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HKDSE Mathematics: Core Formulas and Geometric Theorems

Flashcard formulas, geometric properties, algebraic identities, and statistics for HKDSE Mathematics Compulsory Part (香港中學文憑考試數學科必修部分).

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Laws of Indices and Rational Exponents

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Laws of Indices and Rational Exponents

a^{m} \cdot a^{n} = a^{m + n}

\frac{a ^{m}}{a ^{n}} = a^{m - n}

(a^{m})^{n} = a^{mn}

(ab)^{n} = a^{n} b^{n}

a^{- n} = \frac{1}{a ^{n}}

a^{m / n} = n a^{m}

(

a > 0

Quadratic Formula and the Discriminant (

Δ

)

For

a x^{2} + b x + c = 0

(

a \neq = 0

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

. The discriminant is

Δ = b^{2} - 4 a c

. If

Δ > 0

: 2 distinct real roots;

Δ = 0

: 1 double real root;

Δ < 0

: no real roots.

Relations Between Roots and Coefficients (Vieta's Formulas)

α

and

β

are the roots of

a x^{2} + b x + c = 0

, then: Sum of roots

α + β = - \frac{b}{a}

, Product of roots

α β = \frac{c}{a}

. Also

α^{2} + β^{2} = (α + β)^{2} - 2 α β

Coordinates of Vertex and Axis of Symmetry of Parabola

For

y = a x^{2} + b x + c

: Axis of symmetry is

x = - \frac{b}{2 a}

. Vertex is

V (- \frac{b}{2 a}, c - \frac{b ^{2}}{4 a})

. In completed square form

y = a (x - h)^{2} + k

, vertex is

(h, k)

Remainder Theorem and Factor Theorem

Remainder Theorem: When polynomial

P (x)

is divided by

a x - b

, the remainder is

R = P (\frac{b}{a})

. Factor Theorem:

a x - b

is a factor of

P (x)

if and only if

P (\frac{b}{a}) = 0

Laws of Logarithms and Change of Base Formula

lo g_{a} (x y) = lo g_{a} x + lo g_{a} y

lo g_{a} (\frac{x}{y}) = lo g_{a} x - lo g_{a} y

lo g_{a} (x^{k}) = k lo g_{a} x

lo g_{a} a = 1

lo g_{a} 1 = 0

. Change of base:

lo g_{a} b = \frac{l o g _{c} b}{l o g _{c} a}

Arithmetic Sequence: General Term and Sum

General term:

T_{n} = a + (n - 1) d

. Sum of first

n

terms:

S_{n} = \frac{n}{2} (a + l) = \frac{n}{2} (2 a + (n - 1) d)

, where

a

is the first term,

d

is common difference,

l

is the last term.

Geometric Sequence: General Term, Finite Sum, and Sum to Infinity

General term:

T_{n} = a r^{n - 1}

. Finite sum (

r \neq = 1

S_{n} = \frac{a ( r ^{n} - 1 )}{r - 1} = \frac{a ( 1 - r ^{n} )}{1 - r}

. Sum to infinity (

∣ r ∣ < 1

S_{\infty} = \frac{a}{1 - r}

Equations of a Straight Line and Perpendicularity

Slope

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}} = tan θ

. Point-slope form:

y - y_{1} = m (x - x_{1})

. Slope-intercept form:

y = m x + c

. Parallel lines:

m_{1} = m_{2}

. Perpendicular lines:

m_{1} m_{2} = - 1

Perpendicular Distance from Point

(x_{0}, y_{0})

to Line

A x + B y + C = 0

d = \frac{∣ A x _{0} + B y _{0} + C ∣}{A ^{2} + B ^{2}}

Equation of Circle and its Characteristics

Standard form:

(x - h)^{2} + (y - k)^{2} = r^{2}

with center

(h, k)

and radius

r

. General form:

x^{2} + y^{2} + D x + E y + F = 0

with center

(- \frac{D}{2}, - \frac{E}{2})

and radius

r = (\frac{D}{2})^{2} + (\frac{E}{2})^{2} - F

Sine Formula and Cosine Formula in General Triangle

Sine Formula:

\frac{a}{s i n A} = \frac{b}{s i n B} = \frac{c}{s i n C} = 2 R

(where

R

is circumradius). Cosine Formula:

c^{2} = a^{2} + b^{2} - 2 ab cos C

, or

cos C = \frac{a ^{2} + b ^{2} - c ^{2}}{2 ab}

Formulas for the Area of a Triangle

A r e a = \frac{1}{2} bh = \frac{1}{2} ab sin C = \frac{ab c}{4 R} = r s = s (s - a) (s - b) (s - c)

(Heron's Formula, where

s = \frac{a + b + c}{2}

r

is inradius,

R

is circumradius).

Core Trigonometric Identities

sin^{2} θ + cos^{2} θ = 1

tan θ = \frac{s i n θ}{c o s θ}

1 + tan^{2} θ = sec^{2} θ = \frac{1}{c o s ^{2} θ}

. Quadrant signs (ASTC rule): Q1 (All

+

), Q2 (

{

Sine

}

+

), Q3 (

{

Tan

}

+

), Q4 (

{

Cos

}

+

Circle Theorems: Angle at Center and in Semicircle

1. Angle at center is twice angle at circumference subtended by the same arc:

∠ A O B = 2∠ A P B

. 2. Angle in a semicircle is a right angle:

∠ A P B = 9 0^{\circ}

A B

is a diameter.

Circle Theorems: Angles in Same Segment and Cyclic Quadrilateral

1. Angles in the same segment are equal:

∠ A P B = ∠ A QB

. 2. Opposite angles of a cyclic quadrilateral are supplementary:

∠ A + ∠ C = 18 0^{\circ}

. 3. Exterior angle of cyclic quadrilateral equals its interior opposite angle.

Circle Theorems: Tangent Properties and Alternate Segment

1. Tangent is perpendicular to radius at point of contact (

T ⊥ r

). 2. Two tangents from an external point are equal in length. 3. Alternate segment theorem: angle between tangent and chord equals angle in alternate segment.

Permutations and Combinations (without repetition)

Permutations (order matters):

P_{r}^{n} = \frac{n !}{( n - r )!}

. Combinations (order does not matter):

C_{r}^{n} = (r n) = \frac{n !}{r ! ( n - r )!}

. Properties:

(r n) = (n - r n)

(r n) + (r + 1 n) = (r + 1 n + 1)

Circular Permutations

Number of circular permutations of

n

distinct objects:

P_{c i r c} = (n - 1)!

. If reflections are considered identical (e.g. beads on a necklace):

\frac{( n - 1 )!}{2}

Addition Rule and Mutually Exclusive Events in Probability

General addition rule:

P (A \cup B) = P (A) + P (B) - P (A \cap B)

. If

A

and

B

are mutually exclusive (

A \cap B = \emptyset

P (A \cup B) = P (A) + P (B)

Conditional Probability and Independent Events

Conditional probability:

P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}

(

P (B) > 0

). Events

A

and

B

are independent if and only if

P (A \cap B) = P (A) \cdot P (B)

, or equivalently

P (A ∣ B) = P (A)

Measures of Dispersion: IQR and Standard Deviation

Range

= Max - Min

. Interquartile Range

I QR = Q_{3} - Q_{1}

. Population standard deviation

σ = \frac{1}{N} \sum_{i = 1}^{N} (x_{i} - μ)^{2} = \frac{\sum x _{i}^{2}}{N} - μ^{2}

Linear Transformation Effect on Statistics (

y = a x + b

)

New Mean

\overset{y}{ˉ} = a \overset{x}{ˉ} + b

. New Median

= a (Median) + b

. New Range

= ∣ a ∣ (Range)

. New

I QR = ∣ a ∣ (I QR)

. New Standard Deviation

σ_{y} = ∣ a ∣ σ_{x}

. New Variance

Var (y) = a^{2} Var (x)

Standard Score (

z

-score)

z = \frac{x - μ}{σ}

. Represents the number of standard deviations a data value

x

lies above or below the mean

μ

. A higher

z

-score indicates a relatively better performance.

3D Trigonometry: Angle between a Line and a Plane

The angle between a line

L

and a plane

Π

is defined as the acute angle between

L

and its orthogonal projection

L^{'}

on the plane

Π

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