MATHEMATICS ADMISSIONS TEST

1 MATHEMATICS ADMISSIONS TEST For candidates applying for MATHEMATICS , Computer Science or one of their joint degrees at OXFORD UNIVERSITY and/or IMPERIAL COLLEGE LONDON and/or UNIVERSITY OF WARWICK November 2020 Time Allowed: 2 hours Please complete the following details in BLOCK CAPITALS. You must use a pen. Surname Other names Candidate Number M This paper contains 7 questions of which you should attempt 5. There are directions throughout the paper as to which questions are appropriate for your course. A: Oxford Applicants: if you are applying to Oxford for the degree course: MATHEMATICS or MATHEMATICS & Philosophy or MATHEMATICS & Statistics, you should attempt Questions 1,2,3,4,5. MATHEMATICS & Computer Science, you should attempt Questions 1,2,3,5,6. Computer Science or Computer Science & Philosophy, you should attempt 1,2,5,6,7. Directions under A take priority over any directions in B which are relevant to you.

2 B: Imperial or Warwick Applicants: if you are applying to the University of Warwick for MATHEMATICS BSc, Master of MATHEMATICS , or if you are applying to Imperial College for any of the MATHEMATICS courses: MATHEMATICS , MATHEMATICS (Pure MATHEMATICS ), MATHEMATICS with a Year Abroad, MATHEMATICS with Applied MATHEMATICS /Mathematical Physics, MATHEMATICS with Mathematical Computation, MATHEMATICS with Statistics, MATHEMATICS with Statistics for Finance, you should attempt Questions 1,2,3,4,5. Further credit cannot be obtained by attempting extra questions. Calculators are not permitted. Question 1 is a multiple choice question with ten parts. Marks are given solely for correct answers but any rough working should be shown in the space between parts. Answer Question 1 on the grid on Page 2. Each part is worth 4 marks. Answers to questions 2-7 should be written in the space provided, continuing on to the blank pages at the end of this booklet if necessary.

3 Each of Questions 2-7 is worth 15 marks. _____ FOR OFFICE Q1 Q2 Q3 Q4 Q5 Q6 Q7 USE ONLY each part of the question on pages 3-7 you will be givenfivepossible answers, justone of which is correct. Indicate for each partA-Jwhich answer (a), (b), (c), (d), or(e) you think is correct with a tick (!) in the corresponding column in the table show any rough working in the space provided between the parts.(a)(b)(c)(d)(e) square has centre (3,4) and one corner at (1,5). Another corner is at(a)(1,3),(b) (5,5),(c)(4,2),(d) (2,2),(e) (5,2). is the value of 10(ex x) (ex+x) dx?(a)3e2 26,(b)3e2+ 26,(c)2e2 36,(d)3e2 56,(e)e2+ sum1 4 + 9 16 + + 992 1002equals(a) 101(b) 1000(c) 1111(d) 4545(e) largest value achieved by 3 cos2x+ 2 sinx+ 1 equals(a)115,(b)133,(c)125,(d)149,(e) line is tangent to the parabolay=x2at the point (a,a2) wherea >0. The areaof the region bounded by the parabola, the tangent line, and thex-axis equals(a)a23,(b)2a23,(c)a312,(d)5a36,(e) of the following expressions is equal to log10(10 9 8 2 1)?

4 (a) 1 + 5 log102 + 4 log106,(b) 1 + 4 log102 + 2 log106 + log107,(c) 2 + 2 log102 + 4 log106 + log107,(d) 2 + 6 log102 + 4 log106 + log107,(e) 2 + 6 log102 + 4 cubic has equationy=x3+ax2+bx+cand has turning points at (1,2) and (3,d)for somed. What is the value ofd?(a) 4,(b) 2,(c)0,(d)2,(e) following five graphs are, in some order, plots ofy=f(x),y=g(x),y=h(x),y=dfdxandy=dgdx; that is, three unknown functions and the derivatives of the firsttwo of those functions. Which graph is a plot ofh(x)?(a)(b)(c) 22 2xy 22 55xy 22510xy(d)(e) 22 55xy 22 the range 90 < x <90 , how many values ofxare there for which the sum toinfinity1tanx+1tan2x+1tan3x+..equals tanx?(a) 0(b)1(c)2(d) 3(e) a square with side length 2 and centre (0,0), and a circle with radiusrandcentre (0,0). LetA(r) be the area of the region that is inside the circle but outside thesquare, and letB(r) be the area of the region that is inside the square but outside thecircle. Which of the following is a sketch ofA(r) +B(r)?

5 (a)(b)(c)121234r121234r1251015r(d)(e)122 468r122468rTurn functionsf(n) andg(n) are defined for positive integersnas follows:f(n) = 2n+ 1,g(n) = question is about the setSof positive integers that can be achieved by applying,in some order, a combination offs andgs to the number example asgfg(1) =gf(4) =g(9) = 36,andffgg(1) =ffg(4) =ff(16) =f(33) = 67,then both 36 and 67 are inS.(i) Write out the binary expansion of 100 (one hundred).[Recall that binary is base2. Every positive integerncan be uniquely written as asum of powers of2, where a given power of2can appear no more than once. So,for example,13 = 23+ 22+ 20and the binary expansion of13is1101.](ii) Show that 100 is inSby describing explicitly a combination offs andgs thatachieves 100.(iii) Show that 200 is not inS.(iv) Show that, ifnis inS,then there is only one combination of applyingfs andgs inorder to achieven.(So, for example, 67 can only be achieved by applyinggthengthenfthenfin that order.)

6 (v) Letukbe the number of elementsnofSthat lie in the range 2k6n <2k+ thatuk+2=uk+1+ukfork>0.(vi) Letskbe the number of elementsnofSthat lie in the range 16n <2k+ thatsk+2=sk+1+sk+ 1fork> over9 This page has been intentionally left blank10 Turn overIf you require additional space please use the pages at the end of the IN MATHEMATICSMATHEMATICS & STATISTICSMATHEMATICS & PHILOSOPHYMATHEMATICS & COMPUTER SCIENCE ScienceandComputer Science & Philosophyapplicants should turn topage is a sketch of the curveSwith equationy2 y=x3 x. The curve crosses thex-axis at the origin and at (a,0) and at (b,0) for some real numbersa <0 andb > curve only exists for 6x6 and forx> . The three points with coordinates( , ), ( , ), and ( , ) are all on the curve.( , )(b,0)( , )( , )(a,0)xy(i) What are the values ofaandb?(ii) By completing the square, or otherwise, find the value of .(iii) Explain why the curve is symmetric about the liney= .(iv) Find a cubic equation inxwhich has roots.

7 (Your expression for the cubicshould not involve , , or ). Justify your answer.(v) By considering the factorization of this cubic, find the value of + + .(vi) LetCdenote the circle which has the points ( , ) and ( , ) as ends of a down the equation ofC. Show thatCintersectsSat two other points andfind their commonx-co-ordinate in terms of .12 Turn over13 This page has been intentionally left blank14 Turn overIf you require additional space please use the pages at the end of the IN MATHEMATICSMATHEMATICS & STATISTICSMATHEMATICS & PHILOSOPHY & Computer Science,Computer ScienceandComputer Science & Philos-ophyapplicants should turn to page 20.(i) A functionf(x) is said to beeveniff( x) =f(x) for function is said tobeoddiff( x) = f( x) for allx.(a) What symmetry does the graphy=f(x) of an even function have?What symmetry does the graphy=f(x) of an odd function have?(b) Use these symmetries to show that the derivative of an even function is anodd function, and that the derivative of an odd function is an even function.

8 [You should not use the chain rule.](ii) For 45 < <45 , the lineLmakes an angle with the liney=xas drawnin the figure below. LetA( ) denote the area of the triangle which is bounded bythex-axis, the linex+y= 1 and the +y= 1xyA( )11 (a) Let 0< <45 .Arguing geometrically, explain whyA( ) +A( ) =12.(b) For 0< <45 , determine a formula forA( ).(c) Sketch the graph ofA( ) against for 45 < <45 .(d) In light of the identity in part (ii)(a), what symmetry does the graph ofA( )have?(e) Without explicitly differentiating, explain whyd2Ad 2= 0 when = over17 This page has been intentionally left blank18 Turn overIf you require additional space please use the pages at the end of the and Adam agree to relieve the boredom of the school holidays by eating sweets,but their mother insists they limit their consumption by obeying the following rules. Miriam eats as many sweets on any day as there have been sunny days during theholiday so far, including the day in question.

9 Adam eats sweets only on rainy days. If daykof the holiday is rainy , then he eatsksweets on that example, if the holiday is eight days long, and begins rainy , sunny , sunny , .. , thenthe tally of sweet consumption might look like this:Day12 3456 78 TotalWeatherR S S R R S S RMiriam01 2223 4418 Adam10 0450 0818In this case, Miriam and Adam eat the same number of sweets in total.(i) If the holiday has 30 days, 15 of which are sunny and 15 rainy , what arrangementof sunny and rainy days would lead Miriam to eat the greatest number of sweetsin total, and what arrangement would lead to the least number? Give the numberof sweets that Miriam eats in each case.(ii) Show that, in the two cases mentioned in part (i), Adam eats the same number ofsweets as Miriam.(iii) Suppose, in a sequence of sunny and rainy days, we arrange to swap a rainy daywith a sunny day that immediately follows it. How does the total number of sweetseaten by Miriam change when we make the swap?

10 What about the total numberof sweets eaten by Adam?(iv) If the holiday has 15 sunny days and 15 rainy days, must Miriam and Adam eatthe same number of sweets in total? Explain your over21 This page has been intentionally left blank22 Turn overIf you require additional space please use the pages at the end of the IN COMPUTER SCIENCEMATHEMATICS & COMPUTER SCIENCECOMPUTER SCIENCE & PHILOSOPHY cancellation of the Wimbledon tournament has led to a world surplus of tennis balls,and Santa has decided to use them as stocking fillers. He comes down the chimney withnidentical tennis balls, and he findsknamed stockings waiting for (n,k) be the number of ways that Santa can put thenballs into thekstockings;for example,g(2,2) = 3, because with two balls and two children, Miriam and Adam,he can give both balls to Miriam, or both to Adam, or he can give them one ball each.(i) What is the value ofg(1,k) fork>1?(ii) What is the value ofg(n,1)?(iii) If there aren>2 balls andk>2 children, then Santa can either give the firstball to the first child, then distribute the remaining balls among allkchildren, orhe can give the first child none, and distribute all the balls among the remainingchildren.