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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 multiple-choice 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 2-6 require extended responses. Each question is worth 15 marks.


Navigate between the questions using [ Next button image ] and [ Back button image ] 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.


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QUESTION 1 Answered

For ALL APPLICANTS.

A.   How many real solutions x are there to the equation x|x| + 1 = 3|x|?
  1. 0,
  2. 1,
  3. 2,
  4. 3,
  5. 4.
[Note that |x| is equal to x if x ≥ 0, and equal to − x otherwise.]



B.   One hundred circles all share the same centre, and they are named C1, C2, C3, and so on up to C100. For each whole number n between 1 and 99 inclusive, a tangent to circle Cn crosses circle Cn+1 at two points that are separated by a distance of 2. Given that C1 has radius 1, it follows that the radius of C100 is
  1. 1,
  2. 2,
  3. 10,
  4. 10,
  5. 100.


C.   The equation x2 − 4kx + y2 − 4y + 8= k3  − k is the equation of a circle
  1. for all real values of k.
  2. if and only if either −4 < k < −1 or k > 1.
  3. if and only if k > 1.
  4. if and only if k < −1.
  5. if and only if either −1 < k < 0 or k > 1.


D.   A sequence has a0 = 3, and then for n ≥ 1 the sequence satisfies an = 8(an-1)4. The value of a10 is
  1. 2(220)3
    ,
  2. 6(220)3
    ,
  3. 3(220)2
    ,
  4. 18(220)2
    ,
  5. 6(220)2
    .


E.   If the expression ( x + 1 + 1x )4 is fully expanded term-by-term and like terms are collected together, there is one term which is independent of x. The value of this term is
  1. 10,
  2. 14,
  3. 19,
  4. 51,
  5. 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
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G.   For all real n, it is the case that n4 + 1 = (n2 + 2n + 1)(n22n + 1). From this we may deduce that n4 + 4 is
  1. never a prime number for any positive whole number n.
  2. a prime number for exactly one positive whole number n.
  3. a prime number for exactly two positive whole numbers n.
  4. a prime number for exactly three positive whole numbers n.
  5. a prime number for exactly four positive whole numbers n.



H.   How many real solutions x are there to the following equation? log2(2x3 + 7x2 + 2x + 3) = 3log2(x + 1) + 1
  1. 0,
  2. 1,
  3. 2,
  4. 3,
  5. 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 p1. 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 p2. Which of the following is correct?
  1. p1 = 193512 and p2 = 63256
  2. p1 = 201512 and p2 = 55256
  3. p1 = 243512 and p2 = 13256
  4. p1 = 247512 and p2 = 9256
  5. p1 = 13 and p2 = 13


J.   The real numbers m and c are such that the equation x2 + (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?
  1. 0,
  2. 1,
  3. 2,
  4. 3,
  5. 4.

QUESTION 2 Answered

For ALL APPLICANTS.

i)  Suppose x, y, and z are whole numbers such that x2 − 19y2 = z. Show that for any such x, y and z, it is true that

(x2 + Ny2)2 − 19(2xy)2 = z2
where N is a particular whole number which you should determine.

ii)  Find z if x = 13 and y = 3. Hence find a pair of whole numbers (x, y) with x2 − 19y2 = 4 and with x > 2.

iii)  Hence find a pair of positive whole numbers (x,y) with x2 − 19y2 = 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 x2 −25y2 = 1 with x > 1.

v)  Find a pair of positive whole numbers (x, y) with x2 − 17y2 = 1 and with x > 1.

QUESTION 3 Answered

For ALL APPLICANTS.

i)  Sketch y = (x2 −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 x-axis or the y-axis.

ii)  Without calculating the integral explicitly, explain why there is no positive value of a such that a0(x2 − 1)n dx = 0 if n is even.

If n > 0 is odd we will write n = 2m − 1 and define am > 0 to be the positive real number that satisfies am0(x2 − 1)2m−1 dx = 0, if such a number exists.

iii)  Explain why such a number am exists for each whole number m ≥ 1

iv)  Find a1.

v)  Prove that 2 < a2 < 3

vi)  Without calculating further integrals, find the approximate value of am when m is a very large positive whole number. You may use without proof the fact that 20(x2 − 1)2m−1 dx < 0 for any sufficiently large whole number m.

QUESTION 4 Answered

For Oxford applicants in Mathematics / Mathematics & Statistics / Mathematics & Philosophy, OR those not applying to Oxford, ONLY.

i)  Sketch the graph of y = x
x4
for x ≥ 0, and find the coordinates of the turning point.


ii)  Describe in words how the graph of y = 4x + 1 − x − 1 for x ≥ −
14
is related to the graph that you sketched in part (i). Write down the coordinates of the turning point of this new graph.


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 x-axis 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

For ALL APPLICANTS.

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 ( a1 , a2 , a3 ), then if first Alice picks door 2, the arrangement after the host has swapped items could be either ( a2 , a1 , a3 ) or ( a1 , a3 , a2 ). 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 a2 , whereas in the latter she would get items a2 and a1. 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

For Oxford applicants in Computer Science / Mathematics & Computer Science / Computer Science & Philosophy ONLY.

This question is about influencer networks. An influencer network consists of n influencers denoted by circles, and arrows between them. Throughout this question, each influencer holds one of two opinions, represented by either a △ or a ◻ in the circle. We say that an influencer A follows influencer B if there is an arrow from B to A; this indicates that B has ability to influence A.

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    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.
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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.
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iii)  A partial network of influencers (without opinions for B1,...,B8 ) 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 B1, B2,...,B8 had opinion ◻ at the start. Justify your answer.
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iv)  You are given two influencer networks, N1 and N2, with disjoint sets of influencers shown below. Both are eventually stable. Suppose one of the influencers from network N2 follows the influencer X from the network N1. Is the resulting network guaranteed to be eventually stable? Justify your answer.
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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.

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