1. fmr oekqu m%fjs. ld, m%ia;dr

ld,h

m%fjs.h

u

t1

v

t2

(1) ;ajrKh ^fyda ukaokh& 12 tt

uv

(2) úia:dmkh 122

1ttuv

vaTv 2

1. ¥usrshla ksYap,;djfha isg a ms-2 taldldr ;ajrKhla iys;j m%fjs.h v ms-1 jk f;la .uka lr t;eka isg tu ksh; m%fjs.fhkau hus ld,hla p,kh fjhs' bkamiq a ms-2 taldldr ukaokfhka ksYap,;djhg m;a jk w;r .ukg .; jQ ld,h T

s fjs' p,s;h i|yd m%fjs. ld, jl%hla we|

( i ) ksh; m%fjs.fhkau .uka l< ld,h fidhkak' ( ii ) .uka l< iusmQ¾K ¥r m

nj fmkajkak'

m%fjs.h

ld,h

v

T -2t t 0

a a

a;ajrKh

t

0

0

t

vt

a

v

( i ) ksh; m%fjs.fhkau .uka l< ld,h

tT 2a

vT

2

m%fjs.h

ld,h

v

T -2t t 0

a a

t

a

vt

( ii ) .uka l< iusmQ¾K ¥r

tv 2

1 tTv 2 tv 2

1

tTvvt 2

.uka l< iusmQ¾K ¥r tTvvt 2

tTv

by;ska" a

vt ksid

.uka l< iusmQ¾K ¥r

a

vTv

vaTa

v

2. fmr oekqu idfmalaI m%fjs.h ms<sn| uQ,O¾uh

BCvCAvBAv ,,,

m%fjs. ;%sfldaKh

B

C

A

v u

A kejla u km h-1 taldldr m%fjs.fhka W;=frka fldaKhla kef.kysrg jk osYdjg hd;%d lrhs' kef.kysfrka fldaKhla ol=Kg jk osYdjg v km h-1 taldldr m%fjs.fhka hd;%d lrk ;j;a B kejl isÜk ksrSlaIlfhl=g A kej W;=re osYdjg .uka lrkakd fia fmfka' B kejg idfmalaIj A kefjs p,ss;h i|yd m%fjs. ;%sfldaKh w|skak' ta khska nj fmkajkak'

sinsin uv

2.

fmdf<dj - E EAv , u

EBv , v

EAv , v EBv ,

idfmalaI m%fjs.h ms<sn| uQ,O¾uh

BEvEAvBAv ,,,

v

B kefjs isÜk ksrSlaIlfhl=g A kej W;=re osYdjg .uka lrkakd fia fmfka' BAv ,

?

m%fjs. ;%sfldaKh w|suq'

u

u

BEvEAvBAv ,,,

v?

v

E

B

A

u

u

2

ihska iQ;%fhka

sin

2sin

vu

sincos

vu

cossin vu

3.

g

u

2

3 2

iuìfuys msysá O ,laIHhl isg P wxY=jla u

wdrusNl m%fjs.fhka iy wdfrdayKfhka m%lafIam flf¾' P wxY=j O ,laIHfha isg ;sriaj g jvd ¥rla p,kh úfuka miq iuìfuys m;s; fjs kus m%lafIamK fldaKh i|yd iq¥iq w.hka ks¾Kh lrkak'

R

cosv

sinv

isriaj by<g 2

2

1atuts 2

2

1sin0 TgTu

tksid ;sria mrdihg wkqrEm ld,h T

2

2

1sin0 TgTu sin

2

g

uT

;sriaj fhdouq' uts cosuR sin2

g

u

sincos22

g

u

2sin2

g

u

g

uR

2

3 2

2sin2

g

u

g

u

2

3 2

2

32sin

3sin

3

22

3

46

4,

6

2

4.

P

fmr oekqu

v

tkacsu H w cjfhka l%shd lrhs kus PvH

ksjsgkaf.a fofjks kshuh o fhosh yel'

amF

4. ialkaOh M jk fudag¾ r:hla ;srig wdk;shlska hq;a iDcq fm;la Tiafia by<g tys

tkacsu H kw cjfhka l%shd lrïka R N m%;sfrdaOhlg tfrysj .uka lrhs'

Mg

P R N

tkacsfus cjh kwH

wH1000

amF fudag¾ r:fha p,s;hg fhoSfuka

PvH

v

HP

1000

MaMgRP sin

fudag¾ r:fha m%fjs.h Wmrsu jQ úg tys ;ajrKh Y+kHh fjs'

MaMgRP sin

v

HP

1000

MaMgRv

H sin

1000

MaMgRHv

sin1000

túg sin1000

MgRHv

tksid r:fha Wmrsu m%fjs.h HMgR

vsin

1000

5. fmr oekqu osYdk; f¾Ld LKav Ndú;fhka ffoYsl

ksrEmKh flf¾'

ffoYsl wdl,kh

C B

a

b

ba

O A

;s;A .=Ks;h fyda wosY .=Ks;h'

fyda fyda wNsY+kH ffoYslh úg a b 0ba

cos baba

a

b

tksid 0ba

ba

2aaa

cabacba .. abba

wfkla úg

5. OA iy OB f¾Ld LKavj,ska iy m%Nskak ffoYsl ms<sfj,ska ksrEmKh fjs' fuu ffoYsl ;Dma; lrhs kus" ffoYsl i|yd ;s;a .=Ks;h Ndú;fhka OA iy OB f¾Ld LKav ,usnl nj fmkajkak'

a b

baba

C B

a

b

ba

O A

baOC

baBA

baba oS we;' 22

baba

babababa ..

bbabbaaabbabbaaa ........

2222 .2.2 bbaabbaa

0.4 ba

0. ba

ba

6. fmr oekqu n, ;=kl iu;=,s;;djh hgf;ys tlsfkl

iam¾Yj fyda ikaê lr fyda we;s jia;= mej;sh yels wdldr

fujeks .eg¿ úi|SfusoS

hus osYdjlg moaO;sh u; n, úfNaokh

moaO;sfha fldgila u; n, úfNaokh

>Q¾K ie,lSu

úfYaIs; l%u f,i

,dïf.a m%fushh fhoSu

moaO;sfha fN!;sl/cHdï;sl .=K ie,lSu

ihska" fldaihska" fldgs wdoS iQ;% fhoSu

n, ;%sfldaK ie,lSu

6. nr w jk taldldr AB oKavl A fl<jr wp, ,laIHhlg wijq lr we;' ¨yq wÑ;kH ;ka;=jl fofl<jr Bg iy Ag isriaj by<ska we;s C ,laIHhlg iusnkaO lr oKav iu;=,s;;djfha ;nd we;' fuysoS ABC iumdo ;%fldaKhl YS¾I jk fia msysghs' oKav u; l%shd lrk n, ksrEmKh jk mrsos n, ;%sfldaKhla bosrsm;a lrkak' tkhska fyda wka whqrlska fyda ;ka;=fjs wd;;sh fidhkak'

A

B

C

w

T

T

R

G

oKav u; l%shd lrk n,

oKafvs nr w ;ka;=fjs wd;;sh T

AysoS m%;sl%shdj R

n, ;=kls'

l%shd f¾Ld tal,laIHh fjs'

n, ;%sfldaKh th ACD fjs'

D

w

T

R

n, ;%sfldaKh

A

C

D

60o G

B

w

AC

T

CD

R

AD

ACD

T

CD

w

AC

AC

wCDT

ABC iy GBD ;%sfldaK i|yd AC iy DG iudka;rhs' G hkq AB ys uOH ,laIHh

w

T

R

A

C

D

60o G

B

AC

wCDT

D hkq BC ys uOH ,laIHh

tksid

D hkq BC ys uOH ,laIHh

BCCD2

1

CDAC 2

wT2

1

AC2

1

7. fmr oekqu ksheos wjldYhl A iy B isoaê

nysIaldrl jkafka BA úghs' túg

0 BAP

iuQysl f,i ksrjfYaI BA úghs' túg

1 BAP

iajdh;a; isoaê jkafka tl isoaêhla is¥ ùu flfrys wfkla isoaêh n, fkdmdk úghs'

A iy B isoaê iajdh;a; isoaê kus

BAP AP ABP BP

wiusNdjH iusNdú;djh tla isoaêhla is¥ ù we;s úg ;j;a isoaêhla is¥ ùfus iusNdú;djhhs'

A isoaêh is¥ ù we;s úg B isoaêh is¥ ùfus iusNdú;djh ABP

AP

ABP

7. ksheos wjldYhl A iy B isoaê i|yd " iy fjs' A iy B isoaê ( i ) nysIaldr fjs" ( ii ) iajdh;a; fjs oehs ks¾Kh lrkak'

25.0Ap 4.0Bp 55.0 BAp

BAPBPAPBAP ksid BAP 4.025.055.0

55.04.025.0 BAP

1.0 BAP 0

nysIaldr fkdfjs'

A iy B isoaê iajdh;a; isoaê kus

BAP

BP

BAP

AP

BAPBPAP

;jo BAPBPAPBAP

25.0Ap 4.0Bp

1.0 BAP

4.025.0 BpAp

1.0

BAP BpAp

ksid A iy B isoaê iajdh;a; fjs'

8. A iy B hkq ksheos wjldYhl isoaê folla hehs .ksuq' A isoaêh is¥ ù we;s úg B isoaêh is¥ ùfus iusNdjH iusNdõ;djh i|yd m%ldYkhla ,shkak'

A isoaêh is¥ ù we;s úg B isoaêh is¥ ùfus iusNdú;djh ABP

AP

ABP

8. ¥usrshla kshï; fjs,djg msg;a ùfus iusNdõ;djh 0.96 nj;a th kshï; fjs,djg .ukdka;hg <Õd ùfus iusNdõ;djh 0.84 nj;a oS we;' ;j¥rg;a ¥usrsh kshï; fjs,djg msg;a ùu iy th kshï; fjs,djg .ukdka;hg <Õd ùu hk isoaê folu is¥ ùfus iusNdõ;djh 0.72 nj;a okshs' th

( i ) kshï; fjs,djg msg;a ù we;akus kshï; fjs,djg .ukdka;hg <Õd ùfus iusNdõ;djh"

( ii ) kshï; fjs,djg .ukdka;hg <Õd ù we;akus th kshï; fjs,djg msg;a ùfus iusNdõ;djh fidhkak'

my; f,i isoaè i,luqq'

¥usrsh kshï; fjs,djg msg;a ùu - A

oS we;s f;dr;=re wkqj

th kshï; fjs,djg .ukdka;hg <Õd ùu - B

96.0Ap 84.0Bp

72.0 BAp

( i ) kshï; fjs,djg msg;a ù we;akus kshï; fjs,djg .ukdka;hg <Õd ùfus iusNdõ;djh

ABP

AP

ABP

96.0

72.0

4

3

oS we;s f;dr;=re wkqj 96.0Ap 84.0Bp

72.0 BAp

( ii ) kshï; fjs,djg .ukdka;hg <Õd ù we;akus kshï; fjs,djg msg;a ùfus iusNdõ;djh

BAP

BP

ABP

84.0

72.0

7

6

9. fmr oekqu wiuQys; o;a; iuQyhl ud;h" uOHia:h iy

uOHhkh

nxxxx ,,,, 321 o;a; i|yd uOHhkh

ud;h jeâu jdr .Kkla we;s o;a;h

uOHia:h o;a; wjfrdayK fyda wdfrdayK ms<sfj,g ieliQ úg uOHfha we;s o;a;h

uOHhkh o;a;j, idudkHh

n

xxxxx n

321

9. hus W;aijhlg wuq;a;ka meïKs fudag¾ r:j, j¾Kh wkqj tajd ldKav mylg fjka lr we;' th talud; jHdma;shla jk w;r tys uOHhkHh iy uOHia:h ms<sfj,ska 3 iy 4 fjs' tu jHdma;sh ks¾Kh lrkak'

o;a; m%udKh 5

uOHia:h 4

4 x y z w

fuys wzyx 4

uOHhkh 3 35

4

wzyx

4 x y z w fuys wzyx 4

35

4

wzyx 11 wzyx

4x x i|yd iq¥iq w.hka 4,3,2,1x

kus 1x

1

1 10 wzy

1y9 wz

4 1 1 4 5

talud; jHdma;shla fkdfjs

2y 8 wz

4 x y z w fuys wzyx 4

kus 1x 10 wzy

2y 8 wz

4 1 2 4 4

talud; jHdma;shls

2y 7 wz

kus 2x

wz 4

ksid úh fkdyel' tksid jHdma;sh

4 1 2 4 4

10. fmr oekqu iuQys; o;a; iuQyhl ud;h" uOHia:h iy

uOHhkh

mx;sh

( a, b )

mx;s ,l=K

ixLHd;h

1f1x

2x

nx

2f

nf

uOHhkh

n

kkk xf

N 1

1

n

kkfN

1

uOHia:h iuQys; o;a; i|yd uOHia:h wh;a jk mx;sh fidhkak'

uOHia:h

1

2m

m

m FN

f

cLm

L - uOHia:h wh;a jk mx;sfha my< udhsu'

mc - uOHia:h wh;a mx;sfha ;ru' mf - uOHia:h wh;a mx;sfha ixLHd;h'

1mF - uOHia:h wh;a mx;shg my< mx;sfha iuqpsÑ; ixLHd;h

n

k

kfN

1

ud;h iuQys; o;a; i|yd ud;h wh;a jk mx;sh fidhkak'

ud;h

21

1cLM

L - ud;h wh;a jk mx;sfha my< udhsu'

c - ud; mx;sfha ;ru'

1 - ud; mx;sh iy thg my< mx;sfha ixLHd; w;r fjki'

2 - ud; mx;sh iy thg by< mx;sfha ixLHd; w;r fjki'

10. my; oelafjk ksrSlaIK iys; jHdma;sfha uOHhkHh 31.8 ls'

5 - 15 15 - 25 25 - 35 35 - 45 45 -55

16 a 30 22 20

iq¥iq mrsKdukhla ie,lsfuka a ys w.h ks¾Kh lr jHdma;sfha ud;h fidhkak' mx;sh mx;s w.h ixLHd;h

5 - 15 16

15 - 25 a

25 - 35 30

35 - 45 22

45 -55 20

10

20

30

40

50

a ys w.h fkdokshs iq¥iq mrsKdukh

20 xy

20 xy

mx;sh x ixLHd;h y f y

5 - 15 16

15 - 25 a

25 - 35 30

35 - 45 22

45 -55 20

10

20

30

40

50

20 xy

-10

0

10

20

30

-160

0

300

440

600 yf

1601340

1180ny

1180

20xyn

1180208.31

n

1180208.31

8.118.11

1180 n 100

10020223016 a

10088 a

12 a

ud;h

21

1cLM

mx;sh x ixLHd;h

5 - 15 16

15 - 25 12

25 - 35 30

35 - 45 22

45 -55 20

10

20

30

40

50

ud; mx;sh

22302

82

12301

181

25L 10c

188

181025M

26

181025 923.31

End of Paper Two Part A