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Free Practice Questions for Castilla-La Mancha PAU Applied Mathematics for Social Sciences II (Matemáticas Aplicadas a las Ciencias Sociales II - UCLM 2026)

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Key Facts: Castilla-La Mancha PAU Applied Mathematics for Social Sciences II (Matemáticas Aplicadas a las Ciencias Sociales II - UCLM 2026) Exam

90 min

PAU Math CCSS II exam duration

Tribunal PAU Universidad de Castilla-La Mancha

0–10

Grading scale (minimum 4.0 to weight)

Normativa Académica UCLM

EUR 52.99

Standard PAU UCLM Access Phase registration fee

Universidad de Castilla-La Mancha 2026

100

Practice calculation questions in this assessment bank

OpenExamPrep

Prepare for the UCLM 2026 PAU Applied Mathematics for Social Sciences II exam with 100 worked practice questions on matrices, linear programming, calculus, and statistics. This English-language multiple-choice bank is a study adaptation of the official written PAU paper, not a replica of it.

Sample Castilla-La Mancha PAU Applied Mathematics for Social Sciences II (Matemáticas Aplicadas a las Ciencias Sociales II - UCLM 2026) Practice Questions

Try these sample questions to review concepts for the Castilla-La Mancha PAU Applied Mathematics for Social Sciences II (Matemáticas Aplicadas a las Ciencias Sociales II - UCLM 2026) exam. Each question includes a detailed explanation. Start the interactive quiz above for the full 100+ question experience with AI tutoring.

1Given the objective function Z = 3x + 4y to be optimized over the feasible region with vertices A(0,0), B(5,0), C(3,4), and D(0,6), what is the maximum value of Z?
A.25
B.24
C.15
D.21
Explanation: Evaluating the objective function Z = 3x + 4y at each vertex: Z(0,0) = 0; Z(5,0) = 3(5)+4(0) = 15; Z(3,4) = 3(3)+4(4) = 9 + 16 = 25; Z(0,6) = 3(0)+4(6) = 24. The maximum value is 25 at the point (3,4).
2Calculate the area of the feasible region bounded by the constraints x ≥ 0, y ≥ 0, and x + y ≤ 6.
A.36 sq units
B.18 sq units
C.12 sq units
D.9 sq units
Explanation: The feasible region is a right triangle bounded by the coordinate axes and vertices (0,0), (6,0), and (0,6). The base b = 6 and height h = 6, so the area is A = (b · h)/2 = (6 · 6)/2 = 18 square units.
3To minimize the function Z = 2x + 5y over the region defined by the vertices (1,5), (2,2), (6,1), and (4,7), at which vertex is the minimum achieved?
A.(6,1)
B.(1,5)
C.(2,2)
D.(4,7)
Explanation: Evaluating Z = 2x + 5y at the vertices: Z(1,5) = 2(1)+5(5) = 27; Z(2,2) = 2(2)+5(2) = 14; Z(6,1) = 2(6)+5(1) = 17; Z(4,7) = 2(4)+5(7) = 43. The minimum value of Z is 14 and is reached at vertex (2,2).
4Find the coordinates of the intersection vertex of the boundary lines x + y = 10 and 2x - y = 5.
A.(3, 7)
B.(4, 6)
C.(6, 4)
D.(5, 5)
Explanation: Adding both equations to eliminate y: (x + y) + (2x - y) = 10 + 5 => 3x = 15 => x = 5. Substituting x = 5 into the first equation: 5 + y = 10 => y = 5. The intersection vertex is (5, 5).
5Given the profit function F(x,y) = 5x + 2y defined over a polygon with vertices A(0,4), B(3,5), C(6,1), and D(2,0), calculate the maximum profit.
A.32
B.25
C.10
D.8
Explanation: Evaluating F(x,y) = 5x + 2y at each vertex: F(0,4) = 8; F(3,5) = 5(3)+2(5) = 25; F(6,1) = 5(6)+2(1) = 32; F(2,0) = 5(2)+2(0) = 10. The maximum profit is 32, obtained at point C(6,1).
6Determine whether the point (3,2) belongs to the feasible region defined by the inequalities 2x + y ≤ 8 and x + 3y ≥ 7.
A.No, because it violates the first inequality.
B.Yes, because it satisfies both inequalities simultaneously.
C.No, because it violates the second inequality.
D.No, because it violates both constraints.
Explanation: For (3,2): 1st inequality: 2(3) + 2 = 6 + 2 = 8 ≤ 8 (Satisfied). 2nd inequality: 3 + 3(2) = 3 + 6 = 9 ≥ 7 (Satisfied). Since both constraints are satisfied, the point (3,2) belongs to the feasible region.
7Maximize Z = x + 3y subject to x ≥ 0, y ≥ 0, x ≤ 4, y ≤ 5.
A.19
B.4
C.15
D.12
Explanation: The feasible region is the rectangle with vertices (0,0), (4,0), (4,5), and (0,5). Evaluating Z = x + 3y at the vertices yields Z(0,0)=0, Z(4,0)=4, Z(4,5)=4+3(5)=19, and Z(0,5)=15. The maximum value of 19 is achieved at (4,5).
8A small company manufactures product X and product Y. Product X requires 2 hours of labor and 1 kg of raw material per unit. Product Y requires 1 hour of labor and 3 kg of raw material per unit. The company has up to 100 hours of labor and 150 kg of raw material available. Which set of inequalities defines the feasible region for production (x ≥ 0, y ≥ 0)?
A.2x + y ≤ 100; x + 3y ≤ 150
B.x + 2y ≤ 100; 3x + y ≤ 150
C.2x + 3y ≤ 100; x + y ≤ 150
D.2x + y ≥ 100; x + 3y ≥ 150
Explanation: Labor constraint: 2x + 1y ≤ 100. Raw material constraint: 1x + 3y ≤ 150. Non-negativity constraints: x ≥ 0, y ≥ 0. The correct system of inequalities is 2x + y ≤ 100 and x + 3y ≤ 150.
9Find the maximum of Z = 10x + 15y in an unbounded feasible region defined by x ≥ 0, y ≥ 0, and x + y ≥ 4.
A.60
B.40
C.It does not exist (the function is unbounded above in the region).
D.0
Explanation: The region extends indefinitely into the first quadrant as x and y increase (e.g., points like (100, 100) are feasible). Since the objective function has positive coefficients (10 and 15), Z can increase without bound. Therefore, no finite maximum exists.
10Consider the objective function Z = ax + y. For what value of parameter a does the function achieve its maximum simultaneously at the vertices (2,4) and (5,1) of a feasible region?
A.a = 1
B.a = -1
C.a = 2
D.a = 0.5
Explanation: For the maximum to be achieved at both vertices, the value of Z must be equal at both points: Z(2,4) = Z(5,1) => 2a + 4 = 5a + 1 => 3a = 3 => a = 1.

About the Castilla-La Mancha PAU Applied Mathematics for Social Sciences II (Matemáticas Aplicadas a las Ciencias Sociales II - UCLM 2026) Exam

Comprehensive question bank for the Castilla-La Mancha PAU Applied Mathematics for Social Sciences II exam (UCLM 2026). Features 100 worked calculation practice questions covering linear programming, matrix algebra, differential and integral calculus, probability, and inferential statistics.

Exam sponsor: University of Castilla-La Mancha PAU Tribunal. The requirements and fees below concern the certification or admission exam, separate from our free practice resources.

Assessment

The exam consists of 4 main problem-solving blocks evaluated over 90 minutes (1.5 hours): Linear Programming & Feasible Regions (25%), Matrices & Systems of Linear Equations (25%), Analysis of Functions & Optimization (25%), and Probability & Inferential Statistics (25%).

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 52.99 base registration fee for PAU Access Phase (ordinary sitting, set by Universidad de Castilla-La Mancha).

Exam sponsor website

Fees, eligibility, and exam policies can change. Confirm them with the exam sponsor before applying or paying.

Official sources

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 Programming & Feasible Regions

Formulation of two-variable linear programming models, graphical inequality constraints, convex feasible regions, vertex evaluation, and objective function optimization.

25%

Matrices & Systems of Linear Equations in Social Context

Matrix algebra operations, determinants, inverse matrices, systems of linear equations, Gauss elimination method, Rouché-Capelli theorem, and economic input-output models.

25%

Analysis of Functions, Optimization & Marginal Calculus

Limits, continuity, derivatives, marginal cost/revenue analysis, monotonicity, local extrema, business profit optimization, definite integrals, and area under curves.

25%

Probability, Normal Distribution & Inferential Statistics

Random events, conditional probability, Bayes' theorem, standard normal distribution Z-scores, sampling distributions, and confidence intervals for population means and proportions.

Preparing for the Castilla-La Mancha PAU Applied Mathematics for Social Sciences II (Matemáticas Aplicadas a las Ciencias Sociales II - UCLM 2026) 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 exam consists of 4 main problem-solving blocks evaluated over 90 minutes (1.5 hours): Linear Programming & Feasible Regions (25%), Matrices & Systems of Linear Equations (25%), Analysis of Functions & Optimization (25%), and Probability & Inferential Statistics (25%).
  • Time limit: 90 minutes (1.5 hours)
  • Exam / certification fees: EUR 52.99 base registration fee for PAU Access Phase (ordinary sitting, set by Universidad de Castilla-La Mancha). 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-La Mancha PAU Applied Mathematics for Social Sciences II (Matemáticas Aplicadas a las Ciencias Sociales II - UCLM 2026): Suggested Study Strategy

1Master the precise plotting of feasible regions by evaluating all extreme vertices in the objective function Z = ax + by.
2Practice calculating determinants and inverse matrices by adjugates, as well as solving systems using Cramer's rule and the Rouché-Capelli theorem.
3Review marginal optimization (Profit = Revenue - Cost) by finding zeros of the first derivative and confirming the maximum with the second derivative.
4Memorize the standardizations Z = (X - μ)/σ and the use of the N(0,1) normal table together with the continuity correction for the binomial approximation.
5Correctly apply the confidence interval formulas for population means x̄ ± z_(α/2) * (σ / √n) and proportions p̂ ± z_(α/2) * √(p̂(1-p̂)/n).

Frequently Asked Questions

What is the structure of the PAU Applied Mathematics for Social Sciences II exam in Castilla-La Mancha (UCLM)?

The exam lasts 90 minutes and consists of practical and calculation problems grouped into four main blocks: Linear Programming, Matrices and Systems, Functional Analysis and Calculus, and Probability and Superior Statistics.

How is the access mark calculated in Castilla-La Mancha?

It is graded from 0 to 10 points. A minimum mark of 4.0 in the PAU Access Phase is required to combine with the Bachillerato GPA (60% Bachillerato + 40% PAU Access Phase >= 5.0 to pass).

Which calculators are allowed in the UCLM exam?

Non-programmable scientific calculators without graphic capability, symbolic calculation (CAS), or data transmission connectivity are permitted.

In which language is the exam offered at the University of Castilla-La Mancha?

The official exam is written and sat in Castilian (Spanish).

Are these 100 questions adapted to the official UCLM 2026 curriculum?

Yes, the 100 questions contain detailed, solved numerical calculations adapted to the 2º Bachillerato specifications matrix in Castilla-La Mancha.

Is this practice bank in the same format as the real Castilla-La Mancha PAU Math CCSS II exam?

No. The official Castilla-La Mancha PAU Math CCSS II paper is a 90-minute written examination in Spanish consisting of open-ended, semi-constructed, and short-answer questions — not multiple choice. This bank is an English-language multiple-choice adaptation designed to test and reinforce the core concepts and skills of the 2º Bachillerato curriculum.