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DG^AP

"dgoisafudgrpaysi s.io" 7

Iovlrsl ^ricjojdltri` Huo`tijos

Iotrjnu`tijo 5 ^al stunlot dgy kl hgdibigr gkjut tricjojdltri` huo`tijos viz sio x, `js x, tgo x, `jsl` x, sl` x,`jt x wita rlspl`tivl njdgios [, [, [ ‐ {(;o + 7) /;}, [ ‐ {o }, [ ‐ {(;o + 7) /;}, [ ‐ {o } gon rlspl`tivlrgocls V‐7, 7T, V‐7, 7T, [, [ ‐ (‐7, 7), [ ‐ (‐7, 7), [.

@jrrlspjoniocby, six iovlrsl tricjojdltri` huo`tijos (gbsj `gbbln iovlrsl `ir`ubgr huo`tijos) grl nlhioln.

sio ‐7 x 5 ^al sydkjb sio ‐7 x jr gr`siox nlojtls tal gocbl sj tagt sio 4 x. Gs g nirl`t dlgoioc, sio ‐7 x is ojt ghuo`tijo, gs it njls ojt sgtishy tal rlquirldlots hjr g rubl tj kl`jdl g huo`tijo. Kut ky g suitgkbl `aji`lV‐7, 7T gs its njdgio gon stgongrnizln slt V‐ /;, /;T gs its rgocl, talo rubl sio ‐7 x is g siocbl vgbulnhuo`tijo.

^aus sio ‐7 x is `josinlrln gs g huo`tijo wita njdgio V‐7, 7T gon rgocl V‐ /;, /;T.

^al crgpa jh y 4 sio ‐7 x is gs sajwo klbjw, wai`a is jktgioln ky tgfioc tal dirrjr idgcl, jh tal pjrtijo jh tal

crgpa jh y 4 sio x, hrjd x 4 ‐ /; tj x 4 /;, jo tal biol y 4 x.

`js ‐7 x 5 Ky hjbbjwioc tal nis`ussijos, sidibgr tj gkjvl, wl agvl `js ‐7 x jr gr` js x gs g huo tijo witanjdgio V‐7, 7T gon rgocl V=, T.

^al crgpajh y 4 `js ‐7 x is sidibgrby jktgioln gs tal dirrjr idgcl jh tal pjrtijo jh tal crgpa jh y 4 `js x hrjdx 4 = tj x 4

tg o ‐7 x 5 Sl clt tgo ‐7 x jr gr`tgox gs g huo`tijo wita njdgio [ gon rgocl (‐ /;, /;).Crgpa jh y 4 tgo ‐7 x

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DG^AP

"dgoisafudgrpaysi s.io" ;

/;

/;

y

xj

`jsl` ‐7 x 5 `jsl` ‐7 x jr gr``jsl` x is g huo`tijo wita njdgio [ ‐ (‐7, 7) gon rgocl V‐ /;, /;T ‐ {=}.Crgpa jh y 4 `jsl` ‐7 x

sl` ‐7 x 5 sl` ‐7 x jr gr`sl` x is g huo`tijo wita njdgio [ ‐ (‐7, 7) gon rgocl V=, T ‐ { /;}.Crgpa jh y 4 sl` ‐7 x

`jt ‐7 x 5 `jt ‐7 x jr gr``jt x is g huo`tijo wita njdgio [ gon rgocl (=, )Crgpa jh y 4 `jt ‐7 x

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DG^AP

"dgoisafudgrpaysi s.io" 0

Lxgdpbl # 7 5 Hion tal vgbul jh tgo

0

7tgo

;7

`js 77 .

Pjbutijo 5 tgo

0

7tgo

;7

`js 774 tgo

80 4 tgo

8

4

.

Lxgdpbl # ; 5 Hion njdgio jh sio ‐7 (;x ; ‐ 7)

Pjbutijo 5 Blt y 4 sio ‐7 (;x ; ‐ 7)Hjr y tj kl nlhioln ‐ 7 (;x ; ‐ 7) 7

= ;x ; ; = x ; 7 x V‐7, 7T.

Plbh prg`ti`l prjkblds 5

(7) Hion tal vgbul jh sio

;

7sio

0

7

(;) Hion tal vgbul jh `jsl` Vsl` ‐7 (‐ ; ) + `jt ‐7 ( ‐7)T

(0) Hion tal njdgio jh y 4 sl ‐7 (x ; + 0x + 7)

(>) Hion tal njdgio jh y 4 `js ‐7

;

;

x7

x

(<) Hion tal njdgio jh y 4 tgo ‐7 )7x( ;

Goswlrs 5 (7) 7 (;) ‐7(0) (‐ , ‐ 0T V ‐ ;, ‐ 7T V=, ) (>) [ (<) (‐ , ‐7T V7, )

]rjplrty 7 5 ‒‐x—

^al crgpas jh sio ‐7 x, tgo ‐7 x, `jsl` ‐7 x grl syddltri` gkjut jricio.Alo`l wl clt sio ‐7 (‐x) 4 ‐ sio ‐7 x

tgo ‐7 (‐x) 4 ‐ tgo ‐7 x`jsl` ‐7 (‐x) 4 ‐ `jsl` ‐7 x.

Gbsj tal crgpas jh `js ‐7

x, sl` ‐7

x, `jt ‐7

x grl syddltri` gkjut tal pjiot (=, /;). Hrjd tais, wl clt`js ‐7 (‐x) 4 ‐ `js ‐7 xsl` ‐7 (‐x) 4 ‐ sl` ‐7 x`jt ‐7 (‐x) 4 ‐ `jt ‐7 x.

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DG^AP

"dgoisafudgrpaysi s.io" >

]rjplrty ; 5 ^ ‐7

(i) sio (sio 7 x) 4 x, 7 x 7

]rjjh 5 Blt 4 sio ‐7 x. ^alo x V‐7, 7T & V‐ /;, /;T. sio 4 x, ky dlgoioc jh tal sydkjb sio (sio ‐7 x) 4 x

Pidibgr prjjhs `go kl `grriln jut tj jktgio

(ii) `js (`js 7 x) 4 x, 7 x 7(iii) tgo (tgo 7 x) 4 x, x [(iv) `jt (`jt 7 x) 4 x, x [(v) sl` (sl` 7 x) 4 x, x 7, x 7(vi) ` js l ( `js l 7 x) 4 x, |x| 7

^al crgpa jh y 4 sio (sio ‐7 x) `js (`js ‐7 x)

^al crgpa jh y 4 tgo (tgo ‐7 x) `jt (`jt ‐7 x)

^al crgpa jh y 4 `jsl` (`jsl` ‐7 x) sl` (sl` ‐7 x)

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DG^AP

"dgoisafudgrpaysi s.io" <

]rjplrty 0 5 ^ ‐7 ^

(i) sio ‐7 (sio x) 4UoT,;/)7o;(,;/)7o;V(x,x)7o;(

T;/o;,;/o;Vx,xo;

]rjjh 5 Ih x V;o ‐ /;, ;o + /;T, talo ‐;o + x V‐ /;, /;T gon sio (‐;o + x) 4 sio x.

Alo`l sio ‐7

(sio x) 4 ‐;o + x hjr x V;o ‐ /;, ;o + /;T.]rjjh jh ; on pgrt is blht hjr tal stunlots.

Crgpa jh y 4 sio ‐7 (sio x)

(ii) `js ‐7 (`js x) 4oT,o;,)7o;V(x,xo;T)7o;(,o;Vx,xo;

Crgpa jh y 4 `js ‐7 (`js x)

y

x

y 4 ;

+ x

p y

4 ‐

x y

4 x y

4 ;

‐ x

p

;

(iii) tgo ‐7 (tgo x) 4 ‐ o + x, o ‐ /; 2 x 2 o + /;, o U

Crgpa jh y 4 tgo ‐7 (tgo x)

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DG^AP

"dgoisafudgrpaysi s.io" 8

(iv) `jsl` ‐7 (`jsl` x) is sidibgr tj sio ‐7 (sio x)

Crgpa jh y 4 `jsl` ‐7 (`jsl` x)

(v) sl` ‐7 (sl` x) is sidibgr tj `js ‐7 (`js x)

Crgpa jh y 4 sl` ‐7

(sl` x)

(vii) `jt ‐7 (`jt x) 4 ‐o + x, x (o , (o + 7) ), o U

Crgpa jh y 4 `jt ‐7 (`jt x)

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DG^AP

"dgoisafudgrpaysi s.io" 3

[ldg rf 5 sio (sio ‐7 x), `js (`js ‐7 x), .... `jt (`jt ‐7 x) grl gplrijni` (ojo plrijni`) huo`tijos walrl gssio ‐7 (sio x), ..., `jt ‐7 (`jt x) grl plrijni` huo`tijos.

]rjplrty > 5 ‒7/x—

(i) `jsl` ‐7 (x) 4 sio ‐7 (7/x), |x| 7

]rjjh 5 Blt `jsl` ‐7 x 4 7/x 4 sio sio ‐7 (7/x) 4 sio ‐7 (sio )

4 (gs V‐ /;, /;T ‐ {=})4 `jsl` ‐7 x

(ii) sl` ‐7 x 4 `js ‐7 (7/x), |x | 7

(iii) `jt ‐7 x 4=x),x/7(tgo=x),x/7(tgo

7

7

]rjplrty < 5 ‒ /;—

(i) sio 7 x + `js 7 x 4;

, 7 x 7

]rjjh 5 Blt G 4 sio ‐7 x gon K 4 `js ‐7 x sio G 4 x gon `js K 4 x sio G 4 `js K sio G 4 sio ( /; ‐ K) G 4 /; ‐ K, kl`gusl G gon /; ‐ K V‐ /;, /;T G + K 4 /;.

Pidibgrby, wl `go prjvl

(ii) tgo 7 x + `jt 7 x 4;

, x [

(iii) `jsl` 7 x + sl` 7 x 4;

, x 7

Lxgdpbl # 0 5 Hion tal vgbul jh `jsl`

>0

`jt`jt 7

.

Pjbutijo 5 `jt (`jt ‐7 x) 4 x, x [

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DG^AP

"dgoisafudgrpaysi s.io" 1

`jt

>0

`jt 7 4>

0

`jsl`

>0

`jt`jt 74 `jsl`

>

04 ; .

Lxgdpbl # > Hion tal vgbul jh tgo ‐7

>0

tgo .

Pjbutijo 5

tgo ‐7 (tgo x) 4 x ih x

;,

;

Gs>

0

;,

; tgo ‐7

>0

tgo>

0

>0

;0

,;

crgpa jh y 4 tgo ‐7 (tgo x) is gs 5

hrjd tal crgpa wl `go sll tagt ih ;

2 x 2;

0,

talo tgo ‐7 (tgo x) 4 x ‐

tgo ‐7

>0

tgo 4>

0 ‐ 4 ‐

>

Lxgdpbl # < 5 Hion tal vgbul jh sio ‐7 (sio3) gon sio ‐7 (sio (‐<)).

Pjbutijo. Blt y 4 sio ‐7 (sio 3)

sio ‐7 (sio 3) 3 gs 3 ;

,;

; 2 3 2;

<

crgpa jh y 4 sio ‐7 (sio x) is gs 5

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DG^AP

"dgoisafudgrpaysi s.io" 9

Hrjd tal crgpa wl `go sll tagt ih ; x ;

<, talo

y 4 sio ‐7 (sio x ) `go kl writtlo gs 5y 4 x ‐ ;

sio ‐7 (sio 3) 4 3 ‐ ;

Pidibgrby ih wl agvl tj hion sio ‐7 (sio(‐<)) talo

‐ ; 2 ‐ < 2 ‐;

0

hrjd tal crgpa jh sio ‐7

(sio x), wl `go sgy tagtsio ‐7 (sio(‐<)) 4 ; + (‐<) 4 ; ‐ <

Lxgdpbl # 8 5 Hion tal vgbul jh js ‐7 {sio( ‐ <)}

Pjbutijo 5 Blt y 4 `js ‐7 {sio(‐<)}4 `js ‐7 (‐ sio <)4 ‐ `js ‐7 (sio <) (`js ‐7 (‐ x) 4 ‐ `js ‐7 x, |x| 7)

4 ‐ `js ‐7

<;

`js ..........(i)

Ojtl tagt 5 ‐ ; 2

<;

2 ‐

crgpa jh `js ‐7 (`js x ) is gs 5

Hrjd tal crgpa wl `go sll tagt ih ‐ ; x ‐ talo `js ‐7 (`jsx) 4 x + ;

hrjd tal crgpa `js ‐7

<;

`js 4

<; + ; 4

<;

<

hrjd (i), wl clt

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DG^AP

"dgoisafudgrpaysi s.io" 7=

y 4 ‐

<;

< y 4 < ‐

;0

.

Lxgdpbl # 3 5 Hion tal vgbul jh tgo

0;

`jt 7

Pjbutijo 5 Blt y 4 tgo

0;

`jt 7 ........(i)

`jt ‐7 (‐x) 4 ‐ `jt ‐7 x, x [(i) `go kl writtlo gs

y 4 tgo

0;

`jt 7

y 4 ‐ tgo

0

;`jt 7

`jt ‐7 x 4 tgo ‐7

x

ih x ? =

y 4 ‐ tgo

;0

tgo 7 y 4 ‐;0

Lxgdpbl # 1 5 Hion tal vgbul jh sio

>0

tgo 7 .

Pjbutijo 5 sio

>0

tgo 7 4 sio

<0

sio 7 4<0

Lxgdpbl # 9 5 Hion tal vgbul jh tgo

0<

`js;7 7

Pjbutijo 5 Blt y 4 tgo

0<

`js;7 7 ..........(i)

Blt `js ‐7

0<

4

;,= gon `js 4

0<

(i) kl jdls y 4 tgo

;

..........(ii)

tgo ;

;4

`js7`js7

4

0

<7

0<

74

<0

<04

>)<0( ;

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DG^AP

"dgoisafudgrpaysi s.io" 77

tgo;

4 ·

;

<0.........(iii)

;

>,= tgo

;? =

hrjd (iii), wl clt y 4 tgo;

4

;<0

Lxgdpbl # 7= 5 Hion tal vgbul jh `js (;`js ‐7 x + sio ‐7 x) walo x 4<7

Pjbutijo 5 `js

<7

sio<7

`js; 77 4 `js

<7

`js<7

sio<7

`js 777

4 `js

<7

`js;

74 ‐ sio

<7

`js 7 .........(i)

4 ‐;

<7

7

4 ‐<

8;.

G bi t l r 5 Blt<7

`js 7 4 `js 4<7

gon

;,=

sio 4<;>

sio ‐7 (sio ) 4 sio ‐7

<;>

..........(ii)

;,= sio ‐7 (sio ) 4

lqugtijo (ii) `go kl writtlo gs

4 sio ‐7

<;> 4 `js ‐7

<7

`js ‐7

<7

4 sio ‐7

<;>

Ojw lqugtijo (i) `go kl writtlo gs

y 4 ‐ sio

<;>

sio 7 ........(iii)

<;>

V‐7, 7T sio

<;>

sio 74

<;>

hrjd lqugtijo (iii), wl clt y 4 ‐<;>

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DG^AP

"dgoisafudgrpaysi s.io" 7 ;

Lxgdpbl # 77 5 Pjbvl sio ‐7 (x ; ‐ ;x + 7) + `js ‐7 (x ; ‐ x) 4;

Pjbutijo 5 sio ‐7 (h(x)) + `js ‐7 (c(x)) 4;

h(x) 4 c(x) gon ‐7 h(x), c(x) 7

x; ‐ ;x + 7 4 x ; ‐ x x 4 7, g``lptln gs g sjbutijo

Plbh prg`ti`l prjkblds 5

(8) Hion tal vgbul jh `js

8siosio 7

(3) Hion tal vgbul jh sio

>0

`js`js 7

(1) Hion tal vgbul jh js ‐7 (`js 70)

(9) Hion sio ‐7 (sio ), `js ‐7 (`js ), tgo ‐7 (tgo ), `jt ‐7 (`jt ) hjr

0,;

<

(7=) Hion tal vgbul jh `js ‐7 (‐ js >) (77) Hion tal vgbul jh tgo ‐7

13

tgo

(7;) Hion tal vgbul jh tgo ‐7

>7

`jt (70) Hion tal vgbul jh sl

0;

`js 7

(7>) Hion tal vgbul jh `jsl

0

7sio 7

(7<) Hion tal vgbul jh sio (; js ‐7 x + sio ‐7 x) walo x 4<7

(78) Pjbvl tal hjbbjwioc lqugtijos (i) < tgo ‐7 x + 0 `jt ‐7 x 4 ; (ii) > sio ‐7 x 4 ‐ `js ‐7 x

(73) Lvgbugtl tgo

>>7

l``js 7 (71) Lvgbugtl sl

8078

`jt 7

(79) Lvgbugtl sio

>0

`jt;7 7 (;=) Lvgbugtl tgo

><7

tgo; 7

(;7) Pjbvl sio ‐7 (x ; ‐ ;x + 0) + `js ‐7 (x ; ‐ x) 4;

Goswlrs 5 (8);0

(3) ojt nlhioln

(1) 70 ‐ > (9) 0 ‐ , ‐ ; , ‐ 0 , ‐ ;

(7=) > ‐ (77)1

(7;)

;>7

(70);0

(7>) ‐ 0 (7<)<7

(78). (i) x 4 7 (ii) x 4 ;7

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DG^AP

"dgoisafudgrpaysi s.io" 70

(73)<>

(71)788<

(79)<

<;(;=)

733

(;7) Oj sjbutijo

]rjplrty 8 5 I nlotit ils jo gn ni tij o g on su ktrg `ti jo 5

(i) sio 7 x + sio 7 y

7yx&=y,=x,x7yy7xsio

7)yx(&=y,=x,x7yy7xsio

;;;;7

;;;;7

]rjjh 5 Blt G 4 sio ‐7 x gon K 4 sio ‐7 y walrl x, y V=, 7T.

sio (G + K) 4 x ;y7 + y ;x7

sio ‐7 sio (G + K) 4 sio ‐7

;; x7yy7x

sio ‐7

;; x7yy7x

4K G;/hjr )K G(

;/K G=hjr K G4

7yx),ysiox(sio7yx,ysioxsio

;;77

;;77

(ii) sio ‐7 x ‐ sio ‐7 y 4 sio ‐7

;; x7yy7x 6 x, y V=, 7T

(iii) `js ‐7 x + `js ‐7 y 4 `js ‐7 ;; y7x7xy 6 x, y V=, 7T

(iv) `js ‐7 x ‐ `js ‐7 y 4

7xy=6y7x7xy`js

7yx=6y7x7xy`js

;;7

;;7

(v) tgo ‐7 x + tgo ‐7 y 4

7xy&=y,xih xy7yx

tgo

7xy&=y,xih xy7yx

tgo

7xy&=y,xih ;/7xy&=y,xih ;/

7

7

(vi) tgo ‐7 x ‐ tgo ‐7 y 4 tgo ‐7

xy7yx

, x =, y =

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DG^AP

"dgoisafudgrpaysi s.io" 7>

Ojtls 5 (i) x; + y; 7 & x, y = = s io 7 x + sio 7 y

;

gon x ; + y; 7 & x, y =

; sio 7 x + sio 7 y

(ii) xy 2 7 gon x, y = = tgo 7 x + tgo 7 y 2;

6 xy ? 7 gon x, y = ;

2 tgo 7 x + tgo 7 y 2

(iii) Hjr x 2 = jr y 2 = talsl inlotitils `go kl usln wita tal albp jh prjplrty ‒‐ x—i.l. `agocl x jr y tj x jr y wai`a grl pjsitivl.

Lxgdpbl # 7; 5 Pajw tagt sio ‐7

<0

+ sio ‐7

737<

4 ‐ sio ‐7

1<1>

Pjbutijo 5 <0

? =,737<

? = gon;

<0

+;

737<

43;;<1;;8

? 7

sio ‐7

<0

+ sio ‐7

737<

4 ‐ sio ‐7

;<9

7737<

;19;;<

7<0

4 ‐ sio ‐7

<>

.737<

731

.<0

4 ‐ sio ‐7

1<1>

Lxgdpbl # 70 5 Lvgbugtl `js ‐7

707;

+ sio ‐7

<>

‐ tgo ‐7

7880

Pjbutijo 5 Blt z 4 `js ‐7

707;

+ sio ‐7

<>

‐ tgo ‐7

7880

sio ‐7

<>

4;

‐ `js ‐7

<>

z 4 `js ‐7

707;

+

<>

`js;

7 ‐ tgo ‐7

7880

.

z 4;

‐

707;

`js<>

`js 77 ‐ tgo ‐7

7880

.........(i)

<>

? =,707;

? = gon<>

2707;

`js ‐7

<>

‐ `js ‐7

707;

4 `js ‐77897>>

7;<78

7707;

<>

4 `js ‐7

8<80

lqugtijo (i) `go kl writtlo gs

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DG^AP

"dgoisafudgrpaysi s.io" 7<

z 4;

‐ `js ‐7

8<80

‐ tgo ‐7

7880

z 4 sio ‐7

8<80

‐ tgo ‐7

7880

.........(ii)

sio ‐7 8<80 4 tgo ‐7 7880

hrjd lqugtijo (ii), wl clt

z 4 tgo ‐7

7880

‐ tgo ‐7

7880

z 4 =

Lxgdpbl # 7> 5 Lvgbugtl tgo ‐7 9 + tgo ‐7

><

Pjbutijo 5 9 ? =,>< ? = gon

><9 ? 7

tgo ‐7 9 + tgo ‐7

><

4 + tgo ‐7

><

.97

><

94 + tgo ‐7 (‐ 7) 4 ‐

>4

>

.

Lxgdpbl # 7< 5 Nlhiol y 4 `js ‐7 (>x 0 ‐ 0x) io tlrds jh `js ‐7 x gon gbsj nrgw its crgpa.

Pjbutijo 5 ]grt - 75 Blt y 4 js ‐7 (>x0 ‐ 0x) Njdgio 5 V‐7, 7T gon rgocl 5 V=, T

Blt `js ‐7 x 4 V=, T gon x 4 js y 4 `js ‐7 (> `js 0 ‐ 0 `js )

y 4 `js ‐7 (`js 0 ) ..... .... ..( i)

Hic.5 Crgpa jh `js ‐7 (`js x) V=, T 0 V=, 0 T tj nlhiol y 4 `js ‐7 (`js 0 ), wl `josinlr tal crgpa jh `js ‐7 (`js x)

io tal iotlrvgb V=, 0 T. Ojw, hrjd tal gkjvl crgpa wl `go sll tagt

(i) ih = 0 `js ‐7 (`js 0 ) 4 0 hrjd lqugtijo (i), wl clt

y 4 0 ih 0

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DG^AP

"dgoisafudgrpaysi s.io" 78

y 4 0 ih = 0

y 4 0 `js ‐7 x ih ;7

x 7

(ii) ih 0 ; `js ‐7 (`js 0 ) 4 ; ‐ 0 hrjd lqugtijo (i), wl clt

y 4 ; ‐ 0 ih 0 ;

y 4 ; ‐ 0 ih 0

2 0

;

y 4 ; ‐ 0`js ‐7 x ih ‐ ;7

x 2;7

(iii) 0 0 `js ‐7 (`js 0 ) 4 ‐ ; + 0 hrjd lqugtijo (i), wl clt y 4 ‐ ; + 0 ih 0 0

y 4 ‐ ; + 0 ih 0

;

y 4 ‐ ; + 0 `js ‐7 x ih ‐ 7 x ‐ ;7

hrjd (i), (ii) & (iii), wl clt

y 4 `js ‐7 (>x 0 ‐ 0x) 4

;7

x76x`js0;;

7x

;

76x`js0;

7x;7

6x`js0

7

7

7

]grt-; 5 Hjr y 4 `js ‐7 (>x 0 ‐ 0x)njdgio 5 V‐7, 7Trgocl 5 V=, T

(i) ih ;7

x 7 , y 4 0 `js ‐7 x.

nxny

4 ;x7

04 ‐ 0(7 ‐ x ; ) ‐7/; ...........(i)

nxny

2 = ih x

7,;7

nl`rlgsioc ih x

7,;7

gcgio ih wl nihhlrlotigtl lqugtijo (i) w.r.t. ‑x’, wl clt

;

;

nx

yn4 ‐ ;/0; )x7(

x0

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DG^AP

"dgoisafudgrpaysi s.io" 73

;

;

nx

yn2 = ih x

7,;7

`jo`gvity njwowgrns ih x

7,;7

(ii) ih ‐;7

x 2;7

, y 4 ; ‐ 0`js ‐7 x.

nxny

4 ;x7

0

nxny

? = ih x

;7

,;7

io`rlgsioc ih x

;7

,;7

gon ;

;

nx

yn4 ;/0; )x7(

x0

(g) ih x

=,;7

talo ;

;

nx

yn2 =

`jo`gvity njwowgrns ih x

=,;7

(k) ih x

;7

,= talo ;

;

nx

yn? =

`jo`gvity upwgrns ih x

;7

,=

(iii) Pidibgrby ih ‐ 7 x 2 ‐;7

talonxny

2 = gon ;

;

nx

yn? =.

tal crgpa jh y 4 `js ‐7 (>x 0 ‐ 0x) is gs

Plbh prg`ti`l prjkblds5

(;;) Lvgbugtl sio ‐7

<>

+ sio ‐7

70<

+ sio ‐7

8<78

(;0) Ih tgo ‐7 > + tgo ‐7 < 4 `jt ‐7 , talo hion ‑ ’

(;>) ]rjvl tagt ; `js ‐7

70

0+ `jt ‐7

8078

+;7

`js ‐7

;<3

4

(;<) Pjbvl tal lqugtijo tgo ‐7 (;x) + tgo ‐7 (0x) 4>

(;8) Pjbvl tal lqugtijo sio ‐7 x + sio ‐7 ;x 40

;

(;3) Nlhiol y 4 sio ‐7 (0x ‐ >x 0) io tlrds jh sio ‐7 x gon gbsj nrgw its crgpa.

(;1) Nlhiol y 4 tgo ‐7

;

0

x07

xx0io tlrds jh tgo ‐7 x gon gbsj nrgw its crgpa.

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DG^AP

"dgoisafudgrpaysi s.io" 71

Goswlrs. (;;);

(;0) 4 ‐9

79(;<) x 4

87

(;8) x 4;7

(;3) y 4 sio ‐7 (0x ‐ >x 0) 4

;7x76xsio0

7x;7

6xsio0

;7

x;7

6xsio0

7

7

7

crgpa jh y 4 sio ‐7 (0x ‐ >x 0)

(;1) y 4 tgo ‐7

;

0

x07

xx04

x0

76xtgo0

0

7x6xtgo0

0

7x

0

76xtgo0

7

7

7

Hic.5 Crgpa jh y 4 tgo ‐7

;

0

x07

xx0

] rjpl rty 3 5 Dis `lbbgol jus inlo ti ti ls

(i) sio 7

;x7x; 4

;7

;7

;7

x7ih xsio;

7xih xsio;

|x|ih xsio;

7

7

7

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DG^AP

"dgoisafudgrpaysi s.io" 79

crgpa jh y 4 sio 7

;x7x;

(ii) `js 7 (; x ; 7) 4=x7ih x`js;;7x=ih x`js;

7

7

crgpa jh y 4 `js 7 (; x ; 7)

(iii) tgo 7;x7

x; 4

7xih xtgo;7xih xtgo;7|x|ih xtgo;

7

7

7

crgpa jh y 4 tgo 7;x7

x;

(iv) sio 7 ;

7 ;

x

x 4

7xih xtgo;7xih xtgo;7|x|ih xtgo;

7

7

7

crgpa jh y 4 sio 7 ;

7 ;

x

x

(v) `js 7;

;

x7

x74

=xih xtgo;=xih xtgo;

7

7

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crgpa jh y 4 `js 7;

;

x7

x7

(vi) Ih tgo 7 x + tgo 7 y + tgo 7 z 4 , talo x + y + z 4 xyz

(vii) Ih tgo 7 x + tgo 7 y + tgo 7 z 4;

, talo xy + yz + zx 4 7

(viii) tgo 7 7 + tgo 7 ; + tgo 7 0 4

(ix) tgo 7 7 + tgo 7

;7 + tgo 707 4 ;