Chij Prelim Em 1 2009

Class

Register Number

Name

CHIJ SECONDARY (TOA PAYOH) PRELIMINARY EXAMINATION 2009 SECONDARY FOUR (SPECIAL / EXPRESS)

MATHEMATICS PAPER 1

4016/1 2 September 2009 2 hours

Candidates answer on the Question Paper. Additional materials: Geometrical instruments

READ THESE INSTRUCTIONS FIRST Write your name, register number and class on all the work you hand in. 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 clips, highlighters, and 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 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 the answer to three significant figures. Give answers in degrees to one decimal place. For π , use either your calculator value or 3.142, unless the question requires the answer in terms of π .

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 80.

2

This document consists of 16 printed pages including the cover page. [Turn over

Mathematical Formulae Compound interest r   Total amount = P 1 +   100 

n

Mensuration Curved surface area of a cone = π rl Surface area of a sphere = 4π r 2 Volume of a cone =

1 2 πr h 3

Volume of a sphere = Area of triangle ABC =

4 3 πr 3

1 ab sin C 2

Arc length = rθ , where θ is in radians 1 2 Sector area = r θ , where θ is in radians 2 Trigonometry a b c = = sin A sin B sin C a 2 = b 2 + c 2 − 2bc cos A

. Statistics Mean =

Standard deviation =

∑ fx ∑f ∑ fx ∑f

2

 ∑ fx  −  ∑ f   

2

chijsectp.4S/E.prelim.emath1.2009

3

Answer all the questions showing all your working clearly. 1

Evaluat 4.23 + 0.6 21 .3 × 0.39

(a)

, lea vin g you r ans wer cor rect to 3 dec ima l pla ces. (b)

1 , leaving your answer in standard form. 2.5 ×10 + 8.3 ×10 5 6

Answer [1] (b)…………………… ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ chijsectp.4S/E.prelim.emath1.2009

[1]

4

___ ___ ___ ___ ___ ___ ___ ___ __ 2

−3

 8  ÷   125 

0

(a)

y Simplify   3

(b)

Given that 2(16 3 x +2 ) = 8 4−5 x , find x.

Answer (a)……………………

[1]

(b)……………………

[2]

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3

Tw o nu mb ers h and k, wri tten as pro duc ts of thei r pri me fact ors, are h = 22 × 72

and

k = 2 × 33 × 7 .

Find th

(a) (b)

What is the smallest positive integer n for which 30n is a multiple of k?

Answer [1] (b)…………………… ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ chijsectp.4S/E.prelim.emath1.2009

[1]

6

___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ __ 4

In one o the hou cans at damage percent

[3] ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ chijsectp.4S/E.prelim.emath1.2009

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___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ __A nsw er ….. … … … … … … … … 5

One hu have a cm 3 of (a) (b)

write 0 Calculate the volume of one drop, in m 3 , giving your answer in standard form.

Answer [1] (b)…………………… ___ ___ ___ ___ ___ chijsectp.4S/E.prelim.emath1.2009

[2]

8

___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ __ $12 6 is divi ded bet wee n Jam es and Ton y in the rati o x : 5. Wri te do wn an exp ress ion, in ter ms of x, chijsectp.4S/E.prelim.emath1.2009

9

for Jam es’ shar e.

[2] ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ __A nsw er … … … … … … … chijsectp.4S/E.prelim.emath1.2009

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… … 7

The dia sector o opened covered terms o

120° 6 cm

10 cm

(a) (b)

the area the perimeter, of the paper portion.

Answer (a)……………………

[2]

(b)……………………

[2]

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___ ___ ___ ___ _ 8

(a)

The line 5 x = y +14 passes through the point (3 p,7 p ) . Find the value of p.

(b)

Find the equation of the line passing through the points (−1, 3) and (7, −2) , leaving your answer in the form ax + by = c , where a ,b and c are integers. The distance between the points A ( 1, 2k ) and B ( 1 − k , 1) is 10 k +4 .

(c)

Find the possible values of k.

Answer [1] (b)……………………

[2]

(c) …………………...

[3]

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___ ___ ___ ___ ___ __ 9

In the d is the d is the tangent to the circle at E. DE produced meets CB produced at F. (a) (b)

Prove t Given that CD = 13cm and DE = 5 cm, find FD.

C

D B E A

F

Answer (a) chijsectp.4S/E.prelim.emath1.2009

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…………………………………………………… ……………………………… …………………………………………………… …………………………………………. …………………………………………………… …………………………………… [2] (b)…………………… [2] ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ __ 10

Given a

1,

2 5 13 34 , , , , …………. 3 8 21 55

Write down the next term in the sequence. (b)

x

One of the terms in the sequence is y . Find in terms of x and y, the term which comes chijsectp.4S/E.prelim.emath1.2009

14

x

immediately after y .

Answer [1] (b)…………………… ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ __ 11

Che rise rec ord ed ove ra peri od of two mo nths chijsectp.4S/E.prelim.emath1.2009

[1]

15

the nu mb er of mat he mat ics que stio ns don e by 30 stud ents in her clas s. 36

51

41

57

22

61

54

55

32

40

39

61

52

49

28

43

52

54

25

32

49

60

57

48

59

60

45

43

20

52

(a)

Draw an ordered stem and leaf diagram for the data. [2]

(b)

State the mode.

(c)

State the median.

Answer (b)…………………… (c)…………………… ___ ___ chijsectp.4S/E.prelim.emath1.2009

[1] [1]

16

___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ __ Given

12 (a)

ex pre ss y in ter ms of x, 2x + y

(b)

find the numerical value of x −3 y

Answer (a)……………………

[1]

(b)……………………

[1]

chijsectp.4S/E.prelim.emath1.2009

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___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ __ 13

Cone A (a)

Ca lcu lat e the vol um e of the sec on d Co ne B wh ich has a chijsectp.4S/E.prelim.emath1.2009

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rad ius thr ice tha t of Co ne A an d hal f the hei ght of Co ne A. (b)

Cone C is similar to Cone A. Given that Cone C has a volume of 432 cm 3 , find the ratio of the base area of Cone C to the base area of Cone A.

Answer [2] (b)………………….. ___ ___ ___ ___ chijsectp.4S/E.prelim.emath1.2009

[2]

19

___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ _

If y is i

14

and x = the equ

(a) (b)

the positive value of x when y = 25.

Answer (a)……………………

[1]

(b)……………………

[1]

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___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ __ 15

In the fig ure , AB CD is a squ are. E is a poi nt on AB pro duc ed suc h that BE = BF Prove that triangles ABF and CBE are congruent.

chijsectp.4S/E.prelim.emath1.2009

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Answer ……………………………………………………………………………………………

…………………………………………………………………………………………………… …………………………………………………………………………………………………… …………………………………………………………………………………………………[2] ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ chijsectp.4S/E.prelim.emath1.2009

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___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ __ 16

The rat n-sides (a) (b)

Find th Hence, calculate the sum of interior angles of this polygon.

Answer (a)……………………

[2]

(b)……………………

[1]

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___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ _

In the d 17

cm. It i and ∠ ADB = 90°, find (a)

the valu

the exa

the area

B 1 cm (b) (c)

Answer [1] (b)……………………

[1]

(c) …………………...

[1]

___ ___ ___ ___ chijsectp.4S/E.prelim.emath1.2009

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___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ _ 18

Sol ve the foll owi ng sim ulta neo us equ atio ns:

(a) 1 x −y + 3 = 0 , 2

7 y − 11 = 6 x.

chijsectp.4S/E.prelim.emath1.2009

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Answer x = …………………

(b)

Given that 4 +

y =……………………

[2]

Answer (i)…………………….

[1]

x 1 < 3 x −2 , 3 6

(i) solve the inequality, (ii) write down the smallest integer value of x.

(ii) x =……………… ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ chijsectp.4S/E.prelim.emath1.2009

[1]

26

___ ___ ___ ___ ___ ___ ___ _ 19

The dia manufa its new drink. It is made up of two identical hemispheres and a right cylinder. The radii of the hemispheres are 10.5 cm each and the cylinder has a radius of 3.5 cm and a height of 15 cm. ( π =

22 ) 7

15 cm

10.5 cm

Find (a) (b)

the volume of the container, the total surface area of the container.

chijsectp.4S/E.prelim.emath1.2009

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___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ _ 20

Answer (a)……………………

[2]

(b)………..…………...

[3]

Speed in m/s

10 v 0

5

10

15

Time in seconds

(a)

The dia over a p retarded at this s uniform distanc

Find th

(b)

Distance in m Speed in of m/s Find the speed the particle after 15 seconds.

(c)

Complete the corresponding distance-time graph. 10 v chijsectp.4S/E.prelim.emath1.2009

0

5

10

15

Time in seconds

28

Ti me in sec on ds 0 5 10 15

[2] Ans wer (a) … … chijsectp.4S/E.prelim.emath1.2009

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… … … … … … (b)………..…………...

[1] [2]

___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ _ 21

X

B

C

E 4a

A

4b

ABCD is a parallelogram. E is a point on BD such that BE =

1 BD . 4

chijsectp.4S/E.prelim.emath1.2009

D

30

(a)

(b)

Given t terms o (i)

BD

(ii)

BE

(iii)

AE

Find the ratio of (i)

area of ∆BEX : area of ∆DEA ,

(ii)

area of ∆AEB : area of parallelogram

ABCD .

Answer (a) (i)............................. [1] [1]

(iii)………………….

(ii) … … … … … … …. [1]

(b)(i)………………….

[1]

(ii) …………………

[2]

___ ___ ___ ___ ___ ___ ___ ___ chijsectp.4S/E.prelim.emath1.2009

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___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ __ 22 (a) Sketch the graph of y = (3 − x) (1 + x), showing the x and y intercept. Answer (a)

y

x Y

[2] 5 2

1 4

(b) Sketch the graph of y = ( x − ) 2 − 4 ,showing x and y intercepts.

Answer (b)

chijsectp.4S/E.prelim.emath1.2009

X

32

[2]

___________________________________________________________________________________ 23 In trian

∠ABC

(a)

Constru [2]

(b)

(c)

In triangle ABC, construct (i) the bisector of angle BAC,

[1]

(ii) the perpendicular bisector of the line AC,

[1]

the bisector of angle BAC meets the perpendicular bisector of AC at point P. Measure and write down the length of AP. Answer (a) and (b)

chijsectp.4S/E.prelim.emath1.2009

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B

10 cm

C [1]Answer(c).................................

… … … … … … … … … … … … … …..

En d of chijsectp.4S/E.prelim.emath1.2009

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Pa per 1 … … … … … … … … … … … … … …. CH IJ E MA TH PR ELI M PA PE R1 200 9 64 2 1(a) ( x +1) 1(b)1 3 10 −7 4a

×

2(a)

2(b)

14b

x=

3 5

15

27 y3

x

1.512

BF=BE (given)

∠ABF = ∠CBE

BC= AB (square)

=

1 9

16a

∴∆ABF = ∆CBE n =15

3a

4.2

16b

3b

63

17a

4

4.08% −7 3.6 × 10 g 4 ×10 −15 m 3

17b

34.0

17c

1.50 cm 2

18a

x = 4,

5a 5b 6 7a

12 x x +5

28 π

18bi 18bii

2340 degrees −

3 4

y=5 17 x >1 19 x= 2

chijsectp.4S/E.prelim.emath1.2009

S A S

35 7b 8a

12 + 9 1

1 3

3 4

π

8b

5x +8y

8c

1 K= − 5

9a

∠ CED

= 29

= 90

( ∠ in a semicircle) = ∠FDC ( common angle)

19a 19b

5430 cm 3 2330 cm 2

20a

V=4

20b

14 m/s

21ai ii iii

4(b-a) b-a 3a +b

21bi ii 22

1:9 1:8 Sketch of graph

23

AP =6.1 to 6.25

∠CDE ∠CEO = ∠FCD

9b 10a 10b 11a 11b 12a 12b 13a 13b

∴∆CDE is similar to ∆ FDC 33.8 cm 89 144

x +y x +2 y

52 49 y=

3 x 16

5 4608 9:16

chijsectp.4S/E.prelim.emath1.2009