Test Doc

- 145 - Ajith Wimanga Wijesinghe
07
07

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.
A
B−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 06la
3( + )
wdldrfhk
i|yd 3
+
wNHqykfh
= (1
fidhkak'
wd.kaâ i
lrhs' y
iu
ixlS¾K i
fidhkak'
= 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. uQ
0
0 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 − 12
úi|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
+ wdldrf
1 −
0
0 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 3
Ok 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
.

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.

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

=
§ 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

+ − 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)
=
4
1
( , )
(1, 3)
(−2, 7)
6.
7.
8.
5
5
5
5
5
5
10
5
5
5
5

= 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

(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

(− ) =
= → =
+ = −
=
− + = 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

(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
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

(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
la o wfkla fldgia j,ska
01 m%Yakh ne.ska
= 3 = 16 48
la o ;j;a fldgilska
m%Yak 02 la o wjika
fldgiska m%Yak 01 la
3! = 6 = 96 576
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

+
( )
+
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) = 0
0 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 ) =
2
3
=
cos + sin
13.
2
10
5
5
5
5
5
5
5
5
10
10

= 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
2
3
− sin
= −
√
= 2 − = +
√
= + = = √7
= 2 2
−
3
2
= −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 )
/3
2 /3
/6
0/3
/3 /3
5
20
5
5
5
5
5
5
5
10
5

túg = ±
√
∙
= ±1/√2
ys j¾. uQ, →
√
+
√
fyda
√
−
√
(b) (i) =
1
0
0 0
=
1 − −
0 −
0 0
=
1
0
0 0
1 − −
0 −
0 0
=
1 0 0
0 1 0
0 0 9
=
1 − −
0 −
0 0
1
0
0 0
=
1 0 0
0 1 0
0 0 1
= =
( ) =
=
=
1 0 0
0 1 0
0 0 1
=
= =
1 0 0
− 0
− −
(ii) =
=
=
= + +
+ +
= =
1 0
0 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.
5
5
5
5
5
5
5
5
5
10

(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
∫ ∫
∫
∫
∫
∫
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5

= (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
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5

(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
1
1− 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'
(1 − ) −2 − − = 0
(1 − ) −3 − = 0 3
(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.
5
5
5
5
5
5
5
5
5
5
5

(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
−
−
5
5
5
5
5
5
5
5
10

( ) = 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.
10
5
5
5
5
5
5
5
5
5

= +
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
=
=
=
=
5
5
5
5
5
5
5
5
5
5
5
5
5
5

+ − 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.
5
5
5
5
5
5
5
5
5
5
5
5
5

(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 31
5
5
5
5
5
55
5
5
5
5
5
5
5
5
5
5
5

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
5
5
5
5
5
5
5
5
5
5
5
5

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