Q16 style problems thread (2 Viewers)

[Flatuitous]

Here’s why this problem is a total legend and why you should give it a crack.

1. It’s Ancient Tech (4,000 Years Old!)

The question calls it "Newton’s Method," but the logic actually dates back to Babylonia, circa 1700 BC. Before calculators existed, ancient surveyors used this exact method to calculate square roots for building diagonal walls. It’s arguably the world’s first computer algorithm.

2. It’s the "Source Code" of Engineering

Planning on doing Computer Science or Engineering at uni? Pay attention.

Computers don’t magically know what sqrt{2} is. They can’t store infinite decimals. When you code Math.sqrt(2), the computer runs an iterative loop almost identical to this problem. It’s a sneak peek into Numerical Analysis, the maths that powers everything from game physics to rocket trajectories.

View attachment 51396

you're brilliant vu hung
please keep posting these types of posts

 

[jane1820]

you're brilliant vu hung

fetus what abt me? 🥹

 

[Flatuitous]

fetus what abt me? 🥹

nah

 

[vuhung]

Hi everyone!

Here is a fascinating problem for you to try today. We are diving into Complex Dynamics and the famous Mandelbrot Set.

Why is this relevant for university studies?

This problem touches on concepts fundamental to higher-level STEM courses:

• Engineering & Physics: It introduces Stability Analysis, determining whether a dynamic system remains stable (bounded) or diverges out of control.
• Computer Science: Understanding iterative algorithms and recursion is essential for modelling complex systems.
• Pure Mathematics: This is a classic entry point into Chaos Theory and Complex Analysis.

The Mission:
Your goal is to prove the "Escape Criterion" for the Mandelbrot sequence, using mathematical induction and the triangle inequality.

This is not a so challenging problem but requires strict logic, and it is excellent practice for the Extension 2 exam.

Share your solution or thoughts in the comments below!

 

[vuhung]

Let’s be real. The HSC Extension 2 exam is already a horror movie. You walk in, you battle with Complex Numbers, you survive the Intergations, and then you hit the dreaded Question 16. The place where dreams go to die and where proofs suddenly feel like primary school arithmetic.

But today, we’re raising the stakes.

I found a viral image of Srinivasa Ramanujan’s famous series for 1/pi and thought, "Hey, this looks absolutely terrifying. Let’s make high school students prove it for marks."

At first glance, you might think the hardest part of this problem is that you don't speak Hindi.

But don't worry, the language of "algebraic pain" is universal. Ramanujan didn't need a translator, and neither do you. You just need to trust the factorials.

Why attempt this beast?

1. It’s a flex. This series converges so fast that the first term alone gives you pi to 6 decimal places. You’ll prove why.
2. It’s actually doable. Despite looking like a chaotic mess of numbers, everything cancels out beautifully.
3. It’s perfect Q16 training. It combines Series, Inequalities, and Complex Numbers into one monster.

If you can handle this, the actual HSC will look like a warm-up.

 

[justletmespeak123]

@Trebla why did you combine all the posts into this disgusting long post?

 

[vuhung]

 

[WeiWeiMan]

View attachment 51810

I believe you made a slight typo perhaps

n rational does NOT imply sqrt(n) rational. e.g. sqrt(2) irrational

ifl the paragraph should read:

Proving the irrational of π^2 makes the statement stronger, since if π^2 were irrational, then π would also be irrational (as the square root of an irrational number) (do NOT prove this). Thus, by proving π^2 is irrational, we also confirm the irrationality of π.

 

[ottowatta]

View attachment 51810

My brother are you seriously in year 11, cos if you are, I am cooked beyond return bru

 

[Burnt_Out]

My brother are you seriously in year 11, cos if you are, I am cooked beyond return bru

he's not (i think)

 

[ottowatta]

he's not (i think)

thank fucking goodness I woulda necked myself right then and there

 

[vuhung]

Just bringing back some pre-2010-era Q16 energy to the forums. If your brain isn't melting, are you even doing Ext 2?

 

[Flatuitous]

Hi everyone!

Here is a fascinating problem for you to try today. We are diving into Complex Dynamics and the famous Mandelbrot Set.

Why is this relevant for university studies?

This problem touches on concepts fundamental to higher-level STEM courses:

• Engineering & Physics: It introduces Stability Analysis, determining whether a dynamic system remains stable (bounded) or diverges out of control.
• Computer Science: Understanding iterative algorithms and recursion is essential for modelling complex systems.
• Pure Mathematics: This is a classic entry point into Chaos Theory and Complex Analysis.

The Mission:
Your goal is to prove the "Escape Criterion" for the Mandelbrot sequence, using mathematical induction and the triangle inequality.

This is not a so challenging problem but requires strict logic, and it is excellent practice for the Extension 2 exam.

Share your solution or thoughts in the comments below!

View attachment 51466

Are some of the questions you posted here not available elsewhere?

I can't seem to find this question and the ramanujan question in your Last Resorts booklet

 

[jane1820]

Are some of the questions you posted here not available elsewhere?

I can't seem to find this question and the ramanujan question in your Last Resorts booklet

π =22/7
therefore, π^2 = 484/49

 

[Flatuitous]

π =22/7
therefore, π^2 = 484/49

jane im gna implode

 

[vuhung]

Are some of the questions you posted here not available elsewhere?

I can't seem to find this question and the ramanujan question in your Last Resorts booklet

Thanks! Re-added to the booklet.

 

[vuhung]

I believe you made a slight typo perhaps

n rational does NOT imply sqrt(n) rational. e.g. sqrt(2) irrational

ifl the paragraph should read:

Proving the irrational of π^2 makes the statement stronger, since if π^2 were irrational, then π would also be irrational (as the square root of an irrational number) (do NOT prove this). Thus, by proving π^2 is irrational, we also confirm the irrationality of π.

Thanks, fixed and updated.

 

[vuhung]

I can't seem to find this question and the ramanujan question in your Last Resorts booklet

How did you find the section "Fundamentals Review" in the booklet?

 

[WeiWeiMan]

How did you find the section "Fundamentals Review" in the booklet?

ur booklets seem lwk goated. i kirkenuinely wish i had these when I was doin 4u

 

[Flatuitous]

How did you find the section "Fundamentals Review" in the booklet?

I definitely enjoy how your booklets extend a touch out of syllabus with in syllabus scaffolding

I see the appeal of certain oos techniques and methods like the cross product and jensen's inequality

though, I do feel like some of them (e.g. weighted am-gm, newton's identities) feel less like exam techniques and some of them feel too niche to be applicable for exams


If I could, I would recommend adding this "majorising" tech as a secret tip for when symmetric LHS and RHS have the same degree in each term (not exactly too sure how to phrase it)
For example, to prove a^3 + b^3 + c^3 >= a^2b + b^2c + c^2a, you cycle am-gm

a^3 + a^3 + b^3 >= 3a^2b (from 3-var am-gm)
b^3 + b^3 + c^3 >= 3b^2c
c^3 + c^3 + a^3 >= 3c^2a

Summing those inequalities, we have proven our wanted inequality to be true

Another example:
Prove a^3/b + b^3/c + c^3/a >= a^2 + b^2 + c^2

a^3/b + a^3/b + b^2 >= 3a^2
b^3/c + b^3/c + c^2 >= 3b^2
c^3/a + c^3/a + a^2 >= 3c^2

Again, summing those inequalities will prove our wanted inequality