TOLC-I maths: topics and worked exercises

End of September 2026: the autumn session of the TOLC-I (the Italian engineering admission test; October-November, remaining places) is open, and anyone who registers now sits it in four to eight weeks. For those in Year 5, starting to work on TOLC-I maths now — without waiting for the maturità (Italy’s school-leaving exam) — means arriving ready for the winter-spring sessions of early 2027, when there will be time to prepare calmly without overlapping with study for the state exam. I have already written about it in general in the complete guide to the TOLC-I: here I do not repeat it, I concentrate only on the section that decides the score — maths — with the topics to know and ten worked exercises in the style of the test.

Why maths counts for more than the other three sections put together

The TOLC-I has four subject sections: Maths (20 questions, 50 minutes), Logic (10/20), Sciences (10/20), Verbal comprehension (10/20) — 50 questions in 110 minutes, plus an English section (30 questions, 15 minutes) with no penalty and no weight on the main score. Maths alone is worth 20 points out of 50 — 40% of the maximum score — with the same scheme as all sections: +1 correct, 0 omitted, −0.25 wrong.

There is a second reason not to neglect it: in many universities the score of the maths section alone (regardless of the total) determines whether an OFA (an additional learning requirement, to be made up in the first year) is triggered — always check the specific threshold in the call for your degree course, because it varies from university to university. It is worth arriving prepared for this reason alone: anyone who enters with an open OFA starts the first year with one more exam to clear, just while getting used to the university pace — I discuss it in detail in how to make up the maths OFA in the first year.

The pace: 2.5 minutes per question

Fifty minutes for twenty questions makes 2.5 minutes per question on average, but not all questions cost the same: a first-degree equation is solved in thirty seconds, an analytic geometry problem with several steps can take four or five. The strategy that works is to scan all twenty questions first, answer at once those you recognise at a glance, and leave for a second pass those that need more calculation. When you find yourself stuck on a long procedure and time is short, it pays to omit and move on: a 0 costs less than a −0.25, and you can recover that time on a simpler question left behind.

The syllabus topics

The official CISIA syllabus (to be rechecked every year on cisiaonline.it, because it can change) covers:

• Arithmetic and algebra: integers, rationals, reals; absolute value; powers and roots; logarithms and exponentials; algebraic calculation; polynomials and factorisation.
• Equations and inequalities: first and second degree (or reducible to them), fractional, with radicals; first-degree systems.
• Euclidean geometry: segments, angles, lines, geometric loci, properties of plane figures, areas and volumes of solids.
• Analytic geometry: lines, parabolas, circles, distances between points and from a line.
• Elementary functions: domain, linear, quadratic, exponential, logarithmic functions.
• Trigonometry: fundamental relations, graphs, basic trigonometric equations.
• Statistics and elementary probability: permutations, combinations, simple probability calculation, variance.

No calculus: derivatives and integrals are outside the syllabus. Good news especially for those coming from a technical institute or a non-science high school.

How to organise your study, starting now

If you are sitting the autumn session, you have four to eight weeks. In that time there is no point redoing the whole high-school maths programme: what makes sense is a cold, timed practice test, to see which of the seven blocks of the syllabus weigh most on your current score, and then working in priority order on those — normally algebra and inequalities first (they are the basis of almost everything else), then analytic geometry and functions, finally trigonometry and statistics, which weigh fewer questions but are recovered more quickly with a targeted review of the formulas. Anyone aiming instead at the winter-spring 2027 window has months ahead: in that case it is worth integrating the study of TOLC-I maths into the ordinary Year 5 review, instead of treating it as a separate programme — a good part of the topics (equations, inequalities, analytic geometry, trigonometry) are also school-test topics at that time of year.

Ten worked exercises

Each question has 5 options as in the real test. Strategy given after each solution.

1. Algebra. Solve $2x-5=3x+1$. (a) $x=-6$ (b) $x=6$ (c) $x=-4$ (d) $x=4$ (e) no solution.

$2x-3x=1+5\Rightarrow -x=6\Rightarrow x=-6$. Answer (a). Strategy: isolate $x$ by bringing like terms to the same side, without changing a sign through distraction.

2. Inequality with absolute value. Solve $|x-3|<2$. (a) $x<1$ (b) $x>5$ (c) $1<x<5$ (d) $-1<x<5$ (e) $x<-1$ or $x>5$.

$|x-3|<2⟺-2<x-3<2⟺1<x<5$. Answer (c). Strategy: turn the absolute value at once into a double inequality, avoiding reasoning by cases when the inequality is "$<$" (with "$>$" you would instead need the union of two intervals, as in (e)).

3. First-degree system. Solve $\{\begin{array}{l}x+y=10\\ 2x-y=5\end{array}$. (a) $x=3,y=7$ (b) $x=5,y=5$ (c) $x=7,y=3$ (d) $x=4,y=6$ (e) $x=6,y=4$

Adding the two equations: $3x=15\Rightarrow x=5$, so $y=10-5=5$. Answer (b). Strategy: add the equations when the coefficients of one variable are opposite or easily made so, instead of substituting at once — it is faster under pressure.

4. Euclidean geometry. A right-angled triangle has legs of $6$ cm and $8$ cm. How long is the hypotenuse? (a) $10$ cm (b) $12$ cm (c) $14$ cm (d) $9$ cm (e) $\sqrt{48}$ cm

$\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10$. Answer (a). Strategy: recognising the Pythagorean triple 6-8-10 (a multiple of 3-4-5) saves the square root calculation.

5. Analytic geometry. Find the distance between the points $A(1,2)$ and $B(4,6)$. (a) $3$ (b) $4$ (c) $5$ (d) $7$ (e) $\sqrt{20}$

$d=\sqrt{(4-1{)}^2+(6-2{)}^2}=\sqrt{9+16}=\sqrt{25}=5$. Answer (c). Strategy: the same 3-4-5 triple as in the previous exercise — it is worth memorising, it comes up often.

6. Exponentials. Solve $3^{2x-1}=27$. (a) $x=1$ (b) $x=2$ (c) $x=3$ (d) $x=\frac{3}{2}$ (e) $x=14$

$3^{2x-1}=3^3\Rightarrow 2x-1=3\Rightarrow x=2$. Answer (b). Strategy: write both sides as a power of the same base before equating the exponents — it is quicker than moving to logarithms when the base already coincides. Option (e) is the trap for those who divide 27 by 2 instead of recognising $27=3^3$.

7. Logarithms. What is the domain of $f(x)=\log (x-2)$? (a) $x>0$ (b) $x\ge 2$ (c) $x>2$ (d) $x<2$ (e) $x\ne 2$

The argument of the logarithm must be strictly positive: $x-2>0\Rightarrow x>2$. Answer (c). Strategy: the domain of logarithms is the most frequent mistake in the test — the argument is strictly greater than zero, never $\ge$.

8. Trigonometry. What is $\sin 30^{\circ }+\cos 60^{\circ }$? (a) $0$ (b) $\frac{1}{2}$ (c) $1$ (d) $\sqrt{3}$ (e) $\frac{\sqrt{3}}{2}$

$\sin 30^{\circ }=\frac{1}{2}$ and $\cos 60^{\circ }=\frac{1}{2}$, so the sum is $1$. Answer (c). Strategy: memorising the special values of $0^{\circ }$, $30^{\circ }$, $45^{\circ }$, $60^{\circ }$, $90^{\circ }$ saves you from rebuilding the triangles each time, with a clear time saving.

9. Probability. From a 40-card Italian (Neapolitan) deck (4 suits of 10 cards) you draw a card at random. What is the probability that it is a court card (knave, knight or king)? (a) $\frac{1}{40}$ (b) $\frac{3}{10}$ (c) $\frac{1}{4}$ (d) $\frac{3}{4}$ (e) $\frac{1}{10}$

There are 3 court cards per suit, so $3\cdot 4=12$ out of 40: $\frac{12}{40}=\frac{3}{10}$. Answer (b). Strategy: count first the favourable cases for one suit and then multiply, and simplify the fraction before comparing it with the options — the distractors often come from counting one suit only, like (e) which corresponds to the $\frac{4}{40}$ of someone who thinks of a single court card per suit.

10. Quadratic inequality. Solve $x^2-x-6>0$. (a) $-2<x<3$ (b) $x<-2$ or $x>3$ (c) $x<-3$ or $x>2$ (d) $-3<x<2$ (e) no solution

The roots of $x^2-x-6=0$ are $x=-2$ and $x=3$ (indeed $(x+2)(x-3)=x^2-x-6$), parabola opening upwards. The inequality "$>0$" holds outside the roots. Answer (b). Strategy: for a parabola opening upwards, "$<0$" is always the interval between the roots, "$>0$" is always outside — memorising this rule saves you from having to prove it again each time with a sign study. Options (c) and (d) are for those who get the signs of the roots wrong.

Typical mistakes

Recheck every answer twice when time allows: a sign or calculation error costs −0.25, exactly like not knowing the topic.

Test yourself with the dedicated practice paper

On the site you will find the TOLC-I practice test, with structure and scoring identical to the CISIA test: the correct answer is free, the guided step-by-step solution comes with the first lesson. For the general strategy of the test — timing, study plan, the other three sections — there is the complete guide to the TOLC-I; for the logic of the scoring and when it pays to guess, How the TOLC score is calculated; if your maths score is not enough to avoid the OFA, how to make it up in the first year. For a wider review, maths and admission tests. If you prefer a guided path instead of preparing on your own, there is the TOLC-I preparation course.

If you would like a hand working on these topics before your session, the first introductory call is free and lasts half an hour.