9709 AS & A Level Mathematics: past papers
719 papers from 2002 to 2025, with 386 questions sorted by syllabus topic.
Algebra Paper 2 · 7 questions
Modulus graphs and equations1
Sketch two graphs, then solve an inequality.
Modulus inequalities3
Solve a modulus inequality, then the largest integer satisfying a related one.
Sketch two graphs, then solve an inequality.
Solve an inequality involving the modulus.
Polynomials: division, factor and remainder theorems4
Divide a polynomial by a quadratic, then find exact real roots of a related equation.
Use the factor theorem to find coefficients, factorise, then solve an equation in cot.
Factor and remainder theorems, then a cosec equation.
Polynomial division, then a definite integral of the form a + b ln 2.
Logarithmic and exponential functions Paper 2 · 5 questions
Laws of logarithms2
Solve a logarithmic equation exactly.
Use logarithms to rearrange an exponential equation.
The functions eˣ and ln x1
Show a value cannot occur on a logarithmic curve, then find the equation of a tangent.
Exponential equations and inequalities2
Use logarithms to rearrange an exponential equation.
Solve an exponential equation using logarithms.
Exponential models and linear forms1
Constants from a straight-line graph of ln y against x.
Trigonometry Paper 2 · 8 questions
Reciprocal functions and their identities4
Solve an equation in tan and sec over a given range.
Use the factor theorem to find coefficients, factorise, then solve an equation in cot.
Factor and remainder theorems, then a cosec equation.
Solve an equation in sec and tan.
Compound-angle formulae2
Write a trigonometric expression in R cos(θ ± α) form, then solve an equation.
Prove an identity, an exact value, and integrate a trigonometric product.
Double-angle formulae1
Prove a trigonometric identity, solve an equation and integrate.
The R-form (harmonic form)2
Write a trigonometric expression in R cos(θ ± α) form, then solve an equation.
Write an expression in R-form and solve a related equation.
Differentiation Paper 2 · 10 questions
Derivatives of eˣ, ln x and trigonometric functions3
Show a value cannot occur on a logarithmic curve, then find the equation of a tangent.
Gradient of an exponential curve at a point.
Gradient of an exponential curve, an intercept and an area.
The product and quotient rules4
Find the coordinates of the stationary points of a curve.
Gradient of an exponential curve at a point.
Stationary point of a logarithmic function, located and found by iteration.
Exact gradient at an intercept and the exact maximum point.
Implicit differentiation1
Implicit differentiation, a gradient, and showing there is no vertical tangent.
Parametric differentiation2
Normal to a parametric curve.
Parametric differentiation: where the gradient is 1.
Stationary points and tangents4
Maximum points on a trigonometric curve, then an exact area.
Find the coordinates of the stationary points of a curve.
Show a value cannot occur on a logarithmic curve, then find the equation of a tangent.
Exact gradient at an intercept and the exact maximum point.
Integration Paper 2 · 11 questions
Integrating eˣ, 1/x and trigonometric functions4
Maximum points on a trigonometric curve, then an exact area.
Set up an iteration for where two curves meet, use it, then find an area between the curves.
A function value and an integral.
Gradient of an exponential curve, an intercept and an area.
Integration using trigonometric identities2
Prove a trigonometric identity, solve an equation and integrate.
Prove an identity, an exact value, and integrate a trigonometric product.
Exact definite integrals and logarithms5
Show an integral result with exponentials, locate a constant and find it by iteration.
Show the value of an integral that gives a logarithm, and find a whole-number constant.
Trapezium rule, exact area with a logarithm, and over- or under-estimate.
Polynomial division, then a definite integral of the form a + b ln 2.
Exact area under a reciprocal-type curve.
The trapezium rule1
Trapezium rule, exact area with a logarithm, and over- or under-estimate.
Numerical solution of equations Paper 2 · 5 questions
Locating roots: sketches and sign changes4
Show an integral result with exponentials, locate a constant and find it by iteration.
Sketch graphs, form an equation for where they meet and solve it by iteration.
Stationary point of a logarithmic function, located and found by iteration.
Sketch graphs, locate a root and iterate.
Fixed-point iteration5
Show an integral result with exponentials, locate a constant and find it by iteration.
Set up an iteration for where two curves meet, use it, then find an area between the curves.
Sketch graphs, form an equation for where they meet and solve it by iteration.
Stationary point of a logarithmic function, located and found by iteration.
Sketch graphs, locate a root and iterate.