PI-MATHS 13th September al
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
.
. ...
$
od ey
·
terminating : 5 =
0 6 . or =
5 8
.
↳ AVB =
Gu : x cAou = B]
. 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
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
.
. ...
$
od ey
·
terminating : 5 =
0 6 . or =
5 8
.
↳ AVB =
Gu : x cAou = B]
. 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
