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
2024 x-ens-maths__psi

26 maths questions

Q1 Matrices Matrix Norm, Convergence, and Inequality View
Give a necessary and sufficient condition on $R_u$ for $\mathbb{M}_n(u) = \emptyset$ and give an example of $u$ for which this equality holds.
Q2 Matrices Linear Transformation and Endomorphism Properties View
Show that $\mathbb{M}_n(u) \neq \{0_n\}$.
Q3 Matrices True/False or Multiple-Select Conceptual Reasoning View
Show that the following three assertions are equivalent
(i) $R_u = +\infty$,
(ii) $\mathbb{M}_n(u) = \mathscr{M}_n(\mathbb{C})$,
(iii) $\mathbb{M}_n(u) \neq \emptyset$ and $\forall A \in \mathbb{M}_n(u), \forall B \in \mathbb{M}_n(u), A + B \in \mathbb{M}_n(u)$, and give an example of a sequence $u$ satisfying these three assertions and such that $u_k \neq 0$ for every $k \in \mathbb{N}$.
Q4 Sequences and Series Matrix Exponentials and Series of Matrices View
Let $A \in \mathscr{M}_n(\mathbb{C})$. Show the equivalence of the following two assertions
(i) $A \in \mathbb{M}_n(v)$ for every sequence $v = (v_k)_{k \geqslant 0}$ of $\mathbb{C}$ satisfying $R_v > 0$.
(ii) $A$ is nilpotent (that is, there exists $k \in \mathbb{N}^*$ such that $A^k = 0_n$).
Q5 Sequences and Series Matrix Exponentials and Series of Matrices View
Show that for every integer $m \geqslant 0$, we have $$D_{u^{(m)}} = D_u$$
Q6 Sequences and Series Matrix Exponentials and Series of Matrices View
Let $v = (v_k)_{k \geqslant 0}$ be another sequence of complex numbers. Show that $$\mathbb{M}_n(u) \cap \mathbb{M}_n(v) \subset \mathbb{M}_n(u+v) \cap \mathbb{M}_n(u \star v)$$
Q7 Matrices Matrix Algebra and Product Properties View
We assume in this question that $0 < R_u \leqslant 1$. Let $A \in \mathbb{M}_n(u)$ and $B \in \mathbb{M}_n(u)$ be two symmetric matrices such that $AB = BA$. Show that $AB \in \mathbb{M}_n(u)$.
Q8 Groups Subgroup and Normal Subgroup Properties View
Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ such that $\mathbb{M}_n(u) \neq \emptyset$. Let $A \in \mathbb{M}_n(u)$. Show that $\mathscr{V}(A)$ is nonempty.
Q9 Roots of polynomials Divisibility and minimal polynomial arguments View
Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ such that $\mathbb{M}_n(u) \neq \emptyset$. Let $A \in \mathbb{M}_n(u)$. Let $$m = \min\{k \in \mathbb{N} \mid \exists P \in \mathscr{V}(A) \text{ with } \deg(P) = k\}$$ Show that there exists a unique polynomial $p \in \mathbb{C}[X]$ satisfying the three conditions
(i) $p \in \mathscr{V}(A)$,
(ii) $\deg(p) = m$,
(iii) $p$ is monic.
Q10 Invariant lines and eigenvalues and vectors Annihilating or minimal polynomial and spectral deductions View
Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ such that $\mathbb{M}_n(u) \neq \emptyset$. Let $A \in \mathbb{M}_n(u)$. Let $P \in \mathscr{V}(A)$. Show that $\varphi_A$ divides $P$.
Q11 Invariant lines and eigenvalues and vectors Compute eigenvalues of a given matrix View
Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ such that $\mathbb{M}_n(u) \neq \emptyset$. Let $A \in \mathbb{M}_n(u)$. Show that the roots of $\varphi_A$ in $\mathbb{C}$ are exactly the eigenvalues of $A$.
Q12 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ such that $\mathbb{M}_n(u) \neq \emptyset$. Let $A \in \mathbb{M}_n(u)$. Show that if $A \in \mathscr{M}_n(\mathbb{R})$ then $\varphi_A$ has real coefficients (that is, $\varphi_A \in \mathbb{R}[X]$).
Q13 Polynomial Division & Manipulation View
We denote by $\lambda_1, \cdots, \lambda_\ell$ the eigenvalues of $A$, with $\lambda_i \neq \lambda_j$ if $i \neq j$. We denote by $m_1 \geqslant 1, \cdots, m_\ell \geqslant 1$ the multiplicities of $\lambda_1, \cdots, \lambda_\ell$ respectively as roots of $\varphi_A$. Thus we have $$\varphi_A(X) = (X - \lambda_1)^{m_1} \cdots (X - \lambda_\ell)^{m_\ell}$$ with $m = m_1 + \cdots + m_\ell$.
Show that the map $$T : P \in \mathbb{C}_{m-1}[X] \mapsto \left(P(\lambda_1), P'(\lambda_1), \cdots, P^{(m_1-1)}(\lambda_1), \cdots, P(\lambda_\ell), P'(\lambda_\ell), \cdots, P^{(m_\ell-1)}(\lambda_\ell)\right) \in \mathbb{C}^m$$ is an isomorphism and deduce that there exists a unique polynomial $Q \in \mathbb{C}_{m-1}[X]$ such that $$\forall i \in \llbracket 1; \ell \rrbracket, \forall k \in \llbracket 0; m_i - 1 \rrbracket, Q^{(k)}(\lambda_i) = U^{(k)}(\lambda_i)$$
Q14 Sequences and Series Recurrence Relations and Sequence Properties View
We denote by $\lambda_1, \cdots, \lambda_\ell$ the eigenvalues of $A$, with $\lambda_i \neq \lambda_j$ if $i \neq j$. We denote by $m_1 \geqslant 1, \cdots, m_\ell \geqslant 1$ the multiplicities of $\lambda_1, \cdots, \lambda_\ell$ respectively as roots of $\varphi_A$. We set $u(A) = Q(A)$ where $Q$ is the unique polynomial in $\mathbb{C}_{m-1}[X]$ satisfying $\forall i \in \llbracket 1; \ell \rrbracket, \forall k \in \llbracket 0; m_i - 1 \rrbracket, Q^{(k)}(\lambda_i) = U^{(k)}(\lambda_i)$.
Let $P \in \mathbb{C}[X]$. Show that $u(A) = P(A)$ if and only if $$\forall i \in \llbracket 1; \ell \rrbracket, \forall k \in \llbracket 0; m_i - 1 \rrbracket, P^{(k)}(\lambda_i) = U^{(k)}(\lambda_i)$$
Q15 Matrices Matrix Algebra and Product Properties View
Let $\alpha \in \mathbb{C}$ such that $|\alpha| < R_u$. Show that $$u(\alpha I_n) = U(\alpha) I_n$$
Q16 Matrices Matrix Entry and Coefficient Identities View
We assume in this question only that $n = 2$. Determine $u(A)$ in the following case: $$A = \begin{pmatrix} \alpha & \gamma \\ 0 & \beta \end{pmatrix}$$ where $\alpha, \beta$ and $\gamma$ are fixed real numbers with $\alpha \neq \beta$ and $\{\alpha, \beta\} \subset D_u$. We will express the coefficients of $u(A)$ in terms of $\alpha, \beta$ and $\gamma, U(\alpha)$ and $U(\beta)$.
Q17 Matrices Matrix Algebra and Product Properties View
Let $B \in \mathbb{M}_n(u)$.
(a) Show that there exists a polynomial $R \in \mathbb{C}[X]$ such that $$u(A) = R(A) \text{ and } u(B) = R(B).$$ (b) We assume that $AB \in \mathbb{M}_n(u)$ and $BA \in \mathbb{M}_n(u)$. Show that $$A\, u(BA) = u(AB)\, A$$
Q18 Matrices Matrix Algebra and Product Properties View
Let $v = (v_k)_{k \geqslant 0}$ be another sequence of $\mathbb{C}$ such that $A \in \mathbb{M}_n(v)$. We assume in this question only that the values $\lambda_1, \cdots, \lambda_\ell$ are real. Show that $$(u \star v)(A) = u(A)\, v(A)$$ (after having justified that $A \in \mathbb{M}_n(u \star v)$).
Q19 Roots of polynomials Divisibility and minimal polynomial arguments View
Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ such that $\mathbb{M}_n(u) \neq \emptyset$. Let $A \in \mathbb{M}_n(u)$. We assume throughout this part that $A$ is diagonalizable in $\mathscr{M}_n(\mathbb{C})$ and we denote by $\lambda_1, \cdots, \lambda_\ell$ its eigenvalues with $\lambda_i \neq \lambda_j$ if $i \neq j$. Show that $$\varphi_A(X) = (X - \lambda_1) \cdots (X - \lambda_\ell).$$
Q20 Matrices Projection and Orthogonality View
Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ such that $\mathbb{M}_n(u) \neq \emptyset$. Let $A \in \mathbb{M}_n(u)$. We assume throughout this part that $A$ is diagonalizable in $\mathscr{M}_n(\mathbb{C})$ and we denote by $\lambda_1, \cdots, \lambda_\ell$ its eigenvalues with $\lambda_i \neq \lambda_j$ if $i \neq j$. For every $k \in \llbracket 1; \ell \rrbracket$ we define the polynomial: $$Q_k^A(X) = \prod_{j=1, j\neq k}^{\ell} \frac{X - \lambda_j}{\lambda_k - \lambda_j}$$
(a) Show that $$u(A) = \sum_{k=1}^{\ell} U(\lambda_k) Q_k^A(A).$$
(b) Show that for every $k \in \llbracket 1; \ell \rrbracket$, $Q_k^A(A)$ is a projection whose image and kernel we will specify.
(c) Deduce that $$\sum_{k=1}^{\ell} Q_k^A(A) = I_n.$$
Q21 Matrices Diagonalizability and Similarity View
Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ such that $\mathbb{M}_n(u) \neq \emptyset$. Let $A \in \mathbb{M}_n(u)$. We assume throughout this part that $A$ is diagonalizable in $\mathscr{M}_n(\mathbb{C})$ and we denote by $\lambda_1, \cdots, \lambda_\ell$ its eigenvalues with $\lambda_i \neq \lambda_j$ if $i \neq j$. Let $B \in \mathscr{M}_n(\mathbb{C})$ be an invertible matrix. Show that $$u(BAB^{-1}) = B\, u(A)\, B^{-1}.$$
Q22 Matrices Diagonalizability and Similarity View
Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ such that $\mathbb{M}_n(u) \neq \emptyset$. Let $A \in \mathbb{M}_n(u)$. We assume throughout this part that $A$ is diagonalizable in $\mathscr{M}_n(\mathbb{C})$ and we denote by $\lambda_1, \cdots, \lambda_\ell$ its eigenvalues with $\lambda_i \neq \lambda_j$ if $i \neq j$. Let $D \in \mathscr{M}_n(\mathbb{C})$ be a diagonal matrix and $S \in \mathscr{M}_n(\mathbb{C})$ be an invertible matrix such that $A = SDS^{-1}$.
(a) Show that $u(D)$ is diagonal and that $$\forall i \in \llbracket 1; n \rrbracket, [u(D)]_{i,i} = U([D]_{i,i}).$$ (b) Deduce an expression for $u(A)$.
Q23 Matrices Linear Transformation and Endomorphism Properties View
In this part, we assume that $n \geqslant 4$. Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ satisfying the condition $(C^\star)$: $R_u > 1$.
Let $H \in \mathscr{M}_n(\mathbb{C})$ be the matrix given by $$H = \begin{pmatrix} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & 0 \\ \vdots & & \ddots & \ddots & 1 \\ 0 & \cdots & \cdots & 0 & 0 \end{pmatrix}.$$
(a) Determine the polynomial $\varphi_H$ in this case.
(b) Let $A = H + \alpha I_n$ where $\alpha \in \mathbb{C}$ is such that $|\alpha| < R_u$. Show that $$u(A) = \sum_{k=0}^{n-1} \frac{U^{(k)}(\alpha)}{k!} H^k$$ and deduce that $$u(A) = \begin{pmatrix} U(\alpha) & \frac{U^{(1)}(\alpha)}{1!} & \frac{U^{(2)}(\alpha)}{2!} & \cdots & \frac{U^{(n-1)}(\alpha)}{(n-1)!} \\ 0 & U(\alpha) & \frac{U^{(1)}(\alpha)}{1!} & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & \frac{U^{(2)}(\alpha)}{2!} \\ \vdots & & \ddots & \ddots & \frac{U^{(1)}(\alpha)}{1!} \\ 0 & \cdots & \cdots & 0 & U(\alpha) \end{pmatrix}.$$
Q24 Matrices Eigenvalue and Characteristic Polynomial Analysis View
In this part, we assume that $n \geqslant 4$. Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ satisfying the condition $(C^\star)$: $R_u > 1$.
Let $G \in \mathscr{M}_n(\mathbb{C})$ be the matrix defined by $$G = Y\, {}^t Z$$ where $Y, Z \in \mathscr{M}_{n,1}(\mathbb{R})$ are two column vectors such that ${}^t Y Y = {}^t Z Z = 1$.
(a) Show that $G$ has rank 1 and give its image.
(b) Show that $0$ and ${}^t Z Y$ are the only eigenvalues of $G$.
(c) Deduce that $G \in \mathbb{M}_n(u)$.
(d) Determine $\varphi_G$ when ${}^t Z Y \neq 0$.
(e) Deduce that if ${}^t Z Y \neq 0$ then $$u(G) = U(0) I_n + \frac{U({}^t Z Y) - U(0)}{{}^t Z Y} G.$$ (f) Determine a simple expression for $u(G)$ when ${}^t Z Y = 0$.
Q25 Matrices Matrix Power Computation and Application View
In this part, we assume that $n \geqslant 4$. Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ satisfying the condition $(C^\star)$: $R_u > 1$.
Let $F \in \mathscr{M}_n(\mathbb{C})$ be the matrix defined by $$[F]_{k,j} = \frac{1}{\sqrt{n}} \omega^{(k-1)(j-1)} \text{ for all } (k,j) \in \llbracket 1; n \rrbracket^2,$$ where $\omega = e^{-2\pi i/n}$ (here $i$ denotes the usual complex number satisfying $i^2 = -1$).
(a) Show that $F$ is invertible and that $F^{-1} = \bar{F}$.
(b) Show that $F^2 \in \mathscr{M}_n(\mathbb{R})$.
(c) Deduce that $F^4 = I_n$ and that $F \in \mathbb{M}_n(u)$.
(d) Deduce that $$\begin{aligned} u(F) = & \frac{1}{4}\left(U(1)(F + I_n) - U(-1)(F - I_n)\right)(F^2 + I_n) \\ & + \frac{i}{4}\left(U(i)(F + iI_n) - U(-i)(F - iI_n)\right)(F^2 - I_n) \end{aligned}$$
Q26 Matrices Matrix Power Computation and Application View
In this part, we assume that $n \geqslant 4$. Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ satisfying the condition $(C^\star)$: $R_u > 1$.
Suppose that for all $k \in \mathbb{N}$, $u_k = \mathbb{P}(X = k)$ where $X$ is a random variable taking values in $\mathbb{N}$.
(a) Suppose that $X$ follows a binomial distribution with parameters $(N, p)$. Verify that $u$ satisfies condition $(C^\star)$ and find a simple expression for $u(A)$ for all $A \in \mathbb{M}_n(u)$.
(b) Suppose that $X$ follows a geometric distribution with parameter $p \in ]0,1[$. Verify that $u$ satisfies condition $(C^\star)$ and show that $$u(A) = p\left(I_n - (1-p)A\right)^{-1} A$$ for every diagonalizable matrix $A \in \mathbb{M}_n(u)$.