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A bit late to the game but I recently gave a variant of this question to my top year 9 class. Understandably they struggled and requested an explanation, so posting here in case it helps any future students:

Is this essentially Thale's theorem? ... here's how I proved it without vectors ...

This should have you sorted for Q11:

This might help you for 10 d):

I think this is what you need?

ohhhh … that step is to show why you can start the summation from r=1 instead of r=0 (notice the subtle difference in where the sigma starts).

The second line differentiates both sides wrt x. The purpose of doing that is to get the exponents to match what we are being asked to prove.

Oh wait, I think my first idea does work ... Just needed a few little extra steps ...

Unfortunately nothing jumps out ... I did a quick crack at induction but no luck ...

yeah sorry I think differentiation isn’t the way to go … 🧐

Oh actually I think there’s still an issue with the x^r instead of x^(r-1) on the RHS … hmmmm

Could you differentiate both sides? And then note that for the RHS, when r=0 that term drops off? I think that gets you there …

Try this ...

Does this help? It's a similar question and I reckon if you use the same approach, you should be able to work it out ...

These rectangles and curves questions pop up quite a bit in relation to inequality proofs, so it's good to get comfortable with the technique. Here's a few similar questions I've seen on this front:

P(B | A) = P(B and A) / P(A) ... in this case, let A be winning the first season, and B be winning the second season

Funnily enough I got exactly 94 for my extension 2 HSC mark. From memory I had a crack at everything, but there was one question where I was just stabbing in the dark. This was back when Extension 2 HSC exams only had 8 questions, so I’d say I felt good about 7/8 of the questions to end up with 94 …

Yes good point well presented!

Oooooh yes very good point! ... What about this approach? (I squared top and bottom to simplify instead) ... Or does this approach also have an issue?

Is it 1? ... My instinct was to try and get that x out of the denominator ...