Complex Numbers Mathematics Extension 2 Hsc Examination With Answer Key
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8HSC by Topic 1995 to 2006 Complex Numbers
Page 1
Mathematics Extension 2 HSC Examination
Topic: Complex Numbers
06
2a
Let z = 3 + i and w = 2 – 5i. Find, in the form x + iy,
2
(i)
z
1
HSC
(ii)
z w
1
w
(iii)
1
z
(i)
Express
3 - i in modulus-argument form.
06
2b
1
7
(ii)
Express ( 3 - i)
in modulus-argument form.
1
HSC
7
(iii)
Hence express ( 3 - i)
in the form x + iy.
2
3
06
2c
Find, in modulus-argument form, all solutions of z
= –1.
3
HSC
06
2d
The equation |z – 1 – 3i| + |z – 9 – 3i|corresponds to an ellipse in the Argand
diagram.
HSC
(i)
Write down the complex number corresponding to the centre of the ellipse.
1
(ii)
Sketch the ellipse, and state the lengths of the major and minor axes.
3
(iii)
Write down the range of values of arg(z) for complex numbers z corresponding
1
to points on the ellipse.
05
2a
Let z = 3 + i and w = 1 – i. Find, in the form x + iy ,
(i)
2z + iw
1
HSC
(ii)
z w
1
6
1
(iii)
w
Let β = 1 - i 3
05
2b
(i)
Express β in modulus-argument form.
2
HSC
5
(ii)
Express β
in modulus-argument form.
2
5
(iii)
Hence express β
in the form x + iy.
1
05
2c
Sketch the region on the Argand diagram where the inequalities
|z - z | < 2 and |z − 1| ≥1 hold simultaneously.
3
HSC
05
2d
Let l be the line in the complex plane that passes
α
through the origin and makes an angle
with the
HSC
α
positive real axis, where 0 < arg(z
) <
. The point P
1
represents the complex number z
, where
1
α
0 < arg(z
) <
. The point P is reflected in the line l
1
to produce the point Q, which represents the
complex number z
. Hence |z
| = |z
|.
2
1
2
α
(i)
Explain why arg(z
) + arg(z
) = 2
.
2
1
2
2
(ii)
Deduce that z
z
= |z
|
(cos 2
α
+ i sin 2
α
)
1
1
2
1
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