1. ## BOS trial

Was stuck on a question:
Sketch the locus defined by arg(z^2+z)=pi/3, stating its equation.
Obviously this will turn into the hyperbola with the positive branch, but how do we find the equation?

Also, anyone know when the bored of studies trial will be held this year, of is it held already?

2. ## Re: BOS trial

ive never seen this questions before; but this is how i would solve it. How is it "obviously" a hyperbola btw?
argz+argz+1=pi/3
take tan of both sides, or sin or cos or whatever i dont think it matters
for tan actually: tan(a(z+1)+a(z))= sqr3
tan(arg(z))=Im/Re=y/x
tan(arg(z+1))=y/x+1
let tan(arg(z+1))=a and tan(arg z) = b

we have a+b/(1-ab)=sqr3 (angle formula)
(y/x+y/(x+1)) / (1-y^2/(x^2+x))=3^0.5
just manipulate and u should get an answer; im pretty sure this is incorrect but meh. It gives a shape that looks like at tilted hyperbola btw.

what's the bos trial?

3. ## Re: BOS trial

Originally Posted by mrbunton
what's the bos trial?
ban

4. ## Re: BOS trial

If

$\arg(z+z^2) = \pi/3$,

then

$\tan \frac\pi 3 = \frac{\mathrm{Im}(z^2+z)}{\mathrm{Re}(z^2+z)}$

But since $\tan\pi/3 = \tan -2\pi/3$,

$\tan \frac\pi 3 = \frac{\mathrm{Im}(z^2+z)}{\mathrm{Re}(z^2+z)}$

also gives the locus of $\arg(z+z^2) =-2\pi/3$.

To get around this we can require $z^2 +z$ to be in the first quadrant, i.e. $\mathrm{Re}(z^2+z) > 0$ and $\mathrm{Im}(z^2+z) > 0$.

Setting $z = x + iy$, this gives:

$\frac{y\left(2x+1\right)}{\left(x+y\right)\left(x-y\right)+x}=\sqrt{3}, \quad \left(x+y\right)\left(x-y+1\right)+2xy\ >\ 0$

5. ## Re: BOS trial

ban
what? is that banned?

6. ## Re: BOS trial

Originally Posted by mathsbrain
what? is that banned?
whoosh

was a joke my dude

7. ## Re: BOS trial

BOS trial typically gets held later this year closer to HSC so it has not already been done.

8. ## Re: BOS trial

Thanks for the fun!

9. ## Re: BOS trial

Originally Posted by mathsbrain
Also, anyone know when the bored of studies trial will be held this year, of is it held already?
There are plans underway to hold them this year in early October. Stay tuned for more details.

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