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Key Facts: PAU Mathematics II (Galicia) Exam

Exam Administrator: Comisión Interuniversitaria de Galicia (CIUG)

Target Audience: 2º Bachillerato Science & Technology track students in Galicia

Exam Duration: 90 minutes (1.5 hours)

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

Registration Fee: EUR 63.67 ordinary (50% discount for general large family; full exemption for special large family/disability/terrorism victims)

Galicia PAU Mathematics II (CIUG 2026) evaluates 2º Bachillerato linear algebra, differential calculus, integral calculus, and 3D analytical geometry over a 90-minute examination. This multiple-choice bank is a study adaptation of the official written PAU paper, designed to reinforce core mathematical problem-solving skills.

Sample PAU Mathematics II (Galicia) Practice Questions

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

1Calculate the inverse of the matrix A = [[1, 2], [3, 4]].
A.[[-2, 1], [1.5, -0.5]]
B.[[2, -1], [-1.5, 0.5]]
C.[[-4, 2], [3, -1]]
D.[[1, -0.5], [-1.5, 2]]
Explanation: The determinant of A is det(A) = 1*4 - 2*3 = -2. The inverse formula for a 2x2 matrix is A^(-1) = (1/det(A)) * [[d, -b], [-c, a]]. Substituting yields A^(-1) = (-1/2) * [[4, -2], [-3, 1]] = [[-2, 1], [1.5, -0.5]].
2Compute the determinant of the 3x3 matrix A = [[1, 0, 2], [2, 1, 1], [0, 3, -1]].
A.8
B.-4
C.12
D.4
Explanation: Expanding det(A) along the first row gives 1*(1*(-1) - 3*1) - 0 + 2*(2*3 - 0*1) = 1*(-4) + 2*(6) = -4 + 12 = 8.
3Determine the rank of the matrix A = [[1, 2, 3], [2, 4, 6], [1, 1, 1]].
A.2
B.1
C.3
D.0
Explanation: Notice that the second row is twice the first row (R2 = 2*R1), so R2 is linearly dependent. The first and third rows are linearly independent since (1,2,3) is not a scalar multiple of (1,1,1). Therefore, the rank of matrix A is 2.
4Solve the matrix equation A * X = B for X, where A = [[2, 1], [1, 1]] and B = [[5], [3]].
A.X = [[2], [1]]
B.X = [[1], [2]]
C.X = [[3], [-1]]
D.X = [[5], [3]]
Explanation: The determinant det(A) = 2*1 - 1*1 = 1, so A^(-1) = [[1, -1], [-1, 2]]. Multiplying A^(-1) by B yields X = [[1*5 - 1*3], [-1*5 + 2*3]] = [[2], [1]].
5According to the Rouché-Fröbenius theorem, a linear system with n unknowns A*X = B has a unique solution (compatible determinado) if and only if:
A.rank(A) = rank(A|B) = n
B.rank(A) = rank(A|B) < n
C.rank(A) != rank(A|B)
D.rank(A) = n regardless of rank(A|B)
Explanation: By the Rouché-Fröbenius theorem, a system is compatible if rank(A) = rank(A|B). When this common rank equals the number of variables n, the system has a unique solution (compatible determinado).
6For which values of parameter k is the matrix A = [[k, 1, 1], [1, k, 1], [1, 1, k]] singular (non-invertible)?
A.k = 1 and k = -2
B.k = 0 and k = 2
C.k = -1 and k = 2
D.k = 3 only
Explanation: Calculating the determinant yields det(A) = k^3 - 3k + 2 = (k + 2)(k - 1)^2. Setting det(A) = 0 yields roots k = 1 (double root) and k = -2. Thus A is singular when k = 1 or k = -2.
7Use Cramer's Rule to solve for x in the system: x + y = 3, 2x - y = 3.
A.x = 2
B.x = 1
C.x = 3
D.x = -1
Explanation: The coefficient determinant is Delta = 1*(-1) - 1*2 = -3. The determinant for x is Delta_x = 3*(-1) - 1*3 = -6. By Cramer's Rule, x = Delta_x / Delta = -6 / -3 = 2.
8If A is a 3x3 matrix with det(A) = 4, what is the value of det(2A)?
A.32
B.8
C.12
D.24
Explanation: For any n x n matrix A and scalar c, det(c*A) = c^n * det(A). For n = 3 and c = 2, det(2A) = 2^3 * det(A) = 8 * 4 = 32.
9If A is an invertible matrix with det(A) = 5, what is det(A^(-1))?
A.0.2 (or 1/5)
B.-5
C.5
D.-0.2
Explanation: The determinant of the inverse matrix satisfies det(A^(-1)) = 1 / det(A). Given det(A) = 5, det(A^(-1)) = 1/5 = 0.2.
10Classify the homogeneous system: x + 2y - z = 0, 2x - y + 3z = 0, 3x + y + 2z = 0.
A.Compatible indeterminado (infinitely many solutions)
B.Compatible determinado (only trivial solution x=y=z=0)
C.Incompatible (no solutions)
D.Compatible determinado with x=1, y=2, z=3
Explanation: Every homogeneous system is compatible. Evaluating the coefficient determinant det(A) = 1*(-2-3) - 2*(4-9) - 1*(2+3) = -5 + 10 - 5 = 0. Since det(A) = 0, rank(A) < 3, so the system has infinitely many solutions (compatible indeterminado).

About the PAU Mathematics II (Galicia) Exam

The Galicia PAU Mathematics II (Matemáticas II) examination assesses 2º Bachillerato competencies in Linear Algebra (matrices, determinants, inverse matrices, matrix equations, systems of linear equations, Rouché-Fröbenius 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: Comisión Interuniversitaria de Galicia (CIUG). 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 >= 5.0 to pass).

Exam / Certification Fees

EUR 63.67 ordinary registration fee for PAU (50% discount EUR 31.84 for general large family status; full exemption for special large family, disability, or terrorism victims).

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, determinant properties, matrix rank, classification of linear systems (Rouché-Fröbenius 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 (Galicia) 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: 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 63.67 ordinary registration fee for PAU (50% discount EUR 31.84 for general large family status; full exemption for special large family, disability, or terrorism victims). 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 (Galicia): Suggested Study Strategy

1Master matrix rank determination and parameter discussion using the Rouché-Fröbenius 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 distance calculations.
5Ensure clear step-by-step mathematical reasoning when solving equations and geometric projections.

Frequently Asked Questions

What is the structure of the Galicia PAU Mathematics II exam?

The official exam is a 90-minute written examination administered by CIUG (Comisión Interuniversitaria de Galicia) consisting of open-ended problem-solving exercises across linear algebra, differential calculus, integral calculus, and 3D analytical geometry.

Is a calculator permitted in the PAU Mathematics II exam in Galicia?

Scientific calculators are permitted provided they do not have graphic display, symbolic manipulation (CAS capabilities), programmable memory, or networking capabilities, following CIUG regulatory guidelines.

How is the exam scored and what is the passing threshold?

The exam is marked on a 0–10 scale. A minimum mark of 4.0 in the Access Phase is required to combine with the Bachillerato GPA (weighted 60% Bachillerato + 40% PAU Access Phase >= 5.0 to pass).

Is this practice bank in the same format as the official PAU Mathematics II (Galicia) exam?

No. The official CIUG PAU Mathematics II paper is a written examination with open-ended development problems. This 100-question multiple-choice bank is a study adaptation designed to thoroughly test and reinforce problem-solving skills across the 2º Bachillerato Science & Technology track curriculum.