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Class Register Number Name
CHIJ SECONDARY (TOA PAYOH)
PRELIMINARY EXAMINATION 2010 SECONDARY 4 (SPECIAL/EXPRESS)
MATHEMATICS 4016/01 Paper 1 31 Aug 2010
2 hours Candidates answer on the Question Paper.
INSTRUCTIONS TO CANDIDATES Write your name, register number and class 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 ALL questions. Write your answers in the spaces provided in 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 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 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
This document consists of 17 printed pages including the cover page.
[Turn over]
chijsectp.4S/E.prelim.emath1.2010
2
Mathematical Formulae
Compound interest
Total amount = nrP
+100
1
Mensuration
Curved surface area of a cone = rl
Surface area of a sphere = 24 r
Volume of a cone = hr 231
Volume of a sphere = 334 r
Area of triangle ABC = Cab sin21
Arc length = r , where is in radians
Sector area = 221 r , where is in radians
Trigonometry
Cc
Bb
Aa
sinsinsin==
Abccba cos2222 +=
Statistics
Mean = ffx
Standard deviation = 22
ffx
ffx
chijsectp.4S/E.prelim.emath1.2010
3
Answer ALL questions.
1 (a) (i) Evaluate ( )4.34.76.5
11.35+
.
(ii) Using your answer to part (a), simplify ( )( )11.35
4.34.76.5
+
y.
Express your answer as a fraction in terms of y.
Answer (a)(i) .............. [1]
(ii) .............. [1]
(b) The weight of 735000 pebbles is 1 tonne. Each pebble has the same weight. Calculate the weight, in tonnes, of 1 pebble. Give your answer in standard form correct to 2 significant figures.
Answer (b) .............. [1] _____________________________________________________________________________ 2 Factorise completely 33 82 abba .
Answer .............. [2]
_____________________________________________________________________________
chijsectp.4S/E.prelim.emath1.2010
4 3 The number 84, written as the product of its prime factors, is 73284 2 = .
(a) Express 1350 as a product of its prime factors.
(b) Find the highest common factor of 84 and 1350. (c) Find the smallest positive integer value of n for which 1350 n is a perfect cube.
Answer (a) .............. [1]
(b) .............. [1]
(c) .............. [1]
_____________________________________________________________________________
4 A salesman sold a machine for $5220 and made a profit of 16% on the cost price. If he sells it for $4140, calculate the percentage loss incurred by the salesman.
Answer ................ % [2] _____________________________________________________________________________
chijsectp.4S/E.prelim.emath1.2010
5 5 Two special offers by a supermarket are shown below. Cherries Strawberries Usual: $3 per 100g Usual: $1.50 per 100g Special: $2 per 100g Special: $1.20 per 100g
Mrs Lim says that the offer on cherries is better. Explain, showing your working clearly,
why Mrs Lim says that.
Answer ..... ... ... [2]
_____________________________________________________________________________
6 Given that
=
=
3612
,2013
BA and
=
1001
I , evaluate
(a) A2, (b) the matrix C such that AB + C = BI , where I is the identity matrix.
Answer (a) .............. [2]
(b) .............. [2]
_____________________________________________________________________________
chijsectp.4S/E.prelim.emath1.2010
6
7 The plan of a house is drawn to a scale of 1 : 500.
(a) A rectangular room has an actual length of 19.1 m.
Find the length, in centimetres, of this room on the plan.
(b) The kitchen is represented by a floor area of 2.13 cm2 on the plan.
Find the actual floor area, in square metres, of the kitchen in the house. Answer (a) . cm [1] (b) m2 [2] _____________________________________________________________________________
8 (a) Simplify 12
74
932
nnn and express your answer with positive indices.
(b) Given that 164 3 = m , find the value of m.
Answer (a) .............. [1]
(b) m = ........... [1] _____________________________________________________________________________ 9 (a) Solve the simultaneous equations
chijsectp.4S/E.prelim.emath1.2010
7
xyyx433462=
=
Answer (a) x = .......
y = ........ [3]
(b) (i) Solve the inequality 5413
>+ xx .
(ii) Hence write down all the integer values of x which satisfy both the
inequalities 5413
>+ xx and 012 +x .
Answer (b)(i) .............. [1]
(ii) .............. [1] _____________________________________________________________________________ 10 The diagram shows a pyramid whose volume is 81cm3.
The area of the base is 27 cm2 and the height is x3 cm.
chijsectp.4S/E.prelim.emath1.2010
8 (a) It is given that 27 = k3 , where k is a rational number. Find k.
(b) Using your answer to part (a), find the value of x.
Answer (a) k = ......... [1]
(b) x = ............ [3] _____________________________________________________________________________ 11 (a) Each interior angle of a regular polygon is 135.
Find the number of sides of the polygon. (b) A polygon has n sides. Three of its exterior angles are 50, 60 and 120. The remaining exterior angles are each 10. Calculate the value of n.
Answer (a) .............. [1]
(b) n = ........... [2] _____________________________________________________________________________ 12 The containers shown in the diagram are geometrically similar. The circular bases of the
larger and smaller containers have radii R1 and R2 respectively. The cost of painting the surface of the larger container is 16 times the cost of painting the surface of the smaller container.
x3 cm
27 cm2
chijsectp.4S/E.prelim.emath1.2010
9
(a) Given that the cost of painting is proportional to the area painted, find the ratio 2
1
RR
.
(b) The circular top of the larger container has a circumference of 32 cm. Calculate the circumference of the circular top of the smaller container. (c) The capacity of the smaller container is 0.65 litres. Find the capacity of the larger container. Give your answer in cubic centimetres.
Answer (a) .............. [1]
(b) ........... cm [1]
(c) .................. cm3 [2] _____________________________________________________________________________ 13
6=y
cbxxy ++= 22
0 3 P
B A
y
x
R1 R2
chijsectp.4S/E.prelim.emath1.2010
10
In the diagram, the line 6=y intersects the curve cbxxy ++= 22 at A and B.
The curve intersects the xaxis at P and at (3, 0). Find
(a) the values of b and c, (b) the coordinates of B and P, (c) the minimum point of the curve cbxxy ++= 22 .
Answer (a) b = ......................
c = ................. [2]
(b) B ( ............ , ............ )
P ( ............ , ............ ) [2]
(c) ( ............ , ............ )
[1] _____________________________________________________________________________ 14 On the given diagrams, sketch the graphs of
(a) x
y 1= (b) xy =31
[2]
chijsectp.4S/E.prelim.emath1.2010
11 Answer (a) (b) (c) Sketch the graph of 2)2(1 += xy , showing clearly the intercepts on the x and y
axes. Write down the coordinates of the maximum point of the curve.
Answer (c) [2] Answer (c) ( ............ , ............ ) [1] ___________________________________________________________________________
y
x 0
y
x 0
y
x 0
chijsectp.4S/E.prelim.emath1.2010
12
15 OABC is a parallelogram such that OA= a and
OC = c . X is a point on OA such that
3XA = 2OA and the diagonal AC cuts the line XB at Y.
Express, as simply as possible, in terms of a and/or c
(a) AC ,
(b) XB ,
(c) YC .
Find the value of
(d) ,BCYofAreaAXYofArea
(e) ,XBAofAreaOBXofArea
(f) .OBCofAreaOBXofArea
Answer (a) .............. [1]
(b) .............. [1]
(c) .............. [2]
(d) .............. [1]
(e) .............. [1]
(f) .............. [2] _____________________________________________________________________________
B
X
A
O C
Y
a
c
chijsectp.4S/E.prelim.emath1.2010
13 16 X and Y are two sets such that XY . Given that ( ) 16=Xn , ( ) 5=Yn and ( ) 8' =Xn ,
(a) draw a clearly labelled Venn diagram to represent the universal set, and sets X and Y .
[2] (b) Using the Venn diagram, find (i) ( )'Yn , (ii) n ( ).
Answer (a)
Answer (b) (i) .............. [1]
(ii) .............. [1] _____________________________________________________________________________ 17 The braking distance of a truck is directly proportional to the cube of its speed. When the speed is x metres per second, the braking distance is 8 m. When the speed is decreased by 50%, find
(a) the braking distance, (b) the percentage decrease in the braking distance.
Answer (a) .............. m [1]
(b) .............. % [1] _____________________________________________________________________________ 18 Jenny bought a large multipack of bubble gum for her son.
chijsectp.4S/E.prelim.emath1.2010
14 She told her son that if he had 5 packets of bubble gum each week, the multipack would last 8 weeks. (a) How many weeks would the multipack last if her son had 8 packets of bubble
gum per week? (b) If her son had p packets of bubble gum per week, the multipack would last n
weeks. Write down a formula connecting p and n.
Answer (a) .............. weeks [1]
(b) .............. [1] _____________________________________________________________________________
19 The diagram shows three points P, Q and R on horizontal ground.
P is 9 km due east of R and Q is due south of R. The distance between P and Q is 14 km. Find the bearing of Q from P.
[Diagram is not drawn to scale]
Answer ................ [2] _____________________________________________________________________________
9 km
14 km
P
Q
R
South
East
chijsectp.4S/E.prelim.emath1.2010
15 20 (a) The lengths, to the nearest centimetre, of 24 worms have been recorded in the
table below. (i) Write down the median length of the worms. (ii) Write down the range of the lengths of the worms.
(b) The lengths of 7 other worms, all of different lengths, have a median of 8 cm and a range of 10 cm. One of the worms is 10 cm long. Write down a possible list of the lengths of these 7 worms. Assume that the values of the lengths are integers.
Answer (a) (i) ..................... cm [1]
(ii) ................. cm [1]
(b) ................. [2] _____________________________________________________________________________
Length of worm, in cm 5 6 7 8 9
Frequency 3 7 5 6 3
chijsectp.4S/E.prelim.emath1.2010
16 21 (a) In the answer space below, construct triangle ABC where 4=AB cm, 6=BC cm and = 90ABC .
(b) Measure and write down the length of AC. (c) Using the length of AC from part (b), find the value of 13 and leave your
answer correct to one decimal place.
Answer (a) [2]
Answer (b) ...................... cm [1]
(c) ................ [2] _____________________________________________________________________________
chijsectp.4S/E.prelim.emath1.2010
17 22 The diagram below shows the speedtime graph of a particle over a period of 20 seconds.
(a) Find (i) the speed of the particle at time t = 2, (ii) the total distance travelled in the 20 seconds, (iii) the acceleration when time t = 15.
(b) Sketch the distancetime graph for the same journey.
Answer (a) (i) cm/s [1] (ii)...... cm [2] (iii)... cm/s2 [1]
Answer (b)
[2] _____________________________________________________________________________
End of Paper
5 12 20
Distance (centimetres)
Time (t seconds) 0
Speed (centimetres per second)
Time (t seconds)
40
100
5 12 20 0
chijsectp.4S/E.prelim.emath1.2010
18 E Maths Sec 4E Prelim Examination (Paper 1) 2010 (Answers)
1ai
321
, 0.03125 11a 8 sides 14a
x10 5 5 10
y
10
5
5
10
14b
x10 5 5 10
y
10
5
5
10
16a 20ai 7 cm 20aii 4 cm 21b 7.2 cm 21c 3.6 cm 22ai 76 22aii 1190 22aiii 7.5
1aii
y32
, y03125.0
1
11b 16
1b 6101.4 12a 4 : 1 2 ( )( )babaab 222 + 12b 8 3a 23 532 12c 41600 3b 6 13a
13b 13c
8=b , 6=c B(4, 6) , P(1, 0) (2, 2)
3c 20 14c (2, 1) 4 8% 15a AC = c  a 5 Cherries %67.33=
Strawberries %20= Discount for cherries is higher.
15b XB = 32
a + c
6a
=
40192A
15c AXY is similar to CBY
6b
=
91852
C 15d 15e
4/9 1/2
7a 3.82 15f 1/3 7b 53.25 16bi 19 8a
n32
16bii 24
8b 4=m 17a 1 9a x = 3 , 3=y 17b 87.5 % 9bi 9bii
3