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Exam Review

Key Facts: PAU Applied Math ACS II (Aragón) Exam

90 min

Exam Duration

UNIZAR PAU Commission

0–10

Grading Scale

Gobierno de Aragón

4.0 / 10

Min. Access Phase Mark

PAU Regulations

60 / 40

GPA / PAU Weighting

Spanish University Access Regulations

The Aragón PAU Applied Mathematics for Social Sciences II exam (Matemáticas Aplicadas a las Ciencias Sociales II 2º Bachillerato) is administered by UNIZAR / PAU Organising Commission of Aragón. 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 UNIZAR syllabus.

Sample PAU Applied Math ACS II (Aragón) Practice Questions

Try these sample questions to review concepts for the PAU Applied Math ACS II (Aragón) exam. Each question includes a detailed explanation. Start the interactive quiz above for the full 100+ question experience with AI tutoring.

1Given matrices A = [[2, -1], [3, 4]] and B = [[1, 5], [-2, 0]], calculate the matrix C = 2A - 3B.
A.[[1, -17], [12, 8]]
B.[[1, -17], [0, 8]]
C.[[7, 13], [0, 8]]
D.[[1, 13], [12, 8]]
Explanation: Compute 2A = [[4, -2], [6, 8]] and 3B = [[3, 15], [-6, 0]]. Subtracting 3B from 2A yields C = [[4-3, -2-15], [6-(-6), 8-0]] = [[1, -17], [12, 8]].
2If matrix A has dimensions 2 x 3 and matrix B has dimensions 3 x 4, what are the dimensions of the product matrix C = A * B?
A.4 x 2
B.2 x 4
C.The product A * B is undefined.
D.3 x 3
Explanation: Matrix multiplication is defined when the number of columns of the first matrix (3) equals the number of rows of the second matrix (3). The resulting product matrix C has the number of rows of A (2) and the number of columns of B (4), so its dimensions are 2 x 4.
3Calculate the matrix product A * B for A = [[1, 2], [0, -1]] and B = [[3, 1], [4, 2]].
A.[[3, 2], [0, -2]]
B.[[7, 3], [-4, -2]]
C.[[11, 5], [4, 2]]
D.[[11, 5], [-4, -2]]
Explanation: Perform row-by-column multiplication: c11 = 1*3 + 2*4 = 11; c12 = 1*1 + 2*2 = 5; c21 = 0*3 + (-1)*4 = -4; c22 = 0*1 + (-1)*2 = -2. Thus A * B = [[11, 5], [-4, -2]].
4Which of the following identities is true for any two invertible square matrices A and B of the same size?
A.(A * B)^T = A^T * B^T
B.(A + B)^T = A^T * B^T
C.det(A + B) = det(A) + det(B)
D.(A * B)^T = B^T * A^T
Explanation: The transpose of a product of matrices reverses the order of multiplication: (A * B)^T = B^T * A^T.
5For what value of k is the matrix A = [[k, 4], [2, k-2]] singular (non-invertible)?
A.k = -2 and k = 0
B.k = -4 and k = 2
C.k = 4 and k = -2
D.k = 2 and k = 4
Explanation: A matrix is singular if its determinant equals 0. det(A) = k*(k - 2) - 4*2 = k^2 - 2k - 8 = 0. Factoring gives (k - 4)(k + 2) = 0, so k = 4 or k = -2.
6Calculate the determinant of the 3x3 matrix A = [[1, 0, 2], [3, -1, 1], [2, 1, 4]].
A.-5
B.-3
C.3
D.5
Explanation: Expanding along the first row: det(A) = 1 * det([[-1, 1], [1, 4]]) - 0 + 2 * det([[3, -1], [2, 1]]) = 1*(-5) + 2*(5) = -5 + 10 = 5.
7If A is a 3 x 3 matrix with det(A) = 4, what is the value of det(2 * A^(-1))?
A.2
B.1/2
C.8
D.1
Explanation: For an n x n matrix, det(k M) = k^n * det(M). Here n = 3, so det(2 A^(-1)) = 2^3 * det(A^(-1)). Since det(A^(-1)) = 1 / det(A) = 1/4, we have det(2 A^(-1)) = 8 * (1/4) = 2.
8Find the inverse of the 2x2 matrix A = [[3, 5], [1, 2]].
A.[[1/3, 1/5], [1, 1/2]]
B.[[2, -5], [-1, 3]]
C.[[-2, 5], [1, -3]]
D.[[3, -5], [-1, 2]]
Explanation: For A = [[a, b], [c, d]], det(A) = ad - bc = 3*2 - 5*1 = 1. The inverse is A^(-1) = (1/det(A)) * [[d, -b], [-c, a]] = [[2, -5], [-1, 3]].
9If matrix A is invertible and A * X = B, how is the unknown matrix X isolated?
A.X = A^(-1) * B
B.X = A * B^(-1)
C.X = B * A^(-1)
D.X = B / A
Explanation: To isolate X when A is multiplied on the left (A * X = B), left-multiply both sides by A^(-1): A^(-1) * (A * X) = A^(-1) * B => I * X = A^(-1) * B => X = A^(-1) * B. Order matters in matrix multiplication!
10Solve the matrix equation [[2, 1], [5, 3]] * X = [[4, 2], [1, 0]].
A.[[11, 6], [26, 14]]
B.[[11, 6], [-18, -10]]
C.[[-11, -6], [26, 14]]
D.[[4, -2], [-1, 6]]
Explanation: Multiplying on the left by A^(-1) = [[3, -1], [-5, 2]] gives X = A^(-1) * B = [[3, -1], [-5, 2]] * [[4, 2], [1, 0]] = [[11, 6], [-18, -10]].

About the PAU Applied Math ACS II (Aragón) Exam

The Aragón 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: PAU Organising Commission of Aragón / UNIZAR. 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 Aragón / UNIZAR featuring four 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 75.00 ordinary registration fee (Access Phase plus two voluntary subjects); EUR 30.93 per additional voluntary subject (UNIZAR 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 (Aragón) 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 Aragón / UNIZAR featuring four 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 75.00 ordinary registration fee (Access Phase plus two voluntary subjects); EUR 30.93 per additional voluntary subject (UNIZAR 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 (Aragón): Suggested Study Strategy

1Practice solving matrix equations step by step: isolate X as X = A^(-1)(C - B) or X = (C - B)A^(-1), ensuring correct matrix multiplication order.
2Master linear programming vertex evaluation: draw constraint boundaries, find intersection vertices algebraically, and test all vertices in the objective function.
3Drill piecewise function continuity and differentiability: set left and right limits equal at boundary points to solve for unknown parameters.
4Review optimization problem setup: write the single-variable function to maximize or minimize, find critical points via f'(x) = 0, and verify with f''(x).
5Memorize standard normal distribution Z-table lookup procedures and confidence interval formulas for both means and proportions.

Frequently Asked Questions

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

The official PAU exam organized by UNIZAR is a 90-minute written examination consisting of four exercises worth 2.5 points each. It covers linear algebra, linear programming, analysis/optimization, and probability/statistics. Students must show clear, step-by-step mathematical reasoning for full credit.

What calculators are allowed in the Aragón PAU exam?

Scientific, non-programmable, non-graphical calculators are permitted. The calculator must not handle symbolic algebraic operations (CAS) or store text/formulas in internal memory.

How is the final university access score calculated in Aragón?

The Access Grade (Nota de Acceso) combines 60% of the student's 2-year Bachillerato GPA and 40% of the compulsory PAU phase average, provided the student obtains at least a 4.0 in the PAU phase and a combined grade of at least 5.0 out of 10.

How are confidence intervals tested in the Aragón PAU exam?

Students are typically given a sample mean or sample proportion along with a standard deviation or sample size and a confidence level (such as 95% or 99%). They are required to find the critical value Z_(α/2), calculate the margin of error, write the confidence interval, or determine the minimum sample size required for a specified maximum error.