Maths Première Study Sheet 2026

Maths Spécialité Première — Complete Revision Guide 2026

Première Spécialité Maths (Year 12 / 11th grade equivalent) deepens the foundations from Seconde and introduces powerful tools essential for Terminale and higher education.

1. Algebra

Quadratic Equations

A quadratic function f(x) = ax² + bx + c (a ≠ 0) has vertex form f(x) = a(x − α)² + β with α = −b/(2a).

Discriminant: Δ = b² − 4ac determines the roots:

• Δ > 0: two distinct roots x₁ = (−b − √Δ)/(2a), x₂ = (−b + √Δ)/(2a). Factored: a(x−x₁)(x−x₂).
• Δ = 0: one double root x₀ = −b/(2a). Factored: a(x−x₀)².
• Δ < 0: no real roots; the sign of f(x) equals the sign of a for all x.

Sign rule: the quadratic has the sign of a outside the roots and the opposite sign between them.

Sequences

Arithmetic sequence with common difference r: u(n) = u(0) + nr. Sum of first (n+1) terms: S = (n+1)(u(0)+u(n))/2.

Geometric sequence with ratio q: u(n) = u(0) × q^n. Sum (q ≠ 1): S = u(0)(1−q^(n+1))/(1−q).

To study variations of a general sequence, examine the sign of u(n+1) − u(n).

2. Analysis — Derivatives

Derivative and Tangent

The derivative f'(a) = lim[h→0] (f(a+h)−f(a))/h is the slope of the tangent line at point (a, f(a)).

Tangent equation: y = f'(a)(x − a) + f(a).

Key Derivatives

Rules: (u+v)' = u'+v', (ku)' = ku', (uv)' = u'v+uv', (u/v)' = (u'v−uv')/v².

Variation and Extrema

f' > 0 on interval I means f is strictly increasing. f' < 0 means strictly decreasing. If f'(a) = 0 with a sign change, f has a local extremum at a.

Exponential Function

The unique function with f' = f and f(0) = 1. Properties: e^(a+b) = e^a×e^b, e^x > 0 always, exp is strictly increasing. Derivative: (e^(u(x)))' = u'(x)×e^(u(x)). Key equation: e^a = e^b ⟺ a = b.

3. Geometry

Dot Product

u⃗ · v⃗ = ||u⃗||×||v⃗||×cos(θ). In coordinates: u⃗ · v⃗ = x_u×x_v + y_u×y_v. Orthogonality: u⃗ · v⃗ = 0.

Al-Kashi formula: BC² = AB² + AC² − 2×AB×AC×cos(Â). Generalizes Pythagoras.

Trigonometry

Unit circle: angle θ in radians corresponds to point (cos θ, sin θ). Key identity: cos²θ + sin²θ = 1.

Addition formulas: cos(a±b) = cos(a)cos(b) ∓ sin(a)sin(b), sin(a±b) = sin(a)cos(b) ± cos(a)sin(b).

Double angle: cos(2a) = 2cos²(a)−1, sin(2a) = 2sin(a)cos(a).

Remarkable values: cos(π/3) = 1/2, sin(π/3) = √3/2, cos(π/4) = sin(π/4) = √2/2.

4. Probability

Conditional Probability

P(B|A) = P(A∩B)/P(A). Compound formula: P(A∩B) = P(A)×P(B|A). Use weighted probability trees where each path's probability is the product of branch probabilities.

Total probability: P(B) = Σ P(A_i)×P(B|A_i) over a partition {A₁, …, A_n}.

Independence

A and B are independent if P(A∩B) = P(A)×P(B), equivalently P(B|A) = P(B).

Random Variables

Expectation: E(X) = Σ x_i×P(X=x_i). Variance: V(X) = E(X²)−[E(X)]². Standard deviation: σ = √V(X). Linearity: E(aX+b) = aE(X)+b, V(aX+b) = a²V(X).

5. Python Programming

Lists, sequential search, applications to sequences (computing terms by recurrence, threshold search with while loops), and derivatives (numerical approximation of f'(a) using the limit definition).