Year 11 Mathematics Advanced Resources
We have collected a variety of worksheets and resources created by our tutors for Year 11 Mathematics Advanced.
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In Mathematics Advanced, you will deepen your understanding of complex mathematical ideas and sharpen your problem-solving abilities. This course is designed for you if you’re ready to push your limits, explore advanced topics, and apply your learning to real-world challenges. Get ready to engage with algebra, functions, trigonometry, calculus, and probability in ways that make sense for your future studies and career goals.
Contents
- Algebraic Techniques
- Equations and Inequalities
- Functions
- Trigonometry
- Further Functions
- Introduction to Calculus
- Probability
- Exponential and Logarithmic Functions
- Trigonometric Functions
- Discrete Probability Distributions
Algebraic Techniques
In this unit, you will build your confidence in handling algebraic expressions and equations. You’ll learn how to simplify expressions, expand and factorise polynomials, and work with surds, laying a solid foundation for more complex mathematical problems.
Content
Index laws
Master the laws of indices so that you can simplify expressions involving powers easily.
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Zero and negative indices
Learn how to work with zero and negative indices to simplify and manipulate algebraic expressions.
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Fractional indices
Explore fractional indices and learn to convert between radical and exponential forms.
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Simplifying algebraic expressions
Develop your skills in breaking down and simplifying complex algebraic expressions.
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Expanding algebraic expressions
Practise expanding expressions using the distributive law and multiplying polynomials.
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Binomial products
Recognise patterns in binomial products and learn how to expand them effectively.
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Special products
Identify and use special product formulas, such as perfect squares and differences of squares, to simplify your work.
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Factorisation
Learn to factorise algebraic expressions by extracting common factors and grouping terms.
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Factorisation by grouping in pairs
Practise grouping terms in pairs to extract common factors and simplify expressions.
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Factorising trinomials
Learn how to factorise quadratic trinomials into the product of two binomials.
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Further trinomials
Extend your factorisation techniques to handle more complex trinomials.
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Perfect squares
Understand how to factorise and expand expressions that involve perfect square trinomials.
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Difference of two squares
Recognise the pattern of the difference of two squares and factorise expressions accordingly.
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Mixed factorisation
Combine various factorisation methods to simplify even the most complex expressions.
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Simplifying algebraic fractions
Learn techniques to simplify algebraic fractions, making them easier to work with.
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Operations with algebraic fractions
Perform addition, subtraction, multiplication, and division with algebraic fractions confidently.
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Substitution
Use substitution effectively by replacing variables with numerical values to evaluate expressions.
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Simplifying surds
Master the process of simplifying radical expressions and managing irrational numbers.
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Operations with surds
Perform arithmetic operations with surds, ensuring accuracy and understanding.
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Rationalising the denominator
Learn how to eliminate surds from the denominator of fractions through rationalisation.
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Equations and Inequalities
This unit will empower you to solve a variety of equations and inequalities. You will explore linear, quadratic, exponential, and absolute value scenarios, learn to solve systems of equations, and understand how to interpret solutions both graphically and algebraically.
Content
Equations
Develop techniques to solve both linear and non-linear equations using clear, step-by-step methods.
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Inequalities
Learn how to solve and graph inequalities on the number line so that you can visualise solutions.
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Absolute value equations and inequalities
Master the process of solving equations and inequalities that involve absolute value expressions.
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Exponential equations
Tackle equations where the unknown appears in the exponent, expanding your problem-solving toolkit.
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Quadratic equations
Explore different methods to solve quadratic equations, ensuring you understand the reasoning behind each approach.
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Solving quadratic equations by factorisation
Learn how to factorise quadratic equations to find their solutions efficiently.
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Solving quadratic equations by completing the square
Practise rewriting quadratic equations in completed square form to reveal their solutions.
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Solving quadratic equations by quadratic formula
Use the quadratic formula to determine the real and complex roots of quadratic equations with confidence.
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Formulas and equations
Develop your ability to manipulate and solve equations derived from various formulas.
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Simultaneous equations
Learn to solve systems of equations with two variables by applying effective methods.
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Linear simultaneous equations
Apply substitution and elimination techniques to solve systems of linear equations.
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Non-linear simultaneous equations
Expand your skills to solve systems that include at least one non-linear equation.
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Simultaneous equations with three unknown variables
Challenge yourself by solving systems with three variables using advanced methods such as substitution and elimination. (Extension topic)
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Functions
This unit introduces you to the central concept of functions. You will learn how to work with different types of functions, understand their notation, and interpret their graphs. By exploring both the algebraic and graphical aspects of functions, you will build a strong foundation for tackling more complex mathematical models.
Content
Functions
Grasp the idea of a function as a relation where each input is paired with a unique output.
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Function notation
Learn how to use function notation to evaluate and interpret functions for various inputs.
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Domain and range
Determine the domain and range of functions from equations or graphs, ensuring you understand all possible inputs and outputs.
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Properties of functions
Explore key characteristics like continuity, increasing/decreasing behaviour, and symmetry, so you can better understand how functions behave.
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Linear functions
Examine linear functions and their graphs, and learn how to interpret slope–intercept form in practical terms.
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The gradient of a straight line
Calculate and interpret the gradient (slope) of a line, helping you understand rates of change.
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Finding a linear equation
Develop your ability to formulate the equation of a line given specific conditions.
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Parallel and perpendicular lines
Understand the conditions that make lines parallel or perpendicular and learn how to find their equations.
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Quadratic functions
Study the properties and graphs of quadratic functions to see how they model real-world phenomena.
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Axis of symmetry
Learn to find the axis of symmetry in a parabola, which is key to understanding its graph.
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The discriminant
Use the discriminant to determine the nature of the roots of quadratic equations, giving you insight into their solutions.
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Finding a quadratic equation
Practise determining the equation of a quadratic function based on its key characteristics.
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Cubic functions
Explore cubic functions and their graphs, and learn how they differ from linear and quadratic functions.
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Polynomial functions
Extend your understanding to higher-degree polynomials and their general behaviours, preparing you for advanced topics.
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Inverse functions
Learn how to find the inverse of a function and understand the special relationship between a function and its inverse.
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Composite functions
Practise forming and evaluating composite functions, so you can understand how functions interact with one another.
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Intersection of graphs
Develop the skills to find the intersection points of graphs both algebraically and visually.
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Trigonometry
In this unit, you will expand your understanding of trigonometry beyond right-angled triangles. You will learn to apply trigonometric ratios, the sine and cosine rules, and area formulas to solve more complex problems, while also exploring the concept of radians.
Content
Trigonometric ratios
Learn to define and use sine, cosine, and tangent ratios in right-angled triangles so that you can solve problems effectively.
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Finding a side of a right-angled triangle
Practise calculating unknown sides using trigonometric ratios, making you confident in applying these concepts.
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Finding an angle in a right-angled triangle
Discover how to determine unknown angles using inverse trigonometric functions.
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Applications of trigonometry
Apply your trigonometric skills to solve practical problems and see the real-world relevance of what you learn.
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The sine rule
Learn how to use the sine rule to solve non–right-angled triangles effectively.
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The cosine rule
Apply the cosine rule to find unknown sides or angles in triangles where the sine rule isn’t applicable.
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Area of a triangle
Master the techniques to calculate the area of a triangle using trigonometric methods.
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Mixed problems
Challenge yourself with complex problems that require you to combine different trigonometric principles.
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Introduction to radians
Understand the radian as a measure of angles and learn how to convert between degrees and radians.
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Length of an arc
Learn to calculate the length of an arc when using radian measure.
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Area of a sector
Learn to determine the area of a sector using your understanding of radians.
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Further Functions
In this unit, you will dive deeper into the world of functions. You will explore advanced topics such as hyperbolas, absolute value functions, and the equations of circles. You’ll also study transformations and learn how translations, reflections, and dilations affect function graphs.
Content
The hyperbola
Examine the properties and graphs of reciprocal functions, and understand how they differ from other function types.
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Absolute value functions
Learn to work with absolute value functions and understand how their graphs are formed.
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Circles and semicircles
Discover how to write equations for circles and semicircles and interpret their graphical representations.
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Reflections of functions
Investigate how reflecting a function over an axis changes its graph, enhancing your understanding of symmetry.
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Combined transformations
Apply multiple transformations at once and see how they collectively affect a function's graph.
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Introduction to Calculus
Welcome to the world of calculus! In this unit, you will begin exploring the concept of the derivative as a way to measure change. You will learn how to differentiate functions from first principles and apply a variety of rules to solve problems related to rates of change.
Content
Gradient of a curve
Understand the idea of a gradient at a specific point on a curve and how it represents the rate of change.
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Differentiability
Learn how to determine where a function is differentiable and why this property is important.
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Differentiation from first principles
Develop a deep understanding of differentiation by deriving the derivative of basic functions using its definition.
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Rules of differentiation
Practise applying the power, sum, and constant multiple rules to differentiate functions with ease.
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Derivatives and indices
Learn to differentiate functions that involve indices, expanding your ability to handle exponential expressions.
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Tangents and normals
Find the equations of tangents and normals to curves, an essential skill for understanding the geometry of graphs.
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Chain rule
Master the chain rule to differentiate composite functions effectively.
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Product rule
Learn how to differentiate products of functions using the product rule.
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Quotient rule
Develop your skills in differentiating quotients of functions by applying the quotient rule.
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Rates of change
Explore how to calculate both average and instantaneous rates of change, linking calculus to real-world situations.
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Second derivative and concavity
Understand and calculate the second derivative to analyse the concavity of functions and identify points of inflection.
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Probability
This unit will equip you with the tools to understand and analyse uncertain events. You will learn to use set notation and Venn diagrams, master probability rules, and explore counting techniques like permutations and combinations.
Content
Set notation and Venn diagrams
Learn to represent events using set notation and Venn diagrams, making complex relationships clearer.
Worksheets:
Relative frequency
Understand how experimental probability is interpreted as relative frequency through your own investigations.
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Theoretical probability
Develop the ability to calculate probabilities based on equally likely outcomes.
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Addition rule of probability
Learn how to use the addition rule to combine probabilities for mutually exclusive and non–mutually exclusive events.
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Multiplication rule of probability
Master the multiplication rule to calculate the probability of independent or dependent events occurring together.
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Probability trees
Use tree diagrams as a visual tool to represent sequences of events and calculate their probabilities.
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Conditional probability
Learn how to calculate the probability of an event given that another event has already occurred.
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Permutations and combinations
Practise counting outcomes using permutations and combinations to solve complex probability problems.
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Exponential and Logarithmic Functions
In this unit, you will explore exponential and logarithmic functions, which are key to modelling real-world growth and decay. You’ll learn about the constant e, how to differentiate exponential functions, and how to solve equations using logarithms.
Content
Exponential functions
Explore the properties and behaviours of exponential functions and see how they model growth and decay.
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Euler’s number, e
Understand the significance of Euler’s number (e) and its role in exponential functions.
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Differentiation of exponential functions
Learn how to differentiate exponential functions, including those with the base e, to solve rate of change problems.
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Logarithms
Develop a solid understanding of logarithms, the inverse of exponential functions, and learn how to work with them.
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Logarithm laws
Master the laws of logarithms to simplify and manipulate logarithmic expressions effectively.
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Change of base formula
Learn to use the change of base formula to evaluate logarithms in different bases.
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Logarithmic functions
Investigate the properties and graphs of logarithmic functions, linking them back to exponential models.
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Exponential equations
Practise solving exponential equations by applying logarithms to isolate the unknown.
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Applications of exponential functions
See how exponential functions are used to model real-world scenarios like compound interest, population growth, and radioactive decay.
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Trigonometric Functions
This unit takes your trigonometry skills to the next level by introducing you to trigonometric functions and their graphs. You will work with angles of any magnitude, explore identities, and learn how to analyse transformations in trigonometric graphs.
Content
Angles of any magnitude
Learn how to apply trigonometric ratios to angles beyond 90° so that you can handle a wider range of problems.
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Trigonometric identities
Practise proving and using fundamental trigonometric identities to simplify and solve equations.
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Radian measure
Deepen your understanding of radian measure and its advantages over degrees in certain calculations.
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Graphs of trigonometric functions
Learn to sketch and interpret the graphs of sine, cosine, and tangent functions, helping you visualise periodic behaviour.
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Amplitude, period, and phase shift
Analyse how changes in amplitude, period, and phase shift affect the graphs of trigonometric functions.
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Trigonometric equations
Develop techniques to solve trigonometric equations over specified intervals.
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Applications of trigonometric functions
Learn how to apply trigonometric functions to model periodic phenomena you encounter in real life.
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Discrete Probability Distributions
In this unit, you will investigate discrete probability distributions to understand how probabilities are assigned to distinct outcomes. You will learn to calculate important statistics like the mean, variance, and standard deviation, and apply these concepts using the binomial distribution.
Content
Random variables
Learn to define discrete random variables and understand how their distributions work.
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Discrete probability distributions
Practise constructing and interpreting probability distribution tables and graphs for various scenarios.
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Mean or expected value
Calculate the expected value of a discrete random variable, so you know what outcome to anticipate on average.
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Variance and standard deviation
Determine the variance and standard deviation of a distribution to measure how spread out the outcomes are.
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Binomial distribution
Explore the binomial distribution and see how it applies to situations with a fixed number of independent trials.
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Applications of discrete probability distributions
Learn to apply discrete probability distributions to solve practical problems in various fields.