01 fN!;sl úoHdj
02 - ridhk úoHdj
07 - .◊;h
08 - lDIs úoHdj
09 - „j úoHdj
10 - ixhqla; .◊;h
11 - Wiia .◊;h

Curriculum
Assessment
Evaluation

Teaching

Learning

NETS

w'fmd'i' ^Wiia fm<& úNd.h
jr®I 2011 iy bka miqj meje;afjk úNd. i|yd
m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

1 fjΩu
01 - fN!;sl úoHdj
02 - ridhk úoHdj
07 - .◊;h
08 - lDIs úoHdj
09 - „j úoHdj
10 - ixhqla; .◊;h
11 - Wiia .◊;h
		

Curriculum
Assessment
&
Evaluation

Teaching

Learning

NETS

mr®fhaIK yd ixjr®Ok YdLdj
cd;sl we.hSï yd m¯laIK fiajdj
Y%S ,xld úNd. fomdr®;fïka;=j

ishÆ u ysñlï weú˙◊

w'fmd'i'^Wiia fm<& úNd.h
jr®I 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak
1 fjΩu

Y%S ,xld úNd. fomdr®;fïka;=j
m%:u uqøKh - 2010

ISBN 978-955-668-020-1

mr®fhaIK yd ixjr®Ok YdLdj
cd;sl we.hSï yd m¯CIK iajdj
Y%S ,xld úNd. fomdr®;fïka;=j

uQ,H wkq.%yh
f,dal nexl= wOHdmk wdxYsl ixjr®Ok rduqj iy jevigyfka (ESDFP)
wOHdmk wdxYsl ixjr®Ok m%∞kh (ESDG) u.sks'

								

uqøKh ( Y%S ,xld úNd. fomdr®;fïka;=j'

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

- ii -

úNd. flduid˙ia ckrd,af.a m◊úvh
lsishï mdGud,djlska ,nd.;a ±kqu yd wjfndaOh uekSug úúO l%ufõo Wmfhda.S lr .efkhs'
ta w;=˙ka ,sÅ; úNd. jr®;udkfha Ndú; jk jvd;a m%p,s; l%uhhs' bf.kqï b.ekaùï l%shdj,sfha
ldr®hlaIu;dj by< kexùu iy iïmsKaä; we.hSï l%ufõoh w;r iïnkaO;djla tu.ska f.dvkef.k
nj;a" idOk uÜgï ;SrKh jk nj;a ie,ls,a,g .efkhs'
jir 2007 kj wOHdmk m%;i
s xialrKj,g wkql,
+ j ixfYdaOkh lrk ,o úIh ksrf
® oaY 2009 jifr®
° 12 fY%a◊h i|yd y∫kajd fokq ,eî h' tu isiqka w'fmd'i'^Wiia fm<& úNd.h i|yd m%:u j;djg fmkS
isákqfha 2011 jifr® ° h' ta wkqj ixfYdaê; úIh ksr®foaYj,g wkql+, j 2011 iy bka miq j meje;afjk
w'fmd'i'^Wiia fm<& úNd. i|yd fmkS isák isiqkaf.a m%fhdackh i|yd m%Yak m;% jHQyh iy uQ,dlD;s
m%Yak we;=<;a lr ilia lrk ,o ix.%yhls fuh'
w'fmd'i'^Wiia fm<& úIh ksr®foaYj, úIh ikaOdrh ixfYdaOkh ùu;a iu. Bg WÑ; m˙† m%Yak
m;%j, jHQyfha o fjkialï is≥ ls¯ug o wjYH úh' ta wkqj úIh ldKav f.dkq lr" tla tla ldKav ;=<
m%Yak m;% jHQyfha wkkH;dj /l .ekSug W;aidy ord we;' úoaj;a lñgq u.ska fuu jHQy ;SrKh lrk
,o w;r" tu.ska uQ,dlD;s m%Yak o ilia fl˙◊'
wOHdmk ñkq ï iy we.hq ï ms < s n | úYa j dih iy wjOdkh jeä fjñka mj;s k hq . hl
w'fmd'i'^Wiia fm<& uÜgfï ° tu we.hqï ldr®hh flfrkafka flfia ±hs ishÆ mdr®Yaj ±kqj;a ls¯u
jeo.;a h' w'fmd'i'^Wiia fm<& úNd.h m%Odk jYfhka iy;sllrKh wruqKq lr.;a idOk m¯CIKhls'
tfy;a úYajúoHd, m%;smdok fldñIka iNdj iy fjk;a cd;sl fukau cd;Hka;r Wiia wOHdmk wdh;k
úiska fuu úNd.fha m%;sM, u; úYajúoHd,j,g isiqka f;dard .efkk ksid fuys f;a¯ï m¯CIKhl
.;s.=K o .eíj we;' tneúka fuu lD;sfhys ±lafjk m%Yak m;% jHQy iy uQ,dlD;s m%Yak m˙YS,kh ls¯u
uÛska isiqkag fuu úNd.fha ° uqyqK °ug is≥jkafka l=uk wdldrfha we.hSulg o hkak ms<sn| j ukd
wjfndaOhla ,nd.; yels h' tu ksid isiqka úNd.hg iQ∞kï ls¯fï ° mdi,aj, ú≥y,am;sjrekag yd
.=rejrekag;a ta yd iïnkaO u.fmkajkakkag;a mdi,a m%cdjg;a fuh ufydamld¯ jkq we;'
fuu Wmfoia ix.%yfhys ˚ fldgfiys úNd.h iïnkaO fmd≥ f;dr;=re o" ˚˚ fldgfiys m%Yak
m;%j, jHQy iy iajNdjh o we;=<;a jk w;r ˚˚˚ fldgfiys tla tla úIhh i|yd uQ,dlD;s m%Yak we;=<;a
fõ'
fuys we;=<;a m%Yak m;% jHQy iy uQ,dlD;s m%Yak ilia ls¯fï ° iyfhda.h ≥ka cd;sl wOHdmk
wdh;kfha wOHCI ckrd,a we;=Ω ldr®h uKav,hg o" wOHdmk m%ldYk flduid˙ia ckrd,a we;=Ω ldr®h
uKav,hg o ishÆ u úIhj, md,l m¯CIljrekag o iïm;a mqoa., uy;au uy;aókag o Y%S ,xld úNd.
fomdr®;fïka;=fõ ishÆ u ks,Od¯kag o fmd; uqøKh i|yd uQ,H wkq.%yh ±lajQ f,dal nexl= wOHdmk
wdxYsl ixjr®Ok m%∞kfha ^ESDG& ldr®h uKav,hg o" fuh uekúka uqøKh lr≥ka Y%S ,xld úNd.
fomdr®;fïka;=fõ uqøKd,dêm;s we;=Ω ldr®h uKav,hg o fuu ldr®hh idr®:l lr .ekSfuys ,d ±lajQ
∞hl;ajh w.h lrñka udf.a lD;{;dj mqolrñ'

									
									
2010'06'22
mr®fhaIK yd ixjr®Ok YdLdj"
Y%S ,xld úNd. fomdr®;fïka;=j'

wkqr t†˙isxy
úNd. flduid˙ia ckrd,a

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

- iii -

WmfoaYl;ajh
(
		

wkqr t†˙isxy uhd
úNd. flduid˙ia ckrd,a

fufyhùu yd ixúOdkh
(
		
		
		
		

.hd;%s wfí.=Kfialr ñh
úNd. flduid˙ia ^mr®fhaIK yd ixjr®Ok&
tÉ'f–'tï'iS'ta' chiqkaor uhd
úNd. flduid˙ia ^mdi,a úNd. ixúOdk&

		
		

î' ik;a mQ‚; uhd
úNd. flduid˙ia ^mdi,a úNd. - we.hSï&

úIhh iïnkaëlrKh yd ixialrKh (
		

f–'ta'f–'wdr®' chfldä uhd
iyldr úNd. flduid˙ia

		
		

ufkdañ fifkúr;ak ñh
iyldr úNd. flduid˙ia

iyh iïnkaëlrKh
(
		
		
		
		

B' l=,fialr uhd
ksfhdacH úNd. flduid˙ia
à' nd,isxyï ñh
iyldr úNd. flduid˙ia

		
		

;urd úodkm;srK ñh
iyldr úNd. flduid˙ia

m˙.Kl msgq ieliqu
(
		

fla'mS'ã' wkqId u≥jka;s †idkdhl ñh
o;a; igyka l%shdlre

				
				
msg ljrh ksr®udKh
(
wixl iór °md,a uhd
		
o;a; igyka l%shdlre

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

- iv -

we;=<; msgq
									

msgq wxlh

										

˚ fldgi
		

úNd.h yd iïnkaO fmd≥ f;dr;=re ---------------------------

1		

˚˚ fldgi
		

m%Yak m;% jHQyh yd iajNdjh ----------------------------------- 6		

˚˚˚ fldgi
		
uQ,dlD;s m%Yak ----------------------------------------------------- 7
		
^01& fN!;sl úoHdj ---------------------------------------------- 9
		
^02& ridhk úoHdj --------------------------------------------- 30
		
^07& .◊;h ------------------------------------------------------ 46
		
^08& lDIs úoHdj ------------------------------------------------- 92
		
^09& „j úoHdj -------------------------------------------------- 109
		
^10& ixhqla; .◊;h ------------------------------------------- 122
		
^11& Wiia .◊;h ----------------------------------------------- 169
		
weuqKqu 01 ------------------------------------------------------- 184
		

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

-v-

˚ fldgi
wOHhk fmd≥ iy;sl m;% ^Wiia fm<& úNd.h
úNd.h yd iïnkaO fmd≥ f;dr;=re
1'

ye¢kaùu
w'fmd'i'^Wiia fm<& úNd.h Y%S ,xldfõ fcHIaG oaú;Shsl wOHdmkfha wjidk iy;sllrK
úNd.hhs' fuh m%Odk jYfhka iy;sllrK úNd.hla jYfhka meje;ajj
q o úYajúoHd,j,g" fjk;a
Wiia wOHdmk wdh;kj,g iy wOHdmk úoHdmSGj,g iq≥iaika f;dard.ekSu;a fuu úNd.fha
m%;sM, u; mokïj is≥ lrk neúka fuh f;a¯fï úNd.hla jYfhka o Ndú; fõ'
tfiau uOHu uÜgfï /lshd ,nd.ekSu i|yd o fuu úNd.fha m%;M
s , uQ,l
s iq≥iqlula jYfhka
ie,fla' w'fmd'i'^Wiia fm<& úNd.h 12 iy 13 fY%a◊j, úIh ksr®foaY mokï fldg f.k „j úoHd"
fN!;Sh úoHd" jd◊c yd l,d hk úIh Odrd y;r hgf;a isxy," fou< iy bx.%Sis hk udOH
;=fkkau meje;afõ'

2'

úNd.h i|yd b,aÆï ls¯u
2'1

mdi,a wh≥ïlrefjda

úNd.hg wh≥ïlrk wjia:dj jk úg Wiia fm< mka;s meje;ajQ ld,fhka 80]l meñ”u
iïmQr®K ù ;sîu w;HjYH fõ'
úNd.hg wh≥ï ls¯u i|yd iq≥iqlï iïmQr®K l< mdi,a wh≥ïlrejkaf.a kï" b,aÆï
lrkq ,nk úIh yd udOHh hk f;dr;=re o iu. ú≥y,am;s úiska wh≥ïm;a b†˙m;a l< hq;= h'
tys° wh≥ïlrejkaf.a ku Wmamekak iy;slfha ;sfnk wdldrhg ksjer†j iïmQr®K ls¯u;a úIhh"
úIh wxl yd udOHh ksjer†j igyka ls¯u;a ms<sn| j ±ä wjOdkhla fhduq l< hq;=h'

2'2

fm!oa.,sl wh≥ïlrefjda

mdi,a wh≥ïlrejl= jYfhka fmkS fkdisg m<uqjk j;djg fm!oa.,sl wh≥ïlrejl=
jYfhka fmkS isák yd l,sk
s a fmkS isá tla úIhhla fyda fjkia lr úNd.hg fmkS isàug wfmaCId
lrk wh≥ïlrejka mdi, mokï lr.;a we.hSï jevigykg úl,amhla jYfhka l%shd;aul jk
we.hSï yd jHdmD;s bgq ls¯u i|yd ;ud wh;a m<d;a wOHdmk fomdr®;fïka;=fõ ,shdm†xÑ ù ;sìh
hq;= h' ,shdm†xÑfha ° ;ud fj; ,nd fok mdi,g iïnkaO ù we.hSï yd jHdmD;s iïmQr®K lr tfia
l< njg ú≥y,am;s úiska fokq ,nk iy;slfhys tla msgm;la m<d;a wOHdmk fomdr®;fïka;=jg o
;j;a msgm;la Y%S ,xld úNd. fomdr®;fïka;=jg o túh hq;= h' jhi wjq'21 iïmQr®K jQ wh≥ïlrejka
tu wjYH;dfjka ksoyia fõ' jhi wjq'21g wvq tfia ,shdm†xÑ ù fkdue;s wh≥ïlrejka úNd.hg
fmkS isàu i|yd kqiq≥iaika f,i ie,fla'
úNd. fomdr®;fïka;j
= úiska m<lrkq ,nk mqj;am;a oekaùula u.ska fm!oa.,sl wh≥ïlrejka
i|yd wh≥ïm;a le|jk wjia:dfõ ° tfia b†˙m;a ùug n,dfmdfrd;a;= jk wh≥ïlrejka ksis m˙†
iïmQr®K l< wh≥ïm;%hla kshñ; úNd. .dia;= f.jd ,nd.;a ˙isÜ m;%h o iu. Y%S ,xld úNd.
fomdr®;fïka;=j fj; mqj;am;a ±kaùfuys i|yka m˙† kshñ; †kg fmr ,shdm†xÑ ;emEf,ka
túh hq;= h'

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

-1-

3'

úIh f;dard .ekSu
wOHdmk wud;HdxYh úiska ksl=;a fldg we;s wxl 2009$16 yd 2009'05'18 †ke;s
zw'fmd'i'^Wiia fm<& úIh ixfhdack yd úYaj úoHd, m%fõYh i|yd jQ úIh ixfhdackZ hk
pl%f,aLh ^weuqKqu 1& wkqj 2011 jifr® isg fuu úNd.h meje;afõ'
w'fmd'i'^Wiia fm<& i|yd kj úIh ksr®foaY 2009 jifr® 12 jk fY%a◊h i|yd y∫kajd ≥ka
w;r" tu úIh ksrf
® oaY mokï fldg f.k m%:u j;djg w'fmd'i'^Wiia fm<& úNd.h meje;afjkqfha
2011 jifr®° h' by; i|yka 2009$16 pl%f,aLfha wvx.= úêúOdk wkqj „j úoHd" fN!;Sh úoHd"
jd◊c yd l,d hk úIh Odrd i|yd w∞< úIhhka f;dard .; hq;= h'
w'fmd'i'^Wiia fm<& úNd.h i|yd m%Odk úIhhka ;=klg fmkS isáh hq;= w;r Bg wu;rj
úYajúoHd, m%fõYh i|yd wh≥ï ls¯ug n,dfmdfrd;a;= jk isiqka zidudkH fmd≥ m¯CIKhZ m%Yak
m;%fhka o iu;a úh hq;= h' úYajúoHd, m%fõYh i|yd i,ld fkdneÆj o zidudkH bx.%i
S Zs m%Yk
a m;%hg o
isiqkaf.a wNsu;h m˙† fmkS isáh yelsh'

3'1

w'fmd'i'^Wiia fm<& úNd.h i|yd wkqu; úIh

ishÆ u úIh Odrd i|yd YsIHhka úiska f;dard.; hq;= úIh ixfhdack ms<sn| fldkafoais
weuqKqu ˚ys we;=<;a wxl 2009$16 pl%f,aLfhys i|yka fõ'
w'fmd'i'^Wiia fm<& úNd.h i|yd wkqu; úIhhka iy úIh wxl my; ±lafõ' úNd.h
i|yd b,aÆï ls¯fï ° w∞< úIh wxl Ndú; l< hq;= h'

úIhh

úIh wxlh

fN!;sl úoHdj
ridhk úoHdj
.◊;h		
lDIs úoHdj
„j úoHdj
ixhqla; .◊;h
Wiia .◊;h
isú,a ;dCIKfõoh
hdka;%sl ;dCIKfõoh
ú≥,sh" bf,lafg%dksl iy f;dr;=re ;dCIKfõoh
wdydr ;dCIKfõoh
lDIs ;dCIKfõoh
ffcj iïm;a ;dCIKfõoh
f;dr;=re yd ikaksfõok ;dCIKh
wdr®:ßl úoHdj
N+f.da, úoHdj
foaYmd,k úoHdj
;r®l Ydia;%h yd úoHd;aul l%uh
Y%S ,xld b;sydih
bka†hd b;sydih
hqfrdamd b;sydih
kQ;k f,dal b;sydih
.Dy wdr®:ßl úoHdj
ikaksfõokh yd udOH wOHhkh
jHdmdr ixLHdkh
jHdmdr wOHhkh
.sKqïlrKh
nqoaO Or®uh
yska≥ Or®uh

01
02
07
08
09
10
11
14
15
16
17
18
19
20
21
22
23
24
25
25A
25B
25C
28
29
31
32
33
41
42

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

-2-

l%sia;= Or®uh
bia,dï		
fn!oaO YsIagdpdrh
yska≥ YsIagdpdrh
bia,dï YsIagdpdrh
.%Sl yd frdau YsIagdpdrh
l%sia;shdks YsIagdpdrh
Ñ;% l,dj
kr®;kh ^foaYSh&
kr®;kh ^Nr;&
ix.S;h ^fmr†.&
ix.S;h ^lr®Kdgl&
ix.S;h ^ngysr&
kdgH yd rx. l,dj ^isxy,&
kdgH yd rx. l,dj ^fou<&
kdgH yd rx. l,dj ^bx.%Sis&
isxy,		
fou<		
bx.%Sis 		
md,s		
ixialD;
wrdì		
uef,a		
m%xY		
cr®uka		
reishka
yska°		
Ök		
cmka		

43
44
45
46
47
48
49
51
52
53
54
55
56
57
58
59
71
72
73
74
75
78
79
81
82
83
84
86
87

w'fmd'i'^Wiia fm<& úNd.h i|yd Wla; úIhhka w;=frka wOHdmk wud;HdxYh uÛska
ksl=;a lr we;s 2009$16 pl%f,aLhg wkqj m%Odk úIhhka ;=kla f;dard .; yelsh' tu m%Odk
úIhhkag wu;rj fuu úNd.fha ° my; úIhhka fol i|yd o b†˙m;a úh hq;= h'
* idudkH fmd≥ m¯CIKh ^12&
Y%S ,xldfõ úYajúoHd,hlg wNHka;r YsIHhl= jYfhka we;=<;a ùu i|yd fuu úIhh
i|yd ;SrKh lrkq ,nk wju ,l=Kq uÜgula ,nd ;sîu w;HjYH fõ' tla wjia:djl ° fuu
iq≥iqlu ,nd ;sîu miq wjia:djl úYajúoHd, m%fõYh i|yd w∞< lr .; yelsh' fuu úIhh
i|yd ,nd .kakd ,l=Kq" Z ,l=K .Kkh ls¯u i|yd w∞< lr .kq fkd,efí'
* idudkH bx.%Sis ^13&
fuu úIhh w'fmd'i'^Wiia fm<& m%Odk úIhhla fkdfõ' idudkH bx.%Sis i|yd ,nd .kakd
,l=Kq fyda idur®:Hh fyda úYajúoHd, m%fõYh i|yd Wmfhda.s lr .kq fkd,efí' tfy;a fuu
úIhh i|yd ,nd .kakd m%;sM,h w'fmd'i'^Wiia fm<& úNd. iy;slfha fjku igyka flfr®'

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

-3-

4'

fY%a◊ ;SrKh ls¯u

		
tla tla úIhh i|yd ,nd.kakd o< ,l=Kq wkqj my; i|yka wdldrhg fY%a◊ ;SrKh flfr®'

,l=Kq mrdih
75 - 100
65 - 74
50 - 64
35 - 49
00 - 34
5'

fY%a◊h
A

B C S F -

úYsIag idur®:Hh
wêiïudk idur®:Hh
iïudk idur®:Hh
idudkH idur®:Hh
wiu;a

-

Distinction Pass
Very Good Pass
Credit Pass
Ordinary Pass
Fail

mdi, mokï lr.;a we.hSu
5'1

wruqK

12 yd 13 fY%a◊j, bf.kqu ,nk isiqkaf.a w'fmd'i'^Wiia fm<& úNd.fha ° ueksh fkdyels
yelshd" ksmqK;d wd†h bf.kqï b.ekaùï l%shdj,sh is≥flfrk w;r;=r uek ne,Su;a ≥n,;d yd
m%n,;d y∫kdf.k ≥n,;d we;s isiqka isá;a kï ta i|yd m%;sfmdaIK jevigyka l%shd;aul ls¯u;a
fuys wruqK fõ' fuu jevms<sfj< hgf;a mka;s ldurfha ° bf.kqu ,nk úIhhka i|yd we.hSï
is≥ lrkq ,nk w;r isiqka úiska is≥lrkq ,nk jHdmD;s folla o we.hSug Ndckh flfr®'

5'2

we.hSï is≥flfrk wdldrh

5'2'1

mka;s ldurfha bf.kSug Ndckh flfrk úIhhka i|yd lrkq ,nk we.hSu
^w&
mka;s ldurfha bf.kqu ,nk iEu úIhhla i|ydu jdrhlg we.hSï wjia:d ;=kla
u.ska we.hSug Ndckh flfr®' ^fï i|yd y∫kajd ° we;s we.hSï m%fNao Ndú; l<
hq;=h'&
^wd&
mka;s ldurfha bf.kqu ,nk m%dfhda.sl l%shdldrlï iys; úIhhka ^fN!;sl
úoHdj" ridhk úoHdj" „j úoHdj" lDIs úoHdj" .Dy wdr®:ßl úoHdj" ix.S;h iy
kegqï jeks úIhhka& i|yd wju jYfhka jdrhlg tla m%dfhda.sl we.hSulaj;a
is≥ l< hq;= w;r fiiq we.hSï wjia:d fol i|yd fjk;a we.hSï m%fNao fhd∞
.; yelsh'
^we&
jdrhlg we.hSï wjia:d ;=k ne.ska 12 fY%a◊fha jdr ;=k i|yd we.hSï wjia:d
09la o" 13 fY%a◊fha 1 yd 2 jdrh i|yd we.hSï wjia:d 06la o jYfhka wjqre≥ fol
i|yd we.hSï 15la is≥ flfr®'
^wE&
fuu we.hSï wjia:d 15ys ,l=Kqj, idudkHh" 13 jk fY%a◊fha fojk jdrh
wjidkfha ° Y%S ,xld úNd. fomdr®;fïka;=j fj; f.kajd .kq ,efí' fuu ,l=Kq
wkqj ;SrKh lrkq ,nk ksmqK;d uÜgï w'fmd'i'^W'fm<& m%;sM, f,aLkfha
fjku ;Srhl my; i|yka wdldrhg we;=<;a flfr®'

		

mdi, mokï
lr.;a
we.hSï ,l=Kq

ksmqK;d uÜgï

9" 10
8
6" 7
4" 5
1" 2" 3

úYsIag uÜgfï ksmqK;dj
- (Excellent Level Competency)
by< uÜgfï ksmqK;dj
- (High Level Competency)
iïudk uÜgfï ksmqK;dj - (Credit Level Competency)
wdikak ksmqK;dj
- (Near Competency)
ksmqK;d uÜgug <Ûd fkdjQ - (Not reached to Competency Level)

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

-4-

5'2'2

jHdmD;s u.ska we.hSu

		
w'fmd'i'^W'fm<& y∞rk iEu isiqjl=u jHdmD;s folla ksu l< hq;= h' bka tlla
flaj, jHdmD;shla jk w;r wfkl lKavdhï jHdmD;shla úh hq;= h'
5'2'2'1 flaj, jHdmD;sh
^w&

flaj, jHdmD;sh ilia ls¯u i|yd .e<fmk ud;Dldjla f;dardf.k th
mdif,a jHdmD;s lñgqj u.ska wkqu; lr.; hq;= h'

^wd&

12 fY%a◊fha 1 yd 2 jk jdr ;=< ° fuu jHdmD;sh ksu l< hq;= w;r
mka;sNdr fyda úIhNdr fyda .=rejrhl= fï i|yd u. fmkajkq we;'

^we&

jHdmD;sh iïnkaOfhka t†fk∞ flfrk lghq;= fCIa;% fmd;l igyka
fldg th jHdmD;sh Ndr .=rejrhd ,jd iy;sl lrjd .; hq;= h'

^wE&

fuu jHdmD;sh ls¯u ;=<ska isiqjdg iyc oCI;d m%or®Ykh ls¯ug yelsùu"
Rcqj úúO wdh;k iu. yd iudch iu. iïnkaO;d mj;ajd .ekSu
iy úúO mqoa.,hka iu. lghq;= ls¯u ms<sn| j wjfndaOh ,nd .ekSug
wjia:dj ,nd°u wfmaCId flfr®'

5'2'2'2 lKavdhï jHdmD;sh
^w&
lKavdhï jHdmD;sh i|yd mdif,a isiqka 10 fokl=f.ka hqla; lKavdhula
mdif,a jHdmD;s lñgqj u.ska kï l< hq;= h'
^wd&

fuu lKavdhu mdif,a úoHd" jdKsc yd l,d hk wxY ;=k ksfhdackh
jk m˙† ilia úh hq;= h'

^we&

lKavdhï jHdmD;sh i|yd o lKavdhu uÛska iq≥iq ud;Dldjla jHdmD;s
lñgqjg b†˙m;a fldg wkque;sh ,nd .; hq;= h'

^wE&

12 fY%◊
a h 3 jk jdrh iy 13 fY%◊
a h 1 jk jdrh we;=<; ° fuu jHdmD;sfha
lghq;= ksu l< hq;= h'

^b&

lKavdhï yeÛSfuka lghq;= ls¯u" úúO wdh;k yd mqoa.,hka iu. lghq;=
ls¯u iy fCIa;%fha lghq;= ms<sn| j wjfndaOhla ,nd .ekSug wjia:dj
,nd°u fuu jHdmD;sh uÛska n,dfmdfrd;a;= fõ'

5'2'3
jHdmD;s we.hSu
			
ksr®Kdhl 05la hgf;a jHdmD;s we.hSug Ndckh flfrk w;r jHdmD;s Ndr
.=reNj;d$ lñgqj úiska tla tla ksrK
® dhlh wkqj isik
q a lghq;= lr we;s wdldrh wOHhkh
fldg ,l=Kq ,nd°u is≥ lrkq we;'
			
jHdmD;shla i|yd uqΩ ,l=Kq 20la ,nd fok w;r th ,l=Kq 10g m˙jr®;kh fldg
úNd. fomdr®;fïka;=jg f.kajd .kq ,efí'
			
fuu jHdmD;s foflys wjidk idudkH ,l=Kq wkqj ;SrKh lrkq ,nk ksmqK;d
uÜgu o" 5'2'1 ^wE& ys i|yka wdlD;shg wkqj w'fmd'i'^Wiia fm<& m%;sM, f,aLkfha we;=<;a
flfr®'

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

-5-

˚˚ fldgi
w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk
úNd. i|yd m%Yak m;% jHQyh

m%Yak ixLHdj

ms<s;=re iemhsh hq;= m%Yak ixLHdj

* m%Yak iajNdjh

m%Yak ixLHdj

ms<s;=re iemhsh hq;= m%Yak ixLHdj

50

50

3

3

4

4

4

6

4/6

02' ridhk úoHdj

2

1

5

50

50

3

3

4

4

4

3

2/3

07' .◊;h

3

3, 4

10

10

3

4

10

10

4

7

5/7

08' lDIs úoHdj

2

1

5

50

50

3

3

4

4

4

6

4/6

09' „j úoHdj

2

1

5

50

50

3

3

4

4

4

6

4/6

10' ixhqla; .◊;h

3

3, 4

10

10

4

7

5/7

3

4

10

10

4

7

5/7

11' Wiia .◊;h

3

3, 4

10

10

4

7

5/7

3

4

10

10

4

7

5/7

7

5/7

ms<s;=re iemhsh hq;= m%Yak ixLHdj

* m%Yak iajNdjh

5

C fldgi

m%Yak ixLHdj

ld,h ^meh&

1

4

ms<s;=re iemhsh hq;= m%Yak ixLHdj

2

m%Yak ixLHdj

ms<s;=re iemhsh hq;= m%Yak ixLHdj

01' fN!;sl úoHdj

úIh yd
úIh wxlh

* m%Yak iajNdjh

m%Yak ixLHdj

B fldgi

jrK ixLHdj

A fldgi

* m%Yak iajNdjh

B fldgi

ld,h ^meh&

A fldgi

˚˚ m;%h

* m%Yak iajNdjh

˚ m;%h

4

3

2/3

* m%Yak iajNdjh
1' nyqjrK 				
2' fjk;a jdia;úl			
3' jHQy.;				
4' wr®O jHQy.;
5' rpkd
6' m%dfhda.sl

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

-6-

˚˚˚ fldgi
uQ,dlD;s m%Yak
w'fmdi'^Wiia fm<& úNd.h oaú;Shsl wOHdmkh wjidkfha meje;afjk wjidk iy;sllrK úNd.
hhs' fuu úNd.fha m%;sM, u; úYajúoHd," fjk;a Wiia wOHdmk wdh;k iy wOHdmk úoHdmSG jeks
wdh;kj,g isiqka f;dard .ekSu o is≥ flfrk neúka fuys f;a¯fï m¯CIKhl ,CIK o wvx.= fõ'
ta wkqj w'fmd'i'^Wiia fm<& úNd.fha m%Yak m;% ilia ls¯fï ° idOk m¯CIKhl ;sìh hq;=
,CIK flfrys jeä wjOdkhla fhduq l< hq;= w;r" fuys m%;sM, u; úYajúoHd, yd fjk;a Wiia wOHdmk
wdh;k i|yd isiqka f;a¯u o is≥ lrk neúka ta ms<sn|j o wjOdkh fhduq lr we;'
ta wkqj YsIH idOkh we.hSu i|yd ,sÅ; úNd. l%ufha ° m%Odk jYfhkau rpkd m¯CIK iy
jdia;úl ^úIhnoaO& m¯CIK Ndú; flfr®' rpkd iajrEmfha ms<;
s r
= l ° m¯CIlhdf.a mqo.
a , noaO;dj n,mdk
w;r jdia;úl iajNdjfha m%Yakhla i|yd tla ksjer† ms<s;=rla muKla ;sîu ksid th iïmQr®Kfhkau
mqoa., noaO;dfjka f;dr fõ' fuu rpkd iy jdia;úl hk wka; fol w;rg .efkk §jHQy.; flá
ms<s;=re iemhSfï jr®.fha m%Yak¶ flfrys jr®;udk m¯CIK fCIa;%fha jeä wjOdkhla fhduq ù ;sfí'
jHQy.; m%Yakhla uÛska" †h hq;= ms<s;=r m%Yakfhka lshefjk hï hï iSud ;=< ±ä fia md,kh flfr®' ta
wkqj ms<s;=rem;a ,l=Kq ls¯fï ° m¯CIljreka w;fr® ,l=Kq °fï ix.;;dj wdrCId lr .; yelsjk ksid"
jdia;úl yd jHQy.; iajrEmfha m%Yakj,g m¯CIK fCIa;%fha jeo.;a ia:dkhla ysñ fõ'
ta wkqj w'fmd'i'^Wiia fm<& úNd.fha m%Yk
a m;% ilia ls¯fï ° o w;HjYH wjia:dj, ° muKla rpkd
m%Yak Ndú; flfrk w;r jdia;úl jr®.fha m%Yak ilia ls¯u flfrys jeä wjOdkhla fhduq flfr®' tfiau
w'fmd'i'^Wiia fm<& úNd.fha m%Yk
a m;% ilia ls¯fï ° yqfola iaurKh u; muKla fkd/|S wjfndaOh" Ndú;h"
úYaf,aIKh" ixYaf,aIKh yd we.hSu jeks by< udkisl yelshd uek nef,k wdldrfha m%Yak ilia ls¯u
flfrys wjOdkh fhduq ù we;' yels;dla ≥rg m%dfhda.l
s isoê
a j,g w∞< j m%Yk
a ilia flfrk w;r isik
q f
a .a
hula meye†,sj f;areï.ekSu" W.;a uQ,Or®u fjk;a iudk wjia:djl ° fhd∞.ekSu" .egÆ úi£u" ;r®lkh"
kj fhdackd$ie,iqï b†˙m;a ls¯u" lreKq ikaikaokh ls¯u" NdIdj uekúka Ndú;d ls¯u iy woyia
meye†,sj b†˙m;a ls¯u wdoS yelshd uek nef,k wdldrhg;a tu m%Yak ilia flfrkq we;'
fuu ˚˚˚ jk fldgfiys" m%Yk
a m;% jHQyh yd ,l=Kq ms˙kefuk wdldrh iu. uQ,dlD;s m%Yk
a we;=<;a
lr we;s w;r tajd wdor®Y m%Yk
a m;% fkdfõ' tneúka m%Yk
a m;% ilia ls¯fï ° m%Yk
a j, wka;r®.; wkqfldgia
ixLHdj iy ,l=Kq nr ;efnk wdldrh m%Yak ilia ls¯u i|yd mokï lr .efkk úIh fldgia wkqj
wjia:dkql+, j fjkia úh yels h'
fuys i|yka uQ,dlD;s m%Yakj,g wu;rj jdia;úl yd rpkd m¯CIK i|yd Ndú; flfrk"
8jk msgqfjys i|yka ishÆ u m%Yak jr®. we;=Ω fjk;a m%Yak jr®. o w'fmd'i'^Wiia fm<& úNd.fha
m%Yak m;%j,g wjia:dkql+,j we;=<;a ls¯fï whs;sh úNd. flduid˙ia ckrd,a i;=h'

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

-7-

jdia;úl$úIh noaO m¯CIK (Objective Tests)
iemhSfï jr®.h
(Supply Type)

f;a¯fï jr®.h
(Selection Type)
wE£fï jr®.h
(Matching Type)

fojrK jr®.h
(Alternative Type)

nyqjrK jr®.h
(MCQ Type)

m%Yak jr®.h - (Question Type)
wiïmQr®K m%ldYk jr®.h - (In Complete Statement Type)
m%;sfIaOd;aul jr®.h - (Negative Type)
iïnkaëjrK jr®.h - (Combined Response Type)
fyd|u ms<s;=re jr®.h - (Best Answer Type)
nyq ms<s;=re jr®.h - (Multiple Response Type)
fmd≥ ms<s;=re jr®.h - (Common Response Type)
wdfoaY jr®.h - (Substitution Type)
hq., j.ka;s jr®.h - (Double Statement Type)
m%ldYh yd fya;=j jr®.h - (Assertion & Reason Type)

		

rpkd m¯CIK (Essay Tests)
rpkd jr®.fha m%Yak
(Essay Type Questions)
újD; m%;spdr
jr®.h
(Open Response Type)

jHQy.; rpkd m%Yak
(Structured Essay Type Questions)

wejqrE m%;spdr
jr®.h
(Controlled Response
Type)

wdlD;sh
u.ska
jHQy.;
ls¯u
(Structuring
by using
subsections)

iemfhk
f;dr;=re u.ska
jHQy.;
ls¯u
(Structuring
by information
provided)

ldr®hh
u.ska
jHQy.;
ls¯u
(Structuring
by function)

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

-8-

01 - fN!;sl úoHdj
m%Yak m;% jHQyh
˚ m;%h

- ld,h ( meh 02 hs'
jrK 5 ne.ska jQ nyqjrK m%Yak 50 ls' m%Yak ish,a,g u ms<s;=re iemhsh hq;= h'
tla m%Yakhlg ,l=Kq 02 ne.ska uqΩ ,l=Kq 100 ls'

˚˚ m;%h - ld,h ( meh 03 hs'
fuu m%Yak m;%h jHQy.; rpkd yd rpkd jYfhka fldgia follska iukaú; h'
A fldgi - jHQy.; rpkd jr®.fha m%Yak y;rls' m%Yak ish,a,g u ms<s;=re
iemhsh hq;= h' tla m%Yakhlg ,l=Kq 10 ne.ska ,l=Kq 40 ls'
B fldgi - rpkd jr®.fha m%Yak yhls' m%Yak y;rlg ms<s;=re iemhsh hq;= h'
tla m%Yakhlg ,l=Kq 15 ne.ska ,l=Kq 60 ls'
˚˚ m;%h i|yd uqΩ ,l=Kq } 100
wjidk ,l=K .Kkh ls¯u

(

˚ m;%h
} 100
˚˚ m;%h
} 100
wjidk ,l=K } 200 fi 2 } 100

˚ m;%h
ie,lsh hq;=hs (
* ishÆ u m%Yak i|yd ms<s;=re imhkak'
* ksjer† fyda jvd;a .e<fmk fyda ms<s;=r f;darkak' ^úNd.fha ° ms<s;=re iemhSu i|yd nyqjrK
lv∞ishla imhkq ,efí'&
(g = 10 N kg−1)
1.

ma,dkala ksh;fhys udkj,g iudk udk we;s rdYsh jkafka"
^1& Yla;sh hs'
^2& laIu;dj hs'
^4& jHjr®;h hs'
^5& fldaKsl .uH;dj hs'

2.

m˙mQr®K jdhqjl ksrfmalaI WIaK;ajh fo.=K l< úg tys mrudKqj, jr®. uOHkH uQ, fõ.h
jeäjk idOlh jkafka"
^1& √2
^2& √3
^3& 2
^4& 3
^5& 4

3.

fvdma,r® wdprKh ksid èjks ;rx.hl ixLHd;h fjkia ùu r|d mj;skqfha"
^A) ks¯laIlhdf.a isg m%Njhg we;s ≥r u; h'
^B) jd;h ;=< èjks m%fõ.h u; h'
^C) m%Njh iy ks¯laIlhd w;r idfmalaI m%fõ.h u; h'
by; i|yka m%ldYj,ska i;H jkafka"
^1& (A) mu◊'
^2& (B) mu◊'
^3& (C) mu◊'
^4& (B) iy (C) mu◊'
^5& (A), (B) iy (C) hk ish,a, u h'

4.

0 °C ys we;s ialkaOh 1'5 kg jk whsia l=Üáhla 50 W l wkjr; YS>%;djlska r;a lrkq ,efí' whsiaj,
ú,hkfha úYsIag .=ma; ;dmh 3'0 Ÿ 105 J kg−1 kï" whsia iïmQr®Kfhka u †h ùug .;jk ld,h meh
^1& 150'00 ls'
^2& 100'00 ls'
^3& 3'75 ls'
^4& 3'50 ls'
^5& 2'50 ls'

5.

f,airhla u.ska ksmofjk wdf,dalfha ,dlaIKsl ms<sn| j lrk ,o my; m%ldY i,ld n,kak'
(A) th úoHq;a pqïnl fjhs'
(B) th iupd¯ fjhs'
(C) th taljr®K fjhs'
by; m%ldYj,ska"
^1& (A) muKla i;H fõ'
^2& (B) muKla i;H fõ'
^3& (A) iy (B) muKla i;H fõ'
^4& (B) iy (C) muKla i;H fõ'
^5& (A), (B) iy (C) hk ish,a, u i;H fõ'

^3& fldaKsl ixLHd;h hs'

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

-9-

6.

kefkda ;dlaIKh iuÛ iïnkaê; †. m˙udK m%dka;rh ógr"
^1& 0'1 − 10 fõ'
^2& 10 − 1000 fõ'
−9
−7
^4& 10 − 10 fõ'
^5& 10−15 − 10−13 fõ'

7.

fok ,o jia;=jla u; iïm%hqla; iM, n,hla ls%hd fkdlrhs' tu jia;=j ms<sn|j lrk ,o my;
i|yka m%ldY i,ld n,kak'
(A) th ksYap,j ;sìh yelsh'
(B) th taldldr m%fõ.fhka p,kh fjñka mej;sh yelsh'
(C) th jD;a;dldr m:hla Tiafia p,kh fjñka mej;sh yelsh'
by; i|yka m%ldYj,sk"a
^1& (A) muKla i;H fõ'
^2& (C) muKla i;H fõ'
^3& (A) iy (B) muKla i;H fõ'
^4& (A) iy (C) muKla i;H fõ'
^5& (A), (B) iy (C) ish,a, u i;H fõ'

8.

.s,aÆï ;dmlhla u.ska c,h hï m%udKhla kegùug t1 ld,hla .;fjhs' tu m%udKhg iudk c,
m%udKhla fjk;a .s,aÆï ;dmlhla u.ska kegùug t2 ld,hla .; fjhs' .s,aÆï ;dml folu
Ndú;fhka tu m%udKhg iudk c,h m%udKhla kegùug .;jk ld,h"
^n∫kaj, ;dm Od˙;d iy m˙irhg is≥jk ;dm ydksh fkdi,ld y˙kak'&
t +t
t1 t2
^1& 1 (t1 + t2)
^2& t1 + t2
^3&
^4& √ t1 t2
^5& 1t t 2
 1 2
t1 + t2
2

9.

jr®.M,h 0'15 m2 jk uq≥jl we;s øj mg,hl mDIaGßl Yla;sh jkafka"
^øjfha mDIaGßl wd;;sh 5 N m−1&
^1& 0'75 J
^2& 1'50 J
^3& 2'25 J
^4& 3'00 J

10.

^3& 10−4 − 10−2 fõ'

^5& 6'00 J

fmkajd we;s cd,fha tla tla m%;sfrdaOhl w.h R kï"
A iy B w.% w;r iul m%;sfrdaOh jkafka"
^3& 3R
^2& 2R
^1& R
^5& 5R
^4& 4R

R
R

R
R

A

R
B

11.

f,day oKavla 0 °C isg 100 °C f;la r;a lrkq ,enQ úg tys †. 0'05] lska jeä fjhs' f,dayfha fr®Çh
m%idrK;dj jkafka"
^1& 5 × 10−3 °C−1 ^2& 5 × 10−4 °C−1 ^3& 5 × 10−5 °C−1 ^4& 5 × 10−6 °C−1 ^5& 5 × 10−7 °C−1

12.

wdk; mDIaGhla u; l=Üáhla ia:ß;sl iu;=,s;;dfõ mj;S' my; l=uk rEmh u.ska tys ksjer† ksoyia
jia;= n, igyk ksrEmKh flfrhs o@
R

^1&

^2&

F

R

F

^5&

R

W

13.

F

R

^3&
W

W

^4&

F

R

W

F

W

ixhqla; wkaùlaIhl uqΩ úYd,l n,h 15 ls' tys wjfkf;ys úYd,l n,h 5 la fõ' wjidk
m%;sìïnh úYo oDIaáfha wju ≥˙ka ;efkkafka kï Wmfkf;ys kdNs ≥r jkafka"
25 cm
25 cm
25 cm
25 cm
^5&
^4&
^3&
^2&
25 cm
^1&
3
2
14
10

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

- 10 -

14.

ksYap,;dfjka .uka wrUk jia;=jla m<uqjk 50 m ≥r m%udKh ksh; ;ajrKhlska o B <Û 200 m
taldldr m%fõ.hlska o ;j;a 30 m la ukaokhlska o .uka fldg wjidkfha ksi,;djg m;a fõ'
.; jQ uqΩ ld,h ;;amr 30 ls' jia;=j w;a lr .;a Wm˙u m%fõ.h jkafka"
^1& 12'0 m s−1
^2& 8'7 m s−1
^3& 6'7 m s−1
^4& 6'0 m s−1
^5& 2'7 m s−1

15.

ir, wkqjr®;S p,s;hl fhfok wxY=jl p,s; iólrKh a = − 4x u.ska fokq ,efí' fuys a hkq
;ajrKh jk w;r x úia:dmkhhs' fuu ir, wkqjr®;S p,s;fha ixLHd;hg iudk ixLHd;hlska
p,s;jk ir, wj,ïnhl †. l=ula o@
^1& 3'0 m
^2& 2'5 m
^3& 2'0 m
^4& 1'5 m
^5& 0'4 m

16.

tla tla m%;sfrdaOh 1 Ω ne.ska jk m%;sfrdaO 7 l cd,hla wNHka;r
m%;sfrdaOh fkd.sKsh yels 4 V neg˙hla iu. iïnkaO lr we;s
wdldrh rEmfha ±lafõ' m˙m:fha .,k I Odrdfõ w.h"
^3& 2'0 A
^2& 1'5 A
^1& 0'5 A
^5& 3'5 A
^4& 3'0 A

I
1Ω

4V

1Ω

1Ω

1Ω

1Ω
1Ω

1Ω

17.

.=re;ajc fÈa;%hl X ,laIHfha isg Y ,laIHhla olajd wxY=jla p,kh fjhs' wxY=j u; fjk;a n,
ls%hd fkdlrkafka kï tys .=re;ajdlr®IK úNj Yla;sh fjkiaùu ms<sn|j lr we;s my; l=uk
m%ldYh $ m%ldY i;H fõo@
(A) th X iy Y w;r .uka udr®.h u; rπ mj;S'
(B) th pd,l Yla;s fjkia ùug iudk fõ'
(C) th wxY=fjys ialkaOh u; rπ fkdmj;S'
^3& (A) iy (B) mu◊'
^2& (B) mu◊'
^1& (A) mu◊'
^5& (A), (B) yd (C) ish,a,u h'
^4& (B) iy (C) mu◊'

18.

ksYap,;dfõ isg fy<k ,o .,la ;;amr 8 l ° ìu m;s; fjhs' wjidk ;;amrfha ° ., .uka l<
≥r jkafka"
^5& 70 m
^4& 75 m
^3& 160 m
^2& 245 m
^1& 320 m

19.

rEmfha fmkajd we;s m˙† tal,s; wjldYhl X ,laIHfha + Q wdfrdamKhla ;nd we;' my; i|yka
m%ldYj,ska wi;H jkafka l=ula o?
^1& Y ys iy Z ys úoHq;a fÈa;% ;Sj%;dj, úYd,;aj iudk fõ'
Y
r
^2& Y ys iy Z ys úNj iudk fõ'
+Q
^3& Y ys úNjh Ok fjhs'
X
^4& Y isg Z olajd wdfrdamKhla f.k hdfï° ldr®hhla fkdflfrhs'
r
^5& Z ys ° úoHq;a fÈa;%h ZX Tiafia ls%hd lrhs'
Z

20.

21.

^1& rEmfha fmkajd we;s" jd;h mqrjk ,o iudka;r
;yvq Od˙;%lfha Od˙;dj C fjhs' mdr - úoHq;a ksh;h
k = 2 jk mdr - úoHq;a øjHhlska tys y˙ wvla ^2& rEmfha
fmkajd we;s m˙† msr jQ úg tys Od˙;dj"
3
5
^3& 2 C
^2& 3 C
^1& 3C
4C
^5& C
^4&
3

d

jd;h
(1) rEmh

jd;fha ;nd we;s jr®;kdxlh √2 jk mdroDYH ù≥re ;yvqjla u;g
rEmfha ±lafjk m˙† taljr®K wdf,dal lsrKhla m;kh fjhs' lsrKh
AC mDIaGfhka mQr®K wNHka;r mrdjr®;kh ùug (θ) m;k fldaKfhys
;sìh hq;= wju w.h jkafka"
^3& 45°
^2& 60°
^1& 90°
^5& 0°
^4& 30°

(2) rEmh

C

75°
A

θ

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

- 11 -

d
2
d
2

B

22.

wNHka;r l=yrhla iys;" ialkaOh m jk ,S l=Üáhla ieye,aÆ ;ka;=jla
uÛska c, n∫kl m;=,g iïnkaO lr we;s wdldrh rEmfha ±lafõ' c,fha
iy ,Sj, >k;aj ms<sfj<ska d iy ρ fõ (d > ρ ). ;ka;=fjys wd;;sh 1 mg kï
2
l=yrfhys m˙udj jkafka"
d-ρ m
^1& 0
^2&
^3& m
ρ
d
^4& 3m
2d

^5&

( )
( ρ)
1

3
2d

m

23.

m˙mQr®K jdhqjl ujq, 5la ia:ßr;dmS ixfldapkhl ° jdhqj u; 250 J l ldr®hhla is≥ lrhs' tys
we;sjk wNHka;r Yla;s fjki jkafka"
^1& 50 J
^2& −150 J
^3& 250 J
^4& −250 J
^5& 750 J

24.

≥r (x) iuÛ úoHq;a úNj úp,kh (V) m%ia;drfhys olajd we;' x g wkqj
wkqrEm úoHq;a fÈa;% ;Sj%;dj (E) ys úp,kh jvd;a fyd¢ka ksrEmKh
jkafka"

V

0

A B C

^1&
25.

x

E

E

E

E

A

B C

^2&

x

A B

^3&

C

x

A

B

C

x

E

A B

C

x
A B C

^4&

rEmfha fmkajd we;s m˙† ;sria ;,hl ;nd we;s l=Üáhla u; 90 N l
n,hla ls%hd lrhs' l=Üáh iy ;,h w;r ia:ß;sl yd .;sl >r®IK ix.=Kl
ms<sfj<ska 0'4 iy 0'3 ls' l=Üáh u; ;,h u.ska we;s lrkq ,nk >r®IK
n,fha úYd,;ajh jkafka"
^1& 0
^2& 30 N
^3& 40 N
^4& 45 N
^5& 78 N

x

^5&
90 N
60°

10 kg

26.

jejl c, mDIaGfha isg 0'4 m my<ska isák udΩjl= foi" kdNsh ≥r 3 m jk W;a;, ldphla ;=<ska
n,kq ,efí' c, mDIaGhg 0'2 m la by<ska ldph ;nd we;af;a ldpfha m%Odk wÈh u; udΩjd isák
wdldrhg h' ks¯laIlhdg udΩjd ±lsh yels jkafka" ^c,fha jr®;kdxlh 4$3 fõ'&
^1& w;d;a;aúlj iy ldphg 0'6 m my<sk.s
^2& ;d;a;aúlj iy ldphg 0'6 m my<sks'
^3& w;d;a;aúlj iy ldphg 0'6 m by<sk.s
^4& ;d;a;aúlj iy ldphg 0'43 m my<sk.s
^5& w;d;a;aúlj iy ldphg 3 m my<sk.s

27.

ialkaOh ms<sfj<ska 2m iy m ne.ska jk X iy Y wdfrdams; f.da,
folla rEmfha fmkajd we;' iudka;r f,day ;yvq fol w;r úNj
wka;rhla fh°fuka tajd .=re;ajh hgf;a jeàu hka;ñka j<lajd
we;' ;yvq tlsfklg ióm jk fia p,kh l< úg"
^1& X iy Y ;j≥rg;a ksYap,j mj;S'
^2& X iy Y tlu ;ajrKhlska by<g p,kh ùug mgka .kS'
^3& X iy Y tlu ;ajrKhlska my<g p,kh ùug mgka .kS'
^4& Y g jvd jeä ;ajrKhlska X by<g .uka ls¯ug mgka .kS'
^5& Y g jvd jeä ;ajrKhlska X my<g .uka ls¯ug mgka .kS'

X

2m

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

- 12 -

m Y

28.

F ys w.h 1 ùug A, B, C i|yd ;sìh hq;= ksjer† ;dr®lsl w.hka

ms<sfj<ska ljfr® o@
^1& 0" 0" 0
^2& 0" 1" 0
^4& 1" 0" 1
^5& 1" 1" 0

A
B

^3& 1" 0" 0

F

C

29.

250 Hz, 300 Hz, 400 Hz iy 500 Hz ixLHd; ckl Ndú;fhka tla fl<jrla ixjD; iy fofl<jru
újD; jd; l|kaj, wkqkdo †. fiùug wjYH WmlrK Tng imhd we;' jd;h ;=< èjks m%fõ.h
330 m s−1 kï fláu wkqkdo †. jkafka"
^1& 8'25 cm
^2& 16'5 cm
^3& 33 cm
^4& 66 cm
^5& 132 cm

30.

rEmfha ±lafjk wdldrhg tlsfkl iamr®Yj we;s A iy B jia;= folla
iqug wdk; ;,hla u; ksYap,j ;nd ksoyia flfrhs' B u.ska A u;
we;s lrkq ,nk n,h"
^1& 0
^2& 15 N
^3& 35 N
^4& 50 N
^5& 100 N

31.

(

)

rEmfha fmkajd we;s m˙† >k;ajh ρ jk øjhla U k<fhys wvx.=
fõ' B ys wrh A ys wrh fuka fo.=Khls' A ys wrh r fõ' A ndyqfõ
øj uÜgu h Wilska by< kexjQfha kï" øjfha .=re;ajdlr®IK úNj
Yla;sfha jeäùu"
3 h2 π r2ρ g
^2&
^3& 5 h2 π r2ρ g
^1& 0
8
^4&

33.

3 kg

A

B

30°

>k;ajh ρ jk .a,si¯ka mqrjd we;s Wi n∫kla ;=<g" >k;ajh d jk øjHhlska ;kd we;s ialkaOh
m jk l=vd f.da,hla uq∞ y˙kq ,efí' f.da,h wdka; m%fõ.h ,nd .kakd úg f.da,h u; l%shd lrk
≥iai%dú;d n,h"
d
mgd
mgρ
ρ
^1&
^2&
^3& mg 1 ^4& mg 1 +
^5& mg
d
ρ
ρ
d

(

32.

7 kg

5 h2 π r2ρ g
8

^5&

)

A

B

h

5 h2 π r2ρ g
4

we† ;ka;=jl †." ialkaOh" wd;;sh" >k;ajh iy hx udmdxlh l, M, T, ρ iy E u.ska ms<sfj<ska
ksrEmKh flfrhs'
;Sr®hla ;rx.j, m%fõ.h wkqmd;h fokq ,nkafka
;ka;=j i|yd"
wkajdhdu ;rx.j, m%fõ.h
Tρl
Tρ
Tρl
EM
EM
^1&
^2&
^3&
^4&
^5&
Tρl
EM
Tρ
EM
EM

√

√

√

√

34.

úNjudk m¯ÈKhl ° úNjudk lïìfha 60 cm l †.la yryd fldaIhla iïnkaO l< úg
.e,ajfkdaógrfha W;al%uhla fkd±lafjhs' fldaIh yryd 6 Ω Wm m:hla iïnkaO l< úg lïìfha
50 cm †.l ° ix;=,k ,ÈHhla ,efí' fldaIfha wNHka;r m%;sfrdaOh jkafka"
^1& 0'5 Ω
^2& 0'6 Ω
^3& 1'2 Ω
^4& 1'4 Ω
^5& 1'5 Ω

35.

c, lrduhlska isriaj my<g 1'0 m s−1 l wdrïNl fõ.hlska c,h .,hs' lrdufha yrialv jr®.M,h
10−4 m2 ls' c, m%jdyh Tiafia mSvkh ksh; hehs o .e,Su wkjr; hehs o i,lkak' lrduhg 0'15 m
my<ska m%jdyfha yrialv jr®.M,h jkafka"
^1& 5 × 10−5 m2
^2& 2'5 × 10−5 m2 ^3& 5'8 × 10−5 m2 ^4& 6'7 × 10−4 m2 ^5& 1 × 10−4 m2

36.

ksoyfia N%uKh jk fïihl flakaøfha isg x ≥rlska m ialkaOhla ;nd we;' fïih iy ialkaOh
w;r Wm˙u >r®IK n,h mg fjhs' fïifha flda◊l m%fõ.h taldldrj jeä lrkq ,efí' ialkaOh
,siaiSug mgka .kakd úg fïifha fldaKsl m%fõ.h"
g
mg
mgx
g
g
^1& 2x
^2&
^3&  2x
^4&  2
^5&  3m x
x

�

�

�

�

�

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

- 13 -

37.

wj;, ldphl kdNsh ≥r 25 cm o W;a;, ldphl kdNsh ≥r 30 cm o fõ' tajd tlsfkl iamr®Yj ;nd
ixhqla; ldphla ilia lr we;' fuu ixhqla; ldpfha kdNsh ≥rg iudk kdNsh ≥r iys; ldphla
Ndú;fhka my; i|yka l=uk wlaIs f∞aIh ksjer† lr.; yels o@
^1& 150 cm g jvd wE; we;s jia;= meye†,sj ±l .; fkdyelsùu
^2& 150 cm g jvd <Û we;s jia;= meye†,sj ±l .; fkdyelsùu
^3& 30 cm g jvd wE; we;s jia;= meye†,sj ±l .; fkdyelsùu
^4& 30 cm g jvd <Û we;s jia;= meye†,sj ±l .; fkdyelsùu
^5& 28 cm g jvd <Û we;s jia;= meye†,sj ±l .; fkdyelsùu

38.

we,aµd Èhùï iy îgd Èhùï u.ska
ixLHdj"
^1& 4
^2& 5

39.

238
92

U"

206
82

Pb njg m;a fjhs' tjeks CIhùul ° uqla; jk we,aµd wxY=

^3& 6

^4& 8

†. ;ka;=jl ;Sr®hla ;rx.hla ol=Kg .uka lrk úg hï
fudfyd;l le,öu ^1& rEmfha olajd we;' l=vd máhla rEmfha
fmkajd we;s m˙† ;ka;=fjys fyd|ska .eg.id we;' tu máfha
úia:dmkh (y)" ld,h (t) iuÛ fjkiajk wdldrh jvd;au
fyd†ka ksrEmKh jkafka"

^1&

y

^2&

y

^1& rEmh

^3&

t

y
t

t

y

^4&

^5& 10

^5&

y

t

t

40.

jd;h ;=< èjks fõ.hg jvd jeä fõ.hlska èjksh ksmojk m%Njhla .uka lrk úg" ksmofjk
;rx. fmruqK yevh"
^1& ;, wdldrfha fõ'
^2& f.da,Sh fõ'
^3& is,skavrdldr fõ'
^4& fla;= wdldrfha fõ'
^5& mrdj,hsl fõ'

41.

pqïnl talO%ejhla ^monopole) mj;S hehs is;uq' rEmfha ±lafjk m˙†
talO%ejh (m) wE; ≥rl isg t = 0 ° p,s;h wdrïN lr" ksh; fõ.hlska
p,kh ù t = t0 jk úg" iqms˙ ikakdhl øjHhlska ;kd we;s uq≥jla
;=<ska .uka lrhs' uq≥j ;=< fm%a˙; Odrdj (I) ksjer†j ±lafjkafka l=uk
m%ia;drfhka o@

m

I

I

I

^1&

^2&
t0

^3&

t

t0

t

I

I

^4&

^5&
t0

t

t0

t

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

- 14 -

t0

t

42.

43.

rEmfha ±lafjk m˙† pqïnl fÈa;%hla ;=< iuñ;sl pqïnls; nurhla ^top)
wyfia mdfjñka mej;Sug ie,iaúh yelsh' moaO;sh ia:dhS iu;=,s;;dfjys
mj;ajd .ekSu i|yd tys iuñ;sl isria wÈh jgd nurfha neóu (spin)
w;HjYH jkafka nurfha flda◊l .uH;djh u.ska"
^1& nurfhys pqïnls; nj m%n, lrk ksid h'
^2& pqïnlfha W;a;r O%ejfha pqïnl fÈa;%fha m%n,;dj jeä lrk ksid h'
^3& úlr®IK n,h m%n, lrk ksid h'
^4& nurh ;=< pqïnl i%dj >k;ajh jeä flfrk ksid h'
^5& lrkï fkd.id nurfha †Ydk;sh wdrlaId lrk ksid h'
fyda,a wdprKh wOHhkfha° rEmfha fmkajd we;s mrs† >klu
d iy wdfrdamK >k;ajh n jk PQRS ;yvqjla" i%dj >k;ajh
B jk pqïnl fÈa;%hl ;sriaj ;nd we;' B ys †Ydj rEmfha
m˙† isriaj my<g ls%hd lrhs' rEmfha fmkajd we;s †Ydjg I
Odrdjla ;yvqj ;=<ska .,hs' jdyl >DK wdfrdams; jk w;r"
jdylj, wdfrdamKh q fõ' fydaa,a fjda,aàh;dfjys †Ydj yd
úYd,;ajh jkafka"
^1& RS me;a; isg PQ me;a; olajd E/B h'
^2& RS me;a; isg PQ me;a; olajd BI/ndq h'
^3& PQ me;a; isg RS me;a; olajd E/B h'
^4& PQ me;a; isg RS me;a; olajd BI/ndq h'
^5& PS me;a; isg QR me;a; olajd BI/ndq h'

S
N

N

m%n,
pqïNlhl
W;a;r O%ejh
R

S
I

Q
B

P

d

I

44.

n - kd,Sh ikaê lafIa;% wdprK g%dkaisiagrhla ^n - channel junction field effect transistor) ms<sn|j
lrk ,o my; i|yka m%ldY i,ld n,kak'
(A) g%dkaisiagrh ieuúgu l%shd;aul jkafka oajdr-m%Nj fjda,aàh;dj (VGS) Ok f,ig h'
(B) VGS wvq l< úg g%dkaisiagrh ;=<ska .,k Odrdj wvqfjhs'
(C) ydhs; m%foaYfha m<, u.ska g%dkaisiagrh ;=<ska .,k Odrdj md,kh fjhs'
by; m%ldY w;=˙ka i;H jkafka"
^1& (A) mu◊'
^2& (B) mu◊'
^3& (A) iy (C) mu◊'
^4& (B) iy (C) mu◊'
^5& (A), (B) iy (C) ish,a, u'

45.

rEmfha olajd we;s S - R ms<sfmd, (flip - flop) i|yd my; ° we;s l=uk i;H;d
j.=j ksjer† o@
^m%∞kh °ug fmr mej;s Q m%;s∞kfha ;dr®lsl uÜgu Qold u.ska ksrEmKh
flfrk w;r" m%∞kh ≥ka miq tys ;dr®lsl uÜgu Qnew u.ska ksrEmKh flfrhs'&
^1& S

^4&

46.

0
0
1
1

R
0
1
0
1

S
0
1
0
1

R
0
0
1
1

Qnew
0
0
0
Qold

Qnew
0
0
1
wúksYaÑ;hs

^2& S
0
0
1
1

^5& S
0
0
1
1

R
0
1
0
1

R
0
1
0
1

Qnew
Qold
0
1

wúksYaÑ;hs

S Q
R Q

^3& S
0
0
1
1

R
0
1
0
1

Qnew
Qold
0
1
Qold

Qnew
Qold
1
0
wúksYaÑ;hs

b,lal úNjh 62'0 kV uÛska X - lsrK k<hla ;=< X - lsrK ksmojkq ,nhs' tla .egqulg miq
bf,lafg%dakhla ;u m;s; Yla;sfhka y˙ wvla ydks lr .kS kï" ksmofjk X - lsrK fmdafgdakfha
wju ;rx. wdhduh jkafka" (hc = 1240 eVnm)
^1& 4 × 10−2 nm
^2& 6 × 10−2 nm
^3& 8 × 10−2 nm
^4& 4 × 10−3 nm
^5& 8 × 10−3 nm
w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

- 15 -

47.

;rx. wdhduh λ jk f,air® loïnhla ;=< ialkaOh m jk mrudKqjla ksh; m%fõ.fhka m%;súreoaO
†Ydjg .uka lrhs' f,air® loïnfhka mrudKqj tla fmdafgdakhla wjfYdaIKh lr.kafka kï" tys
fõ.h fjkiaùu fokq ,nkafka,
h
h
^1& 4 h
^2& 2 h
^3& h
^4&
^5&
4λm
λm
 m
 m
2λm
λ
λ

48.

rEmfha fmfkk m˙† i%dj >k;ajh B jQ taldlr pqïnl fÈa;%hla wrh
a jQ is,skavrdldr m˙udjla ;=< talrdYsù mj;S' R ksh; YS>%;djlska
B ys w.h wvq fõ' b ≥˙ka ;nd we;s bf,lafg%dakhla u; we;sjk
È◊l ;ajrKh jkafka"
(e = bf,lafg%dksl wdfrdamKh, m = bf,lafg%dakfha ialkaOh)
eRb
eRb
" ol=Kg ^2&
" jug
2m
2m
eRb
^4& 2am " jug
^5& 0
(1&

49'

b

eRb
^3& 2am " ol=Kg

rEmfha ±lafjk m˙† taldldrj Ok f,i wdfrdamKh jQ uq≥jl wCIh
Tiafia fm%dafgdakhla m%lafIamKh lrkq ,efí' my; i|yka m%fõ. (v) - ld,
(t) m%ia;dr i,ld n,kak'
v

v

v
0

0

t
(A)

a

t

0

p
+

+

+

+

+

+

+

+

t
(C)

(B)

.=re;ajdlr®IK n, fkdi,ld y˙kafka kï" ld,h (t) iuÛ fm%dafgdakfha m%fõ.fha úp,kh (v) jvd;a u
fyd¢ka ksrEmKh lrkq ,nkafka l=uk m%ia;dr$ h o@
^1& (A)
^2& (B)
^3& (C)
^4& (A) iy (B)
^5& (A), (B), (C) ish,a, u
50'

wrh a jk f.da,hl tall m˙udjl taldldrj we;s wdfrdamKh ρ fõ' rEmfha
fmfkk m˙† wrh b jk l=yr f.da,hla by; f.da,h ;=< ksr®udKh ù
we;' f.da,fha flakaøh yd l=yrfha flakaøh w;r ≥r c fõ' l=yrh ;=< iEu
,CIHhlu úoHq;a fCIa;% ;Sj%;dfjys úYd,;ajh jkafka"
^1& 0
^4&

ρ c2

3 bε0

^2&

ρb

3ε0
ρc
^5&
3 ε0

^3&

a

c b

ρa

3ε0

***

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

- 16 -

*
*

01 - fN!;sl úoHdj
˚˚ m;%h

A fldgfia ishÆ u m%Yakj,g ms<s;=re imhkak'
B fldgfia m%Yak y;rlg muKla ms<s;=re imhkak'

A fldgi - jHQy.; rpkd
(g = 10 N kg-1)
1'

m¯CIKd.drfha° ù≥rej, >k;ajh fiùug me;a;l †. 3 cm muK yd ialkaOh 60 g muK jk ù≥re
>klhla Tng imhd we;'
(a) (i)
		

>klfha me;a;l †. (ℓ) 1 ] g jvd jeä ksrjoH;djlska uekSug úoHd.drfha° Tn Ndú; lrk
WmlrKh l=ula o@

			
		 (ii) ù≥re >klfha m˙udj i|yd m%ldYkhla ℓ weiq˙ka ,shkak'
			
			
(b) >Qr®K uQ,Or®uh Ndú; lr" ù≥re lene,af,a ialkaOh fiùug my; ±lafjk whs;u Tng imhd
we;'
		
• ógr flda≥j
		
• msys∞rh
		
• 20 g, 50 g, 100 g mä lÜg,h" kQ,a len,s
(i)
		

>klfha ialkaOh by<u ksrjoH;dfjka ksr®Kh ls¯u i|yd" ° we;s mä w;=rska Tn f;dard
.kafka l=uk mäh o@ Tfí f;dard .ekSug fya;=j meye†,s lrkak'				

			

mäh

(

			

fya;=j

(

						
(ii)

m<uqj" ógr flda≥j msys∞rh u; ;eìh hq;= h' ógr flda≥fõ l=uk ia:dkh" msys∞rh u; ;eìh
hq;= ±hs Tn fidhd .kafka flfia o@

			
			
			
		 (iii) ialkaOh fidhd .ekSu i|yd Tn Ndú; lrk ieleiafï kï lrk ,o rEm igykla w¢kak'
		 by; ° we;s whs;u muKla Ndú; lrkak'

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

- 17 -

(iv) ù≥re l=Üáfha ialkaOh yd f;dard .kakd ,o mäfha ialkaOh ms<sfj<ska m yd M f,i
i,lkak' fuys° ,nd.efkk ≥r ñkqï (ℓ1, ℓ2) by; (b) (iii) ys w¢k ,o rEm igyk u;
i,l=Kq lr m, M, ℓ1 yd ℓ2 w;r iïnkaOh ±lafjk m%ldYkhla ,shkak'
					
		 (v) by; (a) (ii) iy (b) (iv) ys m%ldYk Ndú; lr" ù≥rej, >k;ajh (dg) i|yd m%ldYkhla
,shkak'
					
(c) (i)

c, n∫kla imhd we;akï" ù≥re >klfha m˙udj .Kkh ls¯fuka f;drj ù≥rej, >k;ajh
(dg) fiúh yelsh' msys∞rfha isg ù≥re >klhg we;s ≥r wp,j ;nd .ksñka dg fiùug
,nd.; hq;= wu;r ñkqu (ℓ3) ljfr® o@

			
			
		 (ii) ℓ1, ℓ2, ℓ3 iy c,fha >k;ajh dw weiqfrka" dg i|yd m%ldYkhla ,nd .kak'
			
			
			
			
			
											
2. rEmfha fmkajd we;s m˙† ;=kS taldldr ù≥re k<hla ;=<
ix;Dma; c, jdIamfhka msreKq jd; m˙udjla" ri†h l|la
uÛska isr fldg we;' fuu WmlrKh Ndú; fldg" iq≥iq
m%ia;drhla we£u u.ska" ldur WIaK;ajfha° c,fha ix;Dma;
jdIam mSvkh ^P& fiùug Tng kshuj we;'

;=kS ù≥re
k<h

c,h
ri†h

^a& isrù we;s jd; m˙udj" jdIamj,ska ix;Dma; ù we;s nj Tn oek.kafka flfia o?
		
^b& ri†h lf|a †. cm j,ska h1 f,i o jdhqf.da,Sh mSvkh ri†h cm j,ska H f,i o ldur WIaK;ajfha°
c,fha ix;Dma; jdIam mSvkh ri†h cm j,ska P f,i o .kak'
									
		
		
k<fha my; msysgqï ^1 iy 2& i,lkak' 		
					
h1

1 msysgqu
							
		
		
msysgqï fofla° jd;fha mSvkh i|yd
		
m%ldYk ri†h cm j,ska ,shd olajkak'
		
		

1 msysgqu

h1

2 msysgqu

: jd;fha mSvkh =

2 msysgqu : jd;fha mSvkh =

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

- 18 -

^c&					
									

L

l

			

h

h/

θ

		

		

k<fha wdk;sh (θ), 0° isg 90° f;la úp,kh ls¯u u.ska k<fha isrù we;s jd;fha mSvkh fjkia
l< yelsh'
(i)

			
		

by; fmkajd we;s msysgqfï° isrù we;s jd;fha mSvkh i|yd m%ldYkhla" ri†h cm j,ska ,shd
olajkak'
jd;fha mSvkh =

(ii) isrù we;s jd;h i|yd fndhs,a kshuh fh°fuka h, H, P iy ℓ iïnkaO flfrk m%ldYkhla
,shd olajkak' (ℓ = jd; lf|a †. cm j,ska )

			
			
			
		 ^iii& iq≥iq úp,H folla w;r m%ia;drhla we£u u.ska" c,fha ix;Dma; jdIam mSvkh (P) ksr®Kh l<
hq;=j we;' by; (c) (ii) ys ,nd .;a m%ldYkh H, P, ℓ, h / iy L weiqfrka kej; ilikak'
			
			
			
		 ^iv& P ksr®Kh ls¯u i|yd Tn m%ia;drfhka Wlyd .kakd rdYs fudkjd o ?
			
		

^v&

P ksr®Kh ls¯u i|yd H fidhd .; hq;= fõ' H uekSu i|yd Tn Ndú; lrk WmlrKh
l=ula o@

			
^d& flá ri†h l|la Ndú; lr .ksñka fï m¯laIKh l< yels o@ Tnf.a ms<s;=r i|yd fya;=
fokak'
		
		
3. m¯CIK.drfha° jd;fha èjks m%fõ.h fiùfï m¯CIKhla i|yd fofl<jr újD; wkqkdo k<hla"
c,h ms˙ Wi ù≥re irdjla" wdOdrlhla iy 500 Hz iriq,la Tng imhd we;'
(a) fuu m¯CIKh i|yd Tng wjYH jk wu;r WmlrKh l=ula o@
		

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

- 19 -

(b) fuys° m¯CIKh ls¯u i|yd Tn by; WmlrK ilia lrk whqre ±lafjk rEm igykla
w¢kak'

(c) ^⁄&
		

k<fha wdka;fYdaOkh i,lkafka kï" fuu m¯CIKfha° wkqkdo wjia:d wju jYfhka
fldmuK ixLHdjla .; hq;= o@ tajd y∫kajkak'

			
			
			
		 ^⁄⁄& tu wkqkdo wjia:d ,nd.kakd wdldrh meye†,sj olajkak'
			
			
			
(d) fuu m¯CIKfha° Tn ,nd.kakd ñkqï ljfr® o@ ^ℓ1 iy ℓ2 f,i .ksuq'&
		
		
(e) k<fha wdka;fYdaOkh e f,i f.k jd;fha èjks m%fõ.h v fiùu i|yd wjYH m%ldYk ℓ1" ℓ2
ñkqï yd e weiqfrka ,shkak'
		
		
		
(f) by; (d) fldgfia° ,nd.;a ñkqï ms<sfj<ska 16'5 cm iy 50'5 cm kï v iy e ksr®Kh lrkak'
		
		
		
		
(g) k<fha †. 80 cm kï B<Û wkqkdo wjia:dj Tng ,nd .; yels o@ Tnf.a ms<s;=rg fya;=
olajkak'
		
		

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

- 20 -

4. úNjudk ieleiaula Ndú;fhka fldaIhl wNHka;r m%;sfrdaOh (r) fiùug Tng kshuj we;' ta i|yd
my; whs;u Tng imhd we;'
úNjudkhla" 2V welshqñf,agrhla" m%;sfrdaO fmÜáhla" ir®mk h;=rla" ueo ìx≥ .e,ajfkdaógrhla"
fmakq h;=rla iys; wdrCIl m%;sfrdaOhla" iaúÉÑhla
^a& fuu m¯CIKh is≥ ls¯u i|yd Tn Ndú; lrk úNjudk ieleiafï iïmQr®K m˙m:h rEm
igykl we| olajkak'
		

^b& m¯CIKd;aul ieleiafï ishÆ fldgia ksjer†j iïnkaO fldg we;a±hs m¯CId ls¯u i|yd Tn
wkq.ukh lrk ls%hd ms<sfj< ,shd olajkak'
		
		
		
		
^c& m%;sfrdaO fmÜáfha m%;sfrdaOh ^R& fjkia lrñka úNjudk lïìfha ix;=,k †. ^l& ukskq ,efí'
m%;sfrdaOh wkka; jQ úg ix;=,k †. l0 f,i .kak' m%ia;drhla we£fuka" wNHka;r m%;sfrdaOh (r)
fiùu i|yd Tn Ndú; lrk m%ldYkh jHq;amkak lrkak'
		
		
		
^d& m%ia;drh we£u i|yd Tn úúO o;a; ,CIH ,nd .kafka flfia o@
		
		
^e& m%;sfrdaO fmÜáfha m%;sfrdaOh b;d l=vd w.hl mj;ajd .ekSu iq≥iq ke;' fuhg fya;=j l=ula o@
		
		
^f&

fï m¯CIKfha° Tng ,efnk m%ia;drfha o< igykla" wCI meye†,sj kï lr w¢kak'

0
				

0

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

- 21 -

^g& Tn m%ia;drh weiqfrka r fidhd .kafka flfia o@
		
^h& fldaIfha úoHq;a .dul n,h 2 V g jvd iqΩ jYfhka jeä jQfha kï" Tng fuu m¯CIKh idr®:lj l<
yels o@
		
Tnf.a ms<s;=rg fya;= fokak'
		
		
		
			

**

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

- 22 -

B fldgi - rpkd
(g = 10 N kg-1)

5. ;r, m%jdyhla i|yd nZkQ,s iólrKh
			
2
1
ρ
ρ
		 P + v + gh = ksh;hla
2
		
							
f,iska †h yelsh' fuys ishÆ u ixfla;j, iqmqre≥ f;areu we;'
iólrKfha we;s tla tla moh y∫kajkak'
ihsmkhla (siphon) hkq Ndckhl wvx.= øj bj;a ls¯fï Wml%uhls'
>k;ajh ρ jk c,fhka ms¯ we;s P n∫kla rEmfha fmkajd we;'
taldldr yrialvla we;s T kï k<hl ^ihsmkh& D fl<jr uymg
weÛs,af,ka jidf.k isg wdrïNfha° k<h c,fhka mqrjkq ,efí'
bkamiq uymg weÛs,a, bj;a ls¯fuka c,h uq∞ y˙kq ,efí'

C
A
P

h1

v1

T

B

h2

v

D

c,h" ≥iai%dù fkdjk wiïmSvH øjhla f,i yd c, m%jdyh wkdl+, f,i Wml,amkh lrkak'
(a) k<fha my< fl<jfrys (D) c, m%jdyfha fõ.h v kï" k<fha m%fõYhg (B) hï;ñka we;=<;ska
c,fha fõ.h l=ula o@
(b) n∫fka yrialv jr®.M,h" k<fha yrialv jr®.M,h fuka 100 .=Khla kï" n∫fka ksoyia c,
mDIaGh my<g .uka lrk fõ.h v1 i|yd m%ldYkhla v weiqfrka ,shd olajkak'
		 v1 << v ksid fuhska miq v1 = 0 f,i .kak'
(c) (i) A f.ka wdrïN ù D f.ka wjidk jk wkdl+, fr®Ldjla i,ld" v i|yd m%ldYkhla" iq≥iq
ksh;hla yd rEmfha ° we;s mrdñ;shla weiqfrka jHq;amkak lrkak'
		 (ii) h2 = 45 cm kï" v ksr®Kh lrkak'
		 (iii) h2 = 0 kï" ihsmk l%shdj is≥fkdjk nj fmkajkak'
(d) (i) c, lf|a by<u ia:dkfha (C) c,fha mSvkh PC i|yd m%ldYkhla ρ, g, h1 , h2 iy
jdhqf.da,Sh mSvkh P0 weiqfrka jHq;amkak lrkak'
		 (ii) †f.ka jeä [(h1 + h2) w.h úYd, fõ'] k<hla Ndú; lf<a kï PC = PjdIam jk úg ihsmk l%shdj
kj;S' fuys PjdIam hkq c,fha WIaK;ajfha° c,fha ix;Dma; jdIam mSvkhhs' fuhg fya;=j
l=ula o@			
		 (iii) 30°C ° P jdIam= 4'0 kPa kï" ihs m k l% s h dj ks h udldrfhka is ≥ ùu i|yd (h1 + h2)g ;sìh
		 hq;= Wm˙u w.h ksr®Kh lrkak' ( P0 = 10 5 Pa, ρ =10 3 kg m−3)
											
6. kshudldrfhka lmk ,o †huka;shla ^1& rEmfha fmkajd we;'
†huka;shla †≥,kafka mQr®K wNHka;r mrdjr®;kh ksid"
th ;=<ska meñfKk wdf,dalfha ;Sj%;dj m%n,j jr®Okh jk
neúks'
jd;fha ;nd we;s †huka;shl u;=msg AB mDIaGhg i m;k
fldaKhla iys;j taljr®K wdf,dal lsrKhla ^2& rEmfha
fmkajd we;s whq˙ka m;kh fõ' B <Ûg th r jr®;l fldaKhla
iys;j jr®;kh ù †huka;sfha CD wdk; mDIaGhg j°'

^1& rEmh

		Tnf.a .Kkh ls¯ï i|yd my; ° we;s o;a; Ndú; lrkak'
sin 5°
sin 7'5°
sin 10°
sin 23°

=
=
=
=

0'0870
0'1305
0'1737
0'3920

sin 24° =
sin 42° =
sin 80° =

0'4000
0'6667
0'9800

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

- 23 -

												
(a) †huka;s - jd; w;=re uqyqK; i|yd wjê fldaKh ksr®Kh
lrkak' †huka;sj, jr®;kdxlh 2'5 fõ'
(b) (2& rEmfha fmkajd we;s cHdñ;sl igyk Ndú; lrñka" CD
wdk; mDIaGh u; lsrKfha m;k fldaKh i 1 i|yd m%ldYkhla
θ iy r weiqfrka fidhkak' fuys θ hkq CD mDIaGh ;srig ork
wdk;shhs'

A

i

B
r
i1

C

θ

D
(c) (i) i = 80° i|yd r ksr®Kh lrkak'
90° - r
		 (ii) CD mDIaGh u.ska fuu wdf,dal lsrKh hka;ñka mQr®K
^2& rEmh
wNHka;r mrdjr®;khlg ,laùug kï" θ g ;sìh hq;=
wju ^θwju& w.h ksr®Kh lrkak'
		 (iii) 80° fyda Bg jvd l=vd i w.hj,ska hq;= j AB mDIaGh u; m;kh jk ishÆ u wdf,dal lsrK CD
		 mDIaGfhka mQr®K wNHka;r mrdjr®;khg n∫ka jk nj tkhska fmkajkak'
		 (iv) θ < θwju kï l=ula is≥jkafka o@
(d) (i) (2& rEmfha fmkajd we;s cHdñ;sl yevhg iudk θ = θwju jQ ù≥re jHqyhla i|yd CD mDIaGfhka
mQr®K wNHka;r mrdjr®;khg n∫ka ùug kï lsrKhlg ;sìh hq;= Wm˙u i m;k fldaKh
ksr®Kh lrkak' ^ù≥rej, jr®;kdxlh } 1'5&
^⁄⁄& tkhska" †huka;sh fuka ;ekQ ù≥re jHqyh †,sfikafka ke;s nj" fya;= olajñka wfmdaykh
lrkak'
7. ^a& f.da,Sh øj nqnq,l wu;r mSvkh i|yd m%ldYkhla" øjfha mDIaGßl wd;;sh ^T& yd nqnqf,a wrh ^r&
weiqfrka ,shd olajkak'
								
		
^b& isria U k<hla ;=< >k;ajh ρ jQ øjhla wka;r®.;
fõ' fuu k<fha tla fl<jrla jd;hg újD;j we;s
w;r ^1& rEmfha fmkajd we;s m˙† wfkla fl<jfr®
inka mg,hla id∞ we;' inka mg,fha yevh fjkia
l< yelsjk m˙† inka mg,h wvx.= ndyqfõ mSvkh
úp,kh l< yelsh'
		 (i) nqnqf,a wrh ^r& iy U k<fha øj uÜgï w;r
Wfiys fjki ^h& w;r .=◊;h" ksh;hla nj
fmkajkak'
		 (ii) by; ^i& fldgfia ksh;fha w.h 1'23 × 10−5 m2
kï inka ødjKfha mDIaGßl wd;;sh fidhkak'
^U k<fha we;s øjfha >k;ajh 800 kg m−3 f,i
i,lkak'&

inka mg,h
rnr®
ngh

h
la,smh
øjh

^1& rEmh

jd;h
									
^c& ±ka inka mg,h bj;a lr ndyqfõ tu wkqrEm fl<jr
uqød ;nkq ,efí' bkamiq" ^2& rEmfha fmkajd we;s m˙†
wNHka;r úIalïNh 0'7 mm jQ" c,fha .s,ajk ,o flaYsl
k<hlg U k<fha tu ndyqj iïnkaO lrkq ,efí' ±ka
flaYsl
k<h
U k<h mSvkudkhla f,i ls%hd lrhs' flaYsl k<fha
újD; fl<j˙ka jd;h fifuka we;=Ω lrk úg mSvk
udkfha øj uÜgïj, wka;rh wdrïNfha ° 9'1 cm f;la
jeä úh' wk;=rej 4'0 cm f;la wvq ù kej; 9'1 cm f;la
øjh
h1
c,h
jeä úh'
^2&
rEmh
		 ^c,fha >k;ajh } 1000 kg m−3&
^i& mSvkudk øj uÜgï w;r wka;rh by; i|yka l< m˙† úp,kh jkafka wehs ±hs meye†,s
lrkak'
w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

- 24 -

		
		

^ii& c, uÜgfï isg flaYsl k<fha my< fl<jrg we;s .eUqr ^h1& .Kkh lrkak'
^iii& tkhska" c,fha mDIaGsl wd;;sh fidhkak'

8. my; f√oh lshjd" wid we;s m%Yakj,g ms<s;=re imhkak'
lΩ l=yrhla (black hole) hkak kùk .=re;ajdlr®IK jdofha b;d is;a .kakd yd mq≥u iy.;
mqfrdal:kj,ska tlla jqj o tys uQ,sl woyi ksõfgdakshdkq uQ,Or®u mokï fldgf.k jQj;a f;areï
.; yelsh' ialkaOh M yd wrh R jk N%uKh fkdjk f.da,Sh ;drldjla i,lkak' fuu ;rej i|yd
tys mDIaGh u;° úfhda. m%fõ.h (v),
		 v = 2GM , ................. (1), f,iska m%ldY l< yelsh' fuys G hkq idr®j;% .=re;ajdlr®IK
ksh;hhs' R
úfhda. m%fõ.h (v), wdf,dalfha fõ.h (c) g jvd wêl jqjfyd;a tjeks ;rejlska úfudapkh jk
wdf,dalh kej;;a th lrd wdmiq ye¯ug ie,fik nj wdOqksl ;drld úoHd{hl= jQ fcdaka ñfp,a
kï mQcl;=ud (Rev. John Mitchell) 1783 ° m%ldYhg m;a lf<ah' wm ±ka y∫kajk lΩ l=yr mej;sh
yels nj fhdackd l< uq,au mqoa.,hd jkafka ñfp,ah' ialkaOh M jk ;rejl" wrh R hïlsis wjê
wrhlg jvd l=vd fyda iudk jqjfyd;a th lΩ l=yrhla f,i l%shd lrk nj by; v i|yd jk
m%ldYkh ikd: lrhs' by; v i|yd jk iïnkaO;djfha v = c fh¥ úg tu wjê wrh i|yd jk
m%ldYkhla ks;e;skau ,efí' 1916 ° ld,a YajdÜiaphs,aâ (Karl Schwarzschild), whskaiaghskaf.a idudkH
idfmaÈ;djdoh fhdod .ksñka ±ka YajdÜiaphs,aâ wrh (Schwarzschild radius) f,iska ye¢kafjk
tu wjê wrh Rs i|yd m%ldYkhla jHq;amkak lf<ah' wdf,dalfha pd,l Yla;sh 1 m c2 fkdjqk;a iy
2
GM
lΩ l=yrhla wdikakfha .=re;ajdlr®IK úNjh ir,j − R u.ska fokq fkd,enqj;a (1) iólrKfha
v = c wdfoaY ls¯fuka ,efnk m%ldYkhu YajdÜiaphs,aâ Rs i|yd ,nd .;af;ah' by; i|yka l< fodaI
fol tlsfklska ydksmQrKh ùu ksid fuh is≥fõ' lΩ l=yrhla jgfldg we;s wrh Rs jQ f.da,Sh
mDIAGh §isoaê laIs;sch¶ (event horizon) f,iska ye¢kafõ' fuu f.da,h we;=<;ska wdf,dalhg t<shg
taug fkdyels ksid th ;=< is≥jk isoaê wmg fkdfmfka'
Okq rdYsh kue;s ;drld uKav,fha †Ydjg mD:súfha isg wdf,dal jr®I 26"000la muK wE;ska jQ
wmf.a ÈSr m: ukaodls◊fha (Milky way galaxy) flakaøfha w;s úYd, lΩ l=yrhla mj;sk njg
idÈs ;drld úoHd{hska úiska fidhd f.k we;' ukaodls◊ flakaøfha wê úfNaok m%;sìïn u.ska
fkdfmfkk jia;=jla jgd wêl fõ.j,ska ;re .uka .kakd nj wkdjrKh ù we;' fuu p,s;
úYaf,aIKh ls¯u u.ska ;drld úoHd{hskayg tla tla ;refõ llaIfha ld,djr®;h T yd wrh r wkqudk
l< yelsh' fuu o;a; Wmfhda.s lr f.k fkdfmfkk jia;=fõ ialkaOh m .Kkh l< yelsh' fuu
lΩ l=yrfha ialkaOh 5'4 × 10 36 kg ke;fyd;a wmf.a iQr®hhdf.a ialkaOh fuka ñ,shk 2'7 l .=Khls'
iQr®h ialkaOhg jvd 10 9 .=Khla blaujQ fjk;a úYd, lΩ l=yr" wfkl=;a ukaodls◊j, flakaøj,
mj;sk nj mr®fhaIK u.ska wkdjrKh ù we;' ishÆ u m%udKj, lΩ l=yrj, ks¯ÈKd;aul yd
ffioaOdka;sl wOHhk" fN!;sl úoHdfõ iy ;drld úoHdfõ mr®fhaIK fÈa;%j, jeo.;a ldr®h Ndrhla
;j≥rg;a bgq lrhs'
(a) (i) §isoaê Ès;sch¶ hkafkka woyia jkafka l=ula o@
		 (ii) th tf,i wr®: olajd we;af;a wehs@
(b) (i) v i|yd oS we;s ^1& jk iólrKh jHq;amkak lrkak'
		 (ii) ;drldj iE° we;s øjHfha >k;ajh ksh; hehs Wml,amkh lr" úfhda. m%fõ.h" ;refõ wrh
		 R g iudkqmd;sl jk nj fmkajkak'
(c) (i) G, M yd c weiqfrka Rs i|yd m%ldYkhla ,shd olajkak'
		 (ii) ksõfgdakshdkq .=re;ajh wdOdrfhka Rs i|yd m%ldYkh ,nd .ekSfï° tlsfklska ydksmQrKh
		 jk, wm Ndú; l< fodaI fol fudkjdo@
(d) (i) iQr®h ialkaOh fuka f;.=Khl ialkaOhla we;s ±jqKq ;rejla yels<S lΩ l=yrhla idohs' tys
		 isoê
a Ès;scfhys wrh ksr®Kh lrkak'
			 (G = 6'66 × 10 −11 N m2 kg −2, iQr®h ialkaOh = 2'0 × 10 30 kg, c = 3'0 × 10 8 m s−1)
w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

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		 (ii) lΩ l=yrfha wrh YajdÜiaphs,aâ wrhg (Schwarzschild radius) iudk kï tys >k;ajh fldmuK o@
			 ^ π = 3 f,i .kak'&
(iii) wmf.a ukaodls◊fha flakaøfha mj;sk lΩ l=yrfha isoaê Ès;scfha wrh fldmuK fõ o@
(e) isoaê Ès;sch yryd X - lsrKj,g fyda γ - lsrKj,g úfhda. úh yels o@ Tnf.a ms<s;=rg fya;=
fokak'
(f) wmf.a ukaodlsKsfha flakaøfha mj;sk lΩ l=yrfha ialkaOh (m) i|yd m%ldYkhla th jgd wrh r
jk fia m˙N%uKh jk ;rejl ld,djr®;h T, r iy G weiqfrka jHq;amkak lrkak'
(g) mD:ßúfha isg wmf.a ukaodls◊fha flakaøfha we;s lΩ l=yrhg we;s ≥r km j,ska .Kkh lrkak'
(h) wmf.a iQr®hhd hï úêhlska yels<S lΩ l=yrhla f,ig m;a jqjfyd;a mD:ßúfha llaIh u; tu
isoaêfhka jk n,mEu l=ula fõ o@
		 ^lΩ l=yrhla f,ig yels<Sug ;rï úYd, ialkaOhla wmf.a iQr®hhdg ke;'&
9. (A) fldgig fyda (B) fldgig fyda muKla ms<s;=re imhkak'
												
(A) (a) wNHka;r m%;sfrdaOh fkdie,lsh yels úoHq;a .dul n,h
10V jk neg˙hla u.ska úp,H úNj wka;rhla (V0) ,nd
.ekSu i|yd R úp,H m%;sfrdaOhla (R = R1+R2) iïnkaO l<
m˙m:hla ^1& rEmfha ±lafõ'
			 (i) fuu m˙m:h u.ska V0 i|yd ,nd .ekSug wfmaCId l<
yels úNj wka;r mrdih l=ula o@
			 (ii) R2 iuÛ V0 úp,kh jk wdldrh ±lafjk o< rEm
igykla w¢kak'

10 V

R

R1
R2

V0

^1& rEmh

		

(b) V0 i|yd l=vd úNj wka;r mrdihla ,nd .ekSu ms◊i YsIHhl= by; m˙m:h úlrKh lr
we;s wdldrh my; ±lafõ' V0 w.h" R2 iuÛ fr®Çhj úp,kh fõ hehs YsIHhd n,dfmdfrd;a;=
fjhs'
											
			 (i) úlrKh lrk ,o m˙m:fhka V0 i|yd ,nd.;
R1 900 Ω
yels úNj wka;r mrdih l=ula o@
10 V
R
			 (ii) l=vd fjda,aàh;d ,nd .ekSfï° by; ^1& rEmfhys
R3
i|yka m˙m:hg jvd ^2& rEmfhys ±lafjk m˙m:h
R2
R4 100 Ω V0
Ndú; ls¯fï we;s jdish l=ula o@
^2& rEmh
			 (iii) R = 10 kΩ kï R2 = 5 kΩ ys ;nd.;a úg V0 i|yd
,efnk w.h fidhkak'
				 (1) tu.ska R2 iuÛ V0 úp,kh jk wdldrh ±lafjk o< igykla w¢kak'
				 (2) R2 iuÛ V0 úp,kh fr®Çh o@ Tnf.a ms<s;=r meye†,s lrkak'
			 (iv) R2 iy V0 w;r fr®Çh;dj jeä †hqKq ls¯u i|yd R3 yd R4 i|yd ms<sfj<ska 90 kΩ iy
10 kΩ hk w.h Ndú; ls¯ug YsIHhl= ;SrKh lr we;' 				
				 (1) V0 i|yd kj mrdih i|yka lr" R2 = 5 kΩ kï V0 .Kkh lrkak'
				 (2) tu.ska YsIHhdg jvd fyd| fr®Çh;djla ,nd .ekSug yels jQfha o@ Tnf.a ms<s;=r meye†,s
				
lrkak'
				 (3) by; (iv) ys i|yka m˙m:h R0 Ndr m%;sfrdaOlhg ksr®†Iag fjda,aàh;d iemhSu i|yd Ndú;
				
flfrk úg R0 ys w.h iEuúgu 10 kΩ g jvd wvqúh hq;= hehs Tyq ksr®foaY lrhs' Tn thg
				
tlÛ jkafka o@ Tfí ms<s;=r meye†,s lrkak'

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

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(B) (a) (i) R m%;sfrdaOhla yryd V úNj wka;rhla fhd∞ we;'
V ys w.h l=vd ∆V m%udKhlska jeä l< úg m%;sfrdaOh
yryd Odrdj I isg I + ∆I olajd jeä fõ' ∆V = (∆I)R nj
fmkajkak'

			 (ii) ° we;s m˙m:fha VBE = 0'7 V iy g%dkaisiagrfha Odrd ,dNh
100 ls' IE = IC f,i i,ld
			 (1) VE
(2) IE
(3) VC
(4) IB w.h
				 fidhkak'
				 by; wjia:dfõ° mdou - ix.%dyl ikaêh miq keUqre nj
o fmkajkak'
		

RC
IB

VC

IE
RE

4V

10 V
4'7 kΩ
IC
VE
3'3 kΩ

(b) mdou i|yd iemhqï fjda,aàh;dj l=vd ∆VB m%udKhlska jeä lf<a hehs is;uq'

			 (i) VBE ys fjkia ùu fkdi,ld ye˙h yels kï" mdou fjda,aàh;dj jeäùu u.ska we;sjk
úfudapl fjda,aàh;dfõ (VE) fjkiaùu (∆VE) fidhkak'
			 (ii) tu.ska úfudapl Odrdfõ fjkiaùu (∆I E) i|yd m%ldYkhla ,shkak'
			 (iii) ix.%dylfha fjda,aàh;dfõ fjkiaùu (∆VC) i|yd m%ldYkhla ,shkak'
			 (iv) fuu fjkiaùu jeäùula kï + ,l=K o" fkdtfia kï - ,l=K o Tn ^iii& ys ,nd.;a
m%ldYkh b†˙fhka fhdokak'
			 (v) tkhska jr®Olfha fjda,aàh;d ,dNh i|yd m%ldYkhla ,shkak'
			 (vi) fmd≥ úfudapl jr®Olfha fjda,aàh;d ,dNh i|yd idOdrK m%ldYkhla wfmdaykh ls¯ug
(v) ys m%ldYkh Ndú; lrkak'
10. (A) fldgig fyda (B) fldgig fyda muKla ms<s;=re imhkak'
(A) fmg%,a tka‚ul tla is,skavrhla ;=< ^1& rEmfha fmkajd we;s m˙† jdhq iy fmg%,a ñY%Khla
^2& rEmfha olajd we;s ;dm.;sl l%shdj,sh yryd hhs'
		 msiagkh by<g hk úg jdhq ñY%Kh blauka iïmSvkhlg ^e
f& n∫ka fõ' B <Ûg ñY%Kh b;d
CIKhlska –j,kh ù mSvkfha úYd, jeäùula ^f
g& we;s lrhs' bkamiq msiagkh my< nisk
w;r ñY%Kh blauka m%idrKhlg ^g
h& ,lafõ' wjidkfha ° WKqiï
q jd;h msg;g fkrfmk
^h
e& w;r is,skavrh ;j;a pl%hla wdrïN ls¯u i|yd iQ∞kï fõ'

P

g

f

msi
agkh
Piston

h
e
^2& rEmh

V

^1& rEmh
		
				
		
		 (a) ^⁄& e
f (ii) f
g (iii) g
h (iv) h
e
				 by; tla tla wjia:dfõ° is≥jk ;dm.;sl ls%hdj,s my; ° we;s wjia:d w;=˙ka l=ulg
wh;a jkafka ±hs i|yka lrkak'
				 ksh; mSvk" ksh; m˙ud" ifudaIaK m%idrK$iïmSvk" ia:ßr;dmS m%idrK$iïmSvk

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

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mSvkh / M Pa

6.0

g

5.0
4.0
3.0

h

f

2.0

e

1.0
0

1.0

2.0

3.0

^3& rEmh

4.0

m˙udj$× 10-4m3

(b) ^3& rEmfha ±lafjkafka" tla is,skavrhla iys; tka‚ula i|yd ir, lrk ,o P - V jl%hls'
^jl% fldgia wdikak jYfhka ir, fr®Çh f,i i,ld we;'&
			 (i) g isg h olajd jdhq ñY%%Kh u.ska lrk ,o ldr®hh"
			 (ii) e isg f olajd jdhq ñY%%Kh u; lrk ,o ldr®hh"
			 (iii) efgh pl%hla ;=< jdhqj uÛska lrk ,o iM, ldr®hh m%udKh (W)
			
.Kkh lrkak'
		

		 (c) (i)
			
			 (ii)
				
		

f isg g olajd jdhq ñY%Kh u.ska wjfYdaIKh lrk ,o ;dm m%udKh Q1 kï"
f isg g olajd wNHka;r Yla;sfha fjkia ùu ( ΔUf
) fldmuK o@
g
h isg e olajd jdhq ñY%Kh u.ska uqla; lrk ,o ;dm m%udKh Q2 kï"
h isg e olajd wNHka;r Yla;sfha fjkia ùu ( ΔUh
) fldmuK o@
e

W
(d) (i) tka‚fï ;dm ldr®hlaIu;dj (e) kï rdYshla, e = Q f,i wr®: olajd we;'

				 Q1, Q2 uÛska muKla (e) i|yd m%ldYkhla ,shkak'

1

			 (ii) Q1 = 3'5 × 103 J kï (e) m%;sY;hla f,i olajkak'

			 (iii) fmg%,aj, ;dm oyk w.h 3'5 × 103 J g−1 kï tla pl%hl° oykh jk fmg%,a ialkaOh
				fidhkak'
		 (e) is,skavr 4 la iys; tka‚u ;;amrhg jg 50 lska N%uKh fõ kï tka‚fuka ckkh jk
		 laIu;dj .Kkh lrkak'
(B) 1. (a)

wmf.a iQr®hhdf.a wrh 7'0 × 105 km jk w;r tys mDIaGßl WIaK;ajh 5800 K fõ'
iQr®hhd lDIaK jia;=jla f,i yeisfrkafka hehs Wml,amkh lrkak'
				 (i) iQr®hhdf.ka úlsrKh jk uqΩ laIu;dj .Kkh lrkak'
−8
					 ^iafgµdka ksh;h σ = 6'0 × 10 W m−2 K−4; 584 = 1'0 × 107 f,i .kak'&
			 (b)
				
			

iQr®hhdf.a uOHfha WIaK;ajh 2'0 × 107 K fõ'

(i) iQrh
® hdf.a uOHfha fumuK by< WIaK;ajhla mej;Su b;du;a wjYH jkafka wehs@
(ii) iQr®hhdf.a uOHfha we;=<;ska γ lsrK úfudapkh fõ' kuq;a iQr®h mDIaGh jeäuk;aj
úlsrKh lrkafka wfOdarla; (IR)" oDYH yd mdrcïnq, (UV) wdf,dalhhs' iQr®hhd
uOHfha cks; jk fuu flá ;rx. wdhduj,ska hq;a úlsrKj,g is≥jkafka l=ula o@

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			 (c)

§ks,a iqms˙ fhdaO¶ ;rejlg 30 000 K mDIaGßl WIaK;ajhla we;' ;rej lDIaK jia;=jla f,i
yeisfrkafka hehs Wml,amkh lrkak'
				 (i) fuh úlsrKh lrk m%Odk ;rx. wdhduh fldmuK o@ fuu wdf,dalh oDYH o@
					 fuu ;rej ks,a meyefhka oDYHudk ùug fya;=j meye†,s ls¯ug Tnf.a ms<s;=r Ndú;
					 lrkak' ^ùka ksh;h = 3'0 × 10−3 m K&
			 (ii) fuu ;rej úlsrKh lrk uqΩ CIu;dj wmf.a iQr®hhd úlsrKh lrk uqΩ laIu;dj
					 fuka 105 .=Khla kï fuu ;refõ wrh ksr®Kh lrkak'
												
		 2. tlla 300 K WIaK;ajhl o wfkl 6 000 K WIaK;ajhl o
mj;ajdf.k we;s úYd, ;yvq folla ^P iy Q& rEmfha fmkajd
we;s m˙† tlla <Û wfkl ;nd we;' úYd, ;yvq fol w;r
y˙ ue†ka l=vd ;yvqjla ^R& ;nd we;' ;dm iu;=,s;;djg
m;ajQ miq l=vd ;yvqj wh;a lr.kakd WIaK;ajh fldmuK o@
			 l=vd ;yvqj lDIaK jia;=jla f,i yeisfrkafka hehs
Wml,amkh lrkak' ^648¼ } 5'045&

P
300 K

R

Q
6 000 K

***

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

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02 - ridhk úoHdj
m%Yak m;% jHQyh
˚ m;%h

- ld,h ( meh 02hs'
jrK 5 ne.ska jQ nyqjrK m%Yak 50ls' m%Yak ish,a,g u ms<s;=re iemhsh hq;= h'
tla m%Yakhlg ,l=Kq 02 ne.ska uqΩ ,l=Kq 100ls'

˚˚ m;%h - ld,h ( meh 03hs'
fuu m%Yak m;%h A, B yd C jYfhka fldgia ;=klska iukaú; h'
A fldgi - jHQy.; rpkd jr®.fha m%Yak y;rls' m%Yak ish,a,g u ms<s;=re
iemhsh hq;= h' tla m%Yakhlg ,l=Kq 100 ne.ska ,l=Kq 400ls'
B fldgi - rpkd jr®.fha m%Yak ;=kls' m%Yak follg ms<s;=re iemhsh hq;= h'
tla m%Yakhlg ,l=Kq 150 ne.ska ,l=Kq 300ls'
C fldgi - rpkd jr®.fha m%Yak ;=kls' m%Yak follg ms<s;=re iemhsh hq;= h'
tla m%Yakhlg ,l=Kq 150 ne.ska ,l=Kq 300ls'
˚˚ m;%h i|yd uqΩ ,l=Kq 1000 fi 10 } 100
wjidk ,l=K .Kkh ls¯u

(

˚ m;%h
} 100
˚˚ m;%h
} 100
wjidk ,l=K } 200 fi 2 } 100

˚ m;%h
ie,lsh hq;=hs (
* ishÆ u m%Yak i|yd ms<s;=re imhkak'
* ksjer† fyda jvd;a .e<fmk fyda ms<s;=r f;darkak' ^úNd.fha ° ms<s;=re iemhSu i|yd nyqjrK lv∞ishla
imhkq ,efí'&
idr®j;% jdhq ksh;h" R
wej.dâfrda ksh;h" NA

= 8.314 J K−1 mol−1
= 6.022 × 1023 mol−1

1' 3p ldlaIslhl we;s bf,lafg%dakhla úia;r lrkq ,nk ksjer† lafjdkagï wxl l=,lh my; i|yka
tajdhska l=ula o@
_
n

_
l

(1) 4
(2) 3
(3) 2
(4) 3
(5) 3

2
1
2
1
1

m
_l
-1
-1
-2
2
-2

m
_s
0
+½
-1
-½
+½

2' S2O32− whkh jvd;a u fyd¢ka ksrEmKh lrkq ,nkafka my; i|yka l=uk jHqyh u.ska o@

O

S

S

O:

O

(5) :O + O:

S

S

: : : :

S

(4) :O

: :: :

:S

O:

: :: :

O:

++

: :: :

S

(3) :O

: :: :

O:

:S

O:

: :: :

3'

S

+

: : : :

S

(2) O

: : : :

O:

: : : :
:
: :: :

(1) O

O:

NOCl wKqj iïnkaOfhka my; i|yka m%ldYj,ska i;H fkd jkafka l=ula o@
(1) NOCl wKqfõ yevh flda◊l fõ'
(2) khsÜrcka mrudKqfõ uqyqïlrKh sp2 fõ'
(3) NOCl wKqfõ uOH mrudKqfõ bf,lafg%dak hq.,j, wjldY ieliqu flda◊l fõ'
(4) NOCl j, N yd O w;r π nkaOkhla mj;S'
(5) khsÜrckaj, sp2 uqyqï ldlaIslhla yd laf,da¯kaj, 3p mrudKql lddlaIslhla w;sÉPdokh
ùfuka N − Cl nkaOkh iEfoa'
w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

- 30 -

4' f,dayl cd,fha laIdr f,dayj, mrudKq iïnkaOfhka my; i|yka tajdhska l=ula i;H fõ o@
(1) f,dayl cd,fha ≥r®j, jeka vZ jd,aia n,j,ska muKla tajd tla fldg ;nd .efka'
(2) tajdfha mrudKql l%udxl T;af;a ixLHd fõ'
(3) tajdfha ‘d’ ldlaIslj, bf,lafg%dak fkdue;'
(4) tajd m%n, Tlaisydrl jk whk idohs'
+
(5) tajd ic, whk idok w;r tajd w;=˙ka Li wju f,i ic,SlrKh ù we;'
5' wdjr®;s;d j.=fõ 17 jk ldKavfha µaf,da¯ka isg whäka olajd we;s uQ,øjH iïnkaOfhka my; i|yka
m%ldYj,ska l=ula wi;H fõ o@
(1) tajd oaú mrudKql iyixhqc wKq idohs'
(2) tajdfha jdhquh mrudKq bf,lafg%dak ,nd.ekSfï° Yla;sh msglrhs'
(3) tajd idok yhsv%hsv c,fha †hù m%n, wï, idohs'
(4) tajdhska laf,da¯ka Tlaiswï, idohs'
(5) tajd tlsfkl w;r ixfhda. idohs'
6' X iy Y hk uQ,øjHj, uq,a whkSlrK Yla;s ;=k (kJ mol ) my; i|yka m˙† fõ'
−1

uQ,øjHh
X
Y

m<uqfjks
528
1095

fofjks
7340
2370

;=kafjks
11850
4660

X iy Y uQ,øjH ms<sfj<ska my; i|yka tajdhska l=ula úh hq;= o@
(1) Be iy O
(2) Li iy Be (3) Na iy Mg (4) Li iy C

(5) B iy Ne

7' IUPAC kdulrK l%uhg wkql+,j my; ixfhda.fha kduh jkafka"

H

C

CH2
C =

C

CH2

CH2

O

=

H

=

O

C

CH3

OH

(2) 4-methyl-4-formyl-5-hexenoic acid.
(4) 4-oxo-4-methyl- 5-hexenoic acid.

(1) 4-formyl-4-methyl-5-hexenoic acid.
(3) 4-formyl-4-methyl-5-enehexanoic acid.
(5) 4-methyl-4-formyl-5-enehexanoic acid.

H
p

C ≡C

CH2

CH2

CH2

q

r

s

=

8' my; ° we;s ixfhda.fha p, q, r, s iy t f,i ,l=Kq lr we;s yhsv%cka mrudKq i,lkak'
O
C

H
t

my; ±lafjk hq., w;=˙ka jvd;a wdï,sl yhsv%cka mrudKq hq.,h l=ula o@
(5) s iy t
(4) p iy t
(3) r iy s
(2) p iy s
(1) p iy q
9' a
e
d
c
b
CH3 C ≡ C CH2 CHO
hk ixfhda.h iïnkaOfhka my; ° we;s j.ka;sj,ska l=ula i;H o@
(1) b, c yd e ldnka mrudKq sp uqyqïlrKhg Ndckh ù we;
(2) a, b, c, d yd e ldnka mrudKq tlu ir, fr®Ldjl msysgd we;
(3) cde fldaKh 109'5° la muK fõ
(4) abc fldaKh 120° la muK fõ
(5) a, b yd d ldnka mrudKq sp3 uqyqïlrKhg Ndckh ù we;

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10' A kï ldnksl ixfhda.h olajk m%;sl%shd lsysmhla my; ° we;'
Br
CH3 - CH - CH2Br

X
idkaø H2SO4

Br2 / CCl4

Z

A

HBr

Ni / H2

Y

ms<sfj<ska X, Y, Z úh yelafla"
(1) CH3 CH2 CH2 OH, CH3 CH2 CH2 Br iy CH3 CH2 CH3.
Br

iy CH3 CH2 CH3.

(2) CH3 CH2 CH2 OH, CH3 CH CH3
OH

Br

(3) CH CH CH , CH CH CH iy CH3 CH2 CH3.
3
3
3
3
OH

(4) CH3 CH CH3 , CH3 CH2 CH2 Br iy CH3 CH2 CH3.
OH

Br

iy CH3 CH = CH2.

(5) CH3 CH CH3 , CH3 CH CH3

11' yhsfv%dlais,a whkhla NIauhla f,i l%shdlrk úg" ia:dhS M,hla ,nd°u i|yd my; olajd we;s
ixfhda.fha l=uk ia:dk m%;sl%shd lrkafka ±hs n,dfmdfrd;a;= úh yels o@
Br H
H H
H
(1) a yd e

C

h
g

f

C

C

d

Cb H

H e Hc H a
H
(3) b, d yd f
(2) a, d yd e

(4) e yd d

(5) a, c, e yd g

12' m%;sl%shdjl w;rue†fha wdxYsl jHqyh (Partial structure) my; ±lafõ'
C

C

+

C

H

fujeks wdxYsl jHqyhla ,eîug yelshdjla we;af;a"
(1) kshqla,sfhdaµs,sl wdfoaY m%;sl%shdjl° mu◊'
(2) bf,lafg%dµs,sl wdl,k m%;sl%shdjl° mu◊'
(3) wefrdaueál bf,lafg%dµs,sl wdfoaY m%;sl%shdjl ° mu◊'
(4) kshqla,sfhdaµs,sl wdfoaY m%;sl%shdjl° iy wefrdaueál bf,lafg%dµs,sl wdfoaY m%;sl%shdj,°
mu◊'
(5) kshqla,sfhdaµs,sl wdfoaY" bf,lafg%dµs,sl wdl,k iy wefrdaueál bf,lafg%dµs,sl wdfoaY
m%;sl%shdj,° h'
13' ødjKhla th yryd .uka lrk úlsrKfhka 90]la wjfYdaIKh lrhs' tu ødjKfha wjfYdaIl;dj
l=ula o@
(3) 1.0
(2) 0.90
(1) 0.10
(5) by; lsisjla fkd fõ
(4) 9.0
14' m%n, úoHq;a úÉf√oHhl 0.10 mol dm−3 c,Sh ødjKhl ikakdhl;dj 1.25 × 104 μS cm−1 fõ' c,h u.sska
tu ødjKh oi .=Khla ;kql l<fyd;a we;s jk ødjKfha ikakdhl;djfha mej;sh yels jvd;a u
iïNdjH w.h l=ula o@
(3) 1.45 × 103 μS cm−1
(2) 1.25 × 103 μS cm−1
(1) 1.05 × 103 μS cm−1
−1
4 μ
−1
4 μ
(5) 1.45 × 10 S cm
(4) 1.25 × 10 S cm
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15' ødjKj,ska iudk m˙ud ñY% l< úg by< u iajdrCIl Od˙;djla ,nd fokafka my; i|yka l=uk
ødjK hq.,h u.ska o@
(1) 0.10 mol dm−3 CH3COOH iy 0.10 mol dm−3 CH3CO2Na
(2) 0.50 mol dm−3 CH3COOH iy 0.10 mol dm−3 CH3CO2Na
(3) 0.10 mol dm−3 CH3COOH iy 0.50 mol dm−3 CH3CO2Na
(4) 0.50 mol dm−3 NH4OH iy 0.50 mol dm−3 NH4Cl
(5) 0.10 mol dm−3 NH4OH iy 0.10 mol dm−3 CH3COOH
16' iïu; wjia:dfõ° yd WIaK;ajh 25 °C ys ° Ag+(aq), Fe2+(aq) iy Fe3+(aq) hk whkj, W;amdokhg
w∞< .síia Yla;s úmr®hdi my; olajd ;sfí'
whkh

ΔG °/ kJ mol−1

Ag+(aq)

+77.1

Fe (aq)

− 85.0

Fe3+(aq)

− 10.7

2+

=

iïu; wjia:dfõ° yd 25°C ys ° my; i|yka fldaIfha iuia; m%;sl%shdj i|yd .síia Yla;s fjki
(kJ j,ska) fldmuK o@
Pt(s)/Fe3+(aq, 1.00 mol dm−3), Fe2+(aq, 1.00 mol dm−3) Ag+ (aq, 1.00 mol dm−3)/Ag (s)
(1) -151. 4
(2) -18.6
(3) -2.8
(4) 18.6
(5) by; lsisjla fkdfõ
17' my; i|yka m%ldYj,ska l=uk tajd i;H fõ o@
(a) m%;sl%shdjla wjidkfha° W;afm%arlh ridhksl f,i fjkia fkdfõ'
(b) W;afm%arlh m%;sl%shdjl tka;e,amsh fjkia fkd lrhs'
(c) W;afm%arlh m%;sl%shdjl iu;=,s; ,CIHh fjkia fkd lrhs'
(d) W;afm%arlh" my;a il%shk Yla;shla iys; udr®.hla Tiafia m%;sl%shdjla is≥ùug i,ihs'
(1) (a) yd (b) mu◊
(2) (b) yd (c) mu◊
(3) (a), (b) yd (c) mu◊
(4) (a), (b) yd (d) mu◊
(5) (a), (b), (c) yd (d) ish,a, ksjer†h
18' oaú NdIañl wï,hl 0.400 g l wvx.= ødjKhla uqΩukska W∞iSk ls¯u i|yd idkaøKh 0.100 mol dm−3
NaOH ødjKhlska 40.00 cm3 m˙udjla wjYH fõ' wï,fha wdikak idfmaCI wKql ialkaOh
fldmuK o@
2
2
2
2
(1) 5.00 × 10
(2) 1.00 × 10
(3) 2.00 × 10
(4) 3.00 × 10
(5) 4.00 × 10
19' WIaK;ajh 200 °C ys ° my; m%;sl%shdfõ iu;=,s;;d ksh;h 1.6 × 103 fõ'
H2(g) + Br2(g)
2HBr(g)
−2
1.0 × 10 mol HBr(g) m%udKhla ixjD; Ndckhla ;=< 200 °C g r;a l< fyd;a iu;=,s;;djfha° tys
we;s H2(g) m%udKh fldmuK o@
(1) 80 × 10−2 mol
81
(4) 10 × 10−2 mol
81

(2) 1 × 10−2 mol
81
(5) 40 × 10−2 mol
81

(3) 20 × 10−2 mol
81

20' my; i|yka l=uk lr®udka;hl° fvd,uhsÜ (CaCO3.MgCO3) wuqøjHhla f,i Ndú; fõ o@
(1) isfuka;s
(2) laf,da¯ka
(3) CId,l
(4) hQ˙hd
(5) fidaähï ldnfkaÜ

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21' R-COONa hk CId,lh ;sîu ksid ldr®ñl wmøjH moaO;shl ridhksl Tlaiscka m%udKh ^COD&
b;d by< w.hla ù we;' ^wmøjH uyck c, udr®.j,g neyer ls¯fï° ta i|yd Y%S ,xld m%ñ;s w.hg
wkqj COD w.h wvq l< hq;= h'& wmøjH uyck c, udr®.j,g neyer ls¯fï° COD w.h wvq l<
fkdyels l%uh jkafka"
(1) jd;kh ls¯u h'
(2) ixY=oaO c,h úYd, m%udKhla Ndú; lr ;kql ls¯u h'
(3) (R-COO)2Ca ,jK f,i wjfCIam ls¯u h'
(4) idudkH ÆKq ^NaCl) wmøjHfhys †h ls¯u h'
(5) wmøjH wêfYdaIKh lrk øjH wvx.= l=ΩKla yryd heùu h'
22' my; ° we;s l=uk øjHh" wï, jeis ^yhsârcka whk& iu. m%;sl%shdfjka mdxY= moaO;sh yryd hk
c,fha fidaähï wêfYdaIK wkqmd;h ^SAR& jeä lrhs o@

(

SAR

=

(1) fvd,uhsÜ
23' >k
(1)
(3)
(5)

+

Na
√Ca2+ + Mg2+

idkaøKh mmol dm-3 u.ska

(2) le,aihsÜ

)

(3) fµ,aâiaµdr®

(4) is,sld

(5) .%ekhsÜ

f,day i,afµagj, c,fha ødjH;dj ksjer† j fmkakqï lrkafka my; l=uk mámdáh u.ska o@
BaSO4 > SrSO4 > CaSO4 > MgSO4
(2) BaSO4 < SrSO4 < CaSO4 < MgSO4
≈
>
≈
BaSO4 SrSO4 CaSO4 MgSO4
(4) BaSO4 > SrSO4 > CaSO4 ≈ MgSO4
BaSO4 < SrSO4 ≈ CaSO4 < MgSO4

24' my; ±lafjk l=uk m%ldY wi;H fõ o@
(a) ;d;aúl jdhq idïm,hl mSvkh" tys m%udKh" WIaK;ajh yd m˙udj iys; m˙mQr®K jdhq
idïm,hl mSvkhg jvd ieuúg u wvqfõ'
(b) ;d;aúl jdhq idïm,hl m˙udj" tys m%udKh" WIaK;ajh yd mSvkh iys; m˙mQr®K jdhq idïm,hl
m˙udjg jvd ieuúg u wvqfõ'
(c) ;d;aúl jdhqjl iïmSvH;d idOlh mSvkh u; rπ mj;S'
(d) b hk jeka vZ jd,aia ksh;fha m˙fïh SI tallh m3 fõ'
(1) (a) yd (b) mu◊'
(2) (b) yd (c) mu◊'
(3) (a), (b) yd (c) mu◊'
(4) (a), (b) yd (d) mu◊'
(5) (a), (c) yd (d) mu◊'
25' my; ±lafjk l=uk m%ldYh$m%ldY i;H fõ o@
(a) ksh; mSvkh hgf;a° Ok tkafg%dms úmr®hdihla iys; ;dm∞hl m%;sl%shdjla ieuúg u iajhxisoaO
fkdjkq we;'
(b) Ok tkafg%dms úmr®hdihla iys; m%;sl%shd ieuúg u iajhxisoaO jkq we;'
(c) RK tka;e,ams úmr®hdi iys; m%;sl%shd ieuúg u iajhxisoaO j is≥ fõ'
(d) ksh; mSvkh yd Ok tkafg%dms úmr®hdi iys; ;dm∞hl m%;sl%shd iajhxisoaO j is≥ fõ'
(1) (a) mu◊'
(2) (d) mu◊'
(3) (a) yd (b) mu◊'
(4) (c) yd (d) mu◊'
(5) (a), (b) yd (d) mu◊'
26' wdrïNfha ° l,dm ˚ ys ;snQ X kï ødjlhla tlsfkl iu. ñY% fkd jk l,dm ˚ yd l,dm ˚˚ w;r
jHdma; lrkq ,efí' l,dm ˚ isg l,dm ˚˚ fj;g ksiaidrKh lrkq ,nk X ys m%udKh iïnkaOfhka
my; ±lafjk l=uk m%ldY i;H fõ o@
(a) th l,dm ˚ ys m˙udj u; rπ mj;S'
(b) th l,dm ˚˚ ys m˙udj u; rπ mj;S'
(c) th l,dm foflys m˙ud wkqmd;h u; rπ fkd mj;S'
(d) th m¯CIKh mj;ajk WIaK;ajh u; rπ fkd mj;S'
(1) (a) yd (b) mu◊'
(2) (a) yd (d) mu◊'
(3) (b) yd (c) mu◊'
(4) (b) yd (d) mu◊'
(5) (c) yd (d) mu◊'
27' my; i|yka wjfCIamj,ska l=ula jeämqr wefudakshd ødjKh fh°fuka øjKh fõ o@
(1) fí˙hï i,afµaÜ
(2) whka^˚˚˚& yhsâfrdlaihsâ
(3) ksl,a^˚˚& yhsâfrdlaihsâ
(4) le,aishï Tlaif,aÜ
(5) ue.akSishï yhsâfrdlaihsâ
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28' fmd,af;,aj,ska inka ksIamdok l%shdj,sh i|yd my; i|yka m%;sl%shdj,ska l=ula iïnkaO fõ o@
(1) >kSNjkh
(2) c,úÉf√okh
(3) yhsârckSlrKh
(4) TlaislrKh
(5) tiag¯lrKh
29' my; i|yka tajdhska l=ula fid,afõ l%uh yd iïnkaO fkd fõ o@
(1) CaCO3

(2) NH4Cl

(3) NaCl

(4) (NH4)2C2O4

(5) Ca(OH)2

30' NaA idkaøKh 0.05 mol dm-3 jk f;la NaA ,jKh 0.10 mol dm-3 yhsâfrdlaf,da˙la wï,fha øjKh
lrk ,°' HA i|yd pKa = 6.0. by; ødjKfhka 25.0 cm3 fldgila idkaøKh 0.10 mol dm-3 fidaähï
yhsâfrdlaihsâ ødjKhla iu. wkqudmkh ls¯fuka ,nd .kakd wkqudmk jl%h my; i|yka l=uk
o< igykla u.ska ksrEmKh lrhs o@

(1) pH

14
12
10
8
6
4
2
0

(2) pH

5 10 15 20 25 30 35 40

14
12
10
8
6
4
2
0

(3)

pH

14
12
10
8
6
4
2
0

(4)

5 10 15 20 25 30 35 40

•

pH

14
12
10
8
6
4
2
0

pH

14
12
10
8
6
4
2
0

5 10 15 20 25 30 35 40

0.10 mol dm-3 NaOH
ødjK m˙udj$cm3

0.10 mol dm-3 NaOH
ødjK m˙udj$cm3

(5)

5 10 15 20 25 30 35 40

0.10 mol dm-3 NaOH
ødjK m˙udj$cm3

0.10 mol dm-3 NaOH
ødjK m˙udj$cm3

5 10 15 20 25 30 35 40

0.10 mol dm-3 NaOH
ødjK m˙udj$cm3

wxl 31 isg 40 f;la m%Yakj,g Wmfoia (
wxl 31 isg 40 f;la tla tla m%Yakh i|yd ° we;s (a), (b), (c) iy (d) hk m%;spdr y;r w;=frka tlla
fyda jeä ixLHdjla fyda ksjer† h' ksjer† m%;spdrh$m%;spdr ljfr® ±hs f;dard .kak'
(a) iy (b) muKla ksjer† kï ^1& u; o
(b) iy (c) muKla ksjer† kï ^2& u; o
(c) iy (d) muKla ksjer† kï ^3& u; o
(d) iy (a) muKla ksjer† kï ^4& u; o
fjk;a m%;spdr ixLHdjla fyda ixfhdackhla fyda ksjer† kï ^5& u; o
W;a;r m;%fhys ±lafjk Wmfoia m˙† ,l=Kq lrkak'

31' wdjr®;hl jfï isg ol=Kg hdu iïnkaOfhka my; l=uk j.ka;sh$j.ka;s i;H o@
(a) øjdxlh wvqfõ'
(b) mrudKqfõ m%udKh wvqfõ'
(c) f,dayuh ,CIK wvqfõ'
(d) úoHq;aRK;dj wvqfõ'
w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

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32' m%udKh 0'1nm yd 1μm mrdifha msysgk wxY= ie,l+ úg my; l=uk m%ldYh$m%ldY i;H fõ o@
(a) bf,lafg%daksl lr®udka;j, ° ldnka kefkda - áhqí Ndú; lrhs'
(b) f,â u.ska o µq,¯k iE†h yel'
(c) ;Ska; lr®udka;fha ° ldnka kefkda - áhqí Ndú; fõ'
(d) c,h mú;% ls¯fï ° ldnka kefkda - áhqí Ndú; lrhs'
33' CH3COCH2CH2COOCH3 hk ixfhda.h iïnkaO j my; ±lafjk ljr j.ka;sh$j.ka;s i;H fõ o@
(a) th c,Sh NaOH iu. r;al< úg CH3OH ,nd foa'
(b) th fí%ä m%;sldrlh iu. ;eô,s meye;s wjfCIamhla foa'
(c) .%Skdâ m%;sldrlh iu. we,aflakhla ,nd foa'
(d) fµa,sx ødjKh iu. r;al< úg .fvd,a r;= wjfCIamhla ,nd foa'
34' CH3CH2CH = CH2
CH3CH = CHCH3
by; ±lafjk m˙jr®;kh is≥ ls¯ug .; yels ksjer† mshjr jkafka"
(a) ^⁄& idkaø' H2SO4 ^⁄⁄& H2O ^⁄⁄⁄& idkaø H2SO4 / r;a ls¯u
(b) ^⁄& idkaø' H2SO4 ^⁄⁄& 1700C g r;a ls¯u ^⁄⁄⁄& H2O
(c) ^⁄& HBr ^⁄⁄& uOHid¯h KOH
(d) ^⁄& HBr / fmfrdlaihsv ^⁄⁄& uOHid¯h KOH
35' m%;sl%shd YS>%;d iïnkaOfhka my; l=uk m%ldYh$m%ldY i;H fõo@
(a) ridhksl m%;sl%shdjl° úúO m%;sl%shl bj;a jk idfmaCI YS>%;d m%;sl%shlj, iafgdhslsfhdñ;sl
ix.=Kl u; rπ mj;S'
(b) úoHq;aridhksl m%;sl%shdjl YS>%;dj Odrdj u; rπ mj;S'
(c) jdhquh l,dmfha is≥jk Y=kH fm< m%;sl%shdjl YS>%;dj" w∞< m%;sl%shlfha wdxYsl mSvkh fjkia
ls¯u u.ska fjkia l< yels h'
(d) m%;sl%shdjl YS>%;dj flfrys n,mdk ishÆ u idOlj, n,mEu il%shk Yla;sh u.ska ;SrKh
lrhs'
mSvkh /atm

36' ° we;s rEmigyk u; my; i|yka m%ldY mokï fõ' WIaK;ajh iu.
C
c,fha jdIam mSvkh úp,kh jk wdldrh fmkakqï lrkq ,nkafka l=uk
A
m%ldYh$m%ldY u.ska o@
1.0
B
(a) A ,CIHh u.ska c, jdIam ksrEmKh l< yels h'
D
(b) B g wkqrEm WIaK;ajhg jvd by< WIaK;ajhl ° øj c,h fkd mj;S'
(c) C ,CIHh u.ska c,fha ;dmdxlh ksrEmKh lrhs'
0.0
WIaK;ajh /K
(d) CD jl%h øj - jdIam iu;=,s;;dj ksrEmKh lrhs'
37' bf,lafg%dav iïnkaOfhka my; ° we;s m%ldY w;=˙ka ksjer† m%ldY$m%ldYh l=ula o@
(a) bf,lafg%dav iu;=,s;;djl°" TlaislrK yd TlaisyrK m%;sl%shd iudk YS>%;dj,ska is≥fõ'
(b) iu;=,s;;djfha mj;sk bf,lafg%davhlg" tlS iu;=,s; úNjhg jvd Ok jQ úNjhla fh¥ úg
bf,lafg%davfha iuia; TlaislrK m%;sl%shdj is≥fõ'
(c) iu;=,s;;djfha mj;sk bf,lafg%davhlg" tlS iu;=,s; úNjhg jvd RK jQ úNjhla fh¥ úg
bf,lafg%davfha iuia; TlaisyrK m%;sl%shdj is≥fõ'
(d) ,jK fia;=jla u.ska tlsfklg iïnkaO lrk ,o frfvdlaia bf,lafg%dav folla lïìhla
u.ska ndysrj iïnkaO l<fyd;a jvd RK úNjhla iys; bf,lafg%davfha° TlaisyrK m%;sl%shdjla
is≥ fõ'
38' m˙mQr®K jdhqjl jr®. uOHkH uQ, m%fõ.h iïnkaO my; ° we;s m%ldY i,lkak' tajdhska ksjer†
m%ldYh$m%ldY jkafka"
(a) th ;dm.;sl WIaK;ajhg wkqf,dauj iudkqmd;sl fõ'
(b) th idfmaCI wKql ialkaOfha jr®. uQ,hg m%;sf,dau f,i iudkqmd;sl fõ'
(c) th" uOHkH wKql pd,l Yla;sh we;s wKqjl fõ.hg iudk fõ'
(d) th moaO;sfha mSvkfhka iajdh;a; fõ'

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

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39' jeämqr ;kql HCl wï,h tla ls¯fuka øjKh l< yels ,jKh$,jK jkafka"
(b) whka^˚˚˚& yhsâfrdlaihsâ
(a) fí˙hï i,afµaÜ
(c) ksl,a^˚˚& yhsâfrdlaihsâ
(d) le,aishï Tlaif,aÜ
40' my; i|yka l%shdj,sh$l%shdj,s i|yd ∆G yd ∆H hk fol u RK jkafka"
(a) >k NaCl øjKh ls¯u
(b) øj fmÜfrda,shï oykh ls¯u
(c) wï,hla NIauhla u.ska W∞iSklrKh
(d) by< WIaK;ajfha ° le,aishï ldnfkaÜ úfhdackh

•

wxl 41 isg 50 f;la m%Yakj,g Wmfoia (
wxl 41 isg 50 f;la tla tla m%Yakh i|yd m%ldY fol ne.ska b†˙m;a lr we;' tu m%ldY hq.,hg
fyd¢ka u .e<fmkqfha my; j.=fjys ±lafjk m˙† ^1&" ^2&" ^3&" ^4& iy ^5& hk m%;spdrj,ska ljr
m%;spdrh ±hs f;dard W;a;r m;%fhys WÑ; f,i ,l=Kq lrkak'
m%;spdrh
(1)
(2)
(3)
(4)
(5)

41'

42'

m<uqjeks m%ldYh
i;H h'
i;H h'
i;H h'
wi;H h'
wi;H h'

fojeks m%ldYh
i;H jk w;r" m<uqjekak ksjer† j my∞ fohs'
i;H jk kuq;a m<uqjekak ksjer† j my∞ fkdfohs'
wi;H h'
i;H h'
wi;H h'

m<uqjk m%ldYh
fojk m%ldYh
ldur WIaK;ajfha ° wefudakshd jdhqj c,fha vhsT+laiscka fuka fkdj wefudakshd jdhqj
øjKh jk m%udKh vhsTlaisckaj, m%udKhg NH4 whkh ,nd°u i|yd c,h iu. m%;sl%shd
lrk w;r c,h yd iuyr wefudakshd wKq w;r
jvd fndfyda úYd,h'
yhsârcka nkaOk idohs'
1 jk ldKavfha uQ,øjH fuka fkdj 2jk khsÜrhsv we;s lrkafka oaú ixhqc legdhk
ldKavfha uQ,øjH ish,a, khsÜrhsv iE°u iE†h yels uQ,øjH mu◊'
i|yd N2 iu. m%;sl%shd lrhs'

43'

HF j, ;dmdxlh HCl j, ;dmdxlhg jvd
by< h'

øj yhsârcka µaf,darhsâ O%eùh fõ'

44'

Tlaiscka fuka fkdj i,aµr®j,g SF6 hk
ia:dhS µaf,darhsvh iE†h yels h'

i,aµr®" Tlaisckaj,g jvd wvqfjka úoHq;aRK
fõ'

45'

46'

47'

48'

s iy p f.dKqj, ´kEu ldKavhl m<uqjk s iy p f.dKqj, ´kEu ldKavhl m<uqjk
uQ,øjHfha .=K" f.dKqfõ wfkla uQ,øjHj, uQ,øjHh ldKavfha wfkla uQ,øjHj,g jvd
.=Kj,g jvd fjkia fõ'
m%udKfhka l=vd jk w;r tajdfha úoHq;aRK;dj
by< fõ'
t;fkdhs,a laf,darhsâ iy c,h w;r m%;sl%shdj" t;s,a laf,darhsâj, laf,da¯ka iy ldnka w;r
t;s,a laf,darhsâ yd c,h w;r m%;sl%shdjg jvd nkaOkh iyixhqc fõ'
myiqfjka is≥fõ'
CH3-NH2 j,g jvd CH3CONH2 m%n, NIauhls' CH3CONH2 ys N u; we;s tlir bf,lafg%dak
hq.,h fm%dafgdakhlg ∞hl ls¯u fu;s,a
weókaj,g jvd wmyiq h'
2+
2+
+
2 ppm Mg ødjKhl Mg idkaøKh 2 ppm K
Mg2+ whkfha ujq,sl wdfrdamKh K+ whkfha
+
ødjKhl K idkaøKhg jvd jeäfõ'
ujq,sl wdfrdamKhg jvd úYd, h'

49'

úoHq;a ksê.; ls¯u Yla;sh ;ekam;a lrk
ridhksl l%shdj,shla i|yd W∞yrKhla fõ'

úoHq;a wdf,amKfha° wdf,amK ;gdlfhys
WIaK;ajh by< hdfï m%jk;djla mj;S'

50'

mD:ßú jdhqf.da,fha vhsTlaiscka m%udK ixhq;sh
tys vhskhsg%cka m%udK ixhq;shg jvd y;r
.=Khla muK l=vd h'

ldur WIaK;ajfha° vhsTlaiscka wKqj,
uOHkH fõ.h mD:ßúfha úfhda. m%fõ.h
blaujd hhs'

***
w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

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02 - ridhk úoHdj
*
*

˚˚ m;%h

A fldgfia m%Yak ish,a,g u ms<s;=re imhkak'
B fldgfika m%Yak folla o C fldgfika m%Yak folla o ne.ska f;dardf.k m%Yak y;rlg ms<s;=re
imhkak'
A fldgi - jHQy.; rpkd

1. (a) ó;hs,a whsfidihfkaÜ (CH3NCO) hkq úI jdhqjls' 1984 foieïnr® udifha ° bka†hdfõ fndamd,a ys
fuu jdhqj y†is wk;=rlska jdhqf.da,hg ksoyia ùu ksid mqoa.,hska 3000 lg muK urKh f.k
≥kafkah'
		
my; olajd we;s CH3NCO wKqfjys uQ,sl iels,a, mokï lrf.k my; m%Yakhg ms<s;=re
(1)
(2)
imhkak'
				
H

- -

H-C-N-C-O
(1)

(2)

H
			
		
(i) jvd;a ms<s.; yels Æúia jHqyh w|skak'
		

			
			
			
		

(ii) C , C yd N mrudKq jgd yevh VSEPR jdoh Ndú;fhka wfmdayKh lrkak'

		

(1)

(2)

			
			
			
			
			
			
			
		

(iii) iïm%hqla; jHqy we|" CH3NCO ys iïm%hqla; uqyqfï ia:dhS;djhg wvqfjkau ∞hl jk yd
jeäfhkau ∞hl jk jHqy y∫kd .kak' Tfí f;a¯u i|yd fya;= fláfhka olajkak'

			
			
			
			
			
			
		
		

(iv) C , C yd N mrudKqj, uqyqïlrK olajkak'
(1)

(2)

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

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(v) by; (i) ys jHqyfha yevh lgq igykl we| tys s nkaOk" p nkaOk yd tlir hq.,a
olajkak'
		

			
			
			
			
		

(vi) by; (i) jHqyfha s C − N yd s C − H nkaOk iE°ug ∞hl jk uqyqï yd mrudKql ldlaIsl
y∫kd .kak'

			
(b) jryka ;=< olajd we;s ,laIK mokï lrf.k my; olajd we;s ≤ wdfrdayK wkqms<sfj,g
ilikak' Tfí f;a¯u i|yd fya;= fláfhka olajkak'
		 (i) MgCO3, BaCO3, SrCO3 ^úfhdack WIaK;ajh&
		

			
			
			
			
		

		

(ii) NO2 , NO2 (khsg%cka mrudKqfõ úoHq;a >DK;dj)
−

+

			
			
			
2.

A kï uQ, øjH s f.dkqjg wh;a fõ' th isis,a c,h iu. fiñka m%;sls%hd lrñka wjr®K B kï jdhqjla
msglr C kï ødjKhla ,ndfoa' A uQ, øjHh N2 jdhqj iu. m%;sls%hd lr D kï iq≥ >khla idohs' C
ødjKh yryd CO2 jdhqj nqnq,kh l< úg" E kï iq≥ wjlafIamhla iE° th kej; øjKh ù meye†,s
ødjKhla ,ndfoa' E 900 ° C g r;al< úg F kï iq≥ >kh iEfoa' th ldnka iu. 2000 °C g r;al<
úg G ixfhda.h idohs' G ixfhda.h c,h iu. yhsfv%dldnka jdhqjla jk jd◊c jákdlula we;s
H ,ndfoa' iq≥ meye;s >khla jk E myka isΩ m˙laIdjg ,lal< úg .fvd,a r;= jr®Khla ,ndfoa'
^A, B, C, D, E, F, G yd H hk wlaIr" ixfhda. fyda uQ, øjH fyda mrudKqj, ixfla; fkdfõ'&
(a) A isg H olajd ixfhda. y∫kd f.k tajdfha ridhksl iQ;% my; fldgqj, ,shkak'

A

B

C

D

E

F

G

H

			

		

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

- 39 -

(b) A yd N2 w;r m%;sls%hdj;a F yd ldnka w;r m%;sls%hdj;a i|yd ;=,s; iólrK ,shkak'

		
		
		

(c) E, F, G yd H ys ldr®ñl Ndú;hla ne.ska olajkak'			
		

			

		
3. (a) jdhqjl iïmSvH;d idOlh" Z i|yd .◊;uh m%ldYkhla iólrKhl wdldrfhka ,shkak' tu
iólrKfhys Ndú; lr we;s ishÆ u mo y∫kajkak'
		
		

			

		
		
(b) by; (a) fldgfiys ±lajQ iólrKh Ndú;fhka jdhqjl >k;ajh i|yd iïnkaO;djla wfmdayKh
lrkak'
		
		

			

		

			

		

			

		
(c) wd.ka yd yhsv%cka jdhqj,ska muKla iukaú; jdhq ñY%Khl wd.kays ujq, Nd.h 0.650 fõ' mSvkh
2
1.50 bar yd WIaK;ajh 3.00 × 10 K ys we;s jdhq ñY%Khl >k;ajh .Kkh lrkak' .Kkfha°
Tn Ndú; l< Wml,amk olajkak'
		
idfmalaI mrudKql ialkaO ( Ar = 40, H = 1			
		

			

		

			

		

			

		

			
w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

- 40 -

		
(d) (i) m˙mQr®K jdhq i|yd jk m˙mQr®K jdhq iólrKh yd wKql pd,l iólrKh Ndú;fhka jdhq
wxY=jla i|yd jr®. uOHkH uQ, m%fõ.h ^√u2 & i|yd ujq,sl ialkaOh wka;r®.; m%ldYkhla
jHq;amkak lrkak'
			
				
				
				
		 (ii) He mrudKqj i|yd 25°C ° jr®. uOHkH uQ, m%fõ.h" m s−1 hk tallj,ska .Kkh lrkak'
			 ^He = 4, ñ≥ï ^úfhda.& m%fõ.h 1.1 × 103 m s−1)
		

				
				
				
		

(iii) mD:ßú jdhq f.da,fha He b;d my< idfmaCI iq,N;djlska mej;Sug fya;=jk tla jeo.;a
idOlhla olajkak'

		

				
				
				
				
4. (a) (i) we,aäyhsâ yd lSfgdak hk folu olajk ,dlaI◊l m%;sls%hd jr®.h kï lrkak'
			
		

		

(ii) weisge,aäyhsâ (ethanal), yhsv%cka ihkhsâ iu. m%;sls%hdfjka iEfok M,fha jHqyh w¢kak'

			
			
		

(iii) by; (ii) fldgfiys Tn ±lajQ jHqyhg w∞< ixfhda.fha m%ldY ilS%h wdldr mj;S o hkak
olajkak' Tfí ms<s;=r i|yd fya;= olajkak'

			
			
			
		

(iv) by; (ii) fldgfiys ±lajQ m%;sls%hdj i|yd hdka;%Khla b†˙m;a lrkak'		

			
			
			

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

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(b) (i)

ridhksl m¯laIKhla Ndú;fhka weisge,aäyhsâ yd weisfgdaka (propanone) fjka lr y∫kd
.kafka flfia±hs úia;r lrkak'			

			
		

(ii) by; (b) (i) fldgfiys ±lajQ m¯CIKfha ridhksl moku meye†,s lrkak' 			

			
			
		 (iii) weisfgdaka" 2,4-dinitrophenylhydrazine iu. m%;sl%shd ls¯fuka iEfok M,fha jHqyh yd jr®Kh
,shkak'			
		
			

(c) (i)

ldnkavfhdlaihsâ c,fha †h l< úg iEfok ixfhda.h kï lrkak'		

			
		

(ii) by; (c) (i) ys i|yka l< m%;sl%shdjg w∞< ;=<s; ridhksl iólrKh olajkak'

			

		 (iii) ldnksl m%;sl%shd hdka;%K ms<sn| Tfí ±kqu Ndú;fhka by; (c) (ii) fldgfiys ±lajQ m%;sl%shdj
i|yd hdka;%Khla fhdackd lrkak'				
			

			
			

**

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

- 42 -

B fldgi - rpkd
5. (a) (i) is,ajr® - is,ajr® laf,darhsâ bf,lafg%davfha rEm igykla we| tys we;s ishÆ jeo.;a fldgia
kï lrkak'
		 (ii) bf,lafg%dav iu;=,s; m%;sl%shdj ,shkak'
		 (iii) fuu bf,lag%davfha bf,lafg%dav úNjh rπ mj;sk lreKq folla i|yka lrkak'
		 (iv) is,ajr® - is,ajr® laf,darhsâ bf,lafg%davfha m%dfhda.sl Ndú;h i|yd tla W∞yrKhla fokak'
(v) m%dfhda.slj ikaikaokd;aul bf,lafg%davh f,i iïu; yhsv%cka bf,lafg%davhg jvd fuh
Ndú; ls¯fï fhda.H;dj meye†,s lrkak'
(b) A ixfhda.h B yd C njg c,Sh ødjKhl ° my; m˙† ;dm úfhdackh fõ'
			
2A(aq)
2B(aq) + C(aq)
		 by; m%;sl%shdfõ pd,l wOHhkhl ° ,nd.;a o;a; my; ±lafõ'
		 fok ,o ksh; WIaK;ajhl ° A(aq) ødjK folla i|yd wdrïNl idkaøK ; 1.0 × 10−2 mol dm−3 yd
2.0 × 10−2 mol dm−3 fõ' ødjKfhka A(aq) bj;aùfï wdrïNl YS>%;d ms<sfj<ska 1.2 × 10−5 mol dm−3 s−1
		 yd 2.4 × 10−5 mol dm− 3 s−1 úh' ñks;a;= 9'5l ld,hl ° ødjKfha we;s A(aq) idkaøKh
0.5 × 10−2 mol dm−3 isg 0.25 × 10−2 mol dm−3 olajd wvq ù we;'
		
^i& by; m%;sl%shdfõ YS>%;dj i|yd .◊;uh m%ldYkhla ,shkak'
		 ^ii& by; o;a; Ndú;fhka"
			 (1) A(aq) g idfmaCIj m%;sl%shdfõ fm<"
			 (2) YS>%;d ksh;h
			 .Kkh lrkak'
		 ^iii& (1) m%;sl%shdfõ wr®O „j ld,h"
			 (2) ødjKfha b;s˙j we;s A(aq) idkaøKh 2.4 × 10−3 mol dm−3 isg 1.20 × 10−3 mol dm−3 olajd
wvqùug .;jk ld,h
			 ksr®Kh lrkak'
		 ^iv& (1) m%;sl%shdfõ uOHkH fõ.h .Kkh lrkak'
			 (2) ridhksl m%;sl%shdjl YS>%;d kshuh ksjer† j fiùu i|yd uOHkH fõ.h ishÆ m%;sl%shd
i|yd Ndú; fkdlrkafka uka o@
		
^v& f;dard.;a ld, lsysmhl° by; m%;sl%shdfõ uksk ,o b;s˙ A(aq) idkaøKj, o;a; Tng ,nd °
we;' jdlH ;=klg fkdjeäj Ndú; lr" tu o;a; wdY%fhka by; m%;sl%shdfõ wdrïNl YS>%;dj
yd fudfyd;l YS>%;dj .Kkh lrk wdldrh fláfhka úia;r lrkak'
(c) (i) rjq,a kshuh .◊;uh m%ldYkhl wdldrfhka olajkak'
		 (ii) m<uqj 100°C g r;a lrk ,o idkaø fidaähï laf,darhsâ ødjKhlska l=vd m˙udjla tu
WIaK;ajfhau we;s fkdlvjd r;a lrk kgk c, ;dmlhlg tl;= lrk ,°' CI◊lj keàu
k;r ù flá ld,hl° kej; ødjKh kgk ,°' fuu ks¯CIKh rjq,a kshuh Ndú;fhka
meye†,s lrkak'
6. (a) ;r®uhsÜ m%;sl%shdfõ° ksoyiajk Yla;sh keõj, úYd, tall jk wjrfm;s mEiaiSfï° Ndú; lrhs'
;r®uhsÜ m%;sl%shd i|yd w∞< iïu; ;;a;aj m%;sl%shdj my; ±lafõ'
			 2 X(s) + Fe2O3(s)
X2O3(s) + 2 Fe(s)
			 fuys X hkq c,Sh fidaähï yhsfv%dlaihsâ yd c,Sh yhsfv%dlaf,da˙la wï,h iu. m%;sl%shdfjka
vhsyhsv%cka jdhqj ,ndfok p f.dkqfõ uQ, øjHhls' X2O3(s) yd Fe2O3(s) ys 25°C ° iïu; W;amdok
1
1
tka;e,ams ms<sfj<ska −1675 kJ mol− yd − 825 kJ mol− fõ'
			 (i) X uQ,øjHh y∫kd.kak'
(ii) 25°C ° ;r®uhsÜ m%;sl%shdj i|yd DH° .Kkh lrkak'
		 (iii) ;r®uhsÜ m%;sl%shdfõ 25°C ° iïu; tkafg%dms úmr®hdih" DS° 15'2 J K−1fõ' 25°C ° ;r®uhsÜ
m%;sl%shdj i|yd iïu; .síia Yla;s fjki DG° .Kkh lrkak'
(iv) fl%dañhï f,dayh 5'20 g la tys øjdxlfha° ú,hkh ls¯u i|yd wjYH X m%udKh ^ujq,j,ska&
.Kkh lrkak'
				 (Cr(s) ú,hk tka;e,amsh 14'6 kJ mol−1' idfmaCI mrudKql ialkaOh Cr = 52.0)

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

- 43 -

(b) 500 K we;s ixjD; Ndckhl is≥jk my; ±lafjk jdhq l,dm iu;=,s;h i,lkak'
		 A(g) + B(g)
2C(g)
		 A, B yd C hk ixfla; u.ska ksrEmKh lrkafka" jdhqj, ridhksl iQ;% fõ' A(g) yd B(g) jdhqj,
wdrïNl m%udK 2.0 mol ne.ska jqjo wdrïNfha ° C(g) ke;'
(i) 500 K ° by; m%;sl%shdfõ iu;=,s;;d ksh;h Kp 4'0 fõ' iu;=,s;;djfha° ú>gkh jQ A(g)
m%;sY;h .Kkh lrkak'
(ii) fjk;a WIaK;ajhl° A(g) ys ú>gk m%;sY;h 40] fõ kï tu WIaK;ajfha° by; m%;sl%shdj
i|yd Kp .Kkh lrkak'
7' (a)

		
		
		
		
		

3d wdka;˙l f,dayhla jk M ys ic, i,afµaghla c,fha †h l< úg oïmdg A ødjKhla ,nd foa'
idkaø wefudakshd ìx≥ jYfhka oïmdg A ødjKhg tl;= l< úg ,d fld< mdg wjfCIamh C iE°
jeämqr idkaø wefudakshdys øjKh ù oïmdg ødjKh D ,ndfoa' ,d fld< mdg wjfCIamh (C) c,Sh
NaOH ys ødjH ù fld< mdg ødjKh E ,ndfoa' fld<mdg ødjKhg H2O2 tl;= lr r;al< úg
lymdg ødjKh F iEfoa' F ødjKh ;kql H2SO4 iu. ms˙hï l< úg ;eô,s mdg ødjKh G ,nd
foa' F ødjKh" SO2 iu. NdIañl udOHfha ° m%;sl%shd lr" ødjKfha NdIañl m%n,;dj u; ,d fld<
mdg wjfCIamh C fyda fld< mdg ødjKh E ,nd foa'
(i) by; ks¯CIK u.ska M wdka;˙l uQ, øjHh y∫kdf.k tys bf,lafg%daksl úkHdih yd iq,N
TlaislrK ;;a;ajh$;;a;aj olajkak'
(ii) A, B, C, D, E, F yd G ys we;s úfYaI y∫kd.kak'
(iii) F yd G iE°ug yd F NdIañl ;;a;aj hgf;a SO2 iu. m%;sl%shdjg w∞< ;=<s; whksl iólrK
olajkak'
(iv) G ødjKfha we;s úfYaIfha yd (i) ys y∫kd.;a wdka;˙l uQ,øjHh M ys m%fhdackhla
olajkak'
(v) E ys IUPAC kduh olajkak'

(b) (i) úoHq;a pqïnl úlsrK hkqfjka Tn woyia lrkafka l=ula o@
		 (ii) ;rx. wdhduh wdikakj 325nm jk iQr®hdf,dalh u.ska ysre ms<siaiqï we;sfõ' ;rx. wdhduh
325 nmg w∞< úlsrKfha my; ≤ .Kkh lrkak'
−34
			
(C = 3 × 108 ms−1, h = 6.63 × 10 Js)
			
(1) úlsrKfha ixLHd;h
			
(2) úlsrKfha fmdafgdakhl Yla;sh
			
(3) fmdafgdak ujq,hl Yla;sh
			
(4) Yla;sh 1.0 × 10−3 J g wkqrEm fmdafgdak .Kk
		 (iii) fuu úlsrKh úoHq;a pqïnl jr®Kdj,sfha l=uk m%foaYhg wh;a fõ o@

C fldgi - rpkd
8. (a) jdhqf.da,fha 20] la Tlaiscka o 0'03] la ldnkavfhdlaihsâ o we;' jdhqf.da,h ixjD; moaO;shla
f,i Wml,amkh lrñka yd wvx.= jdhqj, idfmaCI wKql ialkaO tf,iu i,lñka my; i|yka
≤ .Kkh lrkak'
		 ^⁄& jeis c,fha ødú; Tlaiscka m%udKh (mg dm-3 u.sk)a
		 (ii) jeis c,fha wdikak pH w.h
−
−
−
			
Tlaiscka i|yd" KH = 1.3 × 10 8 mol dm 3 Pa 1.
−7
−
−
			
ldnkavfhdlaihsâ i|yd" KH = 3.8 × 10 mol dm 3 Pa 1.
			
ldfndksla weisâ i|yd" pKa = 6.37, pKa = 10.33.
1
2
			
idfmaCI mrudKql ialkaO : C = 12, O = 16.		
		

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

- 44 -

(b)
				
M

A
N

B

1. F
2. G

Ca(HCO3)2(l)
		
		
		
		

fmdiafmaÜ
CH
fmdfydr (s) 2 2(g)
L
I
K
J

+ CO2(g)

C

jeämqr

F

E

E

H

D

úrxck
l=vq (s)

.e,Sï igyfka olajd we;s fmdiafmaÜ fmdfydr" úrxck l=vq yd C2H2 ksIamdokh i,lkak'
(i) ;%sfldaKfha we;s wdrïNl øjH A" i|yd iajdNdúlj we;s iq,N iïm;l ku i|yka lrkak'
(ii) fldgq yd jD;a; ;=< w∞< øjH i|yd ridhksl iQ;% tajdfha fN!;sl wjia:d iu. ,shkak'
(iii) l%shdj,sfha M yd N ;;a;aj olajkak'

9. (a) Na2A ys idkaøKh 0'05 mol dm−3 jk f;la Na2A ,jKh c,fha øjKh lrk ,°'
		 (i) by; ødjKfhka 25'00 cm3 l fldgila 0'10 mol dm−3 jk HCl ødjKhla iu. wkqudmkh
lrk ,°' wjYH jk HCl m˙udj .Kkh lr" wkqudmk jl%fha lgq igykla w¢kak' jl%h
f,an,a lrkak' ^H2A ys Pka1 yd Pka2 w.hhka ms<sfj,ska 1'0 yd 7'1 fõ'&
		 (ii) by; ,jK ødjKfhka 1.0 dm3la pH w.h 7'4la jk iajdrCIl ødjKhla njg m;a ls¯ug
wjYH HCl m%udKh wjYH iólrKh$iólrK idOkh lrñka .Kkh lrkak'

CH2

H

=

N

C

=

N

O

-

[

H

CH2

-

								
(b) jd◊c nyqwjhjlhl ∞ufhka fldgila rEmfha
±lafõ' th ix.Kk nyqwjhùlrKfhka
iEfok M,hls' tu nyqwjhjlh idok ;ekqï
tallfha jHqyh w¢kak'

C

]

O
(c) (i) jdhqf.da,fha we;s ´fidaka úhk ;=kS ls¯ug laf,da¯ka iys; ixfhda. jeämqr ∞hl jk nj
meye†,s lreKls' by; m%ldYh iq≥iq ridhksl m%;sl%shd Ndú;fhka myokak'
		
(ii) jr®;udkfha wï, jeis ms<sn| j wjOdkh fhduq ù we;' kuq;a NIau jeis .ek wjOdkhla
fhduq fkdlrkafka ukao hkak olajkak'
		 (iii) úoHq;a wdf,amkfha° f,day whk u.ska m˙ir ¥IKh is≥úh yels wjia:d myla olajkak'
10. (a) A ixfhda.h (C4H10O) idkaø H2SO4 iu. r;als¯fï° B ixfhda.h (C4H8) iEfoa' Lksc wï,hla we;s
úg B c,h iu. m%;sl%shd lr C ixfhda.h idohs' C ixfhda.h A ys iudjhúlhla fõ' C idkaø
H2SO4 iu. r;al< úg myiqfjka B iEfoa' uD≥ ;;a;aj hgf;a° C wdï,sl K2Cr2O7 iu. m%;sl%shd
fkdlrhs'
		
(i) A" B yd C y∫kdf.k w∞< m%;sl%shd olajkak'
		 (ii) A yd C fjkalr y∫kd .; yels ridhksl m¯CIKhla yd tys ks¯CIK olajkak'
		 (iii) my; m%;sl%shdfõ D yd E ys jHqy olajkak'
			

A

PCC/CH2Cl2

D

;

A

K2Cr2O7 / H

+

E

			 * PCC = pyridinium cholorochromate
		 (iv) A yd C ys mej;sh yels m%ldY iudjhúl;djg w∞< iudjhúlj, jHqy olajkak'
(b) we,alSk" uOHidr yd we,als,a fya,hsv ms<sn| Tfí ±kqu Ndú;fhka my; m˙jr®;kh l< yels
wdldrhla olajkak'
CH3
			

CH3 CH = CH2

CH3 CH O CH2 CH2 CH3

***

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

- 45 -

07 - .◊;h
m%Yak m;% jHQyh
˚ m;%h - ld,h ( meh 03 hs'
			 fuu m%Yak m;%h fldgia follska iukaú; h'
			 A fldgi - m%Yak oyhls' m%Yak ish,a,g u ms<s;=re iemhsh hq;= h' tla
m%Yakhlg ,l=Kq 25 ne.ska ,l=Kq 250 ls'
			 B fldgi - m%Yak y;ls' m%Yak mylg ms<s;=re iemhsh hq;= h' tlaa
m%Yakhlg ,l=Kq 150 ne.ska ,l=Kq 750 ls'
			 ˚ m;%h i|yd uqΩ ,l=Kq 1000 fi 10 } 100
˚˚ m;%h - ld,h ( meh 03 hs'
			 fuu m%Yak m;%h fldgia follska iukaú; h'
			 A fldgi - m%Yak oyhls' m%Yak ish,a,g u ms<s;=re iemhsh hq;= h' tla
m%Yakhlg ,l=Kq 25 ne.ska ,l=Kq 250 ls'
			 B fldgi - m%Yak y;ls' m%Yak mylg ms<s;=re iemhsh hq;= h' tla
m%Yakhlg ,l=Kq 150 ne.ska ,l=Kq 750 ls'
			 ˚˚ m;%h i|yd uqΩ ,l=Kq 1000 fi 10 } 100
wjidk ,l=K .Kkh ls¯u
			
				

(

˚ m;%h
˚˚ m;%h
wjidk ,l=K

}
}
}

100					
100
200 fi 2 } 100

ie,lsh hq;=hs (

*

*

fuys .◊;h ˚ yd ˚˚ m%Yak m;%j, A fldgi i|yd uQ,dlD;s m%Yak 30 ne.ska o
B fldgi i|yd uQ,dlD;s m%Yak 21 ne.ska o we;=<;a lr we;' tfia jqjo
úNd.fha ° fokq ,nk m%Yak m;%h ilia flfrkqfha by; i|yka m%Yak m;%
jHqyhg wkql+,j h'
úNd.fha ° A fldgig m%Yak m;%fha u tla tla m%Yakh i|yd imhd we;s bv
m%udKh ;=< ms<s;=re iemhsh hq;= h'

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

- 46 -

Mathematics I: Part A - 18 11 2010

07 - .Ks;h
I m;%h i|yd uQ,dlD;s m%Yk
a
A fldgi
1. U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} hkq ir®j;% l=,lhla hehs .ksu'q
A = {1, 4, 7, 10} , B = {1, 2, 3, 4, 5} yd C = {2, 4, 6} hkq U ys Wml=,l ;=kla hehs .ksu'q
iqmr
q e≥ wxlkfhka my; oelafjk oE ksrK
® h lrkak'

(i) A′ ∪ C , (ii) B − C ′, (iii) C ys n, l=,lh'
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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2. tla;rd k.rhl ñksiqka 300 la z,laforKZ mqj;a m; lshjk nj;a" ñksiqka 200 la
z,lamykZ mqj;a m; lshjk nj;a" ñksiqka 50 la mqj;a m;a folu lshjk nj;a
ióCIKhlska wkdjrKh úh'
ióCIKhg ,la lrk ,o ñksik
q f
a .ka fldmuK .Kkla fuu mqj;a m;a foflka
(i) hg;a ms˙fihska tlla j;a"
(ii) y˙hgu tlla
lshjkafka oehs fidhkak'
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

- 47 -

Mathematics I: Part A - 18 11 2010

3. R = {( m, n) : m < n} hkq Z u; wr®: oelafjk iïnkaOhla hehs .ksu'q
fuu iïnkaOh
(i) mrdjr®;"S
(ii) iuñ;Sh"
(iii) ixl%duH
oehs ksrK
® h lrkak'
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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1
, x ∈ IR yd x ≠ − 1 hk Y%;
s i,luq'
1+ x
g º f (x) yd tys mrdih fidhkak'
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

4. f ( x) = 3 − x 2 , x ∈ IR yd g ( x) =

'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

- 48 -

Mathematics I: Part A - 18 11 2010

5. P( x) ≡ x 3 + kx 2 − 2 x + 1 hehs .ksuq; fuyss k ∈ IR fõ' ( x − k ) hkafkka P(x) fn¥úg ,efnk
fYaIh k fjhs' k i|yd úh yels ishÆ w.hka fidhkak'
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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6. x 2 + bx + c = 0 jr®.c iólrKfha uQ, α yd β hehs .ksu;q fuys c ≠ 0 fõ'

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1
1
yd β + uQ, jk jr®.c iólrKh fidhkak'
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α

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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

- 49 -

Mathematics I: Part A - 18 11 2010

7. ishÆ x i|yd 4 x( x + k ) − k (3k − 4) m%ldYkh Ok jkfia k ys w.h l=,lh fidhkak'
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8.

x2 + x − 3
( x − 2)( x 2 + 2)

ys Nskk
a Nd. fidhkak'

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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

- 50 -

Mathematics I: Part A - 18 11 2010

9. log 2 3 × log 3 4 × log 4 5 × log 5 6 × log 6 7 × log 7 8 = 3 nj fmkajkak'
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10. isiqjl= i;=j Y=oaO .Ks;h ms<sn| fjkia fmd;a 6 la yd jHjyd˙l .Ks;h ms<sn| fjkia
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tl <Û"
(ii) Y=oO
a .Ks;h ms<n
s | fmd;a ish,a, tl <Û
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

- 51 -

Mathematics I: Part A - 18 11 2010

11. ERROR hk jpkfha wl=re" jrlg ish,a, u f.k iE†h yels fjkia ixlrK .Kk
fidhkak'
ERROR hk jpkfha wl=rej,ska" wl=re ;=k ne.ska f.k iE†h yels fjkia f;dard .ekSï
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14

3

12.  2 x +  ys oaúmo m%idrKfha idOdrK moh ,shd olajkak'
x

iudk ix.=Kl iys; wkqhd; mo y˙hg u folla we;s nj fmkajkak'
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

- 52 -

Mathematics I: Part A - 18 11 2010

13. .Ks; wNHqyk uQ,Or®uh fhdod .ksñka" ´kEu n Ok ksÅ,hla i|yd
n(2n − 1)(2n + 1)
nj idOkh lrkak'
3
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12 + 3 2 + 5 2 + L + (2n − 1) 2 =

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14.

n

∑ r yd
r =1

n

n
1
2
i|yd
iïu;
iQ
;
%
fhdod
.ks
ñ
ka
"
r
∑
∑ r (r + 1) = 3 n(n + 1)(n + 2) nj idOkh
r =1
r =1

lrkak'
n

ta khsk"a

∑ r (r − 1) fidhkak'
r =1

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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

- 53 -

Mathematics I: Part A - 18 11 2010

15. ksYp
a dhlj, .=K fhdod .ksñka
1
1
1
cos θ
sin θ 1 = (1 − cos θ )(1 − sin θ )(sin θ − cos θ ) nj fmkajkak'
2
cos θ sin 2 θ 1
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16. kx + 2 y = 0
2x + 4 y = 1
iólrK moaO;sh kHdi wdldrfhka m%ldY lrkak; fuys k hkq ;d;a;ú
a l ksh;hls'
ta khsk"a moaO;shg wkkH úi∫ula mej;Su i|yd k g ;sìh hq;= w.hka fidhkak'
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

- 54 -

Mathematics I: Part A - 18 11 2010

 2 1
 yd
17. A = 
 − 3 2

 3 2
 hehs .ksu'q
B = 
−1 1

iqmqre≥ wxlkfhka" A T , B T , AB yd B T A T fidhd (AB )T = B T A T nj i;Hdmkh
lrkak'
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 2 3
 hehs .ksu'q
18. A = 
 0 1
A 2 − 3A + 2I = O nj fmkajkak' fuys I hkq 2 jk .Kfha tall kHdih o" O hkq 2 jk
.Kfha Y=kH kHdih o fõ'

ta khsk"a A −1 fidhkak'
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

- 55 -

Mathematics I: Part A - 18 11 2010

19. A, B yd C ,CIHj, LKavdxl ms<sfj<ska (−2, 3 ), (1, 1 ) yd (1, 4) fjhs' B ,CIHfha isg AC
fr®Ldjg we¢ ,ïnfha iólrKh iy tu ,ïnfha wäfha LKavdxl fidhkak'
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20. x − 2 y + 7 = 0 yd 3 x + y − 2 = 0 fr®Ldj, f√ok ,CIHh yd (1, 1) ,CIHh Tiafia hk
fr®Ldfõ iólrKh fidhkak'
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

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Mathematics I: Part A - 18 11 2010

21. x − 2 y + 7 = 0 yd 2 x − y + 2 = 0 fr®Ld w;r fldaKj, iuÉf√ol x-wCIh f√okh lrk
,CIHj, LKavdxl fidhkak'
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22. (−4, − 3 ) yd (−6, − 1) ,CIH hd lrk ir, fr®Ld LKavh úIalïNhla f,i we;s jD;a;fha
iólrKh ,nd.kak'
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

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Mathematics I: Part A - 18 11 2010

23. x = 5 fr®Ldj x 2 + y 2 − 4 x − 8 y + 11 = 0 jD;a;hg iamr®Ylhla nj fmkajkak' iamr®Y
,CIHfha LKavdxl fidhkak'
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24. A(−6, 0) ,CIHh

x 2 + y 2 − 4 x + 2 y − 44 = 0

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s k nj fmkajd

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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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Mathematics I: Part A - 18 11 2010

 x cot x 
25. lim 
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x→0 1 + cos x 
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% dj 1.6 m 2 /min fjhs; fuys
r hkq neÆkfha wrh ógrj,ska fjhs' r =

2
jk úg wrh jeäùfï iS>;
% dj fidhkak'
π

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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

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Mathematics I: Part A - 18 11 2010

27. y = e −2t cos 3t kï

d2 y
dt

2

+4

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+ 13 y = 0 nj fmkajkak'
dt

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28. x = 1 jk úg f ( x) = xe − x Y%;
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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Mathematics I: Part A - 18 11 2010

29. y = e 2 x − 4 x + 8 jl%h y-wCIh yuqjk ,CIHfha ° jl%hg we¢ iamr®Ylfha iólrKh
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30. f ( x) = 3 x 5 + 6 x 4 − 4 x 3 + 1 Y%;
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

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10 11 2010

B fldgi
1. (a) A, B yd C hkq fok ,o S l=,lhl Wml=,l ;=kla hehs .ksu'q
fhdod .kq ,nk l=,ljdofha tk ù„h kshu meye†,sj m%ldY lrñka" iqmr
q e≥ wxlkfhka
′
(i) A − (A ∩ B ) = A ,
(ii) ( A ∩ B) − (A ∩ C ) = A ∩ ( B − C )
nj idOkh lrkak'
(b) t<fUk w'fmd'i' ^Wiia fm<& úNd.fha° b;sydih úIh i|yd isiqka 96 l lKavdhula
fmkS isàug wfmaCId lr;s' fuu isiqka 96 fokdf.ka" isiqka 43 fofkla w;sfr®l
mka;sj,g iyNd.s fj;s , isiqka 38 fofkla .eyekq <uhska fj;s" isiqka 18 fofkla
kej; fmkS isák isiqka fj;s. kej; fmkS isák isiqka 11 fofkla muKla w;sfr®l
mka;sj,g iyNd.s fj;s' kej; fmkS isák isiqka 10 fofkla .eyekq <uhska fj;s's
.eyekq <uhska 7 fofkla w;sfr®l mka;sj,g iyNd.s fj;s' w;sfr®l mka;sj,g iyNd.s
jk isik
q f
a .ka 6 fofkla úNd.hg kej; fmkS isák .eyekq <uhska fj;s.s
ms˙ñ <uhska lS fofkla
(i) w;sfr®l mka;j
s ,g iyNd.s fj;a oehs"
(ii) úNd.h i|yd kej; fmkS isá;a oehs"
(iii) w;sfr®l mka;j
s ,g iyNd.s fkdù m<uqjk jrg úNd.hg fmkS isá;a oehs
fidhkak'

2. (a) Ζ + = {1, 2, 3, L } hehs .ksu'q
‘ x ≥ 2 y u kï muKla xR y fjhs’, f,i R iïnkaOh Ζ + u; wr®: oelafjhs'
R iïnkaOh
(i) mrdjr®;"S
(ii) iuñ;Sh"
(iii) ixl%duH
oehs ksrK
® h lrkak'
(b) ‘(a + b ) brÜfÜ u kï muKla aRb fjhs’" f,i R iïnkaOh Ζ u; wr®: oelafjhs'
R hkq Ζ u; ;=,H;d iïnkaOhla nj fmkajkak'
[1] yd [2] ;=,H;d mka;s ,shd olajkak'
[1] ∩ [2] fidhkak'
25
, x ∈ [1, 9] f,i wr®: olajd ;sfnhs'
3x − 2
(i) f wvq jk Y%;
s hla nj fmkajd tys mrdih fidhkak'
(ii) f tal-tal nj fmkajkak'
(iii) f −1 mj;sk nj fmkajd th m%ldYs; f,i fidhkak'

(c) f Y%;
s hla f : x →

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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3. (a) ( x − 1) ka yd ( x − 3) ka p (x) idxl ^kdhl ix.= Klh 1& >kc nyq moh fn¥ úg ,efnk
fYaI ms<f
s j<ska 7 yd 13 fjhs' ( x − 1)( x − 3) ka p (x) fn¥ úg ,efnk fYaIh fidhkak'
;jo" p (2) = 6 kï" ( x − 1)( x − 3) ka p (x) fn¥ úg ,efnk ,íêh fidhkak'
ta khsk"a p (x) ,nd.kak'
(b) f : IR → IR hkq f ( x) = 2 x + 1 u.ska wr®: oelafjk Y%;
s h hehs .ksu'q
f (x)
(i) tal-tal"
(ii) u;g
oehs ksrK
® h lrkak'
1
;jo" g : IR − {0} → IR hkq g ( x) =
u.ska wr®: oelafjk Y%;
s h hehs .ksu'q
x
iyjiu yd mrdih olajñka gxf yd fxg ixhq; Y%;
s tl tlla m%ldYs; f,i fidhkak'

4. (a) f ( x) ≡ ( x + 1) 2 + k ( x + 1) + k hehs .ksu'q
f ( x) = 0 iólrKhg ;d;a;ú
a l uQ, ;sfnk m˙† k ys ishÆ ;d;a;ú
a l w.hka fidhkak'
k = −2 úg" f (x) ys iuñ;s wCIh yd wvq;u w.h fidhkak'
ta khsk"a k = −2 úg" y = f (x) ys m%i;
a drh w¢kak'
(b) Q( x) ≡ 2 x 2 + ax − 7 yd P( x) ≡ (2 x − 1)Q( x) + b hehs .ksuq; fuys a yd b hkq ;d;a;ú
a l
ksh; fjhs' Q(x) hkq brÜfÜ Y%;
s hla o" x j,ska P(x) fn¥ úg , efnk fYaIh 12 o
kï" a yd b ys w.hka fidhkak'
x ys talc idOlj, .=◊;hla f,i P(x) m%ldY lrkak'

5. (a) ax 2 + bx + c = 0 jr®.c iólrKfha uQ, α yd β hehs .ksuq; fuys c ≠ 0 fjhs' bα +
yd bβ +

c
β

c
α

uQ, jk jr®.c iólrKh fidhkak'

(b) (2 + x) n ys oaúmo m%idrKh ,shd olajkak; fuys n hkq Ok ksÅ,hls'
(i) by; m%idrKfha iudk ix.=Kl iys; wkqhd; mo folla ;sfnhs kï (n + 1) hkak
3 ys .=Kdldrhla nj fmkajkak'
n = 8 úg fuu ix.=Klfha w.h fidhkak'
(ii) x i|yd iq≥iq w.hka folla f;dard .ksñka n brÜfÜ úg
3 n − 1 = n C1 2 n−1 + nC3 2 n−3 + L + nC n−1 2 nj fmkajkak'

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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6. (a) ksYp
a dhlj, .=K fhdod f.k
a3

b3

c3

a
b
c
= −(a − b)(b − c)(c − a )(a + b + c) 2
b+c−a a+c−b a+b−c

nj fmkajkak'
(b) 8 x − y + z = 5
2x − y + z = 2
− 2x + 4 y = 7
iólrK moaO;sh kHdi wdldrfhka m%ldY lrkak'
by; (a) fldgi yd l%dur® kshuh fhdod f.k" iólrK moaO;sh úi|kak'
7. (a) ‘EXAMINATION’ jpkfha
(i) wl=re ish,a, u"
(ii) wl=re y;rla muKla
fhdodf.k iE†h yels fjkia ixlrK .Kk fidhkak'
(b) mqoa.,hska 15 fokl= tla;rd wdh;khlg wÆf;ka n|jdf.k we;'
fuu kjlhska 15 fokd" wxYhlg mia fokd ne.ska jk m˙† fjkia wxY ;=klg
wkqhl
q ;
a l< yels wdldr .Kk fidhkak'
 2 1
 yd f ( A) = A 2 − 5A + 7I hehs . ksuq; fuys I hkq 2 jk .Kfha tall
8. (a) A = 
 − 1 3
kHdih fjhs' f ( A) = O nj fmkajkak; fuys O hkq 2 jk . Kfha Y= kH kHdih
fjhs'
ta khsk"a A-1 fidhkak'

ffjl,aml
s l%uhla u.ska Tfí ms<;
s r
= i;Hdmkh lrkak'
1 −1 1 
 − 3 2 1




(b) A =  1 − 2 3  yd B =  − 9 5 2  hehs .ksu'q
 2 1 − 3
 − 5 3 1




BA wd.Kkh lrkak'

ta khsk"a
x− y+z =2
x − 2 y + 3z = 2
2 x + y − 3z = 2
iólrK moaO;sh úi|kak'

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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 1 1
 hehs .ksu'q P T P yd PP T fidhkak; fuys P T hkq P ys fmrΩu fjhs'
9. P = 
 − 1 1
P ys m%;f
s ,dauh jk P −1 wfmdaykh lrkak'

P −1 fhdod .ksñka (P T ) −1 fidhkak' fhdod .kq ,nk ´kEu m%;M
s ,hla meye†,sj m%ldY
l< hq;h
= '
1 1
 jk m˙† 2 jk .Kfha B úlr®K kHdihla fidhkak'
PBP T = 
1 1
 3 1
 hehs .ksu'q A = 2I + 2PBP −1 nj fmkajkak'
A = 
 1 3
ta khsk"a P −1 AP hkq úlr®K kHdihla nj fmkajkak'

10. (a) .◊; wNHqyk uQ,Or®uh fhdod .ksñka ´kEu n Ok ksÅ,hla i|yd
 n(n + 1) 
1 +2 +3 +L+n = 

 2 
3

3

3

3

2

nj fmkajkak'

ta khsk"a
(i) 13 + 2 3 + 33 + L + (2n) 3 ,
(ii) 13 + 33 + 53 + L + (2n − 1) 3
fidhkak'
(b)

3x + 2
ys Nskk
a Nd. fidhkak'
x( x + 1)( x + 2)
n

ta khsk"a

∑ u r fidhkak; fuys

u r hkq

r =1

5
8
11
+
+
+ L fY%K
a f
s ha r fjks moh fjhs'
1.2.3 2.3.4 3.4.5

by; fY%◊
a h wNsid¯fõ o@ Tfí ms<;
s r
= ikd: lrkak'

log c b
nj idOkh lrkak; fuys a, b yd c Ok jk w;r 1 g iudk fkdfjhs'
log c a
log a b log b c log c a = 1 nj fmkajkak'
ta khsk"a log 5 32 log 4 7 log 49 125 ys w.h ,nd.kak'

11. (a) log a b =

(b) (4 + 3 x)12 ys oaúmo m%idrKfha jeä;u ix.=Klh ,nd.kak'

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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12. ms˙ñ <uhska 5 fofkla yd .eyekq <uhska 5 fofkla isák isiqka 10 fokl=f.ka hq;a
lKavdhulg rjqï fïihla jgd jdäùug is≥j we;'
(i) ms˙ñ <uhska ishÆ fokdu tl <Û jdä fj;a kï"
(ii) .eyekq <uhska fofokl= tl <Û jdä fkdfj;a kï"
(iii) úfYaI;
s ms˙ñ <uhl= yd úfYaI;
s .eyekq <uhl= tl <Û jdä fj;a kï"
(iv) úfYaIs; ms˙ñ <uhl= yd úfYaIs; .eyekq <uhl= tl <Û jdä fkdjk w;r .eyekq
<uhska fofokl= tl <Û jdä fkdfj;a kï"
(v) tla úfYaIs; .eyekq <uhl= jdäùu m%;slafIam lrk w;r wfkla .eyekq <uhska
y;rfokd" .eyekq <uhs fofokl= tl <Û jdä fkdjk fldkafoaish hgf;a ms˙ñ
<uhska miafokd iu. jdäùug tlÛ kï
wdikj, jdäls¯u fjkia wdldr fldmuK .Kklg l< yels oehs fidhkak'

13. (i) ´kEu n Ok ksÅ,hla i|yd
n(n + 1)
1+ 2 + 3 + L + n =
nj idOkh lrkak'
2
(ii) ´kEu r ksÅ,hla i|yd
(2r + 1) 3 − (2r − 1) 3 ≡ 2 + 24r 2 nj fmkajkak'
n

ta khsk"a

n

∑ r 2 = 6 (n + 1)(2n + 1) nj idOkh lrkak'
r =1

by; (i) yd (ii) fhdod .ksñka"
1.4 + 4.7 + 7.10 + L fY%◊
a fha m<uqjk mo n j, ftlHh fidhkak'
by; fY%◊
a h wNsid¯ o@ Tfí ms<;
s r
= ikd: lrkak'

14. n hkq Ok ksÅ,hla hehs .ksu'q (1 + x) n ys oaúmo m%idrKh ,shd olajkak'
(1 + x) n ys oaúmo m%idrKfha x ys T;af;a n, moj, ftlHh A hehs o" x ys brÜfÜ n,
moj, ftlHh B hehs o .ksu'q
(i) B 2 − A 2 = (1 − x 2 ) n ,

(ii) 4 AB = (1 + x) 2 n − (1 − x) 2 n
nj idOkh lrkak'
(a) (2n + 1) n yd (2n − 1) n ys oaúmo m%idrK ,nd.kak; fuys n hkq 1 g jvd jeä ksÅ,hls'
(2n + 1) n > (2n − 1) n + 2 n n n nj wfmdaykh lrkak'

(b) x ys wdfrdayK n, weiqfrka (2 − 3 x) 5 + (2 + 3 x) 5 ys oaúmo m%idrKh ,nd.kak'
ta khsk"a oYuia:dk y;rlg ksrjoHj (1.97) 5 + (2.03) 5 ys w.h fidhkak'

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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15. (a) f.j,a ñ,g .kakl= 1990 ckjd˙ 1 fjks †k ia:djr fmd,S W.ila u; remsh,a 40000 la
.kS' iudk jdr® I s l jd˙l 20 lska th f.úh hq;= w;r" m<uqjk jd˙lh 1990
wjidkfha° f.ùug kshñ;h' fmd,S wkqmd;h jirlg 10% fha ia:djr jk w;r
jdr®Islj fmd,sh .Kkh lrkq ,efí' tla tla jd˙lfha w.h wdikak remsh,g
fidhkak'
dP
= aP − b iólrKh u.ska
dt
wdor®Ykh lrhs; fuys P hkq t ld,fha ° laIqø „ùka .Kk o" a yd b hkq Ok ksh; o
fjhs'
1
(i) P = (e at + b) hkq by; iólrKfha úi∫ula nj fmkajkak'
a
(ii) by; (i) ys ° we;s P ys m%i;
a drh t g tfrysj w¢kak'
(iii) laIqø „j úoHd{hd „j .ykh ms<sn|j ks¯CIK folla is≥ lr t = 1 úg P = 1 yd
t = 2 úg P = 3 nj fidhdf.k we;'
e 2 a − 3e a − 2b = 0 nj fmkajkak'

(b) laIqø „j úoHd{hl= laIqø „ù m%cdjl „j .ykh

16. a1 x + b1 y + c1 = 0 yd a 2 x + b2 y + c 2 = 0 u.ska fokq ,nk ir, fr®Ld fol w;r fldaK
iuÉf√olj, iólrK ,nd.kak'
4 x + y + 3 = 0 yd x + 4 y − 3 = 0 ir, fr®Ld fol w;r fldaK iuÉf√ol u;
iup;=ri%hl mdo folla msysghs' iup;=ri%fha tla YSr®Ihla ^0, 2& fjhs' tjeks iup;=ri%
folla ;sfnk nj fmkajd tu iup;=ri% foflys mdoj, iólrK fidhkak'
17. (a) (i) jD;a; folla tlsfklg m%,ïnj f√okh jkafka"
(ii) jD;a;hla ;j;a jD;a;hl m˙êh iuÉf√okh lrkafka
ljr wjYH;d hgf;a oehs ,nd.kak'
(b ) S1 ≡ x 2 + y 2 + 2 x + 2 y − 1 = 0 jD;a;fha yd
Tiafia S = 0 jD;a;h hhs'

l ≡ x + 2 y = 0 fr®Ldfõ f√ok ,CIH

(i) S = 0 jD;a;h yd S 2 ≡ x 2 + y 2 + 10 y − 9 = 0 jD;a;h tlsfklg m%,ïnj f√okh
fjhs kï S = 0 jD;a;h fidhkak'
(ii) S = 0 jD;a;h" S 3 ≡ x 2 + y 2 + 2 x + 2 y − 7 = 0 jD;a;fha m˙êh iuÉf√okh lrhs
kï S = 0 jD;a;h fidhkak'
18. S1 ≡ x 2 + y 2 − 2 x + 4 y = 0 yd S 2 ≡ x 2 + y 2 − 10 x + 20 = 0 jD;a; fol ndysr f,i
tlsfklg iamr®Y jk A kï ,CIHfha LKavdxl fidhkak'
P hkq" P isg S1 = 0 g w¢k ,o iamr®Ylfha †." P isg S 2 = 0 g w¢k ,o iamr®Ylfha †.
fuka nd.hla jk m˙† jk ,CIHhla kï" P , CIHfha m:h" A ,CIHh yryd hk
jD;a;hla nj fmkajkak'
S 3 = 0 jD;a;hla u.ska S 2 = 0 jD;a;h A yd B(4, 2) ,CIHj,° f√okh flfr® hehs .ksu'q
(i) S 3 = 0 jD;a;fha flakøa fha m:h ir, fr®Ldjla nj fmkajkak'
(ii) S 3 = 0 jD;a;fha C flakøa fha LKavdxl" CB g CA ,ïn jk úg fidhkak'

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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19. (a) lim

x →0

tan 2 x
w.hkak'
1 − cos(tan x)

(b) m%u,
Q Or®u u.ska cos 3 x ys jHq;m
a kakh fidhkak'

dy y sin y + x(2 − xy )
nj fmkajkak'
=
dx x 3 + cos y − x sin y

(c) e xy + sin y − x 2 = 0 hehs ° we;s úg"

20. (a) my; oelafjk oE tl tlla x úIhfhka wjl,kh lrkak:
(i)

(3 x 2 − 1) 1 + x 2
x

2

(b) x = a cos 3 θ ,

θ weiqfrka

,

2

(ii) e −2 x cos( x ln x) ,

y = a sin 3 θ hehs .ksuq; fuys a hkq ksh;hls'

d2 y
dy
yd
fidhkak'
dx
dx 2

ta khska fyda wka l%uhlska fyda 3 xy

21 (a) m%u,
Q Or®u u.ska

dθ 2

d2 y
dx 2

=

dy  dy

 x − y  nj fmkajkak'
dx  dx


a kakh fidhkak'
x ys jHq;m

(b) x = tan θ wdfoaYh fhdod .ksñka"
d2 y

(iii) x sin x

d2 y
dx

2

+

2 x dy
1
+
y = 0 iólrKh
2 dx
1+ x
(1 + x 2 ) 2

+ y = 0 iólrKhg W!Kkh l< yels nj fmkajkak'

2t
1− t2
(c) jl%hla x =
yd y =
hk mrdñ;sl iólrK u.ska ° ; f
s nhs; fuys t
1+ t
1+ t2
hkqq mrdñ;shls' t = 1 g wkqrEm ,CIHfha° jl%hg we¢ iamr®Ylfha yd wNs,ïnfha
ldàishdkq iólrK fidhkak'

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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07 - .Ks;h
II m;%h i|yd uQ,dlD;s m%Yk
a
A fldgi
1.

x

≥ 0 jk x ys ishÆ u w.hka fidhkak'
x −4
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2

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> 8 imqrd,kq ,nk x ys ishÆ u w.h l=,lh fidhkak'
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1

- 69 -

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1

- 70 -

5. my; oelafjk talc m%lu
% K .egÆj úi|kak:
y − 2x ≥ 0 ,
x ≤ 5,
y ≥1
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1

- 71 -

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j.=fõ ° we;:
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rEmjdySkS hka;h
%

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425
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2008
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25

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450
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úl=Kk ,o m%udK
18
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whs;u folu i|yd 2009 (2008 = 100) g ñ, yd m%udK idfmaCIl fidhd Tfí m%;sM,
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

- 72 -

9. cd,hl fldgila my; ° ;sfnhs:
2

A
2
1

6

C

1

B
2

D
4
E

3

3

cd,fha tla tla isoê
a h i|yd läkï isoê
a ld,h yd m%udo isoê
a ld,h fidhkak'
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π
kï (1 + tan A)(1 + tan B) = 2 nj fmkajkak'
4
π
ta khsk"a tan = 2 − 1 nj fmkajkak'
8
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

- 73 -

11. ABC ;%sfldaKhla i|yd" iqmqre≥ wxlkfhka tan

A
= sin( B + C ) kï ABC hkq iDcqfldaKl
s
2

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s ldaKhla nj fmkajkak'
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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7

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3

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2

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x +1
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

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1

- 76 -

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35 – 44
45 – 54
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

- 77 -

19. A yd B hkq P( A) = 0.4 yd P( B) = 0.25 jk m˙† jQ isoê
a fõ'
A yd B iajdh;a; kï
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(ii) tl isoê
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1
2
1
, P( B | A) = yd P( A | B) = jk m˙† jQ isoê
a folla fõ'
4
3
2
P( A ∪ B) fidhkak'
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20. A yd B hkq P( A) =

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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

- 78 -

4
1
5
, P( A ∩ B ′) = yd P( A | B) =
jk m˙† jQ isoê
a folla fõ'
7
3
14
P( A ∩ B) yd P( B | A) fidhkak'
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21. A yd B hkq P( A) =

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22. wdh;khla" tlsfklg fjkia jHdmD;s foll lghq;=j, iajdh;a;j kshe,S isà' tla tla
jHdmD;sh kshñ; ld,hg ksu ls¯ug 0.3 l iïNdú;djla we;' wvq jYfhka tla
jHdmD;shlaj;a kshñ; ld,hg ksu ls¯fï iïNdú;dj fidhkak'
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

- 79 -

23. X úúla; iiïNdù úp,Hhl iïNdú;d ialkaO Y%;
s h
P[ X = x] = cx 2 ; x = 0, 1, 2, 3 yd 4 fõ'

(i) c ys w.h"
(ii) P[0.5 < X ≤ 3]
fidhkak'
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24. tla u,a,l iq≥ fnda, 2 la yd lΩ fnda, 1 la we;' fojk u,a,l iq≥ fnda, 1 la yd lΩ
fnda, 2 la we;' tla tla u,af,ka tla fnda,hla iiïNdù f,i f;dardf.k tu fnda,h
wfkla u,a,g oukq ,efí' m<uqjk u,af,a we;s wfmaCIs; iq≥ fnda, .Kk fidhkak'
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

- 80 -

25. X ika;;sl iiïNdù úp,Hhl iïNdú;d >k;aj Y%;
s h
0≤ x≤2
 k,

f ( x) = k (2 x − 3), 2 < x ≤ 3
0,
otherwise
tfia fkdjk úg


fõ'
(i) k ys w.h"
(ii) P[1 ≤ X ≤ 2.5]
fidhkak'
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26. X ika;;sl iiïNdù úp,Hhl iïNdú;d >k;aj Y%;
s h

 1
2
108 x(6 − x) , 0 ≤ x ≤ 6

f ( x) = 
 0,
tfiaffkdjk
kdjk js
g
tfia
úg


fõ'
X ys úp,;dj fidhkak'
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

- 81 -

27. tla;rd wdh;khl ishÆ u ,sms h;=re,shkh lrkq ,nkafka wdOqksl h;=re,shkakl= jk
ksu,a úisks' Tyq úiska h;=re,shkh lrkq ,nk ´kEu ,smshl fodaI tlla fyda jeä
m%udKhla ;sîfï iïNdú;dj 0.3 ls' tjeks ,sms 4 lska hq;a iiïNdù kshe†hl fodaI rys;
,sms tlla fyda folla ;sîfï iïNdú;dj fidhkak'
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28. ñks;;
a j
= lg 3.4 l idudkH iS>%;djlska md˙fNda.slhska iqms˙ fj<|ie,lg iiïNdùj
meñfKa' meñfKk md˙fNda.slhska ixLHdj mqjdfidka jHdma;shla wkq.ukh flfr® hehs
Wml,amkh lrñka" ´kEu ;;a m r 30 l ld, iSudjl °" md˙fNda.slhska tlaflkl= fyda
jeä .Kkla meñ”fï iïNdú;dj fidhkak'
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

- 82 -

29. iqmr
q e≥ wxlkfhka" X ~ N (a, a 2 ) fõ; fuys a Ok fõ'

3 

P[ X < 0] iy P  X > a  fidhkak'
2 

'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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 0.3 0.7 
 fõ' wdrïNl
30. udr®fldaõ oduhl tla mshjr ixl%uK iïNdú;d kHdih 
 0.5 0.5 
iïNdú;d ffoYslh (0.2 0.8) kï" mshjr follg miq iïNdú;d ffoYslh fidhkak'

'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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Mathematics I I: Part B

B fldgi
1. (a) y = x − 2 + x − 4 yd y = 2 x − 3 ys m%i;
a dr tlu rEm igykl w¢kak'
ta khsk"a x − 2 + x − 4 > 2 x − 3 jk x ys w.hka fidhkak'
(b) E =

x 2 − 4x + 3

hehs .ksu;q fuys x ;d;a;ú
a l fjhs'
x2 − x − 2
(i) E wr®: oelafjk"
(ii) E > 0 jk
x ys w.hka fidhkak'

2. tla;rd lr®udka; Yd,djl A yd B jr®. foll ú≥,s whs;u ksmojkq ,efnhs' jr®. folu
i|yd fudagrhla yd g%dkaiafmdaurhla hk w;HjYH fldgia folla Ndú; lrkq ,efí'
A jr®.fha whs;u tl tlla i|yd fudagr 3 la yd g%dkaiafmdaurhla wjYH fjhs' B jr®.fha
whs;u tl tlla i|yd fudagr 2 la yd g%dkaiafmdaur 3 la wjYH fjhs' uqo,a fjkals¯fï
iSudjka fya;=fjka uilg imhk uqΩ fldgia .Kk fudagr 100 lg yd g%dkaif
a mdaur
200 lg iSud flfrhs' udkj iïm;a ySk;dj ksid A jr®.fha udisl ksIamdokh" whs;u
50 lg iSud flfrhs' A jr®.fha whs;uhla úls”fuka remsh,a 3000 l ,dNhla lr®udka;
Yd,dj Wmhk w;r th B jr®.fha whs;uhla i|yd remsh,a 2000 la fjhs'
(i) lr®udka; Yd,dfõ udisl ,dNh Wm˙u lr.ekSfï wruqKska by; wjia:dj fr®Çh
m%lu
% Khla f,i iQ;% .; lrkak'
(ii) m%ia;d˙l l%uh fhdod .ksñka" Wm˙u ,dNhla ,nd .ekSug uilg tla tla jr®.fhka
ksmoúh hq;= whs;u .Kk fidhkak'
fuu wjia:dfõ ° fuu ,dNh fidhkak'
(iii) A jr®.fha ´kEu whs;uhlska ,efnk ,dNh remsh,a 2000 g wvq jQfha kï" túg m%Yia;
úi∫u fidhkak'
3. X k.rfha yd Y k.rfha msysá fnda;,a ls¯fï hka;%d.dr folla fhdod f.k A, B yd C
isis,a îu jr®. ;=kla iud.ula ksmojhs' †klg fnda;,a .Kk weiqfrka tla tla
hka;d% .drfha Od˙;dj my; j.=fõ ° we;:
X ys° we;s hka;d% .drfha °
Y ys° we;s hka;d% .drfha °

A jr®.h
3000
1000

B jr®.h
1000
1000

C jr®.h
2000
6000

X k.rfha yd Y k.rfha msysá hka;%d.dr fol l%shdlrùu i|yd úhou †klg ms<sfj<ska
remsh,a 500000 la yd rems h ,a 400000 la fjhs' A, B yd C jr®. ;=k i|yd i;shlg wju
b,aÆï fnda;,aj,ska ms<sfj<ska 6000, 4000 yd 12000 fjhs' ishÆ b,aÆï imqrd,ñka
ksIamdok ms˙jeh wju lr .ekSug tla tla hka;%d.drh i;shlg l%shd lrúh hq;= †k .Kk
ks r ® K h ls¯ug m˙md,khg ´kElr we;' Tn m˙md,k lñgq f õ idud‚lhl= hehs
Wml,amkh lrñka by; wjia:dj fr®Çh m%lu
% Khla f,i iQ;% .; lrkak'
m%i;
a d˙l l%uh fhdod .ksñka
(i) YlH fmfoi olajkak"
(ii) m%Yia; úi∫u i|yd .egÆj úi|kak'

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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Mathematics I I: Part B

4. (a) tla;rd rgl .Dy úhoï ms<sn| ióCIKhl °" wjqre≥ ;=kl tlu i;sh ;=<°" tla tla
wjqreoao i|yd idudkH mjq,l jefhys fjkiaùu my; oelafõ:
whs;uh
iSks
mdka
f;a
ls˙
ngr®

ñ,g .kakd ,o
m%udKh
lsf,da.E% ï 2 hs
f.ä 5 hs
lsf,da.E% ï 1 hs
,Sgr 10 hs
lsf,da.E% ï 0.5 hs

1 jir
20
21
18
13
21

ñ, ^tall tllg&
2 jir
22
23
21
13
32

3 jir
57
32
24
22
41

(i) 1 jir moku f,i fhdod .ksñka" 2 jir yd 3 jir i|yd or®Yldxl .Kkh
lrkak'
(ii) fuu whs;u mokï lrf.k is≥l< ióCIKfhka" wjqre≥ ;=k ioyd lEu úhofuys
fjkiaùïj, idOdrK ;lafiarejla ksrEmKh jkafka oehs fya;= olajñka m%ldY
lrkak'
(b) 2001 isg 2008 olajd ld,h ;=< wr®;dm,a wiajekak m%udKh fgdka 1000 tajdfhka" my;
j.=fõ ° we;:
jir
or®Ylh
2003=100

2001

2002

2003

2004

2005

2006

2007

2008

97

102

100

112

128

121

145

149

(i) odu mdo idfmaCIl l=,lhla fmkaùug" by; f;dr;=re m˙jr®;kh lrkak'
(ii) 2004 ° wr®;dm,a wiajekak m%udKh fgdka 587 000 la hehs ° we;akï" 2006 isg
2008 olajd tla tla jir i|yd wr®;dm,a wiajekak m% u dKh ^wdikak fgdka
1000 g& .Kkh lrkak'
5. jHdmD;shla l%shdjka wglska iukaú; fjhs' l%shdj iïmQr®K ls¯fï ld, yd ióm fmr is≥ùï
iïnkaO;d my; oelafjk m˙† fjhs:
l%h
s dj
A
B
C
D
E
F
G
H

iïmQrK
® ls¯fï ld,h ^i;s&
5
7
6
3
4
2
6
5

ióm fmr is≥ùï l%h
s djka
A
B,C
C
D
E,F

(i) iq≥iq cd, igyk w¢kak'
(ii) läkï wdrïNl yd wjidk ld," m%udo wdrïNl yd wjidk ld," yd bmsÆï ld,
.Kkh lrkak'
(iii) ljr l%h
s djka wjê oehs y∫kajd fokak'
(iv) E l%shdj i;s 3 lska m%udo flfrhs kï th jHdmD;sh wjidk ls¯fï ld,hg n,mdkafka
flfia o@
(v) F l%shdj i;s 3 lska m%udo flfrhs kï th jHdmD;sh wjidk ls¯fï ld,hg n,mdkafka
flfia o@
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Mathematics I I: Part B

6. tla;rd ksIamd†;hl ksIamdok l%shdj,sfha mshjr wdor®Ykh lrk CPA– cd,hla my;
igyfkys oelafjhs:
G

2
A

3
B

1

2
wdrïNh

C
4

4

4
H

L

6

2
E

4

K

D

3
3

5

8

1
F

I

6
J

8

M
2

8

9
wjidkh

7

3

tla tla pdmh" th ksrEmKh lrk l%shdj olajk wCIrhlska o" tu l%shdj iïmQr®K ls¯ug
wjYH †k .Kk olajk Ndrhlska o i,l=Kq lr we;'
(i) CPA–cd,hl isoê
a hla hehs lSfuka woyia flfrkafka ljrla oehs m%ldY lrkak'
isoê
a hla i|yd läkï isoê
a ld,h yd m%udo isoê
a ld,h wr®: olajkak'
(ii) we;eï wjia : dj, ° C P A –cd,hl wkq j Hdc l% s h djla wvx.= lrka f ka ukaoehs
fláfhka meye†,s lrkak'
by; fmkajd we;s cd,fhaa iq≥iq l%shd i|yka lrñka Tfí meye†,s ls¯u ksor®Ykh
lrkak'
(iii) cd,fha tla tla isoaêh i|yd läkï isoaê ld,h yd m%udo isoaê ld,h .Kkh
lrkak' ksIamdok l%shdj,sfha fláu iïmQr®K ls¯fï ld,h m%ldY lr wjê fm;
fidhkak'

π  
π 

7. (a) y = tan x +  cot x −  hehs .ksu'q
12  
12 

1 + y = 2( y − 1) sin 2 x nj idOkh lrkak'

π  
π  1

ta khska" x ys ishÆ w.hka i|yd" tan x +  cot x −  g
yd 3 w;r w.hka .;
12  
12  3

fkdyels nj fmkajkak'
3
(b) sin 6 θ + cos 6 θ ≡ 1 − sin 2 2θ ir®jiduHh idOkh lrkak'
4
8
4
(c) sin x − cos 2 x + cos 8 x = 0 iólrKh úi|kak'
8. (a) [0, π ] mrdih ;=< cos 3θ = cos 2θ imqrd,kq ,nk θ ys ishÆ w.hka fidhkak'
cos 3θ − cos 2θ hkak cos θ weiqfrka m%ldY ls¯fuka yd idOl fjka ls¯fuka fyda
2π
4π
fjk;a l%uhlska fyda cos
yd cos
hkq 4 x 2 + 2 x − 1 = 0 jr®.c iólrKfha uQ,
5
5
nj fmkajkak'
2π
ta khsk"a cos
ys w.h fidhkak'
5
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1

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Mathematics I I: Part B

(b)

a l
3 sin 2 x + cos 2 x hkak R sin( 2 x + α ) wdldrfhka m%ldY lrkak; fuys R yd α ;d;a;ú
fjhs'
7π
11π
−
≤x≤
i|yd y = 3 sin 2 x + cos 2 x ys m%i;
a drh w¢kak'
12
12
π
π
− ≤ x ≤ i|yd y = − 3 sin 2 x + cos 2 x ys m%i;
a drh o w¢kak'
6
3

9. (a) ABC ;%f
s ldaKhla i|yd ihska kshuh" iqmr
q e≥ wxlkfhka m%ldY lrkak'
ABC ;% s f lda K hl" A Tiafia hk uOHia : fha †. m jk w;r th" AB yd A C iu.
ms<f
s j<ska θ yd φ fldaK idohs'
(i) 2m(sin θ − sin φ ) = a (sin B − sin C ) ,
A
θ −φ 
(ii) 2m sin 
 = (b − c) sin
2
 2 
nj idOkh lrkak'
(b) fldaihsk kshuh fhdod f.k" iqmr
q e≥ wxlkfhka"
A (a − b + c)(a + b − c)
tan 2 =
nj idOkh lrkak'
2 (b + c − a )(a + b + c)

10. (a) Nskk
a Nd. fhdod .ksñka

8

∫ (3x 2 + 1)(3x + 1) dx fidhkak'

(b) fldgia jYfhka wkql,k l%uh fhdod .ksñka

π
,
2
.Kkh lrkak'

(c ) x = 0, x =

∫e

−x

cos 2 xdx fidhkak'

y = sin x yd y = cos x u.ska imr®hka;.; fmfofiys jr®.M,h

1

2

11. (a) u = − x wdfoaYh fhdod .ksñka

2

3 −x
∫ x e dx w.hkak'
0

(b)

∫

x
x2 − 4

dx fidhkak'
8

ta khsk"a

∫

5

x ln( x 2 − 4)
x2 − 4

dx w.hkak'
1
2

(c) x = cos 2θ wdfoaYh fhdod .ksñka

∫
0

1+ x
dx w.hkak'
1− x

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

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Mathematics I I: Part B

12. (a) iqmr
q e≥ wxlkfhka ;%mi
s dN kshuh m%ldY lrkak'
x ys olajd we;s w.hka i|yd" f (x) Y%;
s hl w.hka oYuia:dk follg ksjer†j my;
oelafjk j.=fõ ° we;:
x
f(x)

– 1.0
7.00

– 0.5
5.00

0.0
3.50

0.5
4.00

1.0
5.50

1.5
6.00

2.0
6.50

by; j.=fõ ° we;s f;dr;=re yd ;%msidN kshuh fhdod.ksñka y = f (x) jl%h hg yd
x-wCIfhka by<" x = – 1.0 yd x = 2.0 w;r jr®.M,h i|yd wdikak w.hla ,nd.kak'
(b) iqmr
q e≥ wxlkfhka" isïika kshuh m%ldY lrkak'
1

h = 0.2 yd isïika kshuh fhdod .ksñka

x

∫ x 3 + 10 dx

i|yd wdikak w.hla oYuia:dk

0

y;rlg ksjer†j ,nd.kak'
13. úúO jhiaj,°" we≥u frda.h m<uq jrg u;=jQ frda.Ska .Kk ms<sn| f;dr;=re iuyrla
my; oelafjk j.=fõ ° we;:
m<uq jrg u;=jQ jhi
frda.k
S a .Kk

0 – 10
14

10 – 20
?

20 - 30
27

30 – 40
?

40 – 50
15

(10 – 20) yd (30 – 40) jhia ldKavj,g wkqrEm frda.Ska ms<sn| o;a; j.=fjys i|yka
fkdfjhs' flfia kuq;a" jHdma;sfha uOHia:h yd ud;h ms<sfj<s k a 25 yd 24 nj fidhd
f.k we;'
(i) i|ykaj fkdue;s ixLHd; .Kkh lrkak'
(ii) jHdma;f
s ha uOHkHh yd iïu; wm.ukh ,nd.kak'
(iii) úp,k ix.=Klh yd l=ál;d ix.=Klh fidhd jHdma;f
s ha yevh y∫kajd fokak'
14. ks¯CIK n iys; l=,lhl uOHkHh yd úp,;dj wr®: olajkak'
{x1 , x 2 , L x n } ks¯CIK n iys; l=, lhl uOHkHh yd úp,;dj .Kkh lr we;' x 1 ys
w.h ifodia nj miqj fidhd f.k we;s w;r th x1′ u.ska m%;i
s :
a dmkh l<hq;= fjhs'
fuu fodaIh ksjer† ls¯u i|yd úp,;djg l<hq;= ieliSu
x ′ − x + 2T 
1

( x1′ − x1 ) x1′ + x1 − 1 1
 nj fmkajkak; fuys T hkq {x1 , x 2 , L x n } uq,a
n
n


o;a; l=,lfha ftlHh fjhs'
tlu lr®udka;hg wh;a A yd B lr®udka;Yd,d foll fiajlhska ms<sn| f;dr;=re my;
fokq ,efnhs:
fiajlhska .Kk
uilg fiajlhl=f.a idudkH fõ;kh
^remsh,aj,sk&a
uilg fõ;kj, úp,;dj
^remsh,a 2 j,sk&a

A lr®udka;Yd,dj
50

B lr®udka;Yd,dj
100

12000

8500

9

16

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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Mathematics I I: Part B

(i) udisl fõ;k f,i jvd jeä m%udKhla f.jkq ,nkafka A yd B j,ska l=uk
lr®udka;Yd,dj o@ Tfí ms<;
s r
= ikd: lrkak'
(ii) mqoa., fõ;kj, jvd jeä úp,H;djla we;af;a l=uk lr®udka;Yd,dfõ oehs
fidhkak'
(iii) B lr®udka;Yd,dfõ tla fiajlhl=f.a fõ;kh remsh,a 10000 fjkqjg remsh,a 12000
hehs jer† f,i jdr®;d lr ;sfnhs'
B lr®udka;Yd,dfõ ksjer† lrk ,o úp,;dj fidhkak'
fuu ksjer† ls¯u (ii) ys Tfí ms<;
s r
= g n,mdkq ,efí o@ Tfí ms<;
s r
= ikd: lrkak'
15. ks¯CIK n ys yrd;aul uOHkHh wr®: olajkak'
(a) ≥ï˙hla m<uq lsf,daógr 40 mehg lsf,daógr 48 l idudkH fõ.fhka o" fofjks
lsf,daógr 80 mehg lsf,daógr 64 l idudkH fõ.fhka o Odjkh jk w;r" udr®.fha
kv;a;= lghq;= ksid ñks;a;= 15 la mehg lsf,daógr 20 l idudkH fõ.fhka .uka lr
wjidk jYfhka b;s˙ ≥r m%udKh mehg lsf,daógr 25 l idudkH fõ.fhka wdjrKh
flfrhs' ≥ï˙fha iïmQr®K .uk i|yd idudkH fõ.h mehg lsf,daógr 4 5 la kï"
≥ï˙h Odjkh lrk uqΩ ≥r fidhkak'
(b) tla;rd jirla i|yd rgl ck.ykfha tla;rd kshe†hl fm!oa.,sl Okh my;
oelafjk j.=fõ we;s jHdma;f
s hka fokq ,efnhs:
fm!oa.,sl Okh
0
2000
5000
10000
15000
20000
25000
50000

- 2000
- 5000
- 10000
- 15000
- 20000
- 25000
- 50000
yd by<

mqo.
a , ixLHdj
(000,000)
19
26
74
41
16
8
5
1

uqΩ fm!oa.,sl Okh
(000m)
2.4
7.8
55.5
49.2
25.7
16.8
15.0
6.3

o;a; ksor®Ykh ls¯ug f,darkaia jl%hla we| th fhdod f.k
(i) wvq;u Okj;a mqoa.,hska 30% fokdf.a mdúÉÑh i|yd we;s uqΩ fm!oa.,sl Ok
m%;Y
s ;h"
(ii) fm!oa.,sl Okh ish,af,ka y˙ wvla m˙yrKh lrkq ,nk jvd;au Okj;a mqo.
a ,
m%;Y
s ;h
weia;fïka;= lrkak'
16. (a) A yd B hkq P( A | B) = 0.6, P( B | A′) = 0.4 yd P( A | B ′) = 0.5 jk m˙† jQ isoê
a folla
hehs .ksu'q
(i) P( A), P( B) yd P( B | A) fidhkak'
(ii) A yd B iajdh;a; fõo@ Tfí ms<;
s r
= ikd: lrkak'

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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Mathematics I I: Part B

(b) ksIamdok fm<la fodaI iys; whs;u 8% l m%udKhla ksmojk w;r thska y;frka
tll m%udKhla m%;l
s f
a Iam flfrhs'
ksIm
a dok fmf<ka whs;u ;=kla iiïNdù f,i f;dard .efka kï
(i) m<uq whs;uh m%;l
s f
a Iam ls¯fï"
(ii) fojk whs;uh m%;l
s f
a Iam ls¯fï"
(iii) whs;u lsij
s la m%;l
s f
a Iam fkdls¯fï"
(iv) wvq jYfhka tla whs;uhla fodaI iys; ùfï
iïNdú;dj fidhkak'
17. idmamqjla fm'j' 9.00 g újD; l< fudfydf;a isg meh t we;=<; thg meñ◊ md˙fNda.l
s
ixLHdj N(t) hehs .ksuq' N(t) g uOHkHh λt iys; mqjdfidka jHdma;shla we;ehs Wml,amkh
lrkak'
(i) idmamqj m'j' 5.00 g jik úg md˙fNda.slhka 20 fofkla thg meñK ;sîfï iïNdú;dj
fidhkak'
(ii) T hkq m<uq md˙fNda.slhd idmamqjg meñ◊ fj,dj hehs .ksuq' P(T > t ) fidhd ta khsk"a
T g >d;Sh jHdma;h
s la ;sfnk nj fmkajkak'
(iii) fm'j' 10.00 f;la lsisu md˙fNda.slfhla idmamqjg meñK fkdue;' fm'j' 10.00 yd
fm'j' 11.00 w;r m<uq md˙fNda.l
s hd meñ”fï iïNdú;dj fidhkak'
18. ´kEu fj<| fiajlfhl= i;shl° .uka lrk ≥r" uOHkHh lsf,daógr 800 yd iïu;
wm.ukh lsf,daógr 90 iys;j m%u; f,i jHdma; ù we;s nj iud.ul jdr®;d fmkajd
fohs' i;shl° lsf,daógr 600 g jvd wvq ≥rla .uka lrk fj<| fiajlhka ≥r®j, f,i
ldr®h idOkh lrkakka jYfhka fj<| l<ukdlre i,lkq ,efí'
(i) iiïNdù f,i f;dard .kakd ,o fj<| fiajlfhl= lsf,daógr 710 g jvd jeä tfy;a
lsf,daógr 935 g jvd wvq ≥rla .uka lr ;sîfï iïNdú;dj fidhkak'
(ii) iud.u fj<| fiajlhka 200 la fiajfha fhdojhs kï tla;rd i;shl° fldmuK
m%udKhla ≥r®j, f,i ldr®h idOkh lrkafka hehs wfmaCId flfr® o@
(iii) fj<| fiajlhskaf.ka 1% muKla blaujd hd yels hehs wfmaCId lrk" i;shl° .uka
lr we;s lsf,daógr .Kk weia;fïka;= ls¯ug fj<| l<ukdlre leue;af;ka isà'
Tyqf.a weia;fïka;j
= .Kkh lrkak'
t

1 − 1000
19. tl tll wdhqld,h" iïNdú;d >k;aj Y%s;h f (t ) =
e
, t ≥ 0 iys; T iiïNdù
1000
úp,Hhla jk" ú≥,s nqnÆ
q ;=kla ldurhl iúlr we;'
(a) P[T ≤ t ] fidhd ta khsk"a P[T > t ] wd.Kkh lrkak'
(b) ú≥,s nqnÆ
q ;=ku tljr mdúÉÑ lrkafka hehs Wml,amkh lrñka
(i) ú≥,s nqnÆ
q ;=ku"
(ii) ú≥,s nqnÆ
q folla muKla"
(iii) ú≥,s nqnÆ
q tlla muKla
meh 1300 lg miqj;a oe,aùfï iïNdú;dj fidhkak'
^oYuia:dk ;=klg ksjer†j exp(-1.3) = 0.273 nj Tng Wml,amkh l< yelsh'&

meh 1300 lg miqj ldurh w∫reùfï iïNdú;dj wfmdaykh lrkak'

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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Mathematics I I: Part B

20. .Dy◊hla A yd B o;afnfy;a jr®. foflka tlla" tla tla udifha° ñ,hg .ekSfï
mqreoaol ksr;j isà' tla udihl ° weh A jr®.h ñ,° .;af;ys kï" bkamiq udifha°
B jr®.h ñ,° .ekSfï iïNdú;dj 0.1 la fõ' weh tla udihl° B jr®.h ñ,° .;af;a kï"
bkamiq udifha° A jr®.h ñ,° .ekSfï iïNdú;dj 0.05 la fõ' n fjks udifha° weh ñ,°
.kakd o;afnfy;a jr®.h Xn hehs .ksu'q
(i) {Xn : n = 0, 1, 2, . . .} u.ska udr®fldaõ oduhla ksrEmKh lrk nj fmkajd tla mshjr
ixl%uK iïNdú;d kHdih fidhkak'
(ii) .Dy◊h ckjd˙ udifha° A jr®.h ñ,° .;af;ys kï" udr®;= udifha° o tu jr®.hu
ñ,° .ekSfï iïNdú;dj fidhkak'
(iii) ckjd˙ udifha° A jr®.h ñ,° .ekSug .Dy◊hg we;s iïNdú;dj 0.4 kï" tu
jr®.hu uehs udifha° o ñ,hg .ekSfï iïNdú;dj fidhkak'
(iv) uehs udifha° A jr®.h ñ,hg .;af;a hehs ° we;s úg" ckjd˙ udifha° tu jr®.h
ñ,° f.k ;sîfï wiïNdjH iïNdú;dj fidhkak'
(v) †.= ld,Skj jr®. fol w;=˙ka tla tla jr®.h fldmuK ld,dka;rhlg jrla weh
ñ,° .kafka o@
21. f,d˙ ˙h≥frla uykqjr" .d,a, iy ymq;f,a k.rj,g nvq fnod y˙hs' Tyq tla tla
†kfha° tl k.rhlg hk kuq;a wkqhd; †kj,° tlu k.rhg hkq fkd,efí' fok ,o
†kl° Tyq uykqjrg .sfha kï" thg miq †fkys .d,a,g fyda ymq;f,ag hEu iufia is≥fõ'
fok ,o †kl° .d,a,g Tyq .sfha kï" bkamiq †kfha° uykqjrg hEfï NjH;dj ymq;f,ag
hEu fuka fo.=Khla fõ' fok ,o †kl° Tyq ymq;f,ag .sfhys kï" bkamiq †kfha°
.d,a,g hEfï NjH;dj uykqjrg hEu fuka ;=k.
a K
= hla fjhs'
(i) fuu wjia:dj ioyd tla mshjr ixl%uK iïNdú;d kHdih ,shkak'
(ii) lsishï i∫od †khl° Tyq uykqjrg .sfha kï" nodod †kfha° .d,a,g hEfï
iïNdú;dj fidhkak'
(iii) †.= ld, mrdihl°" uykq j r" .d,a , iy ymq ; f,a k.rj,g hEfï wkqmd;h
18 : 21 : 16 nj fmkajkak'

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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08 - lDIs úoHdj
m%Yak m;% jHqyh
˚ m;%h

- ld,h ( meh 02hs'
jrK 5 ne.ska jQ nyqjrK m%Yak 50ls' m%Yak ish,a,g u ms<s;=re iemhsh hq;= h'
tla m%Yakhlg ,l=Kq 02 ne.ska uqΩ ,l=Kq 100ls'

˚˚ m;%h - ld,h ( meh 03hs'
fuu m%Yak m;%h jHQy.; rpkd yd rpkd jYfhka fldgia follska iukaú; h'
A fldgi - jHQy.; rpkd jr®.fha m%Yak y;rls' m%Yak ish,a,g u ms<s;=re
iemhsh hq;= h' tla m%Yakhlg ,l=Kq 100 ne.ska ,l=Kq 400ls'
B fldgi - rpkd jr®.fha m%Yak yhls' m%Yak y;rlg ms<s;=re iemhsh hq;= h'
tla m%Yakhlg ,l=Kq 150 ne.ska ,l=Kq 600ls'
˚˚ m;%h i|yd uqΩ ,l=Kq 1000 fi 10 } 100
wjidk ,l=K .Kkh ls¯u

(

˚ m;%h
} 100
˚˚ m;%h
} 100
wjidk ,l=K } 200 fi 2 } 100

˚ m;%h
ie,lsh hq;=hs (
* ishÆ u m%Yak i|yd ms<s;=re imhkak'
* ksjer† fyda jvd;a .e<fmk fyda ms<s;=r f;darkak' ^úNd.fha ° ms<s;=re iemhSu i|yd nyqjrK
lv∞ishla imhkq ,efí'&

•

wxl 1 - 3 olajd m%Yakj,g ms<s;=re iemhSu i|yd my; ±lafjk f;dr;=re f.dkqj Ndú; lrkak'
A.
B.
C.
D.
E.

1'

jr®Idm;k wdldrh
ks˙; †. fudaiu
Bidk †. fudaiu
wka;r® fudaiu - ˚
wka;r® fudaiu - ˚˚
jdiq<s

P.
Q.
R.
S.
T.

jr®Idm;kh ,efnk ld,h
iema;eïnr® isg foieïnr®
ckjd˙ isg udr®;=
uehs isg iema;eïnr®
cqks" cQ,s
ckjd˙" fmnrjd˙

by; iïnkaO;d w;=˙ka jr®Idm;k wdldrh yd tu jr®Idm;kh ,efnk ld,h ms<sn| j ksjer†
iïnkaOh jkafka"
^1& AP h'
^2& AR h'
^3& BQ h'
^4& CS h'
^5& DP h'

2' wka;r® ksjr®;k wNsid¯ l,dmh Y%S ,xldfjka W;=re foig úia:dmkh jkafka"
^1& A u.sks'
^2& B u.sk's
^3& C u.sks'
^4& D u.sks'
^5& E u.sks'
3'

Y%S ,xldfõ úh<s l,dmhg m%Odk jYfhka jr®Idj ,efnkafka"
^2& B u.sks'
^3& C u.sks'
^4& D u.sks'
^1& A u.sks'

^5& E u.sks'

4' mdi,a f.j;af;a msysgd we;s ld,.=◊l tallh ;=< jr®Idudkhla iú ls¯fï° my; i|yka lreKq
i,ld ne,sh hq;= nj ;SrKh lrk ,°'
A - jr®Idudkfha by< fl<jr fmdf<dj uÜgfï isg 30 cm la by<ska isák fia iú l< hq;= h'
B - taldld¯ iu;,d fCIa;%hl iú l< hq;= h'
C - f.dvke.s,s" Ydl wd†fhys Wifuka isõ .=Khlg jvd wE;ska iú l< hq;= h'
by; i|yka lreKq w;=˙ka jr®Idudkh iú ls¯u iïnkaOfhka jvd;a ie,lsh hq;= lreKq jkafka"
^1& A mu◊'
^2& A yd B mu◊'
^3& A yd C mu◊'
^5& A, B yd C ish,a, u h'
^4& B yd C mu◊'
5'

mfia jhkh fCIa;%fha° ks¯CIKh l< YsIHfhla th je,s iys; mila nj ks.ukh lf<ah' tu mi
iE≥Kq ud;D øjHfhys nyq, j wvx.= úh yels Lksch jkafka"
^1& fvd,uhsÜ h' ^2& lafjdaÜia h' ^3& uhsld h'
^4& wemghsÜ h' ^5& fµ,aiamdr® h'
w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

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6' mdxY= CIqø „ùka ms<sn| is≥l< wOHhkl° ,nd.;a .e,Sï igykla my; ±lafõ'
khsgc
% ka
C
u< foay A
wefudakshd B
khsfÜ%Ü
jdhqj
by; A, B yd C l%shdj,s jkafka ms<sfj<ska"
^1& khsà%lrKh" khsà%lrKh iy khsà%yrKh fõ'
^2& khsà%yrKh" khsà%lrKh iy wefudakSlrKh fõ'
^3& wefudakSlrKh" khsà%yrKh iy khsà%lrKh fõ'
^4& khsà%yrKh" wefudakSlrKh iy khsà%lrKh fõ'
^5& wefudakSlrKh" khsà%lrKh iy khsà%yrKh fõ'
7'

tla;rd mila ms<sn| m%ldY lsysmhla my; ±lafõ'
A - pH w.h 7g jeäh'
B - legdhk yqjudre ixlSr®Kfha jeä m%udKhla ^15]g jvd& Na+ we;'
C - Y=Ial yd wr®O Y=Ial m%foaYj, fuu mia we;'
by; mi"
^1& wdï,sl mils' ^2& W∞iSk mils' ^3& ,jK mils' ^4& CId¯h mils' ^5& NdIañl mils'

8'

mdi,a bvfï ìï fldgil m%d:ñl ìï ieliSug fmr;a m%d:ñl ìï ieliSfuka miqj;a isiqka úiska
my; i|yka mdxY= ,CIK m¯CId lrk ,°'
A - jhkh
B - jHqyh
C - i;H >k;ajh
D - oDYH >k;ajh
E - wyUq rΩ nj
F - iújr;dj
fuu ,CIK w;=frka m%d:ñl ìï ieliSfuka miqj fjkialï wfmaCId l< yels mdxY= ,CIK jkafka"
^1& A, B, C yd D h'
^2& A, C, D yd E h'
^3& B, C, D yd E h'
^4& B, D, E yd F h'
^5& C, D, E yd F h'

9' mfia fmdaIl iq,n;dj ms<sn| m%ldY lsysmhla my; ±lafõ'
A - wêl wdï,sl mil w;HdjYH fmdaIl ;snqK o Ydlj, fmdaIl W!k;d we;s úh yels h'
B - wêl wdï,sl mil jefvk Ydlj,g hlv wjfYdaIKh l< fkdyels h'
C - ldnksl øjH wvx.= mil fmdaIl r|jd .ekSfï yelshdj jeä h'
by; m%ldYj,ska ksjer† jkafka"
^2& B mu◊'
^3& A yd B mu◊'
^1& A mu◊'
^4& A yd C mu◊'
^5& B yd C mu◊'
10' ù j.dfõ° f.dùka ksr®foaYs; m%udKhg jvd ridhksl fmdfydr fh°u ksid úúO .egÆ u;= ù ;sfí'
tf,i ksr®foaYs; m%udKhg jvd hQ˙hd fh°u ksid we;s úh yels n,mEï lsysmhla my; ±lafõ'
A - c,dYj, iqfmdaIKh jeäùu'
B - mi wdï,sl ùu'
C - Ydlj, m<sfndaO .%dyS;dj jeäùu'
fuu m%ldYj,ska jvd;a ksjer† jkafka"
^1& A mu◊'
^2& B mu◊'
^3& A yd B mu◊'
^5& B yd C mu◊'
^4& A yd C mu◊'
11' 15 ( 10 ( 0 fmdfydr ñY%Kfhka 1000 kg la ieliSu i|yd wjYH jk ;%s;aj iqmr® fmdiafµaÜ ^45] P2O5&
m%udKh wdikak jYfhka"
^1& 450 kg ls'
^2& 333 kg ls'
^3& 222 kg ls'
^4& 45 kg ls'
^5& 22 kg ls'
12' úh<s l,dmfha fiajfha kshq;= lDIslr®u WmfoaYl uyf;l= ish m%foaYfha f.dùkag ;u j.dìï i|yd
Y=kH ìï ieliSu is≥ lrk f,i Wmfoia fok ,°' fuys° wkq.ukh l< hq;= ksjer† l%shdudr®.h
jkafka"
^1& iS iEfuka wk;=rej md;a;s ilia ls¯u h'
^2& fmdf<dj is≥re lr îc isgqùu h'
^3& mi iu;,d lr md;a;s ilia ls¯u h'
^4& j,j,a lmd îc isgqùu h'
^5& jeá iy ldKq f,ig md;a;s ilia ls¯u h'

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

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13' jir lsysmhla mqrkaj ;snQ ;u f.dv bvu j.d ls¯ug woyia l< f.dú uyf;la ìï ilia ls¯ug ;eá
k.==,la fhd∞ .;af;ah' fï i|yd ;eá k.=, fhd∞ .ekSug m%Odk fya;=j jYfhka ±laúh yelafla"
^1& mia leg fmdä l< yels ksid h'
^2& mfia ±ä ia;r leãug yels ksid h'
^3& j,a me<Eá md,kh l< yels ksid h'
^4& we,s iy jeá ieliSug yels ksid h'
^5& fMkaøSh fldgia ñY% jk ksid h'
14' fCIa;% fnda. j.djlg c, iïmdokh ls¯fï° c, iïmdok wjYH;dj ksr®Kh ls¯u i|yd ±k .; hq;=
mdxY= idOl jkafka"
^1& mfia mj;sk f;;uk m%;sY;h" fCIa;% Od˙;dj yd oDYH >k;ajh hs'
^2& mfia mj;sk f;;uk m%;sY;h" fCIa;% Od˙;dj yd uQ, uKav, Wi hs'
^3& mfia mj;sk f;;uk m%;sY;h" fCIa;% Od˙;dj yd iújr;djh hs'
^4& mfia jhkh" jHqyh yd oDYH >k;ajh hs'
^5& mfia jhkh" jHqyh yd iújr;djh hs'
15' tla;rd f.dúfhl= ;u t<jÆ j.djg c,h iemhSug úiqreï c, iïmdok moaO;shla msysgqjk ,°' tys°
Tyq c, fmdïmhla Ndú; lrkq ,eîh' Tyq c, fmdïmh Ndú; ls¯fï m%Odk wruqK jkafka"
^2& c,fha úNj Yla;sh jeä ls¯u h'
^1& c,h fmdf<dj u;=msgg .ekSu h'
^3& c,fha pd,l Yla;sh jeä ls¯u h'
^4& c,fha m˙udj jeä ls¯u h'
^5& c,fha nr jeä ls¯u h'
16' jeis ld,fha° j.d ìul ;sfnk fnda. Ydl ue,ù m;% ,d fld< meye ù ye,S hkq ±lsh yels úh' tu
Ydl .,jd ne¨ úg uQ, moaO;sfha l=vd uq,a ue¯ ;sfnkq ≥gq w;r uq,a úys° we;s .eUqr b;d wvq nj o
fmkq◊' fuu ;;a;ajhg m%Odk jYfhka fya;= jkafka tu mfia"
^2& fmdgEishï W!k;djh hs'
^3& mi wdï,sl ùu hs'
^1& khsg%cka W!k;djh hs'
^4& c,jykh ≥r®j, ùu hs'
^5& hlv úI ùu hs'
17' Ydl wNsckkh ms<sn| j m%ldY lsysmhla my; ±lafõ'
A - wNsckkh u.ska ksmojk kj m%fNao yeuúgu iajNdúl m˙irhg WÑ; fõ'
B - fnda.j, wiajkq úNj;dj jeä l< yelafla wNsckkfhka mu◊'
C - úlD;s u.ska m%fõ◊l úp,;d we;s l< yels fõ'
by; m%ldYj,ska ksjer† jkafka"
^2& B mu◊'
^3& A yd B mu◊'
^1& A mu◊'
^4& A yd C mu◊'
^5& B yd C mu◊'
18' neis,ia ;=˙ka‚fhkaisia (Bacillus thuringiensis) nelaà˙hdfõ lDñkdYl .=Kh wvx.= cdkh fjka
lrf.k jdylfhl= u.ska Ydlfha m%fõ◊ øjH ;=<g we;=Æ fldg kj Ydl ksIamdokh lr ;sfí' fuu
ffcj ;dCIK l%uh y∫kajkafka"
^1& úlD;s wNsckkh f,i hs'
^2& fouqyqï wNsckkh f,i hs'
^4& cdk kj ixfhdackh f,i hs'
^3& ffoysl laf,dak úp,;dj we;s ls¯u f,i hs'
^5& m%;sixfhda‚; DNA ;dCIKh f,i hs'
19' c, frdams; j.dfõ° udOHfha fmdaIl iq,n;dj m%udKj;a±hs fidhd .ekSug Ndú; lrkafka"
^1& pH w.h hs'
^2& CEB w.h hs' ^3& EC w.h hs' ^4& BOD w.h hs' ^5& FC w.h hs'
20' >k udOHj, ksr®mdxY= j.dj ms<sn| m%ldY lsysmhla my; ±lafõ'
A - fldyqn;a „jdKqyrKh ls¯ug fu;ï fidaähï fhd∞.; yels h'
B - fldyqn;aj,g oyhshd wÛ=re ñY% fldg j.d ñY%Kh ieliSu u.ska c, jykh †hqKq lr .;
yels h'
C - ;jdka me< iE°fï° ;jdka ;eáfha we;s udOHhg iDcqju fmdaIl ñY%Kh fhdokq ,efí'
by; m%ldYj,ska jvd;a ksjer† jkafka"
^1& A mu◊'
^2& B mu◊'
^3& A yd B mu◊'
^4& A yd C mu◊'
^5& B yd C mu◊'

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

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21' rEmfha olajd we;af;a w;= lene,a,l uq,a weoaoùu i|yd ilik ,o ir, m%pdrl jHqyhls'
fuys uq,a weoaoùu m%jr®Okh ls¯u i|yd wkq.ukh lrkq ,nk Wml%u jkafka"
^1& m;% fCIa;%M,h wvq ls¯u yd wdr®ø;dj jeä ls¯u h'
^2& wdr®ø;dj jeä ls¯u yd WIaK;ajh by< kexùu h'
^3& m;% fCIa;%M,h wvq ls¯u yd WIaK;ajh by< kexùu h'
^4& mi f;;a ls¯u yd wdr®ø;dj wvq ls¯u h'
^5& WIaK;ajh yd wdr®ø;dj wvq ls¯u h'
22' Ydl c, W!k;djlg uqyqK≥ka úg m;%j, mQáld jefia' tfia jkafka md,l ffi,j,
Y=k;dj wvq ùfuks' fï i|yd n,mdk fydr®fudakhla yd Lksc whkhla jkafka ms<sfj<ska"
++
+
++
^2& TlaiSk yd K h'
^3& ABA yd Ca h'
^1& TlaiSk yd Ca h'
+
++
^4& ABA yd K h'
^5& .snr,Ska yd Ca h'

•

wxl 23 - 24 olajd m%Yakj,g ms<s;=re iemhSu i|yd my; f;dr;=re Ndú; lrkak'
mf;da, j.djla ks¯CIKh l< isiqka my; i|yka f;dr;=re igyka lr f.k ;snq◊'
ks¯CIK
A - mf;da, lr,a wdjrKh jk fia fmd,s;ska fhd∞ ;sì◊'
B - ydks l< m;%j, kdrá b;s˙ ù ±,la fia †iaúh'

23' A
^1&
^3&
^5&

ks¯CIKh u.ska j<lajd .; yelafla"
wjq,lfmdard lDñ ydksh hs'
frda.jdyl ydksh hs'
ymd lk lDókaf.a ydksh hs'

^2& b,a ueiaidf.a ydksh hs'
^4& tms,elaakd lDñ ydksh hs'

24' B ks¯CIKhg w∞< ydksh is≥ lr we;af;a"
^1& úo hqI Wrd fndk lDñfhls'
^3& imd lk fyda úld lk lDñfhls'
^5& frda. jdyl lDñfhls'

^2& iQrd hqI Wrd fndk lDñfhls'
^4& m;% yl=Ωjk lDñfhls'

25' f.dú uyf;la ;u l=Uqfr® rd;%S ld,fha lel=K f;,a mkaoï ±,aùu;a" †jd ld,fha r;= yd ly jr®K
u,a ;gqjla ilid l=Uqr mqrd ls˙n;a biSu;a lrk ,°' fuu.ska f.dúhd n,dfmdfrd;a;= jkafka"
^1& rd;%S ld,hg l=Uqrg wdf,dalhla ,nd °u h'
^2& l=Uqfr® lDñ m<sfndaO md,kh ls¯u h'
^3& rd;%S ld,fha meñfKk óhka m,jd ye¯u h'
^4& fCIa;%h w,xldr ls¯u h'
^5& l=Uqrg meñfKk m<sfndaO m,jd ye¯u h'

•

wxl 26 yd 27 m%Yakj, olajd we;s (A), (B), (C), (D) hk m%;spdr y;r w;=˙ka ksjer† m%;spdr ljfr®±hs
f;dard bkamiq ksjer† wxlh f;darkak'
1' A, B, D hk m%;spdr muKla ksjer† kï ^1& u; o
2' A, C, D hk m%;spdr muKla ksjer† kï ^2& u; o
3' B, C, D hk m%;spdr muKla ksjer† kï ^3& u; o
4' A, B, C hk m%;spdr muKla ksjer† kï ^4& u; o
5' C iy D hk m%;spdr muKla ksjer† kï ^5& u; o
W;a;r m;%fhys ±lafjk Wmfoia m˙† ,l=Kq lrkak'

26' fydafudmafgrd f.da;%fha ,CIK jkafka"
(A) mQr®j mshdm;a taldldrj >k ù we;'
(B) wmr mshdm;a mg,uh hs'
(C) mQr®K rEmdka;rKhla fmkajhs'
(D) úo hqI Wrd fndk fu!L Wmdx. we;'
27' mhsfr;%ï ms<sn| ksrjoH j.ka;s jkafka"
(A) fïjdfha m%djr®;kh ^fkdkeiS mej;Su& jeäh'
(B) lmqre Ydlfhka ksiaidrKh lr.kS'
(C) Ydl iïNjhla we;s ldnksl lDñkdYlhls'
(D) fuh m%n, lDñ kdYlhla f,i l%shdlrk kuq≥ CISrmdhS i;=ka flfrys th t;rï úI∞hS fkdfõ'
w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

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•

wxl 28 -30 m%Yakj,g ms<s;=re iemhSu i|yd my; j.=fõ i|yka f;dr;=re Ndú; lrkak'
A
ng±,a,
wegjrd
bΩla

B
md;Skshï
we,sf.agr® me<Eáh
ie,aùkshd

C
.srdm,d
nÆk.=g
lvqmyr

28' A ldKavhg wh;a jkafka"
^1& ;DK jr®.hg wh;a j,a me<Eá h'
^2& m<,a m;% jr®.hg wh;a j,a me<Eá h'
^3& mka jr®.hg wh;a j,a me<Eá h'
^4& ;DK yd mka jr®.j,g wh;a j,a me<Eá h'
^5& ;DK iy m<,a m;% jr®.j,g wh;a j,a me<Eá h'
29' by; j,a me<Eá ldKav w;=˙ka wdl%uKYS,S wd.ka;=l j,a me<Eá ldKavh$ldKav jkafka"
^1& A mu◊'
^2& B mu◊'
^3& C mu◊'
^5& B iy C mu◊'
^4& A iy B mu◊'
30' by; ldKav w;=˙ka jdr®Isl j,a me<Eá wh;a ldKavh$ldKav jkafka"
^2& B mu◊'
^1& A mu◊'
^4& A yd B mu◊'
^5& A yd C mu◊'

^3& C mu◊'

31' úh<s l,dmfha udia lkakfha fmnrjd˙ 15 - udr®;= 15 w;r ° wiajkq fk<d.; yelsjk m˙† BG 300
ù m%fNaoh j.d ls¯ug iq≥iq ld,h jkafka"
^1& fkdjeïnr® 10 - 30 w;r ìu ilid foieïnr® uq, jems¯u h'
^2& Tlaf;danr® 25 - fkdjeïnr® 15 w;r ìu ilid fkdjeïnr® ueo jems¯u h'
^3& Tlaf;danr® 10 - 30 w;r ìu ilid fkdjeïnr® uq, jems¯u h'
^4& iema;eïnr® 20 - 30 w;r ìu ilid Tlaf;danr® ueo jems¯u h'
^5& foieïnr® 10 -30 w;r ìu ilid ckjd˙ w. jems¯u h'
32' ùj, fnd,a îc we;sùu iïnkaO m%ldY lsysmhla my; ±lafõ'
A - mqrela mKq ydksh u.ska
B - l,xlh úh<S hdu u.ska
C - l,xlh u; fjk;a ù m%fNaohl mrd. ;ekam;a ùu u.ska
D - mqIam yg.kakd wjêfha khsg%cka yd fmdgEishï wvx.= fmdfydr fh°u u.ska
by; m%ldY w;=˙ka jvd;au ksjer† jkafka"
^1& A iy B mu◊'
^2& C iy D mu◊'
^3& A, B iy C mu◊'
^4& A, B iy D mu◊'
^5& A, C iy D mu◊'
33' wdydr jr®. lsysmhl 100g l wvx.= fmdaIH m∞r®: m%udK my; j.=fjys ±lafõ'
wdydr jr®.h
A
B
C

fm%daàk ^.%Eï&
19
6'8

fïo ^.%Eï&
81
14
0'5

ldfndayhsfâ%g ^.%Eï&
78'2

by; A, B, C wdydr jr®. i|yd W∞yrK jkqfha ms<sfj<ska"
^1& Öia" t<ls˙ yd m˙mamq fõ'
^2& ngr®" .juia yd lel=Ω iy,a fõ'
^4& ó ls˙" m˙mamq yd fidahd fndaxÑ fõ'
^3& W!re uia" lv, yd lel=Ω iy,a fõ'
^5& fmd,a" n, udÆ yd mdka fõ'
34' tkaihsóh ≥Uqre meye ùug ,la fkd jk wdydr ldKavh jkafka"
^1& wem,a" wU iy fmhdr®ia h'
^2& w,sfmar" mefmd,a iy b≥Kq flfi,a h'
^3& wÆ flfi,a" jïngq iy nKavlald h'
^4& ñ†" fmar iy f∞vï h'
^5& wr®;dm,a" wkakdis iy fndaxÑ h'
w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

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35' m,d jr®. wdydr i|yd ieliSfï° fndfydaúg isyskaj ,shd fmd,a o tl;= lr ue,aÆula f,i ms<sfh<
lr.kq ,efí' fuys we;s fmdaI”h jeo.;alu jkafka"
^1& fmd,a tl;= ls¯u u.ska m,dj, we;s úI øjH úI rys; øjH njg m;a ls¯u h'
^2& m,dj, we;s Lksc øjH" fmd,a ls˙j, †hùfuka Y¯rhg wjfYdaIKh ùu myiq ls¯u h'
^3& m,dj, we;s fïo ødjH úgñk Y¯rhg wjfYdaIKh ls¯ug bv ie,eiaùu h'
^4& isyska j ,shQ neúka m,dj, we;s c, ødjH úgñk msg;g meñ”u myiq ls¯u h'
^5& fmd,aj, we;s fldf,iagfrda,a uÜgu wvqlr Y¯rhg ys;lr ;;a;ajhg m;a ls¯u h'
36' hï NdKavhla u; md˙fNda.sl b,aÆula we;s ùug n,mE yels idOl lsysmhla my; ±lafõ'
A - md˙fNda.sl wjYH;dj
B - fhojqïj, ñ,
C - l%h Yla;sh
by; idOl w;=˙ka ksjer† jkafka"
^2& C mu◊'
^3& A iy B mu◊'
^1& A mu◊'
^4& A iy C mu◊'
^5& B iy C mu◊'
37' rch hï NdKavhla u; Wm˙u ñ, kshu ls¯u u.ska uQ,sl jYfhka n,dfmdfrd;a;= jkafka"
^1& NdKavh i|yd jeä b,aÆula we;s ls¯u h'
^2& ksIamdolhd wdrCId ls¯u h'
^3& md˙fNda.slhd wdrCId ls¯u h'
^4& rch ue†y;a ù NdKav ñ,° .ekSu h'
^5& ksIamdolhdg ;u NdKav i|yd jeä ,dNhla ,nd°u h'
38' f.dúfhla ;uka i;= uqof,ka bvula ñ<° .ekSfuka wk;=rej tys ñ˙ia j.djla i|yd fj<|fmdf<ka
ridhksl fmdfydr ñ,° f.k fhdohs' fuys° ork ,o ms˙jeh jr®. ms<sfj<ska"
^1& ia:djr ms˙jeh iy uqÆ ms˙jeh fõ'
^2& úp,H ms˙jeh iy ia:djr ms˙jeh fõ'
^3& ia:djr ms˙jeh iy úp,H ms˙jeh fõ'
^4& úp,H ms˙jeh iy uqÆ ms˙jeh fõ'
^5& uqÆ ms˙jeh iy úp,H ms˙jeh fõ'
39' úh<s l,dmfha nyq,j olakg ,efnk †la j,s.hla iys; ;e,a, iy fud,a,sh fyd¢ka jr®Okh jQ .j
/,la W,dlñka isá w;r bka iuyr i;=ka ìu je;s¯ isák nj ks¯CIKh ls¯ug isiqkag yels úh' tu
ks¯CIK u.ska m˙ir WIaK;ajh flfrys"
^1& i;=kaf.a yeis¯fï fjkialï úoyd olajhs'
^2& i;=kaf.a ldhsl l%shdj,Skaf.a fjkialï úoyd olajhs'
^3& i;=kaf.a rEm úoHd;aul fjkialï úoyd olajhs'
^4& i;=kaf.a yeis¯fï fjkialï yd ldhsl l%shdj,Skaf.a fjkialï úoyd olajhs'
^5& i;=kaf.a yeis¯fï fjkialï yd rEm úoHd;aul fjkialï úoyd olajhs'
40' ukd md,khla iys; f.dúfmd<l isák foaYSh ls˙ .jhskaf.a ksIamdok yelshdj †.=ld,Skj jeä †hqKq
lr .ekSug f.dúfmd< ysñhdg wjYHj we;' fï i|yd .; yels jvd;a fhda.H l%shdudr®.h jkafka"
^1& i;=ka jrKh ls¯u hs'
^2& i;=ka iydNssckkh ls¯u hs'
^3& i;=kag fyd¢ka fmdaIKh ,nd °u hs'
^4& Wiia j˙.fha i;=ka iu. fouqyqï ls¯u hs'
^5& i;a;aj fi!LHh jeä †hqKq ls¯u hs'

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

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41' úoHd.drhl fµdau,Ska ødjKj, .jhdf.a wdydr udr®. moaO;sfha ixlSr®K wdudYfha fldgia fjka
fjkaj nyd ;snQ kuq;a tu Ndckj, f,an, .e,ù ;snq◊' tu Ndckj, ;snQ wdudY fldgiaj, mska;+r
my; ±lafõ'

A
B
C
by; mska;+rj, ,CIK weiqfrka tla tla n∫kg w∞< f,an, wkqms<sfj<ska"
^1& ú;xisldj" nyqkeñh" rEukh iy cGrdYh fõ'
^2& cGrdYh" ú;xisldj" rEukh iy nyqkeñh fõ'
^3& nyqkeñh" ú;xisldj" rEukh iy cGrdYh fõ'
^4& cGrdYh" nyqkeñh" rEukh iy ú;xisldj fõ'
^5& rEukh" ú;xisldj" nyqkeñh iy cGrdYh fõ'

D

42' ms˙is≥ j˙.j, i;=ka we;s lrk wNsckk l=l=Ω f.dúfmd<l fldgq wxl X ys uia ,nd .ekSu i|yd
we;s lrkq ,nk iq≥ meye;s ;ks lru, iys; i;a;= o" fldgq wxl Y ys ì;a;r ,nd .ekSu i|yd we;s
lrkq ,nk iq≥ meye;s i;a;= o isá;s' fuu X yd Y fldgqj, isák l=l=Ω j˙. úh yelafla ms<sfj<ska"
^2& ifilaia iy n%yaud h'
^1& ifilaia iy f,f.daka h'
^3& wdr®'whs'wdr®' iy f,f.daka h'
^4& wdr®'whs'wdr®' iy ksõyeïmaIhr® h'
^5& ifilaia iy fldaÖka h'
43' ì;a;r iy uia ksIamdokh i|yd iq≥iq" †kla jhie;s l=l=Ω megjqka ldKav folla ,nd.;a f.dúhl=g
m<uq i;s 03 i|yd wdydr ñ,g .ekSug wjYH ù kï Tyq úiska ñ,° .; hq;= wdydr jkafka"
^1& l=l=Ω megjqkaf.a i,dlh iy jr®Ol i,dlh fõ'
^2& jr®Ol i,dlh iy ì;a;r ouk lsls<shkaf.a i,dlh fõ'
^3& fn%dhs,r® wjika i,dlh iy jr®Ol i,dlh fõ'
^4& l=l=Ω megjqkaf.a i,dlh iy fn%dhs,r® wdrïNl i,dlh fõ'
^5& jr®Ol i,dlh iy wjika i,dlh fõ'
44' jd◊c uÜgfï l=l=Ω jHdmd˙lhka ±kqj;a ls¯fï jevigykl° mY= ixjr®Ok ks,Od˙ uy;d úiska
lsls<shkaf.a ±lsh yels ,CIK lsysmhla fmkajd fok ,°'
A - l%shdYS,S nj
B - Yla;su;a fomd iy f.ù we;s kshfmd;=
C - l=vd ;o ^rΩ& Worhla iys; nj
D - flá fydg iy †˙u;a nj
E - úh<s l=vd cïnd,shla iys; nj
by; ,CIK w;=˙ka fyd¢ka ì;a;r ouk lsls<shkaf.a ,CIK jkafka"
^1& A, B yd C mu◊'
^2& A, B yd D mu◊'
^3& A, C yd D mu◊'
^5& B, D yd E mu◊'
^4& B, C yd D mu◊'
45' i;a;aj f.dúfmd<la m¯CId ls¯ug meñ◊ mY= ffjoHjrfhl= igyka lr.;a lreKq lsysmhla my;
±lafõ'
A - iuyr ls˙ fokqkaf.a nqre,a, b†óu yd tu i;=kaf.ka f∞jd.;a ls˙j, leá;s iajNdjhla
mej;Su'
B - iuyr .j fokqka mdo fldr .iñka weú°u yd Tjqkaf.a uqLfhka fl< je.s¯u'
C - l=l=Ω fldgqfõ f,a ñY% mdpkh yd n∫kaj, lEu b;s˙ ù ;sîu'
by; A, B yd C frda. ,CIKj,ska wkqudk l< yels frda. jkqfha ms<sfj<ska"
^1& ueiaghsàia" .j rla;md;h iy /kslÜ h'
^2& ls˙ WK" l=r yd uqL frda.h iy fldlaisäfhdaisia h'
^3& ueiaghsàia" l=r wdidokh iy /kslÜ h'
^4& ueiaghsàia" l=r yd uqL frda.h iy fldlaisäfhdaisia h'
^5& ls˙ WK" weka;%elaia frda.h iy l=reΩ WK h'
w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

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•

46'
47'

48'
49'
50'

wxl 46 - 50 olajd m%Yk
a i|yd m%ldY fol ne.ska ° we;' tla tla m%Yakh iïnkaOfhka ° we;s m%ldY
hq.,hg fyd¢kau .e<fmk ms<s;=r my; ms<s;=re mfyka f;dard ms<s;=remf;ys Wmfoia m˙† ,l=Kq
lrkak'
m%;spdrh
m<uqjeks m%ldYh
fojeks m%ldYh
(1)
i;H jk w;r" m<uqjekak ksjer† j my∞ fohs'
i;H h'
(2)
i;H jk kuq;a m<uqjekak ksjer† j my∞ fkdfohs'
i;H h'
(3)
wi;H h'
i;H h'
(4)
i;H h'
wi;H h'
(5)
wi;H h'
wi;H h'
m<uqjeks m%ldYh
nv b˙Û= îch tal îc m;%S fõ'
m%frdayKh i|yd wjYH idOl m%Yia;j ,efnk
úg îc m%frdayKh ùug we;s yelshdj îc
iqma;;dj f,i ye¢kafõ'
iajjkaOH;dj ksid Ydlj, iajmrd.kh
is≥fkdfõ'
ir, m%pdrlh ;=<° w;= len,sj, uq,a we°u
blauka fõ'
mgl frdamKfhka ,efnk Ydlj, úp,;djhka
we;s fkdfõ'

fojeks m%ldYh
nv b˙Û= îcfha îc m;% tlla msysgk w;r th
jr®:ßldj f,i y∫kajhs'
.nvd ;=< CO2 idkaøKh wvq ls¯fuka îc
„jH;dj wdrCId lr.; yelsh'
iajjkaOH;dj hkq mrd. m%frdayKfhka miq mrd.
kd<fha jeãu nd, ùuhs'
by< wdr®ø;dj yd WIaK;ajh w;= len,sj, uq,a
we°u fõ.j;a lrhs'
ffoyssl laf,dak úp,;dj u.ska m%fõ◊l úp,;d
,nd .; yelsh'

***

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

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08 - lDIs úoHdj
*
*

˚˚ m;%h

A fldgfia m%Yak ish,a,g u ms<;
s =re imhkak'
B fldgfika m%Yak y;rlg muKla ms<;
s =re imhkak'
A fldgi - jHQy.; rpkd

1'

° we;s rEm igyfka ±lafjkafka .jhdf.a wdydr udr®. moaO;sh hs' ta weiqfrka my; m%Yakj,g ms<s;=re
imhkak' ^A fldgfia ˚ - ˚˚˚ olajd m%Yakj, ms<s;=rej,g w∞< bx.%Sis wl=r rEmigyfkka f;dard
,shkak'&
Y

X

W

U

T
P

R

V

Q

S

(A) (i) wdydr hdka;%sl „r®Khg Ndckh jk ia:dk folla olajkak'

a.
b.

(ii) wdydr m%Odk f,i CIqø„ù „r®Khg Ndckh jk ia:dkh l=ula o@
(iii) wdydr" tkaihsóh „r®K l%shdj,shg Ndckh jk m%Odk ia:dkh$ia:dk fudkjd o@
(iv) fuu wdydr udr®. moaO;sfha §S¶ ia:dkhg iuldr®h jk W!rdf.a wdydr udr®. moaO;sfha
fldgi kï lrkak'

(v) W ia:dkfha fmdaIl wjfYdaIKh jk m%Odk l%uh kï lrkak'
(B) .jhdf.a wdydr udr®. moaO;sfha is≥jk „r®K l%shdj,s folla my; ±lafõ'

a. msIagh

ia:dkh - 1

tkaihsuh - 1
ia:dkh - 3
b. o< ;ka;=

ia:dkh - 2
fuda,afgdaia

.a¨fldaia

tkaihsuh - 2

jdIamYS,S fïo wï, › jdhq

(i) a l%shdj,sfha 1 yd 2 ia:dk kïlrkak'
ia:dkh - 1

ia:dkh - 2

(ii) by; a l%shdj,shg ∞hljk tkaihsu fol kï lrkak'
tkaihsuh - 1

tkaihsuh - 2

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

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(iii) by; b „rK l%shdj,sh is≥jk ia:dkh - 3 kï lrkak'
(iv) b l%shdj,sfha „r®Khg fya;=jk m%Odk CIqø„ù ldKavh l=ula o@
(v) W!rdf.a „r®K moaO;sh ;=< b l%shdj,sh is≥jk ia:dkh kï lrkak'
(C) f.dúfmd< i;=ka i|yd fokq ,nk wdydr jr®. iy tajdg W∞yrK lsysmhla my; j.=fõ
±lafõ'
wdydr jr®.h
idkaø wdydr
f;;a o< wdydr
úh<s o< wdydr
ixrCIKh l< o< wdydr

P
Q
R
S

W∞yrK

W fya" ihsf,a–
X fya" ms≥re
Y fmd,a mqkaklal=" iy,a ksjqvq" fud,Eiia
Z fviafudaähï" n%efla˙hd" fifÜ˙hd

(i) by; j.=fõ ±lafjk P, Q, R, S wdydr jr®.j,g .e<fmk W∞yrK ldKavh W, X, Y
iy Z w;=˙ka f;darkak'
(a)

P

(c)

R

(b)
(d)

Q
S

(ii) idkaø wdydr yd o< wdydr w;r m%Odk fjkialï folla olajkak'

a.
b.

(D)

fnda. j.dfõ° Ndú; lrkq ,nk úúO lDIs ridhk øjH yd tajdfha Ndú; ±lafjk j.=jla my;
±lafõ'
X1
X2

X
fldmr® iekafvd 4
ldfndmshqrdka

Y1
Y2

X3 MCPA

Y3

X4 ROUND UP
X5 mhsrf;%dhsv

Y4
Y5

Y
ù j.djg fhdok f;dard kik j,akdYlhls'
fCIa;%hl we;s ishΩu j,ame<Eá
ur®okhg fhd∞.kakd j,akdYlhls'
Ydlj,ska ksiaidrKh lrkq ,nk
jvd;a wdrCIdld¯ lDñkdYlhls'
iakdhq úI we;s lrk lDñkdYlhls'
m;% mq,a,s frda.h ur®okhg
fhd∞.kakd †,SrkdYlhls'

by; f;dr;=re u; X yd Y .<mkak'

(i) X1

(ii) X2

(iii) X3
(iv) X4
(v) X5

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

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2' (A) YsIHfhla mdxY= úYaf,aIK m¯CIKhl°" W≥fka úhΩ mia .%Eï 100l my; i|yka whk m%udKhka
we;s nj fidhd .kakd ,°'
Mg2+
} ñ,siul 4'5
2+
Ca
} ñ,siul 6'0
K+
} ñ,siul 3'6
H+
} ñ,siul 1'5
2SO4
} ñ,siul 0'8
(i) by; o;a; Wmfhda.S lrf.k tu mia idïm,fha legdhk yqjudre Od˙;dj .Kkh lrkak'

(ii) mfiys legdhk yqjudre Od˙;djfha jeo.;alï folla olajkak'

a.
b.

(iii) by; mia idïm,fha NIau ix;Dma;sh .Kkh lrkak'

(iv) mil NIau ix;Dma;sh .Kkh ls¯fï jeo.;alu l=ula o@
(v) mil wdï,sl;djh wvq ls¯u i|yd mig fh†h yels øjH folla ,shkak'

a.
b.

(B) ;u f.dúfmdf<a <s|la lmk f.dúfhl= tu N+ñfhys mia úh<s úg ;o nj;a f;;a jQ úg uv iys;
wef,k nj;a" ógrhla muK .eUqrg yE¯fï° ;srejdK .,a ;Ügqjla we;s nj;a ks¯CIKh lrk
,°'
(i) tu N+ñfha we;s mia ldKavh kï lrkak'
(ii) f.dúhd úiska <s| lmk ,o m%foaYh wh;a †ia;%slalh úh yelafla l=ula o@
(iii) tu f.dúfmdf<a j.d ls¯ug iq≥iq fnda. jr®. folla kï lrkak'

b.

a.

(iv) fuu f.dúfmdf<ys fnda. j.d ls¯fï° u;=úh yels m%Odk .egÆ folla ,shkak'

a.
b.

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

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(C) ksire jQ bvul fnda. j.d ls¯ug we;s fhda.H;dj ms<sn| m¯CId ls¯fï° my; lreKq fy,sorõ
úh'
1' N+ñh u| nEjqï iys; h'
2' fndfyda ia:dkj, Wvq mia fia° f.dia we;'
3' mi W.% f,i ydhkhg ,la ù we;'
4' ksire f,i jevqKq j,ame<Eá nyq, h'
(i) §W.% f,i mi ydhkhg ,la ù we;¶ hkafkka l=ula woyia flfrhs o@
(ii) fuu mi ixrCIKh ls¯u i|yd wkq.ukh l< yels mdxY= ixrCIK l%u y;rla ,shkak'

a.

b.
c.
d.
(iii) fujeks fCIa;%hl is≥ úh yels mdxY= Ldok wdldr ;=kla ,shd olajkak'

a.

b.
c.
(iv) j.dj i|yd fuu N+ñh ieliSfï° fhd∞.; yels ìï ieliSfï l%uhla kï lrkak'
(v) fuu fCIa;%fha t<jÆ j.d lrkafka kï ta i|yd WÑ; c, iïmdok l%uhla kï lrkak'
(vi) fuu fCIa;%fha j,a ur®okh i|yd fhd∞ .; yels iq≥iq l%u folla kï lrkak'

a.

b.
3'

(A) (i) iy,a i|yd iemhqï Y%s;h QS = 4P - 50 f,i ° we;' th weiqfrka my; iemhqï f,aLkh
iïmQr®K lrkak'
iy,a 1kg l ñ,
15
25
35
45

iemhqï m%udKh kg
'''''''''''''''''''''
'''''''''''''''''''''
'''''''''''''''''''''
'''''''''''''''''''''

(ii) fuu iemhqï f,aLkhg w∞< iemhqï jl%h m%ia;drhla f,i b†˙m;a lrkak'

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

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(iii) rch u.ska ù j.dj i|yd fmdfydr iykdOdr ,nd ≥kafka kï fuu iemhqï jl%hg we;sfjk
n,mEu m%ia;drhla u.ska olajkak'

(iv) by; (ii) fldgfia ,nd.;a iemhqï jl%h Tiafia" iemhqu m%idrKh ùug iy ixfldapkh ùug
n,mdk m%Odk fya;= folla i|yka lrkak'

a.

b.
(B) (i) my; ±lafjk ms˙jeh ldKav w;r ±lsh yels m%Odk fjkialu ne.ska olajkak'
1'

ia:djr ms˙jeh iy úp,H ms˙jeh

2'

idudkH ms˙jeh iy wdka;sl ms˙jeh

(ii) fn%dhs,r® l=l=Ω f.dúfmd<l uqΩ ms˙jeh my; wdldrfhka ° we;'
C = 15,000 + 250 y
fuys C jQ l,S uqΩ ms˙jeh ^remsh,a& jk w;r y jQ l,S ksIamdok tall .Kk ^kg j,ska& fõ'
fuys ia:djr ms˙jeh fldmuK o@

fuys wdka;sl ms˙jeh fldmuK o@

(iii) wdjia:ßl ms˙jeh wr®: olajkak'

(C) (i) ≥IafmdaIKh hkq Y%S ,xldfõ ±lsh yels m%Odk;u fmdaIK .egÆjls' ≥IafmdaIKfha m%Odk
wdldr fol i|yka lrkak'

a.

b.

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

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(ii) whäka W!k;djh ksid we;s úh yels m%Odk;u frda.S ;;a;ajh i|yka lrkak'
(iii) by; frda.S ;;a;ajh u.yrjd .ekSu i|yd rch u.ska wkq.ukh lr we;s l%shdms<sfj; l=ula o@
(iv) †hjeähd frda.S ;;a;aj Y%S ,xldfõ mdi,a orejka w;r iq,N nj uE; ld,fha° lrk ,o
ióCIKj,ska fy,s ù we;' fï ;;a;ajhg fya;= ù we;s m%Odk lreKq folla i|yka lrkak'

a.

b.
(D) (i) f.dúfhla ;u jhi wjqre≥ 10la jk fmd,a j.dfõ w;=re fnda.hla f,i wkakdis j.d ls¯ug
woyia lr ta i|yd wjYH Wmfoia .ekSug m<df;a lDIslr®u WmfoaYl uy;d yuqúh'
a. WmfoaYl uy;d f.dúhdf.a woyi wkqu; fkdlrk ,°' thg m%Odku fya;=j l=ula úh
yels o@

b. fmd,a j.dfõ w;=re fnda. f,i j.d l< yels fjk;a fnda. folla kï lrkak'
1'
2'
(ii) ù† fnda. j.dfõ ° nyq,j u Ndú; lrk .a,s˙iSähd ^Gliricidia maculata& Ydlfhka mig
,efnk m%Odk jdis folla i|yka lrkak'

a.

b.
(iii) mig ,efnk jdisj,g wu;rj uE; ld,fha .a,s˙iSähd Ydlh fjk;a m%Odk úl,am
m%fhdackhla i|yd o Ndú; lrhs' th l=ula ±hs i|yka lrkak'

(iv) Y%S ,xldfõ nyq,j fhd∞ .kq ,nk j.d rgd folla kï lrkak'

a.

b.
4'

(A) t<jÆ yd OdkH fndfydaúg m%pdrKh lrkqfha îc u.sks' udΩ ñ˙ia" ;lald,s" l=rlalka" mf;da,"
jegfldΩ" nKavlald" ù yd nvb˙Û= Bg W∞yrK lsysmhls'
(i) by; i|yka fnda. jr®. tal îc m;%S yd oaú îc m;%S f,i jr®. lrkak'

a. tal îc m;%S

b. oaú îc m;%S

(ii) by; i|yka fnda. jr®. w;=˙ka wfOdafN!u m%frdayKhla fmkajkafka l=uk fnda.j, îc jr®. o@
(iii) jr®Ol ksfYaOl mej;Su ksid ;lald,s îc iqma;;djfha mj;S' th bj;a lrkafka flfia o@
w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

- 105 -

(iv) óg wu;rj îc iqma;;djg fya;=jk fjk;a idOl folla olajd Bg w∞< fnda.j,g W∞yrK
by; fnda. w;=˙ka f;dard ,shkak'

a.
b.
(v) îc u.ska Ydl m%pdrKh ls¯fuka ,efnk jdis y;rla ,shkak'

a.
b.
c.
d.

(B) (i) my; j.=j iïmQr®K lrkak'

a.
b.
c.

fnda.h
wr®;dm,a
ù
mefmd,a

frda.h
''''''''''''''''''''''''''''''''''''''''''''''
fld<mdÆ frda.h
uq≥ka .eg frda.h

frda. ldrlhd
Phytophthera infestans
'''''''''''''''''''''''''''''''''''''''''''''''
'''''''''''''''''''''''''''''''''''''''''''''''

(ii) my; ±lafjk W!k;d ,CIKj,g m%Odk jYfhka fya;=jk Ydl fmdaIlh w∞< ysia;ek u;
i|yka lrkak'

a. oï meye;s m;% iys; l=re jQ Ydl

b. ,má m;% iïmQr®Kfhkau ly meye ù hdu

c. fïrE m;%j, ∞rh ms,i
s aiS m;%fha ms,siaiqï ,m
olakg ,eîu
d. m;%j, kdrá yer b;s˙ m%foaY y˙;CIh ùu

e. w.%ia:fha wka;r® mr®j fláù l=vd m;% rdYshla we;sùu
(C) YsIHfhla mdi,a f.dúfmdf<ys we;s mf;da, j.djl iuyr m;% is≥re jk f,i ld oud ;sfnk nj
ks¯CIKh lf<ah'
(i) by; i|yka ydksh is≥lrkq ,nk m%Odk lDñhd kï lrkak'
(ii) tu lDñhd wh;a jk f.da;%h l=ula o@
(iii) by; lDñ f.da;%hg úfYaIs; jQ ,CIK folla olajkak'

a.
b.

(iv) by; lDñhd md,kh l< yels l%uhla i|yka lrkak'
(v) a. by; lDñ ydkshg wu;rj mf;da, j.dfõ olakg ,efnk m%Odk lDñ ydkshla kï
lrkak'

b. Tn kï l< lDñ ydksh md,kh ls¯u i|yd fhd∞ .; yels ridhksl fkdjk md,k
l%uhla kï lrkak'

**
w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

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B fldgi - rpkd
5' (i)
(ii)

(iii)

(a)
(b)

foaY.=◊l idOl fjkiaùu ksid lDIslr®u fCIa;%fha we;súh yels m%Odk .egÆ fudkjd o@
tu .egÆ uÛyrjd f.k lDIsldr®ñl lghq;= idr®:l lr .ekSug wkq.ukh l< yels
l%shdudr®. ljfr® o@
(a) f.dúhl= ;u fCIa;%fha j.d ls¯ug fmr mdxY= m¯CIdjla ls¯fuka ,o f;dr;=re lDIs
WmfoaYljrhdg fmkajk ,≥j Tyq úiska mig yqKq ^CaCO3& fh°ug f.dúhdg Wmfoia
fok ,°' fuu Wmfoia ,nd°ug fya;= l=ula úh yels o@
(b) fuu Wmfoi wkq.ukh ls¯fuka ,nd.; yels jdis fudkjd o@
úh<s l,dmfha i;a;aj md,k f.dúfmd<l fud,a,sh iy /,sjegqKq ;e,a,la iys; úYd, m%udKfha
ls˙fokqka olakg ,enq◊'
(a) tu ls˙fokqkaf.a j˙. ldKavh l=ula úh yels o@
(b) tu ls˙fokqka wys;lr foaY.=◊l ;;a;aj,g Tfrd;a;= °ug olajk m%;spdr iy wkqjr®;k
úia;r lrkak'

6' (i)

lDIs úoHd .=re;=ñhla ;u mka;sh lKavdhï ;=klg fn∞ my; i|yka m¯CIK ls¯ug Wmfoia
≥kakdh'
1 - lKavdhu - mfia c, m%;sY;h fiùu'
2 - lKavdhu - mfia fCIa;%Od˙;dj fiùu'
3 - lKavdhu - mfia ia:ßr ue,ùfï wxlh fiùu'
(a) m<uq lKavdhu úiska tu m¯CIKh ls¯ug .;hq;= WmlrK kï lrkak'
(b) fojk lKavdhu úiska mfia fCIa;%Od˙;dj fiùu i|yd .;hq;= mdGdxl ljfr®±hs
olajkak'
(c) f;jk lKavdhu úiska ia:ßr ue,ùfï wxlh .Kkh lrkafka flfia±hs meye†,s
lrkak'

(ii)

f.j;a;la ks¯CIKhg .sh isiqka ilia l< jdr®;djlska Wmqgd.;a f;dr;=re lsysmhla my;
±lafõ'
X - fCIa;%fha fudkrl=vqïìh" jïngq" we;a wä" l=mamfïkshd" wegjrd" ñ˙ia" ;lald,s"
jegfldΩ wd° Ydl ;sîu'
Y - Y%ñlhska ly j.dfõ ms≥re we;s¯u" l=mamfïkshd w;ska b†¯u yd wegjrd Ydl u;g
†hrhla biSu hkd° ldr®hj, fh° isàu'
(a) X ys i|yka f;dr;=re wkqj tu f.j;af;a we;s j,ame<Eá jr®.SlrKh lr olajkak'
(b) tu j,ame<j, meje;au ;yjqre lr .ekSug j,ame<Eá i;= wkqjr®;k fudkjd o@
(c) Y ys i|yka f;dr;=re wkqj j,a uçk l%u jr®. lr olajkak' tu l%uj, jdis yd wjdis
,shkak'

7' (i)

wfma%,a wÆ;a wjqre≥ iuhg uia ksIamdokh i|yd fn%dhs,r® l=l=Ω jHdmdrhla wdrïN ls¯ug
f.dúhl= wfmaCId lrhs'
(a) ta wkqj †kla jhie;s l=l=Ω megjqka f.dúfmd<g /f.k wd hq;af;a l=uk ld,fha o@
(b) fn%dhs,r® i;=ka we;s ls¯fï ° i,ld ne,sh hq;= m%Odk lreKq úia;r lrkak'

(ii)
8' (i)
(ii)

jr®;udk Y%S ,xldfõ lDIs lr®udka;h k.d isgqùu i|yd we;s wNsfhda. úia;r lrkak'
(a)

fylaghdr foll muK N+ñhl" uOH m˙udK ls˙.j jHdmdrhla wdrïN ls¯ug
n,dfmdfrd;a;=jk f.dúfhl= úiska .jhska we;s l< hq;= iq≥iqu l%uh úia;r lrkak'
(b) tu l%uh ;=< ls˙fokqka 20fofkl= i|yd iq≥iq ksjdihl ìï rEmigykla w¢kak'
lDIs ksIamdok wf,úlrK l%shdj,sfha ldr®hCIu;dj jeä lr .; yels wdldr úia;r lrkak'

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

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9' (i)
(ii)

(a)
(b)

§lDIs ksIamdok wf,úlrKh¶ y∫kajkak'
wfkl=;a NdKav yd fiajd wf,úlrK l%shdj,shg jvd lDIs ksIamdok wf,úlrK l%shdj,sh
ixlSr®K fõ' fufia ùug fya;= W∞yrK iys;j meye†,s lrkak'
ls˙.j ksjdi iE°fï ° ie,lsh hq;= lreKq i|yka lrkak'

10' úh<s l,dmfha f.dúfhl= úh<s ñ˙ia ksIamdokh i|yd wlalr foll muK N=ñNd.hl ñ˙ia j.d
ls¯ug woyia lrhs'
(i)
ksfrda.S ñ˙ia j.djla ,nd.ekSug ìï ieliSfï isg fnda. ixia:dmkh olajd wkq.ukh l< hq;=
l%shdudr®. meye†,s lrkak'
(ii) fuu j.dj idrj;aj mj;ajd .ekSu i|yd mdxY= fmdaIK uÜgu ys;lrj mj;ajd .ekSug taldnoaOj
Ydl fmdaIl l<ukdlrK ^IPNS& ixl,amh fhd∞.kakd whqre meye†,s lrkak'
(iii) miq wiajkq ;dCIKh wju lrñka fuu ñ˙ia md˙fNd.slhd fj; heùfï° wiajkq fk,k
wjia:dfõ isg fuu f.dúhd úiska wkq.ukh l< hq;= l%shdudr®. úia;r lrkak'

***

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

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09 - „j úoHdj
m%Yak m;% jHQyh
˚ m;%h - ld,h ( meh 02hs'
			 jrK 5 ne.ska jQ nyqjrK m%Yak 50ls' m%Yak ish,a,g u ms<s;=re iemhsh hq;= h'
			 tla m%Yakhlg ,l=Kq 02 ne.ska uqΩ ,l=Kq 100ls'
˚˚ m;%h - ld,h ( meh 03hs'
			 fuu m%Yak m;%h jHQy.; rpkd yd rpkd jYfhka fldgia follska iukaú; h'
			 A fldgi - jHQy.; rpkd jr®.fha m%Yak y;rls' m%Yak ish,a,g u ms<s;=re
					 iemhsh hq;= h' tla m%Yakhlg ,l=Kq 100 ne.ska ,l=Kq 400ls'
			 B fldgi - rpkd jr®.fha m%Yak yhls' m%Yak y;rlg ms<s;=re iemhsh hq;= h'
					 tla m%Yakhlg ,l=Kq 150 ne.ska ,l=Kq 600ls'
			 ˚˚ m;%h i|yd uqΩ ,l=Kq 1000 fi 10 } 100
wjidk ,l=K .Kkh ls¯u
			
				

(

˚ m;%h
} 100						
˚˚ m;%h
} 100
wjidk ,l=K } 200 fi 2 } 100

˚ m;%h
ie,lsh hq;=hs (
* ishÆ u m%Yak i|yd ms<s;=re imhkak'
* ksjer† fyda jvd;a .e<fmk fyda ms<s;=r f;darkak' ^úNd.fha ° ms<s;=re iemhSu i|yd nyqjrK lv∞ishla
imhkq ,efí'&
1'

l,ams;hla hkq"
^1& m¯ÈKd;aul j ikd: l< fkdyels woyils'
^2& ks¯ÈKhla meye†,s ls¯u i|yd b†˙m;a lrkq ,nk woyils'
^3& ishÆ u m¯ÈKd;aul idÈsj,ska ;yjqre jk woyils'
^4& ms<s.;a uQ,Or®uj,g mgyeks woyils'
^5& m¯ÈKhlska ;yjqre fkdjQ ks.ukhls'

2'

.a,hsfldcka iy msIagh hk folu"
^1& .a¨fldaiaj,ska iE° we;'
^2& Rcq fr®Çh nyq wjhjl fõ'
^3& m%Ndixiaf,aIKfha M, fõ'			
^4& whäka u.ska ks,a meyefhka jr®K .ekafõ'
^5& i;=ka u.ska ixÑ; lrkq ,nk wdydr øjH fõ'

3'

tkaihsuj, ;rÛld¯ ksfYaOl hkq"
^1& Wmia;rhg ne£ m%;sl%shd ksfYaOkh lrk wKq fõ'
^2& tkaihsufha il%sh ,ÈHhg ne£ m%;sls%hdj ksfYaOkh lrk wKq fõ'
^3& tkaihsufha il%sh ,ÈHh yer fjk;a ia:dkj,g ne£ m%;sl%shdj ksfYaOkh lrk wKq fõ'
^4& Wmia;rhg iudk ridhksl m%;sl%shdj, fhfok wKq fõ'
^5& jHQyd;aul j tkaihsuhg iudk jk wKq fõ'

4'

c,fha l=uk .=Kdx.hla „ùka ;=< WIaK;ajh fjkia ùu wju ls¯u i|yd jvd;a u Wmld¯ fõ o@
^1& mqΩ,a WIaK;aj mrdihla ;=< c,h øjhla f,i mej;Su'
^2& c, wKq ;j;a fndfyduhla wKq iuÛ yhsv%cka nkaOk iE°u'
^3& c, wKq ixila; n, u.ska tlsfklg ne£ mej;Su'
^4& c,hg wêl úYsIag ;dm Od˙;djla ;sîu'
^5& c,hg wêl jdIamSlrKfha .=ma; ;dmhla ;sîu'
w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

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NH2

5'
		
-

O

O

O

O-

O-

O-

HC

O - P - O - P - O - P - O - CH2
H

O

		

N C C
N
CH
C
N
N

H

H
H
OH OH

my; i|yka tajd w;=˙ka ljrla u.ska by; i|yka wKql jHqyh fyd¢kau úia;r fõ o@
^1& kshqla,sfhdaghsvhla 				
^2& ATP wKqjla
^3& NAD wKqjla					
^4& DNA j, kshqla,sfhdaghsvhla
^5& iytkaihsu wKqjla
6'

my; i|yka jHqy - lD;H iïnkaO;dj w;=˙ka ljrla jer† o@
^1& kHIaáldj
rRNA ixiaf,aIKh
^2& f.d,a.s ixlSr®Kh
i%dùh wdYhsld iE°u
^3& ma,diau nkaOk
ffi, w;r øjH yqjudrej
^4& .a,hsTlaisfidau
,smsv ldfndayhsfâ%g njg m˙jr®;kh
^5& rΩ wka; ma,diaóh cd,sldj úIyrKh

7'

my; i|yka tajd w;=˙ka W!kkfha m%dlal,dj ˚ ys° is≥ fkdjkafka l=ula o@
^1& iEu jr®Kfoayhlau oaù.=Kkh ù jr®KfoaydxY folla we;s ùu'
^2& jr®Kfoay flá ùu'
^3& iucd; jr®Kfoay iudka;r j ilia ùfuka Wmd.u iQ;s%ld ixlSr®Khla iE°u'
^4& iucd; jr®Kfoayj, jr®KfoaydxY w;r fldgia yqjudre ùu'
^5& kHIaá mg,h fkdfmkS hEu'

8'

my; i|yka ,CIK w;=˙ka thg b†˙fhka ° we;s „ù ldKavhg wkkH jQ ,CIKh i|yka j we;af;a
l=ul o@
		 ,CIKh		
„ù ldKavh			
^1& iÛ, jr,a
Tiafghsla;sfhaia
^2& mCIau
is,sfhdfmdard
^3& fmäif,a˙hd
tlsfkdvr®fïgd
^4& nysIaiels,a,
wdf;%dfmdavd
^5& mshdm;a
wdfõia 		
9'

ksjer† j.ka;sh f;darkak'
^1& mCISka p,;dmSka h'
^2& wmr .=o j,s.h fldavdgd ,CIKhls'
^3& fkufgdavdjka iSf,dañlhka h'		
^4& weka;fidajdjkaf.a nqyqnd iy fuvqid wdldr we;'
^5& ishÆ u ma,eáfy,añka;djka mrfmdaIs;hka h'

10' my; ±lafjkafka fgfrdmhsgdj, ,ÈK lSmhls'
		 A - meye†,s ikd, mgl
		 B - iajdëk ckaudKq Ydlh
		 C - iajdëk îcdKq Ydlh
		 D - tal.=K îcdKq
		 E - p, m%ckl ffi,
n%fhdmhsgdj, ±lsh fkdyelafla by; i|yka ljr ,CIK o@
^1& A iy B 			
^2& A iy C 			
^4& B iy D 			
^5& B iy E

^3& A iy E

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

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11' my; i|yka tajd w;=˙ka wfkHdakHdOdrh i|yd ksoiqkla jkafka l=ula o@
^1& wU .ila u; jefjk Wvjeähd Ydl 		
^2& ;dmi ll=Ωjd u; „j;ajk uqyq≥ u,
^3& u≥rejd ;=< isák Wuchereria bancrofti lSg wjia:d ^4& .il l|la u; jefjk y;=
^5& rks, Ydl uq,a ;=< „j;ajk Rhizobium
12' ñksidf.a Yajikh iïnkaO j ksjer† m%ldYh f;darkak'
^1& tla Yajik pl%hla i|yd .;jk ld,h ;;amr 4 - 5la muK fõ'
^2& „j Od˙;dj o< jYfhka 6 dm3 ls'
^3& wdYajdifha ° m%dÖrh bys,a fõ'
^4& wkqYajdikd,sld mCIauOr ia;ïNsl wmsÉPofhka wdia;rh fõ'
^5& yqiau .ekSfï YS>%;dj ieuúg u wksÉPdkq. j md,kh fõ'
13' maf,dahu m˙jykh iïnkaOfhka wr®;dm,a Ydlj, wmdhkhla f,i l%shd ls¯ug fndfyda ≥rg bv
we;af;a my; i|yka l=uk wjhjh o@
^1& m˙K; m;%h			
^2& jDoaO m;%h			
^3& jr®Okh jk m;%h
^4& N+.; l|			
^5& m%frdayKh jk wdlkaOh
14' maf,dahufhys øjH m˙jykfha° is≥jk wjia:d myla my; olajd we;'
		 A - fmfkar k< ;=<g c,h úirKh ùu
		 B - m;% ffi, u.ska iSks ksmoùu
		 C - iypr ffi, fmfkar k< ;=<g iSks ner ls¯u
		 D - Ydl m;%fha ffi,fhka ffi,hg iSks m˙jykh ls¯u
		 E - l| my<g iSks .uka ls¯u
maf,dahu m˙jykh meye†,s lrk ksjer† wkqms<sfj< f;darkak'
^1& A, B, C, D, E 			
^2& B, A, D, C, E			
^4& B, D, A, C, E			
^5& D, B, A, C, E

^3& B, D, C, A, E

15' Wvq ndyqfjys jQ r;= reêrdKqjla TlaisckSlD; ùu i|yd wfOdaCIl Ysrdj Tiafia fmkye,a, lrd .uka
lrkqfha my; i|yka l=uk udr®.h Tiafia o@
^1& ndyq-YSr®I Ysrdj
W;a;r uyd Ysrdj
yDoh
mqmaMqiSh Ysrdj
^2& fmd≥ YSr®IfmdaIS Ysrdj
W;a;r uyd Ysrdj
yDoh
mqmaMqiSh Ysrdj
^3& ndyq-YSr®I Ysrdj
W;a;r uyd Ysrdj
yDoh
mqmaMqiSh Ouksh
^4& ukH Ysrdj
W;a;r uyd Ysrdj
yDoh
mqmaMqiSh Ouksh
^5& W;a;r uyd Ysrdj
yDoh
mqmaMqiSh Ysrdj
16' ñksidf.a yDoh ms<sn| ksjer† m%ldYh f;darkak'
^1& jï lr®◊ldfõ yD;a fmaYsh ol=Kq lr®◊ldfõ yD;a fmaYshg jvd >klñka jeä h'
^2& ;=Kav lmdg" tajdfha fmaYs ;ka;= u.ska jefia'
^3& ol=Kq lr®◊ldjg Tlaiscka rys; reêrh ,efnkqfha W;a;r iy wOr uyd Ysrdj,ska mu◊'
^4& yD;a r–cq u.ska lr®◊l-fldaIsl lmdg msáld fmaYsj,g iïnkaO lrhs'
^5& ls¯gl ixirKh ixia:dksl ixirKh iu. iïnkaO fõ'
17' fhda.Ü ksIamdokfha° my; i|yka Èqø„ùka w;=˙ka ljrla Ndú; lrkq ,efí o@
^1& Lactobacillus bulgaricus			
^2& Streptomyces griseus
^3& Saccharomyces cerevisiae			
^4& Acetobacter aceti
^5& Bacillus subtilis
18' ridhksl iajhxfmdaIS nelaà˙hd Yla;sh iy ldnka ,nd .kafka wkqms<sfj<ska"
^1& wdf,dalh iy ldnkavfhdlaihsâ u.ska h'
^2& wdf,dalh iy ldnksl ixfhda. u.ska h'
^3& wldnksl ixfhda. TlaislrKh iy ldnkavfhdlaihsâ u.ska h'
^4& wldnksl ixfhda. TlaislrKh iy ldnksl ixfhda. u.ska h'
^5& ldnksl ixfhda. TlaislrKh iy ldnkavfhdlaihsâ u.ska h'
w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

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19' my; i|yka fydar®fudak w;=˙ka isks∫ iy yD;a fmaYs hk fol u u; l%shdlrkqfha l=ula o@
^1& Tlaisfgdaiska			
^2& fkdar®weâ˙k,ska 		
^3& ADH
^4& weâ˙k,ska			
^5& fldr®áfida,a
20' kshqfrdakhl l%shd úNjh iïnkaOfhka jer† j.ka;sh f;darkak'
^1& th we;sùu i|yd foay,Sh W;af;achla wjYH h'
^2& uh,skSlD; kshqfrdakj, l%shd úNj we;sjkqfha /kaúhr® .egj, muKs'
^3& l%shd úNjfha" ke.Sfï l,dj kshqfrdakh ;=<g Na+ .,d hEu ksid is≥fõ'
^4& l%shd úNjhl ° kshq˙f,udfjys O%eùh;dj m%;sjr®; fõ'
^5& ta i|yd .;jk ld,h 10 ms la muK fõ'
21' ñksidf.a m%;Hdkqfõ.S iakdhq moaO;sfha W;af;ackh ùu
^1& wefia l◊ksldj úia;drKh lrhs'			
^3& wdka;%sl hqIh i%djh ùu ksfYaOkh lrhs'
^5& fmkye,s jd;dY%h ùfï YS>%;dj jeä lrhs'

^2& fÄgh i%djh ùu W;af;ackh lrhs'
^4& reêr mSvkh jeä lrhs'

22' khsg%ckSh nysiai%djh ms<sn| jer† j.ka;sh f;darkak'
^1& khsg%ckSh nysiai%djh is≥jkqfha fm%daàk iy kshqla,sla wï, m˙jD;a;sfhks'
^2& ishÆ u i;a;ajhskaf.a khsg%cka wvx.= wdrïNl wmøjHh wefudakshd fõ'
^3& hQ˙hd ixiaf,aIKh lrk m%Odk ia:dkh jDlalh fõ'
^4& hQ˙la wï,h ksmoùfuka tla wKqjlska khsg%cka mrudKq y;rla nysiai%djh ls¯ug yelsfõ'
^5& l%shákska khsg%ckSh nysiai%djfha wjidk M,hls'
23' p,kh ms<sn|j jer† .e<mSï
^1& lIsld p,kh
^2& mÈu p,kh
^3& oaúmdo ixprKh
^4& ksød ikakuk p,kh
^5& m%Ndjr®;S p,kh

hq., f;darkak'
Chlamydomonas
Paramecium
mrúhd
Mimosa m;%
mqIm
a msmSu

24' >kdldr fmd† f,i ieliqKq f.da,dldr nelaà˙hd y∫kajkafka"
^1& iafg%mafgdfldl=i f,i h' 			
^2& iagemsf,dfldl=i f,i h'
^3& ämaf,dfldl=i f,i h' 			
^4& idiSkd f,i h'
^5& p;=Ialh f,i h'
25' iamr®Ydjr®;khg ksoiqk jkafka my; i|yka l=ula o@
^1& wdf,dalh foig kefuk iQr®hldka; mqIamhla
^2& Sonneratia jdhqOr uq,aj, jr®Okh
^3& Mimosa m;%j, yels<Su
^4& je,a f∞vï myqrej, oÛr .eiSu
^5& îc me<hl l| wdf,dalh foig keóu
26' mgl frdamK m¯ÈK i|yd wdrïNl Ydl fldgila jYfhka wju f,i iq≥iq jkafka my; i|yka
tajdhska l=ula o@
^1& w.%ia: úNdclh			
^2& mdr®Yaúl úNdclh		
^3& m;% fldgia
^4& fmd;a;			
^5& l,,h
27' ñksidf.a Y=ldKq .nvd lrk m%Odk ia:dkh jkafka"
^1& Y=l%Or kd,sld h'		
^2& wmsjDIKh h'			
^4& úir®cl m%Kd,h h'		
^5& Y=l% wdYhsld h'

^3& Y=l% kd,h h'

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

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28' ñksidf.a Y=ldKq ckkh ms<sn| jer† m%ldYh f;darkak'
^1& th wdrïN jkqfha fh!jk W∞fjks'
^2& m%d:ñl Y=l%dKqq ffi, ksmoùu i|yd Y=l%dKqud;D ffi, W!kk úNdckhg Ndckh fõ'
^3& ffi, úNdckhg Ndckh fkdù m%dlaY=l% Y=l%dKq njg úlikh fõ'
^4& fgiafgdiafgfrdka Y=l%dKqckkh hdukh lrhs'
^5& iïmQr®K l%shdj,sh i|yd udi folla muK .;fõ'
29' udkj CISrKh ms<sn| jer† j.ka;sh f;darkak'
^1& ls˙ ksIamdokh ia;k .%ka:ßj, .r®;sld ;=< is≥fõ'
^2& Biag%cka CISrKh W;af;ackh lrhs'
^3& <orejdf.a ls˙ Wrdîu CISrKh mj;ajd .ekSu i|yd w;HjYH fõ'
^4& fm%da,elaáka ls˙ ksIamdokh wdrïN lrhs'
^5& Tlaisfgdaiska ls˙ úir®ckh m%jr®Okh lrhs'
30' udkj .r®NdIh ms<sn| i;H m%ldYh f;darkak'
^1& tys ì;a;sfhys jeä fldgila iE° we;af;a tkafvdfuÜ˙hfuks'
^2& ixfiapkh idudkHfhka is≥jkqfha th ;=< h'
^3& l,,nkaOfhka i%djh lrk fm%diag.a,ekaäka u.ska tys ixfldapk ksfYaOkh lrhs'
^4& l,,nkaOh iE°ug tys ì;a;sh ∞hl fkdfõ'
^5& Tlaisfgdaiska u.ska fm%arKh jk tys ixfldapk Ok mqkr®fmdaIS hdka;%Khla u.ska hdukh fõ'
31' m%fõ◊ fla;fha fldafvdak ;=kla weuhsfkda wï, i|yd fla;h fkdimhhs' fuu fldafvdak"
^1& DNA u.ska RNA njg m%;sf,aLkh fkdfõ'
^2& tRNA cdkj, muKla wvx.= h'
^3& wl%sh cdkj, muKla wvx.= h'		
^4& fm%daàk ixiaf,aIKfha° kej;=ï ix« imhhs'
^5& fm%daàk ixiaf,aIKfha° wdrïNl ix« imhhs'
32' cdk laf,daklrKh jvd;au fyd¢ka úia;r lrkafka my; i|yka tajd w;=˙ka ljrla o@
^1& fjkia m%Njj,ska ,nd .;a DNA wKq mrSCIK k< ;=< tl;= ls¯fuka m%;sixfhda‚; DNA wKq
iE°u
^2& i„ù ffi, ;=<g ndysr DNA we;=Ω ls¯u u.ska m%fõ◊l j úlrKh l< „ùka we;s ls¯u
^3& nelaà˙hdjlg ndysr cdkhla we;=Ω ls¯u u.ska fm%daàkhla ksmoùu
^4& jdylhlg ndysr cdkhla we;=Ω fldg" Odrl ffi,j,g m˙Kduh lr Odrlhd ;=< ndysr cdkh
nyq.=Kkh is≥ ls¯u
^5& ndysr cdkhla we;=Ω ls¯u u.ska m%fõ◊lj úlrKh l< nelaà˙hd ud†,shla we;s ls¯u
33' fN!ñl Ydlj, m˙Kduh ù we;s my; i|yka ,ÈK w;=˙ka ljrla fN!ñl „ú;h i|yd jQ
wkqjr®;khla f,i ie,lsh fkdyels o@
^1& ixfiapkh i|yd ndysr c,fha wjYH;dj ke;sùu
^2& Ydlh ;=< c,h iy ødjH m˙jykh i|yd ikd, mgl msysàu
^3& l|ka iy uq,aj, oaù;Shsl >kùu we;sùu
^4& m%Ndixiaf,aYKh i|yd úfYaIKh jQ wjhj f,i m;% we;sùu
^5& ÈqøîcdKqq iq<Ûska jHdma; jk mrd. f,i m˙Kduhùu
34' b˙Û= m%fNaohl tla cdkhl m%uqL we,S,h u.ska îcj,g oï jr®Kh ,nd fok w;r tys oaú;aj ks,Sk
iajrEmh u.ska wjr®K îc we;s lrhs' fjk;a cdkhla jr®Kl ixiaf,aIKh ksfYaOkh lrhs' cdk
fol u flfrys úIuhq.aul jk b˙Û= Ydl folla uqyqï l< úg m%cks;fha jr®Kj;a îc iys; Ydl iy
wjr®K îc iys; Ydl 3(13 wkqmd;fhka ìysfõ' fuu m%fõ◊ rgdj úia;r ls¯u i|yd jvd;a u iq≥iq
jkafka my; i|yka tajd w;=˙ka ljrla o@
^1& nyqwe,S,;djh			
^2& wkqmQrl wNsnjkh		
^3& ks,Sk wNsnjkh
^4& m%uqL wNsnjkh			
^5& cdk m%;snoaOh

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

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^1&
^2&
^3&
^4&
^5&

35'
m%fõ◊ m%;sixfhdack"
ixfiapkfha ° we;sfõ'
W!kk úNdckfha ° we;sfõ'
wkQkk úNdckfha ° we;s úh yelsh'
jr®Kfoayj, iajdëk úhqla;sh m%;sM,hls'
oaù.=Kl „ùka ;=< muKla we;s fõ'

36' mqoa.,hl= jr®KdkaO jQ kuq;a Tyqf.a mshd idudkH flfkla úh' tu mqoa.,hdf.a ifydao˙hf.a mqf;l=
jr®KdkaO flfkl= ùug we;s iïNdú;dj fldmuK o@ Tyqf.a ifydao˙hf.a iajdñmqreIhd idudkH
mqoa.,hl= hehs i,lkak'
^1& 1
^2& 0'75
^3& 0'5
^4& 0'25
^5&
0
37'
my; i|yka ljr j.ka;sfhka OchOd¯ úfYaIhla fyd¢ka u meye†,s fõ o@
^1& m%cdjl fjk;a úfYaI úYd, ixLHdjl meje;aug wjia:dj i<ik neúka" ixrÈK m%h;akj,°
m%uqL;djh ,nd ° we;s úfYaIhls'
^2& hïlsis úfYaIhla „j;ajk m%foaYh tu úfYaIh wdrÈd ls¯u u.ska ;j;a úfYaI fndfyda ixLHdjla
wdrÈdùug fya;=jk ;rï úYd, jk úfYaIhls'
^3& ixrÈK iïmcdkkh iy ls%hdldrlï i|yd mqoa.,hska tla/ia lrùug ixfla;hla jk ckm%sh
úfYaIhls'
^4& mqoa.,hska lKavdhula úiska ixrÈK iïmcdkkh iy l%shdldrlï i|yd f;dard .kq ,nk
úfYaIhls'
^5& iSñ; N+f.da,Sh m%foaY ;=< muKla isák úfYaIhls'
38' jr®;udkfha jvd;a u wjOdkhg ,lajk W.% f.da,Sh md˙i˙l .egÆj l=ula o@
^1& f.da,Sh WIaK;ajh by< hEu
^2& ´fidaka ia;rh ydhkh ùu
^3& ldka;drlrKh
^4& wï, jeis 			
^5& >k wmøjH tla/ia ùu
39' foaYiSud yryd wka;rdhld¯ wmøjH m%jdykh md,kh lrkafka my; ljrla o@
^1& ndi,a iïuq;sh					
^2& CITES
^3& fudkaá%h,a ikaOdkh 			
^4& udfmd,a iïuq;sh
^5& lsfhdaf;da ikaOdkh
40' t˙f;%duhsiska m%;s„jlh n,mdkafka"
^1& nelaà˙hdj, fm%daàk ixiaf,aIKhg h' 		
^3& nelaà˙hdj, ffi, mg,hg h'			
^5& nelaà˙hdj, DNA m%;s.=kKhg h'
÷

^2& nelaà˙hdj, ffi, ì;a;shg h'
^4& jhsri m%;sj,s; ùu flfrys h'

wxl 41 isg 50 f;la m%Yakj, ° we;s m%;spdr w;=˙ka tlla fyda Bg jeä .Kkla fyda ksjer† h' ljr
m%;spdrh$m%;spdr ksjer† o hkak m<uqfjka u úksYaph lr .kak' bkamiq ksjer† wxlh f;darkak'
A, B, D hk m%;spdr muKla ksjer† kï				
''''''''''''''''''''''''''''' 1
A, C, D hk m%;spdr muKla ksjer† kï				
''''''''''''''''''''''''''''' 2
A iy B hk m%;spdr muKla ksjer† kï				
''''''''''''''''''''''''''''' 3
C iy D hk m%;spdr muKla ksjer† kï				
''''''''''''''''''''''''''''' 4
fjk;a lsishï m%;spdrhla fyda m%;spdr ixfhdackhla fydaa ksjer† kï
''''''''''''''''''''''''''''' 5
1
A, B, D

ksjer† h'

2
A, C, D

ksjer† h'

3
A, B

ksjer† h'

Wmfoia ieflúka
4
C, D

ksjer† h'

5
fjk;a lsishï m%;spdrhla fyda
m%;spdr ixfhdackhla fyda ksjer† h'

41' my; i|yka iïnkaO;d w;=˙ka ljrla$ljr tajd ksjer† o@
(A) wl%uj;a f,i >k jQ ffi, ì;a;s			 ia:q,fldaKdia:r
(B) fldkaä%ka					 iqixys; wia:ß
(C) fld,ecka ;ka;=					 w¯h, mglh
(D) µhsì%fkdacka 					 reêrh
(E) wka;r®ia:dms; u~,				 iakdhq mglh
w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

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42' ójk Ydlj,ska újD; îcl Ydl m˙Kdufha° we;s jQ mshjr f,i ie,lsh yelafla my; i|yka tajd
w;=˙ka ljrla$ljr tajd o@
(A) mqx ckaudKqYdlh mrd.hlg iSudùu
(B) cdhd ckaudKqYdlh uydîcdKqOdksh ;=< /£u
(C) îcdKqYdlh oaù.DyS ùu
(D) mqx ckaudKqYdlhg fmdaIKh ,nd .ekSug Wml%uhla f,i mrd. k<h we;s ùu
(E) l,,h îchla ;=< wka;r®.; ù jHdma;s tallhla iE°u
43' 43 jk m%Yakh my; ° we;s i;a;aj ldKavj, iels<s u; mokï ù we;'
		 ^a) wfk,svd
^b) wdf;%dfmdavd ^c) weïµsìhd
^d) fudÆiald
^e) tlsfkdvr®fïgd
by; ° we;s i;a;aj ldKavj, ±lsh yels iels,s wdldr wkqms<sfj<ska nysIaiels,a," øjia:ß;s iels,a,
iy wNHka;r iels,a, f,i olafjkqfha my; i|yka l=uk ixl,kh$ixl,k u.ska o@
(A) d a e
(B) b a c
(C) b c d
(D) e a c
(E) e a d
44' wdlshd wêrdcOdksfha my; i|yka ,CIK w;=˙ka ljrla$ljr tajd ;sfío@
(A) ffi, ì;a;sj, fmmaáfvd.a,shslEka we;'
(B) tla RNA fmd,sufr®ia jr®.hla muKla we;'
(C) idudkHfhka olakg ,efnkafka wdka;sl m˙irj, h'
(D) iafg%mafgduhsiskaj,g ixfõ°;djla fkdfmkajhs'
(E) ip, wdldr iq,n h'
45' my; ±lafjk ljr fydar®fudakh$fydar®fudak yhsfmd;e,ui u.ska ixiaf,aIKh jkafka o@
(A)
(B)
(C)
(D)
(E)

TRH
GnRH
ACTH
ADH
LH

46' ñksidf.a jDlaldKqj ms<sn| i;H j.ka;sh$j.ka;s f;darkak'
(A) jeämqr we;s jDlaldKq wdldrh ndysl jDlaldKq h'
(B) wú≥r ixj,s; kd,sldj wksjdr®h c, m%;sfYdaIKh is≥jk ia:dkhhs'
(C) fmdafvdihsg we;af;a fndauka m%djrfhys msg; ì;a;sfhys h'
(D) H+ i%djh jkqfha ú≥r ixj,s; kd,sldfjys h'
(E) fykaf,a mqvqfjys wjfrdayK ndyqfõ ° K+ m%;sfYdaIKh is≥fõ'
47' my; i|yka ljr tlla$tajd wl%sh m%;sYla;shg W∞yrKhla$W∞yrK fõo@
(A) ujf.a l,, nkaOh yryd .uka l< m%;sfoay u.ska wÆ; Wmka <orejl= ;=< we;sjk m%;sYla;sh
(B) irïm frda.h je<∫kq mqoa.,hl= ;=< we;sjk m%;sYla;sh
(C) ksfrda.s mqoa.,hl=g fmda,sfhda tkak; tkak;a ls¯fuka miqj we;sjk m%;sYla;sh
(D) c,NS;sld frda.h je<∫kq n,a,l= úiska ymd lk ,o mqoa.,hl=g tkak;a ls¯fuka miqj we;sjk
m%;sYla;sh
(E) hï mqoa.,hl= ;=< uef,a˙hd frda.hg tfrysj we;s jQ m%;sYla;sh
48' msfhdmhsgd jxYfha ±lsh yelafla my; i|yka ,ÈK w;=˙ka ljrla o$ljr tajd o@
(A) fndfyduhl jr®Ol foayh wjq,amdiqj" jDka;h iy ;,i f,i úfNaokh ù we;'
(B) y˙;,j ;=< laf,dafrdms,a a iy c wvx.= fõ'
(C) uQ,sl jYfhka ñ˙†fha „j;a fõ'
(D) µhsfldì,ska jr®Klh wvx.= fõ'
(E) „jk pl%fha lYsldOr ffi, ke;'

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

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49' fldl= mKq wdidok md,kh lr .; yelafla my; l=uk l%uh$l%u u.ska o@
(A) t<suyfka u,my ls¯fuka je<lS isàu u.sks'
(B) ñksia u<my fmdfydr f,i Ndú; ls¯fuka je<lS isàu u.sks'
(C) wdydr wkqNj ls¯ug fmr w;a inka oud fyd¢ka fida∞ .ekSu u.sks'
(D) mdjyka me<°u u.sks'
(E) wdydr iEu úg u jid ;eîu u.sks'
50' fmd,a uhsgdjd ms<sn| i;H j.ka;sh$j.ka;s f;darkak'
(A) fïrE f.äj,g fukau <má f.äj,g o ydks lrhs'
(B) fmd,a f.ä m%jdykh ls¯u u.ska jHdma; fõ'
(C) mshú weig meye†,s j †ia fõ'
(D) foaYSh l%u u.ska idr®:l j md,kh l< yelsh'
(E) f.dla w;=j, mdoia:fha nyq, j isáhs'

***

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

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09 - „j úoHdj
*
*

˚˚ m;%h

A fldgfia ishÆ u m%Yakj,g ms<s;=re imhkak'
B fldgfika m%Yak y;rlg muKla ms<s;=re imhkak'
A fldgi - jHQy.; rpkd

1. (A) (i) ATP wKqjl l=uk ,CIKh ksid th Yla;s jdylhla f,i l%shd ls¯ug iq≥iq fjhs o@

(ii) Tlaisldrl fmdiafmdrhs,SlrKh hkafkka woyia jkafka l=ula o@
(iii) Tlail
s drl fmdiafmdrhS,SlrKh is≥jk ia:dkh kï lrkak'
(iv) Tlaisldrl fmdiafmdrhs,SlrKh i|yd Yla;s m%Njh jkafka l=ula o@
(B) (i) wêrdcOdks ;=kg i„úka we;=<;a ls¯fï° Ndú;hg .kakd m%Odk ,CIK ;=kla kï lrkak'

(a)
(b)
(c)

(ii) my; i|yka „ù ldKav wh;a jk wêrdcOdks kï lrkak'
ldKavh
(a) ihfkdnelaà˙hd

wêrdcOdksh

(b) Methanococcus
(c)

lsá%äfhduhsfldagd

(d) is,sfhdfµdard
(C) (i) mCISkaf.a iy Wr.hskf
a .a olakg ,efnk ndysr jHqyd;aul wdfõ◊l ,CIKhla kï lrkak'

(ii) fldkaä%la;sfhaihl= iy Tiafghsla;sfhaihl= fjka fldg y∫kd .; yels ndysr ,CIKhla
fokak'
(iii) l+ve,a,l=f.a iy Nereis f.a olakg ,efnk ndysr ,CIKhla ne.ska Ndú; fldg Tjqka w∞<
jr®.j,g wkqhqla; lrkak'
i;a;ajhd
l+ve,a,d

oDYH ndysr ,CIK

jr®.h

Nereis

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

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2. (A) (i) „ùkaf.a ±lsh yels m%Odk úIufmdaIS fmdaIK wdldr ;=k kï lrkak'

(ii) my; i|yka tla tla „úhdf.a fmdaIK wdldrh i|yka lrkak'
Nitrosomonas
Aspergillus

.................................................

.................................................

(B) wr®;dm,a ffi,j, c, úNjh ksr®Kh ls¯u i|yd my; i|yka m¯CIKh is≥lrk ,°'
wr®;dm,a wdlkaOhlska 60 mm †. tlsfklg iudk ;Sre y;rla lmd .kakd ,°' tla wr®;dm,a

;Srejla wdi%e; c,h iys; m¯CId k<hl .s,ajQ w;r wkslajd fjkia idkaøKj,ska hq;a iqfl%daia
ødjKj, .s,ajk ,°' fuu k< A, B, C iy D f,i kï lrk ,°' meh follg miq wr®;dm,a ;Sre
msg;g f.k kej; †. uksk ,°' tu o;a; my; j.=fjys ±lafõ'
k<h

meh follg miq wr®;dm,a ;Srefõ †. (mm)

A

60

C

58

B

D

65
62

(i) wr®;dm,a ffi,j, c, úNjhg iudk c, úNjhla iys; ødjKhla we;af;a l=uk k<fha o@
(ii) wdi%e; c,h we;af;a l=uk k<fha o@
(iii) A isg D olajd k<" tajdfha wvx.= ødjKj, c, úNjfha wjfrdayK wkqms<sfj<ska ,shkak'
(iv) (a) reêr .skSu hkq l=ula o@
(b) reêr .skSu i|yd m¯CIKd.drhl idudkHfhka Ndú; lrkq ,nk WmlrKh l=ula o@

(C) (i) i;a;ajhskag nysiai%djh w;HjYH jkafka ukao@

(ii) khsg%ckSh nysiai%dù M,h f,i wefudakshd ksmoùfuys jdishla i|yka lrkak'
(iii) ñksidf.a" †.= fykaf,a mqvq orkafka l=uk jDlaldKq jr®.h o@
(iv) ñksia jDlal ldhsl úoHdfõ ° w;sm˙Y%djKh hkqfjka woyia jkafka l=ula o@
(v) ñksia uq;% kshe†hl fm%daàk ;sîfuka idudkHfhka fmkakqï lrkafka l=ula o@
w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

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3. ^A) (i) CISrKh hkq l=ula o@
(ii) msáls˙ °ug jvd uõls˙ °fï jeo.;alula i|yka lrkak'
(iii) ms˙ñkaf.a j|Ndjhg fya;=ldrl úh yels Y=l%fhys W!k;djla i|yka lrkak'
(iv) ñksidf.a Y=l% wdYhsl ;r,fha m%Odk lD;H folla i|yka lrkak'
(a)
(b)
(v) ñksidf.a Y=l% ksIamdokh md,kfha° bkaysìkaj, ldr®hNdrh l=ula o@

(vi) m%iQ;sfhys ° Tlaisfgdaiska ksoyia ls¯fï Ok m%;sfmdaIS md,kfha ldr®hNdrh l=ula o@

(B) (i) fm%daàk ixiaf,aIK l%shdj,sfha° my; i|yka tla tla RNA jr®.fha m%Odk lD;Hh i|yka
lrkak'
RNA jr®.h

lD;Hh

mRNA
rRNA
tRNA

(ii) tkaihsuhla i|yd fla;h imhk cdkhl ,CIH úlD;shla is≥jqjfyd;a by; i|yka l=uk RNA
jr®.h fjkiafõ o@

(C) (i) ,hsfldmhsgdj, ±lsh yels wdldrfha uydîcdKqOdkshl isg wdjD; îcl äïnhla olajd

m˙Kduh ùfï ° is≥ ù we;s jeo.;a fjkialï folla i|yka lrkak'
(a)
(b)

(ii) (a) oaú;aj ixfiapkh hkq l=ula o@
(b) oaú;aj ixfiapkfha m%Odk jdish l=ula o@

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

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4. (A) (i) md˙i˙l ksfla;khla hkq l=ula o@

(ii) yd (iii) m%Yak mokï ù we;af;a my; ° we;s ñ˙†h m˙ir moaO;shl l,ams; wdydr cd,h u; h'
†hldjd
¨,d

;Kfld< ldmhd

fm;shd
idudkH ldmhd
i;a;aj ma,jdx.

c,c lDóka
c,c uydYdl

Ydl ma,jdx.

(ii) tla fmdaIS uÜgulg jvd we;=<;a jk „úhd$„ùka kï lrkak'
(iii) by; m˙ir moaO;sfhka c,c uydYdl bj;a l< fyd;a idudkH ldmhdf.a .yK >k;ajhg
l=ula is≥fõ o@

(B)(i)

†,Srj, olakg ,efnk ,CIK lSmhla my; j.=fõ m<uq ;Srefõ i|yka fõ' j.=fõ i|yka †,Sr
jxYj, tu ,CIK ;sfí kï (√) i<l=fKka o" ke;skï (×) i<l=fKka o olajkak'
,CIKh
lsáä
% fhduhsfldagd ihsf.duhsfldagd weiaflduhsfldagd
jdiia:dkh fndfyda úg c,c fõ'
talffi,Sh idud‚lhska isáh
yel'
p,îcdKq idohs'

(C) (i)

uQ,sl ffi, hkq fudkjd o@

(ii) ñksidf.a ±kg is≥ lrk uQ,sl ffi, m%;sldr i|yd tu ffi, ,nd .kakd jHqyhla kï
lrkak'

**

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

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B fldgi - rpkd
5' ^a& fm%daàkj, idudkH ixhq;sh iy jHqyh úia;r lrkak'

^b& iq≥iq W∞yrK foñka i„ùka ;=< yuqjk úúO fm%daàk jr®. tajdfha lD;Hh wkqj ,ehsia;= .;
lrkak'
^c& .=KydkslrKh ^denatured& jQ tkaihsu tajdfha idudkH l%shdld¯;ajh fkdolajkafka wehs@

6' ^a& ñksia lxld, fmaYs ;ka;=jl idfldñhrhl jHqyh fláfhka úia;r lrkak'

^b& udkj lxld, fmaYs ;ka;=jla ixfldapkh jkafka flfia ±hs fláfhka meye†,s lrkak'

7' ^a& maf,dahu mglfha idudkH jHqyh úia;r lrkak'

^b& maf,dahu m˙jykfha mSvk m%jdy l,ams;h meye†,s lrkak'

^c& maf,dahu m˙jykh iy ffY,u m˙jykh w;r we;s m%Odk fjkialï fudkjd o@
8' ^a& jdhq ¥IKh hkafkka woyia lrkqfha l=ula o@

^b& m˙irh u; jdhq ¥Ilj, n,mEï fláfhka úia;r lrkak'

9' ^a& m%fhdak hkq fudkjd o@

^b& m%fhdak u.ska .jhskag we;s lrk frda.hla kï lrkak'
^c& CIqø„ùka u.ska we;sjk frda.
^i&

md,kh ls¯ug;a"

^ii& iqj ls¯ug;a
we;s idudkH l%u úia;r lrkak'
10' flá igyka ,shkak'
^a& tkaihsu l%shdld˙;ajh flfrys n,mdk ndysr idOl
^b& wNsnjkh

^c& ñksia yfï m%;s.%dyl

***

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

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10 - ixhqla; .◊;h
m%Yak m;% jHQyh
˚ m;%h - ld,h ( meh 03 hs'
fuu m%Yk
a m;%h fldgia follska iukaú; h'
A fldgi - m%Yak oyhls' m%Yak ish,a,g u ms<s;=re iemhsh hq;= h' tla
m%Yakhlg ,l=Kq 25 ne.ska ,l=Kq 250 ls'
B fldgi - m%Yak y;ls' m%Yak mylg ms<s;=re iemhsh hq;= h' tlaa
m%Yakhlg ,l=Kq 150 ne.ska ,l=Kq 750 ls'
˚ m;%h i|yd uqΩ ,l=Kq 1000 fi 10 } 100
˚˚ m;%h - ld,h ( meh 03 hs'
fuu m%Yk
a m;%h fldgia follska iukaú; h'
A fldgi - m%Yak oyhls' m%Yak ish,a,g u ms<s;=re iemhsh hq;= h' tla
m%Yakhlg ,l=Kq 25 ne.ska ,l=Kq 250 ls'
B fldgi - m%Yak y;ls' m%Yak mylg ms<s;=re iemhsh hq;= h' tla
m%Yakhlg ,l=Kq 150 ne.ska ,l=Kq 750 ls'
˚˚ m;%h i|yd uqΩ ,l=Kq 1000 fi 10 } 100
wjidk ,l=K .Kkh ls¯u

(

˚ m;%h
˚˚ m;%h
wjidk ,l=K

}
}
}

100
100
200 fi 2 } 100

ie,lsh hq;=hs (

*

*

fuys .◊;h ˚ yd ˚˚ m%Yak m;%j, A fldgi i|yd uQ,dlD;s m%Yak 30 ne.ska o
B fldgi i|yd uQ,dlD;s m%Yak 21 ne.ska o we;=<;a lr we;' tfia jqjo
úNd.fha ° fokq ,nk m%Yak m;%h ilia flfrkqfha by; i|yka m%Yak m;%
jHqyhg wkql+,j h'
úNd.fha ° A fldgig m%Yak m;%fha u tla tla m%Yakh i|yd imhd we;s bv
m%udKh ;=< ms<s;=re iemhsh hq;= h'

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

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10 - ixhqla; .Ks;h
I m;%h i|yd uQ,dlD;s m%Yk
a
A fldgi
1. α yd β hkq ax 2 + bx + c = 0 jr®.c iólrKfha uQ, fõ; fuys c ≠ 0 fõ'
α
β
yd
nj fmkajkak'
acx 2 − (b 2 − 2ac) x + ac = 0 jr®.c iólrKfha uQ,
β
α
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2. f ( x) ≡ ax 3 + bx 2 − 2 x + c hehs .ksu'q
(i) f (x ) hkak ( x 2 + x) u.ska fn¥úg ,efnk fYaIh 6( x + 1) o"
(ii) ( x − 1) hkq f (x ) ys idOlhla o
kï" túg a, b yd c ys w.hka fidhkak'
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

- 123 -

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4.

x
< −2 wiudk;dj imqrd,kq ,nk x ys ;d;a;ú
a l w.h ish,a, u wvx.+ l+,lh
2x − 1
fidhkak'
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- 124 -

 3 − 1
 hehs .ksu'q
5. A = 
1 1 
.Ks; wNHqyk uQ,Or®uh fhdod .ksñka iEu n Ok ksÅ,hla i|ydu
2 + n − n 
 nj fmkajkak'
A n = 2 n−1 
n
2
−
n


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6. ksYp
a dhlj, .=K fhdod .ksñka
a b c
b c a = − (a + b + c)(a 2 + b 2 + c 2 − ab − bc − ca) nj fmkajkak'
c a b
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

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7. (3 + 2 x) n ys oaúmo m%idrKfha x ys yd x 2 ys ix.+Kl ms< f
s j<ska a1 yd a 2 hehs . ksu;q
fuys n hkq Ok ksÅ,uh or®Ylhla fõ' a 2 = 2a1 kï n ys w.h fidhkak'
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.Kk fidhkak'
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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- 126 -

9.

2 + 3i
ixlSrK
® ixLHdj λ (1 + i ) wdldrfhka m%ldY lrkak; fuys λ ;d;a;ú
a l fõ'
5+i
4

 2 + 3i 
ta khsk"a 
a l nj fmkajkak'
 ;d;a;ú
 5+i 
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 x tan x 
10. lim 
 fidhkak'
x→0 1 − cos x 
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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d2 y

dy
= 0 nj fmkajkak'
dx
dx
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11. 0 < x < 1 i|yd y = sin

−1

x kï" (1 − x 2 )

2

−x

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12. f ( x) = 2 x 3 + 3 x 2 − 12 x + 5 Y%s;fha yereï ,CIH fidhkak'
ta khsk"a f (x ) Y%;
s h jeä jk x ys mrdih ksrK
® h lrkak'
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1

13. fldgia jYfhka wkql,kh fhdod .ksñka

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2 x +1

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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

- 131 -

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19. x − 2 y − 2 = 0 fr®Ldj iu.

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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

- 132 -

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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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23. y = x + 3 fr®Ldfõ;a" flakøa h (3, 4) yd wrh 2 5 jk jD;a;fha;a f√ok ,CIHj,
LKavdxl fidhkak'
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24. flakaøh ( − 1, 6) jk S = 0 jD;a;h x 2 + y 2 − 6 x − 4 y + 11 = 0 jD;a;h ndysrj iamr®Y
lrhs' S = 0 jD;a;fha iólrKh fidhkak'
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25. A yd B ,CIH ms<sfj<ska x yd y wCI u; msysghs' AB ys †. ksh;hla jk f,i A yd B
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s j<ska x yd y wCI u; úp,kh jk úg" AB ys uOH ,CIHfha m:h jD;a;hla nj
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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27. P hkq x 2 + y 2 − 4 = 0 jD;a;h u; úp,H ,CIHhla fõ' N hkq P ,CIHfha isg y wCIhg
we¢ ,ïnfha wäh fõ' M hkq NP ys uOH ,CIHh k ï M ys m:h b,smaihla nj
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28. [0, π] mrdih ;+< sin θ + sin 5θ = sin 3θ iólrKfha úi∫ï fidhkak'
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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a+b b+c c+a
=
=
fõ; fuys a, b yd c i|yd iqmr
q e≥ f;areï
5
7
6
we;' ihska kshuh fhdod .ksñka sin A, sin B yd sin C hkq iudka;r fY%V
a h
ß l wkqhd; mo
nj fmkajkak'
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29. ABC ;%sfldaKhla i|yd"

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30. tan −1

1
1 π
+ tan −1 =
nj fmkajkak'
2
3 4

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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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B fldgi
1. f ( x) ≡ x 2 + 2kx + k + 2, x ∈ IR hehs .ksu;q fuys k hkq ksh;hls'
(a) (i) x ys ishÆ w.hka i|yd f (x) Ok jk k ys w.h mrdih fidhkak'
(ii) iólrKfha uQ, α yd β kï (α − β ) 2 = 4 (k 2 − k − 2) nj fmkajkak'
(iii) f ( x) = 0 iólrKfha uQ, w;r fjki 4 jk m˙† jQ k ys w.hka ksrK
® h lrkak'
α
β
(b) k ≠ − 2 kï" 1 + yd 1 +
uQ, jk jr®.c iólrKh fidhkak'
β
α

4x 2 + 8
1
, x ∈ IR , x ≠ − hehs .ksu'q
2x + 1
2
− 8 yd 4 w;r f(x) mej;sh fkdyels nj fmkajkak'
(i) f ( x) = −8 ,
(ii) f ( x) = 4
jk m˙† jQ x ys w.hka fidhkak'
ta khsk"a f (x) ys m%i;
a drh w¢kak'

2. (a) f ( x) =

(b) y = x 2 − 1 jl%fha m%i;
a drh w¢kak'
a drh wfmdaykh lrkak'
y = x 2 − 1 ys m%i;
tu rEm igyfkysu y = x 2 − 7 ys m%i;
a drh w¢kak'
ta khska fyda fjk;a wdldrhlska fyda"

x 2 − 7 > x 2 − 1 wiudk;dj úi|kak'

 p q
 kHdih AA T = I imqrd,hs; fuys I hkq 2 jk .Kfha tall kHdih fõ'
3. (a) A = 
 r s
(i) p = 0 kï" túg s = 0 yd q 2 = r 2 = 1,
(ii) p ≠ 0 kï" túg ( s = p, r = −q ) fyda ( s = − p, r = q )
nj fmkajkak'
4 
 3
 hehs .ksu'q
(b) M = 
 − 2 − 3
λ 1 yd λ 2 hkq det (M − λ I ) = 0 ys úi≥ï fõ" fuys I hkq 2 jk .Kfha tall kHdih
fõ' λ 1 yd λ 2 fidhkak'
(M − λ i I )x i = O , i = 1, 2 jk whq˙ka ksYY
a k
+ H xi ;Sr ffoYslhla fidhkak" fuys O
hkq Y+kH ;Sr ffoYslh fjhs'

P 2 × 2 = (x 1

x ) hehs .ksu'q

λ
P −1MP =  1
 0

2

0 
 nj i;Hdmkh lrkak'
λ 2 

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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1

x

4. (a) ksYp
a dhlj, .=K fhdod .ksñka 1 x 2
1 x3

x2
x 4 ys idOl ish,a, u fidhkak'
x6

1 − 2 2


(b) C =  1 − 1 2  hehs .ksu'q
0 1 1



C3 − C 2 − C = I nj fmkajkak; fuys I hkq 3 jk .Kfha tall kHdih fõ'
ta khsk,a C ys m%;f
s ,dauh fidhkak'

my; oelafjk iólrK moaO;sh kHdi wdldrfhka ,shd olajkak:
x − 2 y + 2z = 1

x − y + 2z = 0
y + z = −2
ta khsk,a iólrK moaO;sh úi|kak'
5. (a) ishÆ u n Ok ksÅ,uh w.hka i|yd" .Ks; wNHqyk uQ,Or®uh fhdod .ksñka"
33n − 26n − 1 hkak 676 ka fnfok nj idOkh lrkak'
(b)

2 x 2 + 3x − 3
ys Nskk
a Nd. fidhkak'
( x + 1)( x + 2)

ta khska fyda fjk;a wdldrhlska fyda"
2

3

2  1  11  1 
24  1 
a f
s ha n fjks moh jk u n hkak
 +
  +
  + . . . fY%K
2.3  4  3.4  4 
4.5  4 
2
u n = n + (v n − v n−1 ) f,i ,súh yels jk m˙† v n fidhkak'
4
n

Sn =

1 1 + 2n  1 
−
a f
s ha uq,a mo n ys ftlHh fõ'
  nj fmkajkak; fuys S n hkq fY%K
6 3(n + 2)  4 

lim S n fidhkak'

n →∞

6. (a) ´kEu a yd b ;d;a;ú
a l ixLHd folla i|yd a 2 + b 2 ≥ 2ab nj idOkh lrkak'
´kEu a, b yd c Ok ixLHd ;+kla i|yd a 3 + b 3 + c 3 ≥ 3abc nj wfmdaykh lrkak'
ta khsk"a (a + b + c)(ab + bc + ca) ≥ 9abc nj idOkh lrkak'

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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(b) p ( x) ≡ x 4 + ax 3 + bx 2 − 2 x − 4 nyqmohg ( x − 1) yd ( x + 2) hk idOl ;sfnhs'
(i) a = 3 yd b = 2 nj fmkajkak'
(ii) p (x) ys idOl mQrK
® f,i fidhkak'
(iii) p (x) Ok jk m˙† x ys w.hka fidhkak'

7. (a) 0, 1, 2, 3, 4 yd 5 hk ixLHdxl yfhka" jrlg fjkia ixLHdxl ´kEu .Kkla f.k
ixLHd fldmuK .Kkla iE†h yels o@
(b) ‘ENGINEERING’ hk jpkfha wCIr ish,a, u f.k fjkia ms<sfh, ls¯ï fldmuK
.Kkla iE†h yels o@
fïjdhska fldmuK .Kkla ;=< E wCIr ;=k tl <Û yd fldmuK .Kkla ;=< tajd
m<uqj tl <Û ;sfnhs o@
(c) (1 + x 2 )(1 + x) n = a0 + a1 x + a 2 x 2 + a3 x 3 + . . . + a n+ 2 x n+ 2 hehs .ksu'q
a0 , a1 yd a 2 hkq iudka;r fY%V
a h
ß l wkqhd; mo ;=kla kï n ys úh yels w.hka fidhkak'
n ys tla tla w.h i|yd m%ldYkfha mQrK
® m%idrKh x ys n, weiqfrka ,shd olajkak'
8. (a ) a , b yd c hkq 1 g iudk fkdjk ´kEu Ok ;d;a;aúl ixLHd ;=kla kï" túg
log a c = log a b × log b c nj idOkh lrkak'
1
nj wfmdaykh lrkak'
log b a =
log a b
x = log a bc , y = log b ca yd z = log c ab kï xyz = x + y + z + 2 nj idOkh lrkak'
(b) n ≥ 2 kï"
ta khsk"a

n 2 + 2n + 1
A
B
C
≡
+
+ jk wdldrhg A, B yd C fidhkak'
n!
(n − 2)! (n − 1)! n!

∞
2 2 32 4 2
1
+
+
+ . . . = 5e − 1 nj fmkajkak; fuys e = ∑ fõ'
1!
2! 3!
r =0 r!

1
hehs .ksu;q fuys u yd v folu ksYY
a k
+ H yd ;d;a;ú
a l fõ'
u + iv
(i) u ksh;hlaj v úp,kh jk úg"
(ii) v ksh;hlaj u úp,kh jk úg
z ixlSrK
® ixLHdj" wdr®.ka igyfkys jD;a;hla u; msyg
s k nj idOkh lrkak'

9. (a) z =

by; ,nd.kakd ,o jD;a; foflys flakaø tlsfklg ,ïn wp, fr®Ld folla u;
msyg
s k nj o idOkh lrkak'

 z − 2i 
(b) Re
 = 0 kï túg z ys m:h" wrh 5 iys; jD;a;hla nj idOkh lrkak'
 z+4 
tys flakøa h fidhkak'
 z − 2i 
Im
 = 0 kï" z − 1 ys wvq;u w.h fidhkak'
 z+4 

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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1 − cos(sin x)
w.hkak'
x →0
sin x

10. (a) lim

(b) y = ( 1 + sin −1 x ) sin −1 x hehs .ksu'q
(1 − x 2 )

d2 y
dx 2

−x

dy
hkak x j,ska iajdh;a; nj idOkh lrkak'
dx

ta khsk"a n = 1, 2, 3, 4 i|yd x = 0 °

dn y
dx n

fidhkak'

(c) wrh a jQ jD;a ; hl mdolhla ;= < iDcq f lda K di% h la wka ; r® . ; lr we;a f ;a
iDcqfldaKdi%fha mdo folla jD;a; mdolfha udhsï wrhhka †f.a o" iDcqfldaKdi%fha
tla YSr®Ihla jD;a;fha pdmh u; o msysgk wdldrhg h' iDcqfldaKdi%fha Wm˙u
jr®.M,h fidhkak'

11. (a) m%u,
Q Or®u Wmfhda.S lr .ksñka y =
(b) f ( x) =

1
(1 + x) 2

ys jHq;m
a kakh fidhkak'

ax + b

hehs .ksu;q fuys a yd b ;d;a;ú
a l ixLHd fõ' ( − 2, − 1) ,CIHh f (x) ys
1+ x2
yereï ,CIHhla kï" a yd b ys w.hka ksrK
® h lrkak'
m<uqjk jHq;m
a kakh ie,lSfuka y = f (x) ys m%i;
a drh w|skk
a '
f ( x) = k iólrKhg m%Nk
s k
a ;d;a;ú
a l uQ, mej;Sug k i|yd úh yels ishÆ w.hka
wfmdaykh lrkak'

12. (a) (0, π ) m%dka;rh
sin x
ie,lSfuka
x
π
0<x≤
i|yd
2

ms˙jid y = x cos x − sin x Y%;
s fhys m<uqjk jHq;m
a kakfha ,l=K
hkak x ys wvq jk Y%;
s hla nj fmkajkak'
2x
≤ sin x < x nj wfmdaykh lrkak'
π

(b) jl%hl mrdñ;s l iólrK y = a (t 2 − a 2 ) yd x = 2a (t + 2a ) u.ska ° we;; fuys t
;d;a;aúl jk w;r a ks h ;hla fõ' t = t 0 g wkqrEm ,CIHfha° jl%hg w¢k ,o
wNs,ïnfha iólrKh fidhkak'
t = t1 g yd t = t 2 g wkqrEm ,CIHj,° jl%hg w¢k ,o wNs,ïn f√okh jk ,CIHh"
jl%h u; fjhs' t1t 2 = 2 nj fmkajkak'
(c) x > 0 i|yd y =

x 2 (1 + x 2 ) x
3

(1 + x

3

kï"

dy
fidhkak'
dx

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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13. (a) Nskk
a Nd. fhdod .ksñka

x 2 − 3x + 4

∫ ( x − 1) 2 ( x 2 + 1) dx

fidhkak'

(b) x 2 + y 2 = 2ay hehs ° we;; fuys a hkak x yd y j,ska iajdh;a; fõ'
dx
fidhkak'
y = tx wdfoaYh fhdod .ksñka ∫
y
(c) x úIhfhka cos x ln tan
π
2

ta khsk"a

∫

π
3

14. (a) t = tan

x
wjl,kh lrkak'
2

x

sin x ln tan dx w.hkak'
2


x
wdfoaYh fhdod .ksñka"
2

dx

∫ 1 + sin x

ta khska fyda fjk;a l%uhlska fyda

∫

fidhkak'

2 cos x + 3 sin x
dx fidhkak'
1 + sin x

(b) fldgia jYfhka wkql,kh fhdod .ksñka

∫ sin

a

(c) iq≥iq wdfoaYhla fhdod .ksñka"

∫

−1

x dx fidhkak'

a

f ( x)dx =

0

∫

f (a − x)dx nj fmkajkak; fuys a hkq

0

Ok ksh;hls'
1

ta khsk"a

∫

x 2 (1 − x)dx w.hkak'

0

15. (a)

6x + 5

∫ (3x + 1) 2 dx

fidhkak'

(b) I = ∫ cos(a ln x ) dx yd J = ∫ sin (a ln x )dx hehs .ksu;q fuys a hkq ksYY
a k
+ H ;d;a;ú
a l
ixLHdjla fjhs'
(i) x Okjk úg I − aJ = x cos(a ln x ) ,
(ii) x iDKjk úg I + aJ = x cos(a ln x ) .
nj fmkajkak'
x ys tla tla m%dka;rfha° I yd J w;r ;j;a iïnkaOhla ,nd.kak'
ta khsk"a I yd J wkql, w.hkak'
π
3

(c) 1 + t cos x = 0 wdfoaYh fhdod .ksñka

tan x

3

∫ 1 + cos xdx = ln 2

nj fmkajkak'

0

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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16. (a) PQR ;% s f lda K hl PQ, QR yd RP mdo ms<s f j<s k a x − 3 y + 5 = 0 , y − x − 3 = 0 yd
3 x − y − 1 = 0 iólrK u.ska fokq ,efí' QR g ,ïnj P ,CIHh Tiafia hk fr®Ldj;a
PR g iudka;rj Q ,CIHh Tiafia hk fr®Ldj;a S ,CIHfha° yuqfjhs' QS yd PS j,
iólrK fidhkak'
ta khsk"a PQSR frdïnihla nj fmkajkak'
(b) fla k a ø h (–3, –1) ,CIHfhys jQ o" A = (−2, 1) ,CIHh Tiafia hkakd jQ o S = 0
jD;a;fha iólrKh fidhkak'
S = 0 jD;a; hg A ,CIHfha° w¢k ,o iamr®Ylh O uQ, ,CIHh Tiafia hk nj
fmkajkak'
uQ, ,CIHfha isg S = 0 jD;a;hg w¢k ,o fojk iamr®Ylfha iamr®Y ,CIHh jk Q ys
LKavdxl fidhkak'
OA hkak OQ g ,ïn nj fmkajkak'
17. ´kEu ir, fr®Ldjl i ó l r K h x cos α + y sin α = p wdldrfhka ,súh yels nj
fmkajkak; fuys p ≥ 0 yd 0 ≤ α < 2π fjhs'
ta khska" lx + my + n = 0 u.ska fokq ,nk ir, fr®Ldj S ≡ x 2 + y 2 − r 2 = 0 u.ska fokq
,nk jD;a;hg iamr®Ylhla ùu i|yd wksjdr®h yd m%udKj;a wjYH;dj fidhkak'
S = 0 jD;a;hg w¢k ,o t1 yd t2 iamr®Yl y-wCIhg iudka;r fjhs' O uQ, ,CIHh yryd
ir, fr®Ld folla we| we;af;a bka tla fr®Ldjla T1 ,CIHfha° t1 iamr®Ylh o" wfkla
π
fr®Ldj T2 ,CIHfha° t2 iamr®Ylh o yuqjk m˙† h' T1Oˆ T2 =
kï" T1T2 hkq S = 0
2
jD;a;hg iamr®Ylhla nj idOkh lrkak'
18. OA yd OB ir, fr®Ld foll iólrK ms<sfj<ska x + 2 y = 0 yd x − 2 y = 0 fjhs' OA g
iudka;rj P ,CIHfha isg w¢kq ,nk fr®Ldj M ,CIHfha° OB g yuqjk w;r" OB g
iudka;rj P ,CIHfha isg w¢kq ,nk fr®Ldj L ,CIHfha° OA g yuqfjhs' OA g ,ïnj
L ,CIHh Tiafia w¢k ,o fr®Ldj" OB g ,ïnj M ,CIHh Tiafia w¢k ,o fr®Ldjg
Q ,CIHfha° yuqfjhs' Q ,CIHfha Lkavdxl" P ,CIHfha LKavdxl weiqfrka fidhkak'
y 2 = 8 x u.ska fokq ,nk mrdj,h u; P ,CIHh úp,kh fõ kï" Q ,CIHfha m:h o
mrdj,hla nj fmkajd" tys kdNsh" kshdul wCIh yd YSrI
® h fidhkak'
19. (a) iqmr
q e≥ wxlkfhka" ABC ;%f
s ldaKhla i|yd ihska kshuh m%ldY lrkak'
tu wxlkfhkau a sin( B − C ) + b sin(C − A) + c sin( A − B) = 0 nj idOkh lrkak'
(b) f ( x) = 4(sin 4 x + cos 4 x) hehs .ksu'q
f ( x) = 3 + cos 4 x nj idOkh lrkak'
ta khska fyda fjk;a l%uhlska fyda" x ≤

π
i|yd f ( x) = 4(sin 4 x + cos 4 x) ys o<
2

igykla w¢kak'

y = f (x) jl%fhka yd" x = ±

π
yd y = 2 u.ska fokq ,nk ir, fr®Ldj,ska imr®hka;
2

jr®.M,h .Kkh lrkak'

1
1
1
3
(c) tan −1   + tan −1   = tan −1   + tan −1   nj idOkh lrkak'
2
3
4
5
w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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20. (a) iqmr
q e≥ wxlkfhka" ABC ´kEu ;%f
s ldaKhla i|yd fldaihsk iQ;h
% m%ldY lrkak'
4
4
4
2 2
tu wxlkfhkau ABC ;%f
s ldaKhla i|yd a + b + c = 2c (a + b 2 ) kï"
π
3π
C=
fyda
fyda nj fmkajkak'
4
4

π  
π 

(b) y = tan x +  cot x −  hehs .ksu'q
12  
12 

1 + y = 2( y − 1) sin 2 x nj idOkh lrkak'

π  
π 

ta khsk"a x ys ´kEu ;d;a;ú
a l w.hla i|yd tan x +  cot x −  m%ldYkh
12  
12 

1
yd 3 w;r w.hka fkd.kakd nj fmkajkak'
3
21. (a) iqmr
q e≥ wxlkfhka" ABC ;%f
s ldaKhla i|yd ihska kshuh m%ldY lr idOkh lrkak'
ta khsk"a tu wxlkfhkau
 A
 B−C 
 A
 B−C 
(b + c) sin   = a cos
 yd (b − c) cos  = a sin 
 nj idOkh lrkak'
2
 2 
2
 2 
ABC ;%f
s ldaKhla i|yd fldaihsk iQ;h
% wfmdaykh lrkak'
(b) x i|yd úi|kak:
(i) sin 2 x + 4(cos x − sin x) = 4 ,
(ii) tan −1 2 x − tan −1 x = cot −1 4(1 − x 2 ) .

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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10 - ixhqla; .Ks;h
II m;%h i|yd uQ,dlD;s m%Yk
a
A fldgi
1. iDcq u.l taldldr u m%fõ.fhka meo hk mdme†lrejl= ksi,;dfõ we;s fudagr® r:hla miq
lrhs' ta fudfydf;a°u tu †Ydjgu fudagr® r:h p,s;h wrUd taldldr a ;ajrKfhka
iajlSh jeä;u m%fõ.h jk v ( > u ) g <Ûdfjk f;la .uka lrhs' mdme†lref.a iy
fudagr® r:fha p,s; i|yd m%fõ.-ld, m%i;
a dr tlu igykl w¢kak'
u2
fudagr® r:h" mdme†lreg msgqmiska isák ld,h we;=<; Tjqka w;r Wm˙u ≥r
nj
2a
wfmdaykh lrkak'
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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2. u km h −1 Wm˙u fõ.hlg <Ûdúh yels fudagr® fndaÜgqjlg v (< u 2 ) km h −1 ksh;
fõ.fhka jhU †Ydjg hd;%d lrk kejla we,a,Sug wjYHj we;' wdrïNfha° kej" fudagr®
fndaÜgqjg ol=fKka ia:dk.;j we;' iajlSh Wm˙u m%fõ.fhka fudagr® fndaÜgqj p,kh fõ
kï" kej we,a,u
S i|yd th p,kh úh hq;= †Ydj" m%fõ. ;%f
s ldaKhla weiqfrka fidhkak'
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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3. p,kh jk wxY=jl r msysgqï ffoYslh t ld,fha ° r = (5 + 20t )i + (95 + 10t − t 2 ) j u.ska
fokq ,efí· fuys i yd j g iqmr
q e≥ f;areï we;'
(i) wxY=fõ wdrïNl m%fõ.h fidhkak'
(ii) t = T ld,fha°" p,s;fha wdrïNl †Ydjg ,ïn †Ydjla Tiafia wxY=j p,s;fjñka
mejf;hs' T ys w.h o" t = T ys° wdrïNl msyà
s fï isg wxY=jg ≥r o fidhkak'
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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4. >r®IK ix.=Klh µ jk rΩ ;sria ;,hla u; ialkaOh 2m jk l+[a[hlg p,kh ùug
ksoyi we;' ;srig α fldaKhlska wdk; l+[a[fha iqug uqyqKf;ys jeä;u nEjqï
fr®Ldjla †f.aa my<g ialkaOh m jk wxY=jla reyqghs' l+[[
a fha ;ajrKh fidhkak'
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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- 146 -

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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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10. †. ógr 2 la jk Æyq wú;kH ;ka;=jl tla fl<jrla iS,sul wp, ,CIHhlg iïnkaO
lr we;s w;r wfkla fl<jr ia l ka O h 2 kg jk wxY=jla orhs' wxY=j isria ;,hl
π
foda,kh fjhs' háw;a isrfia isg ;ka;=fjys fldaKsl úia:dmkh
jk úg wxY=fõ
3
m%fõ.h 3 m s −1 kï" tu fudfydf;ys° ;ka;=fjys wd;;sh fidhkak'
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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11. ialkaOh m jk wxY=jla" iajNdúl †. a iy m%;Hia:;d udmdxlh λ jk m%;Hia: ;ka;j
= l
tla fl<jrlg iïnkaO lr we;' ;ka;=fjys wfkla fl<jr iqug ;sria ;,hla u; jQ
wp, O ,laIHhlg iïnkaO lr we;' wdrïNfha ° wxY=j O ,laIHfha isg 2a ≥rlska
π ma
ksYap,;dfõ ;nd uqod y˙kq ,efí'
ld,hla wxY=j ir, wkqjr®;S p,s;hl
2 λ
mj;sk nj fmkajkak'
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12. iqug ;sria ;,hla u; jQ ialkaOh m jk P wxY=jla iajNdúl †. l iy m%;Hia:;d
udmdxlh 2mg jk ieye,aÆ m%;Hia: ≥kakl tla fl<jrlg iïnkaO lr we;s w;r
≥kafkys wfkla fl<jr ;,h u; wp, O ,CIHhlg iïnkaO lr we;' wdrïNfha °
≥kak fkdwe° mj;sk w;r" OP †Ydj †f.a l g jvd wvq úia:dmkhla wxY=jg ° th
2g
x = 0 iólrKh imqrd,k nj
uqody˙kq ,efí' t ld,hl°" wxY=fõ x úia:dmkh" &x& +
l
idOkh lrkak'

gl
kï" p,s;fha úia;drh fidhkak'
2
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wxY=fõ jeä;u m%fõ.h

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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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13. úYd,;a j 2P, 3P iy 4P jk n, ;=kla ms<sfj<ska me;a;l †. a j k ABC iumdo
;%sfldaKhl AB, BC iy C A mdo †f.aa l%shdlrhs' iïm%hqla;fha úYd,;ajh iy †Ydj
fidhkak'
iïm%hl
q ;
a fha l%h
s d fr®Ldj AC yuqjk ,laIHhg A isg we;s ≥r o fidhkak'
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14. Pi + 3Pj , − 2 Pi − Pj yd Pi − 2 Pj n, ;=kla ms<f
s j<ska msyg
s ï
q ffoYsl 2i + 5 j , 4 j yd
− i + j jk ,CIH ;= kl° l%h
s d lrhs· fuys i yd j g iqmr
q e≥ f;areï we;' wkqrEm ,CIHj,
LKavdxl olajñka" n," ixrpl wdldrhg rEm igykl ksrEmKh lrkak' >Qr®Kh 10P
jk hq.u
a hlg moaO;sh ;=,H jk nj fmkajd" hq.u
a fha w; olajkak'
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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15. iudk" taldldr AB i y B C nr o~q folla B ys ° iq u gj ikaê lr we;' moaO;sh
A ,laIHfhka t,a,d we;s w;r AB iy BC o~q tlsfklg ,ïn jk m˙† o~qj, uOH
,laIH Æyq wú;kH ;ka;j
= lska iïnkaO lr we;' AB oKav háw;a is r i iu. θ
1
fldaKhla idok m˙† moaO;sh iu;=,;
s ;dfõ mj;S kï" tan θ = nj fmkajkak'
3
;jo" B ys ° m%;l
s h
s% dj BC †Ydj Tiafia nj fmkajkak'
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16. C ys° W Ndrhla ork iqugj ikaê l< ieye,aÆ o~q y;rlska
iukaú; rduq iels,a,la rEmfha oelafjhs' AB, BC iy AC o~q
†.ska iudk fõ' D g isriaj my<ska B isák fia o" AD iy
BC o~q ;sriaj isák fia o rduq iels,a, D wp, ,CIHfha °
iqugj wiõ lr we;s w;r" iqug wdOdrlhla u; B ksi,j
we;' rduq iels,a, isria ;,hl iu;=,s;;dfõ mj;S' AC, BC
i y A B o~q u; l% s h dlrk n," fnda wxlkh fhd∞ f.k
m%i;
a d˙lj fidhkak'

D

A

B

C
W

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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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17. iqug kdoe;a;la u;ska hk †. 4l jk ieye,aÆ wú;kH ;ka;=jla" nr w jQ taldlr oKavl
fofl<jrg iïnkaO lr we;s w;r oKav iu;=,s;;dfõ we;' nr w jQ Ndrhla oKafâ
tla fl<jrlg iïnkaO l< úg ;ka;=j" kdoe;a; u;ska l ≥ rla ir®mKh ùfuka miq
moaO;sh kej; iu;=,;
s ;dfõ mej;sh yels nj fmkajkak'
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18. Wi 40 cm yd m;=f,a wrh 6 cm jk taldldr iDcq jD;a;dldr is,skavrhla" tys wlaIh
;,hg ,ïn jk m˙† rΩ ;sria ;,hla u; ;nd we;' is,skavrh iy ;,h w;r >r®IK
ix.=Klh 0.4 la fõ' ;sria ;,h fifuka we< lrk úg is,skavrh m%:ufhka ,siaihs o
ke;skï fmrf<hs o hkak ksrK
® h lrkak'
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1
2
11
19. A iy B isoaê P( A) = , P( B) =
iy P( A|B ′) =
jk m˙† fõ· fuys B ′ hkq B ys
3
5
20
wkqmr
Q l isoê
a h fõ'
(i) P( A ∩ B),
(ii) P( A ∪ B),
(iii) P( A′|B )
fidhkak'
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20. A yd B isoê
a P( A) = x + 0.2, P( B) = 2 x + 0.1 iy P( A ∩ B) = x jk m˙† fõ'
(i) P( A ∪ B) = 0.7 kï x ys w.h fidhkak'
(ii) A yd B isoê
a iajdh;a; nj i;Hdmkh lrkak'
(iii) P( A | B ′) fidhkak· fuys B′ hkq B ys wkqmr
Q l isoê
a h fõ'
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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8
1
1
a folla hehs .ksu'q
, P( B) = iy P( A|B ) = jk m˙† jQ isoê
15
3
5
(i) isoê
a folu is≥ùfï"
(ii) isoê
a folska tlla muKla is≥ùfï"
(iii) isoê
a folska tllaj;a is≥ fkdùfï
iïNdú;dj fidhkak'
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21. A yd B hkq P( A) =

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22. tl tll jdyk tlla muKla kj;d ;eìh yels r:.d, folla ù:ßhl we;' i∫od
fm'j' 9.00 g r:.d,la ysij
a mej;Sfï iïNdú;dj 0.3ls'
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s la ysij
a fkd;sîfï"
(ii) r:.d,j,ska wvq jYfhka tllaj;a ysij
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- 155 -

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1
o" ud,d
6

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o fõ' isoê
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7
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(ii) Tjqkf
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WmlrKj,ska 30% la" yd C ldKavfha ishÆ WmlrKj,ska 50% la ne.ska WmlrK
ldKav ;=kl l%shdld˙;ajh m¯CId lrhs' ldKav ;=f kys tl tll WmlrKhla
l%h
s dúrys; ùfï iïNdú;d ms<f
s j<ska 0.002, 0.003 iy 0.005 fõ'
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s dúrys; ùfï iïNdú;dj fidhkak'
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iïNdú;dj fidhkak'
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25. 5, 6, 13, 5, 10, 13, 3, x, y hk ks¯CIK 9 ys uOHkHh 8 o" ud;h 5 o fõ'
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(ii) 12 f,i ksjer† lf<a kï
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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27. isiqka 30 fokl=f.a jdr m¯CIKfha ,l=Kqj, iuqys; ixLHd; jHdma;sh my; j.=fõ °
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ixLHd;h

48 – 55
5

56 – 63
12

64 – 71
10

72 – 79
3

jHdma;f
s ha uOHkHh" uOHia:h yd ud;h fidhkak'
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28. n, m%foaY tl tll wdikak jYfhka wf<ú ks,Od¯ka tlu ixLHdjla isák n, m%foaY
10 la iud.ulg we;' wjidk udih ;=<° ,nd .;a wf<ú weKjqï ^oyia .Kkska& my;
oelafõ(
150, 130, 140, 150, 140, 300, 110, 120, 140, 120.
wf<ú o;a;j,
(i) wka;r® p;=r:
® l mrdih"
(ii) uOHkH wm.ukh
fidhkak'
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

- 158 -

29. isiqka lKavdhula úNd.hl° ,nd.;a ,l=Kqj, uOHkHh yd iïu; wm.ukh ms<sfj<ska
42 yd 15 fõ' uOHkHh 50 yd iïu; wm.ukh 20 jk m˙† fr®Çh m˙udKhla fh°fuka
,l=Kq kej; ilia lr we;' úNd.fha° ,l=Kq 54 la ,nd.;a isiqjl=f.a ixfYdaê; ,l=K
fidhkak'
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30. l=vd lDóka 50 lf.a †. X, fikaáógrj,ska yd nr Y, .%Eïj,ska ukskq ,efí' tu rdYsj,
tl;=j iy jr®.j, tl;=j my; ° we;'
50

∑ xi = 212,
i =1

50

∑ xi2 = 902.8,
i =1

50

∑ yi = 261 yd
i =1

50

∑y

2
i

= 1457.6 .

i =1

jvd jeä úp,k we;af;a †f.ys o@ ke;akï nfrys o@ Tfí ms<;
s r
+ ikd: lrkak'
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

- 159 -

B fldgi
1. (a) iS>%.dó ≥ï˙hla, A ≥ï˙h ia:dkhl isg tys B<Û kej;=u jk B ≥ï˙h ia:dkh lrd
.uka lrhs' ≥ï˙h ia:dk fol w;r ≥r d km fjhs' ≥ï˙fha taldldr ;ajrKh yd
ukaokh ms<sfj<ska f km s–2 yd λ f km s–2 fjhs; fuys λ Ok ksh;hls' ≥ï˙hg .; yels
iy mj;ajd f.k hd yels Wm˙u m%fõ.h v km s–1 fjhs' ≥ï˙h, A ≥ï˙h ia:dkfha isg
ksYp
a ,;dfjka wdrïN lr wju ld,hla ;=< B ≥ï˙h ia:dkhg <Ûd ù tys k;r lrhs'
(i) ≥ï˙fha p,s;h i|yd m%fõ.-ld, m%i;
a drh w|skk
a '
2
v 
1
ta khska, d ≥
1 +  kï, B ≥ï˙h ia:dkh lrd <Ûdùug ≥ï˙hg .;jk uqΩ
2f  λ
d
v  1
+
ld,h
1 +  nj fmkajkak'
v 2f  λ
v2 
1
(ii) d <
a drfha wjYH fjkia ls¯ï is≥lr ta khsk,a
1 +  kï, m%fõ.-ld, m%i;
2f  λ
fuu wjia:dfõ° B ≥ï˙h ia:dkh lrd <Ûdùug ≥ï˙hg .;jk uqΩ ld,h
2d 
1
1 +  nj fmkajkak'
f  λ
≥ï˙fhys idudkH fõ.h fidhkak'

(b) u m%fõ.fhka kef.kysr †Ydjg .uka lrk ;eke;a;l=g kef.kys˙ka W;+rg α iq Ω
fldaKhla idok †Ydfjka iq<Û yuk nj oefkhs' Tyq W;=re †Ydjg 2u m%fõ.fhka
.uka lrk úg" kef.kys˙ka W;=rg β iqΩ fldaKhla idok †Ydfjka iq<Û yuk nj
oefkhs' wjia:d folu i|yd tlu rEm igykl m%fõ. ;%f
s ldaK w|skk
a '
ta khsk"a iq<fÛa †Ydj fidhkak'
2. (a) kejla iDcq uqyq≥ .uka u.l u km h–1 taldldr fõ.fhka hd;%d lrhs' jrdhl isg tu
.uka uf.ys wdikak;u ,CIHh jk A g we;s ≥r a km fjhs' A ,CIHhg <Ûdùug
fmr" jrdfha isg b ( > a) km ≥˙ka kej we;s úg" th w,a,d .ekSu i|yd fndaÜgqjla
jrdfhka msg;afjhs' kejg ,Ûdùu i|yd fndaÜgqjg ;sìh hq;= wvq;u taldldr fõ.h
au
km h–1 nj idOkh lrkak'
b
fndaÜgqfjys .uka u. rEm igykl w|skk
a '
au 

fndaÜgqjg v km h–1  u > v >
a ; ia:dk foll° kej
 fõ.hlska hd yels kï" ksYÑ
b 

w,a,d .ekSug thg yels nj;a" tu ia:dk fol i|yd fndaÜgqjg .;jk ld,"
meh

2 b 2 v 2 − a 2u 2
u2 − v2

lska fjkia jk nj;a idOkh lrkak'

(b) nr W jk fudagr® r:hlg H Wmrsu cjhla we;' ;srig α fldaKhlska wdk; nEjqula
†f.a by<g fudagr® r:h .uka lrk úg tys Wm˙u fõ.h v fjhs' tu nEjqu †f.au
my<g th .uka lrk úg tys Wm˙u fõ.h 2v fjhs' wjia:d foflys° u fudagr® r:fha
p,s;hg tfrys m%;sfrdaO iudk hehs Wml,amkh lrñka W yd α weiqfrka m%;sfrdaOh
fidhkak'

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

- 160 -

3. (a) t = 0 ld,fha° A ,CIHhl isg ksYap,;dfjka uqody, neÆkhla, f taldldr ;ajrKhlska
isriaj by< k.S' t = T ld,fha° A ,CIHfha isg u m%fõ.fhka wxY=jla .=re;ajh hgf;a
isriaj by<g m%lafIam flfr®' neÆkfha yd wxY=fõ p,s; i|yd m%fõ.-ld, m%ia;dr
tlu rEm igykl w¢kak'
wxY=j hdka;ñka neÆkh iamr®Y lrhs kï" by; m%i;
a dr fhdod .ksñka
u = T ( f + f ( f + g ) ) nj idOkh lrkak'
(b) O wp, iqug uq≥jla Tiafia hk †. l jk ieye,aÆ wú;kH ;ka;=jl fofl<jrg
ms<sfj<ska ialkaO m yd 2m jk A yd B wxY= folla iïnkaO lr we;X B wxY=j O g
my<ska isriaj msysgk w;r ;ka;=j fkdnqre,aj mj;sk f,i B iu. tlu uÜgfï B g
wE;ska A wxY=j msysghs' ksh; fldaKsl m%fõ.hla yd B flakaøh iys; ;sria jD;a;dldr
.uka u.la A f.jd hk f,i OA g ,ïnj A wxY=j m%lafIam flfr®' AO : OB wkqmd;h
yd A wxY=fõ fldaKl
s m%fõ.h fidhkak'
4. (a) fmdf<dj u; O ,CIHhl isg ;srig α fldaKhlska wdk;j wxY=jla .=re;ajh hgf;a"
u fõ.fhka" O isg a ≥rla wE;ska msysá ;dmamhlg ,ïn ;,hl m%lafIam flfr®' O isg
x ;sria ≥rlska we;s úg wxY=j y Wilska msysghs kï, gx 2 sec α = 2u 2 ( x sin α − y cos α )
nj fmkajkak'
wxY=j" ;dmamh Wäka hdka;ñka f.dia" ;dmamfha isg d ≥rla wE;ska msysá ,CIHhl°
ad
tan α nj fmkajkak'
fmdf<dj u; m;s; fjhs kï" ;dmamfha Wi
a+d
wxY=j <Ûdjk Wm˙u Wi a, d yd α weiqfrka fidhkak'
(b) ieye,aÆ wú;kH ;ka;=jla iS,sul iúlr we;s iqug lmamshla u;ska hk w;r tla
fl<jrlg ialkaOh M jk wxY=jla o" wfkla fl<jrg ieye,aÆ iqug lmamshla o
iïnkaO lr we;' ieye,aÆ wú;kH fjk;a ;ka;=jla fojk lmamsh u;ska hk w;r tys
tla fl<jrla ialkaOh m1 jk wxY=jla o" wfkla fl<jr ialkaOh m2 jk wxY=jla o
4
1
1
ord isà' moaO;sh ksYap,;dfjka uq∞ yefr® kï"
fjf;d;a ialkaOh M
=
+
M m1 m2
jk wxY=j ksYp
a ,;dfõ mj;sk nj fmkajkak'
5. wxY= .egqïj, fhd∞.kq ,nk
^i& .uH;d ixia:;
ß s kshuh"
^ii& ksõgkaf.a m%;Hd.;s kshuh
m%ldY lrkak'
iqug" m%;Hia:" iudk P, Q yd R wxY= ;=kla" iqug ;sria fïihla u; msysá A, B y d C
,CIHj, ms<sfj<ska ksYap,;dfõ ;nd we;' B hkq AC ys uOH ,CIHh fõ' P wxY=j fïih
†f.a u m%fõ.fhka m%lafIam flfrk w;r Q wxY=j iu. th iDcq f,i .efghs' P wxY=j
.egqï foll;a" R wxY+j tla .egqul;a fhfohs kï e > 3 − 2 2 nj fmkajkak; fuys e hkq
m%;Hd.;s ix.+Klh fjhs'

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6. ieye,aÆ m%;Hia: ;ka;=jl tla fl<jrla iS,sul wp, ,CIHhlg iïnkaO lr we;' tys
wfkla fl<jrg iu;=,s;;dfõ t,af,ñka we;s wxY=jla iïnkaO lr we;s w;r tu.ska
;ka;+fõ l ú;;shla we;s flfrhs' ld,h t = 0 °" isriaj háw;g 2 gl m%fõ.hla °fuka
wxY=fõ iu;=,;
s ;dj ì|fy,kq ,efnhs'
(i) ;ka;f
= õ Wmrsu ú;;sh 3l nj"
7π l
(ii)
ld,hlg miqj ;ka;j
= nqre,a jk nj"
6 g
3l
(iii) ;ka;f
= õ iajdNdúl †.
blaujd hf;d;a wxY=j iS,f
s ï fkdj†k nj"
2
3l
(iv) ;ka;f
= õ iajdNdúl †.
g jvd jeä kï" wxY=jg Wm˙u Wig <Ûdùug
2
7π  l

ld,hla wjYH nj
 3+

6  g

idOkh lrkak'
7. wrh a jk iqug iDcqjD;a;dldr l=yr is,skavrhla tys wCIh ;sria jk f,i iúlr we;'
O hkq is,skavrfha wCIh u; ,CIHhla hehs .ksuq' O ,CIHh Tiafia is,skavrfha wCIhg
,ïn isria ;,hl is,skavrfha we;=Ω mDIaGfha msysá" O iu. tlu uÜgfï jk A ,CIHhl
isg" ialkaOh m jk P iqug wxY=jla isriaj háw;g 10ag fõ.fhka m%lf
a Iam flfrhs'
P wxY=j mDIaGfha my<;u ,CIHh fj; <Ûdjk úg ksYap,;dfõ we;s ialkaOh 2m jk Q
1
iqug wxY=jla iu. th iDcqj .efghs' m%;Hd.;s ix.+Klh
fjhs'
2
(i) .egqfuka fudfyd;lg miq Q wxY=fõ m%fõ.h fidhkak'
(ii) OQ Wvq isri iu. θ iqΩ fldaKhla idok úg Q wxY=j yd is,skavrfha we;=Ω mDIaGh
w;r m%;l
s h
s% dj fidhkak'
1
OQ Wvq isri iu. cos −1   fldaKhla idok úg Q wxY=j mDIaGfhka bj;a jk nj
3
wfmdaykh lrkak'
13a
(iii) O Tiafia hk ;sria uÜgfï isg by<g Q wxY=j <Ûdjk Wm˙u Wi
nj
27
fmkajkak'
8. ialkaOh M jk iqug l+[a[hla ;sria wp, iqug ;,hla u; ;nd we;' l+[a[fha nEjqï
iqug uqyqK;a fol" tl tlla ;srig θ fldaKhlska wdk; fjhs' ialkaO ms<sfj<ska m1 yd
m2 jk P yd Q wxY= folla" tl tl uqyqK; u; tlla jk m˙† jeä;u nEjqï fr®Ld †f.a
isriaj my<g rEgkq ,efnhs'
(i) l+[[
a h ;sria ;,hg iúlr we;akï" P yd Q wxY=j, ;ajrK fidhkak'
(ii) l +[a[hg ;sria ;,h u; p,kh ùug ksoyi we;akï" wxY= foflys yd l+[ a [ fha
;ajrK fidhkak'
wxY= fol u; l+[[
a fha m%;l
s h
s% d ,nd.kak'

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9. (a) ;srig α fldaKhlska wdk;j wxY=jla .+re;ajh hgf;a u fõ.fhka m%lafIam flfr®'
t ld,hlg miqj wxY=fõ isria yd ;sria m%fõ. ixrpl i|yd m%ldYk o" wxY=fõ msysàu o
,nd.kak'

R 2 
2

m%CIsm;
a fha jeä;u Wi h yd ;sria mrdih R kï" u = 2 g h +
nj fmkajkak'

16h 

(b) iajlSh tka‚u ksh; H kW cjhlska l%shdlrkq ,nk ialkaOh M kg jk f,d˙hl
Wm˙u fõ.h ;sria udr®.hl° u ms–1 fjhs' by;° fuka tu cjfhkau tka‚u l%shdlrk
úg" f,d˙fha Wm˙u fõ.h ;srig α fldaKhlska wdk; udr®.hl° by<g v m s–1 fjhs'
f,d˙fhys p,s;hg tfrys m%;f
s rdaOh ksh; fjhs'
1000 H (u − v) = Mguv sin α nj fmkajkak'
10. (a) ffoYsl foll ;s;a ^w†Y& .=◊;h wr®: olajkak'
a yd b hkq a + b yd a − b ys ;s;a .=◊;h Y=kH jk" ksYY
a k
= H ffoYsl folla hehs
.ksu'q a yd b ffoYsl úYd,;ajfhka iudk nj fmkajkak'
iudka;rdi%hl úlr®K tlsfklg ,ïn kï th frdïnihla jk nj ffoYsl l%u
fhdod .ksñka idOkh lrkak'
(b) ;,h isria taldldr jD;a;dldr ;eáhlg iajlSh C flakaøh Tiafia ;eáfha ;,hg
,ïnj hk ;sria iqug wCIhla jgd lrleùug ksoyi we;' PQR iumdo ;%sfldaKhla
jk fia ;eáfha .eáfha msysá P, Q yd R ,CIHj,° ms<sfj<ska nr p, q yd r jk wxY=
;=kla iúlr we;' iu;=,s; msysàfï° P yd Q ,CIH C Tiafia hk isria fr®Ldfõ tlu
me;af;a fjhs' CP isri iu. idok iqΩ fldaKh fidhkak'
s ul° ;eáh iu;=,;
s ;dfõ mj;sk nj wfmdaykh lrkak'
p = q = r kï ´kEu msyà
11. (a) iqug f,i ikaê l< ieye,aÆ o~q mylska
iukaú; rduq iels,a,la rEm igyfkys oelafjhs'
fuu rduq iels,a, wp, A ,CIHfha° iqug
f,i wiõ lr we;s w;r C ,CIHfha°
W Ndrhla orhs' rduq iels,a," AC ;sria yd AB
isria jk fia isria ;,hl iu;=,s;;dfõ ;nd
we;af;a AC g iudka;r †Ydjlg B ys° fhdok
,o P n,hla u.sk's
P ys úYd,;a j h o, A ys m%;sl%shdfõ ;sria yd
isria ixrpl o fidhkakX

A

o

C

60
o

30

W

D
30o

P
fnda wxlkh fhdod f.k" rduq iels,a, i|yd
B
m%;Hdn, rEm igykla w|skk
a '
ta khska, wd;;s yd f;rmqï fjkafldg olajñka" o~qj, m%;Hdn, W weiqfrka ksr®Kh
lrkak'
(b) tl tll nr w jk PQ, QR, RS yd SP iudk taldldr o~q y;rla PQRS frdïnihla
jk wdldrhg iqug f,i ikaê lr we;' QRS fldaKh 2 tan −1 2 jk f,i PS yd
RS j, uOH ,CIH ieye,aÆ wú;kH ;ka;=jlska hd lr" th P ,CIHfhka isria
;,hl t,a,d iu;=,;
s ;dfõ ;nd we;'
(i) R yd S ikaêj, m%;l
s h
s% d fidhkak'
(ii) ;ka;f
= õ wd;;sh 4w nj fmkajkak'
w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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12. flakaøfhys 2α fldaKhla wdm;kh lrk taldldr jD;a;dldr pdmhl .=re;aj flakaøfha
msyà
s u fidhkak'
ta khsk"a taldldr wr®OjD;a;dldr
(i) pdmhl"
(ii) wdia;rhl
.=re;aj flakøa h ksrK
® h lrkak'
wrh 2a yd ialkaOh m jk taldldr wr®OjD;a;dldr
wdia ; rhlg ñgla iïnka O lr rEmfha fmka j d we;s
wdldrhg jia;=jla ilia lr we;' †. 4 a yd ialkaOh m jk
taldldr oKavlska iy wrh a yd ialkaOh m jk taldldr
wr®OjD;a;dldr pdmhlska ñg iukaú;h' jia;=fõ .=re;aj
flakøa h fidhkak'
jia;=fõ ;,h isria jk m˙† wr®OjD;a;dldr wdia;rfha
m˙êh iq u g ;s r ia ;,hla iu. ia m r® Y j jia ; = j
iu;=,s;;dfõ ksYap,j ;sìh yelsh' ñfgys iDcq fldgi
3π
isri iu. θ fldaKhla idohs kï" tan θ =
nj
2(9π − 1)
idOkh lrkak'
13. (a) tlu †.ska hq;a" nr ms<sfj<ska W 1 yd W2 (> W1 ) jk AB yd BC taldldr o~q folla
B ys° ksoyia f,i tlg ikaê lr we;' A yd C rΩ ;sria fïihla u; mj;sk fia
moaO;sh isria ;,hl ;nd we;'
s ;dj mej;sh yels nj fmkajkak;
(W1 + W2 ) tan θ ≤ (3W1 + W2 ) µ u kï muKla iu;=,;
fuys µ hkq fïih yd o~q w;r >r®IK ix.+Klh o" θ hkq o~q tl tlla isri iu.
idok fldaKh o fõ'
(b) ABC iqΩflda” ;%sfldaKhl ms<sfj<ska BC, CA yd AB mdo Tiafia l%shdlrkq ,nk F,
λ F y d λ2F n, ;+klska moaO;shla iukaú; fjhs; fuys λ hkq Ok ksh;hls'
iïm%hl
q ;
a fha úYd,;ajh yd †Ydj fidhkak'
iïm%hl
q ;
a hg D ,CIHfha° AC yuqfjhs kï" AD ys †. fidhkak'
iïm%hl
q ;
a h ABC ys ,ïn flakøa h Tiafia hhs kï"
1
λ
λ2
+
=
nj fmkajkak'
cos A cos B cos( A + B)
14. (a) >k taldldr iDcq jD;a;dldr fla;j
= l .=re;aj flakøa fha msyà
s u fidhkak'
>k taldldr iDcq jD;a;dldr fla;=jl wdOdrlfha wrh yd Wi ms<sfj<ska a yd h fjhs'
tlu ;sria uÜgul msysá wp, iqug kdoe;s folla u;ska hk l †.ska hq;a ieye,aÆ
wú;kH ;ka;=jl tla fl<jrla fla;=fõ YSr®Ihg o" wfkla fl<jr wdOdrlfha
m˙êh u; ,CIHhlg o iïnkaO flfr®' fla;=fõ wCIh ;sria jk m˙† fla;=j
iu;=,s;;dfõ mj;S kï" h 2 (l − h)(l + h − 2d ) = 4a 2 (h − d ) 2 nj idOkh lrkak ; fuys
d hkq kdoe;s fol w;r ≥r fõ'

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(b) iqug f,i ikaê lr we;s iudk †.ska hq;a ieye,aÆ taldldr o~q y;lska iukaú; rduq
iels,,
a la rEm igyfka oelafõ'
100 N

100 N
C

B

P
A

60o

D

E

rduq iels,a, A ys° iqug f,i wiõ lr we;s w;r tl tlla 100 N jk nr folla
B yd C ys ;nd we;' AE, ED yd BC o~q ;sria jk m˙† ;srig 60o fldaKhla idok
†Ydfõ D ys° fhdok ,o P n,hla u.ska rduq iels,a, isria ;,hl iu;=,s;;dfõ ;nd
we;' P ys úYd,;ajh fidhkak'
fnda wxlkh fhdod f.k" wd;;s yd f;rmqï fjkafldg olajñka BC, CD yd ED o~qj,
m%;Hdn, ksrK
® h lrkak'
15. taldldr wr®Of.da,dldr lfnd,l .=re;aj flakøa fha msyà
s u fidhkak'
ta khsk,a taldldr >k wr®Of.da,hl .=re;aj flakøa h ksrK
® h lrkak'
wrh 2a jk taldldr >k wr®Of.da,hlska wrh a jk wr®Of.da,dldr fmfoila wr®Of.da,
foflys flakaø iumd; jk m˙† bj;a ls¯fuka jia;=jla idod we;' jia;=fõ .=re;aj
flakøa h ksrK
® h lrkak'
jl% mDIaGh iqug isria ì;a;shla yd rΩ ;sria ;,hla iu. iamr®Y jk m˙† jia;=j ;nd
we;' jia;=fõ jl% mDIaGh yd rΩ ;sria ;,h w;r >r®IK ix.=Klh µ fõ' jia;=fõ ;,
mDIaGfha wdk;sh l=ula jqj;a" jia;j
= iu;=,;
s ;dfõ mj;S kï 112 µ ≥ 45 nj fmkajkak'
16. †. 4a yd nr 2w jk taldldr bksu.l my< fl<jr rΩ ;sria fmdf<djla u; ksi,j
we;s w;r bksuf.ys by< fl<jr iqug isria ì;a;shlg fya;a;= ù ksi,j we;' háw;a
isri iu. θ fldaKhla idoñka bksu. isria ;,hl iu;=,s;;dfõ mj;S' bksu. yd
fmdf<dj w;r >r®IK ix.=Klh µ fjhs'
7
(a) µ ≥ tan θ fjf;d;a nr 6w jk ñksil=g wdrCIs;j bksuf.ys uq≥kg ke.sh yels nj
8
fmkajkak'

3
π
yd µ <
hehs .ksu'q
8
6
(i) ñksidg wdrCIs;j bksuf.ys uq≥kg ke.Su i|yd bksu.g fh°ug wjYH wju n,
hq.u
a h 8aw nj fmkajkak'
(ii) w Ndrhla Wiq,df.k Tyqg bksu. †f.a ke.sh yels Wm˙u ≥r fidhkak'

(b) θ =

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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17. (a) ieye,aÆ o~qq y;rla tajdfha fl<jrj,° ksoyia f,i ikaê lr BD kï ieye,aÆ
miafjks oKavla u.ska ABCD iudka;rdi%hl wdldrhg ;nd we;' BAD fldaKh α,
AD = a yd AB = b fjhs' úYd,;ajh P jk iudk yd m%;súreoaO n, folla A yd C ys
° ms<sfj<ska CA yd AC †f.a fh°fuka rduq lÜgqj iqug ;sria ;,hl ksYap,;dfõ
;nd we;' AB yd AD o~qj, n, fidhkak'
BD ys f;rmqu P

a 2 + b 2 − 2ab cos α
a 2 + b 2 + 2ab cos α

nj idOkh lrkak'

(b) wrh a yd nr w jk taldldr jD;a;dldr ;eáhlg" iajlSh ;,hg ,ïnj iajlSh
flakaøh jk O Tiafia hk wp, wCIhla jgd ksoyia f,i ye¯ug yelsh' ABCD
iup;=ri%hla jk m˙† ;eáfha .eáh u; msysá A, B, C yd D ,CIHj,g nr w, 2w, 3w
yd 4w jk wxY= y;rla ms<sfj<ska iïnkaO lr we;' AC isri iu. θ fldaKhla
idok úg ;eáfha iu;=,s;;dj mj;ajdf.k hdu i|yd wjYH hq.aufha >Qr®Kh
fidhkak'
ta khsk"a n, hq.u
a hla fhdokafka ke;akï ;eáfha iu;=,;
s msyà
s u fidhkak'
18. (a) ta l dldr jD;a ; dldr wdia ; rhls k a " iup;= r i% d ldr is ≥ rla lmd bj;a l r we;'
iup;=ri%dldr is≥frys tla úlr®Khla wdia;rfha wrhla fjhs' is≥r iys; wdia;rfha
a
.=re;aj flakaøh" wdia;rfha flakaøfha isg
≥rlska msyg
s k nj fmkajkak ; fuys
4π − 2
a hkq wdia;rfha wrh fjhs'
(b) †. 4a jk AB ieye,aÆ oKavla A ys° újr®;kh lr we;s w;r B g w Ndrhla iïnkaO
lr we;' C yd D ,CIHj,° iqug uq≥ folla iú lr we;; fuys C ,CIHh A g isriaj
by<ska 2a Wilska o" D ,CIHh A g isriaj my<ska a .eUqrlska o msysghs' AB ;sria jk
m˙† oKav iu;=,s;;dfõ mj;ajd .kq ,nkafka tla fl<jrla B ,CIHhg wEª ieye,aÆ
wú;kH ;ka;=jla" C ;=<ska o" D ;=<ska o hjd wk;=rej ;ka;=fõ wfkla fl<jr
oKafâ ueo ,CIHhg o iïnkaO ls¯fuks' ;ka;=fõ wd;;sh o" A ys m%;sl%shdj o
fidhkak'
19. (a) fmÜáhl ir®jiu fnda, 6 la we;s w;r tajdhska 5 la lΩ o" wfkl iq≥ o fjhs' wks,a
yd ìu,a w;=re udre l%uhg fmÜáfhka iiïNdù f,i jrlg fnda,hla f;a¯fï
l%Svdjl fhfoa' iq≥ fnda,h m<uqfjka f;darkq ,nk mqoa.,hd l%Svdfõ ch.%dylhd fjhs'
wks,a l%Svdj wdrïN lrhs' we°ï tl tll ° .kq ,nk fnda,h m%;sia:dmkh
fkdflfrkafka hehs Wml,amkh lrñka" ìu,a l%Svdj †kSu ±laùu i|yd rela
igykla f.dvk.kak'
ta khsk"a ìu,a l%v
S dj †kSfï iïNdú;dj fidhkak'
tla tla W;aidyfhka miqj fnda,h m%;sia:dmkh lrk m˙† l%Svdj kej; wr®: olajd
;sfnhs'
wks,a
(i) Tyqf.a m<uq W;aidyfha°"
(ii) Tyqf.a f;jeks W;aidyfha°"
(iii) wjidkfha°
l%v
S dj †kSfï iïNdú;dj fidhkak'
ta khsk"a ìu,a wjidkfha° l%v
S dj †kSfï iïNdú;dj fidhkak'
Tfí ms<;
s r
= iajdh;a; l%uhla u.ska i;Hdmkh lrkak'

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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(b) jr®I wjidk úNd.fha° isxy," bx.%Sis yd b;sydih úIh ;=k i|yd wu,a yd lu,a
isiqka fofokd ,nd.;a ,l+Kq o" tla tla úIhh i|yd fh†h hq;= nr o my; oelafjk
j.=fõ oelafjhs(
úIh
wu,af.a ,l+Kq
lu,af.a ,l+Kq
nr

isxy,
80
64
2

bx.%i
S s
72
82
x

b;sydih
46
40
3

wu,af.a yd lu,af.a ,l=Kj, N˙; uOHkH tlu kï x ys w.h fidhd" N˙; uOHkH
,l+K .Kkh lrkak'
20. (a) rCIK iud.ula ˙he≥rka my<" idudkH yd by< wjodkula we;s ˙he≥rka f,i
jr®.SlrKh flfr®' iud.u Tjqkaf.a jdr®;dj, oekg 25% la my<" 60% la idudkH
yd 15% la by< wjodkula we;s ˙he≥rka isák nj weia;fïka;= flfr®' tjeks
˙he≥rkaf.ka" jirlg" fok ,o wk;=re .Kkla is≥ùfï iïNdú;d my; j.=fõ
° we;(

jirlg wk;=re .Kk

0
1
2
3
≥4

my<
x
0.01
0.00
0.00
0.00

wjodku
idudkH
0.93
y
0.01
0.00
0.00

by<
0.74
0.10
z
0.01
0.00

(i) x, y yd z i|yd iq≥iq w.hka fidhkak'
(ii) iiïNdù f,i f;dard .kakd ,o ˙he≥frl=f.ka jir we;+<; wk;=rla is≥fkdù
;sîfï iïNdú;dj fidhkak'
(iii) A f.ka jir we;=<; wk;=rla is≥fkdù ;sfí kï Tyq by< wjodkula we;s
˙he≥rl= ùfï iïNdú;dj fidhkak'
(iv) B f.ka tjeks jir 4 la we;+<; wk;=rla is≥fkdù ;sfí kï Tyq my< wjodkula
we;s ˙he≥rl= ùfï iïNdú;dj fidhkak'
(b) isiqka 20 fokl= .Ks; úIhfha f.or jev i|yd .;lrk ,o ld,j, ^ñks;a;+j,ska&
ftlHh yd jr®.j, ftlHh ms<f
s j<ska 320 yd 5840 fjhs'
(i) isiqka 20 fokd .Ks; úIhfha f.or jev i|yd .;lrk ,o ld,j, jHdma;sfha
uOHkHh yd iïu; wm.ukh .Kkh lrkak'
(ii) fjk;a isiqjl= .Ks; úIhfha f.or jev i|yd .;lrk ,o ld,h tlal< úg
uOHkHh fjkia fkdjk nj fidhd f.k we;' iïu; wm.ukh wvq jk nj
fmkajkak'
(iii) fjk;a isiqka 10 fokl= .Ks; úIhfha f.or jev i|yd .;lrk ,o ld,j,
^ñks;a;+j,ska& ftlHh yd jr®.j, ftlHh ms<sfj<ska 130 yd 2380 fjhs' isiqka
30 fokdu .kakd ,o ld,j, uOHkHh yd iïu; wm.ukh fidhkak'

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

- 167 -

21. (a) bf,lafg%dksl moaO;shlg iajdh;a; f,i l%shd lrkq ,nk R, S yd T kï fldgia
3 la ;sfnhs' jirl ld,hla R, S yd T fldgia l%shd ls¯fï iïNdú;d ms<sfj<ska 0.95"
0.90 yd 0.93 fjhs'
(i) moaO;sh l%shd ls¯u i|yd wvq jYfhka fldgia 2 la l%shd l< hq;= jk m˙† moaO;sh
ie,iqï lr we;akï jirl ld,hla moaO;sh l%h
s d ls¯fï iïNdú;dj fidhkak'
(ii) moaO;sh l%shd ls¯u i|yd wvq jYfhka tla fldgila l%shd l< hq;= jk m˙†
moaO;sh kej; ie,iqï lr we;akï jirl ld,hla moaO;sh l%shd ls¯fï
iïNdú;dj fidhkak'
(b) úNd.hl° .Ks;h i|yd iu;aùfï ,l+K 30 fjhs' úNd.h iu;a isiqkaf.a ,l+Kqj,
jHdma;h
s my; oelafjk m˙† fjhs'
,l+Kq m%dka;rh
30 – 40
40 – 50
50 – 60
60 – 70
70 – 80
80 – 90

ms˙ñ
5
10
15
30
5
5

isik
q a .Kk
.eyekq
15
20
30
20
5
0

(i) úNd.h iu;a
(α ) ms˙ñ <uhska 70 fokdf.a"
(β ) .eyekq <uhka 90 fokdf.a
,l=Kj
q , uOHkHh yd iïu; wm.ukh fidhkak'
(ii) wiu;a ms˙ñ <uhska 30 fokd o we;=Ωj ishÆ u ms˙ñ <uhskaf.a ,l=Kqj, uOHkHh
yd iïu; wm.ukh ms<sfj<ska 48 yd 21.5 fjhs' wiu;a .eyekq <uhska 10 fokd o
we;=Ωj ishÆ u .eyekq <uhska i|yd wkqrEm w.hka ms<f
s j<ska 50 yd 15 fjhs'
úNd.h iu;a fkdjQ
(α ) ms˙ñ <uhska 30 fokdf.a"
(β ) .eyekq <uhska 10 fokdf.a
,l=Kj
q , uOHkHh yd iïu; wm.ukh fidhkak'
(iii) (α ) úNd.h iu;a"
(β ) úNd.h iu;a fkdjQ
isik
q f
a .a ,l=Kj
q , uOHkHh yd iïu; wm.ukh wfmdaykh lrkak'

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

- 168 -

11 - Wiia .◊;h
m%Yak m;% jHQyh
˚ m;%h - ld,h ( meh 03hs'
			 fuu m%Yak m;%h fldgia follska iukaú; h'
			 A fldgi - m%Yak oyhls' m%Yak ish,a,g u ms<s;=re iemhsh hq;= h' tla
m%Yakhlg ,l=Kq 25 ne.ska ,l=Kq 250 ls'
			 B fldgi - m%Yak y;ls' m%Yak mylg ms<s;=re iemhsh hq;= h' tlaa
m%Yakhlg ,l=Kq 150 ne.ska ,l=Kq 750 ls'
			 ˚ m;%h i|yd uqΩ ,l=Kq 1000 fi 10 } 100
˚˚ m;%h - ld,h ( meh 03hs'
			 fuu m%Yak m;%h fldgia follska iukaú; h'
			 A fldgi - m%Yak oyhls' m%Yak ish,a,g u ms<s;=re iemhsh hq;= h' tla
m%Yakhlg ,l=Kq 25 ne.ska ,l=Kq 250 ls'
			 B fldgi - m%Yak y;ls' m%Yak mylg ms<s;=re iemhsh hq;= h' tla
m%Yakhlg ,l=Kq 150 ne.ska ,l=Kq 750 ls'
			 ˚˚ m;%h i|yd uqΩ ,l=Kq 1000 fi 10 } 100
wjidk ,l=K .Kkh ls¯u
			
				

(

˚ m;%h
˚˚ m;%h
wjidk ,l=K

}
}
}

100					
100
200 fi 2 } 100

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHQyh yd uQ,dlD;s m%Yak

- 169 -

Higher Mathematics I: Part - 30 12 2010

11 - Wiia .Ks;h
I m;%h
ie,lsh hq;h
= s(
*
*

A fldgfia ishÆ u m%Yk
a j,g ms<;
s r
= e imhkak'
B fldgfia m%Yk
a mylg muKla ms<;
s r
= e imhkak'
A fldgi

1. mdi,l jr®K m%odk W;aijhl° ms˙ñ <uhs k a 12 fokl= g " 10 fokl=g yd 15 fokl= g
ms<f
s j<ska meis mka≥" l%slÜ yd fydlS i|yd jr®K m%odk lrk ,°' ms˙ñ <uhska 12 fokl=g
fydlS i|yd muKla jr®K m%odk lr we;' l%slÜ yd fydlS hk folu i|yd ms˙ñ <uhska
lsisjl=g jr®K m%odk lr fkdue;' ms˙ñ <uhska fofokl=g l%slÜ yd meis mka≥ hk folu
i|yd jr®K m%odk lr we;' l%Svd ;=fkka wvq jYfhka tlla i|yd jr®K m%odk lr we;s uqΩ
ms˙ñ <uhska .Kk ksrK
® h lrkak'
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

2. R hkq IR × IR u; (a, b) R (c, d ) ⇔ a + d = b + c u.ska wr®: olajd we;s iïnkaOh hehs .ksu'q
R hkq IR × IR u; ;=,H;d iïnkaOhla nj fmkajkak'
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

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Higher Mathematics I: Part - 30 12 2010

x +3
1 
1 
3. f : IR −   → IR −   hkak f ( x) =
u.ska wr®: olajd we;ehs .ksu'q
2 x −1
2
2
f −1 mj;sk nj fmkajkak' f −1 fiùfuka f;drj" f −1 = f nj fmkajkak'
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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a2 b2 c2
a c b
+ 2 + 2 ≥ + + ≥ 3 nj idOkh lrkak'
2
c b a
b c a
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

4. a, b yd c Ok kï"

'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

- 171 -

Higher Mathematics I: Part - 30 12 2010

5. x ′ = Ax m˙Kdukfhka xy-;,fha msyá
s (x, y) ,CIHh (x ′, y ′) ,CIHhg wkqrEmKh flfr®;
2 1
 x
 x′ 
 , x =   yd x ′ =   fõ'
fuys A = 
 3 2
 y
 y′
by; m˙Kdukh u.ska
(i) x + y = 0 ir, fr®Ldj wkqrEmKh flfrk ir, fr®Ldfõ iólrKh"
(ii) y = mx ir, fr®Ldj th u;gu wkqrEmKh flfrk m˙† m ys w.hka
fidhkak'
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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(1+ 3i) 3
6.
= − i nj idOkh lrkak'
(1− i) 6
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

- 172 -

Higher Mathematics I: Part - 30 12 2010

 x3 − 8
,
x ≠ 2 kuz
,
kï"

 x−2
7. f ( x) = 
2,
x = 2 kuz
,
kï"


Y%;
s h x = 2 ys ° ika;;sl fkdjk nj fmkajkak'
s h ika;;sl ùu i|yd f (x) g wjYH fjkialï is≥ lrkak'
x = 2 ys° Y%;
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,
 x, x ≥ 1 kuz
kï"
8. f ( x) =  2
,
kï"
 x , x < 1 kuz
Y%;
s h x = 1 ys° wjl,H fkdjk nj idOkh lrkak'

'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

- 173 -

Higher Mathematics I: Part - 30 12 2010

9. φ (x) hkq x ys ishÆ ;d;a;ú
a l w.hka i|yd φ ( x) + φ (a − x) = b jk wdldrfha x ys wkql,H
Y%;
s hls; fuys a (> 0) yd b hkq ksh; fõ'
a

∫ φ( x)dx =
0

ab
nj fmkajkak'
2

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3a
yd r = 4a cos θ u.ska ms< f
s j<ska fokq ,nk Γ1 yd Γ2
2
π
jl% tlu rEm igykl w¢kak; fuys a > 0 yd 0 ≤ θ ≤ fõ.
2
3a ≤ 2r ≤ 8a cos θ u.ska fokq ,nk fmfoi rEm igyfkys w∫re lrkak'
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

10. O%ejl LKavdxl weiqfrka r =

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**
w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

- 174 -

Higher Mathematics I: Part B- 30 12 2010

B fldgi
11. (a) tla;rd mdi,l 12 jk fY%aKsfha mka;sj, isiqka 120 fokl= isák w;r thska 35 fofkla
.eyekq <uhska fj;s' YsIH kdhl kdhsldjka ishÆ fokd we;=Ωj isiqyq 65 fofkla
l%Svdj,g iyNd.s fj;s' .eyekq <uhskaf.ka 13 fokl= l%Svdj,g iyNd.s fkdjk w;r
.eyekq <uhs k a 7 fofkla YsIH kdhsldfjda fj;s' YsIH kdhl kdhsldjka fkdjk
isiqka 38 fofkla l%Svdj,g iyNd.s fj;s' fuu f;dr;=re fjka igykl ksrEmKh
lrkak'
(i) YsIH kdhlhska fkdjk kuq;a l%v
S dj,g iyNd.s jk"
(ii) l%v
S dj,g iyNd.s fkdjk"
(iii) YsIH kdhlhska jk
ms˙ñ <uhska .Kk fidhkak'
(b) A, B yd C hkq S ir®j;% l=,lhl Wm l=,l hehs .ksu'q
fhdod .kq ,nk l=,l ùcfha kshu ish,a, u meye†,s f,i m%ldY lrñka,
{( A ∪ B ) ∩ C } − ( A ∩ B ∩ C ) = {( A ∩ C ) − B } ∪ {( B ∩ C ) − A }
nj idOkh lrkak'

x 2 + 2ax + b
( Ax + B) 2
hehs
.ks
u
;
q
fuys
a,
b
∈
IR
yd
a
≠
0
fjhs
'
f(x)
–
k
hka
k
x 2 +1
x 2 +1
wdldrfhka ,súh yels m˙† k i|yd k1 yd k2 m%Nk
s k
a ;d;a;ú
a l w.h folla ;sfnk nj
fmkajkak; fuys A yd B hk tajd x j,ska iajdh;a; fjhs'
[(1− k r ) x + a] 2
(i) r = 1, 2 i|yd f ( x) − k =
,
r (1− k )( x 2 +1)
r
(ii) (1 – k1)(1 – k2) = − a2
nj fmkajkak'
x ys ishÆ ;d;a;aúl w.h i|yd" f (x) hkak k1 yd k2 w;r msyg
s k nj wfmdaykh
lrkak'

12. (a) f ( x) =

(b) (b − c)(b + c − a ) 3 + (c − a )(c + a − b) 3 + (a − b)(a + b − c) 3 hkak idOlj,g fjka
lrkak'
13. ´kEu Ok ksÅ,uh or®Ylhla i|yd o uqjdjr® m%fïhh fhdod .ksñka cos 5θ yd sin 5θ
i|yd m%ldYk cos θ yd sin θ weiqfrka ,nd.kak'
ta khsk"a tan 5θ i|yd m%ldYkhla tan θ weiqfrka ,nd.kak'
π 
tan 5θ = 0 iólrKh i,lñka" x 2 − 10 x + 5 = 0 iólrKfha uQ, tan 2   yd
5
 2π 
tan 2 
 nj idOkh lrkak'
 5 
π 
 2π 
ta khsk"a sec 4   + sec 4 
 = 112 nj fmkajkak'
5
 5 

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

- 175 -

Higher Mathematics I: Part B- 30 12 2010

14. (a) wrh a jk f.da , hla ;= < ;eìh yels" Wm˙u m˙udjla we;s iDcq jD;a;dldr
4 3
is,k
s v
a rhl Wi fidhkak' is,k
s v
a rfha Wm˙u m˙udj
π a 3 nj fmkajkak'
9
d 2 y dy
− − 2 y = sin x iólrKfha úi∫ula jk m˙† a yd b
(b) y = a cos x + b sin x hkak
dx 2 dx
ksh; fidhkak'
(c)

dy
+ 2 y = e x wjl, iólrKfha idOdrK úi∫u fidhkak'
dx

π

15. (a)

x

∫ 1 + sin x dx

w.hkak'

0

(b) f ( x) =

1− x2
2

hehs . ksuq; fuys θ hkak x f.ka iajdh;a; fõ' x2 moh

1 − 2 x cos θ + x
olajd f (x) ys uelaf,da¯ka m%idrKh 1 + 2 x cos θ + 2 x 2 cos 2θ nj fmkajkak'

16. (a) S ≡ y 2 − 4ax = 0 mrdj,h u; jQ m%Nk
s k
a ,CIH folla P(ap 2 , 2ap ) yd R(ar 2 , 2ar )
hehs .ksu'q PR fr®Ldfõ iólrKh ir,;u wdldrfhka fidhkak'
PR hkq S = 0 mrdj,fhys kdNSh cHdhla kï" p yd r u.ska ;Dma; úh hq;= wjYH;dj
,nd.kak'
QT o S = 0 mrdj,fhys kdNSh cHdhls' PQRT p;=ri%fha úreoaO me;s hq., tl tlla
S = 0 mrdj,fha kshdul wCIh u;° f√okh jk nj fmkajkak'
(b) b,smaihl kdNs w;r ≥r tall 8 la jk w;r" kshdul wCI w;r ≥r tall 18 ls '
m%Odk wCI wkqidrfhka b,smi
a fha iólrKh fidhkak'
17. (a) a, b yd c hkq ksh; o" a 2 + b 2 > c 2 o kï" a cos θ + b sin θ = c iólrKh ; Dma;
lrk 0 iy 2π w;r α yd β m%Nk
s k
a w.h folla mj;sk nj fmkajkak'
α − β
cos 2 
 2

2

c

nj idOkh lrkak'
= 2
 a + b2

(b) f (θ ) = cos 6 θ + sin 6 θ hehs .ksu;q fuys θ ;d;a;ú
a l fõ' A + B cos kθ wdldrfhka
f (θ ) m%ldY lrkak; fuys A, B yd k ksh; fõ'
ta khsk"a 4(cos 6 θ + sin 6 θ ) − 2 sin 4θ − 5 = 0 iólrKh úi|kak'
(c) tan −1 x + tan −1 2 x =

π
iólrKh úi|kak'
4

***

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

- 176 -

Higher Mathematics II: Part A

11 - Wiia .Ks;h
II m;%h
ie,lsh hq;h
= s(
*
*

A fldgfia ishÆ u m%Yk
a j,g ms<;
s r
= e imhkak'
B fldgfia m%Yk
a mylg muKla ms<;
s r
= e imhkak'
A fldgi

1. OAB ;s%fldaKfhys OA iy OB mdo u.ska ms<sfj<ska a iy b ffoYsl ksrEmKh flfr® hehs
.ksu'q
a = a yd b = b kï" w†Y .=K;
s h Ndú;fhka"
(i) c = ab + ba ffoYslh AOB fldaKfhys wNHka;r iuÉf√olhg iudka;r nj;a"
(ii) c yd ba – ab ,ïn nj;a
fmkajkak'
ta khsk"a AOB fldaKfhys ndysr iuÉf√olhg iudka;r jQ ffoYslhla ,shd olajkak'
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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2. Oxyz wCI moaO;sh wkqnoaOfhka" A, B iy C ,CIHj, LKavdxl ms<sfj<s k a (1, 0, 0),
(0, 2, 0) iy (0, 0, 3) fõ' úYd,;aj P iy 2 P jk n, ms< f
s j<ska A iy B ys° BA iy AC
†Yd Tiafia l%shdlrhs' moaO;sh O ys° R ;ks n,hlg iy >Qr®Kh G jk n, hq.auhlg ;=,H
fõ kï R iy G fidhkak'
R.G .Kkh lr" moaO;sh ;ks n,hlg W!kkh l< yels oehs ksrK
® h lrkak'
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

- 177 -

Higher Mathematics II: Part A

3. me;a;l †. a jk taldldr iup;=ri% ;yvqjla" øjhl iïmQr®Kfhka .s,ajd we;af;a tys
me;s folla ;sria jk m˙† yd tys ;,h isri iu. θ fldaKhla idok m˙† fõ' ;yvqfõ
flakaøh ksoyia øj mDIaGfha isg H .eUqrlska we;' ;yvqj u; l%shdlrk uqΩ øj f;rmqu
fidhkak'
mSvk flakøa fha msyg
s u
q ksrK
® h lrkak'
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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4. a iy b hkq ksh; ffoYsl hehs o" r(t) hkq" ld,h t jk úg" O wp, uQ, ,CIHhla
wkqnoaOfhka P ,CIHhl msyg
s ï
q ffoYslh hehs o .ksu'q
(i) a × b ys †Ydj ms<n
s |j a ys fyda b ys †Ydj weiqfrka l=ula lsj yels o@
d
2
( r − a ) = 2(r − a) . r& nj fmkajkak'
(ii)
dt
r& = b × (r − a) hehs ° we;akï" r − a ksh;hla nj wfmdaykh lrkak'
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

- 178 -

Higher Mathematics II: Part A

5. A yd B ;=kS jD;a;dldr ;eáfollg tlu ialkaOh yd tlu >klu we;' A yd B ys
>k;aj ms<sfj<ska ρ1 yd ρ 2 (> ρ1 ) fjhs' R1 yd R2 hkq ms< f
s j<ska A yd B ys wrhka
hehs .ksuq' I1 yd I 2 hkq ;eáj,g ,ïnj flakøa Tiafia jQ tajdfha wCI jgd ms< f
s j<ska
R
ρ
I
A yd B ys wjia:;
ß s >QrK
® hehs .ksu'q 2 weiqfrka 2 yd 2 fidhkak'
R1
ρ1
I1
I1 > I 2 nj wfmdaykh lrkak'
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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6. ia l ka O h M jk fudagr® r:hla" tys tka‚u Wm˙u cjfhka l%shd lrñka" iu;,d iDcq
udr®.hla Tiafia u i;; m%fõ.hlska .uka lrhs' ld,h t = 0 úg" tka‚u l%shdúrys; fldg
tu udr®.h Tiafiau .uka ls¯ug i,iajkq ,enqfõ kï" t ld,hl° tys m% f õ.h v,
u
v=
, (t ≥ 0) u.ska fokq ,nhs'; fuys α hkq ksh;hls'
1+α t
M, u iy α weiqfrka" t = 0 ° cjh fidhkak'
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

- 179 -

Higher Mathematics II: Part A

7. i;sfha †kl fm' j' 9.00 yd fm' j' 10.00 w;r;=r" idudkHfhka ≥rl:k mKsúv 6 la
wf<ú l<ukdlrejl=g ,efí' fm' j' 9.00 yd fm' j' 9.30 w;r;=r Tyqg y˙hgu ≥rl:k
mKsúv 2 la ,eîfï iïNdú;dj fidhkak'
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8. X ika;;sl iiïNdù úp,Hhg f ( x) = kx (2 − x), 0 ≤ x ≤ 2 iïNdú;d >k;aj Ys;
% h we;'
3
(i) k = nj fmkajkak'
4
(ii) P( X ≤ 1) fidhkak'
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w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

1

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Higher Mathematics II: Part A

9. X ~ U [2, 4] hehs .ksu'q Var( X ) =

1
nj fmkajkak'
3

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10. X ~ N (10, 64) hehss .ksu'q P( X < a ) = 0.95 jk m˙† a ys w.h fidhkak'
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**
w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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- 181 -

Higher Mathematics II: Part B

B fldgi
11. n, moaO;shla" O ,CIHfha° R ;ks n,hlg iy >Qr®Kh G jk n, hq.auhlg ;=,H fõ'
moaO;sh O′ ,CIHfha ° R ′ ;ks n,hlg iy > r
Q K
® h G ′ jk hq.u
a hlg ;=,H fõ kï
R ′ iy G ′ i|yd m%ldYk jHq;
q m
a kak lrkak'
moaO;sh ;ks n,hlg W!kkh ùu i|yd wksjdr®h wjYH;dj R . G = 0 nj fmkajkak'
msyg
s ï
q ffoYsl ck, ai yd bj iys; ,CIHj,° ms<f
s j<ska Li, Mj yd Nk n, ;=kla l%h
s dlrhs;
fuys a, b iy c hkq ksYaYk
= H ksh; jk w;r i, j yd k g iqmqre≥ wr®:h we;' O uQ,
,CIHfha° ;=,H" R ;ks n,h yd hq.u
a fha G >QrK
® h fidhkak'
(i) moaO;sh ;ks n,hlg W!Kkh l< yels o@ Tfí ms<;
s r
= ikd: lrkak'
(ii) moaO;sh lsishï ,CIHhl° R ′ kï ; ks n,hlg yd >QrK
® h λR ′ jQ n, hq.u
a hlg
2
2
2
;=,H fõ kï λ ( L + M + N ) = aMN + bNL + cLM nj fmkajkak'
12. wrh a j k jD;a ; dldr wdia ; rhla tys ;,h i s r i i u . θ fldaKhla idok m˙†
iïmQr®Kfhka ;r,hl .s,ajd we;' ;r,fha ksoyia mDIaGfha isg H .eUqrlska wdia;rfha
flakøa h msyg
s hs kï wdia;rh u; l%h
s d lrkq ,nk uqΩ ;r, f;rmqu fidhkak'
(i) ;, mDIaGh isri iu. θ fldaKhla idok m˙† yd ;,mDIaGhg my<ska jl% mDIaGh
msysgk m˙† wrh a jk taldldr >k wr®Of.da,hla øjhl iïmQr®Kfhka .s,ajd we;'
;, mDIaGfha O flakaøh ksoyia mDIaGfha isg H .eUqrlska msysgk úg jl% mDIaGh u;
iïm%hl
q ;
a øj f;rmqfuys ;srig wdk;sh ksrK
® h lrkak'
(ii) H wp,j θ úp,kh fjoa° jl% mDIaGh u; iïm%hl
q ;
a øj f;rmqfuys l%h
s d fr®Ldj
O Tiafia hk úg" θ ksrK
® h lrkak'
13. wp, isria nghla ;=< jQ i;; jdhq Odrdjla isriaj my<g u m%fõ.fhka yud hhs'
ngfhys jQ O ,CIHhl isg ialkaOh m jk lDñfhla V m%fõ.fhka isriaj by<g mkS'
mg .=re;ajc n,hg w;sfr®lj" lDñhdg idfmaCIj jdhq Odrdfõ m%fõ.h fuka mk jdrhla
jk jd; m%;f
s rdaOhla lDñhdg ú|Sug is≥fjhs; fuys k hkq Ok ksh;hls'
isriaj by<g iy isriaj my<g hk wjia:d i|yd lDñhdf.a p,s; iólrK ,shkak'
iq≥iq p,s; iólrK ie,lSfuka"
(i) lDñhdf.a m%fõ.h Y=kH ùug .;jk ld,h"
(ii) O ,CIHh fj; lDñhd <Ûdjk úg Wf.a m%fõ.h
fidhkak'
14. tl tll ialkaOh m jQ ir®jiu P1, P2 yd P3 wxY= ;=kla" iqmqre≥ wxlkfhka" F =

GmM

:

r2
.=re;ajc kshuhg wkql+,j tlsfklg wdlr®IKh lrhs' t ld,fha ° P1, P2 yd P3 wxY=
tajdfha ialkaO flakaøh jk G uQ, ,CIHh wkqnoaOfhka ms<sfj<ska msysgqï ffoYsl
r1, r2 yd r3 jk ,CIHj, we;ehs .ksu'q
(i) r1 + r2 + r3 = 0 nj fmkajkak'
(ii) P1 i|yd p,s; iólrKh ,shd olajkak'
(iii) wxY= hq., ;=ku w;r ksh; iy iudk d ≥˙ka msyg
s k m˙† moaO;sh p,kh fjhs kï"
3Gm
r&
r1 nj fmkajkak'
r1&
1 =
d3
(iv) r1 = a1 cos(ω t + ε ) hkq by; (iii) ys ° we;s p,s; iólrKfhys úh yels úi∫ula nj
fmkajd" ω ksrK
® h lrkak; fuys a1 hkq ksh; ffoYslhla o, ω yd ε hkq ksh; o fõ'

w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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Higher Mathematics II: Part B

15. ialkaOh m yd wrh a jk taldldr jD;a;dldr uq≥jl wjia:ß;s >Qr®Kh" uq≥fõ ;,hg
,ïnj tys flakøa h Tiafia hk wCIh jgd fidhkak'
uq≥fõ ;,hg ,ïnj m˙êh u; msysá ,CIHhla Tiafia hk wCIhla jgd uq≥fõ wjia:ß;s
>QrK
® h wfmdaykh lrkak'
iajlSh ;,h isria jk m˙†" ia l ka O h m yd wrh a jk taldldr jD;a;dldr uq≥jla
,siaiSulska f;drj rΩ ;sria ;,hla u; fmrf<hs' tys m%fõ.h V jk úg ;sria ;,fha isg
a
Wilska msyá
s A iqug kdoe;a;la iu. uq≥j .efghs' .egqug fudfyd;lg fmr, A jgd
5
uq≥fõ fldaKl
s m%fõ.h ksrK
® h lrkak'
uq≥j CIKslj kdoe;af;ka bj;a fjhs kï 81V 2 ≥ 80 ga nj fmkajkak'
uq≥j kdoe;a; miqlr hhs kï, V g .; yels w.hka fidhkak'
16. iïNdú;d ialkaO Y%s;h P( X = x) = p (1 − p ) x −1 , x = 0, 1, 2, ... , 0 < p < 1 jQ .=fKda;;
a r
jHdma;shla, t†fkod „ú;fha wjia:djla i|yd fhdod.; yelafla l=uk fldkafoais hgf;a
oehs m%ldY lrkak'
E ( X ) fidhkak'
∞

[ ∑ xr x =
x =1

r
(1 − r ) 2

m%;M
s ,h Tng Ndú; l< yelsh.]

1
ls' hï
5
;eke;a;l=" Ndú;fha fkdfh° we;s ≥rl:k l=áhla fidhd .ekSug fmr" W;aidy l< hq;=
≥rl:k l=á .Kk X hehs .ksu'q
(i) X ys iïNdú;d ialkaO Y%;
s h ,shd olajkak'
(ii) E ( X ) fidhkak'
(iii) ≥rl:k weu;=ula ,nd .ekSug n,dfmdfrd;a;= jk hï ;eke;a;l=g Ndú;fha fh°
fkdue;s ≥rl:k l=áhla fidhd .ekSug yelsjkafka yhjk W;aidyfha° ùfï
iïNdú;dj fidhkak'
5
(iv) X ys úp,;dj
hehs ° we;s úg" Y = 5 X − 4 jk, Y ys uOHkHh iy úp,;dj
16
fidhkak'

tla;rd k.rhl ´kEu ≥rl:k l=áhla Ndú;fha fh° ;sîfï iïNdú;dj

17. .=jka f;dgqfmd<lska iiïNdùj f;dard.kq ,nk .=jka hdkhla" meh x g jvd jeä ld,hla
1
( x − 10) 2 , 0 ≤ x ≤ 10 fõ'
m%udo ùfï iïNdú;dj
100
lssisu .=jka hdkhla" kshñ; ld,hg l,ska .=jka.; fkdjk nj;a" meh 10 g jvd jeä
ld,hla m%udo ùu is≥fkdjk nj;a Wml,amkh lruq' iiïNdù f,i f;dard.kq ,nk
.=jka hdkhla m%udo jk meh .Kk X u.ska oelafõ hehs .ksu'q
(i) iuqÉÑ; jHdma;s Y%;
s h fidhd, ta khsk,a X ys uOHia:h fidhkak'
10
(ii) X ys >k;aj Y%;
s h fidhd E ( X ) =
nj fmkajkak'
3
(iii) iiïNdùj f;dard.kq ,nk .=jka hdkhla meh 2 l yd meh 3 l w;r ld,hla m%udo
ùfï iïNdú;dj wd.Kkh lrkak'
tjeks .=jka hdkd 3 l iiïNdù kshe†hla .kq ,efí' tu hdkd 3 ka hg;a ms˙fihska tla
hdkhlaj;a meh 2 l yd 3 l w;r ld,hla m%udo ùfï iïNdú;dj fidhkak'

***
w'fmd'i'^Wiia fm<& úNd.h - 2011 iy bkamiqj meje;afjk úNd. i|yd m%Yak m;% jHqyh yd uQ,dlD;s m%Yak

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