4e 2009 zhonghua prelim 2 maths paper 1

ZHONGHUA SECONDARY SCHOOLPRELIMINARY EXAMINATION 2

2009

Secondary 4 Express/ 4 Normal Academic/5 Normal Academic

Subject : Mathematics Paper 1 (4016/01) Date : 14 Sep 2009

Level : Sec 4E/4NA/5NA Time : 07 45 – 09 45

Classes: 4E1, 4E2, 4E3, 4E4, 4E5, 4E6, 4N1, 5N1 Duration : 2 hours

Name : ________________________________________( )

Class: Sec _______

READ THESE INSTRUCTIONS FIRST

Write your name, class and register number in the spaces provided.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.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 π, use either your calculator value or 3.142, unless the question requires the answer in terms of π.

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 80.

For Examiner’s Use

This document consists of 15 printed pages (including this cover page)

Mathematical Formulae

Compound Interest

Total amount =

Mensuration

Curved surface area of a cone =

Surface area of a sphere =

Volume of a cone =

Volume of a sphere =

Area of triangle ABC =

Arc length = , where is in radians

Sector area = , where is in radians

Trigonometry

Statistics

Mean =

Standard deviation =

2

Answer all the questions

1 The value of a car decreased by 22% between 2007 and 2009. In 2009, the car was valued at $48 000. Find its value in 2007.

Answer $............................................ [2]

2 A portable hard drive has a storage capacity of 2 terabytes. If a photograph takes up about 4.8 megabytes of storage, how many such photographs can be stored in the portable hard drive? Give your answer correct to 2 significant figures.

Answer ……………………………… [2]

3 Four wooden poles of length 54cm, 114cm, 84cm and 60cm are cut into shorter pieces that are all the same length. If no wood is left over, what is the smallest possible number of pieces that can be obtained from cutting the 4 wooden poles?

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Answer ………………………………. [2]4 Two of Preema Supermarket’s special offers are shown below.

Minced Beef Minced Pork Usual: $3 per 500g Usual: $2.50 per 500g

Special: $2 per 500g Special: $1.50 per 500g

Mrs Tan says that the offer on minced pork is better. Explain, showing your working clearly, why Mrs Tan says that.

Answer ..………………………………………………………………………………...

…………………………………………………………………………………………...

…………………………………………………………………………………………...

…………………………………………………………………………………………...

…………………………………………………………………………………………… [2]

5 Which of the following graphs below best fit the following statements:a) The number of people unemployed is still increasing but by less each month.b) The price of gold has fallen steadily over the last few months.

Answer a) Graph…………………….. [1]

4

Time

Graph 1

Time

Graph 4

Time

Graph 3

Time

Graph 2

b) Graph …………………… [1]6 Factorise

Answer ……………………………….. [2]

7 Given that , find the value of .

Answer ………………………………. [2]

8 A bank offers loans at an interest of 8% per annum, compounded monthly. Ms Kaur took a $40 000 loan and makes a monthly payment of $2 800 at the end of each month. Find the amount she still owes the bank after the third payment. Give your answer correct to the nearest dollar.

5

Answer $………………………….. [3]9 Solve

Answer x = ……… or x = …………. [3]

10 Solve the simultaneous equations

Answer x =…………. , y = ………….. [3]

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12 Solve the inequality < and show your solution on the number line in the answer space.

Answer [3]

12 a) Simplify

b) Given that 343 , find the value of k.

Answer a) …………………………… [2]

b) k =..……………………… [2]

7

13 An area of 16cm2 on a map represents an actual area of 0.25km2. Calculatea) the area on the map, in cm2, which represents an actual area of 1.25km2.b) the actual distance, in km, represented by a length of 5.4cm on the map.

Answer a)……………………… cm2 [2]

b) ……………………… km [2]

14 i) Find the values of p, q and r in the number sequence below:

3 3 = 1 (5 + 7) 3 = p

(7 +9 +11) 3 = q (9 + 11 + 13 + 15) 3 = r

: :

: (a + ……. + m + ………+ b) 3 = 529

ii) What special name is given to the answers in part (i)? iii) Write down the 5th line of the sequence in part (i). iv) Find the value of m where m is the value found exactly in the middle of the series

from a to b.

Answer i) p = …...., q = ..…., r = ……. [2] ii) ……………………………..[1]

iii) …………………………………………………………………………………………. [2]

iv) m = ……………………… [1]

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15 In the diagram below, ABC is a triangle in which A is the point , AB = 9 units and C is the point .

a) Write down the coordinates of B.b) Given that the length of AC is units, find the value of k.c) Find the value of .

Answer a) [1]

b) k =..……………………. [2] c) =………………….. [2]

9

x

y

O BA

C

16 a) Given that

i) calculate , leaving your answer correct to 2 decimal places.ii) find the coordinates of A if B is the point (2,5)

Answer i) …..…………………units [1]

ii) [2]

b) In the diagram given below, a and =b.

i) Mark and label clearly on the diagram, the vector such that =2a - b.

ii) Express in the form ma + nb.

Answer b ii) …………………………. [1]

10

O

B

A

R• [1]

17 In the diagram, A, B, C and D are points on the circumference of the circle.Given that AC = BD = 7cm, BC = 2cm, AB = cm, prove thati) AB is the diameter of the circle,ii) ∆ABC is congruent to ∆BAD.

Answer i)………………………………………………………………………………...

…………………………………………………………………………………………...

…………………………………………………………………………………………...

…………………………………………………………………………………………...

……………………………………………………………………………………………

……………………………………………………………………………………………

……………………………………………………………………………………………

…………………………………………………………………………………………… [3]

ii) ………………………………………………………………………………………..

…………………………………………………………………………………………...

…………………………………………………………………………………………...

…………………………………………………………………………………………...

……………………………………………………………………………………………

……………………………………………………………………………………………

…………………………………………………………………………………………… [3]

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j

E

A

B

C

D

18 The graph shows the speed-time graph of a car during a period of 50 seconds.

The distance travelled in the first 20 seconds is 220 m.(a) Find the value of the maximum speed v..(b) Sketch the distance-time graph of the car on the axes given in the answer space below.

[3]

Answer a) ………………………m/s [2]

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Time (seconds)0 3520 50

Speed (m/s)v

10

Time (seconds)

Distance (metres)

19 The waiting time in minutes for 20 patients at two clinics are shown below.

Clinic A

Time (minutes)

No. ofpatients

1 6 4 5 4

Clinic B

Mean = 14 mins

Standard Deviation = 3.28 mins

a) For Clinic A, calculatei) the mean waiting time,ii) the standard deviation.

b) Compare, briefly, the waiting time for the two clinics.

Answer a) i) ………………… mins [1]

ii) …………….……mins [2]

b) ……………………………………………………………………………………………..

…………………………………………………………………………………………….…..

……………………………………………………………………………………………..…..

…………………………………………………………………………………………………

…………………………………………………………………………………………………. [2]

20 a) i) On the axes given below, sketch the graph of , showing

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clearly the intercepts on the x- and y- axes.ii) Write down the equation of the line of symmetry.

[2]

Answer ii) ………………………….. [1]

b) i) Express in the form . ii) Hence, sketch the graph of on the axes below, showing

clearly the coordinates of the maximum or minimum point.

[3]

Answer i) ……………………………. [1]21 Triangle ABC is drawn in the answer space below.

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x

y

O

x

y

O

a) Construct the perpendicular bisector of AC and mark and label clearly the point P where this bisector meets the line AC.

b) Complete the sentence in the answer space below.c) Construct the angle bisector of angle BCA and mark and label the point Q where this

angle bisector meets the perpendicular bisector of AC.d) Measure and write down the bearing of Q from B.

Answer a) and c)

[4]

b) The areas of ∆ABP and ∆CBP are equal because …………………………………...

…………………………………………………………………………………………...

………………………………………………………………………………………..… [2]

d) …………………………............. [1]

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A

B

C

No Answers1. $61 538.46

2.

410 000 or photographs

3. 2 54 114 84 603 27 57 42 30

9 19 14 109 + 19 + 14 + 10 = 52 pieces

4. % discount on minced pork = 40%

% discount on minced beef = %

The percentage discount on minced pork is higher. Hence the offer on pork is better.

5. (a) Graph 2(b) Graph 4

6.

7.

8.Amt at end of 1st mth = $

= $40266.67

Amt at end of 2nd mth =

= $37716.44

Amt at end of 3rd mth =

= $35149.22After 3rd payment, amt owed = - 2800

= $32349.22 = $32349

9. or o.e.

ZHONGHUA SECONDARY SCHOOL2009 Preliminary Examinations 2Answers for Paper 1

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No Answers

10. , 2y

11. and

12. (a)

(b)

13. (a)

(b)

14. (i) , , (ii) Perfect squares / Square numbers(iii)(iv)

15. (a) (3, 0)(b)

(c)= 0.625

16. (a) i) 9.49 units(ii)

=

Coordinates of A are (7, 8)

17

No Answers

(b) i) Vector on grid provided.(ii)

17. (i)

By the converse of Pythagoras’ Theorem, ABC is a right-angled with ACB = 90º.Points A, B and C lie on the circumference of the circle,and hence, AB must be the diameter with ACB as the right angle in a semi-circle.

(ii) ACB =ADB = 90º (Angles in the same segment)AB is a common side to both ABC and BAD.AC = BD (given)Hence, ABC is congruent to BAD.(RHS congruency)

18. (a) Maximum value of v = 22(b) Answer on given grid.

19. (a) i) Mean = 17 minutesii) Standard deviation = 4.88 minutes

(b) The mean waiting time for clinic B is shorter and the spread of the waiting time for clinic B is also shorter.

20. (a) i) Answer on grid provided.

ii) Line of symmetry is x = 1.

(b) i)ii) Answer on grid provided.

21. (a) Answer on the diagram provided.(b) Both triangles have the same height.

Their bases are equal in length (AP = CP)(c) Answer on the diagram provided.(d) The bearing of Q from B is 282º

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