1.
1. Given that
and
, find
(i)
,
[1]
(ii)
arg
[1]
(iii)
[2]
(iv)
[2]
Show the complex numbers
clearly labelling
and arg
2. (a)
and
.
Express Z =
on the same Argand diagram,
[2]
(nov2003)
in modulus argument form. Hence find in their
simplest form the moduli and arguments of the numbers,
(i)
,
.
(ii)
(b)
(i)
[6]
Shade the area represented on an Argand diagram by
│
│
.
[2]
(ii) Sketch the locus of Z if
arg(Z – 1) – arg(Z + 1)
3. Given that
[3]
(june2004)
, find
(i)
and arg ,
(ii)
in the form
[2]
, where
represents the conjugate of
and
and
are real numbers.
[2]
(nov2004)
4. The complex numbers and
are given by
and
respectively, where and are real and positive. Given that
(i)
, find
[1]
(ii)
arg
(iii)
the exact values of
[1]
and .
[6]
The complex number
can be expressed in the form
are real. Find the value of
given that
where
and
[5]
(nov2007)
1.
The complex number
(a)
(b)
(c)
and it is given that
State the relationship between
(i)
│ │ and │ │,
(ii)
arg
and arg
Given that
are real numbers.
find
(b)
(c)
in the form
where
and
[2]
The points P, Q and R in an Argand diagram represent the complex numbers
and respectively.
(i)
State the kind of quadrilateral that OPRQ is, where O is the origin.
[1]
(ii)
Find the area of OPRQ.
[3]
1. The complex number
(a)
[2]
and it is given that
State the relationship between
(i)
│ │ and │ │,
(ii)
arg
and arg
Given that
are real numbers.
find
[2]
in the form
where
and
[2]
The points P, Q and R in an Argand diagram represent the complex numbers
and respectively.
(i)
State the kind of quadrilateral that OPRQ is, where O is the origin.
[1]
(ii)
Find the area of OPRQ.
[3]
1. The complex number
Find
in the form
It is given that
(a)
Find
(b)
If
and the complex number
is such that
and sketch it on an Argand diagram.
[5]
and
and
in the form
, obtain the exact values of the modulus and argument of
[3]
[4]
The complex number
(a)
(b)
Express in the form
Find,
(i)
modulus of
(ii)
argument of
.
where
and
are real.
[2]
[5]
Paper 1 Pure Mathematics 3 hours (120 marks)
1 Indices and proportionality
2 Polynomials
3 Identities, equations and inequalities
4 The modulus function
5 Graphs and coordinate geometry in two dimensions
6 Vectors (1)
7 Functions
8 Sequences and series
9 Series expansions
10 Plane trigonometry
11 Trigonometrical functions
12 Logarithmic and exponential functions
13 Differentiation
14 Integration
15 First order differential equations
16 Numerical methods
17 Complex Numbers (1)