OMPT practice worksheets

The OMPT is the online mathematics placement test used by more than thirty universities, nearly all of them in Europe (omptest.org lists them). These are practice sheets for every level from OMPT-A to OMPT-G, chapter by chapter, and a mixed paper for each — 30 sheets, 900 questions, every one with a full worked answer and the common mistakes named. Free to print.

The seven levels are different SUBJECT LISTS, not seven difficulties of the same paper: OMPT-B adds trigonometry and integration to OMPT-A, while OMPT-E drops the calculus entirely and adds statistics, probability and counting. Practising the wrong one is wasted work, so start from the level your programme names.

Every topic on every syllabus. Each mathematics topic omptest.org lists for a level is asked on that level’s sheets. Where a sheet serves several levels and a question is not on all of their syllabuses, a label on the question says so: “OMPT-G only”, or “Not on OMPT-E”.

You choose how long each sheet is. Every sheet carries 30 questions and a control at the top sets it to 10, 20 or all of them — for a five-minute drill or a full period. The answers follow your choice, and it is remembered across the sheets.

The OMPT is open-answer: you type the answer rather than choosing one. So instead of explaining wrong options, each answer names the mistakes students actually make and shows the value each one leads to — if your answer matches one of them, you can see immediately where it went wrong.

Download all 30 sheets (PDF)

OMPT-A

Numbers, algebra, linear formulas, systems of equations, quadratics, functions, exponentials and logarithms, and differentiation. The test itself: 52 questions in 120 minutes.

330 questions across 11 sheets

Every topic on the OMPT-A syllabus is asked on these sheets.

• NumbersIntegers — the sign rules, division with a remainder, the greatest common divisor and the least common multiple — fractions and decimals, powers and roots, absolute value and the order of operations, all exact and without a calculator.30 questions
• AlgebraVariables, like terms and substitution, expanding brackets and the notable products, factorising, the exponent rules and surds, and algebraic fractions: adding, multiplying and dividing them, and splitting one into partial fractions.30 questions
• Linear formulas and equationsFormulas — evaluating one, working back from it and making a letter the subject — the slope, intercepts and equation of a line, parallel lines and where two lines meet, linear equations with brackets and with fractions, and linear inequalities.30 questions
• Systems of linear equationsTwo linear equations in two unknowns, by substitution and by elimination, and how many solutions a system has — one, none or infinitely many — with the general solution of a dependent system.30 questions
• Quadratic equationsThe parabola — its vertex, completing the square, and its shifts and stretches — solving by factorising and by the formula, the discriminant, where a parabola meets a line or another parabola, and quadratic inequalities.30 questions
• Functions: domain, range, asymptotes and limitsFunction rules and intervals, the domain and the range, fractional functions — their asymptotes, perforations and equations — long division of polynomials, oblique asymptotes, and limits.30 questions
• Functions: powers, roots, inverses and polynomialsPower and root functions and their transformations, equations with powers and with roots, inverse functions, and higher-degree polynomials: the degree, the leading coefficient, and polynomial equations and inequalities.30 questions
• Exponential functions and logarithmsExponential and logarithmic graphs — through two points, their asymptotes, an intercept and their transformations — the logarithm rules and change of base, and exponential and logarithmic equations, including the ones that need a substitution.30 questions
• DifferentiationThe difference quotient and the slope it closes in on, the derivatives of powers and of the standard functions, and the sum, product, chain and quotient rules, with composite functions.30 questions
• Applications of derivativesThe tangent line, where a function increases and decreases, stationary points and the extremes on a closed interval, the second and higher derivatives, and inflection points.30 questions
• OMPT-A mixed paperEvery OMPT-A topic together.30 questions

OMPT-B

Everything in OMPT-A, plus trigonometry and integration. The test itself: 64 questions in 150 minutes.

450 questions across 15 sheets

Every topic on the OMPT-B syllabus is asked on these sheets.

• NumbersIntegers — the sign rules, division with a remainder, the greatest common divisor and the least common multiple — fractions and decimals, powers and roots, absolute value and the order of operations, all exact and without a calculator.30 questions
• AlgebraVariables, like terms and substitution, expanding brackets and the notable products, factorising, the exponent rules and surds, and algebraic fractions: adding, multiplying and dividing them, and splitting one into partial fractions.30 questions
• Linear formulas and equationsFormulas — evaluating one, working back from it and making a letter the subject — the slope, intercepts and equation of a line, parallel lines and where two lines meet, linear equations with brackets and with fractions, and linear inequalities.30 questions
• Systems of linear equationsTwo linear equations in two unknowns, by substitution and by elimination, and how many solutions a system has — one, none or infinitely many — with the general solution of a dependent system.30 questions
• Quadratic equationsThe parabola — its vertex, completing the square, and its shifts and stretches — solving by factorising and by the formula, the discriminant, where a parabola meets a line or another parabola, and quadratic inequalities.30 questions
• Functions: domain, range, asymptotes and limitsFunction rules and intervals, the domain and the range, fractional functions — their asymptotes, perforations and equations — long division of polynomials, oblique asymptotes, and limits.30 questions
• Functions: powers, roots, inverses and polynomialsPower and root functions and their transformations, equations with powers and with roots, inverse functions, and higher-degree polynomials: the degree, the leading coefficient, and polynomial equations and inequalities.30 questions
• Exponential functions and logarithmsExponential and logarithmic graphs — through two points, their asymptotes, an intercept and their transformations — the logarithm rules and change of base, and exponential and logarithmic equations, including the ones that need a substitution.30 questions
• DifferentiationThe difference quotient and the slope it closes in on, the derivatives of powers and of the standard functions, and the sum, product, chain and quotient rules, with composite functions.30 questions
• Applications of derivativesThe tangent line, where a function increases and decreases, stationary points and the extremes on a closed interval, the second and higher derivatives, and inflection points.30 questions
• Trigonometry: angles and trianglesRight-angled triangles — Pythagoras, the angle sum, and sine, cosine and tangent — radians, large and negative angles, the unit circle and its symmetry, exact values, the addition and double-angle formulas, and the sine and cosine rules.30 questions
• Trigonometric functionsThe graphs of sine, cosine and tangent — period, amplitude and equilibrium — and the inverse functions arcsin, arccos and arctan, with what each one gives back.30 questions
• IntegrationAntiderivatives of the standard functions, definite integrals, the area under a graph and between two graphs, and volumes of revolution about either axis.30 questions
• Integration techniquesIntegration by substitution and by parts — repeated, and the integral that comes back round — the squares of sine and cosine, and quotients: by a logarithm, by long division and by partial fractions.30 questions
• OMPT-B mixed paperEvery OMPT-B topic together.30 questions

OMPT-C

Numbers, algebra, linear formulas and basic geometry: lines, angles, triangles and Pythagoras. No quadratics, no functions, no calculus. The test itself: 30 questions in 120 minutes.

150 questions across 5 sheets

Every topic on the OMPT-C syllabus is asked on these sheets.

• NumbersIntegers — the sign rules, division with a remainder, the greatest common divisor and the least common multiple — fractions and decimals, powers and roots, absolute value and the order of operations, all exact and without a calculator.30 questions
• AlgebraVariables, like terms and substitution, expanding brackets and the notable products, factorising, the exponent rules and surds, and algebraic fractions: adding, multiplying and dividing them, and splitting one into partial fractions.30 questions
• Linear formulas and equationsFormulas — evaluating one, working back from it and making a letter the subject — the slope, intercepts and equation of a line, parallel lines and where two lines meet, linear equations with brackets and with fractions, and linear inequalities.30 questions
• Geometry: angles, triangles and circlesAngles at lines, in triangles and in polygons, Pythagoras, the perimeter and area of a triangle, and the circle: its circumference, area, arcs and sectors.30 questions
• OMPT-C mixed paperEvery OMPT-C topic together.30 questions

OMPT-D

The OMPT-B chapters, with limits and asymptotes added to functions, plus geometry: lines, circles, vectors and parametric curves in coordinates, and calculations with triangles and circles. Numbers, algebra, linear formulas and systems are listed as prerequisite chapters. The test itself: 21 questions in 180 minutes.

540 questions across 18 sheets

Every topic on the OMPT-D syllabus is asked on these sheets.

• NumbersIntegers — the sign rules, division with a remainder, the greatest common divisor and the least common multiple — fractions and decimals, powers and roots, absolute value and the order of operations, all exact and without a calculator.30 questions
• AlgebraVariables, like terms and substitution, expanding brackets and the notable products, factorising, the exponent rules and surds, and algebraic fractions: adding, multiplying and dividing them, and splitting one into partial fractions.30 questions
• Linear formulas and equationsFormulas — evaluating one, working back from it and making a letter the subject — the slope, intercepts and equation of a line, parallel lines and where two lines meet, linear equations with brackets and with fractions, and linear inequalities.30 questions
• Systems of linear equationsTwo linear equations in two unknowns, by substitution and by elimination, and how many solutions a system has — one, none or infinitely many — with the general solution of a dependent system.30 questions
• Quadratic equationsThe parabola — its vertex, completing the square, and its shifts and stretches — solving by factorising and by the formula, the discriminant, where a parabola meets a line or another parabola, and quadratic inequalities.30 questions
• Functions: domain, range, asymptotes and limitsFunction rules and intervals, the domain and the range, fractional functions — their asymptotes, perforations and equations — long division of polynomials, oblique asymptotes, and limits.30 questions
• Functions: powers, roots, inverses and polynomialsPower and root functions and their transformations, equations with powers and with roots, inverse functions, and higher-degree polynomials: the degree, the leading coefficient, and polynomial equations and inequalities.30 questions
• Exponential functions and logarithmsExponential and logarithmic graphs — through two points, their asymptotes, an intercept and their transformations — the logarithm rules and change of base, and exponential and logarithmic equations, including the ones that need a substitution.30 questions
• DifferentiationThe difference quotient and the slope it closes in on, the derivatives of powers and of the standard functions, and the sum, product, chain and quotient rules, with composite functions.30 questions
• Applications of derivativesThe tangent line, where a function increases and decreases, stationary points and the extremes on a closed interval, the second and higher derivatives, and inflection points.30 questions
• Trigonometry: angles and trianglesRight-angled triangles — Pythagoras, the angle sum, and sine, cosine and tangent — radians, large and negative angles, the unit circle and its symmetry, exact values, the addition and double-angle formulas, and the sine and cosine rules.30 questions
• Trigonometric functionsThe graphs of sine, cosine and tangent — period, amplitude and equilibrium — and the inverse functions arcsin, arccos and arctan, with what each one gives back.30 questions
• IntegrationAntiderivatives of the standard functions, definite integrals, the area under a graph and between two graphs, and volumes of revolution about either axis.30 questions
• Integration techniquesIntegration by substitution and by parts — repeated, and the integral that comes back round — the squares of sine and cosine, and quotients: by a logarithm, by long division and by partial fractions.30 questions
• Geometry: angles, triangles and circlesAngles at lines, in triangles and in polygons, Pythagoras, the perimeter and area of a triangle, and the circle: its circumference, area, arcs and sectors.30 questions
• Geometry: lines and circles in coordinatesDistances between points, perpendicular lines and the angle between two lines, the distance from a point to a line, the equation of a circle, its tangents, and where a circle meets a line or another circle.30 questions
• Geometry: vectors and parametric curvesVectors — combining them, the vector between two points and its length — lines given by a vector, parametric curves and their slope, and Lissajous figures.30 questions
• OMPT-D mixed paperEvery OMPT-D topic together.30 questions

OMPT-E

Most of OMPT-A — but no differentiation, no inequalities, systems by substitution only and no higher-degree polynomials — plus descriptive statistics, probability, combinatorics, random variables and the binomial and normal distributions. The test itself: 40 questions in 120 minutes.

390 questions across 13 sheets

Every topic on the OMPT-E syllabus is asked on these sheets.

• NumbersIntegers — the sign rules, division with a remainder, the greatest common divisor and the least common multiple — fractions and decimals, powers and roots, absolute value and the order of operations, all exact and without a calculator.30 questions
• AlgebraVariables, like terms and substitution, expanding brackets and the notable products, factorising, the exponent rules and surds, and algebraic fractions: adding, multiplying and dividing them, and splitting one into partial fractions.30 questions
• Linear formulas and equationsFormulas — evaluating one, working back from it and making a letter the subject — the slope, intercepts and equation of a line, parallel lines and where two lines meet, linear equations with brackets and with fractions, and linear inequalities.30 questions
• Systems of linear equationsTwo linear equations in two unknowns, by substitution and by elimination, and how many solutions a system has — one, none or infinitely many — with the general solution of a dependent system.30 questions
• Quadratic equationsThe parabola — its vertex, completing the square, and its shifts and stretches — solving by factorising and by the formula, the discriminant, where a parabola meets a line or another parabola, and quadratic inequalities.30 questions
• Functions: domain, range, asymptotes and limitsFunction rules and intervals, the domain and the range, fractional functions — their asymptotes, perforations and equations — long division of polynomials, oblique asymptotes, and limits.30 questions
• Functions: powers, roots, inverses and polynomialsPower and root functions and their transformations, equations with powers and with roots, inverse functions, and higher-degree polynomials: the degree, the leading coefficient, and polynomial equations and inequalities.30 questions
• Exponential functions and logarithmsExponential and logarithmic graphs — through two points, their asymptotes, an intercept and their transformations — the logarithm rules and change of base, and exponential and logarithmic equations, including the ones that need a substitution.30 questions
• Descriptive statistics: data, tables and graphsTypes of data and the four measurement scales, frequency tables — relative and cumulative frequency, and the mean and median of the data in one — and reading a frequency distribution off a bar chart.30 questions
• Descriptive statistics: centre, spread and quantilesThe mean, median and mode, the range, variance, standard deviation and interquartile range, percentiles and quartiles, outliers and their effect on the mean, and the shape of a distribution.30 questions
• ProbabilitySets, sample spaces and events, the probability rules — complement, union, intersection and difference — conditional probability, independence and contingency tables, and sampling from a large population.30 questions
• Combinatorics, random variables and distributionsThe counting principle, permutations and combinations, random variables and their distributions, expected value, the binomial distribution, and the normal distribution with z-scores.30 questions
• OMPT-E mixed paperEvery OMPT-E topic together.30 questions

OMPT-F

Algebra, linear formulas, quadratics, functions, exponentials and logarithms, and differentiation. No numbers chapter and no systems. The test itself: 21 questions in 180 minutes.

270 questions across 9 sheets

Every topic on the OMPT-F syllabus is asked on these sheets.

• AlgebraVariables, like terms and substitution, expanding brackets and the notable products, factorising, the exponent rules and surds, and algebraic fractions: adding, multiplying and dividing them, and splitting one into partial fractions.30 questions
• Linear formulas and equationsFormulas — evaluating one, working back from it and making a letter the subject — the slope, intercepts and equation of a line, parallel lines and where two lines meet, linear equations with brackets and with fractions, and linear inequalities.30 questions
• Quadratic equationsThe parabola — its vertex, completing the square, and its shifts and stretches — solving by factorising and by the formula, the discriminant, where a parabola meets a line or another parabola, and quadratic inequalities.30 questions
• Functions: domain, range, asymptotes and limitsFunction rules and intervals, the domain and the range, fractional functions — their asymptotes, perforations and equations — long division of polynomials, oblique asymptotes, and limits.30 questions
• Functions: powers, roots, inverses and polynomialsPower and root functions and their transformations, equations with powers and with roots, inverse functions, and higher-degree polynomials: the degree, the leading coefficient, and polynomial equations and inequalities.30 questions
• Exponential functions and logarithmsExponential and logarithmic graphs — through two points, their asymptotes, an intercept and their transformations — the logarithm rules and change of base, and exponential and logarithmic equations, including the ones that need a substitution.30 questions
• DifferentiationThe difference quotient and the slope it closes in on, the derivatives of powers and of the standard functions, and the sum, product, chain and quotient rules, with composite functions.30 questions
• Applications of derivativesThe tangent line, where a function increases and decreases, stationary points and the extremes on a closed interval, the second and higher derivatives, and inflection points.30 questions
• OMPT-F mixed paperEvery OMPT-F topic together.30 questions

OMPT-G

Linear formulas through logarithms, plus sequences and series, trigonometry, and geometry from angles and triangles to lines and circles in coordinates — and a physics and a chemistry module. The test itself: 26 questions in 180 minutes.

390 questions across 13 sheets

Every mathematics topic on the OMPT-G syllabus is asked on these sheets. Not on them: the Physics (Mechanics) and Chemistry modules, which are not mathematics.

• Linear formulas and equationsFormulas — evaluating one, working back from it and making a letter the subject — the slope, intercepts and equation of a line, parallel lines and where two lines meet, linear equations with brackets and with fractions, and linear inequalities.30 questions
• Systems of linear equationsTwo linear equations in two unknowns, by substitution and by elimination, and how many solutions a system has — one, none or infinitely many — with the general solution of a dependent system.30 questions
• Quadratic equationsThe parabola — its vertex, completing the square, and its shifts and stretches — solving by factorising and by the formula, the discriminant, where a parabola meets a line or another parabola, and quadratic inequalities.30 questions
• Functions: domain, range, asymptotes and limitsFunction rules and intervals, the domain and the range, fractional functions — their asymptotes, perforations and equations — long division of polynomials, oblique asymptotes, and limits.30 questions
• Functions: powers, roots, inverses and polynomialsPower and root functions and their transformations, equations with powers and with roots, inverse functions, and higher-degree polynomials: the degree, the leading coefficient, and polynomial equations and inequalities.30 questions
• Exponential functions and logarithmsExponential and logarithmic graphs — through two points, their asymptotes, an intercept and their transformations — the logarithm rules and change of base, and exponential and logarithmic equations, including the ones that need a substitution.30 questions
• Trigonometry: angles and trianglesRight-angled triangles — Pythagoras, the angle sum, and sine, cosine and tangent — radians, large and negative angles, the unit circle and its symmetry, exact values, the addition and double-angle formulas, and the sine and cosine rules.30 questions
• Trigonometric functionsThe graphs of sine, cosine and tangent — period, amplitude and equilibrium — and the inverse functions arcsin, arccos and arctan, with what each one gives back.30 questions
• Trigonometric equationsEvery solution in one turn of an equation in sine, cosine or tangent, including the ones with a double angle and the ones that are quadratic in a trigonometric function.30 questions
• Geometry: angles, triangles and circlesAngles at lines, in triangles and in polygons, Pythagoras, the perimeter and area of a triangle, and the circle: its circumference, area, arcs and sectors.30 questions
• Geometry: lines and circles in coordinatesDistances between points, perpendicular lines and the angle between two lines, the distance from a point to a line, the equation of a circle, its tangents, and where a circle meets a line or another circle.30 questions
• Sequences and seriesTerms of arithmetic and geometric sequences, and their sums: finite, in sigma notation, and for ever.30 questions
• OMPT-G mixed paperEvery OMPT-G topic together.30 questions

Which OMPT are you sitting?

OMPT is not one test. There are seven, and they differ in SYLLABUS as well as in difficulty, so practising the wrong one is wasted work. Your university names the level it wants and sets its own pass mark — check your programme page before you start.

Syllabuses, question counts and times as omptest.org publishes them, read in October 2026.

Every answer on these pages, and the value given for every common mistake, was re-derived by an independent method before the page was built. Nothing here is published on the strength of the code that produced it.