CCC Tam Lee Lai Fun Memorial Secondary School
AL Pure Mathematics Informal Summary
AL Pure Mathematics Informal Summary
Limit of Functions ........................................................................................... 1
Differentiation ................................................................................................. 2
Applications of Differentiation ....................................................................... 4
Indefinite Integral ............................................................................................ 6
Definite Integral ............................................................................................ 10
Applications of Definite Integrals ................................................................. 13
Limit of Sequences ........................................................................................ 14
Binomial Theorem......................................................................................... 17
Polynomials ................................................................................................... 18
Inequalities .................................................................................................... 20
Complex Numbers......................................................................................... 21
System of Linear Equations .......................................................................... 27
Coordinate Geometry .................................................................................... 29
Limit of Functions
 Two Important Limits
sin x
1
x 0
x
1.
lim
(also, lim
x 0
2.
 1
lim 1    e
x 
 x
tan x
 1)
x
x
Sandwich Theorem:
f ( x)  lim h( x)  L, then lim g ( x)  L .
If f ( x)  g ( x)  h( x) and lim
x a
x a
xa
 Continuity
f ( x) is continuous at x  a

P. 1
lim f (a  h)  lim f (a  h)  f (a)
x  0
x 0
CCC Tam Lee Lai Fun Memorial Secondary School
AL Pure Mathematics Informal Summary
Differentiation
 Definition (First Principle)
f ( x)  lim
h 0
f ( x  h)  f ( x )
h
 Formula of Differentiation
1.
2.
3.
4.
5.
6.
7.
8.
9.
d n
x  nx n 1
dx
d x
e  ex
dx
d
1
ln x 
dx
x
d
sin x  cos x
dx
d
cos x   sin x
dx
d
tan x  sec 2 x
dx
d
sec x  sec x tan x
dx
d
1
sin 1 x 
dx
1  x2
d
1
tan 1 x 
dx
1  x2
 Rules of Differentiation
1. Product Rule:
d
du
dv
(uv)  v
u
dx
dx
dx
2. Quotient Rule:
du
dv
u
d u
( )  dx 2 dx
dx v
v
3. Chain Rule:
d
d
du
f u  
f u  
dx
du
dx
v
e.g.
d
1
ln( x 2  1)  2
 2x
dx
x 1
P. 2
CCC Tam Lee Lai Fun Memorial Secondary School
AL Pure Mathematics Informal Summary
 Logarithmic Differentiation
Take ln before differentiate:
e.g.
yx
1
x
1
3
f ( x)  x ( x  1)
e.g.
1
ln x
x
ln x
(ln y )  (
)
x
1
 x  1  ln x
1 dy x

y dx
x2
2
3
1
2
ln f ( x)  ln x  ln( x  1)
3
3
1
1
2
f ( x) 

f ( x)
3 x 3( x  1)
ln y 
f ( x) 

dy 1  ln x 1x

x
dx
x2
2
( x  1)  2 x 13
 x ( x  1) 3
3 x( x  1)
(3 x  1)
2
1
3 x 3 ( x  1) 3
 Leibniz’s Theorem
[ f ( x) g ( x)]( n )  C0n f ( x) g n ( ( x) )  C1n f
( (x1) g)
n
)
( x( ) 1 ......
 Cnn f
OR
n
[ f ( x ) g ( x( n)) 
] 
r 0
C ( rf ) ( x )n( g r
n
r
where C0n  1, C1n  n, C2n 
)
(x )
n(n  1) n n(n  1)(n  2)
, C3 
, etc
2
3 2
 Differentiability
f ( x) is differentiable at x  a if
lim
h 0
f ( a  h)  f ( a )
f ( a  h)  f ( a )
 lim
h 0
h
h
Note: differentiable  continuous
not continuous  not differentiable
P. 3
n
( x) g (( x))
CCC Tam Lee Lai Fun Memorial Secondary School
AL Pure Mathematics Informal Summary
Applications of Differentiation
 L’Hospital’s Rule
For
0

f ( x)
f ( x)
or , lim
 lim
0
 xa g ( x) xa g ( x)
For 0 ,1 , etc, take ln first
e.g. lim(sin x) x
x 0
e.g. lim x ln x
y  (sin x) x
x  0
ln x
x  0 1/ x
1/ x
 lim
2
x  0 1/ x
 lim  x
ln y  x ln(sin x)
 lim
ln( lim y )  lim x ln(sin x)
x 0
ln(sin x)
x 0
1
x
1
cos x
sin
x
 lim
x 0
1
 2
x
2
 x cos x
 lim
x 0
sin x
x
 lim(
)(  x cos x)
x  0 sin x
0
 lim
x 0
0



e.g. lim


x 1 x  1
ln x 

x
x 0
1
x ln x  ( x  1)
x 1
( x  1) ln x
x(1/ x)  ln x  1
 lim
x 1 ( x  1) / x  ln x
ln x
 lim
x 1 1  1/ x  ln x
1/ x
 lim
x 1 1/ x 2  1/ x
1

2
 lim
lim y  e 0  1
x  0
 Proving Inequalities
To prove f ( x)  g ( x) , let F ( x)  f ( x)  g ( x) and find F ( x)
e.g. Prove ln x  x 1 for all x  0 .
Let F ( x)  ln x  x  1
1
1
x
When F ( x)  0 , x  1
F ( x) 
x
1
f ' ( x)
--------- 0 ++++++
 F ( x) is max. at x  1
 F ( x)  F (1) for all x  0
ln x  x  1  ln1  1  1
ln x  x  1
P. 4
CCC Tam Lee Lai Fun Memorial Secondary School
AL Pure Mathematics Informal Summary
 Curve Sketching
Steps:
(1) Find f '( x) and f ''( x)
─ fully simplify and factorize them
─ take ln may be useful
(2) Draw table
e.g.
includes values of x
(1) at which f ( x)  0 and f ( x)  0
(2) at which f ( x), f ( x ) or f ( x)
is undefined
(3) the expression inside absolute value
sign is zero
x 1
e.g. f ( x) 
x( x  2)
include x  1
3
x
f ( x )
--------------
f ( x)
+++++++++++++++++++++++
f ( x)
0
++++++++++
2
f ( x)  0
f ( x)  0
increasing
decreasing
f ( x)  0
Concave upward
f ( x)  0
Concave downward
(3) identify vertical asymptote(s) x  k
f ( x)

 m  lim
x 
x
find oblique asymptote(s) y  mx  b by 
b  lim f ( x )  mx
x 

(4) Draw and label asymptotes/ extreme points/point of inflexions/
intercepts
Then draw the curve
P. 5
CCC Tam Lee Lai Fun Memorial Secondary School
AL Pure Mathematics Informal Summary
Indefinite Integral
 Definition:
If
d
F ( x)  f ( x), then
dx
 f ( x)dx  F ( x)  C
 Integration Formula
1.
 kdx  kx  C
2.
n
 x dx 
3.
 x dx  ln x  C
4.
8.
 e dx  e  C
 sin xdx   cos x  C
 cos xdx  sin x  C
 sec xdx  tan x  C
 sec x tan xdx  sec x  C
9.

10.
x
5.
6.
7.
x n1
C
n 1
1
x
x
2
1
a2  x2
2
dx  sin 1
x
C
a
1
1
x
dx  tan 1  C
2
a
a
a
Rules of Partial Fractions
e.g.
(1)
(2)
(3)
(4)
5x  3
A
B


( x  1)( x  2) x  1 x  2
2x 1
A
Bx  C

 2
2
( x  1)( x  1) x  1 x  1
3x  2 A B
C
  2
2
x ( x  1) x x
x 1
x 1
A
Bx  C Dx  E

 2

2
2
( x  2)( x  1)
x  2 x  1 ( x 2  1) 2
P. 6
(Linear factor)
(Quadratic factor)
(Repeated linear factor)
(Repeated quadratic factor)
CCC Tam Lee Lai Fun Memorial Secondary School
AL Pure Mathematics Informal Summary
 Method of Substitution
e.g.
 x( x
2
 1)5 dx
  x  u5 
 x( x
du
2x
1 5
u du
2
1
 u6  C
12
1
 ( x 2  1)6  C
12
 1)5 dx
2
1
( x 2  1)5 d ( x 2  1)

2
1
 ( x 2  1)6  C
12


OR
 Integration of Rational Functions ( 
P( x)
dx )
Q( x)
─ do long division if deg P( x)  deg Q( x)
─ then use partial fractions
x3  x  2
 x  3 dx
( x 2  3x  10)( x  3)  32

dx
x 3
32
  x 2  3x  10 
dx
x 3
1
3x 2
 x3 
 10 x  32 ln x  3  C
3
2
e.g.
e.g.
5x  1
dx
 x 1
5
3
(2 x  1) 
0
2 dx
2 2
completing square
x  x 1
5 d ( x 2  x  1) 3
1
  2
 
dx
1
2
x  x 1
2 ( x  )2  3
2
4
1
x
5
3 1
2
1
2)C
 ln x  x  1  
tan (
2
2 3
3
2
2
5
2x 1
 ln x 2  x  1  3 tan 1 (
)C
2
3
e.g.
x
2
 2x
2
1
dx
 5x  3
1
dx
(2 x  1)( x  3)
2
1

7
7


dx
2x 1 x  3
2 1
1
  ln 2 x  1  ln x  3  C
7 2
7
1 2x 1
 ln
C
7
x3

5x  1
dx
 x 1
5
3
(2 x  1) 
2 dx
2 2
x  x 1
5
3
1
 ln x 2  x  1  
dx
0
2
2 ( x  1 )2  5
partial fractions
2
4
5
3
1
 ln x 2  x  1  
dx
2
2
1
5
1
5
(x  
)( x  
)
2 2
2 2
5
3
1
1
1
 ln x 2  x  1   (
)[

]dx
2
2
5
1
5
1
5
(x  
) (x  
)
2 2
2 2
e.g.

x
2
5
3
1
5
1
5
ln x 2  x  1 
[ln x  
 ln x  
] C
2
2 2
2 2
2 5
P. 7
CCC Tam Lee Lai Fun Memorial Secondary School
AL Pure Mathematics Informal Summary
 Integration of Rational Functions of sin x / cos x
e.g.
1
 2  sin x dx
Put t  tan
2dt

2t 1  t 2
2
1 t2
2

dt
2(1  t 2 )  2t
1
 2
dt
t  t 1
1

dt
1 2 3
(t  ) 
2
4
 1
t 
1

tan 1  2   C
3
 3 


2
 2 
x 

2 tan  1 

2
2

tan 1 
C
3
3





1
1 t 2
x
2
t
1
x
x
sin x  2sin cos
2
2
t
1
 2(
)(
)
2
1 t
1 t2
2t

1 t2
x 1
dt  sec 2   dx
2 2
x 1
 (1  tan 2 ) dx
2 2
2dt
dx 
1 t2
 Integration of Irrational Functions
For a2  x2 , try to put x  a sin
For a2  x2 , try to put x  a tan 
For x2  a2 , try to put x  a sec
 Integration by Parts
 f ( x) g ( x) d x
x
2
 f( )x g' ( x) d x
“derivative transfer”
P. 8
x
x
 sin 2
2
2
2
1
t


2
1 t 1 t2
1 t2

1 t2
cos x  cos 2
CCC Tam Lee Lai Fun Memorial Secondary School
AL Pure Mathematics Informal Summary
x2
) ln xdx
2
x2
x2 1
 ln x    dx
2
2 x
2
2
x
x
 ln x   C
2
4
e.g.
 x ln xdx   (
e.g.
e
x
sin xdx   e x (  cos x)dx
 e x cos x   e x ( cos x)dx
 e x cos x   e x (sin x)dx
(by parts twice)
 e x cos x  e x sin x   e x sin xdx
x
 e sin xdx 
e x sin x  e x cos x
C
2
 Reduction Formula
─ usually generated by Integration by Parts
e.g.
I n   sin n xdx
  sin n 1 x(  cos x)dx
  sin n 1 x cos x   (n  1) sin n  2 x(cos x)(  cos x)dx
  sin n 1 x cos x  (n  1)  sin n  2 x cos 2 xdx
  sin n 1 x cos x  (n  1)  sin n  2 x(1  sin 2 x) dx
  sin n 1 x cos x  (n  1)[ I n  2  I n ]
nI n   sin n 1 x cos x  (n  1) I n  2
P. 9
CCC Tam Lee Lai Fun Memorial Secondary School
AL Pure Mathematics Informal Summary
Definite Integral
 Second Fundamental Theorem
b
d
F ( x)  f ( x),  f ( x)dx  F (b)  F (a )
a
dx
If
 Method of Substitution
e.g.
2

1
x( x  1)3 dx
3
  (u  1)u 3du
Put u = x + 1
2
3
  (u 4  u 3 )du
du = dx
2
1
1
 [ u 5  u 4 ]32
5
4
1 5 1 4
1
1
519
 (  3   3 )  (  25   2 4 ) 
5
4
5
4
20
x
u
 Integration by Parts
e.g.

2
0
2
xe x 1dx   x(e x 1 )dx
0
2
 [ xe x 1 ]02   (1)e x 1dx
0
 2e  [e x 1 ]02
 2e  e  e 1
e
1
e
 Reduction Formula
1
e.g. I n   x n 1  x 2 dx
0
1
  xn
0

3
2 2
[(1  x ) ]
dx
3x
3
1 1 n 1
2 2
x
[(1

x
)
]dx
3 0
1
3
3
1
1 1
  x n 1 (1  x 2 ) 2   (n  1) x n  2 (1  x 2 ) 2 dx
3
3 0
0

n  1 1 n2
x (1  x 2 ) 1  x 2 dx

0
3
P. 10
2
3
1
2
CCC Tam Lee Lai Fun Memorial Secondary School
AL Pure Mathematics Informal Summary
n 1
[ I n2  I n ]
3
3I n  (n  1) I n  2  (n  1) I n

In 
n 1
I n2
n2
When n is odd,
When n is even,
n 1
I n2
n2
n3

I n4
n
M
In 
I n2
n 1
I n2
n2
n3

I n4
n
M
In 
I n2
2
I1
5
(n  1)(n  3)...2
In 
I1
(n  2)(n)...5
1
I0
4
(n  1)(n  3)...1
In 
I0
(n  2)(n)...4
) I 3 
) I 2 
 First Fundamental Theorem
d x
f (t )dt  f ( x)
dx  a
e.g.
d x
cos(t 2 ) dt  cos( x 2 )

0
dx
 Sum an Infinite Series by Definite Integrals
n
1
k 1
lim  f ( )   f ( x)dx
0
n 
n n
k 1
e.g.
12
22
n2
lim( 3 3  3 3  ...  3
)
n  n  1
n 2
n  n3
n
k2
 lim  3
 use summation notation
3
n 
k 1 n  k
k 2n 1

3
3
n 
n
k 1 n  k
k
( )2
n
1
 lim  n

n 
k
k 1
1  ( )3 n
n
n
 lim 
 ‘make’
1
n
 ‘make’
k
n
P. 11
CCC Tam Lee Lai Fun Memorial Secondary School
1
 n  dx

k
 x
n
n
1



lim
0
 n k 1
x2
dx
0 1  x3
1 1 1
 
d (1  x 3 )
3
0
3 1 x
1
 [ln 1  x 3 ]10
3
1
 ln 2
3

AL Pure Mathematics Informal Summary
1
 Inequalities on Definite integrals
b
b
1. lf f(x)  g(x) x  [a, b],  a f ( x)dx   a g ( x)dx
2.

b
a
b
f ( x)dx   f ( x) dx
a
P. 12
CCC Tam Lee Lai Fun Memorial Secondary School
AL Pure Mathematics Informal Summary
Applications of Definite Integrals
 Plane Area
3
 4

4
8
Area  2   2 xdx  
( x  1) 2 dx 
0
1
27


or Area  
8
0
1


2 3


y
27 y
 dy
 

1



2
8 


2
 Volume: Disc Method
4 8
 4

Volume     2 x dx  
( x  1)3 dx 
0
1 27


Volume     4 x  x 2  3 dx
3
2
1
 Volume: Shell Method
2
Volume   2 xy dx
Volume   2 x  x 2  3 x  1   x  2   dx
3
0
1
2
 2  x  x(2  x) dx
0
P. 13
CCC Tam Lee Lai Fun Memorial Secondary School
AL Pure Mathematics Informal Summary
Limit of Sequences
 Sandwich Theorem
If
an  bn  cn for all n , and lim an  lim cn  L, then lim bn  L .
n 
n 
n 
1
nn  1.
e.g. Prove lim
n 
1
Let n n  1  h
n  (1  h)n
n  1  nh 
n
n(n  1) 2
h  ...
2
n(n  1) 2
h
2
2
n 1
0  h 
lim
n 
( h  0)
2
0
n 1
 lim h  0
n 
1
 lim n n  lim h
n 
n 
1
 Monotone Convergence Theorem:
Monotonic increasing + bounded above
(an 1  an n)
 Convergent
(an  K )
Monotonic decreasing + bounded below  Convergent
(an 1  an n)
(an  K )
usually proved by Method of Difference or M.I.
P. 14
CCC Tam Lee Lai Fun Memorial Secondary School
e.g.
AL Pure Mathematics Informal Summary
Let {an } be a sequence of positive numbers, where
a1  1 and an 
12an 1  12
an 1  13
(a) Prove that an  3 for all positive integers n.
Try Method of Difference:
an  3 
1 2an 1  1 2
3
an 1  13

12an 1  12  3an  39
an 1  13

9( an  3)
an  13
 need to have an  3
 try M.I.
When n = 1, a1  1  3
 an  3
is true for n = 1
Assume ak  31
ak 1  3 
12ak  12
3
ak  13
=
12ak  12  3ak  39
ak  13
=
9(ak  3)
ak  13
0
 an  3
is true for n = k+1
By M.I., an  3 n
P. 15
CCC Tam Lee Lai Fun Memorial Secondary School
AL Pure Mathematics Informal Summary
an exists.
(b) Prove that lim
n 
Need to prove monotonic increasing,
try method of difference:
an 1  an 
12an  12
 an
an  13
=
12an  12  an 2  13an
an  13
=
an 2  an  12
an  13
=
(an  3)(an  4)
an  13
( ( 0  an  3)
0
 {an }
is monotonic increasing and, by(a), bounded above by 3
 {an }
is convergent
an .
(c) Find lim
n 
an  L
Let lim
n 
lim an 
n 
12 lim an 1  12
n 
lim an 1  13
n 
L
12 L  12
L  13
L2  L  12  0
L  3 or L  4
an  0
bn  b .
lf bn  b for all n, then lim
n 
cn  C .
lf cn  c for all n, then lim
n 
 L  lim an  0
n 
L  3
P. 16
CCC Tam Lee Lai Fun Memorial Secondary School
AL Pure Mathematics Informal Summary
Binomial Theorem
n1
(a  b)n  C0n a n  C1n a b  C2n a n2b2  ...  Cnnbn
n
  C rn a n  r b r
total (n+1) terms
Crn 
r 0
n!
(n  r )!r !
(1  x) n  C0n  C1n x  C2n x 2  ...  Cnn x n
n
  C rn x r
r 0
Differentiation and Integration may help you find properties of Binomial
coefficients.
n
e.g.
n (1  x) n1   r C rn x r 1
r 1
n
Put x = 1,
r C
r 1
e.g.
n
r
 n (2 n1 )
n Cn
(1  x) n 1
r

x r 1  C
n 1
r

1
r 0
Put x = 0, C 
1
n 1
n Cn
2n 1
1
r


Put x = 1,
n  1 r 0 r  1 n  1
n

r 0
C rn
r 1

2n 1  1
n 1
P. 17
CCC Tam Lee Lai Fun Memorial Secondary School
AL Pure Mathematics Informal Summary
Polynomials
q( x)
g ( x) f ( x)
 Division Algorithm
f ( x)  g ( x) q ( x)  r ( x)
Remainder
r ( x)
Quotient
deg [ r ( x) ] < deg [ g ( x ) ]
or r ( x)  0 (that is, f ( x) divisible by g ( x ) )
 Remainder Theorem
When a polynomial f ( x) is divided by (ax  b) , the remainder is f (
b
).
a
 Factor Theorem
When f ( )  0 , ( x   ) is a factor of the polynomial f ( x) .
Generally, if 1 ,  2 , ,  n are the roots of a polynomial f ( x) of degree n ,
then f ( x)  A( x  1 )( x   2 ) ( x   n ) , where A is the leading coefficient.
 Euclidean Algorithm (輾轉相除法)
q( x)
g ( x) f ( x)
r ( x) g ( x)

r ( x)
G.C.D. of f ( x) and g ( x ) = G.C.D. of g ( x ) and r ( x)
 Complex Roots Theorem
Let f ( x) be a polynomial with real coefficients.
If ( p  qi) is a root of f ( x)  0 ( p, q  R ), ( p  qi) is also a root of
f ( x)  0 .
P. 18
CCC Tam Lee Lai Fun Memorial Secondary School
AL Pure Mathematics Informal Summary
Implications:
1. A polynomial of degree  3 can be factorized into real linear and/or
quadratic factors.
2. A polynomial of odd degree has at least one real root.
 Multiple Roots
f ( x) has a multiple root 
f ( )  0 and f ( )  0
 Relation between Roots and Coefficients
Let  ,  ,  be the roots of ax3  bx 2  cx  d  0
b

    

a

c

      
a

d

 

a

Tips:
 2   2   2  (     )2  2(     )
 2  2   2 2   2 2  (     )2  2 (     )
 Transformations of Polynomial Equations
Old Roots
e.g.
e.g.
e.g.
New Roots
3 , 3 , 3
  2,   2,   2
, , 
, , 
, , 
  
,
,
  

  
,
,
 2 2 2

(  d / a ) (  d / a) (  d / a)
,
,
2
2
2


P. 19

Put
y  3x
y  x2
(d / a)
y
x2
CCC Tam Lee Lai Fun Memorial Secondary School
AL Pure Mathematics Informal Summary
Inequalities
 Triangle Inequality
x y  x  y
 Inequalities involving absolute values
x  a  a  x  a
x a 
x  a or
x  a
 A.M.  G.M.
a1  a2 
n
 an
 (a1a2
an )
1
n
(ai  0)
1
OR
1 n
 n n
ai    ai 

n i 1
 i 1 
 Cauchy – Schwarz’s Inequality
(a1b1  a2b2 
 anbn ) 2  (a12  a2 2 
 an 2 )(b12  b2 2 
OR
 n
 n 2  n 2 
2
a
b


i
i

   ai   bi 
 i 1
  i 1  i 1 
Proof:
x  R,
n
 (a x  b )
i 1
i
i
2
 bn 2 )
0
 n 2 2
 n

 n 2
a
x

2
a
b
x

 i 
 i i 
  bi   0
 i 1 
 i 1

 i 1 
n
If
a
i 1
2
 0, a1  a2 
2
 0,   0
i
n
If
a
i 1
i
 an  0, the result is trivial.
2
 n

 n 2  n 2 
2
a
b

4

i
i


  ai   bi   0
 i 1

 i 1  i 1 
 n
 n 2  n 2 
2
a
b


i
i

   ai   bi 
 i 1
  i 1  i 1 
P. 20
CCC Tam Lee Lai Fun Memorial Secondary School
AL Pure Mathematics Informal Summary
Strategies to prove inequalities:
1. By Method of Difference/ Ratio
2. M.I.
3. By Differentiation
4. Use known results
Complex Numbers
 Definitions: i  1 ( i 2  1 )
Let z  a  bi
( a, b  R )
1.
a  Re( z ) ,
2.
z  a 2  b2
3.
z  a  bi
4.
2 Re( z )  z  z
5.
2i Im( z )  z  z
6.
z  zz
b  Im( z )
Imaginary part is not imaginary!
2
 Polar Form
principal argument
z  r (cos  i sin  )
yi
(      )
r z
or
z  r cis
  arg( z )
2
e.g. 1  3i  2 cis( )
3
P. 21
x
CCC Tam Lee Lai Fun Memorial Secondary School
AL Pure Mathematics Informal Summary
Let z1  r1 cis1 , z2  r2 cis2
z1 z2  z1 z2 , arg( z1z2 )  arg( z1)  arg( z2 )  2k
z1 z2  r1r2 cis(1  2 )

z1 r1
 cis(1   2 )
z2 r2
z
z 
z1
 1 , arg  1   arg( z1 )  arg( z2 )  2k
z2
z2
 z2 
(k  0, 1, 2)
 Geometric Relationship
w is purely imaginary  w  ik (k  R)  arg( w)  
z2
is purely imaginary
z1

z2
 ik
z1
(k  R)
z 

 arg  2   
2
 z1 
 arg( z2 )  arg( z1 )  

2
 z2  z1
 Locus Problems
z  (a  bi)  r
z  (a  bi)  z  (c  di)
(a , b)
r
(c , d )
(a , b)
General method: Put z  x  yi
P. 22

2
CCC Tam Lee Lai Fun Memorial Secondary School
AL Pure Mathematics Informal Summary
 DeMoivre’s Theorem
(cos   i sin  )n  cos n  i sin n
(n is any integers)
Roots of z n  r (cos   i sin  ) :
   2 k 
  2k
z  r cos  
  i sin  
n 
n
n
 n
1
n



k  0, 1, 2,
, n 1
e.g. Solve z 5  32
z 5  32 cis 
  2k  
z  32 cis  

5 
5
1
5
 
  2 
  4 
  6 
  8 
z  2 cis   , 2 cis  
 , 2 cis  
 , 2 cis  
 , 2 cis  

5 
5 
5 
5
5
5
5
5 5 
 
 3 
 5 
 7 
 9 
z  2 cis   , 2 cis   , 2 cis 
 , 2 cis 
 , 2 cis 

5
 5 
 5 
 5 
 5 
 
 3
z  2 cis   , 2 cis 
5
 5

 3
 , 2 cis   , 2 cis  

 5
 3 
2 cis 

 5 

 
 , 2 cis   

 5
yi
 
2 cis  
5
2cis 
x
 3 
2 cis   
 5 
 
2 cis   
 5
Hence we can factorize z 5  32 into real linear/quadratic factors
z 5  32

   
   
 3  
 3  
  z  2 cis     z  2 cis    z  2 cis     z  2 cis    z  2 cis    
 5  
 5  
 5  
 5 


  
  
 3   
 3
  z  2 cis     z  2 cis     z  2cis     z  2 cis     z  2cis 
 5   
 5   
 5   
 5




   
  3  
 ( z  2)  z 2  4 z Re cis     4  z 2  4 z Re cis     4 
  5 
  5 




   
 3
 ( z  2)  z 2  4 cos    4  z 2  4 cos 
 5  
 5

P. 23
 
  4
 

 

CCC Tam Lee Lai Fun Memorial Secondary School
AL Pure Mathematics Informal Summary
Matrices & Determinants
 Arithmetic of Matrices
 2 3  3 2   2  3 3  2 




 4 1   1 0   4  1 1  0 
Addition/Subtraction:
 2 3  3  2 3  3
3


 4 1   3  4 3 1 
Scalar Multiplication:
 2 3  3 2   2  3  3  (1) 2  2  3  0 




 4 1  1 0   4  3  1 (1) 4  2  1 0 
Multiplication:
 3 1 2   3 4  3

 

0  1   1 0 5 
4
 3 5 6   2  1 6 

 

Note: In general,
AB  BA
T
T
 2 3
 2 4
Transpose: 
 
,
 4 1   3 1
( AB)T  BT AT
 Determinants
a b
 ad  bc
c d
a b c
d
f a
e
g h i
e
f
h i
b
d
f
c
g i
d
e
g h
( expanded along 1st row )
  
* Can be expanded along any row/ column
  
  
Some properties:
1.
3
1
2
5
4
3 6
6 1 1
2.
3
1
2
1 1
5
4
3
2
4
6
1
2
3
1
2
3  21
2
3
2 5
1
2 5
1
Interchanging two rows gives negative sign
Common factor extracted from a row/ column
P. 24
CCC Tam Lee Lai Fun Memorial Secondary School
3.
3
1
6
2 5  6  2  3
4
3
4
1
3
1
4
2  2  (1) 5  2  4
4
3
3
1
4
0
0
3
4
3
1
 (3)
4.
AL Pure Mathematics Informal Summary
1
3 1
4
3
AA1  I
det( AB)  det( A)  det( B)
 det( A1 ) 
5.
1
det( A)
det( AT )  det( A)
 Inverse
Definition: AB  I  A1  B
Existence:
det( A)  0 
1
a b
1

  a b
c d 
c d
A1 exists
 d  b


 c a 
 e

1
 h
a b c 
 b
1



d
e
f



a b c  h
g h i 


d e f  b

g h i  e

f
i

d f
g i
c

a c

a c
i
c
f
g i
d
( AB)1  B1 A1
P. 25
f
T
d e

g h
a b 


g h

a b 

d e 

R2  2  R1
CCC Tam Lee Lai Fun Memorial Secondary School
AL Pure Mathematics Informal Summary
 Applications in Cooradinate Transformation
“output”
 x   a b  x 
   
 
 y   c d  y 
“input”
transformation matrix
Rotation:
 cos

 sin 
 sin  

cos 
 cos 2
Tips:

sin 2 
Reflection: 

 sin 2  cos 2 

slope  tan 
k
0
Enlargement: 

0 k 
P. 26
cos 2  2 cos 2   1
sin 2  2sin cos
CCC Tam Lee Lai Fun Memorial Secondary School
AL Pure Mathematics Informal Summary
System of Linear Equations
 Notation
 3x  2 y  z  0

e.g. 
yz 2
 2 x  y  3z  1


 3  2 1  x   0 

   
 0 1 1  y    2 

   
 2 1  3  z  1 
( AX  B )
 Finding Unique Solution by Cramer’s Rule
(   0)
3 2 1
e.g.   0
2
x

1
1
1 3
0 2 1
3 0
1
3 2 0
2
1
0 2
1
0
1
2
1
1 3
2
1
1
1

y
2 1 3

z

 & Consistency
B0
B0
unique solution
obtained by
unique trivial solution
  0  Cramer’s Rule
 Method of Inverse Matrix
(0, 0, 0)
 G. E.
no solution (inconsistent)
0
infinitely many solutions
(obtained by G. E.)
P. 27
infinitely many solutions
(obtained by G. E.)
CCC Tam Lee Lai Fun Memorial Secondary School
AL Pure Mathematics Informal Summary
 Gaussian Elimination
*
Goal: 0
0
p0
*
*
*
*
0
p
*
* 
q 
(row-echelon form)
unique solution
e.g.
p0
1
0

0
2
3
2
1
0
2
0
z 0
2
2

6
6 
y 3
2
0 
x  2  6  4
no solution
q0
e.g.
q0
1
0

 0
4
3
3
2
0
0
2
6   no solution
1 
infinite many solutions
e.g.
1
0

0
2
3
2
1
0
0
Let z  t , t  R
2

6t
6 
y
2
0 
6t
x  2  3t  (
)
2
P. 28
CCC Tam Lee Lai Fun Memorial Secondary School
AL Pure Mathematics Informal Summary
Coordinate Geometry
 Angle between two straight lines
tan  
slope m1
m1  m2
1  m1m2

slope m2
 Distance from a point to a line
D
( x1 , y1 )
D
ax1  by1  c
a 2  b2
ax  by  c  0
 Equations of straight lines
Point-slope form:
y  y1
m
x  x1
Slope-intercept form: y  mx  c
 Equations of Circles
Standard form: ( x  h)2  ( y  k )2  r 2
General form x2  y 2  Dx  Ey  F  0
Centre (h , k )  (
D E
,
)
2
2
D
2
E
2
Radius r  ( )2  ( )2  F
 Equations of Parabola
focus

( a , 0)
(a , 0)  focus
 y  2at
y  4ax or 
2
 x  at
 x  2at
x  4ay or 
2
 y  at
P. 29
CCC Tam Lee Lai Fun Memorial Secondary School
AL Pure Mathematics Informal Summary
 Equations of Ellipses
b
b (0 , b 2  a 2 )
( a 2  b 2 , 0)
a
a
x2 y 2

1
a 2 b2
 x  a cos

 y  b sin 
or
 Equations of Hyperbola
(0 , a 2  b2 )
y
b
y x
a
b
x
a
( a 2  b 2 , 0)
y
y
b
x
a
x2 y 2

1
a 2 b2
 x  a sec

 y  b tan 
y2 x2

1
b2 a 2
 x  a tan 

 y  b sec
 Family of Straight Lines / Circles
A1 x  B1 y  C1  k ( A2 x  B2 y  C2 )  0
x 2  y 2  Dx  Ey  F  k ( Ax  By  C )  0
x 2  y 2  D1 x  E1 y  F1  k ( x 2  y 2  D2 x  E2 y  F2 )  0
 Locus Problems
Let the movable point / variable point interested be (x, y).
Try to find an equation connecting x and y.
P. 30
b
x
a
CCC Tam Lee Lai Fun Memorial Secondary School
AL Pure Mathematics Informal Summary
 Equations of Tangents and Chords
( x1 , y1 )
( x1 , y1 )
( x1 , y1 )
Tangent
Equation
:
y1 y  2a( x  x1 )
x1 x y1 y
 2 1
a2
b
x1 x y1 y

1
a 2 b2
Rule: x 2  x1 x
y 2  y1 y
x  x1
)
2
y  y1
y(
)
2
x(
chord
( x1 , y1 )
chord
( x1 , y1 )
( x1 , y1 )
chord
Chord
Equation
:
y1 y  2a( x  x1 )
x1 x y1 y
 2 1
a2
b
P. 31
x1 x y1 y

1
a 2 b2