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Exam Review

Key Facts: Valencian PAU Mathematics II Exam

90 Minutes

Exam duration

Universitats GVA / UPV / UV

EUR 78.37

Base registration fee for Access Phase

Generalitat Valenciana

0–10 Scale

Grading scale for Bachillerato and PAU exams

Spanish Ministry of Education

4 Core Blocks

Linear Systems, Calculus, 3D Geometry, Probability

LOMLOE Mathematics II Syllabus

100 Questions

Practice bank size in OpenExamPrep

OpenExamPrep

Valencian Community PAU Mathematics II (Universitats GVA 2026) is a 90-minute university admission exam covering 2º Bachillerato Science & Technology Mathematics. This bank provides an English-language MCQ study adaptation.

Sample Valencian PAU Mathematics II Practice Questions

Try these sample questions to review concepts for the Valencian PAU Mathematics II 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 = [[1, 2], [0, 1]] and B = [[2, 0], [1, 3]], what is the result of the matrix square (A + B)^2?
A.[[11, 14], [7, 18]]
B.[[9, 12], [5, 16]]
C.[[11, 12], [6, 18]]
D.[[10, 14], [7, 16]]
Explanation: First calculate A + B = [[1+2, 2+0], [0+1, 1+3]] = [[3, 2], [1, 4]]. Next compute (A + B)^2 = [[3, 2], [1, 4]] * [[3, 2], [1, 4]] = [[3*3 + 2*1, 3*2 + 2*4], [1*3 + 4*1, 1*2 + 4*4]] = [[11, 14], [7, 18]]. Note that in general matrix algebra, (A+B)^2 = A^2 + AB + BA + B^2, which equals [[11, 14], [7, 18]] because matrix multiplication is non-commutative.
2For which values of the real parameter 'a' does the matrix A = [[a, 1, 0], [1, a, 1], [0, 1, a]] fail to be invertible?
A.a = 0, a = sqrt(2), a = -sqrt(2)
B.a = 0, a = 1, a = -1
C.a = 1, a = 2, a = -2
D.a = 0, a = 2, a = -2
Explanation: A square matrix fails to be invertible if and only if its determinant is zero. Computing det(A) by expanding along the first row yields det(A) = a*(a^2 - 1) - 1*(a - 0) + 0 = a^3 - a - a = a^3 - 2a = a*(a^2 - 2). Setting det(A) = 0 gives a*(a^2 - 2) = 0, which has solutions a = 0, a = sqrt(2), and a = -sqrt(2).
3Compute the inverse matrix A^(-1) for A = [[1, 0, 2], [2, -1, 3], [4, 1, 8]].
A.[[-11, 2, 2], [-4, 0, 1], [6, -1, -1]]
B.[[-11, -4, 6], [2, 0, -1], [2, 1, -1]]
C.[[11, -2, -2], [4, 0, -1], [-6, 1, 1]]
D.[[-8, 2, 2], [-4, 1, 1], [6, -1, 0]]
Explanation: First compute det(A) expanding by row 1: det(A) = 1*((-1)*8 - 3*1) + 2*(2*1 - (-1)*4) = 1*(-11) + 2*(6) = 1. Since det(A) = 1, A^(-1) equals the transpose of the cofactor matrix adj(A). Cofactors: C11 = -11, C12 = -4, C13 = 6, C21 = 2, C22 = 0, C23 = -1, C31 = 2, C32 = 1, C33 = -1. Transposing the cofactor matrix yields A^(-1) = [[-11, 2, 2], [-4, 0, 1], [6, -1, -1]].
4What is the rank of the matrix A = [[1, 2, 3], [2, 4, k], [3, 6, 9]] as a function of the parameter 'k'?
A.Rank is 1 if k = 6, and rank is 2 if k != 6.
B.Rank is 1 for all real values of k.
C.Rank is 2 if k = 6, and rank is 3 if k != 6.
D.Rank is 3 for all k != 6, and rank is 0 if k = 6.
Explanation: Observe that Row 3 = 3 * Row 1, so Row 3 is always linearly dependent on Row 1. Performing row operations R2 -> R2 - 2*R1 yields the second row [0, 0, k - 6]. If k = 6, the second row becomes [0, 0, 0], leaving only 1 non-zero row, so rank(A) = 1. If k != 6, the second row has a non-zero third entry, giving 2 linearly independent rows, so rank(A) = 2. Rank can never be 3 because all 3x3 minors have determinant zero (since col 2 = 2 * col 1).
5Find the matrix X that satisfies the matrix equation A * X = B, where A = [[2, 1], [5, 3]] and B = [[4, 1], [1, 2]].
A.[[11, 1], [-18, -1]]
B.[[7, 5], [-19, -3]]
C.[[-11, -1], [18, 1]]
D.[[11, -18], [1, -1]]
Explanation: Since det(A) = 2*3 - 1*5 = 1 != 0, A is invertible and X = A^(-1) * B. The inverse A^(-1) = (1/det(A)) * [[3, -1], [-5, 2]] = [[3, -1], [-5, 2]]. Multiplying A^(-1) * B = [[3, -1], [-5, 2]] * [[4, 1], [1, 2]] = [[3*4 + (-1)*1, 3*1 + (-1)*2], [(-5)*4 + 2*1, (-5)*1 + 2*2]] = [[11, 1], [-18, -1]].
6If A is a 3x3 square matrix with determinant det(A) = 4, what is the determinant of the matrix 2 * A^(-1)?
A.2
B.1/2
C.8
D.1
Explanation: Using determinant properties for an n x n matrix, det(k * M) = k^n * det(M). For a 3x3 matrix (n=3), det(2 * A^(-1)) = 2^3 * det(A^(-1)) = 8 * (1 / det(A)). Substituting det(A) = 4 yields det(2 * A^(-1)) = 8 * (1/4) = 2.
7Solve for matrix X in X * A + B = C, where A = [[1, 2], [0, 1]], B = [[1, 0], [2, 3]], and C = [[3, 4], [2, 5]].
A.[[2, 0], [0, 2]]
B.[[2, 4], [0, 2]]
C.[[2, -4], [0, 2]]
D.[[1, 0], [0, 1]]
Explanation: Isolate X * A: X * A = C - B = [[3-1, 4-0], [2-2, 5-3]] = [[2, 4], [0, 2]]. Multiply on the right by A^(-1): X = (C - B) * A^(-1). The inverse of A is A^(-1) = [[1, -2], [0, 1]]. Thus X = [[2, 4], [0, 2]] * [[1, -2], [0, 1]] = [[2*1 + 4*0, 2*(-2) + 4*1], [0*1 + 2*0, 0*(-2) + 2*1]] = [[2, 0], [0, 2]] = 2 * I_2.
8If A is an idempotent square matrix (A^2 = A) and A is neither the zero matrix nor the identity matrix, what are all possible eigenvalues of A?
A.0 and 1
B.1 and -1
C.0 and -1
D.Only 1
Explanation: Let lambda be an eigenvalue of A with non-zero eigenvector v, so A * v = lambda * v. Multiplying by A gives A^2 * v = A * (lambda * v) = lambda * (A * v) = lambda^2 * v. Since A^2 = A, we have A^2 * v = A * v = lambda * v. Therefore lambda^2 * v = lambda * v, which implies (lambda^2 - lambda) * v = 0. Since v != 0, lambda^2 - lambda = 0, giving lambda * (lambda - 1) = 0. The only possible eigenvalues are lambda = 0 and lambda = 1.
9If A and B are 3x3 matrices with det(A) = -3 and det(B) = 5, what is the value of det(A^T * B)?
A.-15
B.15
C.-45
D.-2
Explanation: By properties of determinants, det(M * N) = det(M) * det(N) and det(A^T) = det(A). Therefore, det(A^T * B) = det(A^T) * det(B) = det(A) * det(B) = (-3) * 5 = -15.
10Any square matrix A can be uniquely written as A = S + K, where S is symmetric and K is skew-symmetric. If A = [[2, 5], [1, 4]], find the skew-symmetric component K.
A.[[0, 2], [-2, 0]]
B.[[2, 3], [3, 4]]
C.[[0, 3], [-3, 0]]
D.[[0, 1], [-1, 0]]
Explanation: The unique decomposition of a square matrix A into symmetric part S and skew-symmetric part K is given by S = (1/2)*(A + A^T) and K = (1/2)*(A - A^T). Here A^T = [[2, 1], [5, 4]]. Thus A - A^T = [[2-2, 5-1], [1-5, 4-4]] = [[0, 4], [-4, 0]]. Multiplying by 1/2 yields K = [[0, 2], [-2, 0]].

About the Valencian PAU Mathematics II Exam

The Valencian Community PAU Mathematics II examination (Pruebas de Acceso a la Universidad 2026) is the official university entrance exam for 2º Bachillerato Science and Technology track students in the Valencian Community (administered by UPV, UV, UA, UJI, and UMH). It evaluates mastery in linear algebra, differential and integral calculus, 3D analytical geometry, and probability. Please note: This question bank is an English-language multiple-choice study adaptation designed to help students master the official 2º Bachillerato syllabus concepts, calculations, and problem-solving techniques.

Exam sponsor: Valencian Community PAU Managing Commission / Generalitat Valenciana (Universitats GVA) & Public Universities (UPV, UV, UA, UJI, UMH). The requirements and fees below concern the certification or admission exam, separate from our free practice resources.

Assessment

Written 90-minute standardized exam evaluating Matrices & Linear Systems (25%), Differential & Integral Calculus (35%), 3D Analytical Geometry (25%), and Probability & Random Variables (15%).

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 78.37 base registration fee for compulsory Access Phase in Valencian Community (or ~15.00 € per optional subject in voluntary phase).

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 & Linear Systems

Matrix operations, inverses, determinants, rank of a matrix, Rouché-Capelli theorem, Cramer's rule, and linear systems with parameters.

35%

Differential & Integral Calculus

Limits, continuity, L'Hôpital's rule, derivative rules, tangent/normal lines, extrema, inflection points, optimization, indefinite and definite integrals, FTC, and area calculations.

25%

3D Analytical Geometry & Vector Space

Vector operations, dot/cross products, equations of lines and planes in 3D, line-plane intersections, parallel/skew lines, angles, distances, and symmetric points.

15%

Probability & Random Variables

Basic probability, conditional probability, Bayes' rule, law of total probability, Binomial distribution B(n, p), and Normal distribution N(mu, sigma).

Preparing for the Valencian PAU Mathematics II 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: Written 90-minute standardized exam evaluating Matrices & Linear Systems (25%), Differential & Integral Calculus (35%), 3D Analytical Geometry (25%), and Probability & Random Variables (15%).
  • Time limit: 90 minutes (1.5 hours)
  • Exam / certification fees: EUR 78.37 base registration fee for compulsory Access Phase in Valencian Community (or ~15.00 € per optional subject in voluntary phase). 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

Valencian PAU Mathematics II: Suggested Study Strategy

1Master the Rouché-Capelli theorem: compare rank(A) and rank(A*) against the number of variables to classify parametric linear systems.
2Practice derivative applications: critical points, inflection points, curve sketching, L'Hôpital's rule, and geometric optimization problems.
3Review definite integral calculations and area determination: find intersection points, setup bounding functions, and account for sign changes across intervals.
4Solve 3D geometry problems systematically: vector products (dot, cross, scalar triple), line-plane intersections, perpendicular distances, and symmetric points.
5Understand probability trees, total probability theorem, Bayes' rule, and standardized normal distribution Z-score conversions.

Frequently Asked Questions

What is the format of the local questions in this practice bank?

This question bank provides an English-language multiple-choice study adaptation of the Valencian Community 2nd Bachillerato PAU Mathematics II syllabus, featuring realistic numerical problems, step-by-step solutions, and distractor rationales.

What is the fee for the PAU exam in the Valencian Community?

The base registration fee is EUR 78.37 for the compulsory Access Phase in the Valencian Community (set by Generalitat Valenciana / Universitats GVA), or approximately 15.00 € per subject in the voluntary phase.

What passing score is required on the PAU Mathematics II paper?

Marked on a 0–10 scale. A minimum mark of 4.0 is required in the Access Phase to calculate the overall university entrance grade (60% Bachillerato GPA + 40% PAU Access Phase mark >= 5.0).

How long is the PAU Mathematics II examination?

The exam duration is 90 minutes (1.5 hours).

Which public universities administer the PAU in the Valencian Community?

The examination is managed by the Valencian PAU Managing Commission and administered across public universities in the region: UPV (Valencia), UV (Valencia), UA (Alicante), UJI (Castellón), and UMH (Elche).