Introduction

1. Introduction Today we will look at some complex number questions. These questions will help you with your coursework. If you can not do these questions you must re-read your lecture summaries and the HELM workbooks until you can. U:\1st Year Share\Mathematics for Engineers\Complex Numbers Materials

2. Complex Number Exam Questions Obtain the modulus and argument (in radians) of the complex numbers z1 = 2 + 3i and z2 = -1 – i and write z1 and z2 in the polar form r.ei.θ Hence or otherwise, determine the polar form of the complex numbers a) b)

3. Complex Number Exam Questions Given With Hence With However from the Argand diagram we can determine that Hence the correct root is -3π/4 Therefore

4. Complex Number Exam Questions a) b)

5. Complex Number Exam Questions It is given that the polynomial can be written in the form where q(x) is a quadratic a) Obtain q(x) b) Calculate the roots of q(x) c) Plot the roots of q(x) on an Argand diagram

6. Complex Number Exam Questions a) Obtain q(x) If then Equate coefficients of x x4 : 1 = a i.e. a = 1 x3 : -2 = b i.e. b = -2 x2 : -3 = c – 9a i.e. c = 9a – 3 = 6 Hence

7. Complex Number Exam Questions b) The roots of q(x) are given by the quadratic formula c) Argand Diagram Imaginary Axis, y (1,√5) Real axis, x (1,-√5)

8. Complex Number Exam Questions Use the binomial theorem, or otherwise, to expand the expression De Moivre’s theorem gives that for any positive integer n. Taking n = 4 use the above to show that And obtain an analogous expression for Sin(4θ)

9. Complex Number Exam Questions We know that

10. Complex Number Exam Questions Now we can collect the real and imaginary terms together Equate real parts to give Equate imaginary parts to give Hence

11. Conclusion Today we have looked at some complex number questions. You must make sure that you can do these. If you can not do these questions you must re-read your lecture summaries and the HELM workbooks until you can. Don’t forget your seminars this week!