HONG KONG DIPLOMA OF SECONDARY EDUCATION
EXAMINATION
MATHEMATICS
Compulsory Part
SCHOOL-BASED ASSESSMENT
Sample Assessment Task
Complex Number and Geometry
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
Instructions
1.
This task consists of FOUR parts, Part A to Part D. Students are required to attempt ALL parts.
2.
Numerical answers should be exact or correct to 3 significant figures.
3.
All working must be clearly shown.
Teacher Guideline
1.
This task requires students to
(a)
prove trigonometry formulas ;
(b)
use trigonometric formulas to solve problems in complex numbers ;
(c)
interpret the complex numbers geometrically.
2.
It is anticipated that in general students will be able to complete the task in 1.5 hours. However,
teachers can exercise their professional judgment to adjust the time allowed to cater for the ability
level of their students.
3.
Teachers may feel free to modify the question and the marking guidelines to cater for the ability
level of their students.
4.
Feedback should be provided to students after marking the task. For instance, teachers can discuss
different approaches of handling each part of the task with students.
Complex Numbers
2
A
D
X
T
c
a
b
C
Y
B
Figure 1
Part A
1.
Figure 1 shows the quadrilateral ABCD . Given that
DBY  b .
AB  1 , A BY c , ABD  a , and
(a)
Find the length of AY and BY in terms of c .
(b)
Considering ADX , find the lengths of AX and CY in terms of a and b .
(c)
Considering CBD , find the lengths of XY and CB in terms of a and b .
(d)
Using the above results, find the formulas for cos(a  b) and sin(a  b) .
Part B
2.
Using the results of Question 1(d), show that
(a)
sin 2 x  2sin x cos x and cos 2 x  cos 2 x  sin 2 x .
(b)
sin 3 x  3sin x  4sin 3 x and cos3 x  4cos3 x  3cos x .
Part C
3.
It is known that the complex number i satisfies the condition i 2  1 .
(a)
Expand
 cos x  i sin x 2
(b)
Expand
 cos x  i sin x 4
(c)
(i)
Hence, guess the formula for
(ii)
Is the above formula correct when n  3 ? Verify your assertion.
Complex Numbers
in terms of sin 2x and cos 2x .
in terms of sin 4x and cos 4x .
 cos x  i sin x n .
3
Part D
4.
It is given that the complex number a  ib is represented by the point P(a, b) in the rectangular
coordinate system.
(a)
(b)
a 2  b2 is the distance from the origin to P .
Regarding the complex numbers below, find the corresponding coordinates (in surd form)
and their distances to the origin. Also, mark the points in Figure 2 below.
(i)
cos30  i sin 30
(ii)
 cos30  i sin302
(iii)
 cos30  i sin307
State the geometric features relating to the 3 points (corresponding to the 3 complex numbers
in (a)).
(Hint: “Two of the points and the origin are on the same straight line” is an example of the
said geometric features.)
Figure 2
END OF ASSESSMENT TASK
Complex Numbers
4