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Contrary what people have done here, as far as I know, in the HSC you need to sum infinitely small shells or slices and then turn it into an integral...

Differentiate your answer to check. Generally, to find the integral of odd powers of trig functions you require trig identities and for even powers require integration by parts I believe.

Um, did I read correctly that you tutor Ext 1 mathematics? oO

Nice question. Definitely something that the 2U and even some 3U candidates would have trouble with. Contrary to many peoples' responses here (who fell for the obvious red herring), the two lengths of the triangle and the rectangle are: 6.84m and 63.2m (to 3 sf) respectively to yield the...

Yeah my bad.

P is not a constant lol. You need to use the condition (ie. xy=k) since you are not using parametrics. And Trebla, this is a typical 4U conics question isn't it? It's a bit mean asking it to the prelimers :P

They don't use that notation at HSC level I believe (although you can find Coroneos alternating between it and the latter notation in his earlier books).

You could also use sum of roots = 0 by consider w7=1 as a polynomial equation and by noting that there 7 distinct roots are 1, w, w2, ... , w7.

Why is this in Ext 2?

Yep that was it. It's so much easier than the above algebraic approach :)

Personally I thought that Mechanics was the easiest topic in the 4U course. It's just common sense and logic. I don't know why people find it so difficult.

There's another elegant geometrical approach to this problem (which uses symmetry). I might post it later.

Because it's not a standard induction question. It's from the 1990 4U HSC Paper. It was in Q8 I believe.

Yeah a low Band 6.

Whoops. I shouldn't do maths after a long day. haha

This is from a CSSA trial. The second part is more interesting I think.

Let u=e^x du/dx=e^x = u \therefore { Integral, I } = \int \frac{u}{u^2+u} du \\= \int \frac{1}{u+1}du \\= ln|u+1| + c\\ = ln |e^x+1|+c

Do you prove the fundamental theorem of calculus everytime you calculate an integral?

Didn't you complete your HSC in 2006?

Yeah, it's probably not going to be ask anytime soon but it's still in the course. Also, if you check the 1989 HSC paper, they ask for this method (but they don't ask you to derive the tangential component of acceleration), so it's good learning this way too :) It's a bit like proving...