Oxford MAT Prep: Multiple Choice Questions

1. Oxford MAT Prep: Multiple Choice Questions Dr J Frost (jfrost@tiffin.kingston.sch.uk)

2. Index Click to go to the corresponding section. Comparing Values Sequences Trapezium Rule Number Theory Area/Perimeter Remainder Theorem Logarithms Circles Graph Sketching Calculus Reasoning about Solutions Trigonometry

3. General Points The Oxford MAT paper is the admissions test used for applicants applying to Oxford for Mathematics and/or Computer Science, or to Mathematics at Imperial. It consists of two sections. The first is multiple choice, consisting of 10 questions each worth 4% each (for a total of 40%). The second consists of 4 longer questions, each worth 15% (for a total of 60%). We deal with the first section here. The paper is non-calculator. You need roughly 50% to be invited for interviews. However, successful maths applicants have an average of around 75%. The questions only test knowledge from C1 and C2. You must ensure you know the content of these two modules inside out. You should also keep in mind that the MAT won’t test you on theory you wouldn’t have covered, so should think in the context of what you can do. The multiple choice questions become progressively easier (and quicker) the more you practice. So practice these papers regularly. Even redoing a paper you’ve done before has value. I’ve grouped some of the questions from these papers here by topic, to help you spot some of the common strategies you can use. You can find a huge collection of past papers (many with mark schemes) here: http://www.mathshelper.co.uk/oxb.htm

4. Comparing Values Preliminary Tips • You often have to compare logs. If one gives you a rational number and the other not, then form an inequality and rearrange to see if one is bigger than the other. E.g.:Which is bigger: or ?Suppose . Taking 5 to the power of each side:Since this is true, we must have that . • Make approximations where appropriate. E.g. is roughly 2. • Similarly, form inequalities by considering approximations. E.g. is approximately 2, but slightly less than 2, i.e. we know • Remember that • Remember that when .

5. Comparing values  A  B  C  D

6. Key Points Make sure you know your sin/cos/tan of 30/45/60 degrees. Be comfortable with making estimates. is somewhere between 1.5 and 2. Similarly is just less than 4. Know your laws of logs like the back of your hand!

7. Comparing values  A  B  C  D

8. Key Points Changing bases occasionally allows us to evaluate less obvious logs. For example We can see that Forming inequalities and manipulating them often helps us compare logs. If we’re trying to find which of and are bigger, then:If ,  (by taking 5 to the power of each side) , so we were right.We can use the same technique to show that is bigger.

9. Comparing values  A  B  C  D

10. Key Points Again, realise that is less than 1 when is larger than . The root inside outside and outside the log function has different effects. In (d), we can put on front of the log instead. With (b), we can’t do this. Realise that squaring a number between 0 and 1 makes it smaller. Alternatively: is approximately , because . We can use this value to approximate (b), (c) and (d).

11. Comparing values  A  B  C  D

12. Sequences Preliminary Tips • Know your formulae for the following and be able to apply them quickly: • Sum of an arithmetic series: • Sum of a geometric series: • Infinite sum of a geometric series: • Two series might be interleaved, e.g. Find the sum of each in turn. • In some MAT questions (as well as Senior Maths Challenge) it sometimes helps to consider the ‘running total’ when you have a series that oscillates between negative and positive values. You’ll see an example of this.

13. Sequences  A  B  C  D 2011

14. Key Points With sequences, remember that for the (n+1)th term, we can just replace n with n+1. So just substitute these expressions into the inequality. We can sometimes exploit the fact the questions are multiple choice. Once we simplify to , subbing the four different values in tells us (a) must be the answer. Although of course, we could use a ‘C1’ approach to solve a quadratic inequality.

15. Sequences  A  B  C  D 2010

16. Key Points If you have two interweaved sequences, find formulae for them separately. This means we just want the sum of the first n terms from each of the two. Know your formulae for the sum/infinite sum of a geometric series like the back of your hand.

17. Sequences  A  B  C  D 2009

18. Key Points Sometimes it helps to think about the ‘running total’ as we progress along the sum. Our cumulative totals here are 1, -1, 2, -2, 3, -3, … Alternatively, try to spot when you can pair off terms such that things either cancel or become the same. In this case, 1-2 = -1, 3-4=-1, and so on. Although this makes it harder to spot exactly when we hit 100 in this case. Think carefully about what happens at the end. Looking at the running totals, if the 1st is 1, the 3rd is 2, the 5th 3, then the (2n-1)th gives us n. So when our running total was 100, 2n-1 = 199.

19. Trapezium Rule Preliminary Tips • The Trapezium Rule is:where w is the width of each strip. • Know when the trapezium rule overestimates or underestimates area. Overestimates when line curves upwards. Underestimates when line curves downwards.

20. Trapezium Rule  A  B  C  D 2010

21. Key Points Remember that that the trapezium rule uses equal length intervals. This suggests that the boundaries of the strips have to coincide with the dots in the diagram, otherwise our trapeziums wouldn’t exactly match the function. (a) we can eliminate because multiples of won’t include a . We can eliminate (b) and (c) in a similar way. In general, it’ll be the LCM of the denominators.

22. Trapezium Rule  A  B  C  D 2009

23. Key Points This is all about being proficient with the sum of a geometric series. The question is more a test of your dexterity, not your problem solving skills. By the trapezium rule, we get . So using and and , for the summation in the middle we get . Simplifying, we end up with (b). Occasionally we can absorb terms into the geometric series to make our life easier. Since the first length inside the outer brackets was , it makes much more sense to make and . Be careful in considering the number of points you use in the trapezium rule. If you have regions, then you actually have points, of which are duplicated.

24. Trapezium Rule  A  B  C  D 2008

25. Key Points Thinking about the question visually helps. A function which curves upwards will give an overestimate, and a function which curves downwards gives an underestimate. (d) is the only transform which changes the shape of the curve, giving us a reflection on the y-axis (in the line y=1). A curve for example curving up will now curve down, giving us an underestimate.

26. Number Theory Preliminary Tips • For many problems it helps to find the prime factorisation of a number: • A number is square if and only if the powers in its prime factorisation are even. • Similarly, a number is a cube if and only if its powers are divisible by 3, and son on. • We can get the number of factors a number has by adding 1 to each power. E.g. , so it has factors. • Diophantine Equations are equations where you’re trying to identify integer solutions. There will never be anything too difficult on this front relative to the Maths Olympiad for example, but it may be worth seeing the RZC Number Theory slides on this topic. • Know your divisibility rules (also on the RZC Number Theory slides): A number is divisible by 3 if its digits add up to a multiple of 3, divisible by 4 if its last two digits are divisible by 4, divisible by 6 if it’s both divisible by 2 and 3, and so on.

27. Number Theory  A  B  C  D 2008

28. Key Points These kinds of questions are quite common in Maths Challenge papers. The key is to systematically consider how many times each digit appears for the units digit, and then the tens digit (rather than considering each full number in turn). For the units digit, 1 to 9 is each seen 10 times (0 is seen 9 times, but this doesn’t matter because it doesn’t contribute to the sum). And . Each tens digit, 1 to 9, occurs 10 times. This again gives 450, so our total is 900.

29. Number Theory  A  B  C  D 2008

30. Key Points From the RZC lecture notes, remember that a number is square if all the powers in the prime factorisation are even. The prime factorisation is . Be adept with recognising odd/evenness when considering sums and products. If is even, then consider which of the four statements supports this.E.g. If is even, is even (because ), and since , is even.

31. Number Theory  A  B  C  D 2009

32. Key Points This clearly looks like a binomial expansion! As per the RZC algebra slides, you should always try to spot potential factorisations in number theory problems.Factorising gives Then so . Since and must be positive (which does not include 0), can have any value between and so that remains at least 1.

33. Number Theory  A  B  C  D 2007

34. Number Theory  A  B  C  D 2010

35. Area/Perimeter Preliminary Tips • Remember the ‘’ trick: The diagonal of square is times longer than the side. Similarly, the side is times shorter than the diagonal. • For circles, add key radii at strategic places. • Split up the shape into manageable chunks (e.g. right-angled triangles). You’re likely to be able to use either simple trigonometry or Pythagoras. • You MUST memorise sin 30/cos 30/sin 60/cos 60. Remember also that and , which helps if you’re trying to work out without a calculator. • Don’t forget your circle theorems (although they don’t really feature very prominently in MAT). • Sometimes you can come up with two expressions for the same length. Example: The radius of the big circle is 1. What is the radius of the small circle? r 1 1 ? So

36. Area/Perimeter  A  B  C  D 2012

37. Key Points • As per usual, draw in the radius of the circle at strategic places (in this case, where the triangle touches the circle). • This allows us to divide up the triangle into manageable chunks. • The area of one third of the triangle is • By splitting each isosceles triangle into two right angle ones, then each half of the side of the triangle is . There’s 6 of these half lengths. • Using these A and P, we find (a) is true.

38. Area/Perimeter  A  B  C  D 2012

39. Key Points I initially tried to think how the orientation of the ‘spotlight’ affected the area covered. I realised that pointing it symmetrically at the opposite end maximised the area. My approach was to draw radii from Q and R to the centre (where angle QOR = using circle theorems) and P to the centre. This gave me a sector and two triangles, and a bit of simple geometry led me to (b). However, we could again exploit the fact we have multiple choice to try a specific case. If we were to choose , we have a semicircle and a right-angled triangle, which quickly gives us . This gives us (b).

40. Area/Perimeter  A  B  C  D 2011

41. Key Points A sensible first step is to form equations for the perimeter and area. Say and . Often we can introduce an inequality when equations are involved by using the discriminant. Using substitution: . Then using the discriminant: Alternatively, we could have exploited the fact that the question is multiple choice. Choosing a few possible widths and heights will eliminate the incorrect possibilities.

42. Area/Perimeter  A  B  C  D 2006

43. Key Points If the sides of the triangle are , and , start with by Pythagoras. We could always find the areas of the triangles, but it’s quicker to realise that the area of each triangle is proportional to the area of each implied square around the triangle, say , and . Then , so .

44. Remainder Theorem Preliminary Tips • Nothing much to say here! • The remainder when is divided by is . If is a factor then clearly because there’s no remainder. • You might have to factorise the factor first! E.g. If is divisible by , then and .

45. Remainder Theorem  A  B  C  D 2008

46. Remainder Theorem  A  B  C  D 2006

47. Remainder Theorem  A  B  C  D 2009

48. Key Points • If is a factor then both and is a factor. • By the remainder theorem, and . For the latter, we get or . • However, for the former, the middle term will be positive or negative depending on whether is odd or even. can only be 10 or -15. • is the only case when both are satisfied.

49. Logarithms Preliminary Tips • If you see numbers like 2 and 8 together for example, you should be able to spot that and somehow use that to simplify. The ‘related groups’ you’ll see are usually powers of 2: and powers of 3: . • You need to know how to change the base: • and are the inverse of each other. This means for example that:

50. Logarithms  A  B  C  D 2011