Cameroon GCE A-Level Pure Mathematics with Statistics cheat sheet

Remainder Theorem: the remainder when f(x) is divided by (x − a) is f(a); the remainder on division by (ax − b) is f(b/a).

Factor Theorem: (x − a) is a factor of f(x) if and only if f(a) = 0; use it to find unknown coefficients by equating.

For a rational function, a vertical asymptote occurs where the denominator is zero and the numerator is not; a horizontal asymptote is found from the ratio of leading terms as x → ±∞.

A function has an inverse only if it is one-to-one; restrict the domain (e.g. x ≥ 0 for x²) before inverting, and reflect the graph in y = x.

Composite fg means apply g first then f; fg is defined only where the range of g lies inside the domain of f.

Partial fractions: repeated linear factor (x − a)² needs both A/(x − a) and B/(x − a)²; an irreducible quadratic factor needs a linear numerator (Bx + C).

Modulus inequalities: |x − a| < k is equivalent to a − k < x < a + k; |x − a| > k gives x < a − k or x > a + k.

|f(x)| = g(x) requires g(x) ≥ 0, then solve f(x) = g(x) or f(x) = −g(x) and reject solutions violating the condition.

Arithmetic progression: nth term a + (n − 1)d; sum Sₙ = (n/2)[2a + (n − 1)d] = (n/2)(first + last).

Geometric progression: nth term arⁿ⁻¹; sum Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1; sum to infinity a/(1 − r) exists only when |r| < 1.

The sum to infinity of a GP is used to express recurring decimals as exact fractions, e.g. 0.444… = 0.4/(1 − 0.1) = 4/9.

Convergence of a GP requires |r| < 1; if |r| ≥ 1 the series diverges and no sum to infinity exists.

Binomial expansion for positive integer n: (a + b)ⁿ = Σ C(n,r) aⁿ⁻ʳ bʳ with the general term C(n,r) aⁿ⁻ʳ bʳ.

For non-integer or negative n, (1 + x)ⁿ = 1 + nx + n(n − 1)x²/2! + …, valid only for |x| < 1.

Method of differences: split terms into partial fractions so consecutive terms cancel, leaving only the first and last surviving terms.

Sigma notation manipulation: Σ(ax + b) = aΣx + nb, and Σx² ≠ (Σx)²; handle separately.

Straight line: y = mx + c; gradient m = (y₂ − y₁)/(x₂ − x₁); parallel lines have equal gradients and perpendicular lines satisfy m₁m₂ = −1.

Distance between two points = √[(x₂ − x₁)² + (y₂ − y₁)²]; midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2).

Circle centre (a, b) radius r: (x − a)² + (y − b)² = r²; general form x² + y² + 2gx + 2fy + c = 0 has centre (−g, −f) and radius √(g² + f² − c).

The perpendicular from the centre of a circle to a chord bisects the chord; a tangent is perpendicular to the radius at the point of contact.

Parametric equations: eliminate the parameter to obtain a Cartesian equation, e.g. x = at², y = 2at gives y² = 4ax.

Discriminant test for line–curve intersection: b² − 4ac > 0 two points, = 0 tangent, < 0 no intersection.

Parabola y² = 4ax: focus (a, 0), directrix x = −a; every point is equidistant from focus and directrix.

Identities: sin²θ + cos²θ = 1; tanθ = sinθ/cosθ; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.

Compound angles: sin(A ± B) = sinA cosB ± cosA sinB; cos(A ± B) = cosA cosB ∓ sinA sinB; tan(A ± B) = (tanA ± tanB)/(1 ∓ tanA tanB).

Double angles: sin2A = 2sinA cosA; cos2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A; tan2A = 2tanA/(1 − tan²A).

The factor formulae convert sums to products, e.g. sinP + sinQ = 2sin((P+Q)/2)cos((P−Q)/2).

General solutions: sinθ = k gives θ = arcsin k + 360°n or 180° − arcsin k + 360°n; cosθ = k gives θ = ±arccos k + 360°n.

R-form: a sinθ + b cosθ = R sin(θ + α) with R = √(a² + b²) and tanα = b/a; maximum value is R.

Sine rule a/sinA = b/sinB = c/sinC; cosine rule a² = b² + c² − 2bc cosA; area = ½ab sinC.

For the ambiguous case (two sides and a non-included angle), check whether the supplementary angle is also valid.

d/dx(xⁿ) = nxⁿ⁻¹; product rule (uv)′ = u′v + uv′; quotient rule (u/v)′ = (u′v − uv′)/v².

Chain rule dy/dx = dy/du · du/dx; for parametric curves dy/dx = (dy/dt)/(dx/dt).

d/dx(sinx) = cosx, d/dx(cosx) = −sinx, d/dx(tanx) = sec²x, d/dx(lnx) = 1/x, d/dx(eˣ) = eˣ.

Stationary points occur where dy/dx = 0; classify by the sign of d²y/dx² (negative = maximum, positive = minimum).

∫xⁿ dx = xⁿ⁺¹/(n + 1) + C for n ≠ −1; ∫(1/x) dx = ln|x| + C.

Area between curve and x-axis = ∫|y| dx; area between two curves = ∫(upper − lower) dx; points of intersection give the limits.

Volume of revolution about the x-axis = π∫y² dx; about the y-axis = π∫x² dy.

Definite integrals: ∫ₐᵇ f(x) dx = F(b) − F(a); reversing limits changes the sign, and ∫ₐᵃ f(x) dx = 0.

Magnitude |a| = √(x² + y² + z²); unit vector = a/|a|; scalar (dot) product a·b = |a||b|cosθ, zero for perpendicular vectors.

Vector equation of a line: r = a + tb; two lines are parallel if direction vectors are parallel, and intersect where position vectors coincide.

For a 2×2 matrix M = [[a, b], [c, d]], det M = ad − bc and M⁻¹ = (1/det M)[[d, −b], [−c, a]]; the inverse exists only if det M ≠ 0.

A system Ax = b has a unique solution when det A ≠ 0; if det A = 0 the system has no unique solution (parallel or coincident lines).

Matrix multiplication is not commutative: generally AB ≠ BA, but (AB)⁻¹ = B⁻¹A⁻¹ and (AB)ᵀ = BᵀAᵀ.

The identity matrix I satisfies AI = IA = A; a matrix satisfying M² = I can represent a reflection.

Rotation matrix through angle θ anticlockwise: [[cosθ, −sinθ], [sinθ, cosθ]]; reflection in y = x is [[0, 1], [1, 0]].

i² = −1; for z = a + bi, |z| = √(a² + b²) and arg z = arctan(b/a), choosing the quadrant from the signs of a and b.

Modulus–argument form z = r(cosθ + i sinθ) = re^{iθ}; multiplication multiplies moduli and adds arguments, division subtracts them.

De Moivre's theorem: [r(cosθ + i sinθ)]ⁿ = rⁿ(cos nθ + i sin nθ).

The conjugate z̄ = a − bi satisfies z z̄ = |z|² = a² + b², used to rationalise complex denominators.

Roots of zⁿ = w are equally spaced on a circle of radius |w|^{1/n}, separated by 2π/n in argument.

The nth roots of unity sum to zero and have product (−1)ⁿ⁻¹; they form vertices of a regular n-gon on the unit circle.

Loci: |z − a| = r is a circle centre a radius r; |z − a| = |z − b| is the perpendicular bisector of the segment ab.

For real coefficients, non-real roots occur in conjugate pairs; the quadratic with roots α, β is z² − (α + β)z + αβ = 0.

Addition rule P(A ∪ B) = P(A) + P(B) − P(A ∩ B); multiplication rule P(A ∩ B) = P(A)P(B|A).

Independence: P(A ∩ B) = P(A)P(B), equivalently P(A|B) = P(A); mutually exclusive events satisfy P(A ∩ B) = 0.

Bayes' theorem: P(A|B) = P(B|A)P(A)/P(B); total probability P(B) = ΣP(B|Aᵢ)P(Aᵢ) over a partition.

Binomial: X ~ B(n, p), P(X = r) = C(n,r)pʳ(1 − p)ⁿ⁻ʳ, mean np, variance np(1 − p); requires fixed n, independent trials, constant p.

Poisson: X ~ Po(λ), P(X = r) = e^{−λ}λʳ/r!, mean = variance = λ; use when events occur randomly in time or space with a known average rate.

Poisson approximates binomial when n is large and p small (typically n > 50, np < 5) with λ = np.

Normal: X ~ N(μ, σ²) standardised by z = (x − μ)/σ; the standard normal has mean 0 and variance 1.

For combined normal variables, means add and variances add: if X and Y are independent, Var(X ± Y) = Var(X) + Var(Y), but Var(aX) = a²Var(X).

Mean = Σx/n; for grouped data use class midpoints, and for coded data y = (x − a)/b recover x̄ = a + bȳ and sₓ = |b|s_y.

Median and quartiles come from cumulative frequency: median at n/2, Q₁ at n/4, Q₃ at 3n/4; interpolate within the class if grouped.

Interquartile range = Q₃ − Q₁; an outlier is commonly a value below Q₁ − 1.5IQR or above Q₃ + 1.5IQR (box-plot rule).

Variance s² = Σ(x − x̄)²/(n − 1) for a sample; the computational form is (Σx² − n x̄²)/(n − 1), and s = √s².

Standard deviation is unchanged by adding a constant but multiplied by |b| when data are scaled by b.

Histogram with unequal class widths: frequency density = frequency / class width, and area of a bar represents frequency.

Cumulative frequency curve: read median and quartiles off the vertical axis; the shape of the curve shows skewness.

Correlation coefficient r lies in [−1, 1]; r measures linear association only, and correlation does not imply causation.

Standard error of the mean = σ/√n; when σ is unknown and n is large, use the sample standard deviation s/√n.

A (1 − α) confidence interval for μ is x̄ ± z_{α/2}(σ/√n); for 95% use z = 1.96, for 99% use z = 2.576.

Hypothesis test procedure: state H₀ and H₁, choose significance level, compute the test statistic, compare with the critical value, then conclude in context.

For a single mean with known variance, z = (x̄ − μ)/(σ/√n); for a proportion, z = (p̂ − p)/√(p(1 − p)/n).

A one-tailed test places the whole significance level in one tail; a two-tailed test splits α/2 into each tail.

Reject H₀ when the test statistic falls in the critical region; the p-value is the probability, under H₀, of a result at least as extreme as observed.

Type I error: rejecting H₀ when it is true (probability α); Type II error: failing to reject a false H₀ (probability β); power = 1 − β.

A χ² goodness-of-fit statistic is Σ(O − E)²/E with degrees of freedom (number of categories − 1) minus the number of estimated parameters.