kem nknti v ti ü ati 10 kmw c an kmw c an ; kem nkn ti v ... · emeronti4...

kME NKNitv iTü aTI 1 0 Kwm c an ;

- 205 -

emeronTI4 dMeNaHRsayvismIkartamRkab lMhat;

1. eda HR sa yv ism Ik a r nig RbB n§½v ism Ik a rxa g eRk a mtamR k a b³ k> 2 4 0x y+ − > x> 2 21 4x y< + <

K> 2 2 2 3x y x+ ≤ + X> 2 2 9, 2x y x y+ ≤ + >

g> 2 20 1, 0 1, 1x y x y< < < < + > 2. k > sg ;bnÞa t; : 2D y x= − + nig : 2 2 0D x y′ − − = k ñúg tMru yE tm Yy.

x> Ta jrk KUc Me lI yRb B n§½sm Ik a rta mRk a Pic 2

2 2 0

y x

x y

= − +

− − =

3. eda HR sa yR bB n§½vis m Ik a rxa g eR k a m³

k>4 0

2

2

x

y x

y

+ < ≥ − <

x>4 3 6 0

2 5

3 4 0

x y

y x

x y

+ + ≥ ≥ − − ≥

K>0, 0

5 10

6

4 12

x y

x y

x y

x y

≥ ≥ + ≥+ ≥ + ≥

Kwm c an ; kME NKN itv iTü aTI1 0

- 206 -

cMelIy 1. eda HR sa yv ism Ik a r nig RbB n§½vism Ik a rta mRk a b³

k> 2 4 0x y+ − > - sg;bnÞa t ; 1

2 4 0, 2 4, 22

x y y x y x+ − = = − + = − +

eyIg )a n c MelI yrbs; 2 4 0x y+ − > Ca E pñ k E dl qUt x> 2 21 4x y< + < - sg;rg Vg ; 2 2 1x y+ = nig 2 2 4x y+ =

eyIg )a n t Mbn;cMel Iy KW Ca E p ñ k E dlqUt


- 207 -

K> 2 2 2 3x y x+ ≤ + smm Ulnwg

2 2 2 2

2 2

2 3 0, 2 1 4

( 1) 4

x y x x x y

x y

+ − − ≤ − + + ≤

− + ≤

-s g ;rg Vg; 2 2( 1) 4x y− + =


X> 2 2 9, 2x y x y+ ≤ + > -sg ;rgVg; 2 2 9x y+ = nig b nÞa t; 2x y+ =



- 208 -

g> 2 20 1, 0 1, 1x y x y< < < < + > -sg;rg Vg; 2 2 1x y+ = nig bnÞa t; 0, 1, 0, 1x x y y= = = =


2. k > sg;b nÞa t ; : 2D y x= − + nig : 2 2 0D x y′ − − = k ñúg tMru yE tm Yy eyIg ma n : 2D y x= − +

: 2 2 0D x y′ − − =


- 209 -

x> T a jrk KUc Mel IyR bB n§½s m Ik a rta mRk a Pic 2

2 2 0

y x

x y

= − +

− − =

ta mR k a b D D′∩ )a n (2, 0)I dUce nHR b B n§½sm Ik a rma nrws 2, 0x y= = 3. eda HR sa yR bB n§½vis m Ik a r³

k>4 0

2

2

x

y x

y

+ < ≥ − <

-sg ;b nÞa t; 4 0, 2, 2x y x y+ = = − =

eyIg )a ncMel I yrb s ;R bB n§½v i sm Ik a rCa E pñk qUt

x>4 3 6 0

2 5

3 4 0

x y

y x

x y

+ + ≥ ≥ − − ≥

-sg ;bnÞa t; 4 3 6 0, 2 5, 3 4 0x y y x x y+ + = = − − =


- 210 -

eyIg )a nc MelI yrb s ;R bB n§½v i sm Ik a rCa E pñk qUt

K>0, 0

5 10

6

4 12

x y

x y

x y

x y

≥ ≥ + ≥+ ≥ + ≥

-sg ;b nÞa t; 0, 0, 5 10, 6, 4 12x y x y x y x y= = + = + = + = eyIg )a nc MelI yrb s ;R bB n§½v i sm Ik a rCa E pñk qUt


- 211 -

lMhat;CMBUk4 1. eK[c M Nu cB Ir ( 2)A− nig (5)B . tag C Ca c MN ucE ck k ñúg é n AB ta m pleF ob3 : 2 nig D Ca c MNucE c k eRk Aé n AB tamp leFo b 1 : 3 rk RbE vgCD . 2. rkkUG reda ené nc M nuc M E ckk ñúg nig N Eck eRk AénG gát; AB t ampleF o b 1 : 3 c MeB a HKU cM Nuc ( 1, 3), ( 2, 5)A B− − − rYc rk R bE vgMN . 3. rk sm Ik a rbnÞa t;é nemdüa T ½r rbs;G g át; AB c MeB a H (2, 3)A nig (4,5)B . 4. smIk a ré nbnÞa t; 1l nig 2l KW 3 4 18 0x y+ − = ni g 2 4 0x y− + = ero g Kña . rk tMé lénc MnYn ef r k ebIbnÞa t ; y kx= k a t;ta mcMNuc R bsB Vén 1l nig 2l . 5. rktMé lc MnY nef r a eb IcMNu c TaMg bI ( , 4), ( 2, 6)A a a B+ − nig (7,5)C s ßitenA elIbnÞa t;E tm Yy. 6. r képÞRkL aén R tIe kaNEd l xN Ðe day bnÞat;b I 7 0, 3 2 1 0x y x y+ − = − − = nig 4 3 0x y− + = . 7. k Mnt;p © it nig k a Mrg Vg ;E dl k a t;t a mbIc M Nuc (8, 4), (3, 1)A B − nig (6,8)C . 8. rk sm Ik a rénbnÞa t;Edlk a t;t a mc MNuc (1,5)eh Iyb: Hrg Vg ; 2 2 1x y+ = . 9. k Mnt;R bB n§½vis m Ik a rE dl ma ntMbn;c MelI ysß itk ñúg R tIek a N ABC E dl ma n kMB Ul (2, 0), ( 3, 0)A B − nig (0, 4)C ehI yR Cug é nR t Iek a Nm inKit bB a ©ÚleT. 10. rktMé lé nc M nYnef r a [b nÞa t; 4 1ax y+ = nig ( 3) 2x a y+ − = R sbKña .


- 212 -

11. rktMé lé nc M nYnefr a [b nÞa t; 1 0ax y+ + = nig rg Vg ; 2 2 4 3 0x y x+ + + = b :HK ña . 12. eK[c M Nu cnwg B Ir (2, 0)A nig (0,1)B . rk sMNMuc M Nu cénc MN uc P E dl epÞó g pÞa t; : 2 : 1AP PB = . 13. rkkUG reda ené nc M NucR b sBVrva g rg Vg; 2 2 4 4 0x y y+ − − = nig 2 2 2 0x y x+ − = . 14. k MNt ;tMé lrbs; r [rg V g ; 2 2( 4) ( 3) 4x y− + − = nig 2 2 2x y r+ = minma ncMN ucr Ym.

�kMsanþ cUreR bIka rKi tr bs;Gñk >>> >rk nUv c M el Iyén sMN Yrx ag eR k am ³

mYysMN Yr GnuBa Øat[eR bIeBlE t 1naTI b:ueNÑaH>>> 1. I appear once in a year and twice in a decade. What am I?

2. What are the next three letters to this sequence?

J, F, M, A, M, J, J, A, S, _, _, _.

3. How do you make the number one disappear?

4. Imagine you were surrounded by sharks. How do you get

away?

5. When does red mean go and green mean stop?

6. What appears twice in a week, once in a year and twice in a

leap year but never in a day?

7. How do you make the number seven even?

e bIGñkc MN aye B lt icC ag 15 n aTIe TAe lIsMN YrT aMg G s; m an n ½y f akMrit k ar Kit rb s;Gñk KW man k Mr itx<s; � c Me lI yn wg b g ðaje n Ae so ve PA P aK2 >>>


- 213 -

cMelIy

1. rkR bE vgCD .

eyIg ma n ( 2)A− nig (5)B - C Ca cM NucE ck k ñúg én AB t a mpleF o b3 : 2 eyIg )a n

C

mb nax

m n

+=

+eda y 3, 2, 2, 5m n a b= = = − = ena H

( ) ( )3 5 2 2 11 11, ( )

3 2 5 5Cx C

+ −= =

+

- DCa c MNucE ck eR k Aé n AB ta mp leFo b 1 : 3 eyIg )a n

D

mb nax

m n

−=

−eda y 1, 3, 2, 5m n a b= = = − = ena H

( ) ( )1 5 3 2 11 11, ( )

1 3 2 2Dx D

− −= = − −

−

dUce nH 11 11 777.7

2 5 10CD = − − = =

2. rYcrk R bE vgMN eyIg ma n ( 1, 3), ( 2, 5)A B− − − - M Eck k ñúg é nGg át; AB ta mp leFo b1 : 3 eyIg )a n

( ) ( ) ( ) ( )1 2 3 1 1 5 3 3

, , ,1 3 1 3

5( ,1)4

b a b amx nx my nyM M

m n m n

M

+ + − + − − + + + + +

−

-N E ck eRk AénGg át; AB t a mpleF o b1 : 3 eyIg )a n

( ) ( ) ( ) ( )1 2 3 1 1 5 3 3

, , ,1 3 1 3

1( ,7)2

b a b amx nx my nyN N

m n m n

N

− − − − − − − − − − −

−


- 214 -

dUce nHR bE vg KW 2 2( ) ( )N M N MMN x x y y= − + − 2 21 5 3

( ) (7 1) 652 4 4

MN = − + + − = 3. rk sm Ik a rbnÞa t;é nemdüa T ½r rbs;G g át; AB c MeB a H (2, 3)A nig (4,5)B

-sm Ik a remdüa T ½rénG g át; AB KW Ca s m Ik a rbnÞa t;E dl Ek g nwg AB R tg ;cMNu ck Nþa l -k UG reda enc MNu ck Nþa lAB

2 4 3 5( , ), ( , ), (3, 4)2 2 2 2

A B A Bx x y yI I I

+ + + + -sm Ik a rbnÞa t; ( )L k a t;ta m AB

5 3( ), 3 ( 2)

4 2B A

A A

B A

y yy y x x y x

x x

− −− = − − = −

− −

( ) : 1L y x= + -sm Ik a rbnÞa t;emdüa T ½ré nG g át; AB m a nra g ( ) :M y mx n= + -sm Ik a renHk a t;ta m (3, 4)I eyIg ) a n

( )4 3 , 3 4 1m n m n= × + + = -m üa :g eT o t ( ) ( )M L⊥ ena H 1m m′× = − eda y ' 1m = enaH

1 1, 1m m× = − = − CMn Ys( )1 )a n ( )3 1 4, 7n n− + = =

dUcen H sm Ik a rbnÞa t;emdü a T½ré nG g át; AB K W ( ) : 7M y x= − +


- 215 -

4. rktMé lé nc M nYnef r k

eyIg ma n 1 :l 3 4 18 0x y+ − = nig 2 :l 2 4 0x y− + = -R bsB Vrva g bnÞa t;T a MgB Ir

( )

( )

3 4 18 0 1

2 4 0 2

x y

x y

+ − = − + =

-yk ( ) ( )1 2 2+ × )a n 5 10 0, 2x x− = = CMn Ysk ñúg ( )2 )a n 2 2 4 0 , 3y y− + = =

eyIg )a n { }1 2 2, 3l l N∩ = -bnÞa t; y kx= k a t;ta mcMNu cRb sBVé n 1l nig 2l ena Hk UG reda enc Mnuc N epÞó g pÞa t;s m Ik a r

3, 3 2,

2y kx k k= = × =

dUce nHc MnY nefr 3

2k =

5. rktMé lc MnY nef r a -eyIg ma n ( , 4), ( 2, 6)A a a B+ − nig (7,5)C s ßitenAelIb nÞa t;E t mYy -yk ( ) :L y mx n= + sm Ik a rb nÞa t;k a t;ta m ,B C


- 216 -

-( )L k a t;t a m ( 2, 6)B − ena H ( ) ( )6 2 , 2 6 1m n m n= − + − + = -( )L k a t;t a m (7,5)C ena H ( ) ( )5 7 , 7 5 2m n m n= + + = yk ( ) ( )1 2− )a n 1

9 1,9

m m− = = − CM nYs ( )1 )a n 1 52

2 6,9 9

n n − − + = =

eyIg )a n ( )

1 52:

9 9L y x= − +

-eda y , ,A B C s ßitenAelIbn Þa t;E tm Yy ena H ( , 4)A a a + epÞó g pÞa t;sm Ik a r( )L

1 52 84 ,

9 9 5a a a+ = − + ⇒ =

dUce nHc MnY nefrK W 8

5a =

6. rk é pÞR k La é nR t Iek a NE dlxN Ðeda yb nÞa t;Ta Mg bI eyIg ma n 7 0, 3 2 1 0x y x y+ − = − − = nig 4 3 0x y− + =

-sg ;bnÞa t;T aMg bI


- 217 -

-ta g ABCS Ca R k La é pÞR tIek a NEdlx NÐe da yb nÞa t;Ta Mg bI -H Ca cMeNa lE k g é n A elI BC eyI g )a nR k La é pÞR tIek a NKW

( )1

2ABCS BC AH= ×

-rk R bE vg ,BC AH -C Ca R bsB Vrva g bnÞa t; 4 3 0x y− + = nig 3 2 1 0x y− − = eyIg )a n

3 2 1 0

4 3 0

x y

x y

− − =

− + =

eda HR sa yR bB n§½sm Ik a reyIg )a n 1, 1 (1,1)x y C= = ⇒ -B Ca Rb sB Vrva g bnÞa t; 4 3 0x y− + = nig 7 0x y+ − = eyIg )a n

7 0

4 3 0

x y

x y

+ − =

− + =

eda HR sa yR bB n§½s m Ik a reyIg )a n 5, 2 (5, 2)x y B= = ⇒ -ACa R bsB Vrva g bnÞa t; 3 2 1 0x y− − = nig 7 0x y+ − = eyIg )a n

7 0

3 2 1 0

x y

x y

+ − =

− − =

eda HR sa yR bB n§½s m Ik a reyIg )a n 3, 4 (3, 4)x y A= = ⇒ -R bE vg AH Ca c Mg a yB Ic MNuc (3, 4)A eT Ab nÞa t; 4 3 0x y− + =

eyIg )a n ( )

2 2 2 2

1 3 4 4 3 10( , )

171 4

o oax by cd A L

a b

+ + × − × += = =

+ +

10

17AH =


- 218 -

-R bE vg BC ( ) ( ) ( ) ( )

2 2 2 21 5 1 2

17

C B C BBC x x y y

BC

= − + − = − + −

=

dUcenH 1 1017 5

2 17ABCS

= × = Ék ta é pÞR k La

7. k Mnt;p © it nig ka Mrg Vg ;E dlk a t;ta mbIcM Nuc (8, 4), (3, 1)A B − nig (6,8)C -sm Ik a rrg Vg;T UeT Ama nrag 2 2 0x y ax by c+ + + + = -rg Vg ;k at;tam (8, 4)A ena H ey Ig)a n ( )2 28 4 8 4 0, 8 4 80 0 1a b c a b c+ + × + × + = + + + = -rg Vg ;k at;tam (3, 1)B − ena H e yIg )a n ( ) ( ) ( )223 1 3 1 0, 3 10 0 2a b c a b c+ − + × + × − + = − + + = -rg Vg ;k at;tam (6,8)C ena H ey Ig)a n ( )2 26 8 6 8 0, 6 8 100 0 3a b c a b c+ + × + × + = + + + =

ta m ( ) ( ) ( )1 , 2 & 3 eyIg )a n ( )

( )

( )

8 4 80 0 1

3 10 0 2

6 8 100 0 3

a b c

a b c

a b c

+ + + = − + + = + + + =

-yk ( ) ( )1 2− ey Ig )a n ( )5 5 70 0, 14 0 *a b a b+ + = + + = -yk ( ) ( )3 2− ey Ig )a n ( )3 9 90 0, 3 30 0 * *a b a b+ + = + + = -yk( ) ( )* * *− eyIg )a n 2 16 0, 8b b+ = = − CM nYs ( )* )a n 8 14 0, 6a a− + = = − yk 6, 8a b= − = − C MnYs ( )2 )a n


- 219 -

( ) ( )3 6 8 10 0, 0c c− − − + + = = na MeG a y 2 2 6 8 0x y x y+ − − =

2 2

2 2 2

6 9 8 16 25

( 3) ( 4) 5

x x y y

x y

− + + − + =

− + − =

dUce nHrg Vg;ma np ©it (3, 4)I nig k aM 5r = 8. rk sm Ik a rénbnÞa t;Edlk a t;t a mc MNuc (1, 5)eh Iyb: Hrg Vg ; 2 2 1x y+ =

tag ( ) :L y mx n= + s m Ik a rbnÞa t;k at ;ta m (1, 5)eh Iy b:Hnwg rg Vg ; 2 2 1x y+ = -( )L k a t;t a m (1, 5)ena HeyIg )a n

( )5 1 , 5 1m n m n= × + + = -( )L ehIyb :Hn wg rg Vg ; 2 2 1x y+ = ena He yIg )a n

2 2 1

y mx n

x y

= + + =

ma nrwsE tm YyKt ; ( )

( )

2 2 2 2 22

2 2 2

2 22

1, 2 1

( 1) 2 1 0

' ( 1)( 1) 0

x mx n x m x mnx n

m x mnx n

mn m n

+ + = + + + =

+ + + − =

∆ = − + − =

( ) ( )

( )

2 2 2 22

2 2

1 0

1 0 2

mn m n m n

m n

− − + − =

− + =

-ta m ( ) ( )1 & 2 )a n ( )

( )2 2

5 1

1 0 2

m n

m n

+ = − + =

( )

( )

5 1

( )( ) 1 0 2

m n

m n m n

+ = + − + =


- 220 -

( )

( )

5 1

5( ) 1 0 2

m n

m n

+ = − + =

( )

( )

5 1

5 5 1 0 2

m n

m n

+ = − + =

eda HR s a yR b B n§½eyIg )a n 12 13,5 5

m n= = dUce nHsm Ik a rb nÞa t;ena HK W 12 13

5 5y x= +

9. k Mnt;R bB n§½vis m Ik a rE dl ma ntMb n;c MelI ysß itk ñúg R tIek a N ABC eyIg ma nk MB Ul (2, 0), ( 3, 0)A B − nig (0, 4)C -sg ; cMNu cT a Mg bIel ItMr uy

-E pñk qUtKW Ca t Mbn;c Mel Iy r bs;v ism Ik a r -rk sm Ik a rbnÞa t;é nR Cug nIm Y y² rbs;R t Iek a N -sm Ik a rbnÞa t;k a t;t a m AB ma nra g ( )B A

A A

B A

y yy y x x

x x

−− = −

−

eyIg )a n ( ) ( )1

0 00 2 , 0

3 2y x y l

−− = − =

− −

-smIk a rb nÞa t;k a t;ta m AC ma nrag ( )C AA A

C A

y yy y x x

x x

−− = −

− eyIg )a n


- 221 -

( ) ( )24 0

0 2 , 2 40 2

y x y x l−

− = − =− +−

-sm Ik a rb nÞa t;k a t;ta m BC ma nrag ( )C B

B B

C B

y yy y x x

x x

−− = −

− eyIg )a n

( ) ( )34 0 4

0 3 , 40 3 3

y x y x l−

− = + = ++

eda yE p ñk tMb n;cMe lI ysßit e nAk ñúg R tIek a N ABC eh IyR Cug é nR t Iek a N m inKitb B a© Úlena H³ -E pñk c MelI yrbs ;b nÞa t ; ( )1l k at ;ta m AB sßit enAE p ñk 0y> ena Hey Ig )a n ( )0 1y> -E pñk c MelI yrbs ;b nÞa t ; ( )2l k at ;ta m AC sßitenAE pñk 4y< ena Hey Ig )a n ( )2 4 2y x<− + -E pñk c MelI yrbs ;b nÞa t ; ( )3l k at ;ta m BC sßitenAE pñk 4y< ena Hey Ig )a n ( )

44 3

3y x< +

dUce nHR b B n§½v ism Ik a rK W³ ( )

( )

( )

0 1

2 4 2

44 3

3

y

y x

y x

> < − + < +

10. rktMé lé nc M nYnef r a [b nÞa t; 4 1ax y+ = nig ( 3) 2x a y+ − = R sbKña

eyIg ma n ( )1

14 1, 4 1,

4 4

aax y y ax y l+ = = − + = − +


- 222 -

( )21 2

( 3) 2, ( 3) 2,3 3

x a y a y x y x la a

+ − = − = − + = − +− −

- 1 2&l l R sbKña ka lNa 'm m= 1 :l ma n

4

am = − nig 2 :l ma n 1

'3

ma

= −−

eyIg )a n 21

, ( 3) 4, 3 4 04 3

1, 4

aa a a a

a

a a

− = − − = − − =−= − =

dUcen H 1, 4a a= − = 11. rktMé lé nc M nYnefr a [b nÞa t; 1 0ax y+ + = nig rg Vg ; 2 2 4 3 0x y x+ + + = b:H Kña

eyIg )a nR bsB Vrva g bnÞa t; nig rg Vg ;

2 2

1 0

4 3 0

ax y

x y x

+ + = + + + =

s mm Ul ( )

( )2 2

1 1

4 3 0 2

y ax

x y x

= − − + + + =

yk ( )1 CMn Ys ( )2 ey Ig )a n ( )

( )

22 2 2 2

2 2

1 4 3 0, 2 1 4 3 0

( 1) (2 4) 4 0 *

x ax x x a x ax x

a x a x

+ − − + + = + + + + + =

+ + + + =

-bnÞa t ;b:Hn wg rg Vg ;k a lNa sm Ik a r( )* ma nrwsE t m YyKt; e n a H 2 2

2 2

2 2

2

(2 4) 4 4(1 ) 0

4( 2) 16( 1) 0

( 2) 4( 1) 0

3 4 0

a a

a a

a a

a a

∆ = + − × + =

+ − + =

+ − + =

− + =

4( 3 4) 0, 0,

3a a a a− + = = =

dUce nH 40,

3a a= =


- 223 -

12. rk sMNMu c MN ucé ncM Nuc P E dlepÞó g pÞa t ; : 2 : 1AP PB = eyIg ma n (2, 0)A nig (0,1)B -ta g ( , )P x y

: 2 : 1AP PB = smm Ul nwg 22, 2

1

APAP PB

PB= = =

( ) ( ) ( ) ( )

( )

2 2 22

2 2

2 0

2

P A P AAP x x y y x y

AP x y

= − + − = − + −

= − +

( ) ( ) ( ) ( )

( )

2 2 22

22

0 1

1

P B P BPB x x y y x y

PB x y

= − + − = − + −

= + −

eda y 2AP PB= ena H

( ) ( )

( ) ( )

22 2 2

22 2 2

2 2 2 2

2 2

2 2 1

2 4 1

4 4 4( 2 1)

3 3 4 8 0

x y x y

x y x y

x x y x y y

x y x y

− + = + −

− + = + −

− + + = + − +

+ + − =

2 2

2 2

2

2 2

4 80

3 34 4 8 16 20

3 9 3 9 9

2 4 20 20( ) ( )

3 3 9 3

x y x y

x x y y

x y

+ + − =

+ + + − + =

+ + − = =

dUcenHs M NMu cM Nuc P KW Ca rg Vg ;E dlma np© it 2 4( , )3 3

I − nig k aM 20

3r =

13. rkkUG reda ené nc M NucR b s BVrva g rg Vg; 2 2 4 4 0x y y+ − − = nig 2 2 2 0x y x+ − =

eyIg )a nR bsB Vrva g rgVg ;TaMg BIr


- 224 -

( )

( )

2 2

2 2

4 4 0 1

2 0 2

x y y

x y x

+ − − = + − =

-yk ( ) ( )1 2− ey Ig )a n ( )2 4 4 0, 2 2 *x y x y− − = = + CMn Ysk ñúg ( )1 )a n

( )2 2

2 2

2

2 2 4 4 0

4 8 4 4 4 0

5 4 0, (5 4) 0

40,

5

y y y

y y y y

y y y y

y y

+ + − − =

+ + + − − =

+ = + =

= = −

-c MeB a H 0y = CM nYsk ñúg ( )* )a n 0 2 2x = + = -c MeB a H 4

5y = − CM nYsk ñúg ( )* )a n 4 2

2 25 5

x = − + =

dUce nH rg Vg ;T aMgB IrR bsB VKña R t g;B IrcMN uc 2 4

(2, 0), ( , )5 5

A B − 14. k MNt ;tMé lrbs; r [rg V g ; 2 2( 4) ( 3) 4x y− + − = nig 2 2 2x y r+ = m inma ncMN ucr Ym

-edIm, I[ rg Vg ;T aMg B Irm inma ncMNu cr Ymlu HRt a Et r IJ r′− > E dl ( ), , 0r r r r′ ′ > Ca k a Mrg Vg ;ero g Kña énrg Vg ; ( ) 2 2 2:C x y r+ = nig ( ) 2 2: ( 4) ( 3) 4C x y′ − + − = eh IyE dlma np© i t ero g Kña ,I J

-IJ Ca cMg a yrva g p©ité nrg Vg ;T aMg B Ir eda y ( )0, 0 , (4.3)I J ena H

( ) ( )2 24 0 3 0 5IJ = − + − = nig ' 2r = eyIg )a n , 5 2r IJ r r′− > − >


- 225 -

5 2

5 2

r

r

− >

− < −

( )

( )

7 1

3 2

r

r

> <

dUcen Hed Im,I [rg Vg ;TaMg BIrKµa nc MN ucr Ym k a l Na 7r > rW 0 3r< <


- 226 -

CMBUk5 GnuKmn_ emeronTI1 GnuKmn_ nigRkabénGnuKmn_

lMhat; 1. eKma nG nuKmn_ f k MNt ;elI � eda y 2( ) 4 2f x x x= − + . c MeB a HR Kb; tMél én { }1, 0,1, 2x ∈ − k MNt ;tMé l ( )f x . 2. eKma nG nuKmn_ f k MNt ;elI � eda y 2( ) 2( 1) 1f x x= − + .

KNna 1( 1), 2 (1), ( )

2f f f− − .

3. eKma nG nuKmn_ : 3 2g x x−� . eda y ma nCMnY yB IRk a bt ag 3 2y x= − . k > rk sMNMu rUb Pa B é nG nuKm n_ g R tUvnwg E dnk MNt; { }| 1 2x x∈ − ≤ ≤� x> rk Ednk MNt ;énG n uKmn _ g R t Uvnwg s MNMu rUb Pa B { }| 3 2x y∈ − ≤ ≤�

4. rkE dnk MNt;é nG nu Kmn_ xa g eR ka m³ k> 3( ) 4 3

5

xf x x

x= − −

− x>

2

3( ) 4 3 2

3 2

xf x x

x x= − −

− +

K> 1( ) 5 4

3f x x

x= − +

− X> 3 4( ) 5 4f x x x= − − −

5. eK[G nu Kmn_ ( ) 3, 0f x x x= + ≥ nig 2( ) , 2 3g x x x= − ≤ ≤ . rksMNMur UbP a B é nG nuKmn_ ,f g . 6. eK[G nu Kmn_ ( ) 2 3f x x= − nig 2( ) 2g x x= + . sik Sa PaB ekIn rWc u H é nG nuKmn _ f nig g eda yeR bIp leFo b f

x

∆

∆nig g

x

∆

∆.


- 227 -

7. rk G nuKmn_R ca s c MeB a H G nuKmn_ ( )f x nIm Yy² xa g eRk a m³

k> ( ) 2 3f x x= + x> ( ) 2f x x= − K> 1( ) 3f x

x= − X> 3

( )1

f xx

=−

8. sg ;R k a bt ag G nuKmn _xa g eR k am³ k> 1y x= + x> 1 2y x= − + 9. k > rk G nuKmn_R ca s én G nuKmn_ ( ) 3f x x= + . x> sg ;R ka bta g G nuKmn_ f nig G nuKm n_R ca s 1

f− .

sUmrg;caMGan PaK2 Edlnwgecj pSaynaeBlqab;²enH

sUmGrKuN


- 228 -

cMelIy 1. k MNt ;tMé l ( )f x

eyIg ma n 2( ) 4 2f x x x= − + cMeB a HRKb;t Mél { }1, 0,1, 2x ∈ − ena Hey Ig ) a n

( ) ( )

( ) ( )

( )

( )

2

2

2

2

( 1) 1 4 1 2 1 4 2 7

(0) 0 4 0 2 2

(1) 1 4 1 2 1

(2) 2 4 2 2 4 8 2 2

f

f

f

f

− = − − − + = + + =

= − + =

= − + = −

= − + = − + = −

dUce nH ( 1) 7, (0) 2, (1) 1, (2) 2f f f f− = = = − = − 2. KNna 1

( 1), 2 (1), ( )2

f f f− −

yIg ma n 2( ) 2( 1) 1f x x= − + 2

2

2

( 1) 2( 1 1) 1 8 1 9

(1) 2(1 1) 1 1, 2 (1) 2

1 1 9 11( ) 2( 1) 1 2 12 2 4 2

f

f f

f

− = − − + = + =

= − + = =

− = − − + = × + =

dUce nH 1 11( 1) 9, 2 (1) 2, ( )

2 2f f f− = = − =

3. k > rk sMNMur UbPa B énG nuKmn_ g R t Uvnwg Ednk MNt; { }| 1 2x x∈ − ≤ ≤� eyIg ma n : 3 2g x x−� nig 3 2y x= −

1 2, 2 2 4, 2 2 4

4 2 2, 3 4 3 2 3 2

1 3 2 5, 1 5

x x x

x x

x y

− ≤ ≤ − ≤ ≤ ≥ − ≥ −

− ≤ − ≤ − ≤ − ≤ +

− ≤ − ≤ − ≤ ≤

eyIg )a n 1 5y− ≤ ≤


- 229 -

dUcenHsM NuM rUbP a B é nGnuK mn_ g KW { }| 1 5I x y= ∈ − ≤ ≤� x> rk E dnk MNt;é nG n u Kmn_ g R t Uvnwg sMNMu rUbPa B { }| 3 2x y∈ − ≤ ≤�

eyIg ma n 3 2y− ≤ ≤ smm Ul nwg 3 3 2 2x− ≤ − ≤

3 3 2 2 3, 6 2 1

1 66 2 1, 1 2 6,

2 21

32

x x

x x x

x

− − ≤ − ≤ − − ≤ − ≤ −

≥ ≥ ≤ ≤ ≤ ≤

≤ ≤

dUcenHE dnk MNt; énG nu Kmn_ g KW 1| 32

x x ∈ ≤ ≤

�

4. rkE dnk MNt;é nGnu Kmn_ k> 3( ) 4 3

5

xf x x

x= − −

−

eyIg ma n 3x− ma nn½yk a lN a ( )3 0, 3 1x x− ≥ ≥ 3

5

x

x− ma nn½yk a lNa ( )5 0, 5 2x x− > <

ta m ( )1 nig ( )2 ey Ig )a n 3 5x≤ < dUce nHE d nk MNt ; rbs;G nu Km n_ KW { }| 3 5D x x= ≤ <

x> 1( ) 5 4

3f x x

x= − +

−

eyIg ma n 5 4x− ma nn½yk a lNa ( )4

5 4 0, 15

x x− ≥ ≥ 1

3x− ma nn½yk a l Na ( )3 0, 3 2x x− ≠ ≠

ta m ( )1 nig ( )2 eyIg )a n 4, 35

x x≥ ≠ dUce nHE d nk MNt ; rbs;G nu Km n_ KW { }4

| , 35

D x x x= ≥ ≠


- 230 -

K>2

3( ) 4 3 2

3 2

xf x x

x x= − −

− +

eyIg ma n 3 2x− ma nn½yk a lNa 33 2 0,

2x x− ≥ ≤

2

3

3 2

x

x x− + ma nn½yk a lNa 2 3 2 0, 1, 2x x x x− + ≠ ≠ ≠

dUcenHE dnk M Nt; rb s;G nuK mn_ KW { }3| , 1

2D x x x= ≤ ≠

X> 3 4( ) 5 4f x x x= − − − eyIg ma n 3 5x− ma nn½yR Kb; x ∈ �

4 4 x− ma nn½yk a lNa 4 0, 4x x− ≥ ≤ dUce nHE dnk MNt ; rbs ;Gnu K mn_ KW { }| 4D x x= ≤

5. rk sMNMu rUbP a B é nGnuKmn _ ,f g eyIg ma n ( ) 3, 0f x x x= + ≥ nig 2( ) , 2 3g x x x= − ≤ ≤ - ( ) 3, 0f x x x= + ≥

0, 3 3, ( ) 3, 3x x f x y≥ + ≥ ≥ ≥ dUcenH sM NMu rUbPa B é n f KW { }| 3I y y= ≥ - 2( ) , 2 3g x x x= − ≤ ≤

22 3, 4 9 , 4 ( ) 9

4 9

x x g x

y

− ≤ ≤ ≤ ≤ ≤ ≤

≤ ≤

dUcenH s MNM ur UbPa B é n g KW { }| 4 9I y y= ≤ ≤ 6. sik Sa PaB ek In rWcu Hé nG n u Kmn_ f nig g eda yeR bIpleF o b f

x

∆

∆nig g

x

∆

∆

eyIg ma n ( ) 2 3f x x= − nig 2( ) 2g x x= +


- 231 -

- ( ) 2 3f x x= − snµt; 1 2x x< eyIg)a n 2 1 0x x x∆ = − >

( ) ( )2 1f f x f x∆ = − ey Ig )a n ( ) ( ) ( ) ( )

( )

2 1 2 1

2 1 2 1

2 1

2 1

2 3 2 3

22 0 0

f x f x x xf

x x x x x

x x f

x x x

− − − −∆= =

∆ − −

− ∆= = > ⇒ >

− ∆

dUce nH f Ca G nuKm n_ek In - 2( ) 2g x x= +

snµt; 1 2x x< eyIg)a n 2 1 0x x x∆ = − > ( ) ( )2 1g g x g x∆ = − eyIg )a n

( ) ( )

( )( )( )

2 22 1 2 1

2 1 2 1

2 1 2 12 1

2 1

2

g x g xg x x

x x x x x

x x x xx x

x x

−∆ −= =

∆ − −

+ −= = +

−

( )2 1

gx x

x

∆= +

∆ma nsB aØa t a m ( )2 1x x+

-ebI 2 1 0x x> > ena H 2 1 0x x+ > eyIg )a n 0g

x

∆>

∆

dUce nH g Ca GnuKm n_ek In -ebI 1 2 0x x< < ena H 2 1 0x x+ < eyIg )a n 0

g

x

∆<

∆

dUce nH g Ca GnuKm n_c uH


- 232 -

7. rk G nuKmn_R ca s c MeB a H G nuKmn_ ( )f x k> ( ) 2 3f x x= + - CMn Ys ( )f x eda y y eyIg )a n 2 3y x= + -bþÚr y eT ACa x nig x eT ACa y e yIg )a n 2 3x y= +

32 3, 2 3,

2

xx y y x y

−= + = − =

-CM nYs y eda y 1( )f x− eyIg )a n 1 3

( )2

xf x− −

= dUce nHG nuKmn _Rca sKW³ 1 3

( )2

xf x− −

= x> ( ) 2f x x= − - CMn Ys ( )f x eda y y eyIg )a n 2y x= − -bþÚr y eT ACa x nig x eT ACa y e yIg )a n 2x y= −

2, 2x y y x= − = + -CMnYs y eda y 1( )f x− eyIg )a n 1( ) 2f x x− = +

dUce nHG nuKmn _Rca sKW³ 1( ) 2f x x− = +

K> 1( ) 3f x

x= −

- CMn Ys ( )f x eda y y eyIg )a n 13y

x= −

-bþÚr y eT ACa x nig x eT ACa y e yIg )a n 13x

y= −

1 13,

3x y

y x= + =

+

- CMn Ys y eda y 1( )f x− eyIg )a n 1 1( )

3f x

x

− =+


- 233 -

dUce nHG n uKmn_R ca sKW³ 1 1( )

3f x

x

− =+

X> 3( )

1f x

x=

−

-CMnYs ( )f x eda y y eyIg )a n 3

1y

x=

−

-bþÚr y eT ACa x nig x eT ACa y e yIg )a n 3

1x

y=

−

1 3 3, 1 , 1

3 1

xy y

y x x= − = = +

−

-CMnYs y eda y 1( )f x− eyIg )a n 1 3

( ) 1f xx

− = + dUcenHG n u Kmn_R c a sKW³ 1 3

( ) 1f xx

− = + 8. sg ;R k a bt ag G nuKmn _xa g eR k am³ k > 1y x= +

eyIg )a n 1 , 1

1, 1

x xy

x x

+ ≥ −= − − < −

-s g;bnÞa t; 1 , 1y x x= + ≥ − nig 1, 1y x x= − − ≤ − eyIg )a n


- 234 -

x> 1 2y x= − + eyIg )a n 1 2 , 1 1 , 1

,1 2, 1 3, 1

x x x xy y

x x x x

− + ≥ + ≥ = = − + + < − + <

-sg ;bnÞa t; 1 , 1y x x= + ≥ nig 3, 1y x x= − + < ey Ig )a n 9. k > rk G nuKmn_R ca sén G nuKmn_ ( ) 3f x x= +

-CMnYs ( )f x eda y y eyIg )a n 3y x= + -bþÚr y eT ACa x nig x eT ACa y eyIg )a n 3x y= +

3y x= − -CMnYs y eda y 1( )f x

− eyIg )a n 1( ) 3f x x− = −

dUce nHG n uKm n_R ca sKW ³ 1( ) 3f x x− = − x> s g ;R k a b tag G nuKmn_ f nig G nuKmn_R ca s 1

f−

- sg; ( ) 3f x x= + nig 1( ) 3f x x− = −


- 235 -

sUmrg;caMGan PaK2 Edlnwgecj pSaynaeBlqab;²enH

sUmGrKuN

kem nknti v ti ü ati 10 kmw c an kmw c an ; kem nkn ti v ... · emeronti4...

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