561 Answers Chapter 3

Selected answers are given here. For a full set of worked solutions, see Specialist Maths Dimensions Units 3 & 4 Solutions and Study Guide.
Chapter 3
Exercise 3A
1 a Re(z) = 3 and Im(z) = 2
b Re(z) = 5 and Im(z) = −6
c Re(z) = 2 and Im(z) = 0
d Re(z) = 0 and Im(z) = 3
e Re(z) = 0 and Im(z) = 0
f Re(z) = a and Im(z) = 0
g Re(z) = 0 and Im(z) = b
h Re(z) =
2 and Im(z) = −5
3
i Re(z) = -- and Im(z) = 0
4
j
Re(z) = c and Im(z) = 2d
k Re(z) = 0 and Im(z) =
3
l Re(z) = –-- and Im(z) = 0
5
2
2 a 3i b 5i c 10i d 7i e --i f ai
3
3 a −16 b −4 c −7 d −9 e 1 f i
4 a 3 b 2 c −7 d 7 e 0
2
f 0 g −1 h –-- i 0
5
5 a 4 − 2i b −6 + 3i c 2 d −1
6 a x = 3, y = 5 b x = π, y = −2
3 4
c x = 2, y = 3 d x = --, y = –--
2 3
6
e x = 8 and y = −5 f x = --
14
and y = –-----
5 5
5 –2
1
g x = ----- and y = ----- h x = --
1
and y = –--
11 11
8 4
Exercise 3B
1 a 8 + 8i b 8 + 15i c 14 + i
d 8 − 6i e 1 − i f a + c + (b − d)i
2 a 8 + 11i b −2 + 3i c 2 − 3i
d 19 + 29i e −2 − 20i f 5.5 + 6.7i
g 9 h 16
3 a 3 + 36i b 22 + 29i c 26 − 13i
d −2 + 26i e 16 + 30i f 13
g 24 + 18i h 3 + 7i i 69 + 17 3i
j 5i – 13 2 k 40 − 46 2i l 36 − 12 3i
5
4 a 7 − 3i b −1 + 7i c 1 − 7i
d 5 + 11i e 22 − 7i f 22 − 7i
g 12 − 15i h 8 − 10i i 12 + 8i
j −15 − 10i
5 a 5 b 17 c 74 d 36
6 a 2ac + b 2 + i(2bc − ab) b d − 4d 2 + (4d + d 2 )i
c −5e + 7i d 6g − g 2 i
7 a 6 − 9i b 2 − 11i c 3 + 5i
d −7i e 3 – 2i f −8 − i
g 2 h 0
8 Those complex numbers that are real are equal to their
conjugates.
23 2 22 14
9 a ----- + -----i b ----- – -----i c – 1 1
----- + --i
41 41 17 17 10 5
9 7 4 3
d −7 + 5i e ----- – -----i f ----- + -----i
13 13 25 25
g 3 + 4i h −24 − 70i
5 2 1 2 32 12
10 a ----- – -----i b ----- + -----i c ----- – -----i
29 29 15 15 73 73
1
d -- + 0i
4
11 a 3 − 4i b 5 + 2i c 8 − 2i
7 26
d 8 − 2i e ----- – -----i f 23 − 14i
29 29
7 26
g −7 − 24i h ----- – -----i
29 29
21 17 1 1 1 1
12 a ----- + -----i b ----- + -----i c ----- + -----i
26 26 26 26 26 26
11
15 a 3 b -----
8 8
c –----- d -----
10 13 29
42
16 a -----
24 17
b –----- c -----
17 25 25
17 a x = 0, y = 1 b x = −1, y = 0
c x = 1, y = 2 d x = 2, y = 1
e x = 3, y = −1 f x = −1, y = 4
16
g x = –----- y = h x = y =
⎝
⎛ 13⎠
⎞ 28 17
, ----- -----,
13 10
28 46
i x = -----, y = -----
80
j x = –-----
, y =
15 15
37
6 9
4
18 x = -----, y = ----- 19 x = -----, y =
13 13
25
9 17
22 y = –-----
, x = ----- 23 y = −3, x = 0
10 10
1
–-----
10
36
-----
37
3
-----
25
24 x = 0, y = 1
Answers
561

Answers
Selected answers are given here. For a full set of worked solutions, see Specialist Maths Dimensions Units 3 & 4 Solutions and Study Guide.
Exercise 3C
1
d
– 3 + 2i
Im(z)
c
f
– 1 – i
Im(z)
k
2i
πi
a
2+
3i
e
b
3
3–
4i
Re(z)
3 a
Im(z)
h
Rez + Imz
Im(z)
f
4+
5i
g
Imz
Rez
5 + 3i
Re(z)
g
– 2
h
3i
i
l
i
0
j
π
Re(z)
3 + 2i
2 + i
Re(z)
2
Im(z)
b
b
Im(z)
2z
–3 + 4i
z
a
4+
5i
–7 + i
1 + i
Re(z)
d
z – 1
Re(z)
c
Im(z)
z – 1 = -------------
4– 5i
41
z
c
2 + 5i
4 + 6i
e
Im(z)
1 + i
Re(z)
z 2 = – 9 + 40i
d
Im(z)
Re(z)
−1 + 2i
4 + 3i
−5 − i
Re(z)
(−1 + 2i) − (4 + 3i)
562
Specialist Maths Dimensions Units 3 & 4

Selected answers are given here. For a full set of worked solutions, see Specialist Maths Dimensions Units 3 & 4 Solutions and Study Guide.
e
Im(z) 5 + 12i
d-e
Im(z)
i 5 z
(3 + 2i)(3 + 2i)
i 4 z
z
i 2 z
Re(z)
3 + 2i
Re(z)
i 3 z
f
Im(z)
6 a-c
i 2 z
Im(z)
iz
(3 + 2i)(3 − 2i) = 13
3 + 2i
Re(z)
3 − 2i
Re(z)
i 3 z
z = 1 − i
4 a-b
Im(z)
d-e
i 2 z
Im(z)
i 5 z
0.5z
3z
z = 3 + 2 i
Re(z)
Re(z)
i 3 z
i 4 z
z = 1 − i
c-d
Im(z)
1 3 3 1
7 a Given z = -- – ------i, iz = ------ + --i
2 2 2 2
z = 3 + 2 i
1 3
b Given z = i 3 3 1
-- – ------i, z = ------ – --i
2 2 2 2
–3z
--------
2
−z
Re(z)
1 3
c Given z = -- – ------i, i 2 z = – 1 2 2 2 -- 3
+ ------i
2
5 a-c
i 2 z
−3z
Im(z)
iz
−2z
z
Re(z)
1 3
d Given z = -- – ------i, z = z + i 2 Imz
2 2
Exercise 3D
π
1 a z = 2 cis -- b z = 2 cis ⎛ 3π
–-----⎞
4
⎝ 4 ⎠
2π
c z = 4cis ----- d z = 5 2 cis ⎛ π
–--⎞
3
⎝ 4⎠
i 3 z
π
e z = 4cis0 f z = 2cis --
2
π
g z = 3cisπ h z = 3 cis ⎛–--⎞
⎝ 2⎠
3 3 3
2 a --------- + --i b 2 + 2i c
2 2
1 3
– -- – ------i
2 2
Answers
563

Answers
Selected answers are given here. For a full set of worked solutions, see Specialist Maths Dimensions Units 3 & 4 Solutions and Study Guide.
d – 5 3 5
--------- – --i e −4 f 3 3 – 3i
2 2
c z = 2 7 cis ⎛–π
----- ⎞ d z = 2 5 cis ⎛ 5π
–-----⎞
⎝ 3 ⎠
⎝ 6 ⎠
564
2π π
3 a –----- b –-- c π d 0
3 2
– 3
4 2 – 3i = 7 ; tanθ = --------- ⇒ θ = tan −1 ⎛ 3
------ ⎞ ⇒
2
⎝ 2 ⎠
θ = −0.7137 c
6 a i
3π
–-----
ii – 3π ----- + 2kπ, k = 0, ±1, ±2
4
4
b i
π
–--
6
ii – π -- + 2kπ, k = 0, ±1, ±2, ...
6
c i
π
–--
3
ii – π -- + 2kπ, k = 0, ±1, ±2, ...
3
d i
π
--
π
ii -- + 2kπ, k = 0, ±1, ±2, ...
6
6
e i
2π
–-----
ii – 2π ----- + 2kπ, k = 0, ±1, ±2, ...
3
3
f i
3π
–-----
ii – 3π ----- + 2kπ, k = 0, ±1, ±2, ...
4
4
g i
π
–--
2
ii – π -- + 2kπ, k = 0, ±1, ±2, ...
2
h i
π
--
π
ii -- + 2kπ, k = 0, ±1, ±2, ...
2
2
i i π ii π + 2kπ, k = 0, ±1, ±2, ...
j i 0 ii 2kπ, k = 0, ±1, ±2, ...
π
k i –--
ii – π -- + 2kπ, k = 0, ±1, ±2, ...
2
2
5π
l i –-----
ii – 5π ----- + 2kπ, k = 0, ±1, ±2, ...
6
6
3π
7 a 2 2 cis –----- b 4cis
⎝
⎛ 4 ⎠
⎞ ⎛–--
π ⎞
⎝ 6⎠
c cis ⎛ π
–--⎞ π
d 2 2 cis --
⎝ 3⎠
⎝ ⎛ 6⎠
⎞
e 4 2 cis ⎛ 2π
–-----⎞ f 3 2 cis ⎛ 3π
–-----⎞
⎝ 3 ⎠
⎝ 4 ⎠
π
g 2cis –--
h 4cis
⎝
⎛ 2⎠
⎞ π --
⎝ ⎛ 2⎠
⎞
i 3cisπ j 5cis0
1
k ------
π
cis –-- l cis
3
⎝
⎛ 2⎠
⎞ 2 ---------
3 ⎛ 5π
–-----⎞
3 ⎝ 6 ⎠
3π
8 a z = 3 2 cis ----- b z = cis
⎝
⎛ 4 ⎠
⎞ ------
2 ⎛– 3π
---------⎞
3 ⎝ 4 ⎠
Specialist Maths Dimensions Units 3 & 4
π
e z = 5 2 cis -- f z = cis
⎝ ⎛ 4⎠
⎞ ---------
10 ⎛3π
-----⎞
5 ⎝ 4 ⎠
π
g z = cis -- h z = cis
⎝ ⎛ 4⎠
⎞
⎛ 5π
–-----⎞
⎝ 6 ⎠
9 a 1 + 3i b 1 − i c – 3 2 -- 3 3
– --------- i
2
d – 3 e −2 f 3i
2 -- 3
+ ------i
2
g −5i h 1 −
11 a −33 − 56i c
Exercise 3E
3π
1 a i 6cis ----- ii cis
⎝
⎛ 4 ⎠
⎞ 3 π
-- --
2 ⎝ ⎛ 4⎠
⎞
–5π
b i 2cis --------- ii cis
⎝
⎛ 6 ⎠
⎞ 1 -- ⎛–5π
---------⎞
2 ⎝ 6 ⎠
7π
c i 20cis ----- ii cis
⎝
⎛ 12⎠
⎞ 5 -- ⎛ π
----- ⎞
4 ⎝12⎠
11π
d i 3 2 cis -------- ii cis
⎝
⎛ 12 ⎠
⎞ 3 ---------
2 ⎛5π
-----⎞
2 ⎝12⎠
3 π
e i 5 3 cisπ ii ------ cis --
5 ⎝ ⎛ 2⎠
⎞
π
f i 2 2 cis –-- ii cis
⎝
⎛ 3⎠
⎞ ------
2 ⎛ 2π
–-----⎞
2 ⎝ 3 ⎠
3
2 a 2 3i b −1 − ------i c – 3 3
4 -- 3
+ ------i
4
3 3 5
d --------- – --i e – 3 + 3i f
3 2
g
65
---------
25
– 3 1
------ – --i h –--------- 9 + i . 3 ---------
3
6 2 4 2 4 2
11π
3 a –1 – 3 + ( 1–
3)i
= 2 2 cis ⎛–--------⎞
⎝ 12 ⎠
–1 + 3 ( 1+
3)i
2
b ------------------- – ---------------------- = ------ cis ⎛5π
-----⎞
8 8 4 ⎝12⎠
5π
c 2 3 – 2 + i( 2 3 + 2) = 4 2 cis ⎛-----⎞
⎝12⎠
3 3 3 2 π
d -- + --i = --------- cis --
4 4 4 ⎝ ⎛ 4⎠
⎞
3i
3
------
2
1
+ --i
2

Selected answers are given here. For a full set of worked solutions, see Specialist Maths Dimensions Units 3 & 4 Solutions and Study Guide.
Exercise 3F
2π
1 a 8cisπ b 16cis ⎛–-----⎞
⎝ 3 ⎠
1 π
π
c ----- cis -- d 2 cis --
32 3
⎝ ⎛ 6⎠
⎞
2 a −2 + 2 3i b 128 + 128i
c −128 − 128 3i d – 31 3 + 118i
3 a −2 + 2i b
c
–4 2 – 4 2i d ( 2 3 – 2i) 3
16
4 a –-----
i
b
27
c
d
e
–3 2 – 3 6
f ------------------------------ + i⎛--------------------------
3 2– 3 6 ⎞
16 ⎝ 16 ⎠
1
5 a --i
2
π
6 a modulus = 16, argument = -- + 2πk,
6
k = 0, ±1, ±2, ...
b modulus = 1, argument =
k = 0, ±1, ±2, ...
24 3i
1
c modulus = --, argument = – 2π ----- + 2πk,
8
3
k = 0, ±1, ±2, ...
2
d modulus = ------, argument = – π -- + 2πk,
32
2
k = 0, ±1, ±2, ...
3π
7 a w = 2 cis ----- z = cis
⎝
⎛ 4 ⎠
⎞ π
, 2 2 ⎛–--⎞
⎝ 4⎠
8 a i 8cis ⎛ π
–--⎞
ii –8i
⎝ 2⎠
2187 3
------------------
512
6+
2 6–
2
– 78 125⎛-------------------- + i⎛-------------------⎞⎞
⎝ 4 ⎝ 4 ⎠⎠
27
-----
4
27 3
+ ------------ i
4
2( 3+
1) + i 21 ( – 3)
b i 16cis ⎛2π
-----⎞ ii −8 + 8 3i
⎝ 3 ⎠
–
– π -- + 2πk,
3
2187
-----------i
512
2
3π
b modulus = ------, argument = ----- + 2 πk,
4
4
k = 0, ±1, ±2, ...
1 π
c i -- cis -- ii
8 ⎝ ⎛ 2⎠
⎞ 1 --i
8
2
d i ------
π
cis –-- ii
9 ⎝
⎛ 4⎠
⎞ 1 1
-- – --i
9 9
e i 108 2 cis ⎛ 3π
–-----⎞
ii −108 − 108i
⎝ 4 ⎠
9 2 π
f i --------- cis -- ii
2 ⎝ ⎛ 4⎠
⎞ 9 9
-- + --i
2 2
g i – 1 1
----- – -----i ii – 1 1
----- – -----i
12 12
12 12
h i
1
1
-------- cis0 ii --------
216
216
11 a – 1 = cis = cis
2 -- + ------i ----- -- – ------i
12 2i
13
1
d i ----- cis ⎛ 2π
–-----⎞ ii –-----
1 i 3
– --------
16 ⎝ 3 ⎠ 32 32
2
9 a i ------
3π
cis –----- ii −
2 ⎝
⎛ 4 ⎠
⎞ 1 1
–-- --i
2 2
b i
2 π
-------- cis -- ii +
108 ⎝ 4⎠
1
-------- --------i
108 108
c i
1
--
1
cisπ ii –--
4
4
b 0
3
2
– 3 1
-------- – --------i
256 256
Exercise 3G
1 a (z + 3i )(z − 3i )
b (z + 7i )(z − 7i )
c (z + 2 2i )(z − 2 2i )
d (z 2 − 8i )(z 2 + 8i )
e (3z − 4i)(3z + 4i)
2π
⎝
⎛ 3 ⎠
⎞ 1 , 2
f (z 3 − i)(z 3 i – 3 i + 3
+ i) = (z + i)(z − -------------- )(z − -------------- )
2 2
i – 3 i + 3
(z − i)(z + -------------- )(z + -------------- )
2 2
1 11
d -- ± ---------i e ± 3i
2 2
3
2
⎛ π
–--⎞
⎝ 3⎠
7 3i
2 a 2 ± 2i b −3 ± i c -- ± --------
2 2
Answers
565

Answers
Selected answers are given here. For a full set of worked solutions, see Specialist Maths Dimensions Units 3 & 4 Solutions and Study Guide.
2 2
f ------
2 2
± ------i, –------ ± ------i
2 2 2 2
3 a b = 1 b b = 12 c b = 2 d b = 0
4 a z = −3, ± i b z = 4, −2 ± i
c z = 5, ± i d z = 7, −1 ± 3i
e z = −2 − 2i, −2 ± 2i, −2
f z = 0, −1 ± 3i
5 p = −5 − 4i, q = 7 − i
6 a z = −i b z = ±i, −3
7 a z = −2 + i b z = −2 ± i, 4
8 a (z − 7)(z + 1 − 3i )(z + 1 + 3i )
b z = 7, −1 ±
9 a k = 16 b −2 − 2i, −2 + i, –2
10 a a = 1, b = 10 b z = 0, z = −1 ± 3i
π
4
11 a cis --, cis –----- b 2 cis --, 2cis
c cis – π cis
⎝
⎛ -- 4⎠
⎞ 3π , -----
4
1
--
4 π
d 2 cis --, 2 cis ⎛ 7π
–-----⎞
8 ⎝ 8 ⎠
1
1
--
--
4 π
e 2 cis ⎛–--⎞ 4 , 2 cis ⎛7π
-----⎞
⎝ 8⎠
⎝ 8 ⎠
1
1
--
--
4 3π
f 2 cis ⎛–-----⎞ 4 5π , 2 cis -----
⎝ 8 ⎠ 8
1
1
--
--
4 3π
g 2 cis ⎛-----⎞ 4 , 2 cis ⎛ 5π
–-----⎞
⎝ 8 ⎠ ⎝ 8 ⎠
π
h cis –-- cis
⎝
⎛ 3⎠
⎞ 2π , ⎛-----⎞
⎝ 3 ⎠
π
2
12 a i = cis -- so i3
= cis -- + --------- k = 0, 1, 2 = cis -- ,
5π –π
cis ----- , cis -----
6 2
3i
⎛ 3π⎞ π
⎝ 4 ⎠
4
1
--
⎛ 3π
–-----⎞
⎝ 4 ⎠
π 2kπ
⎝
⎛ 6 3 ⎠
⎞ π
6
1
–π --
1
--
b –4 = 4 cis ----- so (–4i)
3 = 43
cis ⎛–
π 2kπ
----- + --------- ⎞ k
2
⎝ 6 3 ⎠
1
1 1
--
= 0, 1, 2 = 43
–π --
cis ----- , 43
π --
cis -- , 43
–5π cis ---------.
6 2 6
1
--
–5π
c – 3 – i = 2cis⎛---------⎞ so ( 3 – i)
3
⎝ 6 ⎠
1
--
= 23
cis ⎛–
5π 2kπ
--------- + --------- ⎞ k = 0, 1, 2
⎝ 18 3 ⎠
1
--
1
= 23
–5π --
1
cis ⎛---------
⎞ , 23
7π --
cis ⎛-----⎞ , 23
cis ⎛–
17π
------------ ⎞
⎝ 18 ⎠ ⎝18⎠
⎝ 18 ⎠
d 2 – 2 3i = 4cis – π
1
--
----- so ( 2 – 2 3i)
3
3
1
--
= 43
cis ⎛–
π 2kπ
----- + --------- ⎞ k = 0, 1, 2
⎝ 9 3 ⎠
1
1
1
--
= 43
–π --
cis ⎛-----⎞ , 43
,
⎝ 9 ⎠ cis
5π --
----- 43
–7π cis ---------
9 9
1 1 2
e -- – --i = ------cis – -----
π
2 2 2 4
1 1
-- --
so ⎛1
1
-- – --i ⎞3
= ⎛ 2
------ ⎞3
cis⎛–π
2kπ
----- + --------- ⎞ k = 0, 1, 2
⎝2
2 ⎠ ⎝ 2 ⎠ ⎝12
3 ⎠
1 1
-- --
1
So ⎛ 1
-- – --i ⎞3
= ⎛ 2
------ ⎞3
cis –π ----- , ,
⎝2
2 ⎠ ⎝ 2 ⎠ ⎝
⎛ 12⎠
⎞ 2
1
--
⎛------
⎞3
cis⎛
7π -----⎞
⎝ 2 ⎠ ⎝12⎠
1
--
⎛ 2
------ ⎞3
cis⎛–3π
---------⎞ .
⎝ 2 ⎠ ⎝ 4 ⎠
1
–3π
--
f –4 – 4i = 4 2 cis⎛---------⎞ so (–4 – 4i)
3
⎝ 4 ⎠
1
--
= ( 4 2)
3 cis ⎛–
π 2kπ
----- + --------- ⎞ k = 0, 1, 2.
⎝ 4 3 ⎠
1
1
--
--
Hence (–4 – 4i)
3 = ( 4 2)
3 cis ⎛–
π
----- ⎞ ,
⎝ 4 ⎠
1
--
1
--
( 4 2)
3 5π cis ⎛-----⎞ , ( 4 2)
3 cis ⎛–
11π
------------ ⎞
⎝12⎠
⎝ 12 ⎠
1
--
–π
g 6 – 2i = 2 2 cis ⎛-----
⎞ so ( 6 – 2i)
3
⎝ 6 ⎠
1
--
= ( 2 2)
3 cis ⎛–
π 2kπ
----- + --------- ⎞ k = 0, 1, 2.
⎝18
3 ⎠
1
1
--
--
Hence ( 6 – 2i)
3 = ( 2 2)
3 cis ⎛–π
-----⎞
,
⎝18⎠
1
1
-- --
( 2 2)
3 11π cis ⎛--------⎞ , ( 2 2)
3 cis ⎛– 13π
------------ ⎞ .
⎝ 18 ⎠
⎝ 18 ⎠
566
Specialist Maths Dimensions Units 3 & 4

Selected answers are given here. For a full set of worked solutions, see Specialist Maths Dimensions Units 3 & 4 Solutions and Study Guide.
1
--
–2π
h 2 – 6i = 2 2 cis ⎛---------⎞ so (– 2 – 6i)
3
⎝ 3 ⎠
1
--
= ( 2 2)
3 cis ⎛–
2π 2kπ
--------- + --------- ⎞ k = 0, 1, 2.
⎝ 9 3 ⎠
1
1
--
--
Hence (– 2 – 6i)
3 = ( 2 2)
3 cis ⎛–2π
--------- ⎞ ,
⎝ 9 ⎠
1
1
-- --
( 2 2)
3 4π cis ⎛-----⎞ , ( 2 2)
3 cis ⎛– 8π
---------⎞ .
⎝ 9 ⎠
⎝ 9 ⎠
1
--
13 a 1 = cis0 so 14
= cis ⎛0
+ -----⎞
k = 0, 1, 2, 3 = cis0,
π –π
cis -- , cisπ, cis -----
2 2
kπ
2 ⎠
1
--
b –1 = cisπ so – 14
= cis ⎛π
kπ
-- + -----⎞
k = 0, 1, 2, 3.
⎝4
2 ⎠
1
--
Hence 14
π 3π –3π –π
– = cis -- , cis ----- , cis --------- , cis -----.
4 4 4 4
1
π --
c i = cis -- so i4
= cis ⎛π
kπ
-- + -----⎞
k = 0, 1, 2, 3
2
⎝8
2 ⎠
π 5π
= cis -- , cis ----- , cis ⎛–7π
---------⎞ , cis ⎛–
3π
---------⎞ .
8 8 ⎝ 8 ⎠ ⎝ 8 ⎠
⎝
1
--
5π
g – 3 + i = 2 cis ----- so (– 3 + i)
4
6
1
1
--
= cis 24
5π kπ
--
cis ⎛-----
+ -----⎞ k = 0, 1, 2, 3 = 24
cis ⎛5π
-----⎞
,
⎝24
2 ⎠
⎝24⎠
1
--
1
24
17π --
1
cis ⎛--------⎞ , 24
– 19π --
cis ⎛------------
⎞ , 24
cis ⎛–
7π
---------⎞
⎝ 24 ⎠ ⎝ 24 ⎠ ⎝ 24 ⎠
h –5 + 5i = 5 2cis 3π
1
--
⎛-----⎞ so (–5 + 5i)
4
⎝ 4 ⎠
1
--
= ( 5 2)
4 cis ⎛3π
kπ
----- + -----⎞
k = 0, 1, 2, 3
⎝ 4 2 ⎠
1
1
-- --
= ( 5 2)
4 3π cis ⎛-----⎞ , ( 5 2)
4 cis ⎛–
3π
---------⎞
,
⎝ 4 ⎠
⎝ 4 ⎠
1
1
-- --
( 5 2)
4 – π cis ⎛-----
⎞ , ( 5 2)
4 π cis --
⎝ 4 ⎠
⎝ ⎛ 4⎠
⎞ .
Exercise 3H
1 a
Im(z)
Re(z) = 3
3
Re(z)
1
–π --
d –i = cis ----- so – i4
= cis ⎛–π
kπ
----- + -----⎞
k = 0, 1, 2, 3
2
⎝ 8 2 ⎠
b
Im(z)
Re(z) = –2
–π 3π 7π
= cis ----- , cis ----- , cis ----- , cis ⎛–5π
---------⎞
8 8 8 ⎝ 8 ⎠
–2
Re(z)
1
--
e 16 = 16 cis0 so 164
= 2cis ⎛ kπ
0 + -----⎞
k = 0, 1, 2, 3
⎝ 2 ⎠
π
–π
= 2cis0, 2cis -- , 2cisπ =, 2cis -----
2
2
c
Im(z)
Re(z + 2) = 3
1 1
-- --
f –64 = 64cisπ so – 644
= 14
π kπ
– 2 2cis ⎛--
+ -----⎞
k
⎝4
2 ⎠
1
1
--
= 0, 1, 2, 3 = 14
π --
– 2 2cis -- , 14
3π
– 2 2cis ----- ,
4
4
1
1
--
14
– 3π --
– 2 2cis --------- , 14
– π
– 2 2cis -----.
4
4
1
--
Hence – 644
= 2 2cis π -- , 2 2cis3π ----- ,
2 2
2 2cis – --------- 3π , 2 2cis – ----- π .
2 2
d
e
Re(z – 1) = –4
Im(z)
–3
1
Im(z)
3
Re(z + 3i) =
2
Re(z)
Re(z)
3
2
Re(z)
Answers
567

Answers
Selected answers are given here. For a full set of worked solutions, see Specialist Maths Dimensions Units 3 & 4 Solutions and Study Guide.
f
Im(z)
Re(z + 1 – i) = –4
h
Im(z)
3
Re(z)
–5
Re(z)
–4
2 a
Im(z)
i
Im(z)
Re(z)
5
(–2, 2 )
–4
Im(z) = –4
Re(z)
b
Im(z)
j
Im(z)
5
2
Im(z)
5
2
5
2
Re(z)
–2
Re(z)
c
Im(z)
3 a
Im(z)
4 Im(z – 3i) = 1
Re(z)
π
4
Re(z)
d
Im(z)
b
Im(z)
Re(z)
π
4
Re(z)
–5
Im(z + 2i) = –3
e
Im(z)
c
Im(z)
5 Im(z + 1) = 6
Re(z)
π
6
Re(z)
f
Im(z)
d
Im(z)
Re(z)
Im(z – 2 + 3i) = 0
–3
π
6
Re(z)
g
Im(z)
e
Im(z)
(3, –4)
Re(z)
π
3
Re(z)
568
Specialist Maths Dimensions Units 3 & 4

Selected answers are given here. For a full set of worked solutions, see Specialist Maths Dimensions Units 3 & 4 Solutions and Study Guide.
f
Im(z)
b
Im(z)
9
π
Re(z)
–9
–9
9
Re(z)
4 a
Im(z)
c
Im(z)
2
1
π
4
Re(z)
–2
–2
2
Re(z)
b
Im(z)
d
Im(z)
2
π
2
Re(z)
–8
8
8
Re(z)
–8
c
Im(z)
e
Im(z)
–3
π
3
Re(z)
–3
(–2, 1)
–2
–1
Re(z)
(–2, –1)
d
Im(z)
1
f
Im(z)
π
3
Re(z)
(3, 3)
0
3
6
Re(z)
(3, –3)
e
(–2, 3)
Im(z)
π
6
g
Im(z)
–4
Re(z)
Re(z)
(–2, –4) (2, –4)
f
Im(z)
h
–6
Im(z)
11
π
3
(–1, –2)
Re(z)
(2, –9) 2 (2, 9)
Re(z)
–7
5 a
Im(z)
4
i
Im(z)
(–1, 3)
–4
–4
4
Re(z)
(–5, –1)
(–1, –1)
(–1, –5)
(3, –1)
Re(z)
Answers
569

Answers
Selected answers are given here. For a full set of worked solutions, see Specialist Maths Dimensions Units 3 & 4 Solutions and Study Guide.
j
(–14, 4)
Im(z)
(2, 20)
(2, 4)
(2, –12)
(18, 4)
Re(z)
b
Im(z)
(3 – 3 , –2)
(3, 1)
(3, –2)
(3, –5)
Re(z)
(3 + 3 , –2)
6 a
–4
Im(z)
4
4
Re(z)
c
17
(–5, )
4
( – 16 3 , 4) ( – 14 3 , 4)
(–5, 15
4
) (–5, 4)
Im(z)
–4
Re(z)
b
Im(z)
d
Im(z)
(–2, 9
4
)
–2
1
–1
2
Re(z)
( – 11 , 1)
3
(–2, 1)
(–2, – 1 )
4
( – 1 , 1)
3
Re(z)
c
Im(z)
2
8 a
Im(z)
Im(z) = 0
Re(z)
–1
1
Re(z)
d
– 2
Im(z)
3 2
2
b
Im(z)
( 1 , 6 )
2
(–3, 0) ( 1 , 0)
2
(2, 0)
Re(z)
e
6
–
2
3 2
–
2
Im(z)
21
3
6
2
Re(z)
c
– 1 2
Im(z)
52
(3 – , 0)
3
6
65
(3, )
2
(3, 0)
52
(3 + , 0)
3
Re(z)
5
–
2
5
2
Re(z)
65
(3, – )
2
f
21
–
3
Im(z)
d
y
√105
4
2
–7 –3
1
x
– 10
10
Re(z)
–
√105
4
7 a
– 2
Im(z)
9 a It is a line in the Argand plane.
b y = −x
10 a y = −7x − 2 b y = −1
(–4, 1)
(–2, 2)
(–2, 1)
(–2, –0)
(0, 1)
Re(z)
25 3
c y = x − 2 d y = ----- + --x
8 4
e y = −x + 3 f y = −2x + 4
570
Specialist Maths Dimensions Units 3 & 4

Selected answers are given here. For a full set of worked solutions, see Specialist Maths Dimensions Units 3 & 4 Solutions and Study Guide.
11 a
y = –7x x –24
Im(z)
13 a y = 0
Im(z)
–
24
7
Re(z)
y = 0
Re(z)
–24
b
Im(z)
–1
Re(z)
y = –1
x
b 2 y
----------
2
= 1
5
--
⎝ ⎛ 2⎠
⎞ 2 + ----------
3
--
⎝ ⎛ 2⎠
⎞ 2
–
5
2
Im(z)
3
2
5
2
Re(z)
–
3
2
c
Im(z)
y = x – 2
2
Re(z)
c y = 0
Im(z)
y = 0
Re(z)
–2
d
Im(z)
25
8
3
y = x
4
25
8
x
d 2 y
--------------
2
4
= 1 where x <
⎛ 5
------ ⎞ 2 + ----------
3
--
⎝ 2 ⎠ ⎝ ⎛ 2⎠
⎞ 2
–--
3
Im(z)
e
–
25
6
Im(z)
Re(z)
–
5
2
3
2
–
3
2
5
2
Re(z)
3
y = –x
x + 3
3
Re(z)
x
e 2 y
----
2 4
= 1 where x <
2 2 + --------------
( 5) 2
–--
3
f
Im(z)
4
Im(z)
y =–2x
+ 4
–2
Re(z)
2
Re(z)
1
3
4
9
12 a (x + -- ) 2 + y 2 = -- b x 2 + (y − -- ) 2 =
c x 2 10
+ (y + ) 2 64
----- = -----
3 9
1
d (x + ) 2 9
+ (y − ) 2 9
-- -- = -----
8 8 32
11
e (x + ) 2 17
+ (y + ) 2 200
----- ----- = --------
3 3 9
16
f (x − ) 2 + (y − 1) 2 276
-----
= --------
5
25
3
4
9
-----
16
x
f 2 y
---------- = 1
3
--
⎝ ⎛ 2⎠
⎞ 2 + --------------
2
⎛ 5
------ ⎞ 2
⎝ 2 ⎠
( x + 1) 14 2 ( y – 1) ------------------- + ------------------
2
= 1
8 8
15 a y = x − 1, straight line
Im(z)
5
2
–
3
3 Re(z)
2
2
–
5
2
b Circle: centre = ⎛1
--
1
, –-- ⎞ 1
, radius = ------
⎝2
2⎠
2
Answers
571

Answers
Selected answers are given here. For a full set of worked solutions, see Specialist Maths Dimensions Units 3 & 4 Solutions and Study Guide.
Exercise 3I
1 a
c
Im(z)
5
Im(z)
Re(z)
2
Re(z)
d
Im(z)
b
Im(z)
Re(z)
–1
Re(z)
e
Im(z)
c
Im(z)
Re(z)
–
5
2
1
Re(z)
f
Im(z)
d
Im(z)
1
Re(z)
4 Re(z)
3 a
Im(z)
e
Im(z)
π
3
Re(z)
–
1
2
Re(z)
b
Im(z)
f
Im(z)
2π
3
Re(z)
–6
Re(z)
c
Im(z)
2 a
Im(z)
Re(z)
Re(z)
–3
d
Im(z)
b
Im(z)
2
3
Re(z)
Re(z)
572
Specialist Maths Dimensions Units 3 & 4

Selected answers are given here. For a full set of worked solutions, see Specialist Maths Dimensions Units 3 & 4 Solutions and Study Guide.
e
Im(z)
5 a
Im(z)
π
3
9
Re(z)
–9
9
Re(z)
–9
f
Im(z)
b
Im(z)
1
Re(z)
–1
1
Re(z)
–1
4 a
Im(z)
c
Im(z)
3
–2
Re(z)
–3
3
Re(z)
–3
b
Im(z)
d
Im(z)
10
1
Re(z)
–10
10
Re(z)
–10
c
Im(z)
e
Im(z)
(–3, 4)
–4
π
4
Re(z)
–7 –3 1
(–3, –4)
Re(z)
d
π
6
Im(z)
1
f
Im(z)
(4, 1)
Re(z)
3 5
4 Re(z)
(4, –1)
e
Im(z)
g
Im(z)
(–4, 2)
π
3
Re(z)
(0, 17)
(0, –1)
(16, –1)
(16, 1)
Re(z)
(0, –15)
f
Im(z)
h
Im(z)
9
(3, –12)
3
(3, 12)
(–4, –2)
π
6
Re(z)
–9
Re(z)
Answers
573

Answers
Selected answers are given here. For a full set of worked solutions, see Specialist Maths Dimensions Units 3 & 4 Solutions and Study Guide.
i
Im(z)
3
Re(z)
(–6, –3)
(–3, –3)
–3
d
Im(z)
4
(1, –3)
1
(1, 3)
(–3, –6)
π
4
Re(z)
(–1, –2) –2
6 a
b
Im(z)
(–1, 4)
(–5, 2) (3, 2)
(–1, 2)
Im(z)
–1
(2, –2)
(2, –4)
(1, –4) (3, –4)
Re(z)
Re(z)
Revision Questions
Multiple choice
1 B 2 C 3 D 4 D
5 A 6 A 7 E 8 C
9 B 10 C 11 A 12 C
13 B 14 D 15 C 16 D
17 D 18 A 19 A 20 E
Short answer
c
(2, –6)
Im(z)
1
7 11i
z + -- – -----------
⎝
⎛ 2 2 ⎠
⎞ ⎛ 7 11i
z + -- + -----------⎞
⎝ 2 2 ⎠
( – 13 , 1)
4
(–3, 3 )
2
(–3, 1 )
2
( – 11 , 1)
4
(–3, 1)
Re(z)
3 −7 + 24i
4 a z = ±1, ±i
Im(z)
1
d
Im(z)
–1
–1
1
Re(z)
(–5, 3)
(–5, 0)
7 a
– 23
4
(–5, –3)
Im(z)
– 17 4
Re(z)
5 z = −1, 2 − i, 2 + i
6 –512 3 − 512i
8
Im(z)
(–2, 8)
1
π
4
(–6, 4) (2, 4)
(–2, 4)
b
–1
–2
–1
Im(z)
2
1
π
3
1
2
Re(z)
Re(z)
9
–3 –2
Im(z)
3
π
4
2
–2
π
4
2
3Re(z)
Re(z)
–2
c
Im(z)
10
Im(z)
(1, 2)
–1
π
4
(1, 0)
3
Re(z)
– 3
–3
3
Re(z)
(1, –2)
574
11 a −11 + 29i
Specialist Maths Dimensions Units 3 & 4
3
3
12 circle, centre at ( --, 0), radius = --
4
4

Selected answers are given here. For a full set of worked solutions, see Specialist Maths Dimensions Units 3 & 4 Solutions and Study Guide.
13
14 a 5 – 3i, 1 + 3i b 6 – 3i, 2 + 3i
Application Tasks
1 a
Im(z)
(1 – π
2
, 0)
Im(z)
(1, π
2
)
(–1, 0)
(1, – π
2
)
π
3
(1 + π
2
, 0)
Re(z)
π
3
11 a
7
213 + 144i
c i -- + 4i ii -------------------------
2
113
–24 + 12i
d i −3 + 1 ii -----------------------
5
–3
27
y
y = x 3 + 9x 2 + 27x + 27
x
4
2
z 2
2 2
13
z 3
z 1 17
y
y = x 3 + 4x 2 – 3x + 8
2 4 6
Re(z)
b
z' Im(z) 3
6
z' 2 4
z 2
–3 2
–18
x
z' 2 1
z 1
z 3
y
–4 –2 2 4 6
Re(z)
y = x 3 – 2x 2 – 5x + 6
2 ( 2 + 2 3i) 3 = −64
1 3
3 2, -- + ------i, – 1 −2,
2 2 2 -- 3
+ ------i, – 1 2 2 -- 3
– ------i,
2
4
5
6–
2
-------------------
4
Im(z)
(3, 3)
1
-- –
2
3
------i
2
–2 1 3
x
y
–1
y = x 3 + 3x 2 + 6x + 4
x
6
(2, –2) (1, –2) (5, –2)
(3, –2)
Im(z)
5
3
(3, –4)
(3, –7)
π
6
π
6
Re(z)
(8, –2)
π
6
b When there is only one x-intercept and it is not a
point of inflection.
Chapter 4
Exercise 4A
1 a 2sec 2 2x
b
7sec 2 7x
7 a
–1 3 5
–1
–2
Im(z)
Re(z)
(2 + i) 3 = 2 + 11i
c
e
g
6sec 2 3x d – 27sec 2 (– 9x)
– 4sec 2 (–
x) f – 36sec 2 4x
1
--sec 2 x
--
5 5
h
3
--sec 2 x
--
4 4
(2 + i) 2 = 3 + 4i
2 + i
Re(z)
d De Moivre’s theorem holds for fractional powers.
8 a cos 3 θ + 3i sinθcos 2 θ – 3 sin 2 θcosθ
– i sin 3 θ
b cos3θ
+ i sin3θ
9 a i 5 + 6i ii
b i 6 + 5i ii
96 + 80i
-------------------
61
–79 + 119i
--------------------------
61
i
k
5
--sec 2 x
– --
7 7
2
-----sec 2 ⎛ x
–-- ⎞ 6
l -----sec
15 ⎝ 3⎠
2 ⎛ 2x
–-----⎞
25 ⎝ 5 ⎠
2 a 3sec3x tan3x
b – 2cosec2x cot2x
c 18 sec2x tan2x
d 10cosec5x cot5x
e – 21 sec7x tan7x
f 4cosec( – x) cot( – x)
g – 35 sec( – 5x) tan( – 5x)
j
12
-----sec 2 2x
– -----
5 5
Answers
575