# Thread: First Year Mathematics B (Integration, Series, Discrete Maths & Modelling)

1. ## Re: MATH1231/1241/1251 SOS Thread

Originally Posted by Flop21
I've become stuck on this again

[3x + 2]/[(x^2+2x+2)(x-1)] = 1/(x-1) + (bx+c)/(x^2+2x+2).

So I go 3x+2 = A(x^2+2x+2) + (bx+c)(x-1)... right?

A is 1 like they already give us. But I can't seem to find an x value that will allow me to find bx+c?
Sub. x = 0, which easily lets us find C. Then we can find B by subbing (say) x = 2.

2. ## Re: MATH1231/1241/1251 SOS Thread

When doing first order linear differential equations...

How did this turn into this?

3. ## Re: MATH1231/1241/1251 SOS Thread

Originally Posted by InteGrand
Sub. x = 0, which easily lets us find C. Then we can find B by subbing (say) x = 2.
Oh thanks, silly me was getting confused when I was getting c = 0 for some reason. But that's the answer.

4. ## Re: MATH1231/1241/1251 SOS Thread

Originally Posted by Flop21
When doing first order linear differential equations...

How did this turn into this?

Product rule.

5. ## Re: MATH1231/1241/1251 SOS Thread

Originally Posted by leehuan
Actually, whilst I can't deny the reduced difficulty of the HSC exams over the years, if it were to appear now I would be thoroughly displeased at that as well.

I don't know what the HSC was like in 2003 to speak for back then

Thanks for that proof though; that I'll show him as well.
Can you recall whether there have been any contradiction proofs in the HSC more recently than 2003? Surely there should've been (though I can't remember any off the top of my head)?

6. ## Re: MATH1231/1241/1251 SOS Thread

Originally Posted by InteGrand
Can you recall whether there have been any contradiction proofs in the HSC more recently than 2003? Surely there should've been (though I can't remember any off the top of my head)?
Couldn't really off my head either

7. ## Re: MATH1231/1241/1251 SOS Thread

1. Will I lose marks if I just write down the limit in questions asking for 'does this series converge, if so find its limit'?

2. Will I lose marks if I don't show fully working when row reducing in questions where it's just a step, e.g. finding eigenvectors I can just sometimes do it my head and will just write the final form.

8. ## Re: MATH1231/1241/1251 SOS Thread

Originally Posted by Flop21

1. Will I lose marks if I just write down the limit in questions asking for 'does this series converge, if so find its limit'?

2. Will I lose marks if I don't show fully working when row reducing in questions where it's just a step, e.g. finding eigenvectors I can just sometimes do it my head and will just write the final form.
Well if you want to find the limit of a series, unless it's that obvious how do you plan to do it by inspection though

But with 2 I also skip row reductions if it's that obvious. I think my tutor does as well.
I work around it by saying matrix*x = 0 therefore x = my answer

9. ## Re: MATH1231/1241/1251 SOS Thread

Good thing is that they're like we WANT to give you marks. So they don't have mark allocations like in the HSC so that they aren't limiting themselves.

So I reckon if you jumped some steps but didn't jump too excessively to get the right answer you'd get it. (But of course if you jumped too many steps only to get the wrong answer well that's another story)

10. ## Re: MATH1231/1241/1251 SOS Thread

(for the same linearly independence qn)

Another friend wants to know if there's fault in what he did and I can't really communicate anything

11. ## Re: MATH1231/1241/1251 SOS Thread

Originally Posted by leehuan
(for the same linearly independence qn)

Another friend wants to know if there's fault in what he did and I can't really communicate anything
Yeah that's correct.

12. ## Re: MATH1231/1241/1251 SOS Thread

I could probably research it up but I feel as though InteGrand's answers are more comprehensible...

$\text{Prove that }i^i=\exp \frac{-\pi}{2}$

13. ## Re: MATH1231/1241/1251 SOS Thread

Originally Posted by leehuan
I could probably research it up but I feel as though InteGrand's answers are more comprehensible...

$\text{Prove that }i^i=\exp \frac{-\pi}{2}$
$\noindent Need to define complex powers and logarithms to make sense of something like i^i. I'll be slightly brief now (search up things like complex logarithm and powers online for more info). We generally define (for z and c complex and z \neq 0) z^c as e^{c \log z}, where \log is the complex logarithm and is \textsl{multivalued} (so powers are generally multivalued). For complex numbers z = re^{i\theta} (r > 0, \theta \in \mathbb{R}), we define \log z = \ln r + i\theta + 2\pi i n (n\in \mathbb{Z}), where \ln is the usual real logarithm.$

$\noindent So i^i = e^{i\log i}, and \log i = \ln \color{blue}{1}\color{black}{}+ i \color{red}{\frac{\pi}{2}}\color{black}{}+ 2\pi i n = i\frac{\pi}{2} + 2\pi i n \Big{(}since |i| = \color{blue}{1} and \mathrm{Arg}(i) = \color{red}{\frac{\pi}{2}}\color{black}{}\Big{)}. Hence i^i = e^{i\left(i\frac{\pi}{2} + 2\pi i n \right)} = e^{-\frac{\pi}{2} - 2\pi n}, n\in \mathbb{Z}. If we take n = 0, this gives us the \textbf{principal value} (corresponding to the \textsl{principal logarithm} being used -- you can search online for more info on this) of e^{-\frac{\pi}{2}}.$

$\noindent So the key things are that in general complex powers and logarithms are multivalued objects. Also pretty cool, i^i is real!$

14. ## Re: MATH1231/1241/1251 SOS Thread

Originally Posted by leehuan
I could probably research it up but I feel as though InteGrand's answers are more comprehensible...

$\text{Prove that }i^i=\exp \frac{-\pi}{2}$
Spending too much time on mathematical memes ?

15. ## Re: First Year Mathematics B (Integration, Series, Discrete Maths & Modelling)

Is everyone still doing this? I'd very much like to participate.

Here's a problem I've been stuck on for a little while. Any takers?

16. ## Re: First Year Mathematics B (Integration, Series, Discrete Maths & Modelling)

Consider the equation \lambda_1.v_1 + ... \lambda_m.v_m = 0. We need to show that all coefficients are zero. To do this, take the dot product of v_i to both sides to show that for arbitrary i = 1, 2, ... , m, \lambda_i = 0 (by the property of orthogonality). And we're done.

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