Solutions to Cambridge 1st Year NST Maths paper 1A 2019

Hi every one I have finally finished my first set of solutions to a Cambridge mathematics paper the paper in question was the first paper of 2019 which was given to the science students. The solutions are given here and I hope you enjoy reading through them

https://drive.google.com/file/d/1vLNuqpxyomuiHsR8nYJ1P-AGPG-XTSTh/view?usp=sharing

Some general comments 

On the whole I thought this paper was quite challenging certainly because the questions were a lot more tricky and involved than say a typical OU paper such as MS224 (although this years paper by all accounts seems to have been really difficult as the OU are still experimenting with the format for remote exams. My sympathies to those who found it difficult and when it eventually comes out I will have a look at it myself and make comments on it) 

So what was in the paper the first set of questions was a set of 10 which were essentially A level topics and on the whole they were relatively straightforward. Although you do have to have integration techniques and trig identities at the tip of your fingers. There is really no time for much thought and one question involved a tricky integration by parts. Something which this paper seems to love testing people on I think there must have been about 6 or 7 times that I had to use this technique. As a result my grasp of this technique improved a lot and I can almost just write down the answer now whereas before I would struggle. Candidates have to answer all of these questions and a bit like multiple guess questions in the MS224 papers no marks are given for showing the working

The next section is a set of 10 questions from which the candidates can choose up to 5 the last two questions being reserved for those who took a more advanced option. The elite of the elite as it were it must be a bit disappointing to be told right at the outset of your undergraduate career that you are not part of this special group. I can see many potential traitors to our country arising from this slight 😅   Anyway the system is what it is and for us working through these papers we don't have to worry about such distinctions. Unfortunately there was one part of this section which I just could not see what the question was asking and hence I just have to accept that I will remain a pleb for the rest of my life 😅 I digress 

Question 11 had four parts on complex variables. I really enjoyed the first part which involved using De Moivre's theorem to reduce a complex trigonometric identity to a product of powers of cos a and sin b. In the course of this I found a quick way of writing the expansion of cos(nx) or sin(nx) in terms of powers of cos(x) or sin(x) using Pascal's triangle. I have described this in detail in an earlier post and now feel I can tackle this sort of problem with ease. The other parts were straightforward enough but one involved a set of trigonometric identities for the  Tan Function something I am not sure I would remember in an exam/ The final part was a bit difficult to decode what the question was asking but it involved the mapping between two complex variables and I got it out eventually. 

Question 12 was a long tedious question on volume integrals with tricky boundary conditions, It went ok but it is not the sort of question I enjoy and it took me two or three attempts to arrive at the correct answer.

Question 13  was a question on first order differential equations and exact differentials. I generally prefer  second order differential equations but  that will have to wait till the next paper Two parts involved using the integrating factor method and one involved a not so obvious substitution in order to get the final integral out. The last part involved a lot of partial differentiation to show that a function obeyed Laplace's equation in spherical polar coordinates. So just like question 12 above this would have been tedious and I doubt whether I would have got it out in time 

Question 14 was divided into two parts the first part asked you to calculate the variation in the relativistic formula for energy in terms of the variations for mass and energy and then given the velocities of two particles work out the relative mass of these particles wrt to each other. This was OK if a bit novel to me and it was difficult to tell whether I had got the signs correct. The second part of question 14 involved a really tedious exercise in partial differentiation of functions of functions. Whilst in principle this type of question is relatively straightforward in practice ir can get extremely tedious so best avoided in an exam and I found myself losing the will to live

Question 15 was a question on Taylor series and expansions. Again relatively straightforward and tedious you were allowed to use known Taylor expansions but could anyone off the top of their head remember if they ever knew the Taylor expansion for sech(x). I had to derive this myself. 

Question 16 was an unusual one about probability involving the various probabilities of drawing a blue ball from a  bag and then deciding to return it or not with a certain probability and then drawing another ball. Why anyone would think this is something useful to do is beyond me. It certainly wouldn't help you win the lottery😅   I hadn't really done much of this type of question before so it was useful to learn the various formulae for conditional probability and how to draw a probability tree using MATLAB. So whilst the actual activity described in the question is quite pointless, it was quite instructive to work out the various probabilities involved again this would have been really difficult to complete in the time given

Question 17 was a whole bunch of questions on integration some of which seemed impossible at first sight but you just have to persist and I strengthened my integration skills a lot by doing this question

Question 18 was a long boring question on matrices the first part solving a matrix equation for 2x2 matrices. The second part at first sight was quite intimidating as it involved a long introduction to set the scene for testing a hypothesis about the size of worm lengths. It turned out that in it's long winded way the question was taking the examinee through the method of least squares fit. So it turned out to be relatively interesting but it really is not the sort of question that any normal human being can just rattle off under exam conditions

Now we come to the questions for the elite. 

Question 19 This was split in two parts the first question being a question on the convergence or not of various series. This was quite straightforward if you are up to speed with regards to the techniques but this is really the sort of question that belongs in a pure paper not a mathematical methods one. I suppose the idea is to get their students to think a bit about real analysis. But they are better off doing a full blown course on it rather than spending just a couple of lectures or so. I would recommend the OU course M208 as giving a far better understanding of real analysis than this sort of rapid overview.

The second part was my nemesis for this paper it involved relating the Newton Rhapson method to a Taylor expansion and estimating the errors with each iteration. I was unable to understand what was going on. However given that the function was a quadratic equation for which the roots are known exactly I was able to provide an estimate of the error at each stage but the resulting expressions looked nothing like the expressions in the paper. If anyone knows the answer to this question I would be grateful. I can't see the point of applying the Newton Rhapson method to a quadratic equation anyway 

Question 20 This was the final question Yippee and was on Integration the first part was showing how the area of a parabola could be approximated by a series of rectangles under the parabola and showing how in the limit the exact expression could be obtained. I got this question out but again it involved a lot of tricky algebra the thing to do here  as Corporal Jones from Dad's Army would say is 'DON'T PANIC' and just see it through even if at first sight your answer appears to be going nowhere.

The last two parts of the question involved something which I hadn't really come across before namely differentiating functions under an integral sign involving variable limits. Apparently Leibniz derived a formula for this many years ago (what a clever fella that man was 😅 ) and once you know this formula the question was quite straightforward. The final part involved yet another tricky substitution making use of the obvious fact  😅that sin(x) = tan(x)/sec(x)  and the derivatives of both sec(x) and tan(x). 

So what would it take to do really well at this exam essentially a real fluency in techniques of integration, trigonometric identities, hyperbolic functions, their derivatives and identities, Taylor series, differentiation using the product and quotient rules, Being able to partially differentiate functions of functions using the chain rule very quickly and a good deal of persistence. Not for the faint hearted but an exercise well worth doing. I certainly got a lot from this exercise even though I am not particularly fluent at the techniques. I guess like good musicians you have to just practice practice until the techniques just become routine. I look forward to doing the second paper in this series which will cover such topics as vector calculus surface and line integrals, and second order differential equations (yippee my favourite technique at this level) I am not very confident with things like the divergence theorem and surface and line integrals I hope by the time I have finished the second paper I will become more so and no doubt I will learn a whole load of new techniques as well I encourage any one reading this blog to have a go themselves. The last paper took me about 6 months to complete and write up I hope the second one wont take so long