GCSE Further Maths practice, marked as they go.

1. 1Binomial Expansion+

   • Expand (a + b)ⁿ for positive integer n
2. 2Indices+

   • Index laws, including fractional and negative indices and the solution of equations
3. 3Quadratics+

   • Solution of linear and quadratic equations
   • Completing the square
4. 4Coordinate Geometry+

   • Know and use the definition of a gradient
   • Know the relationship between the gradients of parallel and perpendicular lines
   • Use Pythagoras' theorem to calculate the distance between two points
   • The equation of a straight line y = mx + c and y − y₁ = m(x − x₁) and other forms
   • Draw a straight line from given information
5. 5Ratio+

   • Use ratio to find the coordinates of a point on a line given the coordinates of two other points
6. 6Calculus: Introduction+

   • Know that the gradient function dy/dx gives the gradient of the curve and measures the rate of change of y with respect to x
   • Know that the gradient of a function is the gradient of the tangent at that point
   • Differentiation of kxⁿ where n is an integer, and the sum of such functions
   • The equation of a tangent and normal at any point on a curve
7. 7Surds+

   • Manipulation of surds, including rationalising the denominator
8. 8Trigonometry+

   • Sine and cosine rules in scalene triangles; area of a triangle = ½ ab sin C
   • Use of Pythagoras' theorem in 2D and 3D
   • Be able to apply trigonometry and Pythagoras' theorem to 2 and 3 dimensional problems
   • Sketch and use graphs of y = sin x, y = cos x and y = tan x for angles of any size
   • Be able to use the definitions of sin θ, cos θ and tan θ for any positive angle up to 360°
   • Knowledge and use of 30°, 60°, 90° triangles and 45°, 45°, 90° triangles
   • Know and use tan θ = sin θ ÷ cos θ and sin²θ + cos²θ = 1
   • Solution of simple trigonometric equations in given intervals
9. 9Algebraic Manipulation and the Factor Theorem+

   • Factorising
   • Algebraic division
   • Use of the factor theorem for rational values of the variable for polynomials
10. 10Functions+

    • Definition of a function
    • Domain and range of a function
    • Composite functions
    • Inverse functions
    • Drawing and sketching of functions
    • Interpretation of graphs
11. 11Simultaneous Equations+

    • Algebraic and graphical solution of simultaneous equations in two unknowns, where the equations could both be linear or one linear and one second order
    • Algebraic solution of linear equations in three unknowns
12. 12Sequences+

    • Using nth terms of sequences
    • Limiting value of a sequence as n → ∞
    • nth terms of linear sequences
    • nth terms of quadratic sequences
13. 13Inequalities+

    • Solution of linear and quadratic inequalities
14. 14Calculus: Stationary Points+

    • Increasing and decreasing functions
    • Understand and use the notation d²y/dx²
    • Use of differentiation to find maxima and minima points on a curve
    • Using calculus to find maxima and minima in simple problems
    • Sketch and interpret a curve with known maximum and minimum points
15. 15Matrices+

    • Multiplication of matrices
    • The identity matrix I
    • Transformations of the unit square in the x–y plane
    • Combination of transformations
16. 16Circle Geometry+

    • Understand that x² + y² = r² is the equation of a circle with centre (0, 0) and radius r
    • Understand that (x − a)² + (y − b)² = r² is the equation of a circle with centre (a, b) and radius r
    • The equation of a tangent at a point on a circle
17. 17Product Rule+

    • Use the product rule for counting
18. 18Geometric Proof+

    • Knowledge of angle properties of parallel and intersecting lines, triangles, all special types of quadrilaterals and polygons
    • Understand and use circle theorems
    • Understand and construct geometrical proofs using formal arguments
19. 19Algebraic Manipulation and Proof+

    • Manipulation of rational expressions
    • Use and manipulation of formulae and expressions
    • Algebraic proof

Topics follow the AQA 8365 specification. PrepAcer is independent and not affiliated with the exam board.