Public Assessment of the HKDSE Mathematics Examination1.PublicAssessmentThe mode of public assessment of the Hong Kong Diploma of Secondary Education (HKDSE) Mathematics Exam in the Compulsory Part is shown below.

Component Weighting Duration

Public Examination Paper 1 Conventional questionsPaper 2 Multiple-choice questions

65%35%

21 4 hours11 4 hours

The mode of public assessment of the Hong Kong Diploma of Secondary Education (HKDSE) Mathematics Exam in Module 1 (Calculus and Statistics) is shown below.

Component Weighting DurationPublic Examination Conventional questions 100% 21 2 hours

The mode of public assessment of the Hong Kong Diploma of Secondary Education (HKDSE) Mathematics Exam in Module 2 (Algebra and Calculus) is shown below.

Component Weighting DurationPublic Examination Conventional questions 100% 21 2 hours

2.Standards-referencedReportingThe HKDSE makes use of standards-referenced reporting, which means candidates’ levels of performance will be reported with reference to a set of standards as defined by cut scores on the variable or scale for a given subject. The following diagram represents the set of standards for a given subject:

Cut scores

U 1 2 3 4 5

Variable/scale

Within the context of the HKDSE there will be five cut scores, which will be used to distinguish five levels of performance (1–5), with 5 being the highest. The Level 5 candidates with the best performance will have their results annotated with the symbols ∗∗ and the next top group with the symbol ∗. A performance below the threshold cut score for Level 1 will be labelled as ‘Unclassified’ (U).

IV

Exam StrategiesA. GeneralStrategies

1. Beforethestartoftheexamination

• Make sure that the time on your watch matches with that of the examination centre.

• Listen carefully to the invigilator for any errors and changes in the examination papers.

• Read carefully the instructions on the cover of the question-answer book or question book.

• Check carefully whether there are any omitted or blank pages in the examination paper according to the invigilator’s instruction.

2. Duringtheexamination

• Use proper stationery.

– Paper 1: use a pen mainly, but an HB pencil for drawing.

– Paper 2: use an HB pencil.

• Show your work clearly and neatly.

• Do not get stuck on any one of the questions. Skip it and go on to another one.

3. Afteransweringallthequestions

• Do not be tempted to leave early.

• Check whether there are any questions that were missed.

• Go back to questions skipped earlier.

• Check whether there are any careless mistakes or not.

• Do not cross out anything unless you have enough time to replace it correctly.

• Make sure you write your candidate number on the answer book, supplementary answer sheets and multiple-choice answer sheet.

B. SpecificStrategies

1. Paper1(214

hours)

• Allocate a reasonable proportion of time to each section.

Sections Suggested Time Allocation Approximate Time per QuestionA (1) 40 minutes 3 − 5 minutesA (2) 40 minutes 5 – 10 minutes

B 50 minutes 5 – 15 minutes

– In general, spend 5 minutes for every 4 marks.

– Allow 5 minutes for final checking.V

Comparison between HKDSEand HKCEE Syllabuses

1. Topicsremovedfromandaddedtothesyllabus

Section Topicsremoved Topicsadded

NumberandAlgebraStrand

Quadratic equations in one unknown

• Sum of roots and product of roots• Operations of complex numbers

Functions and graphs • Concepts of domains and co-domains of functions

More about graphs of functions • Enlargement and reduction

Exponential and logarithmic functions • Change of base

More about polynomials • G.C.D. and L.C.M. of polynomials

• Operations of rational functions

More about equations • Using a given quadratic graph to solve another quadratic equation

Arithmetic and geometric sequences and their summations

• Properties of arithmetic and geometric sequences

Inequalities and linear programming

• Solving quadratic inequalities in one unknown by the algebraic method

• Solving compound linear inequalities involving ‘or’

Measures,ShapeandSpaceStrand

Locus • Describing the locus of points with algebraic equations

Equations of straight lines and circles

• Possible intersection of two straight lines

• Intersection of a straight line and a circle

DataHandlingStrand

Permutation and combination • Concepts and notations of permutation and combination

II

16Trigonometry

16

Trigonometric Ratios of Angles1.

Fig.16.1

3.SignsofTrigonometricFunctions:

Fig.16.3

cosθ = ac

, tanθ = ba

and

sinθ = bc

wherec a b= +2 2

Reducing Trigonometric Ratios

Table16.2

Reducing Trigonometric Ratios

Table

180°-θ 180°+θ 360°-θsin sin θ -sin θ -sin θcos -cos θ -cos θ cos θtan -tan θ tan θ -tan θ

More about Trigonometry

Trigonometric Identities

1. tanθ θθ

≡ sincos

2.sin2θ +cos2θ ≡1

3.sin(90°-θ )≡cos θ 4.cos(90°-θ )≡sin θ

5.tan ( )tan

901° − ≡θθ

Special AnglesSpecial Angles

θ 0° 30° 45° 60° 90° 180° 270° 360°

sin θ 012

1

2 or

22

3

21 0 -1 0

cos θ 1 32

1

2 or

22

12

0 -1 0 1

tan θ 01

3 or

33

1 3 undefined 0 undefined 0

Table16.1

2.

Fig.16.2

cosθ = xr

, tanθ = yx

and

sinθ = yr

where r x y= +2 2

MoreaboutTrigonometry

More about Trigonometry

Trigonometric Equations1.SimpleTrigonometric

Equations (a)asin θ =b (b)acos θ =b (c) atan θ =b2.OtherTrigonometric

Equations  Examples:

(a) 2 atan θ =sin θ (b) 2cos2θ -3sin θ =0

Graphs of Trigonometric Functions1.(a)y =sinx (b)y =cosx

Fig.16.4 Fig.16.5 (c) y =tanx

Fig.16.62.Foranyrealvalueofx ,-1≤sinx≤1and-1≤cosx≤1.3.Theperiodsofsinx,cosxandtanxare360°,360°and180°respectively.

Transformation on the Graphs of Trigonometric Functions Considerthegraphofy =sinx.1.Translation (a) Vertical:y =sinx+k (b) Horizontal:y =sin(x+k°)2.Reflection Aboutthex-axis:y =-sinx3.ReductionorEnlargement (a) Vertical:y =ksinx (b) Horizontal:y =sin (kx)

Rate

Rate is a comparison between two different kinds of quantities. For two

different quantities x and y, the rate is given by xy

or yx

and it has a unit.

For example, a worker cleans 20 cars in 5 hours.

The cleaning speed = 20 cars5 hours

= 4 cars/hour.

The cleaning time for each car = 5 hours20 cars

= 0.25 hour/car.

Ratio

Ratio is a comparison between two or more quantities of the same kind.

The ratio of a to b is a : b or ab

and it has no unit. The ratio for three or

more quantities such as a : b : c is called a continued ratio.

For example, the weights of A, B and C are 80 kg, 50 kg and 20 kg respectively. Weight of A : Weight of B = 80 : 50 = 8 : 5.Weight of A : Weight of B : Weight of C = 80 : 50 : 20 = 8 : 5 : 2.

A

B

A continued ratio cannot be

expressed as a fraction.

22Determine whether each of the following statements is true or false.

1. Rate is a comparison between two different kinds of quantities.

2. Ratio is a comparison between two or more quantities of the same kind.

3. If 5m = 8n, then m : n = 5 : 8.

4. If a : b = 1 : 2 and b : c = 1 : 3, then a : b : c = 1 : 2 : 3.

5. The speed of a car, say 80 km/h, is an example of rate.

6. The speeds of two cars, say 80 km/h and 90 km/h, can be expressed by the ratio 8 : 9.

(For answers, see the bottom of the page.)

22222222

Suggested Answer (Check Your Progress 22)

1. T 2. T 3. F (m : n = 8 : 5)

4. F (a : b : c = 1 : 2 : 6) 5. T 6. T

213

Rate and Ratio

175

Statistics

• Unlessotherwisespecified,numericalanswersshouldbeeitherexactorcorrectto3significantfigures.

• Thediagramsarenotnecessarilydrawntoscale.

Section A(1)

1. (a) Findtherangeandtheinter-quartilerangeof12,13,15,15,19,20.

(b) Findthestandarddeviationof13,14,17,18,20. (Workingstepsarerequired.) (6marks)

Suggested Solution

(a) Range=20-12=8 1A

Q1=13andQ3=19 1M Inter-quartilerange=19-13

=6 1A

(b) Mean = + + + + =13 14 17 18 205

16 4. 1M

∴ Standarddeviation

=

− + − + − + −( . ) ( . ) ( . ) ( . )13 16 4 14 16 4 17 16 4 18 16 42 2 2 22

220 16 45

+ −( . ) 1M

=2.58(cor. to 3 sig. fig.) 1A

2. Determinethesamplingmethodusedineachofthefollowingcases.

(a) Thereare550computersproducedinafactory.Eachcomputerisassignedadistinctnumberfrom001to550.Tendistinctnumbersaregeneratedrandomlysoas toselect10computers to formasample.

(b) Agroupofchildrenareclassifiedinto4agegroups:A,B,CandD.Differentnumbersofchildrenareselectedfromeachgroup.

We should f ind the mean

first, then find the standard

deviation.

Divide the data into two parts:

12,13,15,15,19, 20

↑ ↑ Q1 = 13 Q3 = 19

First determine whether it is

probability sampling or non-

probability sampling.

(b) P(pass) = − =164125

61125

P(6marks | pass)=P(6marks∩pass)

P(pass)

=

=

112561

125161

1M

(c) P(allanswersareincorrect) =64125

Candidates should recognize all of the possible outcomes when calculating probabilities.

(b) P(pass)=P(6marks)+P(3marks)

= +

=

1125

12125

13125

1M

P(6marks | pass)=P(6marks∩pass)

P(pass)

1M=

=

1125131251

13

1A

1M

(c) P(allanswersareincorrect) = × ×

=

34

34

34

2764

1A

153

More about Probability

For (b), the candidate misses

the case of getting 0 mark

and therefore got an incorrect

P(pass).

For (c), the candidate does not

recognize that there are only

4 choices left and hence the

situation is different from that

of (a)(i).

F o r ( c ) , t h e c a n d i d a t e

recognizes that there are only

4 choices in the quiz on the

second attempt.

For (b), the candidate can

make use of the result in (a)

correctly to find P(pass).

114

Mathematics: Conventional Questions Compulsory Part Book 2

20.ThetruebearingofshipBfromislandAis120°.ThedistancebetweenAandBis120kmatnoon.Iftheshipsailsduenorthwithaspeedof20km/h,find

(a) theshortestdistancebetweentheshipandtheisland;

(b) thetimeatwhichthedistanceistheshortest. Hint 5

21.AgeographerstandsatZandfindsthatthetruebearingofXfromZis315°.Ifhewalks180mwesttoY,hefindsthatthecompassbearingofXfromYisN45°E.

(a) Whatkindoftriangleis∆XYZ?Explainyouranswer.

(b) Findthewidthoftheriver. Hint 6

Section B

22.Inthefigure,∠X=30°,∠Z=68°andXY=4cm.FindthelengthofYZ.

23.Inthefigure,BC=17,AC=12and∠A=72°.Find∠B.

24.Inthefigure,AB=3.5cm,AC=6.3cmand∠C=25°.Findq.

25.Find∠Dif AB = 2 cm and BD = 2 3 cm .

Fig. 17.50

Fig. 17.51

Fig. 17.52

Fig. 17.53

Fig. 17.54

Fig. 17.55

192

Mathematics: Conventional Questions Compulsory Part Book 2

Section A (1)Hint 1 Arrangethedatainascendingorderfirst.

Hint 2 FindthetotalnumberofmembersbeforefindingQ1andQ3.

Hint 3 Trytosubstituteavalueforyandfindthestandarddeviationofthenewsetofdata.

Hint 4 Samshouldchoosethreedigitseachtime.

Section A (2)Hint 5 Usethemethodofcompletingthesquare.

Section BHint 6 ConsiderthechangeofΣ(x-x)2after5compactdiscsareadded.

Hint 7 Substitutex=1.

Hint 8 ComparethestandardscoresinPaperAandPaperB.

Hint 9 Number=Totalnumber×percentage

Hint 10 Firstfindoutthepercentagesbelowandabovethemeanseparately.Thensumuptheresults.

97

Rate and Ratio

Section A(1)

1. (a) Themarkingrate=40worksheets

0.5hour =80worksheets/hour

(b) Themarkingrate

=40worksheets30minutes

=1.33worksheets/hour(cor. to 3 sig. fig.)

2. (a) Thecarriagerate=70×30packs

3vans =700packs/van

(b) Thecarriagerate=70×30×12eggs

3vans =8400eggs/van

3. (a) (i) Thespeed=320km4h

=80km/h

(ii) Thespeed=320×1000m4×60×60s

=22.2m/s(cor. to 3 sig. fig.)

(b) Thedistance=80km/h×1.5h =120km

4. (a) Thedistance=900m× 2030

=600m

(b) Thetimetaken=30minutes× 1050900

=35minutes

5. (a) Thewritingspeed=10pages

(550÷60)hours =1.09page/hour (cor. to 3 sig. fig.)

(b) Thewritingtimeforeachpage

=550minutes10pages

=55minutes/page

6. Thetimeneeded=30minutes×4

4 1+ =24minutes Thetimesaved=(30-24)minutes =6minutes

7. Thetime=35seconds× 85

=56seconds

8. Theflowrate=(2×5×200)m3

8s =250m3/s

In 8 seconds, the volume of water passing

through is (2 × 5 × 200) m3.

9. Thetotaldistancehetravels

=90km/h× 4060

h+70km/h×1 860

h

=139 13km

Thetotaltimehespends

=4060

+

1 860

hour

=14860

hour

Hisaveragespeed

=139 1

3 km

1 4860

hour

=77.4km/h(cor. to 3 sig. fig.)

Average Speed = Total Distance

Total Time

8.

Chapter 22 Rate and Ratio