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Oxbridge Physics Prelims Revision

Reply 60

Bezza

2

Hoofbeat

Anyone offer any thoughts on this complex no's exam question:

=====
Q. Find the cubix function

Unparseable latex formula:

\fontsize{2} Q(z)

such that

Unparseable latex formula:

\fontsize{2} Q(z) = -1

when

Unparseable latex formula:

\fontsize{2} z=0

and

Unparseable latex formula:

\fontsize{2} Q(z)=0

when

Unparseable latex formula:

\fontsize{2} z=1, e^{2i\pi/3}

and

Unparseable latex formula:

\fontsize{2} e^{4i\pi/3}

=====

So let our cubic equation be:

Unparseable latex formula:

\fontsize{2} Az^{3} + Bz^{2} + Cz + D

Firstly, D=-1 as when z=0, Q(z) = -1.

Then I was thinking that

Unparseable latex formula:

\fontsize{2} z=1, e^{2i\pi/3}, e^{4i\pi/3}

are the roots so if we calculate the sum of these we know they must be equal to -B/A

Similarly work out the product of the roots, ie.

Unparseable latex formula:

\fontsize{2} e^{4i\pi}

we know this must be equal to -1/A and thus

Unparseable latex formula:

\fontsize{2} A = e^{-4i\pi}

. Now I don't know how I should progress?

You just want to write it factorised since you have the factors already by knowing the roots

Unparseable latex formula:

\fontsize{4} Q(z) = A(z-1)(z-e^{2i\pi/3})(z-e^{4i\pi/3})

and since Q(z)=-1 when z=0, so A=1.

Reply 61

Hoofbeat

OP

14

Bezza

You just want to write it factorised since you have the factors already by knowing the roots

Unparseable latex formula:

\fontsize{4} Q(z) = A(z-1)(z-e^{2i\pi/3})(z-e^{4i\pi/3})

and since Q(z)=-1 when z=0, so A=1.

Ahhh...so much simpler!

EDIT: Hold on...me being v.stupid....I can't see how you get A=1. As if you set z=0, and Q=-1 you end up with

Unparseable latex formula:

\fontsize{2} -1 = -A \times e^{2i\pi}

now I'm confused!!!!

Reply 62

RichE

15

Hoofbeat

Ahhh...so much simpler!

EDIT: Hold on...me being v.stupid....I can't see how you get A=1. As if you set z=0, and Q=-1 you end up with

Unparseable latex formula:

\fontsize{2} -1 = -A \times e^{2i\pi}

now I'm confused!!!!

but e^(2pi i) =1

Reply 63

Bezza

2

RichE

but e^(2pi i) =1

yeah, just use de moivre's - e^(2pi i) = cos(2pi) + isin(2pi) = 1

Reply 64

shiny

12

Willa

still no cookie shiny....I'm tres upset!

wtf?!

Reply 65

Hoofbeat

OP

14

Bezza

yeah, just use de moivre's - e^(2pi i) = cos(2pi) + isin(2pi) = 1

of course! thanks :smile:

Reply 66

Willa

11

shiny

wtf?!

if it aint there tomorrow i will have to presume they put it in the wrong pigeon hole and some evil person ate it!

Reply 67

~Raphael~

9

Willa

if it aint there tomorrow i will have to presume they put it in the wrong pigeon hole and some evil person ate it!

Or that Shiny never sent it :p:

Reply 68

shiny

12

Willa

if it aint there tomorrow i will have to presume they put it in the wrong pigeon hole and some evil person ate it!

I put £3 in there too! :mad:

Reply 69

Willa

11

shiny

I put £3 in there too! :mad:

I'm sure it'll be there tomorrow...it has to be!

Reply 70

Bezza

2

£3

Expensive pints in cambridge! Willa's going to miss out on the whole ben's cookies experience - you need to have it warm so the chocolate all melts and you forgot to pick up a napkin so have chocolate all over your face :cool:

Reply 71

Willa

11

Bezza

£3

Expensive pints in cambridge! Willa's going to miss out on the whole ben's cookies experience - you need to have it warm so the chocolate all melts and you forgot to pick up a napkin so have chocolate all over your face :cool:

well he decided that a pint would cost £3. I said it would cost me £1.20 but he still sent me £3, so I was gonna buy the pint and something else with the change and send that back to him :p:

Reply 72

Willa

11

IT CAME. IT CAME. The gift from the dark side came! Yay for shiny, you're the bestest!

Mmm I think I should pop it in the oven for a few mins so I can eat it warm.....mmmmmm cookie!

edit: it's warming in the oven as we speak. Gonna eat it on my way to my compsci lecture. Then I'll be able to judge whether or not it beat's Millies cookies! (I'll keep in mind it's a few days old so wont be as good as when fresh)

Reply 73

Willa

11

I've eaten my cookie now. I have to say, ben's cookies are very very good. It wasnt as "more-ish" as millies cookies are for me (that might be a good thing though cos it saves money) but it definitely had a distinct "homemade" like taste to it, which millies seems to lack. I'd say they're both at least equally as good as each other.

Reply 74

~Raphael~

9

Willa

I've eaten my cookie now. I have to say, ben's cookies are very very good. It wasnt as "more-ish" as millies cookies are for me (that might be a good thing though cos it saves money) but it definitely had a distinct "homemade" like taste to it, which millies seems to lack. I'd say they're both at least equally as good as each other.

Dude get over your Oxford-hating and the true flavour will come through!!!! :biggrin:

Reply 75

Willa

11

~Raphael~

Dude get over your Oxford-hating and the true flavour will come through!!!! :biggrin:

no I'm being objective about this, I don't think you have the right to say I'm not being when you have yet to try millies gourmet chocolate cookie. It's definitely at least as nice. But I agree that there's hardly anything between the two, both taste v yummy!

Reply 76

~Raphael~

9

Willa

no I'm being objective about this, I don't think you have the right to say I'm not being when you have yet to try millies gourmet chocolate cookie. It's definitely at least as nice. But I agree that there's hardly anything between the two, both taste v yummy!

I have tried Millie's Cookies, which is why I was able to make my comment i nthe first place ages ago...Millie's is a crappy chain all over the country in every shopping centre! They're nice, but they have an artificial flavour to them...Of course, this is just my opinion, but I think it's worth more than yours :p:

Reply 77

Hoofbeat

OP

14

~Raphael~

I have tried Millie's Cookies, which is why I was able to make my comment i nthe first place ages ago...Millie's is a crappy chain all over the country in every shopping centre! They're nice, but they have an artificial flavour to them...Of course, this is just my opinion, but I think it's worth more than yours :p: