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 x-ens-maths1__mp

16 maths questions

Q1 Matrices Matrix Group and Subgroup Structure View

(a) Show that $O _ { n } ( \mathbb { R } )$ is a subgroup of the group $\mathrm { GL } _ { n } ( \mathbb { R } )$ of invertible matrices.
(b) Show that $O _ { n } ( \mathbb { R } )$ is a compact subset of $\mathcal { M } _ { n } ( \mathbb { R } )$.

Q2 Invariant lines and eigenvalues and vectors Diagonalizability determination or proof View

Let $M$ and $N$ be in $S _ { n } ( \mathbb { R } )$. Show that there exists $U \in O _ { n } ( \mathbb { R } )$ such that $N = U M U ^ { - 1 }$, if and only if $\chi _ { M } = \chi _ { N }$.

Q3 Proof Deduction or Consequence from Prior Results View

Let $\hat { \lambda } = \left( \lambda _ { 1 } \geqslant \cdots \geqslant \lambda _ { n + 1 } \right) \in \mathbb { R } ^ { n + 1 }$ and $\widehat { \mu } = \left( \mu _ { 1 } \geqslant \cdots \geqslant \mu _ { n } \right) \in \mathbb { R } ^ { n }$. Let $x \in \mathbb { R }$. Form $$\widehat { \lambda } ^ { \prime } = \left( \lambda _ { 1 } \geqslant \cdots \geqslant \lambda _ { i } \geqslant x > \lambda _ { i + 1 } \geqslant \cdots \geqslant \lambda _ { n + 1 } \right)$$ by choosing the integer $i \in \{ 0 , \ldots , n + 1 \}$ appropriately. If $x > \lambda _ { 1 }$, we thus have $i = 0$, while if $x \leqslant \lambda _ { n + 1 }$, we have $i = n + 1$. Similarly form $$\widehat { \mu } ^ { \prime } = \left( \mu _ { 1 } \geqslant \cdots \geqslant \mu _ { j } \geqslant x > \mu _ { j + 1 } \geqslant \cdots \geqslant \mu _ { n } \right) .$$ Assume that $\widehat { \lambda }$ and $\widehat { \mu }$ are interlaced. Show that $j \leqslant i \leqslant j + 1$. By examining each of the two cases $j = i$ or $i - 1$, show that $\widehat { \lambda } ^ { \prime }$ and $\widehat { \mu } ^ { \prime }$ are interlaced.

Q4 Roots of polynomials Polynomial evaluation, interpolation, and remainder View

Q5 Roots of polynomials Location and bounds on roots View

Let $\widehat { \mu } = \left( \mu _ { 1 } > \cdots > \mu _ { n } \right) \in \mathbb { R } ^ { n }$. We define the polynomials $$Q _ { 0 } = \prod _ { k = 1 } ^ { n } \left( X - \mu _ { k } \right) \quad \text { and } \quad \forall j \in \{ 1 , \ldots , n \} , \quad P _ { j } = \frac { Q _ { 0 } } { \left( X - \mu _ { j } \right) } .$$ Let $P \in \mathbb { R } [ X ]$ be a monic polynomial of degree $n + 1$.
(a) Show that there exists a unique vector $\left( a , \alpha _ { 1 } , \alpha _ { 2 } , \ldots , \alpha _ { n } \right) \in \mathbb { R } ^ { n + 1 }$ such that $$P = ( X - a ) Q _ { 0 } - \sum _ { j = 1 } ^ { n } \alpha _ { j } P _ { j }$$ (b) Assume that the real numbers $\alpha _ { 1 } , \ldots , \alpha _ { n }$ are all strictly positive. Show that $P$ has $n + 1$ distinct real roots $\lambda _ { 1 } > \cdots > \lambda _ { n + 1 }$, and that $\hat { \lambda } = \left( \lambda _ { 1 } > \cdots > \lambda _ { n + 1 } \right)$ and $\widehat { \mu }$ are strictly interlaced.
(c) Conversely, assume that $P$ has $n + 1$ distinct real roots $\lambda _ { 1 } > \cdots > \lambda _ { n + 1 }$, and that $\widehat { \lambda } = \left( \lambda _ { 1 } > \cdots > \lambda _ { n + 1 } \right)$ and $\widehat { \mu }$ are strictly interlaced. Show that, for all $j \in \{ 1 , \ldots , n \} , \alpha _ { j } > 0$.

Q6 Roots of polynomials Multiplicity and derivative analysis of roots View

Q7 Roots of polynomials Location and bounds on roots View

Let $\widehat { \mu } = \left( \mu _ { 1 } > \cdots > \mu _ { n } \right) \in \mathbb { R } ^ { n }$. We are given integers $m _ { k } \geqslant 1$ for $k = 1 , \ldots , n$. We set $$Q _ { 1 } = \prod _ { k = 1 } ^ { n } \left( X - \mu _ { k } \right) ^ { m _ { k } } \quad \text { and } \quad P _ { j } = \frac { Q _ { 1 } } { X - \mu _ { j } } .$$ Let $\left( a , \alpha _ { 1 } , \alpha _ { 2 } , \ldots , \alpha _ { n } \right) \in \mathbb { R } ^ { n + 1 }$ and let $P \in \mathbb { R } [ X ]$ be defined by the formula $$P = ( X - a ) Q _ { 1 } - \sum _ { j = 1 } ^ { n } \alpha _ { j } P _ { j }$$ (a) Give an expression of $P \wedge Q _ { 1 }$ in terms of the $\mu _ { j }$, the $m _ { j }$ and the set $J$ of indices for which $\alpha _ { j } = 0$.
(b) Assume that the numbers $\alpha _ { 1 } , \ldots , \alpha _ { n }$ are non-negative. Show that all roots of $P$ are real.

Q8 Matrices Determinant and Rank Computation View

Let $r$ and $s$ be two non-zero natural integers. Let $A \in \mathcal { M } _ { r } ( \mathbb { R } ) , B \in \mathcal { M } _ { r , s } ( \mathbb { R } ) , C \in \mathcal { M } _ { s , r } ( \mathbb { R } )$ and $D \in \mathcal { M } _ { s } ( \mathbb { R } )$. We further assume that $A$ is invertible. We consider the matrix $M \in \mathcal { M } _ { r + s } ( \mathbb { R } )$ having the following block form $$M = \left[ \begin{array} { l l } A & B \\ C & D \end{array} \right]$$ Find two matrices $U \in \mathcal { M } _ { r , s } ( \mathbb { R } )$ and $V \in \mathcal { M } _ { s } ( \mathbb { R } )$ such that $$M = \left[ \begin{array} { c c } A & 0 \\ C & I _ { s } \end{array} \right] \cdot \left[ \begin{array} { c c } I _ { r } & U \\ 0 & V \end{array} \right]$$ and deduce that $$\operatorname { det } ( M ) = \operatorname { det } ( A ) \cdot \operatorname { det } \left( D - C A ^ { - 1 } B \right)$$

Q9 Roots of polynomials Eigenvalue-root connection for matrices or linear operators View

Let $M \in S _ { n + 1 } ( \mathbb { R } )$ be a symmetric matrix. We write $M$ in block form $$M = \left[ \begin{array} { l l } A & y \\ { } ^ { t } y & a \end{array} \right]$$ with $a \in \mathbb { R } , y \in \mathcal { M } _ { n , 1 } ( \mathbb { R } )$ and $A \in S _ { n } ( \mathbb { R } )$.
(a) If the spectrum of $A$ is $\operatorname { Sp } ( A ) = \left( \mu _ { 1 } \geqslant \cdots \geqslant \mu _ { n } \right)$, show that there exist $U \in O _ { n + 1 } ( \mathbb { R } )$ and $z \in \mathcal { M } _ { n , 1 } ( \mathbb { R } )$ such that $$U M ^ { t } U = \left[ \begin{array} { c c } \Delta \left( \mu _ { 1 } , \ldots , \mu _ { n } \right) & z \\ t _ { z } & a \end{array} \right]$$ (b) Deduce that there exist non-negative real numbers $\alpha _ { j }$ (for $j = 1 , \ldots , n$ ) such that $$\chi _ { M } = ( X - a ) Q _ { 0 } - \sum _ { j = 1 } ^ { n } \alpha _ { j } \frac { Q _ { 0 } } { \left( X - \mu _ { j } \right) } , \quad \text { where } \quad Q _ { 0 } = \prod _ { k = 1 } ^ { n } \left( X - \mu _ { k } \right) .$$ (c) Show that $\operatorname { Sp } ( M )$ and $\operatorname { Sp } ( A )$ are interlaced.

Q10 Matrices Eigenvalue and Characteristic Polynomial Analysis View

Q11 Matrices Eigenvalue and Characteristic Polynomial Analysis View

For $T = \left( t _ { i j } \right) \in \mathcal { M } _ { n + 1 } ( \mathbb { R } )$, we denote by $T _ { \leqslant n }$ the extracted matrix of size $n$ whose coefficients are the $t _ { i j }$ for $1 \leqslant i , j \leqslant n$. Let $M \in S _ { n + 1 } ( \mathbb { R } )$, and denote $\mathcal { C } _ { M } = \left\{ \operatorname { Sp } \left( \left( U M U ^ { - 1 } \right) _ { \leqslant n } \right) \in \mathbb { R } ^ { n } , \text { for } U \text { ranging over } O _ { n + 1 } ( \mathbb { R } ) \right\}$. We further assume that the eigenvalues of $M$ are distinct. We thus have $\operatorname { Sp } ( M ) = \left( \lambda _ { 1 } > \cdots > \lambda _ { n + 1 } \right)$.
(a) Let $\widehat { \mu } = \left( \mu _ { 1 } > \cdots > \mu _ { n } \right)$ such that $\operatorname { Sp } ( M )$ and $\widehat { \mu }$ are strictly interlaced. Show that $\widehat { \mu }$ belongs to $\mathcal { C } _ { M }$.
(b) Show that $$\mathcal { C } _ { M } = \left\{ \widehat { \mu } = \left( \mu _ { 1 } \geqslant \cdots \geqslant \mu _ { n } \right) , \text { such that } \operatorname { Sp } ( M ) \text { and } \widehat { \mu } \text { are interlaced } \right\} .$$

Q12 Matrices Eigenvalue and Characteristic Polynomial Analysis View

We consider the application $$\begin{array} { r c c c } \operatorname { Diag } _ { n } : & S _ { n } ( \mathbb { R } ) & \longrightarrow & \mathbb { R } ^ { n } \\ M = \left( m _ { i j } \right) & \longmapsto & \left( m _ { 11 } , m _ { 22 } , \ldots , m _ { n n } \right) \end{array}$$ Let $M \in S _ { n } ( \mathbb { R } )$. We study the set $$\mathcal { D } _ { M } = \left\{ \operatorname { Diag } _ { n } \left( U M U ^ { - 1 } \right) , \text { for } U \text { ranging over } O _ { n } ( \mathbb { R } ) \right\} .$$ We first study the case $n = 2$. We then denote $\operatorname { Sp } ( M ) = \left( \lambda _ { 1 } \geqslant \lambda _ { 2 } \right)$. Show that $\mathcal { D } _ { M }$ is the line segment in $\mathbb { R } ^ { 2 }$ whose endpoints are $( \lambda _ { 1 } , \lambda _ { 2 } )$ and $( \lambda _ { 2 } , \lambda _ { 1 } )$.

Q13 Matrices Eigenvalue and Characteristic Polynomial Analysis View

We consider the application $$\begin{array} { r c c c } \operatorname { Diag } _ { n } : & S _ { n } ( \mathbb { R } ) & \longrightarrow & \mathbb { R } ^ { n } \\ M = \left( m _ { i j } \right) & \longmapsto & \left( m _ { 11 } , m _ { 22 } , \ldots , m _ { n n } \right) \end{array}$$ Let $M = \left( m _ { i j } \right) \in S _ { n } ( \mathbb { R } )$. We denote $\operatorname { Sp } ( M ) = \left( \lambda _ { 1 } \geqslant \cdots \geqslant \lambda _ { n } \right) \in \mathbb { R } ^ { n }$. We propose to prove that, for all $s \in \{ 1 , \ldots , n \}$, we have: $$\sum _ { i = 1 } ^ { s } m _ { i i } \leqslant \sum _ { i = 1 } ^ { s } \lambda _ { i }$$ (a) What do you think of the case $s = n$ ?
(b) Express $\sum _ { i = 1 } ^ { n - 1 } m _ { i i }$ in terms of the eigenvalues of the matrix $M _ { \leqslant n - 1 }$ obtained by removing the last row and last column of $M$. Deduce inequality (3) when $s = n - 1$.
(c) By proceeding by induction on $n$, show inequality (3), for all $s \in \{ 1 , \ldots , n \}$.

Q14 Linear transformations View

Q15 Linear transformations View

We denote by $E$ the vector space $\mathbb { R } ^ { 2 }$ equipped with the standard inner product and the canonical basis $\mathcal { B } = \left\{ e _ { 1 } , e _ { 2 } \right\}$. We define a basis $\mathcal { C } = \left\{ \omega _ { 1 } , \omega _ { 2 } \right\}$ of $E$ by $\omega _ { 1 } = e _ { 1 }$ and $\omega _ { 2 } = \frac { 1 } { 2 } \left( e _ { 1 } + \sqrt { 3 } e _ { 2 } \right)$. Let $s_1$ be the orthogonal reflection with respect to $\mathbb{R}\omega_1$ and $s_2$ the orthogonal reflection with respect to $\mathbb{R}\omega_2$. Let $H$ be the set of vectors $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in \mathbb { R } ^ { 3 }$ such that $m _ { 1 } + m _ { 2 } + m _ { 3 } = 0$. We denote by $H ^ { + }$ the subset of $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in H$ such that $m _ { 1 } \geqslant m _ { 2 } \geqslant m _ { 3 }$. We consider the application $$\begin{array} { c c c c } \varphi : & H & \longrightarrow & E \\ & \left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) & \longmapsto & \left( m _ { 1 } - m _ { 2 } \right) \omega _ { 1 } + \left( m _ { 2 } - m _ { 3 } \right) \omega _ { 2 } \end{array}$$ (a) Show that $\varphi$ is a linear isomorphism. Describe $\varphi \left( H ^ { + } \right)$.
(b) Show that, for all $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in H$, we have $$s _ { 1 } \circ \varphi \left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) = \varphi \left( m _ { 1 } , m _ { 3 } , m _ { 2 } \right) \quad \text { and } \quad s _ { 2 } \circ \varphi \left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) = \varphi \left( m _ { 2 } , m _ { 1 } , m _ { 3 } \right) .$$ (c) Let $\widehat { \lambda } = \left( \lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 } \right) \in H$ such that $\lambda _ { 1 } > \lambda _ { 2 } > \lambda _ { 3 }$. We denote by $\mathcal { Q } _ { \widehat { \lambda } }$ the set of $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in H ^ { + }$ such that $m _ { 1 } \leqslant \lambda _ { 1 }$ and $m _ { 1 } + m _ { 2 } \leqslant \lambda _ { 1 } + \lambda _ { 2 }$. Show that $\varphi \left( \mathcal { Q } _ { \widehat { \lambda } } \right)$ is a quadrilateral whose vertices will be described.

Q16 Matrices Eigenvalue and Characteristic Polynomial Analysis View

We denote by $E$ the vector space $\mathbb { R } ^ { 2 }$ equipped with the standard inner product and the canonical basis $\mathcal { B } = \left\{ e _ { 1 } , e _ { 2 } \right\}$. We define a basis $\mathcal { C } = \left\{ \omega _ { 1 } , \omega _ { 2 } \right\}$ of $E$ by $\omega _ { 1 } = e _ { 1 }$ and $\omega _ { 2 } = \frac { 1 } { 2 } \left( e _ { 1 } + \sqrt { 3 } e _ { 2 } \right)$. Let $s_1$ be the orthogonal reflection with respect to $\mathbb{R}\omega_1$ and $s_2$ the orthogonal reflection with respect to $\mathbb{R}\omega_2$. Let $H$ be the set of vectors $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in \mathbb { R } ^ { 3 }$ such that $m _ { 1 } + m _ { 2 } + m _ { 3 } = 0$, $H^+$ the subset with $m_1 \geqslant m_2 \geqslant m_3$, and $\varphi : H \to E$ defined by $\varphi(m_1,m_2,m_3) = (m_1-m_2)\omega_1 + (m_2-m_3)\omega_2$. We consider the application $$\begin{array} { r c c c } \operatorname { Diag } _ { n } : & S _ { n } ( \mathbb { R } ) & \longrightarrow & \mathbb { R } ^ { n } \\ M = \left( m _ { i j } \right) & \longmapsto & \left( m _ { 11 } , m _ { 22 } , \ldots , m _ { n n } \right) \end{array}$$ and $\mathcal { D } _ { M } = \left\{ \operatorname { Diag } _ { n } \left( U M U ^ { - 1 } \right) , \text { for } U \text { ranging over } O _ { n } ( \mathbb { R } ) \right\}$. Let $M \in S _ { 3 } ( \mathbb { R } )$ be a matrix with zero trace. We denote $\operatorname { Sp } ( M ) = \left( \lambda _ { 1 } \geqslant \lambda _ { 2 } \geqslant \lambda _ { 3 } \right)$. We propose to describe $\varphi \left( \mathcal { D } _ { M } \right)$.
(a) Let $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in H$. Let $\sigma$ be a permutation of $\{ 1,2,3 \}$. Show that $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in \mathcal { D } _ { M }$ if and only if $\left( m _ { \sigma ( 1 ) } , m _ { \sigma ( 2 ) } , m _ { \sigma ( 3 ) } \right) \in \mathcal { D } _ { M }$.
(b) Using question 13, show that the intersection $H ^ { + } \cap \mathcal { D } _ { M }$ is contained in $\mathcal { Q } _ { \widehat { \lambda } }$.
(c) Let $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in \mathcal { D } _ { M }$. Show that the segment of $H$ whose vertices are $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right)$ and $\left( m _ { 2 } , m _ { 1 } , m _ { 3 } \right)$ is contained in $\mathcal { D } _ { M }$. One may use question 12. Similarly, show that the segment of $H$ whose vertices are $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right)$ and $\left( m _ { 1 } , m _ { 3 } , m _ { 2 } \right)$ is contained in $\mathcal { D } _ { M }$.
(d) Show that $\mathcal { D } _ { M }$ contains $\mathcal { Q } _ { \widehat { \lambda } }$.
(e) Show that if $\lambda _ { 1 } > \lambda _ { 2 } > \lambda _ { 3 }$ then $\varphi \left( \mathcal { D } _ { M } \right)$ is a hexagon, whose vertices will be determined.