grandes-ecoles

Papers (176)
2025
centrale-maths1__official 40 centrale-maths2__official 36 mines-ponts-maths1__mp 17 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 28 mines-ponts-maths2__pc 23 mines-ponts-maths2__psi 25 polytechnique-maths-a__mp 35 polytechnique-maths__fui 9 polytechnique-maths__pc 27 x-ens-maths-a__fui 10 x-ens-maths-a__mp 18 x-ens-maths-b__mp 6 x-ens-maths-c__mp 6 x-ens-maths-d__mp 31 x-ens-maths__pc 27 x-ens-maths__psi 30
2024
centrale-maths1__official 21 centrale-maths2__official 28 geipi-polytech__maths 9 mines-ponts-maths1__mp 23 mines-ponts-maths1__psi 9 mines-ponts-maths2__mp 14 mines-ponts-maths2__pc 19 mines-ponts-maths2__psi 20 polytechnique-maths-a__mp 42 polytechnique-maths-b__mp 27 x-ens-maths-a__mp 43 x-ens-maths-b__mp 29 x-ens-maths-c__mp 22 x-ens-maths-d__mp 41 x-ens-maths__pc 20 x-ens-maths__psi 23
2023
centrale-maths1__official 37 centrale-maths2__official 32 e3a-polytech-maths__mp 4 mines-ponts-maths1__mp 14 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 21 mines-ponts-maths2__pc 13 mines-ponts-maths2__psi 22 polytechnique-maths__fui 3 x-ens-maths-a__mp 24 x-ens-maths-b__mp 10 x-ens-maths-c__mp 10 x-ens-maths-d__mp 10 x-ens-maths__pc 22
2022
centrale-maths1__mp 22 centrale-maths1__pc 33 centrale-maths1__psi 42 centrale-maths2__mp 26 centrale-maths2__pc 37 centrale-maths2__psi 40 mines-ponts-maths1__mp 26 mines-ponts-maths1__pc 20 mines-ponts-maths1__psi 23 mines-ponts-maths2__mp 22 mines-ponts-maths2__pc 9 mines-ponts-maths2__psi 18 x-ens-maths-a__mp 8 x-ens-maths-b__mp 19 x-ens-maths-c__mp 17 x-ens-maths-d__mp 47 x-ens-maths1__mp 13 x-ens-maths2__mp 26 x-ens-maths__pc 7 x-ens-maths__pc_cpge 14 x-ens-maths__psi 22 x-ens-maths__psi_cpge 26
2021
centrale-maths1__mp 34 centrale-maths1__pc 36 centrale-maths1__psi 28 centrale-maths2__mp 21 centrale-maths2__pc 38 centrale-maths2__psi 28 x-ens-maths2__mp 35 x-ens-maths__pc 29
2020
centrale-maths1__mp 42 centrale-maths1__pc 36 centrale-maths1__psi 38 centrale-maths2__mp 2 centrale-maths2__pc 35 centrale-maths2__psi 39 mines-ponts-maths1__mp_cpge 22 mines-ponts-maths2__mp_cpge 19 x-ens-maths-a__mp_cpge 10 x-ens-maths-b__mp_cpge 19 x-ens-maths-c__mp 10 x-ens-maths-d__mp 13 x-ens-maths1__mp 13 x-ens-maths2__mp 20 x-ens-maths__pc 6
2019
centrale-maths1__mp 37 centrale-maths1__pc 40 centrale-maths1__psi 38 centrale-maths2__mp 37 centrale-maths2__pc 39 centrale-maths2__psi 46 x-ens-maths1__mp 24 x-ens-maths__pc 18 x-ens-maths__psi 9
2018
centrale-maths1__mp 21 centrale-maths1__pc 31 centrale-maths1__psi 39 centrale-maths2__mp 23 centrale-maths2__pc 35 centrale-maths2__psi 30 x-ens-maths1__mp 18 x-ens-maths2__mp 13 x-ens-maths__pc 17 x-ens-maths__psi 20
2017
centrale-maths1__mp 45 centrale-maths1__pc 22 centrale-maths1__psi 17 centrale-maths2__mp 30 centrale-maths2__pc 28 centrale-maths2__psi 44 x-ens-maths1__mp 24 x-ens-maths2__mp 7 x-ens-maths__pc 17 x-ens-maths__psi 19
2016
centrale-maths1__mp 41 centrale-maths1__pc 31 centrale-maths1__psi 33 centrale-maths2__mp 25 centrale-maths2__pc 42 centrale-maths2__psi 17 x-ens-maths1__mp 10 x-ens-maths2__mp 32 x-ens-maths__pc 1 x-ens-maths__psi 20
2015
centrale-maths1__mp 18 centrale-maths1__pc 11 centrale-maths1__psi 42 centrale-maths2__mp 44 centrale-maths2__pc 1 centrale-maths2__psi 14 x-ens-maths1__mp 16 x-ens-maths2__mp 19 x-ens-maths__pc 30 x-ens-maths__psi 20
2014
centrale-maths1__mp 28 centrale-maths1__pc 26 centrale-maths1__psi 36 centrale-maths2__mp 24 centrale-maths2__pc 23 centrale-maths2__psi 29 x-ens-maths2__mp 13
2013
centrale-maths1__mp 3 centrale-maths1__pc 45 centrale-maths1__psi 20 centrale-maths2__mp 32 centrale-maths2__pc 50 centrale-maths2__psi 32 x-ens-maths1__mp 14 x-ens-maths2__mp 10 x-ens-maths__pc 22 x-ens-maths__psi 9
2012
centrale-maths1__pc 23 centrale-maths1__psi 20 centrale-maths2__mp 27 centrale-maths2__psi 20
2011
centrale-maths1__mp 27 centrale-maths1__pc 15 centrale-maths1__psi 21 centrale-maths2__mp 29 centrale-maths2__pc 8 centrale-maths2__psi 28
2010
centrale-maths1__mp 7 centrale-maths1__pc 23 centrale-maths1__psi 9 centrale-maths2__mp 10 centrale-maths2__pc 36 centrale-maths2__psi 27
2014 x-ens-maths2__mp

13 maths questions

Q1 Matrices Matrix Power Computation and Application View
Calculate the exponential of the matrix $M_{p,q,r}$, where $$M_{p,q,r} = \begin{pmatrix} 0 & p & r \\ 0 & 0 & q \\ 0 & 0 & 0 \end{pmatrix}.$$
Q2 Matrices Binary Operation Properties View
We consider the set of square matrices of size 3 that are strictly upper triangular: $$\mathbf{L} = \left\{ M_{p,q,r} \mid (p,q,r) \in \mathbf{R}^3 \right\} \quad \text{where} \quad M_{p,q,r} = \begin{pmatrix} 0 & p & r \\ 0 & 0 & q \\ 0 & 0 & 0 \end{pmatrix}.$$
(a) Show that we define a group law $*$ on $\mathbf{L}$ by setting for $M, N \in \mathbf{L}$: $$M * N = M + N + \frac{1}{2}[M, N]$$ Explicitly determine the inverse of $M_{p,q,r}$.
(b) Determine the matrices $M_{p,q,r} \in \mathbf{L}$ that commute with all elements of $\mathbf{L}$ for the law $*$. Is $(\mathbf{L}, *)$ commutative?
Q3 Matrices Group Homomorphisms and Isomorphisms View
We consider the set of square matrices of size 3 that are strictly upper triangular: $$\mathbf{L} = \left\{ M_{p,q,r} \mid (p,q,r) \in \mathbf{R}^3 \right\} \quad \text{where} \quad M_{p,q,r} = \begin{pmatrix} 0 & p & r \\ 0 & 0 & q \\ 0 & 0 & 0 \end{pmatrix},$$ and the group law $M * N = M + N + \frac{1}{2}[M,N]$ on $\mathbf{L}$.
Show that for all matrices $M, N \in \mathbf{L}$, we have: $$(\exp M) \times (\exp N) = \exp(M * N)$$
Q4 Groups Group Homomorphisms and Isomorphisms View
We consider the set of square matrices of size 3 that are strictly upper triangular: $$\mathbf{L} = \left\{ M_{p,q,r} \mid (p,q,r) \in \mathbf{R}^3 \right\} \quad \text{where} \quad M_{p,q,r} = \begin{pmatrix} 0 & p & r \\ 0 & 0 & q \\ 0 & 0 & 0 \end{pmatrix},$$ and the group law $M * N = M + N + \frac{1}{2}[M,N]$ on $\mathbf{L}$.
Let $M$ and $N$ be two elements of $\mathbf{L}$. Show that $$\exp([M,N]) = \exp(M)\exp(N)\exp(-M)\exp(-N)$$
Q5 Groups Matrix Group and Subgroup Structure View
We consider the set of square matrices of size 3 that are strictly upper triangular: $$\mathbf{L} = \left\{ M_{p,q,r} \mid (p,q,r) \in \mathbf{R}^3 \right\} \quad \text{where} \quad M_{p,q,r} = \begin{pmatrix} 0 & p & r \\ 0 & 0 & q \\ 0 & 0 & 0 \end{pmatrix},$$ and $\mathbf{H} = \{I_3 + M \mid M \in \mathbf{L}\}$, with the group law $M * N = M + N + \frac{1}{2}[M,N]$ on $\mathbf{L}$.
Show that $\mathbf{H}$ equipped with the usual product of matrices is a subgroup of $\mathrm{SL}_3(\mathbf{R})$ and that $$\exp : (\mathbf{L}, *) \rightarrow (\mathbf{H}, \times)$$ is a group isomorphism.
Q6 Groups Matrix Algebra and Product Properties View
We consider two matrices $A$ and $B$ of $\mathcal{M}_d(\mathbf{R})$. We further assume that $A$ and $B$ commute with $[A,B]$.
(a) Show that $[A, \exp(B)] = \exp(B)[A,B]$.
(b) Determine a differential equation satisfied by $t \mapsto \exp(tA)\exp(tB)$.
(c) Deduce the formula: $$\exp(A)\exp(B) = \exp\left(A + B + \frac{1}{2}[A,B]\right)$$
Q7 Groups Matrix Group and Subgroup Structure View
We consider two matrices $A$ and $B$ of $\mathcal{M}_d(\mathbf{R})$ that commute with $[A,B]$. We denote $\mathcal{L} = \operatorname{Vect}(A, B, [A,B])$.
(a) If $M, N \in \mathcal{L}$, show that $[M,N]$ commutes with $M$ and $N$.
(b) Let $G = \{\exp(M) \mid M \in \mathcal{L}\}$. Show that $(G, \times)$ is a group and that the map $$\Phi : \mathbf{H} \rightarrow G, \quad \exp(M_{p,q,r}) \mapsto \exp(pA + qB + r[A,B])$$ is a group homomorphism.
Q8 Groups Matrix Norm, Convergence, and Inequality View
Let $(D_n)_{n \in \mathbf{N}}$ be a sequence of $\mathcal{M}_d(\mathbf{R})$ that converges to $D \in \mathcal{M}_d(\mathbf{R})$. It is therefore bounded: let $\lambda > 0$ be such that for all integers $n \in \mathbf{N}$, $\|D_n\| \leq \lambda$.
(a) Let $k \in \mathbf{N}$. Justify that $\frac{n!}{(n-k)! n^k} \rightarrow 1$ when $n \rightarrow +\infty$ and that if $n \geq k$ (and $n \geq 1$), $$0 \leq 1 - \frac{n!}{(n-k)! n^k} \leq 1$$ Deduce that $$\left(I_d + \frac{D_n}{n}\right)^n - \sum_{k=0}^{n} \frac{1}{k!}(D_n)^k \rightarrow 0 \quad \text{when } n \rightarrow +\infty$$
(b) Show that for all integers $k \geq 1$ and $n \geq 0$, $$\left\|(D_n)^k - D^k\right\| \leq k\lambda^{k-1}\|D_n - D\|$$
(c) Conclude that $\left(I_d + \frac{D_n}{n}\right)^n \rightarrow \exp(D)$ when $n \rightarrow +\infty$.
Q9 Groups Matrix Norm, Convergence, and Inequality View
Let $A$ and $B$ be two arbitrary matrices of $\mathcal{M}_d(\mathbf{R})$.
(a) Let $D \in \mathcal{M}_d(\mathbf{R})$ such that $\|D\| \leq 1$. Show that there exists a constant $\mu > 0$ independent of $D$ such that $$\left\|\exp(D) - I_d - D\right\| \leq \mu \|D\|^2$$
(b) Show that there exists a constant $\nu > 0$, and for all $n \geq 1$ a matrix $C_n \in \mathcal{M}_d(\mathbf{R})$, such that $$\exp\left(\frac{A}{n}\right)\exp\left(\frac{B}{n}\right) = I_d + \frac{A}{n} + \frac{B}{n} + C_n \quad \text{and} \quad \|C_n\| \leq \frac{\nu}{n^2}$$
Q10 Groups Matrix Norm, Convergence, and Inequality View
Let $A$ and $B$ be two arbitrary matrices of $\mathcal{M}_d(\mathbf{R})$. Using the results of questions 8 and 9, deduce that $$\exp(A+B) = \lim_{n \rightarrow +\infty} \left(\exp\left(\frac{A}{n}\right)\exp\left(\frac{B}{n}\right)\right)^n$$
Q11 Groups Higher-Order and Special DEs (Proof/Theory) View
Let $T$ be a strictly positive real number. We denote by $E(T)$ the set consisting of pairs $(u,v)$ of continuous functions on $[0,T]$ with real values.
A Carnot path controlled by $(u,v) \in E(T)$ is a map $\gamma : [0,T] \rightarrow \mathcal{M}_3(\mathbf{R})$ of class $C^1$ solution of the matrix differential equation: $$\left\{\begin{array}{l} \gamma'(t) = u(t)\gamma(t)M_{1,0,0} + v(t)\gamma(t)M_{0,1,0} \\ \gamma(0) = I_3 \end{array}\right.$$ where $M_{1,0,0}$ and $M_{0,1,0}$ are as defined in the first part.
(a) For all $(u,v) \in E(T)$, justify the existence of a unique Carnot path controlled by $(u,v)$.
(b) Show that $\gamma$ satisfies $$\forall t \in [0,T], \quad \gamma(t) \in \mathbf{H}$$ and explicitly calculate, as a function of $t$, $u$ and $v$, the functions $p(t)$, $q(t)$ and $r(t)$ such that $$\gamma(t) = \exp\left(M_{p(t),q(t),r(t)}\right).$$
Q12 Groups Higher-Order and Special DEs (Proof/Theory) View
For all $(\theta, \varphi) \in \mathbf{R}^2$ and $t \in \mathbf{R}$, we define the controls $$u_{\theta,\varphi}(t) = \sin(\theta - \varphi t) \quad \text{and} \quad v_{\theta,\varphi}(t) = \cos(\theta - \varphi t)$$ and we denote $\gamma_{\theta,\varphi}(t) = \exp\left(M_{p(t),q(t),r(t)}\right)$ the Carnot path controlled by $(u_{\theta,\varphi}, v_{\theta,\varphi})$.
(a) We assume $\varphi \neq 0$. Calculate $p(t)$ and $q(t)$ and verify that $$r(t) = \frac{t\varphi - \sin(t\varphi)}{2\varphi^2}$$
(b) Similarly calculate $\gamma_{\theta,0}(t)$.
Q13 Groups Higher-Order and Special DEs (Proof/Theory) View
The Carnot sphere is the set: $$B(1) = \left\{(p,q,r) \in \mathbf{R}^3 \mid \exists (\theta,\varphi) \in [-\pi,\pi] \times [-2\pi,2\pi], \quad \gamma_{\theta,\varphi}(1) = \exp\left(M_{p,q,r}\right)\right\}.$$
We define the functions $f$ and $g$ on $]0, 2\pi]$ by: $$f(s) = \frac{2(1-\cos s)}{s^2} \quad \text{and} \quad g(s) = \frac{s - \sin s}{2s^2}$$
Show that $f$ and $g$ extend by continuity to $[0, 2\pi]$; that $f$ is then a continuous bijection from $[0, 2\pi]$ onto a set to be specified; and that $g$ attains its maximum at $\pi$.