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Key Facts: PAU Mathematics II (La Rioja) Exam

Exam Administrator: Universidad de La Rioja (UR) / Comisión Organizadora de la PAU de La Rioja

Target Audience: 2º Bachillerato Science & Technology track students in La Rioja

Exam Duration: 90 minutes (1.5 hours)

Base Fee: EUR 55.43

Scoring Scale: 0-10 scale (Access Phase passing minimum 4.0)

La Rioja PAU Mathematics II (UR 2026) evaluates 2º Bachillerato linear algebra, differential calculus, integral calculus, and 3D analytical geometry over a 90-minute examination. Base registration fee: EUR 55.43. Graded on a 0-10 scale with a minimum 4.0 needed in the Access Phase. This English-language practice bank provides 100 numerical worked questions covering topics from the syllabus.

Sample PAU Mathematics II (La Rioja) Practice Questions

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

1Given the matrices A = [[2, -1], [3, 4]] and B = [[1, 5], [-2, 0]], calculate the matrix expression 2A - 3B.
A.[[1, -17], [12, 8]]
B.[[1, 13], [0, 8]]
C.[[1, -17], [0, 8]]
D.[[7, -17], [12, 8]]
Explanation: Compute scalar multiplications first: 2A = [[4, -2], [6, 8]] and 3B = [[3, 15], [-6, 0]]. Subtract component-wise: 2A - 3B = [[4-3, -2-15], [6-(-6), 8-0]] = [[1, -17], [12, 8]].
2What is the determinant of the upper triangular 3x3 matrix A = [[1, 2, 3], [0, 4, 5], [0, 0, 6]]?
A.10
B.24
C.0
D.15
Explanation: For any triangular matrix (upper or lower), the determinant equals the product of the diagonal elements. Therefore, det(A) = 1 * 4 * 6 = 24.
3Calculate the determinant of the 2x2 matrix A = [[3, -2], [5, 4]].
A.22
B.2
C.-2
D.7
Explanation: The determinant of a 2x2 matrix [[a, b], [c, d]] is ad - bc. Here det(A) = (3)(4) - (-2)(5) = 12 - (-10) = 12 + 10 = 22.
4Find the inverse of the matrix A = [[2, 1], [5, 3]].
A.[[3, -1], [-5, 2]]
B.[[-3, 1], [5, -2]]
C.[[2, -5], [-1, 3]]
D.[[3, 1], [5, 2]]
Explanation: First find det(A) = 2*3 - 1*5 = 6 - 5 = 1. The inverse of [[a, b], [c, d]] is (1/det(A)) * [[d, -b], [-c, a]]. Thus A^-1 = (1/1) * [[3, -1], [-5, 2]] = [[3, -1], [-5, 2]].
5Solve for matrix X in the matrix equation A * X = B, where A = [[1, 2], [3, 5]] and B = [[4, 1], [2, 0]].
A.[[-16, -5], [10, 3]]
B.[[16, 5], [-10, -3]]
C.[[-16, 5], [10, -3]]
D.[[4, 1], [2, 0]]
Explanation: Multiply both sides by A^-1 on the left: X = A^-1 * B. First, det(A) = 1*5 - 2*3 = -1. A^-1 = (1/-1) * [[5, -2], [-3, 1]] = [[-5, 2], [3, -1]]. Multiply A^-1 * B = [[-5, 2], [3, -1]] * [[4, 1], [2, 0]] = [[-5(4)+2(2), -5(1)+2(0)], [3(4)+(-1)(2), 3(1)+(-1)(0)]] = [[-16, -5], [10, 3]].
6If A is a 3x3 matrix with det(A) = -4, what is the value of det(2A)?
A.-8
B.-32
C.32
D.-16
Explanation: For an n x n square matrix A, det(k * A) = k^n * det(A). For n = 3 and k = 2: det(2A) = 2^3 * det(A) = 8 * (-4) = -32.
7What is the rank of the matrix A = [[1, 2, 3], [2, 4, 6], [0, 1, 5]]?
A.1
B.2
C.3
D.0
Explanation: Notice that Row 2 is 2 times Row 1, so Row 2 is linearly dependent and can be eliminated. Row 1 (1, 2, 3) and Row 3 (0, 1, 5) are linearly independent. Thus, the matrix has 2 linearly independent rows, so rank(A) = 2.
8Calculate the determinant of the 3x3 matrix A = [[1, 0, 2], [-1, 3, 1], [2, 1, 4]].
A.3
B.-3
C.7
D.5
Explanation: Expand along the first row: det(A) = 1 * det([[3, 1], [1, 4]]) - 0 + 2 * det([[-1, 3], [2, 1]]) = 1 * (12 - 1) + 2 * (-1 - 6) = 11 + 2(-7) = 11 - 14 = -3.
9Consider the system of linear equations with parameter k: x + y + z = 1, x + 2y + 3z = 2, x + 4y + k*z = 5. For what value of k is the system inconsistent (incompatible)?
A.k = 7
B.k = 5
C.k = 3
D.k = 0
Explanation: Compute the determinant of the coefficient matrix: det(A) = det([[1, 1, 1], [1, 2, 3], [1, 4, k]]) = k - 7. When k = 7, det(A) = 0 so rank(A) = 2. Row reduction of the augmented matrix A* yields third row (0, 0, 0 | 1), so rank(A*) = 3. By Rouché-Capelli theorem, rank(A) = 2 != rank(A*) = 3, making the system inconsistent (incompatible) when k = 7.
10Use Cramer's rule to find the value of y in the system 2x + y = 5, 3x - 2y = 4.
A.1
B.2
C.3
D.-1
Explanation: The main determinant is det(A) = det([[2, 1], [3, -2]]) = -4 - 3 = -7. The numerator determinant for y replaces column 2 with constant terms: det(A_y) = det([[2, 5], [3, 4]]) = 8 - 15 = -7. By Cramer's rule, y = det(A_y) / det(A) = -7 / -7 = 1.

About the PAU Mathematics II (La Rioja) Exam

The La Rioja PAU Mathematics II (Matemáticas II) examination assesses 2º Bachillerato competencies in Linear Algebra (matrices, determinants, inverse matrices, matrix equations, systems of linear equations, Rouché-Capelli theorem, Cramer's rule), Differential Calculus & Function Analysis (domain, limits, L'Hôpital's rule, continuity, differentiability, Bolzano and Rolle theorems, mean value theorem, asymptotes, monotonic intervals, extrema, inflection points, optimization), Integral Calculus & Area Computation (indefinite integrals, integration techniques, definite integrals, Fundamental Theorem of Calculus, area enclosed by curves), and 3D Analytical Geometry in Space (vectors, scalar and vector products, equations of lines and planes, relative positions, angles, projections, distance formulas).

Exam sponsor: Universidad de La Rioja (UR) / Comisión Organizadora de la PAU de La Rioja. The requirements and fees below concern the certification or admission exam, separate from our free practice resources.

Assessment

Written 90-minute standardized examination featuring problem-solving tasks across Linear Algebra, Differential Calculus, Integral Calculus, and 3D Analytical Geometry.

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 55.43 base registration fee for PAU Access Phase (ordinary sitting, set by Universidad de La Rioja).

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, Determinants & Systems of Linear Equations

Matrix operations, matrix algebra, matrix inversion, determinants properties, matrix rank, classification of linear systems (Rouché-Capelli theorem), and solving systems via Cramer's rule or inverse matrices.

25%

Differential Calculus & Function Analysis

Limits, L'Hôpital's rule, continuity, differentiability, derivative calculations, tangent and normal lines, Rolle's and Mean Value theorems, curve sketching (asymptotes, extrema, inflection points), and real-world optimization problems.

25%

Integral Calculus & Area Computation

Indefinite integrals (substitution, integration by parts, rational functions), definite integrals, Fundamental Theorem of Calculus, and computing areas bounded by curves and axes.

25%

3D Analytical Geometry in Space

Vector operations (dot product, cross product, scalar triple product), line and plane equations in 3D space, relative positions of lines and planes, perpendicular and parallel lines/planes, angles, orthogonal projections, and distance calculations (point-to-plane, line-to-line).

Preparing for the PAU Mathematics II (La Rioja) 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: Written 90-minute standardized examination featuring problem-solving tasks across Linear Algebra, Differential Calculus, Integral Calculus, and 3D Analytical Geometry.
  • Time limit: 90 minutes (1.5 hours)
  • Exam / certification fees: EUR 55.43 base registration fee for PAU Access Phase (ordinary sitting, set by Universidad de La Rioja). 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 (La Rioja): Suggested Study Strategy

1Master matrix rank determination and parameter discussion using the Rouché-Capelli theorem.
2Practice derivative rules and L'Hôpital's rule for indeterminate limit forms (0/0 and ∞/∞).
3Review integration by parts and partial fraction decomposition for definite integral area problems.
4Memorize 3D geometry distance formulas and vector cross product applications for area and volume calculations.

Frequently Asked Questions

What is the structure of the La Rioja PAU Mathematics II exam?

The official exam is a 90-minute written examination consisting of open-ended problem-solving exercises across linear algebra, calculus, integration, and 3D analytical geometry.

What is the grading scale and fee for La Rioja PAU Mathematics II?

It is scored on a 0-10 scale. A minimum of 4.0 in the Access Phase is required to combine with Bachillerato GPA (60% Bachillerato + 40% PAU >= 5.0). The base registration fee is EUR 55.43.

Is a calculator permitted in the PAU Mathematics II exam in La Rioja?

Scientific calculators are permitted provided they do not have graphic display, symbolic manipulation (CAS), programmable memory, or communication capabilities, according to Universidad de La Rioja PAU tribunal guidelines.

Is this practice bank in the same format as the real PAU Mathematics II (La Rioja) exam?

No. The official PAU Mathematics II (La Rioja) 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 numerical calculations of the 2º Bachillerato curriculum.