Annales du bac de maths 2023 en terminale to revise online.

Numerous sample exercises for the 2023 baccalaureate classified by chapter.

These sample exercises allow you to revise the high school baccalaureate in order to prepare yourself in the best conditions.

In addition to all the subjects of the baccalaureate of mathematics of the previous sessions, Mathovore provides you with extracts of subject which target each chapter of the program of terminale.

Numerical functions

Consider the application f from $\,\mathbb{R}$ into $\,\mathbb{R}$ defined by :

if $\,x\in\,%5B0;2%5B\,,\,\,f(x)=x^2(2-x)$ ;

and for any $\,x$ of $\,\mathbb{R}\,,\,f(x+2)=f(x)$ .

1. Study the restriction $\,f_0$ of f to the interval [0;2] and construct the representative curve of $\,f_0$ .

How can we deduce the representative curve of the restriction of f to the interval [2n;2n+2] where n is an element of $\,\mathbb{Z}$ .

2. Show that :

If $\,x\in\,%5B2n;2n+2%5D\,,\,\,f(x)=(x-2n)^2(2n+2-x).$

3. Is f continuous on $\,\mathbb{R}$ ?

4. Is f derivable on $\,\mathbb{R}$ ?

Numerical sequences

Let $\,(U_n)$ and $\,(V_n)$ be the sequences defined for any natural number n by :

$\,U_0=9\,,\,U_{n+1}=\frac{1}{2}U_n-3\,,\,V_n=U_n+6$

1.a. Show that $\,(V_n)$ is a geometric sequence with positive terms.

b. Calculate the sum $\,S_n=\sum_{k=0}^{n}V_k$ as a function of n and deduce the sum $\,S'_n=\sum_{k=0}^{n}U_k$ as a function of n .

c. determine $\,lim_{n\,\to\,+\infty}\,S_n$ and $\,lim_{n\,\to\,+\infty}\,S'_n$ .

2. We define the sequence $\,(W_n)$ by $\,W_n=ln\,V_n$ for any integer n .

Show that the sequence $\,(W_n)$ is an arithmetic sequence.

Calculate $\,S''_n=\sum_{k=0}^{n}W_k$ as a function of n and determine $\,lim_{n\,\to\,+\infty}\,S''_n$

3. Calculate the product $\,P_n=V_0\times \,V_1\,\times \,....\times \,V_n$ as a function of n.

Deduce $\,lim_{n\,\to\,+\infty}\,P_n$ .

Probability:

A player starts a game in which he or she is required to play several games in succession.

The probability that the player loses the first game is 0.2.

The game then proceeds as follows:

If he wins one game, then he loses the next with a probability of 0.05 ;

If he loses one game, then he loses the next with a probability of 0.1.

1) We call :

_E1 is the event “the player loses the first game”;

_E2 is the event “the player loses the second game”;

_E3 the event “the player loses the third game”.

We call X the random variable that gives the number of times the player loses in the first three games.

A weighted tree can be used to help.
a) What are the values taken by X?
b) Show that the probability of the event(X = 2) is equal to 0.031 and that the probability of the event(X = 3) is equal to 0.002.
c) Determine the probability distribution of X.
d) Calculate the expectation of X.
2) For any non-zero natural number n, let _En be the event: “the player loses the n-th game”, $\overline{E_n}$ the opposite event, and let _pn be the probability of the event _En.
a) Express for any natural number n not zero, the probabilities of the events
$E_n\cap\,E_n$ and $\overline{E_n}\cap\,E_n$ as a function of _pn.
b) Deduce that _pn+1 = 0.05 _pn + 0.05 for any non-zero natural number n.
3) Consider the sequence(_un) defined for any natural number n not zero by $U_n=P_n-\frac{1}{19}$ .
a) Show that(_un) is a geometric sequence whose reason and first term must be specified.
b) Deduce, for any non-zero natural number n, _a then _pn as a function of n.
c) Calculate the limit of _pn when n tends to $+\infty$ .

The complex numbers

Find all pairs $\,(z_1,z_2)$ of complex numbers satisfying the conditions :

$\,\{{z_1z_2=\frac{1}{2}\atop\,z_1+2z_2=\sqrt{3}}$ .

Give the trigonometric form of each of the numbers thus obtained.

Complex numbers and trigonometry

Let be the complex numbers $\,z_1=\frac{\sqrt{6}-i\sqrt{2}}{2}$ and $\,z_2=1-i$ .

1. Put into trigonometric form $\,z_1\,,\,z_2\,,\,Z=\frac{z_1}{z_2}$ .

Deduce that :

$\,cos(\frac{\pi}{12})=\frac{\sqrt{6}+\sqrt{2}}{4}$ and $\,sin(\frac{\pi}{12})=\frac{\sqrt{6}-\sqrt{2}}{4}$ .

3. Consider the equation with real unknown x :

$\,(\sqrt{6}+\sqrt{2})cos\,x+(\sqrt{6}-\sqrt{2})sin\,x=2$

a. Solve this equation in $\,\mathbb{R}$ .

b. Place the image points of the solutions on the trigonometric circle.

Differential equations

Consider the differential equation $\,y'-2y=e^{2x}\,(E)$ .

1. Prove that the function u defined on $\mathbb{R}$ by $u(x)=xe^{2x}$ is a solution of (E) .

2. Solve the differential equation $\,y'-2y=0\,\,\,(E_0)$ .

3. Prove that a function v defined on $\,\mathbb{R}$ is a solution of (E) if and only if v-u is a solution of $\,(E_0)$ .

4. Deduce all solutions of equation (E) .

5. Determine the function, solution of (E), which takes the value 1 in 0 .

6. The plane is provided with an orthonormal reference frame $\,(O,\vec{i},\vec{j}).$

Let f be the function defined on $\,\mathbb{R}$ by $\,f(x)=(x+1)e^{2x}$ .

We note C the representative curve of f in the reference frame $\,(O,\vec{i},\vec{j}).$

a. Study the variations of f and draw up its table of variation.

b. Draw C .

The barycenter of weighted points

Consider a triangle ABC in the plane .

1.a. Determine and construct the point G, barycenter of the weighted point system :

$\,\{(A;1)\,;\,(B;-1)\,;\,(C;1)\}$ .

b. Determine and construct the point G’, barycenter of the weighted point system :

$\,\{(A;1)\,;\,(B;5)\,;\,(C;-2)\}$ .

2.a. Let J be the middle of [AB].

Express $\vec{GG'}$ and $\vec{JG'}$ in terms of $\vec{AB}$ and $\vec{AC}$ and deduce the intersection of the lines (GG’) and (AB) .

b. Show that the barycenter I of the weighted point system :

$\,\{(B;2)\,;\,(C;-1)}$ belongs to (GG’) .

3. Let D be any point in the plane and O the midpoint of [CD] and K the midpoint of [OA].

a. Determine three real numbers a, b, c such that K is the barycenter of the weighted point system :

$\,\{(A;a)\,;\,(B;b)\,;\,(C;c)\}$ .

b. Let X be the point of intersection of (DK) and (AC).

Determine the real numbers a’ and c’ such that X is the barycenter of the weighted point system :

$\,\{(A;a')\,;\,(C;c')\}$ .

Geometry in space

We propose to study a model of an air traffic control tower, in charge of monitoring two air routes represented by two straight lines in space.

The space is referred to an orthonormal reference frame $(O;\vec{i},\vec{j},\vec{k})$ of unit 1 km .

The plane $(O;\vec{i},\vec{j})$ represents the ground.

the two
air routes
to be controlled are represented by two straight lines $\,(D_1)$ and $\,(D_2)$ , whose parametric representations are known:

$\,(D_1)\,\,\,\\,{x=3+a\,\\\,y=9+3a\,\\\,z=2\,$

$\,(D_2)\,\,\,\\,{x=0,5+2b\,\\\,y=4+b\,\\\,z=4-b\,$

1.a. Give the coordinates of a vector $\,\vec{u_1}$ director of the line $\,(D_1)$ and a vector $\,\vec{u_2}$ director of the line .

b. Prove that the lines $\,(D_1)$ and $\,(D_2)$ are not coplanar.

2. We want to install at the top S of the control tower, of coordinates S(3;4;0,1), n monitoring device which emits a ray represented by a line noted (R).

Let $\,(P_1)$ be the plane containing S and $\,(D_1)$ and let $\,(P_2)$ be the plane containing S and $\,(D_2)$ .

a. Show that $\,(D_2)$ is secant to $\,(P_1)$ .

b. Show that $\,(D_1)$ is secant to $\,(P_2)$ .

c. A technician states that it is possible to choose the direction of (R) so that this line intersects each of the lines $\,(D_1)$ and $\,(D_2)$ .

Is this statement true? (justify the answer)

Problem on differential equations

Part A

We give a strictly positive natural number n, and consider the differential equation :

$(E_n)\,\,y'+y=\frac{x^n}{n!}e^{-x}$

1. We assume that two functions g and h, defined and derivable on $\mathbb{R}$ , verify, for any real x :

$g(x)=h(x)e^{-x}$

a. Show that g is a solution of if and only if, for any real x :

$\,\,h'(x)=\frac{x^n}{n!}$ .

b. Deduce the function h associated to a solution g of , knowing that f(0)=0.

What is then the function g?

2. Let $\phi$ be a differentiable function on $\mathbb{R}$ .

a. Show that $\phi$ is a solution of if and only if $\phi\,-\,g$ is a solution of the equation :

(F) y’+y=0

b. Solve (F).

c. determine the general solution $\phi$ of the equation .

d. Determine the solution f of the equation verifying f(0)=0 .

Part B

The purpose of this section is to show that:
$\fbox{\lim_{n\,\to\,+\infty}(\sum_{k=0}^{n}\frac{1}{k!})=e}$
1. For all real x, we pose,

$f_0(x)=e^{-x}\,,\,f_1(x)=xe^{-x}$ .

a. verify that is a solution of the differential equation: y’+y=.

b. For any strictly positive integer n, we define the function as the solution of the differential equation y’+y= $f_{n-1}$ checking .

Using Part A, show by recurrence that for any real x and any integer $n\ge\,1$ :

$f_n(x)=\frac{x^n}{n!}e^{-x}$ .

2. For any natural number n, we pose :

$I_n=\int_{0}^{1}\,f_n(x)dx$

a. Show, for any natural number n and for any x element of the interval [0;1], the square :

$0\le\,f_n(x)\le\,\frac{x^n}{n!}$ .

Deduce that $0\le\,I_n\le\,\frac{1}{(n+1)!}$ , then determine the limit of the sequence $(\,I_n)$ .

b. Show, for any non-zero natural number k, the equality :

$\fbox{I_k-I_{k-1}=-\frac{1}{k!}e^{-1}}$ .

c. Calculate and deduce from the above that :

$I_n=1-\sum_{k=0}^{n}\frac{e^{-1}}{k!}$ .

d. Finally, deduce :
$\fbox{\lim_{n\,\to\,+\infty}(\sum_{k=0}^{n}\frac{1}{k!})=e}$
Plane similarities and specialization

The complex plane P is related to an orthonormal reference frame $\,(O;\vec{u},\vec{v})$ .

We denote by s the application which to any point M of P of coordinates (x,y) associates the point M’ of coordinates (x’, y’) such that :

$\,\{{x'=-x-y+2\atop\,y'=x-y-1}$

1. Determine the affix z’ of M’ as a function of the affix z of M .

2. Show that s is a direct plane similarity.

Specify its angle, ratio and center I .

3. Let g be the application that associates to any point M of P the isobarycenter G of the points M, M’=s(M) and M”=s(M’) .

a. Calculate, as a function of the affix z of M, the affixes of the points M” and G.

b. Show that g is a direct plane similarity.

What is its center?

c. Determine the affix of the point Mo such that g(Mo) is the point O .

Plot on a figure the corresponding points Mo, M’o, M”o, as well as the point I, center of the similarity s .

Extracts from the baccalauréat S on arithmetic

Let n be a natural number.

1. find according to the values of n, the remainders of the division of $\,5^n$ by 13 .

2. Deduce that $\,1981^{1981}-5$ is divisible by 13.

3. Show that, for any natural number n greater than or equal to 1, the number $\,N=31^{4n+1}+18^{4n-1}$ is divisible by 13 .

Plane similitudes in speciality

In the oriented plane, consider a direct square ABCD with center O.

Let P be a point of the segment [BC] distinct from B .

We note Q the intersection of (AP) with (CD) .

The perpendicular $\,\Delta$ to (AP) passing through A cuts (BC) at R and (CD) at S .

1. Make a figure.

2. Let r be the rotation of center A and angle $\,\frac{\pi}{2}$ .

a. Specify, justifying your answer, the image of the line (BC) by the rotation r .

b. Determine the images of R and P by r .

c. What is the nature of each of the triangles ARQ and APS?

3. N is the middle of the segment [PS] and M is the middle of the segment [QR].

Let s be the similarity of center A, angle $\,\frac{\pi}{4}$ and ratio $\,\frac{1}{\sqrt{2}}$ .

a. Determine the respective images of R and P by s .

b. What is the locus of the point N when P describes the segment [BC] deprived of B?

c. Prove that the points M, B, N and D are aligned.

Exponential in the S baccalaureate in North America
The purpose of this question is to demonstrate that $\lim_{x\,\mapsto \,+\infty\,}\frac{e^x}{x}=+\infty$ .

The following results will be assumed to be known:
– The exponential function is derivable on R and is equal to its derivative function ;
– ;
– for any real x, we have ;
– be two functions $\Psi$ and $\varphi$ defined on the interval $%5BA;+\infty%5B$ where A is a positive real.
If for any x in $%5BA;+\infty%5B$ , $\Psi\,(x)<\varphi\,(x)$ and if $\lim_{x\,\mapsto \,+\infty\,}\Psi\,(x)=+\infty$ then $\lim_{x\,\mapsto \,+\infty\,}\varphi\,(x)=+\infty$ .
a) consider the function g defined on $%5B0;+\infty%5B$ by $g(x)=e^x-\frac{x^2}{2}$ .
Show that for any x in $%5B0;+\infty%5B$ , $g(x)\geq\,\,0$ .
b) Deduce that $\lim_{x\,\mapsto \,+\infty\,}\frac{e^x}{x}=+\infty$ .
Similarities to the Bac S in Pondicherry in speciality

The complex plane is related to a direct orthonormal reference frame $(O,\vec{u},\vec{v})$ .

We consider the application which to the point M of affix z makes correspond the point M’ of affix such that :
$z'=\frac{3+4i}{5}\overline{z}+\frac{1-2i}{5}$ .

We note x and x ‘, y and y ‘ the real parts and the imaginary parts of z and z ‘ .

Prove that $x'=\frac{3x+4y+1}{5}$ and $y'=\frac{4x-3y-2}{5}$
2. a. Determine the set of points invariant to f.
b. What is the nature of the application f ?
3. Determine the set D of points M of affix z such that z ‘ is real.

4. We try to determine the points of D whose coordinates are integers.

Give a particular solution(x0, y0) belonging to ^Z2of the equation4x – 3y = 2.

Determine the set of solutions belonging to ^Z2of the equation4x – 3y = 2.

5. Consider the points M of affix z = x + iy such that x = 1 and $y\,\in\,Z$ . The point M‘ = f(M) has affix z ‘ .

Determine the integers y such that Re(z ‘ ) and lm(z ‘ ^{‘ )} are integers (one can use the congruences modulo 5).

Geometry in space at the bac S in France :

Let be a strictly positive real and OABC a tetrahedron such that :

– OAB, OAC and OBC are right-angled triangles at O,

– OA = OB = OC = .

We call I the foot of the height from C of triangle ABC, H the foot of the height from O of triangle OIC, and D the point in space defined by :

1. What is the nature of the triangle ABC ?
2. Show that the lines (OH) and (AB) are orthogonal and that H is the orthocenter of triangle ABC.
3. Calculation of OH
a. Calculate the volume V of the tetrahedron OABC and the area S of the triangle ABC.
b. Express OH as a function of V and S, deduce that $OH=a\frac{\sqrt{3}}{3}$ .
4. Study of the tetrahedron ABCD.

The space is related to the orthonormal reference frame $(O,\frac{1}{a}\vec{OA},\frac{1}{a}\vec{OB},\frac{1}{a}\vec{OC})$ .
(a) Show that the point H has coordinates: $(\frac{a}{3},\frac{a}{3},\frac{a}{3})$ .
(b) Show that tetrahedron ABCD is regular (i.e., all its edges have the same length).
(c) Let $\Omega$ be the center of the circumscribed sphere of tetrahedron ABCD.
Show that $\Omega$ is a point on the line (OH) and then calculate its coordinates.
Bacteria and differential equations

Let be the number of bacteria introduced in a culture medium at the time ( being a strictly positive real, expressed in millions of individuals).

The purpose of this problem is to study two models of evolution of this population of bacteria:

– a first model for the instants following the seeding(part A),

– a second model that can be applied over a long period of time(part B).

Part A

In the moments following the seeding of the culture medium,

the growth rate of bacteria is considered to be proportional to the number of bacteria present.

In this first model, we note the number of bacteria at time (expressed in millions of individuals).

The function is therefore a solution of the differential equation:. (where is a strictly positive real depending on the experimental conditions).
1. Solve this differential equation, knowing that .
2. We note the doubling time of the bacterial population.
Show that, for any positive real t: $f(t)=N_o2^{\frac{t}{T}}$ .

Part B

The environment being limited (in volume, in nutritive elements, …), the number of bacteria cannot grow indefinitely in an exponential way. The previous model cannot therefore be applied over a long period of time.

To take these observations into account, the evolution of the bacterial population is represented as follows:

Let be the number of bacteria at time t (expressed in millions of individuals);

the function is a strictly positive and derivable function on $%5B0;+\infty%5B$ which verifies for any of $%5B0;+\infty%5B$ the relation :

$(E)\,\,g(t)=ag(t)%5B1-\frac{g(t)}{M}%5D$

where M is a strictly positive constant depending on the experimental conditions and the real defined in part A.1. (a) Show that if is a strictly positive function verifying the relation (E),
then the function $\frac{1}{g}$ is a solution of the differential equation
$(E')\,\,y'+ay=\frac{a}{M}$

(b) Solve (E ‘ ).
(c) Show that if is a strictly positive solution of (E ‘ ), then $\frac{1}{h}$ verifies (E).
2. It is now assumed that, for any positive real $t,\,g(t)=\frac{M}{1+Ce^{at}}$
where is a constant strictly greater than 1 depending on the experimental conditions.
(a) Determine the limit of at $+\infty$ and prove, for any real positive or zero, the double inequality:.
(b) Study the direction of variation of (one can use the relation (E)).
Prove that there is a unique real positive such that $g(t_0)=\frac{M}{2}$ .
(c) Show that $g''=a(1-\frac{2g}{M})g'$ .
Study the sign of .
Deduce that the rate of increase of the number of bacteria is decreasing from the time defined above.
Express as a function of and .
(d) Knowing that the number of bacteria at time is , calculate the average number of bacteria between time 0 and , as a function of M and .
Probabilities for the S baccalaureate in New Caledonia :

A game consists of drawing three balls simultaneously from an urn containing six white balls and four red balls.

It is assumed that all draws are equiprobable.

If the three balls drawn are red, the player wins 100 €;

if exactly two balls are red, he wins 15 €.

and if only one is red he wins 4 €.

In all other cases, he wins nothing.

Let X be the random variable that takes as values the player’s gain in euros during a game.

1°) Determine the probability law of the random variable X.

2°) For a game, the stake is 10 €. Is the game favorable to the player, i.e. is the mathematical expectation strictly greater than 10?

3°) For the organizer, the game is not sufficiently profitable and he considers two solutions:

or increase the stake by 1 €, thus increasing it to 11 €,

_ or decrease each win by €1, i.e. win only €99, e14 or €3.
What is the most cost-effective solution for the organizer?

Arithmetic in Specialty

Consider two natural numbers, nonzero, x and y prime to each other.

We pose S= x + y and P = xy.

1°) a) Show that x and S are prime to each other, as are y and S.

b) Deduce that S = x+y and P =xy are prime to each other.

c) Show that the numbers S and P have different parities (one even, the other odd).
2°) Determine the positive divisors of 84 and arrange them in ascending order.

3°) Find the prime numbers x and y such that: SP = 84.

4°) Determine the two natural numbers a and b verifying the following conditions:
$\,\{\,a+b=84\\ab=d^3\,.$ with d = gcd(a;b)
(We can put a = dx and b = dy with x and y prime between them).

Complex numbers

1°) One considers the polynomial P of the complex variable z, defined by:
$P(z)=z^3+(14-i\sqrt{2})z^2+(74-14i\sqrt{2})z-74i\sqrt{2}$ .
a) Determine the real number y such that iy is a solution of the equation P(z) = 0.
b) Find two real numbers a and b such that, for any complex number z,
we have $P(z)=(z-i\sqrt{2})(z^2+az+b)$
c) Solve in the set $\mathbb{C}$ of complex numbers, the equation P(z) = 0.
2°) The complex plane is related to a direct orthonormal reference frame $(O,\vec{u},\vec{v})$ .

We will take 1 cm for the graphic unit.
a) Place the points A, B and I of affixes _zA = -7 + 5 i ; _zB = -7 – 5 i and $z_I=i\sqrt{2}$ .
b) Determine the affix of the image of point I by the rotation of center O and angle $-\frac{\pi}{4}$ .
c) Locate the point C with affix _zC = 1 + i. Determine the affix of the point N such that ABCN is a parallelogram.
d) Locate the point D with affix _zA = 1 + 11 i.
Calculate $Z=\frac{Z_A-Z_C}{Z_D-Z_B}$ in algebraic form and then in trigonometric form.
Justify that the lines (AC) and (BD) are perpendicular and deduce the nature of the quadrilateral ABCD.

Probabilities

A computer room in a school is equipped with 25 computers, 3 of which are defective.

All computers have the same probability of being chosen.

Two computers in this room are chosen at random.

What is the probability that both of these computers are defective?

The lifetime of a computer (i.e. the duration of operation before the first failure), is a random variable X that follows an exponential law of parameter λ with λ > 0.

Thus, for any positive real t , the probability that a computer has a lifetime of less than t years, denoted p(X ≤ t), is given by :

$p(X\leq\,\,t)=\int_{0}^{t}\lambda\,e^{-\lambda\,x}dx$ .
1. Determine λ knowing that p(X > 5) = 0.4.
2. In this question, we will take λ = 0.18.
Knowing that a computer has not had a failure in the first 3 years, what is, to the nearest ^10-3, the probability that it will have a lifetime greater than 5 years?
3. In this question, we assume that the lifetime of one computer is independent of the others and that p(X > 5) = 0.4.

a. We consider a batch of 10 computers.
What is the probability that, in this batch, at least one of the computers has a lifetime of more than 5 years?

We will give a value rounded to the thousandth of this probability.

b. What is the minimum number of computers that must be chosen so that the probability of the event “at least one of them has a lifetime greater than 5 years” is greater than 0.999?

Geometry in space

We consider the cube ABCDEFGH of edge length 1 represented below. The graph is not required to be returned with the copy.

The space is related to the orthonormal reference frame $(A,\vec{AB},\vec{AD},\vec{AE})$
1. Show that the vector $\vec{v}$ with coordinates (1; 0; 1) is a normal vector to the plane (BCE).
2. Determine an equation of the plane (BCE).
3. We denote (Δ) the line perpendicular at E to the plane (BCE).
Determine a parametric representation of the line (Δ).
4. Show that the line (Δ) is secant to the plane (ABC) at a point R, symmetric of B about A.
5. a. Show that point D is the barycenter of points R, B and C with coefficients 1, -1 and 2 respectively.
b. Determine the nature and the characteristic elements of the set(S) of points M
of space such as $\,\%7C\,\vec{MR}\,-\vec{MB}+2\vec{MC}\,\%7C=2\sqrt{2}$
c. Prove that points B, E and G belong to the set(S).
d. Show that the intersection of the plane (BCE) and the set(S) is a circle whose radius is specified.

Arithmetic (specialty)

Prove Gauss’ theorem using Bézout’s theorem.

We recall the property known as Fermat’s little theorem:

“If p is a prime number and q is a natural number prime with p, then $q^{p-1}\equiv\,1%5Bp%5D$ “.

Consider the sequence defined for any natural number n not zero by :
_un = 2
ⁿ
+3
ⁿ
+6
ⁿ
-1.
1. Calculate the first six terms of the sequence.
2. Show that, for any natural number n not zero, _one is even.
3. Show that, for any even nonzero natural number n , _one is divisible by 4.

We note (E) the set of prime numbers which divide at least one term of the sequence(_un).
4. Do the integers 2, 3, 5 and 7 belong to the set (E) ?
5. Let p be a prime number strictly greater than 3.
a. Show that: ^6×2p-2 ≡ 3 (modulo p) and ^6×3p-2 ≡ 2 (modulo p).
b. Deduce that _6up-2 ≡ 0 (modulo p).
c. Does the number p belong to the set (E)?

Problem on the exponential

Part A

We consider the function defined on $%5B0;+\infty%5B$ by
$g(x)\,=\,e^x-x-1$ .

1. Study the variations of the function g.
2. Determine the sign of g(x) according to the values of x.
3. Deduce that for all x in [0 ; +∞ [, e
^x
– x > 0.
Part B

Consider the function f defined on [0; 1] by
$f(x)=\frac{e^x-1}{e^x-x}$ .
The curve(C) representative of the function in the plane provided with an orthonormal reference frame is given in the appendix.

This appendix will be completed and handed in with the copy at the end of the test.

We admit that f is strictly increasing on [0; 1].

1. Show that for all x in [0; 1], f (x) Î [0; 1].

2. Let (D) be the line of equation y = x.
Show that for any x in [0; 1], $f(x)-x=\frac{(1-x)g(x)}{e^x-x}$ .
Study the relative position of the line (D) and the curve(C) on [0; 1].

3. a. Determine a primitive of f on [0; 1].

b. Calculate the area, in area units, of the domain of the plane bounded by the curve(C), the line (D) and the lines of equations x = 0 and x = 1.

Part C

Consider the sequence(_un) defined by:
$U_0=\frac{1}{2}$
$U_{n+1}=f(U_n)$ for any natural number n.
1. Construct the first four terms of the sequence on the x-axis, leaving the construction lines visible.

2. Show that for any natural number n, $\frac{1}{2}$ _un≤ _un+1 ≤ 1.

3. Deduce that the sequence(_un) is convergent and determine its limit.

Probabilities

To conduct a survey, an employee interviews random people in a shopping mall.

He wondered if at least three people would be willing to respond.
1. In this question, it is assumed that the probability of a randomly selected person agreeing to answer is 0.1.
The employee interviews 50 people independently.
We consider the events :

A: “at least one person agrees to answer

B: “less than three people agree to answer

C: “three or more people agree to answer”.

Calculate the probabilities of events A, B and C. Round to the thousandth.

2. Let n be a natural number greater than or equal to 3. In this question, it is assumed that the random variable X which, to any group of n people interviewed independently, associates the number of people who agreed to answer, follows the probability law defined by :
$\,\{\,Pour\,tout\,entier\,k\,tel\,que\,0\leq\,\,k\leq\,\,n-1,\,P(X=k)=e^{-a}\frac{a^k}{k!}\,\\P(X=n)=\sum_{k=1}^{n-1}e^{-a}\frac{a^k}{k!}\\Formules\,pour\,lesquelles\,a=\frac{n}{10}.$
a. Show that the probability of at least three people responding is given by :
$f(a)=1-e^{-a}(1+a+\frac{a^2}{2})$
b. Calculate.

Give the rounding to the thousandth.

Does this model give a result similar to the one obtained in question 1?

3. We keep the model from question 2.

We wish to determine the minimum number of people to be interviewed so that the probability that at least three of them will answer is greater than or equal to 0.95.

a. Study the variations of the function f defined on ^R+by $f(x)\,=1-e^{-x}(1+x+\frac{x^2}{2})$

as well as its limit in $+\infty$ .

Draw up its table of variations.

b. Show that the equation f(x) = 0.95 has a unique solution on ^R+, and that this solution is between 6.29 and 6.3.

c. Deduce the minimum number of people to be interviewed.

Geometry and barycenter

The space is related to the orthonormal reference frame $(O,\vec{i},\vec{j},\vec{k})$ .
Consider the plane P of equation2x +y– 2z + 4 =0 and the points A of coordinates (3; 2; 6),
B of coordinates (1; 2; 4), and C of coordinates (4; -2; 5).

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

b. Verify that this plane is the P plane.

2. a. Show that the triangle ABC is rectangular.
b. Write a system of parametric equations of the line Δ passing through O and perpendicular to the plane P.
c. Let K be the orthogonal project of O onto P. Calculate the OK distance.
d. Calculate the volume of the tetrahedron OABC.
3. In this question, consider the weighted point system S ={(O, 3), (A, 1), (B, 1), (C, 1)}
a. Verify that this system has a barycenter, which we will note G.
b. I is the center of gravity of triangle ABC. Show that G belongs to (OI).
c. Determine the distance from G to the plane P.
4. Let Γ be the set of points M in the space verifying :
$\,\%7C\,3\vec{MO}+\vec{MA}+\vec{MB}+\vec{MC}\,\,\%7C=5$ .
Determine Γ.

What is the nature of the set of points common to P and Γ?

The integrals

1. Determine three real numbers a, b, c such that , for all $\,x\in\,%5D0;+\infty%5B$ :

$\,\frac{1}{x(1+x)^2}=\frac{a}{x}+\frac{b}{1+x}+\frac{c}{(1+x)^2}$ .

2. Let $\,X\ge\,1$ .

a. Calculate $\,\int_{1}^{X}\,\frac{dx}{x(1+x)^2}$ .

b. Let f be the function defined on $\,x\in\,%5B1;+\infty%5B$ by $\,f(X)=\int_{1}^{X}\,\frac{ln\,x}{(1+x)^3}dx$

By integrating by parts, calculate f(X) as a function of X .

c. Show that $\,\lim_{X\,\to\,+\infty}\,f(X)=\frac{1}{2}(ln2-\frac{1}{2})$

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