Free Practice Questions for CAPE Pure Maths Unit 2

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Key Facts: CAPE Pure Maths Unit 2 Exam

Paper 01 Questions

CXC CAPE Pure Mathematics Syllabus

90m

Paper 01 Duration

CXC Timetable

Grades I–V

Certifiable Passing Grades

CXC CAPE Grading Policy

30%

Paper 01 Weighting

CXC CAPE Pure Mathematics Scheme

BBD 69.50

Subject Fee (per Unit)

CXC Scale of Fees 2026

3 Modules

Syllabus Structure

CXC Pure Mathematics Syllabus

150 hrs

Guided Learning Hours

CXC Curriculum Guide

CAPE Pure Mathematics Unit 2 is administered across three modules: Complex Numbers and Calculus II, Sequences, Series and Approximations, and Counting, Matrices and Differential Equations. Paper 01 consists of 45 compulsory multiple-choice items (15 per module) completed in 90 minutes, contributing 30% of the total Unit grade. Paper 02 provides 50% through six extended-response questions, and School-Based Assessment or Paper 032 contributes 20%. Results are reported as Grades I to VII. These independent English-language MCQs are a study adaptation for Unit 2 topics, not an official-format simulation or a substitute for complete written proofs and multi-step derivations.

Sample CAPE Pure Maths Unit 2 Practice Questions

Try these sample questions to review concepts for the CAPE Pure Maths Unit 2 exam. Each question includes a detailed explanation. Start the interactive quiz above for the full 100+ question experience with AI tutoring.

1What are the modulus and principal argument of the complex number z = -1 - i√3?

A.Modulus = 2, Argument = -2π/3

B.Modulus = 2, Argument = 2π/3

C.Modulus = 4, Argument = -2π/3

D.Modulus = 2, Argument = -π/3

Explanation: The modulus is |z| = √((-1)² + (-√3)²) = √(1 + 3) = 2. Since both real and imaginary parts are negative, z lies in the third quadrant. The reference angle is arctan(√3/1) = π/3, so the principal argument in (-π, π] is -π + π/3 = -2π/3.

2Simplify the complex expression (3 + 4i) / (1 - 2i) into Cartesian form x + iy.

A.-1 + 2i

B.1 + 2i

C.-1 - 2i

D.-5 + 10i

Explanation: Multiply numerator and denominator by the complex conjugate of the denominator, (1 + 2i): (3 + 4i)(1 + 2i) / ((1 - 2i)(1 + 2i)) = (3 + 6i + 4i + 8i²) / (1 - 4i²) = (3 + 10i - 8) / (1 + 4) = (-5 + 10i) / 5 = -1 + 2i.

3Solve the quadratic equation z² - 4z + 13 = 0 over the complex numbers.

A.z = 2 ± 3i

B.z = -2 ± 3i

C.z = 4 ± 6i

D.z = 2 ± 6i

Explanation: Using the quadratic formula, z = (-(-4) ± √((-4)² - 4(1)(13))) / (2(1)) = (4 ± √(16 - 52)) / 2 = (4 ± √(-36)) / 2 = (4 ± 6i) / 2 = 2 ± 3i.

4Find the square roots of the complex number 3 - 4i.

A.±(2 - i)

B.±(2 + i)

C.±(1 - 2i)

D.±(1 + 2i)

Explanation: Let (x + iy)² = 3 - 4i. Expanding gives x² - y² = 3 and 2xy = -4, so xy = -2 (x and y have opposite signs). Furthermore, x² + y² = |3 - 4i| = √(9 + 16) = 5. Adding x² - y² = 3 and x² + y² = 5 gives 2x² = 8 ⇒ x² = 4 ⇒ x = ±2. Thus y = -2/x = ∓1. The square roots are ±(2 - i).

5Express z = 2√3 - 2i in exponential form r e^(iθ) with principal argument θ ∈ (-π, π].

A.4e^(-iπ/6)

B.4e^(i5π/6)

C.2e^(-iπ/6)

D.4e^(-iπ/3)

Explanation: The modulus is r = √((2√3)² + (-2)²) = √(12 + 4) = 4. Since the real part is positive and the imaginary part is negative, z lies in the fourth quadrant. The reference angle is arctan(|-2| / (2√3)) = arctan(1/√3) = π/6. Thus θ = -π/6, giving z = 4e^(-iπ/6).

6Simplify the quotient (6e^(i 5π/6)) / (2e^(i π/3)) into standard Cartesian form.

A.3i

B.-3

C.3

D.-3i

Explanation: Divide the moduli and subtract the exponents: (6/2) e^(i (5π/6 - π/3)) = 3 e^(i (5π/6 - 2π/6)) = 3 e^(i π/2). In Cartesian form, 3(cos(π/2) + i sin(π/2)) = 3(0 + i(1)) = 3i.

7Using De Moivre's theorem, evaluate (1 + i)¹⁰.

A.32i

B.-32i

C.32

D.-32

Explanation: Convert 1 + i to polar form: r = √(1² + 1²) = √2 and θ = π/4, so 1 + i = √2 e^(iπ/4). Applying De Moivre's theorem: (1 + i)¹⁰ = (√2)¹⁰ e^(i 10π/4) = 2⁵ e^(i 5π/2). Since 5π/2 = 2π + π/2, e^(i 5π/2) = e^(iπ/2) = i. Thus (1 + i)¹⁰ = 32i.

8Using De Moivre's theorem, which expression represents cos(3θ) in terms of powers of cos(θ)?

A.4cos³(θ) - 3cos(θ)

B.3cos(θ) - 4cos³(θ)

C.4cos³(θ) + 3cos(θ)

D.cos³(θ) - 3cos(θ)

Explanation: By De Moivre's theorem, cos(3θ) + i sin(3θ) = (cos(θ) + i sin(θ))³. Expanding the binomial: cos³(θ) + 3i cos²(θ) sin(θ) - 3 cos(θ) sin²(θ) - i sin³(θ). Equating real parts gives cos(3θ) = cos³(θ) - 3 cos(θ) sin²(θ). Replacing sin²(θ) with 1 - cos²(θ): cos(3θ) = cos³(θ) - 3 cos(θ)(1 - cos²(θ)) = cos³(θ) - 3 cos(θ) + 3 cos³(θ) = 4 cos³(θ) - 3 cos(θ).

9Using the identity 2i sin(θ) = z - 1/z where z = e^(iθ), express sin³(θ) in terms of multiple angles.

A.(1/4)(3sin(θ) - sin(3θ))

B.(1/4)(sin(3θ) - 3sin(θ))

C.(1/4)(3sin(θ) + sin(3θ))

D.(1/8)(3sin(θ) - sin(3θ))

Explanation: (2i sin(θ))³ = -8i sin³(θ). Expanding (z - z⁻¹)³ = z³ - 3z + 3z⁻¹ - z⁻³ = (z³ - z⁻³) - 3(z - z⁻¹). Since zⁿ - z⁻ⁿ = 2i sin(nθ), this equals 2i sin(3θ) - 3(2i sin(θ)) = 2i sin(3θ) - 6i sin(θ). Equating: -8i sin³(θ) = 2i sin(3θ) - 6i sin(θ) ⇒ sin³(θ) = (6 sin(θ) - 2 sin(3θ)) / 8 = (1/4)(3 sin(θ) - sin(3θ)).

10What are all three solutions to the equation z³ = 8i in Cartesian form?

A.√3 + i, -√3 + i, -2i

B.1 + i√3, -1 + i√3, -2

C.√3 - i, -√3 - i, 2i

D.2 + i, -2 + i, -2i

Explanation: Write 8i in exponential form: 8i = 8 e^(i (π/2 + 2kπ)). Taking the cube root: z_k = 2 e^(i (π/6 + 2kπ/3)) for k = 0, 1, 2. For k = 0: 2(cos(π/6) + i sin(π/6)) = 2(√3/2 + i/2) = √3 + i. For k = 1: 2(cos(5π/6) + i sin(5π/6)) = 2(-√3/2 + i/2) = -√3 + i. For k = 2: 2(cos(9π/6) + i sin(9π/6)) = 2(0 - i) = -2i.

About the CAPE Pure Maths Unit 2 Exam

CAPE Pure Mathematics Unit 2 is the second of two independent certifiable Units in the Caribbean Examinations Council's Advanced Proficiency Examination in Pure Mathematics. The syllabus covers three modules: Module 1 Complex Numbers and Calculus II, Module 2 Sequences, Series and Approximations, and Module 3 Counting, Matrices and Differential Equations. It prepares students for university programmes in mathematics, engineering, physical sciences, economics, and computing. Paper 01 is a compulsory multiple-choice examination of 45 items (15 per module) completed in 90 minutes, accounting for 30% of the overall Unit mark. These independent English-language MCQs provide focused revision for Paper 01 topics.

Exam sponsor: Caribbean Examinations Council (CXC). The requirements and fees below concern the certification or admission exam, separate from our free practice resources.

Assessment

Paper 01: 45 compulsory multiple-choice items, 15 from each of the three Modules (1 hour 30 minutes, 30%). Paper 02: six compulsory extended-response questions, two per Module (2 hours 30 minutes, 50%). School-Based Assessment Paper 031 or private-candidate Paper 032 (20%).

Time Limit

Paper 01 is 1 hour 30 minutes (90 minutes). Paper 02 is 2 hours 30 minutes. Paper 032 for private candidates is 2 hours.

Passing Score

CXC does not publish percentage grade boundaries. CAPE results are reported as Grades I to VII (Grade I highest); Grades I to V are accepted as passing grades. The final grade combines Paper 01 (30%), Paper 02 (50%), and School-Based Assessment or Paper 032 (20%).

Exam / Certification Fees

Effective May–June 2026: BBD $69.50 per CAPE Unit plus BBD $48.50 candidate entry in member territories. External-territory rates are BBD $139.00 per Unit plus BBD $97.00 entry. Local registrar administrative fees may apply; verify with your school or Local Registrar.

Exam sponsor website

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

Official sources

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.

33%

Module 1: Complex Numbers and Calculus II

Paper 01 draws 15 multiple-choice items here. Topics include complex-number arithmetic, roots, modulus, argument, Argand loci, De Moivre's theorem, and polar and exponential forms; exponential, logarithmic, inverse trigonometric, implicit and parametric differentiation; second and partial derivatives; integration by substitution, parts and partial fractions; reduction formulae; and the trapezium rule.

33%

Module 2: Sequences, Series and Approximations

Paper 01 draws 15 multiple-choice items here. Topics include terms and recurrence relations, convergent and divergent sequences, periodic and oscillating sequences, mathematical induction, arithmetic and geometric series, summation notation, partial sums, the method of differences, Maclaurin and Taylor expansions, the binomial theorem for rational exponents, and root-finding using bisection, linear interpolation, Newton-Raphson and given iterations.

34%

Module 3: Counting, Matrices and Differential Equations

Paper 01 draws 15 multiple-choice items here. Topics include the fundamental counting principle, permutations, combinations, sample spaces, elementary probability rules, and conditional probability; matrix operations, determinants, inverses of 2x2 and 3x3 matrices, Gaussian elimination, and consistency of linear systems; first-order differential equations (separable variables, integrating factors) and practical modelling; and second-order linear differential equations with constant coefficients.

Preparing for the CAPE Pure Maths Unit 2 Exam

What You Need to Know

Passing score: CXC does not publish percentage grade boundaries. CAPE results are reported as Grades I to VII (Grade I highest); Grades I to V are accepted as passing grades. The final grade combines Paper 01 (30%), Paper 02 (50%), and School-Based Assessment or Paper 032 (20%).
Assessment: Paper 01: 45 compulsory multiple-choice items, 15 from each of the three Modules (1 hour 30 minutes, 30%). Paper 02: six compulsory extended-response questions, two per Module (2 hours 30 minutes, 50%). School-Based Assessment Paper 031 or private-candidate Paper 032 (20%).
Time limit: Paper 01 is 1 hour 30 minutes (90 minutes). Paper 02 is 2 hours 30 minutes. Paper 032 for private candidates is 2 hours.
Exam / certification fees: Effective May–June 2026: BBD $69.50 per CAPE Unit plus BBD $48.50 candidate entry in member territories. External-territory rates are BBD $139.00 per Unit plus BBD $97.00 entry. Local registrar administrative fees may apply; verify with your school or Local Registrar. Official sources

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CAPE Pure Maths Unit 2: Suggested Study Strategy

1Master complex number conversions between Cartesian (x + iy), polar (r(cos θ + i sin θ)), and exponential (r e^(iθ)) forms, ensuring the principal argument θ is strictly in the interval (-π, π].

2Remember the key derivative rules for inverse trigonometric functions: d/dx(arcsin x) = 1/√(1 - x²), d/dx(arccos x) = -1/√(1 - x²), and d/dx(arctan x) = 1/(1 + x²).

3For parametric differentiation, always compute the second derivative using d²y/dx² = [d/dt(dy/dx)] / (dx/dt) rather than differentiating dy/dx directly with respect to x.

4For the binomial series (1 + x)^n with rational n, use |x| < 1 to ensure convergence. For (a + bx)^n, first factor out a^n to get a^n(1 + (b/a)x)^n, and use |bx/a| < 1.

5For 3x3 matrices, ensure accuracy when expanding determinants along the row or column containing the most zeros, keeping the checkerboard sign pattern (+ - +) firmly in mind.

6When solving second-order linear differential equations with constant coefficients, classify the auxiliary equation roots carefully: distinct real roots yield sum of exponentials, repeated roots introduce an x factor (A + Bx)e^(mx), and complex conjugate roots α ± iβ yield e^(αx)(A cos(βx) + B sin(βx)).

Frequently Asked Questions

How many questions are on CAPE Pure Mathematics Unit 2 Paper 01?

Paper 01 consists of 45 compulsory multiple-choice items, with exactly 15 questions drawn from each of the three Modules. Each item has four options and is worth 1 mark (45 raw marks, weighted to 90 marks, contributing 30% of the overall Unit grade).

Which calculations can I practise in this bank?

This bank provides independent practice in modulus and argument calculations, De Moivre's theorem powers and roots, complex loci equations, inverse trigonometric derivatives, implicit and parametric tangents and normals, integration by parts and substitution, reduction formula evaluations, volumes of revolution, sequence limits, telescoping series summations, ratio tests for convergence, binomial series approximations, Maclaurin and Taylor expansions, Newton-Raphson iterations, permutations and combinations, 2x2 and 3x3 matrix determinants and inverses, Gaussian elimination consistency checks, first-order separable and integrating-factor differential equations, and second-order linear differential equations.

Can I take CAPE Pure Mathematics Unit 2 before Unit 1?

Yes. The Caribbean Examinations Council designs Unit 1 and Unit 2 as independent, discrete certifiable qualifications. While Unit 1 (Basic Algebra and Functions, Trigonometry, Geometry and Vectors, Calculus I) provides foundational background, candidates may register for Unit 2 independently or take both Units in the same examination series.

Are calculators allowed in the CAPE Pure Mathematics Unit 2 examination?

Yes. CXC regulations permit candidates to use silent, non-programmable scientific calculators for Paper 01 and Paper 02. Graphing calculators, programmable models, and devices with communication capabilities are strictly prohibited.

How are CAPE Pure Mathematics Unit 2 results graded?

Results are reported on CXC's seven-point scale from Grade I (highest) to Grade VII. Grades I to V represent acceptable or better performance; university admission requirements vary by institution and programme. Paper 01 contributes 30%, Paper 02 contributes 50%, and the School-Based Assessment (or Paper 032 for private candidates) contributes 20%.