PracticeVideosBlogFlashcardsEspañol
All Practice Exams

Free Practice Questions for PAU Mathematics II (Castilla-La Mancha)

Exam-style questions and explanations by OpenExamPrep.

✓ No registration✓ No credit card
100+ Questions
100% Free

Loading practice questions...

Same family resources

Explore More Spain Castilla-La Mancha PAU (Prueba de Acceso a la Universidad)

Continue into nearby exams from the same family. Each card keeps practice questions, study guides, flashcards, videos, and articles in one place.

Castilla-La Mancha PAU Applied Mathematics for Social Sciences II

Practice Questions

Castilla-La Mancha PAU Artistic Drawing II

Practice Questions

Castilla-La Mancha PAU Artistic Foundations

Practice Questions

Castilla-La Mancha PAU Biology

Practice Questions

Castilla-La Mancha PAU Business and Business Model Design

Practice Questions

Castilla-La Mancha PAU Chemistry

Practice Questions
Exam Review

Key Facts: PAU Mathematics II (Castilla-La Mancha) Exam

Exam Administrator: University of Castilla-La Mancha PAU Tribunal

Target Audience: 2º Bachillerato Science & Technology track students in Castilla-La Mancha

Exam Duration: 90 minutes (1.5 hours)

Question Bank Size: Exactly 100 high-quality worked calculation questions

Scoring Scale: 0-10 scale (Access Phase passing minimum 4.0)

Castilla-La Mancha PAU Mathematics II (UCLM 2026) evaluates 2º Bachillerato linear algebra, differential calculus, integral calculus, and 3D analytical geometry over a 90-minute examination. This English-language multiple-choice bank is a study adaptation of the official written PAU paper, not a replica of it.

Sample PAU Mathematics II (Castilla-La Mancha) Practice Questions

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

1Given the matrices A = [[3, -1], [2, 4]] and B = [[1, 2], [-3, 0]], calculate the matrix operation 2A - 3B.
A.[[3, -8], [13, 8]]
B.[[5, -8], [13, 8]]
C.[[3, -8], [1, 8]]
D.[[3, 4], [13, 8]]
Explanation: Compute 2A = [[6, -2], [4, 8]] and 3B = [[3, 6], [-9, 0]]. Subtracting elementwise: 2A - 3B = [[6-3, -2-6], [4-(-9), 8-0]] = [[3, -8], [13, 8]].
2Let A = [[1, 2], [0, -1]] and B = [[3, 1], [4, 2]]. Compute the matrix product A · B.
A.[[3, 2], [0, -2]]
B.[[11, 5], [-4, -2]]
C.[[11, 5], [4, -2]]
D.[[7, 5], [-4, -2]]
Explanation: Compute row-column dot products: Row 1: [1*3 + 2*4, 1*1 + 2*2] = [11, 5]. Row 2: [0*3 + (-1)*4, 0*1 + (-1)*2] = [-4, -2]. Thus A · B = [[11, 5], [-4, -2]].
3Calculate the determinant of the 3x3 matrix A = [[1, 0, 2], [2, -1, 3], [4, 1, 0]].
A.9
B.-1
C.5
D.13
Explanation: Using Sarrus' Rule: det(A) = 1*(-1)*0 + 0*3*4 + 2*2*1 - (2*(-1)*4 + 0*2*0 + 1*3*1) = (0 + 0 + 4) - (-8 + 0 + 3) = 4 - (-5) = 9.
4If A is a 3x3 matrix with determinant det(A) = -5, what is the determinant of the scalar matrix 2A?
A.-10
B.-40
C.-20
D.40
Explanation: For an n x n matrix, det(k A) = k^n det(A). Here n = 3 and k = 2, so det(2A) = 2^3 · det(A) = 8 · (-5) = -40.
5Find the inverse of the matrix A = [[2, 5], [1, 3]].
A.[[3, -5], [-1, 2]]
B.[[-3, 5], [1, -2]]
C.[[3, 1], [5, 2]]
D.[[2, -5], [-1, 3]]
Explanation: det(A) = 2*3 - 5*1 = 6 - 5 = 1. The formula for the inverse of [[a, b], [c, d]] is (1/det) [[d, -b], [-c, a]]. Thus A^(-1) = [[3, -5], [-1, 2]].
6Given the matrix A = [[2, 1], [-1, 3]], calculate det(A^T · A).
A.49
B.7
C.14
D.25
Explanation: det(A^T · A) = det(A^T) · det(A) = (det(A))^2. Computing det(A) = 2*3 - 1*(-1) = 6 + 1 = 7. Thus det(A^T · A) = 7^2 = 49.
7Solve the matrix equation A · X = B for X, where A = [[1, 2], [0, 1]] and B = [[5, 6], [3, 4]].
A.[[-1, -2], [3, 4]]
B.[[11, 14], [3, 4]]
C.[[-1, 2], [3, 4]]
D.[[1, 2], [3, 4]]
Explanation: X = A^(-1) · B. The inverse A^(-1) = [[1, -2], [0, 1]]. Multiplying: X = [[1, -2], [0, 1]] · [[5, 6], [3, 4]] = [[1*5 - 2*3, 1*6 - 2*4], [0*5 + 1*3, 0*6 + 1*4]] = [[-1, -2], [3, 4]].
8Classify the system of linear equations: 2x + 3y = 7 and 4x + 6y = 14.
A.Compatible indeterminado (infinitely many solutions)
B.Compatible determinado (unique solution)
C.Incompatible (no solution)
D.Trivial system
Explanation: The coefficient matrix A = [[2, 3], [4, 6]] has rank 1 (since Row 2 = 2 · Row 1). The augmented matrix A|B = [[2, 3, 7], [4, 6, 14]] also has rank 1. Since rank(A) = rank(A|B) = 1 < n = 2 variables, by Rouché-Capelli the system is compatible indeterminado.
9For what value of k does the matrix A = [[1, 2, 1], [2, 4, k], [1, 2, 3]] have rank equal to 1?
A.k = 2
B.k = 6
C.k = 0
D.No such value of k exists
Explanation: For rank(A) = 1, all 2x2 minors must vanish. Looking at the minor formed by columns 1 and 3 in rows 1 and 3: [[1, 1], [1, 3]], its determinant is 1*3 - 1*1 = 2 ≠ 0. Since this 2x2 minor does not depend on k and is non-zero, rank(A) is at least 2 for every value of k. Thus, no value of k makes rank(A) = 1.
10Find all real values of m for which the matrix A = [[m, 1, 0], [1, m, 1], [0, 1, m]] is NOT invertible.
A.m = 0, m = √2, m = -√2
B.m = 0, m = 1, m = -1
C.m = 1, m = -2
D.m = 0, m = 2
Explanation: A is non-invertible iff det(A) = 0. Expanding det(A) along row 1: det(A) = m(m^2 - 1) - 1(m - 0) + 0 = m^3 - m - m = m^3 - 2m = m(m^2 - 2). Setting det(A) = 0 yields m = 0 or m = ±√2.

About the PAU Mathematics II (Castilla-La Mancha) Exam

The Castilla-La Mancha PAU Mathematics II (Matemáticas II) examination assesses 2º Bachillerato competencies in Linear Algebra (matrices, determinants, inverse matrices, matrix equations, systems of linear equations, Rouché-Capelli theorem, Cramer's rule), Differential Calculus & Function Analysis (domain, limits, L'Hôpital's rule, continuity, differentiability, Bolzano and Rolle theorems, mean value theorem, asymptotes, monotonic intervals, extrema, inflection points, optimization), Integral Calculus & Area Computation (indefinite integrals, integration techniques, definite integrals, Fundamental Theorem of Calculus, area enclosed by curves), and 3D Analytical Geometry in Space (vectors, scalar and vector products, equations of lines and planes, relative positions, angles, projections, distance formulas).

Exam sponsor: University of Castilla-La Mancha PAU Tribunal. The requirements and fees below concern the certification or admission exam, separate from our free practice resources.

Assessment

Written 90-minute standardized examination featuring problem-solving tasks across Linear Algebra, Differential Calculus, Integral Calculus, and 3D Analytical Geometry.

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 52.99 base registration fee for PAU Access Phase (ordinary sitting, set by Universidad de Castilla-La Mancha).

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%

Matrices, Determinants & Systems of Linear Equations

Matrix operations, matrix algebra, matrix inversion, determinants properties, matrix rank, classification of linear systems (Rouché-Capelli theorem), and solving systems via Cramer's rule or inverse matrices.

25%

Differential Calculus & Function Analysis

Limits, L'Hôpital's rule, continuity, differentiability, derivative calculations, tangent and normal lines, Rolle's and Mean Value theorems, curve sketching (asymptotes, extrema, inflection points), and real-world optimization problems.

25%

Integral Calculus & Area Computation

Indefinite integrals (substitution, integration by parts, rational functions), definite integrals, Fundamental Theorem of Calculus, and computing areas bounded by curves and axes.

25%

3D Analytical Geometry in Space

Vector operations (dot product, cross product, scalar triple product), line and plane equations in 3D space, relative positions of lines and planes, perpendicular and parallel lines/planes, angles, orthogonal projections, and distance calculations (point-to-plane, line-to-line).

Preparing for the PAU Mathematics II (Castilla-La Mancha) 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: Written 90-minute standardized examination featuring problem-solving tasks across Linear Algebra, Differential Calculus, Integral Calculus, and 3D Analytical Geometry.
  • Time limit: 90 minutes (1.5 hours)
  • Exam / certification fees: EUR 52.99 base registration fee for PAU Access Phase (ordinary sitting, set by Universidad de Castilla-La Mancha). 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 (Castilla-La Mancha): Suggested Study Strategy

1Master matrix rank determination and parameter discussion using the Rouché-Capelli theorem.
2Practice derivative rules and L'Hôpital's rule for indeterminate limit forms (0/0 and ∞/∞).
3Review integration by parts and partial fraction decomposition for definite integral area problems.
4Memorize 3D geometry distance formulas and vector cross product applications for area and volume calculations.

Frequently Asked Questions

What is the structure of the Castilla-La Mancha PAU Mathematics II exam?

The official exam is a 90-minute written examination consisting of problem-solving exercises across linear algebra, calculus, integration, and 3D analytical geometry.

Is a calculator permitted in the PAU Mathematics II exam in Castilla-La Mancha?

Scientific calculators are permitted provided they do not have graphic display, symbolic manipulation (CAS), programmable memory, or communication capabilities, according to UCLM PAU tribunal guidelines.

How is the exam scored?

It is scored on a 0-10 scale. A minimum of 4.0 in the Access Phase is required to average with Bachillerato GPA (60% Bachillerato + 40% PAU >= 5.0).

Is this practice bank in the same format as the real PAU Mathematics II (Castilla-La Mancha) exam?

No. The official PAU Mathematics II (Castilla-La Mancha) paper is a 90-minute written examination in Spanish consisting of open-ended, semi-constructed, and short-answer questions — not multiple choice. This bank is an English-language multiple-choice adaptation designed to test and reinforce the core concepts and skills of the 2º Bachillerato curriculum.