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Concours ECRICOME Prépas

Practice Questions100 questions

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Key Facts: Concours BCE Exam

Grandes Écoles de Management (+ 3 associated)

BCE

~9,500

Annual Registered Candidates

BCE Session 2026

22–29 April 2026

Written Examination Dates

BCE Session 2026

50 € – 200 €

Fee per Selected School (0 € boursiers)

BCE Session 2026

3 to 4 hrs

Duration of Individual Written Papers

BCE

SIGEM

Centralized National Allocation System

SIGEM

The Concours BCE is the premier post-CPGE competition granting admission to 18 member French business schools plus associated schools (HEC Paris, ESSEC, ESCP, EDHEC, emlyon, etc.), with about 5,280 places at the 2026 session. Official tests consist of long-form written dissertations and mathematics problems lasting 3-4 hours each in French. This 100-question bank is an English-language MCQ study adaptation.

Sample Concours BCE Practice Questions

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

1Consider the matrix A in M_2(R) given by A = [[3, 1], [2, 2]]. What are the eigenvalues of A?

A.1 and 4

B.2 and 3

C.-1 and 6

D.0 and 5

Explanation: The characteristic polynomial of A is P(lambda) = det(A - lambda I) = (3 - lambda)(2 - lambda) - (1)(2) = lambda^2 - 5*lambda + 4. Factoring yields (lambda - 1)(lambda - 4) = 0, giving eigenvalues lambda_1 = 1 and lambda_2 = 4. Notice also that the trace is Tr(A) = 3 + 2 = 5 = 1 + 4, and the determinant is det(A) = 6 - 2 = 4 = 1 * 4.

2Let A = [[1, 2], [0, 3]] in M_2(R). What is the entry in the first row and second column of A^4?

A.64

B.80

C.81

D.240

Explanation: Matrix A is upper-triangular with eigenvalues 1 and 3. By induction or diagonalization A = P D P^(-1) with eigenvectors (1, 0)^T for lambda = 1 and (1, 1)^T for lambda = 3, the general power is A^n = [[1, 3^n - 1], [0, 3^n]]. For n = 4, the top-right entry is 3^4 - 1 = 81 - 1 = 80.

3In R^3 equipped with the canonical Euclidean inner product, let F be the plane defined by x + 2y - z = 0. What is the orthogonal projection of the vector v = (2, 4, 2) onto F?

A.(4/3, 8/3, -4/3)

B.(1, 2, -1)

C.(2/3, 4/3, 10/3)

D.(0, 2, 4)

Explanation: A normal vector to F is n = (1, 2, -1), with squared norm ||n||^2 = 1^2 + 2^2 + (-1)^2 = 6. The projection of v onto the orthogonal complement F^perp is p_{F^perp}(v) = ((v . n) / ||n||^2) * n. Computing the dot product: v . n = 2(1) + 4(2) + 2(-1) = 8. Thus p_{F^perp}(v) = (8/6) n = (4/3) (1, 2, -1) = (4/3, 8/3, -4/3). The orthogonal projection onto F is p_F(v) = v - p_{F^perp}(v) = (2 - 4/3, 4 - 8/3, 2 - (-4/3)) = (2/3, 4/3, 10/3). Verification: 2/3 + 2(4/3) - 10/3 = 10/3 - 10/3 = 0.

4Find the minimum value of the objective function f(x, y) = x^2 + 2y^2 subject to the linear constraint x + 2y = 9.

A.18

B.24

C.36

D.27

Explanation: Using Lagrange multipliers, L(x, y, lambda) = x^2 + 2y^2 - lambda(x + 2y - 9). Setting partial derivatives to zero yields 2x - lambda = 0 (hence x = lambda/2) and 4y - 2*lambda = 0 (hence y = lambda/2). Therefore, x = y. Substituting into the constraint gives x + 2x = 9, so 3x = 9, which implies x = 3 and y = 3. The minimum value is f(3, 3) = 3^2 + 2*(3^2) = 9 + 18 = 27.

5Let X be a discrete random variable following a Poisson distribution with parameter lambda = 3. What is the factorial moment E[X(X - 1)]?

A.9

B.6

C.12

D.3

Explanation: For any Poisson random variable with parameter lambda, the k-th factorial moment is E[X(X - 1)...(X - k + 1)] = lambda^k. Here for k = 2, E[X(X - 1)] = lambda^2 = 3^2 = 9. Alternatively, E[X(X - 1)] = E[X^2] - E[X] = (Var(X) + (E[X])^2) - E[X] = (3 + 9) - 3 = 9.

6Compute the value of the improper integral I = int_0^infty x * e^(-2x) dx.

A.1/2

B.1/4

C.1/8

D.1

Explanation: Using integration by parts with u = x (so u' = 1) and v' = e^(-2x) (so v = -1/2 * e^(-2x)), we get I = [-1/2 * x * e^(-2x)]_0^infty + 1/2 * int_0^infty e^(-2x) dx. The boundary term vanishes at both limits by standard growth comparisons. The remaining integral is 1/2 * [-1/2 * e^(-2x)]_0^infty = 1/2 * (0 - (-1/2)) = 1/4. Alternatively, using the Gamma function, I = Gamma(2) / 2^2 = 1! / 4 = 1/4.

7Let f: R^4 -> R^3 be the linear mapping whose matrix in canonical bases is M = [[1, 2, 0, 1], [2, 4, 1, 4], [1, 2, 1, 3]]. What is the dimension of ker(f)?

A.0

B.1

C.2

D.3

Explanation: We determine the rank of M by Gaussian elimination on the rows. Row 2 minus 2*Row 1 gives [0, 0, 1, 2]. Row 3 minus Row 1 gives [0, 0, 1, 2]. Subtracting the modified Row 2 from the modified Row 3 yields a row of zeros. Thus there are exactly 2 non-zero pivot rows, so rank(f) = rg(M) = 2. By the Rank-Nullity Theorem (théorème du rang), dim(ker f) = dim(R^4) - rg(f) = 4 - 2 = 2.

8Let T be a continuous random variable representing the lifetime of a server (in years), modeled by an exponential distribution Exp(lambda = 0.5). What is the conditional probability P(T > 5 | T > 2)?

A.e^(-2.5)

B.e^(-1.0)

C.1 - e^(-1.5)

D.e^(-1.5)

Explanation: The exponential distribution possesses the memoryless property (loi sans mémoire): for all s, t >= 0, P(T > s + t | T > s) = P(T > t). Here s = 2 and s + t = 5, which means t = 3. Therefore, P(T > 5 | T > 2) = P(T > 3) = e^(-lambda * 3) = e^(-0.5 * 3) = e^(-1.5).

9Consider the function f: R^2 -> R defined by f(x, y) = x^3 + y^3 - 3xy. What is the nature of the critical point (1, 1)?

A.Strict local minimum with value -1

B.Strict local maximum with value 1

C.Saddle point (point col)

D.Local minimum with value 0

Explanation: The gradient is grad f(x, y) = (3x^2 - 3y, 3y^2 - 3x). At (1, 1), grad f(1, 1) = (0, 0). The Hessian matrix is H(x, y) = [[6x, -3], [-3, 6y]], so H(1, 1) = [[6, -3], [-3, 6]]. Its determinant is det(H(1, 1)) = 36 - 9 = 27 > 0, and the top-left entry is 6 > 0, confirming that H(1, 1) is positive definite. Hence (1, 1) is a strict local minimum. The function value is f(1, 1) = 1^3 + 1^3 - 3(1)(1) = -1.

10Let X be a random variable with mean mu = 50 and variance sigma^2 = 25. According to the Bienaymé-Chebyshev inequality, what is the sharpest upper bound on P(|X - 50| >= 15)?

A.1/3

B.1/9

C.5/15

D.1/25

Explanation: The Bienaymé-Chebyshev inequality states that for any epsilon > 0, P(|X - mu| >= epsilon) <= Var(X) / epsilon^2. Here Var(X) = 25 and epsilon = 15, so P(|X - 50| >= 15) <= 25 / 15^2 = 25 / 225 = 1/9. In terms of standard deviations, epsilon = 3*sigma (since sigma = 5), yielding the bound 1 / 3^2 = 1/9.

About the Concours BCE Exam

The Concours BCE (Banque Commune d'Épreuves) is the premier competitive entrance examination enabling students from Classes Préparatoires aux Grandes Écoles (CPGE) in the commercial tracks (ECG - filières Mathématiques Approfondies / Appliquées and HGG / ESH, ECT) and literary tracks (B/L and BEL) to enter 18 top French management Grandes Écoles, including HEC Paris, ESSEC Business School, ESCP Business School, EDHEC Business School, and emlyon business school, along with 3 partner institutions (including ESM Saint-Cyr). Administered by the Direction des Admissions et Concours (DAC) of the CCI Paris Île-de-France, the competition encompasses multi-hour long-form written examinations in April, followed by oral admissions interviews and language vivas in June and July, culminating in centralized matching via SIGEM. This question bank provides an English-language multiple-choice study adaptation targeting the shared core CPGE curriculum to reinforce core mathematical derivations, economic theories, sociological frameworks, geopolitical dynamics, and philosophical concepts.

Exam sponsor: Direction des Admissions et Concours (DAC) de la CCI Paris Île-de-France for the Banque Commune d'Épreuves. The requirements and fees below concern the certification or admission exam, separate from our free practice resources.

Assessment

The BCE written examination (épreuves écrites) takes place over 6 to 8 weekdays in late April (22–29 April 2026). Candidates sit 3- to 4-hour papers in Mathematics (Approfondies or Appliquées), Economics-Sociology-History (ESH) or Geopolitics (HGG), General Culture and Philosophy (Culture Générale), Text Synthesis (Synthèse de textes), and Modern Languages (LVA and LVB). Candidates declared admissible (9–12 June 2026) sit on-campus oral exams (oraux: entretien de personnalité and language vivas, plus oral mathematics for HEC Paris) from 15 June to 6 July 2026, concluding with preference ranking via SIGEM from 30 June to 17 July 2026.

Time Limit

Over 30 hours of written testing spread across 6 to 8 examination days (typically 3 to 4 hours per individual paper)

Passing Score

Strictly competitive ranking based on school-specific weighting coefficients; about 9,500 candidates competed for roughly 5,280 places at the 2026 session, spread across the 18 member schools and their associated schools.

Exam / Certification Fees

Variable fee structure charged per school: roughly 50 € to 200 € per chosen school (completely free for state scholarship holders / boursiers, with the exception of ESM Saint-Cyr); candidates typically pay between 500 € and 1,800 € depending on the number of schools selected.

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.

30%

Applied and Advanced Mathematics (Mathématiques appliquées et approfondies)

Linear algebra (matrices, determinants, eigenspaces, reduction of endomorphisms), real analysis (multivariable calculus, Taylor series, Lagrange multipliers), and probability theory (continuous random variables, covariance matrices, central limit theorem).

25%

Economics, Sociology, and History (Économie, Sociologie et Histoire du monde contemporain)

Macroeconomic models (Keynesian multiplier, IS-LM, Solow and endogenous growth models), monetary policy transmission, central banking, social stratification and mobility (Bourdieu, Weber, Durkheim, Boudon), and modern economic history.

20%

Geopolitics and International Relations (Histoire, Géographie et Géopolitique du monde contemporain)

Dynamics of economic globalization, geoeconomic rivalry, US-China competition, European integration and Eurozone monetary rules, energy transition geopolitics, critical raw materials, and international institutions (WTO, IMF, World Bank).

15%

General Culture and Philosophy (Culture Générale et Philosophie)

Epistemology, ethical frameworks, political philosophy (Hobbes, Rousseau, Kant, Rawls), philosophy of technology and nature, and critical evaluation of philosophical arguments.

10%

Modern Languages and Text Synthesis (Langues Vivantes et Synthèse de Textes)

Economic English reading comprehension, precise lexical discrimination, business collocations, stylistic nuances, and multi-source argumentative synthesis.

Preparing for the Concours BCE Exam

What You Need to Know

Passing score: Strictly competitive ranking based on school-specific weighting coefficients; about 9,500 candidates competed for roughly 5,280 places at the 2026 session, spread across the 18 member schools and their associated schools.
Assessment: The BCE written examination (épreuves écrites) takes place over 6 to 8 weekdays in late April (22–29 April 2026). Candidates sit 3- to 4-hour papers in Mathematics (Approfondies or Appliquées), Economics-Sociology-History (ESH) or Geopolitics (HGG), General Culture and Philosophy (Culture Générale), Text Synthesis (Synthèse de textes), and Modern Languages (LVA and LVB). Candidates declared admissible (9–12 June 2026) sit on-campus oral exams (oraux: entretien de personnalité and language vivas, plus oral mathematics for HEC Paris) from 15 June to 6 July 2026, concluding with preference ranking via SIGEM from 30 June to 17 July 2026.
Time limit: Over 30 hours of written testing spread across 6 to 8 examination days (typically 3 to 4 hours per individual paper)
Exam / certification fees: Variable fee structure charged per school: roughly 50 € to 200 € per chosen school (completely free for state scholarship holders / boursiers, with the exception of ESM Saint-Cyr); candidates typically pay between 500 € and 1,800 € depending on the number of schools selected. Official sources

Using Our Practice Resources

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Concours BCE: Suggested Study Strategy

1Master linear algebra proofs and continuous probability calculations thoroughly without reliance on formula sheets, as BCE mathematics papers heavily weight complete algebraic deductions.

2In Economics and Sociology (ESH), memorize concrete dates, seminal authors, and statistical trends to corroborate theoretical arguments (e.g., Solow residuals, central bank balance sheet expansion metrics, Bourdieu's social reproduction statistics).

3For Geopolitics (HGG), build precise spatial and thematic case studies covering critical supply chains (semiconductors, rare earths), energy pipelines, and maritime choke points (Strait of Malacca, Bab-el-Mandeb, Suez Canal).

4In General Culture, practice rigorous conceptual dialectics: define terms precisely in their historical philosophical context (Descartes, Kant, Rousseau, Arendt) and avoid superficial opinion-based commentary.

5Sharpen business and economic English vocabulary: master false friends, economic collocations, and formal journalistic registers from publications like The Economist and Financial Times.

6Use this 100-question practice bank for high-yield diagnostic active recall to identify theoretical gaps before tackling full 4-hour past papers (annales).

Frequently Asked Questions

What is the Concours BCE and which schools participate?

The Concours BCE (Banque Commune d'Épreuves) is the shared competitive examination for admission into 18 leading French management Grandes Écoles (including HEC Paris, ESSEC, ESCP, EDHEC, emlyon, Audencia, Grenoble EM, SKEMA, NEOMA, KEDGE, TBS Education, Rennes SB, Montpellier BS, BSB, EM Strasbourg, ICN BS, Institut Mines-Télécom BS, and SCBS) plus 3 associated schools (such as ESM Saint-Cyr). Candidates are primarily students in CPGE ECG (classes préparatoires économiques et commerciales voie générale), CPGE ECT (voie technologique), or literary prépas (B/L and BEL).

What are the official dates for the 2026 Concours BCE session?

For the 2026 session, registration opens on 8 December 2025 and closes on 12 January 2026. The written examinations (épreuves écrites) take place from Wednesday 22 April to Wednesday 29 April 2026 (weekdays only, with 6 May held as a reserve day). Results of admissibility (admissibilité) are published between 9 and 12 June 2026. Oral examinations (oraux) occur from 15 June to 6 July 2026, followed by the national SIGEM allocation procedure between 30 June and 17 July 2026.

How are registration fees calculated for the BCE?

Unlike concours with a single flat fee, the BCE charges fees on a per-school basis, ranging from approximately 50 € to 200 € for each participating Grande École selected. Candidates with French state scholarship status (boursiers du gouvernement français) benefit from a complete fee waiver (0 €) across all schools, with the sole exception of ESM Saint-Cyr. For non-scholarship students, the total registration cost typically ranges between 500 € and 1,800 € depending on how many schools they apply to.

How does the official BCE exam format differ from this practice question bank?

The official BCE concours consists entirely of long-form, multi-hour written papers in French—such as 4-hour mathematical problem solving, 4-hour dissertations in economics (ESH) or geopolitics (HGG), 4-hour essays in culture générale, 4-hour text syntheses, and 3- to 4-hour modern language papers featuring translation (version and thème) and essay writing. There are no multiple-choice questions on the real exam. This 100-question bank is an English-language MCQ study adaptation engineered to help candidates rapidly review, test, and consolidate foundational concepts, quantitative formulas, historical dates, and economic mechanisms.

What is the difference between Mathématiques Approfondies and Mathématiques Appliquées in CPGE ECG?

In the CPGE ECG reform, students choose either Mathématiques Approfondies (oriented toward pure and theoretical mathematics, rigorous algebraic proofs, and advanced analysis) or Mathématiques Appliquées (oriented toward applied mathematics, probability modeling, linear algebra, and Python numerical algorithms). In the BCE, top schools set distinct papers (e.g., Maths Approfondies HEC/ESSEC vs. Maths Appliquées HEC/ESSEC) with varying coefficients, though both require rigorous calculation and conceptual mastery.

How does the SIGEM matching process work?

SIGEM (Système d'Intégration aux Grandes Écoles de Management) is the centralized preference-matching algorithm that allocates admitted students to French business schools. Following written and oral examinations, candidates submit a single, ranked preference list of schools where they have been declared admitted or placed on the waiting list. The algorithm resolves candidate rankings and school quotas without penalizing candidate preferences.