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For convenience, here's the link to the paper: https://www.boardofstudies.nsw.edu.au/hsc_exams/hsc2011exams/pdf_doc/2011-hsc-exam-mathematics-ext2.pdf. Here are the Sample Answers...

For the chord in a circle problem, see also https://en.wikipedia.org/wiki/Bertrand_paradox_(probability).

$\noindent The solutions to $\tan x + 2 = 0$ for $x$ between $0^{\circ}$ and $360^{\circ}$ are indeed $180^{\circ} + \tan^{-1}(-2)$ and $360^{\circ} + \tan^{-1}(-2)$.$ $\noindent To see this, note that if we don't restrict $x$, then $\tan x + 2 = 0 \iff \tan x = -2$ has a solution $x =...

$\noindent In general though, to find the equation of a tangent line at the point $(a, f(a))$ on the graph $y = f(x)$, all you need to work out is the slope of the tangent line. Once you know this, since you know a point on the line as well (namely $(a,f(a))$), you can write down the equation of...

$\noindent The point $(0, 4)$ does not lie on that curve, because $4^{0} + 1 \neq 4$.$

HINT: Using what you have, obtain a quadratic inequality for ln 2 and solve the inequality. This will get you (both of) the desired bounds.

$\noindent It's the same as how $YX - X = X(Y-1)$ (where $Y = a + b$ and $X = a-b$). In other words, factor out an $(a-b)$ from each term in $(a+b)(a-b) - (a-b)$.$

$\noindent You can use either one, just remember that $\boxed{n\geq 1}$ in the inductive hypothesis. So if you say that $\color{blue}n = 2k-1$, then you should say that $\color{blue}k\geq 1$. If instead you say that $\color{red}n = 2k+1$ in the inductive hypothesis, then you should say that...

Well done! And no worries.

$\noindent You're welcome! And here's a hint: try and write $2^{2k+1}$ in terms of $2^{2k-1}$.$

Also, thanks for posting your attempts!

$\noindent For your step 3, you should show the statement is true for $n = 2k+1$, i.e. the \textbf{next odd number} from your inductive hypothesis. $

$\noindent It's essentially asking you what kinds of statements can be proved using induction. For example, here are two propositions:$ $\noindent \textbf{Proposition 1.} If $f$ and $g$ are differentiable functions, then $\frac{d}{dx}(f(g(x))) = f'(g(x)) g'(x)$.$ $\noindent...

$\noindent \textbf{Hint.} The locus is the straight line parallel to the given lines and halfway between them. Therefore, what is the $y$-intercept of this line, and what is its slope?$

$\noindent It (seeing that the $n$-th term of the sequence is $1 + 2 + \cdots + n$, for $n\geq 1$) is really just by inspection. You can also see it by actually drawing ``triangles'' (hence the name \emph{triangular numbers}''), as follows:$ (1 layer) T1 = 1: • (2 layers) T2 = 3: • •• (3...

Re: HSC 2018 MX2 Marathon Haha, that wasn't the motivation at all.

Re: HSC 2018 MX2 Marathon $\noindent For each real number $\sigma$, let $f(\sigma)$ be the maximum value of $|\log_{2}(1 + x) - x- \sigma|$ for $0\leq x \leq 1$. Find the value of $\sigma$ that minimises $f(\sigma)$.$

Re: Kepler's Laws of Planetary Motion - Total energy of Circular and non-circular orb Why are they calling it 'Advanced' Mechanics if there won't be any calculus? Do they mean 'advanced' as in relative to the Moving About topic (previous syllabus)?

$\noindent \textbf{Hints.}$ $\noindent Let $y(t)$ be the vertical displacement (in m) of the bag from the ground after time $t$ (in s) (so $y(0) = 1$).$ $\noindent \textbf{\color{blue}{(a)}} Let $v > 0$ be the speed the bag was thrown at. Then using the ``$\text{change in position} = ut +...

$\noindent Hint for Q1: recall that in $100$ such experiments, the probability of picking the Ace of Hearts exactly $k$ times is $\binom{100}{k}(0.25)^k (0.75)^{100-k}$.$