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A couple of other ways to think about it (I find that it always helps to have multiple ways) A. Another way of thinking about anything around a circle is "there is only one first choice" which then fixes the circle. So place any person in the circle (the "first person"): 1 way. This fixes...

I hadn't heard of King's Property but a quick Google says that it is \int_a^b f(x) \mathrm{d} x = \int_a^b f(a+b-x) \mathrm{d}x You prove this using integration by substitution. The version that you use for all of these questions is: \int_{-a}^a f(x) \mathrm{d} x = \int_{-a}^a f(-x) \mathrm{d}...

Don't forget you also have another answer if you use the "reverse chain rule" on the cosine function! All three answers will differ by a constant (you can see this by using the double angle formulas for cos(2x)). This doesn't mean that any answer for an integral is correct because there are...

Unfortunately there aren't too many other shared properties of functions with the same definite integral across the same interval. For example, if we take the function defined by f(x) = x and find the integral between 0 and 1, the value is 1/2. But we could find an infinite number of functions...

Just because the definite integrals are the same, doesn't mean the functions themselves are the same e.g. integrating f(x) = x and g(x) = 1-x between 0 and 1 give the same answer, but are different functions. I don't think it's saying that you can easily go from the first line to the second...

The learning pyramid is a good way to think about what techniques are effective. Reading notes is definitely one of the least effective methods of studying because you don't retain a lot of it (and it makes you feel like you've done a lot). My two favourite ways to revise is: Teaching is the...

Probably only a bit further, but we spend more time working on the foundations (e.g. understanding how integration actually works through taking a limit of the sum of areas of rectangles). I don't think complex numbers is much harder than 4U. With vectors, we go further than 4U, going past 3...

I teach first year maths at UNSW, and MATH1131/MATH1141 (which is the first maths course that engineering students have to do) covers almost all of the 4U content, including complex numbers, proofs, vectors and integration. Mechanics is not directly covered. Then, second year maths courses cover...

I prefer Cambridge. It covers the content and worked examples with more depth, and also sorts questions by difficulty. However, it is a bit denser than Fitzpatrick and the difficulty of the questions can sometimes be too much, so just make sure you don't get discouraged when you can't do some of...

If you like maths, you should definitely go for it! It's a fun subject, but will definitely take some hard work. What I've heard from students is that the jump from Extension 1 to Extension 2 is bigger than the jump from from Advanced to Extension 1, so make sure you're aware of this. Since...

Q1a: You want to use the idea: if you have a*b = 0, then either a=0 or b=0. So in this case, you can say that either \sin^{-1} (x) = 0 or \cos^{-1}(x) = 0. You can then try to solve each of these, and my preferred way would be to use a graph. You should get x = 0 and 1. Q1b: This one is a lot...

One way that I've found helpful in getting started is to ask yourself to just do 1 practice question, or just do practice questions for 1 minute. This is a version of the 2-minute rule by James Clear.

The function itself actually doesn't exist so the question isn't really a valid one. For example, if you substitute x = 1, you get f(-1) + f(1) = 8, but if you substitute x=2, you get f(-1) + f(1) = 16. Both of these can't be true at the same time. I think the question is trying to get you to...

You want to isolate the x. But because it's an inequality, it's a little tricky because you can't just use the inverse tangent function without accounting for what your domain is. Remember that the inverse tangent function takes the range (-pi/2,pi/2) One way to do it is to draw a graph. So you...

Getting rid of silly mistakes is definitely one of the best ways to improve your marks. What I found helpful: Do past papers and even homework in exam conditions (timed and quiet, not checking the answer right away) so you're constantly getting used to the high pressure of a real exam. This...

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Hi everyone, It's Gary here 👋 I recently published a HSC Economics Workbook for Year 12 students. It has exam-style multiple choice, short answer and extended response questions, with full explanations for the multiple choice and short answer questions. I wrote these questions to mimic the...