HONG KONG DIPLOMA OF SECONDARY EDUCATION
EXAMINATION
MATHEMATICS
Compulsory Part
SCHOOL-BASED ASSESSMENT
Sample Assessment Task
Complex Number and Geometry
Marking Guidelines
教育局 課程發展處 數學教育組
Mathematics Education Section, Curriculum Development Institute
The Education Bureau of the HKSAR
Assessment Scale
The assessment scale for tasks on Mathematical Investigation is shown in the following table.
Level of
Performance
Very good
Good
Fair
Weak
Marks
Mathematical Knowledge and Investigation Skills
13–16
The student demonstrates a complete understanding of
the underlying mathematical knowledge and
investigation skills which are relevant to the task, and is
consistently competent and accurate in applying them in
handling the task. Typically, the student recognises
patterns, and states a conjecture which is justified by a
correct proof.
9–12
The student demonstrates a substantial understanding of
the underlying mathematical knowledge and
investigation skills which are relevant to the task, and is
generally competent and accurate in applying them in
handling the task. Typically, the student recognises
patterns, draws inductive generalisations to arrive at a
conjecture and attempts to provide a proof for the
conjecture.
5–8
The student demonstrates a basic understanding of the
underlying mathematical knowledge and investigation
skills which are relevant to the task, and is occasionally
competent and accurate in applying them in handling
the task. Typically, the student recognises patterns and
attempts to draw inductive generalisations to arrive at a
conjecture.
1–4
The student demonstrates a limited understanding of the
underlying mathematical knowledge and investigation
skills which are relevant to the task, and is rarely
competent and accurate in applying them in handling
the task. Typically, the student manages to recognise
simple patterns and organises results in a way that helps
pattern identifications.
Marks Mathematical Communication Skills
4
The student communicates ideas in
a clear, well organised and logically
true manner through coherent
written/verbal accounts, using
appropriate and correct forms of
mathematical
presentation
to
present conjectures and proofs.
3
The student is able to communicate
ideas
properly
through
written/verbal accounts, using
appropriate and correct forms of
mathematical presentation such as
algebraic
manipulations
and
geometric deductions.
2
The student is able to communicate
ideas with some appropriate forms
of mathematical presentation such
as
algebraic
formulae
and
geometric facts.
1
The
student
attempts
to
communicate
ideas
using
mathematical
symbols
and
notations,
diagrams,
tables,
calculations etc.

The full mark of a SBA task on Mathematical Investigation submitted should be scaled to 20 marks, of which 16 marks
are awarded for the mathematical knowledge and investigation skills while 4 marks are awarded for the communication
skills.

Teachers should base on the above assessment scale to design SBA tasks for assessing students with different abilities
and to develop the marking guidelines.
Complex Numbers_Marking Guidelines
2
Marking Guidelines
Solution
Performance
Part A
Evidence:
The solutions
A
D
c
Fair:
Some mistakes are found
in the working and not all
the answers are correct
Good:
All answers are correct but
minor mistakes are found in
the working
a
b
C
(a)
Attempted the question but
all answers are incorrect
X
T
1.
Weak:
Y
Very good: All answers and working
are correct
B
sin c 
AY
AY
 sin c 
 AY  sin c
AB
1
cos c 
BY
BY
 cos c 
 BY  cos c
AB
1
(b) In A BD ,
sin a 
AD
AD
 sin a 
 AD  sin a ------------------(1)
AB
1
Regarding the similar triangles XTD and YTB ,
XDT  YBT  b .
 XDA  90  XDT  90  b
------------------(2 )
In ADX , and using the results (1) and (2),
AX
AX
 sin XDA 
 sin(90  b)  AX  sin a cos b
AD
sin a
DX
DX
 cos XDA 
 cos(90  b)  DX  sin a sin b
AD
sin a
 CY  DX  CY  sin a sin b
Complex Numbers_Marking Guidelines
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Marking Guidelines
Solution
(c)
In ABD ,
DB
DB
cos a 
 cos a 
 DB  cos a
AB
1
In CBD , using (1),
CB
CB
 cos b 
 cos b  CB  cos a cos b
DB
cos a
CD
CD
 sin b 
 sin b  CD  cos a sin b
DB
cos a
XY  CD
Performance
---------------(1)
Evidence:
The solutions
Weak:
Attempt the question but
the working is incorrect
Fair:
Some mistakes are found
in the working and not all
the answers are correct
Good:
All answers are correct but
minor mistakes are found in
the working
 XY  cos a sin b
(d) Since AY  AX  XY
sin c  sin a cos b  cos a sin b
sin(a  b)  sin a cos b  cos a sin b
Since CB  CY  BY
cos a cos b  sin a sin b  cos c
 cos(a  b)  cos a cos b  sin a sin b
--------(1)
Very good: All answers and working
are correct
---------(2)
Part B
2.
(a)
Evidence:
The proof
With the results in Question 1(d),
put a = x and b = x into formulas (1) and (2), we get,
sin 2 x  sin x cos x  cos x sin x
sin 2 x  2sin x cos x
and
cos 2 x  cos x cos x  sin x sin x
cos 2 x  cos2 x  sin 2 x
Weak:
Attempt the question but
fail to use the formulas in
Question 1(d)
Fair:
Some mistakes are found in
the working and the proof is
not totally correct
Good:
The proof is correct but
minor mistakes are found in
the working
Very good: The proof is correct
Evidence:
The proof
(b) With the results in Question 1(d),
put a = x and b = x into formulas (1), we get,
sin 3x  sin x cos 2 x  cos x sin 2 x
sin3x  sin(cos2 x  sin 2 x)  cos x(2sin x cos x)
Weak:
Attempt the question but
fail to use the formulas in
Part A(d)
Fair:
Some mistakes are found in
the working and the proof is
not totally correct
Good:
The proof is correct but
minor mistakes are found in
the working
sin 3x  sin x cos2 x  sin3 x  2sin x cos2 x
sin3x  3sin x(1  sin 2 x)  sin3 x
sin 3x  3sin x  4sin x
2
put a = x and b = x into formulas (2), we get ,
cos3x  cos x cos 2 x  sin x sin 2 x
cos3x  cos x(cos2 x  sin 2 x)  sin x(2sin x cos x)
cos3x  cos3 x  cos x sin 2 x  2sin 2 x cos x
Very good: The proof is correct
cos3x  cos3 x  cos x(1  cos2 x)  2(1  cos2 x)cos x
cos3x  4cos3 x  3cos x
Complex Numbers_Marking Guidelines
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Marking Guidelines
Solution
Performance
Evidence:
The solutions
Weak:
Attempt the question but
fail to expand the given
expression
Part C
3.
(a)
 cos x  i sin x 2
 cos2 x  2i cos x sin x  i 2 sin 2 x
 cos2 x  sin 2 x  2i cos x sin x
 cos 2 x  i sin 2 x
(b)
 cos x  i sin x 4
  cos2x  i sin 2x 
 cos 4 x  i sin 4 x
2
by (a)
by (a)
Fair:
Some mistakes are found in
the working and not all
answers are correct
Good:
All answers are correct but
minor mistakes are found in
the working
Very good: All answers and working are
correct
(c)
(i)
Evidence:
1. The formula proposed
2. The verification
 cos x  i sin x n  cos nx  i sin nx
(ii) Yes, it does.
 cos x  i sin x 3
Weak:
 cos3 x  3i cos 2 x sin x  3cos x  i sin x    i sin x 
2
3
 cos3 x  3i cos 2 x sin x  3cos x sin 2 x  i sin 3 x
Fair:
 cos3 x  3cos x sin 2 x  3i cos 2 x sin x  i sin 3 x
 cos3 x  3cos x(1  cos 2 x)  3i(1  sin 2 x)sin x  i sin 3 x
 4cos3 x  3cos x  3i sin x  i 4sin 3 x
 cos3 x  i sin 3 x
Good:
Attempt the question but the
formula proposed is
incorrect
The formula proposed is
correct but the verification
is not completed
The formula proposed is
correct but minor mistakes
are found in the verification
Very good: The formula proposed and
the verification are correct
Part D
4.
(a)
(i)
Let the point be A.
 3 1
Coordinates of A   cos30,sin 30   
, .
 2 2
Distance from the origin to the point = cos2 30  sin 2 30  1
Evidence:
1. The coordinates of points
2. The distances
3. The positions of points in the
coordinate plane
Weak:
Attempt the question but all
the answers are incorrect
Fair:
Not all answers are correct
Good:
All the answers are correct
but minor mistakes in
marking the points are
found
(ii) Let the point be B.
1 3
Coordinates of B   cos60,sin 60    ,
.
2 2 
Distance from the origin to the point B = cos2 60  sin 2 60  1
(iii) Let the point be C.

3 1
Coordinates of C   cos 210,sin 210    
,  .
 2
2
2
Distance from the origin to the point C = cos 210  sin 2 210  1
Complex Numbers_Marking Guidelines
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Very good: All are correct
Marking Guidelines
Solution
4.
Performance
(b)
B
A
60
30
30
C
The following are some of the geometric features required:
Evidence:
Geometric features (other than the one
given in the example
(1) Points A, B and C are on the circle with centre at the origin and the radius
being 1 unit.
Weak:
Attempt the question but
fail to give any feature
Fair:
Give one relevant feature
Good:
Give two relevant features
Let the origin be O.
(2) Points A and C are the end points of the diameter of the circle.
(3) If the circle cuts the positive x-axis and negative x-axis at D and E
respectively, AOD and COE are vertically opposite angles.
Very good: Give three or more relevant
features
Complex Numbers_Marking Guidelines
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