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Sample Concours National ENSA Practice Questions

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

1Evaluate the limit: L = lim_{x -> 0} (e^(2x) - 1 - 2x) / x^2.
A.0
B.1
C.2
D.4
Explanation: Using the Taylor expansion of e^(2x) around 0, e^(2x) = 1 + 2x + (2x)^2/2! + o(x^2) = 1 + 2x + 2x^2 + o(x^2). Subtracting 1 + 2x leaves 2x^2 + o(x^2), so dividing by x^2 yields lim_{x -> 0} (2x^2 / x^2) = 2. Alternatively, applying L'Hôpital's rule twice gives lim_{x -> 0} (2e^(2x) - 2)/(2x) = lim_{x -> 0} (4e^(2x))/2 = 2.
2What is the value of the limit L = lim_{x -> +infinity} x * (sqrt(x^2 + 4) - x)?
A.0
B.1
C.2
D.+infinity
Explanation: Multiplying by the conjugate expression gives sqrt(x^2 + 4) - x = 4 / (sqrt(x^2 + 4) + x). Multiplying by x produces 4x / (x(sqrt(1 + 4/x^2) + 1)) = 4 / (sqrt(1 + 4/x^2) + 1). As x -> +infinity, this approaches 4 / (1 + 1) = 2.
3Let f(x) = (ln(1 + 3x)) / sin(2x) for x != 0, and f(0) = c. For what value of the real constant c is f continuous at x = 0?
A.1/2
B.1
C.3/2
D.2/3
Explanation: For f to be continuous at 0, c must equal lim_{x -> 0} f(x). Using standard equivalences near 0, ln(1 + 3x) ~ 3x and sin(2x) ~ 2x, so lim_{x -> 0} (ln(1 + 3x) / sin(2x)) = lim_{x -> 0} (3x / 2x) = 3/2. Therefore, c must equal 3/2.
4Consider the equation x^3 + 3x - 5 = 0. According to the Intermediate Value Theorem (Théorème des Valeurs Intermédiaires), on which interval does this equation admit a unique real root?
A.[-2, -1]
B.[-1, 0]
C.[0, 1]
D.[1, 2]
Explanation: Let g(x) = x^3 + 3x - 5. The derivative is g'(x) = 3x^2 + 3 > 0 for all real x, so g is strictly increasing and continuous on R. Evaluating at integer points gives g(1) = 1 + 3 - 5 = -1 < 0 and g(2) = 8 + 6 - 5 = 9 > 0. Since g(1) * g(2) < 0, the Intermediate Value Theorem guarantees a unique root strictly within [1, 2].
5What is the equation of the tangent line to the curve y = f(x) = x * e^(-x) at the point with abscissa x = 0?
A.y = x
B.y = -x
C.y = x + 1
D.y = 2x
Explanation: The derivative of f(x) = x * e^(-x) is f'(x) = 1 * e^(-x) - x * e^(-x) = (1 - x)e^(-x). At x = 0, f(0) = 0 and f'(0) = (1 - 0)e^0 = 1. The equation of the tangent line is y - f(0) = f'(0)(x - 0), which simplifies directly to y = x.
6Find the inflection point of the curve defined by f(x) = x^3 - 6x^2 + 9x + 2.
A.(1, 6)
B.(2, 4)
C.(3, 2)
D.(0, 2)
Explanation: The first derivative is f'(x) = 3x^2 - 12x + 9, and the second derivative is f''(x) = 6x - 12. Setting f''(x) = 0 yields 6x - 12 = 0 <=> x = 2. Since f''(x) changes sign at x = 2 (from negative for x < 2 to positive for x > 2), an inflection point occurs at x = 2. Computing f(2) = 2^3 - 6(2^2) + 9(2) + 2 = 8 - 24 + 18 + 2 = 4, giving the point (2, 4).
7Let f(x) = x^3 + 2x - 1. The function f is strictly increasing on R. If g is the inverse function f^(-1), what is the value of g'(2)?
A.1/5
B.1/4
C.1/2
D.5
Explanation: By the inverse function derivative theorem, g'(2) = 1 / f'(g(2)). We first find g(2), which is the solution to f(x) = 2. Clearly, x = 1 gives f(1) = 1^3 + 2(1) - 1 = 2, so g(2) = 1. The derivative of f is f'(x) = 3x^2 + 2, so f'(1) = 3(1)^2 + 2 = 5. Therefore, g'(2) = 1 / 5.
8Evaluate the limit: L = lim_{x -> 0} (1 - cos(x)) / (x * sin(x)).
A.0
B.1/2
C.1
D.2
Explanation: Using standard asymptotic equivalents near 0, 1 - cos(x) ~ x^2 / 2 and sin(x) ~ x. Thus, the denominator x * sin(x) ~ x^2. The limit is lim_{x -> 0} (x^2 / 2) / x^2 = 1/2. Alternatively, L'Hôpital's rule gives lim_{x -> 0} sin(x) / (sin(x) + x cos(x)) = lim_{x -> 0} 1 / (1 + cos(x)) = 1 / (1 + 1) = 1/2.
9Find the oblique asymptote (asymptote oblique) of the curve y = f(x) = (2x^2 + 5x - 1) / (x + 1) as x -> +infinity.
A.y = 2x
B.y = 2x + 1
C.y = 2x + 3
D.y = 2x + 5
Explanation: Performing polynomial long division of 2x^2 + 5x - 1 by x + 1: 2x^2 + 5x - 1 = (x + 1)(2x + 3) - 4. Thus, f(x) = 2x + 3 - 4/(x + 1). As x -> +infinity, -4/(x + 1) -> 0, which proves that the straight line y = 2x + 3 is the oblique asymptote.
10What is the value of the limit L = lim_{x -> 0+} x^(sqrt(x))?
A.0
B.1/e
C.1
D.+infinity
Explanation: Rewrite the expression in exponential form: x^(sqrt(x)) = exp(sqrt(x) * ln(x)). Let u = sqrt(x); then x = u^2 and ln(x) = 2 ln(u), so sqrt(x) * ln(x) = 2u * ln(u). By standard growth hierarchies (croissances comparées), lim_{u -> 0+} u * ln(u) = 0. Therefore, lim_{x -> 0+} exp(sqrt(x) * ln(x)) = exp(0) = 1.

About the Concours National ENSA Exam

The 2026–2027 ministerial circular defines the eligible Baccalauréat routes and preselection process, and the candidate instructions specify a 40-question, three-hour national QCM. This independent English-language MCQ bank expands the concept pool beyond one official form; it is not an official translation and should not be read as reproducing the exact 20/20 item count in every practice session.

Exam sponsor: Réseau ENSA under the Moroccan Ministry of Higher Education. The requirements and fees below concern the certification or admission exam, separate from our free practice resources.

Assessment

Two equal-coefficient MCQ parts: Mathematics and Physics-Chemistry; +2/-1/0 scoring and no calculators.

Time Limit

3 hours

Passing Score

Competitive ranking; no fixed pass mark stated.

Exam / Certification Fees

Not stated in the reviewed official 2026 instructions.

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.

Official: 20 of 40 questions, coefficient equal to Physics-Chemistry

Mathematics

Algebra, analysis, probability, complex numbers, and geometry.

Official: 20 of 40 questions, coefficient equal to Mathematics

Physics-Chemistry

Mechanics, electricity, thermodynamics, waves, nuclear physics, and chemistry.

Preparing for the Concours National ENSA Exam

What You Need to Know

  • Passing score: Competitive ranking; no fixed pass mark stated.
  • Assessment: Two equal-coefficient MCQ parts: Mathematics and Physics-Chemistry; +2/-1/0 scoring and no calculators.
  • Time limit: 3 hours
  • Exam / certification fees: Not stated in the reviewed official 2026 instructions. 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 National ENSA: Suggested Study Strategy

1Practice manual calculation and estimation.
2Use negative-marking strategy carefully.
3Split timed review evenly between the two equal-coefficient parts.

Frequently Asked Questions

How is the official QCM scored?

+2 for a correct response, -1 for a wrong response, and 0 for an omitted response.

Are calculators allowed?

The reviewed official 2026 candidate instructions prohibit calculators.