Baccalauréat mathematics paper Session 2026 — 16 juin 2026 with full solutions

A ship operates a service between two Mediterranean ports. For a crossing, a family can book a cabin as well as a space for a vehicle.

Part A

Among all the families making the crossing, 30 % book a space for a vehicle and, among these, 80 % book a cabin.

It is also known that 75 % of the families making the crossing book a cabin.

A family making the crossing is chosen at random, and we consider the following events:

• V: "the family books a space for a vehicle";
• C: "the family books a cabin".

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

1. • a. Give P(C).
   • b. Copy the weighted tree from the paper and fill in the four blanks. (The tree is shown in the PDF: a first level V / V̄, then C / C̄ on each of the two branches.)
2. Calculate the probability that a family books a space for a vehicle and a cabin.
3. A family has booked a cabin. Determine the probability that it books a space for a vehicle.
4. Determine P_V̄(C). Round the result to 10⁻². Interpret the result in the context of the exercise.

Part B

On this route, booking a cabin and a space for a vehicle are charged on top of the cost of the crossing.

These supplements amount to:

• €100 for a cabin;
• €70 for a vehicle space.

We assume that a family can book at most one cabin and at most one vehicle space.

In addition to these supplements, there may be the price paid for extras (meals, drinks, etc.).

Let X be the random variable which, to each family making the crossing, assigns the price it pays, in euros, for supplements. The probability distribution of X is given below.

Let Y be the random variable which, to each family making the crossing, assigns the price it pays, in euros, for extras. We assume that Y has expectation E(Y) = 104 and variance V(Y) = 1686.

We assume that the variables X and Y are independent.

1. Justify that E(X) = 96 and that V(X) = 3114.

2. As an exceptional offer, the company offers a 40 % discount on supplements and extras. Let Z be the random variable which, to each family making the crossing, assigns the total amount paid by that family for supplements and extras, in euros, after the discount.

   • a. Justify that Z = 0.6(X + Y).
   • b. Deduce that E(Z) = 120 and that V(Z) = 1728.

3. Let n be a non-zero natural number. A sample of n families making this crossing and benefiting from the exceptional discount defined in question 2 is chosen at random. We assume that this choice can be treated as drawing with replacement.

   Let Z₁ be the random variable equal to the total price for supplements and extras paid by the first family, Z₂ that of the second family, and so on up to Zₙ.

   Consider the random variable Mₙ = (Z₁ + Z₂ + ... + Zₙ) / n giving the mean total price for supplements and extras paid by these families.

   We assume that the variables Z₁, Z₂, ..., Zₙ are independent and follow the same probability distribution as the random variable Z.

   • a. Show that the expectation E(Mₙ) equals 120, and that its variance V(Mₙ) equals 1728/n.
   • b. Determine the smallest integer n for which the Bienaymé-Chebyshev inequality allows us to assert that P(114 < Mₙ < 126) ⩾ 0.85. Interpret the result in the context of the exercise.

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

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

• the line (d) with parametric representation:

  x = t, y = −1.5 − t, z = 2 − 2t, where t ∈ ℝ;

• the points A(3, 0, 2) and B(2, 1, −3);

• the plane (P) with equation −x + y − 5z − 0.5 = 0.

  • a. Statement 1: the plane (P) is orthogonal to the line (AB) and passes through the midpoint of the segment [AB].

  • b. Statement 2: the lines (d) and (AB) intersect.

  • c. Consider the point C with coordinates (1.5, −3, −1).

  Statement 3: the measure of the angle ACB, rounded to 10⁻¹, is 70.5°.

2. Titouan and Clotilde are taking part in an escape game. They find themselves in a room with two exit doors, A and B, each protected by a keypad with 8 keys bearing different symbols:

• the keypad on door A uses a code of 3 different symbols that must be entered in order;
• the one on door B uses a code of 4 different symbols that can be entered in any order.

Having failed to obtain any clues, they decide to enter codes at random. Clotilde chooses a code for door A, and Titouan chooses a code for door B.

We assume that all choices of code are equally likely.

Statement 4: Titouan is more likely to open his door than Clotilde.

We study how a heating system installed in a room operates.

This system switches on automatically as soon as the room temperature is less than or equal to 18 °C, and switches off when it reaches 20 °C.

Part A: heating phase

While the room temperature goes from 18 °C to 20 °C, the heating system runs continuously. The room temperature rises gradually.

In this part, the room temperature, in degrees Celsius, at time t, expressed in tens of minutes, is modelled by a function T defined on [0, +∞).

We assume that the function T is:

• differentiable on [0, +∞);
• a solution of the differential equation (E): y' = −0.035y + 0.91, where y is a function of the variable t, defined and differentiable on [0, +∞).

Let T' denote the derivative of the function T.

We assume that at the start of the study, the room temperature is 18 °C. Thus T(0) = 18.

1. Give the solutions of the differential equation (E) on [0, +∞).
2. Deduce that for every real number t in the interval [0, +∞): T(t) = 26 − 8e^(−0.035t).
3. According to this model, determine how long it will take for the room to reach a temperature of 20 °C. Give the result in hours and minutes, rounded to the nearest minute.
4. If a fault in the heating system prevents it from switching off when the room temperature reaches 20 °C, could the room temperature exceed 28 °C according to this model? Justify.

Part B: cooling phase

When the room reaches 20 °C, the heating system switches off and the room cools down.

The room temperature is modelled by the sequence (uₙ) defined by u₀ = 20 and, for every natural number n:

uₙ₊₁ = 0.965uₙ + 0.35 + 0.07e^(−0.1n)

where n is an integer expressing the elapsed time in tens of minutes.

1. Show that u₁ = 19.72.

2. Prove by induction that, for every natural number n, uₙ > 10.

   We assume that the sequence (uₙ) is decreasing.

3. Deduce that the sequence (uₙ) converges.

4. Let ℓ denote the limit of the sequence (uₙ).

   • a. Justify that this limit is a solution of the equation x = 0.965x + 0.35.
   • b. Determine ℓ and interpret the result in the context of the exercise.

5. Recall that the heating system switches on automatically as soon as the room temperature is less than or equal to 18 °C.

   • a. Copy and complete lines 4, 5 and 6 of the Python program below, so that it returns the number of tens of minutes after which the heating system will switch back on.

1.  def marche() :
2.      n = 0
3.      u = 20
4.      while … :
5.          u = …
6.          n = …
7.      return n

• b. Determine the time, in tens of minutes, after which the heating system will switch back on.

Consider the function f defined on the interval (−1, +∞) by:

f(x) = a + b ln(x + 1) / (x + 1),

where a and b are real numbers, and ln denotes the natural logarithm function.

We assume that the function f is twice differentiable on (−1, +∞). We denote by f' its derivative and by f'' its second derivative.

Part A

The aim of this part is to determine the values of a and b so that the graph of the function f on [0, +∞) matches the one drawn in the orthonormal coordinate system of the paper.

(The graph is in the PDF: the curve passes through the point A(0, 1), increases up to a maximum located a little before x = 2, then decreases slowly. The line T_A is the tangent to the curve at the point A(0, 1).)

1. Justify that a = 1.

2. Using the graph:

   • a. Give the value of f'(0). Justify.
   • b. Give the sign of f''(1). Justify.
3. • a. Show that for every real x in (−1, +∞):

   f'(x) = b(1 − ln(x + 1)) / (x + 1)².

   • b. Deduce the value of b.

Part B

For the rest of the exercise, we assume that the function f is defined on (−1, +∞) by:

f(x) = 1 + 4 ln(x + 1) / (x + 1).

1. Justify that the line with equation y = 1 is an asymptote to the graph of the function f.

2. Solve the inequality 1 − ln(x + 1) > 0 on (−1, +∞).

3. We assume that lim f(x) = −∞ as x tends to −1.

   Draw up the complete table of variations of the function f, giving the exact value of its extremum. Justify.

4. Show that the equation f(x) = 1.5 has a unique solution in the interval [2, +∞). Give a value of it rounded to 10⁻¹.

5. • a. Show that the integral of ln(x + 1)/(x + 1) between 0 and 2 equals (1/2)(ln 3)².
   • b. Deduce the area, in square units, of the region bounded by the x-axis, the graph of the function f, the y-axis and the line with equation x = 2.