Advanced Problems in Mathematics

Preparing for University

by Stephen Siklos

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Book Description

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 some other UK universities and many mathematics departments recommend that their applicants practice on the past papers even if they do not take the examination.

Advanced Problems in Mathematics bridges the gap between school and university mathematics, and prepares students for an undergraduate mathematics course. The questions analysed in this book are all based on past STEP questions and each question is followed by a comment and a full solution. The comments direct the reader's attention to key points and put the question in its true mathematical context. The solutions point students to the methodology required to address advanced mathematical problems critically and independently.

This book is a must read for any student wishing to apply to scientific subjects at university level and for anyone interested in advanced mathematics.

This open book is licensed under a Creative Commons License (CC BY). You can download Advanced Problems in Mathematics ebook for free in PDF format (2.0 MB).

Table of Contents

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 x^m(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
 

Book Details

Publisher
Open Book Publishers
Published
2019
Pages
186
Edition
1
Language
English
ISBN13
9781783747764
ISBN10
1783747765
ISBN13 Digital
9781783747771
ISBN10 Digital
1783747773
PDF Size
2.0 MB
License
CC BY

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