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Key Facts: PAU Mathematics II (La Rioja) Exam
La Rioja PAU Mathematics II (UR 2026) evaluates 2º Bachillerato linear algebra, differential calculus, integral calculus, and 3D analytical geometry over a 90-minute examination. Base registration fee: EUR 55.43. Graded on a 0-10 scale with a minimum 4.0 needed in the Access Phase. This English-language practice bank provides 100 numerical worked questions aligned with the syllabus.
Sample PAU Mathematics II (La Rioja) Practice Questions
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1Given the matrices A = [[2, -1], [3, 4]] and B = [[1, 5], [-2, 0]], calculate the matrix expression 2A - 3B.
A.[[1, -17], [12, 8]]
B.[[1, 13], [0, 8]]
C.[[1, -17], [0, 8]]
D.[[7, -17], [12, 8]]
Explanation: Compute scalar multiplications first: 2A = [[4, -2], [6, 8]] and 3B = [[3, 15], [-6, 0]]. Subtract component-wise: 2A - 3B = [[4-3, -2-15], [6-(-6), 8-0]] = [[1, -17], [12, 8]].
2What is the determinant of the upper triangular 3x3 matrix A = [[1, 2, 3], [0, 4, 5], [0, 0, 6]]?
A.10
B.24
C.0
D.15
Explanation: For any triangular matrix (upper or lower), the determinant equals the product of the diagonal elements. Therefore, det(A) = 1 * 4 * 6 = 24.
3Calculate the determinant of the 2x2 matrix A = [[3, -2], [5, 4]].
A.22
B.2
C.-2
D.7
Explanation: The determinant of a 2x2 matrix [[a, b], [c, d]] is ad - bc. Here det(A) = (3)(4) - (-2)(5) = 12 - (-10) = 12 + 10 = 22.
4Find the inverse of the matrix A = [[2, 1], [5, 3]].
A.[[3, -1], [-5, 2]]
B.[[-3, 1], [5, -2]]
C.[[2, -5], [-1, 3]]
D.[[3, 1], [5, 2]]
Explanation: First find det(A) = 2*3 - 1*5 = 6 - 5 = 1. The inverse of [[a, b], [c, d]] is (1/det(A)) * [[d, -b], [-c, a]]. Thus A^-1 = (1/1) * [[3, -1], [-5, 2]] = [[3, -1], [-5, 2]].
5Solve for matrix X in the matrix equation A * X = B, where A = [[1, 2], [3, 5]] and B = [[4, 1], [2, 0]].
A.[[-16, -5], [10, 3]]
B.[[16, 5], [-10, -3]]
C.[[-16, 5], [10, -3]]
D.[[4, 1], [2, 0]]
Explanation: Multiply both sides by A^-1 on the left: X = A^-1 * B. First, det(A) = 1*5 - 2*3 = -1. A^-1 = (1/-1) * [[5, -2], [-3, 1]] = [[-5, 2], [3, -1]]. Multiply A^-1 * B = [[-5, 2], [3, -1]] * [[4, 1], [2, 0]] = [[-5(4)+2(2), -5(1)+2(0)], [3(4)+(-1)(2), 3(1)+(-1)(0)]] = [[-16, -5], [10, 3]].
6If A is a 3x3 matrix with det(A) = -4, what is the value of det(2A)?
A.-8
B.-32
C.32
D.-16
Explanation: For an n x n square matrix A, det(k * A) = k^n * det(A). For n = 3 and k = 2: det(2A) = 2^3 * det(A) = 8 * (-4) = -32.
7What is the rank of the matrix A = [[1, 2, 3], [2, 4, 6], [0, 1, 5]]?
A.1
B.2
C.3
D.0
Explanation: Notice that Row 2 is 2 times Row 1, so Row 2 is linearly dependent and can be eliminated. Row 1 (1, 2, 3) and Row 3 (0, 1, 5) are linearly independent. Thus, the matrix has 2 linearly independent rows, so rank(A) = 2.
8Calculate the determinant of the 3x3 matrix A = [[1, 0, 2], [-1, 3, 1], [2, 1, 4]].
A.3
B.-3
C.7
D.5
Explanation: Expand along the first row: det(A) = 1 * det([[3, 1], [1, 4]]) - 0 + 2 * det([[-1, 3], [2, 1]]) = 1 * (12 - 1) + 2 * (-1 - 6) = 11 + 2(-7) = 11 - 14 = -3.
9Consider the system of linear equations with parameter k: x + y + z = 1, x + 2y + 3z = 2, x + 4y + k*z = 5. For what value of k is the system inconsistent (incompatible)?
A.k = 7
B.k = 5
C.k = 3
D.k = 0
Explanation: Compute the determinant of the coefficient matrix: det(A) = det([[1, 1, 1], [1, 2, 3], [1, 4, k]]) = k - 7. When k = 7, det(A) = 0 so rank(A) = 2. Row reduction of the augmented matrix A* yields third row (0, 0, 0 | 1), so rank(A*) = 3. By Rouché-Capelli theorem, rank(A) = 2 != rank(A*) = 3, making the system inconsistent (incompatible) when k = 7.
10Use Cramer's rule to find the value of y in the system 2x + y = 5, 3x - 2y = 4.
A.1
B.2
C.3
D.-1
Explanation: The main determinant is det(A) = det([[2, 1], [3, -2]]) = -4 - 3 = -7. The numerator determinant for y replaces column 2 with constant terms: det(A_y) = det([[2, 5], [3, 4]]) = 8 - 15 = -7. By Cramer's rule, y = det(A_y) / det(A) = -7 / -7 = 1.
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Verified exam format metadata for La Rioja PAU Mathematics II (Matemáticas II - UR 2026) is pending. The practice questions above remain available while official exam length, timing, passing score, fee, and administrator details are reviewed.