TOLC-I: How Much Maths You Really Need (With Worked Examples)

The maths in the TOLC-I stops at the fourth-year syllabus of the scientific high school: equations and inequalities, the straight line, the parabola and the circle, standard functions, basic trigonometry, probability. No derivatives, no integrals, no limits and no curve sketching. The 20 questions in 50 minutes can all be solved with third- and fourth-year tools — the real difficulty is speed, not level.

There is one question I get asked at almost every first TOLC-I preparation lesson, usually with a thread of anxiety running through it: "but do I need to know derivatives?". No. And nor do you need integrals, limits, curve sketching, complex numbers or matrices. And yet half the students I meet arrive convinced they have to prepare for something like a university exam.

That conviction does real damage: I have seen fourth-year students spend weeks on limits "to get ahead" — a topic that does not come up in the TOLC-I — while getting rational inequalities wrong, which very much do come up. This article is here to put a clear ceiling on what you need to know, and then to prove it to you with four worked examples at the real level of the test.

What is the highest level required?

The ceiling is the maths of the first four years of scientific high school. The CISIA syllabus covers number sets and algebra, Euclidean and coordinate geometry, standard functions, basic trigonometry, combinatorics and probability. Everything that at a scientific high school is done in the fifth year — calculus in particular — stays outside it. A TOLC-I question, taken on its own and without a stopwatch, is an end-of-chapter class-test question, not a university exam question.

That does not mean the test is easy: it means the difficulty lies elsewhere. You have an average of 2 minutes 30 seconds per question, with a penalty of $0.25$ points for every wrong answer. The challenge is fluency of execution on topics you know, not depth on topics you have never seen.

For the full map of what comes up and how often, area by area, see TOLC-I Maths: Key Topics and Priorities — here we are concentrating on the level, not the distribution.

What you do NOT need to study

It is worth being explicit, because every one of these has been raised with me at least once by a student convinced it was needed:

• Derivatives and integrals — no, in any form
• Limits and continuity — no
• Full curve sketching (increasing and decreasing intervals, points of inflection, asymptotes) — no: all you need are the graphs of the standard functions and their transformations
• Ellipses and hyperbolas — the syllabus stops at the straight line, the parabola and the circle
• Complex numbers and matrices — no
• Sum-to-product and product-to-sum formulae — no: of the "advanced" trigonometry, all you need are the addition and double-angle formulae
• Inferential statistics — no: only descriptive statistics (mean, median, mode) and elementary probability

If your revision plan contains any of these, you are spending time on marks the test will never give you.

Four worked examples at the real TOLC-I level

The best way to calibrate your expectations is to see the questions. These four exercises are representative, in type and difficulty, of what you actually find in the maths section: one on algebra, one on coordinate geometry, one on functions, one on trigonometry. Try solving them yourself first, stopwatch in hand: the target is 2-3 minutes each.

Exercise 1 — Algebra: a rational inequality

Solve:

Working.

1. Never multiply both sides by $(x + 1)$: you do not know what sign it has. You study the sign of the numerator and of the denominator separately.
2. Numerator: $x^2 - 9 \geq 0$ when $x \leq -3$ or $x \geq 3$.
3. Denominator: $x + 1 > 0$ when $x > -1$. And $x = -1$ has to be excluded in every case, because it makes the denominator zero.
4. Sign table over the four intervals:

- $x < -3$: numerator $+$, denominator $-$ → fraction negative - $-3 < x < -1$: numerator $-$, denominator $-$ → fraction positive - $-1 < x < 3$: numerator $-$, denominator $+$ → fraction negative - $x > 3$: numerator $+$, denominator $+$ → fraction positive

1. The inequality asks for $\geq 0$: I take the positive intervals and include the zeros of the numerator ($x = -3$ and $x = 3$), never $x = -1$.

Solution:

The typical mistake we see in mock tests: writing $-3 \leq x \leq -1$, including the value that makes the denominator zero. The TOLC-I answer options almost always contain that wrong variant, deliberately.

Exercise 2 — Coordinate geometry: a tangent to a circle

For which values of $q$ is the line $y = x + q$ tangent to the circle $x^2 + y^2 = 8$?

Working.

1. The circle has centre $O(0,\,0)$ and radius $r = \sqrt{8} = 2\sqrt{2}$.
2. The line is tangent when its distance from the centre equals the radius. But the point-to-line distance formula needs the implicit form: from $y = x + q$ I get $x - y + q = 0$. Skipping this conversion is by far the most frequent mistake on this type of question.
3. Distance from the centre: $d = \frac{|1 \cdot 0 - 1 \cdot 0 + q|}{\sqrt{1^2 + (-1)^2}} = \frac{|q|}{\sqrt{2}}$.
4. I impose tangency: $\frac{|q|}{\sqrt{2}} = 2\sqrt{2}$, so $|q| = 2\sqrt{2} \cdot \sqrt{2} = 4$.

Solution:

Alternative check (slower, but useful if you cannot remember the distance formula): substitute $y = x + q$ into the circle, get $2x^2 + 2qx + q^2 - 8 = 0$, and set the discriminant to zero: $4q^2 - 8(q^2 - 8) = 0$, that is $q^2 = 16$. Same answer, about a minute longer — and in the TOLC-I a minute is expensive.

Exercise 3 — Functions: the domain

Find the domain of

Working.

1. Two conditions, to be solved together.
2. The expression under the root must be non-negative: $x + 2 \geq 0$, that is $x \geq -2$. The $\geq$ is right: the root is in the numerator, so it is allowed to be zero ($x = -2$ is admissible).
3. The denominator must not be zero: $x^2 - 9 \neq 0$, that is $x \neq 3$ and $x \neq -3$.
4. Intersection: starting from $x \geq -2$, the value $x = -3$ is already outside, so the only exclusion that counts is $x = 3$.

Solution:

Here too the options are built around the predictable mistakes: anyone who mechanically writes "$x \neq \pm 3$" without intersecting picks the option that excludes $-3$ as well; anyone who uses $x > -2$ instead of $x \geq -2$ picks another. The maths is third-year level — precision is what makes the difference.

Exercise 4 — Trigonometry: counting the solutions

How many solutions does the equation $2\sin^2 x - \sin x = 0$ have in the interval $[0,\,2\pi)$?

Working.

1. Factorise: $\sin x \cdot (2\sin x - 1) = 0$. Never divide by $\sin x$: you would be dividing by something that can be zero, and you would lose solutions.
2. First factor: $\sin x = 0$ gives $x = 0$ and $x = \pi$ ($2\pi$ is excluded from the interval).
3. Second factor: $\sin x = \frac{1}{2}$ gives $x = \frac{\pi}{6}$ and $x = \frac{5\pi}{6}$.
4. In total: $0$, $\frac{\pi}{6}$, $\frac{5\pi}{6}$, $\pi$.

Solution: 4 solutions.

Anyone who divides by $\sin x$ finds only the two solutions of $\sin x = \frac{1}{2}$ and answers "2" — and "2" is reliably among the options. The only tool required is the zero-product property, which you learn in the first two years of high school. Nothing more.

What if I am not coming from a scientific high school?

You can prepare for the TOLC-I from a liceo classico, a liceo linguistico or a technical institute too — but be honest about the gaps. On our paths the typical difference is not algebra, which everyone has done: it is trigonometry and coordinate geometry, which on many tracks are covered lightly or not at all. In that case it is not a question of "revising" but of building two areas from scratch, and that takes more weeks of preparation, not more hours a day.

The first useful step is working out where you stand: tell us about your situation and we will build the plan around the track you are coming from and the date of your test. If the problem lies further upstream — shaky algebra foundations, study method — a course of maths tutoring before the test-specific preparation makes everything else go faster.

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