Maths Seconde Study Sheet 2026

Maths Seconde — Complete Revision Guide 2026

Seconde (Year 11 / 10th grade equivalent) is the gateway to French upper secondary school. The maths program consolidates middle school foundations and introduces key concepts for later specialization.

1. Numbers and Algebra

Number Sets

The main sets: N (natural numbers), Z (integers), D (decimals), Q (rationals: a/b with b ≠ 0), R (all real numbers, including irrationals like √2 and π). Inclusion: N ⊂ Z ⊂ D ⊂ Q ⊂ R.

Intervals and Absolute Value

Interval notation: [a, b] (closed), ]a, b[ (open), [a, +∞[. The absolute value |x| is the distance from x to 0. Distance between a and b: |a − b|.

Key Identities

Three essential identities to memorize:

• (a + b)² = a² + 2ab + b²
• (a − b)² = a² − 2ab + b²
• (a + b)(a − b) = a² − b²

These are used for expanding and factoring algebraic expressions.

Equations and Systems

Linear equation ax + b = 0 has solution x = −b/a. For systems of two equations with two unknowns, use substitution (isolate one variable) or elimination (add equations to cancel a variable).

Powers and Roots

Power rules: a^n × a^m = a^(n+m), (a^n)^m = a^(nm), a^(−n) = 1/a^n. Square root rules: √(ab) = √a × √b, (√a)² = a, √(a²) = |a|.

2. Geometry

Coordinate Geometry

Distance between A(x_A, y_A) and B(x_B, y_B): AB = √((x_B−x_A)² + (y_B−y_A)²). Midpoint: ((x_A+x_B)/2, (y_A+y_B)/2).

Vectors

A vector has direction, orientation, and magnitude. Vector AB has coordinates (x_B−x_A, y_B−y_A). Sum: add coordinates. Scalar product: multiply each coordinate by k. Two vectors are collinear if their determinant x₁y₂ − x₂y₁ = 0 (same direction).

Lines

Cartesian equation: ax + by + c = 0. Slope-intercept form: y = mx + p where m is the slope. Slope from two points: m = (y_B−y_A)/(x_B−x_A). Two lines are parallel if they have the same slope.

3. Functions

Core Concepts

A function f maps each x in its domain to a unique value f(x). The image of x is f(x); an antecedent of y is any x such that f(x) = y.

Reference Functions

• Square f(x) = x²: decreasing on (−∞, 0], increasing on [0, +∞), minimum at 0, parabola shape.
• Inverse f(x) = 1/x: defined for x ≠ 0, decreasing on each interval, hyperbola shape.
• Square root f(x) = √x: defined for x ≥ 0, increasing, concave curve.
• Linear/affine f(x) = ax + b: straight line, increasing if a > 0, decreasing if a < 0.

Graphical Solving

To solve f(x) = k graphically, draw y = k and find intersections. To solve f(x) > k, identify where the curve is above the line.

4. Statistics and Probability

Descriptive Statistics

Mean: x̄ = Σ(n_i × x_i)/N. Median: middle value of ordered data. Quartiles: Q1 (25th percentile), Q3 (75th percentile). Standard deviation: σ = √(Σn_i(x_i−x̄)²/N), measures spread.

Probability

An experiment has a sample space Ω. P(A) is between 0 and 1. P(Ā) = 1 − P(A). If outcomes are equally likely: P(A) = favorable outcomes / total outcomes. General addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Probability trees help organize multi-step experiments.

Sampling

Fluctuation interval at 95% for proportion p with sample size n: approximately [p − 1/√n, p + 1/√n].

5. Python Programming

Basic concepts: variables, if/elif/else conditionals, for loops (range(n) iterates 0 to n−1), while loops. Define functions with def and return. Applications: distance calculations, threshold search, random simulations.