CORE PURE 1
Chapter 1 :
Complex numbers
i F1
O
=
*
- =
a + bi (a DE(R)
, ,
(zEK) · if z = a + bi ,
z = a-bi (complex conjugate
iP = 1 i = -
1
Re(z) =
'Real part of z' = a · if roots of a quadratic equation are a and B ,
then 1z-x)(z B) - =
0
is = -
i
1m(z) =
'Imaginary part of z = b z2 -
(x B)z + +
xB =
0
Chapter 2 :
Argand diagram
Im
A
- =
x +
iy (cartesian form / Z z, = r, (C030 , + iSiNO , ) ,
Ec =
Uc(10902 + iSinOc)
x
| z| 6x
· U2(109(0 (0 , (
+ z , Ec
=
y modulus, r = U, 82) i sin +
O2)
·
+ +
,
N O
arg(z)
=
tan"(π argument , O
> Re ④
= (10610 ,
-
01) + i sin(8 ,
+ O
range : ↑ < arg(z) => ↑
principal argument
-
always sketch !
z , z2) =
/z //Zal
,
arg(z , zc) =
arg(z ,
) +
arg(ze)
z r (c0S0 isinO) (modulus-argument form / 1z , 1
arg/E)
=
Ei
+
= =
arg1z , ) -
arg (zz)
↓ Zal
r (cos0- i sino) =
(cos1 0) -
+
isin) -
Oll
3 3
z ,
= = ,
arg(z , 3) =
3 arg(z , )
loci =
a set of points
e. g
. (i) Iz -
z, ) =
P circle (ii) Iz-z , 1 =
1z-Ec1 perpendicular bisector (iii) arg(z-z , ) =
0 half-line
1z -
5 -
3i) =
3 1z -
31 =
/z + il
arg(z + 3 +
2i) =
arg(z -
( 3 2i))
- -
=
) the
/
midpoint (70,
Im =
1z -
15 +
3i)1 = 3 :
Im
-
X
3
IM
-
> Re
(x 5) (y 3) q
/ -
+ - = O
3 3
..
MH = -
Re X --
2
O 1 3 -
.
-
2)
3
-
y (- 2 )
- = -
3(x 2) -
3
-
·
3( 4
y
= -
+
> Re
O j
>
determining minimum and maximum of arglz) and 121 :
in argand diagram
regions
1z -
(4 + 3i)) = 3 z -
4- 21/2 ,
1z -
4K/z -
61
, 01 arg(z -
2- 2i) !
Im Im
/
·
min arg(z) =
0
h
sin (3) 1 29 rad 02arg(z-2-2i)
arg(z) 2
=
·
max = x .
1z -
4/4/z -
6
3
·
X
5
d =
O 3 z -
4- 2i22
Min
·
(z) =
5 -
3 =
2 (d -
r)
> Re -
2
i
X X
·
max(z) = 5 + 3 =
0(d +
r)
Re
d ↓
Chapter 3 : Series Chapter 5 : Volumes of Revolution
"
1 = n
V= 1
Around 11-axis :
π/ay Arth
· =
N
V =
d cylinder
r =
(n(n + 1)
V= 1
Around y-axis :
·
cone
=
5 TU'h
, =
5n(n +
1)(2n 1) +
V =
πfa) dy
2
P
N
3
=
+ (n + = ↑
V= 1
, Chapter 4 : Roots of Polynomials
Polynomials Ex =
- [: xB =
f & BU =
-
* BUS =
a) + bl + C x +
B xB
all + b( + Cx +
d x +
B +
f B +
BU +
UX XBG
a)" +
b) + C11" + dx +
2 x +
B +
8 + f B +
Bu +
08 + 8x +
B8 +
xuaB8 +
BUS +
USx +
XBS NBUS
Identities :
Finding equation of graph / transformed roots :
x B + = (Ex) -
22 xB e new roots of 1-1 B-1
.
g. ,
x+ B +
0 =
(Ex)" 2xB -
let W = ) -
1
x +
B3 = (2x)" -
3 BEx x =
W + 1
x B3 83 + + =
(Ex)3 -
3[XBEx +
3 XBU then substitute back into original equation
x B + 4 02 +
B38
:
:
(EXB)" -
2 B8 Ex
Chapter 6 :
Matrices
Matrix multiplication -2x) 2x3
multiplication is commutative.
+
: -
NOT
(i)(38 -Y ) =
(ii) AB # BA except Al =
IA =
A
(6 % ) or
-
3 x 0 +
4 x2
associative.
<
Identity matrix I =
2 rows ,
2 columns x 2 rows ,
3 columns -
multiplication is ,
no .
of columns in matrix 1 =
no .
of rows in Matrix 2 (AB)C =
A(BC)
Determinant i Inverse :
M 1,I
=
2 x 2 matrix A =
(9) 2 x 2 Matrix (AB)" =
B A-
det(m) =
=, & = ad-bc A
-
=
defcas ( 9 - ( _
(AB)(AB)" =
1
if det(M) = 0
, singular Matrix, no inverse
if det (M) # 0
, non-singular Matrix ,
has inverse A =· a 3 x 3 Matrix 4 steps !
1) calculate det(A) =
((0 -1-1) -
3/0 -2) +
1(0-8) = -1
P of
9
M 3 x 3 Matrix 2) matrix of minors
,
(a
=
13 I
i
0 4
9 h i e. g. minor of 2 in
- O
M -
-
det (M)
a Ye if
=
= A
ef
n i
-
b
df
+ C
aC
9 i gh
h i
9 - - -
minor of a
minor of b of
minor C 3) changing of signs cofactor , c = I
Simultaneous equations :
4) transpose ,
CT =
i switch rows and columns
&
x 6y 22 21
e g
-
+ -
=
. .
60 i
-
I
6x -
2y
-
z = -
16
Y
=
24
5) A" =
det(A) CT
-
2x +
3y + 52 =
24
if A 4 = V
,
then ( = Av
Consistency :
line
i
. e. same plane
Step 1 :
Find det/M).
if det(M) O
,
then one single solution
.
> if det (M) =
0, then follow Step 2 .
Step 2 :
compare the equations of the planes.
*
If 2 of parallel
i
> (i) or more them are , they
. e. not same line plane
look like these .
g )
5)
le . .
3x-24 +
2 =
parallel
6x -
44
+
22 =
9
Chapter 1 :
Complex numbers
i F1
O
=
*
- =
a + bi (a DE(R)
, ,
(zEK) · if z = a + bi ,
z = a-bi (complex conjugate
iP = 1 i = -
1
Re(z) =
'Real part of z' = a · if roots of a quadratic equation are a and B ,
then 1z-x)(z B) - =
0
is = -
i
1m(z) =
'Imaginary part of z = b z2 -
(x B)z + +
xB =
0
Chapter 2 :
Argand diagram
Im
A
- =
x +
iy (cartesian form / Z z, = r, (C030 , + iSiNO , ) ,
Ec =
Uc(10902 + iSinOc)
x
| z| 6x
· U2(109(0 (0 , (
+ z , Ec
=
y modulus, r = U, 82) i sin +
O2)
·
+ +
,
N O
arg(z)
=
tan"(π argument , O
> Re ④
= (10610 ,
-
01) + i sin(8 ,
+ O
range : ↑ < arg(z) => ↑
principal argument
-
always sketch !
z , z2) =
/z //Zal
,
arg(z , zc) =
arg(z ,
) +
arg(ze)
z r (c0S0 isinO) (modulus-argument form / 1z , 1
arg/E)
=
Ei
+
= =
arg1z , ) -
arg (zz)
↓ Zal
r (cos0- i sino) =
(cos1 0) -
+
isin) -
Oll
3 3
z ,
= = ,
arg(z , 3) =
3 arg(z , )
loci =
a set of points
e. g
. (i) Iz -
z, ) =
P circle (ii) Iz-z , 1 =
1z-Ec1 perpendicular bisector (iii) arg(z-z , ) =
0 half-line
1z -
5 -
3i) =
3 1z -
31 =
/z + il
arg(z + 3 +
2i) =
arg(z -
( 3 2i))
- -
=
) the
/
midpoint (70,
Im =
1z -
15 +
3i)1 = 3 :
Im
-
X
3
IM
-
> Re
(x 5) (y 3) q
/ -
+ - = O
3 3
..
MH = -
Re X --
2
O 1 3 -
.
-
2)
3
-
y (- 2 )
- = -
3(x 2) -
3
-
·
3( 4
y
= -
+
> Re
O j
>
determining minimum and maximum of arglz) and 121 :
in argand diagram
regions
1z -
(4 + 3i)) = 3 z -
4- 21/2 ,
1z -
4K/z -
61
, 01 arg(z -
2- 2i) !
Im Im
/
·
min arg(z) =
0
h
sin (3) 1 29 rad 02arg(z-2-2i)
arg(z) 2
=
·
max = x .
1z -
4/4/z -
6
3
·
X
5
d =
O 3 z -
4- 2i22
Min
·
(z) =
5 -
3 =
2 (d -
r)
> Re -
2
i
X X
·
max(z) = 5 + 3 =
0(d +
r)
Re
d ↓
Chapter 3 : Series Chapter 5 : Volumes of Revolution
"
1 = n
V= 1
Around 11-axis :
π/ay Arth
· =
N
V =
d cylinder
r =
(n(n + 1)
V= 1
Around y-axis :
·
cone
=
5 TU'h
, =
5n(n +
1)(2n 1) +
V =
πfa) dy
2
P
N
3
=
+ (n + = ↑
V= 1
, Chapter 4 : Roots of Polynomials
Polynomials Ex =
- [: xB =
f & BU =
-
* BUS =
a) + bl + C x +
B xB
all + b( + Cx +
d x +
B +
f B +
BU +
UX XBG
a)" +
b) + C11" + dx +
2 x +
B +
8 + f B +
Bu +
08 + 8x +
B8 +
xuaB8 +
BUS +
USx +
XBS NBUS
Identities :
Finding equation of graph / transformed roots :
x B + = (Ex) -
22 xB e new roots of 1-1 B-1
.
g. ,
x+ B +
0 =
(Ex)" 2xB -
let W = ) -
1
x +
B3 = (2x)" -
3 BEx x =
W + 1
x B3 83 + + =
(Ex)3 -
3[XBEx +
3 XBU then substitute back into original equation
x B + 4 02 +
B38
:
:
(EXB)" -
2 B8 Ex
Chapter 6 :
Matrices
Matrix multiplication -2x) 2x3
multiplication is commutative.
+
: -
NOT
(i)(38 -Y ) =
(ii) AB # BA except Al =
IA =
A
(6 % ) or
-
3 x 0 +
4 x2
associative.
<
Identity matrix I =
2 rows ,
2 columns x 2 rows ,
3 columns -
multiplication is ,
no .
of columns in matrix 1 =
no .
of rows in Matrix 2 (AB)C =
A(BC)
Determinant i Inverse :
M 1,I
=
2 x 2 matrix A =
(9) 2 x 2 Matrix (AB)" =
B A-
det(m) =
=, & = ad-bc A
-
=
defcas ( 9 - ( _
(AB)(AB)" =
1
if det(M) = 0
, singular Matrix, no inverse
if det (M) # 0
, non-singular Matrix ,
has inverse A =· a 3 x 3 Matrix 4 steps !
1) calculate det(A) =
((0 -1-1) -
3/0 -2) +
1(0-8) = -1
P of
9
M 3 x 3 Matrix 2) matrix of minors
,
(a
=
13 I
i
0 4
9 h i e. g. minor of 2 in
- O
M -
-
det (M)
a Ye if
=
= A
ef
n i
-
b
df
+ C
aC
9 i gh
h i
9 - - -
minor of a
minor of b of
minor C 3) changing of signs cofactor , c = I
Simultaneous equations :
4) transpose ,
CT =
i switch rows and columns
&
x 6y 22 21
e g
-
+ -
=
. .
60 i
-
I
6x -
2y
-
z = -
16
Y
=
24
5) A" =
det(A) CT
-
2x +
3y + 52 =
24
if A 4 = V
,
then ( = Av
Consistency :
line
i
. e. same plane
Step 1 :
Find det/M).
if det(M) O
,
then one single solution
.
> if det (M) =
0, then follow Step 2 .
Step 2 :
compare the equations of the planes.
*
If 2 of parallel
i
> (i) or more them are , they
. e. not same line plane
look like these .
g )
5)
le . .
3x-24 +
2 =
parallel
6x -
44
+
22 =
9
