Advanced Problems in Mathematics Preparing for University

• Main Author: Siklos, Stephen
• Format: eBook
• Language: English
• Published: Cambridge Open Book Publishers 2019
• Edition: 1
• Subjects: Algebra; Arithmetic; Calculus and mathematical analysis; Children’s, Teenage and Educational; Education; Educational material; Educational: Mathematics and numeracy; Educational: Mathematics, science and technology, general; Elementary Problems; Geometry; Infinity; make sense of the world; Mathematics; Mathematics and Science; mathematics beyond the classroom; Mental Skills; Probability and statistics; Statistics; Study and teaching (Higher); Teaching Methods & Materials; Word Problems

"This new and expanded edition is intended to help candidates prepare for entrance examinations in mathematics and scientific subjects, including STEP (Sixth Term Examination Paper). STEP is an examination used by Cambridge Colleges for conditional offers in mathematics. They are also used by s...

Table of Contents:

• Front Matter Table of Contents About this book STEP Worked Problems Problem 1:: An integer equation Problem 2:: Partitions of 10 and 20 Problem 3:: Mathematical deduction Problem 4:: Divisibility Problem 5:: The modulus function Problem 6:: The regular Reuleaux heptagon Problem 7:: Chain of equations Problem 8:: Trig. equations Problem 9:: Integration by substitution Problem 10:: True or false Problem 11:: Egyptian fractions Problem 12:: Maximising with constraints Problem 13:: Binomial expansion Problem 14:: Sketching subsets of the plane Problem 15:: More sketching subsets of the plane Problem 16:: Non-linear simultaneous equations Problem 17:: Inequalities Problem 18:: Inequalities from cubics Problem 19:: Logarithms Problem 20:: Cosmological models Problem 21:: Melting snowballs Problem 22:: Gregory’s series Problem 23:: Intersection of ellipses Problem 24:: Sketching xm(1—x)n Problem 25:: Inequalities by area estimates Problem 26:: Simultaneous integral equations Problem 27:: Relation between coefficients of quartic for real roots Problem 28:: Fermat numbers Problem 29:: Telescoping series Problem 30:: Integer solutions of cubics Problem 31:: The harmonic series Problem 32:: Integration by substitution Problem 33:: More curve sketching Problem 34:: Trig sum Problem 35:: Roots of a cubic equation Problem 36:: Root counting Problem 37:: Irrationality of e Problem 38:: Discontinuous integrands Problem 39:: A difficult integral Problem 40:: Estimating the value of an integral Problem 41:: Integrating the modulus function Problem 42:: Geometry Problem 43:: The t substitution Problem 44:: A differential-difference equation Problem 45:: Lagrange’s identity Problem 46:: Bernoulli polynomials Problem 47:: Vector geometry Problem 48:: Solving a quartic Problem 49:: Areas and volumes Problem 50:: More curve sketching Problem 51:: Spherical loaf Problem 52:: Snowploughing Problem 53:: Tortoise and hare Problem 54:: How did the chicken cross the road? Problem 55:: Hank’s gold mine Problem 56:: A chocolate orange Problem 57:: Lorry on bend Problem 58:: Fielding Problem 59:: Equilibrium of rod of non-uniform density Problem 60:: Newton’s cradle Problem 61:: Kinematics of rotating target Problem 62:: Particle on wedge Problem 63:: Sphere on step Problem 64:: Elastic band on cylinder Problem 65:: A knock-out tournament Problem 66:: Harry the calculating horse Problem 67:: PIN guessing Problem 68:: Breaking plates Problem 69:: Lottery Problem 70:: Bodies in the fridge Problem 71:: Choosing keys Problem 72:: Commuting by train Problem 73:: Collecting voles Problem 74:: Breaking a stick Problem 75:: Random quadratics Syllabus Back Matter
• P61 Kinematics of rotating target -- P62 Particle on wedge -- P63 Sphere on step -- P64 Elastic band on cylinder -- P65 A knock-out tournament -- P66 Harry the calculating horse -- P67 PIN guessing -- P68 Breaking plates -- P69 Lottery -- P70 Bodies in the fridge -- P71 Choosing keys -- P72 Commuting by train -- P73 Collecting voles -- P74 Breaking a stick -- P75 Random quadratics -- Syllabus
• Intro -- About this book -- STEP -- Worked Problems -- Worked problem 1 -- Worked problem 2 -- Problems -- P1 An integer equation -- P2 Partitions of 10 and 20 -- P3 Mathematical deduction -- P4 Divisibility -- P5 The modulus function -- P6 The regular Reuleaux heptagon -- P7 Chain of equations -- P8 Trig. equations -- P9 Integration by substitution -- P10 True or false -- P11 Egyptian fractions -- P12 Maximising with constraints -- P13 Binomial expansion -- P14 Sketching subsets of the plane -- P15 More sketching subsets of the plane -- P16 Non-linear simultaneous equations -- P17 Inequalities -- P18 Inequalities from cubics -- P19 Logarithms -- P20 Cosmological models -- P21 Melting snowballs -- P22 Gregory's series -- P23 Intersection of ellipses -- P24 Sketching xm(1-x)n -- P25 Inequalities by area estimates -- P26 Simultaneous integral equations -- P27 Relation between coefficients of quartic for real roots -- P28 Fermat numbers -- P29 Telescoping series -- P30 Integer solutions of cubics -- P31 The harmonic series -- P32 Integration by substitution -- P33 More curve sketching -- P34 Trig sum -- P35 Roots of a cubic equation -- P36 Root counting -- P37 Irrationality of e -- P38 Discontinuous integrands -- P39 A difficult integral -- P40 Estimating the value of an integral -- P41 Integrating the modulus function -- P42 Geometry -- P43 The t substitution -- P44 A differential-difference equation -- P45 Lagrange's identity -- P46 Bernoulli polynomials -- P47 Vector geometry -- P48 Solving a quartic -- P49 Areas and volumes -- P50 More curve sketching -- P51 Spherical loaf -- P52 Snowploughing -- P53 Tortoise and hare -- P54 How did the chicken cross the road? -- P55 Hank's gold mine -- P56 A chocolate orange -- P57 Lorry on bend -- P58 Fielding -- P59 Equilibrium of rod of non-uniform density -- P60 Newton's cradle
• STEP -- Worked Problems -- Problems -- Syllabus