Mathematics - Analysis, Complex Numbers and Probability in the ENP Alger Access Contest
Mathematics at the ENP Alger access contest is examined through structured problems - typically one on analysis (sequences, limits, function study, integration), one on complex numbers or arithmetic, and one on probability - drawn from the final-year scientific baccalaureate programme. Marks reward justified reasoning more than the final number, so every limit, derivative sign and probability value needs a theorem or a computation behind it.
How the problems are built
- Analysis problems cascade: part 1 establishes a property of a function or sequence that part 3 needs.
- A standard function study asks for domain, limits at the boundaries, derivative sign, table of variation, asymptotes, then an area obtained by integration.
- Probability questions hang on a concrete experiment (sampling without replacement, defective items, screening tests) and mix a tree diagram with conditional probability.
- Complex numbers usually appear as geometry: modulus, argument, roots of z^n = 1, and transformations.
Results worth rehearsing
- Monotone convergence: increasing and bounded above implies convergent. Bounded alone proves nothing.
- For u_{n+1} = f(u_n): a fixed point, an interval I with f(I) ⊂ I, and |f′| ≤ k < 1 on I give convergence and uniqueness of the limit.
- Standard equivalents at 0: sin x ~ x, tan x ~ x, ln(1+x) ~ x, e^x − 1 ~ x, 1 − cos x ~ x²/2.
- Integration: by parts, by substitution (change the bounds), and comparison (0 ≤ f ≤ g gives ∫f ≤ ∫g).
Worked example
Find lim_{n→∞} Σ_{k=1}^{n} 1/(n+k). Factor 1/n: the sum equals (1/n) Σ_{k=1}^{n} 1/(1 + k/n). Each term is (1/n)·f(k/n) with f(x) = 1/(1+x), continuous on [0,1]. The sum is therefore a Riemann sum for f on [0,1] and converges to ∫₀¹ dx/(1+x) = ln 2 ≈ 0.693. Two separate steps earn the marks: recognising the Riemann sum, then justifying convergence by continuity of f on a closed interval.
Traps that cost marks
- Applying monotone convergence to a sequence that is only bounded.
- Confusing continuity, differentiability and monotonicity: differentiable implies continuous, never the converse.
- Forgetting to convert the bounds after a substitution.
- Writing a limit inside a computation as if it were an ordinary equality.
- Rounding intermediate values inside an induction or a recurrence.
- In probability, confusing P(A|B) with P(B|A), and mishandling "at least one" instead of using the complement.