COMBINED MATHEMATICS

2009 A/l Pure Maths

ksfrdaIa pdñkao
B.Sc(Engineering) University of Moratuwa
 G.C.E. ( Adv. Level ) Examination , August 2009
Scheme of Marking Combined Mathematics Paper I

Part - I
( 01 ) ( a )

x2 + bx + c = 0 uQ,  ,  neúka
 +  = - b yd  = c
32 + 23 = 22 (  +  ) yd ( 3 3 ) ( 22 ) = (  )5
yd ka
32 + 23 = -bc2 yd ( 3 2 ) ( 2 3 ) = c5
 32 , 23 uQ,jk iólrKh
x2 + bc2 x + c5 = 0
----------------------------------------------------------------------------------------------------------- 20
1 55 + 1 c5 + 1
 3 2 +  2  3 = = 2 3

m
 2 3  
1  
5 5
+1 c5 + 1

o
tmßÈ, 23 +  3 2 = = 3 2
3 2
 

.c
c +1
5
y = f,i .ksuq.

da
x
c5 + 1
x = y  wdfoaYfhka,

( ) [ ] in
2
am
c5 + 1 c5 + 1 + c5 = 0
+ bc2
y y
1 1
32 + yd 23 + 3 2 uQ,jk iólrKh
ch

 2
2

sh

c5 y2 + bc2 ( c5 + 1 ) y + ( c5 + 1 )2 = 0
----------------------------------------------------------------------------------------------------------- 30
iro

( b ) f ( x ) nyq jdoh ( x -  ) uÕska fn¥ úg ,íêh Q ( x ) o fYaIh R o f,i .ksuq.

f(x)=(x-)Q(x)+R x = f() = R
-----------------------------------------------------------------------------------------------------------------
.n

f ( x ) , ( x -  ) ( x -  ) ( x -  ) uÕska fn¥ úg fYaIh

w

A ( x -  ) ( x -  ) + B ( x -  ) ( x -  ) + c ( x -  ) ( x -  ) f,i m%ldY l< yels neúka uq,a

w

fldgfia mßÈ
f (  ) = A( -  ) (  -  )
w

f (  ) = B (  -  ) ( -  )
f() = c(-)(-)
f() f() f()
A = (-)(-) , B = , C =
(  -  ) ( -  ) (-)(-)
fYAIh
f() f() f()
(x - ) (x - ) + (x-)(x-)+ ( x - ) ( x - )
(-)(-) (  -  ) ( -  ) (  -  ) ( -  )
wdldrh fõ.
20
ksfrdaIa pdñkao

01 Nirosh Chaminda
 G.C.E. ( Adv. Level ) Examination , August 2009
Scheme of Marking Combined Mathematics Paper I

fYaIfha x ys ix.=Klh

- { f()
(-)(-)
(+) +
(  -
f()
 ) ( -  )
(+) +
f()
(-)(-)
(+) }
f ( x ) = x5 - kx yd = -1 ,  = 1 ,  = 2

fYaIfha x ys ix.=Klh

- { f(-1)
( -1 - 1 ) ( - 1 - 2 ) ( 1 + 2 ) +
f(1)
( 1 + 1 ) ( 1 - 2 ) ( -1 + 2 ) } fyda 0

{ }

m
= - ( -1 ) + k - ( -1 )5 - k
5
= -k + 1 = 0  k =1
2 2

o
----------------------------------------------------------------------------------------------------------- 25

.c
( 02 ) ( a ) P-2 , H -2 , 0 -2 , I - 1 , L - 1 , S - 1 , Y- 1

da
wCIr 10ka l< yels uq: ms<sfh, lsÍï .Kk = 10! = 453600

in
2! 2! 2!
----------------------------------------------------------------------------------------------------------- 10
am
HISY tallhla f,i ie,l+ úg ms<sfh, lsÍï .Kk = 7!
2!  2!
ch

= 1260
sh

tallhla ;=< HISY udreúh yels l%u .Kk = 4! = 24

iro

7!
HISY tl <Õ mj;sk uq¿ ms<sfh, lsÍï .Kk =  4!
2!  2!
.n

= 1260  24 = 30240
w

by; ms<sfh, lsÍï ;=< H wCIrh fomi yd I, S, Y 6!  3! = 180  6 = 1080

w

=
ueo mj;sk ms<sfh, lsÍï .Kk 2!  2!
w

m<uq wdldrfha ixlrK lsÍfï § fojk wdldrfha ms<sfh, lsÍï foj;djla fhfok neúka
ksjerÈ ms<sfh, lsÍï i|yd tla m%udKhla wvq l< hq;=h.
 uq¿ ms<sfh, lsÍï .Kk = 30240 - 1080
= 29160
----------------------------------------------------------------------------------------------------------- 25
wCIr 5 la f;aßh yels fjkia wdldr my; mßÈ fõ.

(i) 
( ii ) 
( iii ) _
ksfrdaIa pdñkao

02 Nirosh Chaminda
 G.C.E. ( Adv. Level ) Examination , August 2009
Scheme of Marking Combined Mathematics Paper I

(i) wdldrfha f;aÍï = 3

C2  5C1 = 15

( ii ) wdldrfha f;aÍï = 3
C1  6C3 = 60

( iii ) wdldrfha f;aÍï = 7

C5  5C1 = 21
uq: f;aÍï .Kk = 15 + 60 + 21 = 96
-----------------------------------------------------------------------------------------------------------
( b ) n yd r Ok ksÅ, jk
15
Pn = n ( n + 1 ) .... ( n + r - 1 )
Pn + 1 = ( n + 1 ) ( n + 2 ) .... ( n + r )
n
Pn + 1 = n ( n + 1 ) ( n + 2 ) .... ( n + r ) = ( n + r ) = ( n + r ) Pn = nPn + rPn
takhska
Pn + 1 - Pn = r n ( )
P
n

m
Pn
n ( r - 1 )! ka fnfok nj § we;s neúka

o
Pn + 1 - Pn , r! uÕska fnfoa.

.c
Pn + 1 - Pn = Kn r! f,i ,shuq. fuys Kn hkq ksÅ,hls.

da
P1 = r! neúka
P2 - P1 = k1 r!

in
P3 - P2 = k2 r!
am
ch

Pn - 1 - Pn - 2 = kn - 2 r!
Pn - Pn - 1 = kn - 1 r!
sh

{  k } r!
n n-1
Pn - P1 = r!
 k i  Pn = 1+
iro

i
r=0 i=1
 wkqhd; Ok ksÅ, r m%udKhl .=Ks;h r! uÕska fnfoa.
.n

----------------------------------------------------------------------------------------------------------- 40
w

n
( 03 ) ( x + y ) =
n

n
Cr xn-r yr n = 1 úg i;H fõ.
w

r=0
w

C0 = 1 = K1
1

n = p úg i;H f,i Wml,amkh lruq.

p
(x+y) =p p

Cr xp+r yr
r=0
( x + y ) uÕska fomi .=K lsÍfuka.
p p
(x+y) p+1
= p

r=0
Cr x p-r+1
y +
r
 Cx
r=0
p
r
p-r
yr + 1

o : m : xp - r + 1 yr ys ix.=Klh = pCr + pCr - 1

p! p!
p
Cr+ pCr - 1 = ( p - r )! r! + ( p - r + 1 )! ( r - 1 )!

ksfrdaIa pdñkao

03 Nirosh Chaminda
 G.C.E. ( Adv. Level ) Examination , August 2009
Scheme of Marking Combined Mathematics Paper I

p!
= ( p - r )! ( r - 1 )!
( p + 1 )!
[ 1
r +
1
p-r+1 }
= = p+1
Cr
( p + 1 - r )! r!
 n = p + 1 úg m%;sM,h i;H fõ.
 .Ks; wNHQyk uQ, O¾uhg wkqj n ys ish¨ Ok mQ¾K ixLHd i|yd m%;sM,h i;H fõ.

-----------------------------------------------------------------------------------------------------------
n
30
x = p yd y = q (x+y) = n n

Cr xn - r yr ys
r=0
wdfoaYfhka

m
n-1
(p+q) = p +
n
 C p q +qn n n-r r n

o
r
r=1 n-1 n-1
(p+q) -p -q =  C p q 

.c
n n n n
r
n-r r
= pq C pn
r
n-r-1
qr - 1
r=1

da
r=1
 ,l=K we;=,; msysgk ish¨u mo Ok ksÅ, neúka

in
( p + q )n - pn - qn pq uÕska fnfoa.
----------------------------------------------------------------------------------------------------------- 15
am
(b) ( 2r + 1 ) 1
Ur = ( 3r - 2 ) ( 3r + 1 )
7r
ch

( 2r + 1 ) A B
( 3r - 2 ) ( 3r + 1 ) = ( 3r - 2 ) - ( 3r + 1 ) f,i .ksuq.
sh

2r + 1  A ( 3r + 1 ) - B ( 3r - 2 )  ( 2r + 1 )  3 ( A - B ) r + A + 2B
iro

3 ( A - B ) = 2 yd A + 2B = 1 A = 7/ 9 , B = 1/ 9
.n

Ur =
( 2r + 1 ) 1 1
= 9 { 1 1 -
1 1
}
w

( 3r - 2 ) ( 3r + 1 ) ( 3r - 2 ) 7r - 1 ( 3r + 1 ) 7r
7r
w

Ur = f(r-1)-f(r)
w

1 1
fuys f ( r ) =
9( 3r + 1 ) 7r
-----------------------------------------------------------------------------------------------------------
n
15
Sn = 
r=1
{ f ( r - 1 ) - f ( r )} = f ( 0 ) - f ( n )

1 1 1
= -
9 9( 3r + 1 ) 7 n
----------------------------------------------------------------------------------------------------------- 15
{ }
;jo, 1
1 1
lim
h 
Sn = lim
h 
-
9 9( 3r + 1 ) 7 n
=1
9
ksfrdaIa pdñkao

04 Nirosh Chaminda
 G.C.E. ( Adv. Level ) Examination , August 2009
Scheme of Marking Combined Mathematics Paper I

1 1
lim =0
n   9( 3n + 1 ) 7 n
-----------------------------------------------------------------------------------------------------------
( 04 ) ( a )
10
- 80 - 18i ys j¾.uQ,h a + bi f,i .ksuq. a yd b hkq ;d;a;aúl ixLHd fõ.
( a + ib )2 = - 80 - 18i  a2 - b2 + i2ab = -80 - 18i
;d;a;aúl yd w;d;a;aúl fldgia iudk lsÍfuka
a2 - b2 = -80 yd 2ab = -18
-9
b = a , a2 - b2 - 80 ys wdfoaYfhka
81
a2 - = -80  a4 + 80a2 - 81 = 0
a2

m
 ( a2 - 1 ) ( a2 + 81 ) = 0  a2 = 1 fyda a2 = - 81
kuq;a a2  - 81 a ;d;a;aúl neúka

o
a = 1 fyda - 1 yd b = - 9 fyda 9

.c
 - 80 - 18i ys j¾.uQ, 1 - 9i , -1 + 9i

da
----------------------------------------------------------------------------------------------------------- 20
4z2 + ( 16i - 4 ) z + ( 65 + 10i ) = 0 úfõplh

in
 = ( 16i - 4 )2 - 16 ( 65 + 10i )
= ( -256 - 128i + 16 ) - ( 1040 + 160i )
am
= ( -1280 - 288i ) = 16 ( -80 - 18i )
- 80 - 18i = ( 1 - 9i )2  [ 4 ( 1 - 9i )]2
ch

j¾.c iñlrKfha úi÷ï

- ( 16i - 4 )  4 ( 1 - 9i )
sh

8 - 52i 20i
= yd
24 8 8
iro

13i
uQ, 1- , 5 i
2 2
.n

----------------------------------------------------------------------------------------------------------- 20
w

( b ) arg ( z + 1 ) = /3 uÕska ±lafjk m:h A ,CIHh wh;a fkdjQ A ,CIHfhka wdrïN ù
w

;d;a;aúl wCIh /3 lska wdk; f¾Ld LKavhls.

w

| z | = 0p | z | wju jkqfha op by; m:hg ,ïNl jk wjia:dfõ § h.

Qz
/
| z |wju = OQ = 3 A
3
2

-----------------------------------------------------------------------------------------------------------
( c ) z3 - 1 = ( z - 1 ) ( z2 + z + 1 )
20
 z3 - 1 = 0 ixlS¾K uQ, hkq z2 + z + 1 = 0 uQ, fõ.
-1  3i
z3 - 1 = 0 ixlS¾K uQ,
2
-1 + 3i 1 - 3 - 2 3i -1 - 3i
 = 2 f,i .ksuq. túg 2 = = 2
4
ksfrdaIa pdñkao
05 Nirosh Chaminda
 G.C.E. ( Adv. Level ) Examination , August 2009
Scheme of Marking Combined Mathematics Paper I

;j o ( -1 - 3i
2 ) 2
=
1 - 3 + 2 3i
4
=
-1 + 3i
2

 z3 - 1 = 0 tla ixlS¾K uQ,hla W kï wfkl W2 fõ.

---------------------------------------------------------------------------------------------------------- 15
2 +  + 1 = 0   + 1 = - 2
(  + 1 )k = ( -2 )k k hkq ksÅ,hls.
k T;af;A ksÅs,hla úg, ( -1 )k = -1
 (  + 1 )k + 2k = 0
----------------------------------------------------------------------------------------------------------- 05
f ( x ) = ( x + 1 )n + x 2n f,i .ksuq. n hkq Ok T;af;a ksÅ,hls.
 f() =0
f ( 2 ) = ( 2 )2n + ( 1 + 2 )n = 4n + ( - )n = ( 3 )n n - n = 0

m
 ( x -  ) yd ( x - 2 ) f ( x ) ys idOl fõ.
( x -  ) ( x - 2 ) .=Ks;h i,luq.

o
= x2 - (  + 2 ) x + 3 = x2 + x + 1

.c
tkï x2 + x + 1 x2n + ( 1 + x )n ys idOlhla fõ.
20

da
-----------------------------------------------------------------------------------------------------------
( 05 ) ( a ) lim f ( x + x ) - f ( x ) Sin ( x + x ) - Sin x
x 0 = lim
x 0
x x
( ) in
2 Cos ( x + x ) Sin x
am
lim 2
= x 0
x
x
( )
ch

Sin
lim
= x 0
2
cos x + ( x
)
x
( ) 2
sh

2
iro

( )
Sin
x
2
lim (
cos x +
x
)
= lim
.n

( )
x 0
x
x 0 2
w

2
( )
Sin
x
2
w

= 1 cos x lim =1
( )
x 0
x
w

= cos x 2
-----------------------------------------------------------------------------------------------------------
g ( x ) = cos ( x ) = Sin ( /2 - x ) = f ( /2 - x ) = f ( y )
15
fuys y = /2 - x fõ.

d g ( x ) = d f ( y ) = d f ( y ) dy = cos ( y ) ( -1 ) dy = -1
dx dx dy dx dx
= - cos ( /2 - x ) = - sin x
----------------------------------------------------------------------------------------------------------- 10
d 1 2x
i) sin [ ln ( 1 + x2 ) ] = cos [ ln ( 1 + x2 )] . 2x = 1 + x2 cos [ ln ( 1 + x2 )]
dx (1+x )
2
15
ksfrdaIa pdñkao

06 Nirosh Chaminda
 G.C.E. ( Adv. Level ) Examination , August 2009
Scheme of Marking Combined Mathematics Paper I

d
ii ) cos [ sin x ] = - sin [ sin ( x )] cos x = - cos x sin [ sin x ]
dx
----------------------------------------------------------------------------------------------------------- 10
(b)i) y = sin k  coses  yd x = cos  k hkq ksh;hls.
dy
= k cos ( k  ) coses  - sin ( k  ) cot  coses 
d
dx
yd = - sin 
d
dy
= - k cos ( k  ) coses2  + sin ( k  ) cot  coses2 
dx
dy
sin2  = - k cos ( k  ) + sin ( k  ) cot 
dx

m
dy

o
( 1 - x2 ) - xy + k cos ( k  ) = 0
dx
15

.c
-----------------------------------------------------------------------------------------------------------
ii ) by; iólrKh x úIfhka wjl,kfhka,

da
d2y dy dy dy d
( 1 - x2 ) 2 - 2x -y-x +k ( cos k  ) =0

in
dx dx dx d dx

{ }
am
dy -1
( 1 - x2 ) d y2 - 3x
2
- y - k2 sin ( k  ) =0
dx dx Sin 
dy
( 1 - x2 ) d y2 - 3x
2
+ ( k2 - 1 ) y = 0
ch

dx dx
----------------------------------------------------------------------------------------------------------- 15
sh

dy
( c ) y ( 1 + x2 ) = 2 , kï ( 1 + x2 ) + 2xy = 0
dx
iro

dy
dx -3
1 =
p ( 3 , /5 ) 25
.n

 p ,CIHfha § iam¾Ylfha iólrKh

w

y - 1/ 5 - 3
w

x - 3 = 25  3x + 25y = 14
w

iam¾Ylh jl%h kej; yuq fõ kï,

2 14 - 3x
1+x 2 =
25  3x3 - 14x2 + 3x + 36 = 0

p ,CIHfha § we¢ iam¾Ylhg jl%h kej; yuqjk ,CIHfha LKavdxl by; >Kc iólrKfhka
ksrEmKh fõ. ñka ,CIH folla p yd iumd; jk neúka ( x - 3 )2 >Kcfha idOlhla fõ.
 b;sß idOlh ( 3x + 4 ) fõ.

Q= [ -4 , 18
3 25 ] 20
ksfrdaIa pdñkao

07 Nirosh Chaminda
 G.C.E. ( Adv. Level ) Examination , August 2009
Scheme of Marking Combined Mathematics Paper I

( 06 ) ( a ) Ik
et
tk
=
dt , t > 0
 k hkq Ok ksÅ,hls. ( k  1)

Ik = ek

t
t
dt =
(
( - 1 ) t d
k - 1 )
e

dt t
1
k-1 dt ( )

-1 et 1 et
= + dt
(k-1) t k-1
( k - 1 ) tk - 1
1 et
I k - Ik - 1 + = ksh;hla
( k - 1 ) tk - 1
----------------------------------------------------------------------------------------------------------- 20
 ( )
2
1-x
I = ex dx t = 1 + x wdfoaYh fhdouq.
1+x

  ( )
et - 1 ( 2 - t )2 1 4 4
 I = dt = e et - dt
t2 t +1

m
t 2

o
 
et 1

.c
Ik = dt neúka I=
{ 4 I2 - 4 I1 + et dt }
tk e

da
et
ka I2 - I1 = ksh;hla - t

I =
1
e
ks{h;hla -
4et
+ et dt
 } in
am
t

=
1
ks{h;hla -
4et
+ et }
ch

e t
sh

e1 + x 4 ( e1 + x )
= - e ( 1 + x ) + ksh;hla
e
30
iro

-----------------------------------------------------------------------------------------------------------
a
(b) I = 
f ( x ) dx a > 0 fõ.
.n

0
w

y = ( a - x ) wdfoaYh fhdouq. dy = - dx yd
w

a a

w

I = - f ( a - x ) dx
a
= f ( a - x ) dx

0
----------------------------------------------------------------------------------------------------------

/2 
/2 
/ 2
20
ka  sin 2k
x dx =  
sin2k ( /2 - x ) dx = cos2k ( x ) dx
0 0 0

k hkq Ok ks Å ,hls .
/2 
/2


Sin x dx =
2k
cos2k x

0
Sin x + cos x
2k


2k
Sin2k x + cos2k x

 dx
/2 0 /2
Sin x 1
 
2k
= dx = dx = /4 30
Sin2k x + cos2k x 2
ksfrdaIa pdñkao 0 0 08
 Nirosh Chaminda
 G.C.E. ( Adv. Level ) Examination , August 2009
Scheme of Marking Combined Mathematics Paper I

( 07 )
ax + by + c = 0 f¾Ldjg ,ïNl f¾Ldj
bx - ay - c1 = 0 wdldrh .kS.
( x0y0 ) yryd hkakd jQ o ax + by + c = 0 f¾Ldjg ,ïN jQ o
f¾Ldj bx - ay - bx0 + ay0 = 0
b ( x - x0 ) - a ( y - y0 ) = 0
x - x0 y - y0
 a = b = t ( f,i .ksuq. )

tkï ( x0 y0 ) yryd hkakd jQ o ax + by + c = 0 f¾Ldjg ,ïN jQ o f¾Ldj u; ´kEu ,CIHhla

[ x0 + at , y0 + bt ] f,i ksrEmKh l< yel. fuys t hkq mrdñ;shls.
----------------------------------------------------------------------------------------------------------- 15
ax + by + c = 0 f¾Ldj wkqnoaO j P  ( x0 y0 ) ,CIHfha m%;sìïNh

m
Q [ x0 + at , y0 + bt ] f,i .ksuq.

[ ]

o
2x0 + at , 2y0 + bt
PQ ys uOH ,CIHh 

.c
2 2
ax + by + c = 0

da
uOH ,CIHh ax + by + c = 0 f¾Ldj u; neúka ( x0 y0 )

{ } { }
in
2x0 + at 2y0 + bt
a +b +c=0
2 2
am
2(ax0 + by0 + c )
t= -
( a2 + b2 )
ch

( x0 + at , y0 + bt )
 ( x0 y0 ) ,CIHfha ax + by + c = 0 u; m%;sìïNh
sh

2(ax0 + by0 + c )
[ x0 + at , y0 + bt ] fõ. fuys t = -
( a2 + b2 )
iro

-----------------------------------------------------------------------------------------------------------
OA ys ,ïN iuÉfþolh
10
.n

x cos  + y sin  - 1 = 0
w

by; f¾Ldj wkqnoaOj O ys m%;sìïNh A neúka

A  [ 2 cos  , 2 sin  ] B
w

y-0 2 sin  - 0
w

 OA ys iólrKh =
x-0 2 cos  - 0 A
tkï y = x tan 
O
AB ys ,ïN iuÉfþolh x - y - 1 = 0
B ,CIHh by; f¾Ldj wkqnoaOj A ys m%;sìïNh neúka
B [ ( 1 + 2 sin  ) , ( -1 + 2 cos  )]
y-0 -1 + 2 cos  - 0
OB ys iólrKh =
x-0 1 + 2 sin  - 0
tkï x ( 2 cos  - 1 ) - y ( 2 sin  + 1 ) = 0
y - 2 sin  -1 + 2 cos  - 2 sin 
AB mdofha iólrKh
x - 2 cos 
=
1 + 2 sin  - 2 cos  40
ksfrdaIa pdñkao

09 Nirosh Chaminda
 G.C.E. ( Adv. Level ) Examination , August 2009
Scheme of Marking Combined Mathematics Paper I
OB ys ,ïN iuÉfþolh
1 + 2 sin 
y - [ cos  - 1/2 ] = [ x - ( sin  + 1/2 )]
-1 + 2 cos 
1
y [ 2 cos  - 1 ] + x [ 1 + 2 sin ] - { ( 2 cos  - 1 )2 + ( 2 sin  + 1 )2 } = 0
2
y [ 2 cos  - 1 ] + x [ 1 + 2 sin  ] - [ 3 - 2 cos  + 2 sin  ] = 0
---------------------------------------------------------------------------------------------------------- 15
OA yd AB ys ,ïN iuÉfþol yuqjk ,CIHh ( L kï ) x cos  + y sin  - 1 = 0 yd x - y - 1 = 0 fþok
,CIHh fõ. by; iólrKh úi£fuka
1 + sin 
x= yd y = 1 - cos 
cos + sin  cos + sin 

[
L  1 + sin  , 1 - cos 
cos + sin  cos + sin  ]

m
L ,CIHh OB ys ,ïN iuÉfþolfha wdfoaYfhka

o
( 1 - cos ) ( 2 cos  - 1 ) ( 1 + sin ) ( 1 + 2 sin  )

.c
cos  + sin  + cos  + sin  - ( 3 - 2 cos  + 2 sin  )

da
( - 1 + 3 cos  - 2 cos2  ) + ( 1 + 3 sin  + 2 sin2  )
- ( 3 - 2 cos  + 2 sin  )

in
= cos  + sin 
am
3 [ cos  + sin  ] + 2 [ sin2 - cos2 ]
= cos  + sin  - ( 3 - 2 cos  + 2 sin  )
ch
sh

= 3 + 2 ( sin  - cos  ) - 3 + 2 cos  - 2 sin 

= 0
iro

OA yd AB ys ,ïN iuÉfþol yuqjk ,CIHh OB ys ,ïN iuÉfþolh u; msysghs.

 ,ïN iuÉfþol ;=k tal ,CIHh fõ.
20
.n

-----------------------------------------------------------------------------------------------------------
( 08 ) Q
w

PQ jD;a; folgy fmdÿ iam¾Ylh f,i .ksuq. r2

w

O1 r O2
T
S1  x2 + y2 + 2g1x + 2f1y + c1 = 0 1
w

P
S2  x2 + y2 + 2g2x + 2f2y + c2 = 0 S1 = 0 S =0 2

PQ f¾Ldj O1O2 f¾Ldj fþokh jk ,CIHh T f,i .ksuq. O1P yd O2Q f¾Ld PQ g ,ïN fõ.
O1T r1
O1PT yd O2QT iufldaKS ;%sfldaK neúka T O2 = r 2
r1 yd r2 hkq ms<sfj,ska S1 = 0 yd S2 = 0 jD;a;hkaf.a wrhka fõ.
T uÕska O1O2 f¾Ldj wNHka;rfhka r1 : r2 wkqmd;hg fnfoa.

T  [ - g1 r2 - g2 r1
r1 + r2 +
- f1 r2 - f2 r1
r1 + r2 ]
wfkla fmdÿ iam¾Ylh o T yryd we¢h yel. 25
ksfrdaIa pdñkao

10 Nirosh Chaminda
 G.C.E. ( Adv. Level ) Examination , August 2009
Scheme of Marking Combined Mathematics Paper I

O1 O2
T 1
r1
r2
R
S1 = 0
S S2 = 0

S1 = 0 yd S2 = 0 jD;a;j,g fmdÿ iam¾Ylh RS O1O2 f¾Ldj T1 yS § fþokh fõ hehs i,luq. O1T1R

yd O2T1S ;%sfldaK iufldaKS neúka
O1T1 r1
1 =
O2 T r2

m
 T1 uÕska O1O2 f¾Ldj r1 : r2 wkqmd;hg ndysr j fnfoa.

o
[ g1 r2 - g2 r1 f1 r2 - f2 r1
]

.c
T 1  r1 - r2 + r1 - r2

da
tkï wfkla fmdÿ iam¾Yl T1 yryd we¢h yelsh.
25
in
-----------------------------------------------------------------------------------------------------------
jD;a; fofla iólrK my; mßÈ m%ldY l< yel.
am
( x - 9 )2 + ( y + 3 )2 = 22 yd ( x + 9 )2 + ( y - 3 )2 = 42
T yd T ,CIHj, LKavdxl
1
ch

[ 94 - 92
2+4
, -3  4 + 3  2
2+4 ]  [ 3 , - 1 ] yd
sh

[ -29+4(-9) , -2(-3)+(43)
 [ 27 , - 9 ] ]
iro

2-4 2-4
T yryd hk f¾Ldj ( )y + 1 = m ( x - 3 ) wdldrh fõ.
.n

tkï y - mx + 3m + 1 = 0
w

f¾Ldj ( x - 9 )2 + ( y + 3 )2 = 22 iam¾Y lrhs kï

w

- 3 - 9m + 3m + 1
= 2 úh hq;=h.
1 + m2
w

( -6m - 2 )2 = 4 ( 1 + m2 )  8m2 + 6m = 0  m = 0 fyda m = -3/4

f¾Ldj ( x + 9 ) + ( y - 3 ) = 4 iam¾Y lrk nj o ie,lsh yel.
2 2 2

 T yryd hk fmdÿ iam¾Yl fol

y+1=0 yd y + 1 = -3/4 ( x - 3 )
tkï y+1=0 yd 4y + 3x - 5 = 0

T1 yryd hk f¾Ldj
y + 9 = m1 ( x - 27 ) wdldrh fõ.
tkï y - m1x + 27m + 9 = 0
- 3 - 9m1 + 27m1 + 9
1
f¾Ldj ( x - 9 )2 + ( y + 3 )2 = 22 iam¾Y lrhs kï, = 2
2
1 + m1

ksfrdaIa pdñkao

11 Nirosh Chaminda
 G.C.E. ( Adv. Level ) Examination , August 2009
Scheme of Marking Combined Mathematics Paper I
2 2
tkï ( 18m1 + 6 )2 = 4 ( 1 + m1 )  80m1 + 54m1 + 8 = 0
2
40m1 + 27m1 + 4 = 0
- 27  272 - 640 - 27  729 - 640
m1 = 80
=
80

- 27  89
=
80
T1 yryd hk fmdÿ iam¾Yl fol

y+9 = ( - 27 + 89
80 ) ( x - 27 ) yd

(
- 27 -  89
)

m
y+9 = 80 ( x - 27 )
----------------------------------------------------------------------------------------------------------- 50

o
( 09 ) ( a ) iïu; wxlkhg wkqj ihska kS;sh.

.c
sin A sin B sin C

da
a = b = c
----------------------------------------------------------------------------------------------------------- 05
in
idOkh :- am
ch
sh

(I) ABC iq¿ fldaKS ( II ) ABC RcqfldaKS ( III ) ABC uydfldaKS

;%sfldaKhls. ;%sfldaKhls. ;%sfldaKhls.

iro

AD = AB sin B = AC sin C AD = AB sin B = AC sin C AD = AB sin (  - B) = AC sin C

.n
w

sin B sin C sin B sin C sin B sin C

 b = c  = c  = c
b b
w

tmßÈ
w

sin A sin C sin A sin B sin C

a = c  a = b = c
----------------------------------------------------------------------------------------------------------- 20
( i ) ACD ;%sfldaKhg ihska kS;sh fhdouq.
D
AD h AD sin 
= sin   h=
2 ( )
sin  +  cos 
h
C
ABD  hg ihska kS;sh fhdouq. 
AD x x sin   B
=  AD = x
sin (  -  ) sin (  -  ) sin (  -  )  E
A
ksfrdaIa pdñkao

12 Nirosh Chaminda
 G.C.E. ( Adv. Level ) Examination , August 2009
Scheme of Marking Combined Mathematics Paper I

x sin  sin 
 h =
sin (  -  ) cos 
---------------------------------------------------------------------------------------------------------- 20
( ii ) d = DE = AD sin (  +  )
x sin  x sin  sin (  +  )
AD = , d = sin (  -  )
sin (  -  )
----------------------------------------------------------------------------------------------------------
(b)(i) sin  - cos  = 1 iólrKh.
10
sin (  - /4 ) = 1 f,i m%ldY l< yel.
2
 idOdrK úi÷ï

m
 - /4 = n + ( -1 )n /4 n = 0 úg  = /2

o
n=1 =

.c
 = n + { 1 + ( -1 )n } /4 n 2
----------------------------------------------------------------------------------------------------------- 20

da
( ii ) tan-1 ( 1/2 ) - tan-1 ( 1/3 ) = sin-1 ( x )
 = tan-1 ( 1/2 ) yd  = tan-1 ( 1/3 )
f,i .ksuq.
in 50
am
túg 1
tan  = 1/2 tan  = 1/3
ch

-
1/2 - 1/3 7
tan  - tan  1
sh

tan (  -  ) = = =
1 + tan  tan  1 + 1/ 2  1/ 3 7
iro

1
sin (  -  ) =
50
.n

( )
w

1
 -  = sin-1 50  -  = sin-1 ( x )
w

x = 1 = 2
w

50 10
25

Scheme of marking
2009
ksfrdaIa pdñkao

13 Nirosh Chaminda