MAT: Section B Tips

1. MAT: Section B Tips Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 2nd November 2013

2. What questions might I expect? Here’s a very coarse breakdown of topics on the last several years of MAT papers (although note that some questions combine different topics) In summary: You can’t predict Q5. They like sequences/functions questions. They love questions combining geometry/coordinate geometry and reasoning about graphs.

3. General Tips based on based on Actively reflect on how a part of a question might refer to a previous part. If they get you to prove something, there’s a reason for it!

4. General Tips On the rare occasion they give you some piece of theory possibly required to solve one of the questions. In this example, it’s not used until part (v). If you haven’t used the tip, then reflect on your answers! based on based on

5. Geometry Tips When asked to find the area of a more complex shape, obviously split it up. But there tends to be an easier way to do so. Whenever one of the edges is an arc, one of your sub-areas will be a sector. In this case, we can split this area up into a triangle and a sector. I HIGHLY recommend going through my Geometry slides – the content on angles and areas (you can possibly ignore the ‘Geometric Proofs’ – it’s more useful for Olympiads). http://www.drfrostmaths.com/resource.php?id=10650

6. Trigonometry Tips MAT QUESTIONS ARE ALWAYS IN RADIANS! So you’ll need to remember your “arc length = r” and “sector area = ½ r2” formulae. Other than that, remember that sin(180 – x) = sin(x), cos(360 – x) = cos(x) and sin(90 – x) = cos(x)

7. Geometry/Arithmetic Series Tips It really isn’t hard to remember the summation formulae for arithmetic and geometric series. SO DO IT. ARITHMETIC SERIES GEOMETRIC SERIES Sum to infinity of convergent geometric series.

8. Graph Tips When two lines TOUCH, both the y-value AND THE GRADIENT are the same. When they INTERSECT, only the y-values are the same. When they touch, equate gradients and y-values. Usually do the former first. Ensure you correctly read in the question where it says ‘intersect’ and where it says ‘touch’.

9. Graph Tips Often questions don’t require lots of algebraic manipulation, but just to ‘reflect graphically’. Here it’s clear they’ve just picked an arbitrarily low number for a. So you imagine the point A being shifted far left, and the effect it would have on the straight line. Since a is the x-intercept of the straight line, and b the x-value of the point of contact, we can ‘see’ we can maximise the area when a = b = -1.

10. Graph Tips The discriminant can often be important: “Find under what conditions the lines touch at two distinct points.” Equate the gradients to form an equation, then find when b2 – 4ac > 0.

11. Algebra Tips Spot when we have an identity rather than an equality. This allows us to compare coefficients. You’ve earlier shown that m = 3b2 – 1. All you need to do here is expand out the RHS, then compare coefficients of the x3 terms, etc.

12. Combinatorics I have a whole series of RZC slides on Combinatorics. But here are the bare bones of what you might need... Slot Filling Approach Coin-based questions When considering the number of possible values in a sequence, consider the number of possibilities in each character position, then multiply together. A classic problem is the number of ways of making up £1 using just 20p, 5p and 1p problems. The key is to first fix a number of 20ps (starting with zero of them), then consider how many possible quantities of 5ps there are (in this case 21) for this fixed number. You needn’t consider the number of 1ps because it just fills up the remaining amount. Then consider one 20p, and so on. You’ll end up with an arithmetic series, which you know how to sum. In one question, you were asked to find the number of possible 4-’digit’ sequences consisting of right and up movements. There’s 2 possibilities for each ‘slot’, so 24 possibilities overall. In another question involving calendar years, using a number in one ‘slot’ left less possibilities in the other slots, because you’d used up a number.

13. Limits On two occasions I’ve seen questions which ask you to approximate the value of an expression when variables become large. The key here is that constant values become inconsequential when combined with a growing variable. 20/40 is close to 21/39, and the error margin becomes smaller and x and y become larger. Remember also that 1/x obviously tends towards 0 as x becomes large. We can use this fact to make terms disappear in a limit.

14. Sequences/Functional Equations They really love these questions in Section B! A typical trick is to replace the xn+1 with the expressions involving xn and yn, then simplifying. Sometimes you can reapply your recurrence relationship to obtain larger values needed.

15. Sequences/Functional Equations We’re comparing parameters from the current and next term in the sequence. Just writing xk+1 in two different ways here (in terms of both Ak and Ak+1 for example), and then comparing coefficients, will do the trick.

16. Q6 and Q7 These questions are for Computer Scientists and Maths & Computer Scientists only. Not much to say here unfortunately. Q6 is always a logic question. Sometimes it’s helpful to do a case analysis: If some person is telling the truth, what does that lead to conclude about the other people? Does this lead to a contradiction? Q7 tends to have an algorithmic flavour. It may be worth reading my separate RZC Computer Science slides. http://www.drfrostmaths.com/resource.php?id=11380 In particular, appreciate recurrence relationships, sometimes which involve two variables. This is covered pretty comprehensively in Section 3 of my Combinatorics slides (which incidentally, I highly recommend Computer Science applicants get to grips with) http://www.drfrostmaths.com/resource.php?id=10390