PhysicsMadrid 2026Question 1
Electric field between two plates
From potential to field, to acceleration and to speed: why it is solved this way
9 chapters · 7 minWatch the lesson →
Physics
PhysicsMadrid 2026Question 2.A
A satellite in circular orbit
From mechanical energy to radius, speed, revolutions and escape: why a single piece of data decides everything
11 chapters · 8 minWatch the lesson →
PhysicsMadrid 2026Question 2.B
Gravitational field of two point masses
Where the field vanishes and how to add two fields that are not parallel: why it is solved this way
11 chapters · 8 minWatch the lesson →
PhysicsMadrid 2026Question 3.A
A wave read from two graphs
The snapshot of the wave and the history of one point: what each graph gives, and why we need both
11 chapters · 8 minWatch the lesson →
PhysicsMadrid 2026Question 3.B
A light ray in a parallel-sided slab
Refraction, reflection and a triangle: why the ray bends, how long it takes and where it comes out again
12 chapters · 9 minWatch the lesson →
PhysicsMadrid 2026Question 4.A
The photoelectric effect on a silver plate
Two lights and two stopping potentials: why they give us the Planck constant and the work function
11 chapters · 8 minWatch the lesson →
PhysicsMadrid 2026Question 4.B
Decay of radium-228
From activity to decay constant, to half-life and to transformed mass: why everything decays with the same exponential
11 chapters · 8 minWatch the lesson →
PhysicsFundamentals
Projectile motion
Why it is solved this way: the complete reasoning for a ball thrown from the ground
11 chapters · 9 minWatch the lesson →
Chemistry
ChemistryMadrid 2026Question 1
Corrosion of copper in nitric acid
Half-equation balancing, cell potential and precipitation: why each step is done this way
10 chapters · 9 minWatch the lesson →
Mathematics
MathematicsMadrid 2026Question 1
Rank with a parameter and a matrix equation
Matrices: why the determinant decides the rank, why a minor is needed, and why we multiply by the inverse on the left
10 chapters · 8 minWatch the lesson →
MathematicsMadrid 2026Question 2
Cheapest triangular garden
Optimisation: why those variables, where the constraint comes from and how we know it is a minimum
10 chapters · 8 minWatch the lesson →
MathematicsMadrid 2026Question 3
Reflection, distance and orthogonal plane
3D geometry: why the plane's normal vector decides the perpendicular, the distance and the orthogonal plane
10 chapters · 9 minWatch the lesson →
MathematicsMadrid 2026Question 4.1
Defective batteries: from the normal to the binomial
Why we standardise, how to use a table that only has positive values, and why a batch of ten is a binomial
10 chapters · 8 minWatch the lesson →
MathematicsMadrid 2026Question 4.2
Events A and B: from one conditional probability to every region
Probability: why a conditional probability is a proportion inside a shrunken world, and how the total is shared among four regions
10 chapters · 8 minWatch the lesson →
MathematicsMadrid 2026Question 5.1
Area under the logarithm
The definite integral: where the limits come from, why integration by parts, and how to tell the result is reasonable
9 chapters · 8 minWatch the lesson →
MathematicsMadrid 2026Question 5.2
Piecewise function: continuous, differentiable and tangent line
Why continuity fixes b, why differentiability fixes a, and what you need to write a tangent line
10 chapters · 8 minWatch the lesson →
Technology and Engineering
Technology and EngineeringMadrid 2026Question 2.2
Tensile test of an aluminium alloy
Yield strength, elongation at break, Hooke’s law and minimum diameter: which length and which units to use in each part, and why
10 chapters · 9 minWatch the lesson →
Technology and EngineeringMadrid 2026Question 3.1
Overhanging beam: reactions, shear force and bending moment
From the equilibrium of the beam to its internal-force diagrams: why it is solved this way
11 chapters · 10 minWatch the lesson →
Technology and EngineeringMadrid 2026Question 3.2
The Carnot refrigerator
From 400 kJ per hour to the motor's power: why a refrigerator is a heat engine in reverse and why it is solved this way
9 chapters · 8 minWatch the lesson →
Technology and EngineeringMadrid 2026Question 4.1
Parallel AC circuit
From the capacitor current to the voltage, the powers and the generator current: why it is solved this way
10 chapters · 9 minWatch the lesson →
Technology and EngineeringMadrid 2026Question 4.2
Digital circuit: truth table and Karnaugh map
From the logic-gate circuit to the minimal sum of products: why we solve it this way
10 chapters · 10 minWatch the lesson →
Technology and EngineeringMadrid 2026Question 5.2
Sensor, actuator and a block diagram with two feedback loops
Where the feedback loop formula comes from and how to reduce the diagram step by step down to Y/R: why we solve it this way
9 chapters · 8 minWatch the lesson →
Technical Drawing
Technical DrawingMadrid 2026Question 1.1
Inverse arc under an inversion
Where the centre of inversion is, what its power is, and why each construction line is the way it is
9 chapters · 9 minWatch the lesson →
General Science
General ScienceMadrid 2026Question 2A
X-linked inheritance
A healthy father and a carrier mother: the Punnett square and why it comes out this way
12 chapters · 9 minWatch the lesson →
General ScienceMadrid 2026Question 3A
A geostationary satellite
The four forces, the gravity that makes it orbit and the angular velocity: why it is solved this way
11 chapters · 9 minWatch the lesson →
General ScienceMadrid 2026Question 3B
A box pushed against friction
Free-body diagram, friction, net force and braking: why it is solved this way
12 chapters · 9 minWatch the lesson →
General ScienceMadrid 2026Question 4B
The phase diagram of CO₂ and the heat needed to warm it
Reading the diagram like a map, naming the changes of state and deriving Q = m·cₑ·ΔT
12 chapters · 11 minWatch the lesson →
Geology and Environmental Science
Geology and Environmental ScienceMadrid 2026Question 1.1
Reading a geological cross-section
Which rock is which, in what order they formed, what each surface tells us and why
13 chapters · 11 minWatch the lesson →
Business Studies
Business StudiesMadrid 2026Question 2
NPV and payback period of two investment projects
Why a future euro is worth less today, and why NPV and payback can pick different projects
11 chapters · 12 minWatch the lesson →
Business StudiesMadrid 2026Question 4
Profit and loss account, return on assets and return on equity
Why the account is a waterfall of profits, what is divided by what in each return, and why interest moves one and not the other
12 chapters · 10 minWatch the lesson →
Lessons marked «Fundamentals» are model problems written for this site. The rest are questions from the official Spanish university entrance exams (PAU), set by the public universities of the Community of Madrid and translated here into English. Each lesson quotes its source.
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