maths

formula of an arithmetic series

- Sn \= n/2 (2a + (n-1)d)
- Sn \= n/2 (a + l) where a is the first term and l is the last term

what is an arithmetic series

the sum of the terms of an arithmetic sequence

nth term of an arithmetic sequence

- Un \= a + (n-1)d
- a \= the first term
- d \= the common difference

nth term of a geometric sequence

- Un \= ar^(n-1)
- a \= first term
- r \= common ratio

formula of first n terms of a geometric sequence

- Sn \= a(1-r^n) / 1-r
- Sn \= a(r^n - 1) / r-1
where r does not equal 1

divergent sequence

the sum of the values tend towards infinity

convergent sequence

- the sum of the values tend towards a specific number
- it is only convergent if |r|

sum to infinity of a geometric series

a / 1-r

series can be shown using sigma notation

recurrence relation of form Un+1 \= f(Un)

- defines each term of a sequence as a function of the previous term
- to find the members of the sequence substitute in n\=1, n\=2 ... using the previous terms given

if Un+1 < Un for all n ∈ ℕ, what is true of the sequence

it is decreasing

if Un+k \= Un for all n ∈ ℕ, what is true of the sequence

- it is periodic
- means that the terms repeat in a cycle
- k \= the order of the sequence (how often the terms repeat)

x^2-y^2

(x+y)(x-y)

rationalising the denominator of e.g. 1/sqrt(b)+a

* (a-sqrt(b) / a-sqrt(b))

using the discriminant to find number of roots

b^2 - 4ac \> 0 has 2 distant real roots
B^2 -4ac \= 0 has on real repeated root
b^2 - 4ac < 0 has no real roots

completing the square to find the turning point

if f(x) \= a(x+p)^2 + q, then the turning point is (-p,q)

using lines to represent < and ≤

< is dotted line ≤ is solid line

where are the asymptotes of y \= k/x

x\=0 and y\=0

y \= f(x) + a

translation up by a units

y \= f(x+a)

translation left by a units

y \= af(x)

stretch vertically by scale factor a

y \= f(ax)

stretch by scale factor 1/a horizontally

y \= -f(x)

reflection in x-axis

y \= f(-x)

reflection in y-axis

calculating the gradient with 2 points

m \= (y2 - y1)/(x2 - x1)

another way to calculate equation of a line

y-y1\=m(x-x1)

equation of line perpendicular to y \= mx

y\= -(1/m)x

distance between (x1,y1) and (x2,y2)

Sqrt ((x2 - x1)^2 + (y2 - y1)^2 )

equation of circle centre (0,0)

x^2 + y^2 \= r^2

equation of circle centre (a,b)

(x-a)^2 + (y-b)^2 \= r^2

centre and radius of x^2 + y^2 + 2fx + 2gy + c \= 0

centre: (-f,-g)
radius: sqrt (f^2 + g^2 -c)

a tangent to a circle is ...... to the radius of the circle at the point of intersection

perpendicular

the perpendicular bisector of a chord will go through.....

the centre of a circle

the angle in a semicircle is always

a right angle

if ∠PRQ \= 90° then R lies on the circle with diameter PQ

find the centre of a circle given any 3 points

-find the equations of the perpendicular bisectors of 2 different chords
-find the coordinates of the intersection of the perpendicular bisectors

factor theorem

if f(p) \= 0 then (x-p) is a factor of f(x)

proof by deduction

starting from known facts or definitions then using logical steps to reach the desired conclusion

proof by exhaustion

breaking the statement into smaller cases and proving each case separately

proof by counter-example

an example that does not work for the statement

which row of pascal's triangle gives the coefficients of the expansion of (a+b)^n

(n+1)th row

n!

n * (n-1) * (n-2) * ... *3 * 2 * 1

nCr

n!/r!(n-r)!

binomial expansion of (a+b)^n with nCr

a^n + nC1*a^n-1*b + nC2*a^n-2*b^2 + ... + nCr*a^n-r*b^r + ... + b^n

if x is small the first few terms in a binomial expansion can be used to find an approximate value for a complicated expression

cosine rule

a^2 \= b^2 + c^2 - 2bcCosA

Sine rule

a/sinA \= b/sinB \= c/sinC

ambiguous case of the sine rule θ

sinθ \= sin(180-θ)

quadrants of CAST diagram θ

first quadrant: A sinθ , cosθ and tanθ are all positive
second quadrant: S only sinθ is positive
third quadrant: T only tanθ is positive
fourth quadrant: C only cosθ is positive

find sin, cos , tan of any positive or negative angle using the corresponding acute angle made with the x-axis, θ

e.g. in third quadrant -sinθ \= sin(180+θ)

sin^2θ + cos^2θ ≡ ?

1

sinθ / cosθ \= ?

tanθ

what is λa

any vector parallel to a

unit vectors along x- and y-axes are denoted by

i and j

magnitude of a (vector)

|a| \= sqrt (x^2 + y^2)

unit vector in the direction of a

a / |a|

gradient of the tangent to the curve

gradient of a curve at a given point

f'(x)\=?

lim h-\>0 (f(x+h)-f(x))/h

f(x) \= ax^n
f'(x)\=?

anx^n-1

if y \= f(x) + g(x) what is dy/dx

f'(x) + g'(x)

finding the stationary point

where f'(x) \= 0

f(x) has a stationary point at x\=a and f''(a) \> 0, is the point a local minimum or maximum

local minimum

if dy/dx \= x^n
y\=?

(1/(x+1))x^(n+1)+c

if f'(x) \= kx^n
f(x)\=

(k/(x+1))x^(n+1)+c

how to find the constant of integration, c

-integrate the function
-substitute the values of a point on a curve or the value of the function at a given point into the integrated function
-solve the equation to find c

if f(x) \= e^x what is f'(x)

e^x

if y \= e^(kx) what is dy/dx ?

ke^(kx)

if loga(n) \= x what is n?

a^x

loga(x) + loga(y) \= ?

loga(xy)

loga(x)-loga(y) \= ?

loga(x/y)

loga(x^k) \= ?

kloga(x)

loga(1/x) \= ?

loga(x^-1) \= -loga(x)

loga(a) \= ?

1

loga(1) \= ?

0

f(x) \= g(x)
loga(f(x)) \= ?

loga(g(x))

y \= ln(x) is a reflection of y \= e^x in which line

y \= x

which equation has a linear graph of log(y) against log(x) with a gradient n and vertical intercept a

y\=ax^n

which equation has a linear graph of log(y) against x with a gradient log(b) and vertical intercept log(a)

y\=ab^x

population

the whole set of items that are of interest

census

observes and measures every member of a population

sample

selection of observations taken from a subset of the population which is used to find out information about the population as a whole

sampling units

individual units of a population

sampling frame

sampling units individually named or numbered to form a list

simple random sample

A sample of size n selected from the population in such a way that each possible sample of size n has an equal chance of being selected.

systematic sampling

the required elements are chosen at regular intervals from an ordered list

stratified sampling

The population is divided into mutually exclusive strata and a random sample is taken from each

quota sampling

An interviewer or researcher selects a sample that reflects the characteristics of the whole population

opportunity sampling

consists of taking the sample from people who are available at the time the study is carried out and who fit the criteria you are looking for

associated with numerical observations

Quantitative

associated with non-numerical observations

qualitative

can take any value in a given range

continuous variable

can take only specific values in a given range

discrete variable

when data is presented in a group frequency table, the specific data values are not shown.

classes

class boundaries

maximum and minimum values in a class

midpoint

average of the class boundaries

The difference between the upper and lower class boundaries

class width

mode or modal class

value or class that occurs most often

median

the middle value when the data values are put in order

(Σx) / n

mean

(Σxf)/(Σf)

mean of data in frequency table