grandes-ecoles

Papers (191)
2025
centrale-maths1__official 40 centrale-maths2__official 42 mines-ponts-maths1__mp 20 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 28 mines-ponts-maths2__pc 24 mines-ponts-maths2__psi 26 polytechnique-maths-a__mp 27 polytechnique-maths__fui 16 polytechnique-maths__pc 27 x-ens-maths-a__mp 18 x-ens-maths-c__mp 9 x-ens-maths-d__mp 38 x-ens-maths__pc 27 x-ens-maths__psi 38
2024
centrale-maths1__official 28 centrale-maths2__official 29 geipi-polytech__maths 9 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 20 mines-ponts-maths1__psi 19 mines-ponts-maths2__mp 23 mines-ponts-maths2__pc 21 mines-ponts-maths2__psi 21 polytechnique-maths-a__mp 44 polytechnique-maths-b__mp 37 x-ens-maths-a__mp 43 x-ens-maths-b__mp 35 x-ens-maths-c__mp 22 x-ens-maths-d__mp 45 x-ens-maths__pc 24 x-ens-maths__psi 26
2023
centrale-maths1__official 44 centrale-maths2__official 33 e3a-polytech-maths__mp 4 mines-ponts-maths1__mp 15 mines-ponts-maths1__pc 23 mines-ponts-maths1__psi 23 mines-ponts-maths2__mp 22 mines-ponts-maths2__pc 18 mines-ponts-maths2__psi 22 polytechnique-maths__fui 23 x-ens-maths-a__mp 25 x-ens-maths-b__mp 24 x-ens-maths-c__mp 20 x-ens-maths-d__mp 20 x-ens-maths__pc 18 x-ens-maths__psi 15
2022
centrale-maths1__mp 48 centrale-maths1__official 48 centrale-maths1__pc 37 centrale-maths1__psi 43 centrale-maths2__mp 32 centrale-maths2__official 32 centrale-maths2__pc 39 centrale-maths2__psi 45 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 24 mines-ponts-maths1__psi 24 mines-ponts-maths2__mp 24 mines-ponts-maths2__pc 19 mines-ponts-maths2__psi 20 x-ens-maths-a__mp 13 x-ens-maths-b__mp 40 x-ens-maths-c__mp 27 x-ens-maths-d__mp 46 x-ens-maths1__mp 13 x-ens-maths2__mp 40 x-ens-maths__pc 15 x-ens-maths__pc_cpge 15 x-ens-maths__psi 22 x-ens-maths__psi_cpge 23
2021
centrale-maths1__mp 40 centrale-maths1__official 40 centrale-maths1__pc 36 centrale-maths1__psi 29 centrale-maths2__mp 30 centrale-maths2__official 29 centrale-maths2__pc 38 centrale-maths2__psi 37 x-ens-maths2__mp 39 x-ens-maths__pc 44
2020
centrale-maths1__mp 42 centrale-maths1__official 42 centrale-maths1__pc 36 centrale-maths1__psi 40 centrale-maths2__mp 38 centrale-maths2__official 38 centrale-maths2__pc 40 centrale-maths2__psi 39 mines-ponts-maths1__mp_cpge 24 mines-ponts-maths2__mp_cpge 21 x-ens-maths-a__mp_cpge 18 x-ens-maths-b__mp_cpge 20 x-ens-maths-d__mp 14 x-ens-maths1__mp 18 x-ens-maths2__mp 20 x-ens-maths__pc 18
2019
centrale-maths1__mp 37 centrale-maths1__official 37 centrale-maths1__pc 40 centrale-maths1__psi 39 centrale-maths2__mp 37 centrale-maths2__official 37 centrale-maths2__pc 39 centrale-maths2__psi 49 x-ens-maths1__mp 24 x-ens-maths__pc 18 x-ens-maths__psi 26
2018
centrale-maths1__mp 47 centrale-maths1__official 47 centrale-maths1__pc 41 centrale-maths1__psi 44 centrale-maths2__mp 44 centrale-maths2__official 44 centrale-maths2__pc 35 centrale-maths2__psi 38 x-ens-maths1__mp 19 x-ens-maths2__mp 17 x-ens-maths__pc 22 x-ens-maths__psi 24
2017
centrale-maths1__mp 45 centrale-maths1__official 45 centrale-maths1__pc 22 centrale-maths1__psi 17 centrale-maths2__mp 30 centrale-maths2__official 30 centrale-maths2__pc 28 centrale-maths2__psi 44 x-ens-maths1__mp 26 x-ens-maths2__mp 16 x-ens-maths__pc 18 x-ens-maths__psi 26
2016
centrale-maths1__mp 42 centrale-maths1__pc 31 centrale-maths1__psi 33 centrale-maths2__mp 25 centrale-maths2__pc 47 centrale-maths2__psi 27 x-ens-maths1__mp 18 x-ens-maths2__mp 46 x-ens-maths__pc 15 x-ens-maths__psi 20
2015
centrale-maths1__mp 42 centrale-maths1__pc 18 centrale-maths1__psi 42 centrale-maths2__mp 44 centrale-maths2__pc 18 centrale-maths2__psi 33 x-ens-maths1__mp 16 x-ens-maths2__mp 31 x-ens-maths__pc 30 x-ens-maths__psi 22
2014
centrale-maths1__mp 28 centrale-maths1__pc 26 centrale-maths1__psi 27 centrale-maths2__mp 24 centrale-maths2__pc 26 centrale-maths2__psi 27 x-ens-maths1__mp 9 x-ens-maths2__mp 16 x-ens-maths__pc 4 x-ens-maths__psi 24
2013
centrale-maths1__mp 22 centrale-maths1__pc 45 centrale-maths1__psi 29 centrale-maths2__mp 31 centrale-maths2__pc 52 centrale-maths2__psi 32 x-ens-maths1__mp 24 x-ens-maths2__mp 35 x-ens-maths__pc 22 x-ens-maths__psi 9
2012
centrale-maths1__mp 36 centrale-maths1__pc 28 centrale-maths1__psi 33 centrale-maths2__mp 27 centrale-maths2__psi 18
2011
centrale-maths1__mp 27 centrale-maths1__pc 17 centrale-maths1__psi 24 centrale-maths2__mp 29 centrale-maths2__pc 17 centrale-maths2__psi 10
2010
centrale-maths1__mp 19 centrale-maths1__pc 30 centrale-maths1__psi 13 centrale-maths2__mp 32 centrale-maths2__pc 37 centrale-maths2__psi 27
2015 centrale-maths2__pc

18 maths questions

QI.A.1 Curve Sketching Sketching a Curve from Analytical Properties View
We denote $\mathcal{D}$ the set of functions from $\mathbb{R}$ to $\mathbb{R}$ of class $\mathcal{C}^{\infty}$ and with compact support. We denote $\varphi$ the function defined by: $$\begin{cases} \varphi(x) = 0 & \text{if } |x| \geqslant 1 \\ \varphi(x) = \exp\left(-\frac{x^2}{1-x^2}\right) & \text{if } |x| < 1 \end{cases}$$
a) Study the variations of $\varphi$. b) Sketch the graph of $\varphi$. c) Show that $\varphi$ is $\mathcal{C}^{\infty}$. d) Show that $\mathcal{D}$ is a vector space over $\mathbb{R}$ not reduced to $\{0\}$.
QI.A.2 Differentiating Transcendental Functions Regularity and smoothness of transcendental functions View
Show that the derivative of every element of $\mathcal{D}$ is an element of $\mathcal{D}$.
QI.A.3 Indefinite & Definite Integrals Convergence and Evaluation of Improper Integrals View
We denote $\varphi$ the function defined by: $$\begin{cases} \varphi(x) = 0 & \text{if } |x| \geqslant 1 \\ \varphi(x) = \exp\left(-\frac{x^2}{1-x^2}\right) & \text{if } |x| < 1 \end{cases}$$
a) Show that $\int_{\mathbb{R}} \varphi(t) \mathrm{d}t$ is a strictly positive real number. b) For every real number $x$, we set $\theta(x) = \frac{\varphi(x)}{\int_{\mathbb{R}} \varphi(t) \mathrm{d}t}$ and, for every non-zero natural number $n$, $\rho_n(x) = n\theta(nx)$.
Show that $$\forall n \in \mathbb{N}^* \quad \int_{\mathbb{R}} \rho_n(x) \mathrm{d}x = 1$$
QI.A.4 Differentiating Transcendental Functions Regularity and smoothness of transcendental functions View
For every function $f$ belonging to $\mathcal{F}_{sr}$ and every non-zero natural number $n$, we set $$\left(f * \rho_n\right)(x) = \int_{\mathbb{R}} f(t) \rho_n(x-t) \mathrm{d}t$$
Let $f$ be a function belonging to $\mathcal{F}_{sr}$. Show that the function $f * \rho_n$ is of class $\mathcal{C}^{\infty}$.
QI.A.5 Indefinite & Definite Integrals Properties of Integral-Defined Functions (Continuity, Differentiability) View
For every function $f$ belonging to $\mathcal{F}_{sr}$ and every non-zero natural number $n$, we set $$\left(f * \rho_n\right)(x) = \int_{\mathbb{R}} f(t) \rho_n(x-t) \mathrm{d}t$$
Let $I$ be the function that equals 1 on the interval $[-1,1]$, and 0 elsewhere. For $n \in \mathbb{N}^*$, we set $I_n(x) = I * \rho_n(x)$. a) For $n \in \mathbb{N}^*$ and $x \in \mathbb{R}$, express $I_n(x)$ in terms of $\varphi$. b) For $n \in \mathbb{N}^*$, show that $I_n$ belongs to $\mathcal{D}$ and study its variations. c) Sketch the graphs of $I_2$ and $I_3$. d) Show that the sequence of functions $(I_n)$ converges pointwise to a function $J$ which we shall determine. Show that $J$ and $I$ are equal except on a finite set of points. e) Does the sequence of functions $(I_n)$ converge uniformly to $J$?
QI.B.1 Groups Algebraic Structure Identification View
We say that a real function $f$ of class $\mathcal{C}^{\infty}$ on $\mathbb{R}$ has rapid decay if $$\forall (n,m) \in \mathbb{N}^2, \lim_{x \rightarrow +\infty} x^m f^{(n)}(x) = \lim_{x \rightarrow -\infty} x^m f^{(n)}(x) = 0$$ We denote $\mathcal{S}$ the set of functions from $\mathbb{R}$ to $\mathbb{R}$ of class $\mathcal{C}^{\infty}$ with rapid decay.
Show that $\mathcal{S}$ is a vector space over $\mathbb{R}$.
QI.B.2 Differentiating Transcendental Functions Regularity and smoothness of transcendental functions View
We say that a real function $f$ of class $\mathcal{C}^{\infty}$ on $\mathbb{R}$ has rapid decay if $$\forall (n,m) \in \mathbb{N}^2, \lim_{x \rightarrow +\infty} x^m f^{(n)}(x) = \lim_{x \rightarrow -\infty} x^m f^{(n)}(x) = 0$$ We denote $\mathcal{S}$ the set of functions from $\mathbb{R}$ to $\mathbb{R}$ of class $\mathcal{C}^{\infty}$ with rapid decay.
Show that if $f$ is in $\mathcal{S}$ then $f^{(p)}$ is in $\mathcal{S}$ for every natural number $p$.
QI.B.3 Groups Subgroup and Normal Subgroup Properties View
We say that a real function $f$ of class $\mathcal{C}^{\infty}$ on $\mathbb{R}$ has rapid decay if $$\forall (n,m) \in \mathbb{N}^2, \lim_{x \rightarrow +\infty} x^m f^{(n)}(x) = \lim_{x \rightarrow -\infty} x^m f^{(n)}(x) = 0$$ We denote $\mathcal{S}$ the set of functions from $\mathbb{R}$ to $\mathbb{R}$ of class $\mathcal{C}^{\infty}$ with rapid decay.
Show that if $P$ is a polynomial function and if $f$ is in $\mathcal{S}$, then $Pf$ belongs to $\mathcal{S}$.
QII.A.1 Indefinite & Definite Integrals Convergence and Evaluation of Improper Integrals View
We call a distribution on $\mathcal{D}$ any linear map $T : \mathcal{D} \rightarrow \mathbb{R}$ which satisfies $$\forall \varphi \in \mathcal{D}, \forall (\varphi_n)_{n \in \mathbb{N}} \in \mathcal{D}^{\mathbb{N}} \quad \varphi_n \xrightarrow{\mathcal{D}} \varphi \Longrightarrow T(\varphi_n) \rightarrow T(\varphi)$$
Show that if $f \in \mathcal{F}_{sr}$ then the map $T_f$ defined by $$\forall \varphi \in \mathcal{D} \quad T_f(\varphi) = \int_{-\infty}^{+\infty} f(x) \varphi(x) \mathrm{d}x$$ defines a distribution on $\mathcal{D}$.
QII.A.2 Indefinite & Definite Integrals Convergence and Evaluation of Improper Integrals View
Let $U$ be the function defined by $$\begin{cases} U(x) = 1 & \text{if } x \geqslant 0 \\ U(x) = 0 & \text{if } x < 0 \end{cases}$$ Justify that $U$ defines a distribution on $\mathcal{D}$.
QII.A.3 Indefinite & Definite Integrals Convergence and Evaluation of Improper Integrals View
Let $a$ be a real number. a) Show that the map $\delta_a$ which associates to every $\varphi \in \mathcal{D}$ the value $\varphi(a)$ is a distribution. b) Using the sequence of functions $(\varphi_n)_{n \in \mathbb{N}^*}$ of elements of $\mathcal{D}$ defined by $$\forall t \in \mathbb{R}, \varphi_n(t) = \begin{cases} \exp\left(\frac{(t-a)^2}{(t-a+1/n)(t-a-1/n)}\right) & \text{if } t \in ]a-1/n, a+1/n[ \\ 0 & \text{otherwise} \end{cases}$$ show that $\forall f \in \mathcal{F}_{sr}, T_f \neq \delta_a$.
QII.B.1 Groups Algebraic Structure Identification View
If $T$ is a distribution on $\mathcal{D}$, we define the derivative distribution $T'$ by $$\forall \varphi \in \mathcal{D}, \quad T'(\varphi) = -T(\varphi')$$ Justify that $T'$ is a distribution on $\mathcal{D}$.
QII.B.2 Integration by Parts Prove an Integral Identity or Equality View
If $T$ is a distribution on $\mathcal{D}$, we define the derivative distribution $T'$ by $$\forall \varphi \in \mathcal{D}, \quad T'(\varphi) = -T(\varphi')$$
Let $f$ be a continuous function from $\mathbb{R}$ to $\mathbb{R}$. If $f$ is of class $\mathcal{C}^1$, show that $(T_f)' = T_{f'}$. Adapt this result to the case where $f$ is piecewise of class $\mathcal{C}^1$.
QII.B.3 Indefinite & Definite Integrals Properties of Integral-Defined Functions (Continuity, Differentiability) View
If $T$ is a distribution on $\mathcal{D}$, we define the derivative distribution $T'$ by $$\forall \varphi \in \mathcal{D}, \quad T'(\varphi) = -T(\varphi')$$ Let $U$ be the function defined by $$\begin{cases} U(x) = 1 & \text{if } x \geqslant 0 \\ U(x) = 0 & \text{if } x < 0 \end{cases}$$ Show that $T_U' = \delta_0$.
QII.B.4 Indefinite & Definite Integrals Properties of Integral-Defined Functions (Continuity, Differentiability) View
If $T$ is a distribution on $\mathcal{D}$, we define the derivative distribution $T'$ by $$\forall \varphi \in \mathcal{D}, \quad T'(\varphi) = -T(\varphi')$$
We consider the map $T$ which associates to every function $\varphi$ of $\mathcal{D}$ the real number $T(\varphi)$ defined by $$T(\varphi) = \int_{-1}^{0} t\varphi(t) \mathrm{d}t + \int_{0}^{+\infty} \varphi(t) \mathrm{d}t$$
a) Show that $T$ is a regular distribution. b) Calculate the derivative of this distribution.
QII.B.5 Indefinite & Definite Integrals Properties of Integral-Defined Functions (Continuity, Differentiability) View
If $T$ is a distribution on $\mathcal{D}$, we define the derivative distribution $T'$ by $$\forall \varphi \in \mathcal{D}, \quad T'(\varphi) = -T(\varphi')$$ If $f$ is an element of $\mathcal{F}_{sr}$ and if $a$ is a real number, we set $$\lim_{x \rightarrow a^-} f(x) = f(a^-) \quad \text{and} \quad \lim_{x \rightarrow a^+} f(x) = f(a^+)$$ The difference $f(a^+) - f(a^-)$, called the jump at $a$, is denoted $\sigma(a)$. a) Let $a_1, \ldots, a_p$ be real numbers such that $a_1 < \ldots < a_p$. Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a piecewise $\mathcal{C}^1$ function. We further assume that $f$ is continuous on $]-\infty, a_1[ \cup ]a_1, a_2[ \cup \ldots \cup ]a_p, +\infty[$. Show that $$T_f' = T_{f'} + \sum_{i=1}^{p} \sigma(a_i) \delta_{a_i}$$ b) Recover by this method the results of questions II.B.3 and II.B.4.b.
QII.C.1 Indefinite & Definite Integrals Properties of Integral-Defined Functions (Continuity, Differentiability) View
We say that the sequence of distributions $(T_n)_{n \in \mathbb{N}}$ converges to the distribution $T$ if $$\forall \varphi \in \mathcal{D}, \lim_{n \rightarrow \infty} T_n(\varphi) = T(\varphi)$$
For $n$ a non-zero natural number, we consider the function $U_n$ zero on the negative reals, affine on the interval $[0, 1/n]$, equal to 1 for reals greater than $1/n$ and continuous on $\mathbb{R}$. a) Show that the sequence of regular distributions $(T_{U_n})_{n \in \mathbb{N}}$ converges to $T_U$. b) Show that $$\forall \varphi \in \mathcal{D} \quad T_{U_n}'(\varphi) = \int_0^{1/n} n\varphi(t) \mathrm{d}t$$ c) Deduce that the distribution $T_{U_n}'$ is regular and give a function $V_n$ such that $T_{V_n} = T_{U_n}'$. d) Sketch $V_n$ for $n = 1, 2, 4$. e) Show that if the sequence of distributions $(T_n)_{n \in \mathbb{N}}$ converges to the distribution $T$, then $(T_n')_{n \in \mathbb{N}}$ converges to $T'$. f) What is the limit of $T_{U_n}'$ as $n$ tends to infinity?
QII.C.2 Indefinite & Definite Integrals Properties of Integral-Defined Functions (Continuity, Differentiability) View
We say that the sequence of distributions $(T_n)_{n \in \mathbb{N}}$ converges to the distribution $T$ if $$\forall \varphi \in \mathcal{D}, \lim_{n \rightarrow \infty} T_n(\varphi) = T(\varphi)$$
For every non-zero natural number $n$, we consider the functions $$\begin{cases} f_n(x) = \dfrac{n}{1 + n^2 x^2} & \\ g_n(x) = nx^n & \text{if } x \in [0,1] \text{ and zero elsewhere} \\ h_n(x) = n^2 \sin nx & \text{if } x \in [-\pi/n, \pi/n] \text{ and zero elsewhere} \end{cases}$$
a) Verify that they belong to $\mathcal{F}_{sr}$. b) Study the variations of the functions $f_n, g_n$ and $h_n$ then sketch their graphs for $n = 1$ and $n = 2$. c) Study the convergence of the sequences of distributions $(T_{f_n}), (T_{g_n})$ and $(T_{h_n})$.