Baccalauréat mathematics paper Session 2026 — 17 juin 2026 with full solutions

In the orthonormal coordinate system (O; i, j, k), consider the points:

A(2, 1, 1), B(3, −2, 0), C(0, −1, 1) and D(0, 0, 2).

(A figure showing the tetrahedron ABCD accompanies the statement in the PDF.)

1. Show that the points A, B and C define a plane.

2. • a. Show that the vector n(1, −1, 4) is normal to the plane (ABC).
   • b. Deduce that a Cartesian equation of the plane (ABC) is: x − y + 4z − 5 = 0.

3. Check that a parametric representation of the line Δ passing through D and orthogonal to the plane (ABC) is given by:

   x = t, y = −t, z = 2 + 4t, with t ∈ ℝ.

4. Show that the orthogonal projection of D onto the plane (ABC) is the point H(−1/6, 1/6, 4/3).

5. • a. Show that the triangle ABC is isosceles at B.
   • b. Show that the area of the triangle ABC is 3√2.
6. • a. Deduce that the volume of the tetrahedron ABCD equals 1.

   Recall that the volume of a tetrahedron is given by V = (1/3) × ℬ × h where ℬ is the area of a base and h the height relative to that base.

   • b. We assume that the distance from the point A to the plane (BCD) equals √2. Deduce the area of the triangle BCD.

7. In this question, let D_k denote the point with coordinates (0, 0, k) where k is a real number.

   • a. Determine the value of k for which the points A, B, C and D_k are coplanar. What is then the orthogonal projection of the point D_k onto the plane (ABC)?
   • b. Can we find a value of k such that A is the orthogonal projection of D_k onto the plane (ABC)?

A pond used for trout farming contains 30,000 litres of water, which is regularly renewed.

To check for the possible presence of pollutants in the pond, regular analyses are carried out.

In the two parts below, we study a case of water contamination by a pollutant using two different models. The two parts are independent.

Part A: discrete model

Let Vₙ denote the volume, in litres, of pollutant present in the pond n hours after the start of the pollution. Thus V₀ = 0.

We assume that for every natural number n, Vₙ₊₁ = 0.995Vₙ + 6.

1. Calculate V₁ and V₂.
2. Copy and complete the program below so that it returns the volume, in litres, of pollutant present in the pond after n hours.

def volume(n):
    v = …
    for k in range(n):
        v = …………
    return v

3. Prove by induction that for every natural number n, Vₙ ⩽ Vₙ₊₁ ⩽ 1200.
4. Justify that the sequence (Vₙ) converges, and calculate its limit.

Part B: continuous model

Let v(t) denote the volume, in litres, of pollutant present in the pond at time t, in hours.

We assume that the function v is defined and differentiable on the interval [0, +∞), and we denote its derivative by v'.

We consider that the pollution starts at time t = 0; thus v(0) = 0. We assume that the function v is a solution of the differential equation:

(E): y' = −0.005y + 6,

where y is an unknown function and y' is the derivative of y.

1. • a. Determine the solutions of the differential equation (E) on [0, +∞).
   • b. Show that for every real t in the interval [0, +∞): v(t) = 1200(1 − e^(−0.005t)).
   • c. Calculate the limit of v at +∞.
   • d. Determine how the function v varies on the interval [0, +∞).

2. A pollutant level above 5 % of the volume of the pond requires a complete clean-out. According to this model, will the owner have to carry out this clean-out?

3. To be able to decontaminate the pond quickly and avoid serious consequences for the fish farm, it is recommended that the volume of pollutant should not exceed 50 litres.

   Determine the exact value of the time after which the volume of pollutant in the pond exceeds 50 litres. Give its value in hours and minutes, rounded to the nearest minute.

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

"Professional musicians" refers to all the women and men who play an instrument professionally.

1. A study was carried out among professional musicians playing various musical genres (Pop, Rap, Rock, Classical, etc.). This study shows that:

• 52 % of professional musicians play Pop;
• among professional musicians who play Pop, 32 % are women;
• 20 % of professional musicians are women.

A person is chosen at random from all professional musicians. We denote:

• O the event: "the person plays Pop";
• F the event: "the person is a woman".

Statement 1: if the chosen person is a woman, then the probability that she plays Pop is 0.832.

2. The proportion of professional musicians within the working population of the cultural sector is estimated at 6.2 %. A sample of 5,000 people is drawn at random and independently from this population. The population considered is large enough for this sampling to be treated as drawing with replacement.

Consider the random variable X which, to every sample defined in this way, assigns the number of professional musicians in the sample.

Statement 2: rounding the result to one decimal place, P(X ⩽ 340) ≈ 0.4.

Statement 3: by applying the Bienaymé-Chebyshev inequality, we can assert that there is more than a 95 % chance that, in this sample of 5,000 workers in the cultural sector, the number of professional musicians lies strictly between 230 and 390.

3. A Pop culture quiz game is offered to 10 people, 4 of whom are professional musicians. Two teams of 5 players compete against each other. Each team is made up of 2 professional musicians and 3 people who are not.

Statement 4: 120 different teams can be formed in this way.

Part A

Let f be a function defined and differentiable on the interval [0, +∞), part of whose graph C_f is given in the paper in an orthogonal coordinate system. Let f' denote its derivative.

We are told that:

• the curve C_f has a tangent parallel to the x-axis at the point C with x-coordinate 1;
• the tangent T_A to the curve C_f at the point A(1/2, 0) passes through the point B(1, e²).

1. Using the graph:
   • a. Give f'(1).
   • b. Give the solution of the equation f(x) = 0 on the interval [0, 3].
2. Determine f'(1/2).
3. Among the three curves C₁, C₂ and C₃ given in the paper, two represent antiderivatives of the function f. State which ones and justify.

Part B

A manufacturer has to create a logo with the shape given in figure 1 of the paper. The dimension of 0.3 cm shown in the figure is a constraint imposed by the client.

The manufacturer chooses to model the outline of this logo using a portion of the graph of the function f from Part A, together with the image of this portion under reflection in the x-axis, on an interval [0.5, α]. The number α is a real number greater than 1 which will be determined in this part.

The whole is represented graphically in an orthonormal coordinate system (figure 2) with a unit of 1 cm.

We assume that the function f is defined on the interval [0, +∞) by:

f(x) = (2x − 1)e^(−2x+3).

1. • a. Check that for every real x in the interval [0, +∞):

   f(x) = e² × (2x − 1)/e^(2x−1).

   • b. Deduce the limit of f at +∞. (You may set X = 2x − 1.)
2. • a. Show that for all x in the interval [0, +∞), f'(x) = (−4x + 4)e^(−2x+3).
   • b. Draw up the complete table of variations of the function f on the interval [0, +∞).

3. Justify that the constraint imposed by the client translates into the equality f(α) = 0.15.

   Show that there is a unique value of α in the interval [1, +∞) satisfying this equality. Give a value of it rounded to one decimal place.

Part C

A key ring is made by cutting a steel plate to the shape of the logo represented by the shaded area in figure 2 of Part B. We are told that this plate is 0.2 cm thick.

1. Consider I, the integral of f between 0.5 and 3.3.

   Using integration by parts, show that the integral I, rounded to one decimal place, equals 3.6.

2. We are told that the volume of the key ring is equal to:

   Area of the logo × thickness of the plate.

   Determine the volume of the key ring in cm³, rounded to one decimal place.