Polynomials Linear Equations in Two Variables
Transcript of 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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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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(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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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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(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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(v) (a - i)8 3 a8 - 8a9i + 8ai9 - i8
3 a8 - i8 - 8ai (a - i)
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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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(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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3 6?<1
966921
692281
8
8
8
8
8
8 x
xx
xx
x
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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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wgbhg bs `beoar oquatbde bs deo varbai`o.
Goeho, oquatbde (b) gas uebquo sd`utbde
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(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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CAEBXGM_CAY
CA^GOCA^BHX
(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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CAEBXGM_CAY
CA^GOCA^BHX
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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CAEBXGM_CAY
CA^GOCA^BHX
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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CAEBXGM_CAY
CA^GOCA^BHX
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