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Feedback on 2013 exam

q.1 Linear algebra - mean mark 62.6%

(a) - Gaussian elimination: Mostly well done. As promised, 2 of the marks were for continuing the elimination until the square on the left was in the form of an identity matrix.

(b) - Echelon form: Most of you did OK on part (i), but remember that the when the question asks for the set of solutions, it means you have to describe all the possibilities. "Freely chosen" means some one else can freely choose, not you.

I was surprised how few of you realised that part (ii) had no solutions. Since the possible behaviours include infinitely many solutions and no solutions, it stands to reason that the two parts would be for the different behaviours.

(c) - Inverse matrices: I thought I had written this question to make it easy for you, by giving you over half the answer (two of the columns in the inverse of A). As you can see from the solutions, the intended answer is quite short, but an awful lot of you ignored that and tried to calculate the inverse from scratch. That approach is time-consuming and error-prone, and very few of you succeeded with it. Although the average mark for question 1 was good, I suspect that doing part (c) the hard way left many of you with less time for the other questions.

In general, do watch out for what exam questions are telling you. If they provide more information than you seem to be using, ask yourself if it might be helpful.

It's also smart to think of examiner psychology. I want to test as much as I can in the 90 minutes, and I don't want to waste time testing the same thing twice. You'd done Gaussian elimination in part (a), so doing an even bigger one in part (c) was maybe not what I wanted.

q.2 Analytic geometry - mean mark 46.0%

Most of you gave correct answers for (a), (b) and - to a lesser extent - (c). If you didn't, it suggests you don't understand the basics of lines and planes, and parametric and equational form.

(d) was a little harder. The answer I wrote for the solutions was a geometric one, using the formula for a normal to two vectors. Most of you who attempted this part took a different approach, looking for the set of all solutions to the two equational forms of the planes. You need to have a good feel for what kind of answer you're looking for: a line, with one parameter, hence one freely chosen variable in the set of solutions. It's also easy to make mistakes doing this, so look for simple checks. Are you sure the line you get is really in both planes?

Since part (d) depended on previous parts, I gave full marks if you did the right calculations with wrong answers to previous parts.

Part (e) was intended to help you by getting you to do a simple check on part (d). It didn't work that way. I don't think anyone spotted a mistake because the check failed. I gave a mark for finding the point, a mark for doing the checks correctly, and a mark if they passed.

(f) was a simple application of a formula and most of you did it OK. However, do be careful with your arithmetic.

(g) was misphrased. I meant to ask you to say which of Q₁ and Q₂, and for that you would have to do the calculation for P too. As stated, you just had to show Q₁ and Q₂ were on different sides, so I gave full marks for that.

Most of you used the formula for the signed distance of a point from a plane. One or two of you couldn't remember the formula and were disappointed that I didn't give it in the question. However, you don't need it, as you see in the solutions.

q.3 Set Theory - mean mark 55.8%

(a): Most of you got parts (i) - (iii), but relatively few got (iv). It's not especially deep, it's just there's some working out to do on subsets.

(b): Again most of you got this, but a lot of you miscounted in (i). Also, remember that the cardinality of an infinite set includes whether it is countable or not.

(c): Again, lots of reasonable answers.

• Remember that "irreflexive" and "antisymmetric" are not simply the negations of "reflexive" and "symmetric".
• Be careful when determining transitivity. I expected good justifications for your answers.

(d): A lot of you gave up here. Although the question looks quite complicated, it's fairly natural if you think about how you might encode exceptions as a mathematical function - invent a new "error" result (the dagger).

Also there was some confusion over single-valuedness, definedness, injectivity and surjectivity. There's no substitute for knowing what they mean.

q.4 Probability - mean mark 55.0%

(a) (i)-(iii): most of you did well on these.

(a) (iv): most of you did the expectation OK, but not the variance. You probably didn't remember the formula.

(a) (v): Lots of good answers, but some of you failed to get your heads round the fact that it was about two rounds of the game.

(a) (vi): Hardly any correct answers for this question on conditional probabilities.

(b) (Poisson): Most of you did fine on part (i), but not on (ii). There were two big issues there. First, it asked about 2-minute intervals and so λ changes from 0.7 to 1.4. Second, you have to calculate e^λ for the new λ. e^1.4 is the square of e^0.7, so approximately 0.25.