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Key Facts: PAU Applied Math ACS II (Galicia) Exam

90 min

Exam Duration

CIUG Regulations

0–10

Grading Scale

Xunta de Galicia / CIUG

4.0 / 10

Min. Access Phase Mark

PAU Access Rules

60 / 40

GPA / PAU Weighting

Spanish Education Ministry & CIUG

EUR 63.67

Ordinary Registration Fee

CIUG Official Tariff

The Galicia PAU Applied Mathematics for Social Sciences II exam (Matemáticas Aplicadas a las Ciencias Sociales II 2º Bachillerato) is administered by CIUG (Comisión Interuniversitaria de Galicia). Lasting 90 minutes, it tests linear algebra & matrix equations, linear programming, real analysis & optimization, probability & Bayes' theorem, and inferential statistics. This 100-question practice set offers a comprehensive English MCQ adaptation of the official 2026 CIUG syllabus.

Sample PAU Applied Math ACS II (Galicia) Practice Questions

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

1Given matrices A = [[3, -1], [2, 4]] and B = [[1, 4], [-3, 2]], compute matrix C = 3A - 2B.
A.[[7, -11], [12, 8]]
B.[[7, 5], [0, 8]]
C.[[7, -11], [0, 8]]
D.[[11, -11], [12, 16]]
Explanation: Multiply matrix A by 3 to obtain 3A = [[9, -3], [6, 12]] and matrix B by 2 to obtain 2B = [[2, 8], [-6, 4]]. Subtracting 2B from 3A entry by entry yields C = [[9-2, -3-8], [6-(-6), 12-4]] = [[7, -11], [12, 8]].
2Calculate the matrix product A * B for A = [[1, 2], [-1, 3]] and B = [[2, 0], [1, -2]].
A.[[4, -4], [1, -6]]
B.[[2, 0], [-1, -6]]
C.[[4, 0], [1, -6]]
D.[[3, -4], [2, -6]]
Explanation: Compute entry (1,1) as 1(2) + 2(1) = 4, entry (1,2) as 1(0) + 2(-2) = -4, entry (2,1) as -1(2) + 3(1) = 1, and entry (2,2) as -1(0) + 3(-2) = -6. The product matrix is [[4, -4], [1, -6]].
3For the matrix A = [[4, -2, 5], [1, 3, 0], [-1, 2, 6]], calculate the sum of the trace of A and the trace of its transpose A^T.
A.26
B.13
C.0
D.39
Explanation: The trace of a square matrix is the sum of its main diagonal elements: Trace(A) = 4 + 3 + 6 = 13. Since taking the transpose does not change diagonal entries, Trace(A^T) = Trace(A) = 13. The total sum is 13 + 13 = 26.
4Given matrix A = [[2, 1], [0, 2]] and matrix B = [[x, y], [0, x]], under what conditions on real numbers x and y does AB = BA hold?
A.For all real numbers x and y
B.Only when x = 0
C.Only when y = 0
D.Only when x = y
Explanation: Multiplying AB gives [[2x, 2y + x], [0, 2x]]. Multiplying BA gives [[2x, x + 2y], [0, 2x]]. Comparing corresponding entries shows 2y + x = x + 2y for all real x and y, so AB = BA holds universally for these matrices.
5Find all real values of k for which the matrix A = [[k-1, 4], [2, k+1]] has a determinant equal to zero.
A.k = 3 and k = -3
B.k = 9 and k = -9
C.k = 1 and k = -1
D.k = 5 and k = -5
Explanation: The determinant is calculated as (k-1)(k+1) - (4)(2) = k^2 - 1 - 8 = k^2 - 9. Setting k^2 - 9 = 0 gives k^2 = 9, so k = 3 or k = -3.
6Calculate the determinant of the 3x3 matrix A = [[1, 0, 2], [2, 1, 1], [0, 3, -1]].
A.8
B.-8
C.4
D.12
Explanation: Using cofactor expansion along the first row: det(A) = 1*(1*(-1) - 3*1) - 0 + 2*(2*3 - 0*1) = 1*(-4) + 2*(6) = -4 + 12 = 8.
7Find the inverse matrix A^(-1) of the matrix A = [[3, 1], [5, 2]].
A.[[2, -1], [-5, 3]]
B.[[-2, 1], [5, -3]]
C.[[3, -1], [-5, 2]]
D.[[2, 1], [5, 3]]
Explanation: The determinant det(A) = (3)(2) - (1)(5) = 6 - 5 = 1. The inverse is (1/det(A)) * [[d, -b], [-c, a]] = [[2, -1], [-5, 3]].
8If A is a 3x3 matrix with det(A) = 4, calculate det(2 * A^(-1) * A^T).
A.8
B.4
C.16
D.2
Explanation: Using properties of determinants for an n x n matrix with n = 3: det(2 * A^(-1) * A^T) = 2^3 * det(A^(-1)) * det(A^T) = 8 * (1/4) * 4 = 8.
9Given matrix A = [[1, 1], [0, 1]], calculate the matrix power A^10.
A.[[1, 10], [0, 1]]
B.[[1, 1], [0, 1]]
C.[[1, 1024], [0, 1]]
D.[[10, 10], [0, 10]]
Explanation: Computing powers: A^2 = [[1, 2], [0, 1]], A^3 = [[1, 3], [0, 1]], and by induction A^n = [[1, n], [0, 1]]. For n = 10, A^10 = [[1, 10], [0, 1]].
10If the matrix A = [[2, x+1], [5, 3]] is symmetric, find the value of x.
A.4
B.5
C.3
D.2
Explanation: A matrix is symmetric if A = A^T, which requires corresponding off-diagonal elements to be equal: x + 1 = 5, yielding x = 4.

About the PAU Applied Math ACS II (Galicia) Exam

The Galicia PAU Applied Mathematics for Social Sciences II (Matemáticas Aplicadas a las Ciencias Sociales II) examination evaluates 2nd Bachillerato students on essential mathematical tools for business, economics, and social sciences. Key topics include matrix algebra, matrix equations, Rouché-Capelli classification of linear systems, 2-variable linear programming, differential analysis, function optimization, definite integral area calculation, conditional probability, Bayes' theorem, normal distributions, and confidence intervals for population parameters.

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

Standardized 90-minute written examination organized by CIUG featuring four core problem blocks: Linear Algebra & Programming, Real Analysis & Optimization, Probability, and Inferential Statistics.

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.

30%

Matrices, Linear Systems & Linear Programming

Matrix operations, transpose, inverse matrix calculation, solving matrix equations (AX + B = C), systems of linear equations (2x2 and 3x3), Rouché-Capelli classification theorem, Cramer's rule, 2D feasible regions, inequality boundaries, vertices, and objective function optimization.

30%

Mathematical Analysis & Optimization

Limits, indeterminate forms, continuity of piecewise functions, derivatives, tangent line equations, monotonicity, local extrema, inflection points, real-world economic optimization problems (revenue/cost/profit), antiderivatives, and definite integral area calculations.

20%

Probability & Tree Diagrams

Sample spaces, compound events, conditional probability, Law of Total Probability, Bayes' Theorem, tree diagrams, and independence of events.

20%

Inferential Statistics & Sampling

Binomial distribution B(n, p), normal distribution N(μ, σ) standardization, sample mean and proportion sampling distributions, confidence intervals for mean and proportion, margin of error, and sample size calculations.

Preparing for the PAU Applied Math ACS 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: Standardized 90-minute written examination organized by CIUG featuring four core problem blocks: Linear Algebra & Programming, Real Analysis & Optimization, Probability, and Inferential Statistics.
  • 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 Applied Math ACS II (Galicia): Suggested Study Strategy

1Practice matrix equations carefully: remember that matrix multiplication is non-commutative, so AX = B implies X = A^(-1)B while XA = B implies X = BA^(-1).
2Systematically identify vertices in 2D linear programming: graph the linear inequalities, solve system of equations for intersection points, and evaluate the objective function at every vertex.
3Check continuity and differentiability at piecewise boundaries: ensure left-hand limit equals right-hand limit, and left-hand derivative equals right-hand derivative.
4Set up economic optimization functions rigorously: define the objective function in one variable (e.g. Profit = Revenue - Cost), find critical points where P'(x) = 0, and confirm max/min using the second derivative test.
5Master normal distribution tables and confidence interval formulas: know how to calculate z_(alpha/2) for common confidence levels (90% -> 1.645, 95% -> 1.96, 99% -> 2.575) and apply E = z_(alpha/2) * (sigma / sqrt(n)) or E = z_(alpha/2) * sqrt(p*(1-p)/n).

Frequently Asked Questions

What is the format of the official Galicia PAU Applied Mathematics for Social Sciences II exam?

The official PAU exam organized by CIUG is a 90-minute written examination consisting of structured questions worth 10 points in total. It covers linear algebra & programming, analysis & optimization, probability, and inferential statistics. Students must provide detailed step-by-step mathematical work.

What calculators are allowed in the Galicia PAU exam?

Non-programmable, non-graphical scientific calculators are allowed. Calculators capable of symbolic algebraic manipulation (CAS), matrix calculations, or storing text in memory are strictly forbidden by CIUG regulations.

How is the university access mark calculated in Galicia?

The Access Mark (Nota de Acceso) combines 60% of the student's 2-year Bachillerato GPA and 40% of the compulsory PAU phase average, provided the student obtains at least a 4.0 in the PAU phase and a final score of at least 5.0 out of 10.

How are confidence intervals and sample sizes tested in CIUG PAU?

Questions typically specify a sample mean or sample proportion along with sample size or standard deviation and a confidence level (e.g. 95% or 99%). Students must determine the critical value z_(α/2), calculate the margin of error, specify the interval, or determine the minimum required sample size for a given margin of error.