Pierce Prelim 2009 Em2

Class Candidate Name

Index Number

_________________________________________

PEIRCE SECONDARY SCHOOL Department of MATHEMATICS GCE O Level Preliminary Examination II for Secondary Four Express and Five Academic Mathematics Paper 2 Wednesday

4016/02

2 September 2009

0800 – 1030

Additional materials: A4 writing paper (8 sheets) Graph paper (1 sheet) Electronic calculator

TIME

2 hours 30 minutes

INSTRUCTIONS TO CANDIDATES Write your name, class and register number on all the work you hand in. Write in dark blue or black pen. You may use a soft pencil for any diagrams or graphs. Show your working on the same page as the the answer. Do notall use staples, paper clips, highlighters, gluerest or of correction fluid. Omission of essential working will result in loss of marks. 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. Angles For , use either your calculator value or 3.142, unless the question requires the answer in terms of . 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 100.

This question paper consists of 12 printed pages, including this cover page.

[Turn over Final Copy by GohCL

2 Mathematical Formulae Compound interest Total amount = P (1 

r n ) 100

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

1 2 r h 3

Volume of a sphere = Area of triangle ABC =

4 3 r 3

1 ab sin C 2

Arc length = r , where  is in radians Sector area =

1 2 r  , where  is in radians 2

Trigonometry a b c   sin A sin B sin C

a 2  b2  c 2  2bc cos A

Statistics

Mean =

Standard deviation =

 fx f  fx 2   fx    f f 

2

2

3 1

In the diagram below, EOB is the diameter of the circle, where O is the centre. The sides of ABC touches the circle at points B and C respectively. A

C X B O E

(a)

(b)

Prove that ABX is congruent to ACX . State clearly the case of congruency.

[3]

Prove that BCE is similar to BXO . Show your working steps clearly. [2]

2

Under the Direct School Admission (DSA) exercise for Niche in Uniformed Groups , Primary Six students have the opportunity to join Peirce Secondary before the release of their PSLE results. The admission criteria is based on 3 categories of abilities. The scores of 4 Primary Six student applicants are as below. Applicant David Esther Martha Caleb

Academic Performance 80 70 80 90

Leadership Qualities 80 60 90 90

Interview Skills 70 90 60 80

(a)

Represent the above information as a 4 x 3 matrix. Name this matrix as A. [1]

(b)

(i) (ii)

(c)

1 1 1 1 1 . 4 Explain what does the elements in BA represent.

Find BA if matrix B 

[1] [1]

The 3 categories Academic Performance, Leadership Qualities and Interview Skills were assigned the weightings 25%, 60% and 15% respectively. Write down a matrix X such that AX will give the overall score for each applicant. Evaluate the matrix AX. [2]

3

4 3

(a)

In a group of 40 people, 22 people have car licenses and 20 people have motorcycle licenses. Among the car motorists, half of them also possess motorcycle licenses. There are 9 people without any driving license. It is given that

 = {people in the group}, C = {those who have car licenses}, M = {those who have motorcycle licenses}. (i)

Represent the above information in a Venn diagram.

[2]

(ii)

Express in set notation the statement “ Car motorists who do not possess a motorcycle licenses”.

[1]

(iii)

(b)

Find the value of n( M  C ' ). Explain fully what does n( M  C ' ) mean. [2]

Copy the following Venn diagram and shade the region represented by X Y '.



4

X

Y

x 2  x  12 x 3  3 as a single fraction in its simplest form. 2 2( x  16) x  4 x 2

(a)

Express

(b)

Given that

(c)

Given that

y  3

[1]

x , express x in terms of y. x9

[3]

[3]

2n 3 m  , find the value of . Leave your answer as a fraction. mn 7 n [2]

4

5 5

Cavenagh and Hillview clinics were always packed with patients waiting to see the doctor and the doctor did not have any break when the clinics were in operation. (a)

On a weekday morning, the average waiting time for a patient at Cavenagh clinic was x minutes. Calculate, in terms of x, the number of patients who visited Cavenagh clinic on a weekday morning if Cavenagh clinic was opened for 4 hours. [1]

(b)

On a weekday morning, the average waiting time for a patient at Hillview clinic was 2 minutes longer than that of Cavenagh clinic. Calculate, in terms of x, the number of patients who visited Hillview clinic on a weekday morning 1 if Hillview clinic was opened for 3 hours. [1] 2

(c)

In one particular week, the combined total number of patients who visited Cavenagh clinic and Hillview clinic was 500, excluding weekends. Form an equation in x and show that it is reduced 10 x 2  25 x  48  0 . [3]

(d)

Solve for x and hence, calculate the number of patients who visit Hillview clinic on a weekday morning. [4]

5

6 6

(a)

Diagram I shows a zinc plate in the shape of sector OABA’. The radius of the 11 sector OA is 9 cm and the reflex angle AOA’ is  rad. The zinc plate is 9 then wrapped around to form the curved surface of a cone OAB as shown in diagram II. A’ O

O 9 cm

11  rad 9

9 cm

B

A B

Diagram I

(b)

A

X

Diagram II

(i)

Find the area of the sector OABA’.

[2]

(ii)

Find the base radius XA of the cone OAB.

[2]

(iii)

Calculate the maximum volume of liquid that can be poured into the cone when it is inverted. [3]

The following water tank is made by joining a hemisphere of 30 cm to an open cylinder. Given that the capacity of the water tank is 255 000 cm3, find the height h of the cylinder. [3]

30 cm

h cm

6

7 7

In the following diagram, ABCD is a parallelogram. The ratio of AD to AH is 2 : 3 and the ratio of HG to GF is 3 : 2 and the ratio of AF to FC is 3 : 1. E is the midpoint of AC and G is the midpoint of DC. H

C

G

D

F E

A

(a)

B

  Given that AH  3q, and AB  p, find in terms of p and/or q, (i)

 AC ,

[2]

(ii)

 HC ,

[1]

(iii)

 HF ,

[2]

(iv)

 DE .

[1]

(b)

State two facts about DE and HF.

[2]

(c)

Find the ratio of Area of ADE (i) , Area of AHF

[1]

Area of ADE . Area of ABCD

[1]

(ii)

7

8 8

Answer the whole of this question on a sheet of graph paper. 8 . x Some of the corresponding values of x and y, correct to 1 decimal place, are given in the following table.

The variables x and y are related by the equation y  x  2 

x y

1 7.0

1.5 4.8

2 4.0

2.5 3.7

3 3.7

4 4.0

5 4.6

6 5.3

7 q

8 7.0

(a)

Calculate the value of q. Leave your answer correct to 1 decimal place.

(b)

Using a scale of 2 cm to represent 1 unit, draw the horizontal x-axis for 0 x8. Using a scale of 2 cm to represent 1 unit, draw a vertical y-axis for 0  y  8 . On your axes, plot the points given in the table and join them with a smooth curve. [3]

(c)

Use your graph to estimate (i) the value of y when x = 4.5,

[1]

(ii)

[2]

the range of values of x for which y  4 .

[1]

(d)

By drawing a tangent, find the gradient of the curve at x = 2.

[2]

(e)

By drawing a suitable line, find the solutions of the equation 3 x 2  10 x  8  0 . [2]

8

9

9

(a)

Mr Tan’s annual income for the year 2008 is $95 000. He was entitled to a personal relief of $15 297. Calculate Mr Tan’s individual income tax for the year of assessment 2009 using the income tax rates table below. [3]

Chargeable Income First $20 000 Next $10 000 First $30 000 Next $10 000 First $40 000 Next $40 000 First $80 000 Next $80 000 First $160 000 Next $160 000 First $320 000 Above $320 000

(b)

Rate(%)

Gross Tax (Payable)

0 3.50 5.50 8.50 14 17 20

0 350 350 550 900 3 400 4 300 11 200 15 500 27 200 42 700

Mr Yip wanted to buy a private property valued at 1.2 million. Currently, banks in Singapore lend customers up to 80% of the property value. Mr Yip was considering a loan package with DBA bank. The period of repayment was 30 years and the amount of loan was 80% of the property value. The terms of the interest rates were as follows : Year 1 – 3% simple interest Year 2 – 3.5% simple interest Year 3 to 30 – 5% simple interest Calculate the total value of the bank loan Mr Yip would be taking from DBA bank, inclusive of interest. Hence, calculate the average amount of money Mr Yip would need to pay the bank every month during the duration of 30 years. [4]

(c)

Mrs Lai invested her savings of SGD 200 000 with AUK bank, an Australian bank. AUK bank offered an interest rate of 5.5% compounded yearly and the period of deposit was 10 years. (i)

Find the principal amount of the account in Australian dollars (AUD) if the exchange rate was AUD 1 = SGD 1.216. [2]

(ii)

At maturity of the deposit, calculate the amount of money Mrs Lai would receive in Singapore currency if the currency exchange rate was AUD 1 = SGD 1.345. [3]

9

10 10

In the diagram below, R, S and T are three points on horizontal ground. RS = 9.8 m and ST = 13.5 m. A vertical flag pole RQ stands at R and the angle of depression of S from Q is 40 o. The bearing of T from S is 055o. Q

N

9.8 m

R

S 13.5 m T

(a)

Given that R is due north of S, calculate the length of RT.

[2]

(b)

Find  RTS .

[2]

(c)

Find the bearing of R from T.

[2]

(d)

Calculate the height of the flag pole QR.

[2]

(e)

Tammy walked along the path ST and stopped at point X to view the flag pole. The point X is where the angle of elevation of Q is the greatest. (i)

Calculate the length of RX.

[2]

(ii)

Hence, find the angle of elevation of Q from X.

[2]

10

11

11

40 boys from class 4A1 were weighed and the results, recorded to the nearest kilogram, were as displayed in the following cumulative frequency curve.

Frequency

Cumulative Frequency Curve for the weight of 40 boys from class 4A1 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0 052

54

56

58

60

62

64

66

68

70

72

74

76

78

80

Weight (kg)

(a)

From the curve, find the interquartile range of the data.

[2]

(b)

Using the graph, find the values of p and q.

[2]

(c)

Weight (x kilograms)

Frequency

54  x  58

3

58  x  62

3

62  x  66

p

66  x  70

6

70  x  74

q

74  x  78

5

Find the mean and the standard deviation of the weights of the boys in class 4A1.

[4]

11

12

(d)

Given that the mean and standard deviation of the boys from another class 4A2 is 60 kg and 7 kg respectively, comment on the distribution of the weights of the boys in both classes. [1]

(e)

To achieve the ideal weight, a boy must be 70 kg and below. (i)

(ii)

Calculate the probability one boy chosen an random from class 4A1 achieved an ideal weight.

[1]

Calculate the probability that if three boys from class 4A1 are chosen at random, at least one boy achieved the ideal weight.

[2]

End of Paper

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