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Free Practice Questions for Castilla y León PAU Applied Mathematics for Social Sciences II

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Key Facts: Castilla y León PAU Applied Mathematics for Social Sciences II Exam

90 min

Exam Time Limit

Comisión Organizadora PAU CyL

0–10

Grading Scale

Junta de Castilla y León

4.0

Min. Access Phase Score

USAL / UVa / UBU / ULE

EUR 91.54

Base Registration Fee

Junta de Castilla y León

5 Blocks

Curriculum Content Areas

2º Bachillerato MACS II Syllabus

The Castilla y León PAU Applied Mathematics for Social Sciences II exam is organized by the public universities of Castilla y León (USAL, UVa, UBU, ULE) and the Junta de Castilla y León. The exam lasts 90 minutes and is scored out of 10 points. Note that local practice questions on this platform are an English-language MCQ study adaptation designed to help students master the 2nd Bachillerato curriculum with step-by-step mathematical explanations.

Sample Castilla y León PAU Applied Mathematics for Social Sciences II Practice Questions

Try these sample questions to review concepts for the Castilla y León 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.

1What is the determinant of the 2x2 matrix A = [[3, 5], [1, 4]]?
A.7
B.17
C.12
D.-7
Explanation: For a 2x2 matrix [[a, b], [c, d]], det(A) = ad - bc. Here, det(A) = (3)(4) - (5)(1) = 12 - 5 = 7.
2Given matrices A = [[2, 1], [0, 3]] and B = [[-1, 4], [5, 2]], what is the sum A + B?
A.[[1, 5], [5, 5]]
B.[[3, -3], [-5, 1]]
C.[[1, 4], [0, 6]]
D.[[-2, 4], [0, 6]]
Explanation: Matrix addition is performed entrywise: [[2 + (-1), 1 + 4], [0 + 5, 3 + 2]] = [[1, 5], [5, 5]].
3If matrix A is of dimension 2x3 and matrix B is of dimension 3x4, what is the dimension of the product matrix AB?
A.2x4
B.3x3
C.4x2
D.The product AB is undefined.
Explanation: When multiplying an (m x n) matrix by an (n x p) matrix, the inner dimensions match (n=3) and the product matrix has dimension (m x p) = 2x4.
4What condition must a square matrix A satisfy for its inverse matrix A^(-1) to exist?
A.det(A) != 0
B.det(A) = 0
C.det(A) > 1
D.A must be a diagonal matrix
Explanation: A square matrix A is invertible (non-singular) if and only if its determinant is non-zero (det(A) != 0).
5What is the transpose of the matrix M = [[1, 2, 3], [4, 5, 6]]?
A.[[1, 4], [2, 5], [3, 6]]
B.[[6, 5, 4], [3, 2, 1]]
C.[[4, 5, 6], [1, 2, 3]]
D.[[1, 2], [3, 4], [5, 6]]
Explanation: The transpose M^T converts rows into columns. Row 1 [1, 2, 3] becomes Column 1, and Row 2 [4, 5, 6] becomes Column 2, yielding [[1, 4], [2, 5], [3, 6]].
6According to the Rouché-Frobenius theorem, a system of linear equations with coefficient matrix A and augmented matrix A|B is consistent and determined (has a unique solution) when:
A.rank(A) = rank(A|B) = n (number of unknowns)
B.rank(A) = rank(A|B) < n (number of unknowns)
C.rank(A) != rank(A|B)
D.rank(A) = 0
Explanation: By Rouché-Frobenius, if rank(A) = rank(A|B) = n, the system has a unique solution (sistema compatible determinado).
7Solve the matrix equation AX = B for X, where A is an invertible square matrix.
A.X = A^(-1)B
B.X = BA^(-1)
C.X = B / A
D.X = A B^(-1)
Explanation: Multiplying both sides on the left by A^(-1) gives A^(-1)AX = A^(-1)B, which simplifies to IX = X = A^(-1)B. Matrix multiplication is non-commutative, so left-multiplication is required.
8Calculate the determinant of the 3x3 matrix A = [[1, 0, 2], [3, 1, 0], [2, 0, 4]].
A.0
B.4
C.8
D.-4
Explanation: Expanding along the second column: det(A) = 1 * det([[1, 2], [2, 4]]) = 1 * (1*4 - 2*2) = 1 * (4 - 4) = 0. Alternatively, Row 3 is twice Row 1 (excluding middle element 0), making rows linearly dependent.
9For what value of k is the system of equations { x + y = 3, 2x + ky = 6 } dependent with infinitely many solutions?
A.k = 2
B.k = -2
C.k = 0
D.k = 1
Explanation: Multiplying the first equation by 2 gives 2x + 2y = 6. For the second equation 2x + ky = 6 to be identical, we must have k = 2.
10What is the inverse of the 2x2 matrix A = [[2, 1], [5, 3]]?
A.[[3, -1], [-5, 2]]
B.[[-3, 1], [5, -2]]
C.[[3, 5], [1, 2]]
D.[[1/2, 1], [1/5, 1/3]]
Explanation: For A = [[a, b], [c, d]], det(A) = ad - bc = 2*3 - 1*5 = 6 - 5 = 1. A^(-1) = (1/det(A)) * [[d, -b], [-c, a]] = [[3, -1], [-5, 2]].

About the Castilla y León PAU Applied Mathematics for Social Sciences II Exam

The Castilla y León PAU Applied Mathematics for Social Sciences II exam (Matemáticas Aplicadas a las Ciencias Sociales II) evaluates 2nd Bachillerato students across Castilla y León (USAL, UVa, UBU, ULE) on key mathematical methods used in social sciences, economics, and business. Core topics include matrices, determinants, matrix equations, systems of linear equations (Rouché-Frobenius theorem), linear programming in two variables, function analysis (limits, continuity, asymptotes), differential calculus and optimization, and probability theory, normal distributions, and inferential statistics (confidence intervals and sample size). Please note: The official PAU exam is a 90-minute written test in Spanish; the 100 questions here are an English-language multiple-choice study adaptation for practice and self-assessment.

Exam sponsor: Comisión Organizadora de la PAU de Castilla y León (USAL, UVa, UBU, ULE). The requirements and fees below concern the certification or admission exam, separate from our free practice resources.

Assessment

90-minute written examination featuring 4 main problem blocks with mandatory and choice elements (graded 0-10, 2.5 pts per block; local questions are an English-language MCQ study adaptation)

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 76.82 base registration fee for PAU Access Phase (plus EUR 10.33 per voluntary subject) set by Junta de Castilla y León.

Exam sponsor website

Reported exam pass rate: High (~85-95% PAU overall pass rate in Castilla y León). This describes exam candidates, not OpenExamPrep users or results from using our resources. 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 and Systems of Linear Equations

Matrix algebra, operations (addition, scalar multiplication, matrix product), matrix transpose, determinant calculation for 2x2 and 3x3 matrices, inverse matrix via adjugate or Gauss-Jordan methods, matrix equations of the form AX + B = C, systems of linear equations, Gauss elimination method, and Rouché-Frobenius theorem for system classification and parameter analysis.

20%

Linear Programming

Formulation of two-variable linear programming problems from economic and social scenarios, construction of system of linear inequality constraints, determination of the feasible region and its vertices (corner points), objective function visualization via level curves, and application of the fundamental theorem of linear programming to find maximum/minimum solutions.

15%

Limits, Continuity, and Asymptotes

Concept of limit of a function, algebraic calculation of limits, resolution of indeterminate forms (0/0, infinity/infinity, 1^infinity), definition and analysis of continuity, points of discontinuity (removable, jump, essential), piecewise-defined functions, and vertical, horizontal, and oblique asymptotes.

20%

Calculus, Derivatives, and Optimization

Concept of derivative as instantaneous rate of change, differentiation rules for polynomial, rational, exponential, and logarithmic functions, equation of the tangent line to a curve, analysis of monotonicity (increasing/decreasing intervals), determination of relative/local extrema (maxima and minima), concavity and inflection points, definite integrals and area calculations, and real-world optimization problems in economic and social contexts.

20%

Probability, Distributions, and Inferential Statistics

Random experiments, sample space, events and operations, axioms of probability, conditional probability, independence of events, total probability theorem, Bayes' theorem, discrete probability distributions (Binomial distribution B(n,p)), continuous probability distributions (Standard Normal distribution N(0,1) and N(mu, sigma)), central limit theorem, sampling distributions, confidence intervals for a population proportion and mean, and sample size calculations.

Preparing for the Castilla y León 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 written examination featuring 4 main problem blocks with mandatory and choice elements (graded 0-10, 2.5 pts per block; local questions are an English-language MCQ study adaptation)
  • Time limit: 90 minutes (1.5 hours)
  • Exam / certification fees: EUR 76.82 base registration fee for PAU Access Phase (plus EUR 10.33 per voluntary subject) set by Junta de Castilla y León. 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

Castilla y León PAU Applied Mathematics for Social Sciences II: Suggested Study Strategy

1Practice matrix arithmetic step-by-step, ensuring accuracy in matrix multiplication and calculating inverse matrices via det(A) and adj(A).
2Master linear programming: carefully plot constraint boundary lines, identify the feasible region shaded area, find exact vertex coordinates, and test vertices in the objective function.
3Review limits and derivative rules for piecewise functions, ensuring you test left- and right-hand limits at boundary points to check for continuity and differentiability.
4Solve optimization word problems by setting up single-variable objective functions, finding first derivatives, solving f'(x)=0, and verifying max/min with second derivatives.
5Drill Z-score calculations for normal distributions, Bayes' theorem tree diagrams, and standard error formulas for confidence intervals (Z_{alpha/2} * sigma / sqrt(n) or Z_{alpha/2} * sqrt(p(1-p)/n)).

Frequently Asked Questions

What is the format of the official Castilla y León PAU Applied Mathematics II exam?

The official Castilla y León PAU Applied Mathematics for Social Sciences II exam is a 90-minute written examination administered in Spanish across examination centers of USAL, UVa, UBU, and ULE. It consists of 4 problem-solving exercises. Note that the 100 questions available on this site are an English-language multiple-choice practice adaptation developed to help students test their calculation skills and theoretical understanding of the official 2nd Bachillerato curriculum.

What is the passing score for Castilla y León PAU Applied Mathematics II?

The exam is graded on a 0–10 scale. In the Access Phase (Fase de Acceso), a minimum score of 4.0 is required to average with the Bachillerato GPA (which counts for 60% of the final university access score, while PAU counts for 40%, requiring a total average of >= 5.0).

What is the registration fee for taking the PAU exam in Castilla y León?

The base registration fee for the PAU Access Phase in Castilla y León is established by the Junta de Castilla y León at EUR 91.54, with fee exemptions or discounts available for large families (familia numerosa), students with disabilities, or victims of terrorism.

Which universities coordinate the PAU exam in Castilla y León?

The exam is coordinated jointly by the four public universities of Castilla y León: Universidad de Salamanca (USAL), Universidad de Valladolid (UVa), Universidad de Burgos (UBU), and Universidad de León (ULE), under the supervision of the Junta de Castilla y León.

Why are the practice questions here in English and in multiple-choice format?

These practice questions serve as an English-language MCQ study adaptation designed for international students, bilingual program candidates, and revision learners seeking to master the calculations and concepts required for the Castilla y León 2nd Bachillerato syllabus.