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100+ Free Catalonia PAU Mathematics II (Matemàtiques II - CIC 2026) Practice Questions
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2026 Statistics
Key Facts: Catalonia PAU Mathematics II (Matemàtiques II - CIC 2026) Exam
90 min
Time limit
Consell Interuniversitari de Catalunya
EUR 110.00+ access phase (EUR 41.30 exam right + EUR 68.70 access phase; plus EUR 13.80 per admission exercise, Generalitat de Catalunya 2026)
Base fee set by Generalitat de Catalunya
Generalitat de Catalunya
0–10
Grading scale (min 4.0 in Access Phase)
CIC Regulations
100
Practice questions available
OpenExamPrep
Catalonia PAU Mathematics II is a 90-minute examination assessing 2nd Bachillerato mathematics across algebra, calculus, 3D geometry, and probability.
Sample Catalonia PAU Mathematics II (Matemàtiques II - CIC 2026) Practice Questions
Try these sample questions to test your Catalonia PAU Mathematics II (Matemàtiques II - CIC 2026) exam readiness. Each question includes a detailed explanation. Start the interactive quiz above for the full 100+ question experience with AI tutoring.
1Given matrices $A = \begin{pmatrix} 2 & -1 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 5 \\ -2 & 0 \end{pmatrix}$, compute the matrix $2A - 3B$.
A.\begin{pmatrix} 1 & -17 \\ 12 & 8 \end{pmatrix}
B.\begin{pmatrix} 1 & 13 \\ 0 & 8 \end{pmatrix}
C.\begin{pmatrix} 7 & 13 \\ 0 & 8 \end{pmatrix}
D.\begin{pmatrix} 1 & -17 \\ 0 & 8 \end{pmatrix}
Explanation: Multiplying matrix $A$ by 2 yields $2A = \begin{pmatrix} 4 & -2 \\ 6 & 8 \end{pmatrix}$, and multiplying matrix $B$ by 3 yields $3B = \begin{pmatrix} 3 & 15 \\ -6 & 0 \end{pmatrix}$. Subtracting component-wise gives $2A - 3B = \begin{pmatrix} 4-3 & -2-15 \\ 6-(-6) & 8-0 \end{pmatrix} = \begin{pmatrix} 1 & -17 \\ 12 & 8 \end{pmatrix}$.
2If $A$ is a $3 \times 3$ square matrix with $\det(A) = -4$, what is the value of $\det(2A)$?
A.-8
B.-32
C.32
D.-16
Explanation: For an $n \times n$ matrix $A$, scaling the entire matrix by a scalar $k$ scales its determinant by $k^n$. Here $n = 3$ and $k = 2$, so $\det(2A) = 2^3 \cdot \det(A) = 8 \cdot (-4) = -32$.
3For any square matrices $A$ and $B$ of the same dimension, which of the following matrix transpose properties is always true?
A.$(AB)^T = A^T B^T$
B.$(A + B)^T = A^T + B^T$
C.$\det(A^T) = -\det(A)$
D.$(AB)^T = B A$
Explanation: Matrix transposition is a linear operator over matrix addition, meaning $(A + B)^T = A^T + B^T$ is always valid. In contrast, product transposition reverses order $(AB)^T = B^T A^T$, and determinant is invariant under transposition $\det(A^T) = \det(A)$.
4Calculate the determinant of the matrix $A = \begin{pmatrix} 1 & 2 & 0 \\ 3 & -1 & 4 \\ 2 & 0 & 1 \end{pmatrix}$.
A.9
B.-21
C.15
D.-11
Explanation: Expanding along the first row or applying Sarrus' rule gives $\det(A) = 1((-1)(1) - (4)(0)) - 2((3)(1) - (4)(2)) + 0 = 1(-1) - 2(3 - 8) = -1 - 2(-5) = -1 + 10 = 9$.
5Find the inverse of the matrix $A = \begin{pmatrix} 3 & 1 \\ 5 & 2 \end{pmatrix}$.
A.\begin{pmatrix} 2 & -1 \\ -5 & 3 \end{pmatrix}
B.\begin{pmatrix} -2 & 1 \\ 5 & -3 \end{pmatrix}
C.\begin{pmatrix} 3 & -5 \\ -1 & 2 \end{pmatrix}
D.\begin{pmatrix} 2 & 1 \\ 5 & 3 \end{pmatrix}
Explanation: The determinant of $A$ is $\det(A) = (3)(2) - (1)(5) = 1$. The inverse formula for a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$ yields $\begin{pmatrix} 2 & -1 \\ -5 & 3 \end{pmatrix}$.
6Use Cramer's rule to solve for $y$ in the linear system $\begin{cases} 2x + 3y = 7 \\ x - y = 1 \end{cases}$.
A.y = 1
B.y = 2
C.y = -1
D.y = 3
Explanation: The coefficient determinant is $\Delta = \begin{vmatrix} 2 & 3 \\ 1 & -1 \end{vmatrix} = -2 - 3 = -5$. Replacing the second column with the constant vector yields $\Delta_y = \begin{vmatrix} 2 & 7 \\ 1 & 1 \end{vmatrix} = 2 - 7 = -5$. Thus $y = \frac{\Delta_y}{\Delta} = \frac{-5}{-5} = 1$.
7Given $A = \begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix}$, compute $A^2$.
A.\begin{pmatrix} 1 & 8 \\ 0 & 9 \end{pmatrix}
B.\begin{pmatrix} 1 & 4 \\ 0 & 9 \end{pmatrix}
C.\begin{pmatrix} 1 & 6 \\ 0 & 9 \end{pmatrix}
D.\begin{pmatrix} 2 & 4 \\ 0 & 6 \end{pmatrix}
Explanation: Multiplying $A$ by itself: $A^2 = \begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix} = \begin{pmatrix} 1(1)+2(0) & 1(2)+2(3) \\ 0(1)+3(0) & 0(2)+3(3) \end{pmatrix} = \begin{pmatrix} 1 & 8 \\ 0 & 9 \end{pmatrix}$.
8What is the rank of the matrix $M = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 0 & 1 & 5 \end{pmatrix}$?
A.1
B.2
C.3
D.0
Explanation: The second row $\text{Row}_2 = (2, 4, 6)$ is exactly twice $\text{Row}_1 = (1, 2, 3)$, making them linearly dependent. The third row $\text{Row}_3 = (0, 1, 5)$ is linearly independent of $\text{Row}_1$. Therefore, there are 2 linearly independent rows, so $\text{rank}(M) = 2$.
9Solve the matrix equation $AX + B = C$ for $X$, given $A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$, $B = \begin{pmatrix} 2 & 0 \\ 1 & 3 \end{pmatrix}$, and $C = \begin{pmatrix} 5 & 4 \\ 2 & 7 \end{pmatrix}$.
A.\begin{pmatrix} 2 & 0 \\ 1 & 4 \end{pmatrix}
B.\begin{pmatrix} 3 & 4 \\ 1 & 4 \end{pmatrix}
C.\begin{pmatrix} 2 & 0 \\ 2 & 4 \end{pmatrix}
D.\begin{pmatrix} 1 & 0 \\ 1 & 4 \end{pmatrix}
Explanation: First isolate $AX$: $AX = C - B = \begin{pmatrix} 5-2 & 4-0 \\ 2-1 & 7-3 \end{pmatrix} = \begin{pmatrix} 3 & 4 \\ 1 & 4 \end{pmatrix}$. Then multiply by $A^{-1} = \begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix}$ on the left: $X = A^{-1}(C-B) = \begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 3 & 4 \\ 1 & 4 \end{pmatrix} = \begin{pmatrix} 3-1 & 4-4 \\ 0+1 & 0+4 \end{pmatrix} = \begin{pmatrix} 2 & 0 \\ 1 & 4 \end{pmatrix}$.
10Determine all values of parameter $k$ for which the matrix $A = \begin{pmatrix} 1 & k & 1 \\ k & 1 & 1 \\ 1 & 1 & k \end{pmatrix}$ is non-invertible (singular).
A.k = 1 and k = -2
B.k = 1 and k = 2
C.k = -1 and k = 2
D.k = 0 only
Explanation: A square matrix is singular when its determinant equals zero. Computing $\det(A) = 1(k-1) - k(k^2-1) + 1(k-1) = -(k-1)^2(k+2)$. Setting $\det(A) = 0$ yields roots $k = 1$ and $k = -2$.
About the Catalonia PAU Mathematics II (Matemàtiques II - CIC 2026) Practice Questions
Verified exam format metadata for Catalonia PAU Mathematics II / Matemàtiques II 2º Bachillerato (Proves d'Accés a la Universitat) is pending. The practice questions above remain available while official exam length, timing, passing score, fee, and administrator details are reviewed.