Po Leung Kuk Choi Kai Yau School
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IBDP HL Maths Booklet
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Po Leung Kuk Choi Kai Yau School
Mathematics
IB Higher Level Mathematics
Booklet 21: Complex Number Part-A
The intention is that these booklets are accumulated over the duration of the course to provide
you with an expanding resource that you will refer back to later on in the course, probably when
you are revising. With that in mind, you are asked to do one of the following: either work in
pencil, erasing your errors as you go, perfecting your solution; or work on rough paper first,
until you are confident that you have the solution correct. In either case, by the time that you
hand the prep sheet in, it should be your ‘final and best effort’ and as such it should be clearly
written and, hopefully correct, with a well documented method. In that way these booklets will be
of benefit to you later on.
Full marks are not necessarily awarded for a correct answer with no working. Answers must be
supported by working and/or explanations. In particular, solutions found from a graphic display
calculator should be supported by suitable working, e.g. if graphs are used to find a solution, you
should sketch these as part of your answer. Where an answer is incorrect, some marks may be
given for a correct method, provided this is shown by written working. You are therefore advised
to show all working.
Unless otherwise stated, leave your answers to 3 significant figures.
Questions are taken from the IB Question Bank
1
1.
Let z1 = a  cos   i sin   and z2 = b  cos   i sin  .
3
3
4
4


3
z 
Express  1  in the form z = x + yi.
 z2 
Working:
Answer:
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(Total 3 marks)
2.
If z is a complex number and |z + 16| = 4 |z + l|, find the value of | z|.
Working:
Answer:
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(Total 3 marks)
2
3.
Find the values of a and b, where a and b are real, given that (a + bi)(2 – i) = 5 – i.
Working:
Answer:
…………………………………………..
(Total 3 marks)
4.
Given that z = (b + i)2, where b is real and positive, find the exact value of b when arg z =
60°.
Working:
Answer:
..........................................................................
(Total 3 marks)
3
5.
The complex number z satisfies i(z + 2) = 1 – 2z, where i  – 1 . Write z in the form z = a +
bi, where a and b are real numbers.
Working:
Answer:
..........................................................................
(Total 3 marks)
6.
The complex number z satisfies the equation
z=
2
+ 1 – 4i.
1– i
Express z in the form x + iy where x, y
.
Working:
Answer:
.........................................................................
(Total 6 marks)
4
7.
Consider the equation 2(p + iq) = q – ip – 2 (1 – i), where p and q are both real numbers.
Find p and q.
Working:
Answer:
.........................................................................
(Total 6 marks)
8.
Let the complex number z be given by
z=1+
i
i– 3
.
Express z in the form a +bi, giving the exact values of the real constants a, b.
Working:
Answer:
.........................................................................
(Total 6 marks)
5
9.
A complex number z is such that
(a)
z  z  3i .
Show that the imaginary part of z is
3
.
2
(2)
(b)
Let z1 and z2 be the two possible values of z, such that z  3.
(i)
Sketch a diagram to show the points which represent z1 and z2 in the complex
plane, where z1 is in the first quadrant.
(ii)
Show that arg z1 =
(iii)
Find arg z2.
π
.
6
(4)
(c)
 zk z
Given that arg  1 2
 2i


 = π, find a value of k.


(4)
(Total 10 marks)
6
10.
Given that (a + i)(2 – bi) = 7 – i, find the value of a and of b, where a, b 
.
Working:
Answer:
.........................................................................
(Total 6 marks)
11.
Given that z  , solve the equation z3 – 8i = 0, giving your answers in the form
z = r (cos + i sin).
Working:
Answer:
.........................................................................
(Total 6 marks)
7
12.
Given that z = (b + i)2, where b is real and positive, find the exact value of b when arg z =
60°.
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(Total 6 marks)
8
13.
Given that | z | = 2 5 , find the complex number z that satisfies the equation
25  15  1  8i.
z
z*
Working:
Answer:
(Total 6 marks)
14.
b
a
and z2 =
where a, b
1 2 i
1i
Calculate the value of a and of b.
The two complex numbers z1 =
, are such that z1 + z2 = 3.
Working:
Answer:
(Total 6 marks)
9
15.
Let z1 and z2 be complex numbers. Solve the simultaneous equations
2z1 + z2 = 7, z1 + iz2 = 4 + 4i
Give your answers in the form z = a + bi, where a, b Є
.
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(Total 6 marks)
10
16.
The complex numbers z1 and z2 are z1 = 2 + i, z2 = 3 + i.
(a)
Find z1z2, giving your answer in the form a + ib, a, b
.
(1)
(b)


The polar form of z1 may be written as  5 , arctan
1
.
2
(i)
Express the polar form of z2, z1 z2 in a similar way.
(ii)
Hence show that
1
π
1
= arctan + arctan .
3
4
2
(5)
(Total 6 marks)
11
17.
(a)
Express the complex number 1+ i in the form
ae
i
π
b
, where a, b
+
.
(2)
n
(b)
 1 i 
 , where n
Using the result from (a), show that 
 2
values.
, has only eight distinct
(5)
(c)
Hence solve the equation z8 −1 = 0.
(2)
(Total 9 marks)
12
18.
The polynomial P(z) = z3 + mz2 + nz −8 is divisible by (z +1+ i), where z
Find the value of m and of n.
and m, n .
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(Total 6 marks)
13
19.


Let z1 = r  cos
π
π
 i sin  and z2 = 1 +
4
4
(a)
Write z2 in modulus-argument form.
(b)
Find the value of r if z1 z 2
3
3 i.
= 2.
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(Total 6 marks)
20.
The complex number z is defined by
2π
2π 
π
π


z = 4  cos  i sin
  4 3  cos  i sin .
3
3 
6
6


(a)
Express z in the form rei, where r and  have exact values.
(b)
Find the cube roots of z, expressing in the form rei, where r and  have exact values.
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(Total 6 marks)
14
21.
(a)
Evaluate (1 + i)2, where i =
1 .
(2)
b)
Prove, by mathematical induction, that (1 + i)4n = (–4)n, where n 
*.
(6)
c)
Hence or otherwise, find (1 + i)32.
(2)
(Total 10 marks)
15
22.
Let z1 =
(a)
6 i 2
, and z2 = 1 – i.
2
Write z1 and z2 in the form r(cos θ + i sin θ), where r > 0 and –
π
π
≤ θ ≤ .
2
2
(6)
(b)
Show that
z1
= cos  + i sin  .
z2
12
12
(2)
(c)
Find the value of
z1
in the form a + bi, where a and b are to be determined exactly in
z2
radical (surd) form. Hence or otherwise find the exact values of cos  and sin  .
12
12
(4)
(Total 12 marks)
16
23.
Let u =1+
3 i and v =1+ i where i2 = −1.
(a)
(i)
Show that
(ii)
By expressing both u and v in modulus-argument form show that
u
π
π

 2  cos  i sin  .
v
12 
 12
(iii)
Hence find the exact value of tan
3 1
3 1
u


i.
v
2
2
π
in the form a  b 3 where a, bЄ
12
.
(15)
17
(b)
Use mathematical induction to prove that for nЄ
1 3 i  2
n
n
+
,
nπ
nπ 

 cos  i sin .
3
3 

(7)
(c)
Let z =
2 vu
2 v u
.
Show that Re z = 0.
(6)
(Total 28 marks)
18
1.
(z + 2i) is a factor of 2z3–3z2 + 8z – 12. Find the other two factors.
Working:
Answer:
...........................................................................
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(Total 3 marks)
2. Let P(z) = z3 + az2 + bz + c, where a, b, and c ϵ
(–3 + 2i). Find the value of a, of b and of c.
. Two of the roots of P(z) = 0 are –2 and
Working:
Answer:
(Total 6 marks)
19
3. The polynomial P(z) = z3 + mz2 + nz −8 is divisible by (z +1+ i), where z
value of m and of n.
and m, nϵ
. Find the
(Total 6 marks)
4.
(a)
Express the complex number 1+ i in the form
ae
i
π
b
, where a, bϵ
+
.
(2)
n
(b)
 1 i 
 , where n
Using the result from (a), show that 
 2
values.
, has only eight distinct
(5)
(c)
Hence solve the equation z8 −1 = 0.
(2)
(Total 9 marks)
20
5.
Let x and y be real numbers, and  be one of the complex solutions of the equation
z3 = 1. Evaluate:
(a)
1 +  + 2;
(2)
(b)
( x + 2y)(2x +  y).
(4)
(Total 6 marks)
6.
(a)
Express the complex number 8i in polar form.
(b)
The cube root of 8i which lies in the first quadrant is denoted by z. Express z
(i)
in polar form;
(ii)
in cartesian form.
Working:
Answers:
(a) ..................................................................
(b) (i) ...........................................................
(ii) ...........................................................
(Total 6 marks)
21
7.
Prove that

 
n
3 i 
3 i

n
is real, where nϵ
+
.
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(Total 6 marks)
8.
Given that z
, solve the equation z3 – 8i = 0, giving your answers in the form
z = r (cos + i sin).
Working:
Answer:
.........................................................................
(Total 6 marks)
22
9.
(a)
Express z5 – 1 as a product of two factors, one of which is linear.
(2)
(b)
Find the zeros of z5 – 1, giving your answers in the form
r(cos θ + i sin θ) where r > 0 and –π < θ  π.
(3)
(c)
Express z4 + z3 + z2 + z + 1 as a product of two real quadratic factors.
(5)
(Total 10 marks)
23
2
10.
Consider the complex number z =
(a)
π
π 
π
π

 cos – i sin   cos  i sin 
4
4 
3
3

π
π 

– i sin 
 cos
24
24


4
(i)
Find the modulus of z.
(ii)
Find the argument of z, giving your answer in radians.
3
.
(4)
(b)
Using De Moivre’s theorem, show that z is a cube root of one, ie z = 3 1 .
(2)
(c)
Simplify (l + 2z)(2 + z2), expressing your answer in the form a + bi, where a and b are
exact real numbers.
(5)
(Total 11 marks)
24
11.
(a)
Prove, using mathematical induction, that for a positive integer n,
(cos + i sin)n = cos n + i sin n where i2 = –1.
(5)
(b)
The complex number z is defined by z = cos + i sin.
1
= cos (–) + i sin (–).
z
(i)
Show that
(ii)
Deduce that zn + z–n = 2 cos nθ.
(5)
25
(c)
(i)
Find the binomial expansion of (z + z–l)5.
(ii)
Hence show that cos5 =
1
(a cos 5 + b cos 3 + c cos ),
16
where a, b, c are positive integers to be found.
(5)
(Total 15 marks)
26
12.
(a)
Use mathematical induction to prove De Moivre’s theorem
(cos + i sin)n = cos (n) + i sin (n), n ϵ
+
.
[No need to do part (a), same as #11]
(b)
Consider z5 – 32 = 0.
(i)

 2π 
 2π  
Show that z1 = 2  cos    i sin    is one of the complex roots of this
 5 
 5 

equation.
(ii)
Find z12, z13, z14, z15, giving your answer in the modulus argument form.
(iii)
Plot the points that represent z1, z12, z13, z14 and z15, in the complex plane.
27
(iv)
The point z1n is mapped to z1n+1 by a composition of two linear transformations,
where n = 1, 2, 3, 4. Give a full geometric description of the two
transformations.
(9)
(Total 16 marks)
13.
Let z = cos  + i sin , for –
(a)
π
π
 .
4
4
(i)
Find z3 using the binomial theorem.
(ii)
Use de Moivre’s theorem to show that
cos 3 = 4 cos3 – 3 cos and sin 3 = 3 sin – 4 sin3.
(10)
28
(b)
Hence prove that
sin 3θ  sin θ
= tan.
cos 3θ  cos θ
(6)
(c)
Given that sin =
1
, find the exact value of tan 3.
3
(5)
(Total 21 marks)
29