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   2025
/  
  
 : 3   :100
:     ,            
           
 -
1.         ,      
:
X₁
4
6
7
8
9
13
5
X₂
15
12
8
6
4
10
3
X₃
30
24
20
14
10
4
2
2.                
   
 -
3. ()         
()     A, B  C     2:3:5  
      A, B  C       2%, 4%  5% 
            ,      
   A  B     
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4.            
 -
5.    ?       
6. (a)            
󰇛 󰇜 󰇛 󰇜 
()  X       :
󰇛 󰇜 󰇛 󰇜
      
 -
7. ()        
()       ?
8.         ,     
  
Easy2Siksha.com
   2025
/  
  
 : 3   :100
:     ,            
           
 -
1.         ,      
:
X₁
4
6
7
8
9
13
5
X₂
15
12
8
6
4
10
3
X₃
30
24
20
14
10
4
2
:󹺔󹺒󹺓       ?
        :
X₁  (       )
X₂  X₃  ()
  :
X₁  X₂  X₃      
             X₂  X₃     
  X₁          
󹵍󹵉󹵎󹵏󹵐  
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X₁
X₂
X₃
4
15
30
6
12
24
7
8
20
8
6
14
9
4
10
13
10
4
5
3
2
󼩏󼩐󼩑    (  )
  ,           
  ,              

,  X₂  X₃    X₁     
        :
:
a=  ()
b₁= X₂   
b₂= X₃   
  a, b₁,  b₂     
󽆛󽆜󽆝󽆞󽆟     ?
  ,            , 
         
       :
a, b₁  b₂          X₁     
 
     ,       :
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  
  
ΣX₂², ΣX₃², ΣX₂X₃, ΣX₁X₂,      
      
              ,   
  
󷄧󼿒  
             ,    
 :
a ≈ 8.38
b₁ ≈ 0.41
b₂ ≈ −0.29
󹵱󹵲󹵵󹵶󹵷󹵳󹵴󹵸󹵹󹵺  


󹺖󹺗󹺕      ( !)
             
󹼧 (8.38)
 X₂  X₃  ,  X₁     8.38 
(         )
󹼧X₂   ( +0.41 )
  :
 X₂  1     ,  X₁   0.41       
 X₃   
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 , X₂  X₁     
󹼧X₃   ( −0.29 )
  :
 X₃  1     ,  X₁   0.29       
  X₂  
 , X₃  X₁    
󼩺󼩻   ?
      :
X₂  X₃    X₁   
            
      X₁     
              :

 
 
   
2.                
   
:   ?
                 
                  
     
    :
                
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 , -    ,     
 ,         -       
                
, ,               
         
    
       "-"   ,    
              :
 
: -     ,     
    
 :    ,     ,   
    
  :             
       ,        
,      "S"        
  
              
    ?
               
        
 :       ,     ,  
  
 :           
 :            
  
 ,         ,       
  
      
,             ""  ? 
                    
                 
     :
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1.  :            
 :
󰇛󰇜


=   (      )
=           
=        (      )
          "--"    
    
2.    :
o   :           
        
o  :       ,   
          
o  :R, Python,  Excel       
        
3.  :            
        ,     
, ,      ,      
      ,            ,
             
    
  
       :
,  -  
,       
 ,           
  "--"       
 
       -   ,        
 
    "S"  ,    
 
   , ,        
    
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               
       
 
                
   ,  ,    ,   "--"  
               
  
 -
3. ()         
()     A, B  C     2:3:5  
      A, B  C       2%, 4%  5% 
            ,      
   A  B     
:  ():    
  ? ( )
              
     0  1   
0   
1   
  
    :
 :     
 :     
             
     
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  
   A  B  :
󰇛󰇜 󰇛󰇜󰇛󰇜󰇛󰇜
:
󰇛󰇜= A  B       
󰇛󰇜= A  B      
 :   
  A  B      , :
󰇛󰇜
      :
󰇛󰇜 󰇛󰇜󰇛󰇜
   ?󰇛󰇜
       :
      
      
    
      ,         ,   
         
   
 :
 A   󰇛󰇜
 B   󰇛󰇜
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,  A      :
 A
A  B 
 ,  B   :
 B
A  B 
      ,        󰇛󰇜󰇛󰇜󰇛󰇜
    ,      
 ,
󰇛󰇜 󰇛󰇜󰇛󰇜󰇛󰇜
󷄧󼿒 ,       
 ():      
          
  
     :

 
 
2
2%

3
4%

5
5%
 = 2 + 3 + 5 = 10
 1:      
  10  :
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   A  :
󰇛󰇜

   B  :
󰇛󰇜

   C  :
󰇛󰇜

 2:    
  :
A   = 2% = 0.02
B   = 4% = 0.04
C   = 5% = 0.05
 3:        
      
󰇛󰇜 󰇛󰇜󰇛 󰇜󰇛󰇜󰇛 󰇜󰇛󰇜󰇛 󰇜
 :
󰇛󰇜 󰇡
󰇢󰇡

󰇢󰇡
󰇢
󰇛󰇜 
󰇛󰇜 
 ,           0.041 
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 4:        A  B   
   :
󰇛 or 󰇜
  :
      ,      A  B   ?”
     
󰇛 󰇜
󰇛󰇜󰇛󰇜
󰇛󰇜
    :
󰇛󰇜
 
󰇛󰇜

 
 :
󰇛 󰇜


󰇛 󰇜


󰇛 󰇜 
4.            
:    ?
 ,       ,      
             
       :
          
      ,          
   
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 ""      ,        
        ,     - 
 
  ,             
  ,          
󰇛󰇜 󰇛󰇛󰇜󰇜
:
=  
󰇛󰇜=    
󰇛󰇜=  
  
        :
  6  ,   ₹60  
      ,    
    ?
6    = .
6    = ₹60.
  = .

,    - ₹60      , ,   
 ₹10           " "

  
             
1.   
         :
󰇛󰇜 󰇛󰇜󰇛󰇜
  :      (   )   ,    
    
Easy2Siksha.com
:        3.5 ,        3.5 ,  
        
2.    
        (  5)  ,     
  
󰇛󰇜 
    :      ₹100  ,       ₹100 

3.    
           ,       
  
󰇛󰇜 󰇛󰇜
:        3.5 ,       
      ,     
4. 
           ,      
    
             
,        
5. - (-   )
     -      (      
 ),     - 
6.    
              
,     
  ,    6        ,       
   3.5   
    ?
Easy2Siksha.com
       :
         
              
  
            
            
                  ,  
         
  
      " "         
 ,          
          -        
            
 
                
     ,       , 
         
 -
5.    ?       
:                
    :
     
   
   
       ,           
󹺩󹺪󹺫                
 
,   :
Easy2Siksha.com
           () 
   ,          ,    
   
         ,    ,  
      
        
       ,      :
   
       ,        
   
 
      ,        
  
  
         
,   ,  
          
        :
󷷑󷷒󷷓󷷔      ,      
  ( )
             :
1.  (μ)
2.   (σ)      
Easy2Siksha.com
          ,       
    
          :
    
  
    
         (       
 )
              
  
                 

1.     
        
 -   ,    ,   -   
          
                , 
    
2.    
    
    =    
               
  :  10%        ,   10% 
       
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3.  =  = 
      
  :
()
( )
(     )
󷷑󷷒󷷓󷷔         
        ,   
 
   
4.    
      
  :
      
     
-         ,      
5.        
 :
       
   X-    
  :
   
   - 
,         ,     
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6.      1   
     
       1 ( 100%)    
       
        :
    (50%)
    (50%)
7.   (68–95–99.7 )    
           
   :
68%  ±1      
95%   ±2    
99.7% ±3      
               
8.       
     :
 (μ)
  (σ)
   :
 
  
   
      
9.    ( )
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    :

     
          ,        

  
10.           
        ,      
  :
 ( )
 ( )
 ( )
  ( )
   
       
        ?
           :
    
    
    
      
         
   "  "  
  
      ,    
     
 =  = 
 6895–99.7    
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     100% 
         
 󷊆󷊇
         ,        ,
               
            
6. (a)            
󰇛 󰇜 󰇛 󰇜 
()  X       :
󰇛 󰇜 󰇛 󰇜
      
: ()   
   :
󰇛 󰇜 󰇛 󰇜

,              

 1:     
        :
󰇛 󰇜 󰇡
󰇢
󰇛󰇜

 2:   
 :
󰇛 󰇜

󰇛󰇜
 :
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󰇛 󰇜

󰇛󰇜
 3:      
   :

󰇛󰇜


󰇛󰇜
,         



   :
󰇛󰇜

󰇛󰇜
      :
󰇛󰇜
󰇛
󰇜

  :
 :
 
 4:    
   :
 =
 =󰇛󰇜
  =
󰇛󰇜
:
 =


 =


  =


   3.33      1.49 
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 ()   
   :
󰇛 󰇜 󰇛 󰇜
        
 1:     
󰇛 󰇜

 2:   
 :
󰇛 󰇜



 :
󰇛 󰇜



 3:      
   :


  :

   2   :

 :
 󰇛󰇜
 ,  
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        (   )    
:
 4:   
   :
 =
 =
 :
 = 2
 = 2
   
  (n=10, p=1/3):
o  ≈ 3.33
o   ≈ 1.49
  (λ=2):
o  = 2
o  = 2
  
           :
    ,         ,  
     ,  3       ,
   (   1.5) 
    ,   (,     )
           2  ,    
 2        2     ,  
          
 
              
                , 
    " "  
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 -
7. ()        
()       ?
::         ?
                , 
               
 -        ,     
  
           ,   :
1.   ()
2.  
              
     ,   -  
()     
1.    ( )
   ?
                
    ,     
    100  ,    100         
  2,000  ,       
        
  
             , 
   , , ,             
 ;     
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   
1.     
 ,       
2.  
     ,      
3.  
          
4.   
       
5.  
   ,     
   
1.    
       ,      
 
2.      
      
3.     
 ,            
   
1.  
       
2.       
        
3.  
     ,    
4.       
  ,         
     ?
   
Easy2Siksha.com
     
  
      
      
2.   
   ?
            ()    
,             
     ,    
  
      1,000            , 
       100      100    
  1,000          
   
1.           
2.    
3. 
4.    
5.        
   
1. 
       
2.   
         
3.   
    
4.    
          
Easy2Siksha.com
   
1.    
         ,     
2.    
    ,     
3.     
      
     ?
    
     
   
    
         (  )
 
 
     
      
 
   


  
  
     

        

()     
        :
             ,     
   ?
   :    
              
      
1.    
Easy2Siksha.com
         
   ,  , , ,      ,      
        
󷷑󷷒󷷓󷷔       ,   
2.  
         
        
         
            
3.   
            
       :


    
  ,              
4.  
           
         
:  ,   , 
5.    
Easy2Siksha.com
          
        
6.   
       :
   
        
             
7.  
          
   ,     
8.   
         :


    

   ,     ,          
 
            
     ,     
  
            ,   
   ,            
Easy2Siksha.com
8.         ,     
  
:   ?
                  
      ,       ,     
        :      
 (    ),     ()       
  
  
1.  
                
 
:    
:
o      
o            
:
o     
o           ,   
 
2.  
,        (    )   
 ,        
:  ,            
 ,        
:
o       
o              
 
:
o        
o     
3.  
Easy2Siksha.com
                

:   1,000       100   ,   
10    
:
o   
o      
:
o       ,       
o       
4.  
     ,    ()    
:          ,   
   ()            
 
:
o           
o      
:
o              
o        
5.  
           
:             
  
:
o    
o     
:
o  
o         
6.  
,                
:           50   50  
 
Easy2Siksha.com
:
o       
o           
:
o           
o     
7.  
         ,     ,  
         
:    ,        ,  
      
:
o        
o       
:
o    
o         
  





, 
     


  
 
,    
 


, 
    
 
, 
 ,  
 
, 
 , 
 
     
    ,  



,    

,     
 
                 
              ,   
   
    ,      
Easy2Siksha.com
    ,      
   ,         
    :              
     
             
       ,      