Cameroon GCE A-Level Pure Math with Mechanics cheat sheet

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

Factor Theorem: (x − a) is a factor of f(x) exactly when f(a) = 0; use it with synthetic or long division to factorise cubics.

For a rational function f(x) = p(x)/q(x): vertical asymptotes where q(x) = 0 (and p ≠ 0), horizontal asymptote y = 0 if deg p < deg q, or y = ratio of leading coefficients if deg p = deg q.

|x| < k with k > 0 is equivalent to −k < x < k; |x| > k is equivalent to x < −k or x > k.

A function has an inverse only if it is one-to-one; the graphs of f and f⁻¹ are reflections in the line y = x, and the domain of f⁻¹ equals the range of f.

Composition: (f∘g)(x) = f(g(x)); the domain of f∘g excludes values where g is undefined or g(x) lies outside the domain of f.

Partial fractions: distinct linear factors give A/(x+a) + B/(x+b); a repeated factor (x+a)² needs A/(x+a) + B/(x+a)²; an irreducible quadratic needs (Cx+D)/(quadratic).

For a quadratic ax² + bx + c, the discriminant b² − 4ac decides two real roots (>0), one repeated root (=0), or no real roots (<0).

Quadratic inequalities: solve the equation first, then test sign in the intervals between roots; for a > 0 the expression is negative only between the roots.

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

Gradient m = (y₂ − y₁)/(x₂ − x₁); parallel lines have equal gradients, perpendicular lines satisfy m₁m₂ = −1.

Point–gradient form: y − y₁ = m(x − x₁); general linear form ax + by + c = 0 has gradient −a/b.

Perpendicular distance from (x₁, y₁) to ax + by + c = 0 is |ax₁ + by₁ + c| / √(a² + b²).

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

A tangent to a circle is perpendicular to the radius at the point of contact; a line meets a circle in 0, 1 or 2 points according to the discriminant of the substituted quadratic.

Parametric curves: dy/dx = (dy/dt)/(dx/dt); the tangent at parameter t uses the point (x(t), y(t)) and that gradient.

A chord of a parabola or circle subtended at a point: use simultaneous equations between the line and curve, then the discriminant to test tangency.

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; infinite sum exists only when |r| < 1 and equals a/(1 − r).

Sigma notation: Σ from r = 1 to n of a constant c is nc; Σ r = n(n+1)/2; Σ r² = n(n+1)(2n+1)/6; Σ r³ = [n(n+1)/2]².

A sequence converges when it approaches a finite limit as n → ∞; a GP converges only if |r| < 1 (or the terms are eventually zero).

Binomial theorem for positive integer n: (a + b)ⁿ = Σ ⁿCᵣ aⁿ⁻ʳ bʳ; the general term is ⁿCᵣ aⁿ⁻ʳ bʳ and ⁿCᵣ = n!/(r!(n−r)!).

Binomial expansion for non-integer or negative n: (1 + x)ⁿ = 1 + nx + n(n−1)x²/2! + …, valid for |x| < 1; replace x by the required expression to find the range of validity.

Method of differences: if a term can be written as f(r) − f(r+1), the sum telescopes to f(1) − f(n+1).

Proof by induction: show true for n = 1, assume true for n = k, then prove true for n = k + 1 using the assumption.

Radians: π radians = 180°; arc length s = rθ and sector area A = ½r²θ with θ in radians.

Identities: sin²θ + cos²θ = 1; tan θ = sin θ/cos θ; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.

Compound angles: sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B; tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B).

Double angles: sin 2A = 2 sin A cos A; cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A; tan 2A = 2tan A/(1 − tan²A).

Forms R sin(θ ± α) or R cos(θ ± α): R = √(a² + b²) and tan α = b/a; use to find maximum and minimum values and solve equations.

General solutions: sin θ = sin α gives θ = nπ + (−1)ⁿα; cos θ = cos α gives θ = 2nπ ± α; tan θ = tan α gives θ = nπ + α.

Small angles: sin θ ≈ θ, tan θ ≈ θ, cos θ ≈ 1 − θ²/2 for θ in radians and small.

To solve a trig equation in a given interval, find the principal value first, then add or subtract the period (π or 2π) until all solutions in the interval are listed.

Definition: log_a x = y means aʸ = x; ln x means log_e x, and eˣ and ln x are inverse functions.

Laws: log(xy) = log x + log y; log(x/y) = log x − log y; log(xⁿ) = n log x; log_a a = 1 and log_a 1 = 0.

Change of base: log_a x = log_b x / log_b a; in particular log_a x = ln x / ln a.

Solving aˣ = b: take logarithms of both sides to get x = ln b / ln a.

Exponential growth and decay: y = Ae^{kt} with k > 0 growth and k < 0 decay; the doubling time is ln 2 / k and the half-life is ln 2 / |k|.

To linearise y = axⁿ take ln y = ln a + n ln x and plot ln y against ln x (gradient n, intercept ln a); for y = abˣ plot log y against x (gradient log b, intercept log a).

Derivative and integral: d/dx(e^{kx}) = ke^{kx}; ∫e^{kx} dx = (1/k)e^{kx} + c; d/dx(ln x) = 1/x for x > 0; ∫(1/x) dx = ln|x| + c.

Power rule: 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).

Derivatives to know: d/dx(sin x) = cos x, d/dx(cos x) = −sin x, d/dx(tan x) = sec²x, d/dx(ln x) = 1/x, d/dx(eˣ) = eˣ.

Implicit differentiation: differentiate each term with respect to x, applying the chain rule to y terms, then solve for dy/dx.

Stationary points occur where dy/dx = 0; classify with the second derivative (d²y/dx² < 0 maximum, > 0 minimum) or by the sign change of dy/dx.

A point of inflexion has d²y/dx² = 0 with a change of concavity; it is not necessarily stationary.

Tangent at (x₁, y₁) is y − y₁ = m(x − x₁) with m = dy/dx at that point; the normal has gradient −1/m.

Rates of change: use the chain rule, e.g. dV/dt = (dV/dr)(dr/dt), to relate connected quantities.

Increasing on an interval where dy/dx > 0, decreasing where dy/dx < 0; use this to sketch curves and locate maxima and minima.

Standard integrals: ∫xⁿ dx = xⁿ⁺¹/(n+1) + c for n ≠ −1; ∫(1/x) dx = ln|x| + c; ∫eˣ dx = eˣ + c; ∫cos x dx = sin x + c; ∫sin x dx = −cos x + c.

Definite integral: ∫ from a to b of f(x) dx = F(b) − F(a); reversing limits changes the sign.

Area between curve and x-axis: integrate |y| dx, treating regions below the axis separately so signed areas are corrected.

Area between two curves: integrate (upper curve − lower curve) between the intersection x-values.

Volume of revolution about the x-axis: V = π∫y² dx; about the y-axis: V = π∫x² dy, with limits on the relevant variable.

Substitution: change dx to (dx/du) du and change the limits to u-values for definite integrals; integration by parts uses ∫u dv = uv − ∫v du.

Trapezium rule with n strips of width h = (b − a)/n: ∫ ≈ (h/2)[y₀ + 2(y₁ + … + y_{n−1}) + yₙ].

Centroid of a plane region under y = f(x) from a to b: x̄ = ∫xy dx / ∫y dx, ȳ = ½∫y² dx / ∫y dx.

Position vector of a point plus direction vector times a scalar gives the line r = a + λb; parallel lines have parallel direction vectors.

Magnitude of ai + bj + ck is √(a² + b² + c²); a unit vector is the vector divided by its magnitude.

Scalar (dot) product: a·b = |a||b|cos θ = a₁b₁ + a₂b₂ + a₃b₃; a·b = 0 means the vectors are perpendicular.

Angle between vectors: cos θ = (a·b)/(|a||b|); the angle between two lines uses their direction vectors.

Vector equation of a plane: r·n = d, where n is a normal; a line meets a plane where (a + λb)·n = d.

Shortest distance from a point P to a line r = a + λb: use the perpendicular foot, or compute |AP × b|/|b| in three dimensions.

If two lines intersect, equate components and solve the resulting simultaneous equations; check the solution satisfies all components.

Displacement, velocity and acceleration are vector quantities: v = dr/dt and a = dv/dt, differentiated component by component.

Constant acceleration formulae: v = u + at; s = ut + ½at²; v² = u² + 2as; s = ½(u + v)t.

Displacement–time graph: gradient gives velocity; velocity–time graph: gradient gives acceleration and area under the graph gives displacement.

Projectile motion: horizontal motion has constant velocity; vertical motion has constant acceleration g (≈ 9.8 m s⁻²) downward.

Time of flight for launch speed u at angle θ from horizontal ground: T = 2u sin θ / g; range R = u² sin 2θ / g; maximum height H = u² sin²θ / (2g).

At the highest point of a projectile the vertical velocity is zero but the horizontal component is unchanged.

Relative velocity: velocity of A relative to B is v_A − v_B; closest approach occurs when the relative position is perpendicular to the relative velocity.

Variable acceleration: use a = dv/dt = v dv/ds and integrate, applying initial conditions to find constants.

Average speed is total distance divided by total time; average velocity is total displacement divided by total time.

Newton's second law: resultant force F = ma, applied along the direction of motion and perpendicular to it.

Weight W = mg acts vertically downward; normal reaction acts perpendicular to the contact surface.

Friction: limiting friction F = μR at the point of slipping; while at rest the friction force only equals the applied force up to μR.

On an inclined plane at angle θ: component of weight along the plane is mg sin θ, perpendicular to the plane is mg cos θ.

Tension is the same throughout a light inextensible string over a smooth pulley; connected particles share the same magnitude of acceleration.

For a particle on a rough inclined plane, motion up the plane gives friction acting down the plane; motion down gives friction acting up the plane.

Impulse = Ft = change in momentum mv − mu; momentum is conserved in collisions when no external force acts.

For a lift or elevator problem, apparent weight R = m(g ± a): plus when accelerating upward, minus when accelerating downward.

A particle is in equilibrium when the vector sum of all forces acting on it is zero.

Resolve forces in two perpendicular directions and set each sum to zero: ΣFₓ = 0 and ΣF_y = 0.

For three non-parallel forces in equilibrium, the lines of action are concurrent (meet at one point) and form a closed triangle.

Lami's theorem: for three concurrent forces, each force is proportional to the sine of the angle between the other two.

On a rough inclined plane, a body is on the point of sliding up when the applied force balances mg sin θ + μmg cos θ.

Equilibrium of a body on a horizontal surface: R = total downward force, and friction ≤ μR unless sliding is about to occur.

Tension in a string supporting a hanging mass equals the weight only if the mass is at rest or moving with constant velocity.

Moment of a force about a point = force × perpendicular distance from the point to the line of action.

Principle of moments: for a body in equilibrium, sum of clockwise moments about any point equals sum of anticlockwise moments.

Choose the point that eliminates the most unknown forces (e.g. a hinge or a support) to simplify the moment equation.

A couple consists of two equal, opposite, parallel forces; its moment is F × d where d is the perpendicular separation, independent of the pivot.

Conditions for equilibrium of a rigid body: resultant force zero and resultant moment about any point zero.

For a uniform rod of length L and weight W, the weight acts at the midpoint L/2; for a non-uniform rod the centre of mass must be found from data.

Tilting or toppling occurs when the line of action of the weight passes outside the base of support; at the point of tilting the normal reaction acts at the edge.

A ladder against a smooth wall: wall reaction is horizontal, ground reaction has both normal and friction components; take moments about the foot to find the wall reaction.