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Sample Concours EMI Practice Questions

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

1What is the limit of the real sequence defined by u_n = ( (n + 3) / n )^(2n) as n tends to infinity?
A.e^6
B.e^3
C.e^2
D.e^9
Explanation: Rewriting the general term gives u_n = (1 + 3/n)^(2n) = exp(2n * ln(1 + 3/n)). Using the asymptotic expansion ln(1 + 3/n) = 3/n + O(1/n^2), the argument becomes 2n * (3/n + O(1/n^2)) = 6 + O(1/n). Taking the limit as n tends to infinity yields exp(6) = e^6.
2What is the sum of the convergent numerical series sum_{n=1}^infty 1 / (n * (n + 1) * (n + 2))?
A.1/2
B.1/4
C.1/6
D.1/12
Explanation: Decomposing the general term into partial fractions yields 1/(n(n+1)(n+2)) = (1/2) * [1/(n(n+1)) - 1/((n+1)(n+2))]. Summing this telescoping series from n=1 to N gives S_N = (1/2) * [1/(1*2) - 1/((N+1)(N+2))]. Taking the limit as N tends to infinity, the second term vanishes and S = (1/2) * (1/2) = 1/4.
3For which real values of the parameter p does the alternating series sum_{n=1}^infty ( (-1)^(n-1) ) / n^p converge conditionally (converges, but does not converge absolutely)?
A.p > 1
B.p <= 0
C.0 < p <= 1
D.p >= 2
Explanation: By Leibniz's alternating series test, sum (-1)^(n-1)/n^p converges whenever the sequence 1/n^p is positive, strictly decreasing, and tends to zero, which holds if and only if p > 0. The series of absolute values is the Riemann harmonic series sum 1/n^p, which converges if and only if p > 1. Therefore, convergence is conditional precisely on the interval 0 < p <= 1.
4What is the radius of convergence R of the complex power series sum_{n=1}^infty ( (2n)! / (n!)^2 ) * z^n?
A.R = 4
B.R = 1/2
C.R = 1
D.R = 1/4
Explanation: Let a_n = (2n)! / (n!)^2. Using d'Alembert's ratio test, we compute the limit ratio: a_{n+1}/a_n = [ (2n+2)! / ((n+1)!)^2 ] / [ (2n)! / (n!)^2 ] = (2n+2)(2n+1) / (n+1)^2 = (4n^2 + 6n + 2) / (n^2 + 2n + 1). As n tends to infinity, this ratio approaches 4. By the Cauchy-Hadamard theorem, the radius of convergence is R = 1 / lim (a_{n+1}/a_n) = 1/4.
5Consider the real function f(x) = ln(1 - x + x^2) defined on (-1, 1). What is the coefficient of x^6 in its Maclaurin power series expansion?
A.-1/2
B.-1/3
C.1/6
D.2/3
Explanation: Notice the identity (1 + x)(1 - x + x^2) = 1 + x^3. Thus, for |x| < 1, ln(1 - x + x^2) = ln(1 + x^3) - ln(1 + x). Expanding both terms via the standard Maclaurin series ln(1 + u) = sum_{k=1}^infty (-1)^(k-1) u^k / k, we have ln(1 + x^3) = x^3 - (x^6)/2 + (x^9)/3 - ... and ln(1 + x) = x - x^2/2 + x^3/3 - x^4/4 + x^5/5 - x^6/6 + ... The coefficient of x^6 in ln(1 + x^3) is -1/2, and in ln(1 + x) it is -1/6. Subtracting them yields (-1/2) - (-1/6) = -1/2 + 1/6 = -2/6 = -1/3.
6What is the exact value of the improper Dirichlet integral I = int_0^infty (sin(x) / x) dx?
A.pi / 2
B.pi
C.1
D.sqrt(pi) / 2
Explanation: The Dirichlet integral can be rigorously evaluated by introducing the parametric integral F(a) = int_0^infty e^(-ax) (sin(x) / x) dx for a >= 0. Differentiating under the integral sign gives F'(a) = - int_0^infty e^(-ax) sin(x) dx = - 1 / (1 + a^2). Integrating with respect to a yields F(a) = - arctan(a) + C. Since F(a) -> 0 as a -> infty, C = pi / 2. Evaluating at a = 0 gives I = F(0) = pi / 2.
7Determine the limit of the multivariable function f(x, y) = (x^3 * y) / (x^4 + y^2) as (x, y) tends to (0, 0).
A.The limit does not exist because directional limits depend on the slope.
B.1/2
C.0
D.1
Explanation: Using the arithmetic-geometric inequality, 2 * x^2 * |y| <= x^4 + y^2, which implies |y| / (x^4 + y^2) <= 1 / (2 x^2) for (x, y) != (0, 0). Multiplying both sides by |x^3| yields |f(x, y)| = |x^3 * y| / (x^4 + y^2) <= |x^3| / (2 x^2) = |x| / 2. As (x, y) approaches (0, 0), |x| / 2 approaches 0. By the squeeze theorem, the limit exists and equals 0.
8For the scalar field f(x, y) = x^2 * y + exp(x * y), what is the directional derivative D_v f(1, 0) in the direction of the unit vector v = (3/5, 4/5)?
A.3/5
B.1
C.2
D.8/5
Explanation: First, compute the gradient of f: grad f = (df/dx, df/dy) = (2xy + y*exp(xy), x^2 + x*exp(xy)). Evaluating at the point (1, 0) gives df/dx(1, 0) = 0 + 0 = 0 and df/dy(1, 0) = 1^2 + 1*exp(0) = 1 + 1 = 2. Thus, grad f(1, 0) = (0, 2). The directional derivative along the unit vector v is D_v f(1, 0) = grad f(1, 0) . v = 0 * (3/5) + 2 * (4/5) = 8/5.
9Calculate the double integral iint_D (x^2 + y^2) dx dy, where D is the upper half-disk defined by x^2 + y^2 <= R^2 and y >= 0.
A.(pi * R^4) / 4
B.(pi * R^4) / 2
C.(pi * R^3) / 3
D.(pi * R^4) / 8
Explanation: Switching to polar coordinates (x = r cos theta, y = r sin theta), the domain D is described by 0 <= r <= R and 0 <= theta <= pi, with the area element dx dy = r dr dtheta. The integrand becomes x^2 + y^2 = r^2. The integral transforms into int_0^pi dtheta * int_0^R r^3 dr = [theta]_0^pi * [r^4 / 4]_0^R = pi * (R^4 / 4) = (pi * R^4) / 4.
10Using Green's theorem (théorème de Green-Riemann), evaluate the line integral oint_C (y^3 dx - x^3 dy) counterclockwise along the circle C: x^2 + y^2 = a^2.
A.0
B.-(3/2) * pi * a^4
C.(3/2) * pi * a^4
D.-3 * pi * a^2
Explanation: By Green's theorem, oint_C (P dx + Q dy) = iint_D (dQ/dx - dP/dy) dx dy, where D is the disk x^2 + y^2 <= a^2. Here P = y^3 and Q = -x^3. Computing the partial derivatives gives dP/dy = 3y^2 and dQ/dx = -3x^2. The integrand is dQ/dx - dP/dy = -3x^2 - 3y^2 = -3(x^2 + y^2). In polar coordinates, this becomes -3 * int_0^{2pi} dtheta * int_0^a r^3 dr = -3 * (2pi) * (a^4 / 4) = -(3/2) * pi * a^4.

About the Concours EMI Exam

The official 2026–2027 notice schedules the written competition for 6 July 2026. DEUG/DEUST applicants must come from Mathematics, Mathematics-Computer Science, or Physics, have a 2024 Baccalauréat and a 2025–2026 diploma, be under 22, and meet the published semester-grade conditions. Licence applicants must hold a fundamental or FST Licence in Mathematics or Mathematics-Computer Science, have a 2023 Baccalauréat, be under 23, and meet the corresponding grade conditions. This is independent English-language MCQ study practice, not an official translation or simulation of the constructed-response papers.

Exam sponsor: École Mohammadia d'Ingénieurs (EMI), Université Mohammed V de Rabat. The requirements and fees below concern the certification or admission exam, separate from our free practice resources.

Assessment

Dossier preselection followed by Mathematics and Physics written papers (coefficient 1 each).

Time Limit

6 hours total

Passing Score

Competitive ranking; no fixed pass mark stated.

Exam / Certification Fees

Not stated in the reviewed official notice.

Exam sponsor website

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

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.

50 questions in this practice bank; official paper weight not published

Mathematics

Analysis and linear algebra.

50 questions in this practice bank; official paper weight not published

Physics

Mechanics, thermodynamics, electromagnetism, and waves.

Preparing for the Concours EMI Exam

What You Need to Know

  • Passing score: Competitive ranking; no fixed pass mark stated.
  • Assessment: Dossier preselection followed by Mathematics and Physics written papers (coefficient 1 each).
  • Time limit: 6 hours total
  • Exam / certification fees: Not stated in the reviewed official notice. Official sources

Using Our Practice Resources

  • Work through all 100 available questions
  • Review every answer and explanation
  • Track weak areas and revisit them
  • Use our AI tutor for tough concepts

Concours EMI: Suggested Study Strategy

1Practice complete three-hour written-paper solutions.
2Review analysis and linear algebra.
3Review mechanics, thermodynamics, electromagnetism, and waves.

Frequently Asked Questions

Is there an oral stage in the 2026 notice?

No oral stage is described; the notice specifies dossier preselection and two written papers.

Is this the CPGE CNC route?

No. This bank covers EMI's distinct parallel DEUG/DEUST/Licence competition.