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Key Facts: Madrid PAU Applied Mathematics for Social Sciences II Exam

Madrid PAU

PAU Organising Commission of Madrid / UCM

Community of Madrid Education Department

90 Mins

Official examination duration

Madrid PAU Structure

Min 4.0

Minimum Access Phase mark to calculate admission grade

Spanish University Regulations

5 Core Blocks

Algebra & LP, Limits, Calculus, Probability, Inference

LOMLOE 2º Bach CCSS II Syllabus

100

Practice questions available in this OpenExamPrep bank

OpenExamPrep

Madrid PAU Applied Mathematics for Social Sciences II tests 2º Bachillerato students on matrix operations, linear programming, calculus applications, probability, and inferential statistics for university entry in Madrid.

Sample Madrid PAU Applied Mathematics for Social Sciences II Practice Questions

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

1Calculate the sum of matrices A = [[2, 5], [1, -3]] and B = [[4, -1], [3, 2]].
A.[[6, 4], [4, -1]]
B.[[8, -5], [3, -6]]
C.[[6, 6], [2, -5]]
D.[[2, 4], [4, 1]]
Explanation: Matrix addition is performed entrywise: A + B = [[2+4, 5+(-1)], [1+3, -3+2]] = [[6, 4], [4, -1]].
2Given A = [[3, -2], [4, 1]], evaluate the matrix 3A - 2I, where I is the 2x2 identity matrix.
A.[[7, -6], [12, 1]]
B.[[7, -6], [10, 1]]
C.[[9, -6], [12, 3]]
D.[[1, -2], [4, -1]]
Explanation: 3A = [[9, -6], [12, 3]] and 2I = [[2, 0], [0, 2]]. Subtracting elementwise gives 3A - 2I = [[9-2, -6-0], [12-0, 3-2]] = [[7, -6], [12, 1]].
3Find the product A * B of matrices A = [[1, 2], [3, 4]] and B = [[2, 0], [1, 5]].
A.[[4, 10], [10, 20]]
B.[[2, 0], [3, 20]]
C.[[3, 10], [10, 20]]
D.[[4, 5], [6, 20]]
Explanation: Row-by-column multiplication gives: (1)(2)+(2)(1) = 4; (1)(0)+(2)(5) = 10; (3)(2)+(4)(1) = 10; (3)(0)+(4)(5) = 20. Thus A * B = [[4, 10], [10, 20]].
4What is the determinant of matrix A = [[5, -2], [3, 4]]?
A.26
B.14
C.-26
D.20
Explanation: For a 2x2 matrix [[a, b], [c, d]], det(A) = ad - bc. Here det(A) = (5)(4) - (-2)(3) = 20 + 6 = 26.
5Calculate the inverse of matrix A = [[3, 5], [1, 2]].
A.[[2, -5], [-1, 3]]
B.[[-2, 5], [1, -3]]
C.[[3, -5], [-1, 2]]
D.[[2, 1], [5, 3]]
Explanation: det(A) = (3)(2) - (5)(1) = 6 - 5 = 1. The inverse is (1/det(A)) * [[d, -b], [-c, a]] = [[2, -5], [-1, 3]].
6Compute the determinant of the 3x3 matrix A = [[1, 2, 0], [0, 3, 4], [2, 0, 1]].
A.19
B.11
C.3
D.-13
Explanation: Expanding along row 1: det(A) = 1 * (3*1 - 4*0) - 2 * (0*1 - 4*2) + 0 = 1(3) - 2(-8) = 3 + 16 = 19.
7If A is a 3x3 square matrix with det(A) = 4, what is det(3A)?
A.108
B.12
C.36
D.64
Explanation: For an n x n matrix A and scalar k, det(k A) = k^n * det(A). Here n = 3 and k = 3, so det(3A) = 3^3 * 4 = 27 * 4 = 108.
8Which identity correctly expresses the transpose of the product of two matrices A and B?
A.(A * B)^T = B^T * A^T
B.(A * B)^T = A^T * B^T
C.(A * B)^T = A * B^T
D.(A * B)^T = (A^T * B)^T
Explanation: The transpose of a matrix product reverses the order of factors: (A * B)^T = B^T * A^T.
9Solve the matrix equation A * X = B for X, where A = [[2, 1], [1, 1]] and B = [[4], [3]].
A.[[1], [2]]
B.[[2], [1]]
C.[[5], [-2]]
D.[[7], [1]]
Explanation: det(A) = 2 - 1 = 1, so A^(-1) = [[1, -1], [-1, 2]]. Then X = A^(-1) * B = [[1*4 - 1*3], [-1*4 + 2*3]] = [[1], [2]].
10Determine the rank of matrix M = [[1, 2, 3], [2, 4, 6]].
A.1
B.2
C.3
D.0
Explanation: Row 2 is exactly twice Row 1 (R2 = 2 R1). Thus, there is only 1 linearly independent row, so rank(M) = 1.

About the Madrid PAU Applied Mathematics for Social Sciences II Exam

The Madrid PAU Applied Mathematics for Social Sciences II (Matemáticas Aplicadas a las Ciencias Sociales II) practice question bank provides 100 high-quality worked calculation and conceptual practice questions for 2º Bachillerato Social Sciences students in Madrid. It covers matrix algebra, linear programming, continuity, differential calculus optimization, probability theorems, normal distributions, and confidence interval estimation.

Exam sponsor: PAU Organising Commission of the Community of Madrid / Universidad Complutense de Madrid (UCM). The requirements and fees below concern the certification or admission exam, separate from our free practice resources.

Assessment

90-minute examination covering 5 core syllabus blocks: Linear Algebra & Linear Programming (25%), Functions, Limits & Continuity (20%), Differential Calculus & Optimization (20%), Probability & Random Variables (15%), and Statistical Inference & Confidence Intervals (20%).

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 93.02 base registration fee for compulsory Access Phase in Community of Madrid (or ~11.63 € per optional subject in voluntary phase).

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%

Linear Algebra & Matrices / Linear Programming

Matrix operations, determinants, inverse matrices, matrix equations, systems of linear equations with parameters, Gauss method, and 2-variable linear programming optimization (feasible regions, convex polygons, vertex evaluation).

20%

Functions, Limits & Continuity

Domain of rational/logarithmic/square-root functions, limits at infinity and at finite points, indeterminate forms, continuity of piecewise functions, removable and non-removable discontinuities, and asymptotes.

20%

Differential Calculus & Optimization

Derivatives, differentiation rules, marginal cost/revenue/profit functions, relative extrema, monotonicity, concavity, inflection points, and optimization of real-world social science models.

15%

Probability & Random Variables

Sample spaces, compound events, conditional probability, Bayes' theorem, Total Probability Theorem, independent events, binomial distribution B(n,p), and normal distribution N(mu, sigma).

20%

Statistical Inference & Confidence Intervals

Sampling distributions, central limit theorem, confidence intervals for population mean and population proportion, margin of error, and minimum sample size calculations.

Preparing for the Madrid PAU Applied Mathematics for Social Sciences II 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: 90-minute examination covering 5 core syllabus blocks: Linear Algebra & Linear Programming (25%), Functions, Limits & Continuity (20%), Differential Calculus & Optimization (20%), Probability & Random Variables (15%), and Statistical Inference & Confidence Intervals (20%).
  • Time limit: 90 minutes (1.5 hours)
  • Exam / certification fees: EUR 93.02 base registration fee for compulsory Access Phase in Community of Madrid (or ~11.63 € per optional subject in voluntary phase). 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

Madrid PAU Applied Mathematics for Social Sciences II: Suggested Study Strategy

1Master matrix operations and Cramer's rule: Practice finding inverse matrices via adjugate matrices and solving 2x2 and 3x3 matrix equations.
2Perfect linear programming graphing: Draw constraint inequalities cleanly, locate all feasible region vertices precisely, and evaluate objective function Z = ax + by at every corner point.
3Analyze piecewise function continuity: Ensure left-hand and right-hand limits equal the function value at boundary points when solving for unknown parameters.
4Apply calculus to business and economics models: Differentiate cost C(x), revenue R(x), and profit P(x) = R(x) - C(x) functions to find production levels that maximize profit or minimize average cost.
5Memorize confidence interval formulas and z-scores: Memorize z_{alpha/2} values for standard confidence levels (90% -> 1.645, 95% -> 1.96, 99% -> 2.575) and practice calculating minimum sample sizes for both means and proportions.

Frequently Asked Questions

What is the registration fee for the PAU exam in the Community of Madrid?

The base registration fee for the compulsory Access Phase in Madrid is EUR 93.02 (or approximately EUR 11.63 per optional subject in the voluntary admission phase).

What minimum score is required to pass the PAU Applied Math CCSS II exam in Madrid?

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

How long is the Madrid PAU Applied Mathematics for Social Sciences II examination?

The official examination duration is 90 minutes (1.5 hours).

Are scientific calculators allowed in the Madrid PAU Math CCSS II exam?

Standard non-programmable, non-graphing scientific calculators without symbolic algebra (CAS) capabilities or wireless connectivity are permitted.

Which university organizing committee sets the PAU in Madrid?

The examination is governed by the PAU Organising Commission of the Community of Madrid, coordinated across public universities including Universidad Complutense de Madrid (UCM), Universidad Autónoma de Madrid (UAM), Universidad Carlos III (UC3M), UPM, URJC, and UAH.

What topics are covered in the Applied Mathematics for Social Sciences II syllabus?

The syllabus spans 5 core thematic blocks: Linear Algebra & Linear Programming (25%), Functions, Limits & Continuity (20%), Differential Calculus & Optimization (20%), Probability & Random Variables (15%), and Statistical Inference & Confidence Intervals (20%).