COMBINED MATHEMATICS

2010 A/l Pure Maths

ksfrdaIa pdñkao
B.Sc(Engineering) University of Moratuwa
 G . C . E . ( Advanced Level ) Examination,
Scheme of Marking August
Combined2010
G.C.E. ( Adv. Level ) Examination , August 2010
Mathematics Paper I

Combined Mathematics I
Scheme Of Marking
p(y-1)
1 . ( a ) y ( p - x ) = p + x x = if y  -1
y+1
x i|yd f ( x ) = 0 ys wdfoaY lsÍfuka,

{ } { }
2
p(y-1) p(y-1)
f ( x ) = x + px + q =
2
+p + q = 0 fjhs. 05
y+1 y+1
i.e p2 ( y - 1 )2 + p2 ( y2 - 1 ) + q ( y + 1 )2 = 0
g ( y ) = ( 2p2 + q ) y2 + 2 ( q - p2 ) y + q = 0. 05
15

m
yd  hkq f ( x )  x2 + px + q = 0 j._c iólrKfha uQ, neúka,

o
 +  = -p yd  = q fjhs. 05

.c
p+ - 
x =  kï túg, y = p -  = -2 -  = 05
2 +  fjhs.

da
 hkq f ( x )  x2 + px + q = 0 ys uQ,hka neúka th x2 + px + q = 0 iólrKh imqrd,hs.

in
am
tneúka y =
2 +  hkak g ( y ) = 0 iólrKh imqrd,hs.

ta khska, hkq g ( y ) = 0 ys uQ,hla fjhs. 05
2 + 
ch


tf,iu, x =  úg y =
2 +  ,efnhs. 05
sh


by; wdldrfha u ;¾lkhla uÕska hkq g ( y ) = 0 ys uQ,hla nj fmkaúh yelsh. 05
iro

2 + 
g ( y ) = 0 hkq j._c iólrKhla neúka
 yd
 hkq f ( y ) = 0 ys uQ, fjhs.
2 +  2 + 
.n
w

25

( 
) ( 
) {  
} ( 
2 +  2 +  )
) ( 
2 2 2
w

+ = + -2
2 +  2 +  2 +  2 +  05
w

{ } ( )
2
-2 ( q - p2 ) q
= -2 2p2 + q 05
2p2 + q

4 ( q2 - 2qp2 + p4 ) - 2q ( 2p2 + q )
=
( 2p2 + q )2

2 ( q2 - 6qp2 + 2p4 )
= 05
( 2p2 + q )2
15
(b) ( y + ax ) ( y + bx ) ( y + cx ) = y + ( a + b + c ) xy + ( ab + bc + ca ) x y + abc x 05
3 2 2 3

= y3 - 3mx2 y + abc x3 05
= y ( y2 - 3mx2 ) + abc x3
10


01
 G.C.E. ( Adv. Level ) Examination , August 2010
Scheme of Marking Combined Mathematics Paper I

= ( x2 + m ) { ( x2 + m )2 - 3mx2 } + abc x3 ( y = x2 + m hehs ,sùfuka )

= ( x2 + m ) ( x4 - mx2 + m2 ) + abc x3 = x6 + abc x3 + m3 05
i . e . ( x2 + ax + m ) ( x2 + bx + m ) ( x2 + cx + m ) = x6 + abc x3 + m3  (1)
05
g ( x ) = x + 16x + 64 g ( x - 2x + m ) , ( x + ax + m ) yd ( x + bx + m ) idOl ;sfnhs kï,
6 3 2 2 2

x6 + 16x3 + 64 = ( x2 - 2x + m ) ( x2 + ax + m ) ( x2 + bx + m ) hehs ,súh yelsh.

by; ( 1 ) iólrKh Ndú;fhka m3 = 64 yd - 2ab = 16 m = 4 yd ab = -8 fjhs.
;j o, -2 + a + b = 0 fjhs. 05 05
( a - b )2 = ( a + b )2 - 4ab = 4 + 32 = 36 a - b =  6
tneúka, a = -2 yd b = 4 fyda a = 4 yd b = -2 fyda fjhs.
05 05
20
(i) g ( x ) = ( x2 - 2x + 4 )2 ( x2 + 4x + 4 )
= ( x2 - 2x + 4 )2 ( x + 2 )2  0 ( ish¨ x i|yd ) 05

m
05
x = -2 , -2 , 1  3i , 1  10

o
( ii ) g ( x ) = ( x - 2x + 4 ) ( x + 2 ) = 0
2 2 2
3i
10

.c
da
2. ( a ) ( i ) ixLHdxl yf;ka, mqkrdj¾;kh iys;j, iEÈh yels ixLHdxl y;f¾ fjkia ixLHd .Kk
= 7  7  7  7 = 2401 05

in
05
( ii ) ixLHdxl yf;ka, mqkrdj¾;kh rys;j, iEÈh yels ixLHdxl y;f¾ fjkia ixLHd .Kk
am
= 7  6  5  4 = 840 05
05
ch

(i) wjia : dfõ §,

tlu ixLHdxlh y;r jdrhla we;s ixLHdxl y;f¾ fjkia ixLHd .Kk = 7 05
sh

tlu ixLHdxlh ;=ka jdrhla o, fjk;a ixLHdxlhla tla jdrhla o we;s ixLHdxl y;f¾
iro

fjkia ixLHd .Kk = 7  6 = 42

4!
.n

ixLHdxl y;f¾ tjeks fjkia ixLHd ms<sfh< l, yels wdldr .Kk = = 4

3!
w

ixLHdxl y;f¾ tjeks fjkia ixLHd uq¿ .Kk = 42  4 = 168 05

w

ixLHdxlhla jdr follg jvd jeäfhka fkdue;s ixLHdxl y;f¾ fjkia ixLHd uq¿
w

.Kk = 2401 - ( 168 + 7 ) = 2226

05 05
20

fjk;a l%uhla : 05
fjkia ixLHdxl y;rla iys; ixLHdxl y;f¾ fjkia ixLHd .Kk = 7  6  5  4 = 840
tlu ixLHdxlh fojdrhla o, fjk;a fjkia ixLHdxl folla tla jdrh ne.ska o we;s ixLHdxl
7 6
y;f¾ fjkia ixLHd .Kk = C1  C2 = 7  15 = 105
4!
ixLHdxl y;f¾ tjeks fjkia ixLHd ms<sfh< l, yels wdldr .Kk = = 12
2!
ixLHdxl y;f¾ tjeks fjkia ixLHd uq¿ .Kk = 105  12 = 1260 05
fjkia ixLHdxl folla tl tlla fojdrh ne.ska we;s ixLHdxl y;f¾ fjkia ixLHd .Kk
7
= C2 = 21
4!
ixLHdxl y;f¾ tjeks fjkia ixLHd ms<sfh, l< yels wdldr .Kk = = 6
2! 2!
02

 G.C.E. ( Adv. Level ) Examination , August 2010
Scheme of Marking Combined Mathematics Paper I
ixLHdxl y;f¾ tjeks fjkia ixLHd uq¿ .Kk = 21  6 = 126 05
ixLHdxlhla jdr follg jvd jeäfhka fkdue;s ixLHdxl y;f¾ fjkia ixLHd uq¿ .Kk
= 840 + 1260 + 126 = 2226 05
20
( ii ) wjia : dfõ §,
T;af;a ixLHdxl folla yd brÜfÜ ixLHdxl folla we;s ixLHdxl y;f¾ fjkia ixLHd .Kk
3 4
= C2  C2 05
ixLHdxl y;f¾ tjeks fjkia ixLHd ms<sfh< l, yels wdldr .Kk = 4! 05
tneúka, T;af;a ixLHdxl folla yd brÜfÜ ixLHdxl folla we;s ixLHdxl y;f¾ fjkia ixLHd
3 4
uq¿ .Kk = C2  C2  4! = 3  6  24 = 432 05
15
ixLHdjla brÜfÜ ùu i|yd tys wjidk ixLHdxlh brÜfÜ úh hq;=h .
4
wjidk ixLHdxlh brÜfÜ jk ixLHdxl y;f¾ fjkia ixLHd .Kk = C1 05
T;af;a ixLHdxl folla yd brÜfÜ ixLHdxlhla f;dard .; yels wdldr .Kk

m
3 3
= C2  C1 05

o
tjeks tla tla f;dard .ekSula ms<sfh< l< yels wdldr .Kk = 3!

.c
tneúka , T;af;a ixLHdxl folla yd brÜfÜ ixLHdxl folla we;s ixLHdxl y;f¾

da
3 3 4
= C2  C1  C1  3! = 3  3  4  6 = 216 05
15

in
n n n n
(b) C0 + C1 x + ..... + Cr xr + ....... + Cn xn
am
= ( 1 + x )n
= ( 1 + x ) ( 1 + x )n - 1
ch

n-1 n-1 n-1 n-1

= ( 1 + x ) ( C0 + C1 x + ..... + Cr xr + .... + Cn - 1xn - 1 )
n-1
= n - 1C0 + ( n-1C0 + n-1C1 ) x + ....... + ( n-1Cr - 1 + n-1Cr ) xr + ......... + Cn-1 xn 05
sh

Cr fjhs. 05
n n-1 n-1
wkqrEm mo j, ix.=Kl iei£fuka r = 1 , 2 , ....... n - 1 i|yd Cr = Cr-1 +
iro

10
n n n n
.n

C0 - C1 + C2 - ..... + ( -1 )n - 1 + ( -1 )n Cn
w

n n-1 n-1 n-1 n

= C0 - ( C0 + C1 ) + (
C1 + n-1C2 ) - ........... + ( -1 )n - 1 ( n - 1Cn - 2 + n - 1Cn - 1 ) + ( -1 )n Cn
w

10
= 1 - 1 + ( -1 )n - 1 + ( -1 )n = 0 05
w

15
i;Hdmkh:
n n n n
( 1 + x )n = C0 + C1 x + ...... + Cr x r + .... + Cn xn ys x = - 1 hehs fh§fuka,
n n n n n
( 1 - 1 )n = C0 + C1 ( -1 ) + ....... + Cr ( -1 )r + ........ + Cn - 1 ( -1 )n - 1 + Cn ( -1 )n fjhs.
n n n n
i.e. C0 - nC1 + C2 - .......... + ( -1 )n - 1 Cn - 1 + ( -1 )n Cn = 0 05
05
n hkq brÜfÜ ksÅ,hla kï,
n n n n n
C0 - C1 + C2 - ......... - Cn - 1 + Cn = 0  ( 1 ) fjhs.
n n n n
( 1 + x ) = C0 + C1 x + ....... + Cr x + ........ + Cn xn ys x = 1 hehs fh§fuka,
n r

n n n n n
( 1 + 1 )n = C0 + C1 + ....... + Cr + ........ + Cn - 1 + Cn  ( 2 ) fjhs. 05
n n n n 05 10
(1)+(2)  C0 + C2 + C4 ....... + Cn = 2n- 1

03
 G.C.E. ( Adv. Level ) Examination , August 2010
Scheme of Marking Combined Mathematics Paper I

3. 4p3 - 6n2 + 4n - 1 = n4 - ( n - 1 )4  ( 1 )
n = 1 úg, ( 1 ) ys j . w . me . = 1 = ( 1 ) ys o . w . me . 05
tneúka n = 1 úg, m%;sM,h i;H fõ.
n = p úg, m%;sM,h i;H hehs Wml,amkh lruq.
tkï, 4p3 - 6p2 + 4p - 1 = p4 - ( p - 1 )4 m%;sM,h i;H hehs Wml,amkh lruq. 05
n = p + 1 hehs .ksuq.
4 ( p + 1 )3 - 6 ( p + 1 )2 + 4 ( p + 1 ) - 1 = 4 ( p3 + 3p2 + 3p + 1 ) - 6 ( p2 + 2p + 1 ) + 4 ( p + 1 ) - 1
= 4p3 + 6p2 + 4p + 1 = p4 + 4p3 + 6p2 + 4p + 1 - p4 = ( p + 1 )4 - p4 05 05
tneúka n = p + 1 úg, m%;sM,h i;H fõ.
.Ks; wNHQyk uQ, O¾uh uÕska ´kEu n Ok ksÅ,hla i|yd m%;sM,h i;H fõ.
05
25
m<uq fldgi Ndú;fhka 4r3 - 6r2 + 4r - 1 = r4 - ( r - 1 )4 = ur - ur - 1 hehs ,súh yelsh. fuys ur = r4 fjhs.
05

m
05

o
r = 1 úg, u1 - u0 = 4 . 1 3 - 6 . 1 2 + 4 . 1 - 1

.c
r = 2 úg, u2 - u1 = 4 . 2 3 - 6 . 2 2 + 4 . 2 - 1

da
.............................................
.............................................

in
05
.............................................
am
r = n - 1 úg, un - 1 - un - 2 = 4 . ( n - 1 )3 - 6 . ( n - 1 )2 + 4 . ( n - 1 ) - 1
r = n úg, u n - un - 1 = 4 . n3 - 6 . n2 + 4 . n - 1
_______________________________________________
ch

 r -6  r +4 r-n
n n n

un - u 0 = 4 3 2 05
sh

r=1 r=1 r=1

 r - 4 r + n + u - u
n

r
n n
iro

4 3
= 6 2
n 0
r=1 r=1 r=1
.n

n ( n + 1 ) ( 2n + 1 ) n(n+1)
= 6 -4 + n + n4 05
w

6 2
w

= n ( n + 1 ) ( 2n + 1 - 2 + n2 - n + 1 ) = n2 ( n + 1 )2 05

 ( )
n 2
n(n+1)
w

r3 =
r=1
2
20
12 + ( 12 + 22 ) + ( 12 + 22 + 32 ) + ( 12 + 22 + 32 + 42 ) + .....
r ( r + 1 ) ( 2r + 1 ) 1 3 1 2 1
vr = 12 + 22 + ....... + r2 = = r + r + r
6 3 2 6
05
05

 v =  r ( r + 1 )6( 2r + 1 )
n


n n

r
n

r
n
1 1 1
= 3
+ 2
+ r 05
r=1
r
r=1
3 r=1
2 r=1
6 r=1

=
1
3 { n(n+1) 2 1
2
+
2 } { n ( n + 16) ( 2n + 1 ) } +
1
6 { n ( n2+ 1 ) }
05

04
 G.C.E. ( Adv. Level ) Examination , August 2010
Scheme of Marking Combined Mathematics Paper I

n(n+1) n(n+1) 2 n ( n + 1 )2 ( n + 2 )
= ( n + n + 2n + 1 + 1 ) =
2
( n + 3n + 2 ) = 05
12 12 12
15


n
lim n ( n + 1 )2 ( n + 2 )
n  vr = lim
n  = 
r=1
12
wkka;hg ftlHh mßñ; fkdjk neúka fYa%Ksh wNsidÍ fkdfõ. 05
05

3 5 9
+ 2 + 2 72 + 2 + ........
1 2
1 +2 2
1 +2 +3 2 1 + 2 + 32 + 42
2

2r + 1

m
wr = 05
1 + 22 + ..... + r2
2

o
05

.c
2r + 1 6 6 6
= = r(r+1) = r - 6 = f ( r ) - f ( r + 1 ) ; fuys f ( r ) =

da
fõ.
{ r ( r + 1 )6( 2r + 1 ) } r+1
05
r

in
05
am
r = 1 úg, w1 = f ( 1 ) - f ( 2 )
r = 2 úg, w2 = f ( 2 ) - f ( 3 )
...............................
ch

............................... 05
...............................
sh

r = n - 1 úg, wn - 1 = f ( n - 1 ) - f ( n )
r = 2 úg, wn = f ( n ) - f ( n + 1 )
iro

_____________________

w
n
.n

Sn = = f(1)-f(n+1) = 6- 6 05
r
n+1
w

r=1
10

( )
w

lim 6 6 6
n  Sn = lim
n  6- = 6 - lim
n  = 6 lim
n  = 0.
n+1 n+1 n+1
w

wkka;hg ftlHh mßñ; neúka fY%aKsh wNsidÍ fõ. 05

05
4. ( a ) z = x + iy hehs .ksuq.
túg, z - a = ( x - a ) + iy yd | z + a | = ( x + a ) + iy fõ.
| z - a | = ( x - a )2 + y2 yd | z + a | = ( x + a )2 + y2 05
| z - a | = | z + a |  ( x - a )2 + y2 = ( x + a )2 + y2
( x - a )2 + y2 = ( x + a )2 + y2  4ax = 0  x = 0 05
tneúka | z - a | = | z + a | imqrd,kq ,nk z ixlS¾K ixLHdfõ m:h wd¾.ka ;,fha w;d;a;aúl
wCIh fõ. 05
15


05
 G.C.E. ( Adv. Level ) Examination , August 2010
Scheme of Marking Combined Mathematics Paper I
fjk;a l%uhla : p(z)

wd¾.ka igyfkys P ,CIHHfhka z ixlS¾K ixLHdj

ksrEmKh jkafka hehs .ksuq. 05

AP = | z - a | yd BP = | z + a | fõ; fuys A yd B
hkq wd¾.ka igyfkys a yd - a ixLHd ksrEmKh
lrkq ,nk ,CIH fõ.
B ( -a ) O A(a)
| z1 - a | = | z1 + a | AP = BP 05

tneúka | z - a | = | z + a | imqrd,kq ,nk P ,CIHh , AB ys ,ïn iuÉfþolh tkï, wd¾.ka ;,fha

w;d;a;aúl wCIh úp,kh fjhs. 05
15

m
z1 z1
(b) | z1 - 2z2 | = | z1 + 2z2 | 
z2 - 2 = z2 + 2 05

o
.c
z
( a ) fldgi wkqj z1 ixlS¾K ixLHdj ksrEmKh lrkq ,nk ,CIHh , wd¾.ka ;,fha w;d;a;aúl

da
2

wCIh u; msysghs. 05

in
i z1
tneúka ixlS¾K ixLHdj ksrEmKh lrkq ,nk ,CIHh , wd¾.ka ;,fha ;d;a;aúl wCIh
am
z2
u; msysghs. 05
ch

i z1 iz
ixlS¾K ixLHdj z 1 = k hehs ,súh yelsh; fuys k ;d;a;aúl fõ. 05
z2
sh

20
z1
iro

(i) z2 ixlS¾K ixLHdj ksrEmKh lrkq ,nk ,CIHh , wd¾.ka ;,fha w;d;a;aúl wCIh u;
z
( )
msysgk neúka arg z1 =   fõ; fuys Ok w.h w;d;a;aúl wCIfha Ok me;a; u; ,CIHhlg
.n

2 2
w

wkqrEm jk w;r RK w.h w;d;a;aúl wCIfha RK me;a; u; ,CIHhlg wkqrEm fõ. 10

w

arg z
z1
( )
=

2
 | arg ( z1 ) - arg ( z2 ) | =

2
z
( )
arg z1 = arg ( z1 ) - arg ( z2 )
w

2 2
05
15
P1 ( z1 + 2z2 )
( ii )
05
)
10 R ( z1 | z1 - 2z2 | = | z1 + 2z2 | OP2 = OP1 fõ.
P ( 2z )
2
;j o,
 RP1 = OP = | 2z2 | = | -2z2 | = OQ = RP2 fõ.
P2 ( z1 + 2z2 ) 
^ =  fõ.
ORP
0 1 2 05

Q ( - 2z2 )


06
 G.C.E. ( Adv. Level ) Examination , August 2010
Scheme of Marking Combined Mathematics Paper I

RP OP | 2z | z 2 iz
tan  = OR1 = = | z 2| = 2 z2 =  z 1 = k
OR 1 1 |k| 2
05 05
OP1 hkak OP2 g ,ïN fkdfõ kï túg,     tan   1  | k |  2 05
4
4
2 tan  |k| 4|k|
tan 2 = = 4 = 2 05
1 - tan2  1- k -4
| k |2

i.e. ^ = tan-1
P1OP 2 ( 4k |-k4| )
2

40

OP1 hkak OP2 g ,ïN fõ kï túg,  =
4

m
2 10
tan  = = 1  |k| = 2  k = 2

o
|k|

.c
10
lim 1 - cos 4x + x sin 3x 1 - cos 4x
+ lim ( sin3x3x )

da
5. ( a ) x 0 = lim
x 0 x 0 3 05
x2 x2

in ( sin3x3x )
2 sin2 2x
= lim + 3 lim
am
x 0 x 0
x2

{ }+3 ( sin3x3x ) = 1
2
lim sin 2x lim 05
= 8
ch

x 0 x 0
2x
sh

= 8 + 3 = 11 lim
x 0 ( sin2x2x ) = 1 05
iro

15
.n

(b)(i) z = tan-1 x  x = tan z

w

y = tan-1 ( 1 + x2 - 1
) ys x = tan z hehs wdfoaYfhka,
w

x
w

y = tan-1 ( 1 + tan2 z - 1
tan z ) ,efnhs. 05

( ( )
)
z
2 sin2
( sectanz z- 1) = tan (1 -sincosz z ) = tan
2
= tan-1 -1 -1
05
05 ( ) ( )
2 sin2
z cos z
2 2

= tan-1 tan z
2 ( ( )) =
z
2
05
dy 1
= 05
dz 2
25


07
 G.C.E. ( Adv. Level ) Examination , August 2010
Scheme of Marking Combined Mathematics Paper I

fjk;a l%uhla : y = tan-1 ( 1 + x2 - 1

x
)
x. 1 . 2x - { 1 + x2 - 1 }
dy 1 2 1 + x2
= . 05
dx x2
( )
2
1+x -1
2

1+ 05
x
1 1 + x2 - 1 1
= = 05
2(1+x - 2
1+x )
2
1+x 2 2 ( 1 + x2 )

dz 1
z = tan-1 x  = 05
dx 1 + x2

dy dy dx 1 ( 1 + x2 ) 1

m
= = = 05
dz dx dz 2 ( 1 + x2 ) 1 2

o
25

.c
da
( ii ) y = em sin -1 x
dy m

in
x úIfhka wjl,kh lsÍfuka = em sin -1 x ,efnhs. 05
dx 1 - x2
am
dy
1 - x2 = my 05
dx
by; iólrKh x úIfhka wjl,kh lsÍfuka,
ch

d 2y 2x dy = m dy ,efnhs.
1 - x2 -
sh

05
dx 2
2 1 - x2 dx dx
iro

d 2y dy dy
( 1 - x2 ) -x = m 1 - x2 = m2 y
dx 2
dx dx
.n

d 2y dy
w

i . e . ( 1 - x2 ) - x - m2 y = 0
dx 2 dx
w

20
w

fuu iólrKh x úIfhka wjl,kh lsÍfuka,

d 3y d 2y d 2y dy dy
( 1 - x2 ) - 2x - x - - m2 = 0 ,efnhs. 05
dx 3
dx 2
dx 2
dx dx

d 3y d 2y dy
( 1 - x2 )3 - 3x - ( 1 + m2 ) = 0
dx dx 2 dx
dy d 3y
x = 0 úg y = 1 , = m yd dx 3 = m ( 1 + m )
2
dx
05 05
15
x 1-x
(c) jD;a;fha wrh o, iup;=r>%fha me;a;l È. o fõ. 05
2 4

( ) x
2
x
= j._ tall x .
2
tneúka jD;a;fha j._M,h =  2
2 4x


08
 G.C.E. ( Adv. Level ) Examination , August 2010
Scheme of Marking Combined Mathematics Paper I
( 1 - x )2
iup;=r>%fha j._M,h = j._ tall 1-x
16
x2
(1-x)2 4
A ( x ) = j._ tall + 05
4x 16
10
dA ( x )
dx =
x
2x
-
(1-x)
8
=
( 4 +  ) x - l
8 =
4+
8 (
x - l
4+ )( )
05
dA ( x )
dx
= 0  x- (
l
4+
= 0)  x =
l
4+
05

l dA ( x ) l dA ( x )
x < úg , < 0 yd x > úg , > 0
4+ dx 4+ dx

m
x = l yS § A ( x ) wju fõ.
4+

o
x = l úg jD;a;fha úYalïNh l

.c
yd iup;=r>%fha me;a;l È.
4+ 4+

da
1
(
1 - l ) =
l
fõ. 05

in
4 4+ 4+ am
ta khska wmg m%;sM,h ,efí.
10
2x Ax + B C D
ch

6. ( a ) ( 1 + x2 ) ( 1 + x )2  1 + x2 + 1 + x + ( 1 + x )2 05
sh

2x = ( Ax + B ) ( 1 + x )2 + C ( 1 + x ) ( 1 + x2 ) + D ( 1 + x2 )
= ( A + C ) x3 + ( 2A + B + C + D ) x2 + ( A + 2B + C ) x + ( B + C + D )
iro

wkqrEm mo j, ix.=Kl iei§fuka,

A + C = 0 , 2A + B + C + D = 0 , A + 2B + C = 2 yd B + C + D = 0 fõ.
.n

by; iólrK l=,lh úi§fuka A = C = 0 , B = 1 and D = -1 ,efnhs.

05 05
w

05 05
2x dx 1
 dx 2= tan-1 x +
w

 2 dx =  2 - + K ; fuys K hkq wNsu; ksh;hls.

(1+x )(1+x)
2
1+x (1+x) 1+x
05 05
w

05
40
(b)(i) bI = b  eax cos bx dx = eax sin bx - a  eax sin bx dx = e ax sin bx - aJ
05 05
i . e . bI + aJ = e sin bx  ( 1 )
ax

10

( ii ) - bJ = -b  eax sin bx dx = eax cos bx - a  eax cos bx dx = eax cos bx - aI

05 05
i . e . aI - bJ = eax cos bx ( 2 )
10

b  ( 1 ) + a  ( 2 )  ( a2 + b2 ) I = eax ( b sin bx + a cos bx )


09
 G.C.E. ( Adv. Level ) Examination , August 2010
Scheme of Marking Combined Mathematics Paper I

eax
i.e. I = ( b sin bx + a cos bx ) 05
( a2 + b2 )
05
a  ( 1 ) - b  ( 2 ) ( a2 + b2 ) J = eax ( a sin bx + b cos bx )

i.e. J = eax ( a sin bx - b cos bx ) 05

( a + b2 )
2

05
1
(c) x3 t + 1 = 0 x3 = -
t
dx
05 3x2 t + x3 dt = 0  = - dt , x = -1 úg, t = 1 yd x = - 1 úg, t = 8 fõ. 05
dx x 3t 2
-1 - dt
2 dx 8 3t 1 8 dt 1 1 1 9

m
8
 x ( x3 - 1 ) =  =  = [ In | 1 + t | ] = ( In 9 - In 2 ) = In
( ) 3 1+t 3 3 2
1
-1 1 1 1 3
- -1

o
t 05 05
05

.c
05

da
30

in
7. ( a ) P0 ( x0 , y0 ) hkq fok ,o f¾Ld fol w;r fldaK iuÉfþolhla u; ´kEu ,CIHhla hehs .ksuq .
| a1 x0 + b1 y0 + c1 | | a2 x0 + b2 y0 + c2 |
am
túg P0 P1 = yd P0 P2 = fõ. 05
a 12 + b 12 a 22 + b 22
ch

a1 x0 + b1 y0 + c1 a2 x0 + b2 y0 + c2
P0 P 1 = P0 P2  05 = 
a 12 + b 12 a 22 + b 22
sh

P0 ( x0 , y0 ) hkq fldaK iuÉfþolhla u; ´kEu ,CIHhla neúka , iuÉfþolhkaf.a iólrK

iro

a1 x + b1 y + c1 a2 x + b2 y + c2
=  f,i ,efnhs. 05
a 12 + b 12 a 22 + b 22
.n

15
w

(b) ( x , y ) yd ( x0 , y0 ) ,CIH w;r ÿr

w

( a2 + b2 ) t2 = | t |  a2 + b2 = 1
w

= ( x - x0 )2 + ( y - y0 )2 = ( at )2 + ( bt )2 =
05 05
10

(c)(a) fldgi Ndú;fhka BAD fldaKfha iuÉfþol

j, iólrK
05 D C
x - 2y + 5 2x - y + 1
=  f,i ,efnhs. 2x - y + 1 = 0
1+4 4+1 E

i.e. x + y - 4 = 0 yd y - x - 2 = 0 fõ.  B
A x - 2y + 5 = 0
05 05
AC hkq tla fldaK iuÉfþolhla jk w;r th
BAD ys iq¿ fldaKhg wkqrEm fõ. 05

tneúka AC yd , AB fyda AD fyda w;r fldaKh g wvq úh hq;=h . fuu fldaKh  hehs
4
.ksuq. AD yd y - x - 2 = 0 uÕska fokq ,nk f¾Ldj ie,lSfuka,
10

 G.C.E. ( Adv. Level ) Examination , August 2010
Scheme of Marking Combined Mathematics Paper I

2-1 1
tan  = = 3 < 1 ,efnhs. 05
2+1
tneúka y - x - 2 = 0 hkq AC f¾Ldjg wkqrEm iólrKh fõ. 05
10

y - x - 2 = 0 yd 2x - y + 1 = 0 úi£fuka A = ( 1 , 3 ) hehs ,efnhs. 05

y1 - 3 x1 - 3
( b ) fldgi Ndú;fhka AC ys iólrKh 05 = = | 2 5 | f,i ,súh yelsh;
1/ 2 1/ 2
fuys ( x1 y1 ) hkq A isg 2 2 ÿrlska AC u; msysá ,CIHhls. ( x1 y1 ) i|yd úh yels hq.,
( 3 , 5 ) yd ( -1 , 1 ) fõ.
frdïnih mq¾K f,i m<uq mdolh ;=< msysgk neúka C = ( 3 , 5 ) fõ. 05
BC f¾Ldj AD g iudka;r ksid BC ys iólrKh 2x - y + = 0 f,i ,súh yelsh;

m
fuys  hkq mrdñ;shls. BC f¾Ldj ( 3 , 5 ) Tiafia hk neúka  = -1 fõ.
tneúka BC f¾Ldfõ iólrKh 2x - y - 1 = 0 fõ. 05

o
20

.c
da
DC f¾Ldj AB g iudka;r ksid DC ys iólrKh x - 2y +  = 0 f,i ,súh yelsh.
fuys  hkq mrdud;shls.

in
DC f¾Ldj ( 3 , 5 ) Tiafia hk neúka  = 7 fõ.
tneúka DC f¾Ldfõ iólrKh x - 2y + 7 = 0 fõ. 05
am
05
ch

1 2
DE = AE tan  = 2 3 =
3
sh

05 05
10
iro

1
ABCD ys j._M,h = 4  AED ys j._ M,h = 4  DE . AE
2
.n

1 2 4
= 4 2 = j._ tall
2 3 3
w

05 05
30
w

8. jD;a; foflys flaJø w;r ÿr d hehs .ksuq.

w

túg d = ( g1 - g2 )2 + ( f1 - f2 )2 fõ.
r1 yd r2 hkq jD;a; foflys wrhka hehs .ksuq.

túg r1 = g12 + f12 - c1 yd r2 = g22 + f22 - c2 fõ.

jD;a; fol w;r wNHka;r f,i tlsfkl iam¾Y ùu i|yd wjYH;dj d = | r1 - r2 | fõ. 05
jD;a; fol ndysr f,i tlsfkl iam¾Y ùu i|yd wjYH;dj d = r1 + r2 fõ. 05
10
( P1 C )2 = ( x1 + g )2 + ( y1 + f )2
P0
( P0 C )2 = g2 + f 2 - c
( P0 P1 )2 = ( P1 C )2 - ( P0 C )2
= ( x1 + g )2 + ( y1 + f )2 - ( g2 + f 2 - c ) 05
P1 ( x1 , y1 )
= x 2 + y 2 + 2gx + 2f y + c C ( -g , -f )
1 1 1 1

 P0 P1 = x12 + y12 + 2gx1 + 2f y1 + c 05 10


11
 G.C.E. ( Adv. Level ) Examination , August 2010
Scheme of Marking Combined Mathematics Paper I

S1  x2 + y2 + 4x - 2y - 5 = 0  C1 = ( -2 , 1 ) yd r1 = 4 + 1 + 5 = 10 05
S2  x2 + y2 - 8x - 6y + 15 = 0  C2 = ( 4 , 3 ) yd r2 = 16 + 9 - 15 = 10 05

C1 C2 = ( 4 + 2 )2 + ( 3 - 1 )2 = 2 10 = r1 + r2 05
tneúka S1 = 0 yd S2 = 0 jD;a; fol ndysr f,i tlsfkl iam¾Y fõ.
15
A hkq C1 C2 ys uOHh ,CIHh neúka A = ( 1 , 2 ) fõ. 05
05
P = ( x1 , y1 ) hehs .ksuq.
2 2
túg P ,CIHfha isg S1 = 0 jD;a;hg we¢ iam¾Ylhl È. x1 + y1 + 4x1 - 2y1 - 5 fõ.
2 2
P ,CIHfha isg S2 = 0 jD;a;hg we¢ iam¾Ylhl È. x1 + y1 - 8x1 - 6y1 + 15 fõ.
2 2 2 2
x1 + y1 + 4x1 - 2y1 - 5 = k2 ( x1 + y1 - 8x1 - 6y1 + 15 ) 05

m
2 2
( 1 - k2 ) x1 + ( 1 - k2 ) y1 + ( 4 + 8k2 ) x1 - ( 2 - 6k2 ) y1 - 5 - 15k2 = 0

o
(i) k = 1 kï 12x1 + 4y1 - 20 = 0 fjhs.

.c
tneúka P ,CIHfha m:h, wkql%uKh - 3 jk 3x + y - 5 = 0 ir, f¾Ldj fõ. 05

da
A ,CIHfha LKavdxl jk ( 1 , 2 ) , 3x + y - 5 = 0 iólrKh imqrd,hs. 05
3-1 1

in
C1 C2 ys wkql%uKh = = fõ.
4+2 3
am
( 3x + y - 5 = 0 ys wkql%uKh ) ( C1 C2 ys wkql%uKh ) = ( -3 )
1
3
= -1 fõ. ( )
ch

tneúka P ,CIHfha m:h , C1 C2 g ,ïnj A Tiafia hk f¾Ldj fõ. 05

20
sh

( ii ) k  1 kï 1 - k2  0 fjhs.
iro

tneúka P ,CIHfha m:h ( 1 - k2 ) x2 + ( 1 - k2 ) y2 + ( 4 + 8k2 ) x - ( 2 - 6k2 ) y - 5 - 15k2 = 0 05

iólrKh uÕska fokq ,nk jD;a;hla fõ.
.n

( 1 , 2 ) LKavdxl by; iólrKfha wdfoaYfhka,

( 1 - k2 ) 12 + ( 1 - k2 ) 22 + ( 4 + 8k2 ) ( 1 ) - ( 2 - 6k2 ) ( 2 ) - 5 - 15k2 = 0 ,efnhs.
w

tneúka jD;a;h A Tiafia hhs. 05

w

10
1 3 12 3 12
w

k = kï x + y + 6x1 - 1 y1 - 35 = 0 ,efnhs.
2 4 4 2 4
jD;a;fha iólrKh S3  x2 + y2 + 8x - 2 y - 35 = 0 fõ. 05
3 3
05

C3 = ( -4, 1
3 )yd r3 = 16 +
1 35
9
+
3
=
5 10
3
fõ. 05

( ) 8 10 05
2
C2 C3 = 8 =
8 +
2
3
3
5 10
= + 10 = r3 + r2 05
3
tneúka S3 = 0 yd S2 = 0 jD;a; fol ndysr f,i tlsfkl A ,CIHfha § iam¾> fõ.
15
12
 G.C.E. ( Adv. Level ) Examination , August 2010
Scheme of Marking Combined Mathematics Paper I

( ) 2 10
2
2 05
C1 C3 = 22 + = 3
3

5 10
= 3 - 10 = r3 - r1 05

tneúka S3 = 0 yd S1 = 0 jD;a; fol wNHka;r f,i tlsfkl A ,CIHfha § iam¾Y fõ.

30
a +b +c
2 2 2
9. ( a ) fldaihska kS;sh : cos C = 05
2ab
05

m
Proof : AB2 = AC2 + AD2
A
= ( BC - DC )2 + ( AC2 - DC2 )

o
c b = BC2 + AC2 - 2BC . DC

.c
= BC2 + AC2 - 2BC . AC cos C
B C c2 = a2 + b2 - 2ab cos C

da
a D
a2 + b2 + c2
(i) ABC hkq iq¿ flda” cos C =

in
2ab 05
;%sfldaKhls. 05
am
A
AB2 = AC2 + BC2
ch

b
c = AC2 + BC2 - 2AC . BC cos C  C = 
c2 = a2 + b2 - 2ab cos C 2
sh

C=D a B
iro

a2 + b2 + c2
( ii ) ABC hkq Rcq flda” cos C =
2ab
;%sfldaKhls. 05
.n
w

A AB2 = AD2 + DB2

w

c = AC2 - DC2 + ( DC + BC )2
b = AC2 + BC2 + 2BC . AC cos (  - C )
w

B = BC2 + AC2 - 2BC . AC cos C

D C a
c =
2
a2 + b2 - 2ab cos C
a2 + b2 + c2
( iii ) ABC hkq uyd flda” cos C =
2ab 05
;%sfldaKhls.
20

cos A cos B cos C b2 + c2 - a2 c2 + a2 - b2 a2 + b2 - c2

(i) + b + c = + + 05
a 2abc 2abc 2abc

2 ( cos A cos B
a
+ b
cos C
+ c ) =
a2 + b2 + c2
abc
05

10


13
 G.C.E. ( Adv. Level ) Examination , August 2010
Scheme of Marking Combined Mathematics Paper I

1 1 3
( ii ) + = a+b+c
a+c b+c

a + b + 2c 3
= a+b+c
(a+c)(b+c)
( a + b + 2c ) ( a + b + c ) = 3 ( a + c ) ( b + c ) 05
( a + b )2 + 3c ( a + b ) + 2c2 = 3 ( ab + bc + ac + c2 )
a2 + b2 - c2 = ab 05
a2 + b2 - c2 1 
cos C = =  C = 05
2ab 2 2
15

{ }

m
3 cos  + 1 sin 
(b) 3 cos  + sin  = 2
2 2

o
( )

.c

= 2 cos cos  + sin  sin 
6 6

da
= 2 cos -  ( 05 )
in
6

= R cos (  -  ) ; fuys R = 2 yd  = fõ. 05
am
6
15
3 cos2  + ( 1 - 3 ) sin  cos  - sin2  = cos  - sin 
ch

( 3 cos  + sin  ) ( cos  - sin  ) = cos  - sin  05

sh

( 3 cos  + sin  - 1 ) ( cos  - sin  ) = 0

3 cos  + sin  = 1 or cos  - sin  = 0 05
iro

cos  - sin  = 0  tan  = 1


tneúka idOdrK úi÷u  = n + fõ; fuys n hkq ksÅ,hls. 05
.n

( )
w

3 cos  + sin  = 1  2 cos  -  = 1

6
w

cos ( 1
- 
2 6 )
= cos 
3
=
w

 
 = 2m  + ; fuys m hkq ksÅ,hls. 05
3 6
20

( c ) ( 1 ) wjia:dj x Ok hehs is;uq.


 = cos-1 x hehs .ksuq. túg 0 yd w;r  fõ. 05
2
x = cos 
tneúka - x = - cos  = cos (  -  ) fõ. 05
fuhska cos-1 ( -x ) =  - cos-1 x ,efnhs.

( 2 ) wjia:dj x RK hehs is;uq.

x = -y hehs ,shuq; fuys y Ok fõ.
( 1 ) wjia:dfjka cos-1 ( -y ) =  -  =  - cos-1 y ,efnhs.


14
 G.C.E. ( Adv. Level ) Examination , August 2010
Scheme of Marking Combined Mathematics Paper I

i.e. cos-1 x =  - cos-1 ( -x ) 05

( 3 ) wjia:dj x = 0 hehs is;uq.

 
túg cos-1 x = cos-1 0 = yd cos-1 ( -x ) = cos-1 0 = fõ.
2 2
fuhska cos-1 x =  - cos-1 ( -x ) .uH fõ. 05 20

o m
.c
da
in
am
ch
sh
iro
.n
w
w
w


15