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Key Facts: Madrid PAU Mathematics II Exam

Madrid PAU

PAU Organising Commission of Madrid / UCM

Community of Madrid Education Department

90 Mins

Official examination time

Madrid PAU Exam Structure

Min 4.0

Minimum mark required in Access Phase to calculate overall admission mark

Spanish University Admission Regulations

5 Syllabus Blocks

Matrices, Differential Calculus, Integral Calculus, 3D Geometry, Probability

LOMLOE 2º Bachillerato Mathematics II Syllabus

100

Practice questions available in this OpenExamPrep bank

OpenExamPrep

Madrid PAU Mathematics II (Matemáticas II) tests 2º Bachillerato students on linear algebra, differential calculus, integral calculus, 3D analytic geometry, and probability distributions for university entry in Madrid.

Sample Madrid PAU Mathematics II Practice Questions

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

1What is the determinant of the 2x2 matrix A = [[2, -1], [4, 3]]?
A.10
B.2
C.14
D.-2
Explanation: The determinant of a 2x2 matrix [[a, b], [c, d]] is computed as ad - bc. For matrix A, det(A) = (2)(3) - (-1)(4) = 6 + 4 = 10.
2If A is a 3x3 matrix with det(A) = 5, what is the determinant of the scalar multiple matrix 2A?
A.40
B.10
C.30
D.20
Explanation: For any n x n matrix A and scalar k, det(k A) = k^n * det(A). Since A is a 3x3 matrix (n = 3) and k = 2, det(2A) = 2^3 * det(A) = 8 * 5 = 40.
3Which condition is both necessary and sufficient for a square matrix A to be invertible?
A.det(A) != 0
B.det(A) = 0
C.A is a symmetric matrix
D.The trace of A is positive
Explanation: A square matrix A is non-singular (invertible) if and only if its determinant is non-zero (det(A) != 0).
4What is the transpose of the 2x3 matrix M = [[1, 2, 3], [4, 5, 6]]?
A.[[1, 4], [2, 5], [3, 6]]
B.[[6, 5, 4], [3, 2, 1]]
C.[[1, 2], [3, 4], [5, 6]]
D.[[4, 5, 6], [1, 2, 3]]
Explanation: The transpose M^T of a matrix M interchanges its rows and columns. Row 1 [1, 2, 3] becomes Column 1 and Row 2 [4, 5, 6] becomes Column 2, yielding a 3x2 matrix [[1, 4], [2, 5], [3, 6]].
5If A * X = B, where A is an invertible square matrix, which matrix expression correctly yields X?
A.X = A^(-1) * B
B.X = B * A^(-1)
C.X = B / A
D.X = A * B^(-1)
Explanation: To solve A * X = B, left-multiply both sides by A^(-1): A^(-1) * (A * X) = A^(-1) * B => (A^(-1) * A) * X = A^(-1) * B => X = A^(-1) * B. Matrix multiplication is non-commutative, so the order matters.
6Calculate the determinant of the 3x3 matrix A = [[1, 0, 2], [3, -1, 1], [2, 0, 4]].
A.0
B.8
C.-4
D.4
Explanation: Expanding along the second column (which has two zeros): det(A) = (-1) * (-1)^(2+2) * det([[1, 2], [2, 4]]) = (-1) * (1 * 4 - 2 * 2) = (-1) * (0) = 0. Notice that Column 3 is twice Column 1, so rows/columns are linearly dependent.
7What is the inverse matrix of A = [[1, 2], [3, 4]]?
A.[[-2, 1], [1.5, -0.5]]
B.[[4, -2], [-3, 1]]
C.[[-0.5, 1], [1.5, -2]]
D.[[1, 0.5], [0.33, 0.25]]
Explanation: det(A) = (1)(4) - (2)(3) = -2. Using the formula A^(-1) = (1/det(A)) * [[d, -b], [-c, a]], we get A^(-1) = (-1/2) * [[4, -2], [-3, 1]] = [[-2, 1], [3/2, -1/2]].
8For what real value of the parameter 'a' is the matrix A = [[a, 1, 0], [1, a, 1], [0, 1, a]] singular?
A.a = 0 or a = sqrt(2) or a = -sqrt(2)
B.a = 1 or a = -1
C.a = 2 or a = -2
D.a = 0 only
Explanation: Compute det(A) by expanding along row 1: det(A) = a * (a^2 - 1) - 1 * (a - 0) = a^3 - a - a = a^3 - 2a = a(a^2 - 2). Setting det(A) = 0 gives a = 0 or a^2 = 2 => a = +-sqrt(2).
9Consider the system of linear equations: { x + y + z = 1; x + 2y + 3z = 2; x + 4y + az = 5 }. For what value of parameter 'a' does the system NOT have a unique solution?
A.a = 7
B.a = 5
C.a = 3
D.a = 0
Explanation: The system has a unique solution iff the coefficient matrix determinant is non-zero. det(A) = |[1,1,1],[1,2,3],[1,4,a]|. Subtract Row 1 from Row 2 and Row 3: |[1,1,1],[0,1,2],[0,3,a-1]| = 1 * ((a-1) - 6) = a - 7. Setting det(A) = 0 yields a = 7.
10What is the rank of the matrix A = [[1, 2, 3], [2, 4, 6], [3, 6, 9]]?
A.1
B.3
C.2
D.0
Explanation: Notice that Row 2 = 2 * Row 1 and Row 3 = 3 * Row 1. All 2x2 and 3x3 minors are zero, but there exist non-zero 1x1 minors (e.g. [1]). Hence, rank(A) = 1.

About the Madrid PAU Mathematics II Exam

The Madrid PAU Mathematics II (Matemáticas II) practice question bank provides 100 rigorous worked calculation and conceptual practice questions for 2º Bachillerato students in the Community of Madrid. It spans matrix algebra, systems of linear equations, differential calculus, integral calculus, 3D geometry, and probability distributions.

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

90-minute examination covering 5 core syllabus blocks: Linear Algebra, Matrices & Systems of Equations (25%), Differential Calculus & Applications (25%), Integral Calculus & Area & Volume (20%), Analytic Geometry in 3D Space (15%), and Probability Distributions & Statistics (15%).

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%

Linear Algebra, Matrices & Systems of Equations

Matrix operations, determinants, matrix inverses, matrix equations, rank, Rouché-Capelli theorem, and systems of linear equations with parameters solved via Cramer's rule.

25%

Differential Calculus & Applications

Limits, continuity, differentiability, L'Hôpital's rule, Bolzano and Mean Value Theorems, tangent lines, relative extrema, concavity, inflection points, asymptotes, and optimization problems.

20%

Integral Calculus & Area & Volume

Indefinite integration (substitution, integration by parts, rational functions), definite integrals (Barrow's rule), area enclosed between curves, and volume of revolution around the x-axis.

15%

Analytic Geometry in 3D Space

Vectors in 3D space, lines and planes equations, relative positions of lines and planes, metric properties including distances, angles, projections, and symmetric points.

15%

Probability Distributions & Statistics

Compound probability, Total Probability Theorem, Bayes' Theorem, independent events, Binomial distribution B(n,p), Normal distribution N(mu, sigma), standardization, and normal approximation to binomial.

Preparing for the Madrid PAU Mathematics 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: 90-minute examination covering 5 core syllabus blocks: Linear Algebra, Matrices & Systems of Equations (25%), Differential Calculus & Applications (25%), Integral Calculus & Area & Volume (20%), Analytic Geometry in 3D Space (15%), and Probability Distributions & Statistics (15%).
  • 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 Mathematics II: Suggested Study Strategy

1Master parameter discussion in systems of equations: use Rouché-Capelli theorem by calculating the determinant of the coefficient matrix and analyzing ranks.
2Practice derivative applications: master L'Hôpital's rule, curve sketching (asymptotes, extrema, inflection points), and optimization setup.
3Sharpen integration techniques: practice integration by parts, partial fractions decomposition for rational functions, and setting up definite integrals for bounded areas.
4Visualize 3D geometry problems: write down clear parametric and implicit equations for lines and planes before computing distances, projections, or intersections.
5Review probability theorems: construct clear tree diagrams or contingency tables for Bayes' theorem and practice standard normal distribution calculations.

Frequently Asked Questions

What is the registration fee for the PAU exam in the Community of Madrid?

The base registration fee for the compulsory Access Phase in Madrid is EUR 93.02 (or approximately EUR 11.63 per optional subject in the voluntary admission phase).

What minimum score is required to pass the PAU Mathematics II exam in Madrid?

The exam is marked on a 0–10 scale. A minimum mark of 4.0 in the Access Phase is required to calculate the admission grade, which combines 60% Bachillerato GPA and 40% PAU Access Phase mark (must be >= 5.0).

How long is the Madrid PAU Mathematics II exam?

The official examination time is 90 minutes (1.5 hours).

Which organising body sets the PAU exam in Madrid?

The exam is organized by the PAU Organising Commission of the Community of Madrid, with university coordination managed by public universities such as Universidad Complutense de Madrid (UCM).

Are scientific calculators permitted in the Madrid PAU Mathematics II exam?

Non-programmable, non-graphing scientific calculators without symbolic calculation (CAS) capability or communication features are permitted under Madrid PAU regulations.