9709 AS & A Level Mathematics: past papers
719 papers from 2002 to 2025, with 386 questions sorted by syllabus topic.
Quadratics Paper 1 · 14 questions
Completing the square and the vertex6
Complete the square, the condition for no real roots, and an exact solution.
Complete the square, then find an inverse and a composite function.
Complete the square to find a vertex, then the area between a parabola and a line.
Minimum point of a quadratic, then its image after a stretch and a translation.
Complete the square, then use the discriminant for a quadratic with no real roots.
Complete the square, solve with it, and sketch the turning point.
Solving quadratic equations3
Complete the square, the condition for no real roots, and an exact solution.
Solve an equation that is quadratic in a power of x.
Complete the square, solve with it, and sketch the turning point.
The discriminant and the nature of roots7
Complete the square, the condition for no real roots, and an exact solution.
Condition for a line and a curve not to meet.
Where a curve meets a line, and the condition for them not to meet.
Complete the square, then use the discriminant for a quadratic with no real roots.
Range of a constant for no real roots, and a quadratic in cos θ.
Where a parabola meets a line, and the condition for a tangent.
Trigonometric equation as a quadratic, and the discriminant to show no solutions.
Simultaneous equations and lines meeting curves5
Find where a curve and a straight line meet.
Condition for a line and a curve not to meet.
Where a curve meets a line, and the condition for them not to meet.
Where two curves meet, then the area of the region between them.
Where a parabola meets a line, and the condition for a tangent.
Functions Paper 1 · 14 questions
Functions, domain and range3
Composite functions, transformations of a curve and constants from a range.
Composite functions, range, whether an inverse exists, and a composite equation.
Inverse function, its domain and range, and a composite function equation.
Composite functions8
Show a function is increasing, decide if its inverse exists, and a composite function.
Complete the square, then find an inverse and a composite function.
Composite functions, transformations of a curve and constants from a range.
Sketch a transformed graph, find an inverse and solve a composite function equation.
Composite functions, range, whether an inverse exists, and a composite equation.
Inverse function, its domain and range, and a composite function equation.
Composite and inverse functions with an unknown constant and domain restrictions.
Inverse and composite functions, when no inverse exists, and a tangent triangle area.
One-one and inverse functions8
Show a function is increasing, decide if its inverse exists, and a composite function.
Complete the square, then find an inverse and a composite function.
Sketch a transformed graph, find an inverse and solve a composite function equation.
Composite functions, range, whether an inverse exists, and a composite equation.
Inverse function, its domain and range, and a composite function equation.
Inverse function: find it, sketch it and describe transformations of a sine graph.
Composite and inverse functions with an unknown constant and domain restrictions.
Inverse and composite functions, when no inverse exists, and a tangent triangle area.
Graphs of a function and its inverse1
Inverse function: find it, sketch it and describe transformations of a sine graph.
Transformations of graphs8
Describe combined transformations and write the equation after a stretch and reflection.
Describe two transformations, in order, that map one graph onto another.
Composite functions, transformations of a curve and constants from a range.
Sketch a transformed graph, find an inverse and solve a composite function equation.
Describe a sequence of graph transformations in the right order.
Minimum point of a quadratic, then its image after a stretch and a translation.
Inverse function: find it, sketch it and describe transformations of a sine graph.
Sketch a stretched and translated graph and write its equation.
Coordinate geometry Paper 1 · 10 questions
Parallel and perpendicular lines3
Rates of change on a curve, then the perpendicular bisector of a chord.
Perpendicular lines, a circle through three points, and a tangent.
Perpendicular bisector of a segment and a circle centred on a point.
The equation of a circle9
Circle from a diameter, the tangent at a point and where two tangents meet.
Points on a circle, the angle at the centre and the area of a segment.
Circle centre in terms of the radius, the radius from a distance, and the gradient of a tangent.
Circle in centre–radius form, the normal at a point and unknown constants.
Values of a constant for which a line touches a circle, then a perpendicular diameter.
Perpendicular lines, a circle through three points, and a tangent.
Circle and tangent: find a constant, the normal and parallel tangents.
Perpendicular bisector of a segment and a circle centred on a point.
Circle through two points: find the centre, then a tangent.
Tangents, chords and intersections7
Circle from a diameter, the tangent at a point and where two tangents meet.
Circle centre in terms of the radius, the radius from a distance, and the gradient of a tangent.
Circle in centre–radius form, the normal at a point and unknown constants.
Values of a constant for which a line touches a circle, then a perpendicular diameter.
Perpendicular lines, a circle through three points, and a tangent.
Circle and tangent: find a constant, the normal and parallel tangents.
Circle through two points: find the centre, then a tangent.
Circular measure Paper 1 · 9 questions
Arc length and sector area8
Circle geometry to show a length ratio, then the exact area of a shaded region.
Perimeter and area of a region made from a sector and a triangle.
Perimeter and area of a shape made of sectors; find the radius and angle.
A shape made of sectors: show a length, find an area, and find a radius from the perimeter.
Perimeter of a shape built from circular arcs, and an exact area.
Arc length and areas of sectors and segments in radians.
Perimeter and areas of sectors and triangles in a cross-section.
Arc length to find an angle, then the area of a triangle.
Segments, chords and composite shapes9
Circle geometry to show a length ratio, then the exact area of a shaded region.
Points on a circle, the angle at the centre and the area of a segment.
Perimeter and area of a region made from a sector and a triangle.
Perimeter and area of a shape made of sectors; find the radius and angle.
A shape made of sectors: show a length, find an area, and find a radius from the perimeter.
Perimeter of a shape built from circular arcs, and an exact area.
Arc length and areas of sectors and segments in radians.
Perimeter and areas of sectors and triangles in a cross-section.
Arc length to find an angle, then the area of a triangle.
Trigonometry Paper 1 · 12 questions
Exact values and the unit circle1
Exact value from an equation with inverse tan and inverse cos.
Graphs and transformations3
Sketch a trigonometric graph with a linear term, count solutions and solve an equation.
Range and sketch of a trigonometric curve, then count solutions of an equation.
Find the constants of a sine curve from its graph and count solutions of related equations.
Inverse trigonometric functions1
Exact value from an equation with inverse tan and inverse cos.
Trigonometric identities6
Prove a trigonometric identity, then use it to solve an equation.
Prove a trigonometric identity, then solve an equation in an interval.
Rewrite a trigonometric equation in terms of sine, then solve it.
Prove a trigonometric identity and use it to solve an equation.
Prove a trigonometric identity and use it to solve an equation.
Prove a trigonometric identity, then solve the related equation.
Solving trigonometric equations9
Sketch a trigonometric graph with a linear term, count solutions and solve an equation.
Prove a trigonometric identity, then use it to solve an equation.
Prove a trigonometric identity, then solve an equation in an interval.
Rewrite a trigonometric equation in terms of sine, then solve it.
Prove a trigonometric identity and use it to solve an equation.
Prove a trigonometric identity and use it to solve an equation.
Range of a constant for no real roots, and a quadratic in cos θ.
Prove a trigonometric identity, then solve the related equation.
Trigonometric equation as a quadratic, and the discriminant to show no solutions.
Series Paper 1 · 20 questions
Binomial expansion8
Constant term in a binomial expansion.
Find a constant from a given coefficient in a binomial expansion.
Full binomial expansion, then use it to find one coefficient of a product.
The term independent of x in binomial expansions.
Equate binomial coefficients from two expansions to find a constant.
Binomial expansion: find a constant so one coefficient is a multiple of another.
Binomial expansion: compare coefficients to find an unknown constant.
Match binomial coefficients to find a positive constant.
Arithmetic progressions7
Linked geometric and arithmetic progressions, a sum to infinity and a partial sum.
Arithmetic and geometric progressions: find a constant, partial sums and a sum to infinity.
Sum the terms of an arithmetic progression that lie in a given range.
Sum of an arithmetic progression and a relationship between partial sums.
Arithmetic and geometric progressions with trigonometric terms, including a sum to infinity.
Sum of the first 50 terms of an arithmetic progression.
Arithmetic progression: find a constant, then the sum of 30 terms.
Geometric progressions8
Linked geometric and arithmetic progressions, a sum to infinity and a partial sum.
Arithmetic and geometric progressions: find a constant, partial sums and a sum to infinity.
Common ratio of a geometric progression and a sum to infinity after removing terms.
Arithmetic and geometric progressions with trigonometric terms, including a sum to infinity.
Geometric progression with trigonometric terms: find an angle, then a sum.
Geometric progressions: first term from the second term and the sum to infinity.
A term of a geometric progression.
Sum to infinity of a geometric progression from two of its terms.
Sum to infinity6
Linked geometric and arithmetic progressions, a sum to infinity and a partial sum.
Arithmetic and geometric progressions: find a constant, partial sums and a sum to infinity.
Common ratio of a geometric progression and a sum to infinity after removing terms.
Arithmetic and geometric progressions with trigonometric terms, including a sum to infinity.
Geometric progressions: first term from the second term and the sum to infinity.
Sum to infinity of a geometric progression from two of its terms.
Differentiation Paper 1 · 20 questions
Gradients and the power rule4
Find a constant from a gradient condition, then integrate to find the curve.
Gradient at a point, then the minimum point of a curve.
Gradient of a chord, then the gradient at a point as a limit.
Differentiate and find and classify a stationary point.
The chain rule1
Differentiate a square-root function, find a stationary point and a normal.
Tangents and normals6
Find a curve from its second derivative, classify stationary points and find a normal.
Areas bounded by a curve and the axes, and by a curve, a tangent and an axis.
Normal to a curve, find f(x) by integration, and whether f is increasing.
Normal to a curve from its gradient function, then the curve by integration.
Differentiate a square-root function, find a stationary point and a normal.
Inverse and composite functions, when no inverse exists, and a tangent triangle area.
Stationary points and increasing functions14
Locate a maximum point, then the exact area under the curve.
Show a function is increasing, decide if its inverse exists, and a composite function.
Rate of change along a curve, then a constant from a minimum point.
Find a curve from its second derivative, classify stationary points and find a normal.
Gradient at a point, then the minimum point of a curve.
Classify a stationary point with the second derivative, then find the curve by integration.
Normal to a curve, find f(x) by integration, and whether f is increasing.
Turning point and intercept of a curve, then the area between the curve and a line.
Where a function is decreasing from its derivative, then find the function from a point.
Stationary points, their nature, and where the function is increasing.
Differentiate a square-root function, find a stationary point and a normal.
Integrate a derivative to find the curve, then its stationary point and where it increases.
Minimum point, area between curves, and a rate of change.
Differentiate and find and classify a stationary point.
Optimisation and rates of change3
Rate of change along a curve, then a constant from a minimum point.
Rates of change on a curve, then the perpendicular bisector of a chord.
Minimum point, area between curves, and a rate of change.
Integration Paper 1 · 18 questions
Indefinite integration9
Find a constant from a gradient condition, then integrate to find the curve.
Find a curve from its second derivative, classify stationary points and find a normal.
Classify a stationary point with the second derivative, then find the curve by integration.
Normal to a curve, find f(x) by integration, and whether f is increasing.
Where a function is decreasing from its derivative, then find the function from a point.
Normal to a curve from its gradient function, then the curve by integration.
Find a curve from its derivative, a power of a linear bracket, and a point on it.
Integrate a derivative to find the curve, then its stationary point and where it increases.
Integrate a derivative and use a point to find the curve.
Integrating (ax + b)ⁿ1
Find a curve from its derivative, a power of a linear bracket, and a point on it.
Areas under and between curves9
Locate a maximum point, then the exact area under the curve.
Area between a curve and a line by integration.
Areas bounded by a curve and the axes, and by a curve, a tangent and an axis.
Complete the square to find a vertex, then the area between a parabola and a line.
Turning point and intercept of a curve, then the area between the curve and a line.
Where two curves meet, then the area of the region between them.
Area under a curve with fractional powers, by integration.
Minimum point, area between curves, and a rate of change.
Exact area between a square-root curve and a line.