VWO Syllabus Selection Exam Mathematics

Syllabus Selection Exam - v2025

• Introduction: The exam has three digital subtests covering fundamental topics in mathematics, physics, and first-year material. Content is based on Dutch VWO pre-university education.

Mathematics

1. Functions and Graphs

• Candidates must recognize and construct compositions of standard functions.
• Standard functions include: polynomial functions (xⁿ), n-root functions (x^1/n), power functions (a^x), logarithms (log_a(x)), exponentials (e^x), natural logs (ln(x)), trigonometric functions (sin(x), cos(x)).
• Candidates should analyze, draw, transform these functions (including compositions), and determine limits, domain, range, asymptotes, symmetry.
• Candidates must understand and find inverses of functions (and compositions).

2. Algebraic Solving

• Candidates can isolate variables in expressions and substitute expressions into functions.
• Simplify expressions, recognize special products, manipulate equations/inequalities with standard functions.
• Find solutions (y=f(x) or y≤f(x)).
• Find roots of functions (f(x)=0) via factorization/decomposition.
• Use the quadratic formula (ax² + bx + c = 0) to find roots.
• Solve systems of linear equations (ax + by = c, dx + ey = f) where a, b, c, d, e, f are constants.

3. Differential Calculus

• Candidates know derivatives of standard functions.
• Apply product, quotient, and chain rule to find derivatives of composite functions.
• Determine first (f'(x), dy/dx, df(x)/dx) and second derivatives (f''(x), d²y/dx², d²f(x)/dx²) of functions.
• Use derivatives to determine increasing/decreasing behavior, extreme values, concavity, convexity, inflection points.
• Derivatives' application: slope of a graph, local tangent/normal lines, optimization problems (distance, velocity, acceleration).

4. Integral Calculus

• Understand integration concept & related terms (limits, definite/indefinite integrals, constant of integration).
• Determine antiderivatives/primitives of standard functions (F(x) = ∫f(x)dx).
• Calculate definite/indefinite integrals of functions with form cf(ax+b)+d (where a, b, c, d are constants).
• Applications: surface/volume calculations (solids of revolution).

5. Trigonometry

• Understand trigonometric functions (sin(x), cos(x), tan(x)) and the unit circle.
• Convert degrees/radians.
• Know exact values for common angles (0, 30, 45, 60, 90 degrees).
• Apply periodicity and symmetry properties of sin, cos, tan.
• Solve equations: sin(x)=c, cos(x)=c, tan(x)=c, sin(f(x))=sin(g(x)), cos(f(x))=cos(g(x)), tan(f(x)) = tan(g(x)) where c is a constant and f(x), g(x) are linear functions of x
• Apply Pythagorean identities (sin²x + cos²x = 1), sum/difference/double angle formulae.
• Derive formulas for sine/cosine waves (f(x) = a + b*sin(cx-d) where a, b, c, d are constants).

6. Geometry

• Determine surface area/perimeter of 2D shapes (triangles, rectangles, circles).
• Calculate volume/surface area of 3D shapes (cubes, pyramids, cylinders, cones).
• Understand properties of lines, triangles, circles, and quadrilaterals to find lengths and angles.
• Use sine/cosine/tangent relations, Pythagorean theorem, and law of sines/cosines.
• Formulate equations for lines/circles and relate slopes of parallel/perpendicular lines.
• Find points where lines and circles intersect. Find distances.

7. Vectors

• Understand vector concept: length/direction.
• Decompose vectors into components, multiply by scalars, add/subtract vectors.
• Use dot product for angle/distance calculations.
• Calculate velocity/acceleration of moving points (path described by time-dependent vector).
• Determine vector equation of a line, convert linear equations to vector equations.
• Find local tangents to parametric curves.
• Determine the center of gravity of 2D shapes.

Physics

1. Fundamentals

• Understand SI base units (meter, kilogram, second, ampere, kelvin, degree Celsius, mole).
• Perform dimensional analysis.
• Know and apply vector quantities (velocity, acceleration, force etcetera). Know their magnitude and direction.
• Work with metric prefixes (micro, milli, kilo, mega, giga).
• Understand symbolic expressions like log, ln, e^-ax, e^ax, a^x, x^a, sin(x), cos(x).
• Work with scientific notation (e.g., 4×10^-8)

2. Mechanics

• Relations between distance, velocity, and acceleration (x,v, and a – diagrams).
• Analyze linear motions, (constant acceleration/deceleration, constant friction force, and free fall).
• Apply Newton's laws.
• Understand forces: gravity, friction, drag, tension, spring forces.
• Calculate these forces using free-body diagrams.
• Principles of Work, Energy, Power, Efficiency.
• Relate Work, Energy, Power with potential energy, kinetic energy, elastic/spring energy.
• Angular velocity/circular motion/period/frequency.
• Analyze circular motions/centripetal and centrifugal force.
• Gravitational force/centripetal force balance.

3. Electricity and Magnetic Fields

• Principles of electricity concepts (current, charge, potential, resistance, conductors/insulators, electrons/ions).
• Apply Ohm's law and circuit rules (parallel and series circuits, Kirchhoff's rules).
• Understand Coulomb’s law, forces between charged particles.
• Understand/calculate magnetic flux in homogeneous magnetic fields.
• Understand Lorentz force on charged particles (magnitude and direction).

4. Vibrations and Waves

• Understand vibrations(period, frequency, amplitude, phase, resonance, damping).
• Simple harmonic motion (mass-spring system).
• Understand wave phenomena (longitudinal/transverse waves, wavelength, speed, phase, speed of sound, and speed of light).

This quiz covers essential topics from the VWO syllabus in mathematics, including functions and graphs, and algebraic solving. Candidates will need to demonstrate their understanding of various types of functions, their transformations, and the ability to simplify and isolate variables in expressions. Prepare to analyze and solve problems effectively to excel in your studies.