[Rahul.S]

(Original post by wcp100)
Do you actually derive the moments of inertia though in physics?

yh you do

my teacher printed me the stuff i need to know.....I think you can access via them through e-AQA or something.

theres also angular motion and tonnes to do with thermodynamics

[desijut]

Ok, now here is STEP II/1998:

my opinion on them:

Spoiler:

Show

1) I'd done it before from the Siklos booklet, but it was a while ago... edit: Well it was a long time ago.... so i missed the hence...
2) Done something similar but not quite that before, still made a meal of it.
3) Nice question, should have maybe stated i used the method of differences and a silly error at the end
12) First part was ok, second part i calculated the wrong thing, which i think i was doing right before i crossed it out....
5) Got stuck on the inductive step... Maybe could have proved it like I proved it for |z1 + z2| =< |z1| + |z2| (but that was unconvincing imo)
7) Very nice question, didnt have time to add a conclusion to the last part

All in all, i think i did well but it could have been an amzing paper if i didnt make silly mistakes. But it is a strong 1 i think (better than i usually do) edit: with the mistake in Q1 it makes it a weak 1

[Farhan.Hanif93]

(Original post by desijut)
Ok, now here is STEP II/1998:

my opinion on them:

Spoiler:

Show

1) I'd done it before from the Siklos booklet, but it was a while ago...
2) Done something similar but not quite that before, still made a meal of it.
3) Nice question, should have maybe stated i used the method of differences and a silly error at the end
12) First part was ok, second part i calculated the wrong thing, which i think i was doing right before i crossed it out....
5) Got stuck on the inductive step... Maybe could have proved it like I proved it for |z1 + z2| =< |z1| + |z2| (but that was unconvincing imo)
7) Very nice question, didnt have time to add a conclusion to the last part

All in all, i think i did well but it could have been an amzing paper if i didnt make silly mistakes. But it is a strong 1 i think (better than i usually do)

Here's my pennies worth:

Spoiler:

Show

Q1 - You forgot/didn't bother to show that if (*) has a solution then n is even. Whether that is OK is up for debate because the question requires you to "deduce" that (*) has no solutions from these earlier parts. You may be able to get away with it for the first equation because it explicitly asks you to show that n^2 is a factor of 54, which is a fact you've based your argument upon but you certainly haven't deduced that (**) has no solutions from the fact that n must be even as requested so you may be punished quite heavily for that.

My mark: 12/20 (being generous - I'm not sure how lenient they would be)

Q2 - As you say, you've made quite the meal of the second part. After choosing a, you could have just noted that you need to choose (preferably small) N s.t. for some integer n (as the denominator is already 5^3). Since you want N to be small, looking around the 60 mark for cubes would have been the next obvious step and you could have chosen N by inspection from there. You've also either made a small error in your division or a typo as there is one too many 9's in your approximation.

My mark: 18/20

Q3 - I think you have put in slightly too much effort in here too, you could just as easily have expressed in partial fractions directly rather than doing it twice. The sum at the end should come to 1, rather than 1/2 (mistake is on the 4th line from the bottom, two equality signs in).

My mark: 19/20

Q12 - it's not one that I bothered doing so I can't really comment.

Q5 - If I'm going to be honest, I don't think this would have got you any marks at all. You don't seem to have used the geometrical interpretation of |z_1 - z_2| to prove the triangle inequality, as asked in the question and as you say, there isn't really much of an inductive step in your proof of the general case. If you need a hint, add to both sides of your n=k assumed inequality and use the triangle inequality you proved at the start.

My mark: 1/20

Q7 - This was mostly fine, might be worth familiarising yourself with the definition of a strictly increasing function (at A-Level, a function f is said to be strictly increasing over a particular interval if for all in that interval. Equivalently, with such functions, (with equality iff ) for each pair of such over said interval). This would have saved you from having to find the turning points. For the last part, you want to find an expression for F'(x) in terms of F(x) and other functions, and deduce it's sign from there.

My mark: 16/20

Overall opinion: Low grade 1/high grade 2 (66 marks) excluding Q12, which probably makes it a comfortable grade 1 overall.

Reading that back, I sound pretty harsh. That wasn't the intention, I promise and it should be said; I'm no examiner so it's possible that I'm way off - this is my genuine opinion. Hope that helps.

[desijut]

(Original post by Farhan.Hanif93)
Here's my pennies worth:

Spoiler:

Show

Q1 - You forgot/didn't bother to show that if (*) has a solution then n is even. Whether that is OK is up for debate because the question requires you to "deduce" that (*) has no solutions from these earlier parts. You may be able to get away with it for the first equation because it explicitly asks you to show that n^2 is a factor of 54, which is a fact you've based your argument upon but you certainly haven't deduced that (**) has no solutions from the fact that n must be even as requested so you may be punished quite heavily for that.

My mark: 12/20 (being generous - I'm not sure how lenient they would be)

Q2 - As you say, you've made quite the meal of the second part. After choosing a, you could have just noted that you need to choose (preferably small) N s.t. for some integer n (as the denominator is already 5^3). Since you want N to be small, looking around the 60 mark for cubes would have been the next obvious step and you could have chosen N by inspection from there. You've also either made a small error in your division or a typo as there is one too many 9's in your approximation.

My mark: 18/20

Q3 - I think you have put in slightly too much effort in here too, you could just as easily have expressed in partial fractions directly rather than doing it twice. The sum at the end should come to 1, rather than 1/2 (mistake is on the 4th line from the bottom, two equality signs in).

My mark: 19/20

Q12 - it's not one that I bothered doing so I can't really comment.

Q5 - If I'm going to be honest, I don't think this would have got you any marks at all. You don't seem to have used the geometrical interpretation of |z_1 - z_2| to prove the triangle inequality, as asked in the question and as you say, there isn't really much of an inductive step in your proof of the general case. If you need a hint, add to both sides of your n=k assumed inequality and use the triangle inequality you proved at the start.

My mark: 1/20

Q7 - This was mostly fine, might be worth familiarising yourself with the definition of a strictly increasing function (at A-Level, a function f is said to be strictly increasing over a particular interval if for all in that interval. Equivalently, with such functions, (with equality iff ) for each pair of such over said interval). This would have saved you from having to find the turning points. For the last part, you want to find an expression for F'(x) in terms of F(x) and other functions, and deduce it's sign from there.

My mark: 16/20

Overall opinion: Low grade 1/high grade 2 (66 marks) excluding Q12, which probably makes it a comfortable grade 1 overall.

Reading that back, I sound pretty harsh. That wasn't the intention, I promise and it should be said; I'm no examiner so it's possible that I'm way off - this is my genuine opinion. Hope that helps.

One thing i also wanted to know is, is that ok to lay out these questions? is it clear enough to read? Or is it too messy?

Spoiler:

Show

1) **** forgot about the "deduce" And although i did this ages ago, i dont remember the 2nd part at all
2) and 3) I was making lots of silly errors so i thought it was best to start again in places (i knew what to do so i knew i wasnt wasting time by doing that) Also on the approximations, i wasnt sure how to handle "Justify the accuracy of your approximations"
12) I would have got a few, not many though. Maybe 5 if i'm lucky?
5) I started but then saw 7 and knew it was straight forward (i thought it was very easy originally)
7) Yep, i did it in 30 minutes, i needed an extra 5-10 minutes to wrap up the question...

When i said high 1 i gave myself full marks on Q1 so take off 8-12 marks would make me low 70's (I was a little more leniant but that's because i gave myself a few more on Q5 and 5-6 on Q7) but i take on board what you said.

[ian.slater]

(Original post by TheJ0ker)
Ok for STEP 2 am I expected to know these things? Its just that that topic is in FP3 which I haven't done...

Have a go at 1997 STEP II Q3. It plays on an idea that two similar expressions have totally different integrals. As it happens, MEI make a great point of it in their FP2.

Also, there are a lot of things that you are not 'required' to know, but it helps if you have seen them before because it saves time. For example, 2007 STEP II Q3 asks you to prove the arctan result that would have helped with your earlier question. So it's not on the I/II syllabus now, but the ability to prove it is.

[fruktas]

(Original post by Rahul.S)
im the only one doing applied....its kind of sik. got alot of mechanics....even some m5

Yeah, my teacher said we are doing astrophysics because it's the easiest for the rest of the class. but he said he wanted to teach applied physics lol. It seems very good!

[TheJ0ker]

(Original post by ian.slater)
Have a go at 1997 STEP II Q3. It plays on an idea that two similar expressions have totally different integrals. As it happens, MEI make a great point of it in their FP2.

Also, there are a lot of things that you are not 'required' to know, but it helps if you have seen them before because it saves time. For example, 2007 STEP II Q3 asks you to prove the arctan result that would have helped with your earlier question. So it's not on the I/II syllabus now, but the ability to prove it is.

Ok thanks I will have a go at it.

[TheJ0ker]

Can someone check this proof by induction, FP1 was a long time ago and I am a bit rusty.

Trying to prove for n>=4

Spoiler:

Show

let n=4
which it true

Let n=k where k>4

assume

let n=k+1



as k>4 and (*) must be true.

[matt2k8]

You have the right induction step, but the answer is not written out well - I'd write something like-

4! = 24 < 16 = 2^4, so the statement is true for n = 4.

Now suppose for some k >4, the statement is true.

Then (k+1)! = (k+1)*k! > (k+1)*2^k > 2*2^k = 2^k+1.

So if it's true for n =k, it's also true for n=k+1- so by induction, it's true for all n >= 4.

[desijut]

(Original post by TheJ0ker)
Can someone check this proof by induction, FP1 was a long time ago and I am a bit rusty.

Trying to prove for n>=4

Spoiler:

Show

let n=4
which it true

Let n=k where k>4

assume

let n=k+1



as k>4 and (*) must be true.

I think this would be a better conclusion

Spoiler:

Show

From


Since




must be true.

[TheJ0ker]

Ok thanks guys.

[jack.hadamard]

What is a Proof -- a general read.


A Sample Proof by Induction.

a) Some notes on induction. b) Some Exercises (Ignore Q7).

Divisibility proofs are often a bit unusual -- e.g. how do you prove that is divisible by for all natural .

[ben-smith]

(Original post by jack.hadamard)
Divisibility proofs are often a bit unusual -- e.g. how do you prove that is divisible by for all natural .

Unless I'm being stoopid, that last one is easy:

Spoiler:

Show

it's the decimal expansion of 110000...1. It's digits add up to 3 ===> divisible by 3. A more explicit proof would involve rewriting the 10s as 9+1 and the binomially expanding.

[jack.hadamard]

(Original post by ben-smith)
Unless I'm being stoopid, that last one is easy:

Spoiler:

Show

it's the decimal expansion of 110000...1. It's digits add up to 3 ===> divisible by 3. A more explicit proof would involve rewriting the 10s as 9+1 and the binomially expanding.

Both of these are perfectly good approaches, but they are more direct in nature.
They also rely on more ``advanced" tools like congruences and the binomial theorem.

By induction, a proof may contain the following.

Spoiler:

Show

Suppose that, for some , you have for a positive integer .

Multiplying by , this equality gives , and it follows that .

[Farhan.Hanif93]

(Original post by desijut)
One thing i also wanted to know is, is that ok to lay out these questions? is it clear enough to read? Or is it too messy?

Yeah I think your presentation is fine, it's good that you leave quite a bit of space between your lines of working as it gives you room to fill those gap if needed; I should really learn to do that.

That said, there were a lot of pages of crossing outs. It must have taken quite a bit of time to write each of those pages so it's not really time efficient to write them out twice in the exam. In general, it would have been much faster to just spot the mistake and try again from there - checking through as you go along is often a good idea.

From what I've noticed, the place you can save the most time is in the amount you write down. Taking Q7i) for example, you could have just jumped from (although it would have been quicker still to notice that this is equivalent to ). Further down the line, in the same question, it would have been fine to go from to directly. Looking at these individual cases, it doesn't seem like a lot but when you consider all the similar bits of working throughout the paper, it would have saved you a lot of time to juggle the algebra in your head first. Still have to be cautious not to exclude any key details, and not make little mistakes so there's an element of balance to be struck - only make these leaps in small steps of algebra that you are completely confident about.

Spoiler:

Show

1) **** forgot about the "deduce" And although i did this ages ago, i dont remember the 2nd part at all
2) and 3) I was making lots of silly errors so i thought it was best to start again in places (i knew what to do so i knew i wasnt wasting time by doing that) Also on the approximations, i wasnt sure how to handle "Justify the accuracy of your approximations"
12) I would have got a few, not many though. Maybe 5 if i'm lucky?
5) I started but then saw 7 and knew it was straight forward (i thought it was very easy originally)
7) Yep, i did it in 30 minutes, i needed an extra 5-10 minutes to wrap up the question...

When i said high 1 i gave myself full marks on Q1 so take off 8-12 marks would make me low 70's (I was a little more leniant but that's because i gave myself a few more on Q5 and 5-6 on Q7) but i take on board what you said.

Spoiler:

Show

Even though you knew what you had to do in 2) and 3), you could still have saved yourself time by finding your mistake rather than rewriting - whenever I do a STEP paper, it's the writing which takes most of the time rather than the "doing" of the question. The question says that you don't need to justify the accuracy of your approximation so you don't need to worry about it - I think for STEP, if they do ask you to do this, all you would have to show is that your value is in the interval of validity and the error in your approximation (generally the modulus of the first neglected term in the expansion) doesn't affect the digits in any of the decimal places you've expanded to.

[bananarama2]

(Original post by Rahul.S)
yh you do

my teacher printed me the stuff i need to know.....I think you can access via them through e-AQA or something.

theres also angular motion and tonnes to do with thermodynamics

All of the stuff I need to know for my engineering interview and none of the stuff I get taught facepalm. I don't think I learnt anything in school. I self taught it before hand, so I know the stuff that would be useful.

[Rahul.S]

(Original post by wcp100)
All of the stuff I need to know for my engineering interview and none of the stuff I get taught facepalm. I don't think I learnt anything in school. I self taught it before hand, so I know the stuff that would be useful.

like a bawwws gaussss

[desijut]

(Original post by Farhan.Hanif93)
Yeah I think your presentation is fine, it's good that you leave quite a bit of space between your lines of working as it gives you room to fill those gap if needed; I should really learn to do that.

That said, there were a lot of pages of crossing outs. It must have taken quite a bit of time to write each of those pages so it's not really time efficient to write them out twice in the exam. In general, it would have been much faster to just spot the mistake and try again from there - checking through as you go along is often a good idea.

From what I've noticed, the place you can save the most time is in the amount you write down. Taking Q7i) for example, you could have just jumped from (although it would have been quicker still to notice that this is equivalent to ). Further down the line, in the same question, it would have been fine to go from to directly. Looking at these individual cases, it doesn't seem like a lot but when you consider all the similar bits of working throughout the paper, it would have saved you a lot of time to juggle the algebra in your head first. Still have to be cautious not to exclude any key details, and not make little mistakes so there's an element of balance to be struck - only make these leaps in small steps of algebra that you are completely confident about.

Spoiler:

Show

Even though you knew what you had to do in 2) and 3), you could still have saved yourself time by finding your mistake rather than rewriting - whenever I do a STEP paper, it's the writing which takes most of the time rather than the "doing" of the question. The question says that you don't need to justify the accuracy of your approximation so you don't need to worry about it - I think for STEP, if they do ask you to do this, all you would have to show is that your value is in the interval of validity and the error in your approximation (generally the modulus of the first neglected term in the expansion) doesn't affect the digits in any of the decimal places you've expanded to.

Oh wow i read it as "You need to justify the accuracy of your approximations" :| so I was thinking i had to justify it...

I think those little long ways i took for say to was just me not pausing for a moment to realise it.

[deejayy]

(Original post by ben-smith)
Unless I'm being stoopid, that last one is easy:

Spoiler:

Show

it's the decimal expansion of 110000...1. It's digits add up to 3 ===> divisible by 3. A more explicit proof would involve rewriting the 10s as 9+1 and the binomially expanding.

Could you also use modular arithmetic ?

Spoiler:

Show

I'm not totally sure if you are allowed to do that with indices when working with modular arithmetic though. Can you?

[ben-smith]

(Original post by deejayy)
Could you also use modular arithmetic ?

Spoiler:

Show

I'm not totally sure if you are allowed to do that with indices when working with modular arithmetic though. Can you?

The thing I wrote with binomial expansion is basically mod arithmetic