Pancyprian Advanced Math study guide
Pancyprian Advanced Math · Work through your exam, one topic at a time.
The Pancyprian Advanced Mathematics Examination (Παγκύπριες Εξετάσεις Προχωρημένων Μαθηματικών) is a written, constructed-response examination used in Cyprus for advanced mathematics placement or certification, typically taken by upper-secondary graduates preparing for university entry or by candidates seeking formal recognition of mathematical competence. It is intended for students who have completed a rigorous mathematics curriculum and can sustain multi-step reasoning without a calculator on symbolic problems. The Pancyprian Advanced Math study guide has 7 sections and about 3,716 words.
01Algebra and Functions
Real functions: domain and range, injectivity, inverse, composition, monotonicity, graph transformations, and polynomial, rational, exponential, logarithmic and absolute-value equations and inequalities. Marks are given for exact set notation and sign tables, not for sketches.
484 words, about 2 min02Calculus: Limits, Continuity and Differentiation
Limits of rational, irrational, exponential, logarithmic and trigonometric expressions; continuity at a point and on an interval; the intermediate value theorem; derivatives as limits of difference quotients; tangent lines, monotonicity, extrema, concavity and optimisation, plus Rolle's theorem and the mean value theorem. If you get 0/0, factorise, multiply by the conjugate, or use standard limits: lim(x→0) sin x / x = 1, lim(x→0)(e^x − 1)/x = 1, lim(x→0) ln(1+x)/x = 1, lim(x→0)(1+x)^(1/x) = e.
475 words, about 2 min03Calculus: Integration and Applications
Antiderivatives from the standard table, integration by linear substitution and by parts, the fundamental theorem, definite integrals and their properties, and areas of plane regions - including regions between two curves and, where your syllabus includes it, volumes of revolution. ∫x^n dx = x^(n+1)/(n+1) + c for n ≠ −1; ∫(1/x)dx = ln x + c; ∫e^x dx = e^x + c; ∫sin x dx = −cos x + c; ∫cos x dx = sin x + c; ∫dx/cos²x = tan x + c.
448 words, about 2 min04Sequences, Series and the Binomial Theorem
Arithmetic and geometric progressions, limits of sequences, sums of finite and infinite series, and the binomial expansion with extraction of a specified term or coefficient. Arithmetic: a n = a₁ + (n − 1)d and S n = (n/2)(a₁ + a n) = (n/2)(2a₁ + (n − 1)d).
426 words, about 2 min05Complex Numbers
Arithmetic in ℂ, conjugate and modulus, the Argand diagram, polar (trigonometric) form, De Moivre's theorem, nth roots, quadratic equations with complex roots, and simple loci. z = a + bi with i² = −1; z = w means equal real parts and equal imaginary parts.
456 words, about 2 min06Vectors
Vector algebra in the plane and in space: components, magnitude, collinearity, position vectors, the scalar (dot) product, angles, perpendicularity, projections and simple geometric configurations such as midpoints and points dividing a segment in a ratio. A vector has magnitude and direction.
431 words, about 2 min07Analytic Geometry and Conics
Lines (slope, equations, parallelism, perpendicularity, distance), circles, and the three conics - parabola, ellipse and hyperbola - as loci with their foci, directrices, axes, asymptotes and eccentricity, plus tangency conditions found by the discriminant. Line: y − y₁ = m(x − x₁) or Ax + By + Γ = 0, with slope m = −A/B and normal vector (A, B).
996 words, about 5 min