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
2013 centrale-maths2__psi

32 maths questions

QI.A Complex Numbers Arithmetic Modulus and Argument Computation View
Let $z \in \mathbb{C}$. We set $z = a + ib$, where $a, b \in \mathbb{R}$.
Let $n \in \mathbb{N}^*$. Determine the modulus and an argument of $\left(1 + \frac{z}{n}\right)^n$ as a function of $a$, $b$ and $n$.
QI.B Complex Numbers Arithmetic Complex Function Evaluation and Algebraic Manipulation View
Let $z \in \mathbb{C}$. We set $z = a + ib$, where $a, b \in \mathbb{R}$.
Deduce that $$\lim_{n \rightarrow \infty} \left(1 + \frac{z}{n}\right)^n = e^z$$
QII.A.1 Matrices Matrix Group and Subgroup Structure View
Let $n \in \mathbb{N}^*$. Let $A \in M_2(\mathbb{R})$ be an antisymmetric matrix. We set $A = \begin{pmatrix} 0 & -\alpha \\ \alpha & 0 \end{pmatrix}$ where $\alpha \in \mathbb{R}$.
Determine a number $\beta_n \in \mathbb{R}_+^*$ such that $$\frac{1}{\beta_n}\left(I_2 + \frac{1}{n}A\right) \in SO_2(\mathbb{R})$$
QII.A.2 Matrices Matrix Entry and Coefficient Identities View
Let $n \in \mathbb{N}^*$. Let $A \in M_2(\mathbb{R})$ be an antisymmetric matrix. We set $A = \begin{pmatrix} 0 & -\alpha \\ \alpha & 0 \end{pmatrix}$ where $\alpha \in \mathbb{R}$.
Determine a real number $\theta_n$ such that $$\frac{1}{\beta_n}\left(I_2 + \frac{1}{n}A\right) = \begin{pmatrix} \cos\theta_n & -\sin\theta_n \\ \sin\theta_n & \cos\theta_n \end{pmatrix}$$
QII.A.3 Matrices Matrix Power Computation and Application View
Let $n \in \mathbb{N}^*$. Let $A \in M_2(\mathbb{R})$ be an antisymmetric matrix. We set $A = \begin{pmatrix} 0 & -\alpha \\ \alpha & 0 \end{pmatrix}$ where $\alpha \in \mathbb{R}$.
Deduce that $E(A)$ exists and that it is a rotation matrix, whose angle we shall specify.
QII.B.1 Matrices Linear Transformation and Endomorphism Properties View
Let $B \in M_3(\mathbb{R})$ be antisymmetric.
a) Show that $\det B = 0$.
b) Show that $\left(\operatorname{Ker} u_B\right)^\perp$ is stable under $u_B$.
c) Deduce that $B$ has rank 0 or 2.
QII.B.2 Matrices Diagonalizability and Similarity View
Let $B \in M_3(\mathbb{R})$ be antisymmetric.
Show that there exist a matrix $P$ of $O_3(\mathbb{R})$ and a real number $\beta$ such that $$B = P \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -\beta \\ 0 & \beta & 0 \end{pmatrix} P^{-1}$$
QII.B.3 Matrices Matrix Norm, Convergence, and Inequality View
Let $B \in M_3(\mathbb{R})$ be antisymmetric. Assume that $$B = P \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -\beta \\ 0 & \beta & 0 \end{pmatrix} P^{-1}$$ for some $P \in O_3(\mathbb{R})$ and $\beta \in \mathbb{R}$.
Show that $|\beta| = \frac{\|B\|_2}{\sqrt{2}}$.
QII.B.4 Matrices Matrix Power Computation and Application View
Let $B \in M_3(\mathbb{R})$ be antisymmetric.
Show that $E(B)$ exists and is a rotation matrix. Specify the value of its unoriented angle as a function of $\|B\|_2$.
QIII.A.1 Matrices Matrix Power Computation and Application View
Let $D \in M_p(\mathbb{K})$ be a diagonal matrix.
Show that $E(D)$ exists and that $E(D) \in GL_p(\mathbb{C})$.
QIII.A.2 3x3 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Let $D \in M_p(\mathbb{K})$ be a diagonal matrix.
Show that there exists a polynomial $Q \in \mathbb{C}[X]$ such that $Q(D) = E(D)$.
QIII.A.3 3x3 Matrices Group Homomorphisms and Isomorphisms View
Let $D \in M_p(\mathbb{K})$ be a diagonal matrix. Let $(\Delta, +)$ be the additive subgroup of $M_p(\mathbb{R})$ formed by diagonal matrices.
Show that $E$ defines a group morphism from $(\Delta, +)$ to $(GL_p(\mathbb{R}), \times)$.
QIII.B.1 3x3 Matrices Matrix Power Computation and Application View
Let $A \in M_p(\mathbb{K})$ be a diagonalizable matrix.
Show that $E(A)$ exists.
QIII.B.2 3x3 Matrices Determinant and Rank Computation View
Let $A \in M_p(\mathbb{K})$ be a diagonalizable matrix.
Show that $\det(E(A)) = e^{\operatorname{tr}(A)}$.
QIII.B.3 3x3 Matrices Matrix Power Computation and Application View
Let $A \in M_p(\mathbb{K})$ be a diagonalizable matrix. Let $x \in \mathbb{C}$.
Show that $E\left(xI_p + A\right)$ exists and that $$E\left(xI_p + A\right) = e^x E(A)$$
QIII.C.1 3x3 Matrices Diagonalizability and Similarity View
Let $A, B \in M_p(\mathbb{K})$ be two diagonalizable matrices. We assume that $A$ and $B$ commute.
Show that there exists $P \in GL_p(\mathbb{C})$ such that $P^{-1}AP$ and $P^{-1}BP$ are diagonal.
(We shall study the restrictions of $u_B$ to the eigenspaces of $u_A$.)
QIII.C.2 3x3 Matrices Matrix Power Computation and Application View
Let $A, B \in M_p(\mathbb{K})$ be two diagonalizable matrices. We assume that $A$ and $B$ commute.
Deduce that $E(A + B)$ exists and that $E(A + B) = E(A)E(B) = E(B)E(A)$.
QIV.A.1 3x3 Matrices Linear Transformation and Endomorphism Properties View
Let $A \in M_p(\mathbb{C})$ and $k \in \mathbb{N}^*$ such that $A^k = 0$ and $A^{k-1} \neq 0$ (we say that $A$ is nilpotent of order $k$).
Show that, for every integer $j$ such that $1 \leqslant j \leqslant k$, $\operatorname{Ker} A^{j-1}$ is strictly included in $\operatorname{Ker} A^j$.
QIV.A.2 3x3 Matrices Linear Transformation and Endomorphism Properties View
Let $A \in M_p(\mathbb{C})$ and $k \in \mathbb{N}^*$ such that $A^k = 0$ and $A^{k-1} \neq 0$ (we say that $A$ is nilpotent of order $k$).
Deduce that $k \leqslant p$.
QIV.B 3x3 Matrices Matrix Power Computation and Application View
Let $A \in M_p(\mathbb{C})$ be nilpotent of order $k$.
Show that $E(A)$ exists. Propose a Maple or Mathematica procedure taking as input a strictly upper triangular matrix $A$ and returning the value of $E(A)$.
QIV.C 3x3 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Let $A \in M_p(\mathbb{C})$ be nilpotent of order $k$.
Show that there exists a polynomial $Q \in \mathbb{C}[X]$ such that $Q(A) = E(A)$.
QIV.D 3x3 Matrices Matrix Power Computation and Application View
Let $A \in M_p(\mathbb{C})$ be nilpotent of order $k$. Let $B \in M_p(\mathbb{C})$. We assume that $A$ and $B$ commute and that $E(B)$ exists.
We admit that, for every integer $i$ between 1 and $p$, $$\lim_{n \rightarrow \infty} \left(I_p + \frac{1}{n}B\right)^n = \lim_{n \rightarrow \infty} \left(I_p + \frac{1}{n}B\right)^{n-i}$$
Show that $E(A + B)$ exists and that $E(A + B) = E(A)E(B)$.
QIV.E 3x3 Matrices Matrix Power Computation and Application View
Let $A \in M_p(\mathbb{C})$ be nilpotent of order $k$. Let $x \in \mathbb{C}$.
Show that $E\left(xI_p + A\right)$ exists and that $E\left(xI_p + A\right) = e^x E(A)$.
QIV.F Matrices Linear Transformation and Endomorphism Properties View
Let $A \in M_p(\mathbb{C})$ be nilpotent of order $k$.
Show that $E(A) - I_p$ is nilpotent.
QV.A.1 Polynomial Division & Manipulation Eigenvalue and Characteristic Polynomial Analysis View
Let $A \in M_p(\mathbb{C})$ and $n \in \mathbb{N}^*$. We denote $$P_n(X) = \left(1 + \frac{X}{n}\right)^n \in \mathbb{C}[X]$$ and $\chi_A$ the characteristic polynomial of $A$ defined by $$\chi_A(X) = \det\left(A - XI_p\right)$$
Show that there exists a unique pair $\left(Q_n, R_n\right) \in \mathbb{C}[X] \times \mathbb{C}_{p-1}[X]$ such that $$P_n = Q_n \chi_A + R_n$$
QV.A.2 3x3 Matrices Matrix Power Computation and Application View
Let $A \in M_p(\mathbb{C})$ and $n \in \mathbb{N}^*$. We denote $$P_n(X) = \left(1 + \frac{X}{n}\right)^n \in \mathbb{C}[X]$$ and $\chi_A$ the characteristic polynomial of $A$, with the Euclidean division $P_n = Q_n \chi_A + R_n$.
Show that $E(A)$ exists if and only if $\lim_{n \rightarrow \infty} R_n(A)$ exists.
QV.A.3 3x3 Matrices Linear Transformation and Endomorphism Properties View
Let $A \in M_p(\mathbb{C})$. Let $k \in \mathbb{N}^*$ and $\lambda_1, \lambda_2, \ldots, \lambda_k$ be the roots of $\chi_A$ pairwise distinct, whose respective multiplicities we denote by $n_1, n_2, \ldots, n_k$.
For every integer $q$ between 1 and $p$, we denote by $J_q$ the matrix of $M_q(\mathbb{C})$ whose coefficients are all zero except those located just above the diagonal which equal 1.
Show that, for every $x \in \mathbb{C}$, for every integer $q$ between 1 and $p$, the family $\left\{\left(xI_q + J_q\right)^i,\ 0 \leqslant i \leqslant q-1\right\}$ is free.
QV.A.4 3x3 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Let $A \in M_p(\mathbb{C})$. Let $k \in \mathbb{N}^*$ and $\lambda_1, \lambda_2, \ldots, \lambda_k$ be the roots of $\chi_A$ pairwise distinct, with respective multiplicities $n_1, n_2, \ldots, n_k$. For every integer $q$ between 1 and $p$, $J_q$ denotes the matrix of $M_q(\mathbb{C})$ whose coefficients are all zero except those just above the diagonal which equal 1.
Let $B = \operatorname{diag}\left\{\lambda_1 I_{n_1} + J_{n_1}, \ldots, \lambda_k I_{n_k} + J_{n_k}\right\}$ be the block diagonal matrix defined by $$B = \begin{pmatrix} \lambda_1 I_{n_1} + J_{n_1} & 0 & \ldots & 0 \\ 0 & \lambda_2 I_{n_2} + J_{n_2} & \cdots & 0 \\ \vdots & \ddots & \ddots & 0 \\ 0 & \ldots & 0 & \lambda_k I_{n_k} + J_{n_k} \end{pmatrix}$$
Show that $\chi_B = \chi_A$.
QV.B.1 3x3 Matrices Matrix Power Computation and Application View
Let $A \in M_p(\mathbb{C})$ and $B = \operatorname{diag}\left\{\lambda_1 I_{n_1} + J_{n_1}, \ldots, \lambda_k I_{n_k} + J_{n_k}\right\}$ as defined in V.A.4. Let $i$ be an integer $\geqslant 1$.
Show that $$B^i = \begin{pmatrix} \left(\lambda_1 I_{n_1} + J_{n_1}\right)^i & 0 & \cdots & 0 \\ 0 & \left(\lambda_2 I_{n_2} + J_{n_2}\right)^i & \cdots & 0 \\ \vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & \left(\lambda_k I_{n_k} + J_{n_k}\right)^i \end{pmatrix}$$
QV.B.2 3x3 Matrices Linear Transformation and Endomorphism Properties View
Let $A \in M_p(\mathbb{C})$ and $B = \operatorname{diag}\left\{\lambda_1 I_{n_1} + J_{n_1}, \ldots, \lambda_k I_{n_k} + J_{n_k}\right\}$ as defined in V.A.4. Let $P$ be a non-zero annihilating polynomial of the matrix $B$.
a) Show that the degree of $P$ is $\geqslant p$.
b) Deduce that the family $\left\{B^i,\ 0 \leqslant i \leqslant p-1\right\}$ is free.
QV.B.3 3x3 Matrices Matrix Power Computation and Application View
Let $A \in M_p(\mathbb{C})$ and $B = \operatorname{diag}\left\{\lambda_1 I_{n_1} + J_{n_1}, \ldots, \lambda_k I_{n_k} + J_{n_k}\right\}$ as defined in V.A.4. We denote $P_n(X) = \left(1 + \frac{X}{n}\right)^n$.
Show that $\lim_{n \rightarrow \infty} P_n(B)$ exists.
QV.B.4 Matrices Matrix Power Computation and Application View
Let $A \in M_p(\mathbb{C})$ and $B = \operatorname{diag}\left\{\lambda_1 I_{n_1} + J_{n_1}, \ldots, \lambda_k I_{n_k} + J_{n_k}\right\}$ as defined in V.A.4.
Deduce that $E(A)$ exists.