Pure Mathematics 1 — Complete Formula Sheet

Topic 1

Quadratics

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Quadratic Formula

x = (−b ± √(b² − 4ac)) ÷ 2a

Solves any quadratic ax² + bx + c = 0

Discriminant

Δ = b² − 4ac

Δ > 0: two real roots | Δ = 0: one repeated root | Δ < 0: no real roots

Completing the Square

ax² + bx + c = a(x + b/2a)² + c − b²/4a

Rewrites a quadratic in vertex form

Sum of Roots

Sum = −b / a

For ax² + bx + c = 0, the roots sum to −b/a

Product of Roots

Product = c / a

For ax² + bx + c = 0, the roots multiply to c/a

Factorising Form

(x + p)(x + q) = x² + (p+q)x + pq

Expand brackets to check factorisation

Vertex Form

y = a(x − h)² + k

Vertex at (h, k); a > 0 opens up, a < 0 opens down

Nature of Roots Summary

Δ = b² − 4ac

Δ > 0: 2 distinct reals | Δ = 0: 1 repeated | Δ < 0: 0 real

Topic 2

Functions

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Function Notation

f(x) = expression

Maps each input x to exactly one output f(x)

Composite Function

fg(x) = f(g(x))

Apply g first, then f. Domain must be suitable.

Inverse Function

f&supminus;¹(x)

Swaps domain & range. f(x) must be one-to-one. f(f&supminus;¹(x)) = x

Vertical Translation

y = f(x) + a

Shifts graph up by a units (a > 0) or down (a < 0)

Horizontal Translation

y = f(x + a)

Shifts graph left by a units (a > 0) or right (a < 0)

Vertical Stretch

y = a f(x)

Stretches vertically by factor a. a > 1: stretch; 0 < a < 1: compress

Horizontal Stretch

y = f(ax)

Stretches horizontally by factor 1/a. a > 1: compress; 0 < a < 1: stretch

Reflection in x-axis

y = −f(x)

Flips graph over the x-axis

Reflection in y-axis

y = f(−x)

Flips graph over the y-axis

Topic 3

Coordinate Geometry

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Distance Formula

d = √((x&sub2; − x&sub1;)² + (y&sub2; − y&sub1;)²)

Distance between points (x&sub1;, y&sub1;) and (x&sub2;, y&sub2;)

Midpoint Formula

M = ((x&sub1;+x&sub2;)/2, (y&sub1;+y&sub2;)/2)

Midpoint of segment joining (x&sub1;, y&sub1;) and (x&sub2;, y&sub2;)

Gradient Formula

m = (y&sub2; − y&sub1;) / (x&sub2; − x&sub1;)

Slope of line through (x&sub1;, y&sub1;) and (x&sub2;, y&sub2;)

Point-Slope Form

y − y&sub1; = m(x − x&sub1;)

Equation of line with gradient m through (x&sub1;, y&sub1;)

Slope-Intercept Form

y = mx + c

Gradient m, y-intercept c

General Form

ax + by + c = 0

Standard linear equation form

Parallel Lines

m&sub1; = m&sub2;

Parallel lines have equal gradients

Perpendicular Lines

m&sub1; × m&sub2; = −1

Perpendicular lines have negative reciprocal gradients

Circle Equation (centre-radius)

(x − a)² + (y − b)² = r²

Circle centred at (a, b) with radius r

Circle General Form

x² + y² + 2gx + 2fy + c = 0

Centre (−g, −f), radius √(g² + f² − c)

Tangent-Radius Property

radius ⊥ tangent

The radius to a point of tangency is perpendicular to the tangent at that point

Topic 4

Circular Measure

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Radians Definition

π rad = 180°

Conversion: radians = degrees × π/180

Arc Length

s = rθ

Length of arc subtended by angle θ (radians)

Sector Area

A = ½ r²θ

Area of sector with angle θ (radians)

Segment Area

A = ½ r²(θ − sin θ)

Area of segment = sector area − triangle area

Small Angle Approximations

sin θ ≈ θ, tan θ ≈ θ, cos θ ≈ 1 − θ²/2

Valid for θ in radians, near zero

Topic 5

Trigonometry

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SOH CAH TOA

sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj

Right-angled triangle definitions

Pythagorean Identity

sin²θ + cos²θ = 1

Fundamental trig identity

Tangent Identity

tan θ = sin θ / cos θ

Relates tan to sin and cos

Exact Values Table

0°: sin=0, cos=1, tan=0 | 30°: sin=½, cos=√3/2, tan=1/√3 | 45°: sin=√2/2, cos=√2/2, tan=1 | 60°: sin=√3/2, cos=½, tan=√3 | 90°: sin=1, cos=0, tan undefined

Memorise these for non-calculator exams

Sine Graph

y = sin x

Period 360° (2π), range [−1, 1], amplitude 1

Cosine Graph

y = cos x

Period 360° (2π), range [−1, 1], amplitude 1

Tangent Graph

y = tan x

Period 180° (π), range ℜ, asymptotes at 90° + 180°n

General Solutions

sin θ = sin α → θ = 180°n + (−1)&supn;α cos θ = cos α → θ = 360°n ± α tan θ = tan α → θ = 180°n + α

n ∈ ℤ (use radians in P1)

Trig Graph Transformations

a sin(bx + c) + d

Amplitude a, period 360°/b, phase shift −c/b, vertical shift d

Topic 6

Vectors

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Vector Notation

a = xi + yj

Components in the i (x) and j (y) directions

Magnitude

|a| = √(x² + y²)

Length of vector a = xi + yj

Unit Vector

&ahat; = a / |a|

Vector of length 1 in direction of a

Vector Addition

a + b = (x&sub1;+x&sub2;)i + (y&sub1;+y&sub2;)j

Add corresponding components

Scalar Multiplication

ka = (kx)i + (ky)j

Multiply each component by scalar k

Parallel Vectors

a = kb

If a is parallel to b, one is a scalar multiple of the other

Collinearity

AB = kBC

Points A, B, C are collinear if vectors between them are parallel

Position Vectors

OA = a, OB = b

Position vector of a point relative to origin O

Vector Between Points

AB = b − a

Vector from point A to point B (position vectors)

Magnitude Between Points

|AB| = √((x&sub2;−x&sub1;)² + (y&sub2;−y&sub1;)²)

Distance between A and B

Topic 7

Series

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AP nth Term

u_n = a + (n − 1)d

Arithmetic progression: first term a, common difference d

AP Sum (first form)

S_n = n/2 (2a + (n − 1)d)

Sum of first n terms of an AP

AP Sum (second form)

S_n = n/2 (a + l)

Sum using last term l

GP nth Term

u_n = arⁿ⁻¹

Geometric progression: first term a, common ratio r

GP Sum (finite)

S_n = a(1 − rⁿ) / (1 − r)

Sum of first n terms of a GP (r ≠ 1)

GP Sum to Infinity

S_∞ = a / (1 − r)

Converges only when |r| < 1

Binomial Expansion

(a + b)ⁿ = Σ_r=0ⁿ ⁿC_r a^n−rb^r

General binomial expansion for positive integer n

Binomial Coefficient

ⁿC_r = n! / (r!(n−r)!)

Also written as (&supn;&subcr;)

Pascal's Triangle

1 1 1 1 2 1 1 3 3 1 1 4 6 4 1

Each row gives coefficients of (a+b)ⁿ

Topic 8

Differentiation

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Power Rule

d/dx (xⁿ) = nxⁿ⁻¹

Fundamental rule for differentiating powers of x

Derivative of Constant

d/dx (c) = 0

The derivative of any constant is zero

Sum / Difference Rule

d/dx (f ± g) = f' ± g'

Differentiate term by term

Gradient of Tangent

m = f'(x)

The derivative gives the gradient of the tangent at any point

Equation of Tangent

y − y&sub1; = f'(x&sub1;)(x − x&sub1;)

Tangent line at point (x&sub1;, y&sub1;)

Equation of Normal

y − y&sub1; = −1/f'(x&sub1;) (x − x&sub1;)

Normal is perpendicular to tangent at (x&sub1;, y&sub1;)

Stationary Points

f'(x) = 0

Solve to find stationary (turning) points

Nature of Stationary Points

f''(x) > 0 → min | f''(x) < 0 → max | f''(x) = 0 → possible inflection

Use second derivative test

Increasing Function

f'(x) ≥ 0

Function is increasing when its derivative is non-negative

Decreasing Function

f'(x) ≤ 0

Function is decreasing when its derivative is non-positive

Optimisation

Set f'(x) = 0, check f''(x)

Find max/min values in applied problems

Topic 9

Integration

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Indefinite Integral (Power Rule)

∫ xⁿ dx = xⁿ⁺¹ / (n+1) + c

For n ≠ −1. Include constant of integration c

Definite Integral

∫_a^b f(x) dx = F(b) − F(a)

Evaluates the integral between limits a and b

Area Under Curve

Area = ∫_a^b y dx

Area bounded by curve y = f(x), x-axis, and lines x=a, x=b

Area Between Two Curves

Area = ∫_a^b (f(x) − g(x)) dx

Area between curves y = f(x) (upper) and y = g(x) (lower)

Curve from Gradient

f(x) = ∫ f'(x) dx

Integrate the derivative to recover the original function

Constant of Integration

f(x) = F(x) + c

Use a boundary condition to find c