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A Few FP2(AQA) Questions

I've got my FP2 exam next week and have found a few questions on past papers which I'm having trouble with so thought I might as well bung them all in one post. Any help much appreciated :smile:

Have tried to include as much of my working as possible so you can see where I'm going wrong etc...

1) It is given that

z= e^{i\theta}

. In part a you had to show that

z + \frac{1}{z} = 2\cos\theta

which is fine, then a similar expression for z^2 for which I got

2\cos 2 \theta

then hence show that

z^2 - z + 2 - \frac{1}{z} + \frac{1}{z^2}=4 \cos^2 \theta - 2 \cos \theta

. Again this was fine, I used the two previous identities and a double angle identity. Here's where I get stuck:

b)Hence solve the quartic equation

z^4-z^3+2z^2-z+1=0

giving the results in the form a +bi.

So obviously I see that this is the same as the previous expression but multipled by z^2 in each term but I'm just not really sure where to go from here..

2) By considering the roots of

z^5 =1

show that

\cos\frac{2\pi}{5}+\cos\frac{4 \pi}{5}+\cos\frac{6\pi}{5}+ \cos \frac {8\pi}{5}=-1

so I've used

z= e^{\frac{2k\pi i}{5}}

with k=0,1,2,3,4 and i can see how my results relate as they are all of the form

\cos \frac{2k\pi}{5} + i \sin\frac{2k\pi}{5}

and with the k values inserted are the values they want (sorry I can't be bothered to write them all out but you get the picture) but I don't really know what to do with regard to showing it equals -1..?

I may have a couple more to add in a bit! But for now, any help with these two would be great, ta

Reply 1

MathsSuze

*Bump* would really appreciate some help :smile:

Reply 2

davros

Original post by MathsSuze

*Bump* would really appreciate some help :smile:

Well for part b) you know that z^2 can't be zero (because then the equation wouldn't be true!) so you can divide by this and compare what you get with the equation you had earlier.

For the next one, z^5 - 1 can be factorised as (z - 1)P(z) where P(z) is a polynomial of degree 4 i.e. the 1st term is z^4. The zeros of P(z) (i.e. the values of z where P(z) = 0) will be the solutions to z^5 = 1 except for z = 1. Now, what do you know about the sum of roots of a polynomial equation in terms of the coefficients of that equation?