1. The problem statement, all variables and given/known data
Quite a long intro to the question so I thought it easier to include it as an image:
2. The attempt at a solution
I can do Q2.3 and get the payoff matrix given when V=4 and C=6.
For Q2.4a I get
For Q2.4b I normalize the payoff matrix to get
Now comes the problems.
For an ESS we must have(*)
By using the normalized matrix we can rewrite these as
Let x = (h,d,b) be our interior ESS, then by (*) we have
2d - 0.5b = 0.5h + d and h - b = 0.5h + d .
The first of these can be rearranged to give h=2d-b while the second can be rearranged to give h=2d+2b. Clearly these can only both be satisfied when b=0. But this contradicts the fact that x = (h,d,b) is an interior ESS. Hence there can be no interior ESS's.
Now that seemed correct to me, but it doesn't tie-in with Q2.4c. This question claims that the only ESS is the pure strategy B. By considering the H-D subgame I get an ESS at (2/3,1/3,0).
Assuming the question is written correctly, where am I going wrong?
Thanks for any help!!
Hi there, i'm a bit stuck on this question:
" Given 3 non-coplanar vectors a, b and c convince yourself that the position vector r of any point in space may be represented by
r = λa + μb + γc
for some real numbers λ, μ and γ.
r.(bxc) = λa.(bxc) ,
r.(axb) = γa.(bxc) ,
r.(cxa) = μa.(bxc) . "
I understand how they get the first one - the cross product of b and c is perpendicular to both of them, so won't contain any b or c components. Hence when you do the dot product you'll be multiplying the b and c bits of r by zero so they disappear. However I don't get the other two...?
i'm trying to find the equation of the locus of z where
I wrote z=x+iy in which case we have
I'm ommiting my working obviously, but i've checked it and i think it's correct. This means that
which is the equation of a circle. This seems correct, but I think the locus is only part of the circle not the whole thing. How would you guys go about tackling this problem?
Hi there, I'm having some trouble with this one :-(
"A particle whose initial mass is m is projected upwards at time t=0 with speed gT, where T is constant At time t its speed is u and its mass has increased to me^(t/T). If the added mass is at rest when it is acquired, show that
d/dt [mue^(t/T)] = -mge^(t/T) .
Deduce that the mass of the particle at its highest point is 2m."
I've derived the differential equation using the impulse-momentum principle, but i'm stuck on the second bit. When I solve the equation I get
u = gT[2e^(-t/T) - 1] .
So what I was planning to do was to find du/dt since the highest point that the particle reaches will occur when du/dt = 0. However with the equation i've found du/dt never does equal zero!
I'm doing the exact same thing right now - got my final exams coming up in a few weeks!
I just used the Heinemann books together with another couple of books entitled Further Pure Mathematics (by Brian Gaulter) and Further Pure Mechanics (by Brian Jefferson). These two cover most material in the pure and mechanics modules, and are useful for a couple of chapters that the Heinemann books fail to explain very well. On the whole though I've found the Heinemann books to be pretty good. The key to passing the exams is to do lots of past papers.
If you're planning on doing maths, physics or engineering at uni then i'd definitely recommend doing further maths. I've done up to M5 in mechanics and am so much better prepared going into a physics degree next year. The physics and maths a-levels alone are pretty pitiful!
Hi there, i'm a bit stuck on the question:
A particle P describes the equianglar spiral with polar coordinates r = ae^θ in such a way that its acceleration has no radial component. Show that
(a) the speed of the particle is proportional to r
(b) the magnitude of the acceleration is proportional to r.
I've sat and fiddled with it for a bit but couldn't seem to get very far. Any help appreciated :-)
if I have f(x) = x√(x^2 + a^2) - x^2 , where a is a constant and is quite a bit smaller than x, how do I find out what it converges to?
If I put in values of x which are really massive compared to a then I get close to zero, which is obvious. But if I put in kind of mid-range values then it seems to settle around another value. For example, I have a=17 and I try x as 500, 1000, 1500, 2000, etc then f(x) stays really close to 144. It's confusing me lol.
Any help appreciated :-)
Yeah thanks, I understand it now i think We have not even discussed Dalton's law of partial pressures in our lessons yet so that's why it confused me a bit. I wish my teacher would teach things first before asking questions on them lol!
Thanks for your help. It's getting late here now so i'll return in the morning to calculate the answers.
this one is bugging me. I've written everything down methodically but i seem to end up with 3 equations and 4 unknowns. I can't see what i'm missing Any help appreciated!!
"Two containers of volumes VA = 6.4x10^-2 m³ and VB = 2.7x10^-2 m³ contain 7 and 3 moles of the same gas, respectively. The containers are communicating via a tube with a valve which is initially shut. The voume of the connecting tube is negligible. The temperatures in the two containers are initially TA = 400k and TB = 600K respectively. The valve then opens and after gas transfers from one container to the other, the two gases reach equilibrium (i.e. the two containers are at the same temperature and pressure).
Find the number of moles in each of the containers and also the pressure and temperature of the gas."
so far ive done C1-C4, M1, M2, S1 and in january im taking FP1 and D1
Haha, I took one look at that D1 book and thought there's no way i'm ploughing through this lot! How are you finding FP1?
Over the summer i've done C4, FP1 and FP2, and i've almost finished M3 now. I'm gonna sit all these in january so i'll only have 2 to do in the summer, plus any resits if necessary. I've actually found M3 to be quite good, really interesting topics (well i find them interesting anyway) on centres of mass of solids and stuff. It seems about the same difficulty as M2 to me, just with different material. I think i'm gonna give M4 and maybe even M5 a go - the books are nice and slim From what i've done so far of the further maths material, the integration in FP2 is the fcuker - hopefully the questions in the exam papers won't be as tough as some of those in the book.
Wow there are lots of people doing further maths in your schools!! Where i'm at, there are only 5 people in my regular A2 math class, and one of them doesn't show up much!
I'm doing physics and ICT with the maths. Wanna do a theoretical physics degree at uni.. maybe Warwick or Imperial with any luck. I'm off to the Warwick open day tomorrow actually You two decided what you're gonna study and where yet?
I'm doing further maths as well, but i am doing it at school. I'm in year 12 though.. so I'm on the normal A level at the moment. How are you finding it so far? Are you in year 12 or 13?
Heya. That's cool, i'm in year 13. So which modules are you working on right now / doing over the rest of this year?
I'm not finding it too difficult so far. Did C1-C3 + M1, M2 and S1 last year and it was all fairly staright-forward. M2 was easily the most difficult of the lot for me, but I was pleasantly surprised by my grade when it came through My worst mark came in S1 simply cos it's far too boring to study!
Are there many other ppl taking it at your school?
just wondering whether there's any other ppl on here taking the further maths A-level too? I'm self-teaching it cos there's not a proper class at my school so it's a bit of a lonely experience! Would be nice to get in touch with others who're doing it
Look forward to hearing!
Ah okay, cool. Just ran through that on paper and I get those now, thanks! And learnt a few little tricks there which i'll try to remember.
Hopefully i'll be able to have a better crack at some of the others now. I think the biggest problem I have is not knowing whether to try integration by parts or by substitution, and if so, what substitution to use. Hopefully it'll become easier the more questions I do.
Once again, thanks.
i'm getting frustrated by a lot of the integration material in my latest maths book. I feel like I understand all the topics but there are some questions which just throw me completely. I'll just list two here:
1. Show that
2. Given that
show that, for n ≥ 2,
The second question obviously involves integration by reduction, but I can't for the life of me see how to work either of them out.
Any help appreciated!
Best when you have fractional surds to get rid of the surd in the denominator. You can do this in this case by multiplying top and bottom by root 3. So we have
Answer = (1/3)√8√3 + √24 + √294
= (1/3)√4√6 + √4√6 + √49√6
= (2/3)√6 + 9√6
...which is what you have, and that's as simplified as it gets. The key thing here is to spot the common surd and collect up all the terms. Your method seems a bit too long-winded and unecessarily fiddly. You did have the right answer though, which is the important thing, just try not to over-complicate it all.
Which exam board will you be studying for? It's usually either Edexcel, AQA or OCR?
I'm doing maths/further maths with Edexcel and I find the Heinemann books, which are designed specifically for Edexcel's syllabus, to be quite good. C1 is a piece of cake, my dog could pass that. Generally i've found all the core modules to be easy, so you should have no trouble self-teaching those. Mechanics 1 was relatively easy too but it gets a lot more difficult in the later modules. I'm doing M3 at the moment and it's awful!
So yeah, if you're doing Edexcel i'd get the Heinemann books. Also, one book which i've found to be useful for all syllabi is "Further Pure Mathematics", by Brian and Mark Gaulter. That covers all the pure further maths modules, but will probably not be of use to you just yet. Might be worth getting later on though, depending on how things go.
The constants a^2 and 2 can be pulled out of the integral to the outside.
Then I used "The Wolfram Integrator" online to obtain this answer.
Sort of cheating I know, but it's amazing.
The answer is in the back of the book, or I can just use my calculator to get it Unfortunately I won't have that luxury in an exam lol
Edit: Never mind, apparently that rule won't work if they are the same function.
Yeah, I don't think the partial fractions thing will work.
This one's confusing because it comes amidst some relatively easy questions, so I wasn't sure if I was missing a simple trick.
Hmm, it's gonna bug me, but hopefully they won't ask anything this tough in an exam!
what's the simplest way of finding the value of the folllowing definite integral, in terms of a (where a is a positive constant)?
I've tried expanding and rearranging the trig part into a form that I know how to integrate, but without success
Hi there, i'm having a bit of trouble with this question:
Fluid flows out of a cylindrical tank with constant cross section. At time t minutes, t>0, the volume of fluid remaining in the tank is V m³. The rate at which the fluid flows in m³/min is proportional to the square root of V.
Show that the depth h metres of fluid in the tank satisfies the differential equation
dh/dt = -k√h , where k is a positive constant.
Here's what i've done so far:
Let the radius of the base of the cylinder be r metres. This gives
dV/dt = -k√V
V = pi*r²h
dh/dV = 1/(pi*r^2)
dh/dt = dh/dV * dV/dt = - k√(h/(pi*r²))
Sorry if all this looks a bit messy, i'm not very good at typing equations on here, hopefully it's legible though.
Anyway, my book says the answer is simply dh/dt = -k√h . I can't for the life of me see what i'm doing wrong though.
Any help appreciated!