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AQA Level 2 Further Mathematics Past Papers

Download AQA Level 2 Certificate in Further Mathematics past papers. Number, algebra, geometry, and calculus beyond GCSE. 22 resources for gifted Year 10-11 mathematicians.

Papers Overview Paper Structure Key Info Key Topics Command Words Grade Boundaries Revision Tips Try AI Tutor ✦

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18 of 18 resources

June 2023

5 files

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AQA Certificate Level 2 Further Mathematics – Mark scheme: Paper 2 Calculator – June 2023

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AQA Certificate Level 2 Further Mathematics – Question paper: Paper 1 Non-calculator – June 2023

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AQA Certificate Level 2 Further Mathematics – Question paper: Paper 2 Calculator – June 2023

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AQA Certificate Level 2 Further Mathematics – Question paper (Modified A4 18pt): Paper 2 Calculator – June 2023

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AQA Certificate Level 2 Further Mathematics – Question paper (Modified A3 36pt): Paper 2 Calculator – June 2023

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June 2022

7 files

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AQA Certificate Level 2 Further Mathematics – Mark scheme: Paper 2 Calculator – June 2022

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AQA Certificate Level 2 Further Mathematics – Question paper: Paper 1 Non-calculator – June 2022

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AQA Certificate Level 2 Further Mathematics – Question paper: Paper 2 Calculator – June 2022

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AQA Certificate Level 2 Further Mathematics – Question paper (Modified A3 36pt): Paper 1 Non-calculator – June 2022

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AQA Certificate Level 2 Further Mathematics – Question paper (Modified A4 18pt): Paper 2 Calculator – June 2022

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AQA Certificate Level 2 Further Mathematics – Question paper (Modified A4 18pt): Paper 1 Non-calculator – June 2022

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AQA Certificate Level 2 Further Mathematics – Question paper (Modified A3 36pt): Paper 2 Calculator – June 2022

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November 2021

6 files

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AQA Certificate Level 2 Further Mathematics – Mark scheme: Paper 1 Non-calculator – November 2021

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AQA Certificate Level 2 Further Mathematics – Mark scheme: Paper 2 Calculator – November 2021

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AQA Certificate Level 2 Further Mathematics – Question paper: Paper 2 Calculator – November 2021

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AQA Certificate Level 2 Further Mathematics – Question paper: Paper 1 Non-calculator – November 2021

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AQA Certificate Level 2 Further Mathematics – Question paper (Modified A4 18pt): Paper 2 Calculator – November 2021

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AQA Certificate Level 2 Further Mathematics – Question paper (Modified A4 18pt): Paper 1 Non-calculator – November 2021

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Beyond GCSE: Calculus, Matrices, and Advanced Algebra for Gifted Secondary Mathematicians

AQA Level 2 Certificate in Further Mathematics (specification 8365) is an enrichment qualification for high-attaining students in Years 10 and 11 who wish to extend their mathematical knowledge beyond GCSE standard in preparation for A-Level Mathematics and Further Mathematics. It is not a GCSE substitute but an additional qualification, typically awarded alongside GCSE Mathematics. The specification covers content from four areas: Number and Algebra, Geometry and Measures, Calculus, and Matrices. In Number and Algebra, students encounter surds and rational exponents, factor theorem and polynomial division, algebraic fractions, and the solution of simultaneous equations where one is non-linear (e.g., a circle and a straight line). Geometry and Measures extends beyond GCSE into circle theorems at depth, the equation of a circle in the form (x – a)² + (y – b)² = r², sine and cosine rules applied in non-right-angled triangles, exact trigonometric values, and use of radian measure. Calculus at this level introduces the concept of differentiation from first principles, differentiation of polynomial functions (dy/dx = nxⁿ⁻¹), finding the gradient at a point, identifying turning points by setting the derivative to zero, and distinguishing between maxima and minima using the second derivative. Integration of polynomial functions and its use in finding areas under curves is also examined. Matrices covers 2×2 matrix arithmetic (addition, scalar multiplication, matrix multiplication), the determinant, and the inverse of a 2×2 matrix, with applications to solving simultaneous equations. The two-paper structure mirrors GCSE: Paper 1 is non-calculator (testing algebraic fluency, exact values, and pure reasoning), while Paper 2 allows a calculator for more computationally demanding problems.

Exam Paper Structure

Paper 1No calculator

Non-Calculator Further Maths

⏱ Timed examination🎯 marks📊 50% of grade

Surds and rational exponentsFactor theorem and polynomial divisionExact trigonometric valuesAlgebraic fractions and simultaneous equationsGeometric proofs with circle theorems

Paper 2Calculator ✓

Calculator Further Maths

⏱ Timed examination🎯 marks📊 50% of grade

Differentiation and integration of polynomialsTurning points and optimisationEquation of a circleMatrix arithmetic and the inverse matrixSine and cosine rules in non-right-angled triangles

Key Information

Exam Board	AQA
Specification Code	8365
Qualification	Level 2 Certificate
Target Audience	High-attaining Years 10–11 students
Paper 1	Non-calculator (algebra, surds, exact trigonometric values)
Paper 2	Calculator (geometry, calculus, matrices, applied problems)
Progression	Excellent preparation for A-Level Mathematics and Further Mathematics
Total Resources	22

Key Topics in Further Mathematics

Topics you need to know

Differentiation from first principles and polynomial differentiationIntegration and area under curvesMatrices (multiplication, determinant, inverse)The equation of a circle and intersectionsFactor theorem and algebraic divisionSurds and irrational numbersAdvanced trigonometry (sine rule, cosine rule, radians)

Exam Command Words

Command word	What the examiner expects
Differentiate	Apply the rule dy/dx = nxⁿ⁻¹ (or first principles) to find the gradient function of an expression
Show that	Prove a given result using full mathematical working — the answer is given, so justification is everything
Find	Determine a numerical or algebraic answer by applying the appropriate method
Prove	Establish a result rigorously using logical mathematical steps
Hence	Use the result from the previous part to solve this part — a direct method may not gain full credit

Typical Grade Boundaries

Grade	Approximate mark needed
A*	88-96%
A	74-87%
B	60-73%
C	46-59%

⚠️ AQA Level 2 Further Mathematics is graded A*–C (enrichment qualification, not a GCSE replacement). AQA publishes session-specific thresholds.

Differentiation Technique, Circle Geometry, and Matrix Operations

Level 2 Further Mathematics tests skills that many GCSE students have never encountered, so a systematic approach is essential. For calculus, begin with differentiation: write down dy/dx = nxⁿ⁻¹ and apply it term by term. To find where the gradient is zero (at turning points), set dy/dx = 0 and solve — you will typically get a polynomial equation. To determine whether the turning point is a maximum or minimum, differentiate again to get d²y/dx² and substitute your x-value: if d²y/dx² > 0, the curve is concave up (minimum); if d²y/dx² < 0, concave down (maximum). For circle theorems and the equation of a circle, practise recognising when a question is about a specific circle: (x – 3)² + (y + 1)² = 25 is a circle centred at (3, –1) with radius 5. To find where a line intersects a circle, substitute the line equation into the circle equation and solve the resulting quadratic — you will need the discriminant to determine whether the line is a tangent (discriminant = 0), secant (two intersections), or misses the circle. Matrix multiplication follows the row-by-column rule: the element in row i, column j of the product equals the dot product of row i of the first matrix and column j of the second. Note that matrix multiplication is not commutative (AB ≠ BA in general). The inverse of a 2×2 matrix [a b; c d] is (1/(ad – bc)) × [d –b; –c a] — provided ad – bc ≠ 0 (a non-zero determinant). If the determinant is zero, the matrix has no inverse, meaning the corresponding system of equations has no unique solution.