HKDSE Maths Formula Sheet
Every key formula from the HKDSE Mathematics Compulsory Part, grouped by topic. Open a topic for full notes, worked examples and practice questions.
Geometry & Trigonometry
Equations of Straight Lines
Slope formula
m = (y₂ − y₁) / (x₂ − x₁)Point-slope form
y − y₁ = m(x − x₁)Two-point form
(y − y₁) / (x − x₁) = (y₂ − y₁) / (x₂ − x₁)Slope-intercept form
y = mx + cIntercept form
x/a + y/b = 1 (x-intercept a, y-intercept b)General form
Ax + By + C = 0: slope −A/B, y-intercept −C/B, x-intercept −C/AParallel / perpendicular
m₁ = m₂ / m₁ × m₂ = −1Distance from a point to a line
Full notes & examples →|Ax₀ + By₀ + C| / √(A² + B²)Basic Properties of Circles
1. Chords & Arcs• Perpendicular line from center to a chord bisects the chord.
• Equal chords are equidistant from the center.
• Equal chords are equidistant from the center.
2. Angles & Arcs• Angle at the center is twice the angle at the circumference.
• Angles subtended by the same arc are equal.
• Angle in a semicircle is 90°.
• Angles subtended by the same arc are equal.
• Angle in a semicircle is 90°.
3. Cyclic Quads• Opposite angles of a cyclic quadrilateral add up to 180°.
• Exterior angle of a cyclic quad equals the interior opposite angle.
• Exterior angle of a cyclic quad equals the interior opposite angle.
4. Tangent Basics• A tangent is perpendicular to the radius at the point of contact.
• Tangents drawn from an external point are equal in length.
• Tangents drawn from an external point are equal in length.
5. Alternate SegmentThe angle between a tangent and a chord equals the angle subtended by the chord in the alternate segment.
Full notes & examples →Centres of Triangles
1. CentroidMedians
Divides each median in a ratio of
Divides each median in a ratio of
2:1 (from vertex to base).2. IncentreAngle Bisectors
Equidistant from all three sides. Center of the inscribed circle.
Equidistant from all three sides. Center of the inscribed circle.
3. CircumcentrePerpendicular Bisectors
Equidistant from all three vertices. Center of the circumscribed circle.
Equidistant from all three vertices. Center of the circumscribed circle.
4. OrthocentreAltitudes
The lines meet at right angles to the opposite sides.
Full notes & examples →The lines meet at right angles to the opposite sides.
Equations of Circles
Standard form
(x − h)² + (y − k)² = r² — centre (h, k), radius rGeneral form
x² + y² + Dx + Ey + F = 0Centre formula
(−D/2, −E/2)Radius formula
r = √((D/2)² + (E/2)² − F) = ½√(D² + E² − 4F)Diameter formEndpoints
(x₁, y₁), (x₂, y₂): (x − x₁)(x − x₂) + (y − y₁)(y − y₂) = 0Tangent testDistance from centre to line
Full notes & examples →= r (less than r: 2 intersection points; greater: none)Locus
Distance formula
PA = √((x − x₁)² + (y − y₁)²) — write every condition with it, then square both sidesEquidistant from A and B
PA = PB ⟹ perpendicular bisector of ABFixed distance r from
C(h, k)(x − h)² + (y − k)² = r² (a circle)Fixed distance d from a line
Ax + By + C = 0|Ax + By + C| / √(A² + B²) = d (a pair of parallel lines)Distance from a point to a line
d = |Ax₀ + By₀ + C| / √(A² + B²)Equidistant from two intersecting lines
Full notes & examples →|A₁x + B₁y + C₁| / √(A₁² + B₁²) = |A₂x + B₂y + C₂| / √(A₂² + B₂²) (a pair of angle bisectors)More about Trigonometry
Sine rule
a / sin A = b / sin B = c / sin CCosine rule (side)
a² = b² + c² − 2bc cos ACosine rule (angle)
cos A = (b² + c² − a²) / (2bc)Area of a triangle
½ab sin C; Heron's formula √(s(s − a)(s − b)(s − c)), s = (a + b + c)/2Identities
sin²θ + cos²θ = 1, tan θ = sin θ / cos θObtuse angles
sin(180° − θ) = sin θ, cos(180° − θ) = −cos θBearingsTrue bearing: clockwise from north, e.g.
Full notes & examples →030°; compass bearing: e.g. N30°E3D Trigonometry
3D Pythagoras (cuboid diagonal)
d = √(a² + b² + c²)Angle between a line and a plane
tan θ = height above the plane ÷ length of the projectionAngle between two planesThe angle between two lines, one in each plane, both perpendicular to the line of intersection at the same point
Projection of a line segmentprojection
= L cos θSine rule
a / sin A = b / sin B = c / sin CCosine rule
a² = b² + c² − 2bc cos AArea of a triangle
Full notes & examples →½ab sin CFunctions & Graphs
Functions and Graphs
1. QuadraticVertex Form:
• Vertex:
• Axis of Symmetry:
• Y-intercept: Let
y = a(x − h)² + k• Vertex:
(h, k)• Axis of Symmetry:
x = h• Y-intercept: Let
x = 02. Exponential
(where
• Passes through
• Horizontal Asymptote:
• Always positive (
y = aˣ(where
a > 0 and a ≠ 1)• Passes through
(0, 1).• Horizontal Asymptote:
y = 0• Always positive (
y > 0).3. Logarithmic
(where
• Passes through
• Vertical Asymptote:
• Domain:
y = logₐ x(where
x > 0 and a ≠ 1)• Passes through
(1, 0).• Vertical Asymptote:
x = 0• Domain:
x > 0.4. Graph Translations• Vertical:
• Horizontal:
y = f(x) ± k• Horizontal:
y = f(x ∓ h)5. Reflections & Stretches• Reflection:
• Stretch:
Full notes & examples →−f(x) or f(−x)• Stretch:
a · f(x) or f(kx)Exponential & Logarithmic Functions
1. Laws of Logarithms• Product:
• Quotient:
• Power:
• Identity:
logₐ(xy) = logₐ x + logₐ y• Quotient:
logₐ(x/y) = logₐ x − logₐ y• Power:
logₐ(xⁿ) = n logₐ x• Identity:
logₐ a = 1 and logₐ 1 = 02. Change of Base
logₐ b = log_c b / log_c a = log b / log a3. Exponential Equations• Same Base:
• Different Base: Take log of both sides:
aᶠ⁽ˣ⁾ = aᵍ⁽ˣ⁾ ⟹ f(x) = g(x)• Different Base: Take log of both sides:
aˣ = b ⟹ x log a = log b ⟹ x = log b / log a4. Logarithmic Equations1. Condense both sides:
2. Convert to exponent:
logₐ f(x) = logₐ g(x) ⟹ f(x) = g(x)2. Convert to exponent:
logₐ f(x) = y ⟹ f(x) = aʸ5. Inverse Graphs• Exponential:
• Logarithmic:
Full notes & examples →y = aˣ (passes through (0, 1), asymptote y = 0)• Logarithmic:
y = logₐ x (passes through (1, 0), asymptote x = 0)More about Graphs of Functions
1. Graphical SolutionsTo solve
1. Plot
2. Find where the curves intersect.
f(x) = g(x) graphically:1. Plot
y = f(x) and y = g(x) on the same axes.2. Find where the curves intersect.
2. Graphical InequalitiesTo solve
Look for the intervals of x where the curve
f(x) > g(x) graphically:Look for the intervals of x where the curve
y = f(x) lies above y = g(x).3. Absolute Value: y = |f(x)|Take the graph of
y = f(x). Any portion below the x-axis (y < 0) is reflected upward to become positive. The parts already above the x-axis stay unchanged.4. Absolute Value: y = f(|x|)Take the graph of
y = f(x) for x ≥ 0 only (right side). Erase the entire left side (x < 0), then mirror the right side to the left.5. Reciprocal AsymptotesFor a rational function like
• Vertical Asymptote (VA):
• Horizontal Asymptote (HA):
Full notes & examples →y = 1/(x − h) + k:• Vertical Asymptote (VA):
x = h• Horizontal Asymptote (HA):
y = kAlgebra
Quadratic Equations in One Unknown
1. The Discriminant
• Δ > 0: 2 distinct real roots
• Δ = 0: 1 repeated real root
• Δ < 0: No real roots
Δ = b² − 4ac• Δ > 0: 2 distinct real roots
• Δ = 0: 1 repeated real root
• Δ < 0: No real roots
2. Sum & Product of RootsFor
• Sum (S):
• Product (P):
ax² + bx + c = 0 with roots α, β:• Sum (S):
α + β = −b / a• Product (P):
αβ = c / a3. Forming New EquationsTo form a quadratic equation with roots α′ and β′:
1. Calculate new Sum:
2. Calculate new Product:
3. Equation:
1. Calculate new Sum:
S′ = α′ + β′2. Calculate new Product:
P′ = α′β′3. Equation:
x² − S′x + P′ = 04. Quadratic InequalitiesFor
• If
• If
Full notes & examples →x² + bx + c ≥ 0 or ≤ 0 with roots r₁ < r₂:• If
≥ 0: Outside range (x ≤ r₁ or x ≥ r₂)• If
≤ 0: Inside range (r₁ ≤ x ≤ r₂)More about Polynomials
1. Remainder TheoremWhen f(x) is divided by
ax − b, the remainder R is:R = f(b/a)2. Factor Theorem
(ax − b) is a factor of f(x) if and only if:f(b/a) = 03. Factorizing CubicsTo factorize
1. Guess a root r such that
2. Divide f(x) by
3. Factorize the quadratic term further.
Ax³ + Bx² + Cx + D:1. Guess a root r such that
f(r) = 0 (try ±1, ±2).2. Divide f(x) by
x − r to get a quadratic quotient.3. Factorize the quadratic term further.
4. HCF & LCM1. Factorize every polynomial completely first.
• HCF: Product of the lowest powers of common factors.
• LCM: Product of the highest powers of all factors.
• HCF: Product of the lowest powers of common factors.
• LCM: Product of the highest powers of all factors.
5. Solving EquationsTo solve
1. Factorize f(x) completely.
2. Set each factor to 0 and solve for x.
Full notes & examples →f(x) = 0:1. Factorize f(x) completely.
2. Set each factor to 0 and solve for x.
More about Equations
1. Linear-QuadraticMake y (or x) the subject of the linear equation and substitute it into the quadratic equation:
ax² + bx + c = 02. FractionalMultiply the entire equation by the LCM of the denominators to clear the fractions.
3. ExponentialLet
u = aˣ to transform:p(a²ˣ) + q(aˣ) + r = 0 ⟹ pu² + qu + r = 04. LogarithmicCondense log terms using laws first:
If
log M + log N = log(MN)If
log M = log N ⟹ M = N5. TrigonometricLet
Full notes & examples →u = sin θ (or cos θ) to transform:pu² + qu + r = 0Inequalities & Linear Programming
1. Linear & Compound• General Form:
• AND: Overlapping region on a number line.
• OR: Combined union of both regions.
ax > b• AND: Overlapping region on a number line.
• OR: Combined union of both regions.
2. QuadraticFor
• If
• If
x² + bx + c > 0 or < 0 with roots r₁ < r₂:• If
> 0: Outside range (x < r₁ or x > r₂)• If
< 0: Inside range (r₁ < x < r₂)3. 2D Linear Inequalities
Ax + By ≤ C or Ax + By > C4. Feasible RegionThe overlapping shaded region formed by a system of multiple 2D linear inequalities.
5. Linear ProgrammingOptimize (maximize or minimize) the Objective Function:
Full notes & examples →P = ax + byArithmetic & Geometric Sequences
1. Arithmetic (AP)General Term:
(where a = 1st term, d = common diff.)
Tₙ = a + (n − 1)d(where a = 1st term, d = common diff.)
2. Geometric (GP)General Term:
(where a = 1st term, r = common ratio)
Tₙ = arⁿ⁻¹(where a = 1st term, r = common ratio)
3. Sum to InfinityFormula:
Full notes & examples →S∞ = a / (1 − r)Variations
1. Direct Variation
y ∝ xⁿy = kxⁿ2. Inverse Variation
y ∝ 1/xⁿy = k/xⁿ ⟺ yxⁿ = k3. Joint Variation
• Direct & Inverse combined:
y ∝ xᵐzⁿ• Direct & Inverse combined:
y ∝ x/z²y = kxᵐzⁿy = kx/z²4. Partial Variation• Partly constant & partly varies directly:
• Linear-constant:
• Two variables:
y = const + dir. as x• Linear-constant:
y = k₁ + k₂x• Two variables:
y = k₁x + k₂z²5. Percentage ChangeCompare ynew to yold after substituting changed values.
Full notes & examples →Statistics & Probability
Measures of Dispersion
1. Range
Range = Maximum − Minimum2. Interquartile Range (IQR)
(where Q₁ = lower quartile and Q₃ = upper quartile)
IQR = Q₃ − Q₁(where Q₁ = lower quartile and Q₃ = upper quartile)
3. Standard Deviation (SD)
SD (σ) = √(Variance)4. Box-and-Whisker PlotVisualizes the 5-number summary:
Min, Q₁, Median, Q₃, Max5. Data TransformationsIf each data point
• Mean / Med / Mode: Multiply by a, add b.
• Range / IQR / SD: Multiply by |a| (adding b has no effect!).
• Variance: Multiply by a².
Full notes & examples →xᵢ → a · xᵢ + b:• Mean / Med / Mode: Multiply by a, add b.
• Range / IQR / SD: Multiply by |a| (adding b has no effect!).
• Variance: Multiply by a².
Permutations and Combinations
1. Permutations
P(n, r) = n! / (n − r)!2. Combinations
C(n, r) = n! / [r!(n − r)!]3. Neighbor Restrictions• Must sit together
• Cannot sit together
• Cannot sit together
4. Selections with "At least"• At least one…
• At least two…
• At least two…
5. Circular Arrangements
Full notes & examples →(n − 1)!More about Probability
1. Addition Law
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)2. Conditional Probability
P(A | B) = P(A ∩ B) / P(B)3. Independence of EventsA and B are independent if and only if:
(equivalently,
P(A ∩ B) = P(A) × P(B)(equivalently,
P(A | B) = P(A))4. Tree Diagrams (AP/GP)• Along branches (AND): Multiply.
• Across branches (OR): Add.
• Across branches (OR): Add.
5. P&C Probability
Full notes & examples →P(Event) = Favorable C(n,r) or P(n,r) / Total C(n,r) or P(n,r)Know the formula but stuck on the question?
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