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ºçßæ$ç³§ýl$Ë$

SAKSHI

çÜ*{™éË$ :1. ax2 + bx + c = 0 A ó̄l¨ Ð]lÆý‡Y çÜÒ$MýSÆý‡×æ Ýë«§éÆý‡×æ Æý‡*ç³… (a, b, c ∈ R). ©° Ð]lÊÌêË$ α, β AÆ‡¬¯]l

C…§ýl$ÌZ b2 – 4ac° "Ñ è̂l„ýS×ìæ' A…sêÆý‡$.

2. ax2 + bx + c = 0 Ð]lÆý‡Y çÜÒ$MýSÆý‡×æ…G) Ð]lÊÌêË Ððl¬™èl¢… = – b/a

¼) Ð]lÊÌêË Ëºª… = c/a

3. α, βË$ Ð]lÊÌêË$V> VýSË Ð]lÆý‡YçÜÒ$MýSÆý‡×æ…x2 – (α + β) x + αβ = 0 Ìôæ§éx2 – (Ð]lÊÌêË Ððl¬™èl¢…) x + Ð]lÊÌêË Ëºª… = 0

4. (x – α) (x – β) > 0 AÆ‡¬™ól x ÑË$Ð]l α, βË ºÄ¶æ$r E…r$…¨.5. (x – α) (x – β) < 0 AÆ‡¬™ól x ÑË$Ð]l α, βË Ð]l$«§ýlÅ E…r$…¨.6. ÔóæçÙ íÜ§é®…™èl…: f(x) = (x – a) × Q(x) + f(a)

7. G) f(x) AMýSÆý‡×îæÄ¶æ$ ç³NÆ>~…MýS çÜÐ]l*çÜ…ÌZ° ç³§éË VýS$×æM>Ë Ððl¬™èl¢… "0' AÆ‡¬™ól f(x)MýS$ (x – 1) JMýSM>Æý‡×ê…MýS….

¼) f(x) AMýSÆý‡×îæÄ¶æ$ ç³NÆ>~…MýS çÜÐ]l*çÜ…ÌZ° »ôæíÜçœ*™éË ç³§éË VýS$×æM>Ë Ððl¬™èl¢…l = çÜÇçœ*™éËç³§éË VýS$×æM>Ë Ððl¬™èl¢… AÆ‡¬ ]̄l f(x)MýS$ x + 1JMýS M>Æý‡×ê…MýS….

8. VýS×ìæ™é¯]l$VýSÐ]l$¯]l íÜ§é®…™èl…: JMýS {ç³Ð]l è̂l ]̄l… p(n)

i) n = 1 MýS$ ii) 1, 2....nËMýS$ °f… AÆ‡¬ ]̄lç³#yýlÌêÏiii) n + 1 MýS*yé °f… AÆ‡¬ ]̄l n A°² ÑË$Ð]lËMýS$ °f….

9. ¨Óç³§ýl íÜ§é®…™èl…: (x + y)n = nC0xn + nC1x

n–1y+ nC2xn–2y2 + ....+ nCrx

n–ryr + ....+ nCnyn

C…§ýl$ÌZ nC0, nC1, nC2....nCnË$ ¨Óç³¨ VýS$×æM>Ë$ nC0 = nCn = 1

10. (x + y)n ÑçÜ¢Æý‡×æÌZ r + 1Ð]l ç³§ýl… Tr+1 = nCr.xn– ryr

JMýS Ð]l*Æý‡$P {ç³Ô¶æ²Ë$1. √3x2 + 9x + 6√3 = 0 Ð]lÆý‡Y çÜÒ$MýSÆý‡×æ Ð]lÊÌêË Ððl¬™èl¢…, Ð]lÊÌêË Ë»êª°² MýS ]̄l$Vö ]̄l…yìl?Sol: Ð]lÆý‡Y çÜÒ$MýSÆý‡×æ… √3x2 + 9x + 6√3 = 0

a = √3, b = 9, c = 6√3

Ð]lÊÌêË Ððl¬™èl¢… b 9 3 3 3

a 3

− − − × ×= = =3

3 3= −

2 2b b 4ac b b 4ac,

2a 2a

− + − − −α = β =



SAKSHI

Ð]lÊÌêË Ëºª…

2. 3 + √5, 3 --& √5 Ð]lÊÌêË$ E ]̄l² Ð]lÆý‡YçÜÒ$MýSÆý‡×æ… Æ>Ä¶æ$…yìl?Sol: α = 3 + √5, β = 3 − √5

α + β =

αβ = (3 + √5) (3 – √5) = (3)2 – (√5)2 = 9 – 5 = 4

Ð]lÆý‡Y çÜÒ$MýSÆý‡×æ…: x2 – (α + β)x + αβ = 0

x2 – 6x + 4 = 0

3. x2 – 4x + 5 = 0 Ð]lÊÌêË çÜÓ¿êÐ]l… ™ðlËç³…yìl?Sol: çÜÒ$MýSÆý‡×æ… x2 – 4x + 5 = 0

a = 1, b = – 4, c = 5

Ñ è̂l„ýS×ìæ ∆ = b2 – 4ac

= (– 4)2 – 4 × 1 × 5= 16 – 20 = – 4

b2 – 4ac < 0 M>ºsìæt Ð]lÊÌêË$ çÜ…MîSÆý‡~ çÜ…QÅË$

4. ÑçÜ¢Æý‡×æÌZ Ð]l$«§ýlÅç³§é°² Æ>Ä¶æ$…yìl?

Sol: ¨Óç³¨ çœ*™é…MýS… 6 M>ºsìæt ÑçÜ¢Æý‡×æÌZ 6 + 1 = 7 ç³§éË$ E…sêÆ‡¬.Ð]l$§ýlÅç³§ýl… 4Ð]l ç³§ýl… AÐ]l#™èl$…¨.

(x + y)n ÑçÜ¢Æý‡×æÌZ Tr+1 = nCrxn – ryr

ÑçÜ¢Æý‡×æÌZ T4 = T 3 + 1 = 6C3 (x)6 – 3 (– 1/x)3

5. 3x3 – 2x2 + x + 2 A ó̄l çÜÐ]l*Ýë°² (x – 1) ó̂l ¿êW… è̂lV> Ð]l ó̂la ÔóæçÙ… MýS ]̄l$Vö ]̄l…yìl?Sol: f(x) = 3x3 – 2x2 + x + 2

f(x) ]̄l$ (x – 1) ó̂l ¿êWõÜ¢ Ð]l ó̂la ÔóæçÙ… f(1) AÐ]l#™èl$…¨∴ f(1) = 3(1)3 – 2 (1)2 + 1 + 2= 3 – 2 + 1 + 2 = 4

6. K çÜ…QÅ §é° Ð]lÆ>Y° MýS…sôæ 132 ™èlMýS$PÐ]l AÆ‡¬ ]̄l B çÜ…QÅ ]̄l$ MýS ]̄l$Vö ]̄l…yìl?Sol: JMýS çÜ…QÅ 'x' A¯]l$Mø…yìl

336C x=

3

1

x

−×+

36C= −

( )71x x−

( )61x x−

3 5+ 3 5+ − 6=

c 6 3

a= =

36=

§é° Ð]lÆý‡Y… = x2

§ýl™é¢…Ô¶æ… {ç³M>Æý‡… x2 = x + 132

x2 – x – 132 = 0x2 – 12x – 132 = 0x2 – 12x + 11x – 132 = 0x (x – 12) + 11 (x – 12) = 0(x + 11) (x – 12) = 0x + 11 = 0 x –12 = 0x = – 11 x = 12

M>Ð]lËíÜ ]̄l çÜ…QÅ 12 Ìôæ§é & 11

Æð‡…yýl$ Ð]l*Æý‡$PË {ç³Ô¶æ²Ë$1. x3 – 3x2 + 4x + k ]̄l$ (x – 2) °ÔóæØçÙ…V> ¿êWõÜ¢ k ÑË$Ð]l ]̄l$ MýS ]̄l$Vö ]̄l…yìl? (June 10)

Sol: f(x) = x3 – 3x2 + 4x + k

f(x) ]̄l$ (x – 2) °ÔóæØçÙ…V> ¿êWõÜ¢ f (2) = 0

f(2) = 23 – 3(2)2 + 4(2) + k = 0= 8 – 3 × 4 + 4 (2) + k = 0= 8 – 12 + 8 + k = 0= 16 – 12 + k = 0= 4 + k = 0k = 4

2. ÑçÜ¢Æý‡×æÌZ Ð]l$«§ýlÅ ç³§éË ]̄l$ MýS ]̄l$Vö ]̄l…yìl?

Sol: ¨Óç³§ýl çœ*™èl… 7 M>ºsìæt ÑçÜ¢Æý‡×æÌZ 7 + 1 = 8 ç³§éË$ E…sêÆ‡¬.M>ºsìæt 4, 5 ç³§éË$ Ð]l$«§ýlÅç³§éË$ AÐ]l#™éÆ‡¬(x + y)n ÑçÜ¢Æý‡×æÌZ Tr+1 = nCrx

n – ryr

ÑçÜ¢Æý‡×æÌZ

T4 = t3 + 1 = 7C3 (3x) 7 – 3 (–1/2x)3

t5 = t4+1 = 7C4 (3x)7 – 4 (–1/2x)4

381

7C x8

= ×

31

7C 81 x8

−= × × ×

14 4

37C 3 x= ( )3 3

1.2 x

−

×

3

71

3x2x

−

71

3x2x

−





SAKSHI

3. x2 – x – 2 < 0 Ýë«¨…^èl…yìl?Sol: x2 – x – 2 < 0

x2 – 2x + x – 2 < 0x (x – 2) + 1 (x –2) < 0(x + 1) (x – 2) < 0

∴ Ýë«§ýl ]̄l çÜÑ$† = {– 1 < x < 2}

4. x2 – 6x + 8 > 0 Ýë«¨…^èl…yìl?Sol: x2 – 6x + 8 > 0

x2 – 4x – 2x + 8 > 0x (x – 4) – 2 (x – 4) > 0(x – 2) (x – 4) > 0

Ýë«§ýl ]̄l x ÑË$Ð]l 2, 4ËMýS$ ºÄ¶æ$r E…r$…¨ M>ºsìæt x < 2 Ìôæ§é x > 4 AÐ]l#™èl$…¨.

5. Æð‡…yýl$ Ð]lÆý‡$çÜ çÜÇçÜ…QÅË Ëºª… 168 AÆ‡¬™ól Ðésìæ° MýS ]̄l$MøP…yìl.Sol: Æð‡…yýl$ Ð]lÆý‡$çÜ çÜÇçÜ…QÅË$ = x, x + 2 A¯]l$MýS$…§é…

§é° Ëºª… x (x + 2) = x2 + 2x

M>° Ðésìæ Ëºª… = 168 CÐ]lÓºyìl…¨x2 + 2x = 168

⇒ x2 + 2x – 168 = 0x2 + 14x – 12x –168 = 0x (x + 14) – 12 (x + 14) = 0(x + 14) (x – 12) = 0x + 14 = 0 x – 12 = 0∴x = –14 ∴x = 12

x = 12 AÆ‡¬ ]̄l Ð]lÆý‡$ççÜ çÜÇçÜ…QÅË$ 12, 14x = 14 AÆ‡¬ ]̄l Ð]lÆý‡$ççÜ »ôæíÜçÜ…QÅË$ – 14, –12

6. ÔóæçÙ íÜ§é®…™é°² °Æý‡Ó_…_, °Æý‡*í³… è̂l…yìl?Sol: °Æý‡Ó^èl¯]l…: x è̂lËÆ>ÕV> E ]̄l² AMýSÆý‡×îæÄ¶æ$ ç³NÆ>~…MýS çÜÐ]l*çÜ… f(x) ]̄l$ (x – a)™ø ¿êWõÜ¢ Ð]l ó̂la ÔóæçÙ… f(a)

°Æý‡*ç³×æ: x è̂lÆý‡Æ>ÕV> VýSË AMýSÆý‡×îæÄ¶æ$ ç³NÆ>~…MýS çÜÐ]l*çÜ… f(x). ©°° (x – a)™ø ¿êWõÜ¢ Ð]l ó̂la¿êVýSçœË… Q(x), R ÔóæçÙ… A ]̄l$MýS$…sôæ

427

7C16x

= ×

3 347C 3 x= × × ( )4

4 4

1

2 x

−×

× 1



SAKSHI

f(x) = (x – a), Q(x) + Rf(a) = R

¯éË$VýS$ Ð]l*Æý‡$PË {ç³Ô¶æ²Ë$1. x4 + 4x3 + 3x2 – 4x – 4 ]̄l$ ÔóæçÙíÜ§é®…™èl… Eç³Äñæ*W…_ M>Æý‡×ê…M>Ë$ MýS ]̄l$Vö ]̄l…yìl? (Ð]l*Ça 08)Sol: f(x) = x4 + 4x3 + 3x2 – 4x – 4 A¯]l$Mö¯]l…yìl

ç³§éË VýS$×æM>Ë Ððl¬™èl¢… = 1 + 4 + 3 & 4 & 4 = 8 & 8 = 0∴ f(x)MýS$ (x – 1) JMýS M>Æý‡×ê…MýS…x çÜÇçœ*™èl VýS$×æM>Ë Ððl¬™èl¢… = 1 + 3 & 4 = 0x »ôæíÜçœ*™èl VýS$×æM>Ë Ððl¬™èl¢… = 4 & 4 = 0çÜÐ]l*¯éË$ M>ºsìæt f(x)MýS$ (x + 1) JMýS M>Æý‡×ê…MýS…f(x)ÌZ° ç³§éË VýS$×æM>Ë$ ¡çÜ$MýS$…sôæ

§ýl™èl¢ çÜÐ]l*çÜ… f(x)MýS$ M>Æý‡×æÆ>Ô¶æ$Ë$(x – 1) (x + 1) (x2 + 4x + 4)

(x – 1) (x + 1) (x + 2)2

= (x – 1) (x + 1) (x + 2) (x + 2)

2. xÌZ JMýS Ð]lÆý‡YçÜÐ]l*çÜ… x – 1, x – 2, x – 3Ë ó̂l ¿êW… è̂lºyìl ]̄l ÔóæÚëË$ Ð]lÆý‡$çÜV> 11, 22, 39. B Ð]lÆý‡YçÜÒ$MýSÆý‡×æ… MýS ]̄l$Vö ]̄l…yìl?

Sol: Ð]lÆý‡Y çÜÐ]l*çÜ… f(x) = ax2 + bx + c A¯]l$MýS$…sôæf(x) ]̄l$ (x – 1)™ø ¿êWõÜ¢ Ð]l ó̂la ÔóæçÙ… f(1) = 11

f(1) = a(1)2 + b(1) + Ca + b + c = 1 ..........(1)

f(x)l ]̄l$ x – 2™ø ¿êWõÜ¢ Ð]l ó̂la ÔóæçÙ… f (2) = 22

f(2) = a (2)2 + b(2) + cf (2) = a (2)2 + b(2) + c= 4a + 2b + c= 4a + 2b +c = 22 .........(2)

f(x) ]̄l$ x–3 ó̂l ¿êWõÜ¢ Ð]l ó̂la ÔóæçÙ… f(3) = 39

f(x) = ax2 + bx + c

x = 1 1 4 3 – 4 – 40 1 5 8 4

x = – 1 1 5 8 4 00 –1 – 4 – 4

1 4 4 0



SAKSHI

f(3) = a(3)2 + b(3) + c= 9a + 3b + c9a + 3b + c = 39 ............(3)

(1), (2)Ë ]̄l$…_a + b + c = 1 ................(1)

4a + 2b + c = 22 ...........(2)– – – –___________________

– 3a – b = – 21 Ìôæ§é

3a + b = 21.........(4)

(2), (3)Ë ]̄l$…_4a + 2b + c = 22.....(2)9a + 3b + c = 39 .....(3)___________________

– 5a – b = – 17 Ìôæ§é

5a + b = 17.......... (5)

(4), (5)Ë ]̄l$…_5a + b = 17.....(5)3a + b = 21 .....(4)

– – – ___________________

2a = – 4 Ìôæ§é ⇒ a = – 4/2 = –2

a = – 2 ]̄l$ (4)ÌZ {ç³†„óSí³… è̂lV>3a + b = 213 (– 2) + b = 21– 6 + b = 21b = 21 + 6 = 27a = – 2, b = 27, c = ?a + b + c = 1 from (1)– 2 + 27 + c = 125 + c = 1c = 1 – 25 ⇒ – 24

∴ Ð]lÆý‡Y çÜÐ]l*çÜ…f(x) = ax2 + bx + c

f(x) = – 2x2 + 27x – 24 Ìôæ§éf(x) = 2x2 – 27x + 24



SAKSHI

3. ÑçÜ¢Æý‡×æÌZ x Ìôæ° ç³§ýl… (íÜ¦Æý‡ç³§ýl…) ]̄l$ MýS ]̄l$MøP…yìl?

Sol: ÑçÜ¢Æý‡×æÌZ n = 8

x = 6x2, y = – 5/x2

(x + y)n ÑçÜ¢Æý‡×æÌZ Tr + 1 = nCr xn – ryr

= 8Cr (6x2)8 – r (– 5/x2)r

= 8Cr (6)8 – r x 16 – 2r (– 5). x–2r

= 8Cr (6)8 – r × (– 5)r × x 16 – 2r × x –2r

= 8Cr (6)8 – r × (– 5)r × x 16 – 4r

x Ìôæ° ç³§ýl… A…sôæx 16 – 4r = x0 M>ÐéÍ.16 – 4r = 016 = 4r ⇒ r = 16/4 = 4

r = 4 ]̄l$ Tr+1ÌZ {ç³†„óSí³õÜ¢

ÑçÜ¢Æý‡×æÌZ x Ìôæ° ç³§ýl…

8C4 6 8 – 4 (– 5)4. x16 – 4 × 4

8C4 × 6 4 × (– 5)4 × x0

8C4 × 6 4 × 54 = 8C4 × 304

4. çÜÒ$MýSÆý‡×æç³# Ð]lÊÌêË$ MýS ]̄l$Vö ]̄l…yìl?

Sol:

( )( )( ) ( )a b x

b x 0x a b a

−⇒ − − =

− −

( )( )( )

a b xb x

x a b a

−⇒ = −

− −

bx⇒ ax bx− −( )( )

abb x

x a b a

+ = −− −

( ) ( )( )( )

x b a b x ab x

x a b a

− − −⇒ = −

− −

x bb x

x a b a− = −

− −

x bx b

x a b a+ = +

− −

x bx b

x a b a+ = +

− −

( )822

56xx

∴ −

( )822

56xx

−

( )822

56xx

−



SAKSHI

b – x = 0 ( Ìôæ§é)

(x – a) (b – a) = a

∴ Ð]lÊÌêË$

5. VýS×ìæ™é ]̄l$VýSÐ]l$ ]̄l íÜ§é®…™èl… Eç³Äñæ*W…_ A° è̂l*ç³…yìl?

(Ð]l*Ça 08, þ¯Œæ 10)

Sol:

n = 1MìS p(n) °f… A° è̂l*§éª….

n = 1 AÆ‡¬¯]l

∴ n = 1MìS p(n) °f….............(1)ii) p(n); nMìS °f… AÐ]l#™èl$…¨.

iii) p(n) ; n + 1MìS °f… A° è̂l*ç³yýl…

™èlÆý‡$Ðé™èl ç³§ýl…

CÆý‡$OÐðlç³#Ë MýSË$ç³V>( )( )1

n 1 n 2+ +

( )( )1

n 1 n 2=

+ +( )1

n n 1+

( )1 1 1 1 n

i.e. ..... .....(2)1.2 2.3 3.4 n n 1 n 1

+ + + + =+ +

n 1 1R.H.S.

n 1 1 1 2= + = =

+ +

( ) ( ) ( )1 1 1

p 1 L.H.S.1 1 1 1 2 2

= + = =+

( ) ( )1 1 1 1 n

p n ...1.2 2.3 3.4 n n 1 n 1

= + + + + =+ +

( )1 1 1 1 n

...1.2 2.3 3.4 n n 1 n 1

+ + + + =+ +

ab,a

b a = + −

ax a

b a∴ = +

−

ax a

b a∴ = +

−

ax a

b a∴ − =

−

( )( )a

b x 1x a b a

− = − −

( )( )a

1 0x a b a

− =− −

( )( )a

b x 1 0x a b a

− − = − −



SAKSHI

A…sôæ p(n) {ç³Ð]l è̂l ]̄l… n + 1 MýS* °f… AÆ‡¬…¨.(1), (2), (3)Ë ]̄l$…_ p(n) A ó̄l {ç³Ð]l è̂l ]̄l… n A°² ÑË$Ð]lËMýS$ °f… AÐ]l#™èl$…¨.

I§ýl$ Ð]l*Æý‡$PË$

1. x2 – 3x + 2= 0 ]̄l$ y = x2 Æó‡Rê_{™èl… çÜàÄ¶æ$…™ø Ýë«̈ … è̂l…yìl?Sol: x2 – 3x + 2= 0 ⇒ x2 = 3x – 2

y = x2 = 3x – 2 A¯]l$Mö¯]l…yìl.y = x2 ; y = 3x – 2 A ó̄l Ð]l{M>Ë Q…yýl ]̄l ¼…§ýl$Ð]l#Ë x l°Æý‡*ç³M>Ë$ x2 – 3x + 2 = 0 MýS$ ÐéçÜ¢Ð]l

Ð]lÊÌêË$ AÐ]l#™éÆ‡¬.y = x2

{V>‹œ §éÓÆ> y = x2, y = 3x –2 çÜÆý‡â¶æÆó‡QË Q…yýl ]̄l ¼…§ýl$Ð]l#Ë$ (1, 1) (2, 4)∴ x2 – 3x + 2= 0 çÜÒ$MýSÆý‡×æ Ð]lÊÌêË$ (1, 2)

( )n 1

n 1 1

+⇒+ +

n 1

n 2

+=+

( )( )( )

( )

22 n 1n 2n 1

n 1 n 2 n 1

++ += =+ + + ( )n 2+

( )( )( )n n 2 1

n 1 n 2

+ ++ +

( ) ( )( ) ( )( )1 1 1 1 1 n 1

.....1.2 2.3 3.4 n n 1 n 1 n 2 n 1 n 1 n 2

+ + + + + = ++ + + + + +

x 0 1 2 3 –1 –2 –3

y 0 1 4 9 1 4 9

y = x2

x 0 1 2 3

y –2 1 4 7

y = 3x – 2



2. x2 – x – 6 = 0 ]̄l$ y = x2 – x – 6 Æó‡Rê_{™èl… çÜàÄ¶æ$…™ø Ýë«̈ … è̂l…yìl?Sol:

(– 4, 10),(–3, 6), (–2, 0), (–1, – 4), (0, – 6)

(1, – 6), (2, –10), (3, –6) (4, 6) ¼…§ýl$-Ð]l#-Ë-̄ ]l$ {V>‹œ-Oò³ _{†…^é-Í. Ðésìæ-° Ð]l$–§ýl$-OÐðl-̄ ]l Ð]l{MýS…™ø MýSÍ-í³™ólÐ]l$ ]̄l-MýS$ M>Ð]l-ÍÞ-̄ ]l Æó‡-Rê-_{™èl… Ð]lçÜ$¢…¨.x - A„ýS… Q…yýl-̄ ]l ¼…§ýl$-Ð]l#-Ë$ (4, 0), (–3, 0)

∴ Ýë«§ýl-̄ ]l çÜÑ$-† -Ñ-Ë$-Ð]l-Ë$ x = – 2, x = 3

x – 4 – 3 –2 –1 0 1 2 3 4x2 16 9 4 1 0 1 4 9 16

–x 4 3 2 1 0 –1 –2 – 3 – 4– 6 – 6 – 6 – 6 – 6 – 6 – 6 – 6 – 6 – 6y +10 + 6 0 – 4 – 6 – 6 – 4 – 0 6

y = x2 – x – 6



SAKSHI

ºçßæ$Oâñæ_eMýS {ç³Ô¶æ²Ë$1. x2 – 3x + kMýS$ x – 2 JMýS M>Æý‡×ê…MýS… AÆ‡¬™ól k ÑË$Ð]l

G) 10 ¼) & 10 íÜ) 2 yìl) 52. nC6 = nC9 AÆ‡¬™ól n ÑË$Ð]l

G) 9 ¼) 6 íÜ) 15 yìl) 33. f(x) ]̄l$ ax + b ó̂l ¿êWõÜ¢ Ð]l ó̂la ÔóæçÙ…

G) f (–b/a) ¼) f (b/a) íÜ) f (a/b) yìl) f (– a/b)

4. K ºçßæ$ç³¨ VýS$×æM>Ë Ððl¬™èl¢… Ô¶æ* ]̄lÅ… AÆ‡¬™ól §é°MìS K M>Æý‡×æÆ>ÕG) x + 1 ¼) x íÜ) – x yìl) x – 1

5. 6C4 ÑË$Ð]lG) 6 ¼) 4 íÜ) 15 yìl) 10

6. y = 2x2 Æó‡Rê_{™èl… ´ùÐ]l# ´ë§éË$G) I, II ¼) II, III íÜ) III, IV yìl) I, IV

çÜÐ]l*«§é¯éË$1) ¼ 2) íÜ 3) G 4) yìl 5) íÜ 6) G

RêäË¯]l$ ç³NÇ…^èl…yìl1. 1 < x < 3 Ýë«§ýl ]̄lË$V> E ]̄l² AçÜÒ$MýSÆý‡×æ… ______

2. x + y ó̂l xn + yn °ÔóæØçÙ…V> ¿êW… è̂lºyýlyé°MìS °Ä¶æ$Ð]l$… ______ AÐéÓÍ.3. f(x) ]̄l$ ax – b = 0 ó̂l ¿êW… è̂lV> Ð]l è̂l$a ÔóæçÙ… ______

4. JMýS ºçßæ$ç³¨ çÜÇçœ*™èl ç³§éË VýS$×æM>Ë Ððl¬™èl¢… »ôæíÜçœ*™èl ç³§éË VýS$×æM>Ë Ððl¬™é¢°MìS çÜÐ]l* ]̄lOÐðl$ ]̄l§é°MìS ______ JMýS M>Æý‡×ê…MýS….

5. x2 – 2x – 15 = 0ÌZ Ð]lÊÌêË Ëºª… ______

6. ÑçÜ¢Æý‡×æÌZ _Ð]lÇ ç³§ýl… ______

çÜÐ]l*«§é¯éË$1) x2 – 4x + 3 < 0 2) n »ôæíÜçÜ…QÅ 3) f (b/a) 4) x + 1 5) – 15 6) 1/x6

61

xx

+



f™èlç³Æý‡^èl…yìl1. nC0 ( ) A) x – 1

2. (a + b)5 ÑçÜ¢Æý‡×æÌZ VýS$×æM>Ë Ððl¬™èl¢… ( ) B) 4

3. f(1) = 0 AÆ‡¬™ól f(x) MýS$ M>Æý‡×ê…MýS… ( ) C) 6

4. 10C2n = 10Cn+4 AÆ‡¬™ól ÑË$Ð]l ( ) D) 1

5. 6x2 –5 = 0 Ð]lÊÌêË Ððl¬™èl¢… ( ) E) 326. √3x2 + 9x + 6√3 = 0 Ð]lÆý‡Y çÜÒ$MýSÆý‡×æ Ð]lÊÌêË Ëºª… ( ) F) 0

çÜÐ]l*«§é¯éË$1. D 2. E 3. A 4. B 5. F 6. C

Quick Review Table

1. ax2 + bx + c = 0 Ð]lÊÌêË$ i)

ii)

2. ax2 + bx + c = 0 MìS Ñ è̂l„ýS×ìæ b2 – 4ac

3. ax2 + bx + c = 0 i) Ð]lÊÌêË Ððl¬™èl¢… = – b/a

ii) Ð]lÊÌêË Ëºª… = c/a

4. G) (x – α) (x – β) > 0 x ÑË$Ð]l α, βË Ð]l$«§ýlÅ E…yýl§ýl$¼) (x – α) (x – β) > 0 x ÑË$Ð]l α, βË Ð]l$«§ýlÅ E…r$…¨.

5. VýS×ìæ™é ]̄l$VýSÐ]l$ ]̄l íÜ§é®…™èl… p(n) A ó̄l {ç³Ð]l è̂l ]̄l…i) n = 1MýS$ ii) 1, 2...n MýS*yé °f… A° è̂l*í³ ]̄l p(n) n A°²

ÑË$Ð]lËMýS$ °f…

6. ¨Óç³§ýl íÜ§é®…™èl… (x + y)n = nC0xn + nC1x

n–1yn + nC2xn–2y2+......+nCrx

n–ryr +

....+nCnyn

7. (x + y)n ÑçÜ¢Æý‡×æÌZ tr + 1 = nCrxn–ryr

22b b 4ac

x b 4ac2a

− − −= = −

2b b 4acx

2a

− + −=

bahupadulu

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