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Key Facts: PAU Matemáticas II (Extremadura) Exam

UEx / Junta

Comisión Organizadora de la PAU de Extremadura

Universidad de Extremadura

90 Mins

Official examination duration

PAU Extremadura Regulations

Min 4.0

Minimum Access Phase mark to combine with Bachillerato GPA

UEx Admission Guidelines

5 Blocks

Syllabus blocks covering algebra, calculus, geometry, and probability

LOMLOE 2º Bachillerato Mathematics II Curriculum

100

Practice questions available in this OpenExamPrep bank

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The Extremadura PAU Mathematics II exam tests advanced secondary mathematical problem solving, rigorous algebraic manipulation, spatial geometry, and probabilistic reasoning.

Sample PAU Matemáticas II (Extremadura) Practice Questions

Try these sample questions to review concepts for the PAU Matemáticas II (Extremadura) 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], [-1, 3]] and B = [[0, -2], [4, 1]], compute the matrix result of 2A - 3B.
A.[[4, 8], [-14, 3]]
B.[[4, -4], [10, 3]]
C.[[2, 3], [-5, 2]]
D.[[4, 8], [10, 9]]
Explanation: Compute scalar multiples: 2A = [[4, 2], [-2, 6]] and 3B = [[0, -6], [12, 3]]. Subtracting component-wise yields 2A - 3B = [[4-0, 2-(-6)], [-2-12, 6-3]] = [[4, 8], [-14, 3]].
2Calculate the determinant of the upper triangular 3x3 matrix A = [[1, 2, 3], [0, 4, 5], [0, 0, 6]].
A.15
B.24
C.0
D.12
Explanation: For any triangular matrix (upper or lower), the determinant is the product of its main diagonal entries: det(A) = 1 * 4 * 6 = 24.
3Find the inverse of the 2x2 matrix A = [[3, 1], [5, 2]].
A.[[2, -1], [-5, 3]]
B.[[3, -5], [-1, 2]]
C.[[-2, 1], [5, -3]]
D.[[1/3, 1], [1/5, 1/2]]
Explanation: The determinant of A is det(A) = 3*2 - 1*5 = 6 - 5 = 1. The inverse formula for a 2x2 matrix [[a, b], [c, d]] is (1/det(A)) * [[d, -b], [-c, a]]. Thus A^-1 = [[2, -1], [-5, 3]].
4Solve the linear system 2x + 3y = 8 and x - y = -1 using Cramer's rule. What is the value of x?
A.x = 2
B.x = -1
C.x = 1
D.x = 3
Explanation: The main determinant is D = |[2, 3], [1, -1]| = -2 - 3 = -5. Replacing the first column with constants yields Dx = |[8, 3], [-1, -1]| = -8 - (-3) = -5. By Cramer's rule, x = Dx / D = -5 / -5 = 1.
5Determine the rank of the matrix M = [[1, 2, 3], [2, 4, 6], [0, 1, 1]].
A.1
B.3
C.2
D.0
Explanation: Row 2 is exactly twice Row 1 (R2 = 2*R1), making R2 linearly dependent. Row 1 [1, 2, 3] and Row 3 [0, 1, 1] are linearly independent because no scalar multiple of [0, 1, 1] gives [1, 2, 3]. Therefore, the maximum number of linearly independent rows is 2, so rank(M) = 2.
6If matrix A has dimensions 2x3 and matrix B has dimensions 3x4, what are the dimensions of the product matrix C = A * B?
A.3x3
B.4x2
C.2x3
D.2x4
Explanation: Matrix multiplication of an (m x n) matrix and an (n x p) matrix produces an (m x p) matrix. Here m = 2, n = 3, and p = 4, so C has dimensions 2x4.
7For what values of the real parameter k is the matrix A = [[k, 1, 1], [1, k, 1], [1, 1, k]] singular (det(A) = 0)?
A.k = 1 and k = -2
B.k = -1 and k = 2
C.k = 0 and k = 3
D.k = 1 and k = 2
Explanation: Computing the determinant det(A) = k(k^2 - 1) - 1(k - 1) + 1(1 - k) = k^3 - 3k + 2. Factoring the polynomial yields (k - 1)^2 * (k + 2) = 0. The roots are k = 1 and k = -2.
8Consider the system with parameter a: x + y + z = 1, x + ay + z = 1, x + y + az = a. For a = 1, how many solutions does the system possess?
A.Unique solution
B.No solution
C.Infinitely many solutions with 2 degrees of freedom
D.Infinitely many solutions with 1 degree of freedom
Explanation: Substituting a = 1 reduces all three equations to x + y + z = 1. This represents a single plane in R^3. The rank of the coefficient matrix and augmented matrix are both 1. With n = 3 variables, the number of free parameters is 3 - 1 = 2 degrees of freedom.
9Given the matrix equation A * X + B = C where A is an invertible square matrix, solve for the matrix X.
A.X = (C - B) * A^-1
B.X = A^-1 * (C - B)
C.X = A^-1 * C - B
D.X = (C - A^-1) * B
Explanation: Subtract B from both sides: A * X = C - B. Left-multiply both sides by A^-1: A^-1 * (A * X) = A^-1 * (C - B) => X = A^-1 * (C - B). Left-multiplication is mandatory because matrix multiplication is non-commutative.
10Using Cramer's rule on the system x + y + z = 6, 2x - y + z = 3, and x + 2y - z = 2, find the value of z.
A.z = 1
B.z = 2
C.z = 3
D.z = 4
Explanation: Determinant D = |[1, 1, 1], [2, -1, 1], [1, 2, -1]| = 1(1 - 2) - 1(-2 - 1) + 1(4 + 1) = -1 + 3 + 5 = 7. Dz = |[1, 1, 6], [2, -1, 3], [1, 2, 2]| = 1(-2 - 6) - 1(4 - 3) + 6(4 + 1) = -8 - 1 + 30 = 21. Thus z = Dz / D = 21 / 7 = 3.

About the PAU Matemáticas II (Extremadura) Exam

The Extremadura PAU Mathematics II (Matemáticas II 2º Bachillerato UEx) practice question bank provides comprehensive preparation for the university entrance exam in Extremadura. The exam evaluates competencies in linear algebra (matrices, determinants, systems of equations), differential calculus (limits, continuity, derivatives, theorems, optimization), integral calculus (integration methods, definite integrals, areas), 3D geometry (vectors, lines, planes, metric problems), and probability theory (Bayes' theorem, binomial and normal distributions).

Exam sponsor: Comisión Organizadora de la PAU de Extremadura (Universidad de Extremadura - UEx / Consejería de Educación de la Junta de Extremadura). The requirements and fees below concern the certification or admission exam, separate from our free practice resources.

Assessment

90-minute written examination evaluating mathematical reasoning and problem-solving skills across algebra, calculus, 3D geometry, and probability.

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 78.26 ordinary registration fee for PAU (50% discount EUR 39.13 for general large family; full exemption for special large family, disability, or terrorism victims).

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%

Algebra & Matrices

Matrix operations, determinants, matrix inversion, systems of linear equations, Rouché-Capelli theorem, and Cramer's rule.

25%

Differential Calculus

Limits, continuity, L'Hôpital's rule, derivatives, Bolzano, Rolle, and Mean Value theorems, curve sketching, and optimization problems.

20%

Integral Calculus

Indefinite integrals, integration techniques (substitution, parts, rational fractions), definite integrals, Fundamental Theorem of Calculus, and area calculations.

20%

3D Analytical Geometry

Vectors, dot and cross products, equations of lines and planes in space, relative positions, distances, and angles.

10%

Probability & Statistics

Conditional probability, total probability theorem, Bayes' theorem, binomial distribution, and normal distribution.

Preparing for the PAU Matemáticas II (Extremadura) 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 evaluating mathematical reasoning and problem-solving skills across algebra, calculus, 3D geometry, and probability.
  • Time limit: 90 minutes (1.5 hours)
  • Exam / certification fees: EUR 78.26 ordinary registration fee for PAU (50% discount EUR 39.13 for general large family; full exemption for special large family, disability, or terrorism victims). 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 Matemáticas II (Extremadura): Suggested Study Strategy

1Master matrix rank determination and parameter discussion using the Rouché-Capelli theorem.
2Practice derivative applications, including L'Hôpital's rule for indeterminate limits and optimization problems.
3Review spatial geometry concepts, particularly line-plane intersections, perpendicularity, and distance formulas.
4Solve area integrals by correctly finding function intersection points and setting up definite integrals.
5Apply Bayes' theorem and tree diagrams accurately in probability word problems.

Frequently Asked Questions

What is the fee for taking the PAU exam in Extremadura?

The ordinary registration fee is EUR 78.26. A 50% discount (EUR 39.13) applies to general large families, and full exemptions apply for special large families, recognized disability (>= 33%), or victims of terrorism.

What passing score is required on the PAU Mathematics II exam in Extremadura?

The exam is graded on a 0–10 scale. A minimum mark of 4.0 in the Access Phase is required to combine with the Bachillerato GPA (60% Bachillerato + 40% PAU Access Phase mark must equal at least 5.0).

What is the duration of the PAU Mathematics II exam?

The examination duration is 90 minutes (1.5 hours).

Which authority conducts the PAU in Extremadura?

The exam is administered by the Comisión Organizadora de la PAU de Extremadura, comprising the Universidad de Extremadura (UEx) and the Consejería de Educación de la Junta de Extremadura.