This rigorous mathematics exam is aligned with the format and rigor of AP Calculus, IB Mathematics (Analysis & Approaches HL), and other advanced pre-university programs. Unlike MCQ-based exams, this resource focuses entirely on proof-based, multi-step open-ended problems — developing the deep reasoning and written mathematical communication skills required in top university entrance exams.
📌 Topics Covered

• Analytic Geometry & Integration (Ex.1 — 6 pts) — implicit curve x³−2x²+xy²+2y²=0, symmetry, domain of abscissas, volume of revolution about the x-axis
• Complex Numbers & Transformations (Ex.2 — 10 pts) — triangle ABC, similitude S, rotation R, compositions R∘S and S∘R, equilateral triangles, circumscribed circle, locus of a point
• Complex Numbers & Conics (Ex.3 — 8 pts) — mappings z'=z−2 and z"=z²−z, collinearity condition, rectangular hyperbola x²−y²−2x=0, polar coordinates, parametric point on hyperbola
• Probability & Sequences (Ex.4 — 7 pts) — urn with blue and red dice, total probability, Bayes' theorem, sequence pₙ of conditional probabilities, minimum n such that pₙ ≥ 0.999
• Calculus, Integration & Sequences (Ex.5 — 7 pts) — f(x)=(1+2ln x)/x², auxiliary function g, IVT proof, sequence Iₙ=∫f(x)dx, squeeze theorem, limit, antiderivative by parts, sum Sₙ and its limit
• Calculus & Functions (Ex.6 — 12 pts) — f(x)=e²ˣ(eˣ−2)², full variation table, curve (C), antiderivative F(x), sign of F, relative position of (C) and (γ), horizontal asymptote y=−4/3, bounded area S(m), limit and graphical construction

✅ What's Included

• Full exam with 6 detailed exercises
• Complete step-by-step solutions for every sub-question

💡 Perfect For AP / IB exam prep · Proof-based problem practice · University entrance preparation · In-depth topic mastery.