Download 1. Let z = x + yi. Find the values of x and y if (1 – i)z = 1 – 3i. Working

1.
Let z = x + yi. Find the values of x and y if (1 – i)z = 1 – 3i.
Working:
Answers:
…………………………………………..
(Total 4 marks)
2.
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)
3.
(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)
4.
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)
1
(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)
5.
Let z1 = a  cos   i sin   and z2 = b  cos   i sin  
4
4
3
3


3
z 
Express  1  in the form z = x + yi.
 z2 
Working:
Answers:
....……………………………………..........
(Total 3 marks)
2
6.
If z is a complex number and |z + 16| = 4 |z + l|, find the value of | z|.
Working:
Answers:
....……………………………………..........
(Total 3 marks)
7.
Find the values of a and b, where a and b are real, given that (a + bi)(2 – i) = 5 – i.
Working:
Answers:
…………………………………………..
(Total 3 marks)
3
8.
(z + 2i) is a factor of 2z3–3z2 + 8z – 12. Find the other two factors.
Working:
Answers:
…………………………………………..........
...........................................................................
(Total 3 marks)
9.
Given that z = (b + i)2, where b is real and positive, find the exact value of b when arg z = 60°.
Working:
Answers:
…………………………………………..
(Total 3 marks)
4
10.
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:
Answers:
…………………………………………..
(Total 3 marks)
11.
(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 < 6  π.
(3)
(c)
Express z4 + z3 + z2 + z + 1 as a product of two real quadratic factors.
(5)
(Total 10 marks)
5
12.
(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)
2
13.
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, i.e. 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)
6
14.
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)
15.
(a)
Prove, using mathematical induction, that for a positive integer n,
(cos + i sin)n = cosn + i sinn 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 = 2cos nθ.
(5)
(c)
(i)
Find the binomial expansion of (z + z–l)5.
(ii)
Hence show that cos5 =
1
(a cos5 + b cos3 + c cos),
16
where a, b, c are positive integers to be found.
(5)
(Total 15 marks)
7
16.
Consider the equation 2(p + iq) = q – ip – 2 (1 – i), where p and q are both real numbers. Find p
and q.
Working:
Answers:
…………………………………………..
(Total 6 marks)
17.
(a)
Use mathematical induction to prove De Moivre’s theorem
(cos + i sin)n = cos(n) + i sin(n), n 
+
.
(7)
(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.
(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)
18.
A complex number z is such that
(a)
z  z  3i .
Show that the imaginary part of z is
3
.
2
(2)
8
(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)
 z1k z 2 


 2i 

 = π, find a value of k.
Given that arg
(4)
(Total 10 marks)
19.
Consider the complex number z = cos + i sin.
(a)
Using De Moivre’s theorem show that
zn +
1
= 2cosn.
zn
(2)
(b)

By expanding  z 

1

z
4
show that
cos4 =
1
(cos4 + 4cos2 + 3).
8
(4)
(c)
Let g(a) =

a
0
cos 4 d .
(i)
Find g(a).
(ii)
Solve g(a) = 1
(5)
(Total 11 marks)
20.
Consider the complex geometric series eiθ
(a)
1 2iθ 1 3iθ
e + e +…
2
4
Find an expression for z, the common ratio of this series.
(2)
(b)
Show that z< 1.
(c)
Write down an expression for the sum to infinity of this series.
(2)
(2)
9
(d)
(i)
Express your answer to part (c) in terms of sin θ and cos θ.
(ii)
Hence show that
cos θ +
1
1
4 cos θ  2
cos 2θ + cos 3θ +…=
2
4
5  4 cos θ
(10)
(Total 16 marks)
21.
Let P(z) = z3 + az2 + bz + c, where a, b, and c 
(–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)
10