saltshaker
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how would you do q12 a for 3u?
BoS trials Maths and Chemistry 2022 (1 Viewer)
- Thread starter Trebla
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Not sure if my method is actually correct. But what I essentially did was
- Define the binomial variable with a known n but an unknown p
- Use this to define the sample proporiton variable & the sample proportion's mean and standard deviation (in terms of p)
- From the question P(phat < 4%) = 2.5%. Using the empirical rule we know this means that phat=4% corresponds to a z-score of -2.
- plug in x=4% z=-2 and mean and standard deviation into z-score formula from formula sheet
- Solve for p (which is the mean of the sample proportion and hence the approx mean unemployment rate)
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Basically what 011235 wrote with one little twist. The expected method involves squaring both sides, which introduces an incorrect solution. You’ll need to pick the correct one with justification.
A lot of questions are based on well known results in the broader mathematics/chemistry field, just constructed in a way that is HSC-friendly. Others can come from taking what would normally be a simple idea/question and “enhancing” it into something harder (usually by hybridising multiple topics). Then there are some (especially in Maths) that are completely made up or “discovered” when just mucking around with random ideas.
Daedalus13
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How do you do Q11e of 4U?
Consider x^2+sqrt(x)+sqrt(x)≥3x. Repeat for y and z, sum the three up and see if you can finish from here
Daedalus13
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Thanks so much! I never would have thought of that first step.Consider x^2+sqrt(x)+sqrt(x)≥3x. Repeat for y and z, sum the three up and see if you can finish from here
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Coming soon… just finalising a few pieces
bro who is ever coming up with this, that's crazy you got itConsider x^2+sqrt(x)+sqrt(x)≥3x. Repeat for y and z, sum the three up and see if you can finish from here
Paradoxica
-insert title here-
While I'm finishing off the pile of marking:
Remarks for Q11, MX2
a) More than a handful of students made the incorrect border transformation of (3π/2, π/2), instead of (-π/2,π/2)
Most students were able to correctly derive the relation between dx and dθ and the transformed border values but either gave up immediately, or were not able to successfully transform the integrand, and then gave up.
A few were able to make significant progress after spotting the part where you find an equivalent integral and then add the two together, but made some minor errors and did not get the correct number at the end.
b) Far too many students attempted to square the relation 2y√(x²+y²) = x²+y² and missed the much cleaner factoring out of √(x²+y²) (if they even correctly expanded the locus equation at all)
A handful did not correctly recall their complex modulus results, and one even wrote |...| = √(...)², a property that only holds for real numbers.
Of the students that did arrive at the correct Cartesian Equation, most of them missed the part where y is forced to be non-negative due to an earlier step on the way to arriving at the Cartesian Equation.
No students solved it via complex-exponential or mod-arg form, which circumvents having to write down messy expressions involving x and y, and avoids the pitfall of missing the part where y is forced to be non-negative. (One student did write some sort of rcisθ stuff on the side but had no bearing on their main incorrect Cartesian result)
c) This was not my sub-question, so the issues with it I can see.
Only a few students correctly justified the absolute value, though the wording of the question implied that the student was "successfully" pulling the trolley up the slope, when the intention was that they were "attempting" to pull it up the slope.
One student even wrote 3 extremely short lines but did justify the absolute value so the ruling (by the author of that sub-question) was full marks.
d) i) Most students only "proved" (being very generous with my words of choice here) the even function integral property, but did not prove the converse.
Of the handful that did write the converse, very few were able to correctly spot that they had to use the Fundamental Theorem of Calculus and differentiate both sides of an odd function to arrive at the conclusion that the original function being integrated must be even.
Most simply ended there claiming that this implies f(x) is even without having used the fact that it is true for all values of "a" (which is what enables you to differentiate in the first place, since you need the identity to be true as "a" varies continuously)
Some students attempted to "prove" the converse by saying "if f(x) is odd...", forgetting for a moment that the negation of "f(x) is an even function" is not "f(x) is an odd function" (a function can be neither even nor odd...)
Other students ended there, re-iterating the statement, having only proven one direction of the implication.
d) ii) A handful of students spotted the obvious and official counter-example of f(x) = sin(x) with a = 2π.
Some students attempted to construct piecewise functions, but upon checking them with a plotter, they were not valid counterexamples.
Other students wrote.... words. That's all I have to say about that.
e) This one is also not my sub-question, but in hindsight, I really think it could have used an intermediate step or any sort of hint.
Only one student got 95% of the way to the correct solution by working backwards, ending with expressions involving f(t) = t²+2√t but missed the official solution's first step which let a = t², b = c = √t to yield f(t) ≥ 3t.
No other students were able to make any significant progress towards the inequality.
It appears to be the case that most students have this unspoken assumption that the substitutions that you are allowed to make for a, b and c in the given inequality must be symmetric/cyclic [ for example, a = g(x), b = g(y), c = g(z) for some function g ].
Nowhere is this assumption granted in the question, and it actively hinders you from being able to prove inequalities in general.
Remarks for Q11, MX2
a) More than a handful of students made the incorrect border transformation of (3π/2, π/2), instead of (-π/2,π/2)
Most students were able to correctly derive the relation between dx and dθ and the transformed border values but either gave up immediately, or were not able to successfully transform the integrand, and then gave up.
A few were able to make significant progress after spotting the part where you find an equivalent integral and then add the two together, but made some minor errors and did not get the correct number at the end.
b) Far too many students attempted to square the relation 2y√(x²+y²) = x²+y² and missed the much cleaner factoring out of √(x²+y²) (if they even correctly expanded the locus equation at all)
A handful did not correctly recall their complex modulus results, and one even wrote |...| = √(...)², a property that only holds for real numbers.
Of the students that did arrive at the correct Cartesian Equation, most of them missed the part where y is forced to be non-negative due to an earlier step on the way to arriving at the Cartesian Equation.
No students solved it via complex-exponential or mod-arg form, which circumvents having to write down messy expressions involving x and y, and avoids the pitfall of missing the part where y is forced to be non-negative. (One student did write some sort of rcisθ stuff on the side but had no bearing on their main incorrect Cartesian result)
c) This was not my sub-question, so the issues with it I can see.
Only a few students correctly justified the absolute value, though the wording of the question implied that the student was "successfully" pulling the trolley up the slope, when the intention was that they were "attempting" to pull it up the slope.
One student even wrote 3 extremely short lines but did justify the absolute value so the ruling (by the author of that sub-question) was full marks.
d) i) Most students only "proved" (being very generous with my words of choice here) the even function integral property, but did not prove the converse.
Of the handful that did write the converse, very few were able to correctly spot that they had to use the Fundamental Theorem of Calculus and differentiate both sides of an odd function to arrive at the conclusion that the original function being integrated must be even.
Most simply ended there claiming that this implies f(x) is even without having used the fact that it is true for all values of "a" (which is what enables you to differentiate in the first place, since you need the identity to be true as "a" varies continuously)
Some students attempted to "prove" the converse by saying "if f(x) is odd...", forgetting for a moment that the negation of "f(x) is an even function" is not "f(x) is an odd function" (a function can be neither even nor odd...)
Other students ended there, re-iterating the statement, having only proven one direction of the implication.
d) ii) A handful of students spotted the obvious and official counter-example of f(x) = sin(x) with a = 2π.
Some students attempted to construct piecewise functions, but upon checking them with a plotter, they were not valid counterexamples.
Other students wrote.... words. That's all I have to say about that.
e) This one is also not my sub-question, but in hindsight, I really think it could have used an intermediate step or any sort of hint.
Only one student got 95% of the way to the correct solution by working backwards, ending with expressions involving f(t) = t²+2√t but missed the official solution's first step which let a = t², b = c = √t to yield f(t) ≥ 3t.
No other students were able to make any significant progress towards the inequality.
It appears to be the case that most students have this unspoken assumption that the substitutions that you are allowed to make for a, b and c in the given inequality must be symmetric/cyclic [ for example, a = g(x), b = g(y), c = g(z) for some function g ].
Nowhere is this assumption granted in the question, and it actively hinders you from being able to prove inequalities in general.
Last edited:
Paradoxica
-insert title here-
Remarks for Q16, MX2
a) The overwhelming majority were able to correctly execute the syllabus cloned, word for word, irrationality proof.
No students used infinite descent, which I was hoping would sneak in from any students who looked at a specific previous BoS Trials paper.
b)
i) Some students wrote down y⁴ when the highest power needed is y² and gave up when they were unable to make progress. Otherwise, most were able to get the factorisation.
ii) Highly dependent on the previous part, so no comments here.
iii) Very few students noticed that the product of roots (which are complex conjugates) is 1, and therefore both have modulus 1, and instead crunched out the lines of working for the modulus squared.
iv) A couple of students fell for the trap of claiming that the sequence Tₙ hits all roots of unity of a particular kᵗʰ root of unity.
This is false in general, e.g. if the given root is the principal 6ᵗʰ root of unity, the rest of the sequence only takes on the 3ʳᵈ roots of unity. (The more general behaviour requires some basic number theory to express for arbitrary roots of unity)
Can't give the mark when they started with a false statement. (From a false statement, you can prove anything, by the Principle of Explosion.)
Students who avoided saying this were in the clear. (stuff like "At most", "possible values", etc. is fine by me.)
Some students were unable to understand the question at all. IDK how much clearer I could make it without writing the first 10 terms.
v) Some students incorrectly claimed Re(z)² = Re(z²) or an error of similar magnitude and hence, arrived at linear recurrences instead of non-linear recurrences. This made it impossible to give any carry-forward error marks, due to how different the resulting induction becomes.
vi) Not many students attempted the induction (due to giving up on one of the previous parts), and of those who did, some failed to realise that they needed to look at the two sequences simultaneously to prove the one of the inequalities.
vii) Nothing much to say for this one, since it all hinges on having understood the previous parts and putting them together.
viii) A couple of students were able to realise that they could scrape a free mark without having done any of the previous parts.
a) The overwhelming majority were able to correctly execute the syllabus cloned, word for word, irrationality proof.
No students used infinite descent, which I was hoping would sneak in from any students who looked at a specific previous BoS Trials paper.
b)
i) Some students wrote down y⁴ when the highest power needed is y² and gave up when they were unable to make progress. Otherwise, most were able to get the factorisation.
ii) Highly dependent on the previous part, so no comments here.
iii) Very few students noticed that the product of roots (which are complex conjugates) is 1, and therefore both have modulus 1, and instead crunched out the lines of working for the modulus squared.
iv) A couple of students fell for the trap of claiming that the sequence Tₙ hits all roots of unity of a particular kᵗʰ root of unity.
This is false in general, e.g. if the given root is the principal 6ᵗʰ root of unity, the rest of the sequence only takes on the 3ʳᵈ roots of unity. (The more general behaviour requires some basic number theory to express for arbitrary roots of unity)
Can't give the mark when they started with a false statement. (From a false statement, you can prove anything, by the Principle of Explosion.)
Students who avoided saying this were in the clear. (stuff like "At most", "possible values", etc. is fine by me.)
Some students were unable to understand the question at all. IDK how much clearer I could make it without writing the first 10 terms.
v) Some students incorrectly claimed Re(z)² = Re(z²) or an error of similar magnitude and hence, arrived at linear recurrences instead of non-linear recurrences. This made it impossible to give any carry-forward error marks, due to how different the resulting induction becomes.
vi) Not many students attempted the induction (due to giving up on one of the previous parts), and of those who did, some failed to realise that they needed to look at the two sequences simultaneously to prove the one of the inequalities.
vii) Nothing much to say for this one, since it all hinges on having understood the previous parts and putting them together.
viii) A couple of students were able to realise that they could scrape a free mark without having done any of the previous parts.
Last edited:
Bumblebee_18
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will an answer sheet or marking criteria ever be released or..?
Life'sHard
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Yeah normally solutions are latexed.
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..... and here are the solutions and student results for the 2022 Mathematics Extension 2 BoS trials! 
Thank you to everyone who participated and for your patience! Thanks especially to the other members of the marking team Paradoxica, dan964 and 1039213 for their hard work in helping to get the marking done.
Please also note that the question paper has been updated (which the solutions are aligned to) with some minor adjustments to a few questions (e.g. split into parts, fixed the mark allocation, edited some wordings). The updated file is attached in my original post of the papers to replace the older version.
As for the results, congrats to the student who scored the top mark of 64 with two other students coming very closely behind! We also had two students score full marks in the multiple choice section.
Once again you are reminded of the following:
Thank you to everyone who participated and for your patience! Thanks especially to the other members of the marking team Paradoxica, dan964 and 1039213 for their hard work in helping to get the marking done.
Please also note that the question paper has been updated (which the solutions are aligned to) with some minor adjustments to a few questions (e.g. split into parts, fixed the mark allocation, edited some wordings). The updated file is attached in my original post of the papers to replace the older version.
As for the results, congrats to the student who scored the top mark of 64 with two other students coming very closely behind! We also had two students score full marks in the multiple choice section.
Once again you are reminded of the following:
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Some general feedback on the sections I marked for Ext2:
- Almost 10% of the attendees left at least one question in the multiple choice unanswered. If you don't know the answer to the question, then just guess! You have a 1/4 chance of getting 1 extra mark versus a guaranteed zero. If this was unintentional, then students are reminded to take care in populating the responses if they choose to do the multiple choice questions out of order.
- For Q15a), a very common response was to show that (2,1,-5) and (2,1,1) had a dot product of zero, therefore the direction vector is parallel to (2,1,-5). This claim is only true in two dimensions! If you imagine a line L1 in three-dimensional space, it is possible for two lines say L2 and L3 to each intersect L1 at 90 degrees, but L2 and L3 need not be parallel to each other. This is because they are perpendicular relative to a particular plane that contains both lines. In other words, the dot product being zero is not enough to uniquely locate a specific direction vector. Stronger responses used addition/subtractions/projections of vectors to solve for λ in order to locate the point of intersection, which then allows the specific direction vector to be found.
chilli 412
oo la la
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safe to say whoever scored that 64 is gonna ace the hsc
wendyredvelvet
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Does anyone have the chemistry BOS trial solutions?? If so are you able to send them thru
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Paradoxica
-insert title here-
Well I guess it goes to show you were not intimidated by my question.4u trial was fun thanks for writing and marking everything
anyway i somehow got a higher score in q16 than i got for all the other questions
Some people think that the unorthodoxy of my question is enough to scare students