Cambridge NST Maths year 1 2019 Paper 2 solutions

 Here is the second in a sequence of my solutions to the Cambridge NST maths papers, This one is for the second of the papers set for the first year students in 2019 the last year before COVID. You can access it here 

https://drive.google.com/file/d/1SSePjGdcGjWatVFNHh5kVzYORGgaBUrD/view?usp=sharing

I have previously published my solutions to the first of the papers and for convenience I post the link again here for those who missed it first time around

https://drive.google.com/file/d/1SSePjGdcGjWatVFNHh5kVzYORGgaBUrD/view?usp=sharing

So there you have it a complete set of answers to the first year papers for NST students at Cambridge for 2019. Most of them will have sat there finals last summer and I hope they did well. 

I apologise in advance for any typos spelling errors etc. A review of the paper follows 

Again just like the first paper this was quite challenging and unfortunately I was unable to answer a question on Parseval's theorem properly. On the whole though I think I got the questions out, but I doubt if I could do well under exam conditions. 

Anyway just like the first paper there were 10 short questions which were relatively straightforward once you decoded what the examiner was getting at. The questions included solving a first order differential equation by the integrating factor method. Deriving a recurrence relation for an Integral (something they love testing people on). Calculating the stationary values of a function f(x,y). A slightly confusing question on pronability I really need more practice at questions involviing conditional probability. A volume integral and a surface integral. All this should take no longer than 30 mins but I suspect I would take a lot longer. Again there is no time to think and in the rush you would probably end up making silly mistakes,

The core of the paper is 10 questions of which answers to five must be submitted. The last two questions are reserved for those students deemed clever enough to do some advanced topics although I didn't think they were particularly difficult. 

Anyway here are the questons 

Question 11 was a geometric one involving the equation of a plane and finding the volume of the parallipped enclosed by 3 vectors. This was relatively straightforward once I had reminded myself of the vector equation of a plane. But there were quite a few parts. Although the question asked for a few diagrams I had the luxury of using MATLAB to draw the relevant pictures/ Not very exciting I must admit

Question 12 . Involved fiinding the stationary points of a function in f(x,y) and drawing a contour plot showing the function and the gradient. This was tedious but relatively straightforward and again I was able to use MATLAB to draw some pretty pictures. 

Question 13 Involved calculating the line integral of a vector function F for various paths and also finding a function satisfying curl F = 0. (a conservative function). I hadn't done a question like this for ages so I had to remind myself of how you go about calculating such things but it was relatively straightforward although finding the conservative function involved a little guesswork so not very satisfactory.

Question 14  Involved some questions on probability density functions. and evaluating their products and change of  variables. It was ok but not a very exciting topic. 

Question 15  This was a set of questions on solving second order differential equations with constant coefficients my favourite topic at this level. The first question was a homogeneous equation with boundary conditions and relatively straightforward to solve. The second part was an inhomogenous equation and whilst finding the complementary function was relatively straightforward. In order to find the particular integral you had use a function of the form  x^n f(x) and increase n until you found one that worked. This took a couple of goes and so would have been quite time consuming under exam conditions how nice of them 😊. The last part involved solving two differential equations simultaneously some what surprisingly they don't seem to teach how to solve such systems using matrices and their eigenvectors unlike the open university courses MST210 or MST224 so this is one occasion wihere the open university is better than Cambridge. Anyway compared to the tedium of the last few questions this was a delight to do. 

Question 16   This question was all about calculating various surface integrals and the flux of a field through a surface. It got a bit fiddly but again was relatively straight forward. Another boring topic though I much prefer solving differential equations 

Question 17 This was a boring question on matrices again pretty straighrforward but you have to know the definitions and again there were so many parts to the question. Give me calculus questions any day

Question 18  This was a question on Fourier series you had to find the Fourier series for cosh(x) then differentiate to get the Fourier series for sinh(x). For this topic you really need to be on top of integration by parts. The last part of the question then asked you to use Parsevals theorem to show that the integral of (cosh(x)-sinh(x))^2 over the interval was sinh(2) . I tried this a couple of times but I couldn't get the expansions to cancel out to leave sinh(2) so unfortunately I was unable to complete the paper properly. However if you integrate the function directly it comes out relatively straightforwardly. 

So the two questions for the so called advanced students were as follows 

Question 19 was on Lagrangian multipliers and you had to find the optimum volume of a cylinder the optimum volume of a cone inscrbed in a sphere and then prove that the Arithmetic mean is >= to the geometric mean. I confess to nor really understanding this topic although I can go through the motions and I find it difficult to tell whether I am finding a minimum or a maximum. For the cone inscribed inside a sphere I found it easier to just finding the maximum volume directly. I'll let you solve this question using Lagrangian Multipliers fot your self. 

Question 20  A relatively straightforward question on solving partial differential equations using separation of variables. Two first order ones and a question on the diffusion equaton. This is really a warm up for what comes next year so you aren't asked to solve the differential equations in spherical or cylindrical coordinate systems or use Lagrange Polynomials or Bessel functions or any of the other exotic functions out there. So a bit boring really 

Overall conclusion is that this was an exercise worth doing to remind myself and extend my mathematical knowledge a bit. I think on the whole I preferred the first paper as it seemed to cover slightly more interesting topics. Apart from the two quesitions on differential equations this paper could be described as worthy but dull. 

The second year papers for this year beckon next and I hope that I find them a bit more interesting than this one. Hopefully I can finish them by June next year

I would urge you to have a go for yourself and I hope you find these solutions useful