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61 24 // 2

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71 54 // 2

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91 15 // 2

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02 15 // 2

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32 36 // 2

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2 // 70 24

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1-

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3

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---------------------------------------------------------------

2 // 70 25

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5-

f ( x ) x x D Rf

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---------------------------------------------------------------

62 07 // 2

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2 // 70 27

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g( kx ) kx , k.g( x ) k( x ) kx k

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f ( x ) mx b , f ( ) m( ) b m bf ( x ) f ( x ) f ( ) f ( )

m bm( ) b m b m

m bx

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1 0 4 4 0> < < a a.

----------------------------------------------------------------

. >) t (h ) t (g y -8

t t t t t t

t ro t ) t (t t t P> + + > +

= = = = + =

2 2 20 8 2 01 81 3

24 0 0 4 2 8 2

. 4 0<

---------------------------------------------------------------

13 59 // 2

---------------------------------------------------------------

-1 x

yxy

=

2 1 0 1 2

1 1 24 2 1

2 4

)

3 2 + = yx 3

4 2 = yx 4

----------------------------------------------------------------

x

yxy

=

2 1 0 1 2

1 1 24 2 1

2 4

)

25 =+ yx 5

24 = yx 4

----------------------------------------------------------------

x

yxy

=

2 1 0 1 2

1 1 24 2 1

2 4

)

x

yxy

=

2 1 0 1 2

1 1 39 3 1

3 9

x

yxy

=

2 1 0 1 2

1 1 552 5 1

5 52


---------------------------------------------------------------

23 59 // 2

---------------------------------------------------------------

513 + =x y . 6 y x -2 5

----------------------------------------------------------------

() (( ) ( ( -3

() ( ( ) ( ( (

----------------------------------------------------------------

1 1< = a) y x -4 3

. (

----------------------------------------------------------------

. 3 =x y 3 = yx -5

----------------------------------------------------------------

-6 x

y/y x , y/ x

= = =

1 0 1

8 016 9 8 0 9

9 19

)

x

y/ /) ( y x , ) ( y/ x

= = =

1 0 17 1 1 1

11 0 7 14 41 4

4

)

x

y/y x , y/ x

= = =

1 0 1

3 012 01 3 0 01

01 101

)

x

y/) ( y x , ) ( y/ x

= = =

1 0 13 1 1 1

102 3 101 011 01

01

)


---------------------------------------------------------------

33 59 // 2

---------------------------------------------------------------

) 6 = yx. 6 y x -7

) 3 2 =x y. 4 y x

) 1 3+ =x y . 3 y x

. y x x

1 =) ( y2

)

3 y x 4.

x013 46 =) ( y

4 3 )

)

.

----------------------------------------------------------------

) ) ( ) ) ( )-8

) ) ( ) ( ))

----------------------------------------------------------------


---------------------------------------------------------------

2 // 115 34

---------------------------------------------------------------

1- abd ( ab )d ab d a b dlog log log log log log logc c c c c c c

= = + = + +

----------------------------------------------------------------

2- log log( ) log log log log log log= = + + + + = 53 3 3 3 3 3 3 3 3 3 3 5 3 ----------------------------------------------------------------

3-

nnn a a a aa a.a.....alog log log log ... log n log

c c c cc c

= = + + + =

upcurlybracketleftupcurlybracketmidupcurlybracketrightupcurlybracketleftupcurlybracketmidupcurlybracketright

----------------------------------------------------------------

4- log log log= = = =31000 1010

3 3 1 310 1010

)2 log log log= = = =2100 1010

2 2 1 210 1010

) 1

log l og log log log+ = = = = =24 25 4 25 1010

2 2 1 210 10 10 1010

) 3

log log log log log+ = = =24 4 4 22

2 3 3 610 10 10 1010

) 4

----------------------------------------------------------------

5- log log log log log log m n= = + = + = +2 218 2 2 32 3 3

2 210 10 10 1010 10

) 1

log log log log log log m n+ = + = + = +5 332 27 2 32 3

5 3 5 310 10 10 1010 10

) 2

----------------------------------------------------------------

6- log log log ( )/ /= = = =532 22

5 5 0 5 2 54 44

----------------------------------------------------------------

7- log log log ( )= = = =416 22

4 4 1 42 22

) 1

log log log log ( )

= = = = =

112 772 2 2 1 249 7

777 7

) 2


---------------------------------------------------------------

2 // 115 35

---------------------------------------------------------------

log log log log ( )= = = = =3

2 3 23 3 38 2 2 1222 2 22 2

2

) 3

log log log log ( )= = = = =54

327 333 3 1 32

3 333

) 4

log log log log log ( )+ = = = = =225 4 100 1010

2 2 1 210 10 10 1010

) 5

log log log log log ( )+ = = = = =34 250 1000 1010

3 3 1 310 10 10 1010

) 6

log log log log ( ) = = = = =24 125 31000 10 3 10 3 1 31

29

) 7

( )

log log log log log log log log= = = = = 33 4 164

4 216 100 2 10 4 2 225 10025

101010

) 8

----------------------------------------------------------------


---------------------------------------------------------------

2 // 117 36

---------------------------------------------------------------

1- ( )x

log x ( ) x= = = = = = =3 3 3

23 2 32 2 29 3 3 3 27 27

9 2 ) 1

x x x+ = + = = 1

21 5 1 5 5 1 ) 2

x x xlog( x ) log( ) x x x x x

x x

( x )( x ) x or x

= = = + =

= = =

6 4 6 2 24 4 6 5 6 0

1

2 3 0 2 3

) 3

xlog log x x= = = =5 10 105 10 23 3 5

) 4

alog log a a a= = = =9 27 279 27 34 4 9

) 5

log log x xxx

= = = = =

162 16 16

2 4 16 4210 2 4

10

) 6

log log x xxx

= = + = = = = ++

248 24 8 24

5 3 3 5 257 5 1 8

7

) 7

nlog log log log n= + = = =1

49 2 494 2 49 9816

2 2 22

) 8

n

n x x n xxlog log log log log x x n x nxa a aa xa

= = = = = =

424 4 3 32 4 4

2 1 ) 9

n ( x )log log log n ( x ) ( n ) n x x nb b b

+ = = = = = +3 38 1 3 3 3

8 1 2 2 1 2 1) 10

z( z )

log z( z ) z z ( z )( z )

z or z z or z

+= + = = + = + =

+ = = = =

3 1 21 3 10 10 3 10 0 5 2 0

10

5 0 2 0 5 2

) 11

z = 5 log 510

.

: .


---------------------------------------------------------------

73 711 // 2

---------------------------------------------------------------

a ) a ( gol) a (

a a

+= = + = + =

= = =

2 2 2 2 263 281 2 6 2 2 2

62

24 61 2 81

21 )

. 2 2+ a 4 = a

5 5 1 4 12 2 2 2 = = = = +t t t gol) t () t ( 2

31 )

. 2 t 2 = t

) x (x

x x x x gol

x ro x ) x () x (

= = = =

= = = +

2 2 2 60 61 6 61 4 6 2

4

2 8 0 2 8

41 )

. 6 x 2 = x

x

x x ) ( ) ( x gol

= = + = = = = =2

12 1 1 201 1

11 11 1 01 01 1 11

1 01

01

51 )

. 1 2 x 11 = x

----------------------------------------------------------------

x x xgol gol gol gol gol gol) x (x x -2 c c c

c c c

= = = = +1 1 1

1 1 ) 0

2 ) 2 3

gol gol gol gol a c ) e ( ec e c ee a a c a cgol gol gol golgol gola c a ca agol c c e ece

= = = = =

agol

a c x gol a cc xac

3 ) = = =

a gol c ca acac

4 ) = =


---------------------------------------------------------------

2 // 117 38

---------------------------------------------------------------

3-

log loglog loglog log

= =3 27 3 27

127 3 27 3

)1

log loglog loglog log

= = = =

2249 7 2 772

7 7 77

) 2

log log log , log (log ) log

log (log (log )) log

= = = = =

= =

38 223 3 8 3 1

3 2 32 22

8 1 03 3 2 3

) 3

----------------------------------------------------------------

4- . ----------------------------------------------------------------

5- alog( a b ) log a log b logb

=

----------------------------------------------------------------

6-

log ( x ) log log ( x ) xx

( x )( x ) x x x x( x )( x ) x or x

+ = + =

+ = = =

+ = = =

82 8 5 2

10 10 10 5

2 22 5 8 3 10 8 3 18 0

6 3 0 6 3

x log log log

x log log log ( )= =

= =

6 8 8 110 10 10

3 1 8 810 10 10

----------------------------------------------------------------


---------------------------------------------------------------

93 711 // 2

---------------------------------------------------------------

=x gol y -7 2

1 + =x gol y 2

.

x

x gol yy

=

1 14 2 1

2 42

2 1 0 1 2

----------------------------------------------------------------

1 ) 45 3 5 64 1 01 92 01 9 25 4= + = = = / / /gol gol ) (gol HP -8

/) (gol ) (gol HP

/ / gol gol

= =

= + = =

21 1101 52 01 5 2

2116 01 21 93 1 01 52

2 )

3 ) 7 7 01 17= = =) ( ) (gol HP . 017

----------------------------------------------------------------


---------------------------------------------------------------

04 821 // 2

---------------------------------------------------------------

-1

) )

----------------------------------------------------------------

/ 3 -2

pi

3774 0

2/ 3pi

pi = =

pi

3 35 1

2 2

x x/x , x . pi pi pi

= = = pi pipi

045 045081 081045 298 171

3 3

/

298 171

. 045 ----------------------------------------------------------------

3 1 . 03 21 063= -3

x . 06 03 2= x

pi pi pi= =

06 081

3 081 06

----------------------------------------------------------------

x/ nim -4 / xnim

pi pi= =

pipi

2 0606 5 2

575 22

----------------------------------------------------------------

2=pi pi -5 02 04x

x

pi

pi pi= =

pi

174 02

4902 01 74

01

3 49 79+ = 3

. 71 . 71 04

----------------------------------------------------------------

-6 mk ) (

/mk

= = pi pi

000063 0001 063 0446 2 0639 8

000136404 0446 2


---------------------------------------------------------------

2 // 133 41

---------------------------------------------------------------

1-

tan( ) tan( ) tan( ) tan( )

cos( ) cos( ) cos( )

sin( ) sin( ) sin( ) sin( )

sin( ) sin( ) sin( )

cos( ) cos( ) cos( ) cos( )

pi pi pi pi pi= = pi = =

= = =


= = =


11 12 32

6 6 6 6 6 3

1300 360 60 60

2

3 4 2

4 4 4 4 4 2

2135 180 45 45

2

7 8 22

4 4 4 4 4 2

----------------------------------------------------------------

2- ( sin cos sin

sin sin

+ = + =

= = =

42 2 21 1

25

4 21 2121

25 25 5

(

----------------------------------------------------------------

3- sin , cos = =2 15 5

sintancos

= = = =

2 1 22

15 5

----------------------------------------------------------------

4- sin ,pi pi pi = = = pi =1 51 22 6 6 6

)


---------------------------------------------------------------

2 // 133 42

---------------------------------------------------------------

tan

,

= pi pi pi pi

= pi = = pi =

3

5 221 2

3 3 3 3

)

cos , pi = < < pi = 30 0 22

)

----------------------------------------------------------------

5- sin( ) sin( ) , cos( ) cos( )

sin( ) sin( ) , cos( ) cos( ) cos( )

pi pi pi pipi + = = pi = =

pi pi pi pi pipi = = pi + = = =

3 32 2

3 3 2 6 6 2

2 2 2 12 2

4 4 2 3 3 3 2

----------------------------------------------------------------

6- cos , orpi pi pi pi = < < = = 12 2 2 3 3

-------------------------------------

7- ,pi pi = =7 116 6

) (,pi pi = = 56 6

) (


---------------------------------------------------------------

2 // 133 43

---------------------------------------------------------------

8-

sinsin cos tan

cos

,

( ) ,

( )

= = =

pi pi pi = = pi + =

pi pi pi = + =

pi pi pi = + =

1

5

1 24 4 4

3

32 4 4

3 7

42 4 4

----------------------------------------------------------------


---------------------------------------------------------------

44 831 // 2

---------------------------------------------------------------

3 1 2 1 2 1 2 13= + = + = + =pi pi) ( ) (nis ) (nis y -1 2 6

)

1 1 18 3 3 3 3 3 2 3= + = + = + = pi pi pi) ( ) (nis ) (soc y 8 2 4 3 4 2 6 4

)

4 4 441 2 2 3 2= + = = pi =pi pi) (nis ) (nis y 3 3 2 3 6 3

)

----------------------------------------------------------------

41 3 = pi/ -2

.

----------------------------------------------------------------

-3

) (nis 081 ) (nis nis . nis

----------------------------------------------------------------

-4

pi) (nis 2

+ pi) (soc soc . soc


---------------------------------------------------------------

2 // 138 45

---------------------------------------------------------------

5-

sin( ) sin( )

cos( ) cos( ) cos( )

pi pi =

pi pi pi= pi =

2 2

3 3

5 5

6 6 6

----------------------------------------------------------------

6- sin( ) sin( ) sin( )

sin( ) sin sinsin sin sin sin

pi pipi = pi = =pi = pi pi

pi pi = pi = =

3 2

4 4 2

4 2 20

4 2 2

----------------------------------------------------------------

7- cos( )pi cos

cos cos( ) cos ( cos ) + pi = + = 0 ----------------------------------------------------------------

8- sin cos

. sinm tan tan( )cos

= = = =

3

303

----------------------------------------------------------------


---------------------------------------------------------------

2 // 151 46

---------------------------------------------------------------

1-

x

sin( x ) / / / / /sin( x ) / / /

pi pi pi pi pi pipi

2 3 50

6 3 2 3 4 6

0 0 5 0 8 1 0 8 0 7 0 5 0

2 0 1 1 6 2 1 6 1 4 1 0

x

sin( x ) / / / / / /sin( x ) / / / /

pi pi pi pi pi pi pipi

7 5 4 3 5 7 112

6 4 3 2 3 4 6

0 5 0 7 0 8 1 0 8 0 7 0 5 0

2 1 1 4 1 6 2 1 6 1 4 1 0

----------------------------------------------------------------

2- y sin( x )

x

y

=

pi pi pi pi

+

12

2

0 2 3 4

0 2 0 2 0

)

y cos( x )

x

y

=

pi pi pi pi

13

2

0 2 3 4

3 0 3 0 3

)

----------------------------------------------------------------

3- T , Max , minf fpi

= = pi = = 2

3 32

)

T ( ) ,Max , ming g= pi = pi = = 1

2 6 1 13

)

T ( ) , Max , minh hpi

= pi = = = 2 4 2 22

)


---------------------------------------------------------------

74 151 // 2

---------------------------------------------------------------

----------------------------------------------------------------

-4

mc/ / /) y (xa M ) (nis y 5 6 5 6 5 6= = = = pi pi2 2

)

) 0 0 0 5 6 0= = = = mc /) y (nim ) (nis y

x

nis yy

pi pipi pi

=

32 0

52 2

0 5 0 5 0

)

----------------------------------------------------------------

b b T ( -5 b

pi pi pi pi= = = = =

pi

4 24 0

2 2

4 21 =) x (soc y 21 = a 21 0) , (A

b b T ( bpi pi

= = = pi = pi =pi

2 2 2 0

2 21 =) x (soc y 21 = a 21 0) , (A

b b T ( bpi pi

= = = pi =pi

1 2 20 4

2 4

211 =) x (nis y 21 = a 21 pi) , (A 2


---------------------------------------------------------------

84 151 // 2

---------------------------------------------------------------

= =pi T -6 pi

21

2 . 1 0] , [

t

/ / / d

3 1 11 0

4 2 4

5 3 0 5 3 0 5 3

)

= = pi = =/ / /) ( ) (soc d t 1

5 3 1 5 3 5 32

)

= =pi T pi

21

2 )

----------------------------------------------------------------


---------------------------------------------------------------

2 // 157 49

---------------------------------------------------------------

1-

x ( )cos( )

x /p / /

= +

= + + =

=

= + + =

2 2 240 45 2 40 45 150

1600 2025 1800 5425

5425 73 6

40 45 73 6 158 6

----------------------------------------------------------------

2- n nn

2

180

n = = = 5 25 180 1085

x ( )(cos )

x ( ) x/

= +

+ =

2 2 210 10 2 10 10 108

2100 100 200 0 3 260 260

----------------------------------------------------------------

3-

sin sin /sin .sin /

/ / /

sin ( )/ /

= =

= =

5 27575 0 93

5 2 5 4 5 4

10 93 68 43

----------------------------------------------------------------

4-

x ( ).cos( )

x ( ) BD/ / /

= +

+ = + = =

2 2 26 11 2 6 11 125

236 121 132 0 57 157 75 7 232 7

----------------------------------------------------------------

5-

hsin h sin

S ( )( )

= = =

= + =

60 10 60 5 310

130 20 5 3 125 3

2


---------------------------------------------------------------

2 // 157 50

---------------------------------------------------------------

6- h

sin h sin

S

= = =

= =

30 12 30 612

15 6 90

----------------------------------------------------------------

7-

h htan x

x tanh h

tan xx tan

h h,

tan tan

h

tan tan

= =

= + =+

+ =

= =

6060

45 100100 45

10060 45

100 100236

1 1 31

45 60 3

----------------------------------------------------------------

8-

sinsin sin/ ymy sin /

xsin x y.sin/ / my

= =

+ = =

1 5115 11511 381

11 1 5

1126 5 26 5 11 159

----------------------------------------------------------------

9-

sin sin sinABC : ACmAC sin

DCADC : tanAC

DC AC tan tan

= =

=

= =

10 45 4525 102

25 10

60

60 102 60 177

----------------------------------------------------------------

10-

sin sin sinx / kmx sin

sin sin siny kmy sin

= =

= =

75 40 753 4 5

3 40

65 40 653 4

3 40


---------------------------------------------------------------

2 // 167 51

---------------------------------------------------------------

1- A B , B C , C A + = + = + =

5 6 4 0 3 4

2 6 4 10 2 6 )

A , C , A B , B C = = + = =

4 10 3 3 7 11 2 22 3 2

0 2 6 15 2 7 0 0 )

B A A B

( A B ) C

A ( B C ) A ( B C ) ( A B ) C

+ = = +

+ + = + =

+ + = + = + + = + +

5 6

2 6

5 6 1 1 6 5

2 6 0 5 4 11

2 5 4 0 6 5

0 1 4 10 4 11

)

.

:

( )

( )

+ +

= + =

+ +

= + =

1 2 3 7 8 9 13 14 15

4 5 6 10 11 12 16 17 18

8 10 12 13 14 15 21 24 27

14 16 18 16 17 18 30 33 36

1 2 3 7 8 9 13 14 15

4 5 6 10 11 12 16 17 18

1 2 3 20 22 24 21 24 27

4 5 6 26 28 30 30 33 36

----------------------------------------------------------------

2- A , B , C

= = =

1 31 2 2 1 1

1 21 3 3 4 1

2 1

----------------------------------------------------------------

3- R P Q ( P Q ) ( )

= = + = + = =

3 2 5 7 8 9 8 9

5 1 6 4 1 3 1 3

4 0 2 9 6 9 6 9

----------------------------------------------------------------

4- x y xx x , y y yx y y

= = = = + = = =

+ = =

2 3 6 33 9 3 3 3 9 3 6 2

3 3 0


---------------------------------------------------------------

2 // 173 52

---------------------------------------------------------------

1- A , B , C= + = = = = =5 8 13 10 10 0 1 0 1

----------------------------------------------------------------

2- A

= = = +

2 3

2 3 2 31 1 7 71

1 2 1 2 1 24 3 7

7 7

B = 0 B .B ( ) ( ( ))= = =6 1 3 2 6 6 0 ----------------------------------------------------------------

3- (

A , A , B , B

AB , ( AB ) , A B

( AB ) A B

= = = =

= = =

1 2 5 2 4 11 3 111 1

3 5 3 1 1 3 1 4

2 5 18 5 17 631 1 1

7 18 7 2 10 37

1 1 1

----------------------------------------

(

A , A , B , B

A B , ( A B ) , A B

( A B ) A B

= = = =

+ = + = + =

+ +

1 2 5 2 4 11 3 111 1

3 5 3 1 1 3 1 4

5 9 2 9 2 911 1 1

4 2 4 5 4 546

1 1 1

----------------------------------------------------------------

4- I I I = = = =

1 0 1 0 1 011

0 1 0 1 0 11 0

----------------------------------------------------------------

5- A A ( A ) A( )

= = = = =

3 5 4 5 3 5 3 51 11 1 1

12 4 2 3 2 4 2 4212 10

2

----------------------------------------------------------------


---------------------------------------------------------------

2 // 173 53

---------------------------------------------------------------

6- A ( ) ( a )( a ) a a

a a ( a )( a ) a or a= + = + + =

= + = = =

20 1 4 1 2 0 4 2 0

26 0 3 2 0 3 2

----------------------------------------------------------------

7-

x x

y y

x

y

= =

=

= = = +

=

13 1 4 3 1 4

1 2 3 1 2 3

11 11

2 1 4 111 1 7 7

1 3 3 5 5 56 1 7

7 7

)

x x

y y

x

y

= =

=

= = =

= =

12 3 0 2 3 0

6 4 1 6 4 1

3 3

4 3 0 31 1 10 10

6 2 1 2 2 2 18 18 10

10 10 5

)

x x

y y

x

y

= =

= = = =

=

11 1 7 1 1 7

5 4 1 5 4 1

4 1 7 29 29 2911

5 1 1 36 36 364 5

)

----------------------------------------------------------------

8-

x x

y y

x

y

= =

= = = =

=

11 1 7 1 1 7

1 1 7 1 1 7

1 1 7 14 7 71 1

1 1 7 0 0 01 1 2

----------------------------------------------------------------

9- X I I X I I I + = = = =

1 02 2

0 1 )

X X

= = =

1 24 1 2 3 2 4

2 1 33 2 4 5 1 3

2 2

)


---------------------------------------------------------------

45 181 // 2

---------------------------------------------------------------

23 2 2 2 2 2= 2 1, -1

.

----------------------------------------------------------------

) 000000 6 5 2 1 110011 } { } , , , { -2 ) 46 26= 1 0,

. 1 0,

. 46 26= . ) ( )

n n

2 2 000 2 2= nupcurlybracketrightupcurlybracketmidupcurlybracketleft

.

----------------------------------------------------------------

01 9 8 7 6 5 4 3 2 1} , , , , , , , , { 9 -3 . 09 01 9= . 9 8 7 6 5 4 3 2 1 0} , , , , , , , , , {

----------------------------------------------------------------

21 2 2 3= ( 6 3 2= ( -4

----------------------------------------------------------------

9 9 7 6 5 4 3 2 1} , , , , , , , { 8 -5 . 846 9 9 8= . 9 7 6 5 4 3 2 1 0} , , , , , , , , {

---------------------------------------------------------------- A B A . FEDCBA -6

... B C

. 69 2 2 2 2 2 3=

----------------------------------------------------------------

a, a -7 3 2

. 22 6 8 8= + +


---------------------------------------------------------------

55 581 // 2

---------------------------------------------------------------

) 021 5= ! ) 84 42 2 4 2= = ! ! ) 27 84 021= -1

----------------------------------------------------------------

r , e , t , u , p , moc moc -2

. 027 6= !

----------------------------------------------------------------

... 2 1 -3

. 01 1 8 9 01= ! ... ----------------------------------------------------------------

9 5 0 1 ( -4

4203 1 6 7 8 9=

8 5 4 1 1 ( ) 8 4 ( 0 1 )

(

25701 4 6 7 8 8= . 67731 4203 25701= +

----------------------------------------------------------------

= = = =! !) ( ) , (P! ! 52 3 -5 52 52

00831 32 42 52 3 5222 3 52

----------------------------------------------------------------

-6

. 3! . 5!

. 027 3 5= ! !

----------------------------------------------------------------

11 01 3 01 6 -7

. 4 11 3 01 6 01 ) , (P ) , (P ) , (P . 4 ----------------------------------------------------------------


---------------------------------------------------------------

65 091 // 2

---------------------------------------------------------------

n -1 k . n k

(. ) 2n n

=

0 + 1 +...+ n

----------------------------------------------------------------

( 0 3) ( 1 2) ( 2 1 ) -2

= + + = + +

7 5 7 5 7 5012 7 01 12 5 53 1

1 2 2 1 3 0

----------------------------------------------------------------

. 5 2 -3

= = .

502 01 2 2

2 .

----------------------------------------------------------------

= + + + + + = + + + + + -4

01 01 01 01 01 01215 1 54 012 012 54 1

01 8 6 4 2 0

----------------------------------------------------------------

. -5

! 7 3 ! !

= =

7 753

4 3 3 .

----------------------------------------------------------------

. 02 1 3 5 -6

= =

02 5002 02 01

1 3 .

----------------------------------------------------------------


Documents

Hale Riazi 2