HSC 2016 MX2 Complex Numbers Marathon (archive) (1 Viewer)

math man · Apr 20, 2016

Re: HSC 2016 Complex Numbers Marathon

NEW
Out of scope of course but still fairly doable!
Q. Using Euler's formula e^iz=cis(z) solve the equation sin(z)=2 where z is complex.

KingOfActing · Apr 20, 2016

Re: HSC 2016 Complex Numbers Marathon

math man said:
NEW
Out of scope of course but still fairly doable!
Q. Using Euler's formula e^iz=cis(z) solve the equation sin(z)=2 where z is complex.

$\frac{e^{iz}-e^{-iz}}{2i} = 2\\e^{iz} - e^{-iz} = 4i\\e^{2iz} - 4ie^{iz} - 1 = 0\\e^{iz} = \frac{4i\pm\sqrt{-12}}{2} \\e^{iz} = i(2\pm\sqrt{3})\\iz = \ln|2 \pm \sqrt{3}| + i(\frac{\pi}{2}+ 2n\pi), n \in \mathbb{Z}\\z = -i\ln{(2\pm\sqrt{3})} + \frac{\pi}{2} + 2n\pi, n \in \mathbb{Z}$

wu345 · Apr 21, 2016

Re: HSC 2016 Complex Numbers Marathon

Paradoxica said:
$$\noindent a) Show that, for any integer $n>0 \\ $\indent$ \prod_{r=1}^{n} 2\cos{\frac{r\pi}{2n+1}} =1 \\ $\indent by considering the roots of $ z^{2n+1} = 1 \\ $b) Hence, prove the following numerical identity:$ \\ $\indent$ \sum_{n=1}^{\infty} \prod_{r=1}^{n} \cos{\frac{r\pi}{2n+1}} = 1$

Isn't it meant to be 2cos 2pi r/2n+1 in part i?

math man · Apr 21, 2016

Re: HSC 2016 Complex Numbers Marathon

wu345 said:
Isn't it meant to be 2cos 2pi r/2n+1 in part i?

No it's right. It requires a sneaky few tricks (conjugate and half angles)

wu345 · Apr 21, 2016

Re: HSC 2016 Complex Numbers Marathon

i thought i worked it out for a sec

Paradoxica · Apr 21, 2016

Re: HSC 2016 Complex Numbers Marathon

wu345 said:
Isn't it meant to be 2cos 2pi r/2n+1 in part i?

If you want a hint, factorise into real quadratic factors, and use magic to isolate the term you want.

Paradoxica · Apr 21, 2016

Re: HSC 2016 Complex Numbers Marathon

wu345 said:
i thought i worked it out for a sec

Alternatively, consider the roots of z⁴ⁿ⁺² = 1

math man · Apr 21, 2016

Re: HSC 2016 Complex Numbers Marathon

wu345 said:
i thought i worked it out for a sec

1. Find all roots
2. After k=n find all conjugate pairs
3. Group all conjugate pair roots together and factorise using conjugate properties
4. Sub z=-1
5. Factorise all the 2s
6. Double angle formula
7. And you should hopefully see it

Paradoxica · Apr 21, 2016

Re: HSC 2016 Complex Numbers Marathon

math man said:
1. Find all roots
2. After k=n find all conjugate pairs
3. Group all conjugate pair roots together and factorise using conjugate properties
4. Sub z=-1
5. Factorise all the 2s
6. Double angle formula
7. And you should hopefully see it

Step 5 is unnecessary.

math man · Apr 21, 2016

Re: HSC 2016 Complex Numbers Marathon

Paradoxica said:
Step 5 is unnecessary.

You get two lots of 2^n so there is like 2ception so it just makes it cleaner

wu345 · Apr 22, 2016

Re: HSC 2016 Complex Numbers Marathon

thanks for the help, I finally got it.
for part b) it's just summing the converging g.p 1/2 +1/4 +1/8+...+1/2n so by the limiting sum of a g.p S=1/2/(1-1/2) which is 1.

Quick Question: What is rep power and what is its significance?

Also - do you just make up these good questions or do you get them from some where?

math man · Apr 22, 2016

Re: HSC 2016 Complex Numbers Marathon

wu345 said:
thanks for the help, I finally got it.
for part b) it's just summing the converging g.p 1/2 +1/4 +1/8+...+1/2n so by the limiting sum of a g.p S=1/2/(1-1/2) which is 1.

Quick Question: What is rep power and what is its significance?

Also - do you just make up these good questions or do you get them from some where?

Most of these questions are just pretty results found probably from uni which are doable for 4u. However, I have created some original questions before for my tutoring clients, and some are nasty lol

porcupinetree · Apr 22, 2016

Re: HSC 2016 Complex Numbers Marathon

wu345 said:
thanks for the help, I finally got it.
for part b) it's just summing the converging g.p 1/2 +1/4 +1/8+...+1/2n so by the limiting sum of a g.p S=1/2/(1-1/2) which is 1.

Quick Question: What is rep power and what is its significance?

Also - do you just make up these good questions or do you get them from some where?

Rep power for a particular user is how many reputation points they can give to another user by clicking the reputation button. ie if person X, who has a rep power of 3, decides to give person Y, person Y's overall rep points will increase by 3.

A particular user's rep power depends upon their post count, how long they've been a member (and perhaps how many rep points they have) afaik.

seanieg89 · Apr 22, 2016

Re: HSC 2016 Complex Numbers Marathon

Here is a one way of proving the fundamental theorem of algebra (given an assumption from slightly outside syllabus).

$Assumption: If $p(z)$ is a polynomial with complex coefficients, then for any positive real number $R$, we can find complex numbers $z_{\max},z_{\min}$ with modulus at most $R$ such that:\\ \\ $|p(z_{\min})|\leq |p(z)|\leq |p(z_{\max})|$ for all $|z|\leq R$.\\ \\ Now suppose for the sake of contradiction that $p$ is a non-constant polynomial with no roots in the complex plane.\\ \\ 1. Show that there exists a complex number $s$ such that $0<|p(s)|\leq |p(z)|$ for all complex $z$. (Hint: The triangle inequality is useful here and in Part 3).\\ \\ 2. Let $q(z)=p(z+s)/p(s)$. What do the assumptions on $p$ tell us about $q$?\\ \\ 3. By considering the lowest degree terms of $q(z)$, show that there exists a $z$ close to $s$ with $|p(z)|<|p(s)|$ \\ \\4. Draw a conclusion.$

Paradoxica · Apr 22, 2016

Re: HSC 2016 Complex Numbers Marathon

porcupinetree said:
Rep power for a particular user is how many reputation points they can give to another user by clicking the reputation button. ie if person X, who has a rep power of 3, decides to give person Y, person Y's overall rep points will increase by 3.

A particular user's rep power depends upon their post count, how long they've been a member (and perhaps how many rep points they have) afaik.

Darn. Now I need to invent a time machine and go back to when BOS first came about, and make my account then.

And then give the details to myself, and then tell him to repeat everything to stabilise the timeline.

HSC 2016 MX2 Complex Numbers Marathon (archive) (1 Viewer)

math man

Member

KingOfActing

lukewarm mess

wu345

Member

math man

Member

wu345

Member

Paradoxica

-insert title here-

Paradoxica

-insert title here-

math man

Member

Paradoxica

-insert title here-

math man

Member

wu345

Member

math man

Member

porcupinetree

not actually a porcupine

seanieg89

Well-Known Member

Paradoxica

-insert title here-

Users Who Are Viewing This Thread (Users: 0, Guests: 1)