GCE A/L Combined Mathematics cheat sheet

Quadratic ax²+bx+c=0 has roots (−b±√(b²−4ac))/(2a); discriminant Δ=b²−4ac gives two distinct real roots (Δ>0), a repeated root (Δ=0), or a conjugate pair (Δ<0).

For roots α, β of a quadratic: α+β=−b/a and αβ=c/a; for a cubic ax³+bx²+cx+d=0 with roots α,β,γ: α+β+γ=−b/a, αβ+βγ+γα=c/a, αβγ=−d/a.

Remainder Theorem: dividing p(x) by (x−a) leaves remainder p(a); Factor Theorem: (x−a) is a factor exactly when p(a)=0.

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

Complex number z=a+bi has modulus |z|=√(a²+b²) and argument arg z=atan2(b,a); product multiplies moduli and adds arguments.

De Moivre: (cos θ+i sin θ)ⁿ=cos nθ+i sin nθ; nth roots of z=r cis θ are r^(1/n) cis((θ+2kπ)/n), k=0,1,…,n−1.

Conjugate roots: real-coefficient polynomials have non-real roots in conjugate pairs, so a cubic with real coefficients always has at least one real root.

Inequalities: |x−a|<k means a−k<x<a+k (k>0); multiplying or dividing by a negative number reverses the inequality sign.

Sum of a geometric series: Sₙ=a(1−rⁿ)/(1−r) for r≠1, and S∞=a/(1−r) converges only when |r|<1.

Identities: sin²θ+cos²θ=1, 1+tan²θ=sec²θ, 1+cot²θ=cosec²θ; these hold for every θ where both sides are defined.

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: sin2θ=2sinθcosθ, cos2θ=cos²θ−sin²θ=2cos²θ−1=1−2sin²θ, tan2θ=2tanθ/(1−tan²θ).

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

General solutions: sinθ=sinα gives θ=nπ+(−1)ⁿα; cosθ=cosα gives θ=2nπ±α; tanθ=tanα gives θ=nπ+α, for integer n.

Sine rule: a/sinA=b/sinB=c/sinC=2R (R = circumradius); cosine rule: a²=b²+c²−2bc cosA.

Area of a triangle = ½ab sinC; also = √(s(s−a)(s−b)(s−c)) with s=(a+b+c)/2.

When solving trig equations in a stated interval, find the principal solution first, then use periodicity to list every solution inside the interval; discard any that make a denominator zero.

Straight line through (x₁,y₁) with gradient m: y−y₁=m(x−x₁); general form ax+by+c=0 has gradient −a/b.

Distance from point (x₁,y₁) to line ax+by+c=0 is |ax₁+by₁+c|/√(a²+b²); parallel lines ax+by+c₁=0 and ax+by+c₂=0 are |c₁−c₂|/√(a²+b²) apart.

Two lines are perpendicular exactly when m₁m₂=−1; the angle θ between lines satisfies tanθ=|(m₁−m₂)/(1+m₁m₂)|.

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

Parabola y²=4ax has focus (a,0), directrix x=−a, vertex at the origin, and latus rectum length 4a.

Ellipse x²/a²+y²/b²=1 (a>b) has foci (±c,0) with c²=a²−b² and eccentricity e=c/a<1.

Hyperbola x²/a²−y²/b²=1 has foci (±c,0) with c²=a²+b², eccentricity e=c/a>1, and asymptotes y=±(b/a)x.

For a conic, the tangent at (x₁,y₁) is obtained by replacing x² with xx₁, y² with yy₁, x with (x+x₁)/2, and y with (y+y₁)/2.

Product rule: (uv)′=u′v+uv′; quotient rule: (u/v)′=(u′v−uv′)/v².

Chain rule: dy/dx=dy/du · du/dx; for example d/dx[sin(kx)]=k cos(kx) and d/dx[e^(kx)]=k e^(kx).

Standard derivatives: d/dx(ln x)=1/x, d/dx(aˣ)=aˣ ln a, d/dx(tan x)=sec²x, d/dx(arcsin x)=1/√(1−x²).

Implicit differentiation: differentiate both sides with respect to x, treating y as a function of x, so d/dx(y²)=2y dy/dx.

Parametric curves: dy/dx=(dy/dt)/(dx/dt), provided dx/dt≠0; the tangent gradient at a parameter value comes from substituting that value.

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

For a curve y=f(x), the tangent at x=a is y−f(a)=f′(a)(x−a); the normal has gradient −1/f′(a).

Small increments: δy≈(dy/dx)δx; related rates use dy/dt=(dy/dx)(dx/dt).

Fundamental Theorem: ∫ₐᵇ f(x)dx=F(b)−F(a) where F′=f; reversing the limits changes the sign of the definite integral.

Standard integrals: ∫xⁿdx=x^(n+1)/(n+1)+C for n≠−1; ∫(1/x)dx=ln|x|+C; ∫e^(kx)dx=e^(kx)/k+C; ∫cos(kx)dx=sin(kx)/k+C.

Substitution: when u=g(x), ∫f(g(x))g′(x)dx=∫f(u)du; change the limits of a definite integral to u-values rather than back-substituting.

Integration by parts: ∫u dv=uv−∫v du; choose u as the function that simplifies when differentiated (e.g. ln x or xⁿ).

Partial fractions let rational integrands such as 1/((x−a)(x−b)) be split into terms of the form A/(x−a)+B/(x−b) before integrating to logarithms.

Area between curve y=f(x) and the x-axis from a to b is ∫ₐᵇ|f(x)|dx; area between two curves is ∫ₐᵇ(upper−lower)dx.

Volume of revolution about the x-axis: V=π∫ₐᵇ y²dx; about the y-axis with x=g(y): V=π∫ x²dy.

Trapezium rule with n strips of width h=(b−a)/n: ∫ₐᵇ f dx ≈ (h/2)[f₀+2(f₁+…+fₙ₋₁)+fₙ].

Position vector of the point dividing AB in the ratio m:n is (n a + m b)/(m+n); the midpoint is (a+b)/2.

Scalar (dot) product: a·b=|a||b|cosθ=a₁b₁+a₂b₂+a₃b₃; a·b=0 exactly when a and b are perpendicular.

Vector (cross) product magnitude |a×b|=|a||b|sinθ equals the area of the parallelogram spanned by a and b; the triangle area is half of this.

Component form of the cross product: a×b=(a₂b₃−a₃b₂, a₃b₁−a₁b₃, a₁b₂−a₂b₁); the result is perpendicular to both a and b.

Line through point with position vector a in direction d: r=a+td; a line and a plane meet where the parameter value satisfies the plane equation.

Plane with normal n through point a: (r−a)·n=0, i.e. r·n=a·n; the perpendicular distance from point p to it is |(p−a)·n|/|n|.

Projection of a onto b has length (a·b)/|b| and vector ((a·b)/|b|²)b; the component of a perpendicular to b is a minus this projection.

A particle moving with position r(t) has velocity v=dr/dt and acceleration a=dv/dt, each differentiated componentwise.

Equilibrium of a particle requires the vector sum of all forces to be zero; resolve into two perpendicular directions and solve the resulting equations.

Friction: limiting friction F=μR, where R is the normal reaction and μ the coefficient of friction; at the point of slipping F=μR, otherwise F≤μR.

Moment of a force about a point = force × perpendicular distance to the line of action; for equilibrium of a rigid body, the sum of moments about any point is zero.

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

Constant acceleration: v=u+at, s=ut+½at², v²=u²+2as, s=½(u+v)t; use these only when acceleration is constant.

Projectile motion: horizontal component u cosα is constant; vertical motion has acceleration −g, giving time of flight 2u sinα/g and range u²sin2α/g on level ground.

Momentum p=mv is conserved in collisions when no external impulse acts; impulse = change in momentum = ∫F dt.

Work done by a constant force = F·d (component along displacement); kinetic energy ½mv², gravitational potential energy mgh, and power = Fv.

Addition rule: P(A∪B)=P(A)+P(B)−P(A∩B); for mutually exclusive events P(A∩B)=0.

Conditional probability: P(A|B)=P(A∩B)/P(B); events are independent exactly when P(A∩B)=P(A)P(B).

Bayes' theorem: P(A|B)=P(B|A)P(A)/P(B), with P(B) often expanded by the law of total probability.

Binomial: P(X=r)=C(n,r)pʳ(1−p)^(n−r), mean np, variance np(1−p); applies to n independent trials with constant success probability p.

Normal distribution: standardise with z=(x−μ)/σ, then use the standard normal table; about 68% of values lie within μ±σ and 95% within μ±2σ.

For a continuous random variable, P(a<X<b) is the area under the density curve; total area equals 1 and P(X=x)=0 for any single value.

Mean of a discrete variable: E(X)=ΣxP(X=x); variance Var(X)=E(X²)−[E(X)]².

Sample statistics: mean x̄=Σx/n and variance s²=Σ(x−x̄)²/(n−1) when estimating a population variance from a sample.