Area and Volume (Solved Questions) Mathematics Notes | EduRev

Topic-wise Tests & Solved Examples for IIT JAM Mathematics

Mathematics : Area and Volume (Solved Questions) Mathematics Notes | EduRev

 Page 1


 
For more notes visit santoshifamily.com or call 8130648819 
  
 uestion    ? ? sin(y
 
) y
 
 
 x
 
 
  
(a) 
   os 
 
                                 ( )    os                                   ( )    os                                   ( ) 
   os 
 
 
           Jam MA 2017 
Solution: 
 egion  oun e   y 
x   x   an  y x y   
 ow we evaluate the integral  y  hanging the or er of integral 
? ? sin(y
 
) y
 
 
 x
 
 
 ? siny
 
? x
 
   
 y
 
 
 ? siny
 
,x-
 
 
 y
 
 
 
 ? sin(y
 
) y y
 
 
 
 
 
,  os y
 
-
 
 
 
   os 
 
 
 uestion      he value of the  ou le integral ? ?
siny
  y
 
 
 y
 
 
 x is  
           Jam MA 2016 
Solution: 
 egion of integration is  oun e   y y   y x an  x   x   
 hanging the or er of integration 
? ?
siny
  y
 
 
 y
 
 
 x ? ?
siny
  y
 x
 
 
 y
 
 
 
 ?
siny
  y
,x-
 
 
 y
 
 
 
 ?
siny
  y
,   - y
 
 
 
 ? siny y
 
 
 
 ,  os y-
 
 
 
   
 uestion     he value of the integral ?vx
 
 y
 
 x y
 
   *(x y)  
 
  x x
 
 y
 
  x + is 
(a)                                                    ( ) 
 
 
                                                ( ) 
  
 
                                              ( ) 
  
 
 
           Jam MA 2013 
 uestion   (a)   valuate ? ? (e
 
)
 
 y
 
  v   
 x
 
   
 
Page 2


 
For more notes visit santoshifamily.com or call 8130648819 
  
 uestion    ? ? sin(y
 
) y
 
 
 x
 
 
  
(a) 
   os 
 
                                 ( )    os                                   ( )    os                                   ( ) 
   os 
 
 
           Jam MA 2017 
Solution: 
 egion  oun e   y 
x   x   an  y x y   
 ow we evaluate the integral  y  hanging the or er of integral 
? ? sin(y
 
) y
 
 
 x
 
 
 ? siny
 
? x
 
   
 y
 
 
 ? siny
 
,x-
 
 
 y
 
 
 
 ? sin(y
 
) y y
 
 
 
 
 
,  os y
 
-
 
 
 
   os 
 
 
 uestion      he value of the  ou le integral ? ?
siny
  y
 
 
 y
 
 
 x is  
           Jam MA 2016 
Solution: 
 egion of integration is  oun e   y y   y x an  x   x   
 hanging the or er of integration 
? ?
siny
  y
 
 
 y
 
 
 x ? ?
siny
  y
 x
 
 
 y
 
 
 
 ?
siny
  y
,x-
 
 
 y
 
 
 
 ?
siny
  y
,   - y
 
 
 
 ? siny y
 
 
 
 ,  os y-
 
 
 
   
 uestion     he value of the integral ?vx
 
 y
 
 x y
 
   *(x y)  
 
  x x
 
 y
 
  x + is 
(a)                                                    ( ) 
 
 
                                                ( ) 
  
 
                                              ( ) 
  
 
 
           Jam MA 2013 
 uestion   (a)   valuate ? ? (e
 
)
 
 y
 
  v   
 x
 
   
 
 
For more notes visit santoshifamily.com or call 8130648819 
  
Solution  
? ? (e
 
)
 
 y
 
  v   
 x
 
   
 ? ? e
  
 y
 
  v   
 x
 
   
 
 ? *
e
  
 
+
v   
 
 x
 
   
 
 ? (
e
 
 
 
e
 v   
 
) x
 
   
 
 ?
e
 
 
 x
 
 
 
 
 
? e
 v   
 x
 
 
 
 
 e
 
 
 
 
 
? e
  
(  z ) z
 
 
           ,put   x z
 
- 
 
 e
 
 
 
 
 
? ze
  
 z
 
 
 
 
 e
 
 
 
 
 
*
ze
  
 
 ?
 
  
 
  +
 
 
 
 
  
 
 
 
 
 
*
  
  
 
 
 
  
 
+
 
 
 
 
 e
 
 
 
 
 
* 
 
 
 
 
 
e
 
 
e
 
 
+ 
 
 e
 
 
 
 
  
 
 
 
e
 
 
 
  
e
 
 
 
  e
 
  
 
 
  
 
 
 
  
(  e
 
  ) 
 uestion   (a)   hange the or er of integration in the  ou le integral ? (? f(x y) y
   
 
  
) x
 
  
  
           Jam MA 2011 
Solution: 
 omain of integration is  oun e   y the following  urves 
y  x y   x
 
 x    x   
point of interse tion of the  urve y   x
 
 an  y  x we get when  
  x
 
  x 
 x
 
 x     
 (x  )(x  )   
 x      
Page 3


 
For more notes visit santoshifamily.com or call 8130648819 
  
 uestion    ? ? sin(y
 
) y
 
 
 x
 
 
  
(a) 
   os 
 
                                 ( )    os                                   ( )    os                                   ( ) 
   os 
 
 
           Jam MA 2017 
Solution: 
 egion  oun e   y 
x   x   an  y x y   
 ow we evaluate the integral  y  hanging the or er of integral 
? ? sin(y
 
) y
 
 
 x
 
 
 ? siny
 
? x
 
   
 y
 
 
 ? siny
 
,x-
 
 
 y
 
 
 
 ? sin(y
 
) y y
 
 
 
 
 
,  os y
 
-
 
 
 
   os 
 
 
 uestion      he value of the  ou le integral ? ?
siny
  y
 
 
 y
 
 
 x is  
           Jam MA 2016 
Solution: 
 egion of integration is  oun e   y y   y x an  x   x   
 hanging the or er of integration 
? ?
siny
  y
 
 
 y
 
 
 x ? ?
siny
  y
 x
 
 
 y
 
 
 
 ?
siny
  y
,x-
 
 
 y
 
 
 
 ?
siny
  y
,   - y
 
 
 
 ? siny y
 
 
 
 ,  os y-
 
 
 
   
 uestion     he value of the integral ?vx
 
 y
 
 x y
 
   *(x y)  
 
  x x
 
 y
 
  x + is 
(a)                                                    ( ) 
 
 
                                                ( ) 
  
 
                                              ( ) 
  
 
 
           Jam MA 2013 
 uestion   (a)   valuate ? ? (e
 
)
 
 y
 
  v   
 x
 
   
 
 
For more notes visit santoshifamily.com or call 8130648819 
  
Solution  
? ? (e
 
)
 
 y
 
  v   
 x
 
   
 ? ? e
  
 y
 
  v   
 x
 
   
 
 ? *
e
  
 
+
v   
 
 x
 
   
 
 ? (
e
 
 
 
e
 v   
 
) x
 
   
 
 ?
e
 
 
 x
 
 
 
 
 
? e
 v   
 x
 
 
 
 
 e
 
 
 
 
 
? e
  
(  z ) z
 
 
           ,put   x z
 
- 
 
 e
 
 
 
 
 
? ze
  
 z
 
 
 
 
 e
 
 
 
 
 
*
ze
  
 
 ?
 
  
 
  +
 
 
 
 
  
 
 
 
 
 
*
  
  
 
 
 
  
 
+
 
 
 
 
 e
 
 
 
 
 
* 
 
 
 
 
 
e
 
 
e
 
 
+ 
 
 e
 
 
 
 
  
 
 
 
e
 
 
 
  
e
 
 
 
  e
 
  
 
 
  
 
 
 
  
(  e
 
  ) 
 uestion   (a)   hange the or er of integration in the  ou le integral ? (? f(x y) y
   
 
  
) x
 
  
  
           Jam MA 2011 
Solution: 
 omain of integration is  oun e   y the following  urves 
y  x y   x
 
 x    x   
point of interse tion of the  urve y   x
 
 an  y  x we get when  
  x
 
  x 
 x
 
 x     
 (x  )(x  )   
 x      
Free coachi 
For more notes visit santoshifamily.com or call 8130648819 
So (    ) an  (    ) are the points of interse tion   
 ow the given integral mo ify  y  hanging the or er of integration we get 
? ? f(x y) x
v   
  
 y
 
  
  ? ? f(x y) x
v   
   
 y
 
 
 
Something skip follow book 
 uestion     hanging the or er of integration of ? ? f(x y) y
 
 
 x
 
 
 
(a) ? ? f(x y) x
 
 
 y
 
 
 ? ? f(x y) x
 
 
 y
 
 
                       ( ) ? ? f(x y) x
 
 
 y
 
 
 ? ? f(x y) x
 
 
 y
 
 
 
( ) ? ? f(x y) x
 
 
 
 y
 
 
 ? ? f(x y) x
  
 
 y
 
 
                    ( ) ? ? f(x y) x
 
 
 y
 
 
 ? ? f(x y) x
 
 
 y
 
 
 
           Jam CA 2011 
Solution: 
  ? ? f(x y) y
 
 
 x
 
 
 
 he region of integration is  oun e   y y   y x x   x   
after  hanging the or er the integration  hanges to 
  ? ? f(x y) x
 
 
 y
 
 
 ? ? f(x y) x
 
 
 y
 
 
 
 uestion     et    e the trianle  oun e   y the y axis the line  y    hen the value of the integral  
?
 os y
y
 x y is 
(a) 
 
 
                                                ( )                                                    ( ) 
 
 
                                                ( )   
 Jam MS 2011 
Solution  
? ?
 os y
y
 x
 
 
 y
 
 
 
 ?
 os y
y
,x-
 
 
 y
 
 
 
 
? os y y
 
 
 
 ,siny-
 
 
 
   
 uestion   ( )   hange the or er of integration in the integral ? ? f(x y) y
v   
 
   
 x
 
 
  
           Jam MA 2010 
Solution: 
 he  omain of the  ou le integration is  oun e   y the  urves y x   y v  x x   x   
Page 4


 
For more notes visit santoshifamily.com or call 8130648819 
  
 uestion    ? ? sin(y
 
) y
 
 
 x
 
 
  
(a) 
   os 
 
                                 ( )    os                                   ( )    os                                   ( ) 
   os 
 
 
           Jam MA 2017 
Solution: 
 egion  oun e   y 
x   x   an  y x y   
 ow we evaluate the integral  y  hanging the or er of integral 
? ? sin(y
 
) y
 
 
 x
 
 
 ? siny
 
? x
 
   
 y
 
 
 ? siny
 
,x-
 
 
 y
 
 
 
 ? sin(y
 
) y y
 
 
 
 
 
,  os y
 
-
 
 
 
   os 
 
 
 uestion      he value of the  ou le integral ? ?
siny
  y
 
 
 y
 
 
 x is  
           Jam MA 2016 
Solution: 
 egion of integration is  oun e   y y   y x an  x   x   
 hanging the or er of integration 
? ?
siny
  y
 
 
 y
 
 
 x ? ?
siny
  y
 x
 
 
 y
 
 
 
 ?
siny
  y
,x-
 
 
 y
 
 
 
 ?
siny
  y
,   - y
 
 
 
 ? siny y
 
 
 
 ,  os y-
 
 
 
   
 uestion     he value of the integral ?vx
 
 y
 
 x y
 
   *(x y)  
 
  x x
 
 y
 
  x + is 
(a)                                                    ( ) 
 
 
                                                ( ) 
  
 
                                              ( ) 
  
 
 
           Jam MA 2013 
 uestion   (a)   valuate ? ? (e
 
)
 
 y
 
  v   
 x
 
   
 
 
For more notes visit santoshifamily.com or call 8130648819 
  
Solution  
? ? (e
 
)
 
 y
 
  v   
 x
 
   
 ? ? e
  
 y
 
  v   
 x
 
   
 
 ? *
e
  
 
+
v   
 
 x
 
   
 
 ? (
e
 
 
 
e
 v   
 
) x
 
   
 
 ?
e
 
 
 x
 
 
 
 
 
? e
 v   
 x
 
 
 
 
 e
 
 
 
 
 
? e
  
(  z ) z
 
 
           ,put   x z
 
- 
 
 e
 
 
 
 
 
? ze
  
 z
 
 
 
 
 e
 
 
 
 
 
*
ze
  
 
 ?
 
  
 
  +
 
 
 
 
  
 
 
 
 
 
*
  
  
 
 
 
  
 
+
 
 
 
 
 e
 
 
 
 
 
* 
 
 
 
 
 
e
 
 
e
 
 
+ 
 
 e
 
 
 
 
  
 
 
 
e
 
 
 
  
e
 
 
 
  e
 
  
 
 
  
 
 
 
  
(  e
 
  ) 
 uestion   (a)   hange the or er of integration in the  ou le integral ? (? f(x y) y
   
 
  
) x
 
  
  
           Jam MA 2011 
Solution: 
 omain of integration is  oun e   y the following  urves 
y  x y   x
 
 x    x   
point of interse tion of the  urve y   x
 
 an  y  x we get when  
  x
 
  x 
 x
 
 x     
 (x  )(x  )   
 x      
Free coachi 
For more notes visit santoshifamily.com or call 8130648819 
So (    ) an  (    ) are the points of interse tion   
 ow the given integral mo ify  y  hanging the or er of integration we get 
? ? f(x y) x
v   
  
 y
 
  
  ? ? f(x y) x
v   
   
 y
 
 
 
Something skip follow book 
 uestion     hanging the or er of integration of ? ? f(x y) y
 
 
 x
 
 
 
(a) ? ? f(x y) x
 
 
 y
 
 
 ? ? f(x y) x
 
 
 y
 
 
                       ( ) ? ? f(x y) x
 
 
 y
 
 
 ? ? f(x y) x
 
 
 y
 
 
 
( ) ? ? f(x y) x
 
 
 
 y
 
 
 ? ? f(x y) x
  
 
 y
 
 
                    ( ) ? ? f(x y) x
 
 
 y
 
 
 ? ? f(x y) x
 
 
 y
 
 
 
           Jam CA 2011 
Solution: 
  ? ? f(x y) y
 
 
 x
 
 
 
 he region of integration is  oun e   y y   y x x   x   
after  hanging the or er the integration  hanges to 
  ? ? f(x y) x
 
 
 y
 
 
 ? ? f(x y) x
 
 
 y
 
 
 
 uestion     et    e the trianle  oun e   y the y axis the line  y    hen the value of the integral  
?
 os y
y
 x y is 
(a) 
 
 
                                                ( )                                                    ( ) 
 
 
                                                ( )   
 Jam MS 2011 
Solution  
? ?
 os y
y
 x
 
 
 y
 
 
 
 ?
 os y
y
,x-
 
 
 y
 
 
 
 
? os y y
 
 
 
 ,siny-
 
 
 
   
 uestion   ( )   hange the or er of integration in the integral ? ? f(x y) y
v   
 
   
 x
 
 
  
           Jam MA 2010 
Solution: 
 he  omain of the  ou le integration is  oun e   y the  urves y x   y v  x x   x   
Free AM 
 
? ? f(x y) y
v   
 
   
 x
 
 
 ? ? f(x y) x
   
 
 y
 
  
 ? ? f(x y) x
v   
 
 
 y
 
 
 
 uestion     et   ? ? xy
v   
 
v   
 
 x
 
 
 y ? ? xy  x
v   
 
 
 y
 
 
  hen using the transformation x r os   
y rsin  integral   is equal to 
(a) ? ? r
 
sin   r
 
 
  
 
 
 
                                           ( ) ? ? r
 
sin   r
 
 
  
 
 
 
 
( ) ? ? r
 
sin   r
 
 
  
 
 
 
                                            ( ) ? ? r
 
sin   r
 
 
  
 
 
 
 
             Jam CA 2010 
Solution  
 egion of integration is  oun e   y  urves  
x v  y
 
 x v  y
 
 y   y   
for first integral   
 uestion    valuate integral ? ?
siny
y
 y
 
 
 
 x
 
 
 
 is 
( )                                ( ) 
 
 
                                      ( )                                             ( )   
  Jam CA 2010 
Solution  
 egion of integration is  oun e   y  urves 
y x y 
 
 
 x   x 
 
 
 
 hanging the or er 
  ? ?
siny
y
 y
 
 
 
 x
 
 
 
 ? ?
siny
y
 x
 
 
 y
 
 
 
 
 ? ?
siny
y
 x
 
 
 y
 
 
 
 
 ? siny y
 
 
 
   
 uestion     he value of 
 
  
? ? e
 
 
 
( 
 
  
 
)
 x
 
  
 y
 
  
 equals                 
(a) 
 
 
                                                ( ) 
 
  
                                              ( ) 
 
 
                                                ( ) 
 
 
 
  Jam MS 2010 
Solution: 
Page 5


 
For more notes visit santoshifamily.com or call 8130648819 
  
 uestion    ? ? sin(y
 
) y
 
 
 x
 
 
  
(a) 
   os 
 
                                 ( )    os                                   ( )    os                                   ( ) 
   os 
 
 
           Jam MA 2017 
Solution: 
 egion  oun e   y 
x   x   an  y x y   
 ow we evaluate the integral  y  hanging the or er of integral 
? ? sin(y
 
) y
 
 
 x
 
 
 ? siny
 
? x
 
   
 y
 
 
 ? siny
 
,x-
 
 
 y
 
 
 
 ? sin(y
 
) y y
 
 
 
 
 
,  os y
 
-
 
 
 
   os 
 
 
 uestion      he value of the  ou le integral ? ?
siny
  y
 
 
 y
 
 
 x is  
           Jam MA 2016 
Solution: 
 egion of integration is  oun e   y y   y x an  x   x   
 hanging the or er of integration 
? ?
siny
  y
 
 
 y
 
 
 x ? ?
siny
  y
 x
 
 
 y
 
 
 
 ?
siny
  y
,x-
 
 
 y
 
 
 
 ?
siny
  y
,   - y
 
 
 
 ? siny y
 
 
 
 ,  os y-
 
 
 
   
 uestion     he value of the integral ?vx
 
 y
 
 x y
 
   *(x y)  
 
  x x
 
 y
 
  x + is 
(a)                                                    ( ) 
 
 
                                                ( ) 
  
 
                                              ( ) 
  
 
 
           Jam MA 2013 
 uestion   (a)   valuate ? ? (e
 
)
 
 y
 
  v   
 x
 
   
 
 
For more notes visit santoshifamily.com or call 8130648819 
  
Solution  
? ? (e
 
)
 
 y
 
  v   
 x
 
   
 ? ? e
  
 y
 
  v   
 x
 
   
 
 ? *
e
  
 
+
v   
 
 x
 
   
 
 ? (
e
 
 
 
e
 v   
 
) x
 
   
 
 ?
e
 
 
 x
 
 
 
 
 
? e
 v   
 x
 
 
 
 
 e
 
 
 
 
 
? e
  
(  z ) z
 
 
           ,put   x z
 
- 
 
 e
 
 
 
 
 
? ze
  
 z
 
 
 
 
 e
 
 
 
 
 
*
ze
  
 
 ?
 
  
 
  +
 
 
 
 
  
 
 
 
 
 
*
  
  
 
 
 
  
 
+
 
 
 
 
 e
 
 
 
 
 
* 
 
 
 
 
 
e
 
 
e
 
 
+ 
 
 e
 
 
 
 
  
 
 
 
e
 
 
 
  
e
 
 
 
  e
 
  
 
 
  
 
 
 
  
(  e
 
  ) 
 uestion   (a)   hange the or er of integration in the  ou le integral ? (? f(x y) y
   
 
  
) x
 
  
  
           Jam MA 2011 
Solution: 
 omain of integration is  oun e   y the following  urves 
y  x y   x
 
 x    x   
point of interse tion of the  urve y   x
 
 an  y  x we get when  
  x
 
  x 
 x
 
 x     
 (x  )(x  )   
 x      
Free coachi 
For more notes visit santoshifamily.com or call 8130648819 
So (    ) an  (    ) are the points of interse tion   
 ow the given integral mo ify  y  hanging the or er of integration we get 
? ? f(x y) x
v   
  
 y
 
  
  ? ? f(x y) x
v   
   
 y
 
 
 
Something skip follow book 
 uestion     hanging the or er of integration of ? ? f(x y) y
 
 
 x
 
 
 
(a) ? ? f(x y) x
 
 
 y
 
 
 ? ? f(x y) x
 
 
 y
 
 
                       ( ) ? ? f(x y) x
 
 
 y
 
 
 ? ? f(x y) x
 
 
 y
 
 
 
( ) ? ? f(x y) x
 
 
 
 y
 
 
 ? ? f(x y) x
  
 
 y
 
 
                    ( ) ? ? f(x y) x
 
 
 y
 
 
 ? ? f(x y) x
 
 
 y
 
 
 
           Jam CA 2011 
Solution: 
  ? ? f(x y) y
 
 
 x
 
 
 
 he region of integration is  oun e   y y   y x x   x   
after  hanging the or er the integration  hanges to 
  ? ? f(x y) x
 
 
 y
 
 
 ? ? f(x y) x
 
 
 y
 
 
 
 uestion     et    e the trianle  oun e   y the y axis the line  y    hen the value of the integral  
?
 os y
y
 x y is 
(a) 
 
 
                                                ( )                                                    ( ) 
 
 
                                                ( )   
 Jam MS 2011 
Solution  
? ?
 os y
y
 x
 
 
 y
 
 
 
 ?
 os y
y
,x-
 
 
 y
 
 
 
 
? os y y
 
 
 
 ,siny-
 
 
 
   
 uestion   ( )   hange the or er of integration in the integral ? ? f(x y) y
v   
 
   
 x
 
 
  
           Jam MA 2010 
Solution: 
 he  omain of the  ou le integration is  oun e   y the  urves y x   y v  x x   x   
Free AM 
 
? ? f(x y) y
v   
 
   
 x
 
 
 ? ? f(x y) x
   
 
 y
 
  
 ? ? f(x y) x
v   
 
 
 y
 
 
 
 uestion     et   ? ? xy
v   
 
v   
 
 x
 
 
 y ? ? xy  x
v   
 
 
 y
 
 
  hen using the transformation x r os   
y rsin  integral   is equal to 
(a) ? ? r
 
sin   r
 
 
  
 
 
 
                                           ( ) ? ? r
 
sin   r
 
 
  
 
 
 
 
( ) ? ? r
 
sin   r
 
 
  
 
 
 
                                            ( ) ? ? r
 
sin   r
 
 
  
 
 
 
 
             Jam CA 2010 
Solution  
 egion of integration is  oun e   y  urves  
x v  y
 
 x v  y
 
 y   y   
for first integral   
 uestion    valuate integral ? ?
siny
y
 y
 
 
 
 x
 
 
 
 is 
( )                                ( ) 
 
 
                                      ( )                                             ( )   
  Jam CA 2010 
Solution  
 egion of integration is  oun e   y  urves 
y x y 
 
 
 x   x 
 
 
 
 hanging the or er 
  ? ?
siny
y
 y
 
 
 
 x
 
 
 
 ? ?
siny
y
 x
 
 
 y
 
 
 
 
 ? ?
siny
y
 x
 
 
 y
 
 
 
 
 ? siny y
 
 
 
   
 uestion     he value of 
 
  
? ? e
 
 
 
( 
 
  
 
)
 x
 
  
 y
 
  
 equals                 
(a) 
 
 
                                                ( ) 
 
  
                                              ( ) 
 
 
                                                ( ) 
 
 
 
  Jam MS 2010 
Solution: 
Free M 
 
  
 
  
? ? e
 
 
 
 
 r   
  
 
 
 
 r
 
 
 
 
 
  
? r e
 
 
 
 
 r
 
 
 
 
 
* e
 
 
 
 
+
 
 
 
 
 
 
 uestion    (a)  in  the area of the smaller of the two regions en lose   etween
x
 
 
 
y
 
 
   an  y
 
 x   
                        ( )  valuate ? ? e
 
 
   
  
 x
   
 
 y
 
 
 
 Jam MS 2010 
Solution  
 rea ? ? x
 (  
 
 
 
)
 
 
 
 y
v 
 
 
v 
 
 
 ? ,x-
 
 
 
v
  
 
 
 
 y
v 
 
v 
 
 
  ? (
 
v 
v  y
 
 y
 
) y
v 
 
 
 
  v [? v  y
 
 y
v 
 
 
 
 
 
y
 
]
 
v 
 
 
  v [
y
 
v  y
 
 sin
  
y
v 
]
 
v 
 
 
 
 
 
 
 
 
v
 
 
 
 
 
 
v   v sin
  
v 
 
 
v 
 
 
  v  
 
 
v  
Solution ( ) 
 
 he region of integration is  oun e   y x   x y   y   y   an  shown in figure 
 hanging the or er 
? ? e
. 
 
   
  /
 y
 
   
 x
 
 
 ? 0 e
 
 e
 
   
 (x  )1
   
 
 x
 
 
 
 ? e
  
(x  )e
  
 x
 
 
 
 e
  
? (xe
  
 e
  
)
 
 
 x 
 
e
 
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