Free · Year 4 · 60 minutes · 20 questions

Maths catch-up practice test for Year 4 of the liceo scientifico.

Do you have a failed maths subject to make up from Year 4? The fourth-year catch-up exam centres on trigonometry, exponentials and logarithms. This is the practice version: 20 questions drawn at random from a bank of 50, on the Year 4 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

Trigonometric identities and related angles
Trigonometric equations and inequalities
Trigonometry: sine, cosine, solving triangles
Exponentials and logarithms
Combinatorics and probability
Complex numbers
Solid geometry

Two worked examples from the test

Question 1 · Matematica · Goniometria

Quanto vale $tan (60°)$ ?

A $\frac{3}{3}$
B $1$
C $2$
D $3$ correct answer
E $2$

Solution

Passo 1 — Riconosco l'angolo notevole. $60°$ ha $sin 60° = \frac{3}{2}$ e $cos 60° = \frac{1}{2}$ .

Passo 2 — Applico la definizione di tangente.

$tan 60° = \frac{sin 60°}{cos 60°} = \frac{3 /2}{1/2} = 3$

Risposta corretta: $3$ . Se avevi scelto $\frac{3}{3}$ , hai calcolato $tan 30°$ invece di $tan 60°$ : ricorda che $tan 30°$ e $tan 60°$ sono uno il reciproco dell'altro ( $\frac{3}{3} = \frac{1}{3}$ ), non lo stesso valore — un modo per non confonderli è ricordare che la tangente cresce insieme all'angolo, quindi $tan 60° > tan 30°$ .

Question 2 · Matematica · Goniometria

Sapendo che $tan x = 2$ e che $x$ è un angolo del terzo quadrante, quanto vale $cos x$ ?

A $- \frac{5}{5}$ correct answer
B $\frac{5}{5}$
C $- \frac{2 5}{5}$
D $- \frac{1}{5}$
E $- \frac{1}{2}$

Solution

Passo 1 — Collego tangente e coseno. Dall'identità $1 + tan^{2} x = \frac{1}{cos ^{2} x} :$

$cos^{2} x = \frac{1}{1 + tan ^{2} x} = \frac{1}{1 + 4} = \frac{1}{5}$

Passo 2 — Estraggo la radice. $cos x = \pm \frac{1}{5} = \pm \frac{5}{5} .$

Passo 3 — Scelgo il segno con il quadrante. Nel terzo quadrante seno e coseno sono entrambi negativi (e la tangente, rapporto dei due, è positiva: coerente con $tan x = 2$ ). Quindi

$cos x = - \frac{5}{5}$

Risposta $- \frac{5}{5}$ . Se avevi scelto $\frac{5}{5}$ , hai dimenticato di controllare il quadrante; se avevi scelto $- \frac{1}{5}$ , ti sei fermato a $cos^{2} x$ senza estrarre la radice.

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

Download the PDF

What the Year 4 catch-up exam really asks

The maths catch-up exam for Year 4 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 trigonometric equation that reduces to an elementary, linear or homogeneous one
An exponential equation or inequality and a logarithmic one, with the domain conditions
Solving any triangle with the sine rule or the cosine rule
A combinatorics or probability exercise

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.

Giving a single solution to a trigonometric equation instead of the full family, with + 2kπ or + kπ depending on the case
Solving a logarithmic equation without the domain and accepting a value that makes the argument negative
Confusing log(a + b) with log a + log b: the logarithm splits a product, not a sum

What order to revise in

The trigonometric formulas come before everything else, but learning them all by heart at once does not work: keep the fundamental ones (addition, double angle, related angles) and derive the rest. Then trigonometric equations and inequalities, then exponentials and logarithms — similar in structure, so best revised one after the other. Trigonometry applied to triangles comes last: it is the part you can recover in an afternoon.

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