Maths edexcel as paper
1
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/m
where m is original gradient
Standard Equation of a Circle (x - a)² + (y - b)² = r²
with centre (a, b) and radius r
Equation of a Circle (fg) x² + y² + 2fx + 2gy + c = 0
, with 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 TriangleThe 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
Trigonometry
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) = 1
tan(θ) = sin(θ) ÷ cos(θ)
Triangle Law for Vector Addition A→B + B→C = A→C
If 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.
1
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/m
where m is original gradient
Standard Equation of a Circle (x - a)² + (y - b)² = r²
with centre (a, b) and radius r
Equation of a Circle (fg) x² + y² + 2fx + 2gy + c = 0
, with 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 TriangleThe 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
Trigonometry
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) = 1
tan(θ) = sin(θ) ÷ cos(θ)
Triangle Law for Vector Addition A→B + B→C = A→C
If 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.
