HSC Model questions 2nd part

Model Questions (Suggestion); Higher Mathematics 2nd

Paper

1| ‡h †Kvb yBwU cÖ‡kœi DËi `vIt 5 2=10

1. K) ev¯Íe msL¨vt

1. gvb wbY©q Kit

i) -16 + 3+-1 - 4- 3 - -1 - 7 ii) -2--6

iii) -1 - 8 + 3 - 1 *** iv) 2 - 6 - 1 - 9

v) 3 - 5 + 7 - 12 vi) -3 - 5

*** vii) 13 + -1 - 4 - 3 - -8 DËit i) 7, ii) 4, iii) 11, iv) 4, v) 7, vi) 8, vii) 7

2. wb‡æi AmgZv¸‡jv cig gvb wPýe¨ZxZ cÖKvk Kit

*** i) x - 2< 5 *** ii) 2x - 3< 7

iii) x - 3 < 7 iv) x < 3

*** v) 513

1

x (x ≠

3

1 ) vi) 2x + 4 < 8

DËit i) -3 < x < 7, ii) -5 < x < 2, iii) -4 < x < 10, iv) -3 < x < 3,

v)3

1

5

2 x ev,

15

4

3

1 x vi) -6 < x < 2

3. wb‡æi AmgZv¸‡jv cig gvb wP‡ýi mvnv‡h¨ cÖKvk Kit

i) 4 < x < 10 *** ii) -2 < x < 6

*** iii) -7 < x < -1 ** iv) 2 x 3

*** v) -1 <2x - 3 < 5 vi) -5 < x < 7

vii) -2 < 3 - x < 8 viii) -8 x 2

DËit i) x - 7< 3, ii) x - 2< 4, iii) x + 4 < 3, iv) x - 5 3,

v) 2x - 5< 3 vi) x - 1< 6, vii) x < 5, viii) x + 3 5.

4. wb‡æi AmgZv¸‡jv mgvavb Ki Ges mgvavb †mU msL¨v‡iLvq †`LvIt

*** i) 3x + 2< 7 * ii) 2x + 1< 3

*** iii) 2x - 5 < 3 * iv) 3x - 4 < 2

v) 2x + 5 < 1 *** vi) 253

1

x

vii) 2x + 3 > 9 viii) x < 4

ix) 2x - 5 < 1 x) 2x + 4 < 6.

DËit i) { x R: -3 < x < 3

5} ii) { x R: -2 < x < 1}

iii) { x R: 1 < x < 4} iv) { x R: 3

2 < x < 1}

v) { x R: -3 < x < -2} vi) { x R: 2

3 < x <

3

5ev

3

5< x <

6

11}

vii) { x R: -6 > x > 3} viii) { x R: -4 < x < 4} ix) { x R: 2 < x < 3} x) { x R: -5 < x < 1}

5. *** i) x - 1 < 10

1 n‡j †`LvI †h, x2 - 1 <

100

21

ii) x - 1 < 2

1 n‡j †`LvI †h, x2 - 1 <

4

5

iii) x - 1 < 2 n‡j †`LvI †h, x2 - 1 < 8

6.** i) a, b R n‡j, †`LvI †h, ab=ab

*** ii) hw` a, b R nq, Z‡e cÖgvY Ki †h, a + ba+b

* iii) a, b R n‡j, cÖgvY Ki †h, a - ba+b

** iv) a, b R n‡j, cÖgvY Ki †h, a - ba-b

7.** i) ‡`LvI †h, 2 GKwU Ag~j` msL¨v|

*** ii) ‡`LvI †h, 3 GKwU Ag~j` msL¨v|

** iii) ‡`LvI †h, 5 GKwU Ag~j` msL¨v|

8. i) cÖgvY Ki †h, aa 2 Ges

22aa

ii) cÖgvY Ki †h, x < a n‡j, -a < x < a ( †hLv‡b a > 0).

iii) ‡`LvI †h, -a a a †hLv‡b a †h †Kvb ev¯—e msL¨v|

iv) hw` a, b R nq, Z‡e †`LvI †h, -(a + b) = -a - b Ges (-a)b = -(ab).

* v) hw` a, b R nq, Z‡e †`LvI †h, (ab)-1 = a-1b-1 (a ≠ 0, b ≠ 0), Ges (-a)(-b) = ab.

* vi) cÖgvY Ki †h, hw` a R nq Z‡e a.0 = 0.

*** vii) hw` a, b, c R, ac = bc Ges c 0 nq, Z‡e cÖgvY Ki †h, a = b.

*** viii) hw` a, b, c R Ges a+b = a+c nq, Z‡e cÖgvY Ki †h, b = c.

ix) hw` a < b Ges b < c nq, Z‡e †`LvI †h, a < c.

** x) hw` a < b nq, Z‡e †`LvI †h, a + c < b + c Ges hw` a > b nq,

Z‡e †`LvI †h a + c > b + c, †hLv‡b a, b, c ev¯—e msL¨v|

1

1| (L) RwUj msL¨v (i) 1. gWyjvm I Av ©‡g›U wbY©q Kit

i) i31 ii) i31 DËit i) 2, 3

2 ii) 2,

3

2. eM©g~j wbY©q Kit

*i) 7 - 30 2 ***ii) 168 ***iii) i2 iv) i2 v) 422 xi

vi) )1(22 xix

DËit i) )235( i ii) )31( i iii) )1( i iv) )1( i

v) )22(2

1 xix vi) )}1()1{(

2

1 xix

3. cÖgvY Ki t i) )1(2

1ii * ii) )1(

2

1ii

iii) 2 ii

4. gvb wbY©q Kit

**i) 3 1 ii) 3 1 ***iii) 3 i ***iv) 3 i ***v) 4 81

vi) 4 169 ***vii) 6 64

DËit i) 1, )31(2

1 ii) -1, )31(

2

1 iii) i , )3(

2

1i

iv) i , )3(2

1i v) )1(

2

3i vi) )1(

2

26i vii) i2 , )3( i

5.*** i) ( iba )( idc ) = iyx n‡j †`LvI †h, ( iba )( idc ) = iyx

*** ii) iyxiba 3n‡j cÖgvY Ki †h, iyxiba 3

*** iii) iyxiba 3n‡j cÖgvY Ki †h,

y

b

x

ayx )(4

22

iv) hw` 122 ba nq, Z‡e †`LvI †h, x Gi GKwU ev¯—e gvb iba

ix

ix

1

1 mgxKiY‡K

wm× K‡i, GLv‡b a I b ev¯—e msL¨v|

*** v) idcibayx :: n‡j

†`LvI †h, 0)()(2)(222222 ybaxybdacxdc

*** vi) ip 12 n‡j cÖgvY Ki †h, 01246 ppp

vii) ip 12 n‡j cÖgvY Ki †h, 01246 ppp

viii) ix 23 Ges iy 23 n‡j, †`LvI †h, 2322 yxyx

** ix) hw` n

n

nxaxaxaax 2

210)1( nq, Z‡e †`LvI †h,

2

531

2

420 .....)(......)( aaaaaa = naaaa 210

6.* i) †`LvI †h, 1

iyx

iyx

*** ii) iyxz Ges 212 zz n‡j cÖgvY Ki †h, 122 yx

** iii) iyxz n‡j 2088 zz Øviv wb‡`©wkZ mÂvic‡_i mgxKiY wbY©q Ki| DËit

90025922 yx

iv) iyxz n‡j 35 z e„‡Ëi e¨vmva© I †K› ª wbY©q Ki| DËit (5, 0) Ges 3

** v) iyxz n‡j 2088 zz Øviv wb‡`©wkZ mÂvic‡_i mgxKiY wbY©q Ki|

DËit 90025922 yx

1| (M) RwUj msL¨v ( )

1. GK‡Ki GKwU KvíwbK Nbgyj n‡j, †`LvI †h,

i) 4)1()1(2222

ii) 4)1()1(242242

* iii) 8)1)(1)(1(222

*** iv) 9)1)(1)(1)(1(10842

*** v) xyyxyxyx 6)()()(22222

2 Model Questions (Suggestion); Higher Mathematics 2nd

Paper cÖkœ-1

*** vi) 16)31()31(44

** vii) 16)1)(1)(1)(1(16884422

viii) n2).....1)(1)(1(84422 Drcv`K ch©š— =

n22

2.*** i) hw` GK‡Ki GKwU KvíwbK Nbgyj nq Ges hw` qpx , 2 qpy ,

qpz 2 nq, Z‡e †`LvI †h, pqzyx 6

222

ii) hw` 0)()()(222222 cbacbacba nq Zvn‡j †`LvI

†h, ca ev, )(2

1cab

iii) hw` 0)()(3232 cbacba nq, Z‡e †`LvI †h, )(

2

1cba ev,

)(2

1acb ev, )(

2

1bac .

*** iv) hw` 0 zyx Ges GK‡Ki GKwU KvíwbK Nbgyj nq Z‡e †`LvI †h,

xyzzyxzyx 27)()(3232

v) GK‡Ki GKwU KvíwbK Nbgyj n‡j, †`LvI †h, ))((2222 babababa

vi) GK‡Ki GKwU KvíwbK Nbgyj n‡j, †`LvI †h,

))()((322333 cbacbacbaabccba

vii) hw` n

n

nxpxpxppxx

2

2

2

210

2)1( nq, Z‡e †`LvI †h,

1

630 3 n

ppp .

*** viii) cÖgvY Ki †h, 22

31

2

31

nn

hLb n Gi gvb 3 Øviv wefvR¨ Ges

ivwkwU = 1 , hLb nAci †Kvb cyY© msL¨v nq|

ix) hw` yxa , yxb , yxc2 nq, Z‡e cÖgvY Ki †h,

)(333333

yxcba

x) hw` )31(2

1a Ges )31(

2

1b nq Z‡e †`LvI †h,

04224 bbaa

cÖkœ 2| ‡h †Kvb yBwU cÖ‡kœi DËi `vIt 5 2=10

2| (K) eûc`x I eûc`x mgxKiY(g~‡ji cÖK…wZ I g~j-mnM m¤úK©)

1. i) `yBwU g~‡ji †hvMdj k~b¨ n‡j, 036916423 xxx mgxKiYwU mgvavb Ki|

ii) g~j¸wj ¸‡YvËi cÖMgb †kªYxfy³ n‡j, 0245226323 xxx mgxKiYwU mgvavb Ki|

** iii) 01317523 xxx mgxKiYwUi GKwU g~j 1 n‡j Aci g~j`ywU wbY©q Ki|

*** iv) `yBwU g~‡ji AbycvZ 3 t 4 n‡j, 02422223 xxx mgxKiYwU mgvavb Ki|

v) GKwU g~j Avi GKwUi wØ¸Y n‡j 04563142423 xxx mgxKiYwU mgvavb Ki|

vi) `yBwU g~‡ji †hvMdj 5 n‡j, 0613323 xxx mgxKiYwU mgvavb Ki|

vii) 0108723 xxx mgxKi‡Yi GKwU g~j 31 n‡j mgxKiYwU mgvavb Ki|

viii) GKwU gyj i1 n‡j 0410105234 xxxx mgxKiYwU mgvavb Ki|

ix) GKwU gyj i1 n‡j 02254234 xxxx mgxKiYwU mgvavb Ki|

DËit i) 2

3,

2

3 , 4 ii)

3

2, 2 , 6 iii) i32 , i32 iv)

2

3 , 2 , 4

v) 4

3,

2

3 ,

3

5 vi)

3

2 , )135(

2

1 vii) 31 , 31 , 5

viii) 1 , 2 , i1 , i1 ix) 21 , 11

2. i) 03 rqxx mgxKi‡Yi g~j¸wj a , b , c n‡j

222)()()( baaccb Gi

gvb wbY©q Ki|

ii) 03 rqxx mgxKi‡Yi g~j¸wj a , b , c n‡j ))()(( cbabacacb

Gi gvb wbY©q Ki|

*** iii) 023 rqxpxx mgxKi‡Yi g~j¸wj a , b , c n‡j

222

111

cba Gi gvb wbY©q

Ki|

* iv) 012323 xx mgxKi‡Yi g~j¸wj , , n‡j 2 Gi gvb wbY©q Ki|

** v) 023 rqxpxx mgxKi‡Yi g~j¸wj a , b , c n‡j

222222

111

baaccb Gi

gvb wbY©q Ki|

*** vi) 023 rqxpxx mgxKi‡Yi g~j¸wj , , n‡j 3 Gi gvb wbY©q Ki|


Paper cÖkœ-1

DËit i) q6 ii) r8 iii) 2

22

r

prq iv) 1 v)

2

22

r

qp vi) rppq 33

3

3. i) p , q g~j` n‡j, †`LvI †h, 0)()(2)(2222222 qpxqpxqp mgxKi‡Yi

g~j¸wj g~j` n‡e|

* ii) hw` a , b , c g~j` Ges a +b + c = 0 nq, Zvn‡j †`LvI †h,

0)()()(2 cbaxbacxacb mgxKi‡Yi g~j¸wj g~j` n‡e|

*** iii) ‡`LvI †h, ba bv n‡j 0)(22222 baxbax mgxKi‡Yi g~j¸‡jv ev¯—e n‡Z

cv‡i bv|

*** iv) k Gi gvb KZ n‡j 04)2()1(2 xkxk mgxKi‡Yi gyj¸‡jv ev Íe I mgvb n‡e?

DËit 10, 2

** v) hw` 086222 bacabxxa mgxKi‡Yi g~j`ywU mgvb nq, Z‡e cÖgvY Ki †h,

xbxac22

4)1( mgxKi‡Yi g~j`ywUI mgvb n‡e|

vi) k Gi gvb KZ n‡j 32)3(2)1(2 kxkxk ivwkwU GKwU c~Y© eM© n‡e| Dt 3, -2

* vii) ‡`LvI†h,22222

2)( bkhkxxah ivwkwU GKwU c~Y©eM© n‡e hw 12

2

2

2

b

k

a

h nq|

*** viii) a , b ev¯—e n‡j †`LvI †h, baxbabx 23)(222 mgxKi‡Yi gyj¸wj ev¯Íe n‡e;

hw` mgxKiYwUi GKwU g~j AciwUi wØ¸Y nq, Zvn‡j cÖgvY Ki †h, ba 2 A_ev, ba 114

4. i) hw` 02 cbxax mgxKi‡Yi GKwU g~j AciwUi e‡M©i mgvb nq Zvn‡j cÖgvY Ki †h,

abcbacca 3322

* ii) hw` 02 cbxax mgxKi‡Yi GKwU g~j AciwUi e‡M©i mgvb nq Zvn‡j cÖgvY Ki †h,

33)()( bcabac .

*** iii) 0)2(6272 pxx mgxKi‡Yi GKwU g~j AciwUi e‡M©i mgvb n‡j p Gi gvb wbY©q

Ki| DËit 6, -1

*** iv) hw` 02 qqxpx mgxKi‡Yi g~j`ywUi AbycvZ m t n nq, Z‡e †`LvI †h,

0p

q

m

n

n

m.

v) hw` 02 cbxax mgxKi‡Yi g~j`ywUi AbycvZ 3 t 4 nq, Zvn‡j cÖgvY Ki †h,

acb 49122 .

vi) hw` 011

2

1 cxbxa mgxKi‡Yi g~j`ywUi AbycvZ 022

2

2 cxbxa mgxKi‡Yi

g~j`ywUi Abycv‡Zi mgvb n‡j, †`LvI †h,

22

2

2

11

2

1

ca

b

ca

b .

vii) hw` 02 qpxx mgxKi‡Yi GKwU g~j AciwUi e‡M©i mgvb nq, Z‡e †`LvI †h,

0)13(23 qpqp

*** viii) hw` 02 qpxx mgxKi‡Yi g~j`yBwU µwgK c~Y© msL¨v nq, Zvn‡j cÖgvY Ki †h,

0142 qp

*** ix) hw` 02 qpxx mgxKi‡Yi g~j ywUi cv_©K¨ 1 nq, Z‡e cÖgvY Ki †h,

222)21(4 qqp

*** x) qxpx

111

mgxKi‡Yi g~j `ywUi AšÍi r n‡j, p ‡K q Ges r Gi gva¨‡g cÖKvk Ki|

DËit 22

42 rqqp

* xi) k Gi gvb KZ n‡j 0)13(3)3(22 kkxxk mgxKi‡Yi g~j`ywU ci¯úi Dëv n‡e?

DËit 4, -1

*** xii) 02 cbxax mgxKi‡Yi g~jØq , n‡j, cÖgvY Ki †h,

22

222 2

)()(ca

acbbaba

*** xiii) hw` 02 cbxx Ges 0

2 bcxx mgxKi‡Yi g~j¸wji g‡a¨ †Kej GKwU aªye‡Ki

cv_©K¨ _v‡K, Z‡e cÖgvY Ki †h, 04 cb .

* xiv) 02 cbxax mgxKi‡Yi g~j`ywUi AbycvZ r n‡j †`LvI †h,

ac

b

r

r22

)1(

xv) 02 cbxax mgxKi‡Yi g~jØ‡qi AbycvZ 4 t 5 n‡j cÖgvY Ki †h, acb 2120

2


Paper cÖkœ-2

2| (L) eûc`x I eûc`x mgxKiY (g~j-mnM m¤úK© I mgxKiY MVb)

1.*** i) hw` 012 qxpx Ges 01

2 pxqx mgxKiY ywUi GKwU mvaviY g~j _v‡K, Zvn‡j

†`LvI †h, 01 qp .

*** ii) hw` 062 kkxx Ges 02

2 kxx mgxKiY ywUi GKwU mvaviY g~j _v‡K Zvn‡j

k Gi gvb¸wj wbY©q Ki| DËit 0, 3, 8

*** iii) hw` 02 cbxax Gi GKwU g~j 0

2 abxcx Gi GKwU g~‡ji wØ¸Y nq, Zvn‡j

†`LvI †h, ca 2 A_ev, 22

2)2( bca

* iv) hw` 02 cbxax Ges 0

2 abxcx mgxKiY ywUi GKwU mvaviY g~j _v‡K, Zvn‡j

†`LvI †h, bac .

* v) 02 qpxx Ges 0

2 pqxx mgxKiY ywUi GKwU mvaviY g~j _vK‡j †`LvI †h,

Zv‡ì Aci ywU gyj 02 pqxx mgxKi‡Yi g~j n‡e|

vi) ‡h kZ© mv‡c‡¶ 011

2

1 cxbxa Ges 022

2

2 cxbxa mgxKiY ywUi GKwU g~j

mvaviY n‡Z cv‡i Zv wbY©q Ki| DËit2

122112211221 )())(( acaccbcbbaba

vii) ‡h kZ© mv‡c‡¶ 011

2

1 cxbxa Ges 022

2

2 cxbxa mgxKiY ywUi ywU g~jB

mvaviY n‡Z cv‡i Zv wbY©q Ki| DËit

2

1

2

1

2

1

c

c

b

b

a

a

viii) 02 qpxx Ges 0

2 pqxx mgxKiY ywUi GKwU gvÎ mvaviY g~j _vK‡j cÖgvY

Ki †h, 01 qp

2.*** i) 01642 xx mgxKi‡Yi g~j`ywU , n‡j

1 Ges

1 g~j wewkó mgxKiYwU

wbY©q Ki| DËit 0253042 xx

ii) 02 cbxax mgxKi‡Yi g~jØq , n‡j 2

Ges 2g~jwewkó mgxKiYwU

wbY©q Ki| DËit 03)2(232223 caabcbacxacbabaxa

*** iii) 02 cbxax mgxKi‡Yi g~jØq , n‡j Giƒc mgxKiY wbY©q Ki hvi g~jØq

1 Ges

1 . DËit 0)()(

22 acxacbcax

iv) 02 abxax mgxKi‡Yi g~jØq , n‡j ba Ges ba g~jØq Øviv MwVZ

mgxKiYwU wbY©q Ki| DËit 022 abxx

* v) 02 baxx mgxKi‡Yi g~jØq , n‡j

2)( Ges

2)( g~jwewkó mgxKiYwU

wbY©q Ki DËit 04)2(22422 baaxbax

vi) GKwU mgxKiY wbY©q Ki hvi g~j`yBwU h_vµ‡g 0143172 xx mgxKi‡Yi g~jØ‡qi

†hvMdj I ¸Yd‡ji mgvb n‡e| DËit 0422892892 xx

*** vii) Ggb GKwU mgxKiY wbY©q Ki hvi g~j`ywU h_vµ‡g 02222 baaxx mgxKi‡Yi g~j`ywUi

mgwó I Aš—id‡ji cig gvb n‡e| DËit 04)(22 abxbax

viii) hw` 0)(4

1 222 baaxx mgxKi‡Yi g~j `ywU , nq, Z‡e cªgvY Ki †h,

0)(2 abxbax mgxKi‡Yi g~j`ywU + I - n‡e|

ix) 02 rqxpx mgxKi‡Yi g~jØq , n‡j

3

1

Ges

3

1

g~jwewkó mgxKiY wbY©q Ki|

DËit 0)3(3223 axacbbxc

x) hw` 02 pqxpx mgxKi‡Yi g~j`ywU , nq, Zvn‡j cÖgvY Ki †h,

2))(( pqpqp Ges qp , qp g~jwewkó mgxKiYwU wbY©q Ki|

DËit 022 pqxx

*** xi) hw` 02 cbxax mgxKi‡Yi g~jØq , nq, Zvn‡j 042

2 abxcx mgxKi‡Yi

g~j`ywU , Gi gva¨‡g cÖKvk Ki| DËit

2 ,

2

** xii) hw` 02 cbxax mgxKi‡Yi g~jØq , nq, Zvn‡j

0)2()1(22 xacbxac Gi g~j`ywU , Gi gva¨‡g cÖKvk Ki| DËit

,


Paper cÖkœ-2

2| (M) wØc`x Dccv`¨

1.*** i)

11

2 32

xx Gi we¯Z…wZ‡Z

10x Gi mnM wbY©q Ki| DËit

4732330

* ii)

15

2 3

x

ax Gi we¯Z…wZ‡Z


4110565a

iii)

10

2 2

x

yx Gi we¯Z…wZ‡Z


43360y

* iv) 78)1()1( xx Gi we¯Z…wZ‡Z

7x Gi mnM wbY©q Ki| DËit 35

2.*** i)

11

24

12

xx Gi m¤cÖmvi‡Y x ewR©Z c`wUi gvb wbY©q Ki|

*** ii)

12

3 12

xx Gi we¯Z…wZ †_‡K x ewR©Z c`wUi gvb wbY©q Ki|

*** iii)

10

6

12

xx Gi we¯Z…wZ‡Z x ewR©Z c`wU †ei Ki Ges Gi gvb wbY©q Ki|

*** iv)

18

2

1

x

xGi we¯Z…wZ †_‡K aª‚eK c`wU †ei Ki Ges Gi gvb wbY©q Ki|

* v)

6

2

2 12

xx Gi m¤cÖmvi‡Y x ewR©Z c`wUi gvb wbY©q Ki|

vi)

10

3

2

2

12

xx Gi we¯Z…wZ‡Z x ewR©Z c`wU †ei Ki Ges Gi gvb wbY©q Ki|

vii)

102

3

4

x

y

y

xGi we¯Z…wZ‡Z y ewR©Z c`wU †ei Ki Ges Gi gvb wbY©q Ki|

viii)

15

2

23

xx Gi we¯Z…wZ‡Z x ewR©Z c` Ges c`wUi gvb wbY©q Ki|

*** ix)

n

xx

21

Gi we¯Z…wZ‡Z x ewR©Z c` Ges Zvi gvb wbY©q Kit

*** x) p Ges q abvZ¥K c~Y© msL¨v n‡j, qp

xx )

11()1( Gi we¯Z…wZ‡Z x gy³ c`wUi gvb wbY©q

Ki|

xi) qp

xx )

4

11()41( Gi we¯Z…wZi †Kvb c`wU me©`v x gy³ _vK‡e?

DËit i) 495 ii) 1760 iii) 6 Zg c` =

27

28 iv) DËit 13 Zg c` = 18564

v) 924 vi) 840 vii) 7 Zg c`,

32

1054

x viii) 6 Zg c`,

115321001

ix) !!

)!2()1(

nn

nn x) !!

)!(

qp

qp xi) )1( q Zgc`

3. i) hw` 5

)2( xa Gi we¯Z…wZ‡Z 3

x Gi mnM 320 nq, Zvn‡j a Gi gvb KZ n‡e?

DËit 2a

*** ii)

nx

23 Gi we¯Z…wZ‡Z

7x I

8x Gi mnM yBwU mgvb n‡j, (n N), n Gi gvb wbY©q Ki|

iii) 34)34( x Gi we¯Z…wZ‡Z µwgK `yBwU c‡ì mnM mgvb n‡j G c` `yBwUi x Gi NvZ wbY©q Ki|

DËit 19

x , 20

x

iv) nx)1( Gi we¯Z…wZ‡Z )1( r Zg c‡ì mnM, )3( r Zg c‡ì mn‡Mi mgvb n‡j, †`LvI

†h, 22 nr (n N),

* v) 14)1( x Gi we¯Z…wZ‡Z )1( r Zg c‡ì mnM, )13( r Zg c‡ì mn‡Mi mgvb n‡j r Gi

gvb wbY©q Ki| DËit 4

vi) nx)1( Gi we¯Z…wZ‡Z wZbwU µwgK c‡ì mn‡Mi AbycvZ 1 t 7 t 42 n‡j n Gi gvb wbY©q

Ki| DËit 55

4.*** i ) 44)1( x Gi w e¯Z … wZ ‡Z 21 Zg c` I 22 Zg c` ` y B w U mg v b n ‡j , x Gi

g v b w bY © q Ki| DËi t

8

7


Paper cÖkœ-2

*** ii) hw` n

xa )3( Gi we¯Z…wZ‡Z cÖ_g wZbwU c` h_vµ‡g b , bx2

21 I

2

4

189bx nq, Zvn‡j a ,

b Ges n Gi gvb †ei Ki| DËit 2a , 7n , 7

2b

* iii) nx)1( Gi we¯Z…wZ‡Z hw` a, b, c, d h_vµ‡g 6ô, 7g, 8g, 9g c` nq, Z‡e cÖgvY Ki †h,

c

a

dbc

acb

3

42

2

.

5. i)

10

2

2

13

xx Gi we¯Z…wZ‡Z ga¨c`wU wbY©q Ki| DËit

5

5

10

5

2

3xC

** ii) n N n‡j

12

n

a

x

x

aGi we¯Z…wZ †_‡K ga¨c` (c`¸‡jv) wbY©q Ki|

DËit )1( n Zg c` =

x

aCn

n 12 , )2( n Zg c` =

a

xCn

n

1

12

iii) †`LvI †h,

n

xx

21

Gi we¯Z…wZi ga¨c`wU

n

n

n)2(

!

)12........(5.3.1

.

*** iv) †`LvI †h,

n

xx

21

Gi we¯Z…wZi ga¨c`wU

n

n

n)2(

!

)12........(5.3.1

.

6.*** i) ‡`LvI †h, 2

1

41

x Gi we¯Z…wZ‡Z r

x Gi mnM 2

)!(

)!2(

r

r.

*** ii) ‡`LvI †h, 2

1

21

x Gi we¯Z…wZ‡Z )1( r Zg c‡ì mnM r

r

r

2)!(

)!2(2

.

7.*** i) )21)(1(

1

xx Gi we¯Z…wZ‡Z

rx Gi mnM wbY©q Ki| DËit 12

1 r

*** ii) cÖgvY Ki †h, 12

)651( xx Gi we¯Z…wZ‡Z

nx Gi mnM

1123

nn.

iii) )1)(1( bxax

x

Gi we¯Z…wZ‡Z

nx Gi mnM wbY©q Ki| DËit

ba

bann

iv) ‡`LvI †h,

x

xn

1

)1(Gi we¯Z…wZ‡Z

nx Gi mnM

n2 (n N).

v) 2

1

32)4321( xxx Gi we¯Z…wZ‡Z

rx Gi mnM wbY©q Ki| DËit 1

8.*** i) hw` 32xxxy nq, Zvn‡j †`LvI †h, ....

432 yyyyx

*** ii) hw` .....432 xxxxy nq, Zvn‡j x †K y Gi kw³i DaŸ©µg avivq cÖKvk Ki|

DËit ....432 yyyyx

*** iii) hw` ....43232 xxxy nq, Z‡e †`LvI †h, .....

16

5

8

3

2

1 32 yyyx

iv) cÖgvY Ki †h, ....)321....)(1(22 xxxx =

....)5.44.33.22.1(2

1 32 xxx

v) n N Ges n

n

nxcxcxccx ...)1(

2

210 n‡j cÖgvY Ki †h,

1

531420 2......... n

cccccc .

vi) 8x n‡j

2

1

81

x †K x Gi kw³i DaŸ©µgvbymv‡i cÂg c` ch©šÍ we Ívi Ki Ges †`LvI

†h,

2

3....

32

5.

24

3.

16

1.

8

1

24

3.

16

1.

8

1

16

1.

8

1

8

11

DËit ....2

.32

5.

24

3.

16

1.

8

1

2.

24

3.

16

1.

8

1

2.

16

1.

8

1

2.

8

11

4

4

3

3

2

2xxxx


Paper cÖkœ-2

3| (‡hvMvkªqx ‡cÖvMÖvg) ‡h †Kvb GKwU cÖ‡kœi DËi `vIt 5 1=5

3| (K) Group A

1.*** i) ‡hvMvkªqx ‡cÖvMÖvg wK? †hvMvkªqx ‡cÖvMÖv‡gi myweav I kZ©vejx Av‡jvPbv Ki|

*** ii) ÒAvaywbK Drcv`b I e›Ub e¨ve ’vq †hvMvkªqx †cÖvMÖvg GKwU Acwinvh© nvwZqviÓ| e¨vL¨v Ki|

* iii) wKfv‡e †hvMvkªqx ‡cÖvMÖv‡gi mgm¨v MVb Kiv nq? Zv we ÍvvwiZfv‡e eY©bv Ki|

2. wbæwjwLZ †hvMvkªqx †cÖvMÖvg‡K †jLwP‡Îi mvnv‡h¨ mgvavb K‡i cÖvwš—K we›`y wbY©q Ki Ges m‡e©v”PKiY Kit

*** i) z = 4x + 6y; kZ©t x + y = 5, x 2, y 4, x,y 0

*** ii) z = 3x + 4y; kZ©t x + y 7, 2x +5y 20, x,y 0

*** iii) z = 2x + 3y; kZ©t x + 2y 10, x + y 6, x 4., x,y 0

* iv) z = 2x + y; kZ©t x + 2y 10, x + y 6, x – y 2, x – 2y 10, x,y 0

*** v) z = 12x + 10y; kZ©t 2x + y 90, x + 2y 80, x + y 50, x 0, y 0.

** vi) z = 5x + 7y; kZ©t x + y 4, 3x + 8y 24, 10x + 7y 35, x 0, y 0.

vii) z = 3x + 2y; kZ©t x + y 1, y – 5x 0, 5y – x 0, x - y -1, x + y 6,

x 3, x,y 0. * viii) z = 3x + y; kZ©t 2x + y 8, 2x + 3y 12, x 0, y 0.

* ix) z = 3x + 4y; kZ©t x + y 450, 2x + y 600, x 0, y 0.

** x) z = 3x + 2y; kZ©t 2x + y 8, 2x + 3y 12, x 0, y 0.

*** xi) z = 12x + 10y; kZ©t 2x - y 90, x - 2y 80, x - y 50, x,y 0.

DËit i) 26 ii) 23 iii) 16 iv) 10

v) 580 vi) 24.8 vii) 15 viii) 12 ix) 1800 x) 13 xi) 380 3. wbæwjwLZ †hvMvkªqx †cÖvMÖvg‡K †jLwP‡Îi mvnv‡h¨ mgvavb K‡i cÖvš—we›`y wbY©q Ki Ges me©wbæKib Kit

*** i) a) z = 2y – x

b) z = - x +y Dfq †¶‡ÎB kZ©t 3y – x 10, x + y 6, x – y 2, x,y 0.

*** ii) a) z = 2x – y b) z = 4x - y Dfq †¶‡ÎB kZ©t x + y 5, x + 2y 8, 4x + 3y 12, x,y 0.

*** iii) z = 2x – y kZ©t x + y 5, x + 2y 8, x,y 0.

iv) z = 3x + 5y kZ©t x 2y + 2, x 6 – 2y, y x, x 6.

* v) z = 4x + 6y kZ©t x + y = 5, x 2, y 4, x,y 0.

vi) z = 3x1 + 2x2, kZ©t x1 + 2x2 4, 2x1 + x2 4, x1 + x2 5, x1, x2 0.

DËit i) (a) -2 (b) -2 ii) (a) 1 (b) 5 iii) -5 iv) 16 v) 20 vi) 3

20

4.*** i) A I B `yB cÖKv‡i Lv‡`ï cÖwZ †KwR‡Z †cÖvwUb I †k¦Zmv‡ii cwigvb I Zvi g~j¨ wbæiƒct

Lv`¨ ‡cÖvwUb cÖwZ †KwR ‡k¦Zmvi cÖwZ †KwR cÖwZ †KwRi g~j¨

A 8 10 40 UvKv

B 12 6 50 UvKv

‣`wbK b~¨bZg cÖ‡qvRb 32 22

me‡P‡q Kg Li‡P Kxfv‡e •`wbK Lv`¨ cª‡qvRb †gUv‡bv hv‡e Zv wbY©q Ki|

*** ii) A I B `yB ai‡Yi Lv‡`ï cÖwZ wK‡jv‡Z †cÖvwUb I d¨vU wbæiƒct

Lv`¨ ‡cÖvwUb d¨vU wK‡jv cÖwZ g~j¨

A 1 3 2 UvKv

B 3 2 3 UvKv


me‡P‡q Kg Li‡P Kxfv‡e •`wbK Lv`¨ cª‡qvRb †gUv‡bv hv‡e Zv wbY©q Ki|

iii) F1 I F2 `yB ai‡Yi Lv‡`ï cÖwZ wK‡jv‡Z wfUvwgb C I D cvIqv hvq wbæiƒct

Lv`¨ wfUvwgb C wfUvwgb D wK‡jv cÖwZ g~j¨

F1 2 3 5 UvKv

F2 5 6 3 UvKv


me‡P‡q Kg Li‡P ‣`wbK wfUvwgb C I D Gi Pvwn`v Kxfv‡e †gUv‡bv hv‡e Zv wbY©‡qi Rb¨ GKwU

†hvMvkªqx †cÖvMÖvg mgm¨v •Zix Ki|

* iv) GK e¨w³ X I Y yB ai‡bi Lv`¨ MÖnY K‡i| wZb ai‡Yi cywó N1, N2, N3 Gi cwigvb, Lv‡`ï g~j¨ I

cywói •`wbK me©wbæ cÖ‡qvRb wbæiƒct

`vg

X Y ‣`wbK b~¨bZg

cÖ‡qvRb 1.00 UvKv 3.00 UvKv

N1 30 12 60

N2 15 15 60

N3 6 18 36

‡hvMvkªqx †cÖvMÖv‡gi mvnv‡h¨ Lv‡`ï Ggb GKwU mgšq wbY©q Ki, hv me©wbæ Li‡P H e¨w³i •`wbK

cÖ‡qvRb †gUv‡e|

(v) wb‡æi cÖ Ë ZvwjKv †_‡K mgvavb †ei Ki Ges me©wbæ e¨‡q cÖ‡qvRbxq cywó mgwš^Z Lv‡`ï m‡e©vrK…ó

mgšq Kit

cÖwZ GK‡Ki g~j¨ (UvKv)

Lv`¨-A Lv`¨-B b~¨bZg GKK cÖ‡qvRb

1.00 2.00

cywó-I 20 8 40

cywó-II 10 10 40

cywó-III 4 12 24

DËit i) A cÖKvi 1 †KwR, B cÖKvi 2 †KwR| ii)A cÖKv‡ii

†KwR Ges B cÖKv‡ii

, †gvU LiP

UvKv|

iii) F1 – 5, F2 – 8. iv) X cÖKvi 3 †KwR, Y cÖKvi 1 †KwR|

v) A cÖKvi 3 GKK, B cÖKvi 1 GKK|


Paper cÖkœ-3

3| (L) Group B

1.*** i) GK e¨w³ 500 UvKvi g‡a¨ Kgc‡¶ 6 Lvbv MvgQv Ges 4 Lvbv †Zvqv‡j wKb‡Z Pvq| cÖwZLvbv MvgQvi

`vg 30 UvKv Ges cÖwZLvbv †Zvqv‡ji `vg 40 UvKv| cÖ‡Z¨K cÖKv‡ii KZLvbv wRwbm wKb‡j †m cÖ`Ë

kZ©vax‡b me©v‡c¶v †ewk msL¨K wRwbm wKb‡Z cvi‡e?

*** ii) GKwU †jvK me©vwaK 100 UvKv e¨q K‡i K‡qKLvbv _vjv I Møvm wKb‡Z Pvb| cÖwZwU _vjv I Møv‡mi g~j¨

h_vµ‡g 12 UvKv I 8 UvKv| Aš—Zt 1 Lvbv _vjv I 8 wUi †ewk Møvm wZwb wKb‡eb bv| Dc‡iv³

UvKvq wZwb †Kvb cÖKv‡ii KZK¸‡jv wRwbm wKb‡j GK‡Î me©vwaK msL¨K wRwbm wKb‡Z cvi‡eb?

iii) GKwU †jvK me©vwaK 500 UvKv e¨‡q K‡qKwU Kvc I †cøU wKb‡Z Pvb| cÖwZ Kv‡ci `vg 30 UvKv I

†cø‡Ui `vg 20 UvKv| Ab~¨b 3 wU †cøU I AbwaK 6 wU Kvc †Kbvi k‡Z© H UvKvq †Kvb cÖKv‡ii

KZ¸‡jv wRwbm wKb‡j wZwb †gvU me©vwaK wRwbl wKb‡Z cvi‡eb?

iv) GK e¨w³ Zvi evMv‡b Kgc‡¶ 12wU bvi‡K‡ji Pviv Ges 4wU Av‡gi Pviv jvMv‡Z Pvb| cÖwZwU

bvi‡K‡ji Pviv I Av‡gi Pvivi g~j¨ h_vµ‡g 20 UvKv Ges 30 UvKv| H e¨w³ 600 UvKvi †ekx e¨q

bv K‡i cÖ‡Z¨K cÖKv‡ii KZ¸‡jv Pviv wKb‡Z cv‡ib hv‡Z †gvU Pvivi msL¨v me©vwaK nq?

*** v) GK e¨w³ 1200 UvKvi gv‡Qi †cvbv wKb‡Z Pvq| 100 iyB gv‡Qi †cvbvi `vg 60 UvKv Ges 100

KvZj gv‡Qi †cvbvi `vg 30 UvKv n‡j, wZwb †Kvb gv‡Qi KZ †cvbv wKb‡Z cvi‡eb hvi †gvU msL¨v

me©vwaK 3000 nq?

DËit i) MvgQv 10 Lvbv, ‡Zvqv‡j 5 Lvbv| ii) _vjv 3 Lvbv, Møvm 8 wU|

iii) 6 Uv Kvc, 16 Uv ‡cøU| iv) bvi‡Kj Pviv 18 wU, Av‡gi Pviv 8 wU|

v) iyB 1000, KvZj 2000|

2.*** i) GKRb dj we‡µZv Av½yi I Kgjv wgwj‡q 500 UvKvi dj wKb‡e| wKš‘ ¸`vgN‡i 12 wUi AwaK ev·

ivL‡Z cv‡i bv| GK ev· Kgjvi `vg 50 UvKv Ges GK ev· Av½y‡ii `vg 25 UvKv| †m cÖwZ ev·

Kgjv I Av½yi h_vµ‡g 10 UvKv I 6 UvKv jv‡f weµq K‡i| †jvKwU †h cwigvY dj †K‡b Zvi meB

wewµ n‡h hvq| Kgjv I Av½yi KZ¸‡jv µh Ki‡j †m m‡e©v‛P jvf Ki‡Z cvi‡e?

*** ii) GKRb eëmvqx Zvi †`vKv‡bi Rb¨ †iwWI Ges †Uwjwfkb wg‡j 100 †mU wKb‡Z cv‡ib| †iwWI †mU

I †Uwjwfkb †mU cÖwZwUi µh g~j¨ h_vµ‡g 40 Wjvi I 120 Wjvi| cÖwZ †iwWI I †Uwjwfkb †m‡U

jvf h_vµ‡g 16 Wjvi I 32 Wjvi| m‡e©v‛P 10400 Wjvi wewb‡qvM K‡i m‡e©v‛P KZ jvf wZwb

Ki‡Z cv‡ib?

iii) GKRb K…lK avb Ges M‡gi Pvl Ki‡Z wM‡q †`L‡jb †h cÖwZ †n±i Rwg‡Z avb I Mg Pv‡li LiP

h_vµ‡g 1200 UvKv Ges 800 UvKv| cÖwZ †n±i Rwg‡Z avb I Mg Pv‡li Rb¨ h_vµ‡g 4 Rb I 6

Rb K‡i kªwg‡Ki cÖ‡qvRb nq| m‡e©v‛P 26 Rb kªwgK wb‡qvM K‡i Ges 4800 UvKv wewb‡qvM K‡i

m‡e©v‛P KZ †n±i Rwg wZwb Pvl Ki‡Z cvi‡eb?

DËit i) Kgjv 8 ev·, Av½yi 4 ev·| ii) 2880 Wjvi iii) 5 ‡n±i|

3. i) GKwU cvYxq •Zixi KviLvbvq yBwU kvLv I Ges II Gi Df‡qB A, B Ges C wZb cÖKv‡ii cvbxq

†evZjRvZ K‡i| kvLv yBwUi •`wbK Drcv`b ¶gZv wbæiƒct

kvLv A cÖKv‡ii cvbxq B cÖKv‡ii cvbxq C cÖKv‡ii cvbxq

I 3000 1000 2000

II 1000 1000 6000

A, B I C cÖKv‡ii cvbx‡qi gvwmK Pvwn`v h_vµ‡g 24000, 16000 Ges 48000 †evZj| I Ges II

kvLvi •`wbK Kvq© cwiPvjbvq e¨q h_vµ‡g 600 UvKv I 400 UvKv| gv‡m †Kvb kvLv KZ w`b Pvjy

ivL‡j Zv me©wbæ Kvh© cwiPvjbvi e¨‡q cvbx‡qi gvwmK Pvwn`v c~iY Ki‡Z cvi‡e? me©wbæ e¨q KZ?

ii) GKwU cÖwZôvb Zv‡ì Drcvw`Z A I B cb¨ n‡Z GKK cÖwZ h_vµ‡g 3 UvKv I 4 UvKv jvf K‡i|

cÖwZwU cb¨ M1 I M2 †gwk‡b •Zix nq| A cb¨wU M1 I M2 †gwk‡b •Zix‡Z h_vµ‡g 1 wgwbU I 2

wgwbU mgq jv‡M Ges B cb¨wU M1 I M2 †gwk‡b h_vµ‡g 1 wgwbU I 1 wgwb‡U •Zix nq| cÖwZ Kv‡Ri

w`‡b M1 †gwkb me©vwaK 72

1N›Uv I M2 †gwk‡b me©vw©aK 10 N›Uv eënvi Kiv hv‡e| A I B cb¨ wK

cwigvb •Zix Ki‡j me©vwaK jvf n‡e? †hvMvkªqx †cÖvMÖv‡gi GKwU g‡Wj •Zix Ki|

iii) GKwU dvg© yBwU cb¨ †Uwej I †Pqvi •Zix K‡i| A I B ‡gwk‡bi mvnv‡h¨ cb¨ ywU‡K cÖwµqvRvZ Kiv nq| A

†gwkb 60 N›Uv ch©š— I B †gwkb 48 N›Uv ch©š— KvR Ki‡Z m¶g| GKwU †Uwej •Zix Ki‡Z A †gwk‡b 4

N›Uv Ges B †gwk‡b 2 N›Uv mgq jv‡M| cÖwZ †Uwe‡j 8 UvKv Ges cÖwZ †Pqv‡i 6 UvKv gybvdv n‡j me©vwaK

gybvdv cvIqvi Rb¨ KqLvbv †Uwej I KqLvbv †Pqvi •Zix Ki‡Z n‡e Zv wbY©q Ki|

DËit i) I cÖwZgv‡m 4 w`b, II cªwZgv‡m 12 w`b, me©wbæ e¨q 7200 UvKv|

ii) A – 150, B – 300. iii) 12 †Uwej, 6 †Pqvi|


Paper cÖkœ-3

4| ‡h †Kvb yBwU cÖ‡kœi DËi `vIt 25=10

4| (K) cive„Ë (Parabola) 1. wb‡Pi cÖwZwU cive„‡Ëi kxl©we›`y, Dc‡K›`ª, Dc‡Kw›`ªK j¤, A¶‡iLv Ges w`Kv‡¶i mgxKiY wbY©q Ki|

* )i 018121032 yxy DËit )2,3( ; )2,

6

13( ;

3

10; 2y ;

6

23x

* )ii 023282 yxy DËit )1,3( ; )1,1( ; 8 ; 1y ; 5x

)iii 07822 xyx DËit )

2

9,4( ; )4,4( ; 2 ; 4x ; 5y

* )iv 04101552 yxx

DËit )40

61,

2

3(

; )

40

41,

2

3(

; 2 ; 0

2

3x ; 0

40

81y

** )v 056432 xyx DËit )2,1( ; )

3

5,1(

;

3

4; 01x ; 073 y

*** )vi 05923052 yxx

DËit )7,3( ; )10

71,3(

;

5

2; 03 x ; 06910 y

*** )vii 06822 xyx DËit )5,4( ; )

2

9,4(

; 2 ; 4x ; 0112 y

)viii 0242 yxx DËit )2,2( ; )

2

3,2( ; 2 ; 02 x ; 052 y

* )ix 056432 xyx DËit )2,1( ; )

3

5,1(

;

3

4; 01x ; 073 y

)x yxy 882 DËit )4,2( ; )4,0( ; 8 ; 04 y ; 04 x

* )xi 05923052 yxx

DËit )7,3( ; )10

71,3(

;

5

2; 03 x ; 06910 y

** )xii 06822 xyx DËit )5,4( ; )

2

9,4(

; 2 ; 4x ; 0112 y

)xiii )1(42

yx DËit )1,0( ; )0,0( ; 4 ; 0x ; 2y

)xiv 582 xy DËit )0,

8

5(

; )0,8

11( ; 8 ; 0y ;

5

18x

* )xv )3(22 xy DËit )0,3( ; )0,

2

5(

; 2 ; 0y ; 072 x

2.*** pxy 42 cive„ËwU )2,3( we›`y w`‡q Mgb Ki‡j Gi Dc‡Kw› ªK j‡¤î •`N©¨ Ges Dc‡K‡›`ªi ’vbvsK

wbY©q Ki| DËit

3

4; )0,

3

1(

3. Giƒc cive„‡Ëi mgxKiY wbY©q Ki hvit

*** )i Dc‡K› ª )2,8( Ges w`Kv‡¶i mgxKiY 92 yx

DËit 02592116)2(2 yxyx

*** )ii Dc‡K› ª )1,1( Ges w`Kv‡¶i mgxKiY 143 yx , Zvi A‡¶iI mgxKiY wbY©q Ki|

DËit 0494244)34(2 yxyx ; 0143 xy

*** )iii Dc‡K› ª )1,1( Ges w`Kv‡¶i mgxKiY 01 yx , cive„‡Ëi A‡¶i mgxKiY Ges

Dc‡Kw›`ªK j‡¤i •`N©¨ I Gi mgxKiY wbY©q Ki|

DËit 0362)(2 yxyx ; 02 yx , 0 yx , 2

)iv Dc‡K› ª )0,(a Ges w`Kv‡¶i mgxKiY 0 cx DËit )2)((2

caxcay

* )v Dc‡K› ª )0,2( Ges w`Kv‡¶i mgxKiY 02 x DËit xy 82

)vi Dc‡K› ª )4,0( Ges w`Kv‡¶i mgxKiY 04 y DËit 0162 yx

4.*** i) †h cive„‡Ëi Dc‡K› ª (3,4) Ges kxl© (0,0) Zvi w`Kv‡¶i mgxKiY wbY©q Ki|

DËit 3x+4y+25=0

ii) †h cive„‡Ëi Dc‡K›`ª (-1,1) Ges kxl© (2,-3) Zvi A¶ I w`Kv‡¶i mgxKiY wbY©q Ki|

DËit 4x+3y+1=0, 3x-4y-43=0

** iii) †h cive„‡Ëi Dc‡K› ª (0,0) Ges kxl© (-2,-1) Zvi w`Kv‡¶i mgxKiY wbY©q Ki|

DËit 2x+y+10=0

5. Giƒc cive„‡Ëi mgxKib wbY©q Ki hvit

i) Dc‡K› ª (-1,3) Ges kxl© (4,3) we› y‡Z| DËit y2+20x-6y-71=0

ii) Dc‡K› ª (-6,-3) Ges kxl© (-2,1) we›`y‡Z| DËit (x-y)2 +38x+26y+41=0

** iii) Dc‡K› ª (2,5) Ges x = 4 †iLvwU Gi kxl© we›`y‡Z ¯úk© K‡i| DËit y2 - 10y+8x-7=0

6. Giƒc cive„‡Ëi mgxKib wbY©q Ki hvit

* i) kxl© (2,3) Ges w`Kv‡¶i mgxKiY y = 6; Gi Dc‡Kw›`ªK j‡¤i •`N©Ï wbY©q Ki|


Paper cÖkœ-4

DËit x2 – 4x+12y-32=0, 12

* ii) kxl© (3,1) Ges w`Kv‡¶i mgxKiY 4x+3y-5=0; DËit (3x-4y)2 – 190x-80y+625=0

* iii) kxl© (4,3) Ges w`Kv‡¶i mgxKiY y = 7; DËit x2 – 8x+16y-32=0

7.* i) Giƒc cive„‡Ëi mgxKiY wbY©q Ki hvi kxl© (4,-3) we›`y‡Z Aew¯’Z Ges w`Kv¶ x-A‡¶i mgvšÍivj

Ges hv (-4,-7) we› y w`‡q AwZµg K‡i| DËit x2 – 8x+16y+64=0

* ii) GKwU cive„‡Ëi mgxKiY wbY©q Ki hvi kxl©we›`y (4,-3) we›`y‡Z Aew¯’Z, Dc‡Kw›`ªK j‡¤î •`N©¨ 4

Ges A¶wU x-A‡¶i mgvš—ivj| DËit (y+3)2 = 4(x-4)

*** iii) y = ax2 + bx + c cive„ËwUi kxl© (-2,3) we›`y‡Z Aew¯’Z Ges GwU (0,5) we›`y w`‡q AwZµg

K‡i| a, b, c-Gi gvb wbY©q Ki| DËit a =2

1, b = 2, c = 5

8.***i) y2 = 8x cive„‡Ëi Dcwiw ’Z †Kvb we›`yi Dc‡Kw›`ªK ~iZ¡ 8; H we›`yi ’vbvsK wbY©q Ki|

DËit 34,6(

*** ii) y2 = 16x cive„‡Ëi Dcwiw ’Z †Kvb we›`yi Dc‡Kw› ªK ~iZ¡ 6; H we›`yi ’vbvsK wbY©q Ki|

DËit 24,2( )

iii) y2 = 9x cive„Ë¯’ †Kvb P we›`yi †KvwU 12 n‡j H we›`yi Dc‡Kw›`ªK `~iZ¡ wbY©q Ki|

DËit 184

1

9. i) y2 = 12x cive„‡Ëi kxl©we›`y I Dc‡Kw›`ªK j‡¤i abvZ¥K w`‡Ki cÖvš—we›`yi ms‡hvRK †iLvi mgxKiY

wbY©q Ki| DËit y = 2x.

* ii) ‡`LvI †h, 0 nmylx †iLvwU axy 42 cive„Ë‡K ¯úk© Ki‡e hw`

2ln am nq|

iii) cive„‡Ëi Av`k© mgxKiY wbY©q Ki|

4| (L) Dce„Ë (Ellipse)

10. wb‡Pi Dce„Ë¸wji cÖwZwUi Dr‡Kw›`ªKZv, Dc‡K› ª, Dc‡Kw›`ªK j‡¤i •`N©¨ Ges Dc‡Kw›`ªK j‡¤i I w`Kv‡¶i

mgxKiY wbY©q Kit (i – iv).

* i) 22525922 yx DËit

5

4, )0,4( ,

5

18, 4x , 254 x

** ii) 124322 yx DËit

2

1, )0,1( , 3 , 1x , 4x

*** iii) 400162522 yx DËit

5

3, )3,0( ,

5

32, w`Kv‡¶i mgxKib

3

25y

*** iv) 400251622 yx DËit

5

3, )0,3( ,

5

32, 3x

3

25x

* v) 14416922 yx Dce„‡Ëi Dc‡K›`ªØ‡qi ’vbvsK Ges w`Kv¶Ø‡qi mgxKiY wbY©q Ki|

DËit )0,7( ;

7

16x

** vi) 13222 yx Dce„‡Ëi Dc‡Kw›`ªK j‡¤i •`N©¨ Ges Dc‡K› ª `ywUi ’vbvsK wbY©q Ki|

DËit 23

2, )0,

6

1(

** vii) 14522 yx Dce„‡Ëi w`Kv¶ yBwUi mgxKiY wbY©q Ki| DËit

2

5y

*** viii) 0110165422 yxyx Dce„‡Ëi Dc‡K› ª yBwU, Dc‡Kw›`ªK j‡¤i •`N©¨,

Dr‡Kw›`ªKZv Ges w`Kv‡¶i mgxKiY wbY©q Ki|

DËit )1,3( , )1,1( ,

5

8,

5

1, 07 x , 03 x

ix) †`LvI †h, 0110165422 yxyx mgxKiYwU GKwU Dce„Ë wb‡`©k K‡i; Gi

Dr‡Kw›`ªKZv, †K› ª, Dc‡K› ª, Dc‡Kw›`ªK j‡¤i •`N©¨ Ges w`Kv‡¶i mgxKiY wbY©q Ki| DËit

5

1,

)1,2( , )1,3( I )1,1( ,

5

8, 03 x


Paper cÖkœ-4

* x) †`LvI †h, 0309522 xyx mgxKiYwU GKwU Dce„Ë wb‡`©k K‡i; Gi Dc‡K› ª `ywUi

¯’vbvsK wbY©q Ki| DËit )0,5( , )0,1(

* xi) †`LvI †h, 0128222 yxyx mgxKiYwU GKwU Dce„Ë wb‡`©k K‡i; Gi

Dr‡Kw›`ªKZv, †K› ª Ges Dc‡K› ª `ywUi ’vbvsK wbY©q Ki|

DËit

2

1e , †K›`ª )1,2( , Dc‡K› ª )3,2( I )1,2(

2.*** i) p Gi gvb KZ n‡j 1422 ypx Dce„ËwU )0,1( we›`y w`‡q hv‡e? Dce„ËwUi Dc‡K‡› ªi

¯’vbvsK, Dr‡K›`ªZv Ges A¶Ø‡qi •`N©¨ wbY©q Ki|

DËit 1p ,

2

3, )0,

2

3( ; e„nrA¶ 2 , ¶z`ª A¶ 1

*** ii) p Gi gvb KZ n‡j, 125

22

y

p

xDce„ËwU )4,6( we›`y w`‡q AwZµg Ki‡e? Dce„ËwUi

Dr‡Kw›`ªKZv Ges Dc‡K‡› ªi Ae ’vb wbY©q Ki| DËit 100p ,

2

3, )0,35(

iii) 125

22

p

yx Dce„ËwU )6,4( we›`y w`‡q AwZµg K‡i| p Gi gvb, Dr‡Kw›`ªKZv Ges Dc‡K‡›`ªi

Ae¯’vb wbY©q Ki| DËit 100p ,

2

3e , )35,0(

* iv) p Gi gvb KZ n‡j, 80422 pyx Dce„ËwU )4,0( we›`y w`‡q AwZµg Ki‡e? Dce„ËwUi

A¶Ø‡qi •`N© Ges Dr‡Kw›`ªKZv wbY©q Ki| DËit 5p ,

5

1e , 54 , 8

3. GKwU Dce„‡Ëi mgxKiY wbY©q Ki hvit

*** i) Dc‡K› ª )4,3( , w`Kv¶ 02 yx Ges Dr‡Kw›`ªKZv

3

1.

DËit 04461401041721722 yxyxyx

* ii) Dc‡K› ª )3,2( , w`Kv¶ 07 yx Ges Dr‡Kw›`ªKZv

3

1.

DËit 029221025522 yxxyyx

* iii) Dc‡K› ª )1,1( , Dr‡Kw› ªKZv

2

1 Ges w`Kv‡¶i mgxKiY 03 yx


iv) Dc‡K› ª )1,1( , w`Kv¶ 02 yx Ges Dr‡Kw› ªKZv

2

1; Gi Dc‡Kw› ªK j¤Î wbY©q Ki|

DËit 04121223322 yxxyyx , 4

* v) Dc‡K› ª )2,0( , Dr‡Kw›`ªKZv

2

1 Ges w`Kv‡¶i mgxKiY 04 y , Zvi Dr‡Kw›`ªK j‡¤i

•`N©Ï wbY©q Ki| DËit 0243422 yyx ; 6

vi) Dc‡K› ª )1,2( , Dr‡Kw› ªKZv

3

1 Ges wbqvg‡Ki mgxKiY 32 yx

DËit 06624484141122 yxxyyx

vii) Dc‡K› ª )3,2( , wbqvg‡Ki mgxKiY 032 yx Ges Dr‡Kw›`ªKZv

3

1

DËit 018684724141122 yxxyyx

viii) Dc‡K› ª g~jwe›`y,

5

4e Ges w`Kv¶ 2x . DËit 6425649

22 yxx

4. Dce„‡Ëi cÖavb A¶ yBwU‡K x I y -A¶ we‡ePbv K‡i Giƒc Dce„‡Ëi mgxKiY wbY©q Ki hvit

*** i) Dr‡Kw›`ªKZv

3

1Ges Dc‡Kw› ªK j‡¤i •`N© 8 . DËit 1

1881

422

yx

ii) Dr‡Kw›`ªKZv

3

2Ges Dc‡Kw› ªK j‡¤i •`N© 5 . DËit 4053620

22 yx

iii) Dc‡Kw›`ªK j‡¤i •`N©¨ 8 Ges Dr‡Kw›`ªKZv

2

1 DËit 642

22 yx

* iv) Dc‡K‡› ªi ’vbvsK )0,3( Ges Dr‡Kw›`ªKZv

3

1; Dce„‡Ëi w`Kv‡¶i mgxKiYI wbY©q Ki|

DËit 6489822 yx , 27x


Paper cÖkœ-4

* v) Dc‡K‡› ªi ’vbvsK )4,0( Ges Dr‡Kw›`ªKZv

5

4; DËit 1

259

22

yx

* vi) Dc‡K› ªØq )0,2( Ges e„nr A¶ 8 GKK| DËit 11216

22

yx

vii) Dc‡K› ªØq )1,1( , )2,2( Ges e„nr A‡¶i •`N© 8 GKK| DËit 123

2

16

22

yx

* viii) e„nr A¶ 12 Ges Dr‡Kw›`ªKZv

3

1. DËit 1

3236

22

yx

ix) Dc‡K› ªØq )0,1( Ges Dc‡Kw› ªK j‡¤i •`N© 3 GKK| DËit 124322 yx

x) Dr‡Kw›`ªKZv

5

4Ges )5,

3

10( we›`yMvgx| DËit 225259

22 yx

* xi) Dc‡K› ªØ‡qi `~iZ¡ 8 Ges w`Kv¶Ø‡qi ga¨Kvi ~iZ¡ 18 . DËit 1809522 yx

*** xii) hv )2,2( Ges )1,3( we›`yMvgx| Gi Dr‡Kw›`ªKZvI wbY©q Ki|

DËit 325322 yx ,

5

2

* xiii) hv )6,1( Ges )0,3( we›`yMvgx| DËit 274322 yx

xiv) hv )4,2( Ges )2,5( we›`yMvgx| DËit 563222 yx

5. i) ‡h Dce„‡Ëi Dc‡Kw›`ªK j¤ e„nr A‡¶i A‡a©K Zvi Dr‡K› ªZv KZ? DËit

2

1

ii) †Kvb Dce„‡Ëi ¶z ª A‡¶i •`N©¨ Zvi Dc‡K›`ªØ‡qi ~i‡Z¡i mgvb Ges Dc‡Kw› ªK j¤^ 10 ; Dce„ËwUi

Dr‡Kw›`ªKZv I mgxKib wbY©q Ki| DËit

2

1; 1002

22 yx

iii) †Kvb Dce„‡Ëi GKwU Dc‡K› ª I Abyiƒc w`Kv‡¶i ga¨Kvi ~iZ¡ 16 BwÂ Ges Dr‡Kw›`ªKZv

5

3;

Dce„‡Ëi cÖavb A¶ yBwUi •`N© wbY©q Ki| DËit 30 BwÂ I 24 BwÂ|

** iv) cÖgvY Ki †h, 5 xy mij‡iLvwU 14416922 yx Dce„Ë‡K ¯úk© K‡i| ¯úk© we›`yi

¯’vbvsK wbY©q Ki| DËit

5

9,

5

16

4| (M) Awae„Ë (Hyparabola)

1.** i) 0199641816922 yxyx Awae„‡Ëi †K›`ª, kxl©we›`y, Dc‡K›`ª Ges Dr‡Kw›`ªKZv

wbY©q Ki| DËit (1,-2); (5,-2); (-3,-2); (6,-2); (4,-2); 4

5e

** ii) 11625

22

yx

Awae„ËwUi kxl©we›`y, Dr‡Kw› ªKZv Ges Dc‡K›`ª wbY©q Ki|

DËit )0,5( ;

5

41e ; )0,41(

*** iii) †`LvI †h, 2822 yx Awae„‡Ëi w`Kv‡¶i mgxKiY 43 x Ges Dc‡Kw› ªK j‡¤î •`N©¨

22

1.

** iv) 125144

22

yx

Awae„‡Ëi Dc‡K‡›`ªi ¯’vbvsK Ges Dr‡Kw›`ªKZv wbY©q Ki|

DËit )0,13( ;

12

13e

*** v) 82322 xyx Awae„‡Ëi A‡¶i •`N© , Dr‡Kw›`ªKZv Ges †K‡› ªi ’vbvsK wbY©q Ki|

DËit 6 ; 32 ;

3

2e ; (1,0)

*** vi) 1169

22

yx

Awae„‡Ëi Dc‡K›`ª yBwUi ’vbvsK I w`Kv¶ yBwUi mgxKiY wbY©q Ki|

DËit )0,5( ;

* vii) 14416922 yx Awae„‡Ëi kxl©, Dc‡K›`ª Ges Dr‡K›`ªZv wbY©q Ki|

DËit )0,4( ; )0,5( ;

4

5

** viii) 400162522 yx Awae„‡Ëi ‡K› ª, Dc‡K› ª Ges Dr‡Kw›`ªKZv wbY©q Ki|

DËit (0,0); )0,41( ;

4

41

* ix) 400251622 yx Awae„ËwUi kxl©we›`y Ges Dc‡K› ª wbY©q Ki| Dt )0,5( ; )0,14(


Paper cÖkœ-4

x) 205422 xy Awae„‡Ëi Dr‡K› ªZv, Dc‡K› ª I w`Kv‡¶i mgxKiY wbY©q Ki|

DËit;

√ )3,0( ;

xi) 0364922 yx Awae„‡Ëi kxl©, Dc‡K›`ª, bvwfj¤ I wbqvg‡Ki mgxKiY wbY©q Ki|

DËit )3,0( ; )13,0( ; 13y ;

√

xii) 0637922 yx Awae„‡Ëi Dc‡K‡›`ªi Ae ’vb I w`Kv‡¶i mgxKiY wbY©q Ki|

DËit )4,0( ; 94 y

2. Awae„‡Ëi mgxKiY wbY©q Ki hvi-

*** i) Dc‡K› ª (1,1), w`Kv‡¶i mgxKiY 2x + y =1 Ges Dr‡Kw› ªKZv 3


*** ii) Dc‡K› ª (1,-8), Dr‡Kw›`ªKZv 5 Ges w`Kv¶ 3x-4y=10.

DËit 0225502411422 xxyyx

3. i) GKwU Awae„‡Ëi A¶ yBwU ¯’vbvs‡Ki A¶ eivei| Awae„ËwU (-2,1) Ges (-3,-2) we› yMvgx n‡j Zvi

mgxKiY wbY©q Ki| DËit 3x2 -5y2 = 7

** ii) GKwU Awae„Ë (6,4) I (-3,1) we›`y w`‡q AwZµg K‡i| Gi †K›`ª g~jwe›`y‡Z Ges Avo A¶ x

A¶ eivei n‡j Zvi mgxKiY wbY©q Ki| DËit 1436

522

yx

iii) g~jwe›`y‡Z †K›`ª wewkó GKwU Awae„Ë (4,0) Ges (5, 2.25) we›`y w`‡q AwZµg K‡i; Awae„ËzwUi

Avo A¶ x A¶ eivei Aew ’Z n‡j Zvi mgxKiY wbY©q Ki|

DËit 9x2 -16y2 =144

4. Awae„‡Ëi A¶ yBwU‡K ’vbvs‡Ki A¶ a‡i Ggb GKwU Awae„‡Ëi mgxKiY wbY©q Ki hvi-

i) Dc‡K› ª )13,0( Ges AbyewÜ A‡¶i •`N©¨ 24 GKK| DËit 114425

22

xy

* ii) Dc‡K› ª `yBwUi ~iZ¡ 16 Ges Dr‡Kw›`ªKZv 2 DËit x2 - y2 = 32

5. i) Dc‡K› ª `yBwUi ¯’vbvsK (4,2) I (8,2) Ges Dr‡Kw›`ªKZv 2 n‡j, Awae„‡Ëi mgxKiY wbY©q Ki|

DËit 13

)2(

1

)6(22

yx

** ii) GKwU Awae„‡Ëi mgxKiY wbY©q Ki hvi Avo A¶ y A¶ eivei, AbyewÜ A¶ x -A¶ eivei,

kxl©Ø‡qi ~iZ¡ 2 Ges Dr‡Kw›`ªKZv 2 DËit 122 xy

5| ‡h †Kvb `yBwU cÖ‡kœi DËi `vIt 5 2=10

5| (K) wecixZ wÎ‡KvYwgwZK dvskb

1. cÖgvY Kit

*** i) 2

11tan

5

2cos

5

4sin

111

ii) 48

1tan2

7

1tan

5

1tan2

111

iii) 5

3cos

2

1

9

2tan

4

1tan

111 iv) 2tan3

2cos

3

1sin

111

v) 11

27tan

3

5cot

5

4cos

111

* vi) 29

2tan

2

5cos

5

13sec

111 ec

vii) 11

27tan

5

3sin

3

5cot

111

* viii) )5cos3(cot4

11ec ***ix)

)3cot5

1(sin4

11

*** x) 2

13sec

23

2tan

11

xi) 632

16cos

3

2cos

11

*** xii) 2

)2cos(sin)sin2(sin11

* xiii) xxx 2)3(cossin)cos(sin11

xiv) 2112)sin2sin( xxx

2. cÖgvY Kit

i) 7

1tan

3

1tan2

11 =4

ii)

x

xecx

2

1cos

2

1tan

211

iii) )2

1tan4sin()

7

1tan2cos(

11

*** iv) )tantan(tan2)tan2tan(3111

xxx


Paper cÖkœ-4

v) 0)(cottan)(cottan)2tan2

1(tan

3111 AAA

* vi) )2(sintan}tan)12{(tan}tan)12{(tan111

*** vii) xxxec1111

tan)cottantan(costan2

viii) xxx )3tan(cot)2(tancot

11

* ix)

cos

coscos

2tantan2

11

ba

ab

ba

ba

* x) x

x

x

xx

1

2sin

2

1

1

1cos

2

1tan

111

*** xi) 2

1

2

21

2

11

1

2tan

1

1cos

1

2sintan2

x

x

x

x

x

xx

xii) xy

yxyx

1tantantan

111

3. cÖgvY Kit

*** i) 2tan3

1tan

5

3sin

2

1

5

1cos

1111

** ii) 29

28tan2cot

13

5cos

2

1

5

3sin

1111

4. cÖgvY Kit

*** i) 15)3(cotcos)2(tansec1212

ec

** ii) 25)3(sectan)4(tansec1212

iii) 36

132)2(tancos)3(cotsec

1212 ec

iv) 1)3(cotsec3)2

1(tancos

1212 ec

** v) 9

2)

3

1(sincos)

3

1(cossin

1212

5. cÖgvY Kit

i) 4

3

4

3tansincoscot

11 **ii) xx 11tansincoscot

** iii) xx 11sincottancos ***iv) xx 11

costancotsin

*** v) y

xy

y

x22

11 2sectancossin

vi)

4

3

4

3costancotsin

11

vii) 2

211

2

1tancoscotsin

x

xx

viii)

2

211

2

1cotsintancos

x

xx

6.* i) xb

b

a

a 1

2

21

2

1tan2

1

1cos

1

2sin

n‡j †`LvI †h,

ab

bax

1

* ii)

z

zec

y

yx

2

1cos

2

1

1

1sec

2

1tan

21

2

211

n‡j

†`LvI †h, xyzzyx

iii) 2

sinsin11

yx n‡j †`LvI †h,

***a) 122 yx b) 111

22 xyyx

iv) hw` zyx

111tantantan nq, Z‡e †`LvI †h, xyzzyx

v) hw` zyx

111coscoscos nq, Z‡e †`LvI †h, 12

222 xyzzyx

*** vi) hw`

b

y

a

x 11coscos nq, Z‡e †`LvI †h,

2

2

2

2

2

sincos2

b

y

ab

xy

a

x

*** vii) hw` )sincos()cossin( nq, Z‡e †`LvI †h,

4

3sin

2

1 1

viii) CBA , 2tan1A Ges 3tan

1B n‡j †`LvI †h,

4

C


Paper cÖkœ-5

5| (L) wÎ‡KvYwgwZK mgxKiYt (mvaviY mgvavb)

mgvavb Kit

1. i) 2cottan22 ii) 2tan

sin1

cos

iii) 04sec32tan2 ** iv) tan3tansec

22

v) 2sin2cossin ** vi) 3cotcos ec

vii) 2tancot viii) sin2tan1tansin2

* ix) 0cos32sin22

DËit i) 4

n ii)

32

n vi)

32

n vii)

8)14(

n

iii) n2 , hLb 3sec iv) 4

n , n hLb

2

1tan

v) 4

n , hLb n Gi gvb k~b¨ A_ev †Rvo msL¨v

viii) 4

n ,

6)1(

nn ix)

2)12(

n , 3

2

n , 3

22

n

2. *** i) cosx + 3 sinx = 2 * ii) 1cossin iii) 2

1cossin

iv) 1sin2cos ***v) 3cos3sin xx *vi) 1sincos xx

DËit i) 12

72

n ,

122

n ii) n2 ,

2)14(

n iii) 12

72

n ,

122

n

iv) n2 , 22 n hLb

5

1cos v) n2 ,

3)16(

n vi) n2 , 2

)14(

n

3. * i) 2

5cos

2

5sin

4

1cos4sin

* ii) cot2x = cosx + sinx.

iii) cos2x + sinx = 1 *** iv) 4sin7coscos

* v) 5sin3cos7cos ** vi) 5coscos3cos2

* vii) sin4cos37sin *** viii) cosx + sinx = cos2x + sin2x.

ix) cosx + sinx = cos2x – sin2x. x) cos6x + cos4x = sin3x + sinx

xi) cosx + cos2x + cos3x = 0 ** xii) sinx + sin2x + sin3x = 0

xiii) 2coscos13sin2sinsin

xiv) 4

33cossin3sincos

33 xxxx

DËit i) 6

)1(3

nn ii)

4

n ,

2)1(

2

nn ,

2

15sin

iii) n , 6

)1(

nn iv)

4

n,

18)1(

3

nn

v) 5

n,

12

7)1(

2

nn vi)

6)12(

n , 8

n

vii) 8

)12(

n , 9

)1(3

nn viii) n2 ,

6)14(

n

ix) 3

2 n,

22

n x)

2)12(

n , 6

)14(

n , 14

)14(

n

xi) 4

)12(

n , 3

2

n xii) 2

n,

3

22

n

xiii) 2

)12(

n , 3

22

n ,

6)1(

nn xiv)

8)14(

n

4.** i) tanx + tan2x + tanxtan2x = 1. **ii) 3 (tanx+tan2x)+tanxtan2x = 1

iii) 32tantan32tantan

iv) tanx + tan3x = 0 ***v) 1tan2tan

vi) tanx + tan2x + tan3x = 0

* vii) tanx + tan2x + tan3x = tanxtan2xtan3x.

DËit i) 12

)14(

n ii) 18

)16(

n iii) 9

)13(

n

iv) 4

n v)

6

n vi)

3

n,

2

11 tnan

vii) 3

n


Paper cÖkœ-5

5| (M) wÎ‡KvYwgwZK mgxKiY (we‡kl mgvavb)t

1. * i) 2

tan222

sec2 xx

; hLb 00

3600 x

ii) 2cos2sin ; hLb -2 2

*** iii) 5)cos(sin42 ; hLb -2 < < 2

*** iv) sec2tancot ; hLb -2 < < 2

* v) tan)31(tan312 ; 00

3600

* vi) cos2sin21cossin4 ; 001800

vii) 22sinsin222 xx ; hLb - < x <

viii) cos3sin22 ; hLb -2 2

DËit i) 4

,

4

3 ii)

2

,

2

3 iii)

3

5,

3

,

3

,

3

5

iv) 6

11,

6

7,

6

,

6

5 v) 450, 300, 2250, 2100.

vi) 6

,

3

2,

6

5 vii)

4

,

2

,

4

3 viii)

3

,

3

5

2. * i) 2cossin xx ; hLb - < x <

*** ii) 1sincos3 xx ; hLb -2 < x < 2

*** iii) 2cossin3 ; hLb -2 < < 2

* iv) 2

1sincos ; hLb - < <

DËit i) 4

ii)

2

3,

6

,

2

,

6

11 iii)

3

4,

3

2 iv)

12

7,

12

3. * i) 22sec4sec ; hLb 0 < < 3600.

* ii) xxxx 3cos5cos7cos9cos ; hLb 4

< x<

4

.

*** iii) 13cos2coscos4 xxx ; hLb 0 < x <

iv) 13sinsin2 ; hLb 20

DËit i) 180, 900, 540, 2700, 1260, 1620, 1980, 2340, 3060, 3420.

ii) 0, 12

,

6

iii) 8

,

3

,

8

3,

3

2,

8

5,

8

7

iv) 4

,

6

,

4

3,

6

7,

6

5,

4

5,

6

11,

4

7


Paper cÖkœ-5

6| w ’wZwe`¨vt 5+5=10

[K I L Gi g‡a¨ Dccv`¨ I AsK wewbgq n‡e]

6| (K) ej ms‡hvRb I wefvRb Ges mgwe› ye‡ji fvimvg¨ (Dccv`¨+AsK)

Dccv`¨t

1.*** i) e‡ji mvgvšÍwiK m~ÎwU eY©bv Ki| GK we›`y‡Z †Kv‡Y wµqvkxj `yBwU e‡ji jwäi gvb I w`K wbY©q

Ki|

*** ii) ‡Kvb wbw ©ó w`‡K GK we› yMvgx `yBwU e‡ji j¤vs‡ki exRMwYZxq mgwó GKB w`‡K G‡ì jwäi

j¤^vs‡ki mgvb- cÖgvY Ki|

2.*** i) e‡ji wÎfyR m~ÎwU eY©bvmn cÖgvY Ki|

A_ev, cÖgvY Ki, hw` GKwU we›`y‡Z Kvh©iZ wZbwU e‡ji gvb I w`K GKB µ‡g M„wnZ †Kvb wÎfy‡Ri wZbwU evû Øviv

wb‡`©k Kiv hvq, Z‡e Zviv mvg¨ve¯’vq _vK‡e|

*** ii) jvwgi Dccv`¨wU eY©bvmn cÖgvY Ki|

iii) cÖgvY Ki †h, †Kvb we› y‡Z wµqviZ wZbwU GKZjxq e‡ji cÖ‡Z¨KwUi gvb Aci ywUi wµqv‡iLvi Aš—

M©Z †Kv‡Yi mvB‡bi mgvbycvwZK n‡j Ges †KvbwUB Aci ywUi jwäi mgvb bv n‡j, Zviv mvg¨ve ’v m„wó

Ki‡e|

(mgm¨vejx)

1. ‡Kvb KYvi Dci wµqviZ yBwU e‡ji jwä Zv‡ì GKwUi mv‡_ mg‡KvY Drcbœ K‡i Ges AciwUi GK

Z…Zxqvsk nq| †`LvI †h, ejØ‡qi gv‡bi AbycvZ 3 t 22

2. *** ‡Kvb we› y‡Z wµqviZ 3P Ges 2P gv‡bi yBwU e‡ji jwä R; cÖ_g ejwUi gvb wØ¸Y Ki‡j jwäi gvbI

wØ¸Y nq| ejØ‡qi Aš—M©Z †KvY wbY©q Ki| DËit 1200.

3. * †Kvb we› y‡Z wµqviZ `yBwU e‡ji e„nËg I ¶y`ªZg jwäi gvb h_vµ‡g F Ges G. cÖgvY Ki †h, ejØ‡qi

wµqv‡iLvi gaëZx© †Kv‡Yi gvb n‡j Zv‡ì jwäi gvb

2sin

2cos

2222 GF n‡e|

4. *** mggv‡bi yBwU ej †Kvb we› y‡Z 2 †Kv‡Y wµqviZ _vK‡j †h jwä Drcbœ nq Zv Zviv 2 †Kv‡Y wµqviZ

_vK‡j †h jwä nq Zvi wØ¸Y| cÖgvY Ki †h, cos = 2cos.

5. †Kvb we› y‡Z wbw`©ó †Kv‡Y wµqviZ P I Q ejØ‡qi jwä 3 Q Ges Zv P e‡ji w`‡Ki mv‡_ 300 †KvY

Drcbœ K‡i| †`LvI †h, P = Q A_ev P = 2Q.

6. *** †Kv‡Y wµqviZ P, Q gv‡bi ejØ‡qi jwä (2m + 1)22

QP , D³ †KvYwU 900 - n‡j jwäi

gvb nq (2m - 1)22

QP | cÖgvY Ki †h,

1

1tan

m

m .

7. *** i) P + Q Ges P - Q ejØq 2 †Kv‡Y wµqvkxj Ges Zv‡ì jwä Zv‡ì Aš—M©Z †Kv‡Yi mgwØLÛK

†iLvi mv‡_ †KvY Drcbœ K‡i| †`LvI †h Ptan = Qtan

ii) P + Q Ges P - Q ejØq †Kv‡Y wµqviZ| Zv‡ì jwä Zv‡ì AšÍM©Z †Kv‡Yi mgwØLÛ‡Ki mv‡_

2

†KvY Drcbœ K‡i| cÖgvY Ki †h, P t Q =

2tan

t

2tan

.

8. ‡Kvb we› y‡Z P Ges 2P gv‡bi yBwU ej wµqvkxj| cÖ_gwU‡K wØ¸Y K‡i wØZxqwUi gvb 8 GKK e„w× Ki‡j

jwäi w`K AcwiewZ©Z _v‡K| P Gi gvb wbY©q Ki| DËit P = 4 GKK

9. P I Q ejØ‡qi Aš—M©Z †KvY ; ej yBwUi Ae ’vb wewbgq Ki‡j Zv‡ì jwä hw` †Kv‡Y m‡i hvq Z‡e

†`LvI †h,

2tan

2tan

QP

QP

.

10. * O we›`y‡Z wµqviZ P I Q ejØ‡qi jwä R; GKwU mij‡iLv Zv‡ì wµqv‡iLv¸‡jv‡K h_vµ‡g L, M, N

we›`y‡Z †Q` Ki‡j, cÖgvY Ki †h,

ON

R

OM

Q

OL

P

11. †Kv‡Y †njv‡bv OA Ges OB evû eivei wµqvkxj h_vµ‡g P I Q ejØ‡qi jwä R ejwU OA Gi w`‡Ki

mv‡_ †KvY Drcbœ K‡i| cwieZ©b n‡q / n‡j Zv‡ì jwä R/ ejwU OA Gi w`‡Ki mv‡_

/ †KvY

Drcbœ K‡i| n‡j, †`LvI †h,

)sin(

)sin(/

/

R

R

12. ABC wÎfy‡Ri mgZ‡j Aew¯’Z O GKwU we›`y| BC, CA, AB evû¸‡jvi ga¨we›`y h_vµ‡g D, E, F n‡j cÖgvY

Ki †h, OD, OF Ges ED Øviv m~wPZ ej¸‡jvi jwä OB Øviv m~wPZ n‡e|

13. *** ABC wÎfy‡Ri CA I CB evû eivei wµqviZ yBwU e‡ji gvb cosA I cosB Gi mgvbycvwZK| cÖgvY

Ki †h, Zv‡ì jwäi gvb sinC Gi mgvbycvwZK Ges Zvi w`K C ‡KvY‡K

2

1(C + B – A) Ges

2

1(C

+ A – B) As‡k wef³ K‡i|

14. * †Kvb we› y‡Z wµqviZ P – Q, P, P + Q gv‡bi wZbwU e‡ji w`K GKB µgvbymv‡i †Kvb mgevû wÎfy‡Ri

evû¸‡jvi mgvš—ivj| Zv‡ì jwä wbY©q Ki|

DËit 3Q GKK, P – Q Gi w`‡Ki mv‡_ 2100 †Kv‡Y|

1


Paper cÖkœ-6

15. ABC wÎfy‡Ri BC , CA , AB evûi mgvš—ivj w`‡K P gv‡bi wZbwU mgvb ej †Kvb we›`y‡Z

wµqviZ Av‡Q| cÖgvY Ki †h, Zv‡ì jwä CBAP cos2cos2cos23 .

16. ** †Kvb we› y‡Z wµqviZ P I Q gv‡bi yBwU e‡ji jwä R Ges P Gi w`K eivei R Gi j¤vsk Q n‡j,

cÖgvY Ki †h, ej yBwUi AšÍM©Z †KvY = 2sin-1

Q

P

2 Ges PQPQR 222 .

17.*** †Kvb we› y‡Z wµqviZ P I Q gv‡bi yBwU e‡ji jwä Zv‡ì AšÍM©Z †KvY‡K GK-Z…Zxqvs‡k wef³ K‡i| †`LvI

†h, Zv‡ì AšÍM©Z †Kv‡Yi cwigvb 3cos-1

Q

P

2 Ges jwäi gvb

Q

QP22

(P > Q).

mewe› y e‡ji fvimvg¨t

18. ABC wÎfy‡Ri j¤^‡K›`ª O we›`y n‡Z BC, CA, AB evûi Dci j¤^ eivei h_vµ‡g wµqvkxj P, Q, R ej

wZbwU mg¨ve¯’vq _Kv‡j, cÖgvY Ki †h, P : Q : R = sinA : sinB : sinC

19. GKZjxq wZbwU ej †Kvb we›`y‡Z wµqviZ n‡q fvimvg¨ m„wó Ki‡Q| Zv‡ì gvb 1, 3 , 2 Gi mgvbycvwZK

n‡j, Zviv G‡K Ac‡ii mv‡_ wK †KvY Drcbœ K‡i wbY©q Ki|

DËit 1500, 1200, 900.

20.*** 4P Ges 3P ej `yBwU O we›`y‡Z wµqvkxj Ges 5P Zv‡ì jwä| †Kvb †Q`K Zv‡ì wµqv‡iLv‡K h_vµ‡g

L, M, N we› y‡Z †Q` K‡i Z‡e †`LvI †h,

ONOMOL

534 .

21.*** P, Q, R ej wZbwU †Kvb wÎfz‡Ri A, B, C kxl© we›`y n‡Z h_vµ‡g Zv‡ì wecixZ evûi j¤^vwfgyLx w`‡K

wµqviZ †_‡K fvimvg¨ m„wó K‡i‡Q| cÖgvY Ki †h, P : Q : R = a : b : c

22. ABC Gi AšÍt‡K›`ª I n‡Z IA, IB, IC eivei h_vµ‡g P, Q, R ej wZbwU wµqviZ †_‡K fvimvg¨ m„wó

K‡i‡Q| cÖgvY Ki †h,

*** i) P : Q : R = cos2

A: cos

2

B: cos

2

C

ii) P2 : Q2 : R2 = a(b + c – a) : b(c + a – b) : c(a + b – c)

23. mgvb •`‡N©ï wZbwU GKZjxq mij‡iLv OA, OB, OC hw` O we›`yMvgx †Kvb mij‡iLvi GKB cv‡k¦© Aew¯’Z

bv nq Ges P, Q, R ej wZbwU hw` D³ †iLv¸‡jv eivei Ggb fv‡e wµqviZ _v‡K †h,

OAB

R

OCA

Q

OBC

P

Zvn‡j †`LvI †h, P, Q, R ejwZbwU fvimvg¨ m„wó Ki‡e|

24. ABCD e„Ë¯’ PZzfy©‡Ri AB, AD eivei h_vµ‡g X I Y ejØq wµqviZ Av‡Q| C n‡Z A Gi w`‡K CA

eivei wµqviZ Z ejwUi Øviv Zv‡ì mgZv i¶v Kiv n‡j, †`LvI †h,

BD

Z

CB

Y

CD

X

25.** GKB Abyf~wgK †iLvi c GKK `~i‡Z¡ Aew¯’Z `yBwU we›`y‡Z l GKK `xN© GKwU mi‚ iwki cÖvš—Øq evav

Av‡Q| Aev‡a Szjv‡bv W GKK IRb wewkó GKwU e¯‘‡K enb K‡i Ggb GKwU gm„b IRb wenxb AvsUv H

iwk¥i Dci w`‡q Mwo‡q hv‡‛Q| †`LvI †h, iwki Uvb 22

2 cl

lW

26.*** ACB myZvwUi `yB cÖvš— GKB Abyf~wgK †iLv¯’ A I B we›`y‡Z Ave× Av‡Q| myZvwUi C we›`y‡Z W

IR‡bi GKwU e¯‘‡K wMU w`‡q evav n‡q‡Q| ABC wÎfy‡Ri evû¸‡jvi •`N© a , b , c Ges Zvi †¶Îdj

n‡j, †`LvI †h, myZvwUi CAAs‡ki Uvb )(4

222bac

c

Wb

27. i¤^mvK…wZ GKLvbv mylg cv‡Zi GKwU avi f~wgZ‡ji mgvš—ivj I GKwU †KvY 1200; i¤^mwUi †K›`ª n‡Z

KY© eivei P I Q ejØq wµqviZ †_‡K Zv‡K Lvovfv‡e †i‡L‡Q; P >Q n‡j †`LvI †h, 22

3QP .

28.***P, Q ejØq h_vKª‡g GKwU bZ mgZ‡ji •`N¨© I f~wgi mgvšÍiv‡j †_‡K cÖ‡Z¨‡KB GKKfv‡e gm~Y Z‡ji

Dci¯’ W IR‡bi e¯— enb Ki‡Z cv‡i| cÖgvY Ki †h, 222

111

WQP .

29.** GKwU †njv‡bv mgZ‡ji f~wg I •`‡N¨©i mgvšÍiv‡j h_vµ‡g wµqvkxj `yBwU c„_K ej P I Q-Gi cÖ‡Z¨‡K

GKvKx W IR‡bi †Kvb e¯‘‡K mgZ‡ji Dci w¯’ifv‡e a‡i ivL‡Z cv‡i| cÖgvY Ki †h, W =

22 QP

PQ

.

30.* l ‣`N©¨ wewkó GKwU myZvi GKcÖvš— †Kvb Lvov †Ìqv‡j AvUKv‡bv Av‡Q Ges Zvi Aci cÖvš— a e¨vmva©

wewkó GKwU mylg †Mvj‡Ki Dci¯’ †Kvb we›`y‡Z mshy³ Av‡Q| †MvjKwUi IRb W n‡j †`LvI †h, myZvwUi

Uvb 2

2

)(

lal

laW

.

31.*** †Kvb we›`y‡Z wµqviZ P, Q, R ej wZbwU fvimvg¨ m„wó K‡i‡Q| P I Q- –Gi Aš—M©Z †KvY P I R-Gi

Aš—M©Z †Kv‡Yi wØ¸Y n‡j cÖgvY Ki †h, R2 = Q(Q – P).


Paper cÖkœ-6

A_ev-6| (L) mgvšÍivj ej (Dccv`¨+AsK)

Dccv`¨)

1.***†Kvb eo e ‘i Dci wµqvkxj `yBwU Amgvb I m „k mgvšÍivj e‡ji jwä I Zvi cÖ‡qvM we›`y wbY©q Ki|

2.*** †Kvb Ro e ‘i Dci wµqvkxj yBwU Amgvb I Am „k mgvšÍivj e‡ji jwe×i gvb, w`K I wµqvwe› y wbY©q

Ki|

mgm¨vejx 1.* i) GKwU †mvRv mylg i‡Wi GKcÖv‡šÍ 10 †KwR IR‡bi GKwU e¯‘ Szjv‡bv n‡j, H cÖvšÍ n‡Z 1 wgUvi `~‡i

Aew¯’Z GKwU LyuwUi Dci iWwU Abyf~wgKfv‡e w ’i _v‡K| LyuwUi Dci Pv‡ci cwigvY 30 †KwR IRb

n‡j iWwUi IRb I •`N©¨ wbY©q Ki| DËit 3 wgUvi, 20 †KwR

** ii) GKwU †jvK GKwU mylg jvwVi GKcÖv‡šÍ GKwU †evSv Kuv‡a enb Ki‡Q| †evSvwUi IRb W Ges †jvKwU

Kvua n‡Z †evSvwUi I †jvKwUi nv‡Zi ~iZ¡ h_vµ‡g a I x n‡j †`LvI †h Zvi Kuv‡ai Dci Pvc W

x

a1 n‡e|

iii) 20 ‡m.wg. `xN© GKwU nvév AB jvwV `yBwU ‡c‡i‡Ki Aew ’Z| †c‡iK yBwUi ~iZ¡ jvwVi •`‡N©ï

A‡a©K| A I B we› y‡Z 2W I 3W †KwR IRb Szwj‡q jvwVUv‡K Ggbfv‡e ’vcb Kiv nj †hb

†c‡iK yBwUi Dci mgvb Pvc c‡o, †c‡iK yBwwUi Ae¯’vb wbY©q Ki|

DËit B n‡Z 3 †m.wg. ~‡i GKwU LyuwU Ges A n‡Z 7 †m.wg. ~‡i Aci LyuwU|

*** iv) GKwU nvjKv `‡Ûi GKcÖvšÍ n‡Z 2, 8, 6 wg. `~i‡Z¡ Aew¯’Z wZbwU we›`y‡Z h_vµ‡g P, Q, R

gv‡bi wZbwU mgvšÍivj ej wµqv Ki‡Q| `ÛwU fvimvg¨ Ae¯’vq _vK‡j †`LvI †h, P : Q : R =

1 : 2 : 3.

2.*** i) ‡Kvb wÎfz‡Ri †K․wYK we›`y¸wj‡Z h_vµ‡g P, Q, R gv‡bi wZbwU mggyLx mgvšÍivj ej wµqviZ Av‡Q|

G‡ì jwä H wÎfz‡Ri fi‡K‡› ª wµqviZ n‡j †`LvI †h, P = Q = R.

** ii) ABC wÎf~‡Ri A, B, C †K․wYK we›`y‡Z h_vµ‡g P, Q, R gv‡bi wZbwU mggyLx mgvš—ivj ej

wµqviZ Av‡Q| Zv‡ì jwä H wÎf~‡Ri AšÍt‡K‡›`ª wµqviZ n‡j †`LvI †h,

(i) P : Q : R = SinA : sinB : sinC (ii) c

R

b

Q

a

P

*** iii) ABC wÎf~‡Ri A, B, C †K․wYK we›`y‡Z h_vµ‡g P, Q, R gv‡bi wZbwU mggyLx mgvš—ivj ej

wµqviZ Av‡Q| Zv‡ì jwä H wÎf~‡Ri j¤^‡K› ªMvgx n‡j, cÖgvY Ki †h,

(i)

(ii) P(b2 + c2 – a2) = Q(c2 + a2 – b2) = R(a2 + b2 – c2)

iv) P, Q, R gv‡bi wZbwU mggyLx mgvš—ivj ej ABC wÎf~‡Ri †K․wYK we›`y‡Z wµqv Ki‡Q| Zv‡ì

mvaviY we›`y hvB †nvKbv †Kb, Zv‡ì jwä hw` me©`vB H wÎf~‡Ri cwi‡K‡› ª wµqviZ nq, Z‡e cÖgvY

Ki †h, (i) C

R

B

Q

A

P

2sin2sin2sin .

(ii) P : Q : R = acosA : bcosB : ccosC.

*** v) O we›`ywU ABC wÎf~‡Ri cwi‡K›`ª Ges AO eivei P gv‡bi ejwU wµqv Ki‡Q| †`LvI †h, B I C

we›`y‡Z wµqviZ P e‡ji mgvš—ivj AskKØ‡qi AbycvZ sin2B : sin2C.

3.*** i) †`LvI †h, P I Q yBwU mgvšÍivj e‡ji Q †K

Q

P 2

†Z cwieZ©b K‡i Q Gi mv‡_ ’vb cwieZ©b

Ki‡j jwäi Ae ’vb GKB _v‡K|

*** ii) `yBwU wecixZgyLx mgvš—ivj ej P I Q (P>Q) Gi cÖ‡Z¨‡Ki gvb hw` mgcwigvb ewa©Z Kiv nq,

Z‡e cÖgvY Ki †h, Zv‡ì jwäi wµqvwe›`y P n‡Z AviI ~‡i m‡i hv‡e|

iii) 12 GKK I 8 GKK gv‡bi yBwU mggyLx mgvšÍivj ej h_vµ‡g †Kvb Abo e ‘i A I B we› y‡Z

wµqv Ki‡Q| Zv‡ì Ae ’vb wewbgq Kiv n‡j, Zv‡ì jwäi wµqvwe›`y AB eivei KZ ~‡i m‡i hv‡e

Zv wbY©q Ki| DËit

*** iv) P Ges Q yBwU mggyLx mgvšÍivj ej| P ejwUi wµqv†iLv mgvšÍivj †i‡L Zvi wµqvwe› y‡K x ~‡i

miv‡j, †`LvI †h, Zv‡ì jwä QP

Px

`~‡i m‡i hv‡e|

*** v) `yBwU wecixZgyLx mgvš—ivj ej P Ges Q (P>Q) h_vµ‡g AI B we›`y‡Z wµqviZ; P Ges Q Gi

cÖ‡Z¨K‡K x cwigv‡Y e„w× Ki‡j †`LvI †h, Zv‡ì jwäwU d ~i‡Z¡ m‡i hv‡e, hLb

.

*** vi) †Kvb Abo e¯‘i A I B we›`y‡Z h_vµ‡g wµqviZ `yBwU mggyLx mgvš—ivj ej P I Q (P>Q) Gi

ci¯ú‡ii Ae ’vb wewbgq Ki‡j, †`LvI †h,Zv‡ì jwäi wµqv we›`y AB eivei d ~i‡Z¡ m‡i hv‡e,

hLb d =

AB.

*** vii) †Kvb e ‘i Ici wµqviZ `yBwU mggyLx mgvš—ivj ej P I Q Gi mv‡_ GKB mgZ‡j b `~i‡Z¡ `yBwU

mgvb S gv‡bi wecixZgyLx mgvšÍivj ej‡K mshy³ Ki‡j, †`LvI †h, wgwjZ ej¸‡jvi jwä

`~i‡Z¡ m‡i hv‡e|

*** viii) P, Q gv‡bi yBwU mggyLx mgvšÍivj e‡ji jwä O we›`y‡Z wµqv K‡i| P †K R cwigv‡Y Ges Q †K

S cwigv‡Y e„w× Ki‡jI jwä O we›`y‡Z wµqv K‡i| Avevi P, Q Gi e`‡j Q, R wµqv Ki‡jI jwä

O we›`y‡Z wµqv K‡i| †`LvI †h, S = R - QP

RQ

2)(.


Paper cÖkœ-6

7| MwZwe`¨vt 5 5=10

[K I L Gi g‡a¨ Dccv`¨ I AsK wewbgq n‡e]

7| (K) †eM I Z¡iY (Dccv`¨+ AsK)

Dccv`¨

1.* ‡e‡Mi mvgvšÍwiK m~ÎwU †jL Ges ci¯úi †Kv‡Y wµqvkxj `yBwU ‡eM u Ges v Gi jwäi gvb I w`K wbY©q

Ki|

2.*** mPivPi ms‡KZgvjvi 2

2

1ftuts m~ÎwU cÖwZôv Ki|

3.*** mPivPi ms‡KZgvjvq cÖgvY Ki †h, fsuv 222

mgm¨vejx (‡eM)

1.*** yBwU †e‡Mi e„nËg jwä G‡ì ¶z`ªZg jwäi n¸Y| †eMØ‡qi gaëZx© †KvY n‡j, jwä‡e‡Mi gvb G‡ì

mgwói A‡a©K nq| †`LvI †h,

)1(2

2cos

2

2

n

n

2.*** i) †mªvZ bv _vK‡j GK e¨w³ 100 wgUvi PIov GKwU b`x muvZvi w`‡q wVK †mvRvmywRfv‡e 4 wgwb‡U cvi

nq Ges †mªvZ _vK‡j H GKB c‡_ †m b`xwU 5 wgwb‡U cvi n‡Z cv‡i| †mªv‡Zi MwZ‡eM wbY©q Ki| DËit

15 wgUvi/wgwbU

*** ii) GKRb †jvK t mg‡q GKwU b`x †mvRvmywR cvox w`‡Z cv‡i Ges 1t mg‡q †mªv‡Zi AbyKz‡j mgvb `~iZ¡

AwZµg Ki‡Z cv‡i| kvš— b`x‡Z †jvKwUi †eM u Ges †mªv‡Zi †eM v n‡j †`LvI †h, t t 1t =

vu t vu

iii) 550 wgUvi cÖ ’ GKwU b`xi †mªvZ N›Uvq 3 wKwg †e‡M cÖevwnZ nq| yBwU †b․Kvi cÖ‡Z¨KwU N›Uvq 5

wKwg †e‡M GKwU †b․Kv ¶z`ªZg c‡_ Ges AciwU ¶z`ªZg mg‡q b`xwU AwZµg Ki‡Z †Pôv Ki‡Q| hw`

Zviv GKB mg‡q hvÎv ïi‚ K‡i Z‡e Zv‡ì Aci cv‡o †cu․Qvevi mg‡qi cv_©K¨ wbY©q Ki|DËit 1

wgwbU 39 †m‡KÛ

3.* ‡Kvb we› y‡Z wµqviZ u I v †eMØ‡qi jwä w ; u Gi w`K eivei w Gi j¤^vs‡ki gvb v n‡j cÖgvY Ki

†h, †eM yBwUi Aš—M©Z †KvY

v

uv 1cos Ges uvvuw 2

22

4. ‡Kvb e ‘ KYvq GKB mv‡_ wµqvkxi wZbwU †eM u , v , w ci¯úi , , †Kv‡Y AvbZ| †`LvI †h,

G‡ì jwä 2

1

222)cos2cos2cos2( wuvwuvwvu

5. GKwU Kbv †Kvb mgZj¯’ GKwU mij‡iLv eivei 3 wgUvi/†m. †e‡M Pj‡Q| 3 †m‡KÛ c‡i Kbvi MwZc‡_i

mv‡_ jwäi w`‡K 4 wg./†m. MwZ ms‡hvM Kiv nj| MwZ ïi‚ nIqvi 5 †m‡KÛ c‡i KbvwU hvÎvwe› y n‡Z

KZ ~‡i _vK‡e? DËit 17 wgUvi

6.* ‡Kvb e›ì n‡Z GKLvbv RvnvR DËi cwðg w`‡K N›Uvq 15 wK.wg. †e‡M hvÎv Kij| GKB mg‡q GKB ¯’vb

n‡Z Aci GKLvwb RvnvR `w¶Y-cwðg w`‡K N›Uvq 12 wK.wg. †e‡M hvÎv Kij| Zv‡ì †eZvi h‡š¿i MÖnb

kw³i mxgv 500 wK.wg. n‡j KZ¶Y Zviv G‡K Ac‡ii mv‡_ †hvMv‡hvM i¶v Ki‡Z cvi‡e?DËit 02.26

N›Uv

7. `yBwU †ijc_ ci¯úi mg‡Kv‡Y Aew¯’Z| GKwU †ijc‡_ N›Uvq 30 wK.wg. †e‡M Pjgvb GKwU Mvwo mKvj

10 Uvq Rskb AwZµg K‡i| H gyûZ© n‡Z Aci †ijc‡_ N›Uvq 40 wK.wg. †e‡M Pjgvb Avi GKwU Mvox

we‡Kj 3 Uvq Rsk‡b †cu․‡Q| KLb G‡ì yiZ¡ b~¨bZg wQj Ges ¶z`ªZg `~iZ¡ KZ wQj?

DËit

5

16N›Uv, 120 wK.wg.

8. GKLvbv w÷gvi c~e©w`‡K N›Uvq u wK.wg. †e‡M Ges wØZxq GKLvbv w÷gvi c~e© w`‡Ki mv‡_ †Kv‡Y DËigyLx

w`‡K u2 †e‡M MwZkxj| cÖ_g w÷gv‡i Aew¯’Z †Kvb hvÎxi wbKU wØZxq w÷gv‡ii w`K DËi-c~e© e‡j g‡b

n‡‛Q| †`LvI †h,

4

3sin

2

1 1


Paper cÖkœ-7

Z¡iYt

9.** GKwU ey‡jU †Kvb †Ìqv‡ji wfZi 2 †m.wg. XyKevi ci A‡a©K †eM nvivq| ey‡jUwU †Ìqv‡ji wfZi Avi

KZ ~i XyK‡e? DËit

3

2 †m.wg.

10. GKwU evN 20 wgUvi ~‡i GKwU nwiY‡K †`L‡Z †c‡q w¯’i Ae¯’v n‡Z 3 wgUvi/eM©-‡m‡KÛ Z¡i‡Y nwiYwUi

cðv‡Z †`․ovj| nwiYwU 13 wgUvi/†m‡KÛ mg‡e‡M †`․ov‡Z _vK‡j KZ¶Y c‡i KZ ~‡i wM‡q evNwU

nwiY‡K ai‡Z cvi‡e? DËit 10 †m‡KÛ, 150 wgUvi

11. GK e¨w³ Zvi 50 wgUvi mvg‡b w¯’ive¯’v n‡Z mylg Z¡i‡Y GKwU evm Qvo‡Z †`‡L mg‡e‡M †`․ov‡Z jvMj

Ges GK wgwb‡U †Kvb iK‡g evmwU ai‡Z cvij| †jvKwUi †eM I ev‡mi Z¡iY wbY©q Ki|

DËit

3

5wg./†m.

36

1wg./†m2.

12.***mgZ¡i‡Y PjšÍ †Kvb we› y 1t ,

2t , 3t mg‡q h_vµ‡g mgvb mgvb µwgK `~iZ¡ AwZµg K‡i| cÖgvY Ki †h,

321321

3111

tttttt .

13.***‡Kvb mij‡iLvq mgZ¡i‡Y PjšÍ †Kvb we› yi Mo‡eM avivevwnK 1t ,

2t , 3t mg‡q h_vµ‡g 1v ,

2v , 3v

n‡j, †`LvI †h,

32

21

32

21

tt

tt

vv

vv

.

14.***‡Kvb mij‡iLvq f mgZ¡i‡Y PjšÍ GKwU KYv t mg‡q s `yiZ¡ I cieZx© t/ mg‡q s/ `yiZ¡ AwZµg K‡i|

†`LvI †h, f = 2

t

s

t

s/(t + t/)

15.* u Avw`‡e‡M PjšÍ KYv cÖ Ë `~i‡Z¡i A‡a©K f mylg Z¡i‡Y Ges Aewkó A‡a©K f1 mylgZ¡i‡Y Mgb K‡i| †`LvI

†h †kl †e‡Mi gvb GKB n‡e hw` KYvwU mg¯— ~iZ¡

2

1(f + f1) mgZ¡i‡Y Mgb K‡i|

16.***i) GKLvbv †ijMvwo GK †÷kb n‡Z †Q‡o 4 wgwbU ci 2 wK‡jvwgUvi `~‡i Aew ’Z Aci ‡÷k‡b _v‡g|

MvwoLvbv Zvi MwZc‡_i cÖ_gvsk x mgZ¡i‡Y Ges wØZxqvsk y mgg›`‡b Pj‡j cÖgvY Ki †h,

411

yx.

*** ii) w¯’ive ’v n‡Z mij‡iLvq Pjš— GKwU e¯‘KYv cÖ_‡g x mgZ¡‡Y I c‡i y mgg›`‡b P‡j| KYvwU hw`

t mg‡q s ~iZ¡ AwZµg K‡i, Z‡e †`LvI †h,

yxs

t 11

2

2

.

* iii) GKLvbv †ijMvwo GK †÷kb n‡Z †Q‡o Ab¨ †÷k‡b wM‡q _v‡g| MvwoLvbv MwZc‡_i cÖ_gvsk f

mgZ¡i‡Y Ges c‡i †eªK cÖ‡qvM K‡i Acivsk f1 mgg›`‡b P‡j| †÷kb yBwUi `~iZ¡ ÔaÕ n‡j ‡`LvI

†h, MvwoLvbv GK ‡÷kb n‡Z Aci †÷k‡b †c․Qvi mgq

1

1 )(2

ff

ffa n‡e|

17. GKwU KYv wbw`©ó †e‡M hvÎv K‡i mgZ¡i‡Y 3 †m‡K‡Û 81 †m.wg. `~iZ¡ AwZµg Kij| AZtci Z¡i‡Yi wµqv

eÜ n‡q †Mj Ges KYvwU cieZx© 3 †m‡K‡Û 72 †m.wg. `~iZ¡ AwZµg Kij| KYvwUi Aw`‡eM I Z¡iY wbY©q

Ki| DËit 30 †m.wg./†m.; 2 †m.wg./†m2.

18.* GKLvbv †ijMvwo GK †÷kb n‡Z †Q‡o Ab¨ †÷k‡b wM‡q _v‡g| MvwoLvbv Zvi MwZc‡_i cÖ_g

m

1Ask

mgZ¡i‡Y, †kl

n

1 Ask mgg›`‡b Ges Aewkóvsk mg‡e‡M P‡j| cÖgvY Ki †h, m‡e©v‛P †eM I Mo‡e‡Mi

AbycvZ

nm

111 : 1.

19. `yBwU KYv GKB mij‡iLvq h_vµ‡g a Ges b mgZ¡i‡Y Pj‡Q| H mij‡iLvi †Kvb wbw`©ó we›`y n‡Z hLb

Zv‡ì yiZ¡ x I y ZLb Zv‡ì †eM h_vµ‡g u Ges v; †`LvI †h, Zviv yBev‡ii AwaK wgwjZ n‡Z cv‡i

bv| hw` Zviv yBevi wgwjZ nq Z‡e Zv‡ì wgwjZ nevi mg‡qi cv_©K¨

))((2)(2 2

bayxvuba

20.***i) GKB jvB‡b `yBLvbv †ijMvwo ci¯ú‡ii w`‡K h_vµ‡g u1 Ges u2 †e‡M AMÖmi n‡‛Q| Zv‡ì ~iZ¡

hLb x, ZLb G‡K Aci‡K †`L‡Z †cj| †eªK cÖ‡qv‡M Drcbœ m‡e©v‛P g›`b h_vµ‡g f1 I f2 n‡j,

†`LvI †h, yN©Ubv Gov‡bv m¤¢e n‡e hw`, xfffufu 211

2

22

2

1 2 nq|

* ii) GKB jvB‡b GKLvbv G·‡cÖm Mvwo Ab¨ GKLvbv gvjMvwo‡K AwZµg Ki‡Q| Zv‡ì †eM hLb

h_vµ‡g u1 Ges u2 ZLb x ~iZ¡ n‡Z G‡K Aci‡K †`L‡Z cvq| Mvwo yBLvbvi m‡e©v‛P Z¡iY I

m‡e©v‛P g›`b h_vµ‡g f1 Ges f2 n‡j †`LvI †h, yN©Ubv Gov‡bv †KejgvÎ m¤¢eci n‡e hw` (u1 –

u2)2 = 2(f1 + f2)x nq|


Paper cÖkœ-7

A_ev-

7| (L) gva¨vKl©‡Yi cÖfv‡e e¯‘i Dj¤ MwZ Ges cÖ‡¶cKt (Dccv`¨+AsK)

Dccv`¨

1.*** GKwU e¯‘KYv u ‡e‡M Ges Abyf~wgi mv‡_ †Kv‡Y wbw¶ß nj| e¯‘ KYvwUi me©vwaK D‛PZv, me©vwaK D‛PZvq

†cu․Qvi mgq, wePiYKvj, Abyf~wgK cvjøv I me©vwaK Abyf~wgK cvjøv wbY©q Ki|

2.* cÖgvY Ki †h, cÖw¶ß e¯‘i wePiYKvj Dnvi me©vwaK D‛PZvq DVvi mgqKv‡ji wØ¸Y|

3.*** cÖgvY Ki †h, evqyk~b¨ ¯’v‡b cÖw¶ß e¯‘ KYvi MwZc_ GKwU cive„Ë|

4.*** †`LvI †h, evqynxb ’v‡b †Kvb cÖw¶ß e ‘i MwZc‡_i mgxKiY y = xtan

R

x1

5.* ‡`LvI †h, f~wgi D‛P †Kvb ’v‡bi wbw`©ó we›`y n‡Z Abyf~wg‡K evqyk~b¨ ’v‡b wbw¶ß e ‘KYvi wePiY c_ GKwU

cive„Ë|

mgm¨vejx

gva¨vKl©‡Yi cÖfv‡e e ‘i Dj¤ MwZt

1. 10 wg./†m. †e‡M DaŸ©Mvgx †Kvb †ejyb n‡Z cwZZ GK UzKiv cv_i 10 †m. c‡i gvwU‡Z coj| hLb cv_‡ii

UzKiv cwZZ nq, ZLb †ejy‡bi D‛PZv KZ? DËit 390 wgUvi

2.** 49 wg./†m. †e‡M GKwU ej‡K Lvov Dc‡ii w`‡K wb‡¶c Kiv nj Ges 2 †m. c‡i GKB we› y n‡Z GKB

†e‡M Aci GKwU ej wb‡¶c Kiv nj| †Kv_vq Ges KLb Zviv wgwjZ n‡e?

DËit wØZxq ejwU wb‡¶c Kivi 4 †m. c‡i 6.117 wgUvi D‛PZvq

3.*** GKwU UvIqv‡ii Pzov n‡Z GKLÛ cv_i x wgUvi wb‡P bvgvi ci Aci GKLÛ cv_i Pzovi y wgUvi wbP n‡Z

†d‡j †Ìqv nj| hw` Df‡qB w¯’ive ’v n‡Z c‡o Ges GKB m‡½ f~wg‡Z cwZZ nq, Z‡e †`LvI †h,

UvIqv‡ii D‛PZv

x

yx

4

)(2

wgUvi|

4.* 5.4 †m‡KÛ hveZ mg‡e‡M Lvov Dc‡ii w`‡K DVevi ci GKwU †ejyb n‡Z GKwU fvix e ‘ c‡o †Mj| hw`

e¯‘wU 7 †m‡K‡Û f~wg‡Z c‡o Z‡e †ejy‡bi MwZ‡eM Ges KZ DPz n‡Z e¯‘wU c‡owQj Zv wbY©q Ki| DËit

88.20 wgUvi/†m; 96.93 wgUvi

5.* mg‡e‡M Lvov DaŸ©Mvgx GKwU D‡ovRvnvR n‡Z GKwU †evgv †Q‡o †Ìqvq 5 †m. c‡i Zv gvwU‡Z c‡o|

†evgvwU hLb gvwU‡Z c‡o ZLb D‡ovRvnv‡Ri D‛PZv wbY©q Ki|

DËit 5.122 wgUvi

6.* GKwU KYv u wg./†m. †e‡M Lvov Dc‡ii w`‡K wb‡¶c Kiv nj Ges t ‡m. c‡i H GKB we›`y n‡Z GKB

Avw`‡e‡M Aci GKwU KYv Dc‡ii w`‡K wb‡¶c Kiv nj| cÖgvY Ki †h, Zviv

g

tgu

8

4222

D‛PZvq

wgwjZ n‡e|

7.*** Lvov Dc‡ii w`‡K wbw¶ß GKwU cv_i 1t Ges

2t mg‡q f~wgi h D‛PZvq Ae ’vb Ki‡j †`LvI †h,

212 tgth |

8.*** Lvov Dc‡ii w`‡K wbw`©ó †e‡M wbw¶ß GKwU KYv t ‡m. mg‡q h D‛PZvq D‡V Ges AviI t1 ‡m‡KÛ c‡i

f~wg‡Z †c․Qvq, Z‡e cÖgvY Ki †h h = 2

1gtt1.

9.* GKwU k~b¨ K~‡ci g‡a¨ GKwU cv_‡ii UzKiv †Q‡o †Ìqvi ci Zv 19.6 wg./†m. †e‡M K ~‡ci Zj‡`‡k cwZZ

nq| UzKivwU †Q‡o †Ìqvi 235

2 †m. c‡i cv_iwUi cZ‡bi kã †kvbv †Mj, k‡ãi †eM wbY©q Ki| DËit

343 wgUvi/†m.

10. GKwU k~b¨ K~‡c GKwU cv_‡ii UzKiv †djv nj Ges 5.3 †m. c‡i UzKivwUi K ~‡ci Zj‡`‡k cZ‡bi kã

†kvbv †Mj| k‡ãi †eM 327 wg./†m. Ges 81.9g wg./†m.2 n‡j K~‡ci MfxiZv wbY©q Ki|DËit 5.54

wgUvi

11.* f~wg n‡Z gy2 wg./†m. †e‡M Lvov Dc‡ii w`‡K cÖw¶ß GKwU i‡KU Zvi e„nËg D‛PZvq D‡V we‡ùvwiZ

nj| i‡K‡Ui cÖ‡qvM we›`y Ges Zv n‡Z f~wg eivei x wgUvi ~‡i f~wg‡Z Aew¯’Z Aci GKwU we›`y‡Z

we‡ùvi‡Yi kã Avm‡Z †h mgq jv‡M Zv‡ì Aš—i

n

1 †m.| †`LvI †h, k‡ãi MwZ‡eM

)(22

yyxn wg./†m.|

12. GKwU k~b¨ K~‡ci g‡a¨ GKwU wXj †djvi t ‡m. c‡i K~‡ci Zj‡`‡k wXj covi kã †kvbv †Mj| hw` k‡ãi

†eM v Ges Ky‡ci MfxiZv h nq, Z‡e cÖgvY Ki †h,

*** i) (2h – gt2) v 2 + 2hgt v = h2g

* ii) Kz‡ci MfxiZv =

)1(2

2

v

gt

gt

, h Gi Zzjbvq v GZ e„nr †h

2

v

h †K AMÖvn¨ Kiv hvq|

** iii) t = v

h

g

h

2

cÖ‡¶cKt

13.* i) GKRb †L‡jvqvo 3.5 wg. D‛PZv n‡Z f~wgi mv‡_ 300 †Kv‡Y 9.8 wg./†m. †e‡M GKwU ej wb‡¶c

K‡i Ges Aci GKRb †L‡jvqvo 2.1 wgUvi DuP y‡Z Zv a‡i †d‡j| †L‡jvqvo yRb KZ ~‡i wQj?DËit

44.10 wgUvi

ii) GKRb †L‡jvqvo 2 wg. D‛PZv n‡Z f~wgi mv‡_ 300 †Kv‡Y 20 wg./†m. †e‡M GKwU ej wb‡¶c Ki‡j

Aci GKRb †L‡jvqvo 1 wgUvi DuPy‡Z Zv a‡i †d‡j| †L‡jvqvo yRb KZ ~‡i wQj? DËit 37 wgUvi


Paper cÖkœ-7

* iii) `yB wgUvi Dci n‡Z 50 wg./†m. †e‡M Ges Abyf~wgK Z‡ji mv‡_ 300 †Kv‡Y wbw¶ß GKwU wµ‡KU ej

f~wg n‡Z 1 wgUvi Dc‡i GKRb †L‡jvqvo a‡i †d‡j| †L‡jvqvoØ‡qi ` ~iZ¡ wbY©q Ki|DËit 219wgUvi

* iv) 80 wg. DuPz GKwU cvnv‡oi P~ov n‡Z GKLÛ cv_i 128 wg./†m. †e‡M Ges Abyf~wgK Z‡ji mv‡_ 300

†Kv‡Y wbw¶ß nj| cv_i LÛwU cvnv‡oi cv`‡`k n‡Z KZ ~‡i f~wg‡Z co‡e Zv wbY©q Ki| DËit

48.1492 wgUvi

v) 60 wg. DuPz GKwU cvnv‡oi P~ov n‡Z GKLÛ cv_i 40 wg./†m. †e‡M Ges Abyf~wgK Z‡ji mv‡_ 300

†Kv‡Y wbw¶ß nj| GwU cvnv‡oi cv`‡`‡k KZ`~‡i wM‡q f~wg‡Z co‡e?

DËit 963.210 wgUvi

vi) GKRb •egvwbK 5000 wgUvi Dci w`‡q N›Uvq 250 wK.wg. †e‡M D‡o hvIqvi mgq GKwU †evgv

bvwg‡q w`j| †m †h ¸nvq AvNvZ Ki‡Z Pvq †mB ¸nv n‡Z Zvi Abyf~wgK yiZ¡ KZ nIqv cÖ‡qvRb?

DËit 2218 wgUvi

14.* i) GKwU e ‘ GKB †e‡M Abyf~wgKZ‡ji mv‡_ yBwU wfbœ †Kv‡Y cÖw¶ß n‡q GKB Abyf~wgK cvj v R

AwZµg K‡i| hw` Zvi ågYKv‡j t1 Ges t2 nq Z‡e †`LvI †h, R = 2

1gt1t2.

*** ii) hw` †Kvb cÖw¶ß e ‘i yBwU MwZc‡_ e„nËg D‛PZv H Ges H1 nq Z‡e †`LvI †h, 14 HHR .

iii) †Kvb wbw`©ó †e‡Mi Rb¨ e„nËg cvjøv D n‡j †`LvI †h, R = Dsin2 Ges G n‡Z cÖgvY Ki †h,

†Kvb Abyf~wgK cvjøv R Gi Rb¨ mvaviYZ yBwU mÂvic_ _v‡K| DcwiD³ yBwU mÂvic‡_ jä e„nËg

D‛PZv h1, h2 n‡j †`LvI †h, D = 2(h1 + h2).

15.***i) u Avw`‡e‡M cÖw¶ß †Kvb KYv KZ…©K jä e„nËg D‛PZv H n‡j †`LvI †h, Zvi Abyf~wgK cvjøv

H

g

uHR

24

2

* ii) ‡Kvb Abyf~wgK Z‡ji Dci ’ GKwU we›`y n‡Z GKwU KYv u †e‡M Ges †Kv‡Y cÖw¶ß nj| Zvi

cvjøv R Ges e„nËg D‛PZv H n‡j cÖgvY Ki †h,16gH2 – 8u2H + gR2 = 0

16. I ( > ) †Kv‡Y ywU e¯‘ wbw¶ß n‡jv| Giv h_vµ‡g 1t I

2t mg‡q GKB Abyf~wgK cvj v

AwZµg Ki‡j, cÖgvb Ki †h,

)sin(

)sin(2

2

2

1

2

2

2

1

tt

tt.

17.* i) GKwU ¸wj 9.8 wg. `~‡i Aew¯’Z 2.45 wg. D‛P GKwU †Ìqv‡ji wVK Dci w`‡q Abyf~wgKfv‡e P‡j

hvq| ¸wjwUi cÖ‡¶c‡e‡Mi gvb I w`K wbY©q Ki|

DËit 5.15 wg/†m.;

2

1tan

1

** ii) GKwU wµ‡KU ej‡K f~wg †_‡K wb‡¶c Kiv n‡j GwU 100 MR `~‡i f~wg‡Z wd‡i Av‡m| Gi

wePiYc‡_i me©vwaK D‛PZv 564

1dzU n‡j Gi wePiYKvj I cÖ‡¶cY †Kv‡Yi gvb wbY©q Ki| DËit

4

15 †m‡KÛ,

3

8cot

1

18.* i) GKwU e¯‘‡K Abyf~wg‡Ki mv‡_ 600 †Kv‡Y Ggbfv‡e cÖ‡¶c Kiv nj †hb 7 wgUvi eëav‡b Aew¯’Z

3.5 wg. D‛P yBwU †Ìqv‡ji wVK Dci w`‡q P‡j hvq| e ‘wUi Abyf~wgK cvj v wbY©q Ki| DËit

37 wgUvi

ii) GKwU e ‘‡K u ‡e‡M Abyf~wgK Z‡ji mv‡_ †Kv‡Y Ggbfv‡e wb‡¶c Kiv nj †hb 2a eëav‡b

Aew¯’Z a cwigvb D‛PZv wewkó `yBwU †Ìqv‡ji wVK Dci w`‡q P‡j hvq| cÖgvY Ki †h,

2cot2

aR .

* iii) GKwU wµ‡KU ej‡K AvNvZ Ki‡j Zv wb‡¶c we›`y †_‡K h_vµ‡g b Ges a `~i‡Z¡ Aew¯’Z a Ges

b D‛PZv wewkó `yBwU †Ìqvj †Kv‡bv iK‡g AwZµg K‡i| †`LvI †h, Gi cvj v R

ba

baba

22

.

19.* i) GKwU Lvov †Ìqv‡ji cv`‡`k n‡Z f~wg eivei x ~i‡Z¡ ‡Kvb we›`y n‡Z 450 †Kv‡Y GKwU e¯‘ wb‡¶c

Kiv nj| Zv †Ìqv‡ji wVK Dci w`‡q †Mj Ges †Ìqv‡ji Aci cvk¦©¯’ y `~i‡Z¡ wM‡q gvwU‡Z coj|

†`LvI †h, †Ìqv‡ji D‛PZv

yx

xy

.

** ii) hw` †Kvb cÖw¶ß e ‘ t mg‡q Zvi MwZc‡_i Dci ’ P we›`y‡Z †c․‡Q Ges t1mg‡q P n‡Z cÖ‡¶c

we›`yMvgx Abyf~wgK Z‡j wd‡i Av‡m, Z‡e †`LvI †h, Z‡ji Dc‡i P Gi D‛PZv

2

1gtt1.

* iii) ‡Kvb cÖw¶ß e ‘ Zvi cÖ‡¶c we›`y n‡Z x Abyf~wgK ~i‡Z¡ Ges y Lvov `~i‡Z¡ Aew¯’Z †Kvb we›`y AwZµg

K‡i| e¯‘wUi Abyf~wgK cvj v R n‡j †`LvI †h, cÖ‡¶c †KvY

xR

R

x

y.tan 1

20.* i) GKwU UvIqv‡ii kxl© j¶¨ K‡i e›`yK n‡Z wbw¶ß GKwU †evgv UvIqv‡ii ga¨we›`y‡Z AvNvZ Ki‡j,

†`LvI †h UvIqvi‡K AvNvZ Kivi mgq †evgvwU Abyf~wg‡K P‡j|

ii) GKwU e¯‘ 2.39 wg./†m. †e‡M f~wgi mv‡_ 0

30 †Kv‡Y wbw¶ß nj| KZ mgq c‡i e¯‘wU wb‡¶c w`‡Ki

m‡½ j¤^fv‡e Pj‡e? GB mg‡q Gi †eM KZ n‡e? DËit 8 .†m.; 9.67 wg./†m.


Paper cÖkœ-7

8| ‡h †Kvb yBwU cÖ‡kœi DËi `vIt 5 2=10

8| (K) we¯Ívi cwigvct

1. wb‡Pi MYmsL¨v wb‡ekb n‡Z Mo eëavb wbY©q Kit

‡kÖwY 0-10 10-20 20-30 30-40 40-50

MYmsL¨v 3 7 11 15 5

DËit 9.59 (cÖvq)

2. msL¨v¸wji cwiwgZ eëavb wbY©q Kit DËit √

3. wb‡Pi MYmsL¨v wb‡ek‡bi cwiwgZ eëavb wbY©q Kit

‡kÖwYe¨vwß 20-30 30-40 40-50 50-60 60-70 70-80

MYmsL¨v 8 10 15 10 9 5

DËit 14.98 (cÖvq)

4. wb‡Pi Dcv‡Ëi Rb¨ cwiwgZ eëavb I †f`v¼ wbY©q Kit

cÖvß b¤i 31-40 41-50 51-60 61-70 71-80 81-90 91-100

QvÎmsL¨v 6 8 10 12 5 7 2

DËit 16.72, 279.558 (cÖvq)

5. wb‡Pi Z_¨ n‡Z cwiwgZ eëavb I †f`vsK wbY©q Kit

mvßvwnK Avq 10-20 20-30 30-40 40-50 50-60 60-70

kÖwgK msL¨v 5 10 15 20 10 5

DËit 13.368, 178.70 (cÖvq)


b¤î 10 20 30 40 50 60 70

QvÎmsL¨v 4 5 10 25 10 6 4

DËit 14.68, 215.50 (cÖvq)

7. cwiwgZ eëav‡bi myweav I Amyweav¸wj wjL|

8. cwiwgZ eëavb I †f`v‡¼i msÁv `vI| wb‡Pi Z_¨ n‡Z cwiwgZ eëavb I †f`v¼ wbY©q Ki|

‡kÖwYe¨vwß 20-24 25-29 30-34 35-39 40-44 45-49

RbmsL¨v 7 10 15 12 10 5

DËit 7.44, 55.35 (cÖvq)

9. wb‡P Øv`k †kÖwYi QvÎ-QvwÎ‡ì D”PZi MwY‡Z cÖvß b¤‡ii mviwY †Ìqv nj| cÖvß b¤‡ii cwiwgZ eëavb I

Mo eëavb wbY©q Ki|

b¤î 20-24 25-29 30-34 35-39 40-44 45-49

QvÎ-QvÎx 7 10 15 13 9 6

DËit 7.38, 16.694 (cÖvq)


‡kÖwY 6-15 16-25 26-35 36-45 46-55 56-65 66-75 76-85

MYmsL¨v 10 20 30 40 50 60 70 80

DËit 19.72, 388.89 (cÖvq)

11. wb‡P DcvË n‡Z cwiwgZ eëavb wbY©q Kit

†kÖwY e¨vwß 200-300 300-400 400-500 500-600 600-700 700-800

MYmsL¨v 12 18 36 24 10 8

DËit 134.63 (cÖvq)

12. wb‡P DcvË n‡Z cwiwgZ eëavb wbY©q Kit

b¤î 5-9 10-14 15-19 20-24 25-29 30-34

QvÎ 15 30 55 17 10 3

DËit 5.759 (cÖvq)

13. ‡f`vsK wK? wb‡Pi Z_¨ n‡Z cwiwgZ eëavb I †f`vsK wbY©q Kit

gvwmK Avq 5-9 10-14 15-19 20-24 25-29 30-34

kÖwgK msL¨v 15 30 55 17 10 3

DËit 6.11, 37.33 (cÖvq)

14. wb‡Pi msL¨v¸wji cwiwgwZ eëavb †f`v¼ wbY©q Kit

6, 10, 9, 12, 21, 24, 25, 15, 16, 22.

DËit 6.39, 40.80 (cÖvq)


cÖvß b¤i 51-60 61-70 71-80 81-90 91-100

QvÎmsL¨v 10 15 20 12 3

DËit 11.119, 123.632 (cÖvq)

16. wb‡Pi wb‡ek‡bi cwiwgZ eëavb Kit

x 10 13 25 30 37 42 45

f 3 7 8 15 10 5 2

DËit 10 (cÖvq)

17. B¯úvnvwb wek¦we`¨vjq K‡j‡Ri 100 Rb Qv‡Îi D”Pv †m.wg.) wb‡¤œi mviYx‡Z cÖ`Ë n‡jv:

D‛PZv 141-150 151-160 161-170 171-180 181-190

QvÎ-QvÎx 5 16 56 19 4

DËit 8.43, 70.99 (cÖvq)

18. Find variance for the data set 11, 13, 15, ........, 25.

Ans: 21

19. 50 Rb Qv‡Îi eq‡mi Mo 22 eQi I cwiwgZ eëavb 4 eQi| wKš‘ 2 Rb Qv‡Îi eqm 25 I 24 eQ‡ii

¯’‡j h_vµ‡g 13 I 11 †jLv nq| Zv‡ì eq‡mi Mo I cwiwgZ eëavb wbY©q Ki|

DËit 22.5 eQi; 3.46 eQi

20. a) cÖ_g n msL¨K †Rvo ¯vfvweK msL¨vi †f`vsK wbY©q Ki| DËit

b) cÖ_g n msL¨K we†Rvo ¯vfvweK msL¨vi †f`vsK wbY©q Ki| DËit


Paper cÖkœ-8

8| (L) m¤¢ve¨Zv(Dccv`¨)

1. m¤¢ve¨Zvi ms‡hvM m~Î (Additional law of Probability):

*** i) eR©bkxj ev wew‛Qbœ NUbvi †¶‡Î m¤¢ve¨Zvi ms‡hvM m~ÎwU eY©bv I cÖgvY Ki|

** ii) AeR©bkxj NUbvi †¶‡Î m¤¢veZvi ms‡hvM m~ÎwU eY©bv I cÖgvY Ki|

2. m¤¢ve¨Zvi ¸Yb m~Î (Multiplication law of Probability):

* i) `yBwU ¯^vaxb NUbvi †¶‡Î m¤¢ve¨Zvi ¸Yb m~ÎwU eY©bv `vI I cÖgvY Ki|

ii) `yBwU Aaxb NUbvi †¶‡Î m¤¢ve¨Zvi ¸Yb m~ÎwU eY©bv `vI I cÖgvY Ki|

3. kZ©vaxb m¤¢ve¨Zv (Conditional Probability):

‡Kvb bgybv RM‡Z A I B yBwU NUbv Ges P(B) > 0 n‡j, B NUbvwU NUvi kZ©vax‡b A NUbvwU NUvi

m¤¢ve¨Zv

P(B)

B)P(A

B

AP

cÖgvY Ki|

4. m¤¢ve¨Zvi c~iK m~Î (Complementary theorem of Probalily):

m¤¢ve¨Zvi c~iK m~ÎwU eY©bv Ki I cÖgvY `vI|

8| (M) m¤¢ve¨Zv(mgm¨vejx/AsK)

1. i) 52 Lvbv Zv‡mi c¨v‡K‡U 4wU †U°v Av‡Q| wbi‡c¶ fv‡e †h †Kvb GKLvbv Zvm †U‡b †U°v bv cvIqvi

m¤¢ve¨Zv KZ?

ii) 52 Lvbvi GK c¨v‡KU Zvm n‡Z niZ‡bi ivRv (K) mwi‡q ivLv nj| Aewkó Zvm¸‡jv fvj K‡i

Zvmv‡bv nj| wbi‡c¶fv‡e GKwU Zvm Uvb‡j †mUv niZb nIqvi m¤¢ve¨Zv wbY©q Ki|

* iii) 52 Lvbv Zv‡mi c¨v‡KU n‡Z GKLvbv Zvm •èfv‡e DVv‡bv nj| ZvmwU (a) jvj †U°v (b) jvj A_ev

†U°v nIqvi m¤¢ve¨Zv KZ?

iv) 52 Lvbv Zv‡mi c¨v‡KU †_‡K †hgb Lywk †U‡b avivevwnKfv‡e 4 Lvbv †U°v cvIqvi m¤¢ve¨Zv wbY©q Ki|

v) 52 Lvbv Zv‡mi c¨v‡KU n‡Z wZbLvbv Zvm Uvbv n‡jv| wZbwU ZvmB ivRv nIqvi m¤¢ve¨Zv KZ?

vi) 52 wU Zv‡mi GKwU c v‡KU n‡Z •èfv‡e GKwU Zvm wb‡j Zv i‚BZb ev ivRv nevi m¤¢ve¨Zv wbY©q Ki|

2.*** i) `yBwU Q°v GK‡Î wb‡¶c Kiv n‡j Zv‡ì bgybv‡¶Î •Zix Ki Ges ywU Qq IVvi m¤¢ve¨Zv wbY©q Ki|

*** ii) GKwU Q°v I `yBwU gy`ªv GK‡Î wb‡¶c Kiv n‡j Zv‡ì bg~bv‡¶ÎwU •Zix Ki Ges Q°vq 4 IVvi

m¤¢ve¨Zv wbY©q Ki|

iii) GKwU mylg gy`ªv cici 3 evi Um Kiv nj| cÖwZwU U‡mB cÖ_g †nW cvIqvi k‡Z© 2 ev Z‡ZvwaK †nW

cvIqvi m¤¢vebv KZ? †Kvb kZ© Av‡ivc Kiv n‡j 2 ev Z‡ZvwaK †nW cvIqvi m¤¢vebv KZ?

iv) ‡Kvb cix¶‡Y GKB mv‡_ GKwU bxj Ges GKwU jvj Q°v wb‡¶c Kiv nj| hw` x bxj QKvq cÖvß

†dvUvi msL¨v Ges y jvj Q°vq cÖvß †dvUvi msL¨v wb‡ ©k K‡i Zvn‡j Dcv`vb (x, y) eënvi K‡i

NUbRMZ s wbY©q Ki|

v) GKwU gy`ª cici wZb evi Um Kiv nj| ch©vqµ‡g gy`ªvwUi †nW Ges †Uj cvevi m¤¢ve¨Zv wbY©q Ki|

vi) GKwU Q°v I yBwU gy`ªv GK‡Î wb‡¶c Kiv n‡j Zv‡ì bgybv‡¶ÎwU •Zwi Ki Ges (a) 2 ywU †nW I

†Rvo msL¨v (b) Q°vq 4 cvevi m¤¢ve¨Zv wbY©q Ki|

3.*** i) P(A) = 3

1, P(B) =

4

3, A I B ¯^vaxb n‡j P(A B) wbY©q Ki|

* ii) hw` P(A) = 3

1, P(B) =

4

3 nq, Z‡e P(A B) KZ? †hLv‡b A I B ¯^vaxb|

*** iii) P(A B) = 3

1, P(A B) =

6

5, P(A) =

2

1n‡j P(B), P(B/), P(A/) wbY©q Ki| A I B wK

¯^vaxb?

iv) P(A) = 2

1, P(B) =

5

1 Ges P

B

A=

8

3 n‡j, P

A

B wbY©q Ki|

** v) GKRb Qv‡Îi evsjvq cv‡mi m¤¢ve¨Zv

3

2; evsjv I A¼ yBwU wel‡q cv‡mi m¤¢ve¨Zv

45

14Ges ywUi †h

†Kvb GKwU‡Z cv‡mi m¤¢ve¨Zv

5

4 n‡j, Zvi As‡K cv‡mi m¤¢ve¨Zv KZ?

** vi) GKRb cix¶v_x©i evsjvq †dj Kivi m¤¢ve¨Zv

5

1; evsjv Ges Bs‡iwR `yBwU‡Z cv‡mi m¤¢ve¨Zv

4

3 Ges

`yBwUi †h †Kvb GKwU‡Z cv‡mi m¤¢ve¨Zv

8

7 n‡j, Zvi †Kej Bs‡iwR‡Z cv‡mi m¤¢ve¨Zv KZ?

*** vii) A I B Gi GKwU A‡¼i mgvavb Ki‡Z cvivi m¤¢ve¨Zv h_vµ‡g

3

1Ges

4

1| Zviv GK‡Î A¼wU

mgvav‡bi †Póv Ki‡j A¼wUi mgvavb wbY©‡qi m¤¢ve¨Zv KZ?

viii) ‡Kvb evwYwR¨K cÖwZôv‡bi wZbwU c‡ì Rb¨ GKRb cÖv_x© Av‡e`b K‡i‡Q| H wZbwU c‡` cÖv_x© msL¨v

h_vµ‡g 3, 4, 2 n‡j H cÖv_x©i Aš—Z GKwU c‡` PvKwi cvIqvi m¤¢ve¨Zv KZ?

*** ix) MwYZ I cwimsL¨vb wel‡q 200 Rb cix¶v_x©i g‡a¨ 20 Rb cwimsL¨v‡b Ges 40 Rb MwY‡Z †dj

K‡i| Dfq wel‡q `kRb †dj K‡i‡Q| wbi‡c¶fv‡e GKRb QvÎ‡K evQvB Ki‡j Zvi cwimsL¨v‡b cvm

I MwY‡Z †dj nIqvi m¤¢ve¨Zv wbY©q Ki|

** x) GKwU K‡j‡R GKv`k †kªYxi 40 Rb Qv‡Îi wfZi 20 Rb dzUej †L‡j, 25 Rb wµ‡KU †L‡j Ges

10 Rb dzUej I wµ‡KU †L‡j| Zv‡ì ga¨ n‡Z •èPq‡b GKRb‡K wbe©vPb Kiv nj| hw` †Q‡jwU

dzUej †L‡j, Z‡e Zvi wµ‡KU †Ljvi m¤¢ve¨Zv KZ?

* xi) 10 †_‡K 30 ch©š— msL¨v n‡Z †h †Kvb GKwU‡K B‛QvgZ wb‡j H msL¨vwU †g․wjK A_ev 5 Gi

¸wYZK nevi m¤¢ve¨Zv wbY©q Ki|


Paper cÖkœ-8

xii) †Kvb Rwi‡c †`Lv †Mj 70% †jvK B‡ËdvK c‡o, 60% ‡jvK msev` c‡o Ges 40% †jvK Dfq

cwÎKv c‡o| wbi‡c¶fv‡e evQvB Ki‡j GKRb †jv‡Ki B‡ËdvK ev msev` covi m¤¢ve¨Zv wbY©q Ki|

4.*** i) GKwU ev‡· wewfbœ AvKv‡ii 6wU mv`v ej, 7wU jvj ej Ges 9wU Kv‡jv ej Av‡Q| Gjv‡g‡jvfv‡e

GKwU ej Zz‡j †bIqv nj| ejwU jvj ev mv`v nIqvi m¤¢ve¨Zv KZ?

** ii) GKwU e¨v‡M 5wU mv`v, 7wU jvj Ges 8wU Kv‡jv ej Av‡Q| G‡jv‡g‡jvfv‡e 3 wU ej Zz‡j †bIqv nj|

ej¸‡jv jvj ev mv`v nIqvi m¤¢ve¨Zv KZ?

iii) GKwU ev‡· 4wU jvj, 5wU bxj Ges 7wU mv`v is Gi ej Av‡Q| •èPq‡b GKwU e‡ji jvj ev mv`v

nIqvi m¤¢ve¨Zv KZ?

** iv) GKwU e¨v‡M 4wU mv`v Ges 5wU Kv‡jv ej Av‡Q| GKRb †jvK wbi‡c¶fv‡e wZbwU ej DVv‡jb| wZbwU

ejB Kv‡jv nIqvi m¤¢ve¨Zv wbY©q Ki|

* v) GKwU _wj‡Z 3wU mv`v Ges 2wU Kv‡jv ej Av‡Q| Aci GKwU _wj‡Z 2wU mv`v Ges 5wU Kv‡jv ej

Av‡Q| wbi‡c¶fv‡e cÖ‡Z¨K _wj n‡Z GKwU K‡i ej †Zvjv nj| yBwU e‡ji g‡a¨ Aš—Z GKwU mv`v

nIqvi m¤¢ve¨Zv wbY©q Ki|

vi) GKwU e¨v‡M 5wU mv`v, 7wU jvj Ges 8wU Kv‡jv ej Av‡Q| hw` wewbgq bv K‡i GKwU K‡i ci ci

PviwU ej Zz‡i †bIqv nq, Z‡e me¸‡jv ej mv`v nIqvi m¤¢ve¨Zv KZ?

vii) GKwU ev‡· 5wU jvj I 10wU mv`v gv‡e©j Av‡Q| GKwU evjK †hgb Lywk Uvb‡j cÖwZev‡ii ywU wfbœ

is‡Oi gv‡e©j cvIqvi m¤¢ve¨Zv KZ?

viii) GKwU e¨v‡M wZbwU Kv‡jv Ges 4wU mv`v ej Av‡Q| •èfv‡e GKwU K‡i 2wU ej Zz‡j †bIqv nj; wKš‘

cÖ_gwU DVv‡bvi ci Zv Avi e¨v‡M ivLv nj bv| wØZxq ejwU mv`v nIqvi m¤¢ve¨Zv KZ?

ix) GKwU ev‡· 5wU jvj I 4wU mv`v wµ‡KU ej Ges Aci GKwU ev‡· 3wU jvj I 6wU mv`v wµ‡KU ej

Av‡Q| cÖ‡Z¨K ev· n‡Z GKwU e‡j ej DVv‡bv n‡j yBwU e‡ji g‡a¨ Kgc‡¶ GKwU jvj nIqvi

m¤^ve¨Zv wbY©q Ki|

* x) `yBwU GKB iKg ev‡·i cÖ_gwU‡Z 4wU mv`v I 3wU jvj Ges wØZxqwU‡Z 3wU mv`v I 7wU jvj ej

Av‡Q| mgm¤¢e Dcv‡q GKwU ev· wbe©vPb Kiv nj| H ev· n‡Z wbi‡c¶fv‡e GKwU ej Uvbv n‡jv,

ejwU mv`v nIqvi m¤¢ve¨Zv wbY©q Ki| hw` ejwU mv`v nq Zvn‡j cÖ_g ev· †_‡K wbe©vwPZ nIqvi

m¤¢ve¨Zv KZ?

xi) GKwU e¨v‡M 1wU UvKv I 3wU cqmv, wØZxq e¨v‡M 2wU UvKv I 4wU cqmv Ges Z…Zxq e¨v‡M 3wU UvKv I

1wU cqmv Av‡Q| jUvwii gva¨‡g GKwU e¨vM evQvB K‡i m¤úyY© wbi‡c¶fv‡e GKwU gy`ªv D‡Ëvjb Ki‡j

†mwU UvKv nIqvi m¤¢ve¨Zv wbY©q Ki|

* xii) GKwU ev‡· 10 wU bxj I 15wU jvj gv‡e©j Av‡Q| GKwU evjK †hgb Lywk Uvb‡j cÖwZev‡i `ywU (a)

wfbœ es‡qi (b) GKB is‡qi gv‡e©j nIhvi m¤¢ve¨Zv KZ?

* xiii) `yBwU _wji GKwU‡Z 5wU jvj Ges 3wU Kv‡jv ej Av‡Q| Aci _wj‡Z 4wU jvj I 5wU Kv‡jv ej

Av‡Q| h_vm¤¢e Dcv‡q GKwU _wj wbe©vPb Kiv nj Ges Zv †_‡K yBwU ej †Zvjv n‡j GKwU jvj, GKwU

Kv‡jv nIqvi m¤¢ve¨Zv wbY©q Ki|

xiv) GKwU e¨v‡M 7wU jvj Ges 5wU mv`v ej Av‡Q| wbi‡c¶ fv‡e 4wU ej †Zvjv n‡j 2wU jvj I 2wU mv`v

nIqvi m¤¢ve¨Zv wbY©q Ki|

DËimg~nt

1. i) 13

12 ii)

17

4 iii)

26

1,

13

7 iv)

270725

1

2. i) 36

1 ii)

6

1 iii)

8

3,

2

1 iv)

4

1 v)

8

1,

6

1

3. i) 6

5 ii)

4

1 iii)

3

2,

3

1,

2

1, A I B ¯vaxb|

iv) 20

3 v)

9

4 vi)

40

33 vii)

2

1 viii)

4

3

ix) 20

3 x)

2

1 xi)

21

11 xii)

10

9

4. i) 22

13 ii)

76

3 iii)

16

11 iv)

42

5 v)

7

5

vi) 969

1 vii)

21

10 viii)

7

4 ix)

27

19 x)

140

61,

61

40

xi) 9

4 xii)

2

1,

2

1 xiii)

504

275 xiv)

33

14

msKj‡b-

‡gvt Avãyi iDd

cÖfvlK(MwYZ)

nvwKgcyi gwnjv wWMÖx K‡jR

m¤úv`bvq-

‡gvt AvRvnvi Avjx

cÖfvlK(MwYZ)

KvUjv wWMÖx K‡jR|


Paper cÖkœ-8

Education

HSC Model questions 2nd part