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