Page 1
DCEB-KDP
1
p,q q o
pq
n,mq2 5n m
pxq
p,q x
q2 5n m px
q
(i) a, x 1a nx a log xa n
(ii)
(i) log log logxy x ya a a
(ii) log log logx ya a a
xy
(iii) xalog log
mxa m
(vi) log 1aa
(v) 1log 0a
QZ
N
Q
(R)
N W Z Q R R=QUQ1
Page 2
DCEB-KDP
2
(Short Answer Questions) (S.A.Q.)
i)1325 ii)
143110
i)2 3
log p qr
ii) 3
2log xy iii) 2 3 4log x y z iv)
128log625
log N N
i) log 2 log 5 ii) log10 2log 3 log 2 iii) 3 log 4 iv) 2 log3 - log 2
(Very Short Answer Questions) (V.S.A.Q.)
34
pq
93log x
i) 2433log ii)
2
116log
- (Essay Questions) (E.Q) 3
5 3
3 2 5
12
(a) 34 (b)
43 (c)
65 (d) 0
7.7
(a) (b) (c) (d)
Page 3
DCEB-KDP
3
(a)23 (b)
1625
(c) 8 (d) 0.04
pq
(a) 3
72 (b)
716 (c)
38 (d) 2
72
10log 0.01
(a) 2 (b) 3 (c) -2 (d) -3
3 2
232 5x _________________
8 2x _________________
102 1024 _________________
log10000 _________________
2log16 2log _________________
11000log _________________
343log125 _________________
2562log _________________
Group A Group B
26. log mx [ ] a) 1
27. log mx [ ] b) 0
28. 1loga [ ] c) -1
29. logaa [ ] d) -2
30. 0.110log [ ] e) 2
f) m log xg) log m + log x
Page 4
DCEB-KDP
4
2220 2 5 11 (1/2 Mark)
2284 2 71 (1/2 Mark)
22 4 (1/2 Mark)
22 5 11 71 15,620 (1/2 Mark)
2. (i) 2
13 1325 5
(1/2 Mark)
=2
2 2
13 25 2
(1/2 Mark)
= 2
5210 (1/2 Mark)
= 0.52
3. (i) 2 3
2 3log log log logp q p q rr
(1 Mark)
2log 3log logp q r (1 Mark)
4. (i) log 2 log 5 log(2 5) (1 Mark)
= log 10
(ii) log 10 + 2 log3 - log 2
2log10 log3 log 2 (1/2 Mark)
210 3log2
(1/2 Mark)
5
10log9
2
(1/2 Mark)
= log 45 (1/2 Mark)
5. 156= 2 x 78
= 2 x 2 x 39
= 2 x 2 x 3 x 13 (1/2 Mark)
= 2 1 12 3 13 (1/2 Mark)
Page 5
DCEB-KDP
5
6. (1 Mark)
0 14
24
34
4 14
3
3750.37510
3
3 3
3 52 5
3
32
38
93log x
9 3x (1/2 Mark)
log x nm n x m (1/2 Mark)
9. (i) 2433log x
243 3x53 3x
5 x
3log 243 5x
3
3 rs
,r s
3 ab
,a b
3b a
(1/2 Mark)
(1/2 Mark)
(1/2 Mark)
(1/2 Mark)
Page 6
DCEB-KDP
6
2 23b a 2a
a (1) (1 Mark)
3a C C 2 23b a
2 23 (3 )b C
2 2 2 23 9 3b C b C 2b
b (2) (1 Mark)
,a b
,a b(1/2 Mark)
(1/2 Mark)
3 (1/2 Mark)
5 3 (1/2 Mark)
5 3 ab
,a b(1/2 Mark)
5 3ab
(1/2 Mark)
ab
5 ab
(1/2 Mark)
5 3ab
(1/2 Mark)
3 (1/2 Mark)
(1/2 Mark)
5 3 (1/2 Mark)
3 2 5 (1/2 Mark)
3 2 5 ab
,a b (1/2 Mark)
3 2 5ab
(1/2 Mark)
Page 7
DCEB-KDP
7
ab
3 ab
(1/2 Mark)
3 2 5ab
2 5 (1/2 Mark)
(1/2 Mark)
(1/2 Mark)
3 2 5 (1/2 Mark)
13) a 14) a 15) c 16) c 17) c
18. 3 2 3 3 3
23 23 5 115 0.1152 5 2 5 10
19. 288 2 logx x
20. 1022 1024 10 log 1024
21. 4 4log10000 log 2 5 4 4log 2 log5
4log 2 4log 5
4 log 2 log 5
22. 2log16 2log 2 log16 log 2
log16 log 4
4
16log4
log 4
23. 0
24.343log log343 log125125
Page 8
DCEB-KDP
8
3 3log 7 log 5
3log 7 3log 5
3 log 7 log5
25.8256 2 2
2 2 2log log 8log
8(1) = 8
26. f 27) g 28) b 29) a 30) c
Page 9
DCEB-KDP
9
'' ''
AB AB A B B A
A, B A,B A B
A B x A x B
A,B A B /A B x x A x B
A,B A-B B-A /A B x x A x B
/B A x x B x A
A B B A
(Short Answer Questions) (S.A.Q.) 1, 2,5 , B= 2,3,4,5A A B A B
( ) / , 7i A x x N x
( ) /ii B x x 'School'
Page 10
DCEB-KDP
10
2: 25A x x 6 15x
1,3,5,7 , 1,2,3,4,6A B ,B A A B
( ) , ,i x y z ( ) , , ,ii a b c d
1, 2,3, 4 , 1, 2,3, 4,5,6,7,8A B ,A B A B
1, 2,3,4,5 , 4,5,6,7A B A B B A ,A B B A
(Very Short Answer Questions) (V.S.A.Q.)
(i)A.
(ii)B.
A P A P
2,4,6,8,10 : 2 1x x n x N
1, 2,3A 3, 4,5B A B
n A B
0, 2, 4 ,A A A A
- (Essay Questions) (E.Q) :A x x
:B x x
:C x x :D x x
( )i A B ( )ii A B ( )iii C D ( )iv A C
3,6,9,12,15,18, 21 , 4,8,12,16, 20A B 2,4,6,8,10,12,14,16 , 5,10,15,20C D
( )i A B ( )ii D B ( )iii A C ( )iv B D
Page 11
DCEB-KDP
11
:A x x 1,2,3A
a) A B b) A B c) A B B d) 0A B
a) A B b) B A c) A B d) A B
1,2,3A
a) 3 b) 8 c) 9 d) 6
, , , , ,V a e i o u , , ,B a i k u V B
a) , , , ,a e i o u b) , , ,a e o u c) ,e o d) , ,a e o
a) 1,0 , ,A B a b b) , , ,A a o B b o
c) 3,6,9 , 6,3,9A B d) 1,3,5 , 3,5,7A B
a)
b) 1, 2,3, ,99,100
c) d)
,A B a) A B b) A B c) A B d) B A
A
a) b) A c) 0 d)
A B a) /x x A x B
b) /x x A x B
c) /x x A x B
d) /x x B x A
A B
Page 12
DCEB-KDP
12
:C x x 8 _________________
5, 25,125,625B _________________
:A x x _________________
,A B A B _________________
_________________
_________________
_________________
' 'n _________________
A B A B A B ,A B_________________
A B _________________
:x x A x B [ ] a) A-B
:x x A x B [ ] b) A B
:x x A x B [ ] c) B-A
:x x B x A [ ] d) A B
A B B A [ ] e) A Bf) A=B
1,2,5A 2,3, 4,5B
1, 2,5 2,3, 4,5A B
1,2,3, 4,5 (1) (1 Mark)
1, 2,5 2,3,4,5A B
2,5 (2) (1 Mark)
( ) / , 7i A x x N x
Page 13
DCEB-KDP
13
1,2,3, 4,5,6A (1 Mark)
( ) /ii B x x 'School'
, , , ,B c h l o s (1 Mark)
2: 25A x x 6 15x
2 25 25x x
5 5x or
5,5 (1) (1 Mark)
5
156 15x x 62
52
x
5 (2)2
(1 Mark)
A (1/2 Mark)
1,3,5,7A 1, 2,3,4,6B
1, 2,3,4,6 1,3,5,7B A
2, 4,6 (1) (1 Mark)
1,3,5,7 1, 2,3, 4,6A B
5,7 (2) (1 Mark)
(1 Mark)
(1 Mark)
( ) , ,i x y z
32 2 8n , , , , , ,x y z x y x z
, , , , ,y z x y z (2 Mark)
Page 14
DCEB-KDP
14
1, 2,3, 4A 1, 2,3, 4,5,6,7,8B
1, 2,3, 4 1,2,3, 4,5,6,7,8A B
1,2,3,4,5,6,7,8 (1) (1/2 Mark)
B
1, 2,3, 4 1, 2,3, 4,5,6,7,8A B
1, 2,3,4 (1) (1/2 Mark)
A
A B B (1/2 Mark)
A B A (1/2 Mark)
1, 2,3,4,5A 4,5,6,7B (1/2 Mark)
1,2,3, 4,5 4,5,6,7 1,2,3A B (1/2 Mark)
4,5,6,7 1, 2,3, 4,5 6,7B A (1/2 Mark)
A B B A (1/2 Mark)
(i)A
1, 2,3,6,7,14, 21, 42A (1/2 Mark)
/A x x (1/2 Mark)
A 2,3,5A
P 2,3,5P
A P (1 Mark)
: 2 1x x n x N
1,3,5,7,9,
(1)
2,4,6,8,10 (2)
2, 4,6,8,10 : 2 1x x n x N
1,2,3A 3, 4,5B
1, 2,3 3,4,5 3A B
A B
12
453
A B = 3
Page 15
DCEB-KDP
15
n A B n A n B n A B (1 Mark)
0,2,4A
0, 2, 4A
(1)
0, 2, 4 0, 2, 4A A
0, 2, 4
(2)A
A A A A
:A x x 2,4,6,7,10, (1/2 Mark)
:B x x 1,3,5,7,9, (1/2 Mark)
:C x x 2,3,5,7,11,13, (1/2 Mark)
:D x x 3,6,9 (1/2 Mark)
( ) 2, 4,6,8,10, 1,3,5,7,9,i A B
1, 2,3, 4,5,6,7,8,9,10
N (1/2 Mark)
( ) 2, 4,6,8,10 1,3,5,7,9,ii A B
(1/2 Mark)
( ) 2,3,5,7,11,13, 3,6,9,iii C D
2,5,7,11,13,
( ) 2, 4,6,8,10, 2,3,5,7,11,13,iv A C
2 (1/2 Mark)
3,6,9,12,15,18, 21 4,8,12,16, 20A B
3,6,9,15,18, 21 (1 Mark)
5,10,15, 20 4,8,12,16, 20D B
5,10,15 (1 Mark)
3,6,9,12,15,18, 21 2, 4,6,8,10,12,14,16A C
Page 16
DCEB-KDP
16
2,3,4,6,8,9,10,12,14,15,16,18, 21 (1 Mark)
4,8,12,16, 20 5,10,15, 20B D
20 (1 Mark)
b b b c c
d a a c
17, 26,35, 44,53,62,71
/ 5 , , 5nx x n N n
51,52,53,.........99
2n
/A B x x A x B
b d a c c
Page 17
DCEB-KDP
17
SSC STUDY MATERIALMathematics Paper - I
2 2 22 5 , 3 5 6, 6 ,x x x x y x
33
1 1 1, , , 212
xx xy
x P(x) x P(x)
1 20 1 2 1..........n n n
n np x a x a x a x a x a n
0 1 2 1, , ,.........., ,n na a a a a 0a o
P(x) kP(k)=0k P(x)Note ax+b b
a , ( )ax b a o
,y ax b x ( , )b oa
x
ba
2 6 9x x 2y ax bx c
x x
2ax bx c ,
Ch-3. (Polynomials)(Important Formulae)
1.
2.
3.
3 , 3 5,x y
2 5 4x x 2 12 3
5x x
3 23 5 3 1x x x 3 17 5
5x x
Page 18
DCEB-KDP
18
2 42
b b acxa
2 4b ac
n p(x) = n
2 ( )k x x k
2ax bx c
i) 2
( )b xa x
ii) 2
ca x
, , 3 2ax bx cx d 3 2 ( ) ( )x x x
i) 2
3
( )b xa x
ii) 3
( )da x
iii) 3
c xa x
0
0
0
Page 19
DCEB-KDP
19
p(x)g(x)q(x) r(x)p(x)=g(x). q(x)+r(x)
3( ) 1p t t p(1), p(-1), p(0), p(2) p(-2)
3( ) 1p t t
3(1) (1) 1 1 1 0p
3( 1) ( 1) 1 1 1 2p
3(0) (0) 1 0 1 1p
3(2) (2) 1 8 1 7p
3( 2) ( 2) 1 8 1 9p
13
2ax bx c
12,3
1 1 52 23 3 3
1 223 3
2 2 ( )ax bx c k x x
2 5 23 3
k x x
23 5 23
x xk
3k 23 5 2x x
3 23 2x x x g(x)x-2 -2x+4g(x)
3 2( ) 3 2, ( ) 2, ( ) 2 4, ( ) ?p x x x x q x x r x x g x
( ) ( ) ( ) ( )p x g x q x r x 3 23 2 ( ) ( 2) ( 2 4)x x x g x x x
Page 20
DCEB-KDP
20
3 23 2 2 4 ( )( 2)x x x x g x x 3 2( ) ( 3 3 2) ( 2)g x x x x x
3 2 22) 3 3 2( 1x x x x x x
3 22x x____________ 2 3x x 2 2x x ____________
2x2x
____________0
2( ) 1g x x x
2 7 10x x
2 27 10 2 5 10x x x x x
( 2) 5( 2)x x x
( 2) ( 5)x x
2 7 10x x
2 0x 5 0x
2x 5x
2
7 ( )2 ( 5)1
xx
2
10( 2) ( 5)1 x
7 5( ) 5 6 7 6p x x x x
i) 5x ii) ( )P x iii)
7 5( ) 5 6 7 6P x x x x
i) 5x
ii) ( )P x
iii)
Page 21
DCEB-KDP
21
2( ) 1p y y
( ) 0p y 2 1 0y
( 1) ( 1) 0y y
1 0y 1 0y
1y 1y
( )p y
2 3 1kx x k
2ax bx c ba
2 3 1kx x ( 3) 1k
3 1k
3k
2 2 23 2 5, 5, 6 5 2x x x x x
2 9x
2( ) 9p x x 2( 3) ( 3) 9 9 9 0p
2 9x 2(3) 3 9 9 9 0p
2 9x
2 3x x
2 3 2 (1 )x x x x 2 3x x 2 (1 ) 0x x 2 0x 1 0x
0,x 0,x 1x
Page 22
DCEB-KDP
22
2 3x x
3 24 8 6 2x x x , ,
3 2ax bx cx d , ,
3
c xa x
4, 8, 6, 2a b c d
6 34 2
ca
3 23 3x x x
3 2( ) 3 3p x x x x
1, 3, 1, 3a b c d
3 2(1) (1) 3(1) 1 3p
1 3 1 3
(1) 0p 3 2( 1) ( 1) 3( 1) ( 1) 3p
1 3 1 3
( 1) 0p 3 2( 3) ( 3) 3( 3) ( 3) 3p
27 27 3 3
( 3) 0p
1, 1, 3 ( )p x
2
3
3 ( )1 ( 1) ( 3)1
b xa x
1( 1) ( 3)
3
3 ( 3)1 1
da x
1( 1) ( 1)( 3) ( 3)(1)
3
11 3 31
c xa x
Page 23
DCEB-KDP
23
2 33 3 5x x x 21x x
3 23 3 5x x x
2 1x x
2 3 21) 3 3 5( 2x x x x x x 3 2x x x
_______________
22 2 5x x 22 2 2x x
_______________
3
x-2
2( 1) ( 2) 3x x x 3 2 22 2 2 3x x x x x 3 23 3 5x x x
4 3 22 3 3 6 2x x x x 2 2
4 3 2( ) 2 3 3 6 2p x x x x x
2 2 ( )p x ( )p x 2( 2) ( 2) 2x x x
2 4 3 2 22)2 3 3 6 2(2 3 1x x x x x x x 42x 24x
__________________ 3 23 6x x x 33 6x x
__________________
Page 24
DCEB-KDP
24
2 2x 2 2x
___________ 0
22 3 1x x 22 2 1x x x
2 ( 1) 1( 1)x x x
( 1) (2 1)x x
1 0x 2 1 0x
1x 12
x
12
2, 2, 12
2( ) 12p x x x
2( ) 12p x x x 2 12y x x
x y
2 12y x x x 2 12x x
2( ) 12p x x x
2 4 3 12x x x
( 4) 3( 4)x x x
( 4) ( 3)x x
x
y
(x,y)
-4
8
(-4,8)
-3
0
(-3,0)
-2
-6
(-2,-6)
-1
-10
(-1,-10)
0
-12
(0,-12)
1
-12
(1,-12)
2
-10
(2,-10)
3
-6
(3,-6)
4
0
(4,0)
5
8
(5,8)
Page 25
DCEB-KDP
25
2 12x x 4x 3x
4x 3x
2 12x x
2( ) 6 9p x x x
2 6 9y x x
x y
2 6 9y x x x 2 6 9x x
2( ) 6 9p x x x
2 3 3 9x x x
( 3) 3( 3)x x x
( 3) ( 3)x x 2 6 9x x
3 0x 3 0x 3x 3x
2 6 9x x
x
y
(x,y)
-3
36
(-3,36)
-2
25
(-2,25)
-1
16
(-1,16)
0
9
(0,9)
1
4
(1,4)
2
1
(2,1)
3
0
(3,0)
4
1
(4,1)
Page 26
DCEB-KDP
26
Part-B
I) 22 3 1x x 2x _______________ ( )
a) -3 b) 3 c) 15 d) 12
3 2( ) 2 4f x x x x k 1x k _____ ( )
a) 9 b) 6 c) -3 d) 2
2( ) 1f x x x 1 1 __________ ( )
a) 0 b) -1 c) 1 d) 2
3 24 6x x x __________ ( )
a) 4 b) -4 c) 1 d) 6
2 3 4y x x __________ ( )
a) (-1,0) b) (2,0) c) (3,0) d) (-2,-6)
__________ ( )
a) 25 7 3x x b) 2 1
2 7 3x x
c) 4 3x d) 2/ 3 162
x x
( )p x ax b __________ ( )
a)ab b)
ba
c) ab
d) ba
3 22 5 14 8x x x __________ ( )
a)52 b) -7 c) 4 d)
25
x-__________ ( )a) 0 b) 1 c) 2 d) -3
2( ) 3 1p m m m ( 1)p __________ ( )
a) 4 b) -1 c) 5 d) 3
y ax b x__________ ( )
a) (0, )ab b) ,a o
b
c) ,b oa
d) , aob
Page 27
DCEB-KDP
27
3 4x x __________ ( )
a) 0,2,4 b) -2,0,2 c) 0,3,-2 d) 1,0,2
II ) 24 8u u ________________
2 4 3x x ________________
3 25 2 24x x x , , ________________
2 6 8y x x xx______________
1x 2px qx r ________________
III ) A B
4 32) 2 23
a x x x
2)7
b
) 2 3c x
3) 2 3 5d x x
2) 5 6e x x 3/ 2 2) 5f x x
) 3 4g x
Answers )
I) 1) b 2) c 3)b 4) d
5) a 6) d 7) b 8) a
9) a 10) c 11) c 12) b
II) 1) 0,-2 2) 3 3) -2 4) 2,4
5) p-q+r
III) 1) d 2) e 3) c 4) b 5) a
Page 28
DCEB-KDP
28
Telugu Medium STUDY MATERIAL 2014-2015Unit IV
1 1 1 2 2 20, 0a x b y c a x b y c
i) 1 1
2 2
a ba b
ii)1 1 1
2 2 2
a b ca b c
iii)1 1 1
2 2 2
a b ca b c
2 6 0x y 4 2 4 0x y
1 1 12, 1, 6a b c 2 2 24, 2, 4a b c 1 1 1
2 2 2
2 1 6, ,4 2 4
a b ca b c
1 1
2 2
a ba b
2 6 0x y 4 2 4 0x y
2 6x y 2(2 2) 0x y
6 2y x 2 2x y
2 2y x
6 2y x 2 2y x
0 6 2 0 6 (0,6)x y 0 2 0 2 2 (0, 2)x y
1 6 2 1 4 (1,4)x y 1 2 1 2 0 (1,0)x y
Pair of Linear Equations in two variables
Prepared by : L. Ravindranath Babu, Headmaster, Z.P. High School, R.V. Palli, Peddiveedu (P), Veeraballi (M)
Page 29
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29
2 6 2 2 2 (2, 2)x y 2 2 2 2 2 (2, 2)x y
3 6 2 3 0 (3,0)x y 3 2 3 2 4 (3,4)x y
4 6 2 4 2 (4, 2)x y 4 2 4 2 6 (4,6)x y
2, 2x y
2 3 8, 4 6 9x y x y
2 3 84 6 9
x yx y
1 1 1
2 2 2
2 1 3 1 8, ,4 2 6 2 9
a b ca b c
1 1 1
2 2 2
a b ca b c
2 3 8x y 4 6 9x y
3 2 8y x 6 4 9y x
2 83
xy
4 96
xy
2 0 8 80, 2.6(0, 2.6)3 3
xx y
4 0 9 90, 1.5 (0, 1.5)6 6
xx y
2 1 8 61, 2 (1, 2)3 3
xx y
4 1 9 51, 0.8 (1, 0.8)6 6
xx y
Page 30
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30
2 3 8 23, 0.6 (3, 0.6)3 3
xx y
4 3 9 33, 0.5 (3,0.5)6 6
xx y
2 4 8 04, 0 (4,0)3 3
xx y
4 6 9 156, 2.5 (6,2.5)6 6
xx y
2 5 8 25, 0.6 (5,0.6)3 3
xx y
9 3 12 0,18 6 24 0x y x y 9 3 12 018 6 24 0
x yx y
1 1 1
2 2 2
9 1 3 1 12 1, ,18 2 6 2 24 2
a b ca b c
1 1 1
2 2 2
a b ca b c
9 3 12 0x y 18 6 24 0x y
3 9 12y x 6 18 24y x
9 123
xy
18 246xy
( 9 0) 12 120, 4 (0, 4)3 3
x y
( 18 0) 24 240, 4 (0, 4)6 6
x y
Page 31
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31
( 9 1) 12 9 12 31, 1( 1, 1)3 3 3
x y ( 18 1) 24 18 24 61, 1( 1, 1)
6 6 6x y
( 9 2) 12 18 12 62, 2 ( 2,2)3 3 3
x y
( 18 2) 24 36 24 122, 2( 2,12)6 6 6
x y
( 9 3) 12 27 12 153, 5 ( 3,5)3 3 3
x y
( 18 3) 24 54 24 303, 5( 3,5)6 6 6
x y
x y
10........(1)x y
4y x
4.........(2)x y
10x y
4x y
1 1 1
2 2 2
1 1 10, ,1 1 4
a b ca b c
Page 32
DCEB-KDP
32
1 1
2 2
a ba b
10x y 4x y
10y x 4y x
1, 1 10 9 (1,9)x y 1, 1 4 5 (1,5)x y
2, 2 10 8 (2,8)x y 2, 2 4 6 (2,6)x y
3, 3 10 7 (3,7)x y 3, 3 4 7 (3,7)x y
4, 4 10 6 (4,6)x y 4, 4 4 8 (4,8)x y
x y
4 2 10 0
3 2 4 0x yx y
6 3 15 02 5 0
x yx y
6 8 4 03 4 2 0
x yx y
2 3 5 04 6 15 0
x yx y
Page 33
DCEB-KDP
33
3 72 3 1x yx y
i) ii) iii) iv)
2 1..............(1)2 3 12............(2)x y
x y
2 12 1
12
x yy x
xy
12 3 122
3 32 122
4 3 3 122
7 3 122
xx
xx
x x
x
7 3 12 2 247 24 37 21
21 37
xxx
x
Page 34
DCEB-KDP
34
x y
3 1 4 22 2
3, 2
y
x y
6 6xy
83 5xy
6 6xy
83 5xy
6 6xyy
3 8 5xy
y
6 6xy y 3 8 5xy y
6 6 0xy y 3 5 8 0xy y
6 6.........(1)xy y 3 5 8........(2)xy y
6 6( 6) 6
66
xy yy x
yx
y 6 63 5 86 6
xx x
18 30 86 6x
x x
18 30 86
xx
18 30 8( 6)x x
18 30 8 48x x
18 8 30 48x x
26 78x
78 326
x
x y
Page 35
DCEB-KDP
35
6 6 23 6 3
y
3, 2x y
x y
10 y x
10x y
10 10 66y x x y
11 11 66x y
11 66x y
66 6......(1)11
x y
2........(2)x y
6x y
6y x 6 2x x
6 2x x
2 6 2x
2 2 6 8x
8 42
x
x y
6 4 2y
4, 2x y
8 5 9.......(1)3 2 4.......(2)
x yx y
Page 36
DCEB-KDP
36
(1) 2.......16 1x oy 18
(2) 5.......15 1x oy 20
______________ (-) x =-2
2x 3 2 2 4y
6 2 4y
2 4 6 10y
10 52
y
2, 5x y
i
ii
x
y
10 220.........(1)x y
15 310.........(2)x y
(i) (2) - (1) x 15 310y
x 10 220y ______________
(-) 5y=90
90 185
y
18y 10(18) 220x
180 220x 220 180 40x
x
Page 37
DCEB-KDP
37
(ii)
2 3 2x y
4 9 1x y
1 12 3 2x y
1 14 9 1x y
1 1,p dx y
2 3 2.........(1)p d
4 9 1.........(2)p d
d
(1) 3....6 9p d 6
(2) ....4 9p d 1
( ) 10 5p
5p 10
12
2
12
p
2 12
3 2d
1 3 2d
Page 38
DCEB-KDP
38
3 2 1 1d
13
d
1 1,p qx y
1 1 1 1,2 3x y
2, 3x y 2 22 4, 3 9x y
4, 9x y
1 1 3
3 3 4x y x y
1 1 1
2 3 2 3 8x y x y
Ans:1 1,
3 3p d
x y x y
344
p 1 1 12 2 8
p d
4 4 3.....(1)p d 1
2 8p d
1 128 4
p d
4 1p d
4 4 1.......(2)p d
(1) (2) 4 4p d 3
4 4p d 1
( ) 8 2p
2 18 4
p
14
p
14 4 34
d
1 4 3d 4 3 1 2d
Page 39
DCEB-KDP
39
2 14 2
d
1 1,
3 3p d
x y x y
1 1 1 1,3 4 3 2x y x y
3 4........(3) 3 2.........(4)x y x y
(3) (4) 3x y 4
3x y 2
6 66 16
x
x
1x 3(1) 4y
4 3 1y
1, 1x y
6 3 6
2 4 5x y xyx y xy
2x yxy
6x yxy
Page 40
DCEB-KDP
40
2 3 93 4 5
x yx y
0.2 0.3 130.4 0.5 2.3
x yx y
3 5 11
x yx y
2 5 03 2 4 0
x yx y
1 1 1
2 2 2
2 1 5, ,3 2 4
a b ca b c
1 1 1
2 2 2
a b ca b c
2 02 2 4 0x y
x y
1 1 1
2 2 2
1 1 2 1, ,2 2 4 2
a b ca b c
1 1 1
2 2 2
a b ca b c
4 6 15 02 3 5 0
x yx y
1 1 1
2 2 2
4 2 6 2 15 3, ,2 1 3 1 5 1
a b ca b c
Page 41
DCEB-KDP
41
1 1 1
2 2 2
a b ca b c
1 1 1
2 2 2
, ,a b ca b c
a) 5 4 8 07 6 9 0
x yx y
1 1 1
2 2 2
5 4 2 8, ,7 6 3 9
a b ca b c
1 1 1
2 2 2
a b ca b c
b) 9 3 12 0x y
18 6 24 0x y
1 1 1
2 2 2
9 1 3 1 12 1, ,18 2 6 2 24 2
a b ca b c
1 1 1
2 2 2
a b ca b c
c) 6 3 10 0x y
2 9 0x y
1 1 1
2 2 2
6 3 3 3 10, ,2 1 1 1 9
a b ca b c
1 1 1
2 2 2
a b ca b c
51
x yx y
x y 5......(1)
x y
1......(2)
( ) 2 6x
6 32
x
3x 3 5y
5 3 2y
3, 2x y
Page 42
DCEB-KDP
42
2x yxy
6x yxy
2x yxy xy
6x yxy xy
1 1 2y x
1 1 6y x
1 1,p dx y
2d p 6d p
4 4 2x y y x
a) b) c) d)
5 3 1110 6 22x y
x y
a) b) c) d)
a) b)
c) d)
a) b) c) d)
a) 1Q b) 2Q c) 3Q d) 4Q
a) 1 1 1
2 2 2
a b ca b c
b) 1 1 1
2 2 2
a b ca b c
c) 1 1
2 2
a ba b
d) 1 1 1
2 2 2
a b ca b c
9 0, 2 0x y x y a) 1 b) 0 c) d) 9
Page 43
DCEB-KDP
43
a) 0ax by b) 0ax by c c) ax by d) xy
2 7, 5x y x y y a) 2 b) 3 c) 5 d) 7
3 2 4x y 0x y a) 3 b) 2 c) 4 d) 1
Page 44
DCEB-KDP
44
Telugu Medium STUDY MATERIAL 2014-2015Unit V
, ,a b c 0a 2 0ax bx c x 2 0 0ax bx c a 2y ax bx c
2 42
b b acxa
2 4b ac
i) 2 4 0b ac ii) 2 4 0b ac iii) 2 4 0b ac
i) 2100 20 1 0x x
Ans : 2100 10 10 1 0x x x
10 10 1 1 10 1 0x x x
10 1 10 1 0x x
10 1 0x 10 1 0x
110
x
ii) 22 7 5 2 0x x
Ans : 22 5 2 5 2 0x x x
2 5 2 2 5) 0x x x
2 5 2 0x x
(Quddratic Equations)
Prepared by : L. Ravindranath Babu, Headmaster, Z.P. High School, R.V. Palli, Peddiveedu (P), Veeraballi (M)
5 2 2 5 2 210x
5 2x x
Page 45
DCEB-KDP
45
2 5 0x 2 0x
52
x 2x
5 , 22
x
iii) 23 4 5 4 12x x
Ans : 23 4 5 4 12 0x x
4x a 23 5 12 0a a 23 9 4 12 0a a a
3 3 4 3 0a a a
3 3 4 0a a
3 0a 3 4 0a 4a x 4 3 0x 3 4 4 0x
7 0x 3 12 4 0x
7x 3 8 0x
83
x
7x 83
l b
2 28l b
28 142
l b
14l b
14 .......(1)b l
40lb b 14 l
Page 46
DCEB-KDP
46
14 40l l
214 40l l 2 14 40 0l l 2 10 4 40 0l l l
10 4 10 0l l l
10 4 0l l
10l 4l l b
10l 4l 14 10b 14 4 10b 10, 4l b 4, 10l b
i) 2 10 9 0x x
2 10 9x x 10x 2ab
2 2. .5 9x x
2 2 22. .5 5 9 5x x 2 22. .5 5 9 25x x
25 16x
5 16x
5 4x
5 4x 5 4
9x
ii) 24 4 3 3 0x x
24 4 3 3x x
22 4 3 3x x
22 2.2 . 3 3x x
23
2 222 2.2 . 3 3 3 3x x
22 3 3 3x
Page 47
DCEB-KDP
47
22 3 0x
2 3 0x
2 3x
32
x
24 4 3 3 0x x 4, 4 3, 3a b c
22 2 24 4 3 4 4 3 4 3 4 3 0b ac
iii)1 1 11 , 4,7
4 7 30x
x x
7 4 114 7 30
x xx x
x 7 x
4 114 7 30x x
11 11
4 7 30x x
11 4 7 30 11x x
30 114 7x x
11
4 7 30x x
2 7 4 28 30x x x 2 3 28 30x x 2 3 28 30 0x x 2 3 2 0x x 2 3 2x x
2 2 , ,3 22
x x
2 22 3 3 32, , 2
2 2 2x x
23 922 4
x
Page 48
DCEB-KDP
48
23 8 92 4
x
23 12 4
x
3 12 4
x
3 12 2
x
3 12 2
x 3 12 2
3 12
3 12
42
22
2x
x y
2 2 180........(1)x y
2 8y x
2 8 180x x
2 2. .4 180x x 2 2 22. .4 4 180 4x x
24 196x
4 196 14x
4 14x 14 4x 18x
18x y 10x 2 8 18y 2 8 10y
Page 49
DCEB-KDP
49
2 144y 2 80y
144y 12y
x
x 360
x 5x
5x 3605x
5x 360 360 1
5x x
1 1360 15x x
1 1 15 360x x
x 5 x
15 360x x
5 1
5 360x x
5 360 5x x
2 5 1800x x 2 5 1800 0x x 2 45 40 1800 0x x x
45 40 45 0x x x
45 40 0x x
45 0x 40 0x
Page 50
DCEB-KDP
50
45x 40x 40x
22 6 3 0x x
2, 6, 3a b c
2 24 ( 6) 4 2 3b ac
36 24 12 0
2 42
b b acxa
( 6) 122 2
6 124
6 4 34
6 2 34
1
2
3 3
2
4
3 32
3 3 3 3,2 2
x
22 3 0x Kx 22 3 0x Kx
2 4 0b ac
2, , 3a b k c 2 4 2 3 0k 2 24 0k
Page 51
DCEB-KDP
51
2 24k
24 4 6 2 6k
k 2 6
x 20 x
4x20 4x
16 x
4 16 48x x
216 64 4 48x x x 2 20 64 48x x 2 20 64 48 0x x 2 20 112 0x x
2 20 112 0x x
1, 20, 112a b c 2 24 ( 20) 4 1 112b ac
100 448 48 0
2 12 08
x x
Page 52
DCEB-KDP
52
2 5 6x x
23 4 3 4 0x x
( 2) 6 0kx x k
( 3) (2 1) ( 5)x x x x
2 22 6 3 5x x x x x 2 22 5 3 5x x x x 2 22 5 3 5 0x x x x
2 10 3 0x x
3 22 2 1x x x
3 2 2 3 33. .2 3. .2 2 2 2x x x x x 3 2 36 12 8 2 2x x x x x
3 3 22 2 6 12 8 0x x x x x 3 26 14 8 0x x x
, 1a a
( 1) 306a a 2 306a a 2 306 0a a
32
22 5 3 0x x
31,2
x x
Page 53
DCEB-KDP
53
1x 22(1) 5(1) 3
2 5 3 5 5 0
32
x 23 32 5 3
2 2
1294 2
15 32
9 15 32 2
9 15 6 15 15 0 02 2 2
32
2 3 10 0x x 2 3 10 0x x
2 5 2 10 0x x x
5 2 5 0x x x
5 2 0x x
5 0x 2 0x
5x 2x
,a b 27.......(1)a b
182.......(2)ab
27a b 27b a
b (27 ) 182a a
227 182a a 2 27 182 0a a 2 14 13 182 0a a a
( 14) 13( 14) 0a a a
( 14) ( 13) 0a a
2182a
14 13a a
Page 54
DCEB-KDP
54
14 0, 13 0a a
14, 13 0a a
27 14b 27 13b 13 14
2 4b ac
22 2 2 1 0x x
2, 2 2, 1a b c 2 24 ( 2 2) 4(2)(1) (4 2) 8 8 8 0b ac
a) 2 4 0b ac b) 2 4 0b ac c) 2 4 0b ac d) 2 4b ac
24 4 1 0x x
a) 4,1 b) 3 1,2 2 c) 0.5, 2.5 d)
1 1,2 2
2 0ax bx c
a) b
a
b) ca c)
bc d)
ab
25 6 1 0x x
a) 6
5
b) 15 c)
56 d)
16
2 9 0x a) 1 b) 2 c) 3 d) 9
2 4 0b ac
2 4b ac
2 4b ac
0a
Page 55
DCEB-KDP
55
(d)
1 2 3, , ,.......... na a a a
2 1 1 2 4 3 1.......... n nd a a a a a a a a
, , 2 , 3 ,................a a d a d a d
* n ( 1)na l a n d
* , , 2 ,.............a a d a d n
2 ( 1)2nns a n d
2n nns a a
a na
* n ns n 1n n na s s
* n ( 1)
2nn ns
( 1)1 2 3 4 ...........2
n nn
* n ( 1) (2 1)
6nn n ns
2 2 2 2 2 ( 1) (2 1)1 2 3 4 .............6
n n nn
* n 2( 1)
2nn ns
2 2( 1)
4n n
2 223 3 3 3 3 1 1
1 2 3 4 ...........2 4
n n n nn
* n 1s
n 2s n 3s
(Progressions)
(Important Formulae)
Page 56
DCEB-KDP
56
22 3 19 (1 8 )S S S
23 1S S
* , ,a b c . .A P 2a cb
' 'b ' ', ' 'a c
' 'r
* 2 3, , , ,.........a ar ar ar
32
1 2 1
............... n
n
a aara a a
* n 1nna ar
* n ( 1) ( 1)
1
n
na rs r
r
(1 ) ( 1)1
n
na rs r
r
* 1as
r
* , ,a b c .G pb ac ' 'b ' 'a ' 'c
5 72, ,3, ,...............2 2
5 72, ,3, ,...............2 2
1 252,2
a a
2 1d a a
5 22
5 42
12
d
( 1)na a n d
5 4a a d
Page 57
DCEB-KDP
57
12 42
5 4a
6 5a a d
142
692
a
7 6a a d
9 12 2
7 5a
94, ,52
2 121, 18 21 3a d a a 81na
( 1)na a n d
81 21 ( 1)( 3)n
81 21 3 3n
81 24 3n
81 25 3n
105 3n
35n
n ( 1)na a n d
11 38a
10 38........(1)a d
16 73a
15 73.......(2)a d
15 7310 38
5 35
a da d
d
Page 58
DCEB-KDP
58
7d
(1) 10(7) 38a
70 38a 38 70a
32a
n ( 1)na a n d
31 30a a d
32 30(7)
32 210
31 178a
2 124, 21 24 3a d a a
78, ?ns n
2 ( 1)2nns a n d
78 2(24) ( 1)( 3)2n n
78 48 3 32n n
156 51 3n n
2156 51 3n n 2156 3(17 )n n
52 17n n 2 17 52 0n n 2 13 4 52 0n n n
( 13) 4( 13) 0n n n
( 4)( 13) 0n n
4 0n 13 0n
4n 13n
5 5 5, , ,...........2 4 8
2
1
55 5 2 14, 52 9 5 22
aa ra
Page 59
DCEB-KDP
59
n 1nna ar
1920a ar
195 12 2
20
52
n 1nna ar
15 12 2
n
52n
1 1 1, , ,.........3 9 27 1
2187
1 1 1, , ,.........3 9 27
2
1
11 1 3 19, 13 9 1 33
aa ra
n 1
2187na
1 12187
nar
11 1 13 3 2187
n
1 13 2187n
3 2187n 73 3n
7n
12187
23
1681
n 1nna ar
Page 60
DCEB-KDP
60
423
a
3 2 (1)3
ar
71681
a
6 16 (2)81
ar
6
3
16(2) 812(1) 3
arar
3 827
r
33 2
3r
23
r
3 2(1)3
ar
32 23 3
a
8 227 3
a
2 273 8
a
94
a
1 894
a a
29 2 34 3 2
a ar
22
39 2 9 4 14 3 4 9
a ar
33
49 2 9 8 24 3 4 27 3
a ar
9 3 2, ,1, ,............4 2 3
Page 61
DCEB-KDP
61
2 112, 15 12 3, 99na d a a a
( 1)na a n d
99 12 ( 1)3n
99 12 ( 1)3n
87 ( 1)3n
29 1n
30n
253a 3 8 5d
n ( 1)na a n d
20 19a a d
253 19( 5)
253 95
20 158a
1 2 3 .......... 1000S
2nnS a l
10001000 1 1000
2S
500 1001
500500
n( 1)
2nn nS
Page 62
DCEB-KDP
62
10001000(1000 1)
2S
1000 10012
500500
1256,2
a r
2 3, , , ,..............a ar ar ar2 31 1 1256,256 , 256 , 256 .........
2 2 2
256, 128,64, 32,...............
, 2, 6x x x x
, 2, 6x x x
32
1 2
aaa a
2 62
x xx x
22 ( 6)x x x
2 24 4 6x x x x 4 4 6x x
4 6 4x x
4 2x
2x
2,2 2, 4,.........
2,2 2, 4,.........
2
1
2 22, 2, 128, ?2 n
aa r a na
1nna ar
1128 2 2
n
164 2
n
11264 2
n
Page 63
DCEB-KDP
63
1264 2
n
1
6 22 2n
162
n
12 1n
13n
125, 5,1,5
125, 5,1,5
2
1
5 125 5
ara
i) ii) iii)
n an
3 600a
2 600........(1)a d
7 700a
6 700.........(2)a d
6 700a d
2 600
4 100
a d
d
25d
(1) 2 600a d
2(25) 600a
Page 64
DCEB-KDP
64
50 600a 600 50a
550a
i)
ii) 10 9a a d
550 9(25)
550 225
775
iii) 2 ( 1)2nns a n d
77 2(550) (7 1)252
s
7 1100 1502
7 12502
7 625
4375
2 2 2, , .........81 27 9 n
n
2
1
54 1162,162 3
aa ra
n 1nna ar
11162
3
n
na
2 2 2, , ,.............81 27 9
2
1
22 2 8127, 3281 27 281
aa ra
Page 65
DCEB-KDP
65
n 1nna ar
12 (3)81
nna
1
11 2162 33 81
nn
1 12162 3 381
n n
2 281 81 3 n 8 2 23 3 n
8 2 2n
8 2 2n
10 2n
5n
n
2
1
9030, 330
aa ra
n 1nna ar
910a ar
930(3)
1920a ar
1930(3)
n 1nna ar
130 (3)n
Page 66
DCEB-KDP
66
x 20, 40, 60............x x x , 20, 40, 60............x x x x
2 1, 20 20a x d a a x x
7 700, 7s n
n 2 ( 1)2nns a n d
77 2 (7 1)( 20)2
s x
7700 2 1202
x
1400 7 2 60x
100 60x 160x
n ( 1)na a n d
4 3a a d
8 7a a d
4 8 24a a
3 7 24a d a d
2 10 24a d
2( 5 ) 24a d
5 12 (1)a d
6 5a a d
10a a ad
6 10 44a a
5 9 44a d a d
2 14 44a d
Page 67
DCEB-KDP
67
2( 7 ) 44a d
7 22 (2)a d
7 22a d
5 12
2 10
a d
d
5d
(1) 5(5) 12a
25 12a 12 25a
13a
13a 2 13 5 8a a d
3 2 13 2(5) 13 10 3a a d
d
x y n 1na a n d
. .A P 99x d . .A P 99y d
99 99 100x d y d
99 99 100x d y d
100 (1)x y
. .A P 999x d . .A P 999y d
. .A P 999 999x d y d
999 999x d y d
x y
Page 68
DCEB-KDP
68
100x y
PART-B
I)
1 5 9 13, , , ,...............3 3 3 3 _____________
a) 4 b) 23 c)
43 d)
34
_____________
a) 5 72, ,3, ,..........2 2 b) 0.2,0.22,0.222,...........
c) 2 3 4, , , ,..........a a a a d) 3, 6, 9, 12,..............
_____________ a) -25 b) -12 c) 0 d) -31
1 32, 26a a 2a _____________ a) 16 b) 14 c) 12 d)
_____________ a) 1,4,9,16,....... b) 1,-1,1,-1,1,.............
c) -4,-20,-100,-500,.......... d) 1 1 1 1, , , ,..............4 16 64 256
19,3
a r 7a _____________
a) 1
243 b) 19 c)
127 d)
181
_____________ a) 10 b) 16 c) 5 d) 4
, 3 , 5 ,.............x b x b x b _____________
a) 13x b b) 8x b c) 11x b d) 7x b
, ,x y z 2y _____________
a) x z b) x z c) 2x z
d) xz
Page 69
DCEB-KDP
69
n 2 5n _____________
a) 9 b) 11 c) 5 d) 7
n 12(0.5)n __________
a) 2 b) 1 c) 1n d) 0.5
, ,a b c _____________
a) b a c b b) b a c c) 2b ac d) 2a cb
II)
2, , 2a a a _____________
1nnt
n
1
9 _____________
3
1, , ,................x yy x x _____________
2 , 4 3 ,6 5 ,............a b a b a b _____________
_____________
n _____________
n 11
3
n
na
_____________
_____________
1 2 3 ...............a a a n_____________
3 3 3 3 3 31 2 3 4 5 ........... 50 _____________
III) A B
n (a) ( 1)1
na rr
n (b) 1nar
n (c) ( 1)a n d
n (d) 2 12n a n d
Page 70
DCEB-KDP
70
n (e) 1 2 16
n n n
(f) 1
2n n
(g) 2a b
I) 1) c 2) a 3)d 4) b 5) a 6) d
7) c 8) c 9) b 10) d 11) d 12) c
II) 1) a 6) ( 1)n n
2) 9
10 7) 19
3)2
5
yx
8) -768
4) 2 2a b 9) a n
5) 50 10) 1625625
III) 1) c
2) b
3) d
4) a
5) f
Page 71
DCEB-KDP
71
1 2,0 , , 2x x 2 1x x
1 20, , 0,y y 2 1y y
1 1 2 2, , ,x y x y 2 22 1 2 1x x y y
1 1 2 2, , ,x y x y 1 2:m m
1 2 2 1 1 2 2 1
1 2 1 2
,m x m x m y m ym m m m
:1k 2 1 2 1,1 1
kx x ky yk k
1 1 2 2 3 3, , , , ,x y x y x y
1 2 3 1 2 3,3 3
x x x y y y
1 1 2 2 3 3, , , , ,x y x y x y
1 2 3 2 3 1 3 1 212
x y y x y y x y y
A s s a s b s c 2a b cs
1 1 2 2, , ,x y x y 2 1
2 1
tan y ymx x
2 1 6 ( 4) 6 4 10x x
(2,0), (0,4)A B 2 2 2AB OA OB
2 22 4
4 16
20
20 4 5 2 5AB
Prepared by :K. Kiran Kumar ReddyZPHS, D.Nandyala
Page 72
DCEB-KDP
72
2,0 , 0, 4A B
2 2 2 22 1 2 1 0 2 4 0AB x x y y
2 22 4 4 16 20
2 22 1 2 1x x y y
1 1 2 2
5,6 (3, 2)x y x y
2 23 ( 5) 2 6
22 2 2(3 5) 4 8 4 64 16 80 16 5 4 5
4 5
1 1 2 2 3 3
4,6 , 2, 2 , 2,5x y x y x y
1 2 3 1 2 3,3 3
x x x y y y
4 2 2 6 ( 2) 5 4 4 11 2 0 9, , ,3 3 3 3 3 3
0,3
2,5 , ,3P Q x x
2,5 , ,3P Q x
PQ 2 1
2 1
2y yx x
3 5 22 22 2x x
2 22
x
1 2x 1 2 1x
1x
Page 73
DCEB-KDP
73
,7 , 1,15x
1 1 2 2
, 7 , 1,15xx y x y
2 22 1 2 1 10x x y y
2 21 15 7 10x
2 21 8 10x
21 64 10x
21 64 100x
21 100 64 36x
1 36 6x ( ) 1 6a x 6 1 5 5x x ( ) 1 6a x 6 1 7 7x x
1 1 2 2
(3,5), (8,10),x y x y
1 2
2 : 3m m
1 2 2 1 1 2 2 1
1 2 1 2
,m x m x m y m ym m m m
2 8 3 3 2 10 3 5,2 3 2 3
16 9 20 15,5 5
25 35,5 5
5,7
( 6,10), (3, 8)A B
1 2:m m 1 2:m m
Page 74
DCEB-KDP
74
1 2 2 1 1 2 2 1
1 2 1 2
, 4,6m x m x m y m ym m m m
1 2 1 2
1 2 1 2
3 ( 6) 8 10, 4,6m m m mm m m m
x
1 21 2 1 2
1 2
3 6 4 3 6 4 4m m m m m mm m
1 1 2 23 4 4 6m m m m
1 27 2m m
1
2
27
mm
( , )x x
1 1 2 2
( , ), (1,0)x xx y x y 2 2
2 1 2 1x x y y
2 21 0x x
2 21 x x
, , 0,3x x 2 20 3x x
22 3x x
,x x
22 2 2(1 ) 3x x x x
2 2 2 21 (3 )x x x x
2 2 2 21 (3 ) 0x x x x
1 3 (1 3 ) 0x x x x 2 2a b a b a b
4 2 ( 2) 0x
4 2 0x
Page 75
DCEB-KDP
75
2 4x
4 22
x
3,5 , 5, 4 , 7,10A B C
( , )D x y , , ,A B C D
AC BD
AC 1 2 1 2,2 2
x x y y
1 1 2 2
3,5 , 7,10x y x y
3 7 5 10 10 15, ,2 2 2 2
BD 5 4,2 2
x y
1 1 2 2
5, 4 , ,x yx y x y
5 4 10 15, ,2 2 2 2
x y
5 102 2
x 4 15
2 2y
5 10x 4 15y
10 5 15x 15 4 19y
( 3, 5), ( 6, 8)A B AB
(a) 3, 5 , 6, 8 ,
Page 76
DCEB-KDP
76
1 2 2 1 1 2 2 1
1 2 1 2
,m x m x m y m ym m m m
1 ( 6) 2 ( 3) 1 ( 8) 2 ( 5),1 2 1 2
6 6 8 10,3 3
12 18,3 3
4, 6
(b) 3, 5 , 6, 8 , 1 2: 2 :1m m
2 ( 6) 1 ( 3) 2 ( 8) 1 ( 5),
2 1 2 1
12 3 16 5,3 3
15 21,3 3
5, 7
1 1 2 2 3 3
1, 1 , 4,1 , 2, 3x y x y x y
1 2 3 2 3 1 3 1 212
x y y x y y x y y
1 1 1 ( 3) 4 3 ( 1) ( 2) 1 12
1 1 1 3 4 3 1 2 22
1 1 4 4 2 42
1 4 8 42
1 02
Page 77
DCEB-KDP
77
, , 2,3 , 4, 1k k k
( , ), (2,3), (4, 1)A k k B C
ABC 1 2 3 2 3 1 3 1 21 ( ) ( ) ( )2
x y y x y y x y y
1 3 ( 1) 2( 1 ) 4( 3)2
k k k
1 3 1 2 2 4 122
k k k
1 4 14 22
k k
1 6 142
k
, ,A B C 10 6 14 02
k
6 14 0k
14 76 3
k
y
1 1 2 2
(5, 6), ( 1, 4)x y x y
y :1k :1k
2 1 2 1,1 1
kx x ky yk k
( 1) 5 ( 4) ( 6),1 1
k kk k
5 4 6,1 1
k kk k
y x 5 0 5 0
1k kk
5k :1 5 :1k
Page 78
DCEB-KDP
78
5k 4 5 60,5 1
20 60,6
260,6
130,3
AB (2,3)P ,A B
( ,0), (0, )A x B y
AB 1 2 1 2,
2 2x x y y
0 0,2 2
x y
,2 2x y
, (2,3)2 2x y
22n
32y
4x 6y
(4,0), (0,6)A B
1 2 1 2,2 2
x x y y
6 2 8 4,
2 2
8 12,2 2
4,6
1 1
4,6x y
2 2
1, 2x y
2 22 1 2 1x x y y
Page 79
DCEB-KDP
79
2 24 1 6 2
2 23 4
9 16
25
5
(8, 4), (6,7), (9,0)P Q R
(8, 4)P 2 2a b
2 28 4
64 16
80
(6,7), (0,0)Q O 2 26 7 36 49 85
(9,0), (0,0)R 2 29 0 81 9
80 81 85
(8,4)P
y
y ( , )P o y (6,5), ( 4,3)A B
2 22 1 2 1PA x x y y
1 1 2 2
0, 6,5yx y x y
2 26 0 5 y
226 5 y
236 5 y
2 24 0 (3 )PB y
1 1 2 2
0, 4,3yx y x y
Page 80
DCEB-KDP
80
2 24 (3 )y
216 (3 )y
2 236 (5 ) 16 (3 )PA PB y y
2 236 (5 ) 16 (3 )y y
2 2(5 ) (3 ) 16 36y y
(5 3 ) (5 3 ) 20y y y y 2 2 ( )( )a b a b a b
(8 2 ) (2) 20y
208 2 102
y
2 10 8 18y
18 92
y
,0 , ( ,0), (0, 3)A a B a C a
,0 , ( ,0), (0, 3)A a B a C a
,0 , ( ,0)A a B a 2 1 2 2x x a a a a 2AB a
( ,0), (0, 3)B a C a 2 2a b
2 2( ) ( 3)a a
2 23a a
24a
2a
BC 2a
(0, 3), ,0C a A a 2 2a b
2 2( 3)a a
2 23a a
24a
2a
Page 81
DCEB-KDP
81
2CA a AB BC CA
, ,A B C
7, 5 , 5,10 , 15,8 , 3, 5A B C D
AC 1 2 1 2,2 2
x x y y
1 1 2 2
7, 3 , 15,8x y x y
7 15 3 8,
2 2
8 5,2 2
54,2
BD 5 3 10 ( 5),
2 2
(5,10), (3, 5)
8 5,2 2
54,2
ABCD ,AB CD BC DA ABCD
ABC
1 2 3 2 3 1 3 1 212
x y y x y y x y y
1 7 10 8 5 8 ( 3) 15 3 102
1 1 2 2 3 3
7, 3 , 5,10 , 15,8x y x y x y
1 7 2 5 8 3) 15 132
1 14 5 11 15 132
Page 82
DCEB-KDP
82
1 14 55 1952
1 1154 154 772 2
ABCD 2 ABC
2, 2 , 2,8A B
2, 2 , 2,8A B
AB
AB a) 1 1 2 2, 2,2 , , 2,8x y x y 1 2: 1:3m m
1 2 2 1 1 2 2 1
1 2 1 2
,m x m x m y m ym m m m
1 2 3 ( 2) 1 8 3 2,1 3 1 3
2 6 8 6 4 14 7, , 1,4 4 4 4 2
b) 1 1 2 2
2, 2 , 2,8x y x y
AB
1 2 1 2,2 2
x x y y
2 2 2 8 10, 0, 0,52 2 2
c) 1 2: 3 :1m m
3 2 1 ( 2) 3 8 1 2,
3 1 3 1
6 2 24 2 4 26 13, , 1,4 4 4 4 2
7 131, , 0,5 , 1,2 2
Page 83
DCEB-KDP
83
(0, 1), (2,1), (0,3)A B C
ABC 1 2 3 2 3 1 3 1 212
x y y x y y x y y
1 0 1 3 2 3 ( 1) 0 1 12
1 0 2 3 1) 02
1 2 42
4
BC 1 2 1 2,2 2
x x y yD
1 1 2 2
1,2 , (0,3)x y x y
2 0 1 3,2 2
2 4,2 2
1,2
AB 0 2 1 1,2 2
E
1 1 2 2
0, 1 , 2,1x y x y
2 0,2 2
1,0
AC 0 0 1 3,
2 2F
1 1 2 2
0, 1 , 0,3x y x y
0 2,2 2
0,1
DEF
1 1 2 2 3 3
1,2 , 1,0 , 0,1x y x y x y
Page 84
DCEB-KDP
84
1 1 0 1 1(1 2) 0(2 0)2
1 1 1 1( 1) 02
1 1 12
1 22
1 22
1, 2 ,4, ), ( ,6)y x ,x y (1, 2), (4, ), ( ,6), (3,5)A B y C x D
ABCD
AC 1 2 1 2,2 2
x x y y
1 1 2 2
1,2 , ,6xx y x y
1 2 6,2 2
x
1 8 1, ,42 2 2
x x
BD 4 3 5,
2 2y
1 1 2 2
4, , 3,5yx y x y
7 5,2 2
y
1 7 5, 4 ,
2 2 2x y
1 7
2 2x 54
2y
1 7x 8 5y
7 1 6x 8 5 3y
6x 3y
Page 85
DCEB-KDP
85
(2, 3), (10, )y y
( , )x y ,x y
x
( 5,7), ( 4, 5), ( 1, 6), (4,5)A B C D int :H ABC ACD ABCD ]
21 1,2 2
d d
2,3 , , , 3, 2x y ,x y
. (1, 4)AB B A
B
( 2, 4), ( 2,1), (6,7), (6, 4)J K L M
BITS
10 5,3 3
( ,0), (0, )a b ba
(7, 2), (5,1), (3, )k k
y ( )D
y x 3
Page 86
DCEB-KDP
a)
i) ii)i.e. l m p
, , ,a e b f c g d h
,c e d f
b) i)
ii)
iii)
iv) 12
c) ,i e ABC AB AC B C
d) AB DC ABCD AC, BD
ACD BAC
BDC ABD
86
Page 87
DCEB-KDP
e)
, ,ABC ABD ABE ' 'AB
( ) ( ) ( )ar ABC ar ABD ar ABE
f) , ,a b c , ,x y z a b cx y z
i) ii)
Ex : ABCD 1 1 1 1, , ,A B C D ABCD i) 1 1 1 1, , ,A A B B C C D D
ii) 1 1 1 1 1 1 1 1
AB BC CD DAA B B C C D D A
1 1 1 1ABCD A B C D
Note :
87
Page 88
DCEB-KDP
. ., , ,i e A P B Q C B ABC PQR
( ) AB BC ACor ABC PQRPQ QR PR
Note :
Note :
. ., , , AB BC ACi e A P B Q C RPQ QR PR
,ABC DEF ,A D B E 0 0180 , 180A B C D E F
A B C D E F
D E C D E F
C F
ABC DEF
Note : Note :
. ., , ,AB BC ACi e A P B Q L RPQ QR PR
. ., AB ACi ePQ PR
A P ABC PQR
090ABC EDC
BCA DCE
88
Page 89
DCEB-KDP
ABC EDC AA
1.5 0.487.6
AB BCED DC h
15 4. .,10 876
i eh
40 15 876h
15 876 328.540
h m
AB
BC
PQ
h
QR
ABC PQR 090B Q
C R
ABC PQR AA 1.65 1.8
5.4AB BCPQ QR PQ
1.65 5.4 4.951.8
PQ m
Note
i) GF KI
G IF K
FHG IHK
89
Page 90
DCEB-KDP
FGH KIH
ii) 080A R 040B P 060C Q
ABC RPQ
Ex : (2) ' 'x
AB ZY ,A Z B Y
AB XZY
127.5 18
AX AB xXZ ZY x
10 2 30 150 2075 10 3
x x xx
10 150x 15x
BC
BC EF 30 (1)AB BC CA
20 (2)DE EF DF
ABC DEF AB BC CADE EF DF
12AB CADE EF DF
12. , 12.DE DFAB CAEF EF
90
Page 91
DCEB-KDP
12 12 12. 30DE DFEF EF
12 30DE EF DF EF EF
12 20 30EF
12 20 830
EF
' 'B AB DB
ED ' 'D x
090EDC EBA
E E ABE CDE
90360
ED CD xEB AB ED DB
1 4 480480 4x x x
x
3 480x 160x
AB
BC
DE h
EF
91
Page 92
DCEB-KDP
ABC DEF 0( 90)
B EC F
AB BCDE EF
4 624h
24 166
xh
ADE B
( )i ABC ADE ( )ii AD AE BE BC DE
( )i ADE B A A
ABC ADE
( ) ED ADiiBC AB
3.84.2 5.7x
3.8 4.2 2.85.7
x
DE
ABCD .AB DC AC BD
OA OBOC OD
AB DC ODC OBA
OCD OAB
OCD OAB
OA OBOC OD
92
Page 93
DCEB-KDP
, ,Ab CD PQ BD ,AB x CD y PQ z
1 1 1x y z
ABD PQD
PQ QDAB BD
(1)z QDx BD
PQB CDB
PQ BQCD BD
(2)z BQy BD
(1) (2) 1z z BQ QD BDx y BD BD
1 1 1zx y
1 1 1x y z
ABC DEF ' 'AX ' 'DY : :AX DY AB DE
ABC DEF
B E
( )ABC DEF 090A B DYE
( & )A BC DY EF
AB DEY AB AXDE DY
: :AX DY AB DE
93
Page 94
DCEB-KDP
ABC PQR
2 2 2( )
( )ar ABC AB BC ACar PQR PQ QR PR
AM BC PN QR AM PN
1( ) 2 . ........(1)1( )
2
BC AMar ABC BC AMar PQR QR PNQR PN
,ABM PQN ( )B Q ABC PQR
090AMB PNQ ABM PQN
(2)AB AMPQ PN
( ) .( )
ar ABC BC ABar PQR QR PQ
.AB ABPQ PQ
(
)
ABC PQRAB BCPQ QR
2
(3)ABPQ
AB BC ACABC PQRPQ QR PR
2 2 2( )
( )ar ABC AB BC ACar PQR PQ QR PR
ABC PQR ( ) ( )ar ABC ar PQR
94
Page 95
DCEB-KDP
2 2 2( )( )
ar ABC AB BC ACar PQR PQ QR PR
( )( ) ( ) 1( )
ar ABCar ABC ar PQRar PQR
2 2 2
1AB BC ACPQ QR PR
2 2 2 2 2 2, ,AB PQ BC QR AC PR
, ,AB PQ BC QR AC PR
ABC PQR
2( ) 81ar ABC cm2( ) 49ar DEF cm
,AX DY
,ABC DEF
,ABC DEF 0( , 90 )AB DEY B E A B DYE
AA AB AXDE DY
4.5 (1)ABDE DY
2( )( )
ar ABC ABar DEF DE
281 81 949 49 7
AB ABDE DE
9 4.5 4.5 7 3.57 9
Dy cmDy
ABC DEF 3 , 4BC cm EF cm ABC 254 cm DEF
95
Page 96
DCEB-KDP
2( )( )
ar ABC BCar DEF EF
ABC , ,BC CA AB , ,D E F ,DEF ABC
DE FA 12
DE AB
1 (1)2
FA
AEDF
A D gm AEDF
,BDEF DCEF
B E C F
ABC DEF
2 2
2
( ) 1 ( sin (1))( ) (2 ) 4
ar DEF DE DE u gar ABC AB DE
ABC DEF AB BCDE EF
B E
22
AB BXDE EY
B E
,AX DY
,BC EF AB BXDE EY
B E
ABX DEY
(1)AB AXDE DY
96
Page 97
DCEB-KDP
2( )( )
ar ABC ABar DEF DE
2
( sin (1))AX u gDY
ABCD
BD .BDEBC BC BCF
23( )4
ar BDE BD
2
223 3( ) 2 2 2 ( )
4 4ar BDE BC BC ar BCF
1( ) ( )2
ar BDE ar BDE
ABC 090B BD AC
( )i ADB ABC
( )ii BDC ABC
( )iii ADB BDC
( )i A A ,ABC ADB
090ADB ABC
ADB ABC (1)
AA ( )ii C C ,ABC BDC
090BDC BAC BDC ABC AA (2)
97
Page 98
DCEB-KDP
ADB ABC BDC ABC ADB BDC
ABC 090B
2 2 2AC AB BC BD AC
A A 090ADB B
ADB ABC
AD ABAB AC
2. (1)AD AC AB
BDC ABC DC BCBC AC
2, (2)DC AC BC
2 2. .AD AC DC AC AB BC 2 2( )AD DC AC AB BC
2 2,AC AC AB BC 2 2 2AC AB BC
ABC C , ,BC a CA b AB c AB C
' 'p (i) pc ab (ii) 2 2 2
1 1 1p a b
12
ar ABC b h
12
a b
98
Page 99
DCEB-KDP
12
ab
1( )2
ar ABC AB CD
12
C P
12
CP
( ) ( )ar ABC ar ABC 1 12 2
ab cp
ab cp abcp
(ii) 2 2 2c a b ABC
abcp
22 2ab a b
p
2 22 2
2
a b a bp
2 2
2 2 2
1 a bp a b
2 2 2
1 1 1p a b
ABC 2 2 2AC AB BC
090B
,PQ AB QR BC
090Q PQR
99
Page 100
DCEB-KDP
2 2 2PR PQ QR PQR
2 2AB BC 2AC
PR AC
ABC PQR AB PQ
BC QR
AC PR
ABC PQR B Q 090Q
090B
ABC 090C CD AB
2
2
BC BDAC AD
ADC ACB
CDB ACB
ADC CDB
AD CD ACCD BD BC
2 . (1)AD CD CD AD BDCD BD
2 2
2 2 2
. ( sin (1))CD AC AC CD AD BD u gBD BC BC BD DB
2
2
AC ADBC DB
,AD BC 2 2 2 2AB CD BD AC
ABD 090ADB ( )AD BC
2 2 2AD AB BD (1)
ACD 090ADC ( )AD BC
2 2 2AD AC CD (2)
100
Page 101
DCEB-KDP
2 2 2 2AB BD AC CD
2 2 2 2AB CD AC BD
ABCD AB BC CD DA
12
OB OD BD
12
OA OC AC
OAB 2 2 2AB OA OB
2 2 24 4 4AB OA OB 2 2 2 2 2 2(2 ) (2 )AB AB AB AB OA OB 2 2 2 2 2 2AB BC CD DA AC BD
ABCD 2 2 2 2OB OD OA OC
PQ BC
AB ' 'P
DC Q
PQ BC PQ AB PQ CD
090A B C D
OPB 2 2 2 (1)OB OP PB
2 2 2 (2)OD OQ QD 2 2 2 (3)OA OP PA 2 2 2 (4)OC OQ QC
2 2 2 2 2 2(1) (2) OB OD OP PB OQ QD ( , )PB QC QD PA 2 2 2 2( ) ( )OP QC OQ PA 2 2 2 2( ) ( )OP PA OQ QC
2 2 ( sin (3) & (4))OA OC u g
101
Page 102
DCEB-KDP
ABC C 2 22AB AC
090C AC BC
2 2 2AB AC BC
2 2AC AC
2 22AB AC
ABC BC ' 'D 13
BD BC
2 29 7AD AB
AE BC ' 'AE BC AE E
1 (1)2
BE CE BC
ADE 090AED
2 2 2AE AD DE 2 2( )AD BE BD
2 21 1( )2 3
AD BC BC
2 2136
AD BC
2 2 21 (2)36
AE AD AB ABC
ABE 090AEB
2 2 2AE AB BE 2
2 12
AB BC
2 214
AB AB ABC
2 23 (3)4
AE AB
102
Page 103
DCEB-KDP
2 2 21 336 4
AD AB AB
2 2 236 27AD AB AB 2 236 28AD AB
2 29 7AD AB
ABC B ' 'AB ' 'BC ' 'D ' 'E 2 2 2 2AE CD AC DE ABE 090B
2 2 2AE AB BE
CBD 090B
2 2 2CD BC BD
2 2 2 2 2 2( ) ( )AE CD AB BC BE BD
2 2AC DE
,ABC DBE
ABD A AC BD 2( ) .i AB BC BD
2( ) .ii AC BC DC2( ) .iii AD BD CD
B B 090ACB BAD
AC BD 090A ABD CBA AB BCBD AB
2 . ( )AB BC BD i
D D 090ACD BAD AC BD 090A
ABD CAD
103
Page 104
DCEB-KDP
AD BDCD AD
2 . ( )AD BD CD ii
(i) (ii) CBA CAD
2 . ( )AC BC AC BC DC iiiDC AC
ABC ' 'B BC D E 2 2 28 3 5AE AC AD
13
BD DE EC BC
' ',D E BC 2. . . 8L H S AE
2 28( )AB BE
ABE 2 28 8 (2 )AB BD 2 28 32AB BD
22 18 32
3AB BC
2 23289
AB BC
2 2. . . 3 5R H S AC AD 2 2 2 23( ) 5( )AB BC AB BD ,ABD ABC
22 2 18 3 5
3AB BC BC
2 2 258 39
AB BC BC
2 23289
AB BC
. . . .L H S R H S
AB
CD
104
Page 105
DCEB-KDP
BD
CE CD DE
CD AB ( )DE AB
2 2 2AC AE CE
AEC 2 2BD CE ABDE
2 212 5 144 25 2 169 169 13AC AC
AB xBC AC
2 6AC AB
2 6x
2BC AC
2 6 2x
2 4x
ABC 2 2 2AB BC AC
22 22 4 (2 6)x x x
2 25 16 16 4 24 36x x x x 2 8 20 0x x 2 10 2 20 0x x x
( 10) 2( 10) 0x x x
10, 2x x
10AB (2 10 4) 24BC (2 10 6) 26AC
105
Page 106
DCEB-KDP
CE AE
,A C AB CD
ABE B 2 2 2BE AE AB 2 215 9
225 81
144
144 12BE
CDE D 2 2 2DE CE CD
2 215 12 225 144 81
81 9DE
12 9 21BE DE
ABC ABC 34
(i) ABC
(ii) BX
(iii) 1 2 3, ,B B B 4B BX
1 1 2 2 3 3 4BB B B B B B B 4B C
106
Page 107
DCEB-KDP
(iv) 3B 4B C BC ' 'D
(v) ' 'D CA AB E
BDE
: 3: 4BD BC DE CA
,C D A E ABC EBD
,EB BD EDAB BC AC
34
BDBC
34
EB BD EDAB BC AC
3 3 3, ,4 4 4
EB AB BD BC ED AC
1 1
53
(i) ABC (ii) BX
(iii) BX
1 2,B B
3B
1 1 2BB B B
2 3B B
(iv) 3,B C
(v) 2B 3,B C
BC ' 'D (vi)BC 1 2,C C
1 1 2CC C C CD (vii) 2C CA
107
Page 108
DCEB-KDP
' 'BA 1A
1 2A BC 5 13
1 2ABC A BC
21 1 2 5
3A B ACBCAB BC AC
112
1 312 2
7BC 3.5AD ABC
BC ' 'D
BD CD
BC E CD CE
E CA BA F
BEF
: 3 : 2BE BC CA EF
ABC FBE
32
FB BE FEAB BC AC
108
Page 109
DCEB-KDP
3 3 3, ,2 2 2
FB AB BE BC EF AC
32
112
5AB 060 , 4.5A AC 54
5.3AB 0 040 , 50A B 37
5.3AB 0 040 , 50A B
ABC AZ
1 2 3 7, , ,.......A A A A
1 1 2 6 7...........AA A A A A
7 ,A B
3A 7BA ' 'AB 1' 'B
1B
1B BC AC 1C 1' 'C
1 1AB C ABC
1 1AB C
109
Page 110
DCEB-KDP
ABC DE BC
AD AEBD CE
DM AC EN AB
,BDE CED DE
,DE BC
ar BDE ar CED
ar ADE
ar BDE ar CEDar ADE ar ADE
1 12 21 12 2
BD EN CE DM
AD EN AE DM
BD CEAD AE
AD AEBD CE
(1) AD AEBD CE
1 1AD AEBD CE
AB ACBD CE
( ) AB ACorAD AE
DE BC B D C E
ADE ABC
110
Page 111
DCEB-KDP
(1)AD AEAB AC
1 1AD AEAB AC
AB AD AC AEAB AC
(2)BD CEAB AC
AD AB AE ACAB BD AC CE
AD AEBD CE
ABC DE ,AB AC AD AEBD CE
DE BC
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(i) 3.9PE 3EQ 3.6PF 2.4FR (ii) 4PE 4.5QE 8PF 9RF (iii) 1.28PQ 2.56PR 1.8PE 3.6PF
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111
Page 112
DCEB-KDP
39 3630 2413 310 2EF
QR
40 845 98 89 9EF QR
180 360128 25645 4532 32EF QR
.LM AB 3 , 2 , ( 2)AL x AC x BM x 2 3BC x
x
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( 3) (2 3) 2 ( 2)x x x x
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5 ( 2) 6( 2) 0x x x
( 2) (5 6) 0x x
2x 65
x
112
Page 113
DCEB-KDP
ABC , , ( 2), ( 2)DE BC AD x DB x AE x ( 1)EC x x 4x
LM CB LN CD AM ANAB AD
ABC LM CB AM ALAB AC
(1)
ACD LN CD
(2)AN ALAD AC
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7.5AB DC y 4.5EF BC x ,x y
AB FE
BAC CEF
ABC CFE
ACB ECF
ABC EFC
AB BC ACEF FC EC
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BC CD BDC BEFBF FE
5 53 4.5 8 4.5
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4.5 5 2.81258
y
113
Page 114
DCEB-KDP
,D E ,AB AC
1ADAD BDBD
1AEAE CECE
AD AEBD CE
DE BC
ABCD AB DC
AO COBO DO
EO AB ABD EO AB
(1)AE BODE DO
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BO AODO CO
AO COBO DO
AB ' 'D
DE BC AE ADCE DB
1AECE
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AE CE
AC E
114
Page 115
DCEB-KDP
7.2AB
AX
AX
1 2 8, ,..........A A A
1 1 2 7 8..........AA A A A A 8A B
5A 8A B AB 1B
1 1: 5 : 3AB BB
8ABA 5 1 3A B A B
51
1 5 8
53
AAABB B A A
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115
Page 116
DCEB-KDP
(i) (ii) (iii)
(ii) (iii)
( )i op r ( )ii op r ( )iii op r
' 'O xy
' 'p OP XY OP XY xy xy
Q OQ OP
' 'O xy OP
OP XY
116
Page 117
DCEB-KDP
' 'O ' 'OP OZ
,A B AP BP ' 'P AB AB ' 'p
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117
Page 118
DCEB-KDP
' 'O ' 'p op r
op op
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OP PQ OQ ,A B
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118
Page 119
DCEB-KDP
PA PB
ABCD , ,P Q R ' 'S AB CD BC DA
AP ASBP BQCR CQDR DS
, ,A B C ' 'D
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AP BP CR DR AS DS BQ CQ
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OAP OBP 12
APO BPO APB
119
Page 120
DCEB-KDP
01 602
030
OAP OA PA
00 3030 OASin
OP
1 52 OP
0 1302
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10OP
' 'O 5OA OB ' 'P 10OP ' 'OP ' 'OP M
M 1 52
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120
Page 121
DCEB-KDP
227
(r) = 7 (h)=
( l )= 2 2 2l r h
2 2l r h
2 27 10
49 100
149 12.2
rl
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m
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222 1 1 77
121
Page 122
DCEB-KDP
222 87
2352 50.287
m
' 'x 2x
2x
2x
2 2( )l r h
2 2(2 )x x
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5x
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4 : 4 : 5
3.14
122
Page 123
DCEB-KDP
( ) 12d
12( ) 62
r
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2 2( )l r h
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36 64
100
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213
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13
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9 16
25
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22rl r
123
Page 124
DCEB-KDP
2r l r
3.14 3 5 2 3
9.42 (5 6)
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103.62
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124
Page 125
DCEB-KDP
20( ) 102
r
2r h
227
10 10
40
280
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125
Page 126
DCEB-KDP
2r h
227
5 5
1.4
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770
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323
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227
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154
126
Page 127
DCEB-KDP
( ) 4.2r
343
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127
Page 128
DCEB-KDP
2772 1000
10 000
2 21:3
r h r h
11:3
3:1
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128
Page 129
DCEB-KDP
2 8 4 4 4 8 4
2 32 16 32
2 80
160
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213
r h
2
13
227
6 6 7
129
Page 130
DCEB-KDP
2 rh
rl
24 r
22 r
2 ( )r r h
( )r l r
23 r
2r h
213
r h
343
r
323
r
1232
130
Page 131
DCEB-KDP
Cos A Cos X A X A X
,ABC XYZ 090B Y
Cos A Cos X AB XYAC XZ
AB ACXY XZ
AB AC KXY XZ
.AB K XY .AC K XZ
2 2 2AB BC AC
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BC KYZ
AB BC ACXY YZ XZ
ABC XYZ A X
B ABC tan 3A ( )i Sin A Cos C Cos A SinC
( )ii Cos A Cos C Sin A Sin C
ABC 090B
131
Page 132
DCEB-KDP
tan 3A
31
31
BCAB
: 3 :1BC AB
3BC K AB K
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22 3K K
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A AB KCos AAC
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12
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32
3 3 1 1 3 1 3 1 4( ) . .2 2 2 2 4 4 4
i Sin A Cos C Cos A Sin C
41
1 3 3 1 3 3( ) . . 02 2 2 2 4 4
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0 01 1( ) , ( ) , 0 90 ,2 2
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12
Sin A B
0 1302
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132
Page 133
DCEB-KDP
12
Cos A B
0 1602
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060 ............(2)A B
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A B 0
0
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11
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2
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1
1
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Sin Cos
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133
Page 134
DCEB-KDP
1 CosSin Sin
sec . . .Co Cot R H S
. . . . .L H S R H S
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11
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Sin
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LHS RHS
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tan tanP R
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134
Page 135
DCEB-KDP
2 2 2PQ QR PR 2 2 27 25QR
244 625QR
2 625 49 576QR
576 24QR
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225 5AC K K
4 45 5
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3 35 5
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060AOB
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135
Page 136
DCEB-KDP
O 030COB
COB 030 BCSinOB
12 6
BC
2 6BC
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0 0 0 0 03 3 1 160 , 30 , 30 , 60 , 90 12 2 2 2
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0 0 0 0 3 3 1 1 3 1 3 1 460 30 30 60 . .2 2 2 2 4 4 4
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41
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0 060 , 30A B ( )Sin A B Sin A Cos B Cos A Sin B
0 0 0 036 54 36 54 0Cos Cos Sin Sin 0 0 0 0 0 0 0 0 0 036 54 36 54 (90 54 ) 54 (90 54 ) 54Cos Cos Sin Sin Cos Cos Sin Sin
054Sin Cos 0 054 54Cos 054Sin (90 )(90 )
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0
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A B CCot
ABC 0180A B C
0180
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0902 2
A B C
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136
Page 137
DCEB-KDP
0( )tan tan 902 2
A B C
( )tan cot2 2
A B C 0tan(90 ) cot
2 4 4 2tan tan sec sec
2 4 2 2. . tan tan tan (1 tan )L H S
2 2tan .sec 2 2 2 2
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1sec .sin sec tancos
A A A AA
sinsec sec tancos
AA A AA
sec tan sec tanA A A A
2 2sec tanA A 1
sec (1 sin )(sec tan ) 1A A A A
2 2 2 2sin cos cos sec 7 tan cotA ecA A A A A
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cos sec 2cos sec
L H S A ecA A ec A A ecA
A A A A
2 2 2 2sin cos cos sec 2sinA A ec A A A 1.
sin A2cos A
1.cos A
2 21 cos sec 2 2ec A A
2 25 cos secec A A
2 2
2 2
2 2
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ec A Aec A AA AA A
137
Page 138
DCEB-KDP
2 25 1 cot 1 tanA A 2 27 tan cotA A
0 0sin 45 cos 45
0 01 1sin 45 , cos 452 2
0 0 1 1 1 1 2 2. 2sin 45 cos 45 22 2 2 2 2
00
1cot 0tan 0
00
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2 2
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324
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sin( ) sin sinA B A B
0 0cos12 sin 78
0 0 0 0 0cos12 sin 78 cos(90 78 ) sin 78
0 0sin 78 sin 78 0( cos (90 ) sin )
0
138
Page 139
DCEB-KDP
0tan 2 cot( 18 ), 2A A A A 0tan 2 cot( 18 )A A
0 0cot(90 2 ) cot( 18 )A A 0( cot (90 ) tan )A A 0 090 2 18A A 0 090 18 2A A
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0 0sin 75 cos65 00 045
0 0 0 0sin 75 sin(90 15 ) cos15 0( sin(90 ) cos ) 0 0 0 0cos 65 cos(90 25 ) sin 25 0( cos (90 ) sin )
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15sin17
C cos C
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215 cos 117
C
2 225 289 225 64cos 1289 289 289
C
64 8cos289 17
C
8cos17
C
2 2sin cos sin cos
2 2sin cos sin cos
139
Page 140
DCEB-KDP
2 2sin cos 2sin 2 2cos sin cos 2sin cos
1 1 2
2 2sin cos sin cos 2
2 2sec 1 cos 1ec
2 2 2 2sec 1 cos 1 tan .cotec
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2
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sec tan 1 sec 1 tansec cot 1 cos 1 cotcon ec
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cos ..................A
1213
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21 tan .......................
2 0 2 0
2 0 2 0
sec 60 tan 60 ........................sin 30 cos 30
0
2 0
2 tan 30 ......................1 tan 30
2 0
2 0
1 tan 45 ......................1 tan 45
0
2 0
2 tan 30 ......................1 tan 30
140
Page 141
DCEB-KDP
0
0
tan 36 ......................cot 54
0 0cos 31 sec59 .........................ec
0 0sin15 .sec75 .......................
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5tan , sec .........................12
x x
25cos , cot ......................7
ec then
0 0 0 0sin 5 cos85 cos5 sin85 ...........................
sec tan P sec tan ..........................
cos cotec P cos cot .......................ec
3sin5
tan .......................
2 0 2 0sin 50 cos 50 .......................
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sin cos ..............
3sin2
..............
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A B ....................A B
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1213
513
sec
141
Page 142
DCEB-KDP
32
3
1312
247
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045
060
030
142
Page 143
DCEB-KDP
30BD
A
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143
Page 144
DCEB-KDP
30 3 3 30
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144
Page 145
DCEB-KDP
(2)3
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145
Page 146
DCEB-KDP
1( 10)3
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146
Page 147
DCEB-KDP
21AB
21x
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(2 )32
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147
Page 148
DCEB-KDP
3 30BD
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148
Page 149
DCEB-KDP
15h BC
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tan BCAB
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149
Page 150
DCEB-KDP
3 6AB
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150
Page 151
DCEB-KDP
* * E P(E)
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151
Page 152
DCEB-KDP
1, 2,3, 4,5,6S
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113
152
Page 153
DCEB-KDP
(ii) ( ) 13n D
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153
Page 154
DCEB-KDP
( ) ( ) 1P G P G
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154
Page 155
DCEB-KDP
152
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155
Page 156
DCEB-KDP
(i) 124 31144 36
(ii) 31 5136 36
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156
Page 157
DCEB-KDP
12( )P E
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157
Page 158
DCEB-KDP
(a) x
(b)
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i
f xx
f
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(iii) i i
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(i) ' 'n 12
n
(ii) ' 'n , 12 2n n
(b)
2n cf
l hf
Prepared byK. Kiran Kumar ReddyZPHS, Dommara Mandyala.
158
Page 159
DCEB-KDP
MEK
16 12 18 8 9 0 5 68 9.77 7
7 1 8 4
2 2
, , , ,5 4 3 2x x x xx
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159
Page 160
DCEB-KDP
P,P P
P,
9 11 13 18 196
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706
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= P
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1, 2,3,................., n
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4,10,15, 5, 2 1,18,23, 27x x
4,10,15, 5, 2 1,18,23, 27x x
8 8, 1 4,52 2
160
Page 161
DCEB-KDP
5, 2 1x x
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3 42
x
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x
,A B C , ,A B C
, ,A B C
i i
i
f xf
25 70 40 65 35 5025 40 35
1750 2600 2500100
6100100
61%
161
Page 162
DCEB-KDP
300if 16850, 350, 35i i i i i if x f d f u 300, 55, 10, 16850, 350, 35i i i i i i if a h f x f d f u
(i) 16850 56.2
300i i
i
f xx
f
(ii) i i
i
f dx a
f
35055300
55 1.2
56.2
(iii) i i
i
f ux a h
f
3555 10300
55 1.2
56.2
*
if
ix 55
i
i
dx
5510
ii
xu i if x i if d i if u
162
Page 163
DCEB-KDP
f
44if f 20i if u f
18, 44 , 20, 2i i ia f f f u f h
18i i
i
f ux a h
f
2018 2 1844f
f
20 2 18 18 044f
f
20 2 0 44 0f f
20 0f
20f
f
‹ ¬OY¼ if
ix
18i id x 182
ii
xu i if u
f
(a)
f
163
Page 164
DCEB-KDP
44if f 752 20i if x f 44 , 752 20i i if f f x f
18i i
i
f xx
f
752 20 1844
ff
752 20 18(44 )f f
752 20 792 18f f
20 18 792 752f f
2 40f
402
f
20f
if
ix
f
f
i if x
164
Page 165
DCEB-KDP
8if i if u
0 1 235, 21, 23, 14, 10l f f f h
1 0
1 0 22f fl h
f f f
23 2135 102 23 21 14
2 2035 10 35 35 1.8 36.846 35 11
40, 10, 80, 37i i ia h f f u
i i
i
f ua h
f
37 3740 10 40 40 4.6 35.480 8
ix
40i id x 4010
ii
xu i if u
l
0f 1f
2f
( )a
61
165
Page 166
DCEB-KDP
53n
53 26.52 2n
60, 7, 22, 10l f cf h
2n cf
l hf
26.5 2260 107
4.560 107
45607
60 6.4
66.4
l
f
26.52n
166
Page 167
DCEB-KDP
x y
60n 60 30
2 2n
10, 20, 20, 5 , 45 60h l f cf x n x y
60 45x y
15.......(1)x y
2 28.5
n cfl h
f
30 (5 )20 10 28.520
x
30 5 28.5 20 8.52
x
25 8.5 2 17x
17 25 8x
8x
8x
8 15 15 8 7y y
8, 7x y
x
y
l
x
( )f
y
55 ( )x cf
25 x
40 x
40 x y
45 x y
167
Page 168
DCEB-KDP
100 502 2n
35, 33, 45, 5l f cf h
2n cf
l hf
50 4535 533
5 535 35 0.76 35.7633
*
l
( )f
( )cf
168
Page 169
DCEB-KDP
1 2 3, , ,............, nx x x x x 1 2 32 7, 2 7, 2 7,............., 2 7nx x x x
2 7x
n 1n
n 1
2n
x 5x
2, , 2a a a a
169