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Key Facts: PAU Applied Mathematics for Social Sciences II (Cantabria) Exam

90 min

Exam time duration for the PAU modality-subject paper

University of Cantabria (UNICAN) PAU Organising Commission

5 Blocks

Core syllabus areas: Matrices, Linear Programming, Analysis, Integrals, Probability

2º Bachillerato Curriculum (LOMLOE), Ministerio de Educación

0–10 scale

Scoring system (minimum 4.0 required in Access Phase)

Cantabria PAU Regulations

EUR 71.09

Base registration fee for PAU Access Phase in Cantabria

University of Cantabria (UNICAN)

100

High-quality practice questions in this OpenExamPrep subject bank

OpenExamPrep

Master the 2026 Cantabria PAU Applied Mathematics for Social Sciences II exam with 100 practice questions covering matrix algebra, linear programming, calculus, integration, and probability. This English-language MCQ bank is a study adaptation of the theory and calculation methods behind the official written exam, not a format simulation.

Sample PAU Applied Mathematics for Social Sciences II (Cantabria) Practice Questions

Try these sample questions to review concepts for the PAU Applied Mathematics for Social Sciences II (Cantabria) 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, 5], [-2, 0]], calculate 2A - 3B.
A.[[3, -17], [10, 8]]
B.[[5, 13], [-2, 8]]
C.[[3, -7], [2, 8]]
D.[[4, 4], [0, 4]]
Explanation: First compute 2A = [[6, -2], [4, 8]] and 3B = [[3, 15], [-6, 0]]. Subtracting componentwise yields 2A - 3B = [[6-3, -2-15], [4-(-6), 8-0]] = [[3, -17], [10, 8]].
2Compute the matrix product AB for A = [[2, 1], [0, 3]] and B = [[4, -2], [1, 5]].
A.[[8, -2], [0, 15]]
B.[[9, 1], [3, 15]]
C.[[8, 1], [3, 15]]
D.[[9, 1], [1, 15]]
Explanation: Multiply rows of A by columns of B: row 1 gives [2(4)+1(1), 2(-2)+1(5)] = [9, 1]; row 2 gives [0(4)+3(1), 0(-2)+3(5)] = [3, 15]. Thus AB = [[9, 1], [3, 15]].
3Calculate the determinant of the matrix A = [[6, -3], [4, 2]].
A.0
B.-24
C.24
D.18
Explanation: For a 2x2 matrix [[a, b], [c, d]], det(A) = ad - bc. Here det(A) = (6)(2) - (-3)(4) = 12 - (-12) = 24.
4Given A = [[1, 2], [3, 4]], compute the matrix sum A + A^T.
A.[[2, 4], [6, 8]]
B.[[1, 5], [5, 4]]
C.[[2, 6], [4, 8]]
D.[[2, 5], [5, 8]]
Explanation: The transpose A^T is [[1, 3], [2, 4]]. Adding A and A^T entrywise yields [[1+1, 2+3], [3+2, 4+4]] = [[2, 5], [5, 8]].
5A company has two stores. Store 1 sells 10 units of product X and 20 of product Y. Store 2 sells 15 of X and 25 of Y. Prices are €5 for X and €8 for Y. Which matrix multiplication represents total revenue per store?
A.[[10, 20], [15, 25]] * [[5], [8]] = [[210], [275]]
B.[[10, 20], [15, 25]] * [[8], [5]] = [[180], [245]]
C.[[10, 15], [20, 25]] * [[5], [8]] = [[200], [275]]
D.[[5, 8]] * [[10, 20], [15, 25]] = [[170, 300]]
Explanation: Represent sales by matrix M = [[10, 20], [15, 25]] (rows = stores, columns = products) and price by vector P = [[5], [8]]. Revenue vector MP = [[10(5)+20(8)], [15(5)+25(8)]] = [[210], [275]] euros.
6Find the determinant of the lower triangular matrix A = [[3, 0, 0], [1, 4, 0], [5, -2, -2]].
A.24
B.-24
C.0
D.-6
Explanation: The determinant of a triangular matrix equals the product of its main diagonal entries: det(A) = 3 * 4 * (-2) = -24.
7Find the inverse matrix A^(-1) for A = [[4, 3], [1, 1]].
A.[[-1, 3], [1, -4]]
B.[[1, 1], [3, 4]]
C.[[1, -3], [-1, 4]]
D.[[4, -3], [-1, 1]]
Explanation: First calculate det(A) = 4(1) - 3(1) = 1. Using the 2x2 inverse formula A^(-1) = (1/det(A)) * [[d, -b], [-c, a]], we get [[1, -3], [-1, 4]].
8Calculate the determinant of the 3x3 matrix A = [[1, 2, 0], [3, -1, 4], [2, 1, 1]] using Sarrus's rule.
A.15
B.-5
C.10
D.5
Explanation: Sarrus's rule: det(A) = (1*-1*1 + 2*4*2 + 0*3*1) - (0*-1*2 + 1*4*1 + 2*3*1) = (-1 + 16 + 0) - (0 + 4 + 6) = 15 - 10 = 5.
9Solve the matrix equation A X = B for vector X, where A = [[2, 1], [3, 2]] and B = [[5], [7]].
A.[[3], [-1]]
B.[[-3], [1]]
C.[[17], [29]]
D.[[1], [3]]
Explanation: det(A) = 2(2) - 1(3) = 1. The inverse A^(-1) = [[2, -1], [-3, 2]]. Multiply X = A^(-1) B = [[2(5) - 1(7)], [-3(5) + 2(7)]] = [[10-7], [-15+14]] = [[3], [-1]].
10Determine the rank of the matrix A = [[1, 1, 1], [2, 2, k], [3, 3, 3]] as a function of the parameter k.
A.Rank is always 3 for any real k
B.Rank is 1 if k = 2, and rank is 2 if k ≠ 2
C.Rank is 2 if k = 2, and rank is 3 if k ≠ 2
D.Rank is 0 if k = 2, and rank is 1 if k ≠ 2
Explanation: Row 3 is 3 times Row 1, so rows 1 and 3 are linearly dependent. If k = 2, Row 2 is 2 times Row 1, making all rows proportional (rank 1). If k ≠ 2, Row 2 is independent of Row 1, so rank is 2.

About the PAU Applied Mathematics for Social Sciences II (Cantabria) Exam

Comprehensive 100-question practice test bank for the Cantabria (Universidad de Cantabria / UNICAN) PAU exam in Applied Mathematics for Social Sciences II (Matemáticas Aplicadas a las Ciencias Sociales II). Covers matrix algebra, linear programming, mathematical analysis, integral calculus, and probability & statistics drawn from the official Real Decreto 243/2022 and Real Decreto 534/2024 saberes básicos as applied by UNICAN.

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 Applied Mathematics for Social Sciences II (UNICAN) runs 90 minutes and evaluates candidate proficiency in social-science mathematics: matrix calculations, linear programming optimization, calculus and economic function analysis, integration of revenue/cost functions, and probability modeling.

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.

20%

Matrices, Determinants & Linear Systems

Matrix operations, matrix equations, inverse matrix calculation, 2x2 and 3x3 determinants, matrix rank, system classification via Rouché-Capelli theorem, Cramer's rule, and input-output production modeling.

20%

Linear Programming & Feasible Regions

Formulation of economic objective functions, system of linear constraints, determination of convex feasible polygonal regions, vertex calculation, and optimization (profit maximization / cost minimization).

20%

Analysis: Limits, Continuity, Derivatives & Optimization

Limits at infinity and points, continuity of piecewise functions (tax tiers, cost structures), differential calculus, marginal cost/revenue, local extrema, and economic profit optimization.

20%

Integration, Definite Integrals & Area Calculations

Indefinite integration rules, initial value problems (total cost from marginal cost), definite integrals, Fundamental Theorem of Calculus, and area enclosed by revenue, cost, or demand curves.

20%

Probability, Conditional Probability, Bayes Theorem & Normal Distribution

Tree diagrams, conditional probability, Law of Total Probability, Bayes' Theorem, Binomial distribution B(n, p), Normal distribution N(mu, sigma), standardization, and normal approximation to binomial.

Preparing for the PAU Applied Mathematics for Social Sciences 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 Applied Mathematics for Social Sciences II (UNICAN) runs 90 minutes and evaluates candidate proficiency in social-science mathematics: matrix calculations, linear programming optimization, calculus and economic function analysis, integration of revenue/cost functions, and probability modeling.
  • 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 Applied Mathematics for Social Sciences II (Cantabria): Suggested Study Strategy

1Master matrix operations and determinant rules, especially verifying whether a matrix is invertible before solving matrix equations of the form AX + B = C.
2Practice drawing linear programming feasible regions accurately on grid axes, identifying vertex coordinates by solving system intersections, and testing all corner points in the objective function.
3Review piecewise function continuity and differentiability conditions thoroughly, as piecewise tax or cost models frequently appear on PAU exams.
4Understand the economic interpretations of calculus: marginal cost is C'(x), total cost is the integral of marginal cost plus fixed costs C(0), and profit is maximum when marginal revenue equals marginal cost.
5Memorize Bayes' Theorem and practice standardizing normal random variables Z = (X - mu) / sigma, including continuity corrections for normal approximations to binomial distributions.

Frequently Asked Questions

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

No. The official Cantabria PAU paper for Applied Mathematics for Social Sciences II is a 90-minute written examination featuring open-ended mathematical problems where candidates must show step-by-step working, sketch feasible regions, and justify conclusions. This bank is an English-language multiple-choice adaptation designed to test core mathematical rules, concepts, and calculation steps efficiently.

Who takes Applied Mathematics for Social Sciences II in Cantabria?

Students completing 2º Bachillerato in the Humanities and Social Sciences (Humanidades y Ciencias Sociales) track take this exam during the PAU Access Phase (Fase General) as their compulsory modality subject, or as an elective subject in the Admission Phase (Fase de Admisión).

What software or calculator is permitted in the Cantabria PAU exam?

In Cantabria PAU exams, non-programmable calculators without graphing, symbolic calculation (CAS), or matrix inversion capabilities are allowed, subject to official UNICAN guidelines.

How is the final PAU score calculated?

The Access Phase grade is calculated as 60% Bachillerato GPA + 40% PAU Access Phase average. Candidates must score at least 4.0 in the Access Phase and obtain an overall grade >= 5.0 to qualify for university admission.

What content blocks are tested on this paper?

The official UNICAN syllabus covers five main areas: Matrix Algebra & Linear Systems, Linear Programming, Mathematical Analysis & Optimization, Integral Calculus & Areas, and Probability & Statistics.