Download FP1 Complex Numbers

Further Pure Mathematics 1
Chapter Assessment
Complex Numbers
1.
The complex number α is given by α = –2 + 5j.
(i) Write down the complex conjugate α*.
[1]
(ii) Find the modulus and argument of α.
(iii) Find
[3]
  *
in the form a + bj.

[3]
2.
(i) Given that w = 1 + 2j, express w², w³ and w4 in the form a + bj.
[5]
(ii) Given that w is a root of the equation z  pz  qz  6 z  65  0 , find the
values of p and q.
[5]
(iii) Write down a second root of the equation.
[1]
(iv) Find the other two roots of the equation.
[6]
4
3.
3
2
(i) Show that z1 = 2 + j is one of the roots of the equation z² – 4z + 5 = 0.
Find the other root, z2.
[3]
1 1 4
(ii) Show that   .
z1 z2 5
[3]
(iii) Show also that Im (z1² + z2²) = 0 and find Re (z1² – z2²).
(iv) Find in the form r (cos   jsin  ) , the complex numbers z1, z1², and z1³.
[4]
[7]
(v) Plot the three complex numbers z1, z1², and z1³ on an Argand diagram.
[3]
© MEI, September 2001
Further Pure Mathematics 1
4.
(a) Solve the equation z² + 2z + 10 = 0.
Find the modulus and argument of each root.
[4]
(b) Complex numbers α and β are given by
5
5 




  2  cos  jsin  ,
  4 2  cos  jsin 
8
8 
8
8


(i)
Write down the modulus and argument of each of the complex
numbers α and β.
Illustrate these two complex numbers on an Argand diagram.
(ii)
[3]
Indicate a length on your diagram which is equal to    , and show
that     6 .
[3]
(iii) On your diagram, draw and label
(A)
the locus L of points representing complex numbers z such that
z   6 ,
[3]
(B)
the locus M of points representing complex numbers z such that
5
arg( z   ) 
.
8
[3]
© MEI, September 2001