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$\noindent When you equate real and imaginary parts, you get $\cos{x} = \frac{1}{\sqrt{2}} \ (\star)$ and $\sin{x} = -\frac{1}{\sqrt{2}} \ (\star \star)$. From $(\star)$, $x = \frac{\pi}{4}$ or $-\frac{\pi}{4}$ for $-\pi < x \leq \pi$. And from $(\star \star)$, $x = -\frac{3\pi}{4}$ or...

$\noindent You don't have to equate real and/or imaginary parts. We know that $x$ is the argument of $5-5i$, and we know that the argument of $5-5i$ is $-\frac{\pi}{4}$ and so $x = -\frac{\pi}{4}.

$\noindent \begin{align*}\quad S_1 &= (x+1)^2-1 \\ \implies T_1 &= (x+1)^2 - 1 \\\\ S_2 &= (x+1)^4 - 1 = T_1 + T_2 \\ \implies T_2 &= S_2 - T_1 \\ &= (x+1)^4 - 1 - (x+1)^2 + 1 \\ &= (x+1)^2\left((x+1)^2 - 1\right) \\\\ S_3 &= (x+1)^6 - 1 = T_1 + T_2 + T_3 \\ \implies T_3 &= S_3 - S_2 \\ &=...

Re: HSC 2018 MX2 Marathon $\noindent By the product of roots, $E^2 + n^2 = \frac{c}{a} \implies an^2 = -aE^2 + c \ (\star)$ and by the sum of roots, $2E = -\frac{b}{a} \implies 2aE^2 = -bE \implies -aE^2 = aE^2 + bE \ (\star \star)$ Substitute $(\star \star)$ in $(\star)$: $an^2 = aE^2 + bE + c

Multiplying by i is just a geometric anticlockwise rotation of pi/2, there is no enlargement or dilation so iz will also have modulus r. As above, an argument would be pi/2 + theta

If you draw a line through B parallel to MD then since the parallel lines divide transversal BC into equal intercepts it also divides transversal BE into equal intercepts. ie. BD = DE But DE = BD = DA + AE (*) And BD = DA + AC (**), so equating (*) and (**) gives DA + AE = DA + AC => AE = AC...

I doubt it, ask InteGrand or Carrotsticks

The syllabus only places restriction |r| < 1 for limiting sum

They are the same thing. But if the question says "At what rate is the volume of the soap decreasing?" then -0.2cm^3/s would be wrong as decreasing at -0.2cm^3/s translates to increasing at 0.2cm^3/s

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No it was 0 < a < 4.

No, it asked for x such that sina = sinb then asked to explain why it gives a minimum value of L.

I proved similarity because a = b and the right angles, so you get the proportion x/5 = (9-x)/7

Will they accept both 880 and 870? Exact k gives 870, but rounded gives 880.

3.75?

Did people actually solve that dL/dx = 0 equation? I just used similar triangles, was much faster.

No it's 1. It stops at first but doesn't change direction. assuming t-axis, then no, it just stops at that instant.

gg

I think there is an error in Q13, that is definitely not the graph of y = tan(3x) and even if we ignored the graph, the required integral is divergent, so just ignore this question. And for Q12, I don't know if I'm hallucinating or there's actually no k in the whole question.

any particular questions?