Complex Numbers Mathematics Extension 2 Hsc Examination With Answer Key Page 4
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8HSC by Topic 1995 to 2006 Complex Numbers
Page 4
1
Find zw and
in the form x + iy.
HSC
w
(i)
Express 1 + 3 i in modulus-argument form.
01
2b
1
10
(ii)
Hence evaluate (1 + 3 i)
in the form x + iy.
1
HSC
01
2c
Sketch the region in the complex plane where the inequalities
3
π
π
|z + 1 – 2i| ≤ 3 and -
≤ arg z ≤
both hold.
HSC
4
3
4
01
2d
Find all solutions of the equation z
= –1.
3
Give your answers in modulus-argument form.
HSC
01
2e
In the diagram the vertices of a triangle ABC are
represented by the complex numbers z
, z
and z
,
HSC
1
2
3
respectively. The triangle is isosceles and right-
angled
at B.
2
2
2
(i)
Explain why (z
– z
)
= –(z
– z
)
.
1
1
2
3
2
(ii)
Suppose D is the point such that ABCD is a
square. Find the complex number, expressed
in
terms of z
, z
and z
, that represents D.
1
2
3
1
01
7a
Suppose that z =
(cos θ + i sin θ) where θ is real.
2
HSC
1
(i)
Find |z|.
3
(ii)
Show that the imaginary part of the geometric series
1
2
sin
θ
2
3
1 + z + z
+ z
+ … =
is
.
1
z
θ
−
5
−
4
cos
2
1
1
1
(iii)
Find an expression for 1 +
cos θ +
cos 2θ +
cos 3θ + … in terms of
2
3
2
2
2
cos θ .
2
00
2a
Find all pairs of integers x and y that satisfy (x + iy)
= 24 + 10i.
3
HSC
π
π
6
00
2c
4
(i)
Let z = cos
+ i sin
. Find z
.
6
6
HSC
6
(ii)
Plot, on the Argand diagram, all complex numbers that are solutions of z
= –1.
00
2d
Sketch the region in the Argand diagram that satisfies the inequality
3
z z + 2(z+ z ) ≤ 0
HSC
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