### Introduction

Welcome to the STEP Support programme. This is the first of 25 Assignments designed to develop your advanced problem-solving skills.

STEP is a challenging examination, and is different in style from A-level, although the mathematical content is the same. STEP questions are longer: they are designed to take around 30 minutes, rather than the typical 10 minutes required for an A-level question.

Do not worry if the STEP problems appear difficult: they are meant to be! However, you should not be daunted. These assignments are designed to help you to develop the skills you need, over time, so that by the time you sit the STEP exam in the summer of Y13 you will feel well-prepared.

### 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 involves manipulating surds and simplifying them. The answers to the last part can be expressed surprisingly neatly!

The **main STEP question (2005 STEP 1 Question 3)** is all about the solutions (or roots) of quadratic equations, and whether an equation has 0, 1 or 2 real roots. As is often the case with STEP questions, the coefficients in the equation are not numerical (they are in terms of parameters such as a, b and c).

The **final question** gives a slightly surprising result. A clear diagram with all the given information shown on it is a good starting point.

### Further reading

An introduction to problem solving can be found here.

For more on Holditch's theorem see these entries in Wolfram Mathworld and Wikipedia.

A discussion on the accuracy of $\pi$ can be found here.

This Plus article discusses a counter-intuitive problem with circles.

### Hints, support and self evaluation

The **“Hints and partial solutions for Assignment 1”** file gives suggestions on how you can tackle the questions, and some common pitfalls to avoid, as well as some partial solutions and answers.

Here is a Worked Video Solution to the STEP question from this assignment.