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Further Maths: Paper 2 (Calculator)
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Indices and Surds
Rewrite negative index as fraction
\[ 3x^{-7} \rightarrow \frac{3}{x^7} \]
Rewrite expressions with negative indices.
Simplify expressions using index laws
\[ a^4 \times a^{-9} \]
Apply index laws to simplify.
Simplify algebraic expressions with indices
\[ \frac{(4cd)^2}{2c^3d} \]
Simplify expressions with powers of products.
Solve index equations
\[ (a^3)^m = (a^2)^{m+4} \]
Solve equations by equating indices.
Simplify expressions with fractional indices
\[ x^{\frac{1}{3}} \times x^{\frac{3}{2}} \]
Add fractional indices when multiplying.
Simplify surd expressions
\[ x^3 \times \sqrt{x} \rightarrow x^n \]
Convert surds to index form and simplify.
Algebraic Manipulation
Factorise expressions with common factors
\[ 12w + 18w^2 \]
Take out the highest common factor.
Expand and simplify brackets
\[ (2x + 1)(x – 2)(x + 3) \]
Expand products of algebraic expressions.
Factorise expressions involving powers
\[ (p+6)^{11} – (p+6)^{10} \]
Identify a common bracket factor.
Simplify algebraic fractions by factorising
\[ \frac{x^2 + 2x – 15}{x^2 – 9} \]
Factorise and cancel common factors.
Multiply and divide algebraic fractions
\[ \frac{6a^2}{5b} \times \frac{10b}{3a} \]
Multiply fractions by factorising.
Add or subtract algebraic fractions
\[ \frac{2}{x} + \frac{3}{x^2} \]
Find a common denominator to add.
Binomial expansion – find coefficients
\[ (1 + ax)^5 \text{ coef relations} \]
Find relationships between coefficients.
Formulae and Equations
Rearrange formulae (subject appears once)
\[ w^2 = 3x + 5 \text{ for } w \]
Make a variable the subject.
Rearrange formulae (subject appears twice)
\[ y = \frac{3x + 2}{x – 4} \text{ for } x \]
Rearrange when subject appears on both sides.
Solve equations with surds or roots
\[ \sqrt[3]{2w – 10} = 4 \]
Solve equations involving roots.
Matrices
Matrix multiplication
\[ 2\begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix}\begin{pmatrix} -1 & 2 \\ 1 & 0 \end{pmatrix} \]
Multiply a scalar by a matrix product.
Solve matrix equations for unknowns
\[ \begin{pmatrix} c & d \\ 1 & 5 \end{pmatrix} – \begin{pmatrix} 2 & 0 \\ 7 & 3 \end{pmatrix} \]
Find unknowns in matrix equations.
Solve matrix multiplication equations
\[ \begin{pmatrix} -5 \\ 7 \end{pmatrix}\begin{pmatrix} a & 2 \end{pmatrix} \]
Find unknowns in matrix products.
Describe transformation from matrix
\[ A = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} \]
Identify the geometric transformation.
Coordinate Geometry
Identify circle centre and radius
\[ (x+7)^2 + (y-4)^2 = 36 \]
Read centre and radius from equation.
Find turning point from completed square
\[ y = 4 – (x – 3)^2 \]
Identify the turning point from completed square form.
Find equation of circle from diameter
\[ A(-2, 5), B(4, 13) \text{ diameter} \]
Find circle equation from diameter endpoints.
Find midpoint of line segment
\[ P(2, 6), Q(8, 10) \]
Find the midpoint of a line segment.
Solve line-circle intersection
\[ x^2 + y^2 = 25, y = x + 1 \]
Find where a line intersects a circle.
Functions
State range of a quadratic function
\[ f(x) = 3x^2 + 2 \]
Find the range given a restricted domain.
Find expression for composite function
\[ f(x) = (x+2)^3, gf(x) = (x+2)^{12} \]
Work backwards to find g(x).
Evaluate composite functions
\[ f^{-1}(x) + gf(x) \]
Find inverse and composite, then combine.
Find range from domain and graph
\[ g(x) = a \times b^x \]
Use points to find constants and range.
Calculus
Differentiate polynomials
\[ y = x^2 + \frac{4}{x^3} \]
Differentiate including negative indices.
Find second derivative
\[ \frac{d^2y}{dx^2} \]
Differentiate twice to find second derivative.
Find gradient at a point
\[ y = x^3 – 5x^2 \text{ at } x = -1 \]
Differentiate and substitute to find gradient.
Use gradient function
\[ \frac{17 – 5x}{10} = \text{gradient} \]
Set gradient function equal to given gradient.
Sequences
Find nth term of linear sequence
\[ 15, 18.5, 22, 25.5, … \]
Find the nth term of arithmetic sequence.
Find nth term of quadratic sequence
\[ -3, 3, 13, 27, … \]
Find nth term using second differences.
Use nth term to find specific term
\[ 42 – 3n \text{ first negative} \]
Find when terms satisfy a condition.
Trigonometry
Use trigonometry to find angles
\[ \sin x, \cos x, \tan x \]
Apply trig ratios to find angles.
Area of triangle using ½ab sin C
\[ \text{Area} = \frac{1}{2}ab\sin C \]
Use the area formula to find angles.
3D trigonometry
\[ \text{Pyramid: angle to base} \]
Find angles between lines and planes in 3D.
Solving trigonometric equations
\[ 2\tan^2 x = 3 \]
Find all solutions in 0° to 360°.
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