AL/2023/10/S-I SCM ®

G.C.E. (A/L) Examination - ixhqla; .Ks;h - w'fmd'i' ^W$fm<& úNd.h - Smart Combined Maths®- Combined Mathamaics -Dilshan Mallawaarachchi G.C.E. (A/L) Examination
G.C.E. (A/L) Examination - ixhqla; .Ks;h - w'fmd'i' ^W$fm<& úNd.h - Smart Combined Maths®- Combined Mathamaics -Dilshan Mallawaarachchi G.C.E. (A/L) Examination
Y%S ,xld úNd. fomd¾;fïka;j =
G.C.E. (A/L) Examination - ixhqla; .Ks;h - w'fmd'i' ^W$fm<& úNd.h - Smart Combined Maths®- Combined Mathamaics -Dilshan Mallawaarachchi G.C.E. (A/L) Examination
,yq;ifg; guPl;irj; jpizf;fsk;
G.C.E. (A/L) Examination - ixhqla; .Ks;h - w'fmd'i' ^W$fm<& úNd.h - Smart Combined Maths®- Combined Mathamaics -Dilshan Mallawaarachchi G.C.E. (A/L) Examination
G.C.E. (A/L) Examination - ixhqla; .Ks;h - w'fmd'i' ^W$fm<& úNd.h - Smart Combined Maths®- Combined Mathamaics -Dilshan Mallawaarachchi G.C.E. (A/L) Examination
Department of Examinations, Sri Lanka
G.C.E. (A/L) Examination - ixhqla; .Ks;h - w'fmd'i' ^W$fm<& úNd.h - Smart Combined Maths®- Combined Mathamaics -Dilshan Mallawaarachchi G.C.E. (A/L) Examination
G.C.E. (A/L) Examination - ixhqla; .Ks;h - w'fmd'i' ^W$fm<& úNd.h - Smart Combined Maths®- Combined Mathamaics -Dilshan Mallawaarachchi G.C.E. (A/L) Examination
-
wOHhk fmdÿ iy;sl m;% ^Wiia fm<& úNd.h" 2023 wf.dai; a =
Genaral Certificate of Examination (Adv. Level) Examination, August 2023 03
ixhqla; .Ks;h I meh ;=khs
Combined Mathematics I
10 S I Three hours

INDEX NUMBER
Wmfoia
a
 them follska iukaú; fõ'
fuu m%Yak m;%h fldgia
.
A fldgi ^m%Y.ak 1 - 10& iy B fldgi ^m%Yak 11 - 17&
 A fldgi
ish¨u m%Yakj,g ms<s;=re imhkak' tla tla m%Yakh i|yd Tfí ms<s;=re imhd we;s bfvys
,shkak' jeämqr bv wjYH fõ kï" Tng wu;r ,shk lvodis Ndú;d l< yelsh'
 B fldgi
m%Yak mylg muKla ms<s;=re imhkak' Tfí ms<s;=re imhd we;s lvodisj, ,shkak'
 kshñ; ld,h wjika jQ miq A fldgfiys ms<s;=re m;%h B fldgfiys ms<s;=re m;%hg Wäka
isák mßos fldgia fol wuqKd úNd. Yd,dêm;sg Ndr fokak'
 m%Yak m;%fhys B fldgi muKla úNd. Yd,dfjka msg;g f.k hdug Tng wjir we;'
 fuu m%Yak m;%fhys g u.ska .=re;ajc ;ajrKh olajhs'
mÍlaIljrekaf.a m%fhdackh i|yd muKs'
^10& ixhqla; .Ks;h I . Part A
fldgi m%Yak wxlh ,l=Kq
Part B
01
02 tl;=j
03
wjidk ,l=Kq
A 04
05
06 Final
07 Marks
08 wjidk ,l=Kq
09
10
11
B
12
13
14
15
16
17
tl;=j
m%;sY;h [fojeks msgqj n,kak'
S M A R T C O M B I N E D M A T H S ® -1-
1
 2

A fldgi

❖ ish¨u m%Yakj,g ms<;

s =re imhkak'

01' 𝑥 2 + 𝑎𝑥 + 𝑏 = 0 ys uQ, 𝜆 yd 𝜇 fõ'

𝑏𝑥 2 − (𝑎2 − 2𝑏)𝑥 + 𝑏 = 0 ys uQ, 𝜆, 𝜇 weiqßka fidhkak'

1
02' 𝑥 4 + 2𝑥 3 − 6𝑥 2 + 2𝑥 + 1 = 0 iólrKh 𝑡 = 𝑥 + wdfoaYfhka úi`okak'
𝑥

𝑙𝑖𝑚 𝑥(𝑒 𝑥 −1)

03. w.hkak'
𝑥→0 1−cos 2𝑥

cos 2𝑥 𝑑2 𝑦 𝑑𝑦
04' 𝑦 = kï" 𝑥 𝑑𝑥 2 + 2 𝑑𝑥 + 4𝑥𝑦 = 0 nj fmkajkak'
𝑥

𝜋
05. 2 tan−1 𝑥 + tan−1 𝑥 3 = úi`okak'
2

1
06. 𝑦 = |2 𝑥 − 1| + |𝑥 − 4| iy 𝑦 = |6𝑥 − 𝑥 2 | m%i:
a dr tlu igykl w`Èkak' tkhska
1
|6𝑥 − 𝑥 2 | ≥ 𝑦 ≥ | 𝑥 − 1| + |𝑥 − 4| ;Dma; lrk m%foaYh w`ÿre lrkak'
2

07' oDv jia;=jla u; l%shdlrk tal;, n, hgf;a th

iu;=,s;j ;sfí kï ta i`oyd wjYH;d olajkak' wrh
5
𝑎 jQ taldldr 𝑊 nr f.da,hla tys jl% mDIaGfha 𝐴
2

,laIHfhka ;ka;=jlska t,a,d we;af;a rEmfha whqre th

iqug isria ì;a;shla iam¾Y jkfiah' ;ka;=j 𝐵 yd 𝐶
iqug wp, lmams folla by<ska f.dia wfkla fl<jf¾
𝑊′ Ndrhla t,a,d ;sfí kï túg moaO;sh iu;=,s;j 𝑤′
13
;sfnk úg§ 𝑊 ′ = 12 𝑊 nj fmkajkak' ;jo f.da,h

yd ì;a;sh w;r m%;sl%shdj fidhkak'^fuys 𝐴𝐵 = 4𝑎 fõ '&

DILSHAN MALLAWAARACHCHI
B.Sc.
 3

08. 𝑂 uQ,hla wkqnoaO 𝑃 yd 𝑄 ,laIH 2l msysgqï ffoYsl 2𝑎 yd 2𝑏 fõ' 𝑂𝑃 yd 𝑂𝑄 ys uOH

,laIH ms<sfj,ska 𝑆 yd 𝑅 fõ' 𝑆𝑄 yd 𝑅𝑃 ys fþok ,laIHh 𝑋 fõ'𝑆𝑋: 𝑆𝑄 = 𝑚 yd
𝑅𝑋: 𝑅𝑃 = 𝑛 kï ⃗⃗⃗⃗⃗
𝑂𝑋 = 𝑎 + 𝑚(2𝑏 − 𝑎) yd ⃗⃗⃗⃗⃗
𝑂𝑋 = 𝑏 + 𝑛(2𝑎 − 𝑏) nj fmkajkak' tu`.ska
𝑚 yd 𝑛 ys w.hka wfmdaykh lrkak'

09. 𝐴𝐵𝐶 ;%sfldaKfha 𝐵𝐶, 𝐶𝐴, 𝐴𝐵 Èf.a 𝑃, 𝑄, 𝑆 n, l%shd lrhs' iïm%hqla; n,h ;%sfldaKfha
mßflakaøh yryd hhs kï 𝑃 cos 𝐴 + 𝑄 cos 𝐵 + 𝑆 cos 𝐶 = 0 nj fmkajkak'

10' ialkaOh 𝑀 𝑘𝑔 jQ laIu;dj 𝐻 𝑊 jQ r:hl p,s;hg ´kEu úgl§ m%;sfrdaOh 𝑅 fõ'

;srig 𝛼 fldaKhlska wdk; ud¾.hl by,g r:fha Wmßu m%fõ.h 𝑉 𝑚𝑠 −1 fõ' fuu
ud¾.fha my<g r:fha Wmßu fõ.h 2𝑉 𝑚𝑠 −1 fõ' 𝑅 ys w.h 𝑀, 𝑔 yd 𝛼 weiqfrka m%ldY
3𝐻
lrkak' th f,i m%ldY l< yels nj fmkajkak'
4𝑉

DILSHAN MALLAWAARACHCHI
B.Sc.
 4
G.C.E. (A/L) Examination - ixhqla; .Ks;h - w'fmd'i' ^W$fm<& úNd.h - Smart Combined Maths®- Combined Mathamaics -Dilshan Mallawaarachchi G.C.E. (A/L) Examination
G.C.E. (A/L) Examination - ixhqla; .Ks;h - w'fmd'i' ^W$fm<& úNd.h - Smart Combined Maths®- Combined Mathamaics -Dilshan Mallawaarachchi G.C.E. (A/L) Examination
Y%S ,xld úNd. fomd¾;fïka;j =
G.C.E. (A/L) Examination - ixhqla; .Ks;h - w'fmd'i' ^W$fm<& úNd.h - Smart Combined Maths®- Combined Mathamaics -Dilshan Mallawaarachchi G.C.E. (A/L) Examination
,yq;ifg; guPl;irj; jpizf;fsk;
G.C.E. (A/L) Examination - ixhqla; .Ks;h - w'fmd'i' ^W$fm<& úNd.h - Smart Combined Maths®- Combined Mathamaics -Dilshan Mallawaarachchi G.C.E. (A/L) Examination
G.C.E. (A/L) Examination - ixhqla; .Ks;h - w'fmd'i' ^W$fm<& úNd.h - Smart Combined Maths®- Combined Mathamaics -Dilshan Mallawaarachchi G.C.E. (A/L) Examination
Department of Examinations, Sri Lanka
G.C.E. (A/L) Examination - ixhqla; .Ks;h - w'fmd'i' ^W$fm<& úNd.h - Smart Combined Maths®- Combined Mathamaics -Dilshan Mallawaarachchi G.C.E. (A/L) Examination
G.C.E. (A/L) Examination - ixhqla; .Ks;h - w'fmd'i' ^W$fm<& úNd.h - Smart Combined Maths®- Combined Mathamaics -Dilshan Mallawaarachchi G.C.E. (A/L) Examination
-
wOHhk fmdÿ iy;sl m;% ^Wiia fm<& úNd.h" 2023 wf.dai; a =
Genaral Certificate of Examination (Adv. Level) Examination, August 2023 03
ixhqla; .Ks;h I meh ;=khs
Combined Mathematics I 10 S I Three hours

▪ m%Yak ish,a,gu ms<s;=re imhkak'

11. (a) 𝑎, 𝑏, 𝑐 ;d;a;aúl ksh; fõ' 𝑎 > 0 iy 𝑏 2 − 4𝑎𝑐 < 0 úg ish¿ ;d;a;aúl 𝑥 w.hka
i`oyd 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 > 0 nj fmkajkak' 𝑥 2 + (𝑎 − 𝜆)𝑥 + (1 − 𝑎 − 𝜆) = 0 j¾.c
iólrKfha úfõplh ,shkak' 𝜆 ys ish¿ ;d;a;aúl w.h i`oyd fuu j¾.c
iólrKhg ;d;a;aúl m%Nskk a uQ, ;sfí kï 𝑎 g .;yels w.h mrdih fidhkak'
fuu iólrKfha uQ, folu Ok úh fkdyels nj fmkajkak'
(𝑏) 𝑓(𝑥) = 𝑎1 𝑥 4 + 𝑎2 𝑥 3 + 𝑎3 𝑥 2 + 𝑎4 𝑥 + 1 jk w;r 𝑎1 , 𝑎2 , 𝑎3 yd 𝑎4 ;d;aúl
ksh;hkah' 𝑓(𝑥) hkak (𝑥 2 − 4) ka yd (𝑥 2 − 1) ka fn¥úg fYaIhka ms<sfj,ska
45 − 12𝑥 yd 3(2 − 3𝑥) f,i § we;' 𝑓(𝑥) Y%s;h 𝑥 2 − 3𝑥 + 2 ka fn¥úg fYaIh
fidhkak'
(𝑐) 𝑓(𝑥) kue;s j¾.c nyqmohla ms<sfj<ska (2𝑥 + 1), (3𝑥 + 1) yd (4𝑥 + 1) ka fn¥úg
fYAIhka ms<sfj,ska 2,3 yd 4 fõ' tu j¾.c nyqmoh fkdfidhd
(2𝑥 + 1), (3𝑥 + 1) yd (4𝑥 + 1) hkq 𝑔(𝑥) = 𝑥𝑓(𝑥) + 1 u.ska fokq ,nk 𝑔(𝑥)
nyqmofha idOl nj fmkajkak' tkhska 𝑔(𝑥) nyqmoh ,shd olajkak'

12. (𝑎) mdi,a YsIHhka m%jdykh lrk È. 𝑙 jk jEka r:hla mdi,a YsIHdjka m%jdykh lrk
È. 2𝑙 jk nia r:hla fj; <`.dfõ' tajd tlu ud¾.hl tlu ÈYdjg p,kh fõ'
nia r:h taldldr 𝑣 m%fõ.fhka p,s; jk w;r jEka r:fha bÈß fl<jr nia r:h
fj; <`.djk fudfydf;a tys m%fõ.h 𝑢 (𝑢 > 𝑣) yd ;ajrKh 𝑎 fõ'jEka r:fhys
fõ.h 2𝑢 olajd jeäjk f;la th taldldrj ;ajrKh ù bkamiq nia r:h miq lrk
f;la tu taldldr fõ.fhka .uka lrhs' nia r:hg idfmalaIj jEka r:fhys p,s;hg
wod< m%fõ. ld, m%i:
a drh w`Èkak' tkhska jEka r:h nia r:h miq lrk f;lau nia
r:hg c, m%ydrhla je§fï wjOdkula mj;sk f,i i,ld YsIHdjka wk;=f¾ mj;sk
𝑢2 +6𝑎𝑙
ld, iSudj (2𝑢−𝑣)2𝑎
nj fmkajkak' 3𝑢2 = 𝑎𝑙 kï jEka r:h taldldr fõ.fhka p,s;
15𝑢+2𝑣)𝑙
jQ ld,h nj ;jÿrg;a fmkajkak'
𝑢(2𝑢−𝑣)6

DILSHAN MALLAWAARACHCHI
B.Sc.
 5

(𝑏) 𝐴 yd 𝐵 hkq ;srig 300 l wdk;shla jQ iqug wdk; ;,hl Wmßu nEjqï f¾Ldj u;
tlsfklg 𝑎 ÿrlska msysá ,laIH folls' 𝐴 ,laIHg by<ska msysá 𝐵 ,laIHfha§ 𝑄
wxY=jla ksi,;djfhka 𝑓 taldldr ;ajrKhla iys;j my<g uqod yßkafka 𝐵 yuqjk
mßÈh' tu fudfydf;a§u 𝐴 ys § 𝑃 wxY=jla 𝑢 m%fõ.fhka ksoyfia ;,h Èf.a by<g
m%lafIamkh lrhs' 𝑃 yd 𝑄 f.a p,s;h i`oyd m%fõ. ld, m%i;
a drhla tlu rEm
igykl w`Èkak'

(𝑖) 𝑃 wxY=j ;u Wmßu úia:dmkh ,eîug fmr 𝑄 wxY=j yuqfõ kï

𝑔 𝑎𝑔
𝑓 ≥ 2 ( 𝑢2 − 1) nj fmkajkak'

𝑔 𝑎𝑔 𝑢
(𝑖𝑖) 𝑓 < 2 ( 𝑢2 − 1) kï 𝑃 f.a m%fõ.h jk úg 𝑄 yuqfõ kï
2

2𝑎𝑔 1
𝑓 = ( 3𝑢2 − 2) 𝑔 nj fmkajkak'

13' (𝑎) tal f¾Çh fkdjQ 𝐴, 𝐵 yd 𝐶 ,laI ;=kl 𝑂 uQ,hla wkqnoaOfhka msysgqï ffoYsl
ms<sfj,ska 𝑎, 𝑏 yd 𝑐 fõ' 𝐸 yd 𝐷 ,laI 𝐴𝐶 yd 𝐶𝐵 f¾Ld u; msysgd we;af;a ms<sfj,ska
1
⃗⃗⃗⃗⃗ = (−5𝑎 + 4𝑏 + 𝑐) nj fmkajd"
𝐴𝐸: 𝐸𝐶 = 2: 3 iy 𝐶𝐷: 𝐷𝐵 = 4: 1 jk mßÈh' 𝐴𝐷 5

⃗⃗⃗⃗⃗ fidhkak'
𝐵𝐸

𝐴𝐷 iy 𝐵𝐸 f¾Ld 𝑥 ys§ yuqfjhs' 𝑎 = (2𝑖 + 3𝑗) , 𝑏 = (3𝑖 − 𝑗) yd 𝑐 = (𝜆𝑖 − 7𝑗)

⃗⃗⃗⃗⃗ iy 𝐵𝐸
nj § we;' 𝜆 𝜖 ℤ fõ' 𝐴𝐷 ⃗⃗⃗⃗⃗ " 𝑖, 𝑗 weiqfrka fidhd 𝐴𝐷 ⊥ 𝐵𝐸 kï 𝜆 ys w.h
.Kkh lrkak'

(𝑏) 𝐴(−3,4), 𝐵(3, −1), 𝐶(2, −2) yd 𝐷(−1, −1) ,laI j,§ l%h
s dlrk

5𝑖 − 𝜆 𝑗 , 𝑖 − 4𝑗 , 𝜇𝑖 + 6𝑗 yd −9𝑖 + 𝑗 u`.ska n, moaO;shla ksrEmKh flf¾'

(𝑖) moaO;sh hq.auhlg W!kkh fjhs kï 𝜆 yd 𝜇 ys w.ho hq.aufha úYd,;ajho

fidhkak'
(𝑖𝑖) 𝜆 = 𝜇 = 3 kï yd m<uq n,h (−1,0) ,laIHhg úia:dmkh lf<a kï" kj n,
moaO;sh iu;=,s; nj fmkajkak'
(𝑖𝑖𝑖) 𝜆 = 𝜇 = 16 kï n, moaO;sh ;ks n,hlg W!kkh jk nj fmkajd" tys
úYd,;ajh ÈYdj yd l%h
s d f¾Ldj fidhkak'

DILSHAN MALLAWAARACHCHI
B.Sc.
 6

𝑠𝑖𝑛4 𝜃 𝑐𝑜𝑠4 𝜃 1 𝑠𝑖𝑛8 𝜃 𝑐𝑜𝑠8 𝜃 1

14. (𝑎) + = 𝑎+𝑏 kï + = (𝑎+𝑏)3 nj fmkajkak'
𝑎 𝑏 𝑎3 𝑏3

1
(𝑏) iïu; wxlkhg wkqj 𝐴𝐵𝐶 ;%sfldaKhla i`oyd" ;%sfldaKfha j¾.M,h △= 2 𝑎𝑐 sin 𝐵

nj idOkh lrkak' 𝐴𝐵𝐶 ;%sfldaKfha 𝐶 fldaKfha wNHka;r fldaK iuÉfþolh 𝐴𝐵

mdoh 𝐷 ys§ yuqfõ' 𝐴𝐷 ∶ 𝐷𝐵 = 2 ∶ 3 fõ'

(𝑖) 2𝑏𝑐 sin 𝐴 + 3𝑎𝑐 sin 𝐵 = 5𝑎𝑏 sin 𝐶

(𝑖𝑖) sin 𝐴 = 3 sin 𝐵 nj fmkajkak'

1 1 1
(𝑐) 𝑡𝑎𝑛−1 (2𝑝+1)2 + 𝑡𝑎𝑛−1 (2𝑝+3) − 𝑡𝑎𝑛−1 (2𝑝+1) = 0 nj fmkajkak'

(𝑑) tl tflys nr 𝑤 ne.ska jQ 𝐴𝐵, 𝐵𝐶, 𝐶𝐷, 𝐷𝐸 iy 𝐸𝐴 iudk taldldr o`vq myla ish
fl<jrj, § iqp, f,i ikaê fldg mxpdi%hl iajrEmh .;a iels,a,la idod we;'
fuys 𝐴 yd 𝐶𝐷 mdofha uOH ,laIH ieye,a¨ wú;kH ;ka;=jlska hd fldg ;sfí' 𝐶𝐷
oKav ;sria jk fiao 𝐴𝐵 iy 𝐵𝐶 o`vq ms<sfj,ska isrig 𝜃 yd ∅ fldaK j,ska wdk;
jk fiao " fuu iels,a, 𝐴 ,laIfhka t,a,d iu;=,s;j we;' 𝐵 ikaêfha m%;sl%h
s dfõ
isria yd ;sria ixrpl fidhkak'

tkhska " ;ka;=fõ wd;;sh fidhkak'

by; iels,af,a yevh fkdfjkiaj mj;ajd .ksñka " ;ka;=j bj;a lr 𝐵 yd 𝐸 ikaê
ieye,a¨ oKavlska hd lrk ,§'

(𝑖) 𝐶 ikaêfha m%;sl%h

s dfõ ;sria yd isria ixrpl fidhkak'

(𝑖𝑖) 𝐵𝐸 oKafâ m%;Hdn,h 𝑤(2 tan 𝜃 + tan ∅) nj fmkajkak' th wd;;shla o

f;rmqulao hkak i`oyka lrkak'

DILSHAN MALLAWAARACHCHI
B.Sc.