Argand Diagram question - paper 1 Q2 (b)

beeroclock

Hi all

Would appreciate if anyone can give a worked solution to question 2 (b):

http://www.examinations.ie/archive/markingschemes/2011/LC003ALP030EV.pdf

The solution is given but Im not show how to derive the answer

Thanks!

MathsManiac

I assume you're wondering about part (i), since the full solution to part (ii) is there.

There are two different ways to do it, as far as I can see:
Firstly, just by looking at the distances from the origin. Since you're told that the modulus of z is bigger than 1, it follows that modulus of z^2 is bigger again, and z^3 is bigger again. So, z has to be the one that's closest to the origin, z^2 next, and z^3 the farthest.

Alternatively, you could think about the arguments. z couldn't possibly be the one on the y-axis, because then z^2 would be on the negative x-axis and z^3 on the negative y-axis. Similarly, z couldn't be the point in the fourth quadrant, since that would put z^2 into the third quadrant. So the only possibility is that z is the one in the second quadrant. So the labelling given in the scheme is the only one possible. (If you draw lines from all the points to the origin, you'll see that this labelling is the only one that makes the angles work properly.)

TheBody

You need to think about what happens when you multiply two complex numbers. Multiplying by a complex number with modulus greater than 1 corresponds to a rotation and a lengthening. Similarly, multiplying by a complex number with modulus less than one corresponds to a rotation and a shortening. So to answer your question, that is why the shortest one must be z. As the Modulus of z is greater than 1, you know that [LATEX]z^2[/LATEX] must be the next longest and then [LATEX]z^3[/LATEX].

beeroclock

MathsManiac wrote: »

I assume you're wondering about part (i), since the full solution to part (ii) is there.

There are two different ways to do it, as far as I can see:
Firstly, just by looking at the distances from the origin. Since you're told that the modulus of z is bigger than 1, it follows that modulus of z^2 is bigger again, and z^3 is bigger again. So, z has to be the one that's closest to the origin, z^2 next, and z^3 the farthest.

Alternatively, you could think about the arguments. z couldn't possibly be the one on the y-axis, because then z^2 would be on the negative x-axis and z^3 on the negative y-axis. Similarly, z couldn't be the point in the fourth quadrant, since that would put z^2 into the third quadrant. So the only possibility is that z is the one in the second quadrant. So the labelling given in the scheme is the only one possible. (If you draw lines from all the points to the origin, you'll see that this labelling is the only one that makes the angles work properly.)

Yes I meant (i) and thank you very much for your answer makes perfect sense