation clearly.
c
}.
~
·5
-·I
.3
z
u
-.
A
x
J
-i
B
-3
.
"
Answer (a)
[lJ
(b)
12.
/\ sample space is made up of the first 8 prime numbers.
(a)
List !he sample space.
(h)
Find !he probability that the number is two-digit number.
(c)
Find the probability th~1 the number is 29.
Answer (a) _S ... {_ __ ____ _ _
(b) _
G
II J
[ ll
_
(c) _ _ _ _
- - 'I
_
_
_
[IJ
13.
cr -
The two 1rianglcs in the diagram arc similar.
(a)
State the angle equal lo unglc AEC.
(h)
(l iven 1hnt AB - 12 cm, BC 3crn and CD - 5c111.
calculate DF..
,,'
f:·
D
E
/
\
,,....
/
...
...
,/
/
; '/"
Bi
i
\
\
I
/
,\I
\ I
\\//
A
Answer
(I I
(a) ~e
(h)
14.
cm
121
Thi;; lih:m-ancJ.lcaf diagram below shows the heights, in 1.:rn, of a group of boys and girls.
Girls
Boys
4
9
8
2
9
Hoys
key
4
I 11
R
2
13
8
0
14
2
3
4
5
6
3
15
2
16
0
3
3
3
3
4
t7
0
5
Girls
key
means 174
(a)
Wnte down the mode for all the children.
(b)
(c)
I'ind the percentage or children that is shorter than l 70 cm.
1613
means 163
W1ite down the median for the boys' height
Answer
(o) - - - - - - -
cm_ [If
(b) _
~[If
(c)
7
_
_
_
_ _ __
_
__:.
o/.,;;_
o
[I)
15
(a)
(h)
Simplify 7y - 41J 1 IO(J. y )].
Factorise completely 6ax +I Oar+ I 2hx + 20by .
121
Answer (a)
(b) _ __
16.
_ _ __
(2)
The force of attracti on. F newtons, between two magnets is 111 versel y propo rti onal to th.:
~quarc of the distance, x centimeters, between them
(i)
When the magnets ore 4 cm apai1, the force is 2 newto ns.
Find the formula connecting F and x .
(ii)
Find the force when the magnets are 2 cm apan .
(h)
pis proportional to q 1 • It is known that p =24 for a particular value ofq.
Find the value o f p when this value of q is halved.
Answer
(a)(i) __
[2]
(a)(ii) - - -- (b)
8
[l]
17.
(a)
On the Venn OiagrJm s ho wn in the answer space. shade the set A
ri
s·
(I)
(b)
(c)
c A=
B-
{integer x: I < x < l 5}
{integers that arc even numbers}
{integers that arc divisible by 3}
(i)
(ii)
List the eleme nts in A n R.
Wrile down n(A·" ti R)
Dc.:scrihe, as accuratt:ly us possible in set-builder notation, the foll o wing sets.
A - /4, 8, 12, ll>, 20 . .. I
B - 18, 16. 24 , 32, 40, ... )
Is Ba proper subset of A? Explain your ans wer.
(i)
(ii)
(iii)
Ans>ter
(b){i)
}
(1)
(b)(ii)
(!)
(c)(i)
r11
(c)(ii)
(I)
(c.;)(iii)
111
End of Paper
9
An B ;
AJ1Swcr all the questions
1.
·nie temperature, in ·c , at night, on three successive days was 6, -9 and r.
Find an expression, in tem1s oft, for
(a)
the difference in temperature between the second and the thinl day,
(b)
the mean !emperaturc for the three days.
Answer (a)
t
i'J
~t -j_
Bi
[I]
.:t.2_
(b)
2
Given that WO, PQ, ru1d RS are parallel, LWOP = 7k and
LSRY =23k, find the value of k.
- - ·- [!)
_3._
W - - - - .......
0
y
Answer
3.
Solve the inequality 7 - 3x > 3x - 7 .
Show your solution on the nwnber line below.
7-3x_ /"
{Cf
/
$Jl
-+
6"A,
,)_t 7
"11
;(_
:<. z):t
A11.ru er:
1
(
i
I
J
-5
-4
.3
-2
-I
()
0
?
2
(A-1)
[2]
.,
3
4
5
4.
The diagram shows a square inscribed in a circle. nie radius of lhe circle is 9 cm .
Find 1he sh<tded area.
7v r 2. -lti<{_ .,. "?~i
- - Yl/
= '1 J.. ·5'"
-
5.
Al
_q )_,~ _cm_:_ [2]
Answer
In a triangle XYZ, given that XY - 12 cm, YZ - 5 cm and ZX= 14 cm.
Oetennine whether the triangle XYl is a right-angled ~ru1gle. Show your working clearly.
Answer
A.j
~
-tj~
,
~
"'
r~
l l -H
,,__ ii., 1
;z. >l >. ;:
<f- i..
(
:: ;'/J,
lC'.( -t j c z.
6.
;f-- :z ;< z... -
.vi I
A hoy scout is plaruung to builc.l ~ tent to camp overnight but unfortunately he lost a wooden
suppon pole RZ and he would be looking for a replacement pole. I le measured the groundsheet
and cover sheets and found that PQ ~ 4 metres and PR= RQ = 2.5 metres.
Whal should be the ideal height of the replacemenl pole?
~ C. ,_ -t-
) ) ::.
IZ2
~
2· .r
R
2.
MJ
-
1-.l M.
p
Answer
3
z
_ _ r.r _
Q
__2!_ mt!
7.
One solution of 2x1 + kx-15 = 0 isx - 3. Find
(a)
the value of k.
(b)
the other solution of the equation.
t:.)
.;i.~3.)i -+-\~l') I
I?) >A 2
-
J+
r.r=-o
3~
= IS
3. (~ ,,. - 5
I'- ~ - I -
A/
11 ... 0
)l -
....
-)(_
Answer (a)
k
(b)
8.
:t -
f
-:J r
•'Y -
(I ]
-i clJ
29·7
. I paces.
I
and correct to 2 dcc1ma
-3.04(9.71- 17.26)
(a)
• Iuate the expression
hva
(b)
Express ·- as a pe1·ccntagc.
(c)
Express
.
-
00
19
37
23~% as a fraction in its simplest fomt.
4
b)
Answer (a) _
(b)
(c)
4
_ _/ .
;LJ
s I 5-:/'fo
)3,,.
_d_
!o
,.,,. 1'/-q%
(IJ
[I)
[I)
9.
A nllp is drawn to scale of I : 200 000.
(a)
Find the actual distanc e, in kilometers. represented by 15.5 cm on the map.
(bl
A nature park covers an area of 150 square kilometers.
Find, in square ccn1ime1crs, the area representing the city on the map.
CA)
I?)
c"" --
~ le"' )
4 fL,,,,,, .l
).
I 0.
OJi> Or-
:;,_
Qt~)'
I ""
8w-
)-001
;::
--;_ru
y.
Altrwer (a)
::;.
,_
_3J_
l2J
IJn
-31·S. _
_ c~ (I]
On the answer space below, hne PQ has been drawn for you. On the same drawing,
(a)
construct a triimgle PRQ with QR • ~ cm and L PQR ~ 120°.
(bl
construct the perpendicular bisector of PQ.
(c)
constn1ct the angle bisector of L PQR.
[21
{I)
LI}
;'.) "\ v..i
(b)
Answers (a), (bl and (cl
;
i
/'
I
a)
rJ.-t>' ~ 4-1
L<9 ""
LI F'& {!_ -{Of,...,).
~«1
Aj
b) ~ CM!1""-<.tt,;, )J.:..i._J
p
Q
5
(. ) 1 ,.,...tt_
(,,£
>oz.,, .
d/.t.1 lj
lo the diagram. the coordinates for the points A. D and C arc {-5, - 2), (-1, - 2) and (1 .1,
respectively.
(a)
Find the gradient of BC.
{b)
Calculate the area of triangle ABC.
(c)
Draw lhc line y = - 3 in the diagrnm and label your equation clearly.
l l.
.,, j
2.
....
(')
f.flCel -tk et'e.~
c:"ol dro...i th._ ~~ Ct·rr< e, f-1<JM 1<d
.. ,
_!_ 1 •
J
'. i J,.
1 _.___..
··-1·~-1
:
"
, ; • •:
Answer {a) _l_
~
•-<'
(b) _ __._/==
2-'12.
1·._r~
(JJ
units'
ii I
A sample space is made up of the first 8 prime numbers.
(a)
List the sample space.
Find the probability that tl1e number is two-digjt numhcr.
(b)
(c)
Find the probability that the n11mber is 29.
Answer (a) S• { '.2.r ~1 .f; J 1_LJ , I~, 17,L1J. (I]
__.L
(b) _ _ __..._ _ _ _ __ (1)
o___ _ _
cc> _ __ _
6
_
(ll
13.
The twu triangles in the diagmm arc similar.
(a)
State the angle equal to angle AEC.
(b)
Given that AB " 12 cm, IlC = 3crn and
calcularc DE.
cfl
I:>)
C7f
-L = s
.r-t~
Hrz
3
~-ti)(
q
T)C
14.
Scm,
c..C>
'
CF
CD~
"'
-f'I( · ( « -/lti&;J l £ )
A
_i_
3
(a) .Ang!~ «"
Ans1ver
::
~
_t,- -I r>C
(b) _
~i
Cf-
_
C.P..i) flJ
4:_ __ c~ [21
The stern-and-leaf diagram below shows the heightS, in cm. of a group of boys and girls.
Boys
1
9
8
2
9
Boys
key
(a)
(b)
(c)
4
I 11
8
Girls
2
13
8
0
14
2
3
4
5
6
3
15
2
16
0
J
J
3
1
4
17
0
5
Girls
key
means 174
16 1 3
mean• 163
Write down the mode for a ll the children.
Write down the meditu1 for the boys' height.
Find !he percentage of children that is shorter than l 70 cm.
c)
x
b)
•o.., /
14-f-f-/lf, 1
.)._
::.
:::
(~f
Answer
7
·.X
(a)
r63
(b)
rv--8-r
(c)
__R._'8_
cm
Ill
~ r11
%
(I]
15.
Simplify 7y-4[3-. I 0(2 - y)J.
Factorise complete! y 6(1)( I· IOay t· 12/Jx + 20by .
(a)
(b)
"-) .:r j ~
Lt( 3-t- ::i.o-
roj J - .vi/
41 ~ - 42
b) _:2a.._ ( 3x. -i '\j)
+ t.H.( ~.?\.-t:fj)
::: ( ?.<l. +4'7) C ?:.X 1 S j)
- ::> < ~ -1 2-'-J) t
16.
°' ~>
s'l. -1 rj)
(a)
The force ofallrnction, F newtons, between two magnets is inver$c)y proponiocml to the
square of the distance, x centimeters, between them
(i)
When the magnets are 4 cm apart, the force is 2 newton.~.
Find the formula c onnocting F and x.
(ii)
Find the force when the magne ts are 2 cm npart.
(b)
pis proponional 10 q 1 • II is known that p = 24 for a particular value ofq .
find the value of p when this va!U1; of q is halved.
gr ;- vi
f;t~ :;;..~
.
.
_7,l_t.1-) ~ t
IL '
';)
Answer
:S2 -
f-(.2.)),.
"'
0> P ,,,
i:
~
r
p :::
r " :s
-t
~z.
4-
3.
:<"
.J.t.r ;. 1~ i s - -
':> ~ )(
GJ ~ ©
:i.4- ~ ilp
32.
~
f
M[
©
3
c.JtJ
7 }:i
3
0
Answer (a)(i)
;\
F:{",. 32
g
(a)(ii)
(b)
p= .il.. -/j/
/l "
3
[21
nev..1ons
--- I\ I
( 1J •
~[ II
A1
J 7.
(a)
On the Venn Diagram sho"'11in1hc answer spa<:c, shade the set A n 1:1'
Al
(I I
(b)
(CJ
6' ~
AlJ =
(integer x: I <x < 15 )
(integers that arc even numbers)
{integers that are divisible by 3} _ ....
(i)
(ii)
List the clements in An fl.
Write down 11(A v IJ)
Describe, as accurately as possible in set-builder notation, the following s.:ts.
(i)
A= {4, 8, 12, 16, 20, ... }
(ii)
8 = {8, 16, 24. 32, 40, ... )
(iii)
Is Ba proper subset of A? Explain your answer
8,
2-,
~I
b
I 2_
t
Answer (b)(i) _l!r- fl =
l __6_1.Jl:_
(1 I
(b)(ii)
(cXi)
Ll~ ' :i__ ;~ -Jtf« ·/
<cXii)
~' l;J.. ' :l
YUJ.
(c)(iii)
End of Paper
9
fl]
Otr<'.
;:i
ti. (lj
/YIJ/f':;_-4--J.l fll
Ml ,.,J_/,/'~ ~ 1IJ
...,...,t/7k
f 4-·
YISIIUN SECONDARY SCIIOOL
We Seek, We Strive, We Soar
END-OF-YEAR EXAMINA'flON 2011
Name : _ _ __ ____ _ __ _____ Reg. No: _____ Ciass: - - - - --
Date: 71h October 2011
Sec 2 Express
MATHEMATICS
..
Duration
I hr 30 mins
J\'lAX MARKS: 60
i ~STlHJCTIONS
TO CANDIDATES:
\\Trite your name, class and index numb<--r on every answer paper.
Write your answers and working on the sc.,,arate answer paper provided.
Wri1e in dark blue or black pen on bo1h sides o f the paper.
You may u...e a pencil for any diagrams or graphs.
l>o not use staples, paper clips, highlighters, g lue or correction fluid.
,\r.o;v,.r;r all questions.
!f '" ' •t !-mg is needed for any qucslion it must be shown with 1he a11swc1·.
Orn ,.-;sion of essential working will rcsull in loss of marks.
Cak.•1!a1ors should be used where appro priate.
If ti ;_· dq.;ree of accuracy is not speci fied in the queslion, and iflhc answer is not exact, give 1hc
aoswer to three significant figures. Give answers in degrees to one docimal place.
Foi
1t,
use either your calculator value or 3.142, unless the question requires the answer in tem1s of
"·
.'.t tlic end of the examination, fasten all work securely together.
Th~
number of marks 1s given in brackets [ ) at the end of each question or part question.
T 11r. tc•:•f of the marks for this paper is (>0.
This document consists of
1
printed pages including the cover page
{Tum 01-er/
J1atliematical Formulae
Compound Jmeresl
Total amount =
11
'
+ _.:._)'
100
Mensuration
Curved surface area of a cone = 1cr!
Surface area of a ~phere
=
4m·'
Volume of a cone= -I w ' It
3
-
. Iurnc o1· asp I1ere = -4 m· 1
\ 'O
3
Statislics
).:
r~
Mean ~ _
.!-'
_
·
'iJ
2
{Tum Over]
/\nswer all the questions
1.(a)
(b)
(i)
Express 132 as tbc product of its prime factor.; in index notation.
(ii)
Hence, fimJ the smnll est positive integer d such that I32d is a perfect square. [ l]
[I j
The I ICF of 150 and xis 15 and their LCM is 450. l'ind th<: srnallc~t possible integer x.
[3]
2.
ln a pentagon AHCDE, the mknor angles A, 0, C. D and E are 3x0
,
(x + y)0 , 5x",
{x + 30)" and (3y - 150) 0 respectively.
(i)
Write down an equation in tenns of x and y, and show that it si111plitics to
x
Q.6.Q_-4y
f3l
10
(ii)
3.
Given that Sl
y = 60, hen ct-, so lve for x and y.
f31
In the d iagram below, a solid metal figure is made up of a cone and a cylinder of radius
I.Sm.The height of the con e and the height of the cylinder arc 7m each.
( Ilse :r = -22 1·or the whole quest .ion)
7
(i)
Find the volume of the figure.
(ii)
The figure is 1m:ltcd on<l formed into a sphere. find Ilic radius of this sphere. (2)
(iii)
Find the surfocc area of this sphere.
(2]
(2)
/
7m
......
-._
-
-
7m
.,,- ·-.._
3
[Turn O»erj
4.
Mr C hia is investigating the number or equilateral triangles and rhC1mbuses formed wher.
a m1mber of dots are positioned equidistant from one another as shown in the figures
below.
• •
• •
•
• •
Figure 1
• •
• • •
Figure 2
Figure 3
Figure I shows 3.dots equally spaced out.
Figure 2 shows 4 dots equally spaced out.
Figure 3 shows 5 dots equally spaced out.
He the n dre w up a table shown below to record some observations.
.. .
..
-
-
~ - ---·
Number of dots
II
formed
!
I
Number of
equi lateral
triangJcs
l
1---1
3
I
4
3
5
2
3
I
Number of
r hombuses
I
I
-
II
formed
.
0
·r
-
I
l'ro duct
-- 1
I
o_ .. ..
1
8
30
I
2
I
1 -.-.
20
!
- -- ·
I'
19
20
.
'.....
---
I
r
i-- --1
q
11
.l
s
I
-------1
I
- --[' --
II
· - --
- -·
t
···-··--·_J
Without further drawing, use the infonnation from the given table,
(a)
(b)
find the value of
(i)
p
(I]
(ii)
q
[!l
write down an expression, in tcnns of11, for
(i)
r
(I J
(ii)
s
[l ]
[ 11
(iii)
4
[Turn Over]
5.
Jn a set of six num \Jcrs, cwo oft hem are x and y.
The mean of the other four numbers isl2.
(i)
Write down the s um of"the six numbers in terms of randy.
(ii)
lf.r is added to the six numbers, the mean will be 11.
If 15 is added
10
[I J
the six numbers, the mean will be 12 .
Formulate two equations and hence, find the values <lf x and y .
6.
[5)
A sales promoter of an electronics gadget store is paid a conunission of 2% for every
il'nc1 touch sold. The seHing price of a &GB and 32Gl3 iPod touch are $328 and $428
respectively.
(i)
In a particular month, the sab promoler sold 76 8(Jn <1nu IOR 320H iPod touch.
Calculate the total commission received by the sales promoter.
(ii)
[21
In another month, the sales promoter sold twice as many J2GB as &GB iPod
touch. If the total commi ssion for that month was S2368, find the number of
3200 iPod touch he sold.
(iii)
[2)
The selling price of each il'od touch is inclusive of a 7% <Sr. find the price of a
[lJ
8GB iPod touch before GST.
(iv)
During a sp<:cial i;ale, the iPod touch was sold at a discnunl of 5%. A OST of7%
is added to give the final price. Calculate the price of a 8GO iPod touch inclusive
f21
of OST during the special sale.
5
{Tum Over]
7(a)
(b)
(i)
Factorise comple tely Sx 1 - 125 .
(ii)
.
l"f
ll cnce, s1111p 1 y
Express as a single fraction in its simplest fonn
>+I
(-:)
8.
1
5x
125 ,---2x 5x 25
.
Given that II =
2> • 2
u i. t- v 1
3h
.
, cx:press um tem1s of R,
1• and
[3]
h.
Andy and Rachel s tarted together at the same constant speed x km/ h on a 7 km walk.
A ller I km, Rachel increased her speed by l kmtb, travelling ar this speed for the
remaining 6 km. Andy walked at the constant sp~-cd x km/h till the e nd
l Ij
(i)
Write down, in tenns of x, the time taken by Andy to complete lhc 7 km w«lk
(i i)
Write down. in tenns or x. the time taken by Rachel to complete the 7 km wHlk.
r11
(iii)
Given lh~t Rachel finished the walk half an hour earlie r than Andy, focm ati
&(Uation in x
(iii)
and show thal it rectuces to x2 + x - 12 = 0 .
[3 J
Calculare the time, i n hours, Andy takes to complete his walk using U1e ch• •sen
value. Ex: plain why you need lo rejt."Ct one or the answers.
6
[2)
[Turn Over]
'>.
Answer the whole of Lb is question on a piece of j!rnph pupcr.
The table below g ive:; some va lues of x and the coms ponding values of y for
1•=x'
2x - 4
(a)
Find the value ofa .
(b)
Using a scale of 2cm to represent I unit on each axis, draw the graph of
[1 1
y = x' - 2x - 4 for - 2 s x ~ 4 .
(c)
(1J
llsing your graph,
=2
(i)
find the values ofx when y
(ii)
solve x ' - 2( - 4=0
[21
(iii)
write down the equation of line ofS)1nmctry
fl l
[2]
End of Paper 2
7
{fom Over}
, I
~lo.)
).
?, )...
J-
f:k
~
131 ;: Jl. x ?i
.
~
cL -=-
l1
xii Cftf J
//
3,x 11
---;~ [f,}1 J
//
~b)
H6F = 15 = ;
;
LOY\ ::. 450 =2
xs-.
7.
;><. ~). X
5
"l
J[ 11\I J
so .,, )_y:.. 1 ><- 5 i-
1
c ll'llJ
x.. -;. :/ "')(~
Y.... :.- 45
//
~i) lOJ..
Cfl'l1
+ 4'J -
110 -=- Sl/-O Ct11 IJ
10)1... t ~ .-- 6bO
Lfy [if\\']
6<> 0 4J (&how11J
l0 /... :::. b bO-
x. -;
~?>co
.
.
- 1'>:9
- - --l~
Lo
.
.
ro Cml1
-;: ~
\
to
~300 - .?OJ :::. bOO
-3~
( I i"'\
""
- 11-()u
'j -:;
q0
• /f (
/l f \ )
Crll 1j
~
UtO
:,;) vo) c{-
~jVIYL ~vol of- wf\IL
I
::
).'-
l Si_
)Y..~ ~ - ~ 1-
:: bb uvf
;;1
cf
gf ~Y~
··
;il-5 '--1' f-
l""'IJ
<fiJ
3
i1 "f. "=:::t
)}. f r3 :; b~
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Index
Number
Class
C<111did1J1e Name
YUi IUA SECONDARY SCI IOOL
END -OF-YEAR Examinati on 20 11
Secondary Two Express
•
4016/01
MATHEMATICS
PAPER I
T IME l hour 30 minutes
-
Candidates answer on the Q uustion Paper
READ THF.SF, INSTRUCTIONS FIRST
Write your name, class and index number on the work you hand in.
Write in dark blue or black pen.
You may use a pencil for any diagrams or graphs.
Answer a ll questions.
If working is needed for any question it must be shown with the answer.
Omission of essential working will result in loss of marks.
You arc expected to use a scientific calculator lo evaluate explici t numerical expressions.
If the degree of accuracy is not specified in the question, and if the answer is not exad, g ive
the answer to three significant figures. Give answers in degrees to one decimal place.
For it, use either your calculator val ue or 3. I 42, unless the question req uires the answer in
terms ofn
The number of marks is given in brackets {)at the end of each question or part question.
The total of the marks for this paper is 60.
For Kxa miner 's Use
Total
This qu"51ion 1>a p cr <CJnsists of 14 printed pages and 0 lin<'<l
t>~ g(•S.
[TURN OVER
Sec l F..xpres.t /;-ncl t"t•t1r Ettlttt 1011 Alac/u_·1na1u·::
Yuhua Seccntlary School
1~:
Answer!!..!.! t he 4ucslions (Total: 60 marks)
I.
(a) Express 420 as the product of its prime factor$, leaving your an~wer in index
notation .
(b) Hence, find the sma lle~l positive integer k, such that product of 420k is a perle•:t
square.
(a). - - -- -(b). k =_ _ _
2.
(a) Simplify p '
-
[21
l IJ
(p+;Xp - ~}
(b) Using your answer to (a), write down the value of 1 3489~ - 13-199 x 13479 with<>Ut
using a calculmor.
(a). (b).
2
[21
- _ [21
YuJrua St'>!'011darvSdrfX.1l
2
1
3.
(iiven that a + b
Sec 2
J:.~e.v;,v;
End.r,•u1 F1,un !II/ I A-lathemattr"' Pl __
20 and oh= 5, calculate the value or (3a - 3b)' - 2(a - h)'.
3.
4
,_
_ _ (3]
Sicnpli fy the following.
(a)
(7 x'y)' z'
21yz
(b) ( 2 - l );..{4y_~)
;r
y
x y
(a).
(b).
3
_ _ _ __
[21
[3 J
hthua SeCQ_niJ!!!:l' ~lroo/
5
Sec l b pre.u £m/. l'e11r fawn WI I Mmlier<
(a) Make T the subject of Che fonnula x
(h) Hence, find the va lue
'"'°' ,,I
i 4T
I 2111
= \/
or T whcn m = 2 and x = I.
(a).
-
(b). T= _ _ ___
6.
_
[2'J
[I )
Oneofthesolutionof 2x' + kx - 15 O is x = 3. Find
(a)
the value of k ,
(b)
the other so luti on of the equation .
(a). k
(b). x"'
4
~--
f2J
_ _ _ __ 121
Yulrua s,~,,ondan· Schvol
-----
7.
The chagram shows the graph of2x ' y · 6.
(a) On the grid below, draw the line y -
~.
(b) u~e your graph to solve the simultaneous equations
2x I Y "" 6
y °",f
_ [I]
(a). show'""n,___
(b).x~ _ _ _ __
·-- -- -- 5
y
=
_ _
Ill
Ill
- - - - --------
1'11/1110
Secondary School
R.
In the figure bch•w,
Sec l
Erpre.~<
t\PQR is snnilar to 1\SQT.
End !ear l'..wm WI I Mathematics f'I
It is given 1ha1 QT
Sun, sr - 7
s
cm, SR = I0 cm and PR - 10 cm. Find the values ofx and y
Q
_ __ _ _ cm[2]
_ _ __ _
6
cm[2]
1'11/tua Secondary School
9.
The diagram shows three se1s A, LI and C and 1: = .4 v JJ •.J (
~
•.
-
I
A
B
-
c
Given that 11(1:) - 45 , 11( An B)
.....
= 1-:11(H n
C) = 6 and n(A) - 11(8) = 20.
(a) Using the given info1mation, write down the number <>f clements in each
subset in the diagram above.
(b) Hence find the value of,~ II r. (Av C)'] .
(a). shown
(b). _ _ __ _ _ __
1
[2]
_ (1]
10.
(a) Given that y is
invcr~dy proportional to (x -
2)' and that .J - 2
~
whw
.1
::; fiuJ
the equation connccti ng x and y.
(b) The scale of map Xis l : x and the scale of map Y is I : y . If the same distance is
represented as 5 cm 011 mapX and 7.5 cm on map Y, calcu late the ratio x : y.
____ i21
(a).
(h). _
•
8
_
_
_ _ _ r21
1'11hull Secondary School
11.
Sec Z Expreu F.nd-1','/lr Ewm 1011 .tfatliPmatin I' I
·1he points A, H ~n<l Con 1hc C3ncsian plane below arc conncc1<!cl hy s1ra1ght lines to
form triangl c A RC.
y
..,...-·7 8(3. a)
I
A(-.5, 1)
C(l,
I)
(a) Given that the gradient of the line CB is 2, find the va lue of a.
(b) Find the area of 1nangle ABC.
(a). a
(b).
9
_
_
_
_ _ __ [2)
_ _ _ _unils2 f2]
YuJwa S'r.co1ularr School
12.
__ ·-
_ -·· - - -
Sc<: 2 EXf!!.!:_tSEnd- }'eor£xnn1 21)_1j ,ft,fa!.!!..t:.''!!_~I~<:! J~:{ ...
Consider the following number pattern:
23 -2=6
3
=1 x 2x 3
3 "" /.4
= 2 x 3x 4
4 3 - 4 = 60
=3 x 4x5
3
(i)
Write down the 5•h line of the sequence.
(ii)
Express 17
(iii)
Hence express ni - n, as a product of three consecutive numbers.
3
-
l7 as a product of' three consecutive numbers and find this value.
(i) .
. [l)
(ii).
_ __ [1)
(iii). _ _ _____ _ __
IO
[1)
Ser. 'l F.1pn'.\S £11d Year F.xam 2011 Mathema11c.< P( _
Y,,lwa Sc'"!" dm y Sr.'!!!'!!__·_
I.l.
A survey was con<luctctl to lin<l the number of passengers in each tax i passing nn ERP
gantry o ne afternoon.
No. of passengers in each taxi
No. of taxis
0
I
I
x
1
,_
2(x + 2)
J
'
19
4
11
(a) Given that the modal number of passengers in each taxi is 3, find the largest possible
valuo:: of'.r.
(b) Find the largest t>ossiblc value of x, given that the median number of' passengers in
each taxi
i~
3.
It
(a) . .r = _ ~
(I)
(b)_x ~ - --
- - - (2]
Sec 2 Express End - Year F.xanr 20/j :\-fat/,r r1;ryn.:;_.fi Pi
Yuhua Seco11dary Schoo/
14.
In the diagram , PQRS is a rhombus. Given 1ha1 PR = 6 cm and QS ~ 10 cm.
find the length QR.
14.
. cm f2)
- -- - - ---- - - -- - - -- - - -- - 15.
In the diagram, ABCDE is a regular polygon. AN is the perpendicular biscctur nl c.D al!d
EM inlcrsccts AN at P. If L APE= 50°and L.DEM= 32° , calculate
(a)
L.NAE
lJ
(b)
A
E
(a). L NAE =-- -(b). L:CEM -
12
-
- ----
-
0
- -·
(2)
., (2)
Yul:uu St•t"f!ndary .Vhool
16.
1'11e diagram shows a spherical ball of internal radius 20 cm is partly filled with water.
Water is poured in to the hall 1hrough a small ho le of negli gible s ize. The radius of the
horizontal water surface is 14 cm.
(a) Calculate the dept h of water in the ball.
(b) If' 500 drops o f' water is req ui~d Lo Iii l the ba ll lo the hri111 and each drop is
approximately 2.5 mi lli litres. ca lculate the original volume of water in the hall to
the nearest whole n umber.
[Vo lume of sphere:
4
m·' ]
3
(a).
(b). ---
13
cm
- --
_
3
l21
cm (2]
Se« l Expre» t.t11J-l'e11r Elam 1011 Ma1!1rm"'"'' f''
Yuhua &condllry Sc/roof
17,
l'he diagram helnw represents patl ofa field where I cm r.:prc;,cnts 0,5 knL Trca<urt' i.;
buried al X Xis equidistanl from BC and AB, Xis equidistant from A and B,
!I
A
(a) By drawmg sui table conslru<:tion lines, indicate on the diagram below th.: lrK"auon
ofX
(b) Measure and writ<: down, in kilometres. the leng1h of DX.
[2]
(a), ShCill'.Jl _
(b), _
_
Check ynur work!
End ofpaper@
·>
14
km
L21
Answer au the <jUestions (fotal: 60 n1arks)
1.
(al Express 420 :is tbc !'>roduct of its prime factors, leaving your 3nswer in iudex
notatioH.
(b) Hence, fond the smallcsi positive integer k, such !hat product of 420k· is a perfect
square.
4:).0
= ::t '.)>( 3 x
5
x. =t
r::J I
- Ml
•5) K; :ix5x1-=
105
(;ii.
(b).
2.
<•>Simplify
:£ l X 3X 5 X: ==]_ . .f21
k= t05
. fl ]
p'-(p+~Xp-~}
(b) Using y<>ur answer 10 (a}, wri<c down Ille value of 13489' - 13499x13479 without
using a calculalOr.
a)
pl_ ( P-t ~) (p - ~ J
; P2- ( r :- (~-·r J -
IV)\
-: p:i_ p:i + ~2
-:;
b)
~
q, .,_ - - fl\
D; 134'0Cl
134-i Q-t ~ = \34-qll
q_ .,. O· l ~~~~-
~
2
4(a ) .
- IV\f
---~--------
= (.;.·?)' = !0 o
I+\
_
,.,
:> _ _
__:
"le__
[2]
~(b~)·===
\ 0=0
=====~{2]
>Mhu.i Sn"f.Jltda2· ~,./
3.
Xe 1 F.tprus F.nd- Y~cH' ~"' }f)//
_
'\fotlit:#tOfics~PI_
1
Given that a + J} • 20 and a/J • 5, calculate 1.he value of (.3a - 3b)' - 2(a - 1>)1 •
( 3 a- 3b) J_ :i Ca -b) 1
' Qa').- 18ab+qb..,-;:la~+4ab -O b 2 - --IY)f
-=
=ta' -r ==1-b') - 14-ab
~
=t(~').+b )-l4ab
7
Ml
:: -=t( ~) - 14-(0) ,
= =ro
4.
A-\
"'
- - - - - -- Simphfy the following.
.:ta
_
-
_ Pl
-
<•l L7 '.2'...2_
I
)'
'
21yz
I
b} l I
<ty :c
(- - - )+(---)
x y
x
y
( '=!-X2 ~J i.Z ~.
a)
~
49x4~2z4
::>1 ~4
:ll ~z
.,.
~~4-~3g
-
IV\)
-Al
3
( ~ -t) ~ ( ~- - ; -)
b)
2~ - 'X.
..:.... :fg)- 'X2
x~
.,,
~ - ?(
)(~
-::
.'.:l~ - x
x~
--1-
;:l~-1-X
x~
x
-
Ml
'.d:~
4'J'>~ X-..
x
-x-,J
(2~-x) C.J~+x)
+x 4z3~-
-
- Ml
(a).
(b).
ltl
.3
-·-
2~+)(
[2]
(31
SE"c 1 ~::_pr~.S.." f.nJ· t•~"r f:X.am ZOii Afotlu.·mt:ticJ Pl
t'itlJ...ua_§econclorySchool
~
(a) Make T the suhJCCI of the fo1111ula x - •/ 4 T .
~ 2111 - l
(b) Hence, find the value of Twhen m = 2 and ~=I.
4) ~ =
aj 4T
)l'Y) - \
X~ " ~
-
Ml
2wi-l
J-
?< 3 (:>vvi - 1
T•
f\ I
4
b)
y.J\1QJ/l yYJ ,
~
t\.vld :A' ::. I
I
T = < 1 )"5(~(1)-1]
4
-; ~- - - Bl
(5.(, : A\lOvV Q,.c~)
4
x 3 (:lrvi-1)
__4_:_·_ _
(a). _
(b).
6.
i:,
5
-- f21
4
_ [IJ
One oflhc soluti on of 2x' + kx - 15 -0 is x = 3. Find
(a)
the value o f k,
(b)
the other solution of the equauon .
~)
')( =3 ,
Q('~)l + 1<::(3)-15 <-0 W'r'l~VJ
1it?>k - 1?
k~-1
t>)
WYleM
fli\I
~o
-Al
j.::. ::- 1
.
?x' -'X- 15::0
(Q.x+S)
cx-2>) ~o
-
Ml
(e.cf)
'J.x+5=0
';(' - ~ ·5
Oi -
:l-'
.2.
AI
()
a.
l(~- J
(b).'X=
4
-J-5
_
_
[2)
f2J
1.
Tlic dianram •hows the l!Taoh of 2;,
----~ -
·t
y
=6.
-·r-1
.
4-
-'
l~~l
j_
-t'.j • ?(
;
. r
-
Bl
;
!-li
j
Ii
(a) (;r. the and ocinw. oraw the •inc y :::: x.
ib) Us.-! your tfTapn co soive chc s1muh.ancous eouauon.~
lx+y •- 6
~;x
-
-B J
- - Bl
(.a\. ~~?WP
ii J
(b'1 . ~ ~--~----
fl)
'j
5
._2,,,,,__ __
[l]
It.
In I.he iigurc below.
1!.PQR
is smular 10
6.SQT. h is given tba• QT• 5cm. sr
cm. SR = I0 cm and PR. = 30 cm fiud the values of x
we '
7
s
,.
Q
a)
&R
6
,. 3 0
~-5
QR -: ,?0 OM
~"
bJ
-
;,)o (.vv1 - \Ocw
PQ ;
o
IOCM -
Bl
30
IQ TS
P& "' 400N1 -
':1 7
rV) I
Ml
40 C.wl - ~(.,v-11. -:: 3?0.M - - A\
,. = _
IO
_ ___ cmf2J
__
y: -~35~--
CM[2J
?
The dia1tr•111 $hows three sets A . fl and C and r. = Av B u C.
I
A
I\~ aw2tfae.d foy 211 el&r'lE'M ~
13
I
t_ _ _
<)ta\W (.O(fe?,tt\_y .
- Al
•YJCDrrect
eievnet'\i3 i'JOtlrj ·
_
6
8 L____ _ _ _ _~
G1vcnHoal
>1(cl = ~S.
{Ur ea:,vi
I !l
nrA r. 8\ w ?, n(8 r\C\ • 6 :mo n(A l - n1Bl ~ 20.
(a) tJ~in@ the given 1nfonnation write do\\'ll tl•c nurnbcr o( clements in c.ach
<ubset m the <.!Ju.'fa.ou above
ob) l lcncc fin<l thc value of •1'.ll !"\ (Av Cl'J
(31. sflOWQ
(2.J
- - - - - ---- -·
'°) - :--g..
: :===========:_:('~1
JO.
(a) Given that y is inverse ly proportional lo (x
2) ' and that y = 22 whenx - 'i. find
9
the equation connecting x aud y.
(b) The scale of map Xis I : x and 1hc scale of map Y is I : y. If lhe same dislance is
represented as 5 cm on map X and 7.5 cm on map Y, calculate the raiio x : y
a)
~(~-.'.l) 2 ~k
\VV)e,vi
!:1 ' ~
I
?(
~5
I
k = 2 ~ Cs·-~) 1.
:. :l, !:>
MI
'6 ('X <)) '=
:.15 -
r.'lap 'X
A\
"
~
Yflap
I -. y
I: X
5 : 5')(
::i ·5:
5~ :. =t-5y
')(
- -=]- .!)
.2_
= ..2....
=t -5_0
-h11
--g- - -:3
y
2
1X : .0
;:.
3 : '2.
~l
'.15
or
vi~---.,
J
(a). _1_~
-
-- -
- --- -
(h).
-- -- - 8
__?;
('x-2 ;
..
'~ 25 _
2 _
[2)
__ _12J
Sec} E.x.preJJ f.nd~ Yt•ur f ..rtJm J_OJ J :\1v1lu"1011c.f. PI
>!_1itun Secv111ar1, School _
11.
The points A. 8 :i1ul Con the (~1r1esian plane below a1"C co11ncclc<I by s1raigh1 li11es 10
fom1 trianelc A !IC.
y
.(- >.I)
C (l -1 )
(a) Given lhal Lhe gradier1\ Oflhe line (.'fl is 2, find
lh~
vuh1c of a.
(b) Find 1he Hrca of triangle ABC
a - c-i) :; '.l
3- I
~ ".2
-
-
ff1l
).
3 =2> -
11\
rfl
6)
Me.a~ .Al'.\8C ::(Gx4-J - (tx6 X.'.:J) -(~-x.)1'4-):-(~.x)A4)
: I()
'
l.{V}itS -
~I
<"'>· a :_SL__ _
<1>>. to
__ _ 121
1.1l'lit';l121
-
l'w'n;o ,<;,.,.,,,,da11 Scliool
1
-- - 12. Co11s1d<'r tlic fol lowing number pmtern:
J'
>= 2·1
1
4 - 4
60
= 2x3x4
= 3x4x5
\i)
Write down the S" line of the sequence.
(11)
l'xpress 17! - 17 a~ a 11roduct of tlu<!e consecutive numhcrs and !ind this value
(in)
I lencc cl<press n' - n. ns ~product of three consccu11vc numbers.
;} b; - (,-:. )\0
i~
;ii)
=5
1::('- 1-::r ; 16
X
b X.1-- - ----
x 11
x 1B -:. A-eQ6 - . Bl
'()~-YI= (r1-1) "f. V)
x( ()+I)
'° D(t'J-l)(Yltl) -~
\') ( V)1__ I
-.. Y)
Bl
Bl
J
~()
('J
.
(i) _Q-b = .'./(0 = 5X6):111
4-~9 b
(I )
(iii). ~(\/l - t) (Ylt-!_T ll l
(ii).
- -- - - - ····- -
•
10
·--- - - ·
;J -
_!!1.'~1 !!!!_,·~d.2_ry S~1l_
11
/\ ~urvcy
R~ntry
W-dS
_ - -- - - _ -~Yee 2 Exp_rt~SS f:nd-~ C..\'um 1£! l_!:f'!!!.!e!!-U.!£.f.!!J
l
_
conduc1ed to !ind the number of passengers in each taxi passinp, an ER.I'
one a her noon.
No of pa~sengers in e;,ch taxi
No. of taxis
0
I
l
x
2
2(x + 2)
3
19
4
11
-~
(a) Giveri that the modal number of passengers in each iaxi is 3. find the lllQ!est possihle
value ofx.
(h) Find the largest possible value of x, given that the median number of pass~ngcts in
each taxi is 3.
l('X+:f)
< 1q
~')'il.j.<1q
".).' < }.5
OY
~
LavQe~i IX -- -=t-
- - 8I
+ ;i(x+:i) = 18+ I l
~- 'X + ') )I' + 4- ; '.lq
t +'X
l
1'01
3 -x "" :Ji+
'?( :. '6
- - .BJ
(a).
x~-:/-
__
111
- ----
{21
( b). ')( =g
tt
~
_ Y:.:hua S!:C<): -dury Sch.oaf
l
f..xprc~s
!11 the diagram, PQRS is a rho111bus. Given that l'R
I •I.
=
End Year
f~xcm
2011
,\;fa11t.-,~~':·I:~.> .'' •
6 cm n11d QS- I0 cm.
find the length QR.
lHZ'l~ ')/+5)
-
Mi
QR ~ J?>t+
~
5-83
LiM
( 3_s¥) - - 1)1
5 . 83' -- - <'Ill
14.
15.
l~M intersects AN at/'.
a) tacl'l
(a)
L.NAE.
(b)
L.CiiM .
i'1ter10(
ff / Al'F: ~ 50° and LDEM = 32°, calculfllc
JJ
av.g1e , (5- ;>) x 180
5
~we·
LNf:I&: lO~
b)
[2 J
--- In the diagram, ABCDE is n re gular polygon. AN is the perpcmliwlar biscdor ., r ' ·o 3n<l
- - - - -- -
0
-::.
-
b
A
Ml
2 "' ?4" - - Fl I
AB//EC
E
L CE n "' I Ro 0-
!06 • : 1 ~ 0
Lc1::rn~ 1~·- "1-2 ·-:
4-.
- - - Ml
3':l"
81
(a) .
54
(h) . __
=:f- _ _ ,_,_..
.° [21
0
121
16.
The diagram shows a spherir.al hall of in1ernal radius 20 cm is partly filled "'ith walct
Watc< is poureJ into the ball througb a small hole of neglieiblc size. Tue radius o fth t
boriiontal water surface 1s l 4 cn1.
(/:C-~..... l~;.~. l
\
I
\.,,
/
I'
./
-........_L~../
I•• Calculatcthcdc0<hufwatcr in the ball .
_ ' {h I II' 500 drops or water is required tu fill the ball to the brim and each drop is
approxirtllltcly 2.5 millilitres, calculate the <1riginal volume ofwaict io ~ball
f Volume of
a)
'Dept\<i
of
spl'lefe =
vlC\0( :
1: irr3 J
j ~Q :_ 14-,_ + :10
~ 34 . 3
b)
Vo\u~
- - - IY) I
°"' ( 3~)
of- wQf-eY Odded = 500
x d
,, I J. '.::>O
M
·5 m.t
t.
-:; !.)'::() CW13
YOllJ.rY)(?
C1ri9i"la1
of- SP'l'Rte ~ ~
vo1~ ~
1T
(:io)3 ~ 33, ?10
3
owi
3
tAla\Q.¥ = 2>3.510ew1 ~-1.:i::oew1
3
r:t I
= ;?JI :>60 ew...
(at _
Ml
_
3_4_.~---m [2]
(b). ~3~!.'l_,_~_{::i_Q_ cm' [2)
I~
}i1h11J 5:-.·o,,clc2 S<ho!_;I
Sc-!c 1 £tprr11 f.nJ t'eor £.ram l.011 Alothe"!oric.f Pl
1
17. ·n1c diagram below represents part of a field where 1 cm represen ts 0.5 km . Tre"'""' 1.
huriccl al X. Xis equidistant from BC and AB. Xis equidistant from A and H.
c
AI ( 4- bi~<4ov ~
_------:--A
-
-
.LCBI))
'
r} l ( _h_
.
01'b'tjov
e7{
A.BJ
(a) By drawing sui table construction lines. indicate on the diagram below the loca11on
of X.
(b) Measure and write down, in kilometres, the lcnr,th of DX.
yY)Q3~11veq l€Mgi-\1 ~
A~a\ l~q-11..-t ~
Dx: ~-q~ (-tO·l°""J
') .q_ ,xo- S
et1ow t.C.f --r·7)
~ l-4-!?~
(a) shown.
(b).
Cl"";k your work!
End ofpaper @
14
I· '+S
(21
_ _ km 121
c
Candidah.: Name
Index
Number
Class
-_J_
•
YlJHUA SECONDARY SCHOOL
EN D-OF-VF.AR Exa mination 20 11
Secondary Two Express
401 6/02
MATl l l!:MATI CS
PAP ER 2
T IM E 2 hours
Candidates answer on the Question Paper
R ~AD
T HESE I NSTRU CTIO NS FIRST
Write ynur name, cl ass and index number on all th.,; work you hand in.
Write in dark blue or black pen.
You may use a pencil for any diagrams or graphs.
Answer all questions.
If working is needed for any question II must be shown with the answer.
Omission of essential work ing w ill r esult in loss of m arks .
You arc expected lo use a scientific calculator to evaluate exp lici t nu merica l expressions.
If the degree of accuracy is not specified in the question, and 1f the answer is not exact, give
th e answer to three ~ignificant figures. Give answers in degrees to one decimal plncu.
For it, use e ither your calculator value or 3. 142, unless the question req uires the answer in
terms of it
At the end o f the examination, fasten all your work securely together
The number of marks is given in brackets (] at the end of each question or part question.
The total o f the marks for this paper is 80.
For F:xa m.iucr ' s Use
This 4uestion paper co n siSl i of 15 l)rinled pages aud 0 linec1 p:l){\'S.
[TURNOVER
J
Sec 2 Fxprrss J:.'nll }'<·ur Etan.• 20 ,'I i\,/atlrrnu.,111, ·, l'i
}'uhua St•c(utda1v Srhonl
Answer all the questions (Total: 80 ma rks)
I.
(a) F.~limatc the value of 2,J&O
1 3v'26
.
\/10
(b) If ,;'1.732 ~ k, lind the value of J0.01732 in tcrmsofk.
(a). _
_
_
. [2]
_ _ _ Ill
(b). _
2.
Expand and simplify the following expressions.
(a) (:lk - 5)(2k + 7)
{b) (a
1 4b)(a
- b) - (a+bY
(a). _ _
(b). _
2
_
_
_ __
_ _ 121
___ 121
Yu/:1111_.'it.~<mJarv :,"E_lroo/
3.
Sec l F.xp1r« End- Year /:..ram 1011 Matht·ma11r.< Pl _
The diagram below shows a circular larget wich diamete r AR of 20 cm.
Whal is the probubility that a dart lands on the
(a) red sector.
(b) green sector.
(c) the smaller circ le, with rndius 2 cm drawn
at !he cenlre of the circular 1:1rgct?
_ _ _ [IJ
(a). _ _ _
(b). _ __
_
_
(c).
'L
(a) Factorise
_
[2]
_
121
3x' - 7 x + 2.
(b) The area ofa 1cc1angle is
(3x'
h :!:_3_)cm2. If the breadth of che reclanglc is
X ·l· 2
- .!_)cm, by using your answer to (a), find its length in tcm1s of xin cm.
(lx
x' + 2x
(a).
(b).
_ _
_
_
_ _ [2]
- - - -- -
cm (2]
Yuhcu.1 se,·undar}' Schoal
5.
In the dignim shown below, the pyramid
has~
slant height of I cm, and a square br1sc
with length of b cm. The length from the centre of the base to the midpoint of~
~ 1dr.
:.of
the base is given as r cm.
/cm
bcm
9.
(a) show that
2
~- =Ii - r', where Vis the volume of the pyramid.
h
(h) (i) Make I the subj ect of the equation given in (a).
(ii) I lence, find the slanted height of the pyramid when V = 48cm 3, r = 3cm and
b ~ 6 cm.
[Volume of pyramid=
~ x base area x height )
.)
(a). shown
(h)(i).
[2]
- - - · [2]
(b)(ii). · - - -
________cm
f~I
- - - - - -- ---------- +M
4
Belly and Rila s1art1:d together on a 7 km walk at the same constant speed oft km/h.
/\lier I km, IJctly increased her ~peed by I km/h and wa lked the remaining 6 km at the
111:w speed. Ri1a walked at a constant speed ofx km/h all the way.
(a) Write down the time,
111
tcm1s ofx, Betty look to complete the 7 km walk .
(b) Given that Betty finished the journey 12 minutes i:arlicr than Rita, fom1 an equation
1
in x and show that it reduces to x + x
30 = 0
(c) Solvethcequat1<>n ,~+x - 30 = 0.
(d) Find, in hours and m i nutes, the 11me taken hy Rita to complete the whole journey.
'-
(c) Calculate Betty's averngc speed. 111 km/h . for the whole j ourney
(a).
111
(b). l?.!l.Qwn
[31
(c). x
(d). (c).
5
,.
_ [2]
h
_ _
min [ I)
km/h [2)
~
fr_
1h'"'''S
' -'e"-c·"o-'-'
ut;;.:
a_,,
ry...:
· S'-'c-"
ho:..:o.c.
l _ _ _ _ __ _ ____;Sf>.£ 2 F:_:g_!!.'!_'SS End-Year Extun 20 I J :\tla~jit·ma!i~;'.l_ ·,::.__
7.
In the diagram below, MJ(f' and L\ABC arc 5imilar. The side AB of M BC 1s dt.,:dd
such that 3AX = 2AB. P and! Q arc points on CA and CR
~uch
that XP and XQ arc
parallel to BC and AC respectively.
A
~
~~
16cm
))
B
(a) Find the ratio of
'-~ p
c
Q
A! .
AB
(b)Given thatXP= \6cm anilXQ = 15 cm, lind the length of AC.
(c) Given that LXAP = 65Q, L.QBX
= 70°, find L'.CQX.
(a). _
_
(b).
· --
_
_
- - --
(c). L.CQX =_
6
_
- fll
cm [2)
__ -·-" [?.)
Y11/ru;1 S,y;andary &hool
8.
_ _ _ Ser ] F '!''""' J::nd- l'r11r f ram 1011 Ma1h,·ma11c< P!__
(a) A model of a new estate has been buih \\·ith the scale of I : 200 000.
(1) The actual length ofa nver in the new c:<talC is 3.5 km . Find the length o f the
ri ver on the model.
(11) The recreational pa rk is represented by an ~rea of I0 cm 2 on the model. Fi nd its
actual area in kni2.
(b) Records have shown that I 0 wo rkers need 15 days to build 25 pavilions in the
park. However, d ue to sho n age o f workers, the company has only 6 workers
lo build the pavilions. Assuming that all workers work <it the same rate. how many
more days' are needed to complete the job'!
(a)(i).
cm [2]
(a)(i i). _ __ _ _ _ _ krn 2 [2]
- - -
- - - - ·
--- - ----- - - --7
(b). -
[3)
Yuhuu Secondary School
•l
Sec] £<pres.< F.nd-l'car /:.mm 101 I
Mathcmu_!~o
i'l
It is given that r. = l c x 1saninteger and 2:::; x < l O},
A
·~
{x : 2x - 1 < 13},
R = (x: xis a prim e number}
(a) List the elements in ~<:l A and B.
(b) Draw a Veru1 diab>Tam to show the relationship of A and R marking the clements
in each region clearly.
(c) List the clements of (11 f'l lJ)'.
(d) If one clement is chosen at random from E:, find the probability that it i5 a
member of (A A /J)'.
Answer (h)
=_
1J =
rl J
(a).A
(b). shown
(c). (Ar. B)' =
__
[l]
·-·- [2J
. (I)
(d).
· --
8
I lJ
Yul1ua Seconda,-, 1Sl'hoo/
I 0.
There ure 12 girls und
Se<· 2 Hxprrss End Year F.ra111 201 I
1
~1a1hf.'uu11i's P2
boys in a group. The probability of selecting a boy from the
group i~ 3
7
(a) Find the value of r.
(b) How many more boys ure needed to join the group so tlrnt the probability of
st:lecting a boy from the group will be~ ?
5
...
(u). x =
(b). -
9
_
[I)
_ 121
_ __ __ Set.
_· _·_
2_fu
~
p•_
>'SS
l'ul1ua Secondarv S<i1ool
11 .
£mi-l't•at Exam 201I ilf.,_1h .-l'IJ:i<.
1•:
The table shows the record of scores by 20 students in a Mathematics test.
54
81
16
51
64
71
70
58
62
68
32
60
43
35
59
64
36
60
62
43
The full score is I00 marks .
(a) Mrs Lee tabulated some scores in the ordered stem-and -leaf diagram helow.
Complete the stem-and -leaf diagram for 1hc remaining 6 students.
Stem
Leaf
3
2 5
4
33
5
6
0022448
7
0 I
8
key: 312 repscscnts 12 marks
(b) Write down the difference between the highest and the lowest score for the test.
{c) Find the median score for the test.
(d) Calculate the mean score of the 20 students .
(e) The passing mark for the test is 50. The scores of another 5 students were
~dded
to the record and the number of students who passed drop to 64 per cent.
How many new students p;1ssed the test?
(a). shown
[2]
(b) .
(!]
----
(c). _ _ __ _____
(d).
(e). _ _
10
__ (i]
(i]
12.
Al a patt y, there ace 9 more women thim men . The numb..:r of men is 20% o f the tot:il
number of people at the party.
lly letting w be the number of women and m he the number of men, lt>nn two
equations to find nut ho w many women and men there are at the party .
....
men women -
11
_ _1 41
13.
/\quadrant is removed frorn a cin:k of radius 20 cm as shown in the diag,ram.
;~·
I :'\
I
\
I
\
I
.: - ----\
'
/_..f-C01
_ _,. ,).
f....... _____
..
·•· 0
(a) Find the arc length of the
~eclor
that remains in terms of 7r.
(h) Ry joining OX to OY. a hollow cone is fom1ed.
(i) Show that the radius, r of the base of the cone is 15 cm.
(ii) Find the curved surface area of the cone in tenns of Jr,
(iii) Find the volume oflhe cone correct to 3 significant figures.
I
,_
t:
f
( Volume o f a cone = 3 ;r r· h ; Curved suriace area o a cone =
1T r I]
.
cm [I]
(a).
(bi).
(bii). -
(biii). .
12
shown
- -
__ _ _ __ [2J
-
-
-
crn
7
3
I I]
cm f21
Yuhua Ser211!_1nrv School ___ _
14
_ _
Sec 2 Hxpre.<.t Emf· Year ExCl_l!I 2011 Mathematics P l _
_
The hollow metal lank s hown in the diagram is used to s tore liquid fuel. The lank is
constructed from a cylinder of radius 5 m and length 8 m wnh two hemispheres of radius
5 m auachcd at each enJ. The tl11ckncss of the metal may be neglected . Take
71
as
3.142.
Sm
\
8m
(a) Calculate the surface area of the outside oft he foci tank .
(h) The outside of the tank is to be painted. The paint is sold in tins, each of which
contams S litres. One litre of paint covers 7 m 2. Find l he number of tins that should
be bought.
(Surface area of sphere -
4.nr 2 )
(a).
(b). _
- -_ - _-_ _
13
m 2 [2]
__
- 121
_J2_1hiu1 Secondary School _
15.
Sec 2 J:,):press J:,nd-Ye1.1r Exa1n 2011 Ma1he111at!S3..fJ
A stone is thrown upwards from the top of a ve11ical di ff. Its height. y metres, above
the ground level during its motion is represented by the equation y
= 6x(7
2x) + 28,
where x seconds is the time after the stone has been thrown.
x
0
I
1.5
2
2.5
3
4
4.5
y
28
58
64
64
58
46
h
- 26
(a}
Whal is the height of the vertical cliff/
(h)
Calculate the value of h .
(c)
Using a scale of2 cm to represent I second on the x-axis and 2 cm to represent
IO metres on the y·ax is, draw the graph of y "' 6x(7 - 2x) + 28 fo(
'-
0 S.
(d}
x-;: 4.5 on the graph paper provided.
[3]
Use your graph to lint!
(i)
the greatest height from the ground level reached by the stone,
(ii)
the time taken for the stone _lo hit the i:,>Tound.
(a}.
(b}. h =
(d}(i}.
( <l}(ii).
14
__ rn(ll
rn
(11
__ rn
fl I
____ sfll
l'ui:un S<co11d"ry Scl1o<>I
Ans\vcr all the questions (Total: 80 marks)
I.
(a) Estimate the value of 2 J&O_+· 3./2§_ .
VlO
(b) If Jt.732
a)
=k , find the value of Jo.O1732
::lflfo -t-3.]"3&""' ::i_Jei + 3J)b
-
318
3Ji0
~
in terms of k.
-
;,J(q)-t3(5) ~ lb ·i::,
;>.
b)
J 0 .o 11-3;) =
Ml
or
lb~ -
J '.=t3:2
100
r.L
::: j1 - ~:P Y.J100 _L
10
-
Ml
Al
\b · ".) OY
..k..
I0
(a).
(b}.
2.
AI
16~- ·- [2]
____ IJi. [! ..]
Expand and sim plify the following expressions.
(a) (3k - 5)(2k + 7)
(b) (a +4b)(a - b) - (a+b)'
a) (3k-5) (~Kt:+)" 6k.2+'.).lk: - 10k-35 -MJ
:: bk 1 +11
k -
3S -
I=)
b) ca+tib) (a-b} - (atb)~
• a,._ ab-t 4ab -4b' - a 2 - .)ab -b). -=
2
- 5b +ab -
1
nJ I
Ft\
(a).
(b).
2
6k"t II ~ - 35 121
-
5b~j 4b --
121
SeC' 1 bprrrJ Fnrl· >'t:flr &am 1010
3.
.~1ath~matu:s
Pl
The diagram bdow shows a circular target with diameter AB of20 cm.
What is the probability that a dart lands on the
(a) red sector,
(b) green sector,
(c) the smaller circle, with radius 2 cm drawn
at the centre of the circular target?
a) p( fe.cl g~e;tvf) -: .iQ_ ' - ' 300
6
(h~~ ~~c.!t>r'
b)
=
BI
igp·-60'.,, 120· - r i l l
P(Gveevi g~,Li)
V\U
• 1~0
_ -L
3&0 ~ 3
c) {)( SlYlll\~ ovcte) ~
"Tf (:i)
2-S
""I
•r
2
1T (10)).. -
: -L
-
,
IYI l
fH
-
fl I
-!....
(b). (c).
4.
6
(a).
[2]
3_
_,
_:z.s
(2)
(a) Factorise 3x' - 1x + 2.
(b) The area of a rectangle is
(Jx' x+2
- 7x + 2)cm
2
.
If the breadth of the rectangle is
~x - )cm, by using your answer to (a), find its length in terms of x in cm.
( x· +2x
1
~)
3X '- t-x
+'J-
~ ( ?>X-lJ (x -d) -
I'll
b) Le.viqt\11 = ( 3X:. }?;' +2 )
X+2..
~
C3X -•) (x
-
-2J
- 'X'\ "2
-=-
'X'(')( - ;i) -
. : (3x -1 \
;('+'.lX )
x
'X (*'+-2-J
..- ~)!=1
-
- Mf
Al
(a). { ? X-
(b) .
3
l) ( X - 2.)
A ( X • 2.)
(2)
CM (2)
Yuhua Secondary ~~f!hool
5
Sec l t-:.rprpss End-Year £.ran!...?.2_!0 ..i..\1cuht!.1•:!!.!.!.1..S _['i
In the digram shown below, the pyr;imid has a slant height of I cm, and a square ba,_,,
with length of b cm. The length from the centre of the base to the midpoint of a ~i<J~ ,)I
the base is given a5 r cm.
bcm
(a) show that
9
,._
1
=(1 -
V
b'
r' , where Vis the volume of the pyramid.
(h) (i) Mal«: I chc subject of the e.quatiovlgiven in (a).
(ii) Hence, find the slanted height of the pyran1id when V = 48cm\ r
b = 6 cm.
(Volume of pyramid-=
a)
heigvito
~xbasearea
x height 1
3 .
j~p_,-,_-r_1_
3 Xb
V=
=3cm and
2
Xje.~yl.
-
- Ml
~v:_ ~ fi~r~
qv-i. ~
J..:'- '(.,,
(snowv!) -
A
I
b""
b ·,:'\)
(),, 2
..:L!..-
b4
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Mi
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-
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(a). shown
fil
(b)(i).
(b)(ii). ..
·· - - [2J
i "' ~~_!l'~ _ _ [2J
-2.. ____cm [2]
- - -- - - - - ·--- -------- --- ------- --4
- - -··
· - - _ _..!..t:;ec_?_Expr<!ss End-Yetu Exam 201() ,~fa1hematics:.~!.
rulu1a Secondary School
6.
Beny and Rita started togclher on a 7 km walk a1 the same c~nstant speed ofx km/h.
After I km, Beny increased her speed by I krn/h and walked the remaining 6 km at the
new spc.ed. Rim walked al a constant speed of x km/h all 1he way.
(a) Write down the lirnc, in terms ofx, Belly took to complete the 7 km walk.
(h) Given chat Belly linishcd the journey 12 minutes earlier than Rica, form an equalion
in x and show that it reduces to < + x- 30 = 0.
2
(c) Solve the equation x~ + x- 30 = 0 .
(d) !'ind, in hours and minutes, the time taken by Rita to complete the whole joumcy.
(e) Calculate Belly's average speed, in km/h, for the whole journey
fl) -l;w')e io\(Q,111
_L
'X
L>n
~ ~t~
+ ...L + ~ ::
x+1
.....L.. _.
:
60
;?(
.:J.
~
6
~+I
--
BI
-
M I
6C)('r+i') + 360.:x +D-X(X+I) -;- 4-:2o(.x+1)
bO'X' +Go+ ?>bOx + l(;)x.i -1-id-X' ,, 4-dO + 4-J-O -
x
l~'X
2
+-\~-X - 360 =-0
'X 2-l- ')(
'X
?I:
;y"l
/\
d)
-::'3
Mi
0
- 30 ;:,.
-5JJ
'X 2-+
(, b'X Wll
-30 fV
/\
1iwie tare.vi~
~ <~O
>9
I
«)
"'°-0
ov x-;-6 CN 11) -11 I
( X-h)(-X+(;)
~"5
€;:-WW:VI
Q,J Tiwe -\ateV1 tlj ~ '.:
( SY1<)vVv)) -
o
1Vi
~ 4-'MlVl _ 151
B+~
"/~h -Ml
f\WNa~e ~peed: i-kWl -:-- \i' \tJ '°&;-kw /Vi - A/
(a). _
~1:Tl
~ -:_fu
_._..._--'-~-l qX(x+r)
(b). shown
(3]
(c). ·"'., __5'--- --
(2)
(d). _ _l_
(e).
h ...~_ min fl]
51i;-
~ h [2]
$.R3(38f- ;
I
-
) uhuu Sf!.conda2· Sc'!>o_
I_
7.
_
In 1hc diagram below, MXP and MBC are similar. The side AB of flARC i~ 1livtdcJ
such tbal 3AX = 2AD. P and Q arc points on CA and CB such thal XI' and XQ arc
parallel to BC and AC respectively.
(a) Find the ra 1io of AX .
AB
(b)Given that XP r 16 cm and XQ
~
15 cm, find the length of AC
(c)Ciiven thal LXA.P .,, 65°. L.QBX - 70°, find LCQX .
--
-!l
-BJ
3
AP _
-- - u
AC
3
nP
~
l'Yll
AP+IS :: 3
3~p
(ec~)
::. df\-9+30
'1-P ~ 9JJ ~
AC• 30 + 15 '
c)
45 CIM · - - B-1
LAC~= l~~bS 0--=)-Q ' :: 4-5 ---fVJ/
%
D
£'.CG!?(
0
I go•_ Lf-5°"135b -
'71
(a). _ _::____
(b).
(c).
6
_ _ [IJ
~ 45
LCQX• _ /35
cm [21
"[21
-1:!!!!.!!.P. Sl'Condary School
8
_ Sec 2
Cxpr~s~li!!!.!.!'~ur
£\am 20/{J
~atlrematic_s
1'1
(a) A model of :i new estate has been built with 1he scale of I . 200 000.
(i) The actual le ngth of a river in the new estate is 3.5 km . Find the length of the
river on the moclel.
(ii ) The recrealional park is represented by an area of I 0 cm 2 on the model. Find i1s
actual area in km 2 .
{b) Records have shoM1 that 10 workers need 15 days 10 build 25 pavilions in the
park. However, due to shortage of workers, the company has only 6 workers
10
build the pavilions. Assuming that all workers work at the same rate, how many
more days arc needed to complete the job?
a)
I ' ~CJQ 000
lc.v.,: Q~ . -
Ml
~9tvl ~ rivv/
t>)
I 0"1'\ " : 4 l<VV\ 2
CM
Made.I ..3
-
. s . '.:- 9 ~ I· 15 C-""1 . -
01
YV) I
~q.e1 avQa = 4 >c 10 • 40 \ctM ' ) - ~I
c)
~Q, 11\J : /\P ·CI\" ~ , 0 ~ l'b . of-~
WD-.. IC
I~: ClO) (I~)
\IVD.: 150 -
w :. b
WVl<?-\1
f\'l
-
In ~
l~O
I
'
D~ 150 ..;..6
~5
o:
lO
=do -
Ml
move ctacjS' -
/:)I
(ai).
I· }5
_ cm 12)
(aii) . ~
-- _ _
(b).
7
I0
2
km (2)
(31
fuln!._~t;_o1!!!2!;:):S<~ -
9.
-
Se£_ 2bpress~ur£-.:am10~1~~~..JI• ~·
- -
P ...
c ={x : xisanintcger and2$x S 10},
It is given that
A - { x : 2x • 1 $ 13},
n =Ix · xis a prime number}
(a) Lbl the clements in scl A and B.
(b) Draw a Venn diagram to show the relationship of A and 8
markin~ the
elements
in each region clearly.
(c) List the elements of (Ar. B)' .
(d) If one element is chosen at random from 6, find the probability that ii is a
member of (Ar. fl)'.
a)
l'X -\
=\3
?(~::}
Ii\ ~
t 2,3,4,51b1,..J
--B l
SI
Answer (b)
!:> 1- CovvQct re1~V11p
Oi- ~ 40Wvl
I\
Bl - ~~
\tit ~V\. .Q.OCN\. s ('.+- .
c) (f'.\nB)
1
~f 4,(;, 1 8 1 C\,IOj -
~ _ l vww~ (ovaNa11) iV no h't1tes
~v~ giv™ fov (a) ~ cc.,)-
(a). A •
B•
\31
J ?•31~1
~
2 13 1
\:;>I
0 I :r
5 1 7-]
(b). shown
(c). (Ar.LI)' "'
5
(d).
_ [l]
[2]
~4,b, 2 1Cl1 lQJ_[I]
_ er·- --- - - - - --- - 8
J_[l]
- [11
- -·· - -
Yuhu~:~<:ondt:.ry_School
I 0.
_ ____
S~!..!:~prK~S l;;nd-Ye:!.C_F.xam 1010 A1athtmu!.Jcs I' ] .
There arc 12 9ii\S and x boys in a group. The probability of selecting a boy from the
group is ~ .
7
(a) Find the value ofx.
(b) How many more boys arc needed to join the group so that the probability of
selecting a boy from the group will be ~?
5
'X
-=-
2-
?<+IL-.
::j-'X ~
:'.J-
3X .\-36
4-'X , 3b
81
b°)
f,lt ·\11e. YIO · of
~+q
9.l+Y
~ ~
5
0011~
-
vJVJO jOivKa:i l:?Q, !:::j
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5'1-t-'+S-;; 84 +~
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~l
-;r __
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(a). _
(b).
9
39
(}.L...__ __
:t
11 I
[2]
..J.u~.\'econd<u}'Schoal
11.
·-~ _
_ S~<:- 2 £xp1·css £nd-Ytar /-J.am ?!!.!.!!...l:f!!!!_emat1cs P2
The table shows the record of scores by 20 students in a Mathcmalics test
54
81
36
51
64
71
70
58
62
68
32
60
43
35
59
64
36
60
62
43
The full score is I00 marks.
(a) Mrs Lee tabulated some scores in the ordered stem-and -leaf diagram below.
Complete the stcnHmd-leaf diagram for the remaining 6 students.
Stem
Leaf
3
2 5 b 6
4
33
5
14
6
0 0 2 2 4 4 8
7
01
Bi> - ~Ht <.?W'I~ ta\?W~
Bq
....
UiWeW_y
cl 'Mc.v'? decl~cl fbv eacfa]
VJV(Mg\y '{-a.btl.lofed
8
3.C.OfQ
J
key: 3j2 represents 32 marks
(b) Write down the difference between the highest and the lowest score for the test.
(c) Find the rn~ian score for the test.
·
(d) Calculate the mean score of the 20 students.
(c) The passing mark for the test is 50. The scores of another 5 students were added
to the record and the number of students who passed drop to 64 per cent.
How many new students passed the test"
b)
3 l - 3 ~ : 4.q -
c) IVledi<Wt "
sct;"9
BI
~ sq .s
-e 1
cl) M&WI ~ IJQg_ ::: 'b5 - t.tl?> or S? -S C.3i4J ?.{)
l) l\b- ~ ~1-{Clet.tj.S
"' 0-b'+ x. 9-S
-:: [h
VJVIO
8f
~a&geG
(2]
(a). shown
Ml
lb- IL!-.,_~ -fH
10
(b).
4-'q
(c).
5'.:J ~
(d).
5Y.i .t.1-.9
(c).
;).-,
(I J
_ ____ [I)
or
55 ·5
[I)
[2)
rt~hu(I S<~condory School
! 2.
Sec 2 Upn.~s:; £ nJ-Ytzar Exattr 10 I0 Ma1ht?nu1ti_cs P?
At ;1 party. there are 9 more women than men . The mnnber of men is 20% of the wrnl
number of people al the party.
By
Jetting w be the number of women and m be the number of men, fom1 two
equations to find 0111 how many women and men there are al the party.
~b
0
ivctn@:
m '°o.'.)(m+wi+q)
0 -~'""' 'r I · 8
YY) , 3 - A-1
YY) "'
8
n1en =
3
12 _[4]
women = -_ _
__ -
II
S~c l
YuhuaSec-0ndn2·Sc/m.ol
IJ
/'~
Exprt•ss I-Ad-Year £ -<om 1010 /.lutl_:_e'!!::.:'1 ,
A quadrant is removed from a circle of rndius 20 cm
tis
shown in the diagram
(a) Find the arc length of the sector 1ha1 remains in terms of tr .
(b) By joining OX 10 OY, a hollow cone is fonned.
(i) Show lh<ll the radius, r oflhe
ba.~e
of the cone is 15 cm,
(ii) Find the curved surfocc area of the cone in terms of
ff •
(iii) Find the volume of the cone correct to 3 significant figures.
[Volume of a cone
.a)
t>i)
~
/WC leMqtv\ ~
1tr r 1 h ; Curved surface area of a cone •
~
x
~Hr r "" 3C! TT
r -:
Cu l'Ve.d
tr
~lT (J.0)-:. 301T ON\
13 I
-
IYl )
J
1t-I
I 5 C>V\ ( &hOVIJV)
S:Ur'fac_e, e.V'<l-3.. -:. 1\ f" R_,
:. 1T(\ 5)C.2D) :: 2001T
;o)
l-1el~vrt- ~ Jso:. I? i.
Vot~'Me
o(- eovi~ = ~
-
r I]
3
::
Ji 1-&
°" - B l
:t
_ IV! l
11 x 15 .,_x
Jr:1s
3 1 I =!- ~ 3 12.Q
12
°""
3.
Ct.i-&f) -
(a).
wrr_
(bi).
shown
T?r I
_
_· cm[IJ
[2]
2
(bii).
'3t:J01[
cm [lj
(biii). -
3 1d-O
cni
1 [21
Sel' 1
~)ecnndtlfYSchool
14.
The hollow metal tank
sho~'tl
~pres1 End~ Yeor
FJam 2010 1tf41rhemo11cs r ;
in the diagram is used to store liquic! fuel. The tank 1s
constructed from a cylirlder of rndius 5 m and length 8 m with two hemispheres of radius
S 111 attached at each e nd. ·ine thickn~-ss of the metal may be neglected. Tukc
1l
as
3. 142
1 - - -- - --
Sm
(a) Calcuhitc the s urface area of the outside of the fuel tank.
(b) TI1e outside of the tank is to be painted. The paint is sold in tins, each of which
contains 5 litres. O ne litre of paint covers 7 rn 1. Find th<.: nurn lier of tins that sho uld
be bought.
(Surface area of sphere ~ 4711' 1)
g11Y'fa«, 8YQ-~ "° 4X11' X 0 ,_ -t
:J X 1f )( 5 J<.. 8; -
565 5G rri). -
b)
VO\ ·
'f poi\l\T ~ec{ .,_
fY)J
Al
?65 .,; =t- :: &) ·1-B f, -
MJ
N~ ot-i'~ ~d~ = ~ · ~.L .:, 5 R
"' 16- '
t3
(a ).
5&5 . ?b -
(b).
l:t
m 2 r2J
[2}
Yuhua Secq,1dary St:f!:!El_. -
15.
-
-
--
-
Sec 1 &P!'_tsS /:,nd· Yl'ar Exam Z~h'f!!:!.~':..::_ I'
_
A stone is thrown upwards from the top of a venical cliff. Its height, y metres, :jbttv
the ground level during its motion is represented by the equation y
where~
seconds is the time after the stone has been thrown.
x
0
I
1.5
2
2.5
)
4
4.5
y
28
58
64
64
5&
46
It
-2Q
(a)
6x(7 - 2x) + 28,
What is
th~
[I]
hei ght of the vertical cliff?
(I]
(b) Calculate the value of h.
(c)
Using a scale of 2 cm to represent I second on the x-ax is and 2 cm to represent
10 metres on the y-ax is, draw the r,raph of y ,. 6x(7 - 2x) + 28 for
0 s x < 4.5 on the graph paper provided.
Pl
(d) Use your graph to linJ
(i)
the greatest height from the p,round level reached by the stone,
r11
(ii)
the time taken for the stone to hit the ground.
(I]
(a) .__2._<0
_
4
~5 ±
_ _ __
(b). Ii -
(d)(i).
(d)(ii).
14
m (11
m (I]
I
4 · I .:!-- I
m [I I
_ _s II I
~'l."}
..
•
F..tmt'f.t £nd- Yro,. FS4m 1010 AlatJ1c•mu1ic.s P1
A
.•· = 6\· (' -1.:) - 28
60
so
40
....
20
\I
I
:o
I
l
\\
Ii
s
J
\
4\
.:I
I
\
1
\
·20
~
30
a) +tei<j'll-\ of chW = 2e· m b)
\ti, 6(4) L:i - J{'f-)] 1-2~
c:)
AI At Jl\-l -
31
?
4WI -
c.owec.t sczile usaj
BJ
i?iY ~ ~d ~ aJ<IS
All {)OiVl-\-S p!ot}-ed (OtVetj- ·
SrYIOot-\1\ ~ (OaWed wvve, ctva.wvi .
di) G1~tesi- l'l~1qV1t = GsM - - B 1
(~cte9t 6 ~r:. l)
;;) T;\l\1e. fo\(Q.>1 it> YQtiCM i-V\~ g<Duvd -o 4 . 1.S ( l\f( Q\;i
.& . ! ~
1)
BI
CHUNG CHENG HIGH SCHOOL tYISHUNJ
\_,
End of Year Examination (2011)
Secondary Tw o Express
Candidate
·Register No
Namo
Class
Mathematics
For examiner's use
Date:
4/10/ 2011
Duration: 2 hour
/100
READ THESE INSTRUCTIONS FIRST
Write your name, register number and class on the space provided above.
Write in dark blue or black pen.
You may use a pencil for any diagrams or graphs.
Do not use staplers, paper clips, highlighters, glue or correction fluid .
Answer all questions.
If working is needed for any question, it must be shown with the answer.
Omission of essential working and units of measurement will result in loss
of marks.
Calculators should be used where appropriate.
Simplify your answers to their simplest form. If the answer is not exact,
give the answer correct to three significant figures or in fraction where
applicable. Give answers in degrees correct to one decimal place.
For 1f, use either your calculator value or 3.142, unless the question
requires the answer in terms of ;r .
The number of marks is given in brackets f] at the end of each question or
part question.
The total of the marks for this paper is 100.
Setter : Poh Eng Hua, Terence
This paper consisb of 13 printed pages. INCLUDING lhe cover page.
CCHY End of Year Examination (2011)"
Secondary 2 Mathematics
pglofl3
I.
(a) Factorise fully 2x 2y , Bx
-1
12y + 3xy 2 •
.
.
. 3x - 1
x+2
{b) So Ive the lollowmg equa11011 - - - 1 = - -· .
4
5
Answ er: (a) ............................................... PJ
(b) .............................................. [3]
2.
An actual length of 5km road is represented by a length o f 20cm on a map.
(a) Find the scale of map in the fonn of 1 : n.
2
(b) Given a circular garden o f area 2.5km is represented on the map. find the radius of the
garden on the m ap.
Answ er: (a) ............................................... [21
(b) ............................................... [3]
CCHY End of Year Exam1nahon (2011)
Secondary 2 Malhemahcs
pg2of l3
3.
A fair coin is first tossed followed by a throw or a fair 6-sided dice aml the outcomes arc noted
(a) In the answer space below, hst the sample space.
(b) Hence, find the probability of getting
(i)
a head followed hy a 'I'.
(ii)
a head followed by an even number.
(iii)
a factor of 4 from the dice.
[2]
Answec: 3(a)
Answer: (b)(i) ........................................... (I J
(b)(ii) ........................................ [I]
(b)(iii) ........................................ (IJ
4.
TI1e data below represent the distance in km travelled hy 20 people from home
lo
their
workplace.
11
5
9
35
43
16
23
27
21
12
3
11
16
5
31
35
11
23
15
2'1
(a) In the answer space hdow, complete the stem-and-leaf diagram .
(h) Find the median distance tTavelled.
(c) Find the probability that a person selected from the survey will take l.:ss than 30km to
travel
to
their workplace .
[.1]
1\nswcr: <>(a)
Stern
L.:af
Answer: (b) ............................................... I I I
( c) ..................................................(21
CCHY End of Year Examination (2011)
Secondary 2 Mathematics
pg3oflJ
5.
(a) S men lake 12 days to build a 6km long road. Assuming every man works at the same rate
and the width of the roads are the same. !low many men are required to build a 21 km long
road in 18 days?
(b) Given that xis inversely proportional to the positive value of J2y- 1 and x
=5 when
y = 5. Find the equation C(lnnccting x and y.
Answer: (a) ............................................... (31
(b} ........ :....................___ ___ ____ ________ (2 )
CCHY End of Year Examination (2011)
Secondary 2 Mathematics
pg4ol 13
6.
(a) Express as a single fraction in ils simplest fonn
. o f the 1ronnu Ia 111:
. -2 =
(b) Makex the subject
y
4
3
2x+ 1 x - 5
f -h
-
x
- .
Answer: (a) ............................................... (3]
(b) ................................................ 13]
CCHY End of Year Examination (2011)
Secondary 2 Mathematics
pg5cf13
7.
DiagrJm below shows a ri ght angled triangle ABC wi t h AC ~ 20cm, BC= 12cm. DF. = 4cm.
(a) Find the length of AR.
(b) Given that tl.AllC is simi lar to 6/IDE, find length BD.
A
R
Answer: (a) ..............................................cm [2)
(b).... . ..................................
lt
(~)Shade the region representing
(Ar. B)"
cm 13]
[3 J
for the following three diai;rJms.
B
CCHY End of Year Examination (2011)
Secondary 2 Malllematics
pg6ofl3
(b)
s- {Redbull , McLaren, l·errnri. Mercedes, Renault, Lonis, Sauber, N issan}
A = { Redbull, Mercedes, Fe1Tari}
8 - (Renault, Ferrari, Redbull}
(i)
(ii)
(iii)
Find n(I;).
Fi nd A f"'I B.
List all possible subsets for II.
r
Answer: (b)(i) ............................................... I I
{b)(ii) .............................................. [2) .
(b)(iii)
...... .... ................................................... . .............. . ............................... ... .. . .13]
9.
The children in a childcare cencrc were surveyed to lind
consumed
in
0111
how many swecls they
a week. 'flie results are shown in lhc table helow.
-~- ~1r9: ;
4___._l_ 4<
_ _ x_$_6
_ ..._
Number of sweeis- -(x-)--'- 0< <
Number of children {()
" _ ~
.
Io
(a) Write down the largest value of k given lhat the modal class is
-6-<~~-·~·5<_ 8-_-L)-8< 6$- -IO_
<•<x
A
$ 8.
(b) Write down the smallest v;ilue of k given that the median class is 2< x < 4.
(c) Find the value of k given that lhc mean numl>er or swcc1 consumed pct wcC'k is 5.
Answer: (a) .............................. ................ (l]
(b) ......................... ...................... [I)
(c) . ..... ...................................... [21
CCHY End of Year ExaminatiOn (2011)
Secondary 2 Mathemalies
pg7ofl3
I
I0.
Given that D.ABC "' !:!.EDC, L.B.!lC = 45", L.ACB = 65° and AE = 12 cm. Find
(a) L.CDE.
(b) length of CE.
Answer: (a) ............................................... [I]
(b) ................................................ [I]
11.
Solve the following simultaneous equations using suhstitution method.
4.x+3y=30,
:!:'. +4 = 2x
3
Answer: x= ....... :........... ,y=................... [4]
CCHY End of Year Examination (2011)
Secondary 2 Mathematics
pg8oflJ
12.
(a) Expand (2x +3y X2y
(b} Si mplify
x 2 - 16
2
- x +x+2
(c) G iven that x2 +
(3x -
y2
3x+ 1) .
+4
+ x_·as a single fraction in its simplest form.
x-2
= I I and xy ~ 5, calculate the value o f
3y) 2.
Answer: (a) ...............................................
l2J
(b) ............................................... (3]
(c) ............................................... [3)
CCHY End of Year Exa mination (2011 )
Se<:ondary 2 Mathe matics
pg9ofl3
13.
·111e figure shows a c ircle of radius I Scm. The poinls A, 8 and Care on lhe circumference of
the circle and the centre o f the circle is at point 0. BC = 8cm and AC •
cm.
./836
(a) Show that 6CAH is a right-angled triangle.
(b) Find area of 6 CAB .
(c) The figure i~ used as n dnit board and a dart is thwwn onto it. Find the probability that the
dart will land out~ i<le 6CALI .
c
A
Answer: (a) ............................................... (21
(b) ............................................... 121
(c) ............................................... [2)
CCHY End of Year Examination (2011)
Secondary 2 Mathematics
pg 10of 13
14.
The diagram below shows lhc garden of a mansion. Given lhal AB = (15
BC = (3x
1 3)
x j '"•
m, CV = ( 4x 9) m , AD= (2x + 5) m and L HAD = 90°.
lxt5
D
c
4x-9
(a) Given Ihat the area of the garden ABCD is 157 _! rn 2, fonn an equation in x and show
.
2
2
thal it reduces to 6x + 27x - 285 = 0.
(b)S()lvethccquation 6x 2 ~ 27x-285= 0 and fincllhc valueofx.
(c) The owner oflhe garden wanled lo put fencing around lhc garden. Given 1hal thl· cc
;t
of
fencing is $5.50 per metre, find the cosl of fencing t.hc garden.
f2f
Answer: (b) x = ........................................ (3]
(c) $ ...........................................£2)
CCHY End of Year Examination (2011)
Secondary 2 Mathemalics
pg l'1 of IJ
15.
Solid /I. is fonned by altaching a hemisphere to a cone. The hemisphere has radius 8
cm and the height of the solid is 30 cm,
(a) Show that the volume of solid A is 810 .?.1t'crn 3 .
3
(b) Find the total surface area ofsoli<l A.
(c) Solid A was melted and was cast into smaller s im ilar solids (solid B) such that the
radius of the hemisphere is 2cm.
(i)
Find the height of solid B.
(ii)
Find the maximum number of solid B that can be formed.
\
'I
/
I '-
\
Jc.;1~
"I ""~ ·. ·
~:----:: ..
solid A
solid B
Answer: 15(a)
(2)
Answer: (b) .............................................cm 2 [ 4)
(c}(i) ........................................cm [1)
(c)(ii) ........ ..... ..... ....... . ........ ... [2]
CCHY End of Year Examination (2011)
Secondary 2 Mathematics
pg l2ofl3
16.
Answer th e whol e of this questitm on a sheet of gra ph paper.
T he table below gives some values of xand the correspond ing values of y for
y=-x 2 +3x+2
I:
132
I-~~~~1~~~~
--l1_ --_12
-'~--1-102 -1~ri=4===~=-~q2i-_-_-~~~. .;.__ _
LI l
(a)
Find the value of q.
(b)
Using a scale of2 cm to I unit on !hex-axis, and a scale of I cm to I unit on the
y-axis, draw the aites for - 2 ::5 x ::5 5 and - 10 Sy s; 6 . I Jenee, plot the graph
ofy = - x 2 13x+2
[5 )
(c)
Draw the line ofsynunctry and write down its cquntion o n the graph.
f2)
(d)
rind the value of y when x = -1.5.
fl I
(e)
Using lhe graph, write down the value/Values of x when v - - 1.
[2)
(f)
Mark the maximum 1>oint of the curve and wrile down its coordinates.
[2]
CCHY End of Year Examinahon (2011)
Second~ry
2 Malhematics
pgl3ofJJ
- -·
\
~
......._ -·r
'
\"fl~~;i/
\
', .. I
,
'\/
CHUNG CHENG HIGH SCHOOL (VISHU>I)
.il <)' )J .f-t
End of Year Examination (2011)
Secondary Two Express
Candidate
Class
Register No
Name
Mathematics
For examiner's use
Date:
4/1 Of 2011
Duration: 2 hour
/100
READ THESE INSTRUCTIONS FIRST
Write your name, register number and class on the space provided above.
Write in dark blue or black pen.
You may use a pencil for any diagrams or graphs.
Do not use staplers, paper clips, highlighters, glue or correction fluid.
Answer all questions.
lf working is needed for any question, it must be shown with the answer.
Omission of essential working and units of measurement will result in loss
of marks.
Calculators should be used where appropriate.
Simplify your answers to their simplest form. If the answer is not exact,
give the answer correct to three significant figures or in fraction where
applicable. Give answers in degrees correct to one decimal place.
For JT, use either your calculator value or 3.142, unless the question
requires the answer in terms of n .
The number of marks is given in brackets [ ] at the end of each question or
part question.
The total of the marks for this paper is 100.
Setter: Poh Eng Hua, Terence
This paper consists of 13 printed pages, INCLUDING the covor page.
CCHY End of Year Examination {2011)
Seconda<y 2 Mathematics
pglofl4
I.
(a) Factorise fully 2x 2 y+ 8~+12y + 3.xy 2 .
.
. 3x - 1
x+2
.
(b)s oI vc tI1c 1o 11 owmg equahon - - - 1 = · ~ .
4
(a)
(b)
5
2x(xy + 4) + 3y(4 + xy)
(Ml]
(2x+yXxy+4)
[Al]
3x--1- 4
=- -
x+2
5
[MI]
15x - 25 = 4x + 8
[Ml]
4
11x= 33
x=3
[AI]
Answer: (a) ............................................... L3 J
(b) ............................................... (31
2.
An actual length of Skm road is represented by a length of 20cm on a map.
(a) Find the scale of map in the fom1 of I : n.
(b) Given a circular garden of area 2.5km 2 is represented on the rnup, find the radius nfthe
garden on the map.
(a)
(b)
20c.m :5km
20:500000
(MIJ
1:25000
(Al l
1 : 0.25/an - length
1cm 2 : 0.0625krn 2 - Arca (Ml]
Arca on map =
2·5
0.0625
= 40crn2
[Ml]
nr 2 = 40
r = 3.57(3sf)
CCHY End of Year Examination (201 1)
[Al]
Sec.ond;iry 2 Mathematics
pg2ofl4
A fair coin is first tossed followed by a throw or a fair 6-sided dice and the o utcom vs 01.; noted.
(a) In the answer space below, list the sample space.
(b) He nce, find the p robabi lity of getting
( i)
a head followed b y a ' I '.
(ii)
u hC3d followed by an eve n number.
(iii)
a factor of 4 from the dice.
A nswer: 3(a) sample space =
3.
{( H ,1),( H .2 ).(H,3 ).(11.4 ).( H,5 ),( H.6 ).(T.1).(T.2 ),(T.3 ),(T.4 ) .(T,5 ),(T,6 )}
[21
(i)
-1
12
IB I)
1
4
-
(ii) -
4.
2
iii) 3
(Bl]
TI1e data · below represent the dista nce in km travelled b y 20 people from ho me '" their
workplace.
1I
5
9
35
43
16
23
27
21
12
3
II
16
5
31
35
II
23
15
24
(a) In the a nswer space below, comple te the stem-and-leaf diagram .
(b) Find the median dista nce travelled.
(c) Find the probability that a person selected fro m the survey w ill take less tha11 • ; n to
travel to their wo rkplace.
(31
Answer: 6(a)
Leaf
Stem
3 5 5 9
0
1112 5()6
2
3 3 4 7
3
5 5
4
* R2 for correct entries in stem-and leaf.
ll 1 for 4 correct entries and above.
*B 1 for including correct key.
(b) I 6km
(BI)
16
(c)P(adults travel < 30km) - -
20
[Bl J
CCHY End of Year Examination (2011}
3
0
4
5
r
rep resents 03 km
[Bl ]
Secondary 2 Mathemalics
P.:} 3
or 14
5.
(a) 5 men take 12 day~ to build 11 6km long road. Ass uming every man works at the same rate
and lhc width of the roads arc the same. I low many men arc required to bui Id a 2 1 km long
road in I & <lays'?
(b) Given that xis inversely proportional to the positive value of
/2.y -1
and x =5 when
y • 5. Find the equation connecting x and y.
12days
5mcn
a)
6km
21
6
21
6
x-
x-
17.Smcn
+
3
2
2
[Ml]
18days
12km -
[Ml)
[A 11
<:< 12 men
I
2 1km
3
x2
11 - men
3
b) x =
12days
k
"2y - 1
k
5= ,lg
k=15
[BI)
15
x =€ - -
IAI]
J2y - 1
CCHY End of Year Examination (2011)
Secondary 2 Mathematics
pg4 of 14
. I~
. .111 .ats snnplcst
.
Iionn - 4 - - 3- .
()E
a xpress as a smg
e 1rac11on
2x+1 x - 5
6.
(b) Make
xthe subject of the formula in: 3.y = ,/c-b.
x
4(x - 5)
a)
=
-
(2x + 1Xx-5)
-
3(2x+1)
(2x+ 1Xx- 5)
[Ml ]
4x -20-6x - 3
(2x+ 1Xx-5)
- 2x - 23
- (2x+ 1Xx -5)
2
[ i\2 j - I mark for.each error term
ic=- h.
b)-= 1 ~-
y
' x
c- h
4
x
4x-(c - b)y 2
(c - b)y 2
x::o: -
4
CCHY End of Year Examination (2011)
IM I]
[Ml]
IAIJ
Secondary 2 Mathematics
pg 5 oll4
7.
Oiagram helow shows a right angled triangle ABC with /lC = 20cm,
BC ~
I 2cm, DE = 4crn.
(a) Find the length of AB.
.
(b) Given that !!.ABC is similar to MDE , find length BD.
A
B
(a)
AB 2 =11C 2
-
BC 2
(Pythagoras theorem)
[Ml]
AB = 16cm
(b)
[Al]
AD DE
AB = BC
AD
-
16
=
[Mi]
1
3
1
AD = 5 - crn
3
[Ml]
1
BD = 16 - 5 3
8.
=
2
10 - m
3
(a) Shade the region representing
[Al]
(A r'I B)' for the following three ·diagrarns.
---~:-...:.
·
(3]
·-. .;
A
CCHY End of Year Examination (2011 ).
Secondary 2 Mathematics
pg6of 14
A
(b) i; -
B
I Redbull, McLaren, Ferrari, Merced es, Renault, Lotus, Sauber, Nissan)
A = {Redbull , Mercedes, Ferrari )
B- {Renault, Ferrari, Re<lbull}
(i)
(ii)
(iii)
Find n(I;).
F ind Ar. B.
l ist all possible subsets for A .
8 - [Blj
Answer: (b)(i) .............................................. ( I )
(b)(ii) ..... {Fe rrari , Rcdbull} - (B2 ].............................
. .... [2]
(b )(iii) ... ( }, { Redbull} , {M e rcedes}, {Ferrari) , (R cdbull, Me rced es} , (Red1',,;L • ··rrari },
{Mercedes. Ferrari }. {Redbull. Men .d
' 'rrari }
- Bl for { }, *82 for all oth er 7 s ubsets incl ud ed , Bl on ly for 4 corr ect subsels
9.
·n,e chi ldren in a childcare centre were surveyed to find out how many s;,\ tel~ they
co ns umed in a week. TI1c results a rc shown in the table below.
Number tlf sweets (x) .,
NWTiber of children (f)
0<.\ S2
k
'- - - (a) Write down the largest value of k given that the mo dal class is 6< x S 8.
(b) Write down the smallest va lue o f k given 1ha1 the median class is 2< x S 4.
(c) Find the value of k given that the mean number of sweet consume.I per week is 5.
a) K "' 24
[B I)
h) k = 23
[13 1)
tj
k + 3 x 19 + 15 x 10 + 7 x 25 + 9 x 6
- -5
60+k
CCHY End of Year Examination (201 1)
[Ml ]
Secondary 2 Mathematics
pg 7 of Jtl
k+336 =5
60+k
k =9
JO.
!All
Given that MBC e 6.EDC , L RAC = 45°, L.ACB "' 65° and A£ ... 12 cm. Find
(a) LCDE.
(b) length of CE.
D
70° - [Bl]
Answer: (a) .................... ........................... [I J
(b) ..............6cm
11.
[BI].................................. [ I J
Solve the following simultancou.~ equations using substitu tion method.
4 x+3y = 30 , -eqn (1)
l + 4 = 2x - t:qn(2)
3
cqn (2) x 3
y = 6x - 12 - eqn (3), sub cqn (3) into eqn (I)
[Ml I
4x+ 3(6x-12) = 30
4 x+ 18x- 36 = 30
IMI]
22x =66
x =3
[A I)
sub into eqn (3)
y=6
CCHY End of Year Examinalton (2011)
[Al)
Secondary 2 Mathematics
pg8of 14
.·
12.
(a) Expantl
(2x + 3yX2y - 3x+1).
(b) S.unpI 1f y
x2
- 16
-x 2
(c) Given that x 2
2
(3x - 3y)
~x+2
1
J
"
.,. -x -I -4 ns a s .mg Ic "1ract1.on .m .its s1.111pIest 1onn.
x- 2
~ 11 and :i.y - 5, calculate the value
or
.
a) (2x-13yX2.v - 3x ·t1}
- 4xy - 6x 2 + 3y + 2x - 9xy+ 6y 2
(Ml]
~ - 5xy - 6x 2 + 6y 2 + 2x + 3y
(Al]
· (x + 4 Xx - 4) . x + 4
b)
( 2 -xXx +1) ..,. x- 2
(Ml ]
(x -i 4Xx - 4) x - 2
=
{2 - xXx + 1) x
x+4
[Ml]
4- x
x +1
c)
(A I]
(3x - 3y) 2 = 9x 2 - 18..\y+ 9y 2
=9x 2 +9y 2 - 18..\y
= 9(11) - 90
=9
CC HY End of Year Examination (2011)
Secondary 2 Mathematics
pg 9of !4
13. ·nie figure shows a circle o f radius 1Scm. The points A, IJ and C arc on rhe circumference of
lhe circle and the centre of the circle is at point 0. BC = 8cm and AC - ,1s36 cm.
(a) Show that !:>CAB is a right-angled triangle.
(b) Find area of /\CAB .
(c) The figure is used as a dart hoard and a dart is thrown onto it. Find the probability that the
dart will land outside D.CAB.
BC 2 + AB 2 =836 +64
c
A
0
15
=900
I
AC 2
= BC 2 + AR 2 (Converse of Pythagoras theorem)
*BI for showing the th0on ·n1,
BI for writing out the stateinrnt,
converse of Pythagora~ theore•n
b) Area of /:\CAB
-
_!
X bClS(! X height
2
.
1
2
[MI]
-
~ - x8x v'836
~
116 cm 2 (3~1)
Tt15 2 -
c)
P( Dart
[All
1
x8x .J836
outside6CAB) = --~2~:n152
[AI)
- 0.836 (3sf)
CCHY End of Year Examination (2011)
[Ml ]
Secondary 2 Mathematics
pg \Oof 14
14.
The diai.'fam below shows the garden of a mansion. Given that AB = (15
!IC = (3x+ 3) m, CD = {4..1
x ) m,
9) m, AD = (2x+ 5) m and L BAD = 90°.
B
15-x
A
2x t 5
D
c
4x-9
(a) Givt:n that th e area of the garden A BCD is 157
2
that it reduces to 6x + 27 x - 285
(b) Solve the equation 6 x 2
1
1
2
m 2, fonn an equati on in x and show
=0 .
27 x - 285 = 0 and find thi: value of x..
(c) The owner o f the garden wanted to put fencing aro L1nd the garden . Gi ven that the cost o r
fencing is $5.50 per metre, find the cost o f fencing the garden.
Answer: I 4( a )
(a)
f2]
1
! x (2x+ 5)x(6+3x) =157
2
2
[MI)
! (6x 2 + 27x+ 30)= 157!
2
2
6x 2 + 27x + 30 = 31 5
6x 2 + 27 x - 285 =0
(b)
6x 2
t
(Proven)
[A 1)
27 x - 285 = 0
{2x+ 19X3x - 15)= 0
(M l]
x • -9.5 (reject), x = 5
(A2]
( c) Perimeter of garden =
=
2x + 5 + 3x + 6 + 3x + 3
* if did not reject, A I
[Ml]
8x+ 14
- 8(5) + 14 = 54m.
<;;ost of fencing= 54 x 5.50 = $ 297.
CCHY End of Year Examination (2011)
(A l]
Secondary 2 Mathematics
pg ll ofl4
15.
Solid A is fonned by attaching a hemisphere to a cone. The hemisphere has radius 8
cm and the height of the solid is 30 cm,
(a) Show that the volume of s<Jlid A is
810 ~it crn.l.
3
(b) Find the total surface area of solid A.
(c) Solid A was melted and was cast into smaller similar solids (solid R) such that the
radius of the hemisphere is 2cm.
Find the height of solid B.
(i)
Find the maximum number of solid B that can be formed.
(ii)
\
/'
;\
I )(r~
( ·' (' · ·::~
~---~h/
4.
solid A
-·· '
solid n
(2)
Answer: 15(a)
VolurneofsolidA
-
2
rr(8)3 +2rc(8)2 x22
3
3
1
1
3
3
[Ml]
=341 - rr + 469 - Jt
=
(b)/=J8
2
2
810 - Jt
3
[Al)
+22 2
l = 23. 4094cm
Total surface area=
=
(Ml)
rrrl + 2rcr 2
2
em
n(8X23.4094)+2n(8 2
(Ml)
)cm
2
[Ml]
= 990.46625cm 2
2
= 990. cm (3sl)
CCHY End of Year Examination (2011)
[Al)
Secondary 2 Mathematics
pg 12 or 14
(c)
2 h
8 30
h =7.5cm
(d) Volume of so lid B
[BI)
~
2
3
7t(2)3 +2x{2)2 x5.5
3
16
22
1t+ 1t
3
3
9 -
38
~ -n: cm 3
3
(Ml]
2
810 n:
3
Max number of solid B - - - =
38
7t
3
=64 solid 13
CCHY End of Year Examination (2011)
(A 1)
Secondary 2 Mathematics
pgl3of l4
16.
Answer t he w hole of th is q u es tion on a sheet of graph pa per .
The table below gives some values of x and the correspond ing values of y for
2
y= -x + 3x+2
I;
I ~8
2
1-!2 I~
I!
2
9
[131]
1;
I~2
I ~8
(a)
Find the value o f q.
(b)
Using a scale of2 cm to I unit on the x-axis, and a scale of I cm to l unit on the
~
2
y-ax is, draw the axes for - 2 ~ x
~
5 and - 10 ~ y
$
6 . Hence, plot the graph
2
ofy= - x +3x+2
BI for correct axes, scales
[SJ
B I for correct range.
B2 for correct plotted points • BI for 4 correct p lotted poi nts
BI for smoothness of curve
(c)
Draw the line of symmetry and write down ics equation on the graph.
x- 1.5
[2)
(B2J
Bl for drawing line ofsymmet1y wi1lt equation
(d)
Find the value ofy when x - -1.5.
y = -4.7
± 0.1
[U2J
8 I for drawing 11·orJ..i111( li11e
(e)
Using the graph, write down the valut:ivalues of.l when y- -1 .
x = -0.8 ± 0.05 '3.8± 0.05
(f)
[2]
( 8 2]
Mark the maximum point of the curve and wri lc down its coord inates.
( 1.5, 4.2)
12)
[D2J
BI for marking maximum poilll.
CCHY End of Year Examination (201 1)
Secondary 2 Malhematics
p!]
14 of 1-1