Free Practice Questions for Ingénieur ITPE

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Key Facts: Ingénieur ITPE Exam

12 h

Total written admissibility duration across 3 papers

Ministère de la Transition écologique

< 5 / 20

Score that may lead to elimination on a written or oral test

Arrêté régissant le concours ITPE

Category A

French State Civil Service Engineering Corps

Décret n° 2005-631

ENTPE

National engineering school in Vaulx-en-Velin

ENTPE Official Portal

The Concours ITPE internal track recruits Category A engineers for the French Ministry of Ecological Transition. It comprises 12 hours of written admissibility tests in PCSI mathematics, MPSI physics, and a policy dossier (note problématique), followed by oral examinations in sciences and an administrative interview. Successful candidates undergo paid engineering training at ENTPE. This bank provides an English-language MCQ study adaptation, not a format simulation.

Sample Ingénieur ITPE Practice Questions

Try these sample questions to review concepts for the Ingénieur ITPE exam. Each question includes a detailed explanation. Start the interactive quiz above for the full 100+ question experience with AI tutoring.

1In the PCSI linear algebra curriculum, consider the 3x3 real matrix A = [[1, 2, 1], [0, 3, 1], [2, 1, 4]]. What is the determinant of A?

A.9

B.7

C.-3

D.15

Explanation: Expanding along the first column or using Sarrus' rule: det(A) = 1*(3*4 - 1*1) - 0 + 2*(2*1 - 1*3) = 1*(12 - 1) + 2*(2 - 3) = 11 - 2 = 9. Determinant calculation is a foundational skill in the ITPE concours interne PCSI mathematics paper. Because det(A) != 0, the matrix A is invertible.

2What are the eigenvalues of the real matrix A = [[4, 2], [1, 3]]?

A.lambda = 2 and lambda = 5

B.lambda = 1 and lambda = 6

C.lambda = 3 and lambda = 4

D.lambda = -2 and lambda = -5

Explanation: The characteristic polynomial of A is P(lambda) = det(A - lambda*I) = (4 - lambda)(3 - lambda) - 2*1 = lambda^2 - 7*lambda + 12 - 2 = lambda^2 - 7*lambda + 10. Factoring gives (lambda - 2)(lambda - 5) = 0, yielding eigenvalues lambda_1 = 2 and lambda_2 = 5. Since A has distinct eigenvalues, it is diagonalizable over R.

3Let E = R_3[X] be the vector space of real polynomials of degree at most 3 (dim E = 4). Consider the linear endomorphism u: E -> E defined by u(P) = P'' - 2*P'. What is the rank of u?

A.3

B.2

C.4

D.1

Explanation: To determine dim(ker(u)), solve P'' - 2*P' = 0. The only polynomial solutions are constant polynomials P(X) = C, since any polynomial of degree d >= 1 would have deg(P') = d - 1 and deg(P'') = d - 2, meaning -2*P' cannot be cancelled by P''. Thus ker(u) = R_0[X] and dim(ker(u)) = 1. By the rank-nullity theorem (theoreme du rang), dim(E) = dim(ker(u)) + rg(u), so 4 = 1 + rg(u), which yields rg(u) = 3.

4Let A = [[1, 1], [0, 1]] in M_2(R). What is the matrix power A^n for any positive integer n?

A.[[1, n], [0, 1]]

B.[[1, 2^n - 1], [0, 1]]

C.[[1, n^2], [0, 1]]

D.[[n, n], [0, n]]

Explanation: We can decompose A as A = I_2 + N, where N = [[0, 1], [0, 0]]. Since N^2 = 0 (N is nilpotent of index 2) and I_2 commutes with N, the binomial theorem gives A^n = (I_2 + N)^n = I_2 + n*N = [[1, n], [0, 1]]. This can also be verified directly by induction.

5Solve the first-order linear differential equation y' + (2*x / (1 + x^2))*y = 1 with initial condition y(0) = 0. What is y(1)?

A.2/3

B.1/2

C.4/3

D.1

Explanation: The integrating factor is exp(integral(2*x / (1 + x^2) dx)) = exp(ln(1 + x^2)) = 1 + x^2. Multiplying the ODE by this factor yields ((1 + x^2)*y)' = 1 + x^2. Integrating both sides gives (1 + x^2)*y = x + x^3 / 3 + C. Using y(0) = 0 gives C = 0, so y(x) = (x + x^3 / 3) / (1 + x^2). Evaluating at x = 1: y(1) = (1 + 1/3) / (1 + 1) = (4/3) / 2 = 2/3.

6Consider the second-order Cauchy problem y'' + 4*y' + 4*y = 0 with initial conditions y(0) = 1 and y'(0) = 1. What is the solution y(x)?

A.(1 + 3*x)*exp(-2*x)

B.(1 + x)*exp(-2*x)

C.exp(-2*x) + 3*exp(2*x)

D.cos(2*x) + sin(2*x)

Explanation: The characteristic equation is r^2 + 4*r + 4 = (r + 2)^2 = 0, which has a double real root r = -2. The general solution is y(x) = (A + B*x)*exp(-2*x). Applying y(0) = 1 gives A = 1. The derivative is y'(x) = B*exp(-2*x) - 2*(A + B*x)*exp(-2*x). Evaluating at x = 0 gives y'(0) = B - 2*A = B - 2 = 1, so B = 3. Hence y(x) = (1 + 3*x)*exp(-2*x).

7What is the Taylor expansion (developpement limite) of f(x) = ln(1 + x)*cos(x) around x = 0 up to order 3?

A.x - x^2 / 2 - x^3 / 6 + o(x^3)

B.x - x^2 / 2 + x^3 / 3 + o(x^3)

C.x - x^2 / 2 - x^3 / 2 + o(x^3)

D.x + x^2 / 2 - x^3 / 6 + o(x^3)

Explanation: At x = 0, ln(1 + x) = x - x^2 / 2 + x^3 / 3 + o(x^3) and cos(x) = 1 - x^2 / 2 + o(x^3). Multiplying the two expansions: f(x) = (x - x^2 / 2 + x^3 / 3)*(1 - x^2 / 2) + o(x^3) = x - x^3 / 2 - x^2 / 2 + x^3 / 3 + o(x^3) = x - x^2 / 2 + (-1/2 + 1/3)*x^3 + o(x^3) = x - x^2 / 2 - x^3 / 6 + o(x^3).

8What is the exact sum of the convergent numerical series sum_{n=1}^infty (n / 2^n)?

A.2

B.1

C.4

D.3/2

Explanation: For |x| < 1, the geometric series sum_{n=0}^infty x^n = 1 / (1 - x). Differentiating both sides gives sum_{n=1}^infty n*x^(n-1) = 1 / (1 - x)^2. Multiplying by x yields sum_{n=1}^infty n*x^n = x / (1 - x)^2. Substituting x = 1/2 gives (1/2) / (1 - 1/2)^2 = (1/2) / (1/4) = 2.

9Evaluate the improper integral I = integral_0^{+infty} x*exp(-3*x) dx.

A.1/9

B.1/3

C.1/6

D.2/9

Explanation: Using integration by parts, set u = x and v' = exp(-3*x), so u' = 1 and v = -exp(-3*x) / 3. Then I = [-x*exp(-3*x) / 3]_0^{+infty} + (1/3)*integral_0^{+infty} exp(-3*x) dx. The boundary term vanishes at both limits. The remaining integral is (1/3)*[-exp(-3*x) / 3]_0^{+infty} = (1/3)*(0 - (-1/3)) = 1/9.

10Let f: M_2(R) -> M_2(R) be the linear map defined by f(M) = M - M^T, where M^T is the transpose of M. What is the rank (dimension of the image) of f?

A.1

B.3

C.2

D.4

Explanation: A matrix M is in ker(f) if and only if M = M^T, which is the subspace S_2(R) of symmetric 2x2 matrices. The dimension of S_2(R) is n*(n + 1) / 2 = 2*3 / 2 = 3. Since dim(M_2(R)) = 4, the rank-nullity theorem gives rg(f) = dim(M_2(R)) - dim(ker(f)) = 4 - 3 = 1. The image consists of skew-symmetric matrices A_2(R), which have dimension n*(n - 1) / 2 = 1.

About the Ingénieur ITPE Exam

The Concours d'Ingénieur des Travaux Publics de l'État (ITPE) is the elite competitive civil service entrance competition recruiting Category A state engineers into the French civil service under the Ministry of Ecological Transition. Formed and graduated at the École Nationale des Travaux Publics de l'État (ENTPE) in Vaulx-en-Velin, ITPE engineers lead major state infrastructure projects, transport networks, environmental protection, urban planning, territorial development, and climate risk prevention. The internal competition provides career progression for public servants across the three civil service branches (FPE, FPT, FPH) and military personnel with at least three years of service. It features rigorous scientific examinations in mathematics and physics based on CPGE grand-école preparatory curricula, alongside public policy analysis through a note problématique dossier. This question bank provides an English-language study adaptation focused on foundational PCSI/MPSI quantitative methods, ecological transition policy, and public engineering governance.

Exam sponsor: Ministère de la Transition écologique, de l'Énergie, du Climat et de la Prévention des risques. The requirements and fees below concern the certification or admission exam, separate from our free practice resources.

Assessment

Written admissibility comprises three written papers: Mathématiques (4 h, coef 4, aligned with the first-year PCSI curriculum), Physique (4 h, coef 4, aligned with the first-year MPSI curriculum), and a Note problématique based on a documentary dossier on a general subject (4 h, coef 2). Admission comprises mathematics and physics oral examinations (each 30 minutes after 30 minutes of preparation) and a general-culture examination from a text (30 minutes after 15 minutes of preparation).

Time Limit

Written admissibility: 12 h total; admission: mathematics oral (30 min preparation + 30 min examination), physics oral (30 min + 30 min), and general-culture examination (15 min + 30 min)

Passing Score

The jury sets an admissibility total of at least 90 points and an admission total of at least 180 points; under the governing order, a mark below 5/20 on a test may lead to elimination

Exam / Certification Fees

No application fee is stated on the ministry's competition page

Exam sponsor website

Fees, eligibility, and exam policies can change. Confirm them with the exam sponsor before applying or paying.

Official sources

Our practice resources: topics covered

We aim to reflect publicly available exam outlines and topic information in our study resources. Coverage, format, and difficulty may differ from the actual exam, and we cannot guarantee that every detail is accurate or current. Confirm exam requirements, fees, and policies with the official exam sponsor.

35%

PCSI Mathematics & Applied Analysis

Linear algebra, matrices, vector spaces, eigenvalues, Taylor expansions, differential equations, definite and improper integrals, series, and probability

35%

MPSI General & Applied Physics

Point and solid mechanics, harmonic oscillators, thermodynamics, heat transfer, electrostatics, electromagnetism, Maxwell equations, wave optics, and fluid statics

20%

Ecological Transition, Public Policy & Infrastructure

Decarbonation, multimodal transport networks, civil engineering structures, water resources, biodiversity preservation, natural and technological risks, and urbanism

10%

State Civil Service Statut & Engineering Project Management

Code général de la fonction publique (CGFP), public procurement basics (Code de la commande publique), engineering ethics, and project management

Preparing for the Ingénieur ITPE Exam

What You Need to Know

Passing score: The jury sets an admissibility total of at least 90 points and an admission total of at least 180 points; under the governing order, a mark below 5/20 on a test may lead to elimination
Assessment: Written admissibility comprises three written papers: Mathématiques (4 h, coef 4, aligned with the first-year PCSI curriculum), Physique (4 h, coef 4, aligned with the first-year MPSI curriculum), and a Note problématique based on a documentary dossier on a general subject (4 h, coef 2). Admission comprises mathematics and physics oral examinations (each 30 minutes after 30 minutes of preparation) and a general-culture examination from a text (30 minutes after 15 minutes of preparation).
Time limit: Written admissibility: 12 h total; admission: mathematics oral (30 min preparation + 30 min examination), physics oral (30 min + 30 min), and general-culture examination (15 min + 30 min)
Exam / certification fees: No application fee is stated on the ministry's competition page Official sources

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Ingénieur ITPE: Suggested Study Strategy

1Thoroughly review first-year CPGE mathematics: prioritize matrix diagonalization, Taylor-Lagrange theorems, ordinary differential equations, and convergence tests for improper integrals

2Work through core CPGE physics derivations: harmonic and damped oscillators, Carnot cycles, entropy balances, Gauss's law, and Faraday's law of induction

3Familiarize yourself with French national ecological transition policies: Stratégie Nationale Bas-Carbone (SNBC), Plan National d'Adaptation au Changement Climatique (PNACC), and the ZAN (Zéro Artificialisation Nette) objective

4Review the fundamentals of the Code général de la fonction publique (CGFP): neutrality, secularism, hierarchical obedience, and civil servant responsibility in public engineering

5Practice multi-step quantitative calculation problems under timed conditions without relying on graphical calculators

Frequently Asked Questions

What is the Concours d'Ingénieur des Travaux Publics de l'État (ITPE)?

It is a competitive entrance examination organized by the French Ministry of Ecological Transition to recruit Category A civil engineers into the corps of ITPE, who serve in government ministries, regional directorates (DREAL, DDTM, DIR), and public research bodies.

Who is eligible for the ITPE concours interne?

The internal competition is open to civil servants and public employees of the three branches of the French civil service (FPE, FPT, FPH) and military personnel with at least 3 years of effective public service as of January 1 of the competition year.

What scientific syllabus is tested on the internal competition?

The mathematics paper is strictly based on the first-year PCSI (Physique, Chimie, Sciences de l'Ingénieur) syllabus, and the physics paper is strictly based on the first-year MPSI (Mathématiques, Physique, Sciences de l'Ingénieur) syllabus, as defined by the Arrêté du 4 avril 2013.

What training do successful candidates receive?

Laureates of the concours interne are appointed as civil servant trainee engineers (élèves ingénieurs) and complete a multi-year engineering curriculum at ENTPE in Vaulx-en-Velin (Lyon metropolitan area), graduating with an accredited state engineering diploma.

What are the passing and elimination criteria?

The jury sets an admissibility threshold of at least 90 points across the written papers and an admission threshold of at least 180 total points. Under the governing order, a score below 5/20 on an individual test may lead to elimination; it is not worded as automatic elimination in every case.