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Key Facts: PAU Mathematics II (Cantabria) Exam

90 min

Exam time duration for the PAU specific paper

University of Cantabria (UNICAN) PAU Organising Commission

4 x 2.5 pts

Four compulsory exercises (Algebra, Analysis, Geometry, Probability), each worth 2.5 of 10 points

UNICAN Matemáticas II Criterios de Corrección, June 2026

0–10 scale

Scoring system (minimum 4.0 in Access Phase to combine with Bachillerato)

Cantabria PAU Regulations

4 Blocks (25% each)

Official content blocks weighted equally: Algebra, Mathematical Analysis, Geometry in 3D, Probability & Statistics

UNICAN Matemáticas II Criterios de Corrección, June 2026

100

High-quality practice questions in this OpenExamPrep subject bank

OpenExamPrep

Master the 2026 Cantabria PAU Mathematics II exam with 100 practice questions covering matrix algebra and linear systems, calculus and optimization, 3D vector geometry, and probability and statistics, each weighted equally at 25% based on the official exam structure. This English-language MCQ bank is a study adaptation of the knowledge behind the official written exam, not a format simulation.

Sample PAU Mathematics II (Cantabria) Practice Questions

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

1Given A = [[4, -1], [2, 3]] and B = [[1, 2], [-3, 0]], compute A + 2B.
A.[[6, 3], [-4, 3]]
B.[[5, 1], [-1, 3]]
C.[[9, 0], [1, 6]]
D.[[6, 3], [8, 3]]
Explanation: First scale B by 2: 2B = [[2, 4], [-6, 0]]. Then add A entrywise: A + 2B = [[4+2, -1+4], [2-6, 3+0]] = [[6, 3], [-4, 3]].
2Let A = [[2, 0], [1, -1]] and B = [[3, 1], [2, 4]]. Compute the matrix product AB.
A.[[7, -1], [8, -4]]
B.[[6, 2], [1, -3]]
C.[[6, 0], [2, -4]]
D.[[6, 2], [1, 5]]
Explanation: Multiply row by column: row 1 gives [2*3+0*2, 2*1+0*4] = [6, 2]; row 2 gives [1*3+(-1)*2, 1*1+(-1)*4] = [1, -3]. So AB = [[6, 2], [1, -3]].
3Find the determinant of the matrix A = [[5, 3], [2, 4]].
A.26
B.-14
C.14
D.-2
Explanation: For a 2x2 matrix [[a,b],[c,d]], det(A) = ad - bc. Here det(A) = 5*4 - 3*2 = 20 - 6 = 14.
4Calculate the determinant of the matrix A = [[1, 2, 3], [0, 1, 4], [5, 6, 0]] using Sarrus's rule.
A.16
B.25
C.40
D.1
Explanation: Sarrus's rule: positive diagonals sum to (1*1*0)+(2*4*5)+(3*0*6) = 0+40+0 = 40, and negative diagonals sum to (3*1*5)+(1*4*6)+(2*0*0) = 15+24+0 = 39. det(A) = 40 - 39 = 1.
5If A is a 3x3 matrix with det(A) = 5, what is det(2A)?
A.40
B.10
C.20
D.13
Explanation: For an n x n matrix and scalar k, det(kA) = k^n * det(A). Here n=3 and k=2, so det(2A) = 2^3 * 5 = 8 * 5 = 40.
6If det(A) = -3 for a square matrix A, what is det(A^T), the determinant of its transpose?
A.3
B.-3
C.-1/3
D.0
Explanation: A fundamental determinant property states det(A^T) = det(A) for any square matrix, since transposing does not change the value of the determinant. So det(A^T) = -3.
7If det(A) = 4 and det(B) = -2 for two 3x3 matrices, what is det(AB)?
A.2
B.8
C.-8
D.6
Explanation: A key property of determinants is det(AB) = det(A) * det(B). Here det(AB) = 4 * (-2) = -8.
8Find the inverse of the matrix A = [[3, 1], [5, 2]].
A.[[3, -1], [-5, 2]]
B.[[2, 1], [5, 3]]
C.[[3, 5], [1, 2]]
D.[[2, -1], [-5, 3]]
Explanation: det(A) = 3*2 - 1*5 = 1. For a 2x2 matrix [[a,b],[c,d]], A^-1 = (1/det(A)) * [[d,-b],[-c,a]] = [[2,-1],[-5,3]].
9Use Cramer's rule to solve the system x + 2y = 5, 3x + 5y = 11 for (x, y).
A.x = -3, y = 4
B.x = 3, y = -4
C.x = 4, y = -3
D.x = 3, y = 4
Explanation: The determinant D = |1,2;3,5| = 1*5-2*3 = -1. Dx = |5,2;11,5| = 25-22 = 3, so x = Dx/D = 3/(-1) = -3. Dy = |1,5;3,11| = 11-15 = -4, so y = Dy/D = -4/(-1) = 4.
10For which value of k does the system x + y + z = 1, 2x + y - z = 0, x - y + kz = 2 fail to have a unique solution?
A.5
B.-5
C.-3
D.3
Explanation: The coefficient determinant is |1,1,1;2,1,-1;1,-1,k| = (k-1) - (2k+1) + (-3) = -k-5. Setting -k-5 = 0 gives k = -5, the value at which the coefficient matrix becomes singular.

About the PAU Mathematics II (Cantabria) Exam

Comprehensive 100-question practice test bank for the Cantabria (Universidad de Cantabria / UNICAN) PAU exam in Mathematics II (Matemáticas II). Covers matrix algebra and linear systems, mathematical analysis, 3D geometry, and probability & statistics drawn from the official Real Decreto 243/2022 and Real Decreto 534/2024 saberes básicos as applied by UNICAN, with content weighting confirmed against the University of Cantabria's official June 2026 grading criteria (Criterios de Corrección).

Exam sponsor: University of Cantabria (UNICAN) PAU Organising Commission. The requirements and fees below concern the certification or admission exam, separate from our free practice resources.

Assessment

The Cantabria PAU exam in Mathematics II (UNICAN) runs 90 minutes and evaluates candidate mastery of matrix algebra and linear systems, calculus (limits, derivatives, integrals, and optimization), 3D vector geometry, and probability and statistics, through four open/semi-constructed exercises weighted equally at 2.5 points each.

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 Access Phase >= 5.0 to pass).

Exam / Certification Fees

EUR 71.09 base registration fee for PAU Access Phase (ordinary sitting, set by Universidad de Cantabria).

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%

Algebra

Matrix operations, determinants, matrix inverse, rank, and classification and solving of linear systems via the Rouché–Capelli theorem and Cramer's rule.

25%

Mathematical Analysis

Limits, continuity, differentiability, derivatives (chain, product, and implicit differentiation), curve sketching (asymptotes, monotonicity, extrema, inflection points), optimization problems, and indefinite and definite integrals including area between curves.

25%

Geometry in 3D

Vector operations (dot, cross, and scalar triple product), lines and planes in space, relative positions, distances, angles, projections, and volumes.

25%

Probability & Statistics

Conditional probability, the Total Probability Theorem, Bayes' theorem, binomial distribution, and normal distribution.

Preparing for the PAU Mathematics II (Cantabria) 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 Access Phase >= 5.0 to pass).
  • Assessment: The Cantabria PAU exam in Mathematics II (UNICAN) runs 90 minutes and evaluates candidate mastery of matrix algebra and linear systems, calculus (limits, derivatives, integrals, and optimization), 3D vector geometry, and probability and statistics, through four open/semi-constructed exercises weighted equally at 2.5 points each.
  • Time limit: 90 minutes (1.5 hours)
  • Exam / certification fees: EUR 71.09 base registration fee for PAU Access Phase (ordinary sitting, set by Universidad de Cantabria). 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

PAU Mathematics II (Cantabria): Suggested Study Strategy

1Master the full algebra toolkit: matrix operations, determinants (Sarrus's rule for 3x3), matrix inverse, and using the Rouché–Capelli theorem and Cramer's rule to classify and solve parametric linear systems.
2Build calculus fluency across limits (including L'Hôpital's rule), derivatives (chain, product, and implicit differentiation), curve analysis (asymptotes, monotonicity, extrema, inflection points), and both indefinite and definite integrals, since optimization word problems combine several of these skills at once.
3Practice 3D vector geometry systematically: dot and cross products, equations of lines and planes, relative positions (parallel, intersecting, skew, coplanar), and distance/angle/volume formulas using the scalar triple product.
4Review probability and statistics tools: conditional probability and tree diagrams, the Total Probability Theorem and Bayes' theorem, and the binomial and normal distributions (including the continuity correction when approximating a binomial with a normal distribution).
5Since the official exam requires full written justification of every step (with the calculator usable only for verification), practice showing complete working, not just final numeric answers, when preparing with this bank.

Frequently Asked Questions

Is this practice bank in the same format as the real Cantabria PAU exam?

No, and it is important to know the difference. The official Cantabria PAU paper for Matemáticas II is a 90-minute exam sat in Spanish and structured around four compulsory open/semi-constructed exercises worth 2.5 points each (Algebra, Mathematical Analysis, Geometry in 3D, and Probability & Statistics), where the Algebra, Analysis, and Probability exercises each offer a choice between two options and the Geometry exercise is a single compulsory task. Candidates must show full written working — a calculator may only be used as an aid and comprobación, and results obtained exclusively from a calculator without written development are not credited. The real exam never uses multiple-choice format. This bank is an English-language multiple-choice study adaptation of the official 2º Bachillerato syllabus — not an official translation, not a past paper, and not a simulation of the exam format. Use it to drill the underlying knowledge and calculation skills quickly, then practise writing out complete, justified solutions with every step shown, because that is what the examiners actually mark.

What is the format of the Cantabria PAU Mathematics II exam?

It is a 90-minute written examination administered at University of Cantabria (UNICAN) designated centers, with four compulsory exercises worth 2.5 points each: Algebra, Mathematical Analysis, Geometry in 3D, and Probability & Statistics. All work must be justified in writing; a calculator may only be used to check results, not to replace shown reasoning.

How is the exam scored for university entrance in Spain?

The exam is graded on a 0–10 scale. In the compulsory Access Phase, a minimum mark of 4.0 is required to combine with Bachillerato GPA (60% Bachillerato + 40% PAU Access Phase >= 5.0). In the Admission Phase, Mathematics II is typically a heavily-weighted subject for STEM, health science, and engineering degree programs.

Which topics carry the most weight?

Based on the University of Cantabria's official June 2026 grading criteria, all four content blocks are weighted equally: Algebra, Mathematical Analysis, Geometry in 3D, and Probability & Statistics each account for 2.5 of the 10 total points (25% each).

How does Matemáticas II differ from Matemáticas Aplicadas a las Ciencias Sociales II in the Cantabria PAU?

Both are 2º Bachillerato mathematics papers administered by UNICAN, but they serve different Bachillerato modalities and cover different content. Matemáticas II is for the Ciencias y Tecnología (Science and Technology) modality and includes 3D vector geometry (lines and planes in space) and more advanced calculus, which Matemáticas Aplicadas a las Ciencias Sociales II (for the Ciencias Sociales modality) does not cover; the latter instead emphasizes financial and social-science applications such as linear programming and statistical inference.

In what language is the Cantabria PAU examination offered?

In Cantabria, official PAU examination papers are provided in Spanish.

What are the registration fees and key dates for the Cantabria PAU?

The standard 2026 registration fee is EUR 71.09 (reduced to EUR 35.55 for large or single-parent families, with additional exemptions for candidates with a disability of 33% or more). For the 2026 ordinary sitting, registration ran May 15–20, with the extraordinary sitting registration running June 19–22.