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This course by Imperial College London is designed to help you develop the skills you need to succeed in your A-level maths exams.

You will investigate key topic areas to gain a deeper understanding of the skills and techniques that you can apply throughout your A-level study. These skills include:

- Fluency – selecting and applying correct methods to answer with speed and efficiency
- Confidence – critically assessing mathematical methods and investigating ways to apply them
- Problem solving – analysing the ‘unfamiliar’ and identifying which skills and techniques you require to answer questions
- Constructing mathematical argument – using mathematical tools such as diagrams, graphs, logical deduction, mathematical symbols, mathematical language, construct mathematical argument and present precisely to others
- Deep reasoning – analysing and critiquing mathematical techniques, arguments, formulae and proofs to comprehend how they can be applied

Over seven modules, covering general motion in a straight line and two dimensions, projectile motion, a model for friction, moments, equilibrium of rigid bodies, vectors, differentiation methods, integration methods and differential equations, your initial skillset will be extended to give a clear understanding of how background knowledge underpins the A -level course.

You’ll also be encouraged to consider how what you know fits into the wider mathematical world.

By the end of this course, you'll be able to:

- Use calculus in kinematics for motion in a straight line
- Use differentiation and integration of a vector with respect to time for motion in two dimensions
- Solve projectile motion problems using both calculus/vector methods and constant acceleration formulae
- Use a standard model for friction
- Calculate moments understanding what they mean and how they might be used
- Solve problems involving parallel and nonparallel coplanar forces
- Apply an understanding of moments to statics problems involving rigid bodies
- Use the Normal distribution as a model for continuous data
- Conduct a hypothesis test of the mean using a Normal distribution
- Use a Normal distribution as an approximation of a Binomial distribution
- Add vectors diagrammatically
- Perform the algebraic operations of vector addition and multiplication by scalars
- Apply vector calculations to problems in pure mathematics
- Use methods for differentiating a function of a function, differentiating a product and differentiating a quotient
- Differentiate trigonometric and inverse trigonometric functions
- Use implicit and parametric differentiation
- Identify integrals that can be dealt with “by sight”
- Use a substitution method to integrate a function
- Use partial fractions to integrate rational functions
- Use the method of integration by parts
- Use the method of separating the variable to solve differential equations
- find the family of solutions for a differential equation

**Module 1: Calculus in Kinematics and Projectile Motion**

- Using calculus for kinematics for motion in a straight line:
- Using calculus in kinematics for motion extended to 2 dimensions using vectors.
- Modelling motion under gravity in a vertical plane using vectors; projectiles.
- Composition of functionsInverse functions

**Module 2: Friction, Moments and Equilibrium of rigid bodies**

- Understanding and using the F≤μR model for friction
- The coefficient of friction motion of a body on a rough surface limiting friction
- Understanding and using moments in simple static contexts.
- The equilibrium of rigid bodies involving parallel and nonparallel coplanar forces

**Module 3: The Normal Distribution**

- Understanding and using the Normal distribution as a model
- Finding probabilities using the Normal distribution
- Conducting statistical hypothesis tests for the mean of a Normal distribution with known, given or assumed variance
- Interpreting the results of hypothesis tests in context

**Module 4: Vectors**

- Using vectors in two dimensions and in three dimensions
- Adding vectors diagrammatically
- Performing the algebraic operations of vector addition and multiplication by scalars
- Understanding the geometrical interpretations of vector calculations
- Understanding and using position vectors
- Calculating the distance between two points represented by position vectors.
- Using vectors to solve problems in pure mathematics

**Module 5: Differentiation Methods**

- Differentiation using the product rule, the quotient rule and the chain rule
- Differentiation to solve problems involving connected rates of change and inverse functions.
- Differentiating simple functions and relations defined implicitly or parametrically

**Module 6: Integration Methods**

- Integrating e^kx, 1/x, sinkx, coskx and related sums, differences and constant multiples
- Integration by substitution
- Integration using partial fractions that are linear in the denominator
- Integration by parts

**Module 7: Differential Equations**

- The analytical solution of simple first order differential equations with separable variables
- Finding particular solutions
- Sketching members of a family of solution curves
- Interpreting the solution of a differential equation in the context of solving a problem
- Identifying limitations of the solution to a differential equation

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