ME 2012 Solved

8/20/2019 ME 2012 Solved

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written permission.

! !

" #

$ %&' !

( )

* "" & + * !' + ! +,+ ++,++ -. /+0

*+ 1+ " 2+ $ *3+

( )

4 ' ++ /+ ! */

+ *+1 " *+1 $ 33

( ) $

56() 7 !'

* * * *

*

*

*

/+ ++ 33*

*/

π = π

= ×

= × =

*

8 9 ++ 33*:

−× =

3 &' 6 ! ; ! &' , &&' , < ! &' ! * &' % ;* 9 ; 9 *,

+ " +/ $ +*/

( ) "

*

<

;

*

;

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8/20/2019 ME 2012 Solved

2/22


written permission.

/ &

, ' , '

" ' $ , '

( )

= &! 8 8* > ' ∆

*

8#

8

*

8#

8

" *

8 >#

8

−

$ ?

( )

1 % & !, &' 0, &' 0

++/ + " /+ $ ++

( )

2 *+, *++ *+, ' % =++, ' ', @

4A/ 1A+ " /2+ $ 4=+

( )

56() *

B5 ' %*

=

* **

*

# *+ +*% +3

* *

* * =++ =*24 0

=+ =+

B 5 +3 =*24 1A+ @*

×= = = −

π × π ×= = =

= × × =

A & ! C !D E ! ! 6 % . 5 6

*EC

*5%

*EC

5% "

**EC

5% $

*3EC

5%

3/ /

*/

8/20/2019 ME 2012 Solved

3/22


written permission.

( )

56() *

% *

'5 E

6=

,

'5% E6 "6 = +

,

5% ' 9*

*

E6" 6 "

* + +

7 & ! 6 9 +,'

+6

= F 6 + ' += =

7 " " +, G= =

* *

6

E6 EC' H 6 '

*5% *5%= =

+ 7 , ! ! !

* " 3 $ 2

( )"

56() " 6 ! 9*

*

3 5%

C

π

" ! *

*

5%

C

π

*

8

G, 38 =

69+, ( ) 4 6 6 = +

6 & &

" ' $

( ) $

56( ) ( ) 4 6 6 = + 6 9 +,

& , &'

'

6

8/20/2019 ME 2012 Solved

4/22


written permission.

* 7 , * * *6 ' I + + − , &

, ,+* *

& !'

D* *

+

D* *

− " $

D 4 4 4

+ +

( )

56( ) J& * * *6 ' I + + =

, ,+* *

& * * *6 ' I ++ + − =

, ,+* *

D + D

* * * *+ + = +

* *

* *G & D

* * * *

+

= = +

+

4 E

8 "

> $ *

# 4 &

$ 3 "&

8)3, >), #)*, )4 8)3, >), #)4, )*

" 8), >)*, #)3, )4 $ 8), >)4, #)*, )3

( )

3 % !& D , ! II , &

% '

$ '

" $, ! ! $ %, ! !

( ) $

/ % ! !', I++3++++*/+++

+−

I++4+++*+*/+++

++ 6 !'

3+ 4+ " *+ $ +

8/20/2019 ME 2012 Solved

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written permission.

( ) "

E6 9 6 I K I

9 */+3+)*/+*+ 9*+ µ

= $ I ,

!&

" !&

$

( ) "

1 *++ I ,

9++**/ /++ & 4+* 0

* 9A20

=2 += " 2** $ *4*4=

( )

56()

*

**

+*++**/ /++

+*C; 3G =2*$ * A2 +3

× ×Π ×

= = =× ×

2 7 L , !&' ( )α , &' ( )τ &' ( )ρ

!' L

α + ρ = τ +ρ + α + τ = " α + ρ = $ +α + ρ =

( ) "

A ! ! ' ' 4*/+@0 & 6 /8 L' ' +A L & /8 9**/A3@0 9*/A24@0 &' +0 B ' ! ! E (

=/ 2A " A $ *1+

( )

56() *8 + 4*/ **/A3 +A */A24 **/A3•

= × − = × − + × −

2A++B@ 0 2AE= =

*+ (

J %( 8 2+ 6 4+M

J %%( 8 *+ 6 **/M

% 33+,

*++ A++ " 21/ $ 1*+

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written permission.

( )

56( ) 7

*

*

2+4+;'

*+"**/

ϕ= = =

ϕ

* 33+ +=*/ A++ = × =

* I &' ω α

? α " ( ) ( )** * α + ω $ *ω

( )

** !D % !' : !' :,

,

+ " +2 $ *+*

( ) $

56() G σ 98

*

' ' ! , , 8

*

σ = + −

$ !& L, ( ) = &

++

=

++ = −

8/20/2019 ME 2012 Solved

7/22


written permission.

*3 " ( ) 6 6 & ) 6 = ≤ ≤

( ) 69+, 6

" ! ) !

" " )! $ !

( ) "

56() J& 6 9 N6NON6N 6 9 + ! !

*/*6 +

6

6→−

03 0* " $ *

( )

56()6

*6 +

6→

−

* *

* *6 + 6 +

6 6*

* * * *

3 *6 63

*

→ →

= = = × =

×

*= " I , ! & !(

I ! */

*

# ! ++=

$ ++/

*324 *32A " */+ $ */1

( ) 56() $ 9 $ K * 6 )

*/ * ++= ++/= − × − *324 =

*1 % 3+ , 3+ 2 +: !

+++= ++4 " ++=* $ +=++

( ) "

56() ( )G $ ∆ = − α

G + 2 $ 3+

∆ ×α = − = −

4/1α =

4/1 2+

πα = ×

++=* α =

8/20/2019 ME 2012 Solved

8/22


written permission. !

*2 % $" , &) ! ; *+ /= + & ! / 1 G

; & ; & 3++

/++ , &

(

3/;, 3/+ 1/;, //+ " A/;, A/+ $ /+;, /++

( ) "

56() ; *+ /= +

/, ; 3/ ;= =

1, ; // ;= =

++ +

;; ; % ; &

%

%

= −

++

;3/ ; /++%= − ×

++

;// ; 3++

%= − ×

8/20/2019 ME 2012 Solved

9/22


written permission. "

( )

56() ' P L, 6 ,

* * * *

*4 4

8 ; 8 ;I I

* *+ + = + +

ρ ρ

* * ? ? , ; + F 8 8= = = G

* *

4

8 ;

*=

ρ

*

4

*8; =

ρ

* * 4 48 = ρ + ρ + ρ

8/20/2019 ME 2012 Solved

10/22


written permission. #

( )

4* ! L /+ % ! 3+E8, ' *, !

2 = " *3 $ 4*

( ) 56() 9 /+ K H ! 3+E8τ =

*H= !

1+E8

ττ = =

, 8

? = τ

44

8

= ?

=

π= =

π τ

4= /+ + 1+

× ×= π ×

/42 == ≈

44 3++ ! +/ ! !' , 2+M % ! ! +*/, ! L,

++= /33 " ** $ /1

( )

56() , ! F * !

+*/

*

F *A

µθ ×π= = =

* 2*41/ =

L 9 ( ) * −

/33 = −

43 !6 3 ! = ! ! ! ' !6 !!' ! ! !

*+

* "

4

+ $

*

3++

*

3++=

8/20/2019 ME 2012 Solved

11/22


written permission.

( ) $

56() J&,#

3 =

! # # #

8!!' # 93 = /

+ A 2

× ×

8!!' # 9= / 3

+ A 2× ×

8!!' # 9= 3 /

+ A 2× ×

8 #9 9 !& !!9 +/

4/ " L*

*

*

' '6 6 3' +

66 + − = !'

( ) ( )' + + ' = =

L

*6 6

*

π

" 6 6

*

π

$ 6 6

*

− π

( )

56()*

*

*

' '6 6 3' +

66+ − = ( )' + +=

( )' =

"'P $5*

*

*

' '' 6 *6 *

6 6= = =

* 4 3*

' '' 6 *6 =66 6

− − −= = − = +

* 4 3*

' '' 6 *6 =66 6

− − −= = − = +

( ) ( ) ( )* 3 4 *= * 6 3 +6 6 6−+ − = * * 3 + 3= =

*= ±

∴ L * * * * *" 6 " 6 " 6+ +

G 6*

4= ' ! L & !

6 *' I 3*6 ' *I /6 ' I

+ + =+ + =

− + =

L 69, '9 I9

' 69, '9 I9 69*, '9 I9+

" % !

8/20/2019 ME 2012 Solved

12/22


written permission.

$ !

( ) "

56() & L

( )

( )( )

6 *' I 3

*6 ' *I / *

6 ' I 4

+ + = −

+ + = −

+ + = −

5 F *, &

' 9 )3

5 6 * F 4, &

' 9 )/

C 3 /

G & L

41

+ / +

/ *+ +

+ + +

= −

& E8

8/20/2019 ME 2012 Solved

13/22


written permission.

" /++3 +× $ /34A1 +×

( ) "

56() +3=

"

+=

+

; */ 0

*/

+3*44 **A

=

α =

=

= µ

αφ = = φ =

− α

9( )"

;

α

φ − α × =*/ +

*A */ +−×

=× ×

9 3 /++ + 0×

4A % , / ! /+ $ /++, +*0& 2M *

6,

4/ 4*3 " 4* $ 4+

( )

56() C "

+ + +=

$C /+H *H" 1/ /A 3/

* *

/++H +* 0 &

+/2/ 4/

β= = = = = × =

= =

=

3+ " ' ! % $ $* !&', & 7** & !'

*

$

$

−

? "

*

$

$

$

*

$

$

−

( ) $

56() ** *7 7= −

*

$

$= − = −

3 ! & I !' ' #' ! +++ % &' !' 3, !' ' , !

*

8/20/2019 ME 2012 Solved

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written permission.

3 * " +/ $ +*/

( ) "

56() ' ' ( )

#δ α

# ; ;α δ α

% &' !' 3 !' ' !'

3* 4/ ' +/ & ++8 */M" &' J& */M" 418, & '

=1: //: " 24: $ 1:

( ) $

34 ) D !D & 7 + I 4+ % ! A3E8, , 6 ! &

333 1A* " AA4 $ **=

( ) "56() !8 +1+1= τ

8 +1+1 + 4+ A3 AA413= × × × =

8 AA43E=

33 *C I L! !' 7 &' &!, , L' ' (

*

"

*

$3

C C

77

8/20/2019 ME 2012 Solved

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written permission.

( ) $

3/

66 '' 6'3+E8H ++E8 3+E8σ = σ = τ =

EP & E8

3+ /+ " =+ $ ++

( )

56() 66 '' 6'3+ E8H ++ E8, 3+E8σ = σ = τ =

# EP 9

*66 '' *

6'*

σ − σ + τ

* *4+ 3+= + /+ E8=

3= & C ( )( )

7

=

+ & !'

( ) = ( ) −= " ( ) −= $ ( ) −= −

( ) $

56()( )

C

−

= − ++

C

− − +

−= −

31 7 6/ 4

4

=

, 5 I & &

*

4

*

*

*

−

"

4

+

+

−

$

/

*

/

( )

56() ( ) ( )/ 4

% + + / 4 4 + 4

− λ− λ = = − λ − λ = =

− λ

* *2 / 4 + 2 * + *, = λ − λ + − = λ − λ + = λ = λ =

( ) *% - +− × =

, *λ =

*

64 4 +

6 +

=

H

*

*

6 6 +

6 6

+ =

=

8/20/2019 ME 2012 Solved

16/22


written permission.

G L &

*

*

−

$$ % &' ! ( "

! " ", & ++*, !D I 7

D

32 6 7 ! " 6 ' ! $5 6 ++E8

21 / " 33 $ **4+

( )

56( ) 7 CP ,

7 77

+/ *+ 4/

7 +2A=/7H 7 +14*7

76 7 ++E8

+2A=/7++E8 7 /332 /B

++

= =

= =

∴ = =

= =

3A % 7 = , & &

B

+=4 +4* " *= $ 3=

( )

& # 7 4+ +=4477 B # +=44B

= =

= ∴ =

$$ % &' # (

! *+8 2++8 &

' #)43

"

=+M

3/M7

8/20/2019 ME 2012 Solved

17/22


written permission.

& & L

+*0 8 #)43

(

#)43

( )8 8 M" ( ) @0 ( ) @0 ( ) @0 B⋅ ( ) @0 B⋅

*+ )**4* **/ *41 ++A4 +A/

2++ 44 A// *=14 +4/3 +A2

#)43

( )8 8 M" ( ) @0 ( ) @ 0 B⋅

2++ 3+ *1=3/ +A/

/+ L

/A3 24 " 1A $ 4A/

( ) "56()

8 L ( ) * = −

( )+* *1=3/ *41= × − 1A=

/ 6 @0

*24 3*A " 433 $ 3=

4

*

3

8/20/2019 ME 2012 Solved

18/22


written permission. !

( )

56() # 6 ( ) 3 = −

( )+* *41 A// *24 = − =

)$ * + , &' (

7 D, & ! & 6 ! !

&' 8 $ '

) 4

3

" /

$ 3

5 ! *7 A

J =

G , *

/* D

)!))) ))) " ))) $ )!)))

( ) "

56()

;

5 J G 1 '

" J G = '

$ 7 G 2 ' H

−

−

−

$ 7 G ∴

/4 % &' A + ',

" D A'

" D

" " ! D

*

4

4 =

3 2 A

+

1 /

3

*

5

=J

=

G

"

/

$

3

7

A

8/20/2019 ME 2012 Solved

19/22


written permission. "

$ " D 1'

( )

56() % &' 7 + ', D A '

)$ * + , &' (

! II 4++8, /++B &' +0 % & II ++8 &' 2+0 2+* " ++2@0B

/3 6

/=B /4*B " 323B $ 3=2B

( ) "

56() * * * * Q & 0 * 9 Q & 0 *

* **

* *

* *

*

*

& )& " *

2+ +++2 /++

*

323B

= − = −

−= × −

=

// 6 II *

A+ /=4 " 33 $ *A

( ) $

56() 7 ' L ,

* * *

* * * * *

* * *

**

; ;

8 8 8 ; ; ;

# # 8 ;

4++ 323 + 2+ *A

/++ ++ 2+

ρ = ρ

× = × = × ×

×= × × =

×

#

/= & !' */L , L

#/+ ! ! !' 6I

/ + " / $ */

(

56() *8 /+L /L= −

8/20/2019 ME 2012 Solved

20/22


written permission. #

*

*

/+ +LH +

L L

6 /+ +L + , L /

= − <

∴ − = =

5

/1 " & & ! (

P RRRRRRRR

" $

(

/2 " ' -- (

' & ' ! & G

'S 6

" ' 7' C , !

$ 'S 6 &'

( $

/A &!

.

$ $& " $ $ $

(

=+ " & & ! (

$ & RRRRRRRRRR &

! " $

(

= ', 8) 7 %&

& #L( G), ! &! $', 5& ' , 6 !

! !& &

J)'

-!

" &

8/20/2019 ME 2012 Solved

21/22


written permission.

$ )'

( "

56() J & '

=* J& L , $ "J 7B @8, 6

; " 8; $ 8

(

56()

=4 "##5"

8( 1 ' 1

>( 1 ' 1 &

#( $! ' ! ( $! ' & &

8, > >, # " 8, # $ #,

( "

56() 8 # '

5 T, *, 4, 3U '

=3 ! !' !! - .- =+: . 3+: !! !! !D L' L' ! -S !!, A=: ! .S !!, 1*: !

!!' ' !!, ! !, !' .

+*22 +443 " +==1 $ +1*+

(

56() 6 '

' =+: 3+:

#! A=: 1*:

& +/1= +*22

( ) +*22

8 6 +443+/1= +*22

∴ = =+

=/ '

& ! L *' *6 +6= −

' 6 !

2 + " * $ 3

(

56() *' *6 +6= −

$ , "J , 7B , @8 , ;

*+ 4+ 3+ /+

4+ 3+ /+ =+

8/20/2019 ME 2012 Solved

22/22


written permission

*

*

'* +*6

6

'+ ' 6 * +*6 +

6

6 +

' *+ + +

= −

< ∴ − =

=

∴ = − =

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ME 2012 Solved