I have completed my own set of worked solutions to the 2022 HSC Mathematics Extension 2 Exam while we wait for the official solutions to be published by NESA. My apologies for publishing these a bit late as I have been busy updating our website. I will also be posting the solutions to Extension 1 and the other exams shortly!

Thoughts on this year’s exam (link to the solutions is at the bottom of the post):

I found most of the multiple choice questions to be relatively easy (for Extension 2). I think the hardest questions in the multiple choice were 5 and 9 as they took the most thinking time for me. However, my initial approaches seemed to work out for these problems. I found 9 particularly an enjoyable question as I like to draw the 3D vector questions out, and, of course, it was similar triangles to the rescue to finish this one! I should say that I am not super confident in my answer to Question 10 as I am not sure if we are meant to consider the downwards journey that would if it is initially projected upwards. Maybe I’ve overthought it and we are only considering the one direction and the answer would then be D. (I’m open to feedback on this one.)

Questions 11 and 12 were predictably very easy for Extension 2, and were a good opportunity for students to demonstrate their mastery of the basics from the course. The induction question in Question 13 was on the more creative and unusual side, but it wasn’t too bad if you just follow through with the induction steps, but some extra algebraic manipulation was necessary, such as square rooting the assumption before bringing it into the induction step.

I felt like a spent a good amount of time staring at 14 (a) (iii) until I realised I just needed to focus on setting up the result from part (ii) and redefine the vectors, but that how it usually goes for me with these kinds of questions! 14 (b) also had a creative induction proof which, again, was not too bad but a little bit of algebra trickiness at the end. Once you see what needs to be done, though, it pretty straightforward. The final part of 14 (b) was also a fun way to prove a definition for Euler’s number.

Questions like 15 (a) always make me wary as I am not super confident with physics, but they are never as bad as I feel like they’ll be! I found 15 (a) relatively easy except for when I drew the diagram wrong at first in part (ii) and got stuck. Luckily a colleague pointed it out and the problem then worked out as I expected, resulting in infinite tension. Be careful where you label your angles! Question 15 (b) was also fine but I nearly made the mistake of setting n to 40 instead of 1/40. Don’t mix up frequency and period! I thought I was stuck on 15 (c) until I realised integration by parts would work after the substitution step. And 15 (d) was relatively easy as long as you recognise the expression can become quadratic.

Question 16 brought us two more 4-mark questions (yay). With 16 (a) I tried a couple different approaches that didn’t pan out before coming back around to the expression involving x_{c} and x_{b} and the fact that x_{c} must be real. At first I though this expression meant I made a mistake before realising that it was actually a way to the solution! Calculating x_{b} as a decimal also gave me more confidence as the answer I got seemed reasonable. I ended up with a lot of working out for 16 (b) but it worked out as expected in the end to solve for v_{0} and I don’t think there’s a much more efficient way to do it. 16 (c) was a little too easy, but I suppose it was an unusual question. If I missed something, let me know! And finally 16 (d) was interesting, as usual. I could determine the answer by inspection, but proving it was a little tricky. I got caught up with a method that just became to complicated to work with, working with the real and imaginary parts of each complex number separately. When I came back to the simultaneous equations and their consequences, working with them as simply z_{1}, z_{2}, and z_{3}, I was able to stumble upon the solution.

How did you all go with this exam? Which questions did you find the most difficult or interesting? Drop a comment or email me and let me know!

You can now watch the worked solutions on our YouTube channel. If you’d like the written solutions, you’ll find them below.

My worked solutions can be found at the following link. If any errors are found, I will correct them and keep a log below. Please comment or email me if you find a mistake!

2022 HSC Mathematics Extension 2 Worked Solutions by Apex Tutoring

Corrections:

- Question 2: changed answer from C to D, although I still feel like the error in logic is that four is not just two squared, it is also negative two squared and the mistake is line 3, the official answer is line 4, D.
- Question 15c: the answer is pi/2 – 1 (previously had 3pi/4 – 1), the error was forgetting the 1/2 in v from the integration by parts.
- Question 16d: I adjusted my proof, as looking back at it I couldn’t for the life of me figure out where I got one of the identities I used, so I changed that part of the proof to match the official answer (with additional explanation).