Free · Year 2 · 60 minutes · 20 questions

Maths catch-up practice test for Year 2 of the liceo scientifico.

Do you have a failed maths subject to make up from Year 2? The second-year catch-up exam centres on radicals, quadratic equations and inequalities. This is the practice version: 20 questions drawn at random from a bank of 50, on the Year 2 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

Radicals and rationalising the denominator
Quadratic equations (including with a parameter)
Linear and quadratic inequalities, including fractions
Linear systems
The line in the Cartesian plane
Geometry: Euclid, Pythagoras, the circle
Basic probability

Two worked examples from the test

Question 1 · Matematica · Radicali

Semplifica il radicale $72$ , portando fuori dal segno di radice tutti i fattori possibili.

A $26$
B $62$ correct answer
C $83$
D $612$
E $362$

Solution

Passo 1 — Scompongo il radicando. Cerco il più grande fattore di $72$ che sia un quadrato perfetto: $72 = 36 \cdot 2$ , e $36$ è un quadrato perfetto ( $6^{2}$ ).

Passo 2 — Separo la radice del prodotto. Uso la proprietà $a \cdot b = a \cdot b$ (valida perché $a, b \geq 0$ ):

$72 = 36 \cdot 2 = 36 \cdot 2 = 62$

Verifica. $6^{2} \cdot 2 = 36 \cdot 2 = 72 :$ il conto torna.

Risposta corretta: $62$ . Se avevi scelto $26$ ( $26$ ), probabilmente hai scambiato il numero che resta fuori dalla radice con quello che resta dentro. Se avevi scelto $362$ ( $362$ ), hai portato fuori $36$ senza fargli la radice quadrata: ricorda che esce $36 = 6$ , non $36$ intero.

Question 2 · Matematica · Radicali

Calcola: $23 + 12 - 27 .$

A $- 53$
B $3$ correct answer
C $33$
D $73$
E $93$

Solution

Passo 1 — Non sono "simili" finché non li semplifico. $12$ e $27$ non sono già multipli di $3$ : li riduco prima.

$12 = 4 \cdot 3 = 23$
$27 = 9 \cdot 3 = 33$

Passo 2 — Sostituisco e somma i coefficienti.

$23 + 23 - 33 = (2 + 2 - 3) 3 = 3$

Risposta corretta: $3$ . Se avevi scelto $33$ ( $33$ ), probabilmente hai semplificato $12$ come $43$ , dimenticando di fare la radice di $4$ (che è $2$ , non $4$ ). Se avevi scelto $73$ ( $73$ ), hai cambiato il segno meno in più davanti a $27$ .

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 · 99 KB, ready to print). You receive the solutions by email after taking it online.

Download the PDF

What the Year 2 catch-up exam really asks

The maths catch-up exam for Year 2 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.

A complete quadratic equation, often with a parameter to discuss
An inequality with fractions or a quadratic one, solved with a sign analysis
A linear system of two equations, by substitution or with Cramer’s rule
The line through two points, or parallel and perpendicular to a given line

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.

In an inequality with fractions, multiplying by the denominator without discussing its sign: it is the quickest way to reverse the result
Adding radicals as if √a + √b were √(a + b)
Picking the wrong intervals once the roots are found: “inside” and “outside” depend on the sign of the coefficient of x² and on the direction of the inequality

What order to revise in

Radicals first, because they appear inside the solutions of quadratic equations and a mistake there spreads. Then quadratic equations, then inequalities — the real stumbling block of the year, worth almost half the exam on their own — and finally systems and the line, which are quick to recover if your calculation holds up.

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