A-level Maths formula sheet

ax² + bx + c = 0 ⇒ x = (−b ± √(b² − 4ac)) / 2a
Discriminant b² − 4ac: > 0 two real roots, = 0 one repeated root, < 0 no real roots

a ≠ 0.

log_a xy = log_a x + log_a y; log_a (x/y) = log_a x − log_a y; log_a x^k = k log_a x
a^x = b ⇔ x = log_a b; e^{ln x} = x; ln e^x = x

x, y > 0; a > 0, a ≠ 1. Change of base, log_a x = log_b x / log_b a, is printed in the Edexcel booklet only.

y = Ae^kx: growth if k > 0, decay if k < 0; d/dx (e^kx) = ke^kx; a^x = e^{x ln a}

fg(x) = f(g(x)): apply g first. f⁻¹ reverses f, and its graph is the reflection of y = f(x) in y = x.

f must be one-to-one on its domain for f⁻¹ to exist; the domain of f⁻¹ is the range of f.

a/sin A = b/sin B = c/sin C; a² = b² + c² − 2bc cos A; Area = ½ab sin C

Side a is opposite angle A.

s = rθ; A = ½r²θ; π radians = 180°

θ in radians.

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

sin 2A = 2 sin A cos A; cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A; tan 2A = 2 tan A / (1 − tan²A)

Derived from the compound-angle formulae with B = A.

a sin θ + b cos θ = R sin(θ + α), where R = √(a² + b²), tan α = b/a

Similar forms exist for R cos(θ ± α); choose the form the question asks for.

xⁿ → nxⁿ⁻¹; e^kx → ke^kx; ln x → 1/x; sin kx → k cos kx; cos kx → −k sin kx; a^x → a^x ln a

Trigonometric results require radians.

dy/dx = (dy/dt) / (dx/dt); d/dx [f(y)] = f′(y) dy/dx

dx/dt ≠ 0.

dy/dx = 0 at a stationary point; d²y/dx² > 0 minimum, < 0 maximum (test further if = 0). Point of inflection: d²y/dx² changes sign.

∫xⁿ dx = xⁿ⁺¹/(n + 1) + c (n ≠ −1); ∫1/x dx = ln|x| + c; ∫e^kx dx = e^kx/k + c; ∫cos kx dx = (1/k) sin kx + c; ∫sin kx dx = −(1/k) cos kx + c; ∫sec²kx dx = (1/k) tan kx + c

Area = ∫_a^b y dx; area between curves = ∫_a^b (upper − lower) dx

Signed area is negative below the x-axis: split at roots for total area.

|ai + bj + ck| = √(a² + b² + c²); unit vector â = a/|a|; distance between points = |b − a|

X ~ N(μ, σ²): Z = (X − μ)/σ ~ N(0, 1)

σ is the standard deviation (the square root of the variance).

Reject H₀ when the p-value ≤ significance level (or the test statistic is in the critical region). Two-tailed tests split the significance level between tails.

Failing to reject H₀ does not prove it is true. State conclusions in context.

v = ds/dt; a = dv/dt = d²s/dt²; s = ∫v dt; v = ∫a dt

Use limits or initial conditions to find constants.

F = ma; W = mg; friction F ≤ μR, with F = μR when sliding or in limiting equilibrium

Take g = 9.8 m s⁻² unless told otherwise.

Moment = force × perpendicular distance from the pivot. In equilibrium: resultant force = 0 and total clockwise moment = total anticlockwise moment.

Horizontal: x = (u cos θ)t. Vertical: y = (u sin θ)t − ½gt²; v_y = u sin θ − gt

No air resistance; the horizontal velocity is constant.