Edexcel AS Pure Maths, AS-Level Pure Maths, pure maths As level

• To use the product rule, differentiate both factors separately and then multiply them together using the product rule formula.
• Quadratic Formula

  
• Domain


  The set of possible inputs for a function.
• Range


  The set of possible outputs of a function.
• Discriminant

  b² - 4ac > 0 then two distinct real roots.b² - 4ac = 0 then one repeated real root.b² - 4ac < 0 then a quadratic function has no real roots.
• Types of Lines for Regions


  If y < f(x) or y > f(x) then the curve y = f(x) is not included in the region, and is represented by a dotted line.If y ≤ f(x) or y ≥ f(x) then the curve y = f(x) is included in the region, and is represented by a solid line.
• Graph Translations

  y = f(x) + a is a translation of the graph y = f(x) by a upwards.y = f(x + a) is a translation of the graph y = f(x) by a to the left.
• Graph Stretches

  y = af(x) is a stretch of the graph y = f(x) by a scale factor of a in the vertical direction.y = f(ax) is a stretch of the graph y = f(x) by a scale factor of 1/a in the horizontal direction.
• Graph Reflections

  y = -f(x) is a reflection of the graph of y = f(x) in the x-axis.y = f(-x) is a reflection of the graph of y = f(x) in the y-axis.
• Gradient of Equation

  m = (y₂ - y₁) ÷ (x₂ - x₁)
• Equation of a Line

  y - y₁ = m(x - x₁)with coords (x₁, y₁)
• Distance Formula

  √((x₂ - x₁)² + (y₂ - y₁)²)from (x₁, y₁) to (x₂, y₂)
• Perpendicular Bisector


  -1/mwhere m is original gradient
• Standard Equation of a Circle

  (x - a)² + (y - b)² = r²with centre (a, b) and radius r
• Equation of a Circle (fg)

  x² + y² + 2fx + 2gy + c = 0with centre (-f, -g) and radius √(f² + g² - c)
• Circle Theorems


  • Tangent to a circle is perpendicular to the radius of the circle at the point of intersection.• Perpendicular bisector of a chord will go through the circle centre.• If triangle forms across the circle, its diameter is the hypotenuse of the right-angled triangle.• Equations of the perpendicular bisectors of two different chords will intersect at the circle centre.
• Factor Theorem


  If f(p) = 0, (x - p) is a factor of f(x)
• Mathematical Proofs


  • State any info/assumptions• Show every step clearly• Make sure every step follows logically from the previous step• Cover all possible cases• Write a statement of proof at the end of your working
• Truth by Exhaustion


  Break the statement into smaller cases and prove each case separately.
• Truth by Counter-Example

  Find one example that does not work for the statement.
• Pascal's Triangle


  The (n + 1)th row of Pascal's triangle gives the coefficients in the expansion of (a + b)ⁿ

  
• Factorial Formula

  n! = n x (n - 1) x (n - 2) x ... x 2 x 1
• Factorials in Pascal's Triangle

  The number of ways of choosing r from a group of n items is:ⁿCᵣ = n! ÷ (r! x (n - r)!)
• Binomial Expansion


  (a + b)ⁿ = (ⁿCᵣ)(aⁿ⁻¹bʳ)(a + b)ⁿ = aⁿ + (ⁿC₁)(aⁿ⁻¹b) + (ⁿC₂)(aⁿ⁻²b²) + ... + (ⁿCᵣ)(aⁿ⁻ʳbʳ) + ... + bⁿ
• Cosine Rule (a²)

  
• Cosine Rule (cos(A))

  
• Sine Rule

  
• Sine Rule Solutions


  Sometimes produces two possible solutions for a missing angle:sin(θ) = sin(180 - θ)
• Sine Graph

  
• Cosine Graph

  
• Tangent Graph

  
• CAST Diagram

  
• Trig Triangles (30, 60, 90)

  
• Trig Triangles (45, 90)

  
• Principal Value


  When you use the inverse trig function on your calculator, the angle you get is the principal value.
• Sine and Cosine Formulae

  sin²(θ) + cos²(θ) = 1tan(θ) = sin(θ) ÷ cos(θ)
• Triangle Law for Vector Addition


  A→B + B→C = A→CIf A→B = a, B→C = b and A→C = c, then a + b = c
• Vector Rules

  • P→Q = R→S, then line segments PQ and RS are equal in length and are parallel.• A→B = -(B→A)• Any vector parallel to the vector a may be written as λa
• Vector Magnitude

  a = xi + yj → |a| = √(x² + y²)
• Unit Vector


  In the direction of a, unit vector isa ÷ |a|