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The end of the full cycle is when it comes back to the 8 m point in the first diagram (image a sine wave). It reaches like halfway between the start and the first peak. This is one-eighth of the way to the end.

Re: MATH1131 help thread We just need to draw a rough diagram to see what projection vector to take and what lengths to use etc. So we don't need to worry about coordinates for our sketch. Just draw a line with the known point on the line, and the given point above the line, and see which...

Re: MATH1131 help thread It suffices to draw a 2D diagram for that, even if the point and line are in some arbitrary n-dimensional space.

I think they want you to estimate how much of a cycle it has travelled visually (I don't see how else to do it). It looks like it has gone half-way towards that first peak, i.e. half-way towards a quarter of a cycle (since the first peak is a quarter of a cycle), which is hence one-eighth of a...

1) For the maximum period, we would assume the wave has travelled as small as possible a fraction of a cycle in the time given. It looks to have travelled one-eighth of a cycle minimally, so the maximum period would be 8 * (0.35 s) = 2.8 s. 2) The frequency is the reciprocal of the period, so f...

Re: HSC 2016 2U Marathon Shorter but less accurate.

Re: HSC 2016 2U Marathon Rewritten in more 2U style: $\noindent Let $f(t)$ be a function with domain $0\leq t \leq 1$. Let $a<b$ be fixed real numbers. Let $f(t)$ be defined as $f(t) = a(1-t) + bt$ for $0\leq t \leq 1$. Find the range of $f(t)$.$

$\noindent Yes the order of summation can essentially be changed due to the laws of arithmetic in $\mathbb{R}$ (or $\mathbb{C}$). Here's a general result: if $a_1,\ldots a_n$ are real numbers and $b_1,\ldots, b_m$ are real numbers (also works for complex numbers), then $\sum _{i=1} ^n \sum...

Re: MATH1131 help thread $\noindent Recall that the length of the cross product of two vectors $\vec{a}$ and $\vec{b}$ in $\mathbb{R}^3$ is given by $\left \| \vec{a} \times \vec{b} \right \| = \left \|\vec{a} \right \| \left \| \vec{b} \right \| \sin \theta$, where $\theta$ is the angle...

Re: HSC 2016 2U Marathon $\noindent Let $f$ be a function with domain $[0,1]$, i.e. $\left \{ t \in \mathbb{R} : 0 \leq t \leq 1 \right \}$ (and codomain $\mathbb{R}$). Let $a,b$ be real numbers with $a<b$. Let $f$ be defined by $f(t) = a\left(1-t)+bt$ for $t\in \left[0,1\right]$. Find the...

Yeah all the topics have some level of rote learning involved, just that some have more than others.

It all depends on the definition of "challenging" or "hard". For example, mindless rote may be considered "easy" for some, whereas logical physics may be considered "easy" for others, where "easy" may be defined as referring to the effort required to put in to get the same amount of marks in a...

In my opinion, the easiest is Moving About (since it is the closest to physics) and the "hardest" is The Cosmic Engine, since it is mostly just rote learning.

That would probably depend on the extent to which you enjoy rote learning (and physics).

I think it was originally posted elsewhere (in Maths Extension 1 forum if I recall correctly), but got moved to this area by a Moderator.

Re: MATH1131 help thread Yes, the parametric vector form for a given line is not unique. This is why we sometimes say "a" parametric form rather than "the" parametric form.

Re: MATH1131 help thread $\noindent Yes, it can be the vector $\overrightarrow{BA}$ or $\overrightarrow{AB}$ or more generally any non-zero scalar multiple of $\overrightarrow{AB}$. This is because whichever non-zero scalar multiple of it we use, we'll end up covering precisely the same line...

$\noindent These proofs will be based on the fact that equal arcs in a circle subtend equal angles at the circumference and centre (and the converse of this).$ $\noindent i) Note that $\angle PAX = \angle QAY$ (vertically opposite angles). So these are two equal angles subtended at the...

It is in the Fitzpatrick textbook I think but I believe you're not expected to know it (if you needed it, it'd be given in the question). I haven't seen it in any recent HSC papers.

Speaking of induction in the HSC, it is common for students to write a sort of mantra at the end of induction proofs in the HSC (or at least was common before, not sure if it still is). Basically an explanation of why induction works or something iirc. I think such a mantra is no longer needed...