Chij Tp 2010 Em Prelim p1

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.

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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


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]

_____________________________________________________________________________


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] _____________________________________________________________________________


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]

_____________________________________________________________________________


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


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.


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


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


10

In the diagram, the line 6=y intersects the curve cbxxy ++= 22 at A and B.

The curve intersects the x-axis 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]


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


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


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.


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


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


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] _____________________________________________________________________________


17 22 The diagram below shows the speed-time 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 distance-time 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


18 E Maths Sec 4E Prelim Examination (Paper 1) 2010 (Answers)

1ai

321

, 0.03125 11a 8 sides 14a

x-10 -5 5 10

y

-10

-5

5

10

14b

x-10 -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

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Chij Tp 2010 Em Prelim p1