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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
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)
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
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)
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)
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)
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
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
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'
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
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
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
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)
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
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
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
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
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
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
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)
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
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
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
__________________
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)
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)
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
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
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)
DCEB-KDP
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
DCEB-KDP
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
DCEB-KDP
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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)
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 ,
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
' 'E ' 'F PQR ,PQ PR EF QR
(i) 3.9PE 3EQ 3.6PF 2.4FR (ii) 4PE 4.5QE 8PF 9RF (iii) 1.28PQ 2.56PR 1.8PE 3.6PF
3.9 3.63 2.4
PE PFEQ FR
4 8
4.5 9
PE PFQE RF
1.8 3.61.28 2.56
PE PFPQ PR
111
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
.LM AB
AL BMAC BC
3 22 2 3
x xx x
( 3) (2 3) 2 ( 2)x x x x
2 22 3 6 9 2 4x x x x x
3 9 4x x
9x
x DE AB (8 9), ( 3), (3 4),AD x CD x BE x CE x
DE AB
AD BECD CE
8 9 3 43
x xx x
28 9 ( 3)(3 4)x x x x 28 9x x 23 4 9x x x 1225 4 12 0x x 25 10 6 12 0x x x
5 ( 2) 6( 2) 0x x x
( 2) (5 6) 0x x
2x 65
x
112
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
AM ANAB AD
.AB CD EF
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
7.5 75 3 54.5 3 45
x x
BEF CD FE
BC CD BDC BEFBF FE
5 53 4.5 8 4.5
x y y xx
4.5 5 2.81258
y
113
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
ADC EO DC
(2)AE AODE CO
BO AODO CO
AO COBO DO
AB ' 'D
DE BC AE ADCE DB
1AECE
( )AD DB
AE CE
AC E
114
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
1 4.5AB 1 2.7BB
115
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
DCEB-KDP
' 'O ' 'OP OZ
,A B AP BP ' 'P AB AB ' 'p
OP XY XY
' 'OP O ' 'p d
OA r
AP ' 'p
2 2 2AP OP OA 2 2AP d r
' 'O ' 'P 10OP d
6OA r
2 2 2 210 6 100 36AP d r
64 8
117
DCEB-KDP
' 'O ' 'p op r
op op
' 'Q 12
OP PQ OQ ,A B
,P A ,P B,PA PB
090OAP OA PA
090OBP
OB PB
,PA PB ' 'P
2 2 2PA OP OA 2 2OP OB OA OB
2PB
118
DCEB-KDP
PA PB
ABCD , ,P Q R ' 'S AB CD BC DA
AP ASBP BQCR CQDR DS
, ,A B C ' 'D
AP BP CR DR AS BQ CQ DS
AP BP CR DR AS DS BQ CQ
AB CD AD BC
' 'P
' 'O ' 'P ' 'OP OA OB OP OP PA PB
OAP OBP 12
APO BPO APB
119
DCEB-KDP
01 602
030
OAP OA PA
00 3030 OASin
OP
1 52 OP
0 1302
Sin
10OP
' 'O 5OA OB ' 'P 10OP ' 'OP ' 'OP M
M 1 52
OP
,A B ,P A ,P B ,PA PB
120
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
222 7 12.27
m
2268.4 m
2268.4 m
268.4 134.2
2
( ) 2d
2( ) 12 2dr
( ) 7h
2 ( )r r h
222 1 1 77
121
DCEB-KDP
222 87
2352 50.287
m
' 'x 2x
2x
2x
2 2( )l r h
2 2(2 )x x
2 24x x
25x
5x
2 24 4r x 2 2 .2rh x x
24 x rl
2. . 5 5x x x
24 x 2: 4 x 2: 5 x
4 : 4 : 5
3.14
122
DCEB-KDP
( ) 12d
12( ) 62
r
( ) 8h
2 2( )l r h
2 26 8
36 64
100
10
213
r h
2
13
3.14 6 6 8
rl
3.14 6 10
188.4
( 3.14)
( ) 6d
6( ) 32
r
( ) 4h 2 2( )l r h
2 23 4
9 16
25
( )r ( ) 3r
22rl r
123
DCEB-KDP
2r l r
3.14 3 5 2 3
9.42 (5 6)
9.42 11
103.62
5( )2
r
14 2 52
9
2 rh
22 5 52 27 2 2
222 57 2
22 59 2 27 2
52
22 5 9 57
227
5 14 2
220
( ) 2.8h
( ) 20d
124
DCEB-KDP
20( ) 102
r
2r h
227
10 10
40
280
88000
( ) 10r
( ) 42h
213
r h
1 22 10 103 7
2
6
42
693 000
1000
693
227
( ) 5r
( ) 9.8h
125
DCEB-KDP
2r h
227
5 5
1.4
9.8
770
( )r
323
r
2 22 3.5 3.5 3.53 7
( )r
( ) 5h
313
r
1 22 3.5 3.5 53 7
2 22 1 223.5 3.5 3.5 3.5 3.5 3.53 7 3 7
22 2 13.5 3.5 3.5 57 3 3
22 7 53.5 3.57 3 3
0.5 4
227
3.5123.5 3
154
126
DCEB-KDP
( ) 4.2r
343
r
4 22 4.2 4.2 4.23 7
( ) 6r ( )h
2r h
22 6 67
h
22 4 226 6 4.2 4.2 4.27 3 7
h
2 1.4 1.4 1.4
4h 3
22
74.2 4.2
74.2 22
16
16
3 3
( ) 21r
22 r
32227
21 21
127
DCEB-KDP
2772 1000
10 000
2 21:3
r h r h
11:3
3:1
3 64a 3 34a 4a
2 lb bh lh
128
DCEB-KDP
2 8 4 4 4 8 4
2 32 16 32
2 80
160
( ) 6r
( ) 7h
213
r h
2
13
227
6 6 7
129
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
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
2 2 2BC AC AB
2 2. .K XZ K XY
2 22 2K XZ K XY
2 2 2K XZ XY
2 2K YZ2 2 .BC K YZ K YZ
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
DCEB-KDP
tan 3A
31
31
BCAB
: 3 :1BC AB
3BC K AB K
ABC 2 2 2AC AB BC
22 3K K
2 23K K 24K
24 2AC K K
3A BC KSin AAC
2 K
32
A AB KCos AAC
2 K
12
C AB KSin CAC
2 K
12
3C AB KCos CAC
2 K
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
ii Cos A Cos C Sin A Sin C
0 01 1( ) , ( ) , 0 90 ,2 2
Sin A B Cos A B A B A B A B
12
Sin A B
0 1302
Sin 030 ...........(1)A B
132
DCEB-KDP
12
Cos A B
0 1602
Cos
060 ............(2)A B
A B 030
A B 0
0
60
2 90A
0
090 452
A
045A 0 045 60B
0 0 060 45 15B 0 045 , 15A B
1 sec1
Cos Co CotCos
11
CosLHSCos
1 Cos
1 1.1 1
Cos CosCos Cos
2
2 2
11
CosCos
2
2
11
CosCos
2
2
1 CosSin
2 2
2 2
1
1
Sin Cos
Sin Cos
1 CosSin
133
DCEB-KDP
1 CosSin Sin
sec . . .Co Cot R H S
. . . . .L H S R H S
secCo Cot K 2
2
11
KCosK
secCo Cot K
22
22
sec 11. .1 sec 1
Co CotKR H SK Co Cot
2 2
2 2
sec 2 sec 1sec 2 sec 1
Co Cot Co CotCo Cot Co Cot
2 2
2 2
sec 1 2 secsec 1 2 sec
Co Cot Co CotCo Cot Co Cot
2 2
2 2
2 secsec sec 1 2 sec
Cot Cot Co CotCo Co Co Cot
2 2
2 2
2 2
sec 1sec 1sec 1
Co CotCo CotCo Cot
2
2
2 2 sec2 sec 2 sec
Cot Co CotCo Co Cot
2
secCot Cot Co
2 sec secCo Co Cot
1sec
CosCot CosSin
Co SinSin
Sin
. .1
Cos L H S
LHS RHS
PQR 7PQ 25PR 090Q
tan tanP R
7PQ 25PR
090Q
134
DCEB-KDP
2 2 2PQ QR PR 2 2 27 25QR
244 625QR
2 625 49 576QR
576 24QR
24tan7
P QRPP PQ
7tan24
R PQRR QR
24 7 576 49 527tan tan7 24 168 168
P R
3tan 4A SinACosA
3tan 4A
4tan3
A
4K 3K
ABC 08 902 2 2AC AB BC
2 2(3 ) (4 )K K 2 2 29 16 25K K K
225 5AC K K
4 45 5
KSinAK
3 35 5
KCosAK
6OA OB
060AOB
AB ' 'O OC
135
DCEB-KDP
O 030COB
COB 030 BCSinOB
12 6
BC
2 6BC
2 6AB BC
0 0 0 060 30 30 60Sin Cos Sin Cos 0 060 30Sin
0 0 0 0 03 3 1 160 , 30 , 30 , 60 , 90 12 2 2 2
Sin Cos Sin Cos Sin
0 0 0 0 3 3 1 1 3 1 3 1 460 30 30 60 . .2 2 2 2 4 4 4
Sin Cos Sin Cos
41
0 0 0(60 30 ) 90 1Sin Sin 0 0 0 0 0 0(60 30 ) 60 30 30 60Sin Sin Cos Sin Cos
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 )
Cos SinSin Cos
0
,A B C ABC tan2 2
A B CCot
ABC 0180A B C
0180
2 2A B C
0902 2
A B C
0902 2
A B C
136
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
2 2
( sec tan 1 sec 1 tansec 1 tan )
2 2sec 1 sec
4 2sec sec . .R H S
. . . . .L H S R H S
sec 1 sin sec tanA A A A
sec 1 sin sec tan sec sec sin sec tanA A A A A A A A A
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
2 2 2
2 2
. . sin cos sin cos 2sin cos
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
2 2
[ cos cot 1cos 1 cotsec tan 1sec 1 tan ]
ec A Aec A AA AA A
137
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
1 1cot 0tan 0 0
' 'O
2 0 2 0 2 02 tan 45 cos 30 sin 60
0 0 03 3tan 45 1, cos30 , sin 602 2
2 2
2 0 2 0 2 0 2 3 32 tan 45 cos 30 sin 60 2(1)2 2
324
34
2
sin( ) sin sinA B A B
0 060 , 30A B
0 0 0sin sin(60 30 ) sin 90 1A B
0 0 3 1 3 1sin sin sin 60 sin 302 2 2
A B
sin( ) sin sinA B A B
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
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
0108 3A 0
00
36
1083 108A A 3
036A
,A B tan cotA B 090A B
0tan cot tan(90 )A B B 0( tan(90 ) cot ) 090A B
090A B
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 )
0 0 0 0sin 75 cos 65 cos15 sin 25
15sin17
C cos C
15sin17
C
2 2sin cos 1C C 2
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
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
2ant
2
1.tan
2 2 2 2
2 2 2 2
sec tan 1 sec 1 tansec cot 1 cos 1 cotcon ec
1
2 2sec 1 cos 1 1ec
cos ..................A
1213
CosA sin ...................A
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
DCEB-KDP
0
0
tan 36 ......................cot 54
0 0cos 31 sec59 .........................ec
0 0sin15 .sec75 .......................
0 0tan 26 . tan 64 .......................
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 .......................
2 0 2 0sec 100 tan 100 ..............................
2 0 2 0cos 75 cot 75 .....................ec
sin cos ..............
3sin2
..............
3cos( )2
A B ....................A B
0 0sin10 .sec80 ...............
0
0
sin18 ...................cos72
1213
513
sec
141
DCEB-KDP
32
3
1312
247
1P
1P
34
045
060
030
142
DCEB-KDP
30BD
A
030BAD
C
060BCD
AD x
CD y
ABD
0tan 30 BDAD
1 303 x
30 3 (1)x
BCD
0tan 60 BDDC
3 301
Vy
3 30y
30 (2)3
y
AC AD DC x y
30 3 301 3
143
DCEB-KDP
30 3 3 30
3
30 3 303
90 303
1203
120
3AC
#°O_ = ‹ ¬°ë#ß Hê~¡°# ° 30
AB x
CD h
BC d ( )AC AB BC x d
030PDA DAB 060PDB DBC
BCD
0tan 60 CDBC
3 hd
3 (1)h d
ACD
0tan 30 CDAC
13
hx d
3h x d
144
DCEB-KDP
(2)3
x dh
33
x d d
3x d d 3x d d 2x d
2xd
x
2xd
AB h
BC d
10CD
( 10)BD BC CD d 060ACB 030ADC
ABC 090B
0tan 60 ABBC
31
hd
3 (1)h d
ABD 090B
0tan 30 ABBD
145
DCEB-KDP
1( 10)3
hd
3 10h d
10 (2)3
dh
3 101 3
d d
3 10d d
3 10d d
2 10d
5d
5BC d
3.AB h d
3 5
5 3AB h
2BD
CD h
BC BD DC
(2 )h
A B
AB x 0 045 , 60BAD BAC
ABD 090B
0tan 45 BDAB
146
DCEB-KDP
21AB
21x
2x
2AB x (1)
ABC 0 090 , 60B BAC
0tan 60 BCAB
(2 )32
h
2 2 3h
2 3 2h
2 1.732 2
2(1.732 1)
2 0.732
1.464h
1.464CD h
AB CD h BD
0 060 , 30ADB CBD
ABD 090ABD
0tan 60 ABBD
303BD
147
DCEB-KDP
3 30BD
30 (1)3
BD
BCD 090BDC
0tan 30 CDBD
1303
3
h
3033
h
303 3
h
10
30h 3
10h
10CD h
BC h A 15AB
045CAB
0tan 45 BCAB
1 15h
15h
148
DCEB-KDP
15h BC
15BC
5 3AB
PCA CAB
ABC
tan BCAB
15
3
5 3
33
tan 3
0tan 60 3
060
060
6BC
AB x 060PCA CAB
ABC
0tan 60 BCAB
63AB
149
DCEB-KDP
3 6AB
63
AB
6 33 3
2 6
33
2 3AB
2 1.732
3.464AB
BC AD x
060CBD
600BD
BCD
060 BCCosBD
12 600
x
300x
300BC AD x
150
DCEB-KDP
* * E P(E)
( ) EP E
* E E E
P(E)+P(E)=1
1
(i) (ii)(iii)
Y, 'B' 'R'
. .Y B R
(i) Y
1( )3
P Y
(ii) R
1( )3
P R
(iii) B
1( )3
P B
(i)(ii)
151
DCEB-KDP
1, 2,3, 4,5,6S
( ) 6n S
(i) 5,6E
E ( ) 2n E
( )( )( )
n EP En S
2
6 3
13
(ii) ' 'F
F 1, 2,3, 4F
( ) 4n F
F ( ) 4( )( )
n FP Fn S
2
6 3
23
(i) (ii) (iii) (iv)
(i)
( ) 3( )( )
n AP An S
39 13
113
152
DCEB-KDP
(ii) ( ) 13n D
( )( )( )
n DP Dn S
13
1
39 3
13
(iii) ( ) 39n S
39
39
1
(iv)
0 039
(i)(ii) (iii)
(i)
( ) 5n R
5( )
17P R
(ii)
( ) 8n T
8( )17
P T
(iii) ( ) 4n G
4( )
17P G
P G
153
DCEB-KDP
( ) ( ) 1P G P G
( ) 1 ( )P G P G
4117
17 4 1317 17
(i) (ii ) (iii) (iv) (v) (vi)
( ) 52n S
(i)
( ) 2n R
( )( )( )
n RP Rn S
2 152 26
(ii) ( ) 12n F
( ) 12( )( )
n FP Fn S
3
52 13
313
(iii)
6
3
52 26
326
(iv)
154
DCEB-KDP
152
(v)
13
1
52 4
14
(vi)
152
E E E E E
( ) ( ) 1P E P E
( ) 1 ( )P E P E
E 1 ( )P E P E
,S R
( ) 0.62P S
( ) 1 ( )P R P S
1 0.62
0.38
(i)(ii)
155
DCEB-KDP
(i) 124 31144 36
(ii) 31 5136 36
( ) 0.05P E E
E
( ) (P E P E
(P E 1 ( )P E
( ) 0.992P E
( ) 1 ( )P E P E
2 152 26
156
DCEB-KDP
12( )P E
3
52 13
313
3 10( ) 1 ( ) 113 13
P E P E
316
12
( ) 0.91P E ( ) 0.09P E
( ) ( ) 1P E P E
( )O P E m 1m
1 1 1P A B P A B
1 1 1P A B P A B
1( )3
P E 2( ) 3P E
1100
157
DCEB-KDP
(a) x
(b)
(i) i i
i
f xx
f
(ii) i i
i
f dx a
f
(iii) i i
i
f ux a h
f
(a)
(b)
1 0
1 0 22f fl h
f f f
(a)
(i) ' 'n 12
n
(ii) ' 'n , 12 2n n
(b)
2n cf
l hf
Prepared byK. Kiran Kumar ReddyZPHS, Dommara Mandyala.
158
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
, , , ,5 4 3 2x x x x x
5 1 6 32 2
3x
8 3 8 243x x
1800 37.5
48
159
DCEB-KDP
P,P P
P,
9 11 13 18 196
P
706
P
= P
70 6 706
PP P P
6 70P P 5 70P
70 145
P
1 1 1 11, , , ,...............,2 3 4 n n
1 1 11, , ,.................,2 3 n
1, 2,3,................., n
1 1 11 1 2 3 ...........
2 31 2 3 ................
i i
i
nf x nf n
1 1 1 ...........1 2 3 ................
nn
2 2.
1 1 12
n nn n n nn
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
DCEB-KDP
5, 2 1x x
5 2 12
x x
3 42
x
3 4 3 4 34
2x x
3 34 4 30x
30 103
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
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
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
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
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
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
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
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
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
