ATAR Notes: Forum

HSC Stuff => HSC Exam Discussion 2021 => Topic started by: RuiAce on November 15, 2021, 05:17:30 pm

Title: Maths Advanced 2021 Solutions
Post by: RuiAce on November 15, 2021, 05:17:30 pm
Last year ever of doing this and looks like I'm starting on a low. Haven't yet figured out how to fully solve Q10 lol. All I know is 1/(2pi) < 1/a and m < 1/a. Haven't figured out the inequalities between 1/(2pi) and m. Edit: Credits to an answer provided below, which I have incorporated.

Multiple choice
(https://i.imgur.com/0e0vjJE.png)

GeoGebra screenshot showing the tangent y=mx being barely above the curve y=cos(x) at x=a.

(https://i.imgur.com/csPKOtA.png)

Q17
(https://i.imgur.com/3vyuDGo.png)

Q25
According to the annuity table, an annuity of $1 at 0.75% interest, per annum compounding, for 8 years, will have a future value of $8.2132. Therefore an annuity of $1000 with everything else the same will have a future value of $8213.20.

Then we simply multiply (1.0125)2 for the future value after the extra 2 years, where no deposit is made, but the money still gains interest from just sitting there. Final answer is $8213.20 x 1.01252 = $8419.81 (nearest cent).

Q28-29
(https://i.imgur.com/BqL0ReW.png)

Q30-32
(https://i.imgur.com/F7xDGhk.png)

Q33-34
(https://i.imgur.com/o5Y9gqS.png)

I make no promises on every solution being accurate. Please point out mistakes and I'll get to them slowly.

Refer to fun_jirachi's solutions for questions I did not do.
Title: Re: Maths Advanced 2021 Solutions
Post by: fun_jirachi on November 15, 2021, 06:01:43 pm
11-14
(https://i.imgur.com/hdIPYYS.jpg)

15-16
(https://i.imgur.com/hftnuVv.jpg)

18-20
(https://i.imgur.com/kcc5H55.jpg)

21-24
(https://i.imgur.com/GlVnN1S.jpg)

26
(https://i.imgur.com/0a4RXIE.jpg)

27
(https://i.imgur.com/lGE9J5x.jpg)
Title: Re: Maths Advanced 2021 Solutions
Post by: henry931 on November 15, 2021, 06:11:23 pm
Q10

Spoiler
For Q10 its 1/2pi < m < 1/a. The second inequality is because m=rise/run=e/a for some 0<e<1 (its not hard to see that 'a' is slightly to the left of 2pi, thus e=am is elightly below 1), so m = e/a < 1/a. The first inequality needs some explaining. Notice that the gradient of cos(x) decreases after x=a (just look in a small interval containing 2pi and a), so the tangent line is above the graph after x=a (at least until x=2pi). Consider the y value of the tangent line and cos(x) at x=2pi to get the inequality 1<m*2pi which implies 1/2pi < m.

(Another similar but graphical way) This value 1/2pi can be thought of as the gradient of a line L: y=x/2pi through (0,0) and (2pi,1). Remember that (2pi,1) lies on cos(x), so L will be below y=mx for x=2pi. So the gradient of L is smaller than that of y=xm, since they both pass through the origin. That is, 1/2pi < m.

The bit about the line being above cos(x) can be made more rigorous by considering their rate of change after x=a.
Title: Re: Maths Advanced 2021 Solutions
Post by: RuiAce on November 15, 2021, 06:33:53 pm
Quote from: henry931 on November 15, 2021, 06:11:23 pm
Q10

Spoiler
For Q10 its 1/2pi < m < 1/a. The second inequality is because m=rise/run=e/a for some 0<e<1 (its not hard to see that 'a' is slightly to the left of 2pi, thus e=am is elightly below 1), so m = e/a < 1/a. The first inequality needs some explaining. Notice that the gradient of cos(x) decreases after x=a (just look in a small interval containing 2pi and a), so the tangent line is above the graph after x=a (at least until x=2pi). Consider the y value of the tangent line and cos(x) at x=2pi to get the inequality 1<m*2pi which implies 1/2pi < m.

(Another similar but graphical way) This value 1/2pi can be thought of as the gradient of a line L: y=x/2pi through (0,0) and (2pi,1). Remember that (2pi,1) lies on cos(x), so L will be below y=mx for x=2pi. So the gradient of L is smaller than that of y=xm, since they both pass through the origin. That is, 1/2pi < m.

The bit about the line being above cos(x) can be made more rigorous by considering their rate of change after x=a.
Thanks. I was really hoping to avoid the need to visualise tangents this way, but I can't see any other approach that would clean things up nicely. I'll incorporate this into my solution.

m < 1/a was something I was able to figure out.
Title: Re: Maths Advanced 2021 Solutions
Post by: RuiAce on November 15, 2021, 06:55:23 pm
Thoughts: Hard is not the correct word. But I don't feel this paper was friendly, especially with COVID causing enough stress as is within the HSC this year. I concur that statistics coverage was pretty spiked this year.

Noteworthy questions I did:
- Q10: Not happy that the answer I settled for was 'fluffy'. It required mostly strong visualisation skills, and I couldn't find a purely algebraic approach.
- Q30: Assuming I'm not wrong, important to understand why P(X<=x) = 0.99. Question itself isn't hard I believe.
- Q33b): Really? Making me differentiate a rational function twice to prove that I obtain a maximum (for the mode)?
- Q33d): A bit shook that I had to use Bayes rule. Makes me worry that my answer is wrong.
- Q34: Not a hard question, but in all honesty it can be easy to miss that the sum of all probabilities is 1, for something like this (also very late in the paper).
Title: Re: Maths Advanced 2021 Solutions
Post by: Nomsie on November 16, 2021, 09:53:56 am
Quote from: RuiAce on November 15, 2021, 06:55:23 pm
Thoughts: Hard is not the correct word. But I don't feel this paper was friendly, especially with COVID causing enough stress as is within the HSC this year. I concur that statistics coverage was pretty spiked this year.

I'd have to disagree, coming from a student who actually had to deal with the HSC this year, it has definitely been harder than in the past. I prepared for math, going through countless past papers/questions especially of high difficulty and I haven't seen questions of this nature. They never have focused on stats/normal distribution so much. A lot of these questions also required very refined skills that completely differ from (my own experience at least) what we have been taught.

Everyone raved about the difficulty of last year, I personally believe this was far more difficult (excluding those few easy marks questions). This is also the general consensus I've heard from multiple schools.
Title: Re: Maths Advanced 2021 Solutions
Post by: Nomsie on November 16, 2021, 09:54:43 am
I genuinely believed if I studied a little more, I could've only gotten an extra like 4 marks or so, definitely 'hard' and difficult.
Title: Re: Maths Advanced 2021 Solutions
Post by: RuiAce on November 16, 2021, 12:29:24 pm
Quote from: Nomsie on November 16, 2021, 09:53:56 am
I'd have to disagree, coming from a student who actually had to deal with the HSC this year, it has definitely been harder than in the past. I prepared for math, going through countless past papers/questions especially of high difficulty and I haven't seen questions of this nature. They never have focused on stats/normal distribution so much. A lot of these questions also required very refined skills that completely differ from (my own experience at least) what we have been taught.

Everyone raved about the difficulty of last year, I personally believe this was far more difficult (excluding those few easy marks questions). This is also the general consensus I've heard from multiple schools.
Just having more statistics questions does not automatically make a paper "hard/difficult".

I consider the added stats a primary reason in why I found the paper unfriendly. I believe this year's cohort have every right to be angry downright p***ed about the paper. But at the end of the day, that just means the paper has been loaded with more relatively foreign material. It makes preparing for this paper significantly less straightforward, and yes there would be fewer resources to facilitate this. Incredibly "stressful" is another word I'd consider appropriate.

But I reserve the word "hard/difficult" for the actual difficulty of the questions themselves. Several of the questions themselves do not require heavily abstract thinking. For the most part, the questions just demanded knowledge of various concepts in the statistics topic. (Obviously not all - there were a few major curveballs. But there were several "conceptually doable" questions.)

I also do not believe that "very refined skills" are needed to do the paper. I respect that many teachers and students don't yet have the skills to do this much statistics (again, no denying the paper was "unfriendly", or another word stronger than that). But the actual skills themselves were, for the most part, not outrageous.

If anything, I would be more willing to call the paper "evil", than to call it "hard/difficult".
Title: Re: Maths Advanced 2021 Solutions
Post by: tywebb on November 17, 2021, 11:50:49 am
Here is the paper: https://educationstandards.nsw.edu.au/wps/portal/nesa/resource-finder/hsc-exam-papers/2021/mathematics-advanced-2021-hsc-exam-pack