AS Further Mathematics₇₃₆₆

Specification7366

7366

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AS Further Mathematics Specification for first teaching in 2017

01 Nov 2018

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Appendix B: mathematical formulae and identities

Students must be able to use the following formulae and identities for AS and A-level further mathematics, without these formulae and identities being provided, either in these forms or in equivalent forms. These formulae and identities may only be provided where they are the starting point for a proof or as a result to be proved.

Pure mathematics

Quadratic equations

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

has roots

\frac{- b \pm \sqrt[]{b^{2} - 4 a c}}{2 a}

Laws of indices

a^{x} a^{y} \equiv a^{x + y}

a^{x} \div a^{y} \equiv a^{x - y}

{(a^{x})}^{y} \equiv a^{x y}

Laws of logarithms

x = a^{n} \Leftrightarrow n = \log_{a} x

for

a > 0

and

x > 0

\log_{a} x + \log_{a} y \equiv \log_{a} (x y)

\log_{a} x - \log_{a} y \equiv \log_{a} (\frac{x}{y})

{k \log}_{a} x \equiv \log_{a} (x^{k})

Coordinate geometry

A straight line graph, gradient

m

passing through

(x_{1}, y_{1})

has equation

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

Straight lines with gradients

m_{1}

and

m_{2}

are perpendicular when

m_{1} m_{2}

-

Sequences

General term of an arithmetic progression:

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

General term of a geometric progression:

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

Trigonometry

In the triangle ABC

Sine rule:

\frac{a}{\sin \begin{matrix}  \end{matrix} A}

\frac{b}{\sin \begin{matrix}  \end{matrix} B}

\frac{c}{\sin \begin{matrix}  \end{matrix} C}

Cosine rule:

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

Area

= \frac{1}{2} a b \sin C

\cos^{2} A + \sin^{2} A \equiv 1

\sec^{2} A \equiv 1 + \tan^{2} A

{cosec}^{2} A \equiv 1 + \cot^{2} A

\sin 2 A \equiv 2 \sin A \cos A

\cos 2 A \equiv {c o s}^{2} A - {s i n}^{2} A

\tan 2 A \equiv \frac{2 \tan A}{1 - ta n^{2} A}

Mensuration

Circumference and Area of circle, radius

r

and diameter

d

C = 2 π r = π d

A = π r^{2}

Pythagoras’ Theorem: In any right-angled triangle where

a

b

and

c

are the lengths of the sides and

c

is the hypotenuse:

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

Area of a trapezium =

\frac{1}{2} (a + b) h

, where

a

and

b

are the lengths of the parallel sides and

h

is their perpendicular separation.

Volume of a prism = area of cross section

\times

length

For a circle of radius

r

, where an angle at the centre of

θ

radians subtends an arc of length

s

and encloses an associated sector of area

A

s = r θ

A = \frac{1}{2} r^{2} θ

Complex numbers

For two complex numbers

z_{1} = r_{1} e^{{i θ}_{1}}

and

z_{2} = r_{2} e^{{i θ}_{2}}

z_{1} z_{2} = r_{1} r_{2} e^{i (θ_{1} + θ_{2})}

\frac{z_{1}}{z_{2}} = \frac{r_{1}}{r_{2}} e^{i (θ_{1} + θ_{2})}

Loci in the Argand diagram:

|z - a| = r

is a circle radius

r

centred at

a

arg

(z - a) = θ

is a half line drawn from

a

at angle

θ

to a line parallel to the positive real axis.

Exponential form:

e^{i θ} = \cos θ + i \sin θ

Matrices

For a 2 by 2 matrix

(\begin{matrix} a & b \\ c & d \end{matrix})

the determinant

∆ = |\begin{matrix} a & b \\ c & d \end{matrix}| = a d - b c

the inverse is

\frac{1}{∆} (\begin{matrix} d & - b \\ - c & a \end{matrix})

The transformation represented by matrix AB is the transformation represented by matrix B followed by the transformation represented by matrix A .

For matrices A , B :

( AB )^–1 = B ^–1 A ^–1

Algebra

\sum_{r = 1}^{n} r = \frac{1}{2} n (n + 1)

For

a x^{2}

b x

c

= 0 with roots

α

and

β

α + β = \frac{- b}{a}

α β = \frac{c}{a}

For

a x^{3}

b x^{2}

c x

d

= 0 with roots

α

β

and

γ

\sum α = \frac{- b}{a}

\sum α β = \frac{c}{a}

α β γ = \frac{- d}{a}

Hyperbolic functions

cosh

x

\equiv \frac{1}{2} (e^{x} + e^{- x})

sinh

x

\equiv \frac{1}{2} (e^{x} - e^{- x})

tanh

x

\equiv \frac{\sinh x}{\cosh x}

Calculus and differential equations

Differentiation

Function	Derivative
$x^{n}$	${n x}^{n - 1}$
$\sin k x$	$k \cos k x$
$\cos k x$	$- k \sin k x$
$e^{k x}$	${k e}^{k x}$
$\ln x$	$\frac{1}{x}$
$f (x) + g (x)$	$f' (x) + g' (x)$
$f (x) g (x)$	$f^{'} (x) g (x) + f (x) g' (x)$
$f (g (x))$	$f' (g (x)) g' (x)$

Function

Derivative

x^{n}

{n x}^{n - 1}

\sin k x

k \cos k x

\cos k x

- k \sin k x

e^{k x}

{k e}^{k x}

\ln x

\frac{1}{x}

f (x) + g (x)

f' (x) + g' (x)

f (x) g (x)

f^{'} (x) g (x) + f (x) g' (x)

f (g (x))

f' (g (x)) g' (x)

Integration

Function	Integral
$x^{n}$	$\frac{1}{n + 1} x^{n + 1} + c, n \neq - 1$
$\cos k x$	$\frac{1}{k} \sin k x + c$
$\sin k x$	$- \frac{1}{k} \cos k x + c$
$e^{k x}$	$\frac{1}{k} e^{k x} + c$
$\frac{1}{x}$	$\ln \| x \| + c, x \neq 0$
$f' (x) + g' (x)$	$f (x) + g (x) + c$
$f' (g (x)) g' (x)$	$f (g (x)) + c$

Function

Integral

x^{n}

\frac{1}{n + 1} x^{n + 1} + c, n \neq - 1

\cos k x

\frac{1}{k} \sin k x + c

\sin k x

- \frac{1}{k} \cos k x + c

e^{k x}

\frac{1}{k} e^{k x} + c

\frac{1}{x}

\ln | x | + c, x \neq 0

f' (x) + g' (x)

f (x) + g (x) + c

f' (g (x)) g' (x)

f (g (x)) + c

Area under a curve

= \int_{a}^{b} y \begin{matrix}  \end{matrix} d x \begin{matrix}  \end{matrix} (y \geq 0)

Volumes of revolution about the

x

and

y

axes:

V_{x} = π \int_{a}^{b} y^{2} d x

V_{y} = π \int_{c}^{d} x^{2} d y

Simple Harmonic Motion:

\ddot{x} = - ω^{2} x

Vectors

|x i + y j + z k| = \sqrt[]{(x^{2} + y^{2} + z^{2})}

Scalar product of two vectors a

= (\begin{matrix} a_{1} \\ a_{2} \\ a_{3} \end{matrix})

and b

= (\begin{matrix} b_{1} \\ b_{2} \\ b_{3} \end{matrix})

(\begin{matrix} a_{1} \\ a_{2} \\ a_{3} \end{matrix}) . (\begin{matrix} b_{1} \\ b_{2} \\ b_{3} \end{matrix}) = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3} = |a| |b| \cos θ

where

θ

is the acute angle between the vectors a and b .

The equation of the line through the point with position vector a parallel to vector b is:

r = a +

t b

The equation of the plane containing the point with position vector a and perpendicular to vector n is:

( r – a ) . n = 0

Mechanics

Forces and equilibrium

Weight = mass

\times g

Friction:

F \leq μ R

Newton’s second law in the form:

F = m a

Kinematics

For motion in a straight line with variable acceleration:

v = \frac{d r}{d t}

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

r = \int v \begin{matrix}  \end{matrix} d t

v = \int a \begin{matrix}  \end{matrix} d t

Statistics

The mean of a set of data:

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

The standard Normal variable:

Z = \frac{X - μ}{σ}

where

X ~ N (μ, σ^{2})

Appendix A: mathematical notation