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CCHY Preliminary Examination 1 (2011) Mathematics /4E5N pg 1 of 15

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Preliminary Examination 1 (2011) Secondary 4 Express / 5Normal (Academic)

Candidate

Name Register No Class

Mathematics (4016) Paper 1 Date : 12 / 7 / 2011 Duration : 2 hour

READ THESE INSTRUCTIONS FIRST Write your name, register number and class on the space provided above. Write in dark blue or black pen. You may use a pencil for any diagrams or graphs. Do not use staplers, 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 and units of measurement will result in loss of marks. Calculators should be used where appropriate. Simplify your answers to their simplest form. If the answer is not exact, give the answer correct to three significant figures or in fraction where applicable. Give answers in degrees correct 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.

Setter : Thong Nai Kee

This paper consists of 15 printed pages, INCLUDING the cover page.

For examiner’s use

/ 80


Mathematical Formulae Compound interest

Total amount =

nr

P

1001

Mensuration

Curved surface area of a cone = rl

Surface area of a sphere = 4r 2

Volume of a cone = 13 r 2 h

Volume of a sphere = 43 r 3

Area of triangle ABC = Cabsin2

1

Arc length = r , where is in radians

Sector area = 2

2

1r , where is in radians

Trigonometry

C

c

B

b

A

a

sinsinsin

a2 = b2 + c2 – 2bc cos A

Statistics

Mean =

f

fx

Standard deviation =

22

f

fx

f

fx


1 The numbers 84 and 360, written as products of their prime numbers, are 84 = 22 × 3 × 7, 360 = 23 × 32 × 5. Find the

(a largest integer which is a factor of both 84 and 360, (b) smallest integer which is an exact multiple of both 84 and 360.

Answer (a) …………..…………………[1]

(b) .. ……………..…………… [1]

2(a) Light travels 1 kilometre in 3.33 microseconds. It takes an estimated time of 8 minutes 20 seconds for light to travel from the Sun to the Earth. Calculate, in kilometres, the distance between the Sun and the Earth. Express your answer in standard form, correct to 2 significant figures.

2(b) Matt sold two houses at $450 000 each. He made a profit of 25 % on the first house.

He lost 25 % on the second house. Find the (i) total cost price of the two houses. (ii) percentage gain or loss, on the combined sale of the two houses.

Answer (a) …………..…………km [2]

(b) (i) $…..….………..…………… [1]

(ii) ……………..……………% [2]


3(a) Solve ff

2

431 .

3(b)(i) Factorise completely 2123416 bbcacab .

(ii) Add all the factors from (i).

Answer (a) …….………..…………… [2]

(b) (i) ……………..…………… [2]

(ii) ……………..…………… [1]

4 The 1- metre mark on the side of a swimming pool is 15 cm above the water level. Water is being pumped into the pool such that the water level is rising at the rate of 0.8 cm per minute. (a) How far is the 1-metre mark above the water level after 15 minutes? (b) How long will it take for the 1-metre mark to be 5 cm below the water level?

Answer (a) …….………..…………… [1]

(b) ……………..…………… [1]

1 m mark

water

15 cm


5(a) Simplify 2

3

2

1

2

3

ba

a

ba

ba

3

34

8

152

1

, leave your answer in positive indices.

5(b) Simplify 2

2

41

510

x

xx

.

Answer (a) …….………..…………… [2]

(b) ……………..…………… [2] 6 9 workers are required to complete a job in 240 days.

After completing half the job, a notice was received to bring forward the date of completion by 30 days. Given that all the workers work at the same rate, find the additional number of workers required to complete the job on time as scheduled.

Answer …….………..…………… [3]


7(a)(i) In the answer space below, sketch the graph of xxy 23 , showing clearly

the intercepts and the turning point.

(ii) Write down the equation of the line of symmetry of xxy 23 . [3]

Answer (a) (i)

Answer: (a) (ii) ………….………………..[1]

(b)(i) Express y = x2 – 8x + 21 in the form y = (x – a)2 + b. (ii) Hence, sketch the graph of y = x2 – 8x + 21 in the answer space below.

Answer: (b)(i) y = ……………….………………..[1] Answer (b)(ii)

[2]

x

y

0

x

y

0


8 The diagram shows a circle with centre O and radius 5 cm. The points A, B and D lie

on the circumference of the circle such that BAO = DAO = 18

5 radians. BCD is

an arc of a circle with centre A and radius AB. (a) Find the length of the chord AD. (b) Find the total area of the shaded regions.

Answer (a) …….………..…………… [2]

(b) ……………..…………… [3] 9(a) The diagram shows a regular six-sided polygon. State the exterior angle of the

polygon.

9(b) Does a regular polygon with an exterior angle of 72 exist? Explain your answer. .

Answer (a) …….………..…………[1]

Answer (b) ……………………………………………………………………..…………...… …………………………………………………………………………….……………. [2]

exterior angle

A

O

D

B

C

5

5 5

18

5


10 In the following diagram, AC = 13 cm, BC = 11 cm, AB = 20 cm. N is a point on BC produced, where CN = 5 cm and AN = 12 cm.

(a) Explain why ANB = 90. (b)(i) Find the area of triangle ABC.

(ii) Write down the value of cosACB.

N

5

11

20

13

12

B

C

A

Answer (a) [2]

Answer (b) (i) …….……..……….. … [1]

(ii) ……………..…………… [1]


11 Given : A = { 2, 3, 4 } and B = { 1, 2, 3, 4 }, answer the following questions.

(a) Write down a possible set C such that A C = B.

(b) List the elements of a set D such that A D = B and A D = A. Answer (a) ……………………….………..…………… [2]

(b) ……………………………..…………..……………[1]

12 In the diagram, ST and SU are tangents to a circle, centre O. The bearing of U from

S is 030 and T is due east of S. Calculate the (a) bearing of U from T, (b) bearing of O from U.

Answer (a) …….………..…………… [2]

(b) ……………..…………… [2]

.

S T

U

O N

30o


13 The grid below shows the positions of the points P, Q and R where

QR =

2

6.

(a) Express

PR as a column vector.

(b) A is the point such that QRAP is a parallelogram. Express

RA as a column vector.

Answer (a) …….………..…………… [1]

(b) ……………..…………… [1]

14(a) The nth term of a sequence is given by 12 2 n . Write down the first 4 terms of the

sequence. (b) The first 4 terms of another sequence are 3, 9, 19, 33. (i) Write down the next term.

(ii) By comparing this sequence with your answers to (a), write down the nth term.

Answer (a) ……………………………….[1]

(b)(i) ……………………………….. [1]

(ii) ……………………………………[1]

R

P

Q . .

.


5 cm

8 cm

12 cm

15 The container, shown in the diagram, is full of water initially. There is a hole at the bottom and water is leaking through the hole at a constant rate.

It takes 1 minute, 3 minutes and 2 minutes for the water level to drop a depth of 5 cm, 8 cm and 12 cm respectively as indicated on the diagram. On the axes in the answer space below, sketch the graph showing clearly how the height of water in the container (h cm) varies with time over the 6 minutes.

Answer [3]

h (cm)

0 1 2 3 4 5 6 time (min)

25 20 15 10 5 0


16 In the diagram, QR is a diameter of the circle, centre O. TQ is a tangent to the circle and TR cuts the circle at S such that TQ = 12 cm and SR = 18 cm.

(a) Prove that TQR and TSQ are similar. (b) Hence, show that TS = 6 cm.

(c) Find the ratio area of TQS : area of RQS.

Answer (a) [2]

Answer (b) [1]

Answer (c) …….………..…………… [2]

R

O

Q

S

T

18 cm

12 cm


17 Find integers b and c such that 4

b +

c

1 =

12

7.

Answer (a) …….………..…………… [2]

18 On any school day, June goes to school by bus or her father’s car. The probability

that she will take the bus to school is 3

2. If she takes the bus, the probability that

she will be late for school is 4

3. If she goes by her father’s car, the probability that

she will be late is 5

2. The possibilities are represented by the tree diagram below.

Find the (a) values of p and of q in the tree diagram, (b) probability that June will be late for school on a certain day, (c) probability that June will not be late for two consecutive days.

Answer (a) p = …….……q = .……… [1]

(b) ……………..…………… [1]

(c) ……………..…………… [2]


19 Two similar solids are of heights 24 cm and 0.32 m. (a) Find the ratio of the height of the smaller solid to the larger solid. Give your answer in

the simplest form. (b) The mass of the smaller solid is 64 g. Find the mass of the larger solid.

Answer (a) …….………..…………… [1]

(b) ……………..…………g [2]

20 The weights (kg) of the children in two classes were taken. Information relating to the results is shown in the tables below.

Class A

Weight(kg) 41 – 45 46 – 50 51 – 55 56 – 60

Frequency 5 9 18 3

Class B

Mean = 52.3 kg

Standard Deviation = 4.6 kg

(a) For Class A, calculate the (i) mean, (ii) standard deviation. (b) Compare briefly the results for the two classes in 2 ways.

Answer (a)(i) ……………………………

(ii) …………………………………[3]

(b) ……..……………………………………………………………………………………….…………… …………………………………………………………………………………………………...…….. …………………………………………………………………………………………………………. ……………………………………………………………………………………………….. …… [2]


21 In sector AOB, OA = OB = 7 cm and chord AB = 5 cm. The side OA is drawn in the answer space below. (a) Complete the two possible sectors. Label the positions of B as B1 and B2. [3] (b) For one of these sectors,

(i) construct the perpendicular bisector of OB, [1] (ii) construct the line which is equidistant from the lines OA and OB, [1]

(c) Hence, locate and mark on the diagram, the point P in the sector which is

equidistant from the points O and B, and equidistant from the lines OA and OB. [1]

Answers

************** End of Paper 1 **************

O A


2011 CCHY Prelim1 EMP1 Answer Key 1. (a) 12 (b) 2520

2 (a) Distance = 6103500 km

= 1.667 x 108 km

~ 1.7 x 108 km

(b) (i) $960000 (b) 6.25%

3(a)

ff

2

431

2 – 3f + 4 = 2f [M1]

5f = 6

f = 5

61 or 1.2 [A1]

3(b)(i) 16ab – 4ac- 3bc + 12b2 = 4a(4b – c) + 3b(4b – c)

= (4b – c)(4a + 3b)

3(b)(ii) 4b – c + 4a + 3b = 4a + 7b - c

4 (a) 3 cm (b) 25 min

5(a) 3

2

5a (b)

6 6 men – 4 h – 2 holes

8 men – 3 h – 2 holes

time = 5

32 h = 7.5 h

7(a)(i)

7(a)(ii) x = 1

1 2

y

x

y = 3x(2 - x) 3

0


7(b)(i) y = (x – 4)2 + 5

7(b)(ii)

8(a) 6.43cm (b) 42.5cm2

9(a) 60

9(b)

72

360= 5

Yes, since

5 exterior angles of the regular polygon can be joined to form 360

72

360 is whole number, therefore, the regular polygon with ext angle 72º exist

(accept any logical explanation)

10(a) since 122 + 52 = 132, i.e. AN2 + NC2 = AC2, by Pythagoras Theorem, ANC = 90

10(b)(i) area of ABC = ½ x 11hlhlih x 12 cm2

= 66 cm2

10(b)(ii) - 5

13

11(a) Possible C = {1}, {1,2}, {1,3},{1,4},{1,2,3}, {1,2,4}, {1,3,4},{1,2,3,4}

11(b) D = {1,2,3,4}

12(a) 330 12(b) 120

13(a) PR =

3

2 13(b) RA =

4

1

14(a) 1, 7, 17, 31 14(b)(i) 51 14(b)(ii) 12 2 n

15

4

y

x

y = x2 – 8x + 21

21

0

5

25 20 15 10 5 0

h (cm)

0 1 2 3 4 5 6 time (min)

12


16(a) RQT = 90 (radius tan)

QST = 90 (rt in semicircle)

RQT = QST

RTQ = QTS (common angle)

TQR is similar to TSQ since the corresponding pairs of angles are equal (proven)

16(b) TQ

TR

SQ

QR

TS

TQ

12

18

6

12 ST

186

1212

ST = 6 TS = 6 cm (shown)

16(c) 1

3

17 c = 3

b

4 =

7

12 -

1

3

= 3

12

= 1

4 b = 1

18(a) p = 1

3 , q =

3

5

18(b) P(late) = 2

3 x

3

4 +

1

3 x

2

5

= 1

2 +

2

15

= 19

30

18(c) P(not late) = 2

3 x

1

4 +

1

3 x

3

5

= 1

6 +

1

5

= 11

30

P(late for 2 consective days) = 11

30 x

11

30

= 121

900

19(a) 3 : 4 19(b) 152 g

20(a)(i) 50.7 kg 20(a)(ii) 4.20 kg

20(b) Class B has a greater spread of weight.

Class A results are more consistent.

Accept any logical answer

CCHY S4/5 EM Prelim 1 P2 2011 Page 1 of 8

Preliminary Examination I Secondary 4 Express/ 5 Normal Academic

Candidate

Name Register No

ELEMENTARY MATHEMATICS PAPER 2 Date: 15th July 2011

(4016/2) Duration: 2 h 30 min

Additional Materials: 1 sheet of graph paper

READ THESE INSTRUCTIONS FIRST

Setter: Ms Lin Xiaoying


CHc CHUNG CHENG HIGH SCHOOL YISHUN

义顺

Write your name, register number and class in the space provided above.

Write in dark blue or black pen.

You may use a pencil for any diagrams or graphs.

Do not use paper clips, highlighters, glue or correction fluid / tape.

Answer all questions.

If working is needed for any question, it must be shown with the answer.

Omission of essential workings and units 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 correct to three significant figures.

At the end of the examination, staple the question papers with the graph paper.




/ 100

Class

[TURN OVER


Mathematical Formulae

Compound interest

Total amount =

nr

P

1001

Mensuration


Surface area of a sphere = 24 r

Volume of a cone = hr 2

3

1

Volume of a sphere = 3

3

4r


1


Sector area = 2

2


Trigonometry

C

c

B

b

A

a

sinsinsin

Acbcba cos))((2222

Statistics

Mean =

f

fx


22

f

fx

f

fx


1. (a) Given that ,66 22 qqpp where ,qp find the value of .qp [2]

(b) Solve the equation )2(7)2(5 xxx [2]

(c) The values of 7 numbers are (x – 1), (x + 2), (x – 2), (x + 8), (x – 1), (x + 5) and (x + 3).

(i) Find the value of x if the sum of the mode and the median of the numbers is 17. [2]

(ii) Hence, find

(i) the mean of the numbers, [2]

(ii) the standard deviation of the numbers. [2]

____________________________________________________________________________________

2. A man buys a few books for $24.50.

(a) Given that the price of each book is $x, write down an expression for the number of

books bought.

[1]

(b) Write down an expression for the number of books that can be bought for the same

amount of money if the price of each book is reduced by $4.

[1]

(c) If 4 more books can be bought at the reduced price, form an equation in x and show

that it reduces to 2x2 8x 49 = 0.

[3]

(d) Solve the equation 2x2 8x 49 = 0, giving your answers correct to 2 decimal places. [2]

(e) Another man who has $110 wants to buy as many of the books as possible at the

original price. Find the number of books that he can buy.

[1]

___________________________________________________________________________________

3. In the diagram, ABCE is a trapezium. A is the point (0, 3), B is the point (1, 1), C is the point (5, 3),

E is the point )2

13,3( hh and D is a point lying on the line AE.

Find

(a) the equation of the line AD, [2]

(b) the coordinates of point E. [2]

Given that

3

2

AE

DE,

(c) find the coordinates of point D, [2]

(d) find the area of ∆ DEC. [2]

(e) find the coordinates of point F where F is the point on the x – axis such that C, D and

F are collinear.

[2]

____________________________________________________________________________________

B (1, 1)

D

E

y

x

A (0, 3) C (5, 3)

B (1, 1)

D

E

y

x

A (0, 3) C (5, 3)


4. The diagram shows three jetties A, B and C at the same ground level along the coast of a bay.

B is 1.48 km along a straight coastline at a bearing of 032 from A. C is 1.92 km from A at a

bearing of 080.

(a) Calculate the distance between B and C. [2]

(b) A boat from C sails due west to reach a point on the coastline between A and B.

Calculate the distance travelled by the boat.

[2]

(c) A helicopter flies in a straight line over A towards B, keeping a constant altitude of 95

metres above the coast. Find its angle of elevation from C when it is nearest to C.

[3]

(d) At the moment when the helicopter is nearest to C, the pilot fires a pistol once.

The sound is recorded by a detector at A and by another at C. If sound travels at 300

m/s, calculate the lapse of time, in seconds, between the detection at A and at C.

[4]

____________________________________________________________________________________

5. The table below shows the distribution of apples in 100 boxes.

Number of apples 100 105 110 115 120 125

Number of boxes 10 10 20 30 15 15

(a) Find the probability that a box chosen at random contains at least 120 apples. [1]

(b) A box is chosen at random. Its contents are counted and replaced.

The box is then placed back among with the rest of the other boxes.

Another box is chosen. Find the probability that

(i) both boxes contain at least 120 apples, [1]

(ii) the total number of apples in the two boxes is 240, [2]

(iii) the first box contains at least 100 apples and the second box contains at least 125

apples.

[1]

____________________________________________________________________________________

N

1.92 km

1.48 km

A

C

B


6. A container in the shape of a circular cone is filled with water to a height of 20 cm. It is given

that the height of the container is 30 cm and the diameter of the circular top is 20 cm.

[Use ].142.3

(a) Calculate the amount of water, in litres, in the container, leaving your answer in terms

of .

[3]

(b) The container is now inverted as shown in the diagram below.

(i) Find the depth, d, of the water. [4]

(ii) Find the radius, x, of the upper surface of the water. [1]

(c) In another spherical container, water is poured in until it reaches a depth of 24 cm measured

from the bottom of the container.

Given that the diameter of the container is 30 cm, find the surface area of the water

that is not in contact with the container.

[3]

____________________________________________________________________________________

30 cm

20 cm

20 cm

30 cm

20 cm

20 cm

24 cm24 cm

x cm

d cm

x cm

d cm


7. A departmental store was having a 2-day sale. All items were sold at a discount of 25%. Items that

were not sold on the first day were given a further discount of 20% from the previous day’s price.

(a) The selling price of a handbag was $141 on the second day of the sale.

Find the original selling price of the bag before the sale.

[2]

(b) On the first day of the sale, Eric saw a mattress and he bought it on hire purchase. The

original selling price was $2699 and he paid a deposit of 20% of the discounted price

followed by 24 monthly instalments of $80.50.

Find the extra cost of buying the mattress on hire purchase as a percentage of the

discounted price.

[3]

(c) On the second day of the sale, Alice bought the same mattress. To pay for it, she

borrowed the whole cost to be repaid at the end of 3 years with compound interest of

6.5% per year. Calculate the amount of interest that Alice had to pay.

[3]

____________________________________________________________________________________

8. (a) In the diagram, a small circle with centre O intersects a large circle at A and B. PBR

and AQR are straight lines and .75BQR

Find, showing your reasons clearly,

(i) ,OAB [2]

(ii) .PBQ [2]

(b) The diagram shows two concentric circles with centre O. The diameters of the circles

are 32 cm and 50 cm respectively. PAQ is a tangent to the inner circle at A.

Calculate

(i) POQ in radians, [2]

(ii) the area of the shaded region. [2]

____________________________________________________________________________________

X Y

O

P A Q

X Y

O

P A Q

Q

A

O

P

B

R

75

Q

A

O

P

B

R

75


9. (a) In the diagram below, ADC is a straight line, BC = 10 cm and AC = 26 cm.

Find

(i) the length of AB, [2]

(ii) ,BAC [2]

(iii) the length of BD. [2]

(b) In triangle PQR, ,27PQR ,90PTQ QT = 3x cm and RT = x cm.

Calculate .QPR

[4]

____________________________________________________________________________________

10. (a) P =

40

12 and Q =

4

10

2

1x

[2]

Find the value of x given that PQ is the identity matrix.

(b) The table shows the number of bottles of soft drinks sold from four vending machines.

Soft drink

Machine P Q R S

A 80 70 35 40

B 25 68 57 15

C 90 42 81 43

D 71 53 66 64

Sale price of

each bottle $0.90 $1.20 $1.50 $1.80

Using matrix multiplications twice, find the total amount collected from the four

vending machines.

[3]

____________________________________________________________________________________

A

B

C

D

A

B

C

D

P

QT R

3x x

27

P

QT R

3x x

27


11. Answer the whole of this question on a sheet of graph paper.

The variables x and y are connected by the equation .512

x

xy

Some corresponding values of x and y are given in the table below.

x 1 1.5 2 3 4 5 6 7 8

y a 4.5 3 2 2 2.4 3 3.7 b

(a) Calculate the value of a and of b. [1]

(b) Using a scale of 2 cm to represent 1 unit on both axes, draw the graph of

512

x

xy for 1 x 8.

[4]

(c) Use your graph to find the value of x when the gradient of the curve 512

x

xy is [1]

equal to zero.

(d) By drawing a tangent, estimate the coordinates of the point P on the curve

where the gradient of the curve is – 2.

[2]

(e) The line y = kx touches the curve 5

12

xxy at point Q . [3]

By drawing a suitable straight line on the same axes, use your graph to find the

positive value of k and the coordinates of the point Q.

(f) By drawing a suitable straight line on the same axes, solve the equation

.01312

3

5

xx

[3]

__________________________________[End of Paper]______________________________________

B

D

E

y

x

A C

B

D

E

y

x

A C


Preliminary Examination I Secondary 4 Express/ 5 Normal Academic

Candidate

Name Register No

ELEMENTARY MATHEMATICS PAPER 2 Date: 15th July 2011




Setter: Ms Lin Xiaoying


CHc CHUNG CHENG HIGH SCHOOL YISHUN

义顺









If the degree of accuracy is not specified in the question, and if the answer is not exact, give the

answer correct to three significant figures. Give answers in degrees to one decimal place.





/ 100

Class

[TURN OVER

Qn 1 – 3: Mrs Lynn Goh Qn 4 – 5: Ms Lin X Y Qn 6 – 7: Ms Tan M M Qn 8 – 9: Mr Vijay Qn 10 – 11: Mr Terence Poh



Compound interest

Total amount =

nr

P

1001

Mensuration




3

1


3

4r


1


Sector area = 2

2


Trigonometry

C

c

B

b

A

a

sinsinsin

Acbcba cos))((2222

Statistics

Mean =

f

fx


22

f

fx

f

fx


1. (a) Given that ,66 22 qqpp where ,qp find the value of .qp [2]

(b) Solve the equation )2(7)2(5 xxx [2]

(c) The values of 7 numbers are (x – 1), (x + 2), (x – 2), (x + 8), (x – 1), (x + 5) and (x + 3).

(i) Find the value of x if the sum of the mode and the median of the numbers is 17. [2]

(ii) Hence, find

(a) the mean of the numbers, [2]

(b) the standard deviation of the numbers. [2]

____________________________________________________________________________________

1. (a)

6or (rejected) 0

0]6))[((

)(6))((

)(6

66

22

22

qpqp

qpqp

qpqpqp

qpqp

qqpp

(b) )2(7)2(5 xxx

5

21or 2

0)75)(2(

0)2(7)2(5

xx

xx

xxx

(c) (i) Rearranging: (x – 2), (x – 1), (x – 1), (x + 2), (x + 3), (x + 5), (x + 8)

Mode = x – 1

Median = x + 2

(x – 1) + (x + 2) = 17

x = 8

(ii) (a) 6, 7, 7, 10, 11, 13, 16

Mean = 7

70

= 10

(ii) (b) Standard Deviation = 210

7

780

= 38.3 (3 s.f)

M1

A1

M1

A1

M1

A1

M1

A1

M1

A1

B2: using calculators

B2: using calculators


2. A man buys a few books for $24.50.

(a) Given that the price of each book is $x, write down an expression for the number of

books bought.

[1]

(b) Write down an expression for the number of books that can be bought for the same

amount of money if the price of each book is reduced by $4.

[1]

(c) If 4 more books can be bought at the reduced price, form an equation in x and show

that it reduces to 2x2 8x 49 = 0.

[3]

(d) Solve the equation 2x2 8x 49 = 0, giving your answers correct to 2 decimal places. [2]

(e) Another man who has $110 wants to buy as many of the books as possible at the

original price. Find the number of books that he can buy.

[1]

___________________________________________________________________________________

(a) No. of books bought =

x

50.24

(b) No. of books bought =

4

50.24

x

(c) 4

50.24

4

50.24

xx

)4(4)4(5.245.24 xxxx

xxxx 164985.245.24 2

098164 2 xx

04982 2 xx (shown)

(d) 04982 2 xx

7.34or (rejected) 34.3

4

4568

)2(2

)49)(2(4)8()8( 2

x

x

x

(e) No. of books bought

= 3385.7

110

= 14.989

14

B1

B1

M1: Forming correct equation

M1: Multiplying by x(x – 4) throughout

M1: Correct simplification

M1

A1

B1


3. In the diagram, ABCE is a trapezium. A is the point (0, 3), B is the point (1, 1), C is the point (5, 3),

E is the point )2

13,3( hh and D is a point lying on the line AE.

Find

(a) the equation of the line AD, [2]

(b) the coordinates of point E. [2]

Given that

3

2

AE

DE,

(c) find the coordinates of point D, [2]

(d) find the area of ∆ DEC. [2]

(e) find the coordinates of point F where F is the point on the x – axis such that C, D and

F are collinear.

[2]

____________________________________________________________________________________

3. (a) Gradient of the line BC = 2

1

15

13

Line AD is parallel to line BC: Same gradient

Equation of the line AD: 32

1 xy

(b) Using )

2

13,3( hhE and A (0, 3),

Alternatively

Use 32

1 xy and )

2

13,3( hhE

Gradient of the line AE = 2

1

3

32

13

h

h

hh 3)32

13(2

hh 367

2

11

64

h

h

Therefore, )4

15,

2

14(E

B (1, 1)

D

E

y

x

A (0, 3) C (5, 3)

B (1, 1)

D

E

y

x

A (0, 3) C (5, 3)

M1

A1

M1

A1


(c) AEDE

3

2

4

33,

2

11

4

33

2

11

4

15

2

14

3

0

3

2

4

15

2

14

)(3

2

)(3

2

D

OD

OD

OEAOOEOD

OEAOOEDO

(d) Area of ∆ DEC =

2

1(base)(height)

=

2

1(DE)(AB)

= 2

1 22 5.13 22 21

= 2

1 25.11 5

= 3.75 units2

(e) F (x, 0)

Collinear => CFkDC

3

5

4

32

13

3

5

3

5

4

33

2

11

xk

xk

25.0k , 19x

F (19, 0)

M1

A1

M1

A1

M1

A1


4. The diagram shows three jetties A, B and C at the same ground level along the coast of a bay.

B is 1.48 km along a straight coastline at a bearing of 032 from A. C is 1.92 km from A at a

bearing of 080.

(a) Calculate the distance between B and C. [2]

(b) A boat from C sails due west to reach a point on the coastline between A and B.

Calculate the distance travelled by the boat.

[2]

(c) A helicopter flies in a straight line over A towards B, keeping a constant altitude of 95

metres above the coast. Find its angle of elevation from C when it is nearest to C.

[3]

(d) At the moment when the helicopter is nearest to C, the pilot fires a pistol once.

The sound is recorded by a detector at A and by another at C. If sound travels at 300

m/s, calculate the lapse of time, in seconds, between the detection at A and at C.

[4]

____________________________________________________________________________________

4. (a) Using Cosine Rule,

440138.1

48cos)92.1)(48.1(292.148.1 222

BC

BC

44.1BC km (3 s. f)

(b) Let the point be D.

) of sum ( 1221048180 ADC

Using Sine Rule,

s.f) (3 km 68.1

682497.1

122sin

92.1

48sin

CD

CD

CD

N

1.92 km

1.48 km

A

C

B

D

32

80

48

E

10

100

0.095 km

Helicopter

N

1.92 km

1.48 km

A

C

B

D

32

80

48

E

10

100

0.095 km

Helicopter

M1

A1

M1

A1


(c) Let the shortest distance be CE.

Looking at ACE,

km 42684.1

48sin92.148sin

CE

ACCE

Let the angle of elevation be .

d.p) (1 8.3

42684.1

095.0tan

(d) Let the location of the helicopter be H.

Using Pyth’s Thm,

s.f) 3 ( sec 473.0

29412795.4766657184.4

3.03.0

timeof Lapse

km/s 0.3 m/s 300 Speed

km 429997155.1

095.0

Thm, sPyth' Using

km 288238385.1

095.0

Thm, sPyth' Using

km 284730764.1

92.1

222

222

222

AHCH

CH

CECH

AH

AEAH

AE

CEAE

M1

A1

M1

M1

M1

M1

A1


5. The table below shows the distribution of apples in 100 boxes.

Number of apples 100 105 110 115 120 125

Number of boxes 10 10 20 30 15 15

(a) Find the probability that a box chosen at random contains at least 120 apples. [1]

(b) A box is chosen at random. Its contents are counted and replaced.

The box is then placed back among with the rest of the other boxes.

Another box is chosen. Find the probability that

(i) both boxes contain at least 120 apples, [1]

(ii) the total number of apples in the two boxes is 240, [2]

(iii) the first box contains at least 100 apples and the second box contains at least

125 apples.

[1]

____________________________________________________________________________________

5. (a) P(at least 120 apples) 10

3

100

30

(b) (i) P(both boxes contain at least 120 apples)

100

9

10

3

10

3

(ii)

P(total number of apples in the two boxes is 240)

= P(1st box: 115 and 2nd box: 125) + P(1st box: 125 and 2nd box: 115)

+ P(1st box: 120 and 2nd box: 120)

80

9

100

15

100

15

100

15

100

302

(iii) P(1st box: at least 100 apples and 2nd box: at least 125 apples)

20

3

100

15

100

100

B1

B1

M1

A1

B1


6. A container in the shape of a circular cone is filled with water to a height of 20 cm. It is given

that the height of the container is 30 cm and the diameter of the circular top is 20 cm.

[Use ].142.3

(a) Calculate the amount of water, in litres, in the container, leaving your answer in terms

of .

[3]

(b) The container is now inverted as shown in the diagram below.

(i) Find the depth, d, of the water. [4]

(ii) Find the radius, x, of the upper surface of the water. [1]

(c) In another spherical container, water is poured in until it reaches a depth of 24 cm measured

from the bottom of the container.

Given that the diameter of the container is 30 cm, find the surface area of the water

that is not in contact with the container.

[3]

____________________________________________________________________________________

30 cm

20 cm

20 cm

30 cm

20 cm

20 cm

24 cm24 cm

x cm

d cm

x cm

d cm


6. (a) Using similar figures,

cm 3

20 Radius

3

40)20(

3

2

30

20

20

D

D

Amount of water

litres 27

8

cm 27

8296

cm 203

20

3

1

3

3

2

(b) (i) Using similar figures,

3

30

30

30

20

2

dx

dx

Amount of water = Volume of big cone – Volume of small cone

s.f) (3 cm 32.31900030

1900030

27

19703

27

30

303

30

3

11000

27

8296

303

13010

3

1

27

8296

3

3

3

2

22

dd

d

d

dd

dx

(b) (ii) s.f) (3 cm 89.83

30

dx

(c) Using Pyth’s Thm,

s.f) 3(cm 452or cm 448.452

)144)(142.3(

area Surface

cm 12

915

2 2

2

222

πr

r

r

D

20 cm

30 cm

20 cm

D

20 cm

30 cm

20 cm

2x cm

d cm

30 cm

20 cm

(30 – d) cm

2x cm

d cm

30 cm

20 cm

(30 – d) cm

15

1515

r

O

9

15

1515

r

O

9

B1

A1

M1

A1

M1

B1

M1

B1

A1

M1


7. A departmental store was having a 2-day sale. All items were sold at a discount of 25%. Items that

were not sold on the first day were given a further discount of 20% from the previous day’s price.

(a) The selling price of a handbag was $141 on the second day of the sale.

Find the original selling price of the bag before the sale.

[2]

(b) On the first day of the sale, Eric saw a mattress and he bought it on hire purchase. The

original selling price was $2699 and he paid a deposit of 20% of the discounted price

followed by 24 monthly instalments of $80.50.

Find the extra cost of buying the mattress on hire purchase as a percentage of the

discounted price.

[3]

(c) On the second day of the sale, Alice bought the same mattress. To pay for it, she

borrowed the whole cost to be repaid at the end of 3 years with compound interest of

6.5% per year. Calculate the amount of interest that Alice had to pay.

[3]

____________________________________________________________________________________

7. (a) Price on first day

25.176$

141$80

100

Price before sales

235$

25.176$75

100

(b) Price of mattress

25.2024$

2699$75.0

20% of discounted price = 85.404$

24 monthly instalments 1932$50.80$24

Hire purchase price 85.2336$85.404$1932$

Required percentage s.f) (3 %4.15%10025.2024

60.312

(c) Price of mattress 40.1619$25.2024$8.0

Total amount 153623.1956$100

5.6140.1619$

3

Interest paid d.p) (2 75.336$40.1619$153623.1956$

A1

M1

A1

M1

A1

A1

M1

A1


8. (a) In the diagram, a small circle with centre O intersects a large circle at A and B. PBR

and AQR are straight lines and .75BQR

Find, showing your reasons clearly,

(i) ,OAB [2]

(ii) .PBQ [2]

(b) The diagram shows two concentric circles with centre O. The diameters of the circles

are 32 cm and 50 cm respectively. PAQ is a tangent to the inner circle at A.

Calculate

(i) POQ in radians, [2]

(ii) the area of the shaded region. [2]

____________________________________________________________________________________

8. (a) (i) line)straight aon s( 10575180 AQB

) ncecircumfereat s2 centreat (

210)(2Reflex

AQBAOB

)point aat s( 150210360 AOB

) isos. of s (base 152

150180

OAB

(ii) segment) opp.in s( 30180 AOBARB

) of s(ext. 1053075 PBQ

(b) (i) 25

16cos AOQ

s.f) (3 rad 876.0)(2 AOQPOQ

(ii) Area of the shaded region

= Area of ∆POQ – Area of sector OXY

s.f) (3 cm 5.74

)16)(16(2

1sin)25)(25(

2

1

2

POQPOQ

Q

A

O

P

B

R

75

105

210 150

30

Q

A

O

P

B

R

75

105

210 150

30

X Y

O

P A Q

25

16X Y

O

P A Q

25

16

M1

A1

M1

A1

M1

A1

M1

A1


9. (a) In the diagram below, ADC is a straight line, BC = 10 cm and AC = 26 cm.

Find

(i) the length of AB, [2]

(ii) ,BAC [2]

(iii) the length of BD. [2]

(b) In triangle PQR, ,27PQR ,90PTQ QT = 3x cm and RT = x cm.

Calculate .QPR

[4]

____________________________________________________________________________________

9. (a) (i) Using Pyth’s Thm,

cm 24

1026 222

222

AB

AB

BCACAB

(b)

27tan3

327tan

xPT

x

PT

d.p) (1 96.2

193.339027180

193.33

27tan3

1

27tan3

tan

QPR

RPT

x

x

PT

xRPT

(ii)

26

10sin

AC

BCBAC

d.p) (1 6.22 BAC

(iii) Area of ∆ABC = ))((

2

1ABBC

s.f) (3 cm 23.9

)24)(10()26)((

))((2

1))((

2

1

BD

BD

ABBCACBD

A

B

C

D

A

B

C

D

P

QT R

3x x

27

P

QT R

3x x

27

M1

A1

M1

A1 M1

A1

B1

M1

A1


10

. (a) P =

40

12 and Q =

4

10

2

1x

[2]

Find the value of x given that PQ is the identity matrix.

(b) The table shows the number of bottles of soft drinks sold from four vending machines.

Soft drink

Machine P Q R S

A 80 70 35 40

B 25 68 57 15

C 90 42 81 43

D 71 53 66 64

Sale price of

each bottle $0.90 $1.20 $1.50 $1.80

Using matrix multiplications twice, find the total amount collected from the four

vending machines.

[3]

____________________________________________________________________________________

10. (a) PQ = I

40

12

4

10

2

1x

=

10

01

104

121 x

=

10

01

8

1

04

12

x

x

(b)

1st Product:

70.341

30.330

60.216

50.280

80.1

50.1

20.1

90.0

64665371

43814290

15576825

40357080

2nd Product:

10.1169

70.341

30.330

60.216

50.280

1111

The total amount is $1169.10

M1

A1

B1

A1

B1


11

.

Answer the whole of this question on a sheet of graph paper.

The variables x and y are connected by the equation .512

x

xy

Some corresponding values of x and y are given in the table below.

x 1 1.5 2 3 4 5 6 7 8

y a 4.5 3 2 2 2.4 3 3.7 b

(a) Calculate the value of a and of b. [1]

(b) Using a scale of 2 cm to represent 1 unit on both axes, draw the graph of

512

x

xy for 1 x 8.

[4]

(c) Use your graph to find the value of x when the gradient of the curve 512

x

xy is [1]

equal to zero.

(d) By drawing a tangent, estimate the coordinates of the point P on the curve

where the gradient of the curve is – 2.

[2]

(e) The line y = kx touches the curve 5

12

xxy at point Q . [3]

By drawing a suitable straight line on the same axes, use your graph to find the

positive value of k and the coordinates of the point Q.

(f) By drawing a suitable straight line on the same axes, solve the equation

.01312

3

5

xx

[3]

__________________________________[End of Paper]______________________________________

11. (a) a = 8 and b = 4.5 B1 (b)

Scale and axes: B1

Plotting of points: B2

Smooth graph and label of eqn: B1

(c) x = 3.4 to 3.6 B1

(d) Drawing of tangent B1

P(2, 3) B1

(f) Solve 013

12

3

5

xx

083

25

12

x

xx

Plot xy3

28

x 0 3 6

y 8 6 4

Solutions are x = 1.10.1 and 6.70.1

(e) 0) intercept -( ykxy

0) (0, through passing line a Draw M1

Q(5, 2.4) A1

48.05

4.2k A1

A1 A1: Plot and label line with eqn

M1

CCHY Preliminary Examination II(2011) Mathematics 4E5N pg 1 of 16

\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\


Candidate


Mathematics 4016/1 Date: 22/08/ 2011 Duration: 2 hour


Setter : Poh Eng Hua, Terence



/ 80

Write your name, register number and class on the space provided above. Write in dark blue or black pen. You may use a pencil for any diagrams or graphs. Do not use staplers, 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 and units of measurement will result in loss of marks. Calculators should be used where appropriate.

Simplify your answers to their simplest form. If the answer is not exact,

give the answer correct to three significant figures or in fraction where

applicable. Give answers in degrees correct 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.


.


1. (a) Find the HCF of 45, 30 and 20.

(b) Three joggers ran in a circular route of 5.4km at a speed of 120m/min, 180 m/min and 270 m/min respectively. Given that the joggers started at 0700, when will the three of them next meet again? 1b) Time to complete one round (mins) : 45 , 30 , 20 [M1] (or equivalent step)

Answer: (a) ……180.[B1]……………[1] (b) .. ……1000..……………..[2]

2. (a) Find the range of x that satisfy the inequality of 2

733

2

35

xx

x.

(b) Use your answer in part (a) and the information that given 62 y , find the

smallest value of xyy

x

2

.

(a) x < 4.5 , x1 [m1] (either one of the compound inequality)

Answer: (a) …… 541 .x …[A1]……[2]

(b) .. …6

55 [A1]..…………….. [1]


3. Tap A can fill up the tank shown below in 3 minutes while Tap B can fill up the tank in 7 minutes. Tap C is used to drain the tank and it can drain the tank completely in 5 minutes.

(a) Given that all the three taps are turned on at the same time, how long does it take for the tank to be filled up completely? (b) Given that Tap A’s flow rate is adjusted so that the tank can be filled up in 5 minutes when all the 3 taps are turned on. Find out how long does it take for Tap A to fill up the tank now.

Tap A Tap B

Tap C

(a) fill the 105

29tank in one min [m1] (or equivalent)

Answer: (a) …3.62 mins (29

183 )….minutes [2]

(b) .. 3.89 mins (9

83 )….minutes [1]


4. (a) Solve the following simultaneous equations,

yxy4

34

2

1 ,

xy 363

(b) A line cut the x-axis at x = 3 and passes through the point (-1, 4). Find the equation of the straight line.

Answer: (a) x = ……….., y = ………….. [2] (b) y = …………………………. [2]


5. The exchange rate of a money changer is shown in the table below

(a) Calculate the amount of Euro dollars received in exchange of S $200.

(b) Calculate the amount of Singapore dollars received in exchange of 25000 Thai bahts.

Answer: (a) €…………….......................[1] (b) S $………..…………………[2]

6. The number series -3, -1, 3, 11, 27, ..… is represented in the table below

(a) Find the value of p and q. (b) Find in terms of k, the value of r and s.

n Value of nth term Difference between nth term and (n-1)th term

1 -3 -

2 -1 2

3 3 4

4 11 8

5 27 16

6 q p

k s r

Answer: (a) p = ………, q=…………… [1]

Foreign Currency

Exchange Rate (SGD to one unit of foreign currency)

Foreign Currency

Exchange rate (SGD to 100 units of foreign currency)

Buy Sell Buy Sell

US $ 1.234 1.351 Thai Baht 4.156 4.356

Euro Dollar(€)

1.983 2.116 Japanese Yen (¥)

1.453 1.475


(b) r =……….., s = …………..[2] 7. (a) A TV was priced at $999. Payment was made with a down payment of $250

and 2 years of monthly instalments of $40. Find the interest charged for the hire purchase.

(b) $1500 is deposited into a bank with an annual interest rate of 1.5% which is

compounded every quarter of the year. Find the interest earned after 2.5 years. Answer: (a) $....................................... [2]

(b) $................................ …… [1]

8. A survey is carried out to find out the number of babies in a household and the

result was shown in the table below.

Number of babies

0 1 2 3 4

Number of household

5 10 10 x 2

(a) Find the largest value of x when the median number of babies in a household is 1. (b) Find the value of x when the median number of babies in a household is 1.5.

Answer: (a) …………………………..… [1]


(b) ……..……………………… [1] 9. (a) From the list of numbers below, write down the irrational numbers.

, 729 , .

. 330 , 3

4 , 1.5

(b) Estimate the value of 501149548

965245818 2

..

..

correct to 1 significant figure.

Show your working steps clearly.

Answer: (a)………………………….. …[1] (b)………………………………[2]

10. (a) Solve 0942 2 xx

(b) Make y the subject of the equation zyx

1112

Answer: (a) …………………………..… [2]


(b) …………………………….. [2] 11. Sound travels 330 metres in 1 second.

(a) Express the speed of sound in kilometres per second, giving your answer in

standard form.

(b) A ship wanted to use ultrasonic sound wave to detect the depth of the sea from the

sea level. It sent out a sound wave and received its echo in 9.5 x 106 microseconds.

Find the depth of the sea from the sea level, give your answer in standard form.

Answer: (a) ……………………….km/s [1]

(b)…………………………...m [2]

12. A, B, C , D, …. are some of the vertices of a regular polygon. Given that reflex ABC= 220o, calculate the number of sides the polygon has.

Answer: ……………………sides [2]

D

A

B

C

220o


13. (a) Simplify 0

4

5

2

1

4

353 2

cab

cba in the form apbqcr where p, q and r are rational numbers.

(b) Solve 10004

5161

221

x

xx

Answer: (a) ……………………………. [2]

(b) …. …………………………. [2]

14. A square pyramid with a base length 30cm has its vertex E directly above point D. Given that the volume of the pyramid is 12000 cm3. (a) Find the height, h, of the pyramid. (b) Given that surface DCE, DAE, ABE and CBE

are right angle triangles , find the total surface area of the pyramid.

Answer: (a) …………………………… [1]

A 30 cm B

C D

E

h cm

30 cm


(b) …. ………………………… [2] 15. ABC is a right- angled triangle in which cm , 1290 BCABC and cm 13AC .

The point D lies on BC produced. Write down, as a fraction, the value of

(a) xsin

(b) ACDcos

(c) x90tan

Answer: (a) …………………………….. [2]

(b) …. ………………………….. [1]

(c) …. ………………………….. [1]

16. The point (1, 1) is marked on each diagram in the answer space. On these diagrams, sketch the graphs of

(a) 21 xy (b) 2

2

xy

Answer:

(a)

[2]

D

A

B C

x

13 cm

12 cm

y

x 0


(b)

[2]

17. A map is drawn to a scale of 1 : 50 000. (a) A road is represented by a line of length 1.8 cm on the map. Calculate the actual length of the road, giving your answer in kilometres.

(b)The actual area of an airport is 3.4 km2. Calculate the area of the airport on the map, giving your answer in square centimetres.

Answer: (a) . .………………….…km [1]

(b) ………………………..cm2 [2]

y

x 0


18. In the diagram below, O is the centre of the circle and OD is perpendicular to CE.

Show your reasons clearly in the answer space below and name a pair of congruent triangles. [2]

E

D

C

O

A

19. PQRST is a pentagon inscribed in a circle with PS as the diameter and centre O.

Given that POT = 65 and RPS = 26, find

(a) PRT,

(b) PQR.

Answer: (a) ………………………….…. [1]

P

Q

R

S

T

O

65

26


(b) ……………………………...[2]

20. Given that

10

32M and

51

50

34

N calculate

(a) NM

(b) 2M2 Answer: (a) ………………………....…. [1] (b) …………………………..... [2] 21. (a) Define the shaded region using set notation.

Answer: (a) ……………………….. …. [1]

A B


(b) Shade the region define by ABB' [2]

22. There are six red cards and three blue cards in a box. (a) David draws two cards from the box at random without replacement. Find the

probability that the two cards are of the same colour.

(b) Joe draws one card at a time from the box at random with replacement, until he gets a blue card. Find the probability that he will be successful exactly on this third draw.

Answer: (a) …………………………... [1]

A B


(b) …………………………… [2]

23. Factorise

(a) xyyx 22 103

(b) 1644 22 yxyx

Answer: (a) ………………………….……….. [2]

(b) …………...……………….…….. [2]

24. The diagram is the distance-time graph for the first 22 seconds of a journey.

(a) Find the distance travelled when time is 20 seconds.

(b) Given that the speed increased uniformly in the first 10 seconds, sketch the speed-time graph for the first 22 seconds on the answer space given below.

Distance (m)

(b) [2] Speed ( m/s )

0 10 16 22 time (s)

0 10 16 20 time (s)

20

36

24


Answer: (a) …………………………..… [1] 25. (a) Using your construction set, construct triangle PQR where base PQ = 7cm,

PR = 7cm and 60PQR . PQ is given below. [2]

(b) Construct a circle which has PQ, PR and QR as its tangents. [2]

(c) Measure the diameter of the circle.

Answer: (c) ………………………..cm [1]

Q P

S


\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\


Candidate


Mathematics 4016/1 Date: 22/08/2011 Duration: 2 hour


Setter: Poh Eng Hua, Terence



/ 80

Write your name, register number and class on the space provided above. Write in dark blue or black pen. You may use a pencil for any diagrams or graphs. Do not use staplers, 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 and units of measurement will result in loss of marks. Calculators should be used where appropriate. Simplify your answers to their simplest form. If the answer is not exact, give the answer correct to three significant figures or in fraction where applicable. Give answers in degrees correct 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.


.


1. (a) Find the HCF of 45, 30 and 20.

(b) Three joggers ran in a circular route of 5.4 km at a speed of 120m/min, 180 m/min and

270 m/min respectively. Given that the joggers started at 0700, when will the three of them

next meet again?

(a) 5345 2

5 HCF

5220

53230

2

(b)

They will meet again 180 min (LCM = 532 22 ) later => 0700 + 0300 = 1000 h

2. (a) Find the range of x that satisfy the inequality of 2

733

2

35

xx

x.

(b) Use your answer in part (a) and the information that given 62 y , find the smallest

value of xyy

x

2

.

(a) 32

35

x

x

x < 4.5 ,

(b) Smallest value of 6

55)6)(1(

6

)1( 22

xyy

x [B1]

Jogger Speed Time taken to complete one round (5.4 km)

A 120 m/min 45120

5400

S

DT min

B 180 m/min 30180

5400

S

DT min

C 270 m/min 20270

5400

S

DT min

[M1]

x

xx

1

2

733

[M1] (either one of the compound inequality)

Answer: 5.41 x [A1]

[B1]


3. Tap A can fill up the tank shown below in 3 minutes while Tap B can fill up the tank in 7

minutes. Tap C is used to drain the tank and it can drain the tank completely in 5 minutes.

(a) Given that all the three taps are turned on at the same time, how long does it take for

the tank to be filled up completely?

(b) Given that Tap A’s flow rate is adjusted so that the tank can be filled up in 5 minutes

when all the 3 taps are turned on. Find out how long does it take for Tap A to fill up

the tank now.

Tap A Tap B

(a) Tap A and Tap B can fill

21

10

7

1

3

1of the tank in 1 min.

Tap C can drain 5

1of the tank in 1 min.

=> In 1 min,

105

29

5

1

21

10of the tank will be filled with water. [M1]

Therefore, amount of time needed = f) s (3 mins 3.62or 29

183

105

29

1

[A1]

(b) Let x be the amount of time taken by Tap A.

Amount of time needed = 5 mins

mins 3.89or 9

83

9

35

35

91

mins 5

5

1

7

11

1

x

x

x

Tap C

[B1]


4. (a) Solve the following simultaneous equations.

yxy4

34

2

1 ,

xy 363

(b) A line cut the x-axis at x = 3 and passes through the point (-1, 4).

Find the equation of the straight line.

(a)

(1) --------- 0162

042

1

4

1

4

34

2

1

xy

xyyxy

(2) --------- 2 363 xyxy

Subst. (2) into (1)

14 Therefore,

12

01642

016)2(2

x

y

yy

yy

(b) (3, 0) and (1, 4) lie on the line.

Gradient = 131

04

and y-intercept = 3 [M1]

Line of equation is y = x + 3 [A1]

[B2]


5. The exchange rate of a money changer is shown in the table below

(a) Calculate the amount of Euro dollars received in exchange of S$200.

(b) Calculate the amount of Singapore dollars received in exchange of 25000 Thai Bahts.

(a) Moneychanger sells Euro dollars

S$2.116 = €1

Amount of Euro dollars received = 116.2

200€ 94.52 (2 d.p) [B1]

(b) Moneychanger buys Thai Bahts.

S$4.156 = 100 Thai Bahts

Amount of Singapore dollars received

1039$

156.4100

25000

6. The number series -3, -1, 3, 11, 27, ..… is represented in the table below

(a) Find the value of p and q.

(b) Find in terms of k, the value of r and s.

(b) r = 52 k ,

s = … 12 k ……….[B2]

Foreign

Currency

Exchange Rate

(SGD to one unit of

foreign currency)

Foreign

Currency

Exchange rate

(SGD to 100 units of

foreign currency)

Buy Sell Buy Sell

US $ 1.234 1.351 Thai Baht 4.156 4.356

Euro Dollar(€) 1.983 2.116 Japanese Yen (¥) 1.453 1.475

n Value of nth term Difference between

nth term and (n – 1)th term

1 -3 -

2 -1 2

3 3 4

4 11 8

5 27 16

6 q p

k s r

[M1]

[A1]

12 22 32 42 52

(a) p = 32, q = 59 [B1]

(b) 32 Term 1 1st a

5a

Therefore, 52 ks

1

1

1

1

1th

th

2

)12(2

22

)52 (52

52 term1

52 term

k

k

kk

kk

k

k

r

r

r

r

n

n

[B2]


7. (a) A TV was priced at $999. Payment was made with a down payment of $250 and 2

years of monthly instalments of $40. Find the interest charged for the hire purchase.

(b) $1500 is deposited into a bank with an annual interest rate of 1.5% which is

compounded every quarter of the year. Find the interest earned after 2.5 years.

(a) Hire purchase price = ($40 × 24) + $250 = $1210 [M1]

Interest charge = $1210 $999 = $211 [A1]

(b) A = 21.1557$100

4

5.1

11500

3

125.2

Interest earned = $1557.21 - $1500 = $57.21 (2 d.p) [B1]

8. A survey is carried out to find out the number of babies in a household and the result was

shown in the table below.

Number of babies 0 1 2 3 4

Number of household 5 10 10 x 2

(a) Find the largest value of x when the median number of babies in a household is 1.

(b) Find the value of x when the median number of babies in a household is 1.5.

(a) Largest value of x = 2 [B1]

(b) Value of x = 3 [B1]

9. (a) From the list of numbers below, write down the irrational numbers.

, 729 , .

33.0 , 3

4, 1.5

(b) Estimate the value of 501.14954.8

965.2458.18 2

correct to 1 significant figure.

Show your working steps clearly.

(b) s.f) 2 correct to number,each off (round 150.9

0.318

501.14954.8

965.2458.18 22

= 0.5


10. (a) Solve 0942 2 xx

(b) Make y the subject of the equation zyx

1112

(a) 0942 2 xx

f) s (3 3.35or 35.1

4

884

)2(2

)9)(2)(4()4()4(

0942

2

2

x

x

x

xx

(b) zyx

1112

xz

xzy

xz

xzy

xz

xz

y

zxy

2

2

2

1

111

11. Sound travels 330 metres in 1 second.

(a) Express the speed of sound in kilometres per second, giving your answer in standard form.

(b) A ship wanted to use ultrasonic sound wave to detect the depth of the sea from the sea level.

It sent out a sound wave and received its echo in 9.5 x 106 microseconds. Find the depth of

the sea from the sea level, give your answer in standard form.

(a) Speed = km/s10 3.3 km/s 33.01000

m/s 330 m/s 330 1- [B1]

(b) Time = 9.5 x 106 microseconds = 9.5 sec

Distance (to and fro) = Speed × Time

= 330 × 9.5 [M1]

= 3135 m

Depth of the sea = m 105675.12

3135 3 [A1]

[M1]

[A1]

Or equivalent: 22 xyxzzy [M1]

[A1]


12. A, B, C, D, …. are some of the vertices of a regular polygon. Given that reflex ABC = 220,

calculate the number of sides of the polygon.

Exterior angle = 40° [M1]

Sum of exterior angles = 360

Therefore, no. of sides = 940

360

[A1]

13. (a) Simplify 0

4

5

2

1

4

353 2

cab

cba in the form apbqcr, where p, q and r are rational numbers.

(b) Solve 10004

5161

221

x

xx

(a) 1

353 2

0

4

5

2

1

4

353 2 cba

cab

cba

2

3

2

5

3

1

2

1

353

2

2

1

353 2

cba

cba

cba

(b) 10004

5161

221

x

xx

33

22

2244

522

52

x

xx

332222 5252 xx [M1]

5.0322 xx [A1]

D

A

B

C

220

[M1]

[A1]


14. A square pyramid with a base length 30 cm has its vertex E directly above point D.

Given that the volume of the pyramid is 12000 cm3.

(a) Find the height, h, of the pyramid.

(b) Given that surface DCE, DAE, ABE and CBE are right angle triangles ,

find the total surface area of the pyramid.

(a) Volume of the pyramid = 12000 cm3

cm 40 Height

12000 height )3003(3

1

12000 height area) base(3

1

(b) Surface Area = )30502

1(2)3030()3040

2

1(2

= 3600 cm2

15. ABC is a right-angled triangle in which cm 12 ,90 BCABC and cm 13AC .

The point D lies on BC produced. Write down, as a fraction, the value of

(a) xsin

(b) ACDcos

(c) x90tan

(a) Using Pythagoras’ Theorem,

cm 51213 22 AB [M1]

13

5sin

AC

ABx [A1]

(b) 13

12coscos

AC

BCACBACD [B1]

(c) 5

22

5

12tan90tan

AB

BCBACx [B1]

16. The point (1, 1) is marked on each diagram in the answer space.

On these diagrams, sketch the graphs of

(a) 21 xy (b) 2

2

xy

CD

E

h cm

30 cm

A 30 cm B

CD

E

h cm

30 cm

A 30 cm B

CD

E

h cm

30 cm

A 30 cm B

CD

E

h cm

30 cm

A 30 cm B

[B1]

[M1]

[A1]

D

A

BC

x

13 cm

12 cm

y

x0

11

y

x0

11


17. A map is drawn to a scale of 1 : 50 000.

(a) A road is represented by a line of length 1.8 cm on the map. Calculate the actual length of

the road, giving your answer in kilometres.

(b) The actual area of an airport is 3.4 km2. Calculate the area of the airport on the map, giving

your answer in square centimetres.

(a) Scale = 1 : 50 000

= 1 cm : 0.5 km

Actual length of road = 1.8 × 0.5 km = 0.9 km [B1]

(b) Area scale = 1 cm2 : 0.25 km2 [M1]

Area on the map = 3.4 0.25 = 13.6 cm2 [A1]

18. In the diagram below, O is the centre of the circle and OD is perpendicular to CE. Show your

reasons clearly in the answer space below and name a pair of congruent triangles. [2]

OD is the common side

OE = OC (radius of circle)

DE = DC (perpendicular line from centre of circle bisect chord) [M1]

ODE ≡ ODC (SSS) [A1]

19. PQRST is a pentagon inscribed in a circle with PS as the diameter and centre O.

Given that POT = 65 and RPS = 26, find

(a) PRT,

(b) PQR.

(a)

5.322

65PRT ( nce)circumfereat 2 centreat

(b) )semicircle ain ( 90 PRS

PSR = 180 26 90 = 64° ) of sum ( [M1]

PQR = 180 64 = 116° segments) oppsite in ( s [A1]

P

Q

R

S

T°

O

65

26

O

A E

C

D


20. Given that M

10

32 and N

51

50

34

calculate

(a) NM

(b) 2M2

(a) NM =

51

50

34

82

50

98

10

32 [B1]

(b) 2M2 = 2

10

32

10

32 [M1]

=

20

68 [A1]

21. (a) Define the shaded region using set notation.

Answer: 'AB [B1]

(b) Shade the region define by ABB '

A B

[B2]

A B

A B


22. There are six red cards and three blue cards in a box.

(a) David draws two cards from the box at random without replacement. Find the probability

that the two cards are of the same colour.

(b) Joe draws one card at a time from the box at random with replacement, until he gets a blue

card. Find the probability that he will be successful exactly on this third draw.

(a) P(two cards are of the same colour)

= P(red, red) + P(blue, blue)

=

8

2

9

3

8

5

9

6

= 2

1 [B1]

(b) P(blue card on third draw)

= P(red, red, blue)

= 9

3

9

6

9

6 [M1]

= 27

4 [A1]

23. Factorise

(a) xyyx 22 103

(b) 1644 22 yxyx

(a) )2)(53(103 22 yxyxxyyx [B2]

(b) 1644 22 yxyx

1644 22 yxyx

2242 yx [M1]

)42)(42( yxyx [A1]

24. The diagram is the distance-time graph for the first 22 seconds of a journey.

(a) Find the distance travelled when time is 20 seconds.

(b) Given that the speed increased uniformly in the first 10 seconds, sketch the

speed-time graph for the first 22 seconds on the answer space given below.

(a) Gradient of line = 2

m 32

21620

24

d

d

[B1]


(b)

Answer :

25. (a) Using your construction set, construct triangle PQR where base PQ = 7cm,

PR = 7cm and 60PQR . PQ is given below. [2]

(b) Construct a circle which has PQ, PR and QR as its tangents. [2]

(c) Measure the diameter of the circle. [1]

20

36

24

0 10 16 22 time (s)

Distance (m)

0 10 16 22 time (s)

Speed (m/s)

2

3

2

60

P Q

R

(c) Diameter = 3.9 – 4.1 cm

(b) could either construct the

perpendicular bisectors or angle

bisectors to obtain (c) because the

PQR is equilateral.

Recall : When we want PR and

PQ to be the tangents to the circle,

we would construct the angle

bisector of the common pt, in this

case QPR

Note :

Area under speed-time graph = distance

Speed (m/s)

2

3

4

10 16 22 0 time(s)

2

Speed (m/s)

2

3

4

10 16 22 0 time(s)

2

Speed (m/s)

2

3

4

10 16 22 0 time(s)

Also accept

2

S

Preliminary Examination II Secondary 4 Express/ 5 Normal Academic

Candidate Answer scheme

Name Register No

ELEMENTARY MATHEMATICS PAPER 2 Date: 25th August 2011




Setter: Mrs L Goh



/ 100

CHUNG CHENG HIGH SCHOOL YISHUN

义顺









Simplify your answers to their simplest form. If the degree of accuracy is not specified in the

question, and if the answer is not exact, give the answer correct to three significant figures. Give

answers in degrees to one decimal place.




Class

[TURN OVER

2


Compound interest

Total amount =

nr

P

1001

Mensuration




3

1


3

4r


1


Sector area = 2

2


Trigonometry

C

c

B

b

A

a

sinsinsin

Acbcba cos))((2222

Statistics

Mean =

f

fx


22

f

fx

f

fx

3

1. (a) (i) Factorise 224 )(81 xxx completely. [3]

(ii) Express

xx

x

2

3

4

22

as a single fraction. [3]

(b) Mr and Mrs Lim wanted to buy a resale flat. The cost of the resale flat consists of the

valuation price (amount paid to the Housing Board), upfront cash (amount paid to the

seller) and the agent’s fees.

(i) They identified a unit in Ang Mo Kio which was valued at $338 000.

The owner of the unit requested for $45 000 above the valuation price. If the

agent’s fee is $3 800 of the total value of the unit, calculate the percentage of

agent’s fee.

[1]

(ii) Mr and Mrs Lim liked the unit very much. However, they would like a x %

reduction from the total of upfront cash and agent’s fees as they are only

comfortable to pay just another 10% above the valuation price.

Calculate the value of x .

[2]

(a) (i) )](9)][(9[ 2222 xxxxxx M1

=[ 10x2 – x ] [ 8x2 + x ] M1

= x2 ( 10x -1 ) ( 8x2 + 1 ) A1

(ii)

2

3

4

22

xx

x M1

= )2)(2(

)2(32

xx

xx M1

= )2)(2(

65

xx

x A1

(b) (i) %992167101.0%100

383000

3800 0.992 % A1

(ii) 33800)380045000(

100

)100(

x M1

100 – x = 69.262295

x = 30.737705

30.7 (3sf) A1

4

2. The diagram shows a rectangular sheet of metal measuring 15 cm by 9 cm. A square

of side x cm is cut out from each corner and the metal sheet is used to make an open

box with base ABCD and height x cm.

(a) Express the length of the side AB and BC in terms of x.

[2]

(b) If 60 cm3 of water is added into the box, the height of the water level will be

11

2cm.

Form an equation in x and show that it reduces to

24 48 95 0x x .

[2]

(c) Solve the equation 24 48 95 0x x .

[2]

(d) Hence, find the capacity of the box.

[1]

(a) AB = ( 15 – 2x) cm A1 BC = ( 9 – 2x ) cm A1

(b) ( 15 – 2x) ( 9 – 2x )

2

3 = 60 M1

135 – 48 x + 4x2 = 40

4x2 – 48 x + 95 = 0 (shown) A1

(c) x=

)4(2

)95)(4(4)48(48 2 M1

= 8

78448

= 9.5 (rejected) or 2.5 A1

(d) Capacity = 10 4 2.5 = 100 cm3 A1

x

x

D C

B A

9

15

5

3. Solutions to this question by accurate drawing will not be accepted.

The diagram above, which is not drawn to scale, shows an isosceles triangle ABC

where A is ( -4, 6), B is (-7, 3) C is (2,6) and AB = BC. The coordinates of D is (14,0)

and AD is parallel to BC. The point E lies on AC and is such that the area of the

triangle ABE is 3

1 the area of triangle ABC.

(a) Find the equation of the line AB [2]

(b) Find the gradient of the line AD. [1]

(c) Show that ABC is 90o. [2]

(d) The point G is such that ABCG is a square. Find the coordinates of G. [2]

(e) Find the coordinates of E. [2]

(a)

74

36

7

3

x

y M1

63993 xy

183 xy A1

(c) AB2 + BC2 = [(-4-(-7)]2 +[-6-3]2 +[6-3]2 + [2-

(-7)]2 = 180

AC2 = [2 – (-4) ]2 + [ 6- (-6)}2 = 180 M1

By the converse of Pythagoras theorem,

AB2 + BC2 = AC2, therefore, ABC is 90o A1

Accept cosine method and gradient method.

(b)

3

1

)4(14

)6(0

m A1

(d) Eqn of CG : 123 xy

Eqn of AG : 3

24

3

1 xy M1

Equate to solve for x and y coordinates

Coordinates of G ( 5, -3 ) A1

Accept logical method G (-4+9, -6 + 3)

y

C (2,6)

D ( 14, 0) x

B (-7, 3)

E

A ( -4, -6)

O

6

(e)

3:1:3

1

ACAE

ABCofArea

ABEofArea

3

1

24

4

x

3

1

66

6

y

x = -2 y = -2 Coordinates of E = ( -2 , -2 )

4. In the diagram, O is the centre of the circle PQRS. TUPA and TRB are tangents to the

circle at P and R respectively. It is also given that RTU = 34 and QP = QR.

(a) Explain why POR = 146. [2]

(b) Calculate

(i) PSR, [1]

(ii) PQR, [1]

(iii) PRQ. [1]

(c) Show that PQ bisects RPT. [2]

(a) oORTOPT 90 ( tan to radius at point of contact) M1

POR = 360o – 90o – 90o – 340 ( sum of s in a quadrilateral) A1

= 146o

(b) (i) PSR = o

o

732

146 ( at centre = 2 at circumference ) A1

(ii) PQR = 180o -73o =107o ( s in opp segment) A1

(iii) PRQ = ooo

5.362

107180

( base s of isos PQR ) A1

A

P

U

T

R

B

S O

S

Q 34

7

(c) oPRQRPQ 5.36 ( base s of isos PQR )

ooo

RPT 732

34180

(base s of isos PRT as PT = RT tangents drawn to a

circle from an external point are equal ) M1

QPU = 73o – 36.5o = 36.5o = RPQ A1

Therefore , PQ bisects RPT.

5. (a) AB =

3

2, CB =

7

5 and DC =

9

8 .

(i) Find the column vector DA . [1]

(ii) Find the value of ABDA 2 . [2]

(i) DA = DC + CB + BA =

5

15

3

2

7

5

9

8 A1

(ii) 482)11()19(

11

19

3

22

5

1522

= 21.9544984 22.0 units

M1 A1

5 (b) OPA and OQC are straight lines and PC intersects AQ at B.

Given that OQ = 8

3OC ,

BC

PB =

5

2 , OP = 8 p and OQ = 6 q ,

express the following vectors as simply as possible in terms of p and/or q .

(i) CP [1]

(ii) PB [1]

(iii) OB [1]

(iv) QB [1]

(c) Find the value of

OCPofarea

OQPofarea

[2]

A

8 p

Q 6 q

P

C O

B

8

(b) (i) OPCOCP

= 3

8(-6q) + 8p

= -16q + 8p = 8 ( -2q + p ) A1

(iii) PBOPOB

= 8p + 7

16( 2q - p )

= 7

8 ( 4q + 5 p ) A1

(ii) PCPB

7

2

= 7

28 ( 2q - p )

= 7

16( 2q - p ) A1

(iv) OBQOQB

= -6q + 7

8( 4q + 5 p )

= 7

10 ( -q + 4 p ) A1

(c)

8

3

AreaofOBC

OQBAreaof

7

5

OCPAreaof

OBCAreaof M1

56

15

7

5

8

3

OCPAreaof

OQBAreaof A1

6. A ship sails from a port W at 08 00 on a bearing of 040 towards port X. It sails at an

average speed of 10 km/h, reaching port X at 10 24. From port X, it sails to port Y

which is 40 km away. After resting for 1

2 an hour, the ship sets sail again to port Z,

which is due north of port X. The figure below shows the route taken by the ship.

(a) Find the distance of

(i) WX, [1]

(ii) WY, [2]

(iii) XZ, [2]

(b) Calculate the bearing of Y from W, to the nearest degree. [2]

(c) Calculate the shortest distance between X and the ship as it sails from Y to Z. [2]

(d) If the ship uses the same speed throughout the whole journey from port W to Z,

find the time it reaches port Z, correct to the nearest minute.

[3]

9

(a) (i) WX = km2410

60

242 A1 (b)

09395673.50

100sin

40

sin o

= 51.84742646o M1

Bearing = 040o + 051.84742646o

= 091.84742646o

092o ( nearest degree) A1

(ii) WY

=o100cos)40)(24(24024 22 M1

=50.09395673 50.1 km A1

(c) sin 30o =40

d M1

d = 20 km A1

(iii) oooZXY 12060180

ooooXZY 3030120180 M1

Since oXYZXZY 30 (isos )

XZ = 40 km A1

(d) ZY =o120cos)40)(40(24040 22

= 69.2820323 km

Total distance = 24 + 40 + 69.2820323

= 133.2820323 km M1

Time taken = 5.010

2820323.133

= 13.82820323 h

= 13 h 50 min M1

Time reached = 0800 +1350 = 2150 A1

7. Figure (i) below is a central section of an ice-cream ‘cornet’ which holds a sphere of

ice-cream of radius 1.9 cm. The ice-cream cornet is made of a thin right circular

cone and a similarly thin cylinder. The cone has a slant height of 11 cmOA OB ,

and a base diameter of 3.8 cm. The cylinder has a height of 2.3 cm and a base

diameter of 3.8 cm. The centre P of the sphere of ice-cream is also the centre of the

base of the cylinder through DC.

W

Y

X

Z

100 30

N

40 40 km

d

10

(a) Calculate

(i) the total length OP of the cornet, [2]

(ii) the volume of the sphere of ice-cream, [1]

(iii) the volume of the air in the cornet when the sphere of ice-cream is in

position

[3]

(b) If the right circular cone is cut open along the side OA, it forms a sector, OABA’ of a

circle with radius OA and centre O, as shown in figure (ii), calculate

(i) the arc length ABA’, [1]

(ii) the angle of the sector, θ, giving your answer in radians, [1]

(iii) the area of the sector OABA’, [2]

(c) the total amount of the material used, in 2cm , to make the ice-cream cornet. [2]

(a) (i) h = 83466658.109.111 22 M1

OP = 2.3 + 10.83466658 13.1 cm A1

(b) (i) Arc length = 2 9.1 =

11.93805208 11.9 cm A1

(ii) Vol =

39.13

4 = 28.73091201

28.7 cm3 A1

(ii) 93805208.11r

11

93805208.11 = 1.085277462

1.09 rad A1

(iii) Vol of cone = 83466658.109.1

3

1 2

= 40.95916621 cm3 M1

(iii) Area of sector =

085277462.1112

1 2 = 65.65928645 M1

65.7 cm2 A1

O

A B

P C D

Figure (i)

3.8 cm

2.3 cm

11 cm

O

A’ A

B

Figure (ii)

11 cm θ

11

Vol of air in cylinder = Vol of cylinder – vol

of hemisphere = 2

3091201.283.29.1 2

= 11.71919 cm3 M1

Total vol of air = 52.67835621 52.7 cm3

A1

(c) Total amt = curved surface of cone + curved surface of cylinder M1

= 3.29.12119.1

= 93.11680625 93.1 cm2 A1

8. The marks scored by 400 candidates in an examination are shown in the cumulative

frequency below.

(a) Find the median mark. ( accept 54 to 55 marks A1) [1]

(b) Find the interquartile range. ( 71 – 46 = 25 marks A2) [2]

(c) Find the 80th percentile. ( accept 75 to 76 marks A1 ) [1]

400

0

300

200

100

20 40 60 100 80

Number

of pupils

Examination mark (m)

12

(d) Given that 240 candidates passed the examination, find the passing mark.

( 51 marks A1 )

[1]

(e) (i) Copy and complete the following frequency table for the above distribution.

Examination mark Number of pupils

0 < x ≤ 20 10

20 < x ≤ 40 40 -10 = 30

40 < x ≤ 60 250 – 40 = 210

60 < x ≤ 80 340 – 250 = 90

80 < x ≤ 100 400 – 340 = 60

[2]

8. (e) (ii) Calculate an estimate of the mean mark of the candidates. [2]

(iii) Calculate the standard deviation of the distribution. [2]

(ii) Mean marks = 400

609090702105030301010 M1

= 58 A1

(ii) Sd = 18.33030278 18.3 marks A2

9. (a) In a game, 3 fair dice are thrown together at the same time. The winner is the player

who obtains three consecutive numbers in any order.

(i) Find the probability of winning the game. [2]

(ii) If one of the dice is replaced by a coin, what is the probability of winning

the game if the winner has to roll two consecutive numbers in any order and

a tail?

[2]

(i) Possibilities = ( 1,2,3 ), ( 2,3,4) , (3,4,5 ) , (4,5,6) M1

P (wins) = 246

1

6

1

6

1

=

9

1 A1

(ii) Possibilities = ( 1,2,T ), ( 2,3,T) , (3,4,T ) , (4,5,T), (5,6,T) M1

P (wins) = 102

1

6

1

6

1

=

36

5 A1

13

(b) The following table shows the nutrition contents of some local food items

sold in a hawker centre.

Contents per 100g of food Serving Size

(g) Food Protein (g) Total Fat (g) Dietary Fibre

(g)

Carbohydrate

(g)

Roti Prata 6.3 9.4 2.5 46.9 64

Carrot cake 0.7 11.9 2.0 14.2 295

Nasi Lemak 6.2 29.4 3.1 38.1 210

The nutrition contents of the three food items may be represented by the matrix.

N =

1.381.34.292.6

2.140.29.117.0

9.465.24.93.6

(i) Write down a matrix S such that the product SN gives the contents of

protein, total fat, dietary fibre and carbohydrate per serving of each food

item.

[2]

Four friends ordered three servings of roti prata, two servings of carrot cakes and

one serving of nasi lemak for breakfast.

(ii) Write down a matrix B such that BSN will give the contents of protein, total

fat, dietary fibre and carbohydrate in this breakfast order.

[1]

(iii) Evaluate BSN, giving each element of the matrix correct to one decimal

place.

[2]

The prices per serving of roti prata, carrot cake and nasi lemak are $1.60, $2 and

$1.80 respectively.

(iv) Write down a matrix P such that BP gives the total cost of the breakfast

order.

[1]

(v) Find the total cost of the order by working out BP. [2]

(i) S =

1.200

095.20

0064.0

A2

ii) B= 123 A1

(iii) BSN = 123

1.200

095.20

0064.0

1.381.34.292.6

2.140.29.117.0

9.465.24.93.6

M1

= 8.2531.230.1501.21 A1

14

(iv) P =

80.1

00.2

60.1

A1 (v) BP = 123

80.1

00.2

60.1

= ( 10.6) M1

Total cost = $ 10.60 A1

10. Answer the whole of this question on a sheet of graph paper.

The variables x and y are connected by the equation

2

32

xxy . Some corresponding

values are given in the following table:

x 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

y 13 5 4.33 4.75 p 6.33 7.24 8.19

(a) Find the value of p. [1]

(b) Taking 4 cm to represent 1 unit on the x-axis and 1 cm to represent 1 unit on the

y-axis, draw the graph of 2

32

xxy for the values of x in the range 0.45.0 x .

[3]

(c) By drawing a suitable tangent to your curve, find the coordinates of the point at

which the gradient of the tangent is equal to –4.

[2]

(d) Find the gradient of the tangent to the curve at the point where 2x . [2]

(e) By drawing a suitable straight line, use your graph to solve the equation

0362 23 xx in the range 0.45.0 x .

[3]

(f) State the range of values of x for 0.45.0 x such that the gradient of the curve is

positive.

[1]

(a) p = 5.48 (d) Plot tangent B1

Gradient = 1.2 0.2 A1

(b) Scale and axes B1 plotting of points B1

Smooth graph and label B1

(e) Plot y = 6 B1

x = 0.9, 2.8 0.1 A2

(c) Plot tangent B1

Coordinates = (1.05 0.05, 4.6 0.1)

(f) x > 1.5 A1

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