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- 145 -
Ajith Wimanga Wijesinghe
07
07
- 146 -
Ajith Wimanga Wijesinghe
1 + (1 + ) + (1 + ) + … … … … … . (1 + ) m%idrKfha ys ix.=Klh ( )!( )!( )! nj
fmkajkak'
− ( − 3) + = 0 j¾. iólrKfha uQ, ;d;aúl ùu i|yd ∈ (−∞, 1) ∪ (9, ∞) nj o
uQ, 2 u Ok ùu i|yd ∈ [9, ∞) nj o fmkajkak' wd.kaâ igyk u; olajd we;s j¾.M,h fokq ,nk mrdih fidhkak'
log (2 + 1) < 1 wkqrEm l=vdu Ok ksÅ,h fidhkak'
mrdñ;shla úg" = 2 cos , = sin uÕska fokq ,nk jl%fha = g wkqrEm ,CIHfha §
we¢ wNs,ïNh jl%h kej; = wkqrEm ,CIHfha § yuqfõ'
2 sin − 8 cos + 3√2 = 0 nj fmkajkak'
+ − 1 = 0 yd + − 6 + 8 + 9 = 0 jD;a; iam¾Y jk ,CIHfha LKAvdxl fidhkak'
= jk ;%sfldaKfha (1 , 3 ) yd (−2, 7) fõ' ,CIHfha LKAvdxl 6 − 8 + 43 = 0 ;Dma; lrk nj fmkajkak'
w.hkak'
(2) = 4 yd (2) = 1 úg" [ hkq ys wjl,Hhhs] → 2 ( ) ( ) = 2 nj fmkajkak'
= cos sin , = asin cos kï , ksÅ, úg ( ) hkak f.ka iajdh;a; úg 4 = 5 nj fmkajkak'
1.
1
07
2.
3.
4.
5.
6.
7.
8.
9.
10.
AB −2 0 4
C (1 + 3√3 )
11. (i)
(ii)
(iii)
(a)
(b)
(c)
(a)
(b)
11.
12.
13.
yd g
w.h .; o fmkajka
yd fidhkak' fidhkak' |5 − 3 | ≥ , , kï
m%Yak 12 k
(i) tla
(ii) ´kE
(iii) tla
m%Yak 06la3( + )wdldrfhk
i|yd 3 +
wNHqykfh= (1fidhkak'
wd.kaâ i
lrhs' y
iu
ixlS¾K ifidhkak' = 12 +(i) =
m%;súreoaO
yels nj o ak' + +tfiau , ,tkhska >kc≥ 2 − 3 jk
ï fldgia ;=k
ka m%Yak yhl
tla fldgfi
Eu fldgilsk
tla fldgisk
la f;dard.; y
m%ldYkfha
ka m%ldY l<+ 2 +. …
hka idOkh + √3) hhs
igyk u;
yd yryd h
udka;rdi%hla
ixLHdj +5 ys j¾. uQ00 0
,l=Kq we;s
yd g i= 0 ys uQ,, + hkq c iólrK ú
k mßÈ jQ y
klska iukaú
lg ms<s;=re i
ia m<uq m%Yak
ka m%Yak ;=k
ka tla m%Yakh
yels wdldr
yd i|
< yels nj f
hkak 7 ka … … … … … … .lrkak'
.ksuq' 2 uQ, ,CIH
hk f¾Ldj u
la jk fia + ldàiSh
uQ, fol + iy =
- 147 -
;s kï ys ;
iudk ,l=Kq
, fõ' fuys 36 − 12úi|kak'
ys w.h mrd
ú; m%Yak m;%
iemhsh hq;=h
kh wksjd¾h
klg jvd jeä
hlgj;a ms<s
fidhkak'
|yd iqÿiq w
fmkajkak' f
fnfok nj . . + ( ), ixlS¾
Hh o hkak
u; ksrEmK
,CIHh f;
h wdldrfhk+ wdldrf1 −00 0
;d;aúl w.
we;akï ±2√ yd ksh− 11 + 2
dih fidhkak
;%hl tla fl
h'
kï"
äfhka Tyqg
<s;=re iemhSu
w.hka f;dar
uys yd
fmkajkak'
=
¾K ixLHd
k 2 ' h
Kh lrkq ,
;dardf.k we
ka ks¾Kh
hka fidhkak−− hhs .
Aj
.h i|yd √ lsisu w
h; fõ' ,2 = 0 iólr
k'
ldgil m%Yak
g ms<s;=re iem
u wksjd¾h k
rd .ksñka 3Ok ksÅ, f
− ( )
tl tll
hkak ixl
,nk ,CIHh
e;' u.ska
lrkak'
k'
.ksuq'
Ajith Wimang
+ g ish
w.hla .; f
weiqßka
rKfha uQ,
k 04 ne.ska w
mhsh fkdye
kï" 3 hkak
fõ' tkhska O
− ( )udmdxlh y
lS¾K ixLH
h msysgkafkao
ka ksrEmKh
ys úl
ga Wijesingh
h¿ ;d;aúl
fkdyels nj+ yd
kï yd
wvx.=h' tu
els kï"
7 + 3(2 )Ok ksÅ,uh
nj .Ks;
yd úia;drh
Hd ksrEmKh
o@
lrkq ,nk
l¾Kj, È.
1
07
he
l
j
u
h
;
h
h
k
.
- 148 -
Ajith Wimanga Wijesinghe
fuys , yd hkq ≠ 1 jk mßÈ jQ ;d;aúl ksh; fõ' = = nj
fmkajkak' hkq 3 jk >kfha talc kHdihhs' tkhska" = jk mßÈ kHdihla fidhkak'
(ii) = kHdih = imqrd,hs kï" = 0 kï = 0 nj fmkajkak'
(a) ( +1) 1+ 2 = nj fmkajkak'
fuys = tan fõ'
sin = (sin + cos ) + (cos − sin ) jk mßÈ yd fidhd wkql,kh w.hkak'
(b) i. ( ) w.hkak'
ii. ln(sec + tan ) = sec nj fmkajd
we.hSug th Ndú;d lrkak'
( )√ we.hSï i|yd = √
wdfoaYh fhdokak'
(a) = (1 + sin ) sin−1 kï" (1 − ) − hkak flfrka iajdh;a; nj
fmkajkak' tkhska = 1, 2, 3, 4 i|yd = 0 § fidhkak'
(b) ( ) = , , ;d;aúl ksh; fõ' (−2, −1) ,CIHh ( ) ys yereï ,CIHhla kï ,
fidhkak' m<uq jHq;amkakh ie,lSfuka = ( ) ys m%ia:drh w¢kak' ( ) = iólrKhg
m%Nskak ;d;aúl uQ, mej;Sug mej;sh hq;= w.h o fidhkak'
(c) ksh;hla o ≠ 0, cos ≠ 0 o úg" = (cos + sin ), = (sin − cos ) kï
ys Y%s; f,i yd fidhkak'
(a) + + = 0 ir, f¾Ldj u; ( ) ,CIHfha m%;sìïNfha LKavdxl fidhkak'
;%sfldaKhl , , YS¾I msysgd we;af;a ms<sfj,ska = , = 2 , = 3 f¾Ld u;h
,ïN iuÉfþolfha iólrKh 3 + − 18 = 0 fõ' f¾Ldj + = 0 ir,
f¾Ldjg iudka;rh' ;%sfldaKfha mdoj, iólrK ,nd.kak'
(b) + = 25 jd;a;fha;a − + 1 = 0 f¾Ldfõ;a fþμ ,CIHh yryd , jD;a; folla
we| we;af;a , jD;a; folu + = 25 f¾Ldj iam¾Y lrk mßÈh' yd ys ixlrk fidhkak'
yd ys fmdÿ iam¾Yl fþokh fkdjk nj o fmkajkak'
(a) = tan + cot − f,i .ksuq' 1 + = 2( − 1) sin 2 nj idOkh lrkak'
tkhska ys ´kEu ;d;aúl w.hla i|yd tan + cot − m%ldYh 1/3 yd 3 w;r
w.hla fkd.kakd nj fmkajkak' (b) i|yd úi|kak' 4 − 4(cos sin ) − sin 2 = 0
cos − sin = 6⁄ ' fuys 0 ≤ cos ≤ yd − ≤ sin ≤ '
(c) ;%sfldaKfha , fldaKhkaf.a ihsk − ( + ) + = 0 iólrKfha uQ, fõ' cos = sin nj fmkajkak'
∫
∫ ∫ 2
-1
∫ 1
0 ∫
0 14.
15.
16.
17.
- 149 -
Ajith Wimanga Wijesinghe
1 + (1 + ) + (1 + ) + … … … … … . (1 + ) m%idrKfha ys ix.=Klh ( )!( )!( )!
nj fmkajkak'
1 + (1 + )(1 + ) +. … … + (1 + ) = ( ) ( ) → .=fKda;a;r fY%aKshla f,i
.;aúg'
= [(1 + ) − 1] = +
+ 1∑= 0 − 1]
= + 1∑= 1
= + 1 úg ys ix.=Klh = =
( )!( )!( )! − ( − 3) + = 0 j¾. iólrKfha uQ, ;d;aúl ùu i|yd ∈ (−∞, 1) ∪ (9, ∞) nj
o uQ, 2 u Ok ùu i|yd ∈ [9, ∞) nj o fmkajkak'
− ( − 3) + = 0
∆ = ( − 3) − 4 = − 6 + 9 − 4 = ( − 1)( − 9) wd.kaâ igyk u; olajd we;s j¾.M,h fokq ,nk mrdih fidhkak'
= 6
1 + 3√3 = −2 + 3 + 3√3
= −2 + 6 + √
= −2 + 6 cos + sin
ixhqla; .Ks;h - I Combined Mathematics - I úi÷ï - 07 A fldgi
1.
+ + − +1 9
A B−2 0 4
uQ, ;d;aúl ùu i|yd" ∆ ≥ 0 úh hq;=h'
< 1 fyda > 9 úh hq;=h'
uQ, ;d;aúl ùug" (−∞, 1) ∪ (9, ∞) úh hq;=h'
uQ, fol yd kï" + = − 3
=
uQ, folu Ok ùu i|yd ∆ ≥ 0
− 3 > 0
> 0
th > 0, > 3, ≥ 9 úh hq;=h'
uQ, fol u ;d;aúl ùug ∈ [ 9, ∞)
2.
C (1 + 3√3 )
3.
5
10
5
5
5
5
5
5
5
5
5
- 150 -
Ajith Wimanga Wijesinghe
=
§ we;s j¾,M,h" | + 2| ≥ 6 yd 0 < | | ≤ uÕska ±lafõ'
log (2 + 1) < 1 wkqrEm l=vdu Ok ksÅ,h fidhkak'
log (2 + 1) < 1
≠ 0 yd ≥ −2
;jo" 2 + <
− − 2 > 0
( − 2)( + 1) > 0
< −1 fyda > 2
wiudk;djg wkqrEm l=vd u ksÅ,h = 3
mrdñ;shla úg" = 2 cos , = sin uÕska fokq ,nk jl%fha = g wkqrEm ,CIHfha §
we¢ wNs,ïNh jl%h kej; = wkqrEm ,CIHfha § yuqfõ'
2 sin − 8 cos + 3√2 = 0 nj fmkajkak'
= 2 cos
= sin
+ = 1 jl%h b,smaihls'
+2 = 0
= = −
= úg" = √2 , = √ , =
= úg
wNs,ïNfha iólrKh" √√ = 2
√2 − 1 = 2√2 − 4
(2 cos , sin ) fuu f¾Ldj u; ksid"
√2 sin − 1 = 4√2 cos − 4
√2 sin − 4√2 cos = −3
2 sin − 8 cos = −3√2
2 sin − 8 cos + 3√2 = 0
= úg" 2 sin − 8 cos + 3√2 = 0
+ + − −1 2
4.
5.
5
10
5
5
5
5
5
5
5
5
5
5
- 151 -
Ajith Wimanga Wijesinghe
+ − 1 = 0 yd + − 6 + 8 + 9 = 0 jD;a; iam¾Y jk ,CIHfha LKAvdxl fidhkak'
≡ + = 1 ≡ (0, 0) = 1 ≡ + − 6 + 8 + 9 = 0 ≡ ( − 3) + ( + 4) = 4 ≡ (3, −4) = 4 = √4 + 3 = 5 = 5 = 4 + 1 = + jD;a; fol ndysrj iam¾Y fõ'
iam¾Y ,CIHh = ,
= jk ;%sfldaKfha (1 , 3 ) yd (−2, 7) fõ' ,CIHfha LKAvdxl 6 − 8 + 43 = 0 ;Dma; lrk nj fmkajkak'
= ksid
( − 1) + ( − 3) = ( + 2) + ( − 7)
10 = −2 − 6 = 4 − 14 + 53
6 − 8 + 43 = 0
w.hkak'
=
= 2
= 2
(3, −4) (0, 0)= 41
( , )
(1, 3) (−2, 7)
6.
7.
8.
5
5
5
5
5
5
10
5
5
5
5
- 152 -
Ajith Wimanga Wijesinghe
= 2 cot − 3 + cot
= 2 ln sin − 3 + 2 ln|sin | + − wNsu; ,CIHhls'
(2) = 4 yd (2) = 1 úg" [ hkq ys wjl,Hhhs ] → 2 ( ) ( ) = 2 nj
fmkajkak'
→ 2 ( ) ( ) = → 2 ( ) ( ) ( ) ( )
= → 2 ( ) ( ) −
( ( ) ( ))
= → 2 (2) −2 → 2 ( ) ( )
= 4 − 2 (2) = 4 − 2 × 1 = 2
= cos sin , = asin cos kï , ksÅ, úg ( )( ) hkak f.ka
iajdh;a; úg 4 = 5 nj fmkajkak'
+ = cos sin + sin cos
= cos sin [cos + sin ] = cos sin
, = (cos sin )(sin cos ) = cos sin
( )( ) =
( )( )
= ( )( )
f.ka iajdhla; ùug" 4 = 5
9.
10.
5
5
5
5
5
5
5
5
5
5
10
- 153 -
Ajith Wimanga Wijesinghe
(i) yd g m%;súreoaO ,l=Kq we;s kï ys ;d;aúl w.h i|yd + g ish¿ ;d;aúl
w.h .; yels nj o yd g iudk ,l=Kq we;akï ±2√ lsisu w.hla .; fkdyels nj
o fmkajkak'
(ii) yd + + = 0 ys uQ, fõ' fuys yd ksh; fõ' , weiqßka + yd
fidhkak' tfiau , , + hkq 36 − 12 − 11 + 2 = 0 iólrKfha uQ, kï yd
fidhkak' tkhska >kc iólrK úi|kak'
(iii) |5 − 3 | ≥ 2 − 3 jk mßÈ jQ ys w.h mrdih fidhkak'
(i) = +
= +
− + = 0
ys ;d;aúl w.h i|yd
∆= − 4
yd ,l=Kq wiudk kï" < 0
−4 > 0
− 4 > 0
∆ > 0
tkï ys ;d;aúl w.h i|yd g ish¿ ;d;aúl w.h .; yel'
yd ys ,l=kq iudk kï"
> 0
túg" ∆= − 4
= − 2√ ( − 2√ )
−2√ < < 2√ úg ∆< 0 fõ'
tkï ;d;aúl w.h g −2√ yd 2√ w;r w.hla .; fkdyel'
(ii) + + = 0 ys uQ, yd kï"
+ = −
=
36 − 12 − 11 + 2 = 0 ys
uQ, , yd ( + ) kï"
+ + + =
2( + ) = → −2 =
= −
( + ) =
ixhqla; .Ks;h - II Combined Mathematics - II úi÷ï - 07 B fldgi
11.
+ +− −2√ 2√
5
5 5
5
5
5
5
5
5
5
5
5
10
10
- 154 -
Ajith Wimanga Wijesinghe
(− ) =
= → =
+ = −
=
− + = 0
6 − + 2 = 0
= ± ( )( )×
= ± √
= √
, = √
+ =
§ we;s iólrKfha uQ, jkqfha"
, √ , √
h'
(iii) = 2 − 3
= |5 − 3 | = 5 − 3 ≤ 5 3⁄−5 + 3 >
ish¿ i|yd |5 − 3 | > 2 − 3 fõ'
tfukau ish¿ ;d;aúl i|yd |5 − 3 | > 2 − 5 fõ'
−
5
5
5
5
5
5
5
10
20
10
- 155 -
Ajith Wimanga Wijesinghe
(a) , , kï fldgia ;=klska iukaú; m%Yak m;%hl tla fldgil m%Yak 04 ne.ska wvx.=h' tu
m%Yak 12 ka m%Yak yhlg ms<s;=re iemhsh hq;=h'
(i) tla tla fldgfia m<uq m%Yakh wksjd¾h kï"
(ii) ´kEu fldgilska m%Yak ;=klg jvd jeäfhka Tyqg ms<s;=re iemhsh fkdyels kï"
(iii) tla tla fldgiska tla m%Yakhlgj;a ms<s;=re iemhSu wksjd¾h kï"
m%Yak 06la f;dard.; yels wdldr fidhkak'
(b) 3( + ) m%ldYkfha yd i|yd iqÿiq w.hka f;dard .ksñka 3 hkak 7 + 3(2 ) wdldrfhka m%ldY l< yels nj fmkajkak' fuys yd Ok ksÅ, fõ' tkhska Ok ksÅ,uh
i|yd 3 + 2 hkak 7 ka fnfok nj fmkajkak'
(c) + +. … … … … … … . . + ( ) = − ( ) − ( ) nj .Ks;
wNHqykfhka idOkh lrkak'
(a) (i)
4 4 4 − 12 m<uq m%Yakh wksjd¾h úg m%Yak 06 f;dard.; yels wdldr'
m%Yak f;dard .; yels wdldr m%Yak f;dard .;
yels fldgia
fldgilska m%Yak
f;dard .ekSu
´kEu fldgilska m%Yak 03 la
03
01
03
tla fldgilska m%Yak 02 la
fjk;a fldgilska m%Yak 01 la = 6 = 9 54
tla fldgilska tla m%Yakhla
ne.ska
01 = 27 27
tla tla fldgiska m<uq m%Yakh wksjd¾h úg m%Yak f;dard .ekSï = 3 + 54 + 27 = 84
(ii) ´kEu fldgilska m%Yak 03 lg jvd jeäfhka f;dard .; fkdyels kï"
m%Yak f;dard .; yels
wdldr
m%Yak f;dard .;
yels fldgia
fldgilska m%Yak f;dard
.ekSu
m%Yak 03la fldgia 02
lska = 3 = 16 48
tla fldgilska m%Yak 03
la o ;j;a fldgilska
m%Yak 02 la o wjika
fldgiska m%Yak 01 la
3! = 6 = 96 576
tla fldgilska m%Yak 02
ne.ska
01 = 216 216
´kEu fldgilska m%Yak 03 jvd jeäfhka f;dard .; fkdyels kï l< yels f;dard
.ekSï = 48 + 576 + 216 = 840
12.
5
5
5
5
5
5
5
5
- 156 -
Ajith Wimanga Wijesinghe
(iii) tla fldgilska tla m%Yakhla j;a wksjd¾h kï"
m%Yak f;dard .; yels
wdldr
m%Yak f;dard .;
yels fldgia
fldgilska m%Yak f;dard
.ekSu
tla fldgilska m%Yak 4
la o wfkla fldgia j,ska
01 m%Yakh ne.ska
= 3 = 16 48
tla fldgilska m%Yak 03
la o ;j;a fldgilska
m%Yak 02 la o wjika
fldgiska m%Yak 01 la
3! = 6 = 96 576
tla fldgilska m%Yak 02
ne.ska
01 = 216 216
tla fldgilska tla m%Yakhla j;a wksjd¾h kï"= 48 + 576 + 216 = 840
(b) 3 = 3(3 ) = 3(9) = 3( + ) f,i .ksuq' = 7 yd = 2 f,i .;a úg"
3 = 3(2 + 7)
= 3(2 ) + 3 7 2
= 3 7 2 ∈ f,i .ksuq' túg"
3 = 3(2 ) + 7
3 + 2 = 7 + 32 + 2
= 7 + 32 + 42
= 7 + 2 (3 + 4) = 7 + 7 ∙ 2
= 7( + 2 ) tkï" 3 + 2 hkak 7 ka fnfoa'
(c) + +. … … … … … … . . + ( ) = − ( ) − ( ) = 1 úg j'me' = =
o'me' = − − = − =
= 1 úg m%;sM,h i;H fõ'
= úg m%;sM,h i;H hhs .ksuq'
+ +. … … … … … … . . + ( ) = − ( ) − ( ) fomigu ( ) tl;= lruq'
∑ − 1
= 0 ∑ − 1
= 0
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
- 157 -
Ajith Wimanga Wijesinghe
+ ( ) + 1( +2)2−1 = − ( ) − ( ) + ( ) = − ( ) − ( ) + ( )( ) = − ( ) − ( ) + ( ) + ( ) = − ( ) − ( ) tkï" = + 1 úgo m%;sM,h i;H fõ' tneúka ish¿ Ok mQ¾K i|yd"
+ +. … … … … … … . . + ( ) = − ( ) − ( ) fõ'
(a) = (1 + √3) hhs .ksuq' 2 , ixlS¾K ixLHd tl tll udmdxlh yd úia;drh
fidhkak'
wd.kaâ igyk u; uQ, ,CIHh o hkak 2 ' hkak ixlS¾K ixLHd ksrEmKh
lrhs' yd yryd hk f¾Ldj u; ksrEmKh lrkq ,nk ,CIHh msysgkafkao@
iudka;rdi%hla jk fia ,CIHh f;dardf.k we;' u.ska ksrEmKh lrkq ,nk
ixlS¾K ixLHdj + ldàiSh wdldrfhka ks¾Kh lrkak' ys úl¾Kj, È.
fidhkak'
= 12 + 5 ys j¾. uQ, fol + wdldrfhka fidhkak'
(b) (i) = 00 0 iy = 1 − −0 −0 0 hhs .ksuq'
fuys , yd hkq ≠ 1 jk mßÈ jQ ;d;aúl ksh; fõ' = = nj
fmkajkak' hkq 3 jk >kfha tall kHdihls' tkhska" = jk mßÈ kHdihla
fidhkak'
(ii) = kHdih = imqrd,hs kï" = 0 kï = 0 nj fmkajkak'
(a) = (1 + √3)
= + √
= cos + sin
| | = 1, =
= cos + sin
= cos − sin + 2sin cos
= cos + sin
2 = 2 cos + sin
|2 | = 2
(2 ) = 23
= cos + sin
13.
2
10
5
5
5
5
5
5
5
5
10
10
- 158 -
Ajith Wimanga Wijesinghe
= 3 cos − sin = 3 cos + sin
= 3, =
1' = − + −
= 2 −
=
, , tal f¾Çh fõ'
ys ixlS¾K ixLHdj ≡ 2 + 3⁄
= 2 cos − sin +3 cos + sin
= 5 cos 23 − sin = −
√
= 2 − = + √
= + = = √7
= 2 2 − 32
= −1 + √3 + 1 + √3
= + √ = +
= = √19
ys j¾. uQ, + kï"
( + ) =
( − ) + 2 = 12 + 5
− = 12
2 = 5
− = 12
4 − 25 = 48 4 − 48 − 25 = 0
(2 + 1)(2 − 25) = 0
2 + 1 ≠ 0 ksid 2 − 25 = 0
=
= ± 5 √2
30
(2 ) /32 /3
/6 0/3/3 /3
5
20
5
55
5
5
5
5
10
5
- 159 -
Ajith Wimanga Wijesinghe
túg = ± √ ∙
= ±1/√2
ys j¾. uQ, → √ + √ fyda √ − √
(b) (i) = 100 0 = 1 − −0 −0 0
= 100 0 1 − −0 −0 0
= 1 0 00 1 00 0 9
= 1 − −0 −0 0 100 0
= 1 0 00 1 00 0 1
= =
( ) =
=
= 1 0 00 1 00 0 1
=
= = 1 0 0− 0− −
(ii) =
=
=
= + ++ +
= = 1 00 1
+ = 1 1
+ = 0
= 0 kï = 1
= 0
+ = 0
= 0
(a) ( +1) 1+ 2 = nj fmkajkak'
fuys = tan fõ'
sin = (sin + cos ) + (cos − sin ) jk mßÈ yd fidhd wkql,kh w.hkak'
∫ 1
0 ∫
0 14.
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- 160 -
Ajith Wimanga Wijesinghe
(b) ( ) w.hkak'
ln(sec + tan ) = sec nj fmkajd
we.hSug th Ndú;d lrkak'
( )√ we.hSï i|yd = √
wdfoaYh fhdokak'
(a) = tan úg"
= tan → = sec
= 0 úg = 0
= 1 úg = 4
( )( ) = ( )( ) sec
= ( )
sin = (sin + cos ) + (cos − sin ) sin ys ix.=' → 1 = − 1
cos ys ix.=' → 0 = + 2
2 = 1 → = 1/2
= −1/2
sin = (sin + cos ) − (cos − sin ) = −
= −
= 2 − ln|sin + cos | = − ln √2
(ln ) = (ln )
= (ln ) − (ln )
= (ln ) −
= (ln ) − ln
= (ln ) − ln ( )
∫
∫ ∫ 2
-1
∫ 1
0 ∫0 ∫
0
∫
0 ∫
0 ∫
0
∫ ∫ ∫∫∫∫
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- 161 -
Ajith Wimanga Wijesinghe
= (ln ) − ln + (ln ) = (ln ) − ln +
= (ln ) − ln + +
[ln(sec + tan )] = × sec tan + sec
= [ ]
= sec
= sec ''''' = sec tan
= √
= sec
= [ln(sec + tan )]
= ln(sec + tan ) +
(b) = √
= ( )( )/√ √( )
= ( )( ) ( )( ) √
= ( ) √ 1
= ( )
− 1 = ( ) ( )
= ( )
= ( )( )
− 1 = ( + 1)/( + 2) 1 ka" =
( )( ) ( )√
= ( )√
( )√ =
= log [sec + tan ] fuys sec = 1 o sec = fõ'
∫∫ ∫ ∫ ∫
∫ 2
-1 ∫2
-1
∫ 5
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- 162 -
Ajith Wimanga Wijesinghe
(a) = (1 + sin ) sin−1 kï" (1 − ) − hkak flfrka iajdh;a; nj
fmkajkak' tkhska = 1, 2, 3, 4 i|yd = 0 § fidhkak'
(b) ( ) = , , ;d;aúl ksh; fõ' (−2, −1) ,CIHh ( ) ys yereï ,CIHhla kï ,
fidhkak' m<uq jHq;amkakh ie,lSfuka = ( ) ys m%ia:drh w¢kak' ( ) = iólrKhg
m%Nskak ;d;aúl uQ, mej;Sug mej;sh hq;= w.h o fidhkak'
(c) ksh;hla o ≠ 0, cos ≠ 0 o úg" = (cos + sin ), = (sin − cos ) kï
ys Y%s; f,i yd fidhkak'
(a) = (1 + sin ) sin kï"
= (1 + sin ) √ + sin 11− 2
= (1 + sin ) √ + √
= (1 + 2 sin ) √
√1 − = 1 + 2 sin 1
kej; úIfhka wjl,kfhka"
√1 − − √ = √
(1 − ) − = 2 2
tkï" (1 − ) − hkak flfrka iajdhla; fõ'
kej; úIfhka wjl,kfhka"
(1 − ) −2 − − = 0
(1 − ) −3 − = 0 3
kej; úIfhka wjl,kfhka"
(1 − ) −2 −3 −3 − = 0
(1 − ) −5 −4 = 0 4 ka"
= 0 úg 1 ka = 1
= 0 úg 2 ka = 2
= 0 úg 3 ka = = 1
= 0 úg 4 ka = 4 = 4 × 2 = 8
15.
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- 163 -
Ajith Wimanga Wijesinghe
(b) ( ) =
( ) = ( ) ( )( )
= −2 úg (−2) = 0
( )
= 0 → −3 + 4 = 0 1
= −2 úg (−2) = −1
= −1
−2 + = −5 2
5 = 20
= 4
= 3
( ) =
( ) = ( ) ( )( )
= ( )
= ( )( )
= ( )( )( )
( ) = 0 úg (2 + )(1 − 2 ) fõ'
= −2 fyda = 1/2
< 2 −2 < < 1/2 1/2 <
( ) − + −
= −2 § Y%s;h wjuhls' wjuh ≡ −1 ⇒ (−2, −1) = 1/2 § Y%s;h Wmßuhls' Wmßuh ≡ 4 ⇒ (1/2, 4) = 0 úg (0) = 3 ⇒ (0, 3) = 0 úg 4 + 3 = 0 → = −3/4 ⇒ (−3/4,0) → ±∞ úg → 0
−
−
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- 164 -
Ajith Wimanga Wijesinghe
( ) = i,luq'
= ( ) o = o úg
( ) = g m%Nskak úi÷ï i|yd yd fþok ,CI úh hq;=hs' ta i|yd"
−1 ≤ ≤ 4 úh hq;=hs' (c) = (cos + sin )
= ( cos + sin − sin ) = cos = (sin − cos )
= (cos − cos + sin ) = sin
= ×
= sin × = tan
=
= ×
= tan ×
= sec ×
=
(a) + + = 0 ir, f¾Ldj u; ( ) ,CIHfha m%;sìïNfha LKavdxl fidhkak'
;%sfldaKhl , , YS¾I msysgd we;af;a ms<sfj,ska = , = 2 , = 3 f¾Ld u;h
,ïN iuÉfþolfha iólrKh 3 + − 18 = 0 fõ' f¾Ldj + = 0 ir,
f¾Ldjg iudka;rh' ;%sfldaKfha mdoj, iólrK ,nd.kak'
(b) + = 25 jd;a;fha;a − + 1 = 0 f¾Ldfõ;a fþμ ,CIHh yryd , jD;a; folla
we| we;af;a , jD;a; folu + = 25 f¾Ldj iam¾Y lrk mßÈh' yd ys iólrK
fidhkak'
yd ys fmdÿ iam¾Yl fþokh fkdjk nj o fmkajkak'
(a) + + = 0
wkql%uh = − /
( , ) yryd § we;s f¾Ldj ,ïN f¾Ldj u; ´kEu ( ) ,CIHhla ie,l+ úg"
= 1
= = f,i .ksuq' mrdñ;shls'
= +
( , )
( , ) + + =
16.
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- 165 -
Ajith Wimanga Wijesinghe
= +
ys m%;sìïNh ( ) kï"
= + , = + fõ'
ys uOH ,laIHh ≡ +
,CIHh = 0 u; ksid
+ [ ]
+ = 0
( + ) = −2[ + + ] = −2
[ ]
tuÕska" , ys m%;sìïNh = − 2 ( )
= − 2 ( )
≡ + 3 − 18 = 0
= + = 0
≡ ( , ) f,i .ksuq'
túg = 0 u.ska ys m%;sìïNh ksid
= − [ ]
= − . [ ]
,CIHh = 2 u; ksid
− [ + 3 − 18] = 2 − ( + 3 − 18) − [4 − 18] = 2 − [4 − 18] 5 − 12 + 54 = 10 − 8 + 36
9 = 18
= 2
≡ 2 + (10) , 2 + (10) ≡ (4, 8)
≡ (2, 2) f¾Ldj ksid"
^ f¾Ldfõ wkq& × (− ) = −1
f¾Ldfõ wkql%uKh = 3
f¾Ldfõ iólrKh
= 3
3 − − 4 = 0
∥ + = 0 f¾Ldj ksid"
ys wkql%uKh = 1
ys iólrKh
= 1
− 8 = − + 4
=
C
B A
= =
= =
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- 166 -
Ajith Wimanga Wijesinghe
+ − 12 = 0
≡ ( ) kï" + = 12 1
= 3 2
4 = 12 → = 3
= 9
≡ (3, 9) ys iólrKh +
− 3 = 7( − 3) 7 − − 18 = 0
(b) = + − 25 = 0 = − − 1 = 0
yd ys fþok ,CIHh yryd jk ´kEu jD;a;hls'
+ − 25 + ( − − 1) = 0
+ + − − 25 − = 0
flakaøh = " wrh = 2⁄ + 25 +
fuu jD;a; + − 25 = 0 f¾Ldj iam¾Y lrhs'
^flakaøfha isg ÿr& = ^wrh&
√ = + 25 +
+25 + =
+ 2 − 575 = 0
( + 25)( − 23) = 0
= −25 fyda
= 23
= −25 úg ≡ + − 25 + 25 = 0
= 23 úg ≡ + + 23 − 23 − 48 = 0
= 0 yd = 0 fþokh jk ksid yd g we;af;a fmdÿ iam¾Yl folls'
;jo yd ys wrhka iudkh'
tneúka yd ys fmdÿ iam¾Yl fþokh fkdfõ'
(a) = tan + cot − f,i .ksuq' 1 + = 2( − 1) sin 2 nj idOkh lrkak'
tkhska ys ´kEu ;d;aúl w.hla i|yd tan + cot − m%ldYh 1/3 yd 3 w;r
w.hla fkd.kakd nj fmkajkak'
(b) i|yd úi|kak'
4 − 4(cos sin ) − sin 2 = 0
cos − sin = 6⁄ ' fuys 0 ≤ cos ≤ yd − ≤ sin ≤ '
(c) ;%sfldaKfha , fldaKhkaf.a ihsk − ( + 5) + = 0 iólrKfha uQ, fõ' cos = sin nj fmkajkak'
=
17.
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- 167 -
Ajith Wimanga Wijesinghe
(a) = tan + cot −
= ×
=
= sin 2 − sin 2 + (2 sin 2 − 1) = 2 sin 2 + 1
2 sin 2 ( − 1) = 1 +
sin 2 = ( )
ish¿ i|yd −1 ≤ sin 2 ≤ 1 ksid
−1 ≤ ( ) ≤ 1
= ( ) +1 ≥ 0 = ( ) −1 ≤ 0
= ( )( ) ≥ 0 =
( )( ) ≤ 0
= ( ) ≥ 0 = ( )( ) ≤ 0
yd m%ldYk folu i;H ùug"
< 1/3 yd > 3 úh hq;=h'
tneúka 1/3 < < 3 w;r w.hla .; fkdyel'
(b) 4 − 4(cos sin ) − sin 2 = 0 (cos − sin ) = 1 − sin 2 ksid
4 − 4(cos − sin ) + (cos − sin ) − 1 = 0
= cos − sin f,i fhdouq'
3 − 4 + = 0
( − 3)( − 1) = 0
= 3 fyda = 1 fõ' ≠ 3 ksid"
= 1 úg cos − sin = 1
√ cos − √ sin = √
cos − = √
cos − = cos
+ = 2 ± " ksÅ,hls'
= 2 ± −
= 2 fyda = 2 −
+ + − 1/3 1
− − +1 3
1/3 3 1
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- 168 -
Ajith Wimanga Wijesinghe
cos − sin =
= cos o = sin f,i .ksuq'
cos = o sin =
− =
= +
cos = cos +
= cos − sin sin
= √
∙ cos −
3 = √3 cos
√3 = cos
3 = (1 − ) 4 = 1
= → = ±
fojk jD;a; mdolfha;a 4 jk jD;a; mdolfha;a kï"
= úh fkdyel'
∴ = fõ'
(c) − ( + ) + = 0
uQ, sin yd sin kï"
sin + sin =
sin + sin =
sin + sin ≠ 0 kï"
sin = 1
= 2
+ + =
+ = 2
= 2 −
cos = cos 2 −
cos = sin
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