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GCE A-Level Sri Lanka: Combined Mathematics Core Theorems & Mechanics

Essential formulas and theorems in Pure Mathematics (polynomials, trigonometry, calculus, complex numbers) and Applied Mathematics (statics, dynamics, vectors) for GCE A-Level Sri Lanka.

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Quadratic Equation Roots and Vieta's Relations

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Quadratic Equation Roots and Vieta's Relations

For

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

(

a \neq = 0

α + β = - \frac{b}{a}

α β = \frac{c}{a}

. The roots are real and distinct if

Δ = b^{2} - 4 a c > 0

, equal if

Δ = 0

, complex if

Δ < 0

Complex Numbers Modulus-Argument Form

z = x + i y = r (cos θ + i sin θ)

, where modulus

r = ∣ z ∣ = x^{2} + y^{2}

and principal argument

θ = Arg (z) \in (- π, π]

De Moivre's Theorem

For any integer

n

(cos θ + i sin θ)^{n} = cos (n θ) + i sin (n θ)

. For rational

n

cos (n θ) + i sin (n θ)

is one of the values.

Cube Roots of Unity and Properties

Solutions of

z^{3} = 1

are

1, ω, ω^{2}

, where

ω = - \frac{1}{2} + i \frac{3}{2}

. Properties:

1 + ω + ω^{2} = 0

and

ω^{3} = 1

Remainder and Factor Theorems for Polynomials

When

P (x)

is divided by

(x - c)

, the remainder is

R = P (c)

. Also,

(x - c)

is a factor of

P (x)

if and only if

P (c) = 0

Principle of Mathematical Induction

1. Verify base case

P (1)

is true. 2. Assume

P (k)

is true for arbitrary

k \geq 1

. 3. Prove

P (k + 1)

is true using the induction hypothesis. Conclude

P (n)

is true for all

n \in Z^{+}

Compound Angle and Double Angle 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

;

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

Triple Angle Trigonometric Identities

sin (3 θ) = 3 sin θ - 4 sin^{3} θ

;

cos (3 θ) = 4 cos^{3} θ - 3 cos θ

;

tan (3 θ) = \frac{3 t a n θ - t a n ^{3} θ}{1 - 3 t a n ^{2} θ}

Standard Limits: Trig and Exponential

lim_{x \to 0} \frac{s i n x}{x} = 1

;

lim_{x \to 0} \frac{t a n x}{x} = 1

;

lim_{x \to 0} \frac{e ^{x} - 1}{x} = 1

;

lim_{x \to 0} \frac{l n ( 1 + x )}{x} = 1

Derivatives of Inverse Trigonometric Functions

\frac{d}{d x} (arcsin x) = \frac{1}{1 - x ^{2}}

;

\frac{d}{d x} (arccos x) = - \frac{1}{1 - x ^{2}}

;

\frac{d}{d x} (arctan x) = \frac{1}{1 + x ^{2}}

Integration by Parts Formula

\int u \frac{d v}{d x} d x = uv - \int v \frac{d u}{d x} d x

. For definite integrals:

\int_{a}^{b} u v^{'} d x = [uv]_{a}^{b} - \int_{a}^{b} u^{'} v d x

Definite Integral Symmetry Identity (King's Property)

\int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x

. More generally,

\int_{a}^{b} f (x) d x = \int_{a}^{b} f (a + b - x) d x

Lami's Theorem for Three Coplanar Forces

If three coplanar forces acting at a point are in equilibrium:

\frac{P}{s i n α} = \frac{Q}{s i n β} = \frac{R}{s i n γ}

, where

α, β, γ

are the angles between the opposite pairs of forces.

Triangle of Forces Theorem

If three coplanar forces acting at a point are in equilibrium, their vectors can be represented completely in magnitude and direction by the sides of a triangle taken in cyclic order.

Laws of Dry Friction (Coulomb Friction)

Static friction

F_{s} \leq μ_{s} R

. Limiting friction

F_{m a x} = μ R

. Angle of friction

λ

tan λ = μ

. Total reaction

S = R 1 + μ^{2}

inclined at

λ

to the normal.

Center of Mass of Standard Uniform Solids

Solid hemisphere of radius

r

\frac{3}{8} r

from flat base. Hollow hemisphere:

\frac{1}{2} r

. Solid right circular cone of height

h

\frac{1}{4} h

from base. Hollow cone:

\frac{1}{3} h

Parallel Axis Theorem for Moments of Inertia

I = I_{G} + M d^{2}

, where

I_{G}

is the moment of inertia about a parallel axis passing through the center of mass

G

M

is total mass, and

d

is the distance between axes.

Kinematics Equations for Constant Acceleration

v = u + a t

;

s = u t + \frac{1}{2} a t^{2}

;

v^{2} = u^{2} + 2 a s

;

s = \frac{u + v}{2} t

Simple Harmonic Motion (SHM) Governing Equation

\overset{x}{¨} = - ω^{2} x

. Velocity at displacement

x

v^{2} = ω^{2} (a^{2} - x^{2})

where

a

is amplitude. Period

T = \frac{2 π}{ω}

; maximum acceleration is

ω^{2} a

at endpoints.

Projectile Motion: Range, Time of Flight and Height

Time of flight

T = \frac{2 u s i n α}{g}

; Maximum height

H = \frac{u ^{2} s i n ^{2} α}{2 g}

; Horizontal range

R = \frac{u ^{2} s i n 2 α}{g}

(maximum at

α = 4 5^{\circ}

R_{m a x} = \frac{u ^{2}}{g}

Newton's Second Law and Conservation of Momentum

Impulse

J = \int F d t = Δ p

. In the absence of external impulsive forces:

m_{1} u_{1} + m_{2} u_{2} = m_{1} v_{1} + m_{2} v_{2}

Newton's Law of Restitution (Collision of Spheres)

Separation speed

= e \times

Approach speed:

v_{2} - v_{1} = - e (u_{2} - u_{1})

, where

e \in [0, 1]

is the coefficient of restitution (

e = 1

perfectly elastic,

e = 0

completely inelastic).

Work-Energy Theorem

Work done by all forces equals change in kinetic energy:

W_{net} = Δ E_{k} = \frac{1}{2} m v_{2}^{2} - \frac{1}{2} m v_{1}^{2}

Circular Motion: Radial and Transverse Acceleration

Radial (centripetal) acceleration:

a_{r} = \frac{v ^{2}}{r} = r ω^{2}

directed towards the center. Transverse (tangential) acceleration:

a_{θ} = r \ddot{θ} = \frac{d v}{d t}

Oblique Impact of an Elastic Particle on a Smooth Plane

Parallel component is unchanged:

v sin ϕ = u sin θ

. Normal component reversed and scaled by

e

v cos ϕ = e u cos θ

. Hence

tan ϕ = \frac{1}{e} tan θ

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