Embed Size (px)
Citation preview
CATHOLIC HIGH SCHOOL
PRELIMINARY EXAMINATION III 2008
Subject : Mathematics 4016/2 Paper : 2
Level : Secondary 4 Date : 10 September 2008
Marks : 100 Time : 11 00 – 13 30
This question paper consists of 9 printed pages, including this cover page.
READ THESE INSTRUCTIONS FIRST
Write your name, class and index number in the spaces provided on the separate
answer booklet.
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.
Calculator 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.
For , use either your calculator value or 3.142, unless the question requires the answer
in terms of .
The number of marks given in brackets [ ] at the end of each question or part question.
The total marks for this paper is 100.
Catholic High School 2008 Preliminary Exam III
2
Mathematical Formulae
Compound interest
Total amount =
nr
P
1001
Mensuration
Curved surface area of a cone = lr
Surface area of a sphere = 2 4 r
Volume of a cone = hr 3
1 2
Volume of a sphere = 3
3
4r
Area of triangle ABC = Cba sin 2
1
Arc length = r , where is in radians
Sector area = 2 2
1r , where is in radians
Trigonometry
C
c
B
b
A
a
sin
sin
sin
Acbcba cos 2 222
Statistics
Mean =
f
xf
Standard deviation =
22
f
xf
f
xf
Catholic High School 2008 Preliminary Exam III
3
1 A confectionery sells small, median and large strawberry cakes and chocolate cakes.
The number of each type of cake that was sold on a particular week are summarised
in the following table.
Chocolate Strawberry
Small 10 12
Medium 18 17
Large 15 24
The prices of each small, medium and large cakes are $8, $15 and $20 respectively
and the matrix P = 20158 can represent the information.
(a) The information on the above table can be represented by a matrix Q.
Write down the matrix Q. [1]
(b) N = PQ, evaluate N and describe the elements in N. [3]
(c) Write down a matrix M such that the multiplication of matrices M and N
gives the total amount of money collected by the confectionery shop on the
particular week. Find this amount of money by matrix multiplication. [2]
2
PQRS is a rectangle with dimensions 15 cm by 10 cm. The two circles touch each
other.
The larger circle, centre A and radius 5 cm, touches the rectangle at three points.
The small circle, centre B and radius x cm, touches the rectangle at two points.
(a) Given that ABC is a right-angled triangle, write down, in terms of x, the
length of BC. [1]
(b) Form an equation in x and show that it reduces to 0100402 xx . [3]
(c) Showing your method clearly, solve the equation 0100402 xx . [3]
(d) Find the area of the trapezium ABDE. [1]
A
B
5 x
15
10
C
D E P Q
R S
Catholic High School 2008 Preliminary Exam III
4
3 The cash price of a car is $128 000 net.
(a) Mr Ang bought the car on hire purchase terms. He paid a deposit of 60%
of the cash price followed by 24 monthly installments of $2 500 each.
Calculate how much more Mr Ang paid for the car in hire purchase
terms compared to cash terms. [3]
(b) Mr Lee bought the car on simple interest loan terms. He paid a down
payment of $50 000 and the balance at the end of 3 years with a simple
interest rate of 5% per year.
Calculate how much more Mr Lee paid for the car in simple interest
loan terms compared to cash terms. [2]
(c) Mr Leong bought the car on compound interest loan terms. He paid a down
payment $30 000 and the balance at the end of 5 years with a compound
interest rate of 2.5% per year.
Calculate how much more Mr Leong paid for the car in compound interest
loan terms compared cash terms. [3]
(d) In 2008 the price of one litre of petrol was $1.98 which is 20% more than
the price in 2007, find the price of one litre of petrol in 2007. [2]
4 (a) Express 22
3
1
9
8
xx as a single fraction in its simplest form. [4]
(b) Given that 2
2
4
5
y
yx
, express y in terms of x. [5]
(c) Guna has x two-dollars notes and y five-dollars notes. The total value of
the note is $100.
(i) Form an equation connecting x and y. [1]
(ii) If Guna has at most 20 two-dollar notes, use the equation in (i),
or otherwise, find the possible number of five-dollars notes. [2]
Catholic High School 2008 Preliminary Exam III
5
5
The points A, B, C, D and E lie on a circle, centre O. AB is a diameter
of the circle. TG is a tangent to the circle at E. 30ˆEDA and 62ˆDEO .
(a) Find, giving your reasons,
(i) EOA ˆ , [1]
(ii) EAD ˆ , [2]
(iii) GED ˆ , [2]
(iv) DCB ˆ , [2]
(b) Given that the radius of the circle is 6 cm, find
(i) the length of ET, [2]
(ii) the area of the shaded region. [3]
T
A
B
D
C
62
30
E
D
O G
Catholic High School 2008 Preliminary Exam III
6
6
Triangle ABC lies on a horizontal ground. AB = 1.8 m, BC = 2.4 m and AC = 3.4 m.
D is on AB produced such that 42ˆDCB . D is due east of A.
(a) Calculate
(i) CBA ˆ , [3]
(ii) the length of BD, [3]
(iii) the bearing of D from C. [2]
(b) A vertical flagpole, CF, stands at C. The angle of elevation of F from
A is 20o. A bird rests at the top of the flagpole.
Find the angle of depression of B viewed by the bird. [3]
7 The dot diagram shows the number of children living in the houses in a certain region.
0 1 2 3 4 5 6
Number of children
(a) Find the mode, median and mean number of children. [3]
(b) Find the standard deviation of the number of children. [2]
(c) A child is chosen at random, find the probability that the child belongs to
the house which has 3 children living in it. [2]
The table below shows the heights of the same group of children living in the region.
Height
in cm
6050 x
7060 x
8070 x
9080 x
10090 x
No. of
children
3
18
18
6
3
(d) Find the mean height of the children. [2]
(e) Find the standard deviation of the heights of the children. [2]
42o
2.4
A D B 1.8
3.4
North
C
F
Catholic High School 2008 Preliminary Exam III
7
8
OABC is a parallelogram. M is the midpoint of OB and Q is the midpoint of OM.
P is the point on OA such that OP = 2PA. OA = 3a and OC = c.
(a) Express, in terms of a and c,
(i) OB , [1]
(ii) OQ , [1]
(iii) CQ , [1]
(iv) MP . [1]
(b) Explain why CQ is parallel to MP . [1]
(c) Write down the value of the ratio of
(i) area of MOP : area of QBC, [2]
(ii) area of MPQ : area of QCM. [1]
C
O
M
A
Q
3a
c
B
P
Catholic High School 2008 Preliminary Exam III
8
9
Diagram I shows a cylindrical tank of radius 50 cm and length 120 cm. The tank is
partially filled with water and placed with its curved surface on a horizontal floor.
Diagram II shows the circular cross-section of the cylinder. O is the centre of the
circle. 2ˆ BOA radians.
(a) Find the area of the shaded region in Diagram II. [3]
(b) Find the area of the internal surface of the tank which is in contact with the
water in Diagram I. [4]
(c) The water in the cylindrical tank is now poured into an inverted cone of
radius 50 cm and of height 120 cm. Find
(i) the volume of the inverted cone, [1]
(ii) the height of the water in the cone. [3]
Diagram I Diagram II
A B
O 2 rad
50
120
50 A
B
Diagram III
120
50
Catholic High School 2008 Preliminary Exam III
9
10 Answer the whole of this question on a sheet of graph paper.
The variables x and y are connected by the equation x
xy22 .
The table below shows some values of x and the corresponding values of y.
x 3 2 1 5.0 1.0 0.1 0.5 1 2
y 8.33 k 1 75.3 99.19 20.01 4.25 3 5
(a) Find the value of k. [1]
(b) Using a scale of 2 cm to represent 1 unit on the x-axis and 1 cm to
represent 2 units on the y-axis, draw the graph of x
xy22
for 23 x . [3]
(c) Use your graph to find the value of p for which the equation px
x 22
has exactly two solutions for 23 x . [1]
(d) Use your graph to find the values of x in the range 23 x for which
0622 x
x . [2]
(e) By drawing a suitable straight line on your graph, find the value of x
satisfies the equation 022 23 xxx for 23 x . [2]
(f) By drawing a tangent, find the x-coordinate of the point P at which the
gradient of the curve x
xy22 at P is 2. [2]
End of Paper
Catholic High School 2008 Preliminary Exam III
10
Mathematics (Paper 2) Answers
1. (a) Q =
2415
1718
1210
(b) N = 831650
The total amount of money collected for chocolate and strawberry cakes respectively.
(c) M =
1
1 1481
1
1831650
2. (a) xBC 10 (b) 2225510 xxx
(c) x = 37.3, 2.68 (d) Area of ABDE = 28.1
3. (a) Amount paid more = $8 800 (b) Amount paid more = $11 700
(c) Amount paid more = $12 878 (d) price = 65.1$
4. (a) 33
2772
xx
x (b)
2
2
1
54
x
xy
(c) (i) 10052 yx
(ii) The possible number of five-dollars notes are 12, 14, 16, 18, 20.
5. (a) (i) 60ˆEOA (ii) 28ˆEAD (iii) 28ˆGED (iv) 148ˆDCB
(b) (i) ET = 10.4 cm. (ii) area of the shaded region = 12.3
6. (a) (i) 2.107ˆCBA (ii) 77.1BD m (iii) The bearing of D from C is 155.2
(b) The angle of depression = 27.3
7. (a) mode = 1 child. median = 2 children. mean = 92.1 children
(b) standard deviation = 1.44 (c) P = 16
5
48
15
(d) mean = 72.5 cm (e) standard deviation = 9.68 cm
8. (a) (i) caOB 3 (ii) caOQ4
1
4
3
(iii) caCQ4
3
4
3 (iv) caMP
2
1
2
1
(b) Because MPCQ2
3 , hence CQ is parallel to MP .
(c) (i) area of MOP : area of QBC = 9:43:22
(ii) area of MPQ : area of QCM.= 2 : 3
9. (a) area of the shaded region = 1360. (b) required area = 14700
(c) (i) volume of the cone = 314100 (ii) 6.96h cm
1
CATHOLIC HIGH SCHOOL PRELIMINARY EXAMINATIONS (3) 2008
SECONDARY FOUR MATHEMATICS
Subject : Mathematics Paper 1
Level : Secondary 4 Date : 12 September 2008
Marks : 80 Time : 0815 – 1015
Name : ____________________________________ ( )
Class : Sec 4 - ____
This question paper consists of 20 printed pages, including this cover page.
INSTRUCTIONS TO CANDIDATES :
Write your NAME, CLASS and INDEX NUMBER in the spaces at the top of this page.
Answer all 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.
You are expected to use an electronic 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
your answer to 3 significant figures. Give answers in degrees to one decimal place.
For , use your calculator value or 3.142, unless the questions requires the answer in term of .
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 80.
For Examiner’s Use Only:
Units
Fractions
Brackets
Accuracy
Qn No.Types
Units
Fractions
Brackets
Accuracy
Qn No.Types Qn No.Types
Others
Geometry
Diagrams
Graphs
Qn No.Types
Others
Geometry
Diagrams
Graphs
For Examiner's Use
80
2
Mathematical Formulae
Compound interest
Total Amount = Pn
r
1001
Mensuration
Curved surface area of a cone = rl
Surface area of a sphere = 4 2r
Volume of a cone = hr2
3
1
Volume of a sphere = 3
3
4r
Area of triangle ABC = Cabsin2
1
Arc length = r , where is in radians
Sector area = 2
2
1r , where is in radians
Trigonometry
C
c
B
b
A
a
sinsinsin .
bccba 2222 cos A
Statistics
Mean =
f
fx
Standard deviation =
22
f
fx
f
fx
3
Answer all the questions in the spaces provided on the Question Paper.
1 Light travels 1 metre in 3.3 nanoseconds.
Find the total distance in metres, that light will travel in 6.6 microseconds.
Answer ___________________ m [1]
2 In Iceland, the highest air temperature recorded is 30.5 °C.
The lowest air temperature recorded is 7.39 °C.
Find
(a) the difference between the two temperatures.
(b) the mean of the two temperatures.
Answer (a) ___________________ C [1]
(b) ___________________ C [1]
3 Find the integer values of x for which 2
154531
xxx .
Answer ______________________ [3]
4
4 23,0,5.2,,3
1,1232
(a) Complete the following table using the list of numbers provided above.
5 A polygon has n sides. Two of its exterior angles are 23º and 85º, while the other 2n
exterior angles are 14º each. Calculate the value of n.
Answer ______________________ [2]
Rational Numbers:
Integers:
[2]
5
6 Solve 100632x .
Answer ______________________ [2]
7 Factorise bxaybyax 4520 .
Answer ______________________ [2]
8 Simplify a
baa3203
, leaving your answer in index form.
Answer ______________________ [2]
6
9 The table shows the population statistics of Singapore from 2005 to 2007.
Total population comprises Singapore residents and non-residents.
(a) Find the number of non-residents in Singapore in 2006, leaving your answer in
standard form.
(b) Calculate the percentage increase in the total population from 2005 to 2007.
Answer (a) ______________________ [1]
(b) ______________________ [2]
10 (a) Express each of the numbers 66 and 168 as product of prime factors.
(b) Find the highest common factor of 66 and 168.
(c) Find the smallest integer value of n for which 66n is a multiple of 168.
Answer (a) 66 = _________________
168 = _________________ [2]
(b) ______________________ [1]
(c) ______________________ [1]
Year Total Population
(Millions)
Singapore Residents
(Millions)
2005 4.27 3.47
2006 4.40 3.53
2007 4.59 3.58
7
11 (a) The first five terms of a sequence are 1, 3, 5, 7 , 9, 11.
Find in terms of n , the n th term of the sequence.
(b) Using the answers from part (a) or otherwise, write down an expression, in terms
of n, for the n th term of the sequence
(i) 1, 9, 25, 49, 81, 121, ……….
(ii) 25, 49, 81, 121, ……….
Answer (a) ______________________ [1]
(b)(i) ______________________ [1]
(ii) ______________________ [1]
12 A box contains 5 red balls, 3 black balls and 1 white balls.
Two balls are taken from the bag at random, without replacement.
Find the probability
(a) that both balls are white,
(b) at least one ball is black.
A third ball is now taken from the box at random.
(c) Find the probability that none of the three balls is red.
Answer (a) ______________________ [1]
(b) ______________________ [2]
(c) ______________________ [2]
8
13 (a) y is inversely proportional to 3x .
9y when 3x .
Find y when 10x .
(b) p is directly proportional to 2q .
q is increased by 50%.
Find the percentage increase in p .
Answer (a) ______________________ [2]
(b) ______________________ [2]
14 ε = { x : x is an integer and 100 x }
A = { x : x is divisible by 3 }
B = { x : x is a prime number }
(a) Draw a Venn diagram to illustrate this information. Insert all elements of ε, A and
B in the Venn Diagram.
Answer (a)
[2]
(b) Write down BAn .
(c) List the elements in the set 'BA .
Answer (b) ___________________________ [1]
(c) 'BA = ____________________ [1]
9
15 In the diagram, A is 2,6 , B is 2,4 and C is k,12 .
y
xA (- 6 , 2) B (4 , 2)
C (12 , k)
0
(a) Given that A, B and C form an isosceles triangle such that AB = BC, show that the
value of k is 8.
Find
(b) the midpoint of AC.
(c) the gradient of line BC.
(d) the equation of the line which passes through the midpoint of AC and is parallel to
BC.
(e) the area of triangle ABC.
Answer (a) ______________________________________________________________
______________________________________________________________
______________________________________________________________
[1]
(b) ( _________ , _________ ) [1]
(c) ______________________ [1]
(d) ______________________ [1]
(e) ________________ units 2 [1]
10
16 (a) Sketch the graph of 221 xy .
Answer (a)
y
x
[2]
(b)
y
xO
The sketch represents the graph of
nxy .
Write down a possible value of n.
[1] Answer (b) n = ________________
(c)
y
xO
1
Write down a possible equation for the
graph
[1] Answer (b) ______________________
11
17
h cm
r cm
Container A Container B
r cm
h
2cm
h
2cm
The containers shown in the diagrams has height h cm.
Their other dimensions are as shown.
The containers are being filled to the brim with water which flows into each one at the
same constant rate.
It takes 2.5 minutes for the water to reach a depth of 2
hcm in container A .
(a) Find the time taken for the water to reach the brim of
(i) container A,
(ii) container B.
Answer (a)(i) ______________ minutes [1]
(ii) ______________ minutes [1]
12
(b) On the grid in the answer space, sketch the graph showing how the depth of the
water in each container varies with time.
Answer (b)
0 20 40
Depth
of
water
(cm)
Time ( minutes)
10 30
h
2
h
[2]
13
18 ABCD is a parallelogram and E is a point on AB.
BD and CE meet at X.
A B
CD
X
E
(a) Prove that triangles BEX and DCX are similar.
Answer (a) In triangles BEX and DCX, _________________________________________
________________________________________________________________
________________________________________________________________
________________________________________________________________
________________________________________________________________
________________________________________________________________
________________________________________________________________
[2]
(b) It is given that ABAE 4
Find the ratio
(i) area of BEX : area of DCX ,
(ii) area of BCX : area of parallelogram ABCD .
Answer (b)(i) ______________________ [1]
(ii) ______________________ [1]
14
19 The diagram is the speed – time graph for the first 20 seconds of a journey.
0 2 4 6 8 10 2012 14 16 18
5
10
15
20
25
Speed
(metres per second)
Time ( t seconds)
(a) Find
(i) the deceleration when 9t ,
(ii) the speed when 17t ,
(iii) the average speed for the last 10 seconds.
Answer (a)(i) _________________ m/s 2 [1]
(ii) __________________ m/s [1]
(iii) __________________ m/s [2]
15
(b) Part of the distance – time for the same journey is shown in the answer space.
Complete this graph.
Answer (b)
0 2 4 6 8 10 2012 14 16 18
20
40
60
80
100Distance Travelled
(metres)
Time ( t seconds)
120
140
160
180
[2]
16
20 The points P and Q are 6,1 and 4,7 respectively.
The point R is such that QR =
4
2.
(a) Find the coordinates of R.
Answer (a) ______________________ [2]
(c) It is given that RS =
h
12.
Find the two possible values of h which will make PQRS a trapezium.
You may use the grid below to help you with your investigation.
P
Q
Answer (c) h = _______ or _______ [2]
17
21 In the diagram, ABCD is a square and DEFG is a rectangle. EAB is straight line.
D C
BAE
F
G
(a) Show that angle ADE = angle CDG.
(b) Prove that triangle ADE is congruent to triangle CDG.
(c) Hence, show that DEFG is a square.
Answer (a) _______________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
[2]
(b) In triangles ADE and CDG, _______________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
[3]
(c) _______________________________________________________________
__________________________________________________________________
__________________________________________________________________
[1]
18
22 The plan of a triangular field has a scale of 1 cm to 50 m .
(a) Express this scale in the form 1 : n .
Answer (a) ______________________ [1]
The diagram below is part of a scale drawing of the field.
Another point, C , is 300m from B on a bearing of 063°.
(b) Complete the map to show position of C.
(c) On the same diagram, using ruler and compasses only, construct
(i) the bisector of angle ABC ,
(ii) the perpendicular bisector of the line AB.
[3]
B
A
N
19
23 At School A, 160 pupils took an English Test.
The diagram below is the cumulative frequency curve for their results.
20
40
60
80
100
120
140
160
20 40 60 80 100Marks
0
Use the graph to find
(a) the interquartile range,
(b) the value of x , if 20% of the students scored x marks and above.
Answer (a) ______________________ [1]
(b) ______________________ [1]
20
At another school, B, 120 pupils took the same English Test.
The diagram below is the box-and-whiskers plot for their results.
15 41 5410 98 Marks
(c) Compare the test results for the two schools in two different ways.
Answer (c) _______________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
[2]
~ END OF PAPER ~
21
Catholic High School
2008 Mathematics Preliminary Examination 3
Paper 1 Answer Key
1. 2000 m
2. (a) 70.2°C (b) 6.4 °C
3. 0, 1 and 2
4. 0,5.2,3
1,1232
1232,0
5. 20
6. 3
11or
3
15 xx
7. bayx 54
8. 62
5
3 ba
9. (a) 5107.8 (b) %49.7
10. (a) 113266
732168 3
(b) 6
(c) 28
11. (a) 12 n (b) 212 n (c) 2
32 n
12. (a) 0 (b) 12
7 (c)
21
1
13. (a) 0.243 (b) 125%
14. (a)
AB
36 92
5
7
4 8 101
0
(b) 1 (c) 10,9,8,6,4,3,1,0
15. (a)
8
62
362
100264
10212422
k
k
k
k
k
2
2
(b) ( 3 , 5) (c) 4
3
(d) 4
32
4
3 xy or 1134 xy
(e) 30 units2
16.
(a)
y
x
( - 2 , 1 )
-3 -1
-3
(b) 2n (c) xy 2
17. (a) (i) 20 min (ii) 40min
(b)
0 20 40
Depth
of
water
(cm)
Time ( minutes)
10 30
h
2
h
18. (a) AAA Similarity
(b)(i) 9 : 16 (ii) 3 : 14
3
19. (a)(i) 5m/s2 (ii) 12.5 m/s (iii) 7.5m/s
(b) 0 2 4 6 8 10 2012 14 16 18
20
40
60
80
100Distance Travelled
(metres)
Time ( t seconds)
120
140
160
180
20. (a) ( 5 , 0) (b) 4h or 10
21.
square. a is
., Since
.& rectangle, a is Since
.
triangletocongruent is triangleSince )c(
)Congruency (A.A.S. triangletocongruent is triangle
line)straight aon angle(adjacent 90
rectangle) a is（90180
)square a is(
) (a)part from (
, and esIn triangl (b)
＝
square) a is （90
rectangle) a is （90 (a)
DEFG
FEDGFGDEDGDE
FEDGFGDEDEFG
DGDE
CDGADE
CDGADE
DEFG DAE
ABCDCDAD
CDGADE
CDGADE
CDGADE
CDGADGADGADE
CDGADGADCADGADEEDG
ADC DC EDG
ABCDADC
DEFG EDG
23. (a) 16 (b) 66 or 67
(c) School B has a higher interquartile range at 39 as compared to that of School A at 16.
School A has a higher median at 55 as compared to that of School B at 41.
A
BC
D
E
O
F
