˘ ˇ - SITAM 3mt 25 3 mt 3 20 3mt 50 mt 3 SITAM ˘ˇ ... x xxx x − ++ + += − ˘

SITA

M

tan2

( )6cos 2sin2

xf x

x x

tan 2( )

6cos 2sin 2

xf x

x x

2 1 , 2

R n n Z

2 1 , 4

R n n Z

2 1 , 2 1 , 2 4

R n n Z R n n Z

2. If A = 1, 2, 3, 4 and B = 1, 2, 3 then the number of into functions that can be defined from A toB is

2

x - 3x

2

x - 3x !"

( ), 0−∞ ( ], 0−∞ ( )0, ∞ R 0−

!"#$% &'&''&'''&('')'!'

&'&''&'''&('')'!'#$%"&'()*+","

* $+ ,-$ , .-$ . +- +$+ ,-$ , .-$ . +++

5 1

4

5 1

8

5 1

2

5 1

* /α 01# 2$ 3+ 4 %4

+

3+"- 4 %4,.# /0α 1234'56"-(7&' / /,&.89:+,"

k 1

sink 1

+ α− α

+k

sink 1

k 1

sink 1

− α+ α

( /# cos cos9 2.3 1 0x x

cos cos9 2.3 1 0x x

n 2

n 2n 2 1

2n

SITA

M

* 1 1cos tanx x

1sin cos x

1 1cos tanx x

1sin cos x

+ + 1

x

12x

!) * ABC " 060A $ #

ABC !"" 060A #$$%&'(')&

2 3 7 0c c 2 3 7 0c c 2 3 7 0c c 2 3 7 0c c

! * + #+ &56(

*+,+-./0 &56( 77

2 − !

* 8 6x y z 2 3 10x y z 2x y z 8

6x y z 2 3 10x y z 2x y z &12345678 3, 10 3, 10 3 0

* 2 3a i j k 2b i j k 3c i j d # " a andb ( , )c d

2 3a i j k 2b i j k 3c i j /0 d * ,a b & 74594:&4;'(34<=( , )c d

41cos30

41sin30

21cos30

21sin30

* #$ # , , ,a b c d #$1# ( ) ( )a b c d× × × =

, , ,a b c d &>0?0-&>( ) ( )a b c d× × × =

a b 0

1 3/2sinh 2

−

log 3 8e

− log 3 8e

+ log 2 18e

+ log 2 18e

−

/ #,$ #1 " % / 1# # $ % / $" %;<=$>?+@ABC"DE;FGH4"I$"-JKL";<AMD5"634"N$+;<O7PC"?+

25 3mt 25 3

mt3

20 3mt 50

mt3

SITA

M

* 2x x 1 0− + = 25 1nx nxn 1

+∑=

2x x 1 0− + = 25

nn

n 1

1x

x=

+ ∑

!

( $ 3 1 1

2 2

ni

3 11

2 2

n

i

+,"

6 6

0$ #$ 2 5 61F i j k

2

2F i j k

$ 1# $* 1# $

1$ #1 0 1 $ #4 3i j k 1 0 1 $ #

6 3i j k 0#% 1#$8

(7QR;LA'1 2 5 6F i j k

"-2 2F i j k

O4"ST5DD6,UV4"

WXY@ ! 4 3i j k / K#"ZWXR@ ! 6 3i j k / K#"Z[-\+UV5WX";Y","

! 6 6!

]5^_ !]5^_ 6]5^_ 6!]5^_

) 9 2b i j k= + − 3c i k= + * a $ # + $#

#11#$ a b c

2b i j k= + − ` 3c i k= + "-a @O]5)*@ !+ a b c -ab,"

3 10 6+ 59 60

! * , ,a b c # #6$1#$ ##1# ",$$## / 1#11

$ ( ) ( ) ( ) ( ) ( ) ( ). . .a b b c b c c a c a a b+ × + + × + + × =

, ,a b c "]7+c/5@ !"+;C"d'eWX5DfC

-"4"NgKh]5 _+( ) ( ) ( ) ( ) ( ) ( ). . .a b b c b c c a c a a b+ × + + × + + × =

± !

9 ( ) ( )1,0 , 0,0P Q= − = ( )3,3 3R = " #1 8 "$ # /

( ) ( )1,0 , 0,0P Q= − = ( )3,3 3R = !"

3 0x y+ = 3

02

x y+ = 3

02

x y+ = 3 0x y+ =

SITA

M

* &&!:&!## $ #/ /

#/

&&!"-:&!,J&.8i'5jkel+ mno"Z".# +pnq7"Z

5 2 5 !

#"11 #/ +1 ( ), ,α β γ $# +

: $ 1 #$ 1 #/:1#

$# 1

O@AB+"( ), ,α β γ 2QY;K#"Z mno+5;/l:"#F@U5;+A'

"+;" /r^: mno"Z+"7K#"ZK##"

1x y z

α β γ− − = 1x y z

α β γ+ + = 1α β γ+ + = 1x y z

α β γ− + =

* ( ) 7f x x= − /&+& &+ #+; | ( )g x =

( ) 7f x x= − "-/&+& &+++; | ( )g x =

! 6 1

2

−

# &4 ( )( )

16 1515 16 12 4 81 1 1 12

1

d x xx x x xdx x

− ++ + + + =−

.#s7e&4 ( )( )( )( ) ( )

16 1515 16 12 4 81 1 1 121

d x xx x x xdx x

− ++ + + + =−

&4 ( )16 12 4 81 1 1 1

1xx x x xx

−+ + + + = −

/;4"&4 ( )( )( )( ) 16

2 4 8 11 1 1 1

1

xx x x x

x

−+ + + + =−

#$##$ $##$ +1 A "- R -t,U"- R @A -t,";4U ##$##$ $##$ +1 A "- R -t,U"- R @A -t,";4/#U # A -t@` R -t@/#U # A -t@/#` R -t@U

$ 0$#22 yx 1

2 2a b

22 yx 12 2l m $ # /,

2 2

2 2

x y1

a b

2 2

2 2

x y1

l m 2"/LDu.#"DvB5""

'"' 3"3

3"' '"3

SITA

M

( "# # #0$ ,6$# 1 #$ +',!+,' /#

+',!/0+,'&@45AB4C&,-DEF4 > G0&&

2 2 2( ) (1 )y mx a m 2 2 21y nx a n

##"" 8#

02 2 2( ) (1 )y mx a m /0

2 2 21y nx a n H@&I0>0J4(I0>0KL4 > 2 2m n 1m 6 1mn

) * 0 #/#/", 2 22 0x xy y $ #5 1

,3 3

8 #2 22 0x xy y H@&.MNL4745AMNO&/0

5 1,

3 3

*.MNL4

H4M4 >P.MNL%5AMNO4 2 0x y 2 0x y 2 0x y 2 0x y

!33| |x xlt a ax a

&;!0#<+= >*? #8 +

33| |x a

x xlt

a a

&;!Q<+= *>*?R

2 3a

2 1a

2a 22

a

1 2 3 4 5 6 ...... 2

2 21 4 1

nltn n n

1

3

1

3

1

5

1

6 1f x x 1 ( )Q x f x x & 1 3 4f

2 2(10) [ (10)]f

1f x x /0 ( )1 ( )Q x f x x= ∀ &/0 1 3 4f 2 2(10) [ (10)]f

! )

9 2 4 2( ) .......0 1 2nf x a a x a x a xn " 1, x R 0

0 .......0 1a a an &+

2 4 20 1 2( ) ....... n

nf x a a x a x a x @Ldw@x R 0 10 ....... na a a &+12,STU ,+S')TU+ S')/TU/0STU4 #+# ')/TUVTU&&C

SITA

M

!

/ ##0 # #/ $# cosy x #1 $ $

$#

cosy x !"

2 2 2 2x y y x 2 2 2 2x y x y 2 2 2 2x y x y

2 2 22

x x yy

# $/ $ / $!!$

11## $## $#8$

#$%& '(#$)*+,-./0123432-5'"678/09:#;< 6=7>?04@4A4BC 401 2 400

400

401 2

+ ,+&/+

,+&/+ -ab,"

( * # $# / $ 8 # #/$@# # $# #MWX.MNL4MNL4YZ5AC!"[')&H\$@#/0MNLYZ5AC$ .MNL]8 ^&_4[')&H\`

28 3 /cm hr 24 3 /cm hr 24 3 /cm hr 8 2/3

cm hr

* / /&!

/+/&!

) ) ) )

) * #$ $ 9#&""" /" 09+6+3

1# 1" #"#H@*ab[8&&"""/0H@9+6 c_a

3

"&&

2

3

3

8

8 2,

3 3

! AB #/ 0ACB#1#1$# $ #* AC

B" # PQR DDDDD8

AB .MNL!" AC B_')>0 H@& d &4;H@& > ACB" PQR ]8 ^&_I0C0%:e&!"

3

2ab

4

3ab

3

4ab

2

3ab

9 , " # 2 1 0x x+ + = 8 0# # 229α 1004α

, 2 1 0x x "]A'+ 229 "- 1004

"] / xy>;4" 2 1 0x x− − = 2 1 0x x+ − = 2 1 0x x+ − = 2 1 0x x+ + =

SITA

M

"

# 8 3 214 56 64 0x x x− + − = # 1#/#

3 214 56 64 0x x x "]A'r7 z4j

# $6/ #$ 4 z4j E#$ d'e+|

# $ z4j > #$ 4 z4j

* +1# 12 14

x a x a

0"1# $ 8# f

2 11

4x a x a

gh>f! !

* , , # # 3 26 11 6 0x x x 2 2

3 26 11 6 0x x x 12%!& , , > 2 2

! )! 6

F)FC1%# , ,.......1 2 9

S S S #/ #/"# 0, ,$"

##/ , ,73 9S S S 1%" # 1S

F)F&'+ 1 2 9, ,.......S S S O" !AMnXAy2e6; 3 7 9, ,S S S e; 1S vB"# /\DAB5,.~2e

9

5P 9 3 !5C 9 4 !

5C

9 3!5P

+ #"# 0,$ ## # # 1# ,e4jV; ;:6d'+ ;UO;AB+vB?""@5VVd'G5D,.~2e

( "# 1 /# "$!

"$!DEF4&4@_! ( !

* $ $ +1 10

2 2 6 11a x ax

!0#

$

102 2 6 11a x ax ,;4AM 4/+e"I%=NQRe"+,"

) * +1 6

2 31 x x x $ $ 14x

62 31 x x x ,;4AM 14x 4"! !

! * 2 2 2 2

.......... 50 1 2n n n nC C C C

8

+ 2 2 2 2

0 1 2 5..........n n n nC C C C ,"

!

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M

#

* /#5 6

2 1

+

6

2 1 ,"2e6+"e/AM -abe) )( ) ))

$+ $" "# #0 ,1#"" , / / $"#

5XjtO2e4""-nXfH-_f2e;U2e4">#&"|"-nXfST&BQ7m5j&.'"+

@ @ @ @

##"$ ,#1#11, # G$11 #0 $ / #1#"" , #11, #

O !AM"]7_"ZU" y;"O-5O;@DmO)jH;#;'DvB" y;Op)jH;#;'D2&.'"+@) @) @) (@)

* "$ # #" /"//# $$ 1# $#"//#0/ "$

KS _KDu /!f2e;UO+tKDu /7=KS^_e"m5j"/ !"!@ !@ @ @!

1#"" , #"#$ #5$ !!!1#"" , !!!+$ , #0 #"#$ #!$ %!&'()*!+,)-.!&'(/#!0123456678 3540- 999: ;999 0<!=2>?@A- BCBDE".F-,)-.!&'(/34560<!=2 678 3540-

1) 12e

2) 2

23e3)

823e

4) 423e

# &4*H #" !8!( !

i5A]E_@_&j*kML!"!8!( _flm&4* + 0H0 1# #1 1(Qj+ *1&E>0&'(')&*kML /nJ_f1 # # $##$ +1 /0 /o`/0 * 12/o` # # $##$ +1 1) /0 /o`/0 * 12/o` # /o*V /o*(` " # /o*(V /o*`

( ##"+ #/!* ( 1)( )3

c rp x r r

##! $

]E_pJ!( 1)

( )3r

c rp x r

#! $

1

2

4

9

1

4

SITA

M

$

*

21

2 11

ni

ni

#"#

2

2 1

1

1

n

n

i

i

]EJ4@_ >

IJ+ IJ_pq>0(

)230

ni

n

9 6

13

i

9 6

13

i

9 6i 9 6i

! "# $# # ,#

#l$lrVsr%&&3t59%!&4@_

!

$ $ 3x

5 6

2 1

x

x x

5 6

2 1

x

x x

J!" 3x ')

15

2

15

4

5

2

5

4

#$ # ! #"#

*G1H9I4% 0J2KL

833

4(

437

4 * C #H # $! $ $ #

>EJ4 12I0&uV 4_sm :K&f

3

20

20

3!

# $#$0$ $ 2 2 6 6 17 0x y x y + #,/ $6# + ##

2 2 6 6 17 0x y x y i>EJvMW_4'(iw-JV-(c& B&4;4&'(')&i>EJ]E_x 2 1 4 2 1

3 2 1 3 2 1

SITA

M

* : $ # # # $#$ 2 2 2 2 0S x y gx fy c

, , ,1 1 2 2

A x y B x y #$5/ 1 0 #1$ C!

1 1 2 2, , ,A x y B x y & 2 2 2 2 0S x y gx fy c i>0J40')tB4C&/0:H4V#]E_x& >f 2 2 22CA CB r 2 2 2CA CB r 2 2 23CA CB r 2 2 2CA CB r

&&6(6#$5/ 1 0# 2 2 6 8 1 0S x y x y /" 0

$#$C! $#$/ #

&&6(6&i>EJ40')tB4C Vi>0J]E_'(yhi>0J&_aC!]E_'(yhi>0J&_a`

3

! 6

2

( * $#$ 2 2 8 4 0x y x y c+ + − + = $ $#$ 2 2 2 4 11 0x y x y+ + + − = + #,

$ $#$ 2 2 6 8 0x y x y k+ − + + = # /, %2 2 8 4 0x y x y c+ + − + = "s+" 2 2 2 4 11 0x y x y+ + + − = "s+e56&'d' /s"-2 2 6 8 0x y x y k+ − + + = "s+e56LDu.#"D:%,"

) 6) ) 6)

* # 1#" 2 4y ax 2,2at at

$ 1#"/ 2,2aT aT

2 4y ax E&012 2 , 2at at B4Cgz B&4;H@E0-v 2 , 2aT aT

gynJ 2 2T , 8 8,T 2 8T 2 8T

) 10# #/0# $#$ $$# 2 4y ax #

2 4y ax E0|EBO_]E_'(yhi>EJQmVmRi~_B4 J 22a

22a 23a 23a

(!9 CC# $" #+ 1* 1BSS θ=

C"-C2e&.8"-@d'WX""- 1BSS θ= + tanθ nsi θ cosθ cotθ

( * AB "# ,1#"22

12 2

yxa b

$ OAB &L#/

8 # #/ $$ #$ ,MN ,1#"

.MNL4OAB QO%&B4CRMWX.MNL4 _\\ *`2 2

2 2 1x y

a b .E&04

AB *k-4` >Q3>04>0R

3e 2

13

e 2

3e

2

3e

SITA

M

( $/## / $$ #$ 0/,1#"

D"E'F7G0 6/! "H0 7 '*IJ+"4,1#"0$ $" 0 $ # $" 0

#$ #$

|EMN&_-0>0H@&_-Eu7\345 .E&04E,1#"0$ ##+" $5/ +

._ cV40')t c(7:4CC45 .E&04:E,1#"0 ,1 +','!+6,'!

+','!+6,'!& 4>0dH@&'(& .E&04: : : :

(

12 31

x dxx x x x

112tan 1 x cx

112tan 1 x cx

112tan 1 x cx

11 2tan 12x cx

74. Match the following

List I List II

1) 2sin4cos

x dxx

A) 2tan ln cos

2x x c

2) 4sin2cos

x dxx

B) cosx + secx + c

3) 3sin2cos

x dxx

C) 3tan

3x c

4) 3sin3cos

x dxx

D) sin 2 3

tan4 2

x xx c+ − +

Select the correct answer

1) 1 , 2 ,3 , 4D C B A→ → → → 2) 1 , 2 ,3 , 4D C A B→ → → →

3) 1 , 2 ,3 , 4C D A B→ → → → 4) 1 , 2 ,3 , 4C D B A→ → → →

( * sec (sec tan )

5(sec tan )

nxdx x x cnx x

− −

(23 11 1an an

2 11

x xT T dxxx

+− −+ =∫+−

2π π 4π 2

π

SITA

M

(( * 2 ,0 2xe dx

2, 00

axe dx a is

2

2a

2a

1

2 a

( #"", $#,, #/ +&8

,,"/"-++3; !T

1

2ee

+ − 1

2ee

+ + 1

1ee

+ + 1

1ee

+ −

() * ,&+ 2 sin

cos1

x dyx

y dx

,&! / 2y is

2 sincos

1

x dyx

y dx

12,&+ >,&! / 2y f

1

3

1

2 !

! 2

2 2xdy ydx

xdy ydx ax y

2

2 2

xdy ydxxdy ydx a

x y

>;4'5jWX.#

2 1tan yxy a Cx

2 1tan xxy a C

y

2 1sin yxy a C

x

2 1sec yxy a C

x

0# $#+1#& ± !Ω & ± !Ω O #6

$0 ,#$$ &#&1#0 1#$ /###P & ± !Ω & ± !Ω !"#$% #&" '(16 23%, 3 6.25%Ω ± Ω ± 16 6.25%, 3 31.25%

3 23%, 16 6.25%Ω ± Ω ± 16 6.25%, 3 23%Ω ± Ω ± * 1# $ % $$8#$ , K@# / #/

$ 01 0 $$# /# ,#/2/#1$ ,

) ! ! *+,- ./012*3 #4 56 #7#489 : *+,;<#= *+6/ /6- 6*3 / / : ! K@>? ! @%*+ < !A!BC 3 #70K/ K/ K/ K/

SITA

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2 sin ( cos )2 1 2V V V

gα α −

2 sin ( cos )1 1 2V V V

gα α −

2 sin ( cos )2 2 1V V V

gα α −

2 cos ( sin )2 1 2V V V

gα α −

#$% !%/!%/ + $ , %@$ :6$ $ , #$% /0 # $1 %/@$&/ /# ,

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1 #/, 1# $$# /", 2a bU

rr= − 0#"#

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4

mg

AK

3

mg

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2

mg

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mg

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water

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wood

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PHYSICS

CHEMISTRY

MATHEMATICS

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Prepared bySri Chaitanya Educational Institutions,Visakhapatnam.

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˘ ˇ - SITAM 3mt 25 3 mt 3 20 3mt 50 mt 3 SITAM ˘ˇ ... x xxx x − ++ + += − ˘