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STEP Support Programme

STEP Support - Assignment 13


This STEP support module practises sketching graphs including consideration of whether they are convex or concave, and using these to determine how many roots of an equation there are. Previous assignments can be found here, but you can do this one without having done the others first.

STEP questions are difficult, they are supposed to be and you should expect to get stuck. However, as you tackle more and more STEP questions you will develop a range of problem solving skills (and spend less time "being stuck").

About this assignment

The assignment is published as a pdf file below. Each STEP Support assignment module starts with a warm-up exercise, followed by preparatory work leading to a STEP question. Finally, there is a warm-down exercise.

The warm up for this assignment introduces the definitions of convex and concave graphs, and also discusses points of inflection.

The main STEP question (2012 STEP 1 Question 2) uses some of these ideas to help sketch some quartics, and use these to determine how many solutions there are of some related equations.

The final question is a problem where all the solutions have to be integers, i.e. a Diophantine equation.

Assignment PDF: 
Hints and Partial Solutions: 

Useful Links

Underground Mathematics: Selected worked STEP questions

STEP Question database

University of Cambridge Mathematics Faculty: What do we look for?

University of Cambridge Mathematics Faculty: Information about STEP

University of Cambridge Admissions Office: Undergraduate course information for Mathematics

Stephen Siklos' "Advanced Problems in Mathematics" book (external link)

MEI: Worked solutions to STEP questions (external link)

OCR: Exam board information about STEP (external link)

AMSP (Advanced Maths Support programme): Support for University Admission Tests (external link)