OMPT practice questions: 14 worked examples for OMPT-A to OMPT-G (2026)

OMPT practice questions look like ordinary school maths problems with one difference: the answer is typed into a box and checked by software, so only the exact final answer counts. Below are 14 worked examples, two for each of the seven modules OMPT-A to OMPT-G, each with the format line of its module, numbered solution steps and the answer in the form the checker accepts. The questions are Examify's own, written to the official chapter lists; they are not past papers, because the test provider does not publish any. Examify is an independent preparation provider and is not affiliated with the OMPT, SOWISO or any university.

How OMPT questions are graded

Every OMPT answer is checked by a computer-algebra system and then reviewed by a mathematician. The answer has to be mathematically correct and unambiguous; a right method with a wrong final answer earns nothing on that question, so there is no partial credit to fall back on. Four rules follow from that.

• Enter exact forms. 3/2 is safe, and so is 1.5 because it terminates, but x = 4/3 rounded to 1.33 is wrong. Simplify first: 6/4 becomes 3/2, √8 becomes 2√2.
• Use a dot as the decimal separator and the variable name the question uses (t, not x, if the question is written in t).
• Only the on-screen OMPT calculator is allowed. Your own calculator stays in the drawer, so practise with the built-in one.
• You may have up to 3 blank A4 sheets, pencils and one eraser, no pens, and you tear the sheets up on camera after submitting.

The rules are on the provider's test rules page. Each example below ends with the answer as you would type it.

OMPT-A practice questions

Format: 52 questions in 120 minutes, about 2.3 minutes per question, 176 topics in 8 chapters (OMPT-A syllabus).

Question 1 (Quadratic equations). Solve 2x² − 7x + 3 = 0.

1. Read off a = 2, b = −7, c = 3 and compute the discriminant: D = (−7)² − 4·2·3 = 49 − 24 = 25.
2. D is positive, so there are two solutions, and √25 = 5.
3. Apply the quadratic formula: x = (7 ± 5) / (2·2) = 12/4 or 2/4.

Answer: x = 3 or x = 1/2. Enter 1/2, not 2/4.

Question 2 (Differentiation). The graph of f(x) = x³ − 6x² + 9x has two points with a horizontal tangent. Find them.

1. A horizontal tangent means f′(x) = 0, so differentiate term by term: f′(x) = 3x² − 12x + 9.
2. Take out the factor 3: f′(x) = 3(x² − 4x + 3) = 3(x − 1)(x − 3).
3. Set each factor equal to zero: x − 1 = 0 or x − 3 = 0.

Answer: x = 1 and x = 3.

These two questions show why OMPT-A is a pace test: each one is a single standard procedure (discriminant, derivative, factorise) that has to be finished and simplified in a little over two minutes, and Functions and Differentiation together hold 69 of the 176 topics.

OMPT-B practice questions

Format: 64 questions in 150 minutes, about 2.3 minutes per question, 226 topics in 10 chapters (OMPT-B syllabus).

Question 3 (Trigonometry). Solve 2·sin(x) = √3 on the interval 0 ≤ x ≤ 2π.

1. Divide both sides by 2: sin(x) = √3/2.
2. The reference angle whose sine is √3/2 is π/3.
3. Sine is positive in the first and second quadrant, so x = π/3 or x = π − π/3.

Answer: x = π/3 or x = 2π/3. Both solutions are needed.

Question 4 (Integration). Evaluate ∫₀² (3x² − 2x) dx.

1. Find an antiderivative term by term: ∫3x² dx = x³ and ∫2x dx = x², so F(x) = x³ − x².
2. Substitute the upper limit: F(2) = 8 − 4 = 4.
3. Substitute the lower limit F(0) = 0 and subtract: 4 − 0.

Answer: 4.

OMPT-B is OMPT-A plus the Trigonometry and Integration chapters at the same pace, so the extra work is in radians, the unit circle and antiderivatives rather than in harder algebra.

OMPT-C practice questions

Format: 30 questions in 120 minutes, about 4 minutes per question, 66 topics in 4 chapters (OMPT-C syllabus).

Question 5 (Linear equations). Solve 4(x − 2) = 2x + 6.

1. Expand the bracket: 4x − 8 = 2x + 6.
2. Move the x-terms left and the numbers right: 4x − 2x = 6 + 8.
3. Simplify: 2x = 14.

Answer: x = 7.

Question 6 (Geometry). A right-angled triangle has legs of 9 cm and 12 cm. Give the hypotenuse and the area.

1. Pythagoras: c² = 9² + 12² = 81 + 144 = 225.
2. Take the square root: c = √225 = 15 cm.
3. The legs are perpendicular, so the area is ½ · 9 · 12.

Answer: hypotenuse 15 cm, area 54 cm².

OMPT-C stays at arithmetic, linear equations and basic geometry, and 4 minutes per question leave time to check a sign or a unit.

OMPT-D practice questions

Format: 21 questions in 180 minutes, about 8.6 minutes per question, 242 topics in 11 chapters (OMPT-D syllabus).

Question 7 (Differentiation, product and chain rule). Differentiate f(x) = x² · e^(3x).

1. This is a product, so use f′ = u′v + uv′ with u = x² and v = e^(3x).
2. u′ = 2x, and the chain rule gives v′ = 3e^(3x).
3. Substitute: f′(x) = 2x·e^(3x) + x²·3e^(3x), then take out x·e^(3x).

Answer: f′(x) = x·e^(3x)·(2 + 3x).

Question 8 (Integration by substitution). Evaluate ∫ 2x·(x² + 1)⁴ dx.

1. Substitute u = x² + 1. Then du/dx = 2x, so du = 2x dx.
2. The integral becomes ∫ u⁴ du = u⁵/5 + C.
3. Substitute u = x² + 1 back in.

Answer: (x² + 1)⁵ / 5 + C.

Both OMPT-D questions combine two rules in one problem (product plus chain, substitution plus power rule), which is why the module gives 8.6 minutes per question and has the largest syllabus of the seven.

OMPT-E practice questions

Format: 40 questions in 120 minutes, 3 minutes per question, 161 topics in 12 chapters, of which chapters 8 to 12 are statistics and probability (OMPT-E syllabus).

Question 9 (Probability). A bag holds 5 red and 3 blue marbles. You draw two without replacement. What is the probability that both are red?

1. On the first draw 5 of the 8 marbles are red, so P = 5/8.
2. After one red is removed, 4 of the remaining 7 are red, so P = 4/7.
3. Multiply the two: (5/8)·(4/7) = 20/56.

Answer: 5/14 (about 0.357). Enter the simplified fraction; 0.357 is rounded.

Question 10 (Random variables, binomial distribution). X follows a binomial distribution with n = 6 and p = 0.5. Compute P(X = 4).

1. The binomial formula is P(X = k) = C(n,k)·p^k·(1 − p)^(n−k).
2. Here C(6,4) = 15, p⁴ = 0.0625 and (1 − p)² = 0.25.
3. Multiply: 15 · 0.0625 · 0.25.

Answer: 15/64 (about 0.234).

The two OMPT-E questions come from the statistics block that holds 41 of the 161 topics; students who prepare from OMPT-A material never see this quarter of the syllabus.

OMPT-F practice questions

Format: 21 questions in 180 minutes, about 8.6 minutes per question, 149 topics in 6 chapters (OMPT-F syllabus).

Question 11 (Functions and inverses). Find the inverse of f(x) = (2x − 1)/(x + 3).

1. Write y = (2x − 1)/(x + 3) and multiply both sides by (x + 3): y(x + 3) = 2x − 1.
2. Expand and collect the x-terms on one side: yx − 2x = −1 − 3y, so x(y − 2) = −(1 + 3y).
3. Divide by (y − 2) and swap the letters.

Answer: f⁻¹(x) = (3x + 1)/(2 − x), for x ≠ 2.

Question 12 (Differentiation, quotient rule). Differentiate f(x) = (x² + 1)/(x − 2).

1. Use f′ = (u′v − uv′)/v² with u = x² + 1 and v = x − 2.
2. u′ = 2x and v′ = 1, so the numerator is 2x(x − 2) − (x² + 1)·1.
3. Expand and simplify: 2x² − 4x − x² − 1 = x² − 4x − 1.

Answer: f′(x) = (x² − 4x − 1)/(x − 2)².

OMPT-F has no trigonometry, integration or statistics; its two questions show the depth it asks instead: rational functions, their derivatives, and the domain restriction stated.

OMPT-G practice questions

Format: 26 questions in 180 minutes, about 6.9 minutes per question, 119 topics in 11 chapters including physics and chemistry (OMPT-G syllabus).

Question 13 (Physics, mechanics). A ball is thrown straight up at 20 m/s. With g = 9.81 m/s², how high does it rise?

1. At the highest point the speed is zero, so use v² = v₀² − 2gh with v = 0.
2. Rearrange to h = v₀²/(2g).
3. Substitute: h = 20²/(2·9.81) = 400/19.62.

Answer: about 20.4 m. A decimal is expected here because g is given as one.

Question 14 (Chemistry, amount of substance). How many moles are in 36 g of water, and how many molecules is that? Take M = 18 g/mol.

1. Use n = m/M with m = 36 g and M = 18 g/mol.
2. n = 36/18 = 2 mol.
3. Multiply by Avogadro's number: 2 · 6.022×10²³.

Answer: 2 mol, about 1.2×10²⁴ molecules.

OMPT-G is the only module with science chapters: they test whether you pick the right formula and carry the units through, on top of the algebra and trigonometry it shares with OMPT-B.

How to use these OMPT practice questions

Work through your module's two questions under three conditions, in this order.

1. Timed. Give yourself your module's minutes per question: 2.3 for OMPT-A or OMPT-B, 3 for OMPT-E, 4 for OMPT-C, 6.9 for OMPT-G, 8.6 for OMPT-D or OMPT-F. Correct in double that time is not yet a pass.
2. In the answer box. Type the final answer into an on-screen input rather than writing it on paper. Examify's free OMPT practice test shows two of the harder questions per module in the real input format, with the worked solution after each.
3. As part of a full mock. Fourteen questions test the procedure, not whether you can hold the pace for 120 or 180 minutes. A full-length mock built by Examify uses the real question count and time limit for each module, from €39.99 in the shop, with a worked solution for every question.

Then score the mock per chapter and study the weak chapters from the topic list on the syllabus pages. The provider's own figure is the reason to do this before booking: students who worked through the learning materials scored 76% on average on their first attempt, those who did not scored 40%, and an attempt costs €240 (€270 for OMPT-D and OMPT-F) either way.