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Re: HSC 2016 2U Marathon $\noindent \textbf{NEW QUESTION}$ $\noindent Suppose that $f$ is a linear function of $x$. Let $a,b\in \mathbb{R}$. Show that $\int _{a}^{a+b}f(x) \text{ d}x = \frac{b}{2}\left(f(a)+f(a+b)\right)$.$

Did he provide a proof or proof sketch of it? If so, can you recall any of the details of it (these may provide some clues)?

$\noindent If we don't want to use dot products or normal vectors, the tedious way to do it would be to convert the plane equation into a parametric form, thereby obtaining two vectors $\bold{u}_1,\bold{u}_2$ that are parallel to the plane. Then, we would need to show that the line's direction...

$\noindent I used the following standard fact: for the plane $ax+by+cz=d$, a normal vector is $\left[ \begin{matrix} a \\ b \\ c \end{matrix} \right] $ (as well as the fact that non-zero two vectors are perpendicular iff their dot product is 0).$

$\noindent The given plane has a normal $\bold{n}=\left[ \begin{matrix} 3 \\ -3 \\ -1 \end{matrix} \right] $. Let $\bold{v} = \left[ \begin{matrix} 2 \\ 1 \\ 3 \end{matrix} \right]$. Then clearly $\bold{v}\cdot \bold{n} = 6-3-3 = 0$, so $\bold{v}$ is perpendicular to the normal vector...

$\noindent No point $P$ in the plane will satisfy $PA+PB=1$. Since the distance from $A$ to $B$ is 2, the minimum value that $PA+PB$ can attain is 2 (this is due to the triangle inequality).$

$\noindent 1. This is just the interior of the triangular region bounded by the lines $y=2x$, $x=0$ and $x=2$, as well as the boundaries. The sketch is in the link below:$ http://www.wolframalpha.com/input/?i=0%3C%3D+y+%3C%3D+2x+and+0%3C%3D+x+%3C%3D+2. $\noindent 2. This turns out to be the...

$\noindent Oh by the way, the average acceleration vector doesn't point towards the centre here. That happens to the \emph{centripetal acceleration} vector, which is the limit of the average acceleration vector as $\Delta t \to 0$. The direction of the average acceleration vector approaches...

$\noindent Remember, the formula $a_c = \frac{v^2}{r}$ gives us the magnitude of the centripetal acceleration vector, \emph{which is by definition the limit of the average acceleration vector as the time increment is made very small}, i.e. $\overrightarrow{a_c}= \lim_{\Delta t\to 0}\frac{\Delta...

$\noindent Whether it had said `show', `verify' or `prove', it wouldn't have made a difference for this question. It's kind of like if they ask us to show that $\int \ln x \text{ d}x = x\ln x -x+c$, it suffices to simply differentiate the R.H.S. and show that we obtain $\ln x$.$ So in summary...

$\noindent Since they've given the answer basically, to show that it's a solution, we just need to check that it satisfies the given equation. If they'd asked us to actually find solutions to the equation, we'd generally have to go through the equation-solving process.$

Aim for 100. :p

$\noindent In general, for any functions $f,g$, if $g$ is the inverse of $f$, then $f$ is also necessarily the inverse of $g$. In this particular case, $f$'s domain is the codomain of $g$ and $f$'s codomain is the domain of $g$. This is something necessary for two functions to be inverses. (It's...

For option B), since its pH is 1.0 (implying a H+ concentration of 0.1 mol/L), and concentration is 0.1 mol/L, it follows that each mole of the acid contains 1 mole of H+, so it is monoprotic. For C), the concentration is 0.05 mol/L, but the H+ concentration is 0.1 mol/L (as the pH is 1.0), so...

Re: HSC 2016 4U Marathon - Advanced Level I think a complex numbers vectors proof of this was in last year's Carrotsticks BOS 4U Trial. It's easy to prove by elementary geometry as well though. We'll give this proof below. $\noindent Let the triangle have vertices $A, B, C$, with...

Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread Nice! Yeah, most people seem to find combinatorics or binomials questions the hardest.

Re: MX2 2016 Integration Marathon $\noindent Let $F_1 (x) = \int \ln (x) \text{ d}x$, and for integers $n\geq 2$, let $F_n (x) = \int F_{n-1} (x) \text{ d}x$. Find a formula for $F_n (x)$ involving $n$. (Your answer may be expressed with the help of Sigma notation, and should contain $n$...

Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread Probably Perms and Combs, judging by popular consensus.

$\noindent 3. This can be proved in a variety of ways. We will show a generalisation. Let $x$ be a fixed real number, and consider the sequence $\{a_n\}$, where $a_n = \left(1+\frac{x}{n}\right)^{n}$. We will show that this is increasing for $1+\frac{x}{n} \geq 0 \Longleftrightarrow n \geq -x$...

$\noindent For the log one, if we can use the fundamental theorem of calculus and chain rule, we can also do it as follows:$ $\noindent Note $L(x) = \int _1 ^ x \frac{1}{t} \text{ d}t \Rightarrow L^\prime (x) =\frac{1}{x}$. Now, fix $a>0$, then we want to show that $L(ax) = L(a) + L(x)$...