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Key Facts: PAU Applied Math ACS II (La Rioja) Exam

90 min

Exam Duration

UR PAU Commission

0–10

Grading Scale

Gobierno de La Rioja

EUR 55.43

Base Registration Fee

UR 2026 Rate

4.0 / 10

Min. Access Phase Mark

PAU Regulations

60 / 40

GPA / PAU Weighting

Spanish University Access Regulations

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

Sample PAU Applied Math ACS II (La Rioja) Practice Questions

Try these sample questions to review concepts for the PAU Applied Math ACS II (La Rioja) 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, -2], [1, 5]] and B = [[2, 4], [-1, 3]], calculate C = 3A - 2B.
A.[[5, -14], [5, 9]]
B.[[5, -14], [1, 9]]
C.[[13, 2], [1, 21]]
D.[[5, 2], [5, 9]]
Explanation: Compute 3A = [[9, -6], [3, 15]] and 2B = [[4, 8], [-2, 6]]. Subtracting 2B from 3A gives C = [[9-4, -6-8], [3-(-2), 15-6]] = [[5, -14], [5, 9]].
2If matrix A has dimensions 3 x 2 and matrix B has dimensions 2 x 4, what are the dimensions of matrix product C = A * B?
A.2 x 2
B.3 x 4
C.4 x 3
D.The product is undefined.
Explanation: The product of a (m x n) matrix and an (n x p) matrix is an (m x p) matrix. Here, m = 3, n = 2, and p = 4, so C has dimensions 3 x 4.
3Calculate the determinant of matrix A = [[4, 7], [2, 5]].
A.6
B.-6
C.34
D.13
Explanation: For a 2x2 matrix [[a, b], [c, d]], the determinant is ad - bc. Thus, det(A) = (4)(5) - (7)(2) = 20 - 14 = 6.
4Find the product matrix A * B for A = [[2, 1], [0, 3]] and B = [[1, 4], [2, -1]].
A.[[4, 7], [6, -3]]
B.[[2, 4], [0, -3]]
C.[[4, 7], [6, 3]]
D.[[3, 5], [2, 2]]
Explanation: Element (1,1) = 2(1)+1(2) = 4; (1,2) = 2(4)+1(-1) = 7; (2,1) = 0(1)+3(2) = 6; (2,2) = 0(4)+3(-1) = -3. So A*B = [[4, 7], [6, -3]].
5Compute the determinant of the 3x3 matrix A = [[1, 2, 0], [0, 3, 1], [2, 1, 4]].
A.13
B.11
C.15
D.7
Explanation: Expanding det(A) along row 1: 1 * det([[3, 1], [1, 4]]) - 2 * det([[0, 1], [2, 4]]) + 0 = 1(12 - 1) - 2(0 - 2) = 11 - 2(-2) = 11 + 4 = 15.
6Which matrix identity is always true for invertible matrices A and B of the same size?
A.(A * B)^(-1) = B^(-1) * A^(-1)
B.(A * B)^(-1) = A^(-1) * B^(-1)
C.(A + B)^(-1) = A^(-1) + B^(-1)
D.det(A + B) = det(A) + det(B)
Explanation: The inverse of a matrix product reverses the multiplication order: (AB)^(-1) = B^(-1)A^(-1).
7Find the transpose matrix A^T for A = [[1, 5, 9], [2, 6, 0]].
A.[[1, 2], [5, 6], [9, 0]]
B.[[2, 6, 0], [1, 5, 9]]
C.[[9, 5, 1], [0, 6, 2]]
D.[[1, 6], [5, 0], [9, 2]]
Explanation: The transpose A^T converts rows into columns. Row 1 [1, 5, 9] becomes column 1, and Row 2 [2, 6, 0] becomes column 2, yielding a 3x2 matrix [[1, 2], [5, 6], [9, 0]].
8Solve the matrix equation X + [[2, -1], [4, 3]] = [[5, 2], [1, 7]].
A.[[3, 3], [-3, 4]]
B.[[7, 1], [5, 10]]
C.[[3, 1], [-3, 4]]
D.[[3, 3], [3, 4]]
Explanation: Subtract A from B: X = [[5-2, 2-(-1)], [1-4, 7-3]] = [[3, 3], [-3, 4]].
9Calculate the inverse matrix A^(-1) for 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 inverse formula for [[a, b], [c, d]] is (1/det)[[d, -b], [-c, a]]. So A^(-1) = [[3, -5], [-1, 2]].
10Solve the matrix equation A * X = B for X, where A = [[1, 2], [0, 1]] and B = [[5, 6], [2, 3]].
A.[[1, 0], [2, 3]]
B.[[5, 6], [2, 3]]
C.[[9, 12], [2, 3]]
D.[[1, 1], [2, 3]]
Explanation: Compute A^(-1) = [[1, -2], [0, 1]]. Then X = A^(-1)B = [[1, -2], [0, 1]] * [[5, 6], [2, 3]] = [[1(5)-2(2), 1(6)-2(3)], [0(5)+1(2), 0(6)+1(3)]] = [[1, 0], [2, 3]].

About the PAU Applied Math ACS II (La Rioja) Exam

The La Rioja 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 Organizadora de la PAU de La Rioja / Universidad de La Rioja (UR). 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 the PAU Organising Commission of La Rioja / Universidad de La Rioja featuring core problem blocks: Matrix Algebra, Linear Programming, Real Analysis & Optimization, and Probability & Inferential Statistics (adapted here to a 100-item English MCQ practice format).

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 55.43 base registration fee (UR 2026)

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%

Matrix Algebra & Systems of Linear Equations

Matrix operations, transpose, inverse matrix calculation, solving matrix equations (AX + B = C), systems of linear equations (2x2 and 3x3), Rouché-Capelli classification theorem, and Cramer's rule.

15%

Linear Programming

Inequality constraints, graphical representation of two-variable feasible regions, vertex coordinate calculation, and objective function optimization (finding maximum or minimum points).

30%

Mathematical Analysis & Optimization

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

25%

Probability & Inferential Statistics

Sample spaces, compound events, conditional probability, Law of Total Probability, Bayes' Theorem, binomial distribution B(n, p), normal distribution N(μ, σ) standardization, sample mean and proportion sampling distributions, confidence intervals, and margin of error calculations.

Preparing for the PAU Applied Math ACS II (La Rioja) 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 the PAU Organising Commission of La Rioja / Universidad de La Rioja featuring core problem blocks: Matrix Algebra, Linear Programming, Real Analysis & Optimization, and Probability & Inferential Statistics (adapted here to a 100-item English MCQ practice format).
  • Time limit: 90 minutes (1.5 hours)
  • Exam / certification fees: EUR 55.43 base registration fee (UR 2026) 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 (La Rioja): Suggested Study Strategy

1Master solving matrix equations step by step: isolate X as X = A^(-1)(C - B) or X = (C - B)A^(-1), ensuring correct order of multiplication.
2Practice linear programming vertex methods: plot constraints, find intersection points, and evaluate the objective function at every vertex.
3Drill piecewise function continuity and differentiability: set left and right limits equal at boundary points to determine unknown parameters.
4Review function optimization problems: construct the objective function, locate critical points via f'(x) = 0, and verify maxima/minima using f''(x).
5Memorize standard normal distribution Z-tables and confidence interval formulas for both sample means and sample proportions.

Frequently Asked Questions

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

The official PAU exam organized by Universidad de La Rioja (UR) is a 90-minute written examination consisting of multi-part calculation exercises covering linear algebra, linear programming, analysis/optimization, and probability/statistics. Students must present clear, step-by-step mathematical work.

What calculators are permitted in the La Rioja PAU exam?

Scientific, non-programmable, non-graphical calculators without symbolic computation (CAS) capabilities or text storage features are allowed during the exam.

How is the university access grade calculated in La Rioja?

The Access Grade (Nota de Acceso) is computed as 60% of the 2-year Bachillerato GPA plus 40% of the compulsory PAU Access Phase mark, provided the student obtains at least 4.0 in the PAU phase and an overall score of at least 5.0 out of 10.

How are confidence intervals assessed in the exam?

Questions test calculating confidence intervals for population means or proportions using given sample statistics, finding critical Z-values for specified confidence levels (90%, 95%, 99%), computing margins of error, and determining minimum sample sizes.