Mathematics Extension 1 HSC thoughts (1 Viewer)

[hogzillaAnson]

I've noticed quite a bit of discussion about Q10 of the multiple choice, the one about the differential equation



and thought it might be helpful to go through the thought process of solving it not by finding y in terms of x (as you would do a 3 mark question, or perhaps in Extension 2...) but in a practical way that you could use under time pressure, and for other similar questions.

First, note that
.

So, the derivative must be both bounded (not infinity) and non-negative. This instantly eliminates A, as the derivative in A tends to infinity as we get to the vertical asymptotes, as well as C because the function in C decreases.

We are left with B and D. D is very tempting because it shows periodic behaviour which is what you'd expect from sin(x). However, let's find some stationary points.


So the two stationary points closest to the origin have y values that differ in magnitude by a multiple of 3. However, in option D, the two stationary points closest to O look like they have the same vertical distance from y=0. In contrast, the asymptotes in option B look like they differ by a factor of 3 from the origin, which is what we need from our solution.

So B is the answer.

Main takeaways: Look carefully at the graph to eliminate options, and don't be tempted by the thing that looks like sin(x) or whatever function they decide to throw in there.

 

[hogzillaAnson]

Q9 was another tricky one (inverse functions). Many people probably paid for it because of the overly simplistic way in which inverse functions are treated in the Year 11 syllabus.

The answer is D. Consider y = -x, with inverse function y = -x (same thing). The function intersects with the inverse everywhere, not just on y=x. So A is eliminated. But of course, if we set x = 0, then y = 0 and so the functions do have one point of intersection with y=x, eliminating B.

C cannot be true either. The two functions are the same so the derivatives are also the same, meaning the tangents are parallel everywhere.

Only D can be true.

For those interested, this is how you would actually prove rigorously that the tangents can never be perpendicular using a uni-level theorem. The complexity of the proof says to me that NESA doesn't intend for us to show that an option works, only that the other three don't work.

Let g(x) be the inverse function of f(x). Suppose that the function and its inverse meet at c. We first establish the derivatives.



The most important thing here is that the derivatives cannot ever be 0. This makes sense as a stationary point on the function would imply a vertical tangent on the inverse, but we are told that both derivatives exist for all real x.

We want to show a contradiction. Assume that the tangents are perpendicular. Then


In the case where , we have

a contradiction. Otherwise, we can assume without loss of generality that and therefore .
By Darboux's theorem, satisfies the Intermediate Value Property. Therefore, there exists some number between c and g(c) such that , a contradiction.

Thus, the tangents are never perpendicular. D is true.

 

[tywebb]

Terry Lee's solutions are up: http://advancedmathematics.com.au/Resources/2022 Ext1.pdf

 

[tgone]

TL 14a is incorrect:


their substitution for (0,1) is done incorrectly, C should equal -2ln2.

Also, 14b has an incorrect conclusion (correct method however)
14d can remove ambiguity from choosing which solution by solving the quadratic eq (1) on the pdf w.r.t sqrt(n) rather than squaring both sides.

 

[tywebb]

Looks like Terry Lee has fixed up 14a. This is why I think it is better to have a link for his solutions rather than a copy of the file here.

Yesterday's pdf was timestamped 26 Oct 2022 at 6:19 pm

The new version (same link) is 27 Oct 2022 at 4:59 am: http://advancedmathematics.com.au/Resources/2022 Ext1.pdf


Not sure about the other bits though.

 

[tgone]

Looks like Terry Lee has fixed up 14a. This is why I think it is better to have a link for his solutions rather than a copy of the file here.

Yesterday's pdf was timestamped 26 Oct 2022 at 6:19 pm

The new version (same link) is 27 Oct 2022 at 4:59 am: http://advancedmathematics.com.au/Resources/2022 Ext1.pdf

View attachment 36822
Not sure about the other bits though.

Yep, I messaged him, looks like he fixed it.

 

[tgone]

I made the same mistake so how many marks could I potentially lose for that?

You would only lose one mark for that, you’ll get a carry through for the final answer being slightly off.

 

[absolute_intellect]

Does anyone know if nesa is lenient in the way they award marks for working? Like does it always have to be a substantial amount of relevant working, or making an attempt at a question is enough? Thanks

 

[notme123]

i had same concerns last year for 3u as well. they mark based on criteria which I'm sure you've seen in marking guidelines before. if the attempt you made had something relevant they may give a mark but depends on how many marks are allocated for the question. they might give more marks for 4 markers than a 2 marker.

 

[tywebb]

itute solutions: https://www.itute.com/wp-content/uploads/2022-NSW-ESA-Mathematics-Extension-1-Solutions.pdf

 

[tywebb]

Matrix Education solutions: https://www.matrix.edu.au/2022-hsc-maths-ext-1-exam-paper-solutions/

 

[sukan]

I know im late to the party but personally i found the exam to be pretty good besides 14 c and d