Complex Numbers Question

To do question 4 on page 78 you need

(a) to be able to draw an Argand Diagram (see page 72) and to plot the complex numbers
z = 1-4i and iz = i(1-4i) on it.
Before you plot the second number you have to multiply out i(1-4i). This works the same way as multiplying say, x(1+4x) would in algebra, but there is a second step you need to do - resolving the i²=-1

(b) To do this you need to remember division of complex numbers (page 68).
10-2i
------
2-3i
You multiply above and below by the complex conjugate of the bottom.
[If z = 1 + 7i, then z¯, the complex conjugate of z = 1-7i]
So you multiply above and below by (2+3i)
(10-2i)(2+3i)
-------------
(2-3i)(2+3i)
If you don't get a real number below (after you have adjusted i²=-1) then you've made a mistake.
Whatever you get for your result, you have to factor out (1+i) in order to find the missing value k. If you don't get something like 2+2i which could be written as 2(1+i) or 3+3i which could be written as 3(1+i) then you have made a mistake.
Note also that they say k∈N which is a further clue - telling you that it is positive and not a fraction.
(c)
(i) for this you need to revise equality of complex numbers (p70).
If you are told 2a + 2i = 6 +bi then you can equate the real parts of the equation
2a = 6 so a = 3
and the imaginary parts
2 = b (Note - leave the i's out of the imaginary equation)
to solve.
Sub in w = 3-4i and solve.
(ii) ¦w¦ is the modulus of w (page 73).
Sub in all the values you are given, and solve the real and imaginary equations using the same rules as above (equate the reals, equate the imaginary parts).
In this case, the solution is much more complicated. You end up with s's and t's in both equations, so you have to solve using simultaneous equations.
The answers turn out to be fractions which look a bit unlikely - but again there is a clue in the question where you are told s,t ∈ R.