Sin (A + B) =

Sin A Cos B + COS A SMD

Sin (A-B) =
sinAcosB -
cos A so B

Cos (A + B) =
Cos A COS B -
Son A SinB

cos (A-B) = COS A COSB + Sin A Sin B

ten (A + 1) = fen A + zen B

I-ton A tenB

ton (A-B) = tan A-ten B

It ten A ten B

Week 1
 Cycle :

Tidale boleh boulang except start and end

A 3 A ,
B , C Circuit
B < D -
Bolch beuleng tapi hous berakhir sama
young
C 3 A C ,

D > Path
As
long v follow the rule True
-
:
as

~
~

A 3 A ,
B , C

B < D

C 3 A C
,

D >

~
 A 3 A ,
B , C

B < D

C 3 A C ,

~ D >
~

Degree : 2

22 23 52 43 33, 23
Edges = , ,

Nodes : 22 3 , 4, 5)
Edges
,

Nodes
f ↓
 Nodes [1 7 :

, ,
3 4
, ,
5)

Edges [1 , 33 [u:
, 13 32 4) [3 23 (2
,
, ,
5)

b
①

Tree 31 ,35
=
33/23 [3 53 32 ,
, 43

(2" - 1)
Height Max Nodes
1 I

2 3

3
7

15

Y 31

So it needs to be 5 minimum
 Binary : Base - Hexadecmal Base : 16
Octal Base
Week z
: o

Decimal Base : o

(x2 1x o xox2" + + + (x2 1x2 0x2+ 0x20

+ +

128 + 64 + 8 + 4
128 + 76

204
204

4x5+ 2x5'+ 145'43x58

500 + 50f5 +]

555 + 3 : 550

559

777 : 2 : 300 remander : 1 M: 2 : 12 remander : O

said : 2 =
1g4 O
:

12 : 2 : 6 O

194 : 2 :
97 8
6 : 2 : J
O

97 : 2 : no I 3 : 2 : / I

1100001001
48 : 2 : M 8 1 : 2 : 0 I

5 :
101

-
:
7 11l

3 : 01/

17 B

11001 : 25

25 X25 : 625

512 256128 643216 & Y 2

I O 01 11000
 6776

Works X
2300
X

ag
2
II log

2x2 x11 XI09

5)

71
-
42

1337X 2021 : 269629]

3) 2021 : 1333 : / remander: 600

1333 : 60d 1 :
remainder : 645

608 645
:
1
:
remander : 43

645 : US : 15 remander : 0
43 ↑
If
highest

2) 6776 -
> 2x7x11 220 > - 22x5X1

Y
X
23388

2Yogy
255 The smallest exponent
X
2847
511 :
2"XII
44 X
: YU
E >

Il Il
Weeken
For example

logio 100 :2
<log a
b:

102 :
100 < al = b

4
log ,-x
3 &, :

7
33()Y
z 347 :

blog y a + loy b
7

77 loga a log
o

logza logzb
7 x7

ad

ab

x-
y : 64
yx
+

= 43
x =
Y

X:2

12
 2
2 = 256

2 %
F
2 = 8

20 :
:3
: 30

log ,
x+
log y
K :g
log z(x) : (

logh(2) log 264 +

log(
by
logp (x) : i x:

logr (b)

64
log y (2) log : > (x)
x+ E =
g - log2() : <

logz (4)
39
(
- Sx
:
18 (: 6

+
A : P(htr)
A : 300 (1
+ 0 .
065)S

A : 300 (1 065)) .

= -300 A 362 3 :
362
0 065
r 6 5%
:
.
=

36
.

= .

t =
3

E
A P(H + r)
:

F (Hr)
+
=

=
log (*) log (1+ r) :

p /
:

log (f) :
log (1+r)
-
Gg v= 0,
15

A : 1, 000 , 000
to log (f)
log (Hr)
= log (1000000) 16 -

90 05: 99
.

log (1-15)
Week 5
1) zx" + Yx -
16 : 0

((x + 8) (x 2) -

2:
-

E x= 2

=
-

G ,
x : 2

2) (2 - x+ 1
11 E3
2
-
b 1 b =
uac

20
It 5
2
1 I 1-4 1 .
.
O
2 .

3) Sey : -
& (Think Logically : 11)
15

ey : 15 > X:
y -
3 + ( 5)
- : - 0

-
- ]X-5 : 15

E + y :
-
d

15+ y2 =
-

DY

y" By + 15: 0

(y 3)(y 5)
-
-

y: S y =5

4) (x + 1) (2x -

u) (ux 12) + : 0

Gc+ 1) (ox" +rus-1Ex-ud

Gr + 1) (Ox2 + &x -40)

Get1)(x2+ x 6) -

Joh +yc3-6x" + sch + > 6 =

(4 + x3 5x2" +x 6 =
-
5) (2x2 -
8)< (3x 27) -
= 0
(2x" 8)"(SX-27))
- = 0

2x28 : 0 > IX" = 8

x2 4 :

7 2 :

3) -

27 : 0 > 7: 9

6) (2-1) (2) =
0 a b"-

: (a b) (a+b)
-

3
x -
(x"+ 11) 6 -

x" 12 (x 1) (x+) =
-

(x -

1) (c)+ 1)(x2)(x+ 2) 22 22 - :
(2-2) (x( 2)
+

O
=

(E 1) (x2)(x 3)
=
-

(Polynomial Division)
1) (x + 2) 1-6 11-6
(2 +
=0

(x 3) -

I 1
=

56 +

(x 1) + (x + 2) = 0 1 -

560
T: - Th : L
2 -

5x + 6

1) =x" + 13) -
2

(ax 1)(x 2) - +

xif 2: -h

gx" y8s + bu
(3x -
0 ((xx 0) -

x =
?
x - 2(ti

b1 b2- you

29

M .
2

:I
Z
6
1) f(x) 5x" Ox +
=
-
11x" -

17-35

201 24
= -
+ 22x -

17

2) fas : (2x1)5 (2+ 2)

f() =
10 (2x-1)" (x + 2)" + &xbi + 2) · (2x -

175

f'll) = 10 81 .
+ 216 ,

=8 10 + 216
= 1026

3) f(x) : e

:
125(x" g) +

: (x g) +
-
2x

: 125x(x" +
9)

u) fac <c :

f' (x) :
checks

5) f() : 40 If : 15

"
f(x) : 40 ( ++ 1) 20
i5
"
f 'G2)
:

= 20 (t+ 17 15-1
 f(u) (16-1673 fix) 60(210)
<

2) fix) (2" 10)

:
-
:
=0 :

f (2) 62 (22-16)
:
~

f (0) : - 163 :
- 4096
iv
(Cai TOP)
- L
flu) (10 16) f"
(2 16)
O Give u Vi
60
:
0 :
-
= : .

(20
222 16 : 0
-
(coriy-value) :
6(2 16)" + 24x2 (x2-16)
-
4, 0 , 4

2 : 14 6x(" - 10)" 0 f(u) 6 (0) :

"
+ 24 . 16 (0)
0
, -4096 , 0
"

Stationary points : - 4
,
0 , 4 (2 16) : 0 :
0

f(0) 6 (-16)2 + 24
2
16 02 (02-15)
Cari-value)
ic -

: 0 = .

72 : 14 =
6 256 +
. 716) f l) > 0 = local

(lawx-value)
min

=
1536-16 :
1520

2) +(2) (2? 0) :
2
f(u) :0

fox) : 2 (2-0) . 2"In (2)

uv

2(2-0) :
0 foc) : 2 (2- D) .h (2) u 2 8 zh()
: = :

f(x) a
2 8 :0 f "( [uvur'] 2 h(2) 2 h (2)
:
-
: 2 vi : :
3
.

f(x) :
ah (a)
2= & : 2 (2h(2) 2h (2) + ((2x D)
-
2h2(27)
g
. .
.

=S :

2[(hm))" + ((ex 8) 2 :
.
22(2))] Second trm =

< First Second Term

local min f(s) : y: (2"-d) Term

: S First term : 2 . (23h(27) "Do : local min

f(x) :x- coxerhe) f(3) -lox + 12 h(s)

:
f () : 2x 10 -
+ 122

f(x) :
2x 10
-
E :
-
21 + Rh (3) : -
7 01.
f (2)
"
: 2 - 12x

0 : 2x-10 +
I f(e) : 4-20 + 12 hacks f "(3) : 2-g
3 ,2
0 = 2x2 103 + 12 =
-
16 + 12 h (2) : -
7 68
.

· - Local minimum

-
7 81 -7 60
.
0 :
(2x 0S(x 2) - -

2 . (y-value) &"(2) : 2 -
E : -1 1 Local maximum
,
.

7 3 X: L
(Find (
:
3 .
the Top

Local mi , Local May 1 .

(X-value)

u) f(x) :+ - G + 3:0

f (x) = 0 (x2 -x+ 1) -

3(2x 1)
-

2 =
2
alb-abl (x" -x + 1)
Quotient :

Rule b2
(x + 3
:
=

alo
& (2)
-
a =3
= O
(c-2 + 1
b:
b' = 2x 1
-

(x2 - x+ 1)2
- = max value
 5) x +y :
14 y = M-x xi 7 .

P = x Y -

: 14-7 :7
P x (( x) Product 7x7
Y
= -

: :

- x
↑ P : = "th
Parabola opens = Max
downward

[2(2) 5() 2(e)] (i) + -

·

[( 27 (i) (177· (ii)

.
+ +

(i) (i) : 43 20 260 · = - + + :

237

7) a(y) b(z) (ii + =

(49) +=
49 + 76 : =
47 a - zb = =
2b 13 xY
z
-
=
a 13
=

a =

=
-

49 + 76 47
: =

ya-db : -52 a =
- k +
-
156 : 5 a = -

39
b : a
= -

37

9) v = x + y" + z?

- C- E)"+ 352
=
249 + 25

=
6
&

Dot product :
Magnodes :

U . V: (1 -20)+
.

(3 .
12-105) Ini Milities)
-
-

26 + 36 305 - - No = 676 + (144-2405 + 300)

: 10-301 =
676 + (444-mov)

·moss

cost : U U
.
10-305 = 1120-205
-
.
I

120 Cultu No 1120-mot .

5x + -
15 : 3 : 0

52 + 45 :0

x= g
=

Product :
0

Cosine Rule Sin Rule

d2b" + c-2bc cos a a
a Gu + 40"-2
: .
64 .
40 ·
cos60

= 40984 / 000
-

2566

:
1156 =
33 7 .

by
Too 60

* *

xl . x cost -
y SMA
* *
Y' x So -y 100

c=
( 2)(E) = -
( -
1 2)(i)
(2 5) (ot vs)
-
-
·
-

2
-
Y
·
(3 -B) z 5 + +

2
=

y (v : -

2) (2) ( 1 25)(E ) + - -

:
( 1) + ( - 3) -
-

- Y
 A :
( %)
: acc (1)
·

I (in)

(
:
z4

(E 7417

x = 29 + 1x x -

+ x 23 +

AxiB AB
6 -7
> :

- :

Ex-39 + Yx2) + Ej Y Y
-

y:

:
- 42.
c: - x
-
39 +
yx23
-
+ =

yX -

-
7
~
15-30 -Y
Y 4

B B: .

(4) (43) (i) :

&G

B2 A :
(2) (1) Logi 101) :
-

= B B A
.
.

160

&9 -gi

-
103
 ais bi =
2

a) M100u : a+ (n 1) b-

b)S1000 : (Unta)
=
3+ (ggg) -
2 = 500 (2001 + 3)
= 3 +
(1990) = 500 (zoon)
=
2001 = 1002 000.

= 12 b :
a) U ath-1) b :
b) Sco : (Un a) +

= 12 + 9 x
1 5(2 12)
= +

= 12 + & 5 +) =

- 5() =

c) S : a + or + witar

= 12 + 6+ 3 E
-

If there aren
people ,
so the number of people that will do

the handshake n-1 but what happens if there is what

people
is

have a handshake? So n-1 and handshake -o can't be in the

same
group
 d!
Permutation Problem aPb :
=
2016
(a b) ! -

:
26 ! 26 ! 26415X24x23 x 22x21x10x19X1DX17
:

26 letters total (20-10) : 161

:
19275223968000

From (0 0) .
- >
(20 10) ,
:
20 to the right + 10 to up =
30 total steps

Combination Permutation
d!
a (b :
(a b): r !
-
apb :b : Combination Order : doesn't matters

3000 : 30045015 Ex A group

:
of Alice Bob, Charlie
, - Bob, Alice ,
chalie

or Permutation Order matters

:

so (20 : the same answer Ex Passcode ABC # ACB

: