Questions

[frog1944]

Hi,

For the HSC, if it says show that the root of the polynomial P(x) x=1 has root of multiplicity 2. I know I can show that P'(1)=P(1)=0, but how do I know that P''(1) doesn't equal zero? Would I have to show that P''(1) doesn't equal zero and thus it isn't multiplicity 3, but only 2?

Thanks
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<Following post merges>

For the HSC if I'm trying to refer to an angle that goes along the x-axis (or y), can I label it \angle symbol PQX (where P and Q are some points, but X is referring to the x-axis). If not, what would I have to do to introduce another point, or just say some point along the x-axis named X?

For the HSC, what specific graphing terminology should we know to describe the features of a graph? I'm aware of the basics; asymptote (vertical, horizontal and oblique), intercepts, inflexion points, stationary points, relative max/min (and global). But one that I learnt the other day which I've never seen before then/been told is the idea of a "cusp". Is there any other similar graphing terminology (in this particular exam they required that you label on the graph a cusp)?

I was attempting to tackle the 2015 Q 16 a i, however I struggled quite a bit. When I eventually looked at the solutions I saw that the way they found the sample space was by saying there are 15C5 ways to place 5 black counters in the grid. I don't quite understand how this is the case? How does this account for any black counters being next to each other and not adding up or what not?

[RuiAce]

Yes, because the question didn't say "at least 2".

At this point, I believe you need to calm down, because you've just chain-posted 4 consecutive questions (which, for the record I will need to merge because this can be considered spam), following many onslaughts of questions over the past 48 hours already. Take a deep breath, think about some of them yourself (because whereas all the questions are good, some of them can be addressed by yourself with a little common sense) and then post all the important ones in one thread at a time.

[frog1944]

Ok, thanks RuiAce  , I'm sorry for the bombardment of questions

[frog1944]

Regarding the question about labelling angles on axes. Whilst in some solutions by individuals I've seen they just refer to it as you've stated. Though, to be certain I advise that you say something along the lines "let there be some point X along the x-axis at a significant distance away" or something of the like, to create a clarity in your proof.

[RuiAce]

You really should introduce the arbitrary point X. A description such as "where X is on the x-axis, to the right of Q" will suffice.
Although, one potential work-around is just to say "let theta be the angle PQ makes with the positive x-axis."

Cusps are rare. As an example, you may like to consider the graph of \( y = x^{2/3} \) to view what one looks like. And also the graph of \( \sqrt{|x|} + \sqrt{|y|} = 1 \).
Essentially, cusps are like corners, except they aren't straight (and rather curved). Generally, a cusp is identified by noticing that the derivative as you approach x from the left and from the right are different.

[frog1944]

Awesome! Thanks RuiAce for the reply, yeah the probability one  makes more sense now, I was just overcomplicating it and being confused