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New Senior Secondary New Senior Secondary Mathematics Curriculum Mathematics Curriculum Public Examination of Extended Part Public Examination of Extended Part HKEAA HKEAA November 2008 November 2008

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New Senior Secondary New Senior Secondary

Mathematics Curriculum Mathematics Curriculum

Public Examination of Extended PartPublic Examination of Extended Part

HKEAAHKEAA

November 2008November 2008

2

Public Examination of Extended PartPublic Examination of Extended Part

� Result of Questionnaires after Third Consultation

� Examination Format

� Sample Paper

� Draft Level Descriptor

� Annotated Examples

M1 & M2

3

Results of Questionnaires Results of Questionnaires

after Third Consultationafter Third Consultation

4

The proposed assessment objectives of the public assessment are The proposed assessment objectives of the public assessment are

in alignment with the aims and objectives of the curriculum. (M1in alignment with the aims and objectives of the curriculum. (M1))

0.0%

10.0%

20.0%

30.0%

40.0%

50.0%

60.0%

1 2 3 4 5 6

Strongly Disagree Strongly Agee

5

The proposed paper structure for the public examination is The proposed paper structure for the public examination is

appropriate. (M1)appropriate. (M1)

0.0%

10.0%

20.0%

30.0%

40.0%

50.0%

60.0%

70.0%

1 2 3 4 5 6

Strongly Disagree Strongly Agree

6

The proposed duration of the public examination is The proposed duration of the public examination is

appropriate. (M1)appropriate. (M1)

0.0%

10.0%

20.0%

30.0%

40.0%

50.0%

60.0%

1 2 3 4 5 6

Strongly Disagree Strongly Agree

7

The proposed assessment objectives of the public assessment are The proposed assessment objectives of the public assessment are

in alignment with the aims and objectives of the curriculum. (M2in alignment with the aims and objectives of the curriculum. (M2))

0.0%

10.0%

20.0%

30.0%

40.0%

50.0%

60.0%

1 2 3 4 5 6

Strongly Disagree Strongly Agree

8

The proposed paper structure for the public The proposed paper structure for the public

examination is appropriate. (M2)examination is appropriate. (M2)

0.0%

10.0%

20.0%

30.0%

40.0%

50.0%

60.0%

1 2 3 4 5 6

Strongly Disagree Strongly Agree

9

The proposed duration of the public examination is The proposed duration of the public examination is

appropriate. (M2)appropriate. (M2)

0.0%

10.0%

20.0%

30.0%

40.0%

50.0%

60.0%

70.0%

1 2 3 4 5 6

Strongly Disagree Strongly Agree

10

Examination FormatExamination Format

Module 1 (Calculus and Statistics)

2 1/2 hours100%Conventional questionsPublic Examination

DurationWeightingComponent

2 1/2 hours100%Conventional questionsPublic Examination

DurationWeightingComponent

Module 2 (Algebra and Calculus)

11

Some Features in Both M1 and M2Some Features in Both M1 and M2

� Compulsory Part + Secondary 1-3 Mathematics Curriculum

� Two Sections� Section A: 8-12 short questions

� Section B: 3-5 long questions

� All the questions to be attempted

� Each question may not carry the same mark

12

Some of the contrasts between the content Some of the contrasts between the content

of the HKDSE Mathematics (M1) and the of the HKDSE Mathematics (M1) and the

present HKALE Mathematics and Statisticspresent HKALE Mathematics and Statistics

� Expectation

� Sampling Distribution

� Point Estimates

� Confidence Interval

13

Some of the contrasts between the content Some of the contrasts between the content

of the HKDSE Mathematics (M2) and the of the HKDSE Mathematics (M2) and the

present HKALE Pure Mathematicspresent HKALE Pure Mathematics

� Vectors in R3

� Vector Product

14

Sample Paper (M1)Sample Paper (M1)

Section A

1. Expand the following in ascending powers of x as far as the term in :

(a) ;

(b) .

2x

xe

2−

xe

x

2

6)21( +

15

3.A political party studied the public view on a certain government policy. A random sample of 150 people was taken and 57 of them supported this policy.

(a) Estimate the population proportion supporting this policy.

(b) Find an approximate 90% confidence interval for the population proportion.

16

5. A manufacturer produces a large batch of light bulbs, with a mean lifetime of 640 hours and a standard deviation of 40 hours. A random sample of 25 bulbs is taken. Find the probability that the sample mean lifetime of the 25 bulbs is greater than 630 hours.

17

7. The random variable X has probability distribution for ,2 and 3 as shown in the following table.

Calculate(a) ,

(b) .

0.30.60.1

321x

)(P xX = 1=x

)(E X

)23(Var X−

)(P xX =

18

Section B12.

(a) Let be a function defined for all . It is given that

where a and b are negative constants and , and .

(i) Find the values of a and b .

(ii) By taking , find .

)f(t 0≥t

8)(f 2 ++=′ btbtaeet

0)0(f =

3(0)f =′ 73.4)1(f =′

5.0−=b )12f(

19

(b) Let be another function defined for all .

It is given that

,

where k is a positive constant. Figure 1 shows a sketch of the graph of against t . It is given that

attains the greatest value at and .

)g(t 0≥t

kttet

−=′10

33)(g

)(g t′

)(g t′ 5.7=t 0)0(g =

20

(i) Find the value of k .

(ii) Use the trapezoidal rule with four sub-intervals to estimate .

(iii) From the estimated value obtained in (b)(ii) and Figure 1, Jenny claims that . Do you agree? Explain your answer.

30 6 9 12

t

)(g t′

)12g(

)12f()12g( >

21

14. The Body Mass Index (BMI) value (in ) of

children aged 12 in a city are assumed to follow a

normal distribution with mean and standard

deviation 4.5 .

(a) A random sample of nine children aged 12 is

drawn and their BMI values (in ) are

recorded as follows:

16.0 , 18.3 , 15.2 , 17.8 , 19.5 , 15.9 , 18.6 , 22.5 , 23.6

(i) Find an unbiased estimate for .

(ii) Construct a 95% confidence interval for .

µ 2m/kg2m/kg

2m/kg

2m/kg

µ

µ

22

(b) Assume . If a random sample of 25 children aged 12 is drawn and their BMI values are recorded,

find the probability that the sample mean is less than 17.8 .

7.18=µ

2m/kg

23

(c) A child aged 12 having a BMI value greater than

25 is said to be overweight. Children aged 12

are randomly selected one after another and their BMI values are recorded until two overweight children are

found. Assume that .

(i) Find the probability that a selected child is overweight.

(ii) Find the probability that more than eight children have to be selected in this sampling process.

(iii) Given that more than eight children will be selected in this

sampling process, find the probability that exactly ten children

are selected.

2m/kg

7.18=µ

24

Draft Level Descriptors (M1)Draft Level Descriptors (M1)

The typical performance of candidates at this level:Level 5� demonstrate comprehensive knowledge and understanding of the

calculus and statistics concepts in the curriculum by applying them successfully at a sophisticated level to a wide range of unfamiliar situations

� communicate and express views and arguments precisely and logically using mathematical language, notations, tables, diagrams and graphs

� formulate mathematical models successfully in complex situations, employ appropriate strategies to arrive at a complete solution, and evaluate the significance and reasonableness of results

� integrate knowledge and skills from different areas of the curriculum in handling complex tasks using a variety of strategies

25

Level 4� demonstrate sound knowledge and understanding of the calculus

and statistics concepts in the curriculum by applying them successfully to unfamiliar situations

� communicate and express views and arguments accurately using mathematical language, notations, tables, diagrams and graphs

� formulate mathematical models successfully in unfamiliar situations, employ appropriate strategies to arrive at a solution, and explain the significance and reasonableness of results

� integrate knowledge and skills from different areas of the curriculum in handling a range of tasks

26

Level 3� demonstrate adequate knowledge and understanding of the calculus

and statistics concepts in the curriculum by applying them successfully to familiar and some unfamiliar situations

� communicate and express views and arguments appropriately using mathematical language, notations, tables, diagrams and graphs

� formulate mathematical models in familiar situations, employ relevant mathematical techniques to arrive at some results, and are aware of the significance and reasonableness of results

� integrate knowledge and skills from different areas of the curriculum in handling mathematical tasks in explicit situations

� apply calculus and statistics techniques in solving simple real-life problems

27

Level 2� demonstrate basic knowledge and understanding of the calculus

and statistics concepts in the curriculum by employing simple algorithms, formulas and procedures in performing routine tasks

� communicate and express relevant views using mathematical language, notations, tables, diagrams and graphs

� define appropriate parameters or variables in simple tasks and employ elementary mathematics techniques to arrive at some results

� use differentiation and integration in handling straightforward and clearly defined problems

� apply some properties of binomial, geometric, Poisson and normaldistributions in simple cases

28

Level 1

� demonstrate elementary knowledge and understanding of the calculus and statistics concepts in the curriculum by performingstraightforward procedures according to direct instructions

� communicate and express simple ideas using mathematical language, notations, tables, diagrams and graphs

� define some parameters or variables that are relevant to the context

� find derivatives and integrals of simple functions

� find probabilities in straight-forward cases

29

Annotated Examples (M1)

30

Level 5 (Q13)

There are 80 operators in an emergency hotline centre. Assumethat the number of incoming calls for the operators are independent and the number of incoming calls for each operator is distributed as Poisson with mean 6.2 in a ten-minute time interval (TMTI). An operator is said to be idle if the number of incoming calls received is less than three in a certain TMTI.

(a)Find the probability that a certain operator is idle in a TMTI.

(b)Find the probability that there are at most two idle operators in a TMTI.

(c) A manager, Calvin, checks the numbers of incoming calls of the operators one by one in a TMTI. What is the least number of operators to be checked so that the probability of finding an idle

operator is greater than 0.9 ?

31

Level 5 (Q13)

Demonstrate

comprehensive

knowledge and

understanding

of the statistics

concepts

Formulate

mathematical

models

successfully

Apply

Poisson and

binomial

probabilities

accurately

32

L is the tangent to the curve at .

(a) Find the equation of the tangent L .

(b) Using the result of (a), find the area bounded by the y-

axis, the tangent L and the curve C .

7: 3 += xyC 2=x

Level 4 (Q9)

33

Level 4 (Q9)

Demonstrate

sound

knowledge and

understanding

of the calculus

concepts

The area of

triangle is

incorrect

Apply

differentiation

to find the

equation of

tangent

Integrate

knowledge of

differentiation

and integration

34

Level 3 (Q6)

Let , where .

(a) Use logarithmic differentiation to express in terms of u and x .

(b) Suppose , express in terms of x .

)2)(1(

32

++

+=

xx

xu 1−>x

x

u

d

d

yu 3=

x

y

d

d

35

Level 3 (Q6)

Demonstrate

adequate

knowledge and

understanding

of the calculus

concepts

The derivative

is incorrect

Apply log

differentiation

and chain rule

36

Level 2 (Q14)

(Refer to Slide 21)

The s.d. of

sample mean

is incorrect

Apply normal

probability in

this simple case

37

Demonstrate

basic

knowledge and

understanding

of the statistics

concepts

Apply binomial

probability in

this simple case

The numerator of

the conditional

probability is

incorrect

38

Level 1 (Q11)

The manager, Mary, of a theme park starts a promotion plan to increase the daily number of visits to the park. The rate of change of the daily number of visits to the park can be modelled by

where N is the daily number of visits (in hundreds) recorded at the end of a day, t is the number of days elapsed since the start of the plan and k is a positive constant.Mary finds that at the start of the plan, and .

(a) (i) Let , find .

(ii) Find the value of k , and hence express N in terms of t .

(b) (i) When will the daily number of visits attain the greatest value?

(ii) Mary claims that there will be more than 50 hundred visits on a certain day after the start of the plan. Do you agree? Explain your answer.

(c) Mary’s supervisor believes that the daily number of visits to the park will return to the original one at the start of the plan after a long period of time. Do you agree? Explain your answer.

(Hint: .)

te

tk

t

N

t 4

)25(

d

d

04.0 +

−= ( )0≥t

ttev

04.041

−+=t

v

d

d

0lim 04.0 =−

∞→

t

tte

10=N 50d

d=

t

N

39

Level 1 (Q11)

Demonstrate

elementary

knowledge and

understanding

of the calculus

concepts

The derivative

is incorrect

Cannot use

(a)(i) to find an

appropriate

substitution

40

Sample Paper (M2)Sample Paper (M2)

Section A

2. A snowball in a shape of sphere is melting with its volume decreasing at a constant rate of 4 . When its radius is 5 cm, find the rate of change of its radius.

4. Find .

13scm

xx

x d1

42∫

41

8. (a) Using integration by parts, find .

(b)

Figure 1

The inner surface of a container is formed by revolving the curve (for ) about the y-axis (see Figure 1). Find the capacity of the container.

xxx dcos∫

π≤≤ x0xy cos−=

42

9.

Figure 2

Let , and .

Figure 2 shows the parallelepiped

OADBECFG formed by , and .

ji 34 +=OA kj += 3OB kji 53 ++=OC

OA OB OC

OA

B

C

D

E

F

G

43

(a) Find the area of the parallelogram OADB .

(b) Find the volume of the parallelepiped

OADBECFG .

(c) If is a point different from C such that

the volume of the parallelepiped formed

by , and is the same as that of

OADBECFG , find a possible vector of .

C′

OA OB CO ′

CO ′

44

12. Let , where .

(a) (i) Find the x- and y-intercept(s) of the

graph of .

(ii) Find and prove that

for .

(iii)For the graph of , find all the

extreme points and show that there are

no points of inflexion.

11

4

1

4)(f −

+−

−=

xxx 1±≠x

)(f xy =

)(f x′

33

2

)1()1(

)13(16)(f

+−

+=′′

xx

xx

1±≠x

)(f xy =

45

(b) Find all the asymptote(s) of the graph of .

(c) Sketch the graph of .

(d) Let S be the area bounded by the graph of , the straight lines ,

and .

Find S in terms of a .

Deduce that .

)(f xy =

)(f xy =

)(f xy = 3=x

( )3>= aax 1−=y

2ln4<S

46

14.

Figure 3

In Figure 3, is an acute-angled triangle, where Oand H are the circumcentre and orthocentre

respectively. Let , , and

H

O

B C

A

a=OA b=OB c=OC h=OH

ABC∆

47

(a) Show that .

(b) Let , where t is a non-zero constant.

Show that

(i) for some scalar s ,

(ii) .

(c) Express h in terms of a , b and c .

)//()( cbah +−

)( cbah +=− t

)()( acbacb +=−++ st

0)())(1( =−⋅−− acabt

48

Draft Level Descriptors (M2)Draft Level Descriptors (M2)

The typical performance of candidates at this level:Level 5� demonstrate comprehensive knowledge and understanding of the

algebra and calculus concepts in the curriculum by applying themsuccessfully at a sophisticated level to a wide range of unfamiliar situations

� communicate and express views and arguments precisely and logically using mathematical language, notations, tables, diagrams and graphs

� provide complex mathematical proofs successfully in a logical, rigorous and concise manner

� integrate knowledge and skills from different areas of the curriculum in handling complex tasks using a variety of strategies

49

Level 4� demonstrate sound knowledge and understanding of the

algebra and calculus concepts in the curriculum by applying them successfully to unfamiliar situations

� communicate and express views and arguments accurately using mathematical language, notations, tables, diagrams and graphs

� provide mathematical proofs successfully in a logical manner

� integrate knowledge and skills from different areas of the curriculum in handling a range of tasks

50

Level 3� demonstrate adequate knowledge and understanding of the algebra

and calculus concepts in the curriculum by applying them successfully to familiar and some unfamiliar situations

� communicate and express views and arguments appropriately using mathematical language, notations, tables, diagrams and graphs

� employ appropriate strategies to provide mathematical proofs� integrate knowledge and skills from different areas of the curriculum

in handling mathematical tasks in explicit situations� apply algebra and calculus techniques in solving simple problems

involving mathematical contexts successfully

51

Level 2� demonstrate basic knowledge and understanding of the algebra and

calculus concepts in the curriculum by employing simple algorithms, formulas and procedures in performing routine tasks

� communicate and express relevant views using mathematical language, notations, tables, diagrams and graphs

� employ routine techniques to arrive at some preliminary results in providing mathematical proofs

� use differentiation and integration in handling straightforward and clearly defined problems

� perform matrices and vectors operations in simple situations andsolves systems of linear equations

52

Level 1

� demonstrate elementary knowledge and understanding of the algebra and statistics concepts in the curriculum by performing straightforward procedures according to direct instructions

� communicate and express simple ideas using mathematical language, notations, tables, diagrams and graphs

� find derivatives and integrals of simple functions

� perform straightforward matrix and vector manipulations

53

Annotated Examples (M2)

54

Level 5 (Q12)

(Refer to Slide 44)

The

presentation

of intercepts

is incorrect

The second

derivative is

positive only

on a certain

range

55

The graphs

of the curve

and

asymptotes

are

correctly

sketched

56

Demonstrate

comprehensive

knowledge and

understanding of

the algebra and

calculus concepts

The limit

notation is not

very appropriate

Can integrate

knowledge

from different

areas of

curriculum

57

Level 4 (Q12)

(Refer to Slide 44)

The

presentation

of intercepts

is incorrect

Complete

the proof

successfully

58

The graphs of

the curve and

asymptotes

are correctly

sketched

59

Demonstrate

sound knowledge

and

understanding of

the algebra and

calculus concepts

The integrand

is not of the

required area

60

Level 3 (Q11)

Let , and .

(a)Let I and O be the identity matrix and zero matrix respectively.

(i) Prove that .

(ii) Using the result of (i), or otherwise, find .

(b) (i) Prove that .

(ii) Prove that D and A are non-singular.

(iii) Find .

Hence, or otherwise, find .

=

101

011

002

A

=

111

110

100

P

=

200

010

001

D

33×OIPPP =+−− 23 2

1−P

APPD1−=

1001)( −D

1001)( −A

61

Level 3 (Q11)

Apply matrix

techniques in

simple parts

62

Demonstrate

adequate

knowledge and

understanding of

the algebra and

calculus concepts

Employ appropriate

strategies, although

not flawless, to

provide

mathematical

proofs

63

Level 2 (Q7)

Solve the system of linear equations

=++

=+−

−=−+

934

13743

4z67

zyx

zyx

yx

.

64

Level 2 (Q7)

Demonstrate

basic knowledge

and understanding

of the algebra and

calculus concepts

Apply Gaussian

elimination to

solve system of

linear equations

Wrongly

introduce two

parameters

65

Level 1 (Q3)

The slope at any point of a curve is given by .

It is given that the curve passes through the point .

Find the equation of the curve.

),( yx )1ln(2d

d 2 += xxx

y

)1,0(

Wrongly apply

integration by

parts rather than

substitution

Demonstrate

elementary knowledge

and understanding of

the algebra and

calculus concepts

66

Thank you