# BOS trial (1 Viewer)

#### mathsbrain

##### Member
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?

Thanks in advanced!

#### fan96

##### 617 pages
If

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

then

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

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

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

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

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

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

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

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#### mathsbrain

##### Member
what? is that banned?

#### BLIT2014

##### The pessimistic optimist.
Moderator
BOS trial typically gets held later this year closer to HSC so it has not already been done.

#### mathsbrain

##### Member
Thanks for the fun!

#### Trebla

##### Administrator
Administrator
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.