PracticeVideosBlogFlashcardsEspañol
All Practice Exams

100+ Free Madrid PAU Mathematics II Practice Questions

Madrid PAU Mathematics II Exam — Matemáticas II 2º Bachillerato Comunidad de Madrid 2026 practice questions are available now; exam metadata is being verified.

✓ No registration✓ No credit card✓ No hidden fees✓ Start practicing immediately
100+ Questions
100% Free

Loading practice questions...

Same family resources

Explore More Spain Madrid PAU (Prueba de Acceso a la Universidad)

Continue into nearby exams from the same family. Each card keeps practice questions, study guides, flashcards, videos, and articles in one place.

Madrid PAU Applied Mathematics for Social Sciences II

Practice Questions

Madrid PAU Art Foundations

Practice Questions

Madrid PAU Artistic Drawing II

Practice Questions

Madrid PAU Biology

Practice Questions

Madrid PAU Business and Business Model Design

Practice Questions

Madrid PAU Chemistry

Practice Questions
2026 Statistics

Key Facts: Madrid PAU Mathematics II Exam

Madrid PAU

PAU Organising Commission of Madrid / UCM

Community of Madrid Education Department

90 Mins

Official examination time

Madrid PAU Exam Structure

Min 4.0

Minimum mark required in Access Phase to calculate overall admission mark

Spanish University Admission Regulations

5 Syllabus Blocks

Matrices, Differential Calculus, Integral Calculus, 3D Geometry, Probability

LOMLOE 2º Bachillerato Mathematics II Syllabus

100

Practice questions available in this OpenExamPrep bank

OpenExamPrep

Madrid PAU Mathematics II (Matemáticas II) tests 2º Bachillerato students on linear algebra, differential calculus, integral calculus, 3D analytic geometry, and probability distributions for university entry in Madrid.

Sample Madrid PAU Mathematics II Practice Questions

Try these sample questions to test your Madrid PAU Mathematics II exam readiness. 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 = [[2, -1], [4, 3]]?
A.10
B.2
C.14
D.-2
Explanation: The determinant of a 2x2 matrix [[a, b], [c, d]] is computed as ad - bc. For matrix A, det(A) = (2)(3) - (-1)(4) = 6 + 4 = 10.
2If A is a 3x3 matrix with det(A) = 5, what is the determinant of the scalar multiple matrix 2A?
A.40
B.10
C.30
D.20
Explanation: For any n x n matrix A and scalar k, det(k A) = k^n * det(A). Since A is a 3x3 matrix (n = 3) and k = 2, det(2A) = 2^3 * det(A) = 8 * 5 = 40.
3Which condition is both necessary and sufficient for a square matrix A to be invertible?
A.det(A) != 0
B.det(A) = 0
C.A is a symmetric matrix
D.The trace of A is positive
Explanation: A square matrix A is non-singular (invertible) if and only if its determinant is non-zero (det(A) != 0).
4What is the transpose of the 2x3 matrix M = [[1, 2, 3], [4, 5, 6]]?
A.[[1, 4], [2, 5], [3, 6]]
B.[[6, 5, 4], [3, 2, 1]]
C.[[1, 2], [3, 4], [5, 6]]
D.[[4, 5, 6], [1, 2, 3]]
Explanation: The transpose M^T of a matrix M interchanges its rows and columns. Row 1 [1, 2, 3] becomes Column 1 and Row 2 [4, 5, 6] becomes Column 2, yielding a 3x2 matrix [[1, 4], [2, 5], [3, 6]].
5If A * X = B, where A is an invertible square matrix, which matrix expression correctly yields X?
A.X = A^(-1) * B
B.X = B * A^(-1)
C.X = B / A
D.X = A * B^(-1)
Explanation: To solve A * X = B, left-multiply both sides by A^(-1): A^(-1) * (A * X) = A^(-1) * B => (A^(-1) * A) * X = A^(-1) * B => X = A^(-1) * B. Matrix multiplication is non-commutative, so the order matters.
6Calculate the determinant of the 3x3 matrix A = [[1, 0, 2], [3, -1, 1], [2, 0, 4]].
A.0
B.8
C.-4
D.4
Explanation: Expanding along the second column (which has two zeros): det(A) = (-1) * (-1)^(2+2) * det([[1, 2], [2, 4]]) = (-1) * (1 * 4 - 2 * 2) = (-1) * (0) = 0. Notice that Column 3 is twice Column 1, so rows/columns are linearly dependent.
7What is the inverse matrix of A = [[1, 2], [3, 4]]?
A.[[-2, 1], [1.5, -0.5]]
B.[[4, -2], [-3, 1]]
C.[[-0.5, 1], [1.5, -2]]
D.[[1, 0.5], [0.33, 0.25]]
Explanation: det(A) = (1)(4) - (2)(3) = -2. Using the formula A^(-1) = (1/det(A)) * [[d, -b], [-c, a]], we get A^(-1) = (-1/2) * [[4, -2], [-3, 1]] = [[-2, 1], [3/2, -1/2]].
8For what real value of the parameter 'a' is the matrix A = [[a, 1, 0], [1, a, 1], [0, 1, a]] singular?
A.a = 0 or a = sqrt(2) or a = -sqrt(2)
B.a = 1 or a = -1
C.a = 2 or a = -2
D.a = 0 only
Explanation: Compute det(A) by expanding along row 1: det(A) = a * (a^2 - 1) - 1 * (a - 0) = a^3 - a - a = a^3 - 2a = a(a^2 - 2). Setting det(A) = 0 gives a = 0 or a^2 = 2 => a = +-sqrt(2).
9Consider the system of linear equations: { x + y + z = 1; x + 2y + 3z = 2; x + 4y + az = 5 }. For what value of parameter 'a' does the system NOT have a unique solution?
A.a = 7
B.a = 5
C.a = 3
D.a = 0
Explanation: The system has a unique solution iff the coefficient matrix determinant is non-zero. det(A) = |[1,1,1],[1,2,3],[1,4,a]|. Subtract Row 1 from Row 2 and Row 3: |[1,1,1],[0,1,2],[0,3,a-1]| = 1 * ((a-1) - 6) = a - 7. Setting det(A) = 0 yields a = 7.
10What is the rank of the matrix A = [[1, 2, 3], [2, 4, 6], [3, 6, 9]]?
A.1
B.3
C.2
D.0
Explanation: Notice that Row 2 = 2 * Row 1 and Row 3 = 3 * Row 1. All 2x2 and 3x3 minors are zero, but there exist non-zero 1x1 minors (e.g. [1]). Hence, rank(A) = 1.

About the Madrid PAU Mathematics II Practice Questions

Verified exam format metadata for Madrid PAU Mathematics II Exam — Matemáticas II 2º Bachillerato Comunidad de Madrid 2026 is pending. The practice questions above remain available while official exam length, timing, passing score, fee, and administrator details are reviewed.