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It also depends on your school of course. Being ranked around top 5 in something like MX2 at a school like James Ruse will obviously still put you in contention for topping the state. If you're at a school that's ranked much lower though and not first internally, it's probably much harder to...

Re: MX2 2016 Integration Marathon $\noindent Let $a,b\in \mathbb{R}$ such that $a+b\sin x$ is always non-zero for $x\in \mathbb{R}$. Find $\int _{0}^{2\pi} \frac{1}{a+b\sin x}\, \mathrm{d}x$.$

Re: MATH1231/1241/1251 SOS Thread $\noindent How did you calculate your value of $A \bold{v}$ $\big{(}$where $\bold{v} = (6,6,3)^{T}$$\big{)}$? You should've gotten a different first entry to what you got (just standard matrix muitplication).$ I think you just typoed or made a silly mistake...

Re: MATH1231/1241/1251 SOS Thread $\noindent Try finding values of scalars $a$ and $b$ (which will depend on $x$ and $y$) such that given any $x,y \in \mathbb{R}$, we have $\begin{pmatrix}x \\y\end{pmatrix} = a \begin{pmatrix}1 \\ 0\end{pmatrix} + b \begin{pmatrix}1 \\ 1\end{pmatrix}$. Then by...

$\noindent Let $B$ be the origin and $C = (c,0)$ on the $x$-axis, where $c$ is a fixed positive number. By a suitable choice of units and since whether the line $L$ is above or below the $x$-axis clearly won't affect the \emph{shape} of the locus, we can just take $L$ to be the line $y = 1$...

$\noindent Before the volume was $10$ mL, and now it is $1000$ mL. So the volume has been multiplied by a factor of $100$, meaning the concentration gets \emph{smaller} by a factor of $100$ (since no new molecules of the substance were added, just water or whatever the solvent was, and...

Re: MATH1231/1241/1251 SOS Thread Let's see what it's like for three elements (say a, b, c). The claim is that any bracketing and ordering of a+b+c results in the same thing (and hence that brackets aren't needed). I.e. Let A = (a+b)+c. The claim is that any bracketing and ordering results in...

Re: MATH1231/1241/1251 SOS Thread Yeah. The inductive hypothesis is basically the given statement (ignoring the lambdas) for a particular integer n.

Re: MATH1231/1241/1251 SOS Thread First note we don't need to worry about the lambdas, suffices to just prove it for v1 + v2 + … + vn (can you see why?). So given that we know (from axioms) that + is an associative and commutative operation, we need to show that any bracketing will lead to...

Re: MATH1231/1241/1251 SOS Thread A zero row being present only means no solutions if the right-hand column in that row has a non-zero entry (in other words, the right-hand column is leading). Here, the right-hand column is non-leading, so solutions exist. Since there is a non-leading column...

Re: Discrete Maths Sem 2 2016 $\noindent When $n=1$, $LHS = (1 +1)^1 = 2 < 3$. For $n>1$, we have that$ $$\begin{align*}\left(1+\frac{1}{n}\right)^n &= \sum _{k=0}^{n} \binom{n}{k}\frac{1}{n^k} \\ &= \sum _{k=0}^{n} \frac{n(n-1)\ldots (n-(k-1))}{k!\cdot n^k} \\ &< \sum _{k=0}^{n}...

You theoretically don't need to come first internally, but probably a lot of (maybe most) state-toppers do come first internally.

Re: MATH1231/1241/1251 SOS Thread Well you'd be imposing initial conditions yourself wouldn't you (like starting at x = 0)? Because the question didn't seem to give any initial conditions. Anyway, if we did end up solving the ODE, we'd see it's something like sin(omega*t), which we know has...

Re: MATH1231/1241/1251 SOS Thread $\noindent Oh. It's a pretty standard SHM fact, kind of surprising you can't assume it I guess. But what was the other method you mentioned? Something about an arcsin? Did it require you to find the solution in the form $R \sin \left(\omega t +\varphi\right)$?$

Re: MATH1231/1241/1251 SOS Thread Why not just use the 2pi/(omega) method? Seems faster.

Maybe as in weighting by assessments' weightings. This is done all the time to find say your final raw mark if you know the weightings of each assessment (and your mark in each assessment).

Re: MATH1231/1241/1251 SOS Thread $\noindent The fact that the period is $\frac{2\pi}{\omega}$ follows from the fact that the general solution of the ODE $\ddot{x} = -\omega^{2} x$ is $x(t) = R \cos \left(\omega t +\varphi\right)$, for arbitrary constants $\varphi, R$. This function has period...

Re: MATH1231/1241/1251 SOS Thread $\noindent To show that $T$ is not linear, we just need to show that it fails to satisfy a condition of linearity. For example, it suffices to come up with an example where $T$ doesn't satisfy $T\left(\alpha x\right) = \alpha T\left(x\right)$. We show an...

Re: MATH1231/1241/1251 SOS Thread Yeah, remember P_n has dimension n+1, not n. So P_4 has dimension 5. So five linearly independent vectors in P_4 would form a basis for it.

$\noindent First we'll find the probability the product is \emph{not} even (i.e. odd). The product will be odd if and only if every roll was an odd number (because the product of a bunch of odd numbers is odd, whereas the product of integers containing at least one even number is even). Since...