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GCE A/L Combined Mathematics: Formula Cards

Flashcards on pure mathematics and mechanics formulae for GCE Advanced Level Combined Mathematics in Sri Lanka: roots of equations, series, complex numbers, calculus, circles, vectors, forces and kinematics. Original study material.

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Sum and product of roots of

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

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Sum and product of roots of

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

α + β = - \frac{b}{a}

and

α β = \frac{c}{a}

α^{2} + β^{2}

in terms of the sum and product

(α + β)^{2} - 2 α β

α^{3} + β^{3}

in terms of the sum and product

(α + β)^{3} - 3 α β (α + β)

Condition for equal roots of

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

b^{2} - 4 a c = 0

Remainder theorem

The remainder when

P (x)

is divided by

(x - a)

P (a)

Factor theorem

(x - a)

is a factor of

P (x)

if and only if

P (a) = 0

General term of

(a + b)^{n}

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

\sum_{r = 1}^{n} r

\sum r^{2}

\sum r^{3}

\frac{n ( n + 1 )}{2}

\frac{n ( n + 1 ) ( 2 n + 1 )}{6}

[\frac{n ( n + 1 )}{2}]^{2}

Sum of the first

n

terms of an AP

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

Sum to infinity of a GP

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

for

∣ r ∣ < 1

Modulus of

z = a + bi

∣ z ∣ = a^{2} + b^{2}

Product of

(a + bi) (c + d i)

(a c - b d) + (a d + b c) i

De Moivre's theorem

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

Cube roots of unity

1, ω, ω^{2}

with

1 + ω + ω^{2} = 0

and

ω^{3} = 1

sin 2 A

cos 2 A

2 sin A cos A

;

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

Maximum value of

a sin θ + b cos θ

a^{2} + b^{2}

Derivative of

ln (a x + b)

\frac{a}{a x + b}

Product rule

\frac{d}{d x} (uv) = u^{'} v + u v^{'}

Quotient rule

\frac{d}{d x} (\frac{u}{v}) = \frac{u ^{'} v - u v ^{'}}{v ^{2}}

Chain rule

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

Second derivative test

f^{'} (x) = 0

and

f^{''} (x) > 0

gives a minimum;

f^{''} (x) < 0

gives a maximum.

Area between curve and the

x

-axis

\int_{a}^{b} y d x

Volume of revolution about the

x

-axis

V = π \int_{a}^{b} y^{2} d x

Standard limit with sine

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

Equation of a circle

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

: centre

(- g, - f)

, radius

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

Length of a tangent from

(x_{1}, y_{1})

to a circle

x_{1}^{2} + y_{1}^{2} + 2 g x_{1} + 2 f y_{1} + c

Dot product

a \cdot b = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3} = ∣ a ∣∣ b ∣ cos θ

Combinations

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

Resultant of two forces

P

and

Q

at angle

θ

R^{2} = P^{2} + Q^{2} + 2 P Q cos θ

Lami's theorem

For three concurrent forces in equilibrium, each force is proportional to the sine of the angle between the other two.

Limiting friction

F = μ R

Acceleration down a smooth slope

a = g sin θ

Acceleration down a rough slope

a = g (sin θ - μ cos θ)

Equations of uniform acceleration

v = u + a t

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

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

Time of flight, range and greatest height of a projectile (level ground)

T = \frac{2 u s i n α}{g}

R = \frac{u ^{2} s i n 2 α}{g}

H = \frac{u ^{2} s i n ^{2} α}{2 g}

Two particles over a smooth pulley, masses

m_{1} > m_{2}

a = \frac{( m _{1} - m _{2} ) g}{m _{1} + m _{2}}

T = \frac{2 m _{1} m _{2} g}{m _{1} + m _{2}}

Power, force and speed

P = F v

Work done by a force at angle

θ

to the displacement

W = F s cos θ

Kinetic and potential energy

\frac{1}{2} m v^{2}

and

m g h

Conservation of momentum for two particles

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

Impulse

I = m (v - u)

Centre of mass of particles on a line

\overset{x}{ˉ} = \frac{\sum m _{i} x _{i}}{\sum m _{i}}

Tension for a particle in a horizontal circle

T = \frac{m v ^{2}}{r}

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