9709 AS & A Level Mathematics: past papers
719 papers from 2002 to 2025, with 386 questions sorted by syllabus topic.
Algebra Paper 3 · 19 questions
Modulus graphs and equations3
Sketch a modulus graph and solve an inequality.
Sketch a modulus graph and solve a modulus inequality.
Sketch a modulus graph and solve an inequality.
Modulus inequalities4
Sketch a modulus graph and solve an inequality.
Sketch a modulus graph and solve a modulus inequality.
Sketch a modulus graph and solve an inequality.
Solve an inequality involving the modulus.
Polynomials: division, factor and remainder theorems6
Factor theorem with a repeated factor, finding unknown coefficients.
Polynomial division, then an exact definite integral.
Coefficients from the factor and remainder theorems, then factorise and solve an inequality.
Find two coefficients of a cubic from the factor and remainder theorems.
Factor theorem to find a constant, then a cubic inequality.
Find two coefficients of a cubic from divisibility and a remainder.
Partial fractions7
Partial fractions, then an exact integral in logarithmic form.
Partial fractions, then a definite integral with a logarithmic answer.
Stationary point, then an integral by substitution and partial fractions.
Split a rational expression into partial fractions.
Partial fractions and a definite integral to a given exact result.
Partial fractions, then integrate to a single logarithm.
Partial fractions, then integrate between limits.
Binomial expansion for any rational n2
Expand an algebraic expression as a series up to x² and state where it is valid.
Binomial expansion for a rational power, up to x².
Logarithmic and exponential functions Paper 3 · 9 questions
Exponential equations and inequalities5
Solve an exponential equation to 3 significant figures.
Solve an equation involving exponentials, giving the answer to 3 decimal places.
Solve an exponential or logarithmic equation to 3 decimal places.
Solve an exponential equation with different bases.
Solve an equation with exponentials and logarithms to 3 decimal places.
Logarithmic equations2
Solve a logarithmic equation to 4 decimal places.
Solve a logarithmic equation exactly.
Exponential models and linear forms2
Find constants from a straight-line graph of ln y against x.
Show a relationship becomes a straight line using logarithms and find its equation.
Trigonometry Paper 3 · 10 questions
Reciprocal functions and their identities2
Solve a trigonometric equation involving cotangent for angles between 0° and 180°.
Turn a tan equation into a quadratic in tan x and solve it.
Compound-angle formulae3
Turn a trigonometric equation into a quadratic in tan x and solve it.
Turn a tan equation into a quadratic in tan x and solve it.
Prove a triple-angle identity and use it to solve an equation.
Double-angle formulae8
Prove a trigonometric identity, then use it for an exact integral.
Solve a trigonometric equation involving cotangent for angles between 0° and 180°.
Turn a trigonometric equation into a quadratic in tan x and solve it.
Prove a trigonometric identity and use it to evaluate an integral.
Prove a triple-angle identity and use it to solve an equation.
Solve a trigonometric equation involving a half angle.
Write a combination of cos and sin in R-form, then solve a double-angle equation.
Solve an equation with cos 2θ and cos θ.
The R-form (harmonic form)2
Write a trigonometric combination as R cos(θ − α), then solve an equation with it.
Write a combination of cos and sin in R-form, then solve a double-angle equation.
Differentiation Paper 3 · 17 questions
Derivatives of eˣ, ln x and trigonometric functions2
Differentiate a secant expression, then solve a differential equation.
Show a derivative result, then solve a separable differential equation.
The product and quotient rules2
Stationary point of a curve made of a trigonometric product.
Stationary point on a trigonometric curve.
Implicit differentiation5
Implicit differentiation, then where the normal is parallel to the y-axis.
Gradient of an implicitly defined curve at a point.
Gradient of an implicitly defined curve at a point.
Implicit differentiation and points where the tangent has a given direction.
Implicit differentiation and points with a horizontal tangent.
Parametric differentiation2
Parametric differentiation and the normal at a given parameter.
Gradient of a parametric curve at a point.
Stationary points and tangents9
Implicit differentiation, then where the normal is parallel to the y-axis.
Exact position of a maximum point, then an enclosed area using a substitution.
Equation for a turning point, check a root lies in an interval, then iterate.
Stationary point, then an integral by substitution and partial fractions.
Find a maximum point, then evaluate an integral.
Maximum point of a trigonometric curve and the exact area under it.
Maximum point of a curve and an area by substitution.
Stationary point of a curve made of a trigonometric product.
Stationary point on a trigonometric curve.
Integration Paper 3 · 17 questions
Integrating eˣ, 1/x and trigonometric functions1
Evaluate an exact definite integral.
Integration using trigonometric identities3
Prove a trigonometric identity, then use it for an exact integral.
Exact value of a definite integral of a product of trigonometric functions.
Prove a trigonometric identity and use it to evaluate an integral.
Exact definite integrals and logarithms7
Partial fractions, then an exact integral in logarithmic form.
Polynomial division, then an exact definite integral.
Partial fractions, then a definite integral with a logarithmic answer.
Evaluate an exact definite integral.
Partial fractions and a definite integral to a given exact result.
Partial fractions, then integrate to a single logarithm.
Partial fractions, then integrate between limits.
Integration by substitution5
Exact value of a definite integral.
Exact position of a maximum point, then an enclosed area using a substitution.
Stationary point, then an integral by substitution and partial fractions.
Maximum point of a curve and an area by substitution.
An exact value from a sine curve, then an area by substitution.
Integration by parts4
Exact value of a definite integral.
Exact value of a definite integral of a product of trigonometric functions.
Find a maximum point, then evaluate an integral.
Maximum point of a trigonometric curve and the exact area under it.
Numerical solution of equations Paper 3 · 9 questions
Locating roots: sketches and sign changes9
Locate a root graphically and by a sign change, then use an iterative formula.
Show a single root graphically, locate it by calculation, then use an iterative formula.
Equation for a turning point, check a root lies in an interval, then iterate.
Sketch graphs to show there is one root, and check an iteration converges.
Locate a root, show an iteration converges to it, and use the iteration.
Locate a root by sketch and sign change, then iterate.
Locate a root and use fixed-point iteration.
Set up an equation for an angle from a segment area and use iteration.
Show there is one root by sketching, then iterate.
Fixed-point iteration8
Locate a root graphically and by a sign change, then use an iterative formula.
Show a single root graphically, locate it by calculation, then use an iterative formula.
Equation for a turning point, check a root lies in an interval, then iterate.
Locate a root, show an iteration converges to it, and use the iteration.
Locate a root by sketch and sign change, then iterate.
Locate a root and use fixed-point iteration.
Set up an equation for an angle from a segment area and use iteration.
Show there is one root by sketching, then iterate.
Rearrangements, staircases and cobwebs1
Sketch graphs to show there is one root, and check an iteration converges.
Vectors Paper 3 · 9 questions
Position vectors, magnitude and unit vectors3
Vectors: a midpoint, the angle between two vectors and the area of a triangle.
Position vectors in a trapezium, where diagonals meet, and an angle by scalar product.
Angle between vectors by scalar product, then a triangle’s area in 3D.
The scalar product and angles7
Vectors: a midpoint, the angle between two vectors and the area of a triangle.
Vector equation of a line, an angle by scalar product and the exact area of a triangle.
Show two lines are skew and find the obtuse angle between them.
Position vectors in a trapezium, where diagonals meet, and an angle by scalar product.
Angle between two lines, then a perpendicular distance from a point to a line.
Angle between vectors by scalar product, then a triangle’s area in 3D.
Two lines in 3D: when they meet, perpendicularity and the point of intersection.
The vector equation of a line3
Vector equation of a line, an angle by scalar product and the exact area of a triangle.
Vector equations of two lines, where they meet, and a point at a given distance.
Two lines in 3D: when they meet, perpendicularity and the point of intersection.
Intersecting, parallel and skew lines5
Show two lines are skew and find the obtuse angle between them.
Position vectors in a trapezium, where diagonals meet, and an angle by scalar product.
Vector equations of two lines, where they meet, and a point at a given distance.
Show two lines do not meet and find the foot of a perpendicular in 3D.
Two lines in 3D: when they meet, perpendicularity and the point of intersection.
Foot of the perpendicular and shortest distances2
Angle between two lines, then a perpendicular distance from a point to a line.
Show two lines do not meet and find the foot of a perpendicular in 3D.
Differential equations Paper 3 · 9 questions
Forming differential equations2
Form and solve a differential equation for a tank filling and draining.
Form and solve a differential equation for a balloon’s radius.
Separating the variables7
Solve a first-order differential equation from an initial condition.
Solve a differential equation with an initial condition.
Differentiate a secant expression, then solve a differential equation.
Solve a separable differential equation with a condition and find a limit.
Solve a separable differential equation and describe the long-term behaviour.
Show a derivative result, then solve a separable differential equation.
Solve a separable differential equation with an initial condition.
Particular solutions with harder integrals4
Solve a first-order differential equation from an initial condition.
Form and solve a differential equation for a balloon’s radius.
Solve a separable differential equation and describe the long-term behaviour.
Solve a separable differential equation with an initial condition.
Modelling with differential equations2
Form and solve a differential equation for a tank filling and draining.
Form and solve a differential equation for a balloon’s radius.
Complex numbers Paper 3 · 14 questions
Arithmetic and conjugates6
Find the exact square roots of a complex number by forming a quartic.
Exact square roots of a complex number by forming a quartic.
Square roots of a complex number.
Complex products, modulus–argument form and a geometric result.
Equate real and imaginary parts to form and solve an equation.
Simplify a complex expression and find its argument.
The Argand diagram, modulus and argument5
Argand diagram: where a line meets a circle, and the greatest argument in a region.
Argument of a complex quotient, and the geometric transformation to its conjugate.
Complex products, modulus–argument form and a geometric result.
Simplify a complex expression and find its argument.
Solve a complex quadratic, then modulus, argument and an equilateral triangle.
Polar and exponential forms1
Complex products, modulus–argument form and a geometric result.
Polynomial equations and square roots6
Find the exact square roots of a complex number by forming a quartic.
Exact square roots of a complex number by forming a quartic.
Square roots of a complex number.
Equate real and imaginary parts to form and solve an equation.
Solve a complex quadratic, then modulus, argument and an equilateral triangle.
Use a complex root of a cubic, then sketch a locus and find the greatest argument.
Loci and regions in the Argand diagram6
Argand diagram: where a line meets a circle, and the greatest argument in a region.
Shade a region of the Argand diagram given by a modulus and an argument inequality.
Describe a region of the Argand diagram with inequalities and find its greatest modulus.
Shade a region on an Argand diagram and find the greatest argument.
Sketch a circular locus and find the least modulus.
Use a complex root of a cubic, then sketch a locus and find the greatest argument.