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Key Facts: PAU Mathematics II (Castilla-La Mancha) Exam
Castilla-La Mancha PAU Mathematics II (UCLM 2026) evaluates 2º Bachillerato linear algebra, differential calculus, integral calculus, and 3D analytical geometry over a 90-minute examination. This English-language multiple-choice bank is a study adaptation of the official written PAU paper, not a replica of it.
Sample PAU Mathematics II (Castilla-La Mancha) Practice Questions
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1Given the matrices A = [[3, -1], [2, 4]] and B = [[1, 2], [-3, 0]], calculate the matrix operation 2A - 3B.
A.[[3, -8], [13, 8]]
B.[[5, -8], [13, 8]]
C.[[3, -8], [1, 8]]
D.[[3, 4], [13, 8]]
Explanation: Compute 2A = [[6, -2], [4, 8]] and 3B = [[3, 6], [-9, 0]]. Subtracting elementwise: 2A - 3B = [[6-3, -2-6], [4-(-9), 8-0]] = [[3, -8], [13, 8]].
2Let A = [[1, 2], [0, -1]] and B = [[3, 1], [4, 2]]. Compute the matrix product A · B.
A.[[3, 2], [0, -2]]
B.[[11, 5], [-4, -2]]
C.[[11, 5], [4, -2]]
D.[[7, 5], [-4, -2]]
Explanation: Compute row-column dot products: Row 1: [1*3 + 2*4, 1*1 + 2*2] = [11, 5]. Row 2: [0*3 + (-1)*4, 0*1 + (-1)*2] = [-4, -2]. Thus A · B = [[11, 5], [-4, -2]].
3Calculate the determinant of the 3x3 matrix A = [[1, 0, 2], [2, -1, 3], [4, 1, 0]].
A.9
B.-1
C.5
D.13
Explanation: Using Sarrus' Rule: det(A) = 1*(-1)*0 + 0*3*4 + 2*2*1 - (2*(-1)*4 + 0*2*0 + 1*3*1) = (0 + 0 + 4) - (-8 + 0 + 3) = 4 - (-5) = 9.
4If A is a 3x3 matrix with determinant det(A) = -5, what is the determinant of the scalar matrix 2A?
A.-10
B.-40
C.-20
D.40
Explanation: For an n x n matrix, det(k A) = k^n det(A). Here n = 3 and k = 2, so det(2A) = 2^3 · det(A) = 8 · (-5) = -40.
5Find the inverse of the matrix A = [[2, 5], [1, 3]].
A.[[3, -5], [-1, 2]]
B.[[-3, 5], [1, -2]]
C.[[3, 1], [5, 2]]
D.[[2, -5], [-1, 3]]
Explanation: det(A) = 2*3 - 5*1 = 6 - 5 = 1. The formula for the inverse of [[a, b], [c, d]] is (1/det) [[d, -b], [-c, a]]. Thus A^(-1) = [[3, -5], [-1, 2]].
6Given the matrix A = [[2, 1], [-1, 3]], calculate det(A^T · A).
A.49
B.7
C.14
D.25
Explanation: det(A^T · A) = det(A^T) · det(A) = (det(A))^2. Computing det(A) = 2*3 - 1*(-1) = 6 + 1 = 7. Thus det(A^T · A) = 7^2 = 49.
7Solve the matrix equation A · X = B for X, where A = [[1, 2], [0, 1]] and B = [[5, 6], [3, 4]].
A.[[-1, -2], [3, 4]]
B.[[11, 14], [3, 4]]
C.[[-1, 2], [3, 4]]
D.[[1, 2], [3, 4]]
Explanation: X = A^(-1) · B. The inverse A^(-1) = [[1, -2], [0, 1]]. Multiplying: X = [[1, -2], [0, 1]] · [[5, 6], [3, 4]] = [[1*5 - 2*3, 1*6 - 2*4], [0*5 + 1*3, 0*6 + 1*4]] = [[-1, -2], [3, 4]].
8Classify the system of linear equations: 2x + 3y = 7 and 4x + 6y = 14.
A.Compatible indeterminado (infinitely many solutions)
B.Compatible determinado (unique solution)
C.Incompatible (no solution)
D.Trivial system
Explanation: The coefficient matrix A = [[2, 3], [4, 6]] has rank 1 (since Row 2 = 2 · Row 1). The augmented matrix A|B = [[2, 3, 7], [4, 6, 14]] also has rank 1. Since rank(A) = rank(A|B) = 1 < n = 2 variables, by Rouché-Capelli the system is compatible indeterminado.
9For what value of k does the matrix A = [[1, 2, 1], [2, 4, k], [1, 2, 3]] have rank equal to 1?
A.k = 2
B.k = 6
C.k = 0
D.No such value of k exists
Explanation: For rank(A) = 1, all 2x2 minors must vanish. Looking at the minor formed by columns 1 and 3 in rows 1 and 3: [[1, 1], [1, 3]], its determinant is 1*3 - 1*1 = 2 ≠ 0. Since this 2x2 minor does not depend on k and is non-zero, rank(A) is at least 2 for every value of k. Thus, no value of k makes rank(A) = 1.
10Find all real values of m for which the matrix A = [[m, 1, 0], [1, m, 1], [0, 1, m]] is NOT invertible.
A.m = 0, m = √2, m = -√2
B.m = 0, m = 1, m = -1
C.m = 1, m = -2
D.m = 0, m = 2
Explanation: A is non-invertible iff det(A) = 0. Expanding det(A) along row 1: det(A) = m(m^2 - 1) - 1(m - 0) + 0 = m^3 - m - m = m^3 - 2m = m(m^2 - 2). Setting det(A) = 0 yields m = 0 or m = ±√2.
About the PAU Mathematics II (Castilla-La Mancha) Practice Questions
Verified exam format metadata for Castilla-La Mancha PAU Mathematics II (Matemáticas II - UCLM 2026) is pending. The practice questions above remain available while official exam length, timing, passing score, fee, and administrator details are reviewed.