hogzillaAnson
Member
- Joined
- Jul 14, 2018
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- HSC
- 2020
- Uni Grad
- 2025
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.
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.