COMBINED MATHEMATICS
2002 A/l Pure Maths
ksfrdaIa pdñkao
B.Sc(Engineering) University of Moratuwa
[ish¨u ysñlï weúßKs.]
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YS% ,xld úNd. fomd¾;fïka;=j /,yq;ifg; ghPl;irj; jpizf;fk ;/Department of Examinations,Sri L anka
wOHhk fmdÿ iy;sl m;% ^ Wiia fm< & úNd.h , 2002 wfm%a,a
fy;tpg; nghJj; juhjug;gj;jpu ( cah; ju )g; ghPl;ir> 2002 Vg;gpuy;
General Certificate of Education ( Adv . Level ) Examination , April 2002
ixhqla; .Ks;h I
10
,ize;j fzpjk I
Combined Mathematics I S I
A
mE ;=khs / ;%d;W kzpj;jpahyk; / Three hours
D
m%Yak yhlg muKla ms<s;=re imhkak.
IN
01. f ( x ) = x2 + 2x + 9 ; x R hehs .ksuq.
(i)
( ii )
iólrKh ,nd .kak.
AM
, hkq f ( x ) = 0 ys uQ, kï, 2 - 1iy 2 - 1 uQ, jYfhka we;s j¾.c
f ( x ) = k iólrKhg, x i|yd yßhg u tla ;d;a;aúl uQ,hla mj;sk fia
CH
jQ k ;d;a;aúl ksh;hl w.h fidhkak.
1
( iii ) ys jeä;u w.h fidhd, th ,efnkakd jQ x ys w.h o fokak.
f(x)
SH
( iv ) f ( x ) = x iólrKhg x i|yd ;d;a;aúl úi÷ula fkdue;s jk fia jQ
;d;a;aúl ksh;fha w.h l=,lh ks¾Kh lrkak.
RO
02.( a ) yßhg u YsIHhka y;r fofkl=f.ka iukaú; mdi,a újdo lKavdhula , iqÿiqlï
,enQ YsIHhka fodf<dia fofkl= w;frka f;dard .ekSug kshñ;h. tu lKavdhu
NI
f;dard .; yels wdldr .Kk fidhkak. wkqr iy Njka iqÿiqlï ,enQ YsIHhka
fodf<dia fokd w;r fõ.
©
(i) wkqr iy Njka fofokdu lKavdhfï isà ,
( ii ) tlaflda wkqr ke;fyd;a Njka lKavdhfï isà ,
( iii ) wkqrj;a Njkaj;a lKavdhfï fkdisà,
hk tla tla wjia:dj i|yd újdo lKavdhu f;dard .; yels wdldr
.Kk fidhkak.
( 7 - 6x
)
13
(b) ys m%idrKh i,lkak.
6x 7
1
(i) x ys brÜfÜ n, fyda ys brÜfÜ n, fyda tu m%idrKfha fkdue;s nj,
1 x
( ii ) x ys ix.=Klh 2002 nj fmkajkak.
02
03.( a ) .Ks; wNHqykh ms<sn| uQ,O¾uh fhdod .ksñka, iEu n Ok ksÅ,hla i|yd,
n! 2 n - 1 nj idOkh lrkak.
n 1
1
k! 2 - 2 n-1
k=1
nj wfmdaykh lrkak.
takhska, e 3 nj fmkajkak; fuys e hkq m%lD;s ,>q .Klj, mdoh fõ.
( b ) y = | 3x - a | iy y = | bx - 2 | ys m%ia:dr tlu o< rEm igykl w¢kak; fuys a yd
b Ok ixLHd fõ. | 3x - a | < | bx - 2 | wiudk;dj imqrd,kakd jQ x ys ish¨u w.h
{
l=,lh x : x >
4
} kï, m%ia:dr Wmfhda.S lr .ekSfuka fyda wka l%uhlska fyda
A
3
a iy b fidhkak.
D
IN
04. z ixlS¾K ixLHdj, z = x + iy, x > 0, y > 0 uÕska fokq ,efí. wd.ka igyfkys
z , 2iz , z + 2iz g wkqrEm ,CIH ms<sfj,ska A , B , C fõ. A , B , C ,CIH i,l=Kq lr,
^ iy tan AOC ^
AM
AOB ks¾Kh lrkak.
(i) C w;d;a;aúl wCIfha msysghs kï, x iy y w;r iïnkaO;djla ,nd .kak.
CH
( ii ) y = 2x kï, z2 ixlS¾K ixLHdj ksrEmKh lrk ,CIHh OC f¾Ldj u;
msysgk nj fmkajkak.
( iii ) | z | 4 iy tan-1
( )
1
2
arg z tan -1 ( 2 ) jk mßÈ jQ z ixlS¾K ixLHdj
SH
ksrEmKh lrk ,CIHhkaf.ka iukaú; fmfoi fjk;a rEm igykl
w÷re lrkak. w÷re l< fldgfia j¾.M,h fidhkak.
RO
d2y dy
05.( a ) y = e4x sin 3x kï, -8 + 25 y = 0 nj fmkajkak.
dx
NI
dx 2
[ ]
dy
[ ]
, d y
2
iy
d3y
dx3 [ ]
fidhkak.
©
dx dx2
x=0 x=0 x=0
( b ) >k f.da,hlska, f.da,fha flaJøh yryd hkakd jQ wCIHhla iys; Rcq jD;a;dldr
is,skavrhla lmkq ,efí. is,skavrfha mßudj, f.da,fha mßudj fuka 1 g jvd
jeä úh fkdyels nj idOkh lrkak. 3
2 x3
06.( a ) iqÿiq wdfoaYhla fh§fuka, x2 - 1
dx wkql,h w.hkak.
1
03
1
(b) fldgia jYfhka wkql,k l%uh Ndú;fhka, x tan-1 x dx wkql,h w.hkak.
2 0
(c) 5x - 4
( 1 - x + x2 ) ( 2 + x )
dx fidhkak.
1
07. u1 a1x + b1y + c1 = 0 iy u2 a2x + b2y + c2 = 0 hkq § we;s iudka;r fkdjk
ir, f¾Ld folls. ys iEu w.hla i|yd u u1 + u2 = 0 ir, f¾Ldj wp,
,CIhhla yryd hk nj fmkajkak.
ABC ;%sfldaKhl iïuqL mdoj,g B, C, yryd w¢kq ,enQ ,ïnj, iólrK ms<fs j,ska
A
x - 4y + 5 = 0 iy 2x - y + 3 = 0 fõ. A ys LKavdxl ( k , -k ) f,i .kq ,enqfõ kï,
AB iy AC f¾Ldj, iólrK o B ys iy C ys LKavdxl o k weiqfrka fidhkak.
D
IN
k úp,kh jk úg, ABC ;%sfldaKfha flaJølh x + 5y - 4 = 0 f¾Ldj u; msysgk nj
idOkh lrkak.
08.
AM
x2 + y2 + 2g1x + 2f1y + c1 = 0 iy x2 + y2 + 2g2x + 2f2y + c2 = 0 jD;a; iam¾Y ùu i|yd
CH
wjYH;djla fidhkak.
tajd iam¾Y fõ kï, iam¾Y ,CIHh 2 ( g1 - g2 ) x + 2 ( f1 - f2 ) y + c1 - c2 = 0 iy
SH
( f1 - f2 ) x- ( g1 - g2 ) y + f1g2 - f2g1 = 0 f¾Ld tl tlla u; msysgk nj idOkh
lrkak.
RO
x2 + y2 - 2x + 4y = 0 iy x2 + y2 - 10x + 20 = 0 jD;a;, tlsfkl ndysr j iam¾Y lrk
nj fmkajd, jD;a; foflys A iam¾Y ,CIHfha LKavdxl fidhkak.
NI
P hkq, P ys isg m%:u jD;a;hg we¢ iam¾Ylfha È., P ys isg fojeks jD;a;hg we¢
iam¾Yl;fha È. fuka k ^ ksh;hla & jdrhla jk fia jQ ,CIHhls. k2 1 kï P
©
ys m:h A yryd jQ jD;a;hla nj idOkh lr k weiqfrka tys iólrKh fidhkak.
09. ABC hkq, b > c mßÈ jQ ;%sfldaKhls. D iy E hkq, A yryd uOHia:h AD jk mßÈ o,
AD , AE uÕska A fldaKh ;%sõfþo lrk mßÈ o BC u; msysá ,CIh fõ. iqÿiq f,i
f;dard .kq ,enQ ;%sfldaK follg ihska kshuh fh§fuka,
A b
cos = nj idOkh lrkak.
3 2c
A (2+k)c
DE : EB = 1 : k kï cos rdYsh g o iudk nj fmkajkak.
3 2kb
04
k = 1 kï A = 900 nj o k = 2 kï A = 1350 nj o wfmdaykh lrkak.
tla tla wjia:dfõ §, a weiqfrka b iy c ks¾Kh lrkak.
D A
IN
AM
CH
SH
RO
NI
©
