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Free Practice Questions for Catalonia PAU Mathematics II (Matemàtiques II - CIC 2026)

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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 review concepts for the Catalonia PAU Mathematics II (Matemàtiques II - CIC 2026) exam. 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) Exam

The Catalonia PAU Mathematics II (Matemàtiques II) exam is the university entrance test for STEM, Health Sciences, and Architecture tracks in Catalonia. It assesses 2nd Bachillerato mathematics across linear algebra, differential and integral calculus, 3D geometry, and probability.

Exam sponsor: Consell Interuniversitari de Catalunya (CIC) / Generalitat de Catalunya. The requirements and fees below concern the certification or admission exam, separate from our free practice resources.

Assessment

The official Catalonia PAU Mathematics II exam is a written problem-solving examination administered in Catalan (with Spanish available upon request), consisting of multi-part analytical and computational problems lasting 90 minutes. This online question bank provides an English-language multiple-choice practice adaptation designed to master the mathematical calculations and theoretical concepts tested on the PAU.

Time Limit

90 minutes

Passing Score

Marked on a 0–10 scale. Minimum 4.0 required in Access Phase to combine with Bachillerato GPA (60% Bachillerato GPA + 40% Access Phase mark >= 5.0).

Exam / Certification Fees

EUR 41.30 examination right + EUR 68.70 access phase + EUR 13.80 per admission-phase exercise (Generalitat de Catalunya, PAU 2026); 50% reduction and full exemption available for eligible groups

Exam sponsor website

Fees, eligibility, and exam policies can change. Confirm them with the exam sponsor before applying or paying.

Our practice resources: topics covered

We aim to reflect publicly available exam outlines and topic information in our study resources. Coverage, format, and difficulty may differ from the actual exam, and we cannot guarantee that every detail is accurate or current. Confirm exam requirements, fees, and policies with the official exam sponsor.

25%

Linear Algebra & Systems of Equations

Matrix operations, determinants, inverse matrices, matrix equations, rank, and resolution of linear systems using Rouché-Capelli Theorem and Cramer's Rule.

35%

Differential & Integral Calculus

Limits, L'Hôpital's rule, continuity, derivatives, tangents, optimization, curve sketching, asymptotes, definite integrals, and area computation.

25%

3D Analytical Geometry

Vectors in R3, dot and cross products, line and plane equations, relative positions, projections, symmetric points, angles, and distances.

15%

Probability & Statistics

Conditional probability, Bayes' Theorem, independent events, Binomial distribution B(n,p), and Normal distribution N(mu, sigma).

Preparing for the Catalonia PAU Mathematics II (Matemàtiques II - CIC 2026) Exam

What You Need to Know

  • Passing score: Marked on a 0–10 scale. Minimum 4.0 required in Access Phase to combine with Bachillerato GPA (60% Bachillerato GPA + 40% Access Phase mark >= 5.0).
  • Assessment: The official Catalonia PAU Mathematics II exam is a written problem-solving examination administered in Catalan (with Spanish available upon request), consisting of multi-part analytical and computational problems lasting 90 minutes. This online question bank provides an English-language multiple-choice practice adaptation designed to master the mathematical calculations and theoretical concepts tested on the PAU.
  • Time limit: 90 minutes
  • Exam / certification fees: EUR 41.30 examination right + EUR 68.70 access phase + EUR 13.80 per admission-phase exercise (Generalitat de Catalunya, PAU 2026); 50% reduction and full exemption available for eligible groups Official sources

Using Our Practice Resources

  • Work through all 100 available questions
  • Review every answer and explanation
  • Track weak areas and revisit them
  • Use our AI tutor for tough concepts

Catalonia PAU Mathematics II (Matemàtiques II - CIC 2026): Suggested Study Strategy

1Master matrix operations and determinant properties, especially Rouché-Capelli analysis of linear systems with parameters.
2Practice derivative-based optimization problems and definite integration for area calculations between curves.
3Understand 3D line and plane vector equations, including finding normal vectors via cross products and metric distance calculations.
4Apply Bayes' Theorem and normal distribution standardizations accurately.

Frequently Asked Questions

What is the Catalonia PAU Mathematics II exam format?

The official exam is a 90-minute written problem-solving test in Catalan/Spanish. This study tool provides an English-language multiple-choice adaptation for practice.

What is the registration fee for Catalonia PAU?

The Generalitat de Catalunya, through the Consell Interuniversitari de Catalunya (CIC), sets the 2026 PAU ordinary fees at EUR 41.30 examination right, EUR 68.70 access phase, and EUR 13.80 per admission-phase exercise. A 50% reduction and a full exemption are available for eligible groups such as large families (família nombrosa general), scholarship holders, and students with a recognised disability of 33% or more.

What passing score is required?

The exam is graded on a 0–10 scale. A minimum score of 4.0 in the Access Phase is required to combine with the Bachillerato GPA.

Who administers the Catalonia PAU exam?

The exam is organized and administered by the Consell Interuniversitari de Catalunya (CIC) / Generalitat de Catalunya.