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1. Help Proof

Prove by induction. In your inductive step, you will need to use the result k² > 2k+1 for integers k > 4 in order to retrieve the perfect square. This result can be proven in a number of ways. Perhaps the shortest approach is as follows: k > 4 k² > 4k k² > 2k + 2k > 2k + 1 (since 2k > 8 > 1)
2. differential equations

The order of the DE is effectively the number of times you need to “integrate” it to get the general solution. Each time you integrate you generate an arbitrary constant which cannot be found unless you know the specific conditions.
3. Can someone explain this to me please?

As time passes the magnitude of v gets smaller around point P before it hits zero (i.e. the lowest possible speed). This suggests the speed is decreasing.
4. Projectile Motion Help

If you assume a ground level projection then when you specify the coordinate axes you have three known points (0,0), (0.5R,H) and (R,0) where R is the range and H is the maximum height. Three points allows you to specify the exact Cartesian equation of the parabola that models the projectile...
5. Same Sex Marriage Debate

I think the user meant when all parties have a choice. In a murder, one party doesn’t exactly get a choice...

7. Maths Help

Moved to Maths Ext1 thread. For future reference, please post these questions in the Maths forum.
8. Stats

The mean is based on the values of the data points. The median is based on the rank of the data points, rather than their value. The former is therefore sensitive to big outliers or extreme values which can distort the measure of the “centre”. The median does not care how big/small the extreme...
9. Another HSC mark moderation question

Decimal places would naturally exist when you apply weightings to percentages in a single assessment. I’m pretty sure schools submit marks with decimal places, though it is not clear how many decimal places.
10. Another HSC mark moderation question

The whole point of the subject is not intended to revolve around a single theory exam. The external HSC exam is only ONE way to assess syllabus outcomes. The subject is supposed to teach and assess a diverse range of skills that are applicable to uni/workplace etc, not just how to answer...
11. Another HSC mark moderation question

What you are suggesting would completely defeat the purpose of having other assessments lol School assessments exist as they can cover skills specified in the syllabus beyond just a theory exam. For example, you can’t claim to be good at Chemistry if you are hopeless at applying theory into...
12. Another HSC mark moderation question

There is not enough information because you haven’t specified how the rest of your cohort performs in both internal vs external. If we assume that the: - shape of the distribution of your cohort‘s internal and external marks are the same - there is no one between you and the other student both...
13. Another HSC mark moderation question

Are you only talking about swapping internal marks? If so, then that is trivial as you and the other person obviously just swap marks. If you’re talking about having internal marks and then swapping places on external marks then that’s a little different.
14. Is the half angle formula ever in the HSC syllabus?

Half angle formula is literally just the double angle formula. The only other half angle thing is t-formula. Either way, you don’t really need to remember them as they are more or less derived from the “Compound angles” section of the formula sheet. If you can derive it from these formulae then...
15. combinations ext math question

Do it sequentially: - There are \binom{12}{6} ways to select 12 people to form the table of 6 - Once those 6 are selected there are 6 left so there are \binom{6}{4} ways to select 6 people to form the table of 4 - Once those 4 are selected there are 2 left so there is \binom{2}{2} way to form...