Free · Year 1 · 60 minutes · 20 questions

Maths catch-up practice test for Year 1 of the liceo scientifico.

Do you have a failed maths subject to make up from Year 1? The first-year catch-up exam centres on algebraic manipulation, factorising and first-degree equations. This is the practice version: 20 questions drawn at random from a bank of 50, on the Year 1 programme, so every attempt is a different test. Scoring works like the TOLC admission tests (+1 correct, −0.25 wrong, 0 blank), and at the end you receive by email the correct answer to every question with your weak points by topic — with an account you see them straight away on this page (the step-by-step solution unlocks with your first lesson).

The catch-up exam is the test an Italian high-school student sits in late August or September to make up a subject failed during the school year; the liceo scientifico is the science-oriented high school.

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

Topics covered by the test

Algebraic manipulation and special products
Factorising
Algebraic fractions and domain restrictions
First-degree equations (with and without fractions) and word problems
Sets and logic
Euclidean geometry: triangles and congruence
Descriptive statistics

Two worked examples from the test

Question 1 · Matematica · Calcolo letterale e prodotti notevoli

Semplifica il monomio: $3 x^{2} y \cdot (- 2 x y^{3})$ .

A $6 x^{3} y^{4}$
B $- 6 x^{2} y^{3}$
C $- 6 x^{3} y^{4}$ correct answer
D $- 6 x^{3} y^{3}$
E $- 5 x^{3} y^{4}$

Solution

Per moltiplicare due monomi devo fare due cose separate: moltiplicare i coefficienti numerici tra loro, e moltiplicare le lettere uguali tra loro somando i loro esponenti.

Passo 1 — Moltiplico i coefficienti numerici. I coefficienti sono $3$ e $- 2$ :

$3 \cdot (- 2) = - 6$

Passo 2 — Moltiplico le potenze di $x$ . Ricorda: quando moltiplichi potenze con la stessa base, gli esponenti si somano (non si moltiplicano). Qui $x^{2} \cdot x = x^{2} \cdot x^{1}$ :

$x^{2 + 1} = x^{3}$

Passo 3 — Moltiplico le potenze di $y$ . Stessa regola, con $y^{1} \cdot y^{3}$ :

$y^{1 + 3} = y^{4}$

Passo 4 — Metto insieme i pezzi. Unisco il coefficiente e le due lettere:

$- 6 x^{3} y^{4}$

Risposta corretta: $- 6 x^{3} y^{4}$ . Se avevi scelto $6 x^{3} y^{4}$ , probabilmente hai dimenticato che un numero positivo per un numero negativo dà un numero negativo — controlla sempre il segno del prodotto dei coefficienti per primo. Se avevi scelto $- 6 x^{2} y^{3}$ , hai ripetuto gli esponenti di partenza invece di somarli: nella moltiplicazione di potenze con stessa base gli esponenti si somano, non restano com'erano.

Question 2 · Matematica · Calcolo letterale e prodotti notevoli

Semplifica l'espressione: $(x + 3)^{2} - (x - 1) (x + 5) + 2 x$ .

A $12 x + 4$
B $- 2 x + 14$
C $2 x^{2} + 12 x + 4$
D $4 x + 14$ correct answer
E $4 x + 4$

Solution

Passo 1 — Sviluppo il quadrato. $(x + 3)^{2} = x^{2} + 6 x + 9$ .

Passo 2 — Sviluppo il prodotto. $(x - 1) (x + 5) = x^{2} + 5 x - x - 5 = x^{2} + 4 x - 5 .$

Passo 3 — Sostituisco, mettendo il prodotto tra parentesi perché c'è un meno davanti.

$x^{2} + 6 x + 9 - (x^{2} + 4 x - 5) + 2 x$

Passo 4 — Tolgo la parentesi cambiando tutti i segni.

$x^{2} + 6 x + 9 - x^{2} - 4 x + 5 + 2 x$

Passo 5 — Sommo i termini simili. I termini in $x^{2}$ si annullano; $6 x - 4 x + 2 x = 4 x$ ; $9 + 5 = 14$ .

$4 x + 14$

La risposta corretta è $4 x + 14$ . Se hai scelto $12 x + 4$ , hai tolto la parentesi cambiando solo il segno del primo termine: il meno davanti alla parentesi cambia il segno di tutti i termini dentro.

In the online practice test you will find 18 more questions like these; the correct answer to every question arrives by email.

Prefer paper? Download the test as a PDF (20 questions, 5 pages · 93 KB, ready to print). You receive the solutions by email after taking it online.

Download the PDF

What the Year 1 catch-up exam really asks

The maths catch-up exam for Year 1 does not fish at random across the whole programme: it is always the same four or five patterns, at textbook difficulty. These are the ones that come back almost every time.

An expression with polynomials and special products to simplify all the way
A factorisation using grouping or a difference of squares
An equation with fractions, with the domain restrictions written out
A geometry problem on the congruence criteria for triangles

The mistakes that cost you the grade

You do not fail the September exam because of the hard topics: you fail it because of three or four mechanical mistakes that repeat in every exercise. On this programme they are these.

Getting the sign wrong when removing a bracket preceded by a minus: it is by far the most frequent mistake and it drags the rest of the exercise down with it
Writing (a + b)² as a² + b², forgetting the double product
Solving the equation with fractions without requiring the denominator to differ from zero, and accepting a solution that should be rejected

What order to revise in

Start again from algebraic manipulation, not from equations: special products and factorising feed into everything else in the programme, and anyone who slips there also gets wrong the exercises they actually knew how to do. Only then move on to equations and algebraic fractions, and leave geometry for last — it is the part most separate from the others.

Is the score not enough?

In 2–3 weeks of targeted revision you can make up the subject: read the guide to the September exam or let us look at your results together — the diagnostic is free.

Book the free diagnostic call Maths tutoring