High school Math syllabus

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43

IIIIIIIV

III

IIIIV

AB B A

44

I4

1.1

1.

1.2

1.3

1.2

2. 2.1

2.2

1.1 1.2

1.

1.3

1.3 4

2.

2.1

2.1

3. 3.1 3.2 n a

3.3

3.1

4.

4.1

4.1

1. 1.1

2. 2.1

3. 3.1 3.2

3.2

4. 4.1

-

45

5.

5.1

5.2 5.3 5.4

5.1

2

46

II4

1. 1.1 1.2

1.1

1.2 / I

2. 2.1 1.

1.1

1.2

1.3

1.4

2. 2.1 2.2

2.1

3. 3.1

3.1

1.

1.1

2.

2.1 2.1

3.

3.1

1.

1.1 1.1

2. 2.1

2.1

1.

2.

-

47

III4

1.

1.1

2.

2.1

2.2

2.1 cot, sec, csc II

3.

3.1

4. 4.1

4.1

5. 5.1 5.2

5.1

1.

1.1 1.2

2. 2.1 2.2

3.

3.1 3.2

3.2

1.

1.1

1.2

2.

2.1

2.2

3.

3.1

3.2

48

IV4

1. 1.1

1.1

2.

2.1

2.2

3.

3.1

4.

4.1

4.2

4.3

4.3

1. 1.1

2.

2.1

2.2

3.

3.1 3.2

1.

1.1

1.1

2. 2.1

3. 3.1

4.

4.1 4.2

4.2

1. 1.1 2. 2.1

3. 3.1

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49

1.

2. 3. 4.

5.

1 2 3 6.

7.

8.

50

IIIIII IV

I

2

1. 1.1

-

51

1.2

2 12

121 +=

2a a=

2a bab +

1.3 ( )3a b+ ( ) ( )2 2a b a ab b+ + ( ) ( )2 2a b a ab b + + ( )2a b c+ +

( ) ( )21 1x x x + + ( ) ( )1 2x y x y+ + 3 3 3 2 2 23 ( )( )x y z xyz x y z x y z xy yz zx+ + = + + + +

1 21 1 12

aba b

a b

= + +

2 2 2 2

1 c

a ba bc c

= + +

5 2 6 3 2+ = +

2 2 12x x x x + + = + 2. 2.1

,a b ,a b 2:3 x 2.2 a b a b+ + 3 2x < 1 1x < 1 2x< < 1 2 3x x <

ncxy = n=1234

52

( )ax ( )ax ( )bx 2 1x + 2 1x x+ + ( )xf ( )ax ( )f a ( )ax ( )xf ( )ax ( )xf a ( )xf ( )ax ( )bx ( ),a ( ),b ( )xf ( )ax ( )bx

I n

1. 1.1 ( )y f x= ( )0y mx b m x x= + = 0, ,m x b m y x 1.2 22,32)( 2 ++= xxxxf 2 4x x +

-

53

( ) ( )y c x a x b= 2y ax bx c= + + ( )2y a x h k= +

1.3 ny x= 1, 2,3, 4n = [ ]1.5,1.5

n ny cx= c ny cx= ( )ny c x h k= +

2. 2.1 ( ) ( )1 2 1n n n n nx a x x a a x a + + + = L n = 2,3,4 3322 ))(( axaaxxax +=++

( ) ( ) ( ) ( )2 33 22 5 6 3 1 1 1f x x x x a b x c x d x= + + = + + + , , ,a b c d ( )1.01f

( ) ( ) ( )( ) ( ) ( ) ( )3 22 5 6 3 1 1 2 1 2 3f x x x x a b x c x x d x x x= + + = + + +

, , ,a b c d ( )xf ( )ax ( )bx ( )( ) ( )( ), , ,a f a b f b f a,b f ( ) ( )( )( )f x q x x a x b= ( ) ( ) ( )1,1 , 2, 3 , 3, 7 ( ) ( ) ( ) ( )1 1 2f x a b x c x x= + +

, ,a b c 12

f

54

)8,13(),5,12(),3,11(

)1213)(1113()12)(11(8

)1312)(1112()13)(11(5

)1311)(1211()13)(12(3)(

++

= xxxxxxxf

( )11.5f ( ) ( ) ( )1,1 , 2, 3 , 3, 7

( ) 2f x a bx cx= + + , ,a b c IV

3. 3.1

2 0ax bx c+ + =

2 5 3 0x x+ + = 2 2 + 3 3 + 1 1 3

1 2 2x x+ =

3.2 n a

nx a= a>0 ( ) 3 22 3 4f x x x x= + + + n 3.3

( ) ( ) ( ) ( ) ( )11 2 21 1 1 mk s sr rk m mf x k x a x a x b x c x b x c= + + + +L L

( ) 4 3 25 21 30 9 7f x x x x x= + + 2x i= + 4. 4.1

( ) ( ) ( )21 2 4 0x x x + > ( ) ( ) ( )3 21 2 1 0x x x x + + > 3 1 0x > 4 22 3 0x x > 1 0

x< 1 1

1x

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55

10 , 0.1,0.2, ,0.9x x = L

0, 1a a> 2x 10x

logax b= xa b= 10 ( ) yxxy logloglog +=

( )log / log logx y x y= ( ) xx loglog =

10 1log log

logax x

a=

logy x= logy x=

1. 1.1

n 1 513 6210 10 10 =

1 1 13 3 32 3 6 =

1 13 210 10<

1 1 1( ) ( ) ( )a b ca b c a b c a b c a b cx x x 2 2 3xa = +

3 3

6

x x

x x

a aa a

+

+ +

56

2. 2.1 2x

10x 3. 3.1 ( ) yxxy logloglog += ( )log / log logx y x y= ( ) xx loglog =

( )( ) log loglog log , log log log , x xm b an a a b aa nb b b c c a bm= = = 3.2 1log log

logax x

a=

10 4. 4.1 xy a= logax y= 1log loga x x= log a =

2

a bab + log log log2 2

a b a b+ + a b= 5. 5.1

5.2 1002 ( )101.18

5.3 5.4

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57

II

/ 1. 1.1 1n na a d+ = + 1n na ra+ = 1n na a n+ = + 21n na a n+ = + ( )1 1n na n a+ = + 1.2 / II 2. 2.1

( )1 1 1

n n n

k k k kk k k

a b a b= = =

+ = + 1 1

n n

k kk k

ca c a= =

=

1

11 2

n n

k kk k

a a+

= =

=

1

n

kk

= 2

1

n

kk

= ( )1

11

n

k k k= +

1. 1.1

58

1.2

1.3 |S| S

10 8 3 5 7 21 4 1 3 6

10+8+3+5+7+21+4+1+3+6 = 68

51 51 50 1.4 A B |A B| = |A| + |B| A, B A B |A B| = |A||B| A, B, C

1 |A B| = |A| + |B| |A B| 2 |A B C| = |A| + |B| + |C| |A B| |B C| |C A| + |A B C| Principle of Inclusion and Exclusion PIE

Inclusion and Exclusion PIE 2. 2.1 n n!

1 n 1 n n!

n k

-

59

)!(!kn

nPnk = 1 k 1 n nkP

50 1

n k nk

1 k 1 n nk

10 310 1 0 A={0,1} n

An = A A . . . A n 2n 2.2

!!( )!

nk

nCk n k

= n k

!!( )!

nk

nCk n k

= k 1 n

nkC

n k

1n kkC+

k 1 n

1n kkC + n k x1 + x2 + + xn = k

1n kkC + 3.

3.1 (x + yn

5

2 1 1xx

+ + x

60

1. 1.1

3 2. 2.1

50 3. 3.1 ABC A 40%B

30%C 30% A 5%B 10%C 8% A

90% 90% 90% 2%

1. 1.1

1

1 nk

kx

n

=

= ( )1

1 2n

nx x xL

( )21

1 nk

kx

n

=

= ( )2

1

1 nk

kx x

n = 2 x=

ix ix

-

61

2. 2.1 0

( ) , , 1,2, ,k kx y k n= L , ( )21

( )n

k kk

e r y rx=

= r 2 2 2 2

1 1

( ) ( 2 ) = 2( ) 1n n

i i i i i ii i

e r y x y r x r r x y r= =

= + +

1

n

i ii

r x y=

= y rx= ( )e r

II

1. 2.

III

62

360 r 0 360

X

1. 1.1

-

63

O

( )2 2cos ,sinB ( )1 1cos ,sinA

1,0

2. 2.1

X 150 , 30 ; 225 , 45 ; 300 , 60 = = = = = =

cos , sin

2.2 r, 0 , 0 360r < < 3.

ABC B ( ), 0c C ( )cos , sinb A b A ABC 1 sin

2c b A

ABC 1 sin2

a c B

ABC 1 sin2

a b C ABC A B ( ), 0c C

( )cos , sinb A b A ( ) ( )2 22 2 2cos sin 2 cosa b A c b A b c bc A= + = +

4. ( )2 1 1 2 1 2cos cos cos sin sin = + 4.1

A,B ( ) ( )1 1 2 2cos ,sin , cos ,sin OAB ( )2 1 1 2 1 2cos cos cos sin sin = +

A B D

C

( )2 2 2 22 22 2

sin cos

2 cos

BC CD BD b A c b A

b c bc A

= + = + = +

64

( )1 2cos + ( )1 2sin + ( )1 2tan + cos2 sin 2 cos

2 sin

2

cos15o 5.

5.1 5.2

1. 1.1 1.2 2. 2.1 2.2 ax by k+ = 3. 3.1 3.2

2

-

65

a

vb

v

1 1 2 2cosa b a b a b = +v v

1 2 2 1sina b a b a b =

v v

1 1 12 2 2

a x b y ca x b y c

+ = + =

c xa yb= +v v v av bv 0 a

vb

v

1. 1.1

1.2

1 2 0 1 ,0 12 1

x y x y +

45

12

21

2. 2.1

66

ar

br

( )1 2,b b

( )1 2,a a

( ) ( )1 2 1 2, , ,a a a b b b= =r r ( ) ( ) 2 22 21 1 2 2 2 cosa b a b a b a b + = + r r r r 1 1 2 2 cosa b a b a b + =

r r

,a b

r r a br r

a b b a = r r r r ( )a b c a c b c+ = + r r r r r r r 2a a a =r r r ( ) ( )2 2 2 2 2ax by a b x y+ + + a b a b+ +r rr r 4, 51, 2

2.2 3. 3.1

1. O ABO0,0 ( )1 2,A a a ( )1 2,B b b 2. ,a b

r r

( )( ) ( ) ( )

2 2 2

22 2 2 21 2 1 2 1 1 2 2 1 2 2 1

1 1 1sin 1 cos2 2 2

1 12 2

OAB a b a b a b a b

a a b b a b a b a b a b

2 = = =

= + + + =

v v v v v v v v

1 1 1 2 1 12 2 1 2 2 2

a b a a b aa b b b b a

= = 1 1 1 1 1 1 12 2 2 2 2 2 2

a c b a b c ba c b a b c b

+ = ++

1 1 1 1

2 2 2 2

ca b a bc

ca b a b=

D C

( )1 2,A a a ( )1 2,B b b

O x

y

( )1 2 2 2 1 1 1 2

1 2 2 1

1 1 1( )( )2 2 2

12

OAB b b a b a b a a

a b a b

= + +

=

-

67

3.2

1 1 12 2 2

a x b y ca x b y c

+ = + =

cv

av

bv

av

bv

0

IV

0 0 1. 1.1 2. 2.1 2.2

( ) ( ){ }1,2,3 0,1, 1 | 0 1t t+ ( ) ( ){ }1,2,3 0,1, 1 | 0 1,0 2s t s t+ 3. 3.1 ( )1 2 3, ,a a a ( )1 2 3, ,b b b

1 1 2 2 3 3 cosa b a b a b a b + + =v v

av

bv

1 1 2 2 3 3a b a b a b+ + 4. 4.1

68

( ) 2 2 22 2 3 3 1 1 22 2 22 3 3 1 1 2

| | | |a a a a a a

A a b a bb b b b b b

= = + +v v v v 4.2 , ,a b c

vv v )( cba vvv

4.3

2

2

2

1 1 1

a ba caab b cbac bc c

++

+

baccbcabaacb

++

+

1. 1.1 2. 2.1 1: 1

2 3x zL y = + = : 2 3E x y z+ =

2.2 3. 3.1 3.2

1 1 1 1

2 2 2 2

3 3 3 3

a x b y c z da x b y c z da x b y c z d

+ + = + + = + + =

d

uv a

vb

v c

v a

vb

v c

v

0

-

69

1.

2. 2.1 Caley-Hamilton 3. 3.1 4. 4.1 4.2

1 22 1

A = A 1

0

01

12

21

12

34

12

21

70

1. 1.1

( )22x y c y c+ - = + 2. 2.1

( ) ( )2 22 2 2x c y x c y a + + + + =

x x hy y k= =

( ) ( )2 22 2 1

x h y ka b + =

2 2

2 2 1x ya b

+ =

2 2

2 2x y ka b

+ =

,x yx yt t

= = 2 2

22 2

x y ta b

+ = 2 2

2 2 1x ya b

+ =

2 2 0ax cy dx ey f+ + + + = ( ) ( )2 2 2 2a x h c y k ah ck f + = +

3. 3.1

( ) ( )2 22 2 2x c y x c y a + + + =

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High school Math syllabus