Polynomials Linear Equations in Two Variables

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CAEBXGM_CAY

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[D@REDCBA@X

  [D@REDCBA@ : Ae a`foirabh oxprossbde dl tgo ldrc

p(x) 3 aexe + ae-1xe-1

+ ae-9xe-9

+ …… + a1x1

+ a;x;

wgoro a;    ; aen a

;, a

1, a

9  ....ae   aro roa` euciors aen oahg pdwor dl x bs a pdsbtbvo betofor bs ha``on a

pd`yedcba`.

Goeho, ae, ae-1, ae-9, aro hdollbhboet dl xe, x

e-1…….. x

;aen aexe, ae-1x

e-1,ae-9x

e-9aro torcs dl tgo pd`yedcba`.

Oxacp`o - (b)   p(x) 3 aexe

+ ae-1 xe-1

+ …. a1  x + a; bs a pd`yedcba` be varbai`o x.

(bb)   p(y) 3 aeye

+ ae-1 ye-1

+ …. a1  y + a;  bs a pd`yedcba` be varbai`o y.

(bbb)   p(z) 3 aeze

+ ae-1 ze-1

+ …. a1  z + a;   bs a pd`yedcba` be varbai o z.

aidvo a`` aro ha``on pd`yedcba`s bl 

Oxacp`o -

(b)   p(u) 39

1u8 - 8u9 + 9y - > bs a pd`yedcba` be varbai`o u.

(bb)9

1u8, -8u9, 9u, -> aro medwe as torcs dl pd`yedcba` aen

9

1, -8, 9, -> aro tgobr hdollbhboets.

  ^R[OX DL [D@REDCBA@X

Foeora` y wo nbvbnon tgo pd`yedcba`s be twd hatofdrbos.

^ypos dl pd`yedcba`s

Iasos ed eucior dl torcs Iason de nofroo

(A) IAXON DE E_CIOY DL ^OYCX :- ^goso aro ld``dws :

(a) Cdedcba` :   [d`yedcba`s gavbef de`y deo torc aro ha``on cdedcba`s. (‗Cded‒ coaes ‗deo‒)

9x, ?x9, -?x9, y, u>

(i) Ibedcba` :   A pd`yedcba` dl twd torcs bs ha``on ibedcba`.

p(x) 3 x+ 1, q(x) 3 x9, r(q) 3 y8; + 1

(h) ^rbedcba` :   A pd`yedcba` dl tgroo torcs bs ha``on a trbedcba`.

p(x) 3 e + x9 +  

q(x) 3   9  + x + x9

(I) IAXON DE NOFYOO :-

Nofroo dl [d`yedcba`s :  ^go gbfgost pdwor dl varbai`o be a pd`yedcba` bs medwe as bt nofroo.

Ldr oxacp`o :

(a)   p(y) 3 9y9 - 8y + 6 bs a pd`yedcba` be tgo varbai`o y dl nofroo 9.

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(i)   q(x) 3   x9   + 18x> + ?x< bs a pd`yedcba` be varbai`o x dl nofroo <.

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(b) @beoar [d`yedcba` :  A pd`yedcba` dl nofroo deo bs ha``on a `beoar pd`yedcba`.

Ox.   p(x) 3 >x + ?

q(y) 3 9y

r(t) 3 t +   9

s(u) 3 8 - u

(bb) Quanratbh [d`yedcba` : A pd`yedcba` dl nofroo twd bs ha``on a quanratbh pd`yedcba`.

Ox.   9x9 + ?

?x9 + 8x +  

x9 +?

9x

(bbb) Huibh [d`yedcba` :  A pd`yedcba` dl nofroo tgroo bs ha``on a huibh pd`yedcba`.

Ox.   >x9 . 9x8 + 1, ?x8 + x 9

9x8 + >x9 + <x + 6

(vb) Ibquanrato pd`yedcba` :  A pd`yedcba` dl ldurtg nofroo bs ha``on a ibquanrato pd`yedcba`.

Ox.   >x> + 9x8 + ?x9 + x + 1

^ypo dl [d`yedba`s Nofroo Oxprossbde

(b)

(bb)

(bbb)

(bv)

(v)

Hdestaet pd`yedcba`s

@beoar pd`yedcba`s

Quanrbtbh pd`yedcba`s

Huibh pd`yedcba`s

Ib-quanratbh pd`yedcba`s

Zord

1

9

8

>

L(x) 3 h, {(x) 3 ;, bs zord pd`yedcba` ]Nofroo edt nolbeonS

L(x) 3 ax + i, a     ;

L(x) 3 ax9

+ ix + h, a     ;

L(x) 3 ax8

+ ix9

+ hx + n a     ;

L(x) 3 ax>

+ ix8

+ hx9

+ nx + o, a     ;

  \A@_O DL [D@REDCBA@X :

Bl p(x) bs a pd`yedcba` be varbai o aen    bs aey roa` eucior, tgoe tgo va`uo ditabeon iy rop`ahbef x iy     be

p(x) bs ha``on va`uo dl p(x) at 3    aen bs noedton iy p( ).

Ldr oxacp`o : Lben tgo va`uo dl p(x) 3 x 8 - < x9 + 11x - < at x 3 -9

  p(-9) 3 (-9)8 - < (-9)9 + 11(-9) -< 3 -7 -9> - 99 - <

  p(-9) 3 - <;

  ZOYD DL [D@REDCBA@X :

 A roa` eucior    bs a zord dl tgo pd`yedcba` p(x) bl p( ) 3 ;

Ox:   p(x) 3 x8 - <x9 + 11x - <

p(1) 3 (1)8 - <(1)9 + 11(1) - < 3 1 - < + 11 - < 3 ;

p(9) 3 (9)8 - <(9)9 + 11(9) - < 3 7 - 9> + 99 - < 3 ;

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CAEBXGM_CAY

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p(8) 3 (8)8 - <(8)9 + 11(8) - < 3 96 - ?> + 88 - < 3 ;

^gus, 1,9 aen 8 aro ha``on tgo zords dl pd`yedcba` p(x).

  YOCABENOY ^GODYOC :

Bl p(x) bs aey pd`yedcba` dl nofroo froator tgae dr oqua` td 1 aen    io aey roa` eucior. Bl p(x) bs nbvbnon iy

x-a. tgoe tgo rocabenor bs oqua` td p(a).

Nbvbnoen 3 Nbvbsdr Ö qudtboet + Yocabenor 

Ox.1   Lben tgo rocabenor wgoe tgo pd`yedcba`

l (x) 3 9x> - <x8 + 9x9 - x + 9 bs nbvbnon iy x + 9.

Xd`.   Vo gavo , (x + 9) 3 ;   x 3 - 9.

Xd, iy rocabenor tgodroc, wgoe l(x) bs nbvbnon iy (x + 9) 3 ;    x 3 - 9 tgo rocabenor bs oqua` td l (-9).

  l(x) 3 9x> - <x8 + 9x9 - x + 9

  l(-9) 3 9(-9)> - <(-9)8 + 9(-9)9 - (-9) + 9

  l(-9) 3 9 Ö 1< - < Ö - 7 + 9 Ö > + 9 + 9

3 89 + >7 + 7 + 9 + 9 3 29

Yoqubron rocabenor 3 29

Ox.9   Nbvbno tgo pd`yedcba` 8x> - >x8 - 8x - 1 iy (x - 1)

Xd`.   Iy `def nbvbsbde

>8

>

88

18>81

98

8>

8>

xxx

xx

xxxx

98

818

xx

xx

xx

xx9

918

1

>>

1>

xx

-?

Goro, tgo rocabenor bs -?. Edw, tgo zord dl (x - 1) bs 1. Xd, puttbef x 3 1 bs p(x), wo soo tgat p(1) 3 8(1) > -

>(1)8 -8(1) - 1

3 8 - > - 8 - 1

3 - ?, wgbhg bs tgo rocabenor.

  LAH^DY ^GODYOC :

Bl p(x) bs pd`yedcba` dl nofroo e     1 aen     bs aey roa` eucior, tgoe   (b)  x - a bs a lahtdr dl p(x), bl p(a) 3 ;

aen  (bb) p(a) 3 ; bl x - bs a lahtdr dl p(x).

[rddl :  Iy tgo rocabenor tgodroc, p(x) 3 (x - a) (x) + p(a).

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CAEBXGM_CAY

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(b)   Bl p(a) 3 ;, tgoe p(x) 3 (x - a) q(x), wgbhg sgdws tgat x - a bs a lahtdr dl p(x).

(bb)   Xbeho x - a bs a lahtdr dl p(x), p(x) 3 (x - a) f(x) ldr saco pd`yedcba` f(x). Be tgbs haso, p(a) 3 (a - a) f(a)

3 ;.

Edto : - (b)   (x + a) bs a lahtdr dl p(x)  p(-a) 3 ;

(bb)   (ax - i) bs a lahtdr dl p(x) 3 p    

 a

i3 ;

(bbb) (ax + i) bs a lahtdr dl p(x) 3   ; 

  a

ip

Ox-1   _sbef lahtdr tgodroc, sgdw tgat (x - 1) bs a lahtdr dl 9x> + 2x8 + <x9 - 11x - <

Xd`.   @ot l(x) 3 9x> + 2x 8 + <x9 - 11x -<

Iy lahtdr tgodroc, (x - 1), x - 1 3 ;   x 3 1 wb`` io lahtdr dl l(x), bl l(1) 3 ;.

Edw, l(x) 3 9x> + 2x8 + <x9 - 11x - <

  l(1) 3 9 Ö 1> + 2 Ö 18 + < Ö 19 - 11 Ö 1 - < 3 9 + 2 < - 11 - < 3 ;

  (x - 1) bs lahtdr dl l(x).

Goeho, (x - 1) bs a lahtdr dl tgo fbvoe pd`yedcba`.

Ox.9   Notorcbeo tgo va`uo dl a ldr wgbhg tgo pd`yedcba` 9x> - ax8 - >x9 + 9x + 1 bs nbvbsbi`o iy 1 - 9x.

Xd`.   @ot p(x) 3 9x> - ax8 - ax9 + 9x + 1. Bl tgo pd`yedcba` z p(x) bs nbvbsbi`o iy (1 - 9x), tgoe (1 - 9x) bs a lahtdr dl 

p(x).

  ;9

1   

  p

  ;19

19

9

1>

9

1

9

19

98>

 a

  9?77

9?;111

77

1;1

9

9

>

>

71<

9   a

aaa

Goeho, tgo fbvoe pd`yedcba` wb`` io nbvbsbi`o iy 1 - 9x, bl a 3 9?.

  ^R[O DL LAH^DYBZA^BDE :-

(b) Lahtdrbzatbde iy tambef dut tgo hdccde lahtdrs

Ox.   ai(a9 + i9 - h9 ) + ih (a9 + i9 - h9) - ha(a9 + i9 - h9)

Xd`.   Vo gavo

ai(a9 + i9 - h9) + ih (a9 + i9 - h9) - ha(a9 + i9 - h9)

3 (a9 + i9 - h9) (ai + ih - ha)

(bb) Lahtdrbzatbde iy frdupbef tgo torcs

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CAEBXGM_CAY

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Ox.   (x9 + 8x)9 - ?(x9 + 8x) - y(x9 + 8x) + ?y

Xd`.   Vo gavo

(x9 + 8x)9 - ?(x9 + 8x) - y(x9 + 8x) + ?y

3 (x9

+ 8x) {(x9

+ 8x) -?} - y{(x9

+ 8x) -?} 3 (x9

+ 8x - ?) (x9

+ 8x - y)

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CAEBXGM_CAY

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(bbb) Lahtdrbzatbde iy cambef a porloht squaro

Ox.   a9 + i9 - 9(ai - ah + ih)

Xd`.   Vo gavo

a9

+ i9

- 9(ai - ah + ih)3 a9 + i9 - 9ai + 9ah - 9ih

3 (a - i)9 + 9h (a - i)

3 (a - i) {(a - i) + 9h} 3 (a - i) (a - i + 9h)

(bv) Lahtdrbzatbde tgo nblloroeho dl twd squaros

Ox.   x7 - y7

Xd`.   Vo gavo

x7 - y7 3 {(x>)9 - (y>)9} 3 (x> - y>) (x> + y>)

3 {(x9)9 - (y9)9 (x> + y>) 3 (x9 - y9) (x9 + y9) (x> + y>)

3 (x - y) (x + y) (x9 + y9) (x> + y>)

3 (x - y) (x + y) (x9 + y9) {( x9)9 + (y9)9 + 9x9y9 - 9x9y9)

3 2x - y) (x + y) (x9 + y9) {(x9 + y9)9 - (   9  xy)9}

3 2x - y) (x + y) (x9 + y9) (x9 + y9 -   9  xy) (x9 + y9 +   9  xy)

(v) Lahtdrbzatbde dl quanratbh pd`yedcba`s iy sp`bttbef tgo cbnn`o torc

Ox.   x9 + 8   8 x - 8;

Xd`.   Be drnor td lahtdrbzo x9 + 8   8 x - 8;, wo gavo td lben twd euciors p aen q suhg tgat p + q 3 8   8  aen pq 3 -

8;. H`oar`y, ?   8   + (-9   8 ) 3 8   8   aen ?   8   Ö -9   8  3 - 8;

Xd. wo wrbto tgo cbnn`o torc 8   x8   as ?   x8   -9   x8

  x9 + 8   x8   - 8;

3 x9 + ?   x8   - 9   x8   - 8;

3 (x9 +   x8?   ) -(   x89   + 8;)

3 (x9 +   x8?   ) (   x89   + 1;   8   Ö   8 )

3 x(x + ?   8 ) -9   8 (x + ?   8 ) 3 (x + ?   8 ) (x - 9   8 )

  A`foirabh Bnoetbtbos :

(b) (a + i)9 3 a9 + 98ai + i9

(bb) (a - i)9 3 a9 - 9ai + i9

(bbb) a9 - i9 3 (a + i) (a - i)

(bv) (a + i)8 3 a8 + 8a9i + 8ai9 + i8

3 a8 + i8 + 8ai (a + i)

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CAEBXGM_CAY

  CA^GOCA^BHX

(v) (a - i)8 3 a8 - 8a9i + 8ai9 - i8

3 a8 - i8 - 8ai (a - i)

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CAEBXGM_CAY

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(vb) a8 - i8 3 (a - i) (a9 + ai + i9)

(vbb)a8 + i8 3 (a + i) (a9 -ai + i9)

(vbbb) a8 + i8 + h8 - 8 aih 3 (a + i + h) (a9 + i9 + h9 - ai - ih - ha)

(bx) (a + i + h)9

3 a9

+ i9

+ h9

+ 9ai + 9ih + 9ha

Xd`vo tgo ld``dwbef :

Ox. (b)   (8x + >y)9

(bb)   (9x + 8y) (9x -8y)

(bbb)  (2x + 9y + z)9

(bv)  (9x + 8y)8

(v)   (x + 8y) (x9 - 8xy + 2x9)

(vb)  (x + y + 9z) (x9 + y9 + >z9 - xy - 9yz - 9zx)

(vbb) Bl x9 +9

1

x3 78. Lben tgo va`uo dl x8 -

8

1

x.

Xd`. (b)   Vo gavo,

(8x + >y)9 3 (8x)9 + 9 Ö 8x Ö >y + (>y) 9 3 2x9 + 9>xy + 1<y9

(bb)   Vo gavo,

(9x + 8y) (9x - 8y)

3 (9x)9 - (8y)9 ]_sbef (a + i) (a - i) 3 a 9 - i9)S

3 >a9 - 2y9

(bbb)   Vo gavo, ]usbef (a + i + h)9 3 a9 + i9 + h9 + 9ai + 9ih + 9haS

(2x + 9y + z)9

3 (2x)9 + (9y)9 + z9 + 9 Ö 2x Ö 9y + 9 Ö 9y xz + 9 Ö 2x Ö z

3 71x9 + >y9 + z9 + 8<xy + >yz + 17xy

(bv)   Vo gavo, ]usbef (a + i)8

3 a8

+ i8

+ 8ai (a + i)S

(9x + 8y)8

3 (9x)8 + (8y)8 + 8 Ö 9x Ö 8y Ö (9x + 8y)

  (9x + 8y)8 3 7x8 + 96y8 + 17xy Ö 9x + 17xy Ö 8y

  (9x + 8y)8 3 7x8 + 96y8 + 8<x9y + ?>xy9

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CAEBXGM_CAY

  CA^GOCA^BHX

(v)   Vo gavo,

(x + 8y) (x9 - 8xy + 2y9)

3 (x + 8y) {(x9 - x Ö 8y + (8y)9S

3 (a + i) (a9

- ai + i9

), wgoro, a 3 x aen i 3 8y3 a8 + i8 ](a + i) (a9 - ai + i9) 3 a8 + i8S

3 x8 + (8y)8 3 x8 + 96y8

(vb)  Vo gavo,

(x + y + 9x) (x9 + y9 + >z9 - xy - 9yz - 9zx)

3 (x + y + 9z) (x9 + y9 + (9z)9 - x Ö y - y Ö 9x - 9z Ö xS

3 (a + i + h) (a9 + i9 + h9 - ai - ih - ha), wgoro a 3 x, i 3 y, h 3 9z

3 a8 + i8 + h8 - 8aih

3 x8 + y8 + (9z)8 - x Ö xy Ö 9z

3 x8 + y8 + 7z8 - <xyz

(vbb) Vo gavo,

911

9

9

9

xx

xx

3   9781

  9

xx   ][uttbef x9 +

9

1

x3 78S

3   711

  9

xx

3   92

1

xx

3   21

xx   ]^ambef squaro rddt dl idtg sbnosS

3   8

8

21

xx   ]Huibef idtg sbnosS

3   6921

81

8

8  

xx

xx

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CAEBXGM_CAY

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3   6?<1

966921

692281

8

8

8

8

8

8 x

xx

xx

x

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CAEBXGM_CAY

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@BEOAOY OQ_A^BDEX BE ^VD \AYBAI@OX

  OQ_A^BDEX :   A statocoet dl oqua`bty dl twd a`foirabh oxprossbdes wgbhg bevd`vo deo dr cdro uemedwe

quaetbty bs medwe as ae oquatbde.   Oxacp`o : 9x - ? 3 98.

 Ae oquatbde gas twd parts. ^go part wgbhg bs de tgo `olt sbno td tgo qua`bty sbfe bs medwe as `olt gaen sbno

(@.G.X.) aen tgo part wgbhg bs edt tgo rbfgt sbno td tgo oqua`bty sbfe bs medwe as rbfgt gaen sbno (Y.G.X.)

Oqua`bty

Hdesbnor ae oquatbde     9x ‘ ? 3 98

@.G.X. Y.G.X.

\AYBAI@O :  ^go uemedwe quaetbtbos uson be aey oquatbde aro medwe as varbai`os. Foeora``y, tgoy aro

noedton iy tgo `ast Oef`bsg a`pgaiots x, y, z oth.

  @BEOAY OQ_A^BDE :   Ae oquatbde be wgbhg tgo caxbcuc pdwor dl varbai`o bs deo bs ha``on a `beoar 

oquatbde.

  @BEOAY OQ_A BDE BE DEO \AYBAI@O :-  Ae oquatbde dl tgo ldrc ax + i 3 ;. wgoro x bs varbai`o, a.i. aro

roa` euciors aen a     ; bs ha``on @beoar oquatbde be deo varbai`o.

XD@_^BDE :   Aey va`uo dl tgo varbai`o tgat satbslbos tgo fbvoe oquatbde bs ha``on a sd`utbde dr rddts dl tgo

oquatbde.

Ox.   Xd`vo : 8x + 9 3 11

Xd`.   8x + 9 3 11

  8x 3 11 - 9   8x 3 2  x 38

23 8

  @BEOAY OQ_A BDEX BE ^VD \AYBAI@OX :

NOLBEBA BDE : Ae oquatbde dl tgo ldrc ax + iy + h 3 ;, wgoro a,i,h aro roa` eucior, a     ;, i    ; aen x.

XD@_^BDE :  @ot ax + iy + h 3 ;, wgoro a,i,h aro roa` eucior suhg tgat a     ; & i     ; tgoe aey pabr dl 

va`uos dl x & y wgbhg satbslbos tgo oquatbde ax + iy + h 3 ; bs ha``on a sd`utbde dl bt.

Edtos : (b)   Xd`utbde dl a `beoar oquatbde bs wrbttoe as drnoron pabr    ),(  

(bb)   Aey oquatbde be twd varbai`o gas at `oast twd sd`utbde.

(bbb)  A `beoar oquatbde be td varbai`o at `oast twd sd`utbde.

Ox.   Xgdw tgat (x 3 1, y 3 1) bs a sd`utbde dl >x - y - 8 3 ;.

Xd`.   Bl wo put x 3 1 aen y 3 1 be tgo fbvoe oquatbde.

wo gavo @.G.X. 3 > Ö 1 - 1 - 8 3 ; 3 Y.G.X.sd, x 3 1, y 3 1 bs a sd`utbde dl >x - y - 8 3 ;

Ox.   Vgbhg dl tgo ld``dwbef oquatbdes gavo a uebquo sd`utbde dr belbebto`y caey sd`utbdes.

(b)  x + 1 3 1<   (bb)  y 3 8x + 9   (bbb)x

y39

Xd`. (b)   ^go fbvoe oquatbde bs x + 1 3 1<

  x 3 < - 1  x 3 ?

 x + ;.y - ? 3;, ....(b)

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CAEBXGM_CAY

  CA^GOCA^BHX

wgbhg bs `beoar oquatbde bs deo varbai`o.

Goeho, oquatbde (b) gas uebquo sd`utbde

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CAEBXGM_CAY

  CA^GOCA^BHX

(bb)   ^go fbvoe oquatbde bs y 3 8x + 9

  8x + (-1)y + 9 3 ; ....(bb)

wgbhg bs `beoar oquatbde be td varbai`o.

Goeho, oquatbde (bb) gas belbebto`y caey sd`utbde

(bbb)  ^go fbvoe oquatbde bs   9x

y

  9x - y 3 ;

  9x + (-1) y + ; 3 ; ….(bbb)

wgbhg bs `beoar oquatbde be twd varbai`o.

Goeho, oquatbde (bbb) gas belbebto`y caey sd`utbde

(A) FYA[G DL A @BEOAY OQ_A BDE BE DEO \AYBAI@OX :

Be drnor td nraw tgo frapg dl `beoar oquatbde be deo varbai`o wo cay ld``dw tgo ld``dwbef a`fdrbtgc.

Xtop-B :   Ditabe tgo `beoar oquatbde.

Xtop-BB :   Bl tgo oquatbde bs dl tgo ldrc ax 3 i, a    ; tgoe p`dt tgo pdbet    

 ;,

a

iaen deo cdro pdbet  

 ,

a

i, wgoro

  bs aey roa` eucior de tgo frapg papor. Bl tgo oquatbde bs dl tgo ldrc ay 3 i, a    ;, tgoe p`dt tgo pdbet

 a

i,;   aen  

a

i,   , wgoro   bs aey roa` eucior de tgo frapg papor.

Xtop-BBB:   Jdbe tgo pdbets pdtton be stop-BB td ditabeon tgo roqubron `beo:-

Yocarm B :   Bl be tgo `beoar oquatbde ax 3 i, a     ; aen wo gavo i 3 ;, tgoe bts frapg bs x - axbs.

Ox.   Nraw tgo frapg dl tgo ld``dwbef `beoar oquatbde x - 9 3 ;.

Xd`.   Vo gavo x - 9 3 ;  x 3 9

H`oar`y, bt ndos edt hdetabe y, sd bts frapg bs a `beo para``o` dl y axbs passbef tgrdufg tgo pdbet (9, ;). Be laht bt

passos tgrdufg ovory pdbet wgdso x-hd-drnbeato bs 9. ^gus wo gavo tgo ld``dwbef tai`o oxgbibtbef tgo hd-

drnbeatos dl pdbets de tgo `beo roprosoeton iy x 3 ;.

[`dttbef aey twd pdbet say (9, ;), (9, >) aen (9, -1) fbvoe iy tgo aidvo tai`o de tgo frapg papor aen jdbebef

tgoc wo ditabe tgo strabfgt as sgdwe lbf.

x 9 9 9

y ; > -1

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(I) FYA[G DL A @BEOAY OQ_A^BDE BE ^VD \AYBAI@O, VGOYO a     ;, i     ;:

Be drnor td nraw tgo frapg dl a `beoar oquatbde ax + iy + h 3 ;, a     ;, i     ;, wo cay ld``dw tgo ld``dwbef

a`fdrbtgc.

Xtop-B :   Ditabe tgo `beoar oquatbde `ot tgo oquatbde io ax + iy + h 3 ;.

Xtop-BB :   Oxpross y be torcs dl x td ditabe y 3 -    

    i

hax.

Xtop-BBB :   [ut aey twd dr tgroo va`uos dl a x aen ha`hu`ato tgo hdrrospdenbef va`uo dl y lrdc tgo

oxprossbde be stop-BB td ditabe twd sd`utbde say11

  ,   aen   ),(99

    bl pdssbi`o tamo va`uo dl x as betofors be

suhg a caeeor tgat tgo hdrrospdenbef va`uo dl y aro a`sd betofors.

Xtop-B\ :   [`dt pdbets   )( ,   11    aen   ),(

99    .

Xtop-\ :   Jdbe tgo pdbets carmon be stop B\ td ditabe a `beo.

^go `beo ditabeon bs tgo frapg dl tgo oquatbde ax + iy + h 3 ;.

Ox.   Nraw tgo frapg dl tgo ld``dwbef oquatbde y - x 3 9.

Xd`.   Vo gavo,

y - x 3 9

y 3 x + 9

wgoe x 3 1, y 3 1 + 9 3 8

wgoe x 3 9, y 3 9 + 9 3 >

wgoe x 3 8, y 3 8 + 9 3 ?

^gus wo gavo tgo ld``dwbef tai`o oxgbibtbef tgo aishbssa aen drnbeatos dl pdbets de tgo `beo roprosoeton iy

tgo fbvoe oquatbde.

[`dttbef tgo pdbets (1, 8), (9, >) aen (8, ?) de tgo frapg papor aen nrawbef tgo `beo jdbebef tgoc wo ditabe

tgo frapg dl tgo `beo roprosoeton iy tgo fbvoe oquatbde as sgdwe lbf.

x 1 9 8

y 8 > ?

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  NBLLOYOE^ LDYCX DL A @BEO :-

X`dpo dl a `beo :

Bl `a `beo camos ae aef`o    wbtg pdsbtbvo nbrohtbde dl x-axbs tgoe taefoet dl tgbs aef`o bs ha``on tgo s`dpo dl a

`beo, bt bs noedton iy c b.o. c 3 tae .

^go oquatbde dl a `beo passbef tgrdufg drbfbe bs y 3 cx. Goro h 3 ; tgoe tgo `beo passos a`ways lrdc drbfbe

c bs ha``on tgo s`dpo dl tgo `beo.

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OWOYHBXO - 1 [D@REDCBA@

1.   Deo dl tgo lahtdr dl tgo oxprossbde x> + 7x bs -

(A) x + 9 (I) x - 9 (H) x9 + 7 (N) x9 + 9x + 9

9.   Deo dl tgo lahtdr dl tgo oxprossbde (a + i)8 - (a - i)8 bs -

(A) a (I) 8a9 - i (H) 9i (N) (a + i) (a - i)

8.   Bl (p(x) 3 9x8 + ?x9 - 8x - 9 bs nbvbnon iy (x - 1) tgoe rocabenor bs -

(A) 9 (I) -1 (H) ; (N) -9

>.   Zord dl tgo pd`yedcba`s 9x> + 2x8 + 11x9 + >x - < bs -

(A) 1 (I) -1 (H) 8 (N) -8

?.   Bl x8 + (x + c `oavos tgo rocabenor 6, wgoe nbvbnon iy (x - 1) dr (x + 1) lben aen c -

<.   Vrbto tgo hdollbhboet dl x9 be oahg dl tgo ld``dwbef -

(b) 9 + x9 + x (bb) 8 - x9 + x8 (bbb)   xx     9

9

( bv)   89   x

6.   \orbly wgotgor tgo ld` dwbef aro zords (rddts) dl tgo pd`yedcba`s, benbhaton afabest tgoc -

(b) p(x) 3 8x + 1, x 38

1(bb) p(y) 3 ?y -

?

>,

7.   Bl x 38

>bs rddt dl tgo pd`yedcba` p(x) 3 <x8 - 11x9 + 2x - 9;, lben tgo va`uo dl a.

2.   Lben tgo rocabenor, wgoe x8 - ax9 + <x - a bs nbvbnon iy (x - a). A`sd lben tgo rocabenor usbef nbvbsbde

cotgdn.

1;.   _sbef lahtdr tgodroc, sgdw tgat (x - 1) bs a lahtdr dl 9x> + 2x8 + <x9 - 11x - <.

11.   _sbef lahtdr tgodroc sgdw tgat (x +   9  ) bs a lahtdr dl (   99   x9 + ?x +   9  )

19.   Lben tgo va`uo dl m ldr wgbhg tgo pd`yedcba` x> - x8 - >x9 - x + m bs nbvbnon iy (x + 8).

18.   Lahtdrbos : x8 - 8x9 - 2x -?

1>.   Lahtdrbos :8

>

9

8   9 xx

1?.   Lahtdrbos : 9x8 - x9 - 18x - <.

[D@REDCBA@ OWOYHBXO -1

1. A   9. H   8. A   >. A   ?.  ; 3 1, c 3 6   <. (b) 1 (bb) -1 (bbb)9

, (bv) ;

6. (b)8

1   , (bb)   ;>     )(   7. 12 2.?2   1;. ;   11. ;   19. -19

18. (x - ?) (x + 1)9 1>.    

 8

>

9

8x

x1?. (x + 9) (x - 8) (9x + 1)

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OWOYHBXO - 9 @BEOAY OQ_A^BDE BE VD \AYBAI@OX.

\OYR XGDY^ AEXVOY ^R[O Q_OX^BDEX :

1.   Be oahg dl tgo ld``dwbef vorbly wgotgor tgo fbvoe va`uo dl tgo x bs a sd`utbde dr edt.

(b)   ,xx

7>8   x 3 19

(bb)   8x + 9 3 11,x 3 8

(bbb)  9x + < 3 x - 7, x 3 - 1>

(bv)  >x - y - 8 3 ;, x 3 -, y 3 1

(v)   8x + ?y - 9 3 ; x 3 9, y 3 -1

9.   Xd`utbde dl tgo oquatbde x - 9y 3 9 bs/aro -

(A) x 3 >. y 3 1 (I) x 3 9, y 3 ; (H) x 3 <, y 3 9 (N) A`` dl tgoso

8.   ^go frapg dl `beo ?x + 8y 3 >, huts y axbs at tgo pdbet -

(A)    

 8

>;,   (I)  

 >

8;,   (H)  

 ;

?

>,   (N)  

 ;

>

?,

>.   Bl x 3 1, y 3 1 bs a sd`utbde dl oquatbde 2ax + 19ay 3 <8 tgoe tgo va`uo dl a bs -

(A) -8 (I) 8 (H) 6 (N) ?

?.   Lben tgroo sd`utbde dl oquatbde : 9x + y 3 6.

<.   Lben tgo va`uo dl m bl x 3 9. y 3 1 bs sd`utbde dl tgo oquatbde 9x + 8y 3 m

6.   (9p - 1, p) bs sd`utbde dl oquatbde 1; x - 2y 3 1?, lben tgo va`uo dl p.

7.   A `beoar oquatbde dl tgo ldrc y 3 cx, a`ways pass tgrdufg tgo -2.   Bl tgo hd-drnbeatos dl a pdbet p aro (-9, ?) tgoe aishbssa & drnbeatos aro -

1;.   ^go frapg dl a `beoar oquatbde be twd varbai`os bs a ..........

11.   X`dp dl tgo beo ax + iy + h 3 ; bs.......

19.   Oquatbde dl x-axbs bs ....... aen oquatbde dl y-axbs bs ............

18.   ^go frapg dl tgo `beo x 3 a bs para` o` td ..............

1>.   ^go frapg dl tgo `beo y 3 i bs para` o` td ...........

@BEOAY OQ_A^BDEX BE VD \AYBAI@OX OWOYHBXO - 9

1. (b) Ed.   (bb) Ros,   (bbb) Ros,   (bv)  Ros,   (v) Ed   9. A   8.  I   >. I   ?.  (;, 6), (1, ?), (9, 8)

<. m 3 6   6.11

9?p   7. Drbfbe,   2. -9, ?   1;. Xtrabfgt `beo   11.

i

a

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19. y 3 ;, x3 ;   18. y-axbs   1>. x-axbs

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OWOYHBXO - 8 @BEOAY OQ_A^BDE BE VD \AYBAI@OX

1.   Lben dut wgbhg dl tgo ld` dwbef oquatbde gavo x 3 9, y 3 1 as a sd`utbde.

(b) 9x + ?y 3 2 (bb) x+ y + > 3 ;

(bbb) 9x - 8y + 6 3 7 (bv) 9x - 8y 3 1

9.   Lben tgo va`uo dl  ‗a‒  sd tgat oahg dl tgo ld``dwbef oquatbde cay gavo x3 1, y3 1 as a sd`utbde.

(b) 8x + ay 3 < (bb) x - y 3 a

(bbb) ?x + 9ay 3 8a (bv) 2ax + 19ay 3 <8

8.   Lben tgo va`uo dl x & y fbvoe oquatbde : 9x - 8y - 8 3 ;,   ;9

1>

8

9   y

x

>.   Nraw tgo frapg dl oahg dl tgo ld``dwbef oquatbdes -

(b) 9(x + 8) -8(y + 1 ) 3 ; (bb) (x - >) - y + > 3 ;

(bbb) x - 9y 3 > (bv) ?x + 8y 3 >

?.   Nraw tgo frapg dl tgo `beo >x - y 3 ? & ?y - >x 3 6 de tgo saco frapg pafor & lben tgo hd-drnbeato dl tgobr 

pdbet dl betorsohtbde.

<.   Nraw tgo frapg dl x - y + 1 3 ; & 8x + 9y - 19 3 ; de + > saco frapg. Ha`hu`ato tgo aroa iduenon iy tgoso

`beo & x-axbs.

6.   ^go sd`utbde dl 8x - 9y 3 ? bs -

7.   Nraw tgo frapg dl tgo oquatbde 8x + 6y 3 1;. Lben wgotgor x 3 1 aen y 3 1 bs tgo sd`utbde dl tgoso oquatbde.

2.   Nraw tgo frapg y 3 - 9x. Xgdw tgat tgo pdbet (9, - ?) bs edt de tgo frapg.

1;.   Bl 9x - 7 3 7, tgoc lben tgo va`uo dl x   9 + x - 6;.

@BEOAY OQ_A^BDEX BE VD \AYBAI@OX OWOYHBXO - 8

1. (b), (bbb), (bv) 9. (b)  a 3 8,   (bb) a 3 ;,   (bbb)  a 3 ?,   (bv) a 3 8.   8.1;

8

1;

91     y.x

6. x 3 1, y 3 -1   1;. 9