We explore the possibility of describing curves with an alternative method, by describing both the x and y coordinates of any point in terms of a 3rd variable (parameter). We also explore differentiation of curves expressed parametrically and a good variety of fully solved examples related, including exam questions.

In this exam, each question is fully solved, including detailed explanations, comments, diagrams, and calculations. It can be useful as a revision tool or as a study guide. Hope it may help!!

It explores the functions sine, cosine, tangent, secant, cosecant and cotangent as well as formulae for the addition/subtraction of 2 angles, double angle formulae and other techniques to re-arrange trigonometric expressions and solve trigonometric equations related. Additionally, it contains a complete study over the differentiation and integration of these functions, and it comes with a wide set of solved (step-by-step) exam questions, chosen from A2 Cambridge past papers. Hope it may be useful, both to students and teachers!!

In this PowerPoint, we explore two numerical methods to find roots of equations: the “sign-change-rule” and the “method of iterations”. These are especially useful when it is very difficult to solve them algebraically. We explain the details of each method and illustrate with worked examples, including exam questions. As always, I expect this ppt to be useful both for students and teachers!

This ppt introduces differential equations of first degree as well as possible algebraic techniques to solve them. It also contains a series of fully solved exam questions related, all taken from the 9709 CIE exams. Each step is explained, enabling this resource to be used, both as teaching guide and a revision tool.

This ppt contains an helpful guide through the topic “Vectors” for the A2 Pure Mathematics syllabus, after the restructure in 2020. It explores the vector equation of a line, properties of vectors and includes also approaches to intersections, distances or angles between points and lines. It finishes with a couple of solved exam questions, where each step is explained carefully to make it easier to be fully understood.

This resource focus on different algebraic methods to differentiate and integrate functions, such as exponentials, logarithmic and trigonometric as well as functions defined parametrically or implicitly. It also covers the “product rule” and the “quotient rule” for differentiation together with the methods of “integration by parts” and "integration with a “substitution”. The last section includes a series of fully solved exam questions related, where each step is carefully described and explained.

This resource can become useful both for students and teachers. It approaches the concept of modulus of a number and extends the idea to the modulus function. It also includes a section with equations and inequalities involving modulus. Hope it may be useful!!

This Powerpoint explores the main properties of exponential and logarithmic functions as well as the differentiation and integration processes for each. It contains explanations and worked exercises, chosen from A2 Cambridge past papers. Hope it may be useful both to students and teachers!!

We continue to explore alternative forms of studying certain geometric curves, as done previously with parametric equations. The concept of differentiation will be studied and some of its applications. This resource contains a wide variety of detailed examples, including fully solved exam questions. I hope it may useful both for students and teachers who wish to explore this topic.

This ppt explores multiple features of complex numbers expressed in algebraic, polar or exponential forms. It includes equations, regions of the plane represented by inequalities with complex numbers as well as geometric problems related. It ends with different exam questions from Cambridge past papers of A2 level. Hope it may be useful both for students and teachers.

This ppt introduces all the main computational properties of the roots of polynomial equations, relating them with the coefficients involved. It also contains a series of fully solved exam questions related, all taken from the 9231 CIE exams. Each step is explained, enabling this resource to be used, both as teaching guide and a revision tool.

We will study the formula for the Binomial expansion, for cases when the index is either negative or fractional. Then, we investigate how to obtain partial fractions from algebraic fractions and how to link these 2 topics as well as how to use them for integration purposes. We finish with a series of fully solved exam questions related, obtained from the CIE examinations.

This resource can be used both by students and teachers. It starts with some key words and definitions and then focus in the division of two polynomials using: - the method of “Equating the coefficients” - the “Algorithm of division” it also mentions the particular cases where we can use the “Remainder and Factor Theorems”. Hope this may help!

This resource approaches the method of “Integration by parts”. I show some worked examples, all taken from past papers of A2 Level (Cambridge) and I try to show how to use the formula provided in the new MF19 formulaire in a practical manner. Hope it may help!!

All questions are solved, trying to show clearly all the steps and deductions made. This includes some diagrams of the “Normal distribution” which were used to help the understanding of some questions. Hope it may be useful!!

This is an informal approach to all the questions in this paper and all answers were fully checked. Everything is handwritten but hopefully it will be easy to understand. Hope it may become useful for students and also for teachers!

In this resource, each question was solved, step-by-step, with detailed explanations and comments to make it easier to be fully understood. Some attention was also given to the presentation and accurate notation was used throughout the paper.

All questions were solved in a detailed and clear manner, showing all relevant steps towards each answer. Hopefully, it will be useful both for students and teachers!