In this paper of “Statistics 2”, each question is fully solved, and all the steps are explained with relevant calculations and/or comments and diagrams.

In this paper of “Pure Mathematics 3”, each question is fully solved, and all the steps are explained with relevant calculations and/or comments and diagrams.

In this paper of “Statistics 1”, each question is fully solved, and all the steps are explained with relevant calculations and/or comments and diagrams.

In this paper of “Pure Mathematics 1”, each question is fully solved, and all the steps are explained with relevant calculations and/or comments and diagrams.

In this paper of “Pure Mathematics 3”, each question is fully solved, and all the steps are explained with relevant calculations and/or comments and diagrams.

In this paper of “Statistics 1”, each question is fully solved, and all the steps are explained with relevant calculations and/or comments and diagrams.

In this paper of “Statistics 2”, each question is fully solved, and all the steps are explained with relevant calculations and/or comments and diagrams.

In this paper of “Pure Mathematics 1”, each question is fully solved and all the steps are explained with relevant calculations and/or comments and diagrams.

In this paper of “Further Pure Mathematics 2”, each question is fully solved, and all the steps are explained with relevant calculations and/or comments and diagrams. Attention was given to the specific requirements of all the topics involved, namely “Complex numbers”, “Hyperbolic functions”, “Properties of matrices”, “Differentiation”, “Integration” and “Differential equations”. Additionally, there were also algebraic techniques which were used and are part of the regular A-Level syllabus for mathematics.

In this paper of “Further Pure Mathematics 1”, each question is fully solved, and all the steps are explained with relevant calculations and/or comments and diagrams. Attention was given to the specific requirements of all the topics involved, namely “Vectors”, “Properties of the roots of polynomials”, “Proof by induction”, “Method of differences”, “Matrices and geometric transformations”, “Polar coordinates” and “Rational functions and their graphs”. Additionally, there were also algebraic techniques which were used and are part of the regular A-Level syllabus for mathematics.

In this PowerPoint, we introduce two different types of numerical sequences (arithmetic progression and geometric progression) as part of the syllabus for Pure Mathematics (AS Level). We begin by detailing the main features for each of these sequences, showing examples as well as the rationale for the algebraic expression which provides the nth term. Additionally, we also show how to find the algebraic expressions for the sum of the first n terms, for both cases. It is important to note that these formulas are provided in most examinations and available to students, hence the way to derive them consists on additional skills for students to acquire. In the last section, we explore different examples of worked exam-style questions, where all steps are clear and detailed. Hopefully, this resource can assist students when preparing for their examinations but also as a complementary teaching tool.

In this PowerPoint, we introduce the Binomial theorem as part of the syllabus for Pure Mathematics (AS Level). In the first section, there are mentions to specific details to use the formula correctly as well as different approaches to calculate the coefficient for a specific power of the variable. In the last section, we explore different examples of worked exam-style questions, where all steps are clear and detailed. Hopefully, this resource can assist students when preparing for their examinations but also as a complementary teaching tool.

In this PowerPoint, we introduce an alternative way of describing the coordinates of the points on the graph of a curve. We explain the relation between the parametric equations of a curve and the corresponding Cartesian equation, also showing how to transition from one to the other. We then approach the concept of differentiation of a curve which is defined parametrically, by showing how to find the gradient in any point, using the information available. In the last section, we explore different examples of worked exam-style questions, where all steps are clear and detailed.

In this PowerPoint, we begin by recalling some of the most common algebraic techniques used to solve equations and inequalities. Then, we show that, for some cases, those methods won’t solve the problem, hence requiring for alternative techniques. We then introduce the “Sign-change rule” which can be used, either to prove the existence of a root of an equation in a given interval as to actually determine such root. Afterwards, we introduce the method of iterations, where we develop a full method to gradually obtain approximations to the root of a given equation. In the last section, we explore different examples of worked exam-style questions, where all steps are clear and detailed. Hopefully, this resource can help students in their preparation for examinations and can also be used as a complementary teaching tool, when delivering these topics.

In this PowerPoint, we explore the topic of “Vectors”. Despite the fact that the concept of plane was removed from the syllabus of the Pure Mathematics (A2 Level) since 2020, this PPT contains references to it as it seems an important feature to learn, and relates with possible continuation of this topic in the curriculum of “Further Mathematics”. We begin with some considerations about the concept of a vector as well as some basic operations between vectors, including the “scalar product” and the “vector product”. The next section includes examples of solved exam style questions, which describe the following situations: • possible intersections between lines and planes • angles between lines and planes • distances between points, lines and planes~ In each example, we can find detailed explanations complemented with key diagrams and tips for similar cases. Hopefully, this resource can help students when revising for their examinations as well as teachers, who can use it a complement to their explanations in lessons.

In this PowerPoint, we introduce the concept of polynomial. We begin by exploring some key definitions, such as the degree of a polynomial and then explore operations between polynomials, with a special emphasis on the division between two polynomials. We introduce two different methods to perform division between two polynomials, namely the algorithm of division and the method of equating coefficients. Afterwards, we explore the remainder and factor theorems, pointing out their advantages and limitations. Additionally, we explore different examples of worked exam-style questions, where all steps are clear and detailed. Hopefully, this resource can help students in their preparation for examinations and can also be used as a complementary teaching tool, when delivering these topics.

In this PowerPoint, we begin by introducing the concept of exponential function and, by exploring its inverse, we introduce the logarithmic function. We then investigate key features of these functions, including domain, range, graphs, algebraic expressions as well as the most important algebraic properties related with logarithms. Afterwards, we investigate the process of differentiating and integrating, both logarithmic and exponential functions, alerting for the need to use relevant operational rules, like the “product and quotient rules” for differentiation or the technique of “integration by parts”. For each case, there are worked examples, guided steps and clear explanations. Additionally, we explore different examples of worked exam-style questions, where all steps are clear and detailed. Hopefully, this resource can help students in their preparation for examinations and can also be used as a complementary teaching tool, when delivering these topics.

In this PowerPoint, we present the concept of the modulus function. We explore it both from algebraic and graphic perspectives, trying to establish relevant connections between the two. We then study equations and inequalities involving modulus and show how to proceed in such cases. We also include the algebraic technique of “squaring both sides” and alert for the situations in which this method can lead to wrong or incomplete answers. Additionally, we explore different examples of worked exam-style questions, where all steps are clear and detailed. Hopefully, this resource can help students in their preparation for examinations and can also be used as a complementary teaching tool, when delivering these topics.

In this PowerPoint, we explore different forms of differentiating and integrating functions. We include the study of trigonometric, exponential and logarithmic functions, whether we wish to differentiate or integrate them. Additionally, we also take a close look on how to differentiate functions which are defined parametrically or implicitly, including the “product rule” and “quotient rule” for differentiation. In the section of “Integration”, apart from the already mentioned functions, we also include slides where we study the methods of “integration by parts” and the method of “integration with a substitution”. For all these cases, there are worked examples with explanations and tips for similar cases. Finally, the last section of this resource includes a series of solved exam-style questions, where we go over some of the typical questions related, by explaining each step taken as well as relevant comments to help understand the rational used in each of those questions. Hopefully, this resource can help students in their preparation for examinations which require differentiation and integration techniques and can also be used as a complementary teaching tool, when delivering these topics.

In this PowerPoint, we begin by deriving the formula for the binomial expansion for cases when the power is not a positive integer. We then explore different examples where the coefficient for a specific power of the variable is requested, including detailed explanations and relevant comments. Afterwards, we explore different algebraic manipulation to decompose rational fractions into specific types of partial fractions, by introducing related worked examples. In the last section, we explore different examples of worked exam-style questions, where all steps are clear and detailed. Hopefully, this resource can help students in their preparation for examinations and can also be used as a complementary teaching tool, when delivering these topics.