MATHEMATICS ADMISSIONS TEST (MAT)
Accessibility Settings
Font Size
Colour Overlay
Contrast
INSTRUCTIONS
Please read these instructions carefully as they include important information about your test and how to navigate the platform. Once you have started your test, you will be able to view these instructions again at any time via the [ ] at the top of the right side of the screen.
DO NOT START THE TEST UNTIL YOU ARE TOLD YOU MAY DO SO.
You have 2 hours 30 mins to complete the test. The timer at the top of the screen tells you how much time you have left. The timer will become yellow when there are five minutes left before the end of the test. Your response will be submitted automatically when the time runs out.
This paper contains 6 questions, and you should attempt all questions required for the course you are applying for.
 Q1 is for ALL APPLICANTS.
 Q2 is for ALL APPLICANTS.
 Q3 is for ALL APPLICANTS.
 Q4 is for Oxford applicants in Mathematics / Mathematics & Statistics / Mathematics & Philosophy, OR those not applying to Oxford, ONLY.
 Q5 is for ALL APPLICANTS.
 Q6 is for Oxford applicants in Computer Science / Mathematics & Computer Science / Computer Science & Philosophy ONLY.
Question 1 is a multiplechoice question with ten parts. Each part is worth 4 marks. Marks are given solely for correct answers, but any rough working should be shown in the space provided in the answer booklet.
Questions 26 require extended responses. Each question is worth 15 marks.
Navigate between the questions using [ ] and [ ] buttons. You can access the questions in any order. Answer the questions in the answer booklet provided to you. Complete your personal details in BLOCK CAPITALS in the answer booklet. Use a BLACK PEN.
Calculators are not permitted.
After you have finished answering a question in the answer booklet, click [ ] for the relevant question. Use [ ] to bookmark the questions you would like to review later. You can see all flagged questions by clicking on [ ] at the bottom of the screen. Click on each individual question's icon to go to the question.
Once you are happy with your answers, you can end the test by clicking on the [ ] at the bottom of the screen.
 To adjust the font size and/or to add colour overlays or high contrast, click on [ ] in the topright corner of the screen.
 To use the highlighter, click on [ ] so that it turns yellow.
 To add and save an annotation to highlighted text click on [ ].
 To show and hide annotation(s), click on [ ].
 To remove highlighting or an annotation, click on [ ] which automatically appears above.
Click [ ] below to begin the test.
Loading your test... ()
QUESTION 1 Answered
A. How many real solutions x are there to the equation xx + 1 = 3x?
 0,
 1,
 2,
 3,
 4.
B. One hundred circles all share the same centre, and they are named C_{1}, C_{2}, C_{3}, and so on up to C_{100}. For each whole number n between 1 and 99 inclusive, a tangent to circle C_{n} crosses circle C_{n+1} at two points that are separated by a distance of 2. Given that C_{1} has radius 1, it follows that the radius of C_{100} is
 1,
 2,
 √10,
 10,
 100.
C. The equation x^{2} − 4kx + y^{2} − 4y + 8= k^{3} − k is the equation of a circle
 for all real values of k.
 if and only if either −4 < k < −1 or k > 1.
 if and only if k > 1.
 if and only if k < −1.
 if and only if either −1 < k < 0 or k > 1.
D. A sequence has a_{0} = 3, and then for n ≥ 1 the sequence satisfies a_{n} = 8(a_{n1})^{4}. The value of a_{10} is

2^{(220)}3,

6^{(220)}3,

3^{(220)}2,

18^{(220)}2,

6^{(220)}2.
E. If the expression ( x + 1 + 1x )^{4} is fully expanded termbyterm and like terms are collected together, there is one term which is independent of x. The value of this term is
 10,
 14,
 19,
 51,
 81.
F. Given that sin(5θ) = 5sinθ  20(sinθ)^{3} + 16(sinθ)^{5} for all real θ, it follows that the value of sin(72^{◦}) is
G. For all real n, it is the case that n^{4} + 1 = (n^{2} + √2n + 1)(n^{2} − √2n + 1). From this we may deduce that n^{4} + 4 is
 never a prime number for any positive whole number n.
 a prime number for exactly one positive whole number n.
 a prime number for exactly two positive whole numbers n.
 a prime number for exactly three positive whole numbers n.
 a prime number for exactly four positive whole numbers n.
H. How many real solutions x are there to the following equation? log_{2}(2x^{3} + 7x^{2} + 2x + 3) = 3log_{2}(x + 1) + 1
 0,
 1,
 2,
 3,
 4.
I. Alice and Bob each toss five fair coins (each coin lands on either heads or tails, with equal probability and with each outcome independent of each other). Alice wins if strictly more of her coins land on heads than Bob’s coins do, and we call the probability of this event p_{1}. The game is a draw if the same number of coins land on heads for each of Alice and Bob, and we call the probability of this event p_{2}. Which of the following is correct?
 p_{1} = 193512 and p_{2} = 63256
 p_{1} = 201512 and p_{2} = 55256
 p_{1} = 243512 and p_{2} = 13256
 p_{1} = 247512 and p_{2} = 9256
 p_{1} = 13 and p_{2} = 13
J. The real numbers m and c are such that the equation x^{2} + (mx + c)^{2} = 1 has a repeated root x, and also the equation (x − 3)^{2} + (mx + c − 1)^{2} = 1 has a repeated root x (which is not necessarily the same value of x as the root of the first equation). How many possibilities are there for the line y = mx + c?
 0,
 1,
 2,
 3,
 4.
QUESTION 2 Answered
i) Suppose x, y, and z are whole numbers such that x^{2} − 19y^{2} = z. Show that for any such x, y and z, it is true that
ii) Find z if x = 13 and y = 3. Hence find a pair of whole numbers (x, y) with x^{2} − 19y^{2} = 4 and with x > 2.
iii) Hence find a pair of positive whole numbers (x,y) with x^{2} − 19y^{2} = 1 and with x > 1.
Is your solution the only such pair of positive whole numbers (x, y) ? Justify your answer.
iv) Prove that there are no whole number solutions (x, y) to x^{2} −25y^{2} = 1 with x > 1.
v) Find a pair of positive whole numbers (x, y) with x^{2} − 17y^{2} = 1 and with x > 1.
QUESTION 3 Answered
i) Sketch y = (x^{2} −1)^{n} for n = 2 and for n = 3 on the same axes, labelling any points that lie on both curves, or that lie on either the xaxis or the yaxis.
ii) Without calculating the integral explicitly, explain why there is no positive value of a such that ∫^{a}_{0}(x^{2} − 1)^{n} dx = 0 if n is even.
If n > 0 is odd we will write n = 2m − 1 and define a_{m} > 0 to be the positive real number that satisfies ∫^{am}_{0}(x^{2} − 1)^{2m−1} dx = 0, if such a number exists.
iii) Explain why such a number a_{m} exists for each whole number m ≥ 1
iv) Find a_{1}.
v) Prove that √2 < a_{2} < √3
vi) Without calculating further integrals, find the approximate value of a_{m} when m is a very large positive whole number. You may use without proof the fact that ∫^{√2}_{0}(x^{2} − 1)^{2m−1} dx < 0 for any sufficiently large whole number m.
QUESTION 4 Answered
i) Sketch the graph of y = √x −
ii) Describe in words how the graph of y = √4x + 1 − x − 1 for x ≥ −
Point A is at (−1, 0) and point B is at (1, 0). Curve C is defined to be all points P that satisfy the equation
 AP  ×  BP  = 1, that is; the distance from P to A, multiplied by the distance from P to B, is 1.
iii) Find all points that lie on both the xaxis and also on the curve C.
iv) Find an equation in the form y = f(x) for the part of the curve C in the region where x > 0 and y > 0. You should explicitly determine the function f(x)
v) Use part (ii) to determine the coordinates of any turning points of the curve C in the region where x > 0 and y > 0.
vi) Sketch the curve C, including any parts of the curve with x < 0 or y < 0 or both.
QUESTION 5 Answered
Alice is participating in a TV game show where n distinct items are placed behind n closed doors. The game proceeds as follows. Alice picks a door i which is opened and the item behind it is revealed. Then the door is shut again and the host secretly swaps the item behind door i with the item behind one of the neighbouring doors, i − 1 or i + 1. If Alice picks door 1, the host has to then swap the item with the one behind door 2; similarly, if Alice picks door n, the host has to swap the item with the one behind door n − 1. Alice then gets to pick any door again, and the process repeats for a certain fixed number of rounds. At the end of the game, Alice wins all the items that were revealed to her.
As a concrete example, suppose n = 3, and if the original items behind the three doors were ( a_{1} , a_{2} , a_{3} ), then if first Alice picks door 2, the arrangement after the host has swapped items could be either ( a_{2} , a_{1} , a_{3} ) or ( a_{1} , a_{3} , a_{2} ). So if Alice was allowed to pick twice, had she chosen door 2 followed by door 1, in the former case she would only get the item a_{2} , whereas in the latter she would get items a_{2} and a_{1}. Alice’s aim is to find a sequence of door choices that guarantee her winning a large number of items, no matter how the swaps were performed.
i) For n = 13, give an increasing sequence of length 7 of distinct doors that Alice can pick that guarantees she wins 7 items.
ii) For any n of the form 2k + 1, give a strategy to pick an increasing sequence of k + 1 distinct doors that Alice can use to guarantee that she wins k + 1 items. Briefly justify your answer.
iii) For n = 13, give a sequence of length 10 of doors that Alice can pick that guarantees she wins 10 items.
iv) For any n of the form 3k + 1, give a strategy to pick a sequence of 2k + 2 doors that Alice can use to guarantee that she wins 2k + 2 items. Briefly justify your answer.
v) (a) For n = 3, give a sequence of length 3 of doors that Alice can pick that guarantees she wins all 3 items.
(b) For n = 5, give a sequence of length 5 of doors that Alice can pick that guarantees she wins all 5 items.
vi) For n = 13, give a sequence of length 11 of doors that Alice can pick that guarantees she wins 11 items.
vii) For any n of the form 4k + 1, give a strategy to pick a sequence of 3k + 2 doors that Alice can use to guarantee that she wins 3k + 2 items. Briefly justify your answer.
viii) For n = 6, is there a sequence of any length of doors that Alice can choose that will guarantee that she wins all 6 items? Justify your answer.
QUESTION 6 Answered
The example in Figure 1 above shows a network with A and B following each other, B and C following each other, and C also following A. In this example, initially B and C have opinion ◻, while A has opinion △. An influencer will change their mind according to the strict majority rule, that is, they change their opinion if strictly more than half of the influencers they are following have an opinion different from theirs. Opinions in an influence network change in rounds. In each round, each influencer will look at the influencers they are following and simultaneously change their opinion at the end of the round according to the strict majority rule. In the above network, after one round, A changes their opinion because the only influencer they are following, B, has a differing opinion, and the network becomes as shown in Figure 2 above.
An influencer network with an initial set of opinions is stable if no influencer changes their opinion, and a network (with initial opinions) is eventually stable if after a finite number of rounds it becomes stable. The network in the above example is eventually stable as it becomes stable after one round.
i) A network of three influencers (without opinions) is shown below. Is this influencer network eventually stable regardless of the initial opinions of the influencers A, B and C ? Justify your answer.
ii) Another network of influencers (without opinions) is shown below. Is this influencer network eventually stable regardless of the initial opinions of the influencers? Justify your answer.
iii) A partial network of influencers (without opinions for B_{1},...,B_{8} ) is shown below. You can add at most six additional influencers, assign any opinion of your choice to the new influencers, and add any arrows to the network to describe follower relationships. Design a network that is eventually stable regardless of initial opinions, and has the property that when it becomes stable A has opinion ◻ if and only if each of B_{1}, B_{2},...,B_{8} had opinion ◻ at the start. Justify your answer.
iv) You are given two influencer networks, N_{1} and N_{2}, with disjoint sets of influencers shown below. Both are eventually stable. Suppose one of the influencers from network N_{2} follows the influencer X from the network N_{1}. Is the resulting network guaranteed to be eventually stable? Justify your answer.
v) (a) Given a network with n influencers, where the arrows are fixed, but you are allowed to assign opinions (△ or ◻) to each influencer, how many possible assignments of opinions is possible?
(b) Given an influencer network and an initial assignment of opinions, explain how you would determine whether the influencer network is eventually stable.
Justify your answer.
Review & Submit
Questions
2 / 3 Answered
Saving test details
Thank You!
Your test has been submitted.
Ready to submit?
Please note that you will not be able to return to this test once you submit it.
You have 2 hours to complete the test. The timer at the top of the screen tells you how much time you have left. The timer will become yellow when there are five minutes left before the end of the test. Your response will be submitted automatically when the time runs out.
This paper contains 6 questions, and you should attempt all questions required for the course you are applying for.
 Q1 is for ALL APPLICANTS.
 Q2 is for ALL APPLICANTS.
 Q3 is for ALL APPLICANTS.
 Q4 is for Oxford applicants in Mathematics / Mathematics & Statistics / Mathematics & Philosophy, OR those not applying to Oxford, ONLY.
 Q5 is for ALL APPLICANTS.
 Q6 is for Oxford applicants in Computer Science / Mathematics & Computer Science / Computer Science & Philosophy ONLY.
Question 1 is a multiplechoice question with ten parts. Each part is worth 4 marks. Marks are given solely for correct answers, but any rough working should be shown in the space provided in the answer booklet.
Questions 26 require extended responses. Each question is worth 15 marks.
Navigate between the questions using [ ] and [ ] buttons. You can access the questions in any order. Answer the questions in the answer booklet provided to you. Complete your personal details in BLOCK CAPITALS in the answer booklet. Use a BLACK PEN.
Calculators are not permitted.
After you have finished answering a question in the answer booklet, click [ ] for the relevant question. Use [ ] to bookmark the questions you would like to review later. You can see all flagged questions by clicking on [ ] at the bottom of the screen. Click on each individual question's icon to go to the question.
Once you are happy with your answers, you can end the test by clicking on the [ ] at the bottom of the screen.
 To adjust the font size and/or to add colour overlays or high contrast, click on [ ] in the topright corner of the screen.
 To use the highlighter, click on [ ] so that it turns yellow.
 To add and save an annotation to highlighted text click on [ ].
 To show and hide annotation(s), click on [ ].
 To remove highlighting or an annotation, click on [ ] which automatically appears above.
Click [ ] below to begin the test.