Complex Numbers Mathematics Extension 2 Hsc Examination With Answer Key Page 5
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8HSC by Topic 1995 to 2006 Complex Numbers
Page 5
00
2e
In the Argand diagram, OABC is a rectangle,
3
where OC = 2OA. The vertex A corresponds to
HSC
the complex number ω.
(i)
What complex number corresponds to the
vertex C?
(ii)
What complex number corresponds to the
point of intersection D of the diagonals OB
and AC?
99
2a
Let z = 3 + 2i and w = –1 + i.
3
Express the following in the form a + ib, where a and b are real numbers:
HSC
2
(i)
zw
(ii)
iw
α
Let
= 1 + i 3 .
99
2b
4
α
α
(i)
Find the exact value of |
| and arg
.
HSC
11
α
(ii)
Find the exact value of
in the form a + ib, where a and b are real numbers.
99
2c
Sketch the region in the Argand diagram where the two inequalities
2
π
|z – i| ≤ 2 and 0 ≤ arg (z + 1) ≤
both hold.
HSC
4
99
2e
The points A and B in the complex plane
4
correspond
to
complex numbers z
and
z
HSC
1
2
respectively. Both triangles OAP and OBQ are
right-angled isosceles triangles.
(i)
Explain why P corresponds to the complex
number (1 + i) z
.
1
(ii)
Let M be the midpoint of PQ. What
complex
number corresponds to M?
2
π
2
π
2
4
99
8a
α
8
Let ρ = cos
+ i sin
. The complex number
= ρ + ρ
+ ρ
is a root of the
7
7
HSC
2
quadratic equation x
+ ax + b = 0, where a and b are real.
2
6
(i)
Prove that 1 + ρ + ρ
+… + ρ
= 0.
(ii)
The second root of the quadratic equation is β. Express β in terms of positive
powers of ρ. Justify your answer.
(iii)
Find the values of the coefficients a and b.
π
2
π
3
π
7
(iv)
Deduce that –sin
+ sin
+ sin
=
.
7
7
7
2
1998
98
2a
Evaluate i
.
2
HSC
18
4
+
i
98
2b
5
Let z =
3
i
−
HSC
(i)
Simplify (18 + 4i) (
3
−
i
)
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