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FM – Paper 1 (AQA Further Maths)
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Factorising
Factorise difference of squares
\[ x^2 – 49 \]
Factorise a² – b² expressions.
Factorise quadratics (coef 1)
\[ x^2 + 5x – 14 \]
Factorise when x² coefficient is 1.
Factorise quadratics (coef > 1)
\[ 6x^2 + 7x – 3 \]
Factorise with leading coefficient > 1.
Factor theorem
\[ x^3 – 5x^2 + 2x + 8 \]
Use factor theorem to factorise cubics.
Surds and Indices
Simplify a surd
\[ \sqrt{72} = 6\sqrt{2} \]
Simplify using largest square factor.
Add and subtract surds
\[ 3\sqrt{5} + 2\sqrt{45} \]
Simplify to like surds then combine.
Expand brackets with surds
\[ (3 + \sqrt{5})(2 – \sqrt{5}) \]
Expand and simplify surd expressions.
Rationalise single surd
\[ \frac{12}{\sqrt{6}} \]
Rationalise single surd denominators.
Rationalise two-term
\[ \frac{3}{2 – \sqrt{5}} \]
Rationalise using conjugate.
Negative indices
\[ \frac{x^3}{x^{-2}} \]
Simplify with negative indices.
Fractional indices
\[ 27^{\frac{2}{3}} = 9 \]
Evaluate fractional powers.
Completing the Square
Complete the square
\[ x^2 – 6x + 7 \]
Write in (x + a)² + b form.
Solve using completed square
\[ (x – 3)^2 – 7 = 0 \]
Solve giving exact answers.
Algebraic Manipulation
Simplify algebraic fractions
\[ \frac{x^2 – 9}{x^2 + 5x + 6} \]
Factorise and cancel.
Add/subtract algebraic fractions
\[ \frac{2}{x-1} + \frac{3}{x+2} \]
Combine into single fraction.
Multiply/divide fractions
\[ \frac{x^2-4}{x-2} \times \frac{1}{x+2} \]
Multiply or divide and simplify.
Equations and Inequalities
Simultaneous equations
\[ y = 2x + 1, \; y = x^2 \]
Solve linear-quadratic pairs.
Quadratic inequalities
\[ x^2 – 5x + 4 < 0 \]
Solve quadratic inequalities.
Equations with fractions
\[ \frac{x}{4} + \frac{3}{x} = 2 \]
Solve equations with algebraic fractions.
Differentiation
Differentiate polynomials
\[ y = 3x^3 – 2x^2 + 5 \]
Find dy/dx using power rule.
Gradient at a point
\[ y = x^3 – 4x \text{ at } (2, 0) \]
Find gradient using differentiation.
Stationary points
\[ y = x^3 – 3x^2 – 9x + 5 \]
Find and classify stationary points.
Equation of tangent
\[ y = x^2 – 3x + 5 \text{ at } (2, 3) \]
Find tangent line equation.
Coordinate Geometry
Gradient and distance
\[ (3, -2) \text{ and } (7, 6) \]
Find gradient and distance between points.
Equation of a line
\[ \text{Through } (4, -1), m = 2 \]
Find line equation from point and gradient.
Parallel/perpendicular lines
\[ y = 3x – 2 \text{ through } (6, 1) \]
Find parallel or perpendicular lines.
Equation of a circle
\[ \text{Centre } (3, -2), r = 5 \]
Write circle equation in standard form.
Tangent to a circle
\[ x^2 + y^2 = 25 \text{ at } (3, 4) \]
Find tangent line to circle.
Trigonometry
Know exact trig values
\[ \tan 60° = \sqrt{3} \]
Exact sin, cos, tan for 0°, 30°, 45°, 60°, 90°.
Pythagorean identity
\[ \cos\theta = \frac{3}{5}, \text{ find } \sin\theta \]
Use sin²θ + cos²θ = 1.
Solve trig equations
\[ 2\sin x = 1 \text{ for } 0° \leq x \leq 360° \]
Find all solutions in range.
Cosine rule
\[ a^2 = b^2 + c^2 – 2bc\cos A \]
Find sides or angles in triangles.
Sequences
Quadratic sequence nth term
\[ 3, 8, 15, 24, … \]
Find nth term of quadratic sequence.
Limiting value
\[ \frac{3n + 2}{n + 5} \text{ as } n \to \infty \]
Find limit of sequence.
Functions
Inverse function
\[ f(x) = 2x + 3, \text{ find } f^{-1}(x) \]
Find the inverse function.
Composite functions
\[ f(x) = 2x + 1, g(x) = x^2, fg(3) \]
Evaluate fg(x) or gf(x).
Matrix Transformations
Transformation matrices
\[ \text{Reflection in } y = -x \]
Write or identify transformation matrices.
Combined transformations
\[ A \text{ then } B = BA \]
Find matrix for combined transformations.
Circle Theorems
Angles with tangents
\[ \text{Tangent} \perp \text{radius} \]
Apply tangent circle theorems.
Angles at centre/circumference
\[ \angle \text{centre} = 2 \times \angle \text{circ} \]
Apply angle theorems in circles.
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