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Key Facts: Madrid PAU Technical Drawing II Exam

90 Mins

Official examination duration

Madrid PAU Regulations

Non-graphic calculators

Calculators without graphic display or symbolic capabilities are allowed

Comunidad de Madrid PAU Guidelines

EUR 93.02

Base registration fee for compulsory Access Phase in Madrid

Comunidad de Madrid Official Gazette

Min 4.0

Minimum Access Phase mark required to calculate university admission mark

Spanish University Admission Framework

100 Questions

Practice questions available in this OpenExamPrep bank

OpenExamPrep

Prepare for the Madrid PAU Technical Drawing II exam with 100 verified practice questions across plane geometry, descriptive geometry, axonometry, standardization, and tangency problems.

Sample Madrid PAU Technical Drawing II Practice Questions

Try these sample questions to review concepts for the Madrid PAU Technical Drawing II exam. Each question includes a detailed explanation. Start the interactive quiz above for the full 100+ question experience with AI tutoring.

1In plane geometry, what defines a homothety (homotecia) with center O and ratio k = -1.5?
A.Every point P and its image P' lie on opposite sides of center O with OP' = 1.5 · OP.
B.Every point P and its image P' lie on the same ray from center O with OP' = 1.5 · OP.
C.The figure is rotated by 90 degrees around center O and scaled by a factor of 1.5.
D.The figure is translated along a vector of length 1.5 units from center O.
Explanation: A homothety with a negative ratio k = -1.5 places the transformed image point P' on the opposite side of the center of homothety O relative to P. The distance from the center satisfies OP' = |k| · OP = 1.5 · OP, resulting in an inverted image scaled by 150%.
2A point P is located 4 cm from the center of homothety O. Under a homothety of ratio k = -2.5, what is the distance between point P and its transformed image P'?
A.10 cm
B.14 cm
C.6 cm
D.16 cm
Explanation: Since k = -2.5, the image point P' lies on the opposite side of O relative to P. The distance from O to P' is OP' = |-2.5| · 4 cm = 10 cm. Because O lies between P and P', the total distance between P and P' is OP + OP' = 4 cm + 10 cm = 14 cm.
3In a plane inversion with center O, point P is at distance OP = 4 cm from O and its inverse point P' is at distance OP' = 9 cm. What is the radius R of the circle of inversion?
A.36 cm
B.6.5 cm
C.6 cm
D.5 cm
Explanation: The power of inversion K is given by K = OP · OP' = 4 cm · 9 cm = 36 cm². Since the radius of the circle of inversion R satisfies R² = K, we have R = √36 = 6 cm.
4What is the inverse of a circle that passes through the center of inversion O?
A.A concentric circle centered at the center of inversion O.
B.A circle passing through O of twice the original diameter.
C.A straight line passing directly through the center of inversion O.
D.A straight line not passing through O, perpendicular to the diameter passing through O.
Explanation: In plane inversion, any circle passing through the center of inversion O transforms into a straight line that does not pass through O. This line is perpendicular to the line connecting O and the center of the original circle.
5A point P is located at distance d = 10 cm from the center of circle C with radius R = 6 cm. What is the radical power of point P with respect to circle C?
A.64 cm²
B.16 cm²
C.100 cm²
D.36 cm²
Explanation: The power of a point P with respect to a circle centered at O with radius R is P(C) = d² - R², where d is the distance OP. Here, P(C) = 10² - 6² = 100 - 36 = 64 cm².
6What geometric locus forms the radical axis of two intersecting circles?
A.The line segment connecting the centers of the two circles.
B.The straight line passing through the two intersection points of the circles.
C.A circle concentric with the larger circle passing through the intersection points.
D.The line perpendicular to the line of centers passing through one of the centers.
Explanation: The radical axis of two circles is the locus of points having equal power with respect to both circles. For two intersecting circles, their intersection points both have zero power with respect to each circle, so the radical axis is the straight line passing through both intersection points.
7In a plane homology (homología), what is the key property of the axis of homology e?
A.It is perpendicular to all lines connecting corresponding points P and P'.
B.It is the line containing the center of homology O and all vanishing points.
C.It is the locus of invariant points; any line intersects its transformed image at a point on e.
D.It divides the plane into two regions with opposite homothety ratios.
Explanation: The axis of homology e consists entirely of double (invariant) points. A fundamental property of planar homology is that any line and its transformed image line always intersect at a point belonging to the axis of homology e.
8An affinity (afinidad) in the plane is a special case of homology. Which element of general homology is situated at infinity in an affinity?
A.The origin of coordinates.
B.The axis of homology e.
C.The vanishing line r_v.
D.The center of homology O.
Explanation: An affinity is a planar homology whose center of homology O is located at infinity. Consequently, lines connecting corresponding points P and P' are all parallel to a fixed direction known as the direction of affinity.
9A line segment AB of length L is divided into golden section (sección áurea) at point C such that AC is the major segment (AC > CB). What is the exact formula for the length of AC?
A.AC = L · (√5 - 1) / 2
B.AC = L · (√5 + 1) / 2
C.AC = L · (√3 - 1) / 2
D.AC = L · (√2 / 2)
Explanation: By definition of the golden ratio, L / AC = AC / CB, which yields the quadratic equation (AC)² + L · AC - L² = 0. Solving for AC gives AC = L · (√5 - 1) / 2 ≈ 0.61803 · L.
10Which three classic triangle centers always lie on the Euler line (recta de Euler) of any scalene triangle?
A.Incenter, Orthocenter, and Circumcenter.
B.Orthocenter, Centroid (Baricentro), and Circumcenter.
C.Incenter, Centroid, and Excenter.
D.Nine-point center, Incenter, and Centroid.
Explanation: The Euler line passes through the Orthocenter (H), Centroid (G), and Circumcenter (O) of any non-equilateral triangle. Furthermore, the centroid divides the segment between the orthocenter and circumcenter in a 2:1 ratio (HG = 2 · GO).

About the Madrid PAU Technical Drawing II Exam

Comprehensive practice exam bank for Madrid PAU Technical Drawing II (Dibujo Técnico II). Designed for 2nd Bachillerato students preparing for university entrance exams in the Community of Madrid, local questions are an English-language MCQ study adaptation covering plane geometry, descriptive geometry (diédrico), axonometric perspective, technical standardization, and tangency problems.

Exam sponsor: PAU Organising Commission of the Community of Madrid / Universidad Complutense de Madrid (UCM). The requirements and fees below concern the certification or admission exam, separate from our free practice resources.

Assessment

Standardized 90-minute practical drafting examination in Madrid testing 5 main domain areas: Plane Geometry & Geometric Transformations (25%), Descriptive Geometry & Orthographic Projection (25%), Axonometric & Perspective Systems (20%), Standardization, Dimensions & Technical CAD (15%), and Intersection & Tangency Problems (15%). Local practice items are an English-language MCQ study adaptation designed for Bachillerato curriculum review.

Time Limit

90 minutes (1.5 hours)

Passing Score

Marked on a 0–10 scale. Minimum 4.0 required in Access Phase to combine with Bachillerato GPA (60% Bachillerato + 40% PAU >= 5.0 to pass).

Exam / Certification Fees

EUR 93.02 base registration fee for compulsory Access Phase in Community of Madrid (or ~11.63 € per optional subject in voluntary phase).

Exam sponsor website

Fees, eligibility, and exam policies can change. Confirm them with the exam sponsor before applying or paying.

Our practice resources: topics covered

We aim to reflect publicly available exam outlines and topic information in our study resources. Coverage, format, and difficulty may differ from the actual exam, and we cannot guarantee that every detail is accurate or current. Confirm exam requirements, fees, and policies with the official exam sponsor.

25%

Plane Geometry & Geometric Transformations

Homothety, similarity, homology, affinity, plane inversion, radical power, triangle geometry, and conic sections (ellipse, parabola, hyperbola).

25%

Descriptive Geometry & Orthographic Projection (Diédrico)

Monge diédrico system: representations of points, lines, and planes; intersections, parallelism, perpendicularity, plane changes, rotations, rebattements (abatimientos), true shape/distance determination, and polyhedra.

20%

Axonometric & Perspective Systems

Isometric, dimetric, and trimetric projections, scale reduction coefficients, Cavalier perspective (perspectiva caballera), and conical linear perspective with vanishing and distance points.

15%

Standardization, Dimensions & Technical CAD

UNE and ISO standards, paper sizes (A-series), line types, First-Angle (European) projection, cross-section hatching, dimensioning principles, normalized scales, and CAD fundamentals.

15%

Intersection & Tangency Problems

Fundamental tangency principles, Apollonius problems, inversion tangency methods, tangent arc connections, polyhedral/conic intersections, and surface developments.

Preparing for the Madrid PAU Technical Drawing II Exam

What You Need to Know

  • Passing score: Marked on a 0–10 scale. Minimum 4.0 required in Access Phase to combine with Bachillerato GPA (60% Bachillerato + 40% PAU >= 5.0 to pass).
  • Assessment: Standardized 90-minute practical drafting examination in Madrid testing 5 main domain areas: Plane Geometry & Geometric Transformations (25%), Descriptive Geometry & Orthographic Projection (25%), Axonometric & Perspective Systems (20%), Standardization, Dimensions & Technical CAD (15%), and Intersection & Tangency Problems (15%). Local practice items are an English-language MCQ study adaptation designed for Bachillerato curriculum review.
  • Time limit: 90 minutes (1.5 hours)
  • Exam / certification fees: EUR 93.02 base registration fee for compulsory Access Phase in Community of Madrid (or ~11.63 € per optional subject in voluntary phase). Official sources

Using Our Practice Resources

  • Work through all 100 available questions
  • Review every answer and explanation
  • Track weak areas and revisit them
  • Use our AI tutor for tough concepts

Madrid PAU Technical Drawing II: Suggested Study Strategy

1Master plane transformation invariant properties (homology axis, affinity direction, inversion center) to quickly deduce transformed geometries.
2Systematically analyze diédrico trace intersections and auxiliary plane changes before attempting complex 3D distance or true-shape calculations.
3Memorize normalized axonometric reduction factors (isometric theoretical factor ~0.816, isometric drawing 1:1, Cavalier Y-axis 0.5 or 0.75).
4Review UNE 1032 and ISO 128 dimensioning rules to avoid losing marks on line overlaps, missing diameter symbols, or redundant dimensions.
5Practice quantitative geometric calculations including focal distances of conic sections, inversion power, radical axis distances, and cone development angles.

Frequently Asked Questions

What is the format of the Madrid PAU Technical Drawing II exam?

The official Madrid PAU exam is a 90-minute practical drafting test comprising geometric construction problems scored out of 10 points. Students solve plane geometry, descriptive geometry (diédrico), axonometry, and standardization tasks using drawing instruments.

Why are these practice questions presented in multiple-choice format?

While the official PAU test requires full paper-and-pencil geometric drafting, this bank translates core geometric theorems, spatial relationship rules, transformation formulas, and standardized technical conventions into multiple-choice questions to build speed and conceptual mastery.

What drawing instruments and tools are permitted during the official exam?

Students are required to bring compasses, set squares (escuadra 45° and cartabón 30°/60°), a millimetric ruler, protractor, technical drawing pencils (such as 2H for construction and HB for final outlines), and non-programmable non-graphing calculators.

How is the passing grade calculated for Madrid PAU Dibujo Técnico II?

The exam is graded on a scale from 0 to 10. In the compulsory Access Phase, a minimum grade of 4.0 is required to calculate the university entrance score (60% Bachillerato average + 40% PAU Access Phase mark >= 5.0). In the Voluntary Phase, marks above 5.0 add weighting up to 0.2 per subject.

What projection system is standard in Spanish PAU technical drawing?

Spanish UNE/ISO standards strictly mandate First-Angle (European) Orthographic Projection. In this system, the top view (planta) is positioned below the front view (alzado), and the left side view (perfil izquierdo) is placed to the right of the front view.