In an exam, if you see this type of question what is the best way to know what to do and should i leave these types of questions last
Pretty sure this question was the last question on the paper that year so that's saying something.
i) is just your ordinary proof. Not much can be said here. (Combining the fractions could've helped make it clearer though)
With ii) you try to start on the LHS, but when doing a reduction formula you always need to look at the final result. Most notably, whether you have I
n, I
n-2 and what not. If you want to manipulate the reduction formula, anything that doesn't "look right" needs to be eliminated somehow. Anything that looks right (or promising), stays.
iii) is actually quite common - evaluating a reduction formula over and over again. Factoring out 2
n would've been the hard part here. But if you really look at your working out and compare it to what you're trying to prove, it shouldn't be as less obvious.
iv) was really hard mostly cause of the summation. It's a bit hard to tell that by the linearity of integrals you can just swap their order. That aside, your result in part (iii) was obviously looking somewhat similar to what C
n was in part (iv), so you have to play with the algebra to make it work.
Seeing that interchange of order is probably something new to add to your toolbox.
Also note how I broke that question up into bits and pieces and built up. That's what you do when things are becoming truly messy.
v) required you to note the similarity between the RHS and part (iv). Provided anything you do doesn't break maths, all you do is try to find the similarity, and then force it out. Like how I did it (introduce the square root, then the integral symbol).
vi) is an example of testing how well 4u students know their algebra. This comes with a lot of experience - algebraic manipulation.
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