AS Level Mathematics formulas

Pure Mathematics 1

Quadratics

• Completing the square: x² + bx + c = (x + b/2)² − (b/2)² + c
• a(x + p)² + q has its turning point at (−p, q): a minimum if a > 0, a maximum if a < 0
• Discriminant b² − 4ac: greater than 0 means two distinct real roots, equal to 0 means one repeated root, less than 0 means no real roots
• Quadratic formula: x = (−b ± √(b² − 4ac))/(2a)
• A line is a tangent to a curve when the quadratic you get after substituting has b² − 4ac = 0
• With critical values p < q and a positive x² term: (x − p)(x − q) < 0 gives p < x < q, and (x − p)(x − q) > 0 gives x < p or x > q
• Disguised quadratic: let u = x² (or √x, or tan x), solve for u, then go back and find x

Functions

• fg(x) = f(g(x)): apply g first, then f
• To find f⁻¹(x): write y = f(x), make x the subject, then swap x and y
• Domain of f⁻¹ = range of f, and range of f⁻¹ = domain of f
• The graph of y = f⁻¹(x) is the reflection of y = f(x) in the line y = x
• y = f(x) + a is a translation of a units in the y-direction; y = f(x + a) is a translation of −a units in the x-direction
• y = af(x) is a stretch of factor a parallel to the y-axis; y = f(ax) is a stretch of factor 1/a parallel to the x-axis
• A quadratic is one-one only when its domain stays on one side of the turning point

Coordinate geometry

• Gradient m = (y₂ − y₁)/(x₂ − x₁)
• Line through (x₁, y₁) with gradient m: y − y₁ = m(x − x₁)
• Perpendicular lines: m₁ × m₂ = −1, so the second gradient is −1/m₁
• Length = √((x₂ − x₁)² + (y₂ − y₁)²); midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)
• Circle with centre (a, b) and radius r: (x − a)² + (y − b)² = r²
• A circle with diameter AB has its centre at the midpoint of AB and radius AB ÷ 2
• Line meets curve: substitute, then b² − 4ac > 0 means two points, = 0 means a tangent, < 0 means they do not meet

Circular measure

• π radians = 180°: multiply degrees by π/180 to get radians, multiply radians by 180/π to get degrees
• Arc length = radius × angle (s = rθ), θ in radians
• Sector area = ½ × radius² × angle (A = ½r²θ), θ in radians
• Perimeter of a sector = 2r + rθ
• Area of the triangle formed by two radii and the chord = ½r² sin θ
• Segment area = sector − triangle = ½r²(θ − sin θ)
• Chord length = 2r sin(θ/2)

Trigonometry

• sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3; sin 45° = cos 45° = 1/√2, tan 45° = 1; sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3
• tan θ = sin θ/cos θ and sin²θ + cos²θ = 1
• Signs: all positive from 0° to 90°, only sin from 90° to 180°, only tan from 180° to 270°, only cos from 270° to 360°
• If α is the principal value: sin x = k gives α and 180° − α; cos x = k gives α and 360° − α; tan x = k gives α and α + 180°. Add or subtract 360° to reach the interval
• Principal values: sin⁻¹ and tan⁻¹ lie between −90° and 90°; cos⁻¹ lies between 0° and 180°
• y = a sin bx and y = a cos bx have amplitude a and period 360°/b (2π/b in radians)

Series

• In (a + b)ⁿ the term containing b^r is ⁿC_r × a^{n − r} × b^r, where ⁿC_r = n!/(r!(n − r)!)
• AP nth term = a + (n − 1)d
• AP sum: S_n = (n/2)(2a + (n − 1)d) = (n/2)(a + l), where l is the last term
• GP nth term = ar^{n − 1}
• GP sum: S_n = a(1 − rⁿ)/(1 − r)
• Sum to infinity: S_∞ = a/(1 − r), only when −1 < r < 1
• For any series, nth term = S_n − S_{n − 1}

Differentiation

• If y = xⁿ then dy/dx = nx^{n − 1}. First rewrite 1/xⁿ as x⁻ⁿ and √x as x^1/2
• Chain rule: dy/dx = dy/du × du/dx, so (ax + b)ⁿ differentiates to an(ax + b)^{n − 1}
• Gradient of the tangent = dy/dx at the point; gradient of the normal = −1 ÷ (gradient of the tangent)
• Increasing function: dy/dx > 0. Decreasing function: dy/dx < 0
• Stationary point: dy/dx = 0. It is a minimum if d²y/dx² > 0 and a maximum if d²y/dx² < 0
• Connected rates of change: dy/dt = dy/dx × dx/dt

Integration

• ∫xⁿ dx = x^{n + 1}/(n + 1) + c, for n ≠ −1
• ∫(ax + b)ⁿ dx = (ax + b)^{n + 1}/(a(n + 1)) + c, for n ≠ −1
• Definite integral from a to b = (value at x = b) − (value at x = a)
• Area between a curve and the x-axis from x = a to x = b = ∫y dx between a and b
• Area between two graphs = ∫(upper y − lower y) dx between the x-values where they meet
• Volume about the x-axis = π∫y² dx; volume about the y-axis = π∫x² dy

Mechanics

Forces and equilibrium

• Weight = mass × g (W = mg), with g = 10 m s⁻²
• A force F at angle θ to a direction has component F cos θ along that direction and F sin θ at right angles to it
• Resultant of perpendicular forces X and Y: magnitude √(X² + Y²), at angle θ to X where tan θ = Y/X
• Equilibrium: the sum of the components in any direction is zero
• Friction F ≤ μR. In limiting equilibrium (about to slide) F = μR, where μ is the coefficient of friction
• On a plane at angle α to the horizontal, weight has components mg sin α down the plane and mg cos α into the plane
• Total contact force = √(R² + F²). By Newton's third law the particle pushes on the surface with an equal and opposite force

Kinematics of motion in a straight line

• v = u + at
• s = ut + (1/2)at² and s = (1/2)(u + v)t
• v² = u² + 2as
• velocity = rate of change of displacement (v = ds/dt); acceleration = rate of change of velocity (a = dv/dt)
• displacement = integral of v with respect to t; velocity = integral of a with respect to t (remember the constant)
• A particle is instantaneously at rest when v = 0; at the top of a vertical throw v = 0 and a = −10 m s⁻² (taking up as positive)
• Total distance: find where v changes sign, then add the sizes of the separate displacements

Momentum

• momentum = mass × velocity (mv), unit kg m s⁻¹ (the same as N s)
• Conservation: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂, with signs for direction
• If the bodies coalesce: m₁u₁ + m₂u₂ = (m₁ + m₂)v
• change in momentum = m × (final velocity − initial velocity)
• A body that rebounds has a change of velocity equal to the sum of the two speeds, not the difference

Newton’s laws of motion

• resultant force = mass × acceleration (F = ma), with F in the direction of a
• weight = mass × g (W = mg), with g = 10 m s⁻²
• Smooth plane at angle θ: acceleration down the plane = g sin θ
• On a plane, the weight has components mg sin θ along the plane and mg cos θ perpendicular to it
• Friction when sliding = μ × normal reaction (F = μR); on a plane R = mg cos θ
• Lift accelerating upwards: R − mg = ma; accelerating downwards: mg − R = ma
• Two particles over a smooth pulley: (heavier weight − T) = m₁a and (T − lighter weight) = m₂a

Energy, work and power

• work done = force × distance × cos θ (W = Fd cos θ), where θ is the angle between force and motion; unit J
• kinetic energy = (1/2) × mass × speed² (KE = (1/2)mv²)
• change in gravitational potential energy = mass × g × change in vertical height (mgh)
• work done by driving force − work done against resistance = gain in KE + gain in PE
• power = work done ÷ time; power = driving force × speed (P = Fv); unit W, and 1 kW = 1000 W
• At constant speed on a level road, driving force = resistance; on a hill going up, driving force = resistance + mg sin α
• To find acceleration from power: driving force = P ÷ v, then resultant force = ma

Probability & Statistics 1

Representation of data

• frequency density = frequency ÷ class width; frequency = frequency density × class width
• mean = Σx ÷ n; for grouped data, mean ≈ Σfx ÷ Σf using class mid-points
• variance = (Σx² ÷ n) − mean²; standard deviation = √variance
• interquartile range = upper quartile − lower quartile
• Cumulative frequency is plotted at the upper class boundary; the median is read at n ÷ 2
• Coding y = x − a: mean of x = mean of y + a, and the standard deviation is unchanged
• Combined mean = (total of all values) ÷ (total number of values)

Permutations and combinations

• n different objects in a line: n! arrangements
• Arrangements of r objects from n: P(n, r) = n! ÷ (n − r)!
• Selections of r objects from n: C(n, r) = n! ÷ (r! × (n − r)!)
• n objects with p alike of one kind and q alike of another: n! ÷ (p! × q!) arrangements
• Objects that must be together: treat as one block, then multiply by the arrangements inside the block
• Not together = total arrangements − arrangements with them together (for two objects)
• 'And' means multiply the numbers of ways; 'or' (separate cases) means add them

Probability

• P(not A) = 1 − P(A); P(at least one) = 1 − P(none)
• P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
• Mutually exclusive: P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B)
• Independent: P(A ∩ B) = P(A) × P(B)
• Conditional: P(A | B) = P(A ∩ B) ÷ P(B)
• For independent events, P(A | B) = P(A)

Discrete random variables

• Σ P(X = x) = 1
• E(X) = Σ x × P(X = x)
• Var(X) = Σ x² × P(X = x) − [E(X)]²
• Binomial: P(X = r) = C(n, r) × p^r × q^{n − r}, where q = 1 − p
• Binomial: mean = np, variance = npq
• Geometric: P(X = r) = q^{r − 1} × p, and mean = 1 ÷ p
• Geometric: P(X > r) = q^r (the first r trials are all failures)

The normal distribution

• z = (x − μ) ÷ σ
• P(Z > z) = 1 − Φ(z), and Φ(−z) = 1 − Φ(z)
• P(a < Z < b) = Φ(b) − Φ(a)
• To find μ or σ: find z from the tables for the given probability, then solve (x − μ) ÷ σ = z; z is negative below the mean
• B(n, p) ≈ N(np, npq) when np > 5 and nq > 5, where q = 1 − p
• Continuity correction: P(X ≥ 25) becomes P(X > 24.5); P(X ≤ 25) becomes P(X < 25.5); P(X = 25) becomes P(24.5 < X < 25.5)