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Re: HSC 2016 3U Marathon It should just be what HeroicPandas and I wrote above.

Re: HSC 2016 3U Marathon $\noindent I'm guessing the Q. is asking to prove the inequality $\frac{a^n + b^n}{2} \geq \left(\frac{a+b}{2}\right)^n$, for $a,b \geq 0$, $n$ a positive integer. This is a special case of the \textsl{power means inequality}, linked below.$...

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread $\noindent Let $f(x)=x^2+5$. At the point where $x=t$ ($t$ a real number), the slope of the tangent is $f^\prime (t) = 2t$, and the $y$-value is $f(t) = t^2 + 5$. So the equation of the tangent is $y = t^2 + 5 + 2t...

Yeah I did this Q. in my head in the early morning so I'm not too surprised that there was an error in part of it haha. I edited initially after reading tywebb's post without writing I saw that post because I believed the main purpose of these threads was just to help answer people's Q's (and...

Other solution as in there are two solutions in that interval, the one I had isn't the right one, and the one you had is. Sure, I'll acknowledge your input in the post. :)

The total of n integers will have a remainder of either 0, 1, …, (n-1) modulo n. This means the possible fractionals parts for the average are just: 0, 1/n, 2/n, 3/n, …, (n-1)/n. E.g. If 1 subject: only can have decimal of 0. If 2 subjects: only can have 0 or 0.5. If 3 subjects: 0, 0.333...

Don't think so (even if it is allowed, apparently it creates a bad impression on markers / makes them think you're lazy. So it's probably "allowed", just bad.).

$\noindent Are you sure you're looking at the right question's answer? If we assume those, we have $x(t) = \frac{32\sqrt{2}}{\pi} \sin \left(\frac{1}{4}t +\frac{\pi}{4}\right)$. Then the RHS here should equal $3$ at $t=1$, but instead it is $\frac{32\sqrt{2}}{\pi} \sin \left(\frac{\pi...

This is indeed equivalent to laters' expression of (21C7)*(14C7), as we can see by writing each of these in factorial form and cancelling stuff from numerators and denominators of adjacent fractions. In terms of laters' main Q. (which was whether the groups are 'labelled' or not), I'd guess we...

$\noindent Since the period is $8\pi$, we have $n=\frac{1}{4}$. Note we can find $a$ quickly by using $v^2 = n^2 \left(a^2 -x^2\right)$. Sub. in the given data of $v=-1$ at $x=3$, so $1 =\frac{1}{16}\left(a^2 -9\right)$. Solving for $a$ yields $\boxed{a = 5}$. Now, we know $x = 3$ when $t=1$. So...

This is a classic and oversimplified explanation that is not quite correct. Your school marks (mainly insofar as the distribution of and relative gaps compared to other students in your cohort) are also important.

Re: First Year Uni Calculus Marathon $\noindent Let $a<b$, $a,b$ real. Let $f\colon \left[a,b\right] \to \mathbb{R}$ be a \emph{continuous} function that satisfies $f(x) \geq 0$ for all $x\in \left[a,b\right]$. Suppose $\int _a ^b f(x)\text{ d}x=0$. Show $f(x)\equiv 0$.$

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread Yeah that's why I said "unless told otherwise". I.e. If nothing else is said, the convention is what was required for this Q. (so in this case compounded monthly, and in general compounded per period of payment, which could be...

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread Yes. The "rate per annum" given needs to be divided by the appropriate number (in this case 12) to get the per-period interest rate. I don't think it's actually ambiguous, it's just that these conventions aren't usually taught...

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread If it just says monthly repayments, then it means compounded monthly too (unless told otherwise).

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread What was your working?

Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread When we expand a 10th power using binomial expansion, there'll be 11 terms. So the middle term is the sixth term (as there will be five terms either side of this, for a total of 11 terms). We don't need to expand everything...

Re: HSC 2016 3U Marathon You'd need to give the reason as related to the congruent triangles part. If you said alternate angles as the reason for the equal angles, then that means you're assuming the lines are parallel, which is what we're trying to prove.

Stars and bars would be if the five objects to be placed (students here) were identical, which is not the case here. One way to do it is as follows. Call the rooms 1, 2, 3. Let U be the set of all possible arrangements without restrictions, and let Sk be the set of all arrangements if room k is...

Re: HSC 2016 4U Marathon - Advanced Level The questions in HSC 4U (technically "Extension 2" of course) papers of today are overall much easier than most of the Q's in these 4U Advanced Marathons.