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Singapore GCE O-Level: Additional Mathematics Formulas & Identities

Essential formulas, identities, calculus rules, and coordinate geometry theorems for Singapore-Cambridge GCE O-Level Additional Mathematics (4049).

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Quadratic Discriminant Conditions

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Quadratic Discriminant Conditions

Δ = b^{2} - 4 a c

. Real & distinct roots:

Δ > 0

; Real & equal roots (tangent):

Δ = 0

; No real roots:

Δ < 0

; Real roots:

Δ \geq 0

Vieta's Formulas for Quadratic Equations

For

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

: Sum of roots

α + β = - \frac{b}{a}

; Product of roots

α β = \frac{c}{a}

Remainder and Factor Theorems

Remainder Theorem:

P (x) \div (a x - b)

leaves remainder

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

. Factor Theorem:

(a x - b)

is a factor of

P (x) ⟺ P (\frac{b}{a}) = 0

Binomial Theorem Expansion

(a + b)^{n} = \sum_{r = 0}^{n} (r n) a^{n - r} b^{r}

, where general term

T_{r + 1} = (r n) a^{n - r} b^{r}

and

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

Laws of Logarithms

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

; Change of base:

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

Pythagorean Trigonometric Identities

sin^{2} θ + cos^{2} θ = 1

1 + tan^{2} θ = sec^{2} θ

1 + cot^{2} θ = csc^{2} θ

Compound Angle Trigonometric Identities

sin (A \pm B) = sin A cos B \pm cos A sin B

;

cos (A \pm B) = cos A cos B \mp sin A sin B

;

tan (A \pm B) = \frac{t a n A \pm t a n B}{1 \mp t a n A t a n B}

Double Angle Formulas

sin (2 A) = 2 sin A cos A

;

cos (2 A) = cos^{2} A - sin^{2} A = 2 cos^{2} A - 1 = 1 - 2 sin^{2} A

;

tan (2 A) = \frac{2 t a n A}{1 - t a n ^{2} A}

R-Formula Transformation:

a cos θ \pm b sin θ

a cos θ \pm b sin θ = R cos (θ \mp α)

, where

R = a^{2} + b^{2}

and

α = arctan (\frac{b}{a})

with

0^{\circ} < α < 9 0^{\circ}

Derivative Product Rule

y = uv

, then

\frac{d y}{d x} = u \frac{d v}{d x} + v \frac{d u}{d x}

Derivative Quotient Rule

y = \frac{u}{v}

, then

\frac{d y}{d x} = \frac{v \frac{d u}{d x} - u \frac{d v}{d x}}{v ^{2}}

Derivative Chain Rule

\frac{d y}{d x} = \frac{d y}{d u} \cdot \frac{d u}{d x}

; for

y = [f (x)]^{n} ⟹ \frac{d y}{d x} = n [f (x)]^{n - 1} f^{'} (x)

Derivatives of Trigonometric Functions

\frac{d}{d x} (sin a x) = a cos a x

\frac{d}{d x} (cos a x) = - a sin a x

\frac{d}{d x} (tan a x) = a sec^{2} a x

Derivatives of Exponential and Log Functions

\frac{d}{d x} (e^{a x}) = a e^{a x}

\frac{d}{d x} (ln (a x + b)) = \frac{a}{a x + b}

Equation of Tangent and Normal to a Curve

Tangent at

(x_{1}, y_{1})

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

where

m = \frac{d y}{d x}

. Normal:

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

(

m \neq = 0

Nature of Stationary Points (Second Derivative Test)

Set

\frac{d y}{d x} = 0

: if

\frac{d ^{2} y}{d x ^{2}} > 0 ⟹

local minimum; if

\frac{d ^{2} y}{d x ^{2}} < 0 ⟹

local maximum; if

= 0 ⟹

test first derivative signs.

Connected Rates of Change

\frac{d y}{d t} = \frac{d y}{d x} \cdot \frac{d x}{d t}

Standard Indefinite Integrals

\int (a x + b)^{n} d x = \frac{( a x + b ) ^{n + 1}}{a ( n + 1 )} + C

(

n \neq = - 1

);

\int \frac{1}{a x + b} d x = \frac{1}{a} ln ∣ a x + b ∣ + C

Integration of Trigonometric Functions

\int cos (a x + b) d x = \frac{1}{a} sin (a x + b) + C

;

\int sin (a x + b) d x = - \frac{1}{a} cos (a x + b) + C

;

\int sec^{2} (a x) d x = \frac{1}{a} tan (a x) + C

Integration of Exponential Functions

\int e^{a x + b} d x = \frac{1}{a} e^{a x + b} + C

Definite Integral and Area Bounded by a Curve

Area between curve

y = f (x)

and

x

-axis from

x = a

x = b

A = \int_{a}^{b} ∣ y ∣ d x

Area Between Two Curves

A = \int_{a}^{b} [y_{upper} - y_{lower}] d x

Kinematics: Displacement, Velocity, Acceleration

Displacement

s

; Velocity

v = \frac{d s}{d t} = \int a d t

; Acceleration

a = \frac{d v}{d t} = \frac{d ^{2} s}{d t ^{2}}

. Instantaneous rest when

v = 0

Distance Travelled vs Displacement in Kinematics

Displacement

= s (t_{2}) - s (t_{1}) = \int_{t_{1}}^{t_{2}} v d t

. Total distance travelled

= \int_{t_{1}}^{t_{2}} ∣ v ∣ d t

(account for turning points where

v = 0

Equation of a Circle in Coordinate Geometry

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

with centre

(h, k)

and radius

r

. General form:

x^{2} + y^{2} + 2 g x + 2 f y + c = 0

with centre

(- g, - f)

and

r = g^{2} + f^{2} - c

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