Further maths
,1
. Number
1 1 Basic-fractions decimals ratio
proportions BIDMA
.
, ,
percentages , , ,
1 2 of
.
The
product rule
counting
-n ! =
nx(n -
1) x (n 2) ...
-
1
↳ for 0 and
only positive integers
EXAMPLE:
4
digit number -
first is
prime ,
number is even
1st Ind 3rd Ht
4 10 10 S
X X X
=
prime
:
2 3 3 7
, , ,
even : 0 2 4 6 8
, , , ,
1
.
3 Surds
↳
addition/subtraction
↳
need to have same base
EXAMPLE :
EXAMPLE : EXAMPLE :
To + V32 - -
Vz Eis-Tz +- 3S
t
↓ ↓ ↓
-x4x5
↓ +
jaxz VExJz 59 / x 59x /3 Jes 3(π6 (3)x
3(45)
= 35 = 452 = 753 : 353 = 3 53 = 2
= 1253
34 + 46 =
>z 75 -
35 =
45 35 -
253 + 1253 =
1553
denominators
rationalising
↳
multiply top by
↳ surd
and bottom
-
difference of two
complex squares
:
EXAMPLE :
25X(5
+ 1)
(5 + 1)
=
65 + 6 + 26 + 25
s + 5 5 - - 1
-
=
2 5 + 4
.
2 ALGEBRA
2 .
1 Basic rules
2 .
I Function
↳ notation :
f(x) :
...
EXAMPLE :
f(x) =
x3 -
2
f(s) =
(s)3 - 2
=
12) -
IT
=
98
, .
2 3 Domain and
Range
↳ Domain : set of
inpur
values it can take (values
EXAMPLE :
f(x) =
3x + 1 -
1
f(x) = x2 domain is infinite
f(x) = O
Range of
ly values)
↳
set
ourpur
:
values
4) domain
dependent on
usually an
inequality
EXAMPLE :
for
f(x) =
3x + 1 -
22x7
range
:
f( 2) - =
3) 2) -
+ 1
S
=
-
f() =
3() + 1
= 22
-
> =
j(x) < 22
2 4 functions
composite
.
↳
Ourpur of first function becomes of second function
input
EXAMPLE :
f(x) =
x2g(x) =
3x + 1
find the of such than
fg(x) gf(x)
=
values
fg(x) = (3x +
1)2g +(x)
=
3 +1
(3x + 1)2 = 3x + 1
qx + 6x + 1 = 3x2 + 1
6x2 + 6x =
O
6x(x + 1)
=
0
x
=
0 or K = -1
2 .
S inverse functions
- notation :
f"(x)
EXAMPLE :
f(x) =
3x -
1
Let =
3x-1 ! must be written
y
+ 1 = 3x
y
11 x
=
make the
subject
f(x) and
11
=
interchange y
Expanding
6
simplifying
-
2 and brackets
,1
. Number
1 1 Basic-fractions decimals ratio
proportions BIDMA
.
, ,
percentages , , ,
1 2 of
.
The
product rule
counting
-n ! =
nx(n -
1) x (n 2) ...
-
1
↳ for 0 and
only positive integers
EXAMPLE:
4
digit number -
first is
prime ,
number is even
1st Ind 3rd Ht
4 10 10 S
X X X
=
prime
:
2 3 3 7
, , ,
even : 0 2 4 6 8
, , , ,
1
.
3 Surds
↳
addition/subtraction
↳
need to have same base
EXAMPLE :
EXAMPLE : EXAMPLE :
To + V32 - -
Vz Eis-Tz +- 3S
t
↓ ↓ ↓
-x4x5
↓ +
jaxz VExJz 59 / x 59x /3 Jes 3(π6 (3)x
3(45)
= 35 = 452 = 753 : 353 = 3 53 = 2
= 1253
34 + 46 =
>z 75 -
35 =
45 35 -
253 + 1253 =
1553
denominators
rationalising
↳
multiply top by
↳ surd
and bottom
-
difference of two
complex squares
:
EXAMPLE :
25X(5
+ 1)
(5 + 1)
=
65 + 6 + 26 + 25
s + 5 5 - - 1
-
=
2 5 + 4
.
2 ALGEBRA
2 .
1 Basic rules
2 .
I Function
↳ notation :
f(x) :
...
EXAMPLE :
f(x) =
x3 -
2
f(s) =
(s)3 - 2
=
12) -
IT
=
98
, .
2 3 Domain and
Range
↳ Domain : set of
inpur
values it can take (values
EXAMPLE :
f(x) =
3x + 1 -
1
f(x) = x2 domain is infinite
f(x) = O
Range of
ly values)
↳
set
ourpur
:
values
4) domain
dependent on
usually an
inequality
EXAMPLE :
for
f(x) =
3x + 1 -
22x7
range
:
f( 2) - =
3) 2) -
+ 1
S
=
-
f() =
3() + 1
= 22
-
> =
j(x) < 22
2 4 functions
composite
.
↳
Ourpur of first function becomes of second function
input
EXAMPLE :
f(x) =
x2g(x) =
3x + 1
find the of such than
fg(x) gf(x)
=
values
fg(x) = (3x +
1)2g +(x)
=
3 +1
(3x + 1)2 = 3x + 1
qx + 6x + 1 = 3x2 + 1
6x2 + 6x =
O
6x(x + 1)
=
0
x
=
0 or K = -1
2 .
S inverse functions
- notation :
f"(x)
EXAMPLE :
f(x) =
3x -
1
Let =
3x-1 ! must be written
y
+ 1 = 3x
y
11 x
=
make the
subject
f(x) and
11
=
interchange y
Expanding
6
simplifying
-
2 and brackets
