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

2021 centrale-maths1__pc

36 maths questions

Q1 Discrete Probability Distributions Binomial Distribution Identification and Application View

What is the distribution of $Y _ { n }$ ? Deduce the expectation and variance of $Y _ { n }$.

Q2 Discrete Random Variables Expectation and Variance of Sums of Independent Variables View

What relation exists between $S _ { n }$ and $Y _ { n }$ ? Deduce the expectation and variance of $S _ { n }$. Justify that $S _ { n }$ and $n$ have the same parity.

Q3 Sequences and Series Evaluation of a Finite or Infinite Sum View

Give without proof the value of $C _ { 3 }$ and represent all Dyck paths of length 6.

Q4 Number Theory Combinatorial Number Theory and Counting View

Let $n \in \mathbb { N }$ and $\gamma = \left( \gamma _ { 1 } , \ldots , \gamma _ { 2 n + 2 } \right)$ be a Dyck path of length $2 n + 2$. Let $r = \max \left\{ i \in \llbracket 0 , n \rrbracket \mid s _ { \gamma } ( 2 i ) = 0 \right\}$. We assume $0 < r < n$ and we consider the paths $\alpha = \left( \gamma _ { 1 } , \ldots , \gamma _ { 2 r } \right)$ and $\beta = \left( \gamma _ { 2 r + 2 } , \ldots , \gamma _ { 2 n + 1 } \right)$. Justify using a figure that $\gamma _ { 2 r + 1 } = 1 , \gamma _ { 2 n + 2 } = - 1$ and that $\alpha$ and $\beta$ are Dyck paths.

Q5 Discrete Probability Distributions Binomial Distribution Identification and Application View

Let $n \in \mathbb { N } ^ { * }$ and let $\gamma = \left( \gamma _ { 1 } , \ldots , \gamma _ { 2 n } \right)$ be a Dyck path of length $2 n$. For $t \in \mathbb { N }$, express $\mathbb { P } \left( A _ { t , \gamma } \right)$ as a function of $n$ and $p$, where $A _ { t , \gamma } = \bigcap _ { k = 1 } ^ { m } \left( X _ { t + k } = \gamma _ { k } \right)$.

Q6 Discrete Probability Distributions Recurrence Relations and Sequences Involving Probabilities View

Let $T$ be the random variable, defined on $\Omega$ and taking values in $\mathbb { N }$, equal to the first instant when the particle returns to the origin, if this instant exists, and equal to 0 if the particle never returns to the origin: $$\forall \omega \in \Omega , \quad T ( \omega ) = \begin{cases} 0 & \text { if } \forall k \in \mathbb { N } ^ { * } , S _ { k } ( \omega ) \neq 0 \\ \min \left\{ k \in \mathbb { N } ^ { * } \mid S _ { k } ( \omega ) = 0 \right\} & \text { otherwise } \end{cases}$$ Show that $T$ takes even values and that, for all $n \in \mathbb { N } , \mathbb { P } ( T = 2 n + 2 ) = 2 C _ { n } p ^ { n + 1 } ( 1 - p ) ^ { n + 1 }$.

Q7 Sequences and Series Recurrence Relations and Sequence Properties View

Using question 4, show $$\forall n \in \mathbb { N } , \quad C _ { n + 1 } = \sum _ { r = 0 } ^ { n } C _ { r } C _ { n - r } .$$

Q8 Sequences and Series Convergence/Divergence Determination of Numerical Series View

Using the random variable $T$, show that the series $\sum _ { n \geqslant 0 } \frac { C _ { n } } { 4 ^ { n } }$ converges.

Q9 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View

Deduce that the power series $\sum _ { n \geqslant 0 } C _ { n } t ^ { n }$ converges uniformly on the interval $I = \left[ - \frac { 1 } { 4 } , \frac { 1 } { 4 } \right]$.

Q10 Probability Generating Functions Deriving moments or distribution from a PGF View

We set, for all $t \in I$, $$f ( t ) = \sum _ { n = 0 } ^ { + \infty } C _ { n } t ^ { n } \quad \text { and } \quad g ( t ) = 2 t f ( t ) .$$ The generating series of $T$ is given by $G _ { T } ( t ) = \sum _ { n = 0 } ^ { \infty } \mathbb { P } ( T = n ) t ^ { n }$, defined if $t \in [ - 1,1 ]$. Using the previous questions, express $G _ { T }$ using $g$ and $\mathbb { P } ( T = 0 )$.

Q11 Continuous Probability Distributions and Random Variables Distribution of Transformed or Combined Random Variables View

We set, for all $t \in I$, $$f ( t ) = \sum _ { n = 0 } ^ { + \infty } C _ { n } t ^ { n } \quad \text { and } \quad g ( t ) = 2 t f ( t ) .$$ Deduce that if $p \neq \frac { 1 } { 2 }$, then $T$ admits an expectation.

Q12 Sequences and series, recurrence and convergence Series convergence and power series analysis View

We set, for all $t \in I$, $$f ( t ) = \sum _ { n = 0 } ^ { + \infty } C _ { n } t ^ { n } \quad \text { and } \quad g ( t ) = 2 t f ( t ) .$$ Show that $\forall t \in I , g ( t ) ^ { 2 } = 2 g ( t ) - 4 t$.

Q13 Sequences and series, recurrence and convergence Closed-form expression derivation View

We set, for all $t \in I$, $$f ( t ) = \sum _ { n = 0 } ^ { + \infty } C _ { n } t ^ { n } \quad \text { and } \quad g ( t ) = 2 t f ( t ) .$$ Deduce that there exists a function $\varepsilon : I \rightarrow \{ - 1,1 \}$ such that $$\forall t \in I , \quad g ( t ) = 1 + \varepsilon ( t ) \sqrt { 1 - 4 t } .$$

Q14 Sequences and series, recurrence and convergence Convergence proof and limit determination View

We set, for all $t \in I$, $$f ( t ) = \sum _ { n = 0 } ^ { + \infty } C _ { n } t ^ { n } \quad \text { and } \quad g ( t ) = 2 t f ( t ) .$$ Show that $\varepsilon$ is continuous on $I \backslash \left\{ \frac { 1 } { 4 } \right\}$. Deduce $$\forall t \in I , \quad g ( t ) = 1 - \sqrt { 1 - 4 t } .$$

Q15 Sequences and series, recurrence and convergence Applied/contextual sequence problem View

We set, for all $t \in I$, $$f ( t ) = \sum _ { n = 0 } ^ { + \infty } C _ { n } t ^ { n } \quad \text { and } \quad g ( t ) = 2 t f ( t ) .$$ Deduce that $\mathbb { P } ( T \neq 0 ) = 1 - \sqrt { 1 - 4 p ( 1 - p ) }$. Interpret this result when $p = \frac { 1 } { 2 }$.

Q16 Discrete Random Variables Existence of Expectation or Moments View

Show that if $p = \frac { 1 } { 2 }$, then $T$ does not admit an expectation.

Q17 Generalised Binomial Theorem View

Justify the existence of a sequence of real numbers $\left( a _ { n } \right) _ { n \in \mathbb { N } }$ such that $$\forall x \in ] - 1,1 \left[ , \quad \sqrt { 1 + x } = 1 + \sum _ { n = 0 } ^ { + \infty } a _ { n } x ^ { n + 1 } , \right.$$ and, for all $n \in \mathbb { N }$, express $a _ { n }$ using a binomial coefficient.

Q18 Number Theory Combinatorial Number Theory and Counting View

Deduce $\forall n \in \mathbb { N } , C _ { n } = \frac { 1 } { n + 1 } \binom { 2 n } { n }$.

Q19 Arithmetic Sequences and Series Telescoping or Non-Standard Summation Involving an AP View

Recall Stirling's formula. Deduce an asymptotic equivalent of $C _ { n }$ as $n$ tends to $+ \infty$.

Q20 Arithmetic Sequences and Series Telescoping or Non-Standard Summation Involving an AP View

From the previous question, recover the result of questions 11 and 16.

Q21 Proof Proof of Set Membership, Containment, or Structural Property View

In $\mathbb { R } [ X ]$ equipped with an inner product $( \cdot \mid \cdot )$, let $\left( V _ { n } \right) _ { n \in \mathbb { N } }$ be an orthogonal system (a sequence of polynomials that is orthogonal and where each $V_n$ is monic of degree $n$). Show that, for all $n \in \mathbb { N }$, the family $\left( V _ { 0 } , V _ { 1 } , \ldots , V _ { n } \right)$ is an orthogonal basis of the vector space $\mathbb { R } _ { n } [ X ]$ of polynomials with real coefficients of degree at most $n$.

Q22 Sequences and Series Inner Product Spaces, Orthogonality, and Hilbert Space Structure on Sequence/Function Spaces View

Let $\left( V _ { n } \right) _ { n \in \mathbb { N } }$ be an orthogonal system in $\mathbb { R } [ X ]$ equipped with an inner product $( \cdot \mid \cdot )$. Let $n \in \mathbb { N }$ and $P \in \mathbb { R } [ X ]$ such that $\operatorname { deg } P < n$. Show that $\left( V _ { n } \mid P \right) = 0$.

Q23 Sequences and Series Inner Product Spaces, Orthogonality, and Hilbert Space Structure on Sequence/Function Spaces View

Let $\left( V _ { n } \right) _ { n \in \mathbb { N } }$ be an orthogonal system in $\mathbb { R } [ X ]$ equipped with an inner product $( \cdot \mid \cdot )$. Let $\left( W _ { n } \right) _ { n \in \mathbb { N } }$ be another orthogonal system. Show that $\forall n \in \mathbb { N } , W _ { n } = V _ { n }$.

Q24 Matrices Determinant and Rank Computation View

For all $n \in \mathbb { N }$, let $G _ { n } = \left( \left( X ^ { i - 1 } \mid X ^ { j - 1 } \right) \right) _ { 1 \leqslant i , j \leqslant n + 1 }$ be the Gram matrix and let $\left( V _ { n } \right) _ { n \in \mathbb { N } }$ be an orthogonal system. Let $Q _ { n } = \left( q _ { i , j } \right) _ { 1 \leqslant i , j \leqslant n + 1 }$ be the matrix of the family $\left( V _ { 0 } , V _ { 1 } , \ldots , V _ { n } \right)$ in the basis $\left( 1 , X , \ldots , X ^ { n } \right)$ of $\mathbb { R } _ { n } [ X ]$. Show that $Q _ { n }$ is upper triangular and that $\operatorname { det } Q _ { n } = 1$.

Q25 Matrices Matrix Algebra and Product Properties View

For all $n \in \mathbb { N }$, let $G _ { n } = \left( \left( X ^ { i - 1 } \mid X ^ { j - 1 } \right) \right) _ { 1 \leqslant i , j \leqslant n + 1 }$, let $G _ { n } ^ { \prime } = \left( \left( V _ { i - 1 } \mid V _ { j - 1 } \right) \right) _ { 1 \leqslant i , j \leqslant n + 1 }$, and let $Q _ { n }$ be the matrix of the family $\left( V _ { 0 } , V _ { 1 } , \ldots , V _ { n } \right)$ in the basis $\left( 1 , X , \ldots , X ^ { n } \right)$. Show that $Q _ { n } ^ { \top } G _ { n } Q _ { n } = G _ { n } ^ { \prime }$, where $Q _ { n } ^ { \top }$ is the transpose of the matrix $Q _ { n }$.

Q26 Matrices Determinant and Rank Computation View

For all $n \in \mathbb { N }$, let $G _ { n } = \left( \left( X ^ { i - 1 } \mid X ^ { j - 1 } \right) \right) _ { 1 \leqslant i , j \leqslant n + 1 }$ and let $\left( V _ { n } \right) _ { n \in \mathbb { N } }$ be an orthogonal system. Deduce that $\operatorname { det } G _ { n } = \prod _ { i = 0 } ^ { n } \left\| V _ { i } \right\| ^ { 2 }$.

Q27 Indefinite & Definite Integrals Convergence and Evaluation of Improper Integrals View

Let $P \in \mathbb { R } [ X ]$ and $Q \in \mathbb { R } [ X ]$. Show that the function $x \mapsto P ( 4 x ) Q ( 4 x ) \frac { \sqrt { 1 - x } } { \sqrt { x } }$ is integrable on $\left. ] 0,1 \right]$.

Q29 Sequences and Series Recurrence Relations and Sequence Properties View

Let $\left( U _ { n } \right) _ { n \in \mathbb { N } }$ be the sequence of polynomials defined by $U _ { 0 } = 1 , U _ { 1 } = X - 1$ and $\forall n \in \mathbb { N } , U _ { n + 2 } = ( X - 2 ) U _ { n + 1 } - U _ { n }$. For all $n \in \mathbb { N }$, show that $U _ { n }$ is monic of degree $n$, and determine the value of $U _ { n } ( 0 )$.

Q30 Sequences and Series Recurrence Relations and Sequence Properties View

Let $\left( U _ { n } \right) _ { n \in \mathbb { N } }$ be the sequence of polynomials defined by $U _ { 0 } = 1 , U _ { 1 } = X - 1$ and $\forall n \in \mathbb { N } , U _ { n + 2 } = ( X - 2 ) U _ { n + 1 } - U _ { n }$. Let $\theta \in \mathbb { R }$. Show that $\forall n \in \mathbb { N } , U _ { n } \left( 4 \cos ^ { 2 } \theta \right) \sin \theta = \sin ( ( 2 n + 1 ) \theta )$.

Q31 Standard Integrals and Reverse Chain Rule Orthogonality and Inner Product Integrals View

Let $( m , n ) \in \mathbb { N } ^ { 2 }$. Calculate $\int _ { 0 } ^ { \pi / 2 } \sin ( ( 2 m + 1 ) \theta ) \sin ( ( 2 n + 1 ) \theta ) \mathrm { d } \theta$.

Q32 Sequences and Series Inner Product Spaces, Orthogonality, and Hilbert Space Structure on Sequence/Function Spaces View

Let $\left( U _ { n } \right) _ { n \in \mathbb { N } }$ be the sequence of polynomials defined by $U _ { 0 } = 1 , U _ { 1 } = X - 1$ and $\forall n \in \mathbb { N } , U _ { n + 2 } = ( X - 2 ) U _ { n + 1 } - U _ { n }$, with the inner product $$( P \mid Q ) = \frac { 2 } { \pi } \int _ { 0 } ^ { 1 } P ( 4 x ) Q ( 4 x ) \frac { \sqrt { 1 - x } } { \sqrt { x } } \mathrm { ~d} x .$$ Deduce that $\left( U _ { n } \right) _ { n \in \mathbb { N } }$ is an orthogonal system and that, for all $n \in \mathbb { N } , \left\| U _ { n } \right\| = 1$. To calculate the value of $( U _ { m } \mid U _ { n } )$, one may perform the change of variable $x = \cos ^ { 2 } \theta$.

Q33 Reduction Formulae Derive a Reduction/Recurrence Formula via Integration by Parts View

For all $n \in \mathbb { N }$, set $\mu _ { n } = \left( X ^ { n } \mid 1 \right)$ with the inner product $$( P \mid Q ) = \frac { 2 } { \pi } \int _ { 0 } ^ { 1 } P ( 4 x ) Q ( 4 x ) \frac { \sqrt { 1 - x } } { \sqrt { x } } \mathrm { ~d} x .$$ Using integration by parts, show $$\forall n \in \mathbb { N } ^ { * } , \quad 4 \mu _ { n - 1 } - \mu _ { n } = \frac { 2 \times 4 ^ { n } } { \pi } \int _ { 0 } ^ { 1 } x ^ { n - 3 / 2 } ( 1 - x ) ^ { 3 / 2 } \mathrm { d } x = \frac { 3 } { 2 n - 1 } \mu _ { n } .$$

Q34 Sequences and Series Evaluation of a Finite or Infinite Sum View

Q35 Sequences and Series Evaluation of a Finite or Infinite Sum View

Let $n \in \mathbb { N }$. Deduce from the previous parts the value of the determinant $$H _ { n } = \operatorname { det } \left( C _ { i + j - 2 } \right) _ { 1 \leqslant i , j \leqslant n + 1 } = \left| \begin{array} { c c c c c c } C _ { 0 } & C _ { 1 } & C _ { 2 } & \ldots & C _ { n - 1 } & C _ { n } \\ C _ { 1 } & \therefore & & \therefore & \therefore & C _ { n + 1 } \\ C _ { 2 } & & \ddots & \ddots & \therefore & \vdots \\ \vdots & \therefore & \ddots & \ddots & & C _ { 2 n - 2 } \\ C _ { n - 1 } & \therefore & \ddots & & \therefore & C _ { 2 n - 1 } \\ C _ { n } & C _ { n + 1 } & \cdots & C _ { 2 n - 2 } & C _ { 2 n - 1 } & C _ { 2 n } \end{array} \right|.$$

Q36 Sequences and Series Functional Equations and Identities via Series View

Q37 Sequences and Series Functional Equations and Identities via Series View

We set, for all $n \in \mathbb { N }$, $$D _ { n } ( X ) = \left| \begin{array} { c c c c c c } C _ { 0 } & C _ { 1 } & C _ { 2 } & \ldots & C _ { n - 1 } & C _ { n } \\ C _ { 1 } & \therefore & & \therefore & \therefore & C _ { n + 1 } \\ C _ { 2 } & & \ddots & \therefore & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & & C _ { 2 n - 2 } \\ C _ { n - 1 } & C _ { n } & C _ { n + 1 } & \ldots & C _ { 2 n - 2 } & C _ { 2 n - 1 } \\ 1 & X & \cdots & X ^ { n - 2 } & X ^ { n - 1 } & X ^ { n } \end{array} \right|$$ and $$H _ { n } ^ { \prime } = \left| \begin{array} { c c c c c c } C _ { 1 } & C _ { 2 } & C _ { 3 } & \ldots & C _ { n - 1 } & C _ { n } \\ C _ { 2 } & \therefore & & \ddots & \ddots & C _ { n + 1 } \\ C _ { 3 } & & \ddots & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & & C _ { 2 n - 3 } \\ C _ { n - 1 } & \therefore & \ddots & & \ddots & C _ { 2 n - 2 } \\ C _ { n } & C _ { n + 1 } & \cdots & C _ { 2 n - 3 } & C _ { 2 n - 2 } & C _ { 2 n - 1 } \end{array} \right|.$$ Deduce that $\forall n \in \mathbb { N } , D _ { n } = U _ { n }$, then determine, for all $n \in \mathbb { N } ^ { * }$, the value of the determinant $H _ { n } ^ { \prime }$.