Practice test — Geometry.
20 questions drawn at random from the whole Geometry programme, mixed as in a real exam sitting: scoring +1/−0.25/0. At the end you receive by email the correct answer to every question and your weak points by topic — with an account you see them straight away on this page (the guided solution unlocks with your first lesson).
Each test mixes every topic of one first-year course at an Italian university, like a real exam sitting.
Test in ItalianThe questions of this simulator are in Italian, as in the real exam.
Topics covered by the test
- Analytic geometry in space (lines and planes)
- Vector spaces and linear maps
- Bilinear and quadratic forms
- Diagonalisation and principal axes
- Conics
- Quadrics
- Euclidean spaces
Two worked examples from the test
La retta passante per e ha come vettore direzione (a meno di multipli):
Solution
Il vettore direzione è proporzionale a . Errore tipico: usare le coordinate di invece della differenza .
La retta : interseca il piano nel punto:
Solution
Sostituisco: Il punto è .
In the online practice test you will find 18 more questions like these; the correct answer to every question arrives by email.
What the Geometry exam is like
The typical written paper asks you to study the relative position of lines and planes in space, classify a conic or a quadric by reducing it to principal axes (eigenvalues of the quadratic form), and discuss the definiteness of a quadratic form as a parameter varies. Compared with Linear algebra, here the calculation serves the geometric figure: a sign error changes the type of conic, not just a number.
Where the exam is lost
The mixed term xy forgotten or badly halved in the matrix of the quadratic form (it must be split in half between the two symmetric positions, never carried over whole). Then, when classifying conics and quadrics, stopping at the sign of the eigenvalues without also checking the determinant of the complete matrix: that is what distinguishes a conic from its degenerate version (two lines instead of a hyperbola).
How to read your score, in short.
What are the questions of the Geometry test like?
20 multiple-choice questions (4 options), drawn at random from a larger bank and spread over the WHOLE course programme together, not one topic at a time: analytic geometry in space (lines and planes), vector spaces and linear maps, bilinear and quadratic forms, diagonalisation and principal axes, conics, quadrics, Euclidean spaces.
Is there a penalty for wrong answers?
Yes: +1 point for a correct answer, −0.25 for a wrong one, 0 for a question left blank — as in the TOLC (the Italian university admission tests). You should answer only when you are reasonably sure: it is the same logic as a real multiple-choice exam.
What does the final score mean?
15 out of 20 or more: you are ready, the programme is solid. 10-14: needs consolidating — review the topics of the questions you got wrong and try again. Under 10: to be reviewed calmly before you sit the test again.
Course still weak?
Let us look at your results together — the diagnostic is free — and I will tell you honestly what it takes to close the gap.