You are Here: Home >< Maths

AQA FP2 De Moivre's questions

Announcements Posted on
Would YOU be put off a uni with a high crime rate? First 50 to have their say get a £5 Amazon voucher! 27-10-2016
1. Final question. from the June 13 paper

Part a is sin4x = 4(cosx)^3sinx - 4cosxsin^3x

I can do the first part of part b but don't know how to work out the other roots

And part c is just lost on me

Thanks
2. (Original post by TheKian)
Final question. from the June 13 paper

Part a is sin4x = 4(cosx)^3sinx - 4cosxsin^3x

I can do the first part of part b but don't know how to work out the other roots

And part c is just lost on me

Thanks
For the first part of (b), you noticed that is a root of the quartic by recognising that it gives .

But the same can be said for any , where is such that . Can you write down all such (between 0 and - why is this enough?), and consequently all the remaining roots of the quartic?

For part (c), write down a quartic with roots by making a substitution for into the quartic, rearranging and squaring suitably. Then recall that you can determine the sum of the roots of a polynomial from it's coefficients.
3. (Original post by Farhan.Hanif93)
For the first part of (b), you noticed that is a root of the quartic by recognising that it gives .

But the same can be said for any , where is such that . Can you write down all such (between 0 and - why is this enough?), and consequently all the remaining roots of the quartic?

For part (c), write down a quartic with roots by making a substitution for into the quartic, rearranging and squaring suitably. Then recall that you can determine the sum of the roots of a polynomial from it's coefficients.
I'm still stuck on part c. When you say write a quartic with roots by maing a substitution for into the quartic. Do you mean ? if so, where do I go from there.
4. (Original post by TheKian)
I'm still stuck on part c. When you say write a quartic with roots by maing a substitution for into the quartic. Do you mean ? if so, where do I go from there.
Not quite. Let t = √u. Then, after some rearrangement followed by careful squaring (i.e. to get rid of the square roots), you will have another quartic in terms of u with roots u = t^2 for each t that is a root of the old quartic.
5. (Original post by TheKian)
Final question. from the June 13 paper

Part a is sin4x = 4(cosx)^3sinx - 4cosxsin^3x

I can do the first part of part b but don't know how to work out the other roots

And part c is just lost on me

Thanks
For part C it is just a disguised FP1 question where you find the relationship between the roots of the polynomial and the coefficients. If you remember that Σα2 = (Σαβ)2 -2Σαβ. Apply the same concept here and, and you will have to spot a symmetry between the roots you will get to the result.
6. (Original post by Farhan.Hanif93)
Not quite. Let t = √u. Then, after some rearrangement followed by careful squaring (i.e. to get rid of the square roots), you will have another quartic in terms of u with roots u = t^2 for each t that is a root of the old quartic.
That sounds like hard work. I think it is easier to let the roots be then form by considering the square of the sum of the roots - you can then use the various relationships between the coefficients and sum, product, etc of the roots.

[Aside: latex still broken - FFS - someone please fix this]
7. (Original post by atsruser)
That sounds like hard work. I think it is easier to let the roots be then form by considering the square of the sum of the roots - you can then use the various relationships between the coefficients and sum, product, etc of the roots.

[Aside: latex still broken - FFS - someone please fix this]
It largely boils down to whether one is required to take the extra step in proving the identity relating the sum of the squares of the 4 roots to the sum and products of the roots at A-Level - I don't recall it as quotable when I did FP2 around 5 years ago but I may have misremembered.

If that is the case, I don't see how this is any easier than squaring both sides of u^2 - 6u + 1 = 4(1-u)√u and rearranging for a quartic that you may simply read the result off from; as opposed to reading two results from the old quartics, proving a relation (which requires squaring, too) and then computing the sum of the squares by substitution into said relation. [Although, I concede that this is probably the less obvious/unexpected approach.]
8. (Original post by Farhan.Hanif93)
It largely boils down to whether one is required to take the extra step in proving the identity relating the sum of the squares of the 4 roots to the sum and products of the roots at A-Level - I don't recall it as quotable when I did FP2 around 5 years ago but I may have misremembered.

If that is the case, I don't see how this is any easier than squaring both sides of u^2 - 6u + 1 = 4(1-u)√u and rearranging for a quartic that you may simply read the result off from; as opposed to reading two results from the old quartics, proving a relation (which requires squaring, too) and then computing the sum of the squares by substitution into said relation. [Although, I concede that this is probably the less obvious/unexpected approach.]
The Vieta's formulas can be assumed in FP1 as well as FP2.

Register

Thanks for posting! You just need to create an account in order to submit the post
1. this can't be left blank
2. this can't be left blank
3. this can't be left blank

6 characters or longer with both numbers and letters is safer

4. this can't be left empty
1. Oops, you need to agree to our Ts&Cs to register

Updated: June 15, 2016
TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Today on TSR

University open days

Is it worth going? Find out here

Poll
Useful resources

Maths Forum posting guidelines

Not sure where to post? Read here first

How to use LaTex

Writing equations the easy way

Study habits of A* students

Top tips from students who have already aced their exams

Chat with other maths applicants