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Sample Concours ISCAE ECT Practice Questions

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1Given the 2 × 2 matrix A = [[4, 3], [2, 5]], what is the determinant of matrix A?

A.26

B.8

C.14

D.11

Explanation: The determinant of a 2 × 2 matrix [[a, b], [c, d]] is computed as ad - bc. For matrix A, det(A) = (4 × 5) - (3 × 2) = 20 - 6 = 14.

2Under what necessary and sufficient condition is an n × n square matrix A invertible?

A.Its determinant is non-zero (det(A) ≠ 0)

B.Its trace is strictly positive (tr(A) > 0)

C.All of its entries along the main diagonal are non-zero

D.Its eigenvalues are all real and positive

Explanation: A square matrix A is invertible (non-singular) if and only if its determinant is non-zero (det(A) ≠ 0). This condition is mathematically equivalent to A having full rank (rank(A) = n) and having trivial kernel {0}.

3Calculate the determinant of the 3 × 3 matrix M = [[2, 1, 0], [1, 3, 2], [0, 1, 4]].

A.12

B.16

C.20

D.24

Explanation: Expanding along the first row: det(M) = 2 × det([[3, 2], [1, 4]]) - 1 × det([[1, 2], [0, 4]]) + 0 × det([[1, 3], [0, 1]]). Evaluating the 2 × 2 minors yields 2 × (12 - 2) - 1 × (4 - 0) = 2 × 10 - 4 = 20 - 4 = 16.

4Consider a 3 × 3 matrix A whose third row is exactly equal to 2 × (Row 1) + 3 × (Row 2). What is the maximum possible rank of matrix A?

A.3

B.0

C.1

D.2

Explanation: Because the third row is a linear combination of the first two rows, the three row vectors are linearly dependent, so det(A) = 0 and the rank cannot be 3. The maximum possible number of linearly independent rows is 2, assuming Row 1 and Row 2 are linearly independent.

5What are the eigenvalues of the matrix B = [[5, 2], [2, 2]]?

A.λ₁ = 5, λ₂ = 2

B.λ₁ = 6, λ₂ = 1

C.λ₁ = 4, λ₂ = 3

D.λ₁ = 7, λ₂ = 0

Explanation: The characteristic equation is det(B - λI) = (5 - λ)(2 - λ) - (2 × 2) = λ² - 7λ + 10 - 4 = λ² - 7λ + 6 = 0. Factoring gives (λ - 6)(λ - 1) = 0, so the eigenvalues are λ₁ = 6 and λ₂ = 1.

6Which of the following statements is a fundamental property of real symmetric matrices in linear algebra?

A.All their eigenvalues are real, and they can be orthogonally diagonalized

B.Their determinant is always strictly positive

C.Their inverse is always equal to their transpose

D.They can never have repeated eigenvalues

Explanation: By the Spectral Theorem, every real symmetric matrix has exclusively real eigenvalues and admits an orthonormal basis of eigenvectors, meaning it can be diagonalized as P D P^T with an orthogonal matrix P.

7Using Cramer's rule, what is the value of x in the system of linear equations: 2x + 3y = 13 and x - y = -1?

A.-2

B.3

C.5

D.2

Explanation: The determinant of the system matrix is D = (2 × -1) - (3 × 1) = -2 - 3 = -5. For x, replace the first column with the constants: D_x = (13 × -1) - (3 × -1) = -13 + 3 = -10. Therefore, x = D_x / D = -10 / -5 = 2 (and y = 3).

8For the multivariable function f(x, y) = 3x²y + 4y³ - 5x, what is the first-order partial derivative with respect to x, ∂f/∂x?

A.6xy + 12y² - 5

B.3x² + 12y²

C.6xy - 5

D.6x - 5

Explanation: To compute ∂f/∂x, treat y as a constant: d/dx[3x²y] = 6xy, d/dx[4y³] = 0, and d/dx[-5x] = -5. Summing these terms gives ∂f/∂x = 6xy - 5.

9According to Schwarz's Theorem (Clairaut's Theorem) on multivariable functions, under what condition are the mixed second-order partial derivatives ∂²f/∂x∂y and ∂²f/∂y∂x equal?

A.The second-order partial derivatives are continuous on an open domain

B.The function f(x, y) is strictly convex everywhere

C.The first-order partial derivatives are both equal to zero

D.The function f(x, y) is homogeneous of degree one

Explanation: Schwarz's Theorem states that if the mixed second partial derivatives of f are defined and continuous on an open set containing a point, then ∂²f/∂x∂y = ∂²f/∂y∂x at that point. Continuity of second partial derivatives guarantees symmetry of the Hessian matrix.

10Let (x₀, y₀) be a critical point of a twice continuously differentiable function f(x, y). If the Hessian determinant det(H) = f_xx · f_yy - (f_xy)² > 0 and f_xx > 0, what is the nature of the critical point?

A.Saddle point

B.Local minimum

C.Local maximum

D.Inconclusive test

Explanation: When det(H) > 0 and f_xx > 0, the Hessian matrix is positive definite, confirming that the function curves upward in all directions and (x₀, y₀) is a strict local minimum.

About the Concours ISCAE ECT Exam

Groupe ISCAE publishes distinct competitions rather than one combined examination. This bank is specifically canonicalized to the 2026 Grande École CPGE ECT route and must not be read as covering the Cycle d'Expertise Comptable or university-degree admission routes. Official preselected ECT candidates sat Gestion II, Culture Française, English Culture, Mathématiques I, Droit, and Économie papers before the oral stage. This independent MCQ bank preserves French and English target-language prompts where they are the assessed skill, but it is not an official translation, paper simulation, or substitute for written analysis and oral preparation.

Exam sponsor: Groupe ISCAE. The requirements and fees below concern the certification or admission exam, separate from our free practice resources.

Assessment

Preselection, six written papers, and a mandatory oral stage.

Time Limit

11 hours of written papers across two days, plus the oral stage

Passing Score

Competitive ranking; no fixed pass mark published.

Exam / Certification Fees

Not published in the reviewed official materials.

Exam sponsor website

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

Official sources

Groupe ISCAE: official exam information

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.

Official: 3h; practice bank: 24 questions

Gestion II

Financial and management accounting, financial mathematics, and management.

Official: 2h; practice bank: 26 questions

Mathématiques I

Analysis, algebra, probability, optimization, and statistics.

Official: 2h; practice bank: 20 questions

Économie

Microeconomics, macroeconomics, policy, trade, and development.

Official: 2h; practice bank: 10 questions

Culture Française

French comprehension, grammar, synthesis, and argument.

Official: 1h; practice bank: 10 questions

English Culture

English business communication and interpretation.

Official: 1h; practice bank: 10 questions

Droit

Legal reasoning and business-law foundations.

Preparing for the Concours ISCAE ECT Exam

What You Need to Know

Passing score: Competitive ranking; no fixed pass mark published.
Assessment: Preselection, six written papers, and a mandatory oral stage.
Time limit: 11 hours of written papers across two days, plus the oral stage
Exam / certification fees: Not published in the reviewed official materials. Official sources

Using Our Practice Resources

Work through all 100 available questions
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Concours ISCAE ECT: Suggested Study Strategy

1Allocate preparation across all six official written papers.

2Practice French synthesis and English tasks in their target languages.

3Prepare oral motivation and analysis separately from MCQ study.

Frequently Asked Questions

Does this bank cover the Cycle d'Expertise Comptable?

No. It is scoped only to the distinct Grande École CPGE ECT competition.

Is there an oral stage?

Yes. Groupe ISCAE describes preselection, written testing, and oral testing for the current competition.

Free Practice Questions for Concours ISCAE ECT

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