Baccalauréat mathematics paper Session 2025 — 17 juin 2025 with full solutions

There are four blood groups in the human species: A, B, AB and O.

Each blood group may carry a rhesus factor. When it is present, the rhesus is said to be positive; otherwise it is said to be negative.

In the French population, it is known that:

• 45 % of individuals belong to group A, and among them 85 % are rhesus positive;
• 10 % of individuals belong to group B, and among them 84 % are rhesus positive;
• 3 % of individuals belong to group AB, and among them 82 % are rhesus positive.

A person is chosen at random from the French population. We denote by:

• A the event "the chosen person has blood group A";
• B the event "the chosen person has blood group B";
• AB the event "the chosen person has blood group AB";
• O the event "the chosen person has blood group O";
• R the event "the chosen person has a positive rhesus factor".

For any event E, we denote by Ē the complementary event of E and by P(E) the probability of E.

1. Copy the weighted tree from the paper, filling in the ten blanks. (The tree is shown in the PDF: four branches A, B, AB, O, then R / R̄ on each.)

2. Show that P(B ∩ R) = 0.084. Interpret this result in the context of the exercise.

3. We are given that P(R) = 0.8397. Show that P_O(R) = 0.83.

4. An individual is called a "universal donor" when their blood can be transfused to anyone without any risk of incompatibility. Group O rhesus negative is the only one with this property.

   Show that the probability that an individual chosen at random from the French population is a universal donor is 0.0714.

5. During a blood drive, a sample of 100 people is chosen from the population of a French town. This population is large enough for the choice to be treated as drawing with replacement.

   Let X be the random variable which, to each sample of 100 people, assigns the number of universal donors in that sample.

   • a. Justify that X follows a binomial distribution, stating its parameters.
   • b. Determine, to within 10⁻³, the probability that there are at most 7 universal donors in this sample.
   • c. Show that the expectation E(X) of the random variable X equals 7.14 and that its variance V(X) equals 6.63 to within 10⁻².

6. During national blood donation week, a blood drive is organised in N French towns chosen at random and numbered 1, 2, 3, …, N, where N is a non-zero natural number.

   Consider the random variable X₁ which, to each sample of 100 people from town 1, assigns the number of universal donors in that sample. The random variables X₂ for town 2, …, X_N for town N are defined in the same way.

   We assume that these random variables are independent and that they all have the same expectation, equal to 7.14, and the same variance, equal to 6.63.

   Consider the random variable M_N = (X₁ + X₂ + … + X_N) / N.

   • a. What does the random variable M_N represent in the context of the exercise?
   • b. Calculate the expectation E(M_N).
   • c. Let V(M_N) denote the variance of the random variable M_N. Show that V(M_N) = 6.63/N.
   • d. Determine the smallest value of N for which the Bienaymé-Chebyshev inequality allows us to assert that P(7 < M_N < 7.28) ⩾ 0.95.

Consider a function f defined on the interval (0, +∞). We assume that it is twice differentiable on (0, +∞). We denote by f' its derivative and by f'' its second derivative.

In an orthogonal coordinate system, the paper shows:

• the graph of f, denoted C_f, on the interval (0, 3];
• the line T_A, tangent to C_f at the point A(1, 2);
• the line T_B, tangent to C_f at the point B(e, e).

We are also told that the tangent T_A passes through the point C(3, 0).

(The figure is in the PDF. The shaded region, used in Part C, is bounded by the curve C_f, the tangent T_B and the lines with equations x = 1 and x = e.)

Part A: reading the graph

Answer the following questions, justifying your answers using the graph.

1. Determine the derivative f'(1).
2. How many solutions does the equation f'(x) = 0 have in the interval (0, 3]?
3. What is the sign of f''(0.2)?

Part B: study of the function f

In this part, we assume that the function f is defined on the interval (0, +∞) by:

f(x) = x(2(ln x)² − 3 ln x + 2)

where ln denotes the natural logarithm function.

1. Solve in ℝ the equation 2X² − 3X + 2 = 0. Deduce that C_f does not intersect the x-axis.
2. Determine, with justification, the limit of f at +∞. You may assume that the limit of f at 0 is 0.
3. We assume that for all x in (0, +∞), f'(x) = 2(ln x)² + ln x − 1.
   • a. Show that for all x in (0, +∞), f''(x) = (1/x)(4 ln x + 1).
   • b. Study the convexity of the function f on the interval (0, +∞) and give the exact value of the x-coordinate of the point of inflection.
   • c. Show that the curve C_f lies above the tangent T_B on the interval [1, +∞).

Part C: area calculation

1. Justify that the tangent T_B has equation y = 2x − e.

2. Using integration by parts, show that the integral of x ln x between 1 and e equals (e² + 1)/4.

3. Let 𝒜 denote the area of the shaded region in the figure, bounded by the curve C_f, the tangent T_B, and the lines with equations x = 1 and x = e.

   We assume that the integral of x(ln x)² between 1 and e equals (e² − 1)/4.

   Deduce the exact value of 𝒜 in square units.

For each of the following statements, say whether it is true or false. Justify each answer. An answer without justification earns no marks.

Space is equipped with an orthonormal coordinate system (O; i, j, k).

1. Consider the points A(−1, 0, 5) and B(3, 2, −1).

Statement 1: a parametric representation of the line (AB) is:

x = 3 − 2t, y = 2 − t, z = −1 + 3t, with t ∈ ℝ.

Statement 2: the vector n(5, −2, 1) is normal to the plane (OAB).

2. Consider:

• the line d with parametric representation x = 15 + k, y = 8 − k, z = −6 + 2k, with k ∈ ℝ;
• the line d' with parametric representation x = 1 + 4s, y = 2 + 4s, z = 1 − 6s, with s ∈ ℝ.

Statement 3: the lines d and d' are not coplanar.

3. Consider the plane 𝒫 with equation x − y + z + 1 = 0.

Statement 4: the distance from the point C(2, −1, 2) to the plane 𝒫 is equal to 2√3.

A team of biologists is studying how the area covered by a seagrass called Posidonia changes over time on the seabed of Alycastre Bay, near the island of Porquerolles.

The area studied covers a total of 20 hectares (ha), and on 1 July 2024, Posidonia covered 1 ha of this area.

Part A: study of a discrete model

For every natural number n, let uₙ denote the area of the zone, in hectares, covered by Posidonia on 1 July of the year 2024 + n. Thus u₀ = 1.

A study of this area established that for every natural number n:

uₙ₊₁ = −0.02uₙ² + 1.3uₙ.

1. Calculate the area that Posidonia should cover on 1 July 2025 according to this model.
2. Let h be the function defined on [0, 20] by h(x) = −0.02x² + 1.3x. We assume that h is increasing on [0, 20].
   • a. Prove that for every natural number n, 1 ⩽ uₙ ⩽ uₙ₊₁ ⩽ 20.
   • b. Deduce that the sequence (uₙ) converges. Let L denote its limit.
   • c. Justify that L = 15.
3. The biologists want to know how long it will take for the area covered by Posidonia to exceed 14 hectares.
   • a. Without any calculation, justify that, according to this model, this will happen.
   • b. Copy and complete the following algorithm so that, at the end of its execution, it displays the answer to the biologists' question.

def seuil():
    n = 0
    u = 1
    while ……………… :
        n = …………
        u = …………
    return n

Part B: study of a continuous model

We wish to describe the area of the studied zone covered by Posidonia over time with a continuous model.

In this model, for a time t, in years, elapsed since 1 July 2024, the area of the studied zone covered by Posidonia is given by f(t), where f is a function defined on [0, +∞) satisfying:

• f(0) = 1;
• f does not vanish on [0, +∞);
• f is differentiable on [0, +∞);
• f is a solution on [0, +∞) of the differential equation (E₁): y' = 0.02y(15 − y).

We assume that such a function f exists; the aim of this part is to find an expression for it. Let f' denote the derivative of f.

1. Let g be the function defined on [0, +∞) by g(t) = 1/f(t). Show that g is a solution of the differential equation (E₂): y' = −0.3y + 0.02.
2. Give the solutions of the differential equation (E₂).
3. Deduce that for all t ∈ [0, +∞): f(t) = 15 / (14e^(−0.3t) + 1).
4. Determine the limit of f at +∞.
5. Solve the inequality f(t) > 14 on the interval [0, +∞). Interpret the result in the context of the exercise.