Results and formulae needed in examinations
The MEI Students’ Handbook is designed for use during the course and contains a large
number of results, and other useful information. Not all of this is available in the booklet
provided for the examinations, currently designated MF2 and published by OCR.
Consequently there are some results which students need for examinations that they will have
to recall or derive. This file contains such results. They are listed by units, rather than by topic
as in the Students’ Handbook. In a few cases the list would be identical to a page in the
Students’ Handbook and in that case the page reference is given instead.
Before you go on to look at the lists, there are a number of points we would like to bring to
your attention.
•
Learning formulae and results is not always the best thing to do. In some cases it is
better practice to know how to derive them and to be prepared to do so should the
need arise.
•
By their nature such lists cannot be completely inclusive. There is almost no limit
to the elementary results that a candidate needs. The results listed are those that
are given in the Students’ Handbook but are not in the examination booklet, MF2.
•
The results to be known for any unit include all those that need to be known for
earlier units in the same strand. Thus a candidate for C3 needs to recall or derive
the results required for C1 and C2.
•
The results to be known for an applied unit include some of those needed for
equivalent pure units (see the Assumed Knowledge section on the unit’s title
page).
•
There is an underlying assumption that students know all the results needed for
Intermediate Tier GCSE.
There is a typesetting error on page 12 of the current printing of the handbook (Fifth edition,
July 2004). The integral of sin 2 x is given at the bottom of the right hand column; the answer
should be 12 ( x − 12 sin 2 x ) and not 12 ( x − 12 sin 2 x ) . This will of course be corrected at the next
printing.
MEI Structured Mathematics:
Formulae and results
C1 results that are not given in the examination booklet (page 1)
Quadratic equations
−b ± b 2 − 4ac
.
2a
Discriminant > 0: 2 real roots
The roots of the equation ax 2 + bx + c = 0 are x =
The discriminant is ( b 2 − 4ac )
Discriminant = 0: 1 repeated root
Discriminant < 0: No real roots
Modulus
x means the positive value of x. For x ≥ 0, x = x ; for x < 0, x = − x .
x < a ⇔ − a < x < a (a > 0)
Binomial coefficients
⎛n⎞
The notations nCr and ⎜ ⎟ are equivalent.
⎝r⎠
n
n
C0 = Cn = 1
Binomial coefficients may be found using Pascal’s triangle:
1
1
1
1
2
1
1
1
3
1
3
4
6
5
10
10
and so on.
1
4
1
5
1
Indices
a m × a n = a m+ n
1
a−m = m
a
MEI Structured Mathematics:
a m ÷ a n = a m−n
1
an = n a
a0 = 1
(a )
m n
= a mn
Formulae and results
C1 results that are not given in the examination booklet (page 2)
Straight lines
The line joining ( x1 , y1 ) to ( x2 , y2 ) has:
y2 − y1
x2 − x1
Gradient
m=
Equation
y − y1 y2 − y1
=
x − x1 x2 − x1
( x2 − x1 ) + ( y2 − y1 )
2
Length
Mid-point
2
⎛ x1 + x2 y1 + y2 ⎞
,
⎜
⎟
2 ⎠
⎝ 2
Other formulae for the equation of a straight line:
Through ( 0, c ) with gradient m:
y = mx + c
Through ( x1 , y1 ) with gradient m:
y − y1 = m ( x − x1 )
Through
( a, 0 ) and ( 0, b ) :
x y
+ =1
a b
Perpendicular lines: The product of their gradients m1m2 = −1
Circles
The circle with centre ( a, b ) radius r has equation ( x − a ) + ( y − b ) = r 2
2
MEI Structured Mathematics:
2
Formulae and results
C2 results that are not given in the examination booklet (page 1)
You are also expected to recall or derive C1 results that are not given in the examination
booklet.
Logarithms and exponentials
⎛x⎞
log a ⎜ ⎟ = log a x − log a y
⎝ y⎠
1
log a n x = log a x
n
log a ( xy ) = log a x + log a y
( )
log a ( x n ) = n log a x
⎛1⎞
log a ⎜ ⎟ = − log a x
⎝ x⎠
log a a = 1
log a 1 = 0
y = a x ⇔ x = log a y
Trigonometric ratios of some angles
θ
0o
30o
45o
60o
90o
180o
sin θ
0
1
2
1
2
3
2
1
0
cos θ
1
3
2
1
2
1
2
0
-1
tan θ
0
1
3
1
-
0
3
Trigonometric identities
sin θ
= tan θ
cos θ
sin 2 θ + cos 2 θ = 1
Triangles
Area = 12 ab sin C = 12 bc sin A = 12 ca sin B
Sine rule:
a
b
c
=
=
sin A sin B sin C
MEI Structured Mathematics:
Formulae and results
C2 results that are not given in the examination booklet (page 2)
Circular measure
2π radians = 360o
Arc length s = rθ
Area of sector A = 12 r 2θ
Trigonometrical relationships
sin ( −θ ) = − sin θ
cos ( −θ ) = cos θ
sin (θ + 90° ) = cos θ
cos (θ + 90° ) = − sin θ
sin (θ + 180° ) = − sin θ
cos (θ + 180° ) = − cos θ
tan ( −θ ) = − tan θ
tan (θ + 180° ) = tan θ
MEI Structured Mathematics:
Formulae and results
C2 results that are not given in the examination booklet (page 3)
Differentiation
Differentiation from first principles
dy
δy
f ( x + δx) − f ( x)
= Lim = Lim
dx δx →0 δx δx→0
δx
dy
is also written as f ′( x)
dx
For y = x n ,
dy
= nx n −1
dx
For y = f ( x) + g( x),
dy
= f ′( x) + g′( x)
dx
Integration
x n +1
+ c, n ≠ −1
n +1
∫ ( f ′( x) + g′( x) ) dx = f ( x) + g( x) + c
n
∫ x dx =
Definite integrals
∫
b
∫
∫
b
a
f ( x)dx gives the area of the region bounded by y = f ( x) , the x-axis and the lines x = a
and x = b .
a
a
b
c
c
f ( x)dx + ∫ f ( x)dx = ∫ f ( x)dx
b
a
b
f ( x)dx = − ∫ f ( x)dx
a
MEI Structured Mathematics:
Formulae and results
MEI Structured Mathematics:
Formulae and results
C3 results that are not given in the examination booklet
You are also expected to recall or derive C1 and C2 results that are not given in the
examination booklet.
Exponentials
y = e x ⇔ x = log e x = ln x
ln ( e x ) = x
eln x = x
Differentiation
Standard derivatives
f ( x)
f ′( x)
e kx
kekx
ln x
1
x
sin kx
k cos kx
cos kx
− k sin kx
Product rule
y = uv,
dy
dv
du
=u +v
dx
dx
dx
Chain rule
y is a function of u and u is a function of x,
MEI Structured Mathematics:
dy dy du
=
×
dx du dx
Formulae and results
C3 results that are not given in the examination booklet (page 2)
Integration
Standard integrals
f ( x)
∫ f ( x)dx (+ a constant)
e kx
1 kx
e
k
1
x
ln x
sin kx
1
− cos kx
k
cos kx
1
sin kx
k
Integration by substitution
y = f (u ), u = g( x),
MEI Structured Mathematics:
∫ f ′ [g( x)] g′( x)dx = f [g( x)] + c
Formulae and results
C4 results that are not given in the examination booklet
You are also expected to recall or derive C1, C2 and C3 results that are not given in the
examination booklet.
Trigonometry
sec2 θ ≡ 1 + tan 2 θ
cosec 2θ ≡ 1 + cot 2 θ
sin 2θ = 2sin θ cos θ
cos 2θ = cos 2 θ − sin 2 θ = 2 cos 2 θ − 1 = 1 − 2sin 2 θ
2 tan θ
tan 2θ =
1 − tan 2 θ
Vectors
⎛ u1 ⎞ ⎛ v1 ⎞
⎜ ⎟⎜ ⎟
⎜ u2 ⎟ . ⎜ v2 ⎟ = u1v1 + u2 v2 + u3v3
⎜u ⎟ ⎜v ⎟
⎝ 3⎠ ⎝ 3⎠
MEI Structured Mathematics:
Formulae and results
MEI Structured Mathematics:
Formulae and results
FP1 results that are not given in the examination booklet
You are also expected to recall or derive relevant C1 and C2 results that are not given in
the examination booklet.
Finite series
n
∑ r = n ( n + 1)
r =1
1
2
Complex numbers
Real-imaginary form
For z = x + jy , Re( z ) = x and Im( z ) = y
mod( z ) = z = x 2 + y 2
arg( z ) = θ where cos θ =
x
y
and sin θ =
and −π < θ ≤ π
z
z
( arg(0) is undefined)
The conjugate of z is z* where z* = x − jy
1
x − jy
z*
2
= 2
= 2
zz* = z ,
2
z x +y
z
Modulus-argument (polar) form
If z = r and arg( z ) = θ , then z = [ r , θ ] = r ( cos θ + jsin θ )
z* = r (cos θ − jsin θ ) ,
1 1
= ( cos θ − jsin θ )
z r
MEI Structured Mathematics:
Formulae and results
Mechanics 1 results that are not given in the examination booklet
No results for Mechanics 1 are given in the examination booklet.
You are expected to be able to recall those on page 14 of the Students’ Handbook.
You are also expected to recall or derive relevant C1 and C2 results that are not
given in the examination booklet.
Particular attention should be given to the constant acceleration formulae.
Written in scalars
(u + v ) t
s = ut + 12 at 2
s=
v = u + at
v 2 − u 2 = 2as
1
2
s = ut + 12 at 2
s = 12 ( u + v ) t
v = u + at
(There is no simple vector equivalent of v 2 − u 2 = 2as .)
Written in vectors
s = vt − 12 at 2
s = vt − 12 at 2
Attention should also be given to the equations for general motion where the
acceleration need not be constant.
Written in scalars
ds
= s
dt
dv
d2s
a=
= v = 2 = s
dt
dt
v=
s = ∫ vdt
v = ∫ adt
These results can also be written using vectors; in that case the symbol r is often used
instead of s for displacement.
MEI Structured Mathematics:
Formulae and results
Mechanics 2 results that are not given in the examination booklet
You are also expected to recall or derive Mechanics 1 results that are not given in the
examination booklet.
Friction
F = μR
F ≤ μR
when sliding takes place
when no sliding takes place
Centre of mass
( ∑ m ) x = ∑ ( mx )
( ∑ m ) r = ∑ ( mr )
Work, energy and power
Work done by a constant force, W = F ( s cos θ ) = ( F cos θ ) s
Kinetic energy = 12 mv 2
Work done against gravity = mgh
Power =
dW
= Fv cos θ
dt
The work energy principle
The total work done by the forces acting on a body is equal to the increase in the kinetic
energy of the body.
MEI Structured Mathematics:
Formulae and results
Mechanics 2 results that are not given in the examination booklet (page 2)
Impulse
Impulse = Ft for constant F
Impulse = change in momentum = mv − mu
Impacts
Newton’s Experimental Law
v2 − v1 = −e ( u2 − u1 )
where e is the coefficient of restitution
e=
velocity of separation v2 − v1
=
velocity of approach u1 − u2
MEI Structured Mathematics:
Formulae and results
Statistics 1 results that are not given in the examination booklet (page 1)
Samples
There are two notations, according to whether the data are grouped or not.
Ungrouped data A sample has n observations of x, x1 , x2 , ... xn .
x=
Sample mean:
∑x
i
n
Sum of squares of deviations: S xx = ∑ ( xi − x ) = ∑ x
2
Grouped data
2
i
(∑ x )
−
i
n
2
= ∑ xi2 − nx 2
A sample has n observations of x, with fi observations of xi .
Sample mean:
x=
∑f
i
=n
∑x f
i i
n
2
Sum of squares of deviations: S xx = ∑ ( xi − x ) f i
=∑x f
2
i i
(∑ x f )
−
i i
n
2
= ∑ xi2 f i − nx 2
For both notations
S xx
n
Mean square deviation:
msd =
Root mean square deviation:
rmsd = msd
Sample variance:
s2 =
Sample standard deviation:
s = s2
S xx
n −1
Some calculators give both rmsd and s; make sure you understand the output from your
calculator.
Coding
If y = a + bx
then y = a + bx and s y2 = b 2 sx2
MEI Structured Mathematics:
Formulae and results
Statistics 1 results that are not given in the examination booklet (page 2)
Selections and arrangements
Number of ways of
*
arranging n unlike objects in line = n !
*
selecting r objects from n unlike objects when the order does not
⎛n⎞
n!
matter
= n Cr = ⎜ ⎟ =
⎝ r ⎠ r !( n − r ) !
selecting r objects from n unlike objects when order does matter
n!
= n Pr =
( n − r )!
*
Conditional probability
P( B A) =
P ( A ∩ B)
P ( A)
Discrete random variables
[The formulae for finding the mean or expectation, E ( X ) , and the variance, Var( X ) , of X
are given in MF2.]
E ( a + bX ) = a + bE ( X ) , Var ( a + bX ) = b 2 Var ( X )
The binomial distribution
For the binomial distribution, B ( n, p ) , the random variable, X, is the number of successes
from n independent trials of a process for which P(success) = p .
P( X = r ) = nCr p r q n − r for r = 0,1, 2, ... , n where q = 1 − p .
MEI Structured Mathematics:
Formulae and results
Statistics 1 results that are not given in the examination booklet (page 3)
The binomial hypothesis test
(A description of the process of hypothesis testing is given on page 18 of the Students’
Handbook.)
Null hypothesis
H0: The probability of the underlying population, p, has a given
value
Alternative hypotheses H1: p ≠ the given value (2-tail test)
p > the given value (1-tail test)
or
p < the given value (1-tail test)
or
Test statistic
Observed number of successes in a sample of size n trials
Critical values
Can be calculated, or derived from cumulative frequency tables
MEI Structured Mathematics:
Formulae and results
Statistics 2 results that are not given in the examination booklet
You are also expected to recall or derive Statistics 1 results that are not given in the
examination booklet.
The χ2 test on a contingency table
The test statistic is X = ∑
2
( fo − fe )
fe
2
.
For a contingency table with r rows and c columns there are ( r − 1)( c − 1) degrees of
freedom.
Product moment correlation and regression
The formulae are given in the examination booklet but you are advised to work with the
forms using sums of squares and sums of products, as given below.
Correlation coefficient: r =
S xy
S xx S yy
Least squares regression line: y − y = b ( x − x ) where b =
MEI Structured Mathematics:
S xy
S xx
Formulae and results
MEI Structured Mathematics:
Formulae and results
Decision Mathematics 1 results that are not given in the examination
booklet
No results are given in the examination booklet.
You are expected to be able to recall or derive those on page 26 of the Students’
Handbook, and those at the top of page 28.
MEI Structured Mathematics:
Formulae and results
Decision Mathematics 2 results that are not given in the examination
booklet
You are also expected to recall or derive Decision Mathematics 1 results that are not
given in the examination booklet.
Decision trees
Decision
node
Chance node
Payoff node
EMV
Expected monetary value
Propositional connectives
∨
or
∧
and
⇔
equivalence
⇒
implies
~
not
Distributive laws
a ∧ (b ∨ c ) = ( a ∧ b ) ∨ ( a ∧ c )
a ∨ (b ∧ c ) = ( a ∨ b ) ∧ ( a ∨ c )
de Morgan’s Laws
~ ( p ∧ q ) ⇔ ~ p∨ ~ q
~ ( p ∨ q ) ⇔ ~ p∧ ~ q
MEI Structured Mathematics:
Formulae and results
Decision Mathematics 2 results that are not given in the examination
booklet (page 2)
Algorithms and vocabulary
See page 28 of the Students’ Handbook for step-by-step descriptions of various algorithms
and also for explanations of standard terms used in linear programming.
MEI Structured Mathematics:
Formulae and results
Decision Computation results that are not given in the examination booklet
You are also expected to recall or derive Decision Mathematics 1 results that are not
given in the examination booklet.
Recurrence relations
First order
For un +1 = aun + f (n)
( a ≠ 1) ,
if f (n) = 0, un = a n u0
otherwise, un = Aa n + particular solution
For un +1 = un + f (n) (i.e. a = 1 in the above relation),
n
un +1 = u0 + ∑ f (i )
i =0
Second order
For un + 2 + aun +1 + bun = f (n)
The auxiliary equation is λ 2 + aλ + b = 0 with roots λ1 and λ2
( λ1 ≠ λ2 )
un = ( An + B ) λ n ( λ1 = λ2 = λ )
If f (n) = 0, un = Aλ1n + Bλ2n
If f (n) ≠ 0 , a particular solution is added to the corresponding expression for un where
f ( n) = 0 .
Algorithms and vocabulary
See page 28 of the Students’ Handbook for step-by-step descriptions of various algorithms
and also for explanations of standard terms used in linear programming.
MEI Structured Mathematics:
Formulae and results
Numerical Methods results that are not given in the examination booklet
You are also expected to recall or derive C1 and C2 results that are not given in the
examination booklet.
You will be given all the results except those below for numerical differentiation.
Numerical differentiation
f ′( x) ≈
f ( x + h) − f ( x)
,
h
f ′( x) ≈
MEI Structured Mathematics:
f ( x + h) − f ( x − h)
2h
Formulae and results