YMA01 International A Level Mathematics: past papers

Write expressions with fractional and negative powers in the form kxⁿ.

Calculus with an expression in powers of x.

Work with indices, rewrite an equation by substitution and solve the quadratic.

Index laws and a surd equation, rationalising the answer.

Write an expression using indices, then differentiate.

Expand surds, rationalise a denominator and a proof.

A quadratic in x³, then the range of k giving no real roots.

Work with indices, rewrite an equation by substitution and solve the quadratic.

Sketch a curve with an asymptote, then the range of a constant for two intersections.

Complete the square, a maximum point, trigonometric graph features and a region by inequalities.

Where two quadratic curves meet, and the condition for one meeting point.

A line meeting a quadratic, and a region described with inequalities.

Sketch a cubic and find when an equation has distinct roots.

Solve a quadratic inequality.

Perimeter and area of a rectangle leading to simultaneous equations.

Complete the square and sketch the quadratic.

A region bounded by lines, described with inequalities.

Find a cubic from given facts, sketch curves and find where they meet.

Describe graph transformations and find an exact intersection.

Find a function from its derivative and a point, then transform a point.

Sketch a curve with an asymptote, then the range of a constant for two intersections.

Sketch transformations of y = f(x) with intercepts.

Sketch a cubic and find when an equation has distinct roots.

Complete the square and sketch the quadratic.

A perpendicular line, where two lines meet, and a distance to the origin.

Line through two points, a point at a given distance and a triangle area.

A region bounded by lines, described with inequalities.

A perpendicular line, where two lines meet, and a distance to the origin.

Gradients and the equation of a perpendicular line through a point.

A quadratic curve, gradients and perpendicular lines.

Sector and triangle in radians: an angle, then area and perimeter.

Angles, areas and arc lengths in radians for a badge design, and its perimeter.

Sector area, trigonometry in radians and a composite area.

Sector and triangle in radians: an angle, then area and perimeter.

Points on a cosine graph and the number of solutions of trigonometric equations.

Complete the square, a maximum point, trigonometric graph features and a region by inequalities.

Trigonometric graph: maximum point and shifts of sin x.

Trigonometric graphs and their transformations.

Angles, areas and arc lengths in radians for a badge design, and its perimeter.

Sector area, trigonometry in radians and a composite area.

Arc length, sector area and the perimeter and area of a garden plan.

Points on a cosine graph and the number of solutions of trigonometric equations.

Trigonometric graph: maximum point and shifts of sin x.

Trigonometric graphs and their transformations.

Sector and triangle in radians: an angle, then area and perimeter.

Angles, areas and arc lengths in radians for a badge design, and its perimeter.

Sector area, trigonometry in radians and a composite area.

Area of a circle segment and a related rate of change.

Arc length, sector area and the perimeter and area of a garden plan.

A quadratic curve, gradients and perpendicular lines.

Differentiate a power curve and find where the gradient takes a given value.

Differentiate a curve and use a tangent condition to find a constant.

Write an expression using indices, then differentiate.

A tangent from a derivative with an unknown, a second derivative, then integrate to find f(x).

First and second derivatives of a cubic and an unknown constant.

A tangent from a derivative with an unknown, a second derivative, then integrate to find f(x).

Differentiate a curve and use a tangent condition to find a constant.

Normal at a point, then integrate f′(x).

A tangent, then integrate to find f(x).

Integrate an expanded expression in powers of x and simplify.

Integrate powers of x.

A tangent from a derivative with an unknown, a second derivative, then integrate to find f(x).

Find a function from its derivative and a point, then transform a point.

Normal at a point, then integrate f′(x).

A tangent, then integrate to find f(x).

Proof by exhaustion about divisibility.

Counterexample about irrational numbers, and a proof about even numbers and multiples of 4.

Prove an inequality and give a counterexample.

Factor and remainder theorems, then polynomial division.

Factor and remainder theorems on a cubic, then factorise fully.

Remainder and factor theorems, factorise a cubic and count real roots.

Circle from a diameter, then the point nearest the x-axis.

A line through a midpoint and the centre, then the equation of the circle.

Circle centre and radius, and two circles that do not meet.

A line through a midpoint and the centre, then the equation of the circle.

Circle centre and radius, and two circles that do not meet.

Binomial expansion terms, then a coefficient to find an unknown.

Binomial expansion and choosing x for an approximation.

Binomial expansion and a coefficient in a product.

Geometric series sum and a bound on a term, and the greatest sum of an arithmetic series.

Number of terms and sum of an arithmetic series.

Arithmetic series: first term, difference and smallest n for a sum.

Geometric series sum and a bound on a term, and the greatest sum of an arithmetic series.

Common ratio and first term of a geometric series from partial and infinite sums.

Geometric models and when two models are equal.

Common ratio and first term of a geometric series from partial and infinite sums.

Trapezium rule for a logarithmic function, and log laws to find a constant.

Solve a logarithmic equation using the laws of logarithms.

Solve a logarithmic equation and find where exponential curves meet exactly.

Solve a logarithmic equation and find where exponential curves meet exactly.

Solve a logarithmic equation using the laws of logarithms.

Solve a tan equation in radians and a trigonometric equation in degrees.

Prove a trigonometric identity and solve an equation in an interval.

Solve a trigonometric equation and a sinusoidal temperature model.

Solve a tan equation in radians and a trigonometric equation in degrees.

Prove a trigonometric identity and solve an equation in an interval.

Solve a trigonometric equation and a sinusoidal temperature model.

Turning point, an x-intercept and the area of a region under a curve.

Stationary point, then an equal-area condition.

Model the surface area of a container and minimise it.

Optimise the dimensions of an open water tank with calculus.

Area of a circle segment and a related rate of change.

Trapezium rule, then use it to estimate related integrals.

Trapezium rule for a logarithmic function, and log laws to find a constant.

Sketch an exponential curve and use the trapezium rule.

Area between a curve and a line segment.

Turning point, an x-intercept and the area of a region under a curve.

Stationary point, then an equal-area condition.

A modulus function: vertex, an inequality and the range of k for two intersections.

Sketch transformed modulus graphs and solve a modulus equation.

Modulus graph, a modulus inequality and a transformation.

Composite functions, range, an inverse and solving f⁻¹(x) = f(x).

Intercept and stationary point of a logarithmic curve, then a range after a transformation.

Inverse functions, show a function is increasing, and the range of a composite.

Composite functions, range, an inverse and solving f⁻¹(x) = f(x).

Inverse and composite of a rational function, then transformations of a point.

Inverse functions, show a function is increasing, and the range of a composite.

A modulus function: vertex, an inequality and the range of k for two intersections.

Modulus graph, a modulus inequality and a transformation.

Composite functions, range, an inverse and solving f⁻¹(x) = f(x).

Inverse and composite of a rational function, then transformations of a point.

Inverse functions, show a function is increasing, and the range of a composite.

Inverse and composite of a rational function, then transformations of a point.

Sketch transformed modulus graphs and solve a modulus equation.

Modulus graph, a modulus inequality and a transformation.

Solve trigonometric equations in radians and degrees.

Prove a tangent identity, then solve an equation in an interval.

Solve trigonometric equations in radians and degrees.

Compound angle identities and a trigonometric equation.

Prove a tangent identity, then solve an equation in an interval.

Solve trigonometric equations in radians and degrees.

Trigonometric identities, R-form, and the maximum value.

Write a sine–cosine combination as R cos(x + α), then its minimum.

Trigonometric identities, R-form, and the maximum value.

An exponential cooling model: initial value, asymptote, a time and a rate.

Turn a straight-line log–log graph into a power law.

Cooling model: find constants and when the cooling rate takes a value.

A logarithmic model: initial value and time to reach a target.

An exponential cooling model: initial value, asymptote, a time and a rate.

Log graph of a model, showing N = abᵗ.

Cooling model: find constants and when the cooling rate takes a value.

Intercepts of a trigonometric curve, a second derivative, and a normal and tangent.

Intercept and stationary point of a logarithmic curve, then a range after a transformation.

Differentiate a trigonometric curve, show a gradient form and find an exact point.

Show a rational form of f(x) and differentiate it.

Where a rational function is increasing.

A projectile model: exact root, maximum and iteration.

Differentiate a trigonometric curve, show a gradient form and find an exact point.

A curve given by an equation in x and y: gradient, normal and a triangle area.

A stationary point, an iteration formula and a root to 3 decimal places.

Intercepts of a trigonometric curve, a second derivative, and a normal and tangent.

Where a rational function is increasing.

Intercept and stationary point of a logarithmic curve, then a range after a transformation.

A normal by differentiation and an area by integration.

A curve given by an equation in x and y: gradient, normal and a triangle area.

A trigonometric integral and an algebraic integral.

A normal by differentiation and an area by integration.

A trigonometric integral and an algebraic integral.

Find constants in a split fraction, then integrate to a logarithm.

Rewrite a rational function and integrate to a logarithm.

A stationary point, an iteration formula and a root to 3 decimal places.

Locate a root of a trigonometric function and use an iteration.

A stationary point, an iteration formula and a root to 3 decimal places.

Locate a root of a trigonometric function and use an iteration.

A projectile model: exact root, maximum and iteration.

Prove a trigonometric inequality by contradiction.

Proof by contradiction about whether an expression is even.

Binomial expansion to x³ and proof by contradiction.

Partial fractions, then an exact integral giving a logarithm.

Integrate an exponential expression, and partial fractions leading to a logarithm.

Area under a parametric curve with partial fractions.

Binomial series, then a numerical approximation.

Binomial expansion with a fractional power: constants, validity and a coefficient.

Binomial expansion to x³ and proof by contradiction.

Points on an implicit curve, then the tangent.

Implicit differentiation and the gradient where the curve meets the y-axis.

Implicit differentiation and a normal.

Normal to a parametric curve, then its Cartesian equation.

Tangent to a parametric curve and where it meets the curve again.

Area under a parametric curve with partial fractions.

Volume of revolution with integration by parts, and a composite solid.

Exact volume of a solid of revolution about the x-axis.

Volume of revolution using a substitution.

Integration by the substitution u = tan x.

Integrate an exponential expression, and partial fractions leading to a logarithm.

Exact definite integral with a trigonometric substitution.

Area under a parametric curve with partial fractions.

Volume of revolution using a substitution.

Volume of revolution with integration by parts, and a composite solid.

Integration by parts between limits.

Rate of change of the radius of a melting sphere.

Rate of change for an expanding sphere, and a differential equation for water depth.

Solve a differential equation with an initial condition.

Rate of change for an expanding sphere, and a differential equation for water depth.

Separable differential equation, long-term limit and a time.

Partial fractions, then an exact integral giving a logarithm.

Rate of change for an expanding sphere, and a differential equation for water depth.

Separable differential equation, long-term limit and a time.

Lines in vector form: coordinates, an angle and points giving a set triangle area.

Position vectors and perpendicular vectors.

Vector equation of a line and unknown coordinates from an angle.

Lines in vector form: coordinates, an angle and points giving a set triangle area.

Position vectors and perpendicular vectors.

Vector equation of a line and unknown coordinates from an angle.

Lines in vector form: coordinates, an angle and points giving a set triangle area.

Vector equations of lines, a point on a line and a triangle area.

Vector equations of lines, a point on a line and a triangle area.

Two tensions on a barge: an angle, then acceleration against resistance.

Acceleration from forces, then a constant that fixes the direction of the resultant.

Add force vectors given by magnitude and bearing.

Constant velocity with position vectors, and a bearing between two particles.

Two boats with constant acceleration vectors: speed, velocity and direction conditions.

Constant velocity from position vectors, then the distance to a fixed point.

Velocity and position vectors, bearings and closest approach.

Speed–time graph for a motorbike that speeds up and slows down.

Friction and resolving at constant speed, then deceleration and distance.

Vertical motion under gravity: launch and impact speeds, and a speed–time sketch.

Particles connected over a pulley: tension, impact speed and time to a height.

Motion under gravity then with resistance, and a speed–time graph.

Vertical motion under gravity: launch and impact speeds, and a speed–time sketch.

Speed–time graph for a motorbike that speeds up and slows down.

Motion under gravity then with resistance, and a speed–time graph.

Two tensions on a barge: an angle, then acceleration against resistance.

Acceleration from forces, then a constant that fixes the direction of the resultant.

Newton’s second law for particles connected vertically.

Motion under gravity then with resistance, and a speed–time graph.

Rough inclined plane: a deceleration, a starting speed and a minimum holding force.

Impulse, motion against friction before a collision, and speed after impact.

Friction and resolving at constant speed, then deceleration and distance.

Connected particles on a rough slope over a pulley.

Particles over a pulley: equations of motion, force on the pulley, impulse and time to rest.

Particles connected over a pulley: tension, impact speed and time to a height.

Newton’s second law for particles connected vertically.

Connected particles on a rough slope over a pulley.

Direct collision: momentum and impulse for a final speed and a mass ratio.

Particles over a pulley: equations of motion, force on the pulley, impulse and time to rest.

Impulse, motion against friction before a collision, and speed after impact.

Momentum and impulse when a string becomes taut.

Particle on a rough slope in limiting equilibrium, and whether it stays at rest when the string breaks.

Rough inclined plane: a deceleration, a starting speed and a minimum holding force.

Particle on a rough slope in limiting equilibrium, and whether it stays at rest when the string breaks.

A uniform rod held by two ropes, with a particle added and about to tilt.

A non-uniform rod on a ledge about to tilt: find its centre of mass.

A uniform rod held by two ropes, with a particle added and about to tilt.

Moments for a non-uniform rod about to tilt.

Modelling, correlation and regression from coded summary data.

Histogram heights, estimates, median, quartiles, skewness and a normal model.

Range, median, quartiles and standard deviation from a stem-and-leaf diagram, then adding data.

Read a box plot, interpolation and outliers, then a normal model.

Back-to-back stem and leaf, quartiles, outliers and box plots.

Histogram heights, estimates, median, quartiles, skewness and a normal model.

Read a box plot, interpolation and outliers, then a normal model.

Histogram, median by interpolation, coding, mean and standard deviation.

Combine two groups' means and standard deviations, then add a constant.

Range, median, quartiles and standard deviation from a stem-and-leaf diagram, then adding data.

Histogram, median by interpolation, coding, mean and standard deviation.

Combine two groups' means and standard deviations, then add a constant.

Range, median, quartiles and standard deviation from a stem-and-leaf diagram, then adding data.

Histogram, median by interpolation, coding, mean and standard deviation.

Union of events and conditional probability.

Three-set Venn diagram, a test for independence and conditional probability.

Venn diagrams, independence and conditional probability.

Union of events and conditional probability.

Tree diagram without replacement, conditional probability and a quadratic in n.

Complete a tree diagram and find a conditional probability.

Venn diagrams, independence and conditional probability.

Three-set Venn diagram, a test for independence and conditional probability.

Venn diagrams, independence and conditional probability.

Modelling, correlation and regression from coded summary data.

Correlation coefficient, a regression line and its meaning, and the effect of rescaling.

Summary statistics, correlation and the regression line.

Modelling, correlation and regression from coded summary data.

Correlation coefficient, a regression line and its meaning, and the effect of rescaling.

Summary statistics, correlation and the regression line.

Discrete distribution: cumulative probability, expectation, variance and a linear change.

Two four-sided dice: probabilities, expectation, variance and a missing value in a cumulative distribution.

Discrete distribution, E(X) and Var(X), and linear transformations.

Discrete distribution: cumulative probability, expectation, variance and a linear change.

Two four-sided dice: probabilities, expectation, variance and a missing value in a cumulative distribution.

Discrete distribution, E(X) and Var(X), and linear transformations.

Normal distribution, a binomial probability, and σ from a percentage.

Normal probabilities, and a mean and standard deviation from two conditions.

Read a box plot, interpolation and outliers, then a normal model.

Normal distribution: standardising and inverse normal.

Normal distribution, a binomial probability, and σ from a percentage.

Normal probabilities, and a mean and standard deviation from two conditions.

Normal distribution: standardising and inverse normal.