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Re: HSC 2016 2U Marathon $Well you asked for \textsl{necessary} conditions, not \textsl{necessary and sufficient} conditions (there's a difference between \textsl{necessary} and \textsl{sufficient} in mathematics and logic etc.). The conditions given by sharoooooo are indeed necessary...

$The expression for $D_{PX}$ should be $|x+1|$ (as you said, \textsl{magnitude} of the difference)$.

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread $Let $z=\cos \alpha + i\sin \alpha$ and $w=\cos \beta + i\sin \beta$. Then $z+w = \left( \cos \alpha + \cos \beta \right)+i \left( \sin \alpha + \sin \beta \right)$.$ $Now, the sum-to-product rules (see link at bottom) state...

Re: HSC 2016 MX2 Combinatorics Marathon $Well letting $X_K$ be the expected no. of tosses to get $K$ H's in a row \Big{(}with $X_1 = \frac{1}{p}$\Big{)}, we can set up the following recurrence:$ $$X_K = \left(X_{K-1}+1 \right)\cdot p + \left(X_{K-1}+1 + X_K \right)\cdot...

Re: HSC 2016 MX2 Combinatorics Marathon Right, misread it, didn't see the "in a row".

Re: HSC 2016 MX2 Combinatorics Marathon $Is the answer just $\frac{K}{p}$?$ $Let the expected number of tosses be $X_K$, noting $X_0 = 0$, $X_1 = \frac{1}{p}$ (expectation of geometric distribution) if one is uncomfortable with $X_0$. Then note that $X_K = \frac{1}{p}+X_{K-1}$ for $K\geq...

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread What equation?

$This is why if one wants to use the ``backward way'', it is a good idea to use the $\Longleftrightarrow$ symbol between lines of working, to make explicit the biconditional nature of the statements (assuming they are indeed biconditional; if they're not, this method won't be valid!).$

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread $Since $\sin E \cos E = \frac{1}{2} \sin 2E$, we have $4 - \tan E = 5 \sin E \cos E \Rightarrow 4 - \tan E = \frac{5}{2} \sin 2E$. Letting $x=\tan E$, we have by the $t$-formulae that $\sin 2E = \frac{2x}{1+x^2}$. Substituting...

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread $So basically, we have $\left(1+2i \right)^3 + p\left(1+2i \right) + q = 0$. Therefore,$ $$\begin{align*}LHS&= 1 + 3\cdot (2i) + 3 \cdot (2i)^2 + (2i)^3 + p + 2ip + q \quad (\text{expanding})\\ &= 1 + 6i -12 - 8i + p + 2ip +q...

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread Since we know 1 + 2i is a root, we can sub. it into that equation, then expand everything and group real and imaginary parts on the LHS, then equate the real and imaginary parts to 0, since the RHS is 0. You'll end up with two...

$Yes, I would not do that in the HSC, despite its validity, because the markers seem to hate it and give 0, as you say.$ $On the topic of proofs in the HSC, if a question asked something like prove $\frac{\sin \theta}{1-\cos \theta}=\frac{1+\cos \theta}{\sin \theta}$, would something like...

$This is indeed logically valid, assuming you are using ``if and only if'' steps, and it is good to make this explicit by using the $\Longleftrightarrow$ symbol between lines.$

$Yeah it can't be 2 since $TL$ is the hypotenuse of triangle $TLE$, so $TL>EL\Rightarrow TL > 6$. And usually in the HSC exams, they do put the vertices of the similar triangles in the right order. But you should just quickly check to make sure that they are indeed in the right order first.$...

$We should write $\triangle CLW$ is similar to $\triangle TLE$ so that the corresponding sides match up as written (i.e. $CL$ corresponds to $TL$, etc.), rather than the way the question wrote it in part (a).$ $Now, let $TL=x$. Then$ $$\begin{align*}\frac{WL}{LC}&= \frac{EL}{LT} \quad...

$He means that if $M$ is the level of maths taken by the person with family wealth $W$, then $M= \frac{k}{W}$ for some positive constant $k$. (This is the mathematical definition of ``inverse relation'' between two variables.)$ $(Seriously speaking, he means that generally, he finds that people...

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread $Let $z=x+iy$, where $x=\Re({z})$ (real part of $z$) and $y=\Im ({z})$ (imaginary part of $z$). Note that $z\bar{z}=|z|^2$. Also, $\Re \left(z-1+3i \right)=x-1$ (since the real part of a sum of complex numbers is the sum of the...

Re: HSC 2016 3U Marathon $Well yes, the binomial theorem $(a+b)^n = \sum _{k=0}^n \binom{n}{k}a^{n-k}b^k$ is expected to be known by students. From this, if we let $a=1$ and $b=x$, we have $a^{n-k}=1^{n-k}=1$ for all $k$, and so $(1+x)^n = \sum _{k=0}^n \binom{n}{k} x^k$ (but you should know...

We would need to use the concept of conservation of angular momentum to explain the physics of an ice skater spinning on the same spot, and this concept is not in the high school syllabus.

The Saturn part is a typo, replace the Saturns with Jupiters (i.e. the moons orbit the same planet).