MATHEMATICS 4016/01 PAPER 1 - · PDF fileMATHEMATICS 4016/01 PAPER 1 Date: 24/08/2011...

1

Name: ______________________________ Index. No: ______ Class:_________

Guangyang Secondary School Guangyang Secondary School Guangyang Secondary School Guangyang Secondary School Guangyang Secondary School Guangyang





G U A N G Y A N G S E C O N D A R Y S C H O O L

PRELIMINARY EXAMINATION 2011 Secondary Four Express / Five Normal Academic

MATHEMATICS 4016/01 PAPER 1 Date: 24/08/2011 Duration: 2 hours Candidates answer on the Question Paper.

READ THESE INSTRUCTIONS FIRST

Write your name, index number and class on all the work you hand in. Write in dark blue or black pen. You may use a pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all questions. 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 required to use a scientific 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 , 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 80.

This question paper consists of 18 printed pages, inclusive of this cover page.

2

Mathematical Formulae

Compound interest

Total Amount = Pn

r

1001

Mensuration

Curved surface area of a cone = rl

Surface area of a sphere = 4 2r

Volume of a cone = hr2

3

1

Volume of a sphere = 3

3

4r

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 .

bccba 2222 cos A

Statistics

Mean =

f

fx

Standard deviation =

22

f

fx

f

fx

For For

Examiner’s Examiner’s use use

3

[Turn over

1 Express 4 millions as a percentage of 12 billions.

Answer ….……………………..… % [1]

2 Factorise completely

(a) 25 – 9z2,

Answer (a)…………..……………….. [1]

(b) 2a2 + 6a – ab – 3b.

Answer (b)…………..………………..[2]

3 A polygon has n sides. If it has 85o, 79o and 43o as three of its exterior angles and

the remaining exterior angles are 17o each, calculate the value of n.

Answer ….…………………….…..… [2]

For For


4

[Turn over

4 A map is drawn to a scale of 1 : 5 000.

(a) Two villages are 3 kilometres apart.

Calculate, in centimetres, their distance apart on the map.

Answer (a) ……………….…… cm [1]

(b) A field occupies 24 square centimetres on the same map.

Find the actual area, in square metres.

Answer (b) ...............…….....…… m2 [1]

5 A bus journey from Terminal A to Terminal B took 4

32 hours.

(a) The bus travelled at an average speed of 58 km/h.

Find the length of the journey.

Answer (a)…………..…………… km [1]

(b) If the last bus arrived at Terminal B at 02 10, find the time the last bus left

Terminal A.

Answer (b)…………..………………. [1]

For For


5

[Turn over

6 (a) The driving distance from New York City to Las Vegas is about 2 570 miles.

If 1 kilometre is approximately 0.621 miles and the driving speed is 90 km/h,

find the time taken required to drive from New York City to Las Vegas.

Answer (a)…………..………………h [1]

(b) Given that 33 814 ounces is equivalent to 1 000 litres and 1 000 cubic metres

is equivalent to 220 gallons. Express 25 gallons in ounces.

Answer (b)…………..……… ounces [1]

7 Find the two possible values of integer x such that 34912 xx .

Answer …………… , ………………. [2]

For For


6

[Turn over

8 (a) Simplify5

222

3

)(2

3

2

a

ab

b

, leaving your answer in positive index form.

Answer (a)…………..……………….. [2]

(b) Given that 64x = (0.5)6, find the value of x.

Answer (b)…………..………………..[2]

For For


7

[Turn over

9 (a) Solve the equation 52

r.

Answer (a)…………..……………….. [1]

(b) Solve the simultaneous equations

3x – 2y = 18,

2x + 3y = – 1.

Answer (b) x = …………..…….……

y = …………….………… [2]

10 A conical container has a vertical height of 6 cm. Water is poured into the empty

container at a constant rate. It takes 24 seconds to fill the container.

After t seconds the depth of the water is d cm.

(a) Find the value of t when d = 3 cm.

Answer (a)…………..……………….. [1]

(b) On the axes in the answer space, sketch the graph showing how the depth

varies during the 24 seconds.

Answer (b)

6 cm

0 6 18 24

Time (t seconds)

12

Depth (cm)

1

2

3

4

5

6

[1] 3 9 15 21

d

For For


8

[Turn over

11 (a) Bernice opened a savings account in a bank that offers 1.08% per annum rate

of interest. She deposited a sum of money in it. Calculate the amount of

money she deposited if she received $22.50 interest in 10 months.

Answer (a) $………..……………….. [2]

(b) Joel’s loan company charges a 2% monthly compound interest on any

outstanding debts. Yan Xun owes the company $4 200 at the end of

January 2011. If he can only pay the company at the end of October 2011,

calculate the amount of interest he has to pay.

Answer (b) $………..……………….. [2]

For For


9

[Turn over

12 ABCD is a trapezium. AB = 35 cm, CD = 23 cm and AD = 20 cm. Angle ABC = 90o.

(a) Find length BC.

Answer (a)…………..…………… cm [2]

(b) Find the area of trapezium ABCD.

Answer (b)…………..…..……… cm2 [2]

(c) Write down the value of cos ADC.

Answer (c)…………..…..…………… [1]

A B

C D

35 cm

23 cm

20 cm

For For


10

[Turn over

13 The value of a car depreciates with years. It is given that the value of a particular

car, $V, is inversely proportional to the square of its age, x years.

(a) Write down a formula connecting V, x and a constant k.

Answer (a)…………..…………….…. [1]

(b) After 2 years, the value of the car becomes $80 000.

Find the value of the car after 3 years.

Answer (b) $…………..…………….. [2]

14 The temperature at the top of the mountain was –18oC. At the same time, the

temperature at sea level was 6oC.

(a) Calculate the difference between these temperatures.

Answer (a)…………..……………. oC [1]

(b) The height of the mountain is 2 600 m. Given that the temperature changed

uniformly with the height, calculate the height above sea level at which the

temperature was 0oC.

Answer (b)…………..……………. m [2]

For For


11

[Turn over

15 In the diagram, A, B, C, D and E lie on the circumference of a circle, centre O.

AD is a diameter of the circle. OB is parallel to DC. Angle CDO = 38o.

Find, stating your reasons clearly,

(a) angle AOB,

Answer (a)…………..……..………. o [1]

(b) angle BAO,

Answer (b)…………..……..………. o [1]

(c) angle BCD,

Answer (c)…………..……..………. o [1]

(d) angle AEB.

Answer (d)…………..……..………. o [1]

A

B

D

E

O

C

38o

For For


12

[Turn over

16 Consider the sequence 13 – 3, 23 – 6, 33 – 9, …

(a) Write down the 6th term of the sequence.

Answer (a)…………..……..………. . [1]

(b) Write down, an expression in terms of n, for the nth term of the sequence.

Answer (b)…………..……..………. . [1]

(c) Evaluate the 15th term of the sequence.

Answer (c)…………..……..………. . [1]

17 (a) Express x2 – 8x + 7 in the form (x – a)2 + b.

Answer (a)…………..……..………. . [1]

(b) Sketch the graph of 782 xxy .

Answer (b)

[2]

y

x 0

For For


13

[Turn over

18 Kevin and Carol took 8 Mathematics tests.

Their results are shown in the tables below.

Kevin

Marks 51 106 1511 2016

Frequency 0 2 2 4

Carol

Mean = 14.25

Standard Deviation = 2.8

(a) For Kevin’s results, calculate

(i) his mean score,

Answer (a)(i)………...……..………… [2]

(ii) the standard deviation of his scores.

Answer (a)(ii)………...……..………… [2]

(b) Compare, briefly, the results for the two pupils.

Answer (b) ……………………………………………………………………… [1]

For For


14

[Turn over

19 It is given that A is the point (– 2, 0), B is the point (7, 0) and

2

3BC .

(a) Find AB .

Answer (a)…………..……..…units [1]

(b) Find the coordinates of the point C.

Answer (b) ( ………… , ..……..…) [1]

(c) Calculate the gradient of AC.

Answer (c)…………..……..………. . [1]

(d) Find the equation of line AC.

Answer (d)…………..……..………. . [2]

For For


15

[Turn over

20 The points P, Q and R on the sides of triangle ABC are such that RQ is parallel

to PC, RP is parallel to QC and RP meets BQ at X.

(a) Name two triangles that are similar to triangle BXP.

Answer (a) Triangles ……..… and .………… [2]

(b) Given that BP = 7 cm, PC = 5 cm and QC = 6 cm, find

(i) length XP,

Answer (b)(i)………...……..…… cm [1]

(ii) exact length AQ,

Answer (b)(ii)………...……..…... cm [2]

(iii) the value of BAC

BRP

triangleof area

triangleof area,

Answer (b)(iii)………...……..……… [1]

(iv) the value of BCQ

ABQ

triangleof area

triangleof area.

Answer (b)(iv)………...……..…........ [1]

A

B C

R Q

P

X

For For


16

[Turn over

21 Gang Wei has 6 coins in his pouch. He has 2 ten-cent coins and 4 twenty-cent coins.

He takes 2 coins at random from his pouch, one after the other.

(a) Complete the tree diagram in the answer space.

Answer (a):

[2]

(b) Find the probability that the total value of the two coins is

(i) twenty cents,

Answer (b)(i)………...………..……… [1]

(ii) at least thirty cents.

Answer (b)(ii)………...………..……... [1]

(c) Gang Wei has a dice. If he would given a chance to draw a third coin only if

he managed to obtain a score that is prime, find the probability that the third

coin he selected will be a twenty-cent coin.

Answer (c)……….......………..……… [2]

( )

First coin

selected

ten-cent

twenty-cent

Second coin

selected

ten-cent

twenty-cent

ten-cent

twenty-cent

2

6

æ

èç

ö

ø÷

( )

( )

( )

( )

For For


17

[Turn over

22 The diagram represents the speed-time graph of a moving particle.

(a) Calculate the speed of the particle when t = 4.

Answer (a)…………..……..………. s [1]

(b) Calculate the distance travelled by the particle in the first 15 seconds.

Answer (b)…………..……..…….. m [1]

(c) Given that the rate of retardation is half the rate of acceleration during the

first 6 seconds, calculate the time at which it stops.

Answer (c)…………..……..………. s [2]

Speed (v) in m/s

Time (t) in seconds 0

16

40

6 15

For For


18

[End

23 The diagram shows points A and B.

Answer (a), (b), (c)

(a) Given that AC is 12 cm and angle ABC = 100o, complete triangle ABC.[2]

(b) Construct the perpendicular bisector of AB. [1]

(c) Construct the bisector of angle ABC. [1]

(d) Mark the point P where these two lines meet.

Measure and write down the length of AP.

Answer (d)…………..……..……. cm [1]

--- End of Paper ---

A

B

Answers

1

30

1

15a 38

2a )35)(35( zz 15b 71

2b )3)(2( aba 15c 109

3 12 15d 19

4a 60 16a 1863

4b 60 000 16b nn 33

5a 159.5 16c 3330

5b 23 25 17a 9)4( 2 x

6a 46.0 17b Parabola through (0, 7); turning point at (4, )9

6b 3842.5 18ai 14.25

7 3, 4 18aii 4.15

8a

2

3

8

27

b

a

18b Carol’s result is more consistent.

Carol’s test scores are less wide spread than Kevin’s.

8b 1 19a 9

9a

5

2

19b (10, 2)

9b x = 4, y = 3 19c

6

1

10a 3 19d

3

1

6

1 xy

10b Curve that passes through (3, 6) and (24, 12) 20a QXR and BQC

11a 2500 20bi 3.5

11b $819.39 20bii 4

7

2

12a 16 20biii

144

49

12b 464 20biv

7

5

12c

5

3

21bi

15

1

13a

2x

kV

21bii

15

14

13b 35 555.56 21c

30

1

14a 24 22a 32

14b 650 22b 528

22c 35

23d 5.3

1






G U A N G Y A N G S E C O N D A R Y S C H O O L

PRELIMINARY EXAMINATION 2011 Secondary Four Express/ Five Normal (Academic)

MATHEMATICS 4016/02 PAPER 2 Date: 25/08/2011 Duration: 2 h 30 min Additional Materials : Answer Paper Graph Paper (1 sheet)

READ THESE INSTRUCTIONS FIRST Write your class, index number and name on all the work you hand in. Write in dark blue or black pen. You may use a pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all questions. If working is needed for any question it must be shown with the answer. Omission of essential working will result in loss of marks. Calculators should be used where appropriate. If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answer to three significant figures. Give answers in degrees to one decimal place.

For , use either your calculator value or 3.142, unless the question requires the answer in

terms of . 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 11 printed pages, inclusive of this cover page.

2

Mathematical Formulae

Compound interest

Total Amount = Pn

r

1001

Mensuration

Curved surface area of a cone = rl

Surface area of a sphere = 4 2r

Volume of a cone = hr2

3

1

Volume of a sphere = 3

3

4r

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 .

bccba 2222 cos A

Statistics

Mean =

f

fx

Standard deviation =

22

f

fx

f

fx

3

Answer all the questions.

1 (a) Solve the equation 11p . [1]

(b) Express as a single fraction in it simplest form

45

2

2

5

xx . [3]

(c) Simplify 152

92

2

uu

u. [3]

(d) Given that k

axmy

2

, express x in terms of a, k, m and y. [2]

2 In 2010, a baker can buy x kilograms of flour with $16.

(a) Write down an expression, in terms of x, for the price of flour he paid for

each kilogram of flour. [1]

(b) Due to economic downturn, the amount of flour that he can buy for the

same amount of money is reduced by 5 kilograms in 2011.

Write down an expression, in terms of x, for the new price of flour he paid

for each kilogram of flour. [1]

(c) The baker needs to pay 20 cents more for one kilogram of flour in 2011

than in 2010.

(i) Write down an equation in x to represent this information, and show

that it reduces to 040052 xx . [2]

(ii) Solve the equation 040052 xx , giving both answers correct to

to 2 decimal places. [3]

(d) Calculate the amount of flour the baker can get if he spends $200 in 2011. [2]

4

3 The container consists of half a cone ABCD with radius OA attached to a triangular

prism ACDPQR as shown in the diagram. Given that PQR is an isosceles triangle

with PQ = PR = 10 cm. DP = 3 cm and RQ = 8 cm.

(a) Find

(i) angle RPQ, [3]

(ii) OD, [2]

(iii) volume of the container. [3]

(b) A cylindrical vessel of radius 3 cm and length 50 mm is completely

filled with water. The water from the vessel is poured into the container.

Calculate the percentage of the container filled. [3]

4

In the diagram, A, B and N are points on a circle with centre M. B, M and C are

points on a circle with centre N. AMNC is a straight line.

Prove that triangles ABN and CBM are congruent. [3]

A

B

O C

D

R

Q

P

8 cm

10 cm

3 cm

M N A

B

C

5

5 The cumulative frequency curve below illustrates the marks obtained, out of 100, by

500 students in GYSS Mock Exam.

(a) Use the graph to find

(i) the median mark, [1]

(ii) the interquartile range, [2]

(iii) the ninetieth percentile mark, [1]

(iv) the percentage of students who score less than 50 marks. [2]

(b) Given that 15% students scored a distinction, find the minimum marks

students must score to get a distinction. [1]

6

(c) The same 500 students sat for their Preliminary Examination. The box

and whiskers diagram below illustrates the marks obtained.

(i) Compare the marks obtained in the Mock Exam and Preliminary

Exam in two different ways. [2]

(ii) Yu Fang said that Preliminary Exam was much difficult than

Mock Exam.

Do you agree? Give a reason for your answer. [1]

6

AODC is a square with sides 2 cm. AB is an arc of a circle with centre D and AD

is an arc with centre O.

(a) Find BD. [1]

(b) Find the area of sector ABD. [2]

(c) Hence find the area of the shaded region. [3]

A O

D C

B

7

7 In the diagram, P, Q and R are three points on level ground such that the bearing of Q

and R are from P are 038 and 118 respectively. Given that QR = 152 m and the

bearing of R from Q is 163o.

(a) Find the bearing of P from R. [2]

(b) Find the length of PR. [3]

(c) Calculate the area of triangle PQR. [2]

(d) Calculate the shortest distance of Q from PR. [2]

(e) Q is the foot of a vertical mast, TQ.

The angle of elevation of the top of the mast, T, from R is 40o.

Calculate the greatest angle of elevation of the top of the mast, T, when

viewed from any point along PR. [3]

N

N

P

Q

R

152 m

163o

8

8 In 2009, Amali and Nabilah decided to start a business. Amali and Nabilah invested

$25 000 and $15 000 respectively. In 2011, their profit made totalled to an amount

of $20 000.

(a) Calculate Amali’s share of profit in 2011, if they decided to divide the

profit in the same ratio as their investment. [1]

(b) Amali bought a car. If he used all his share of profit to pay the deposit, he

still have to pay monthly instalment for 10 years. If the car cost $105 000,

calculate the monthly instalment he had to pay. [3]

(c) The total profit in 2011 was 30% greater than that made in 2010.

Calculate the profit made in 2010. [1]

(d) They projected their profit in 2012 to be $35 000. Calculate this expected

increase in profit as a percentage of the profit made in 2011. [2]

(e) Their successful business story was published in New York Times.

Given that the rate of exchange between US dollars (USD) and Singapore

dollars (S$) was USD1 = S$1.2268.

Calculate their amount of profit made in 2011 reported in New York Times.

[1]

9

9

ATB is a semicircle with centre O. TC is tangent to the semicircle. AOB produced

meets tangent at C. Given that OA = 5 cm and TC = 4.5 cm.

(a) Show that angle TOC is approximately 0.7328 radians, corrected to 4

decimal places. [1]

(b) Calculate the shaded region P. [2]

(c) (i) Find AT. [3]

(ii) Calculate the perimeter of the shaded region Q. [2]

C B A 5 cm

4.5 cm

10

10 (a) = {integers x: }101 x

A = {prime numbers}

B = {integers divisible by 3}

(i) Draw a Venn Diagram to illustrate this information. [1]

(ii) State n(A). [1]

(iii) List the elements contained in the set ')( BA . [1]

(b) In a tuition group, Joann, Joshua and Jaypriyah took a test containing 20

multiple choice questions. Their results, stating the number of questions done

correctly, incorrectly and not attempted, are shown in the table below.

Correct Incorrect Not attempted

Joann 12 3 5

Joshua 18 2 0

Jaypriyah 15 2 x

Marks 2 – 1 0

(i) Write down the value of x. [1]

(ii) Given matrix Q =

0

1

2

and PQ represents the marks obtained by

each pupil, write down the 33 matrix P and describe what is

represented by the elements of Q. [3]

(iii) Find PQ. [1]

(iv) Given that R = 111 , calculate S = RPQ and describe what it shows. [2]

(v) Calculate 3

1S and describe what it shows. [2]

11

11 Answer the whole of this question on a sheet of graph paper. The table gives some values of x and the corresponding values of y, where

31830 xxy .

x 4 3 2 1 0 1 2 3 4

y 38 57 58 47 30 p 2 3 22

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

(b) Using a horizontal scale of 2 cm to represent 1 unit and a vertical scale of

2 cm to represent 10 units, draw the graph of 31830 xxy . [3]

(c) Use your graph to find

(i) the largest value of 31830 xx , [1]

(ii) the possible solutions of 101830 3 xx . [2]

(d) By drawing a tangent, find the gradient of the curve when x = 2. [2]

(e) (i) On the same axes, draw the graph of the straight line 278 xy . [1]

(ii) Find the equation, in the form of 03 baxx , which is satisfied

by the values of x at the points where the two graphs meet. [2]

--- End of paper ----

Answer

1a p = 121 7c Area of PQR = 6790 m2

1b

)45)(2(

1623

xx

x

7d Shortest distance = 107 m

1c

5

3

u

u

7e Greatest angle of elevation = 49.9o

1d

a

mykx

)(

8a $12500

2a $ )

16(

x

8b $770.83

2b $

5

16

x

8c $15 384.62

2ci

100

2016

5

16

xx

8d 75%

2cii x = 22.66 or 66.17 8e USD16302.58

2d x = 22.66 or 66.17 (reject)

Amt of flour = 221 kg (3 sf)

9a )

5

5.4(tan 1TOC

3ai oRPQ 2.47 9b 2.09 cm2

3aii OD = 9.17 cm 9ci 9.34 cm

3aiii Vol of container = 187 cm3 9cii 21.4 cm

3b Percentage = 75.7% 10aii 4

4 Use of RHS / SAS 10aiii {1,4,8,10}

5ai Median mark = 51 10bi x = 3

5aii Interquartile range = 62 – 40 = 22 10bii

P =

3215

0218

5312

P represents the marks scored for each

correct, incorrect and unattempted question

respectively.

5aiii 90th percentile = 73 10biii

PQ =

28

34

21

5aiv 48% 10biv S = (83)

It represents the total marks obtained by the

3 pupils.

5b 69 10bv S

3

1= (27

3

2)

It shows the average marks obtained by the

3 pupils.

5ci Range of marks from Mock Exam is

larger than that of Preliminary Exam.

Median mark from Preliminary Exam is

higher than that of Mock Exam.

Spread of Preliminary Exam marks is

higher than that of Mock Exam.

11a p = 13

5cii Disagree since the median mark from

Preliminary Exam is higher than that of

Mock Exam.

11ci 59 ( 5.0 )

6a BD = 2.83 cm 11cii x = 1.2 1.0 or 3.45 1.0

6b Area of sector ABD = 3.14 cm2 11d Gradient = 5.6~5.5

6c Area of shaded region = 2.00 cm2 11eii 03103 xx

7a Bearing of P from R = 298o

7b PR = 126 m

MATHEMATICS 4016/01 PAPER 1 - · PDF fileMATHEMATICS 4016/01 PAPER 1 Date: 24/08/2011...

Documents

Transcript of MATHEMATICS 4016/01 PAPER 1 - · PDF fileMATHEMATICS 4016/01 PAPER 1 Date: 24/08/2011...