VWO Syllabus Selection Exam Mathematics

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Questions and Answers

What does the first derivative of a function help determine?

The amplitude of a sine wave

The local behavior of a function such as increasing and decreasing (correct)

The surface area of a solid

The sum of angles in a right triangle

Which of the following correctly defines an antiderivative of a function?

It is derived by multiplying the function by its coefficient.

It represents the area under the curve of the function.

It is the slope of the tangent line at a point on the graph.

It is another function whose derivative gives the original function. (correct)

Which of the following describes a standard function?

The sine function is always positive.

The quadratic function has a degree of two. (correct)

The exponential function is its own derivative.

The logarithmic function can only accept positive inputs. (correct)

How can the Pythagorean identity sin²(x) + cos²(x) = 1 be applied?

To derive periodic functions and properties. Signup and view all the answers

What is the primary method to isolate a variable in algebraic equations?

Add and subtract terms from both sides. Signup and view all the answers

Which form represents the general equation of a sine wave?

f(x) = a + b · sin(c · x - d) Signup and view all the answers

Which rule must be applied to find the derivative of a product of two functions?

Product Rule Signup and view all the answers

What determines the period of a trigonometric function?

The frequency of the wave Signup and view all the answers

In calculus, what does integration help calculate?

The area under the curve of a graph Signup and view all the answers

When finding roots of a quadratic equation, which formula should be applied?

Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Signup and view all the answers

Which technique is used to determine inflection points of a function?

Calculating the second derivative and setting it to zero Signup and view all the answers

What is the effect of the derivative at a point where the function intersects with the x-axis?

The derivative equals zero. Signup and view all the answers

If $f(x) = 3x^2 + 5x - 2$, what is the derivative of the function using the power rule?

$6x + 5$ Signup and view all the answers

Which statement is true regarding the integration of standard functions?

The antiderivative of sin(x) is cos(x). Signup and view all the answers

What describes the relationship between a function and its inverse?

The compositions of the function and its inverse yield the original input. Signup and view all the answers

Which of the following involves recognizing special products in algebra?

Simplifying expressions like $(a + b)^2$ to $a^2 + 2ab + b^2$. Signup and view all the answers

Which metric prefix is used to denote one million?

mega- (M) Signup and view all the answers

What is the formula to calculate the area of a triangle?

$\frac{1}{2} \times base \times height$ Signup and view all the answers

What is the relationship between distance, velocity, and acceleration?

Velocity is the derivative of distance with respect to time. Signup and view all the answers

What is the formula for density?

ρ = m/V Signup and view all the answers

Which theorem is used to find the length of a side in a right triangle when the lengths of the other two sides are known?

Pythagorean theorem Signup and view all the answers

To determine the direction of a vector, which of the following concepts is NOT necessary?

Density Signup and view all the answers

In the context of circular motion, what does angular velocity describe?

The rate of change of angular position of an object. Signup and view all the answers

Which of the following represents the principle of conservation of energy?

The total energy of an isolated system remains constant. Signup and view all the answers

What is the result of adding two vectors with the same direction?

A vector with the same direction and doubled magnitude Signup and view all the answers

What type of forces can be analyzed using a free-body diagram?

All types of forces including tension and friction. Signup and view all the answers

What do SI base units provide in scientific measurement?

Consistency and standardization in measurements Signup and view all the answers

Which of the following correctly defines electric potential?

Electric potential is the energy per unit charge. Signup and view all the answers

Which of the following describes the dot product of two vectors?

It produces a scalar quantity Signup and view all the answers

What is represented by the term efficiency in the context of mechanical systems?

The ratio of output work to input work. Signup and view all the answers

What is the typical use of the law of cosines in triangle calculations?

To find the length of one side when the other two sides and the included angle are known Signup and view all the answers

Which of the following measurements is NOT a SI base unit?

Gram Signup and view all the answers

What is the correct formula to calculate electric potential difference in a circuit?

U = I · R Signup and view all the answers

Which law is applied to analyze the current in a circuit when multiple branches are present?

Kirchhoff’s current law Signup and view all the answers

What property describes the force between two electrically charged particles according to Coulomb's law?

Both A and B Signup and view all the answers

Which of the following terms is used to describe the maximum extent of a wave's oscillation?

Amplitude Signup and view all the answers

When calculating magnetic flux through a surface, what is the factor that influences the result?

The angle between the magnetic field and the surface Signup and view all the answers

Study Notes

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).

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Description

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.