[Exercise] - 1a : Hint

Let $n \in \mathbb{N}$, show that $$f(n) = \frac{(2+\sqrt{3})^{1+2n}+(2-\sqrt{3})^{1+2n}+2}{6}$$ is a square.

Hint: MST121 / MS221 math suffices to solve this one, we need to find a function $A(n)$ such that $${A(n)}^2 = \frac{(2+\sqrt{3})^{1+2n}+(2-\sqrt{3})^{1+2n}+2}{6}$$
and
$$f(n) = A(n) \in \mathbb{N}$$
( To be continued. )

Touching complex analysis

Prof. Mattuck is in top form in lecture 6 when he talks about Euler and the beauty of complex numbers. Although it is a lecture in the DE series it can be watched as a stand-alone lecture. So if you are doing MST121, MS221 or M208 it is great fun to watch this lecture. He even touches the field of Complex Analysis when he explains differentiating $e^{i\theta}$. He notes ( jokes ) that time is always a real variable but he isn't so sure when the next Einstein comes around: he may very well decide we need complex time! Anyway, how would you integrate $\int{e^{x}}\cos{x}\ dx$ ? Prof. Mattuck says these integrals are easy if you switch to the complex domain. Finally he solves the beautiful equation $x^n+1=0$ ( as we have seen in MS221, M208 ).



Hopefully, after or during MST209, I will be able to analyze the synchronization of metronomes problem using differential equations some day.

M208-TMA03

M208 TMA03 consists of 6 questions. I had already finished question 3. Today I completed questions 1 and 2. The three questions I have done sofar still only amount to 30% of the marks. None of them was really about linear algebra though, questions 4-6 ( 70% ) definitely are. As mentioned before q3 was about solving 3 simultaneous eqns with 3 unknowns using the inverse of a 3by3 matrix. Question 1 was about 2D geometry with vectors, simple but extremely important. Question 2 was about conics. A parabola this time. The conic stuff in M208 is very easy compared to what I had to learn in MS221. Or maybe it is -because- of MS221 that conics don't scare me anymore. That's how it usually goes. All math looks horrifying difficult at first sight. And then slowly the mist disappears and then it dawns, things get clearer and clearer.

MS221: Grade 2 pass ( Cont. )

( From my Open University home page )

All in all, it has been a few hours since I know the result. I am happy with it, of course I am. I am fully motivated for the courses next year, that's what counts now. I can't wait to make another TMA actually.

Should register asap. Registration for the 2010 courses closes the 16th.

MS221: Grade 2 pass

Initial reaction: happiness
2nd reaction: relief, not the feared grade 4 pass
3rd reaction: disappointment only 3pnts below distinction
4th reaction: doesn't mean a thing because 90% passed at eirher d, 2, 3 or 4
5th reaction: jealousy, 20% or so did get a distinction
Just recording what I felt when I read the pass.

I MUST DO BETTER NEXT TIME.

Time to sign up for M208, MT365.

( I had 82/100 on the examinable component, that's really bad. Yakh. )

MS221 course result ...

... will be available by the 18th december 2009. Why does that take so long. And even after the expiry dates of registering for new courses?

About MS221 exam (3)

A commenter asked if the TMA results count in any way.

Well, yes and no.

Your final result is the lowest of the two!

So a 100 for TMA's and a 40 for exam = 40.

so a 40 for TMA's and a 100 for exam = 40.

At the end of the day only the exam counts, that is if your TMA's were 85+.

About the MS221 exam ( 2 )

Ok, let's compare my earlier prediction with how I see things now

I
2nd order recurrence system (6)
conics (6)
matrix calculation (6)
composite isometry (4)
linear transformation rotation / reflection (4)
differentiation (3)
integration(3)
taylor series (1)
function iteration (1)
logic (1)
complex numbers (4)
number theory (4)
group theory (6)
49

IIconic with xy component ( rotated )
iteration( 4 )
logic ( 8 )
12
So that is 61 or a grade 3 pass.

In a TMA @home situation I would have a 99/100 no doubt about it whatsover.

About the MS221 exam

The exam was held in The Hague, there werea about 20 people taking the MS221 exam. There were 8 exams going on in total. Can't remember which ones. What was the crowd like? I suppose +/- 25% were like me... after-career type of people finally doing / studying what they really like / love. The rest were late 20-ers, early 30-ers going for their first degree, I suppose. MS221 is not a real math-exam, the S says it all: Math Sciences, still there was only one ( 1 ) woman in the group. There was more balance in the other exams though.

The exam itself. Well, it was hard, hard, HARD! Everybody was still writing after three hours. I just finished the last question at 17.30. Wow, did time fly! I didn't know time could go -that- fast. The questions were as expected just as I predicted in an earlier post. 12 + 2 questions in three hours is a lot. No time to check, double-check, evaluate and play around with the exercises as I was used to when doing TMA's.

A distinction is impossible. I underestimated the time-pressure component. There is no time to think all you can do is robotically do the exercise as if on auto-pilot. But you can train for this so I think distinctions will be possible in future exams.

I am confident I'll pass but on a lower grade I had in mind. I am 200% motivated to continue my mathematics study. Either Pure Mathematics (60) + Groups and Symmetry (30) or maybe just Pure Mathematics (60) depending on other commitments.

The question on group theory was really easy, I saw all answers immediately in a flash, but writing all the answers down still took me 10 minutes ( or more ).

Anyway, my WIN of today is that I:
- am confident I'll pass
- am full of motivation to continue ( will register very soon )
- have learned from the experience in order to do better in future exams.

40m before the exam

Getting crowded @ the ORG.
Preparing to leave.
Sort of excited.

2h30 before MS221 exam

2h30 and counting.
Staying calm is definitely a quality.
Preparing at Uni.
My goal was ( is ) a distinction.
Had another look at the specimen exam. In an at home TMA like situation with Mathematica a 90+ score is very well possible.
But with only three hours at the worst possible moment of the day (14.30 - 17.30) and with only a calculator, I am not sure at all.
Thank G_d, we are allowed to use the MST121/MS221 handbook.

Speculating...
To get a grade 4 pass...

33 + 7.
11 / 12 questions @ 3/6 + 1/2 @ 7/14.
That is still a lot.
OR
5/12 questions @ 6/6 + 2/2 @ 5/14.

To get a grade 2 pass...
12 /12 questions @ 4/6 2/2 @ 11/14

More realistically, questions are expected on the following subjects ( expected result )
I
2nd order recurrence system (6)
conics (3)
composite isometry (3)
function iteration (3)
linear transformation rotation / reflection (3)
matrix calculation (3)
differentiation (6)
integrationb (6)
taylor series (0)
complex numbers (3)
number theory (6)
group theory (6)
logic (6)
II
conic with xy component ( rotated )
eigenvalue problem ( 11 )
volume of 3D body
proof by induction ( 17 )

54 + 17 = 71 grade 2 pass

More tomorrow or after the exam.

Result MS221 - TMA04 is in

All TMA's are done now. - Due to the substitution rule which is used at the OU the 87 for TMA02 has been replaced by 91,25.

What's it worth? Well, it means I am placed for distinction but that has to be proved at the exam next tuesday. I haven't got a clue how I'll perform during the exam. It's in the middle of my afternoon dip: 14.30 - 17.30. There isn't a single topic I don't understand that's why I was able to score high TMA's. But an exam has a time limit. 

I am not afraid of the exam because even at a score of 15 I am entitled to a resit and 40 is a grade 4 pass. But, but, but: I need 85 for a distinction. Still three days to go to the exam. 

MS221 - TMA04: Done

I reviewed MS221 - TMA04, made some minor corrections and decided that the A is ready for shipment to T. I nevertheless wait with shipping the TMA ( Cut-off date is 30 sep anyway ) because I haven't received TMA03 back yet. If I get a high enough mark on that one I might even get away with a low score ion TMA04. We'll see.

MS221 exam close by

I copied this from a post by "Tma Machine" :

Here’s how a rough outline of what’s in the 2008 paper (hopefully this won’t constitute copyright infringement!):

Part 1
Question 1: Finding a closed form for a recurrence system
Question 2: Identifying and sketching a conic
Question 3: Stating the rule for some isometries, and for a composite isometry, and then using the double-angle and half-angle formulas to show a result
Question 4: Classifying fixed points of a curve, then sketching the graph of the curve and using graphic iteration construction
Question 5: Identifying basic linear transformations, applying them to a vector, and stating an invariant line for each one
Question 6: Finding eigenvalues, eigenlines and eigenvectors.
Question 7: Differentiation
Question 8: Integration
Question 9: Finding and manipulating Taylor series about 0, and classifying stationary points
Question 10: Finding the modulus and argument of a complex number, converting them from Cartesian to polar form and vice versa, and using the formula for powers of complex numbers.
Question 11: Using Euclid’s Algorithm and working with exponential ciphers.
Question 12: Combining variable propositions, finding a case for which a given proposition is false, and finding the converse of a proposition.

Part 2
Question 13: Looks like it’s about conics, but I haven’t done this one yet.
Question 14: Linear transformations
Question 15: I haven’t done this one yet either, but it looks like it involves differentiation, integration and stationary points.
Question 16: Groups

I sort of expected this, so I am not surprised. But I am not entirely ready for the exam yet. I still have several weaknesses but I am confident I can handle these in time. To be honest I wished the exam was over and done with. It is obviously my objective to score in the 90-100 because that is what I score for TMAs. But I check, doublecheck, re-check my TMAs at least two times. That makes a lot of difference.

MS221 TMA03 - Result

MS221 TMA03 - Result is in... 99.

Lost a point due to a typo. This once again proves that you can literally save points by reviewing, QA-ing, checking whatever you call it. And check your results as much as possible by using a mathematics package like MathCad or your own preferred software, as long as it can do the math.

Sample MS221 - Exam

Had a first browse through the sample MS221 exam today. It is a lot of work for only three hours. Twelve questions which test knowledge of what was in the course. With these twelve questions you can score up to 60% of the total points. Four difficult questions remain for the other 40%. Of the four difficult questions only two will be marked. ( You have to do all four of them because you don't know which will be marked. )

I would say that anything is possible with regards to this exam. I could fail but I could just as well get a high score. It all depends on proper training from now till the exam. I am sure that a lot of exam training will pay off.

But first I have to finish MS221 - TMA04.

MS221 - TMA03 Completed

MS221 - TMA03 completed and ready to be shipped to T. I My prediction for this one is 82 / 100. We'll see. It takes at least five weeks before a result may come back. The cut-off date is August 5.
TMA03 questions
1. Derivatives
2. Graph sketching
3. Indefinite integrals
4. Area and volume calculations
5. Taylor series
6. ( more ) Taylor series

MS221 - TMA03

Started to work on MS221 / TMA03. I completed Q1a and Q2 partially ( steps 1-4 ). I suppose one of the shared experiences of all OU students is that at some point one cognites that starting asap with the TMA's is a crucial succes factor. Resultwise and otherwise.

MS221 TMA02 - Result is in

MS221 TMA02 - Result is in... 87. Very disappointing at first. I hoped for 90+. Looking back at the effort I put in book B I can only be satisfied with the 87 points. Had to do a lot of MathCad work. ( A program I can't get used to. I suppose that I am spoiled by Mathematica. )

VERY OFF-TOPIC: MJ

( His biographer made the right prediction.

I don't think he has it in him to take his own life. I don't see him putting a gun to his head. It'll be an accidental overdose - something like that. ( Stacy Brown, July 2005 )

When MJ announced the London concerts I had the feeling it wasn't Michael but a double. )

MS221 - TMA02 - ( 2 )

Completed question 1 on Sunday. Was surprised by the work required for question 2. Worked yesterday from 8am till 23.30 pm. Mailed the pack to the tutor today. After e-mailing him the pdf version and announcing the pack was to be shipped today.

( Pffhhh. )

Time to study new things again! Book C is on calculus, while book D has introductions to number theory, group theory and logic.