Practice maths class test — Year 4: complex numbers and solids.

15 questions drawn at random from a bank of 25, on the block of the programme your teacher is finishing: every attempt is a different test. No penalty for wrong answers, so answer even if you are not sure. At the end you receive by email the correct answer to every question and your weak topics — with an account you see them straight away here on the page. The step-by-step solution unlocks with your first lesson.

A class test is the written test an Italian high-school class sits after each block of the programme; this practice version is multiple choice.

Note on school types. Complex numbers are Year 4 programme at the liceo scientifico; at the classical and human-sciences high schools they are often reduced to the algebraic form alone, while solid geometry is the same everywhere.

Test in ItalianThe questions of this simulator are in Italian, as in the real exam.

Test prepared by Gaetano Livornese · Aerospace engineer, Politecnico di Milano · maths and physics tutor · updated 13 October 2026

The Year 4 test on complex numbers and solids comes in March-April and puts together two topics that have nothing in common except the textbook chapter: half the test is algebra (complex numbers), the other half is solid geometry. It is worth preparing them as two separate tests, not one.

On the complex side, a frequent mistake concerns the modulus: an equation like |z − (1 + 2i)| = 3 describes a circle in the Argand plane, with centre (1, 2) and radius 3 — not a linear equation to solve for z. It has to be translated by writing z = x + iy and expanding (x − 1)² + (y − 2)² = 9, not manipulated algebraically as if the modulus could be “removed” in one step.

The second, on the trigonometric form: when dividing two complex numbers you divide the moduli and subtract the angles, you do not add them — adding the angles is the rule for multiplication, and swapping the two operations is the most common mistake in this block.

On the geometry side, the third mistake concerns Euler’s formula V − E + F = 2 (vertices minus edges plus faces): applied to a composite solid, for example two pyramids joined at a base, the edges and vertices of the gluing plane must not be counted twice — you recount them on the final solid, otherwise the total does not add up and the formula seems wrong when it is actually the count that is.

To prepare, do one division exercise in trigonometric form and one on Euler’s formula on a composite solid before the practice test: those are the two points where you know the theory but get the calculation wrong.

What a maths test on complex numbers and solids looks like

Operations in algebraic form with the conjugate, powers of i, quadratic equations with a negative discriminant, trigonometric form with De Moivre and nth roots; on the geometry side, volumes and surface areas of solids, including composite ones, and the theorem of the three perpendiculars.

The mistake that costs the grade

Forgetting that i² = −1 and treating i like any letter in products. In nth roots, giving only one: there are n of them, evenly spaced on the circle, and the teacher wants all of them. On solids, using the apothem instead of the height in the pyramid.

How to use this practice test

Take it once cold, without revising and with the timer on: it is there to show you where you stand, not to look good. Then look at your weak topics, revise only those — usually two or three, not the whole chapter — and retake it after a few days. The 15 questions are drawn from a bank of 25, so the second attempt is not the same test and the scores are genuinely comparable.