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Maths Extension 2 2021 Solutions

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RuiAce:
Last set of 4u I'll be doing.

Multiple choice
Q13 a) picture from tywebb belowExplanation: Basically consider the modulus and argument separately.

- Modulus: Since a+bi has modulus greater than 1, i.e. |a+bi| > 1, we also know that |a+bi|1/4 > 1, therefore they should lie outside the unit circle. However, the fourth root of a number greater than 1 will also be smaller than the original number. Therefore, the fourth roots would also be closer to the origin, than a+bi itself is.

- Argument: In general, if all values for \(\arg(a+bi)\) take the form \(\arg(a+bi) = \theta + 2k\pi\) for some integer \(k\), then all values for \( \arg((a+bi)^{1/4})\) take the form \( \arg((a+bi)^{1/4}) = \frac{\theta}{4} + \frac\pi2 k\). This can be proven using the standard algebraic method of explicitly computing fourth roots.

Therefore, we know that the first value for \( \arg((a+bi)^{1/4})\) will be one quarter of the argument of a+bi, which gives the first root. After that, we see that the other values require us to add \( \frac\pi2\) to the argument to obtain all of the other roots.


Q14 b)
Q15

Q16 (a)+(c)Some comments regarding the graph for part c) :
- Asymptotic behaviour occurs as Arg(z) approaches -pi/2 from below. Note that the entire ray corresponding to Arg(z)=-pi/2 belongs in the required region.
- With enough time spent brainstorming, you'll soon realise that none of the 2nd quadrant matters, and all of the 4th quadrant matters. These are the two easier quadrants to address in the graph. For example, in quadrant 2 specifically, Arg(z) is always positive whereas Re(z) is always negative, and hence none of the region will be included.
- The point of inflexion in the third quadrant is really not obvious. I would be so shocked if they penalised people for it. But the asymptote along the ray Arg(z)=-pi/2 is important.
- The ray Arg(z)=pi is NOT included.



Q16 (b)

I make no promises on every solution being accurate. Please point out mistakes and I'll get to them slowly.

Refer to fun_jirachi's solutions for questions I did not do.

fun_jirachi:
11

12


13

14ac

RuiAce:
Terribly sorry. Q16 has been very slow to do, and I still have a) (iii) skipped. I'm also worried I made algebra errors along the way for b). Will post everything for it soon.

tywebb:
Does this help? Found it on BOS.

caleb.clark21:
Damn, I wasn't even close for most of Q16 ahaha

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