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SPM Mathematics Formulas & Essential Concepts

Mathematical formulas, definitions, coordinate geometry rules, financial equations, and theorems for SPM Mathematics.

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Quadratic Equation: General Form and Roots

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Quadratic Equation: General Form and Roots

General Form:

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

Roots:

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

Quadratic Discriminant (

b^{2} - 4 a c

)

•

b^{2} - 4 a c > 0

: Two distinct real roots
•

b^{2} - 4 a c = 0

: Two equal real roots
•

b^{2} - 4 a c < 0

: No real roots

Axis of Symmetry of

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

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

Consumer Mathematics: Simple Interest

I = P r t

where

P =

Principal,

r =

Annual interest rate (decimal),

t =

Time in years.

Consumer Mathematics: Compound Interest

A = P (1 + \frac{r}{n})^{n t}

where

A =

Total accrued amount,

P =

Principal,

r =

Annual rate,

n =

Compounding frequency per year,

t =

Time in years.

Return on Investment (ROI)

ROI = \frac{Total Return}{Initial Capital Invested} \times 100%

Sets: Number of Subsets of Set

A

n (A) = k

, the total number of possible subsets is

2^{k}

Sets: Addition Rule for Two Sets

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

Graph Theory: Sum of Degrees of Vertices

\sum d (v) = 2 E

The sum of degrees of all vertices equals twice the total number of edges.

Graph Theory: Tree Properties

A tree is a simple connected graph with no cycles.
For a tree with

V

vertices, the number of edges is strictly

E = V - 1

Inverse of a

2 \times 2

Matrix

For

A = (a c b d)

A^{- 1} = \frac{1}{a d - b c} (d - c - b a)

, where

a d - b c \neq = 0

Condition for a Singular Matrix

det (A) = a d - b c = 0

A singular matrix has no inverse.

Identity Matrix (

I

) Property

A I = I A = A

for any compatible square matrix

A

Probability: Addition Rule (General)

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

If mutually exclusive:

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

Probability: Multiplication Rule (Independent Events)

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

Probability: Complementary Event (

A^{'}

)

P (A^{'}) = 1 - P (A)

Statistics: Mean of Ungrouped Data

\overset{x}{ˉ} = \frac{\sum x}{N}

\overset{x}{ˉ} = \frac{\sum f x}{\sum f}

Statistics: Variance (

σ^{2}

)

σ^{2} = \frac{\sum x ^{2}}{N} - \overset{x}{ˉ}^{2}

σ^{2} = \frac{\sum ( x - x ˉ ) ^{2}}{N}

Statistics: Standard Deviation (

σ

)

σ = σ^{2} = \frac{\sum x ^{2}}{N} - \overset{x}{ˉ}^{2}

Statistics: Interquartile Range (IQR)

IQR = Q_{3} - Q_{1}

(Third quartile minus First quartile)

Statistics: Data Transformation Effects

• Adding/subtracting

c

σ

and

σ^{2}

remain unchanged.
• Multiplying by

k

: New

σ = ∣ k ∣ σ

, New

σ^{2} = k^{2} σ^{2}

Coordinate Geometry: Gradient (

m

)

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

m = - \frac{y -intercept}{x -intercept}

Coordinate Geometry: Distance Formula

d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

Coordinate Geometry: Midpoint Formula

M = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})

Motion Graphs: Speed-Time Graph

• Gradient

= Acceleration / Deceleration = \frac{Δ v}{Δ t}

• Area under graph

= Total Distance Travelled

Consumer Mathematics: Co-Insurance Compensation

Compensation = \frac{Amount of insurance purchased}{Amount of insurance required} \times Loss - Deductible

Consumer Mathematics: Chargeable Income

Chargeable Income = Total Annual Income - Tax Exemptions - Tax Reliefs

Trigonometry: Sine Rule

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

Trigonometry: Cosine Rule

a^{2} = b^{2} + c^{2} - 2 b c cos A

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

Navigation: Bearings Rule

Measured from True North in a clockwise direction, formatted as a 3-digit number (

00 0^{\circ}

36 0^{\circ}

).
Back bearing

= θ \pm 18 0^{\circ}

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