Free · Year 3 · 60 minutes · 20 questions

Maths catch-up practice test for Year 3 of the liceo scientifico.

Do you have a failed maths subject to make up from Year 3? The third-year catch-up exam centres on absolute value, equations with radicals, functions and conics. This is the practice version: 20 questions drawn at random from a bank of 50, on the Year 3 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

Equations and inequalities with absolute value and with radicals
Higher-degree inequalities and inequalities with fractions
Functions: domain, composition, inverse
The line and families of lines
Parabola and circle
Ellipse and hyperbola
Transformations of graphs

Two worked examples from the test

Question 1 · Matematica · Equazioni e disequazioni con modulo e irrazionali

Risolvi l'equazione $∣2 x - 3∣ = 5$ .

A $x = 4$
B $x = - 4 \lor x = 4$
C $x = - 1 \lor x = 4$ correct answer
D $x = - 1$
E $x = 1 \lor x = 4$

Solution

Devo trovare i valori di $x$ per cui l'espressione dentro il modulo vale $5$ oppure $- 5$ : infatti $∣ A ∣ = k$ (con $k > 0$ ) significa sempre $A = k \lor A = - k$ .

Passo 1 — Primo caso, $2 x - 3 = 5$ .

$2 x - 3 = 5 \Rightarrow 2 x = 8 \Rightarrow x = 4$

Passo 2 — Secondo caso, $2 x - 3 = - 5$ .

$2 x - 3 = - 5 \Rightarrow 2 x = - 2 \Rightarrow x = - 1$

Entrambi i valori sono soluzioni accettabili: il modulo è sempre $\geq 0$ e qui il termine a destra, $5$ , è già positivo, quindi non serve nessuna verifica ulteriore (a differenza dei casi in cui il termine a destra contiene la $x$ ).

Risposta $x = - 1 \lor x = 4$ . Se avevi scelto solo $x = 4$ , probabilmente hai risolto il primo caso e ti sei fermato lì, dimenticando che un'equazione con il modulo si spezza sempre in due equazioni distinte da risolvere entrambe.

Question 2 · Matematica · Equazioni e disequazioni con modulo e irrazionali

Risolvi l'equazione $∣ x - 2∣ = 2 x - 1$ .

A $x = - 1$
B $x = - 1 \lor x = 1$
C $x = 1$ correct answer
D $x = \frac{1}{2}$
Enessuna soluzione

Solution

Perché $∣ x - 2∣ = 2 x - 1$ abbia senso, il termine a destra deve essere $\geq 0$ : il modulo non è mai negativo, quindi qualunque cosa sia uguale a un modulo deve esserlo a sua volta. Questa condizione va sempre verificata quando il modulo è uguagliato a un'espressione con la $x$ .

Passo 1 — Condizione di esistenza.

$2 x - 1 \geq 0 \Rightarrow x \geq \frac{1}{2}$

Passo 2 — Primo caso, $x - 2 \geq 0$ cioè $x \geq 2$ .

$x - 2 = 2 x - 1 \Rightarrow x = - 1$

Non compatibile con $x \geq 2$ né con la condizione $x \geq \frac{1}{2}$ : si scarta.

Passo 3 — Secondo caso, $x - 2 < 0$ cioè $x < 2$ .

$- (x - 2) = 2 x - 1 \Rightarrow - x + 2 = 2 x - 1 \Rightarrow 3 x = 3 \Rightarrow x = 1$

$x = 1$ soddisfa $x < 2$ : coerente con il caso
$x = 1$ soddisfa la condizione $x \geq \frac{1}{2}$ : verificata

Risposta $x = 1$ . Se avevi scelto $x = - 1 \lor x = 1$ , hai risolto correttamente i due casi ma hai dimenticato di controllare la condizione $2 x - 1 \geq 0$ , che scarta proprio $x = - 1$ .

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

Download the PDF

What the Year 3 catch-up exam really asks

The maths catch-up exam for Year 3 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 inequality with absolute value or radicals, with the cases discussed
The domain of a function, often with a fraction and a square root
The parabola through three points, or the tangent to a parabola from an outside point
Going from the general equation of a circle to its centre and radius

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.

Squaring an equation with radicals without requiring the radicand ≥ 0 and the right-hand side ≥ 0, and keeping solutions that are not acceptable
Treating the absolute value with a single case instead of splitting it into its two branches
Getting the signs wrong when going from x² + y² + ax + by + c = 0 to the centre (−a/2, −b/2): the minus is almost always lost

What order to revise in

Absolute values and radicals first, because that is where most points are lost and also the part that is fixed fastest: it is two or three patterns, not a whole chapter. Then functions and domain, the bridge towards Year 5. Conics last, and studied together — parabola, circle, ellipse and hyperbola resemble each other far more than the textbook suggests.

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