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Exam pass rate: About 95% of Bachillerato candidates passed the Aragón PAU Access Phase in the June 2025 ordinary sitting (regional reported results).
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Key Facts: PAU Mathematics II (Aragón) Exam

90 min

Exam Time Limit

UNIZAR PAU Commission

0–10

Grading Scale

Gobierno de Aragón

4.0

Min. Access Phase Mark

Spanish PAU Regulations

EUR 75.00

Ordinary Registration Fee

UNIZAR PAU inscription page 2026

3 Blocks

Curriculum Blocks

2º Bachillerato Mathematics II

The Aragón PAU Mathematics II exam (Matemáticas II) is administered by UNIZAR and the PAU Organising Commission of Aragón for 2nd Bachillerato Science track students. Running 90 minutes, it evaluates linear algebra, differential and integral calculus, and 3D analytical geometry on a 0–10 grading scale. Questions on this platform provide an English MCQ study adaptation with full step-by-step mathematical derivations.

Sample PAU Mathematics II (Aragón) Practice Questions

Try these sample questions to review concepts for the PAU Mathematics II (Aragón) 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, -1], [5, 2]]?
A.1
B.-11
C.11
D.6
Explanation: For a 2x2 matrix [[a, b], [c, d]], the determinant is ad - bc. Here det(A) = (3)(2) - (-1)(5) = 6 - (-5) = 6 + 5 = 11.
2Given matrices A = [[1, 4], [-2, 3]] and B = [[3, -1], [5, 0]], what is the matrix sum A + B?
A.[[4, 3], [3, 3]]
B.[[3, -4], [-10, 0]]
C.[[4, 5], [3, 3]]
D.[[2, 5], [-7, 3]]
Explanation: Matrix addition is performed entrywise: A + B = [[1+3, 4+(-1)], [-2+5, 3+0]] = [[4, 3], [3, 3]].
3Given A = [[2, 0], [1, 3]] and B = [[1, -1], [4, 2]], compute the linear combination 3A - 2B.
A.[[5, -2], [-5, 5]]
B.[[4, 2], [5, 5]]
C.[[4, -2], [11, 13]]
D.[[4, 2], [-5, 5]]
Explanation: Compute 3A = [[6, 0], [3, 9]] and 2B = [[2, -2], [8, 4]]. Then 3A - 2B = [[6-2, 0-(-2)], [3-8, 9-4]] = [[4, 2], [-5, 5]].
4For matrices A = [[1, 2], [3, 0]] and B = [[2, -1], [1, 4]], what is the matrix product A * B?
A.[[7, 4], [-3, 6]]
B.[[2, -2], [3, 0]]
C.[[4, 7], [6, -3]]
D.[[4, 7], [6, 0]]
Explanation: Row-column multiplication: top-left = 1(2)+2(1)=4; top-right = 1(-1)+2(4)=7; bottom-left = 3(2)+0(1)=6; bottom-right = 3(-1)+0(4)=-3. Thus A*B = [[4, 7], [6, -3]].
5If A = [[1, 2], [0, 5]], what is the transpose matrix A^T?
A.[[1, 0], [2, 5]]
B.[[5, 2], [0, 1]]
C.[[1, -2], [0, 5]]
D.[[5, 0], [2, 1]]
Explanation: The transpose of a matrix swaps its rows and columns. The first row (1, 2) becomes the first column, and the second row (0, 5) becomes the second column, yielding [[1, 0], [2, 5]].
6Solve the matrix equation A X = B for column vector X, where A = [[1, 2], [3, 5]] and B = [[1], [2]].
A.[[1], [-1]]
B.[[3], [1]]
C.[[-5], [2]]
D.[[-1], [1]]
Explanation: det(A) = 1(5) - 2(3) = -1. A^-1 = (1/-1) [[5, -2], [-3, 1]] = [[-5, 2], [3, -1]]. Then X = A^-1 B = [[-5(1) + 2(2)], [3(1) - 1(2)]] = [[-1], [1]].
7If the matrix A = [[2, x - 1], [3, 5]] is symmetric, what is the value of x?
A.-2
B.3
C.2
D.4
Explanation: A matrix is symmetric if A^T = A, meaning off-diagonal elements are equal: a_12 = a_21. Here x - 1 = 3 => x = 4.
8Given A = [[1, 0], [2, 1]], find the matrix power A^5.
A.[[1, 0], [2, 1]]
B.[[5, 0], [10, 5]]
C.[[1, 0], [32, 1]]
D.[[1, 0], [10, 1]]
Explanation: A^2 = [[1, 0], [4, 1]], A^3 = [[1, 0], [6, 1]], and by induction A^n = [[1, 0], [2n, 1]]. For n = 5, A^5 = [[1, 0], [10, 1]].
9Using Sarrus' rule, calculate the determinant of the 3x3 matrix A = [[1, 0, 2], [2, -1, 1], [0, 3, 1]].
A.11
B.14
C.8
D.-8
Explanation: det(A) = 1(-1)(1) + 0(1)(0) + 2(2)(3) - [0(-1)(2) + 3(1)(1) + 1(2)(0)] = (-1 + 0 + 12) - (0 + 3 + 0) = 11 - 3 = 8.
10If A is a 3x3 matrix with determinant det(A) = 5, what is the determinant of the matrix 2A?
A.30
B.40
C.20
D.10
Explanation: For an n x n matrix, det(k A) = k^n det(A). Here n = 3 and k = 2, so det(2A) = 2^3 * det(A) = 8 * 5 = 40.

About the PAU Mathematics II (Aragón) Exam

The Aragón PAU Mathematics II (Matemáticas II 2º Bachillerato) exam tests high school students entering Spanish public universities on linear algebra (matrix inverse, determinants, parametric systems via Rouché-Capelli & Cramer), mathematical analysis (limits, L'Hôpital, Rolle/Mean Value theorems, derivative applications, optimization, integration by substitution/parts/partial fractions, Barrow's rule, plane areas), and 3D geometry (vectors, dot/cross products, lines, planes, relative positions, distances, and angles). Please note: The official exam consists of written open-ended problems in Spanish; these 100 practice items form an English-language MCQ study adaptation with detailed derivations for self-assessment.

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

90-minute written examination organized by the PAU Organising Commission of Aragón / UNIZAR featuring multi-step open-ended math calculations (adapted here into a 100-item English MCQ practice suite with step-by-step derivations).

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

Reported exam pass rate: About 95% of Bachillerato candidates passed the Aragón PAU Access Phase in the June 2025 ordinary sitting (regional reported results).. 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.

30%

Linear Algebra & Systems

Matrix operations, determinants (Sarrus, expansion by minors, properties), matrix inverse via adjugate matrix, matrix equations (AX = B, AX + B = C), and classification and resolution of linear systems of equations with parameters using Rouché-Capelli Theorem and Cramer's Rule.

40%

Mathematical Analysis & Calculus

Limits of functions, indeterminate forms (0/0, inf/inf, 1^inf), L'Hôpital's Rule, continuity and Bolzano's theorem, differentiation techniques, tangent lines, Rolle's and Mean Value theorems, curve sketching (asymptotes, extrema, concavity, inflection points), real-world optimization problems, indefinite integrals (u-substitution, integration by parts, partial fractions), Barrow's Rule, and area between intersecting curves.

30%

3D Analytical Geometry

Vector operations in R3 (dot product, cross product, scalar triple product), vector, parametric, continuous, and implicit equations of lines and planes, relative positions of lines and planes (secant, parallel, skew, coincident), metric computations (distances from point to plane, point to line, parallel planes, skew lines; angles between lines, planes, line and plane).

Preparing for the PAU Mathematics 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: 90-minute written examination organized by the PAU Organising Commission of Aragón / UNIZAR featuring multi-step open-ended math calculations (adapted here into a 100-item English MCQ practice suite with step-by-step derivations).
  • 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 Mathematics II (Aragón): Suggested Study Strategy

1Systematically analyze parameter cases in linear systems using the Rouché-Capelli Theorem by setting the determinant of the coefficient matrix to zero.
2Check all necessary hypotheses (continuity on [a, b], differentiability on (a, b)) before applying Rolle's or Lagrange's Mean Value Theorem.
3Master integration by parts using the ALPET rule (Arcsin/Log/Polynomial/Exponential/Trig) and partial fraction decomposition for distinct or repeated real linear factors.
4Use vector cross products to find normal vectors to planes or direction vectors perpendicular to two skew lines.
5Memorize exact distance formulas in R3, such as distance from a point to a plane d(P, pi) = |Ax0 + By0 + Cz0 + D| / sqrt(A^2 + B^2 + C^2).

Frequently Asked Questions

What is the format of the official Aragón PAU Mathematics II exam?

The official PAU Mathematics II exam administered by UNIZAR is a 90-minute written test in Spanish requiring open-ended multi-step problem solving. Students present step-by-step mathematical proofs and derivations. The 100 questions here are an English MCQ study adaptation to test concepts and calculations.

What score is required to pass the PAU exam in Aragón?

The exam is scored on a 0–10 scale. In the Access Phase (Fase de Acceso), candidates must obtain at least 4.0 out of 10. The final University Access Grade (Nota de Acceso) combines 60% Bachillerato GPA and 40% PAU Access Phase average, which must equal at least 5.0.

Which calculators are allowed during the Aragón PAU Mathematics II exam?

Standard scientific non-programmable calculators without graphing, symbolic calculation (CAS), matrix manipulation, or wireless/internet capability are permitted according to UNIZAR PAU commission guidelines.

What main topic blocks are covered in 2nd Bachillerato Matemáticas II?

The 2nd Bachillerato Mathematics II curriculum in Aragón comprises three main blocks: Linear Algebra (matrices, determinants, linear systems), Mathematical Analysis (limits, derivatives, optimization, integrals, areas), and 3D Geometry (vectors, lines, planes, relative positions, metric calculations).