BTS CPI Mathematics Prerequisites Assessment Study Guide

Assessment Overview and Administrative Specifications

The positioning assessment for mathematical prerequisites is designed for apprentices enrolled in the BTS Conception de Produits Industriels (CPI) diploma program under the FORMACTION D'APPRENTIS - Pôle Industrialisation. The examination has an allocated total duration of 2 hours ($2\text{ hours}$). Students are permitted to use a scientific calculator, provided it is set to authorized examination mode (Calculatrice : Autorisée (mode examen)).

The administrative details for the evaluation record the apprentice's family name as CHABAS, the first name as Morgan, and the host company (entreprise d'accueil) as Safran. The official assessment date is set to 08/09/26. The evaluation is scored out of a maximum total of 20 points ($/20$), under the supervision of the referring instructor, Cédric BIEGUET. Evaluation criteria explicitly specify that clarity of mathematical reasoning, neatness of drafting, and full indication of units across all computations are directly taken into account during grading. Students are instructed to leave the cover page blank and turn the page to begin the assessment test.

Laws of Exponents and Scientific Notation Standards

Exponents govern operations involving powers of identical base values. When multiplying two terms with identical bases $a$, the exponents are added together according to the rule:

$a^n \times a^m = a^{n+m}$

When dividing two terms sharing the same base $a$, the exponent in the denominator is subtracted from the exponent in the numerator:

$\frac{a^n}{a^m} = a^{n-m}$

Raising an exponential expression to another exponent requires multiplying the power indices:

$(a^n)^m = a^{n \times m}$

A negative exponent represents the reciprocal of the corresponding positive exponent expression:

$a^{-n} = \frac{1}{a^n}$

Scientific notation expresses numerical values in a standardized compact form composed of a coefficient multiplied by a power of ten:

$a \times 10^p$

In standard scientific notation, the coefficient $a$ must strictly satisfy the boundary condition $1 \le |a| < 10$, where $p$ is an integer representing the order of magnitude.

Fractional Operations and Algebraic Identities

Calculations involving algebraic fractions require establishing common denominators for addition and reciprocal multiplication for division. Adding two fractions with distinct denominators requires finding a common denominator:

$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$

Multiplying two rational expressions involves directly multiplying their respective numerators and denominators:

$\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$

Dividing a fraction by another fraction is equivalent to multiplying the dividend fraction by the inverse (reciprocal) of the divisor fraction:

$\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$

Polynomial expansion relies on fundamental algebraic identities. Expanding the square of a sum yields the first classic polynomial identity:

$(a + b)^2 = a^2 + 2ab + b^2$

Expanding the square of a difference yields the second classic polynomial identity:

$(a - b)^2 = a^2 - 2ab + b^2$

Multiplying the sum and difference of two terms yields the difference of squares identity:

$(a - b)(a + b) = a^2 - b^2$

Quadratic Equations and Discriminant Analysis

A second-degree polynomial equation in standard single-variable form is expressed as:

$ax^2 + bx + c = 0$

where $a$, $b$, and $c$ are constant coefficients with $a \neq 0$. To determine the real roots of a quadratic equation, the discriminant $\Delta$ is computed using the formula:

$\Delta = b^2 - 4ac$

The number and nature of the real solutions depend strictly on the mathematical sign of the discriminant $\Delta$:

When $\Delta > 0$, the quadratic equation possesses two distinct real solutions $x_1$ and $x_2$, defined as:

$x_1 = \frac{-b - \sqrt{\Delta}}{2a}$

$x_2 = \frac{-b + \sqrt{\Delta}}{2a}$

When $\Delta = 0$, the quadratic equation possesses exactly one double real solution $x_0$, defined as:

$x_0 = -\frac{b}{2a}$

When $\Delta < 0$, the discriminant is negative, meaning the quadratic equation has no real solutions (aucune solution réelle).

Affine Functions, Rate of Change, and Linear Intercepts

An affine function defines a linear relationship between an independent variable $x$ and a dependent output $f(x)$, represented mathematically in explicit form as:

$f(x) = ax + b$

In this functional expression, $f(x)$ represents the calculated output (ordinate value), $a$ represents the slope or rate of change (pente / taux d'accroissement), and $b$ represents the y-intercept (ordonnée à l'origine).

Given two distinct points on the straight line, $A(x_1, y_1)$ and $B(x_2, y_2)$, the slope $a$ is determined by calculating the ratio of the vertical change to the horizontal change:

$a = \frac{y_2 - y_1}{x_2 - x_1}$

Once the slope $a$ is determined, the y-intercept $b$ is evaluated by substituting the coordinates of point $A(x_1, y_1)$ into the linear equation:

$b = y_1 - a \times x_1$

Right-Triangle Trigonometry and Geometric Theorems

Trigonometric ratios relate the interior angles of a right-angled triangle to the relative lengths of its sides. For an acute angle $\theta$ within a right triangle, the three primary trigonometric ratios are defined as follows:

The sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse:

$\sin(\theta) = \frac{\text{Opposé}}{\text{Hypoténuse}}$

The cosine function is defined as the ratio of the length of the adjacent side to the length of the hypotenuse:

$\cos(\theta) = \frac{\text{Adjacente}}{\text{Hypoténuse}}$

The tangent function is defined as the ratio of the length of the opposite side to the length of the adjacent side:

$\tan(\theta) = \frac{\text{Opposé}}{\text{Adjacente}}$

In addition to trigonometric ratios, the fundamental geometric relationship between the sides of a right triangle is defined by the Pythagorean Theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the two adjacent leg lengths:

$\text{Hypoténuse}^2 = \text{Côté}_1^2 + \text{Côté}_2^2$

Foundations of Probability Theory

Probability theory quantifies the likelihood of occurrence for specific random events within a well-defined sample space. Under the assumption of equally likely outcomes, the theoretical probability of an event $A$ is calculated as the ratio of favorable outcomes to total possible outcomes:

$P(A) = \frac{\text{Nombre de cas favorables}}{\text{Nombre total de cas possibles}}$

The complementary event of $A$, denoted as $\bar{A}$ (événements contraires), represents the event that $A$ does not occur. The probability of the complementary event is given by:

$P(\bar{A}) = 1 - P(A)$

By fundamental probability axioms, the numerical probability of any valid event is strictly bounded between $0$ and $1$ inclusive:

$0 \le P(A) \le 1$