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Free Practice Questions for Canary Islands PAU Mathematics II / Matemáticas II COPAU ULPGC ULL

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Sample Canary Islands PAU Mathematics II / Matemáticas II COPAU ULPGC ULL Practice Questions

Try these sample questions to review concepts for the Canary Islands PAU Mathematics II / Matemáticas II COPAU ULPGC ULL exam. Each question includes a detailed explanation. Start the interactive quiz above for the full 100+ question experience with AI tutoring.

1Given the matrices A = [[2, 1], [-1, 3]] and B = [[1, -2], [0, 4]], what is the result of the matrix operation 2A - 3B?
A.[[1, 8], [-2, -6]]
B.[[1, 8], [-2, 18]]
C.[[3, -4], [-2, -6]]
D.[[1, 4], [-2, -6]]
Explanation: To calculate 2A - 3B, multiply matrix A by scalar 2: 2A = [[4, 2], [-2, 6]], and multiply matrix B by scalar 3: 3B = [[3, -6], [0, 12]]. Subtracting elementwise gives: 2A - 3B = [[4-3, 2-(-6)], [-2-0, 6-12]] = [[1, 8], [-2, -6]].
2Let A = [[1, 2], [3, 4]] and B = [[2, 0], [1, 5]]. Compute the matrix product A * B.
A.[[4, 10], [10, 20]]
B.[[2, 0], [3, 20]]
C.[[4, 10], [6, 20]]
D.[[3, 10], [10, 20]]
Explanation: Matrix multiplication is computed by taking row-column dot products. Row 1: [1*2 + 2*1, 1*0 + 2*5] = [4, 10]. Row 2: [3*2 + 4*1, 3*0 + 4*5] = [10, 20]. Thus, A * B = [[4, 10], [10, 20]].
3If A is a 3x3 matrix with determinant |A| = -4, what is the determinant of the matrix 3A?
A.-108
B.-36
C.-12
D.108
Explanation: For an n x n matrix A and scalar k, |k A| = k^n * |A|. For a 3x3 matrix (n = 3) and scalar k = 3: |3A| = 3^3 * |A| = 27 * (-4) = -108.
4Calculate the determinant of the matrix A = [[1, 0, 2], [3, -1, 1], [2, 0, 4]].
A.-4
B.4
C.0
D.2
Explanation: Expand along the second column, whose only non-zero entry is a₂₂ = -1 with cofactor sign (+1)^(2+2) = +1: |A| = (-1) · |[1, 2], [2, 4]| = (-1) · (1·4 - 2·2) = (-1) · 0 = 0. The quickest route is to notice that Row 3 = 2 · Row 1, so the rows are linearly dependent and the determinant is automatically 0.
5A sample of n = 64 light bulbs is drawn from a normal population with known standard deviation σ = 40 hours, and the sample mean lifetime is x̄ = 1200 hours. Construct the 95% confidence interval for the population mean μ. (Use z_(α/2) = 1.96.)
A.(1160.0, 1240.0)
B.(1190.2, 1209.8)
C.(1198.8, 1201.2)
D.(1191.8, 1208.2)
Explanation: For a normal population with known σ, the confidence interval for μ is x̄ ± z_(α/2) · σ/√n. The standard error is σ/√n = 40/√64 = 40/8 = 5, so the margin of error is 1.96 · 5 = 9.8. The interval is 1200 ± 9.8 = (1190.2, 1209.8).
6Given A = [[2, 1], [5, 3]], calculate its inverse matrix A^(-1).
A.[[2, -1], [-5, 3]]
B.[[-3, 1], [5, -2]]
C.[[3, -1], [-5, 2]]
D.[[3, 5], [1, 2]]
Explanation: For a 2x2 matrix A = [[a, b], [c, d]], |A| = ad - bc = 2*3 - 1*5 = 1. The inverse is A^(-1) = (1/|A|) * [[d, -b], [-c, a]] = [[3, -1], [-5, 2]].
7Solve for the matrix X in the equation A * X + B = C, assuming matrix A is invertible.
A.X = (B - C) * A^(-1)
B.X = A^(-1) * (C - B)
C.X = A^(-1) * C - B
D.X = (C - B) * A^(-1)
Explanation: Subtract B from both sides: A * X = C - B. Left-multiply both sides by A^(-1): A^(-1) * (A * X) = A^(-1) * (C - B) ⇒ X = A^(-1) * (C - B). Left-multiplication is required because matrix multiplication is non-commutative.
8What is the rank of the matrix M = [[1, 2, 3], [2, 4, 6], [3, 6, 9]]?
A.0
B.2
C.3
D.1
Explanation: Row 2 is 2 * Row 1 and Row 3 is 3 * Row 1. All 2x2 and 3x3 minors equal 0. Since M contains non-zero entries, rank(M) = 1.
9According to the Rouché-Capelli Theorem, a system of 3 linear equations with 3 unknowns is consistent and dependent (compatible indeterminado) if and only if:
A.rank(A) = rank(A|B) = 3
B.rank(A) = 3 and rank(A|B) = 2
C.rank(A) = rank(A|B) < 3
D.rank(A) < rank(A|B)
Explanation: By Rouché-Capelli: a system is consistent if rank(A) = rank(A|B). If this common rank r is strictly less than the number of unknowns n (n = 3), the system is compatible indeterminado (infinitely many solutions).
10Consider the system: x + y + z = 6, 2x - y + z = 3, x + 2y - z = 2. Find the value of x using Cramer's rule.
A.1
B.2
C.3
D.4
Explanation: Coefficient matrix A = [[1, 1, 1], [2, -1, 1], [1, 2, -1]]. |A| = 1(1 - 2) - 1(-2 - 1) + 1(4 - (-1)) = -1 + 3 + 5 = 7. Replacing column 1 with B = [6, 3, 2]^T: |A_x| = |[6, 1, 1], [3, -1, 1], [2, 2, -1]| = 6(1 - 2) - 1(-3 - 2) + 1(6 - (-2)) = -6 + 5 + 8 = 7. Thus x = |A_x| / |A| = 7 / 7 = 1.

About the Canary Islands PAU Mathematics II / Matemáticas II COPAU ULPGC ULL Exam

The Canary Islands PAU Mathematics II (Matemáticas II) examination assesses 2º Bachillerato knowledge across four core blocks: Linear Algebra (matrix algebra, determinants, matrix inverse, system classification via Rouché-Capelli and Cramer's rule), Mathematical Analysis (limits, continuity, differentiability, L'Hôpital's rule, curve sketching, optimization problems, indefinite and definite integrals, area between curves), Geometry in 3D (vector operations, lines and planes in space, relative positions, distances, angles, projections), and Probability & Statistics (conditional probability, Bayes' theorem, binomial distribution, and normal distribution).

Exam sponsor: Canary Islands PAU Organising Commission (COPAU) / ULPGC & ULL. The requirements and fees below concern the certification or admission exam, separate from our free practice resources.

Assessment

Written 90-minute standardized exam divided into Algebra, Mathematical Analysis, Geometry in 3D, and Statistics & Probability blocks. Local practice items are an English-language MCQ study adaptation of the official open/semi-constructed Canary Islands PAU paper, not an official translation or format simulation.

Time Limit

90 minutes (1.5 hours)

Passing Score

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

Exam / Certification Fees

EUR 76.12 base registration fee for PAU Access Phase (set by Gobierno de Canarias / COPAU).

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

Matrix operations, determinants, matrix inverse, rank, and classification/solving of linear systems via Rouché–Capelli and Cramer's rule.

30%

Mathematical Analysis

Limits, continuity, differentiability, L'Hôpital's rule, curve sketching, optimization problems, indefinite and definite integrals, and area between curves.

25%

Geometry in 3D

Vector operations, lines and planes in space, relative positions, distances, angles, and projections.

20%

Probability & Statistics

Conditional probability, Bayes' theorem, binomial distribution, and normal distribution.

Preparing for the Canary Islands PAU Mathematics II / Matemáticas II COPAU ULPGC ULL 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 + 40% PAU >= 5.0 to pass).
  • Assessment: Written 90-minute standardized exam divided into Algebra, Mathematical Analysis, Geometry in 3D, and Statistics & Probability blocks. Local practice items are an English-language MCQ study adaptation of the official open/semi-constructed Canary Islands PAU paper, not an official translation or format simulation.
  • Time limit: 90 minutes (1.5 hours)
  • Exam / certification fees: EUR 76.12 base registration fee for PAU Access Phase (set by Gobierno de Canarias / COPAU). 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

Frequently Asked Questions

Is this practice bank in the same format as the real Canary Islands PAU exam?

No, and it is important to know the difference. The official Canary Islands PAU paper is a 90-minute examination sat in Spanish and built around multi-part written problems that must be solved and justified in full. Under the 2026 PAU design rules agreed by COPAU and the CRUE, open and semi-constructed responses must account for at least 70% of every paper, so the real exam contains no multiple-choice section. This bank is an English-language multiple-choice study adaptation of the same official 2º Bachillerato syllabus — not an official translation, not a past paper, and not a simulation of the exam format. Use it to drill the underlying knowledge and reasoning quickly, then practise writing out complete, justified solutions and showing every step of your working separately, because that is what the examiners actually mark.

What is the format of the Canary Islands PAU Mathematics II exam?

It is a 90-minute written examination administered at ULPGC and ULL designated centers, featuring open and semi-constructed problem-solving tasks across algebra, analysis, geometry, and probability/statistics.

How is the exam scored for university entrance in Spain?

The exam is graded on a 0–10 scale. In the compulsory Access Phase, a minimum mark of 4.0 is required to combine with Bachillerato GPA (60% Bachillerato + 40% PAU Access Phase >= 5.0). In the Admission Phase, it is typically a heavily-weighted subject for STEM degrees.

Which topics carry the most weight?

Mathematical Analysis (limits, derivatives, integrals, optimization) is typically the largest block, followed by Linear Algebra and Geometry in 3D, with Probability & Statistics as a smaller but consistently examined block.

Which universities manage the PAU in the Canary Islands?

The exam is organized jointly by the Canary Islands PAU Organising Commission (COPAU), Universidad de Las Palmas de Gran Canaria (ULPGC), and Universidad de La Laguna (ULL).