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100+ Free PAU Mathematics II (Galicia) Practice Questions
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Key Facts: PAU Mathematics II (Galicia) Exam
Galicia PAU Mathematics II (CIUG 2026) evaluates 2º Bachillerato linear algebra, differential calculus, integral calculus, and 3D analytical geometry over a 90-minute examination. This multiple-choice bank is a study adaptation of the official written PAU paper, designed to reinforce core mathematical problem-solving skills.
Sample PAU Mathematics II (Galicia) Practice Questions
Try these sample questions to test your PAU Mathematics II (Galicia) exam readiness. Each question includes a detailed explanation. Start the interactive quiz above for the full 100+ question experience with AI tutoring.
1Calculate the inverse of the matrix A = [[1, 2], [3, 4]].
A.[[-2, 1], [1.5, -0.5]]
B.[[2, -1], [-1.5, 0.5]]
C.[[-4, 2], [3, -1]]
D.[[1, -0.5], [-1.5, 2]]
Explanation: The determinant of A is det(A) = 1*4 - 2*3 = -2. The inverse formula for a 2x2 matrix is A^(-1) = (1/det(A)) * [[d, -b], [-c, a]]. Substituting yields A^(-1) = (-1/2) * [[4, -2], [-3, 1]] = [[-2, 1], [1.5, -0.5]].
2Compute the determinant of the 3x3 matrix A = [[1, 0, 2], [2, 1, 1], [0, 3, -1]].
A.8
B.-4
C.12
D.4
Explanation: Expanding det(A) along the first row gives 1*(1*(-1) - 3*1) - 0 + 2*(2*3 - 0*1) = 1*(-4) + 2*(6) = -4 + 12 = 8.
3Determine the rank of the matrix A = [[1, 2, 3], [2, 4, 6], [1, 1, 1]].
A.2
B.1
C.3
D.0
Explanation: Notice that the second row is twice the first row (R2 = 2*R1), so R2 is linearly dependent. The first and third rows are linearly independent since (1,2,3) is not a scalar multiple of (1,1,1). Therefore, the rank of matrix A is 2.
4Solve the matrix equation A * X = B for X, where A = [[2, 1], [1, 1]] and B = [[5], [3]].
A.X = [[2], [1]]
B.X = [[1], [2]]
C.X = [[3], [-1]]
D.X = [[5], [3]]
Explanation: The determinant det(A) = 2*1 - 1*1 = 1, so A^(-1) = [[1, -1], [-1, 2]]. Multiplying A^(-1) by B yields X = [[1*5 - 1*3], [-1*5 + 2*3]] = [[2], [1]].
5According to the Rouché-Fröbenius theorem, a linear system with n unknowns A*X = B has a unique solution (compatible determinado) if and only if:
A.rank(A) = rank(A|B) = n
B.rank(A) = rank(A|B) < n
C.rank(A) != rank(A|B)
D.rank(A) = n regardless of rank(A|B)
Explanation: By the Rouché-Fröbenius theorem, a system is compatible if rank(A) = rank(A|B). When this common rank equals the number of variables n, the system has a unique solution (compatible determinado).
6For which values of parameter k is the matrix A = [[k, 1, 1], [1, k, 1], [1, 1, k]] singular (non-invertible)?
A.k = 1 and k = -2
B.k = 0 and k = 2
C.k = -1 and k = 2
D.k = 3 only
Explanation: Calculating the determinant yields det(A) = k^3 - 3k + 2 = (k + 2)(k - 1)^2. Setting det(A) = 0 yields roots k = 1 (double root) and k = -2. Thus A is singular when k = 1 or k = -2.
7Use Cramer's Rule to solve for x in the system: x + y = 3, 2x - y = 3.
A.x = 2
B.x = 1
C.x = 3
D.x = -1
Explanation: The coefficient determinant is Delta = 1*(-1) - 1*2 = -3. The determinant for x is Delta_x = 3*(-1) - 1*3 = -6. By Cramer's Rule, x = Delta_x / Delta = -6 / -3 = 2.
8If A is a 3x3 matrix with det(A) = 4, what is the value of det(2A)?
A.32
B.8
C.12
D.24
Explanation: For any n x n matrix A and scalar c, det(c*A) = c^n * det(A). For n = 3 and c = 2, det(2A) = 2^3 * det(A) = 8 * 4 = 32.
9If A is an invertible matrix with det(A) = 5, what is det(A^(-1))?
A.0.2 (or 1/5)
B.-5
C.5
D.-0.2
Explanation: The determinant of the inverse matrix satisfies det(A^(-1)) = 1 / det(A). Given det(A) = 5, det(A^(-1)) = 1/5 = 0.2.
10Classify the homogeneous system: x + 2y - z = 0, 2x - y + 3z = 0, 3x + y + 2z = 0.
A.Compatible indeterminado (infinitely many solutions)
B.Compatible determinado (only trivial solution x=y=z=0)
C.Incompatible (no solutions)
D.Compatible determinado with x=1, y=2, z=3
Explanation: Every homogeneous system is compatible. Evaluating the coefficient determinant det(A) = 1*(-2-3) - 2*(4-9) - 1*(2+3) = -5 + 10 - 5 = 0. Since det(A) = 0, rank(A) < 3, so the system has infinitely many solutions (compatible indeterminado).
About the PAU Mathematics II (Galicia) Practice Questions
Verified exam format metadata for Galicia PAU Mathematics II (Matemáticas II - CIUG 2026) is pending. The practice questions above remain available while official exam length, timing, passing score, fee, and administrator details are reviewed.