Can VCE'ers compete with HSC Maths?

[kpo2305]

Hey Guys! so like a lot of people currently completing the highest level of maths in the NSW HSC, we are all itching to know who would come out on top? NSW HSCers or VIC VCEers?
Here is a thread to compare and contrast the best of the best and the most challenging of the challenging.
Post any and all questions here and let's see who comes out as supreme!

[kpo2305]

To start of with a relatively simple one here is one from the HSC Volumes course


a) The region bounded by the curve y = (x − 1)(3 − x) and the x-axis is rotated about
the line x = 3 to form a solid. When the region is rotated, the horizontal line
segment l at height y sweeps out an annulus.
                 (i) Show that the area of the annulus at height y is given by .
                 (ii) Find the volume of the solid.

[pi]

I think it's a well established fact that Extension 2 dumps on Specialist Maths haha. Maybe our uni Maths (enhancement studies) students could compete? Not sure if that sort of stuff is allowed in NSW though?

[99.90 pls]

Prove, by means of complex numbers, that arctan(1) + arctan(2) + arctan(3) = pi.

VCE specialist

[brenden]

I think it's a well established fact that Extension 2 dumps on Specialist Maths haha. Maybe our uni Maths (enhancement studies) students could compete? Not sure if that sort of stuff is allowed in NSW though?

totally valid competitors.

[pi]

Prove, by means of complex numbers, that arctan(1) + arctan(2) + arctan(3) = pi.

lol that was on my spesh SAC (although they didn't tell us it equalled pi), not overly difficult

[kpo2305]

Haha, I'm not sure what about enrichment stuff, but everyone and anyone is welcome to contribute!

[pi]

Haha, I'm not sure what about enrichment stuff, but everyone and anyone is welcome to contribute!

Absolutely! I'd like to extend an invitation to HSC students to smash out some question in these threads, plenty unanswered and perhaps plenty that can be answered in more elegant ways!

Recreational Problems (SM level)

Fun questions :)

Have fun mathing!

[cosine]

Prove, by means of complex numbers, that arctan(1) + arctan(2) + arctan(3) = pi.

VCE specialist

This thread is like a math nerd's dream, hence, I could not resist 
Although I do VCE with 99.90pls, once again, I could not resist


We were dealing with angles (arctan(x) being an angle), so I decided to work with complex angles, Arguments.



Say z = 1+i





Say z = 1+2i





Say z = 1+3i






So arctan(1) + arctan(2) + arctan(3) = Arg(1+i) + Arg(1+2i) + Arg(1+3i)

Arg(1+i) + Arg(1+2i) + Arg(1+3i) = 3.14159265359

[cosine]

Is this thread only for 'prove' questions, or any mathematically challenging questions?

[pi]

This thread is like a math nerd's dream, hence, I could not resist 
Although I do VCE with 99.90pls, once again, I could not resist

Nice! Another way would be by using tan(A+B) twice and going via a trig route

[brenden]

Is this thread only for 'prove' questions, or any mathematically challenging questions?

Any mathematically challenging questions!

[TheAspiringDoc]

Find and prove the most space efficient way to pack cannonballs into a box/prism?

[Stevensmay]

Find and prove the most space efficient way to pack cannonballs into a box/prism?

It's 74% but not easily provable, assuming you have a cube. https://en.wikipedia.org/wiki/Kepler_conjecture

idunn says Gauss proved it for a cube but I can't find the proof quickly.

[Conic]

Arg(1+i) + Arg(1+2i) + Arg(1+3i) = 3.14159265359

What did you do here?