Free Practice Questions for GCE A-Level Further Mathematics

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Key Facts: GCE A-Level Further Mathematics Exam

0775

Official CGCEB Subject Code

Cameroon GCE Board Syllabus Regulations

50 MCQs / 90 min

Paper 1 Format & Time Limit

CGCEB Examination Specifications

12,000 FCFA

Standard Subject Registration Fee

Cameroon Ministry of Secondary Education

Grades A–E

Official GCE Advanced Level Pass Range

CGCEB Grading System Guide

Cameroon GCE A-Level Further Mathematics (0775) evaluates elite high-school mathematical proficiency across Advanced Pure Mathematics, Further Mechanics, and Further Probability & Statistics. Tested through Paper 1 (50 MCQs, 90 min) and Paper 2 (Structured, 180 min), this 100-question study bank provides step-by-step mathematical solutions, formulas, and distractor analyses aligned with the official CGCEB syllabus.

Sample GCE A-Level Further Mathematics Practice Questions

Try these sample questions to review concepts for the GCE A-Level Further Mathematics exam. Each question includes a detailed explanation. Start the interactive quiz above for the full 100+ question experience with AI tutoring.

1Evaluate the complex power $(1 + i\sqrt{3})^6$ using De Moivre's theorem.

A.64

B.-64

C.64i

D.-64i

Explanation: Convert $z = 1 + i\sqrt{3}$ to polar form: modulus $r = \sqrt{1^2 + (\sqrt{3})^2} = 2$ and principal argument $\theta = \arctan(\sqrt{3}/1) = \pi/3$. By De Moivre's theorem, $z^6 = r^6 [\cos(6\theta) + i\sin(6\theta)] = 2^6 [\cos(2\pi) + i\sin(2\pi)] = 64(1 + 0i) = 64$.

2Using De Moivre's theorem, express $\cos(4\theta)$ entirely in terms of powers of $\cos\theta$.

A.4\cos^4\theta - 4\cos^2\theta + 1

B.8\cos^4\theta - 8\cos^2\theta + 1

C.8\cos^4\theta - 6\cos^2\theta + 1

D.16\cos^4\theta - 12\cos^2\theta + 1

Explanation: Expanding $(\cos\theta + i\sin\theta)^4 = \cos(4\theta) + i\sin(4\theta)$ via the binomial theorem gives $\cos(4\theta) = \operatorname{Re}[(\cos\theta + i\sin\theta)^4] = \cos^4\theta - 6\cos^2\theta\sin^2\theta + \sin^4\theta$. Substituting $\sin^2\theta = 1 - \cos^2\theta$ yields $\cos^4\theta - 6\cos^2\theta(1-\cos^2\theta) + (1-\cos^2\theta)^2 = 8\cos^4\theta - 8\cos^2\theta + 1$.

3If $\omega = e^{i 2\pi/5}$ is a primitive fifth root of unity, what is the value of the finite sum $1 + \omega + \omega^2 + \omega^3 + \omega^4$?

A.1

B.-1

C.0

D.5

Explanation: The sum of all $n$ distinct $n$th roots of unity is always zero for any integer $n > 1$. Analytically, $1 + \omega + \omega^2 + \omega^3 + \omega^4 = \frac{\omega^5 - 1}{\omega - 1} = \frac{1 - 1}{\omega - 1} = 0$, since $\omega^5 = e^{i 2\pi} = 1$ and $\omega \ne 1$.

4The locus of points $z = x + iy$ in the complex plane satisfying $|z - 3| = 2|z - 3i|$ represents a circle. What are the center and radius of this circle?

A.Center $(1, -4)$, Radius $2\sqrt{2}$

B.Center $(-1, 4)$, Radius 8

C.Center $(3, 3)$, Radius 4

D.Center $(-1, 4)$, Radius $2\sqrt{2}$

Explanation: Squaring both sides yields $(x-3)^2 + y^2 = 4[x^2 + (y-3)^2]$. Expanding gives $x^2 - 6x + 9 + y^2 = 4x^2 + 4y^2 - 24y + 36 \implies 3x^2 + 3y^2 + 6x - 24y + 27 = 0$. Dividing by 3 yields $x^2 + y^2 + 2x - 8y + 9 = 0 \implies (x+1)^2 + (y-4)^2 = 1 + 16 - 9 = 8$. Thus the center is $(-1, 4)$ and radius is $\sqrt{8} = 2\sqrt{2}$.

5Find the three distinct complex roots of the polynomial equation $z^3 = -8i$ in exponential form $r e^{i\theta}$ with $-\pi < \theta \le \pi$.

A.$2e^{-i\pi/6}, 2e^{i\pi/2}, 2e^{-i5\pi/6}$

B.$2e^{i\pi/6}, 2e^{i5\pi/6}, 2e^{-i\pi/2}$

C.$8e^{-i\pi/6}, 8e^{i\pi/2}, 8e^{-i5\pi/6}$

D.$2e^{-i\pi/2}, 2e^{i\pi/6}, 2e^{i5\pi/6}$

Explanation: Express $-8i$ in polar form: $-8i = 8 e^{i(-\pi/2 + 2k\pi)}$. Taking the cube root gives $z_k = 8^{1/3} e^{i(-\pi/6 + 2k\pi/3)} = 2 e^{i(-\pi/6 + 2k\pi/3)}$ for $k = 0, 1, 2$. For $k=0$, $z_0 = 2e^{-i\pi/6}$. For $k=1$, $z_1 = 2e^{i\pi/2}$. For $k=2$, $z_2 = 2e^{i7\pi/6} = 2e^{-i5\pi/6}$.

6Simplify the hyperbolic expression $\cosh(2x) - 1$ using hyperbolic double-angle identities.

A.2\cosh^2(x)

B.2\sinh^2(x)

C.\sinh^2(x)

D.-\sinh^2(x)

Explanation: From the fundamental hyperbolic identity $\cosh(2x) = \cosh^2(x) + \sinh^2(x)$ and $\cosh^2(x) - \sinh^2(x) = 1$, we have $\cosh(2x) = 1 + 2\sinh^2(x)$. Rearranging gives $\cosh(2x) - 1 = 2\sinh^2(x)$ (analogous to Osborn's rule applied to $\cos 2x = 1 - 2\sin^2 x$).

7Evaluate the exact numerical value of the inverse hyperbolic sine $\operatorname{arsinh}\left(\frac{3}{4}\right)$ using its logarithmic definition.

A.\ln(3)

B.\ln(4)

C.\ln(2)

D.\frac{1}{2}\ln(2)

Explanation: The logarithmic definition of inverse hyperbolic sine is $\operatorname{arsinh}(x) = \ln\left(x + \sqrt{x^2 + 1}\right)$. Substituting $x = 3/4$: $\operatorname{arsinh}(3/4) = \ln\left(\frac{3}{4} + \sqrt{\frac{9}{16} + 1}\right) = \ln\left(\frac{3}{4} + \sqrt{\frac{25}{16}}\right) = \ln\left(\frac{3}{4} + \frac{5}{4}\right) = \ln\left(\frac{8}{4}\right) = \ln(2)$.

8Evaluate the definite integral $\int_0^{\ln 2} \sinh^2(x)\,dx$.

A.\frac{15}{16} - \ln 2

B.\frac{15}{32} + \frac{1}{2}\ln 2

C.\frac{7}{16} - \frac{1}{2}\ln 2

D.\frac{15}{32} - \frac{1}{2}\ln 2

Explanation: Using $\sinh^2(x) = \frac{\cosh(2x) - 1}{2}$, the integral is $\int_0^{\ln 2} \left(\frac{\cosh(2x)}{2} - \frac{1}{2}\right)dx = \left[ \frac{\sinh(2x)}{4} - \frac{x}{2} \right]_0^{\ln 2}$. At $x = \ln 2$, $\sinh(2\ln 2) = \sinh(\ln 4) = \frac{e^{\ln 4} - e^{-\ln 4}}{2} = \frac{4 - 1/4}{2} = \frac{15}{8}$. Thus, $\frac{15/8}{4} - \frac{\ln 2}{2} = \frac{15}{32} - \frac{1}{2}\ln 2$.

9Evaluate $\int \frac{1}{\sqrt{x^2 - 9}}\,dx$ for $x > 3$ using a hyperbolic substitution.

A.\operatorname{arcosh}\left(\frac{x}{3}\right) + C

B.\frac{1}{3}\operatorname{arcosh}\left(\frac{x}{3}\right) + C

C.\operatorname{arsinh}\left(\frac{x}{3}\right) + C

D.\frac{1}{3}\operatorname{artanh}\left(\frac{x}{3}\right) + C

Explanation: Let $x = 3\cosh u$ with $u > 0$, so $dx = 3\sinh u\,du$. Then $\sqrt{x^2 - 9} = \sqrt{9\cosh^2 u - 9} = 3\sinh u$. The integral becomes $\int \frac{3\sinh u}{3\sinh u}\,du = \int 1\,du = u + C = \operatorname{arcosh}\left(\frac{x}{3}\right) + C$.

10Calculate the determinant of the $3\times 3$ matrix $A = \begin{pmatrix} 2 & 1 & 0 \\ 0 & 3 & 4 \\ 1 & 2 & 1 \end{pmatrix}$.

A.6

B.-6

C.-10

D.14

Explanation: Expand along the first row: $\det(A) = 2 \begin{vmatrix} 3 & 4 \\ 2 & 1 \end{vmatrix} - 1 \begin{vmatrix} 0 & 4 \\ 1 & 1 \end{vmatrix} + 0 \begin{vmatrix} 0 & 3 \\ 1 & 2 \end{vmatrix}$. Evaluating the $2\times 2$ determinants: $2(3(1) - 4(2)) - 1(0(1) - 4(1)) = 2(3 - 8) - 1(0 - 4) = 2(-5) - (-4) = -10 + 4 = -6$.

About the GCE A-Level Further Mathematics Exam

The Cameroon GCE Advanced Level Further Mathematics (Subject Code 0775) is the most prestigious and challenging secondary school mathematics credential administered by the Cameroon General Certificate of Education Board (CGCEB). Designed for students targeting elite university programs in engineering, physical sciences, actuarial science, and pure mathematics, the syllabus encompasses Advanced Pure Mathematics, Further Mechanics, and Further Probability & Statistics. This 100-question practice bank provides rigorous multiple-choice questions reflecting the exact depth, computational rigor, and pedagogical standards of CGCEB Paper 1 examinations. Format note: this site's practice bank is 100 four-option multiple-choice questions covering the whole official syllabus. Paper 1 of the real examination is genuinely multiple choice (50 compulsory questions), so the format matches that paper, but the bank is a study aid only — it does not simulate the written theory/essay paper(s) or any practical examination, and its length does not describe the official exam.

Exam sponsor: Cameroon General Certificate of Education Board (CGCEB), Buea. The requirements and fees below concern the certification or admission exam, separate from our free practice resources.

Assessment

Official Advanced Level structure for subject code 0775 (Further Mathematics) per the Cameroon GCE Board June 2026 timetable (Form G6): Paper 1: 50 compulsory multiple-choice questions (1 hour 30 minutes); Paper 2: written theory/structured questions (3 hours); Paper 3: written paper (2 hours 30 minutes). Total written time is 7 hours. The Board does not publish per-paper mark weightings for individual subjects.

Time Limit

Paper 1: 1 hour 30 minutes; total written time 7 hours.

Passing Score

Grade E or better (Cameroon GCE Advanced Level grades A, B, C, D and E are passes; O is a subsidiary pass and F is a fail)

Exam / Certification Fees

12,000 FCFA

Exam sponsor website

Reported exam pass rate: 59.90% pass rate in the June 2026 session (Cameroon GCE Board, Performance by Subjects, results released 21 August 2026). This describes exam candidates, not OpenExamPrep users or results from using our resources. 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.

20%

Advanced Pure Mathematics

Comprehensive mastery of advanced algebraic, analytical, and geometric methods: Complex numbers (De Moivre's theorem, trigonometric identities via complex exponentials, roots of unity, and complex loci in the Argand plane), hyperbolic and inverse hyperbolic functions, matrix algebra (determinants, matrix inverses, Gaussian elimination, eigenvalues, eigenvectors, and 2D/3D linear transformations), Maclaurin and Taylor series expansions with limits, summation methods (method of differences, telescoping, C + iS series), first and second order linear ordinary differential equations (integrating factors, complementary functions, particular integrals), polar coordinates (curve sketching, sector areas, arc length, and tangents), 3D vector geometry (lines, planes, skew lines, perpendicular distances, and scalar triple products), and integration reduction formulas for definite and improper integrals.

20%

Further Mechanics

Rigorous physical and mathematical modeling of classical mechanics: Rectilinear motion in resisting media under resistive forces proportional to velocity ($R = kv$) and velocity squared ($R = kv^2$), terminal velocity and vertical trajectories under gravity with resistance, horizontal and vertical circular motion dynamics (conical pendulums, banked tracks, critical string tension, bead detachment angles on smooth spheres), rotational dynamics of rigid bodies (moments of inertia via direct integration, parallel and perpendicular axis theorems), rotational kinetic energy and torque equations ($\tau = I\alpha$), compound pendulums and small oscillation periods, rolling without slipping on inclined planes, harmonic oscillations (undamped, underdamped, critically damped, and overdamped motion), and center of mass calculations for complex laminas, hemispherical solids, and conical bodies.

20%

Further Probability and Statistics

Quantitative modeling of stochastic processes and advanced statistical inference: Continuous probability distributions (probability density functions, cumulative distribution functions, expectations, variances, medians, modes, uniform, exponential, and normal distributions), linear combinations of independent normal variables, discrete distribution models (Poisson processes, geometric distributions, negative binomial distributions, and Poisson/Normal approximations), bivariate continuous distributions (joint PDFs, marginal distributions, conditional densities, conditional expectations, covariance, and Pearson correlation coefficients), probability generating functions (PGFs) and moment generating functions (MGFs) for sums of independent random variables, sampling distributions and the Central Limit Theorem, confidence intervals for single means ($z$ and Student's $t$) and differences between means with pooled variance, and formal hypothesis testing (Type I/II errors, test power, single/two-sample $t$-tests, chi-squared goodness-of-fit tests, and contingency tables).

Preparing for the GCE A-Level Further Mathematics Exam

What You Need to Know

Passing score: Grade E or better (Cameroon GCE Advanced Level grades A, B, C, D and E are passes; O is a subsidiary pass and F is a fail)
Assessment: Official Advanced Level structure for subject code 0775 (Further Mathematics) per the Cameroon GCE Board June 2026 timetable (Form G6): Paper 1: 50 compulsory multiple-choice questions (1 hour 30 minutes); Paper 2: written theory/structured questions (3 hours); Paper 3: written paper (2 hours 30 minutes). Total written time is 7 hours. The Board does not publish per-paper mark weightings for individual subjects.
Time limit: Paper 1: 1 hour 30 minutes; total written time 7 hours.
Exam / certification fees: 12,000 FCFA Official sources

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GCE A-Level Further Mathematics: Suggested Study Strategy

1Master Complex Numbers & De Moivre's Theorem: Practice expanding $\cos(n\theta)$ and $\sin(n\theta)$, expressing $\cos^n\theta$ in multiple angles, and finding $n$th roots of unity along with their geometric representations on the Argand plane.

2Internalize Differential Equation Solutions: Memorize the integrating factor method for first order ODEs and auxiliary equation root classifications (distinct real, repeated, complex conjugate) for second order linear ODEs with complementary functions and particular integrals.

3Work with 3D Vector Geometry & Polar Curves: Practice calculating shortest distances between skew lines, finding line-plane intersection angles, and evaluating polar area integrals $\frac{1}{2}\int r^2\,d\theta$ and arc lengths $\int \sqrt{r^2 + (dr/d\theta)^2}\,d\theta$.

4Model Resisting Media & Rotational Dynamics: Derive velocity-displacement and velocity-time relations using separation of variables for forces $kv$ and $kv^2$. Apply the parallel and perpendicular axis theorems to calculate moments of inertia for compound pendulums and rolling cylinders.

5Master Continuous Distributions & Statistical Hypothesis Testing: Know the formulas for expectation, variance, conditional expectations $E[Y|X=x]$, and PGF/MGF properties. Understand degrees of freedom for $\chi^2$ goodness-of-fit $(k-1-p)$ and contingency tables $(r-1)(c-1)$.

Frequently Asked Questions

What is Cameroon GCE Advanced Level Further Mathematics (0775)?

Cameroon GCE A-Level Further Mathematics (Subject Code 0775) is the highest-level secondary mathematics examination offered by the Cameroon General Certificate of Education Board (CGCEB). It extends beyond ordinary A-Level Mathematics, providing advanced preparation in pure mathematics, classical mechanics, and mathematical statistics for university-level STEM studies.

What is the examination structure of Further Mathematics 0775?

The examination consists of two papers: Paper 1 is a 90-minute multiple-choice paper featuring 50 compulsory questions. Paper 2 is a 3-hour written structured examination containing advanced pure mathematics and mechanics/statistics sections.

How are grades awarded for Further Mathematics in Cameroon?

Candidates are awarded letter grades from A (highest distinction) to E (pass grade), carrying 5 to 1 grade points respectively. An O-grade represents Ordinary Level pass equivalency, while F indicates failure.

What mathematical aids and tables are permitted in the exam hall?

Candidates are permitted to use non-programmable scientific calculators and the official CGCEB Formula Booklet (containing standard mathematical and statistical tables, trigonometric identities, and integration reduction formulas).

Which exam topics do these independent study resources cover?

This 100-question bank covers the full two-year syllabus with exact proportional weighting: 50 Advanced Pure Mathematics questions, 25 Further Mechanics questions, and 25 Further Probability & Statistics questions. Every question features four distinct options and a comprehensive step-by-step mathematical explanation.

Free Practice Questions for GCE A-Level Further Mathematics

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