Detailed notes covering the Paper 1 Syllabus of the 9707 AS Mathematics syllabus, including coloured graphs, illustrated examples, solving methods and common questions,
0: PRELIMINARIES
SPECIAL SETS OF NUMBERS ③ Empty/nunsets :
D or
53
# Universal set : a set that contains all the elements under
1. N =
[1 , 2, 3 , 4 ,
5 ... 3 ,
the set of natural numbers discussion ,
↳ includes & complement set of any set A is written Al
all positive whole numbers
↳
contains all elements not in set A
2
. # =
& ...
3
,
2, 1
,
0 ,
1 , 6 ... 3 ,
the set of integers
⑥ subsets
↳
includes natural numbers , zero and 4 A & B A is a subset of B elements in A
negative , where all
am
numbers are also in B
↳ It set of positive integers ↳ A =
B where both are equal sets As B and BEA
J
-
included
,
zero not
.
↳ Z, ↳ A B
set of negative integers <
denotes that A is a proper subset of B ,
where
A = B and A F B .
. D
3 (x
= : x =
- ; p q + A a+ 03 the set
tiv ing
,
, ,
& PERATIONS ON SETS
of rational numbers
↳
can be fraction AND > intersection of sets Al B is the set of elements
expressed in the
-
numbers that as a ,
a and B
. A B
where both integers found in common to both
o
form pand a are set
B}
booms& Gn
&, and ↳ A1B : x = Aandn =
a is non-zero (or the thing goes
=
uc as
↳ D
·
as
repeating
decimals are either :
: 5 =
0 1666....
. 8 .
18 OR - the union of Jets A and B , AUB is the set of all
↑ 17
x
0 272727 8 A B
elements belonging both sets
=
to
.
. ...
. R
4
is the set of real numbers , including all rational and JET DIFFERENCE
- the difference between two sets
,
A- B
Al B
pr vij
irrational numbers :
can be expressed as a number line
. or ,
are the elements that belong toA but not B
↳ irrational numbers ↳ A -
B =
En : n cAanda + By A B
=
A1B'
cannot be written as fractions
·
i
decimals neither terminate nor repeat
·
un in
·
e .
.
g # = 3 . 141592654 and various surds , -F INTERVALS
The set of real 1. finite intervals
numbers ↑ Venu diagram to ↳ a and b are real numbers where a < b :
@ m
↑ If
IR R -
2
!
represent the way the ·
(a , b) =
En : a < a < b} Open interval
188
organised
-
sets are
E N [ J >
a
I .I
8
A B
Q IR
-
I 1082 N
5 ·
[a b] ,
=
En .
a < US bY CLOSED interval
i
th
< ⑧ ⑧ >
A D
·
[a b)
.
=
En : a = x <
by HALF-OPEN/CLOSED
& + strict inequality
>
JETS O NOTATION
↳ D +
· non-strict inequality
. Infinite intervals
2
& If a is an element of set s a ES ↳If
a is real number , then the set of all numbers to
, a
satisfy
It a is not an element of sets a S aca usa is infinite interval
the inequality or an
.
, .
·
[a , a)
=
En : n,
a}
② 3 (a 8)
representation
J ,
set
[ ⑧ - &
↳ Descriptive :
A = the first five positive odd numbers ] -a
A
· (
-
w
, 9]
↳ Listing all the elements : A =
2 1 ,
3, 5, 7 , 9} only OPEN brackets
< Set builder notation : A =
Ea : his anodd number ,
0 << 103 * M =
( -
0
, g
x) - can beused next
by convention that you can
to &
never reach infinity
,PI-MATHS 13th September al
0: PRELIMINARIES
INEQUALITIES ↳
general properties for a ,
b EIR ,
(ab1 =
191 x1b |
# > real numbers a b and c
* solution sets are
li
, .
+ b + 0
·
a < b and b < c ,
then ac the set of numbers that
3 distribute mode
↳ a + c < b+ C satisfy an equation or
( a + b) = (9) + 1b) and generally CANN OT
10 as
if c is a number inequality c g 34 + 1
b) +
=
·
the real : . .
| a -
19) -
1b)
↳ a b then ac < ba [x : x =
337
SURDS
am
·
if c is a -ve real number : use set builder
↳ notation
a < b ,
then ac > De ↳ an expression containing a root with an irrational
3
solution (non-terminating or
repeating) e .
g.
PHRASE INEQUALITY
3
2 is non-negative u-8 ↳ caws of surds
tiv ing
↑
o is non-positive a j inclusive terms p(n +
q( =
(a(p + q) + up q , .
a ER
I is at least 5 u75 of 'O'or the given #
Jab =
Vat for a b ER and a b > 8
is at most 5 a45 , ,
2
Given
uc asa < b ,
a + c < b + c
-
S
Instead of (C) , apply function flu) to both sides CANNOT :
5 subtraction
x
f(n) =
x + C a+ b = a + or for
od ey
+(a) a + C
~
only distributed over multiplication and division
· =
↓ + (b) = b + C
↳ rationalise denominator
graph of f(u) is increasing by multiplying by the
pr vij
conjugate surd i e" surd w/ reversed operations but
a O even if i is negative constant
a
.
1 · a ,
↓ same magnitude
· inequality sign does NOT need to be
·
25 + 1 has conjugate -
205 + /
2
applied
i
>
flipped when increasing exus are
un in
TO NOTE :
Va =
positive root of a, fa = 3 (one answers
Will only get z answers for IJa , with the
1
Y
1
y 189 , 02 vn form >8
y
=
=
@ m
other increasing Fac =
(2) and (vi) =
u
-
fxn that can be applied
a
th
ABSOLUTE VALUES -MODULUS
↳ The absolute value of a real number n is given by
In) is defined as :
if , 8
E
n3
121 =
as
is
: absolute value of any real number is always non-negative .
↳ Note :
In k 3 =- 3 < U < 3 for the possible
values ofa (to the left or right ofa in distance terms)
NOTE : Es means ' it and only if
=> means implies:
, PI MATHS 9789 29th September '2/
1: QUADRATICS
SOLVING QUADRATICS SKETCHING GRAPH
1. factorisation Important info
↳ 0 ↳ (n , y)
must snow (an b) (an + b) Step y-intercept
-
- =
/ &
( 5 8) ( 2 0)
↳ n-intercept/roots
-
-
, &
,
(-3 =
=
2)
2. completing the square
↳
stationary point co-ordinates
e .
g. 2n
=
+ 8x -
3 =
8
↳ - for f(n) = an + bu + c +(u) = a(u -
4) + k :
more constant to other side ,
↳
am
=
2
2n + On =
3 The line of symmetry is n = n
↳ ofwe to When
equate co-efficient 1 a >8
,
>
n2 + 4n =
↳ Graph is U-shape with minimum point at
divide co-efficient oh by two , square it and
X (u k) ,
add to both sides of equation
tiv ing
(y)" E (i)
= ↳
n + +n + =
+ When a < 8 ,
X
Graph is a I shape with maximum point at
L
make Its a perfect square and compute RIS (n k) ,
(n + 2)
t
=
uc as
final form >
-
a(n + p)2 +
q
REDUCING COMPLEX ERNS TO QUADRATIC FORM
↳
Introduce variable and let n2 then compute and
x
a new u =
where :
od ey
·
turning point =
( -p , q) so Ive directly .
e .
.
g
n -
4vn -
12 = 8
(vi) -
4vn -
12 =
0 CHECK answers when dealing with
Quadratic formula 4 4v 12 10 +
04
pr vij
.
3 b)(vn 2) 0vn
-
(Vn
+ -
=
-
+ =
-
b + y2 -
492 V = 6 Mn = -
2 reject answer as
a MUST be
u =
V 2
X
36 ( 2)
.
29 n
-
non-negative
=
=
n i c
-
= .
i
u
4
=
does not exist
un in
RELATIONSHIP ROOTS X COFFFICIENTS
↳ fractional
equations :
have unknowns at their denominator
↳
given equation an + bu + c = 0 ,
with roots
Esta 4
@ m
2 .
9 .
& and B ,
it follows that
>
B
=
X 1 the LCM of
roots , multiplying by
-
sum of + Eliminate fraction by
a
(SOR) denominators
>
product of roots x
B => 5 "nu-2) +
-
th
,
(POR)
i 2
. .
(n -
x)(n -
1) =
0 given roots are < ,
B .
=
+3) - 3 PERFORM CHECKING
(n + 1) (n + 2) (ah)(3)
- 2 n
- =
=
+
↳ -
4
Quadratic equation can be written (n-2)(2 1)
0-
as -
=
I
v x
n n 20Rn = 1 both equate to 8 but we
(SORI
= =
u n + POR =
0 where coefficient =
1 ,
↓
CANNOT divide by zero
.
reject is extraneous
e g
. . x =
E and
B = 2
,
soR=1 and Por 7
=
n2
. CHECK both roots will
2 satisfy the equation. If the root makes
: equation is
In + 7 =
0
the denominator of one or more fractions 0, - REJECT &
2x2 -
11n + 14 =
54
root is known as extraneous
.
(2n -
7)(n -
2) = 0
Roots n =
= and n =
2
Les avantages d'acheter des résumés chez Stuvia:
Qualité garantie par les avis des clients
Les clients de Stuvia ont évalués plus de 700 000 résumés. C'est comme ça que vous savez que vous achetez les meilleurs documents.
L’achat facile et rapide
Vous pouvez payer rapidement avec iDeal, carte de crédit ou Stuvia-crédit pour les résumés. Il n'y a pas d'adhésion nécessaire.
Focus sur l’essentiel
Vos camarades écrivent eux-mêmes les notes d’étude, c’est pourquoi les documents sont toujours fiables et à jour. Cela garantit que vous arrivez rapidement au coeur du matériel.
Foire aux questions
Qu'est-ce que j'obtiens en achetant ce document ?
Vous obtenez un PDF, disponible immédiatement après votre achat. Le document acheté est accessible à tout moment, n'importe où et indéfiniment via votre profil.
Garantie de remboursement : comment ça marche ?
Notre garantie de satisfaction garantit que vous trouverez toujours un document d'étude qui vous convient. Vous remplissez un formulaire et notre équipe du service client s'occupe du reste.
Auprès de qui est-ce que j'achète ce résumé ?
Stuvia est une place de marché. Alors, vous n'achetez donc pas ce document chez nous, mais auprès du vendeur thaminivijeyasingam. Stuvia facilite les paiements au vendeur.
Est-ce que j'aurai un abonnement?
Non, vous n'achetez ce résumé que pour €10,56. Vous n'êtes lié à rien après votre achat.
