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100+ Free Castilla y León PAU Applied Mathematics for Social Sciences II Practice Questions

Castilla y León PAU Applied Mathematics for Social Sciences II (Matemáticas Aplicadas a las Ciencias Sociales II) practice questions are available now; exam metadata is being verified.

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Key Facts: Castilla y León PAU Applied Mathematics for Social Sciences II Exam

90 min

Exam Time Limit

Comisión Organizadora PAU CyL

0–10

Grading Scale

Junta de Castilla y León

4.0

Min. Access Phase Score

USAL / UVa / UBU / ULE

EUR 91.54

Base Registration Fee

Junta de Castilla y León

5 Blocks

Curriculum Content Areas

2º Bachillerato MACS II Syllabus

The Castilla y León PAU Applied Mathematics for Social Sciences II exam is organized by the public universities of Castilla y León (USAL, UVa, UBU, ULE) and the Junta de Castilla y León. The exam lasts 90 minutes and is scored out of 10 points. Note that local practice questions on this platform are an English-language MCQ study adaptation designed to help students master the 2nd Bachillerato curriculum with step-by-step mathematical explanations.

Sample Castilla y León PAU Applied Mathematics for Social Sciences II Practice Questions

Try these sample questions to test your Castilla y León PAU Applied Mathematics for Social Sciences 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 = [[3, 5], [1, 4]]?
A.7
B.17
C.12
D.-7
Explanation: For a 2x2 matrix [[a, b], [c, d]], det(A) = ad - bc. Here, det(A) = (3)(4) - (5)(1) = 12 - 5 = 7.
2Given matrices A = [[2, 1], [0, 3]] and B = [[-1, 4], [5, 2]], what is the sum A + B?
A.[[1, 5], [5, 5]]
B.[[3, -3], [-5, 1]]
C.[[1, 4], [0, 6]]
D.[[-2, 4], [0, 6]]
Explanation: Matrix addition is performed entrywise: [[2 + (-1), 1 + 4], [0 + 5, 3 + 2]] = [[1, 5], [5, 5]].
3If matrix A is of dimension 2x3 and matrix B is of dimension 3x4, what is the dimension of the product matrix AB?
A.2x4
B.3x3
C.4x2
D.The product AB is undefined.
Explanation: When multiplying an (m x n) matrix by an (n x p) matrix, the inner dimensions match (n=3) and the product matrix has dimension (m x p) = 2x4.
4What condition must a square matrix A satisfy for its inverse matrix A^(-1) to exist?
A.det(A) != 0
B.det(A) = 0
C.det(A) > 1
D.A must be a diagonal matrix
Explanation: A square matrix A is invertible (non-singular) if and only if its determinant is non-zero (det(A) != 0).
5What is the transpose of the matrix M = [[1, 2, 3], [4, 5, 6]]?
A.[[1, 4], [2, 5], [3, 6]]
B.[[6, 5, 4], [3, 2, 1]]
C.[[4, 5, 6], [1, 2, 3]]
D.[[1, 2], [3, 4], [5, 6]]
Explanation: The transpose M^T converts rows into columns. Row 1 [1, 2, 3] becomes Column 1, and Row 2 [4, 5, 6] becomes Column 2, yielding [[1, 4], [2, 5], [3, 6]].
6According to the Rouché-Frobenius theorem, a system of linear equations with coefficient matrix A and augmented matrix A|B is consistent and determined (has a unique solution) when:
A.rank(A) = rank(A|B) = n (number of unknowns)
B.rank(A) = rank(A|B) < n (number of unknowns)
C.rank(A) != rank(A|B)
D.rank(A) = 0
Explanation: By Rouché-Frobenius, if rank(A) = rank(A|B) = n, the system has a unique solution (sistema compatible determinado).
7Solve the matrix equation AX = B for X, where A is an invertible square matrix.
A.X = A^(-1)B
B.X = BA^(-1)
C.X = B / A
D.X = A B^(-1)
Explanation: Multiplying both sides on the left by A^(-1) gives A^(-1)AX = A^(-1)B, which simplifies to IX = X = A^(-1)B. Matrix multiplication is non-commutative, so left-multiplication is required.
8Calculate the determinant of the 3x3 matrix A = [[1, 0, 2], [3, 1, 0], [2, 0, 4]].
A.0
B.4
C.8
D.-4
Explanation: Expanding along the second column: det(A) = 1 * det([[1, 2], [2, 4]]) = 1 * (1*4 - 2*2) = 1 * (4 - 4) = 0. Alternatively, Row 3 is twice Row 1 (excluding middle element 0), making rows linearly dependent.
9For what value of k is the system of equations { x + y = 3, 2x + ky = 6 } dependent with infinitely many solutions?
A.k = 2
B.k = -2
C.k = 0
D.k = 1
Explanation: Multiplying the first equation by 2 gives 2x + 2y = 6. For the second equation 2x + ky = 6 to be identical, we must have k = 2.
10What is the inverse of the 2x2 matrix A = [[2, 1], [5, 3]]?
A.[[3, -1], [-5, 2]]
B.[[-3, 1], [5, -2]]
C.[[3, 5], [1, 2]]
D.[[1/2, 1], [1/5, 1/3]]
Explanation: For A = [[a, b], [c, d]], det(A) = ad - bc = 2*3 - 1*5 = 6 - 5 = 1. A^(-1) = (1/det(A)) * [[d, -b], [-c, a]] = [[3, -1], [-5, 2]].

About the Castilla y León PAU Applied Mathematics for Social Sciences II Practice Questions

Verified exam format metadata for Castilla y León PAU Applied Mathematics for Social Sciences II (Matemáticas Aplicadas a las Ciencias Sociales II) is pending. The practice questions above remain available while official exam length, timing, passing score, fee, and administrator details are reviewed.