1. By Perfectmaths.com Page 1
SPM Additional Mathematical Formulae
Algebra
1.
2
4
2
b b ac
x
a
− ± −
=
2. m n m n
a a a +
× =
3. m n m n
a a a −
÷ =
4. ( )m n mn
a a=
5. log log loga a amn m n= +
6. log log loga a a
m
m n
n
= −
7. log logn
a am n m=
8.
log
log
log
c
a
c
b
b
a
=
9. ( 1)nT a n d= + −
10. [ ]2 ( 1)
2
n
n
S a n d= + −
11. 1n
nT ar −
=
12.
( 1) (1 )
, 1
1 1
n n
n
a r a r
S r
r r
− −
= = ≠
− −
13. , 1
1
a
S r
r
∞ = <
−
Calculus
1. ,
dy dv du
y uv u v
dx dx dx
= = +
2. 2
,
du dv
v u
u dy dx dxy
v dx v
−
= =
3.
dy dy du
dx du dx
= ×
4. Area under a curve = or
b b
a a
y dx x dy∫ ∫
5. Volume generated = 2 2
or
b b
a a
y dx x dyπ π∫ ∫
Geometry
1. Distance = ( ) ( )
2 2
2 1 2 1x x y y− + −
2. Midpoint 1 2 1 2
( , ) ,
2 2
x x y y
x y
+ + 
=  
 
3. A point dividing a segment of a line
1 2 1 2
( , ) ,
nx mx ny my
x y
m n m n
+ + 
=  
+ + 
4. Area of a triangle =
1 2 2 3 3 1 2 1 3 2 1 3
1
( ) ( )
2
x y x y x y x y x y x y+ + − + +
5. 2 2
r x y= +
6.
2 2
ˆ
xi yj
r
x y
+
=
+

2. By Perfectmaths.com Page 2
Statistics
1.
x
x
N
=
∑
2.
fx
x
f
=
∑
∑
3.
2 2
2( )x x x
x
N N
σ
−
= = −
∑ ∑
4.
2 2
2( )f x x fx
x
f f
σ
−
= = −
∑ ∑
∑ ∑
5.
1
2
m
N F
m L C
f
 
− 
= + 
 
 
6. 1
0
100
Q
I
Q
= × 7. i i
i
W I
I
W
=
∑
∑
8.
!
P
( )!
n
r
n
n r
=
−
9.
!
( )! !
n
r
n
C
n r r
=
−
10. ( ) ( )( ) ( )P A B P A P B P A B∪ = + − ∩
11. ( ) , where 1n r n r
rP X r C p q p q−
= = + =
12. Mean/Min, npµ =
13. npqσ =
14.
X
Z
µ
σ
−
=
Trigonometry
1. Length of an arc, s rθ=
2. Area of a sector, 21
2
A r θ=
3. 2 2
sin cos 1A A+ =
4. 2 2
sec 1 tanA A= +
5. 2 2
cosec 1 cotA A= +
6. sin 2 2sin cosA A A=
7.
2 2
2
2
cos2 cos sin
2cos 1
1 2sin
A A A
A
A
= −
= −
= −
8. ( )sin sin cos cos sinA B A B A B± = ±
9. ( )cos cos cos sin sinA B A B A B± =
10. ( )
tan tan
tan
1 tan tan
A B
A B
A B
±
± =
11. 2
2 tan
tan 2
1 tan
A
A
A
=
−
12.
sin sin sin
a b c
A B C
= =
13. 2 2 2
2 cosa b c bc A= + −
14. Area of a triangle =
1
sin
2
ab C