Download Complex Numbers - Uplift North Hills Prep

1
Complex Numbers
1.
Let z = x + yi. Find the values of x and y if (1 – i)z = 1 – 3i.
2.
(a)
Evaluate (1 + i)2 , where i =
(b)
Prove, by mathematical induction, that (1 + i) 4n = (–4)n , where n 
(c)
Hence or otherwise, find (1 + i)32 .
3.
Let z1 =
1 .
*.
[10]
6 i 2
, and z2 = 1 – i.
2
(a)
Write z1 and z2 in the form r(cos θ + i sin θ), where r > 0 and –
(b)
Show that
(c)
Find the value of
π
π
θ  .
2
2
z1


= cos
+ i sin
.
12
12
z2
z1
in the form a + b i, where a and b are to be determined exactly in radical (surd)
z2
form. Hence or otherwise find the exact values of cos
4.
[4]


and sin
.
12
12
[12]
Let z1 = a  cos   i sin   and z2 = b  cos   i sin  .
3
3
4
4


z 
Express  1 
 z2 
3
in the form z = x + yi.
[3]
5.
If z is a complex number and |z + 16| = 4 |z + l|, find the value of | z|.
[3]
6.
Find the values of a and b, where a and b are real, given that (a + bi)(2 – i) = 5 – i.
[3]
7.
Given that z = (b + i)2 , where b is real and positive, find the exact value of b when arg z = 60°.
[3]
8.
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.
9.
The complex number z satisfies the equation
2
z=
+ 1 – 4i.
1– i
Express z in the form x + iy where x, y  .
[3]
[5]
10.
Consider the equation 2(p + iq) = q – ip – 2 (1 – i), where p and q are both real numbers. Find p and q.
11.
Let the complex number z be given by
[6]
z=1+
i
.
i– 3
Express z in the form a +bi, giving the exact values of the real constants a, b.
[6]
12.
A complex number z is such that
z  z  3i .
2
(a)
(b)
(c)
3
.
2
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.
π
(ii) Show that arg z1 = .
6
(iii) Find arg z2 .
Show that the imaginary part of z is
 zk z
Given that arg 1 2
 2i


 = π, find a value of k.


[10]
13.
Given that (a + i)(2 – bi) = 7 – i, find the value of a and of b, where a, b 
.
[6]
14.
, solve the equation z3 – 8i = 0, giving your answers in the form z = r (cos + i sin).
Given that z 
[6]
15.
Given that z = (b + i)2 , where b is real and positive, find the exact value of b when arg z = 60°.
[6]
16.
Given that | z | = 2 5 , find the complex number z that satisfies the equation
25  15  1  8i.
z
z*
7.
The two complex numbers z1 =
[6]
b
a
and z2 =
where a, b
1 2i
1i
, are such that z1 + z2 = 3. Calculate
the value of a and of b.
[6]
18.
19.
20.
The complex numbers z1 and z2 are z1 = 2 + i, z2 = 3 + i.
(a)
Find z1 z2 , giving your answer in the form a + ib, a, b
(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


π
4
Let z1 = r  cos  i sin
.
1
π
1
= arctan + arctan .
2
4
3
π
 and z2 = 1 +
4
(a)
Write z2 in modulus-argument form.
(b)
Find the value of r if z1 z 2
3
[6]
3 i.
= 2.
[6]
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
.
[6]
3
21.
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


22.
(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.
The polynomial P(z) = z3 + mz2 + nz −8 is divisible by (z +1+ i), where z
m and of n.
[6]
and m, n . Find the value of
[6]
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
Hence find the exact value of tan
1 3 i  2
n
(c)
u
π
π

 2  cos  i sin  .
v
12
12 

π
in the form a  b 3 where a, b
12
Use mathematical induction to prove that for n +,
(iii)
(b)
3 1
3 1
u


i.
v
2
2
Let z =
n
.
nπ
nπ 

 cos  i sin .
3
3 

2 vu
2 v u
.
Show that Re z = 0.
24.
(a)
[28]
Express the complex number 1+ i in the form
ae
i
π
b
, where a, b
+
.
n
(b)
 1 i 
 , where n
Using the result from (a), show that 
 2
(c)
Hence solve the equation z8 −1 = 0.
, has only eight distinct values.
[9]
25.
Find, in its simplest form, the argument of (sin + i (1− cos ))2 where  is an acute angle.
26.
Consider w =
[7]
z
where z = x + iy, y  0 and z2 + 1  0.
z 1
2
Given that Im w = 0, show that z
= 1.
[7]
27.
(z + 2i) is a factor of 2z3 –3z2 + 8z – 12. Find the other two factors.
[3]
28.
Let P(z) = z3 + az2 + bz + c, where a, b, and c 
(–3 + 2i). Find the value of a, of b and of c.
[6]
. Two of the roots of P(z) = 0 are –2 and