2.
   2025
/  
  - IV
: 3     :100
:    -    ,        
             
-
1.          on     
  :

4
6
7
8
9
13
5
X₂
15
12
8
6
4
10
3

30
24
20
14
10
4
2
2.             
   
-
3. (a)        
()     A, B  C      2: 3: 5  
         A, B  C      2%,
4%  5%               ,
      A  B     
2.
4.           
-
5.    ?       
6. (a)            
󰇛 󰇜 󰇛 󰇜 
(b)  X    :
󰇛 󰇜 󰇛 󰇜
     
-
7. ()      
()       ?
8.   -           
 
2.
   2025
/  
  - IV
: 3     :100
:    -    ,        
             
-
1.          on     
  :

4
6
7
8
9
13
5
X₂
15
12
8
6
4
10
3

30
24
20
14
10
4
2
:󹺔󹺒󹺓       ?
       :
  ( 
   )
X₂  X₃  (  )
  :
X₂  X₃  X₁     
    
       X₁    
     , X₂  X₃     
󹵍󹵉󹵎󹵏󹵐  
2.



1
4
30
2
6
24
3
7
20
4
8
14
5
9
10
6
13
4
7
5
2
󼩏󼩐󼩑   (  )
  , 
          
  , 
           
  
, 
X₂  X₃    X₁     
          :
:
=  ()
= X₂   
= X₃   
  a, b₁,  b₂    
󽆛󽆜󽆝󽆞󽆟     ?
 , 
             , 
           
-      :
a, b₁,  b₂          X₁      
   
     ,     :
2.
   
 
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  

 , X₂  X₁     
2.
󹼧X₃ (−0.29)  
  :
 X₃ 1   ,  X₁  0.29   ,     X₂  

 , X₃  X₁     
󼩺󼩻  
?
        :
X₁     X₂  X₃   
          
     X₁ 
  
 
      '     :
 
 
 
   
2.             
   
:   ?
               
                    
'    
     :
 ,     
        
 , -   ,     
 ,     '    -    

2.
 -              
, , ,          
       '   
    
       "S-" ,         
 
   :
 : -     ,     
  

 :      ,     , 
        
  :-         ,    
        ,       
 ,      "S"           
          
   
?
    
           
     
 
  :       ,     
,     
 :  
         
 
 :             
  
  ,         ,       
  
      
,             "" 
?        '       
     
    
           
     '    :
1.  :          
  :
2.
󰇛󰇜


=   (       )
=             
=      (      )
         -   "--"  
     
2.    :
o -  :        
         
o - :
      
,   
          
o   :       R, Python,   
 Excel        
3.  :  
       
  
  
   ,      , ,
      ,      
    
  ,          , 
 
                 
  
     :
,  -  
,        
 ,    '        
  "--"        
 
          ,   ,     
 
    "S" ,       
   , ,      
       
    -         
     
 
2.
               
   ,  ,      ,   "--"  
              
 
-
3. (a)        
()     A, B  C      2: 3: 5  
         A, B  C      2%,
4%  5%               ,
      A  B     
:  (a):    
  ? ( -)
                
     0  1    
0    
1    
 
    :
 :  
  
 :   
           
     
  
    A  B :
2.
󰇛󰇜 󰇛󰇜󰇛󰇜󰇛󰇜
:
󰇛󰇜=   A  B    
󰇛󰇜=   A  B   
 :   
  A  B  
 , :
󰇛󰇜
    :
󰇛󰇜 󰇛󰇜󰇛󰇜

 
 ?󰇛󰇜
      :

        

        
   
 
   ,  
       , 
 
 
       
   
:
 A   󰇛󰇜
 B   󰇛󰇜
,  A      :
 
2.
  
 ,  B   :
 B
  
   
  ,        󰇛󰇜󰇛󰇜󰇛󰇜
   , 
   
 
 ,
󰇛󰇜 󰇛󰇜󰇛󰇜󰇛󰇜
󷄧󼿒 ,       
 ():     
 -     
 
     :

 
 
2
2%

3
4%

5
5%
  = 2 + 3 + 5 = 10
 1:    

   10  :
  A  :
2.
󰇛󰇜

  B  :
󰇛󰇜

  C  :
󰇛󰇜

 2:    
 :
A   = 2% = 0.02
B   = 4% = 0.04
C   = 5% = 0.05
 3:        
         
󰇛󰇜 󰇛󰇜󰇛 󰇜󰇛󰇜󰇛 󰇜󰇛󰇜󰇛 󰇜
 :
󰇛󰇜 󰇡
󰇢󰇡

󰇢󰇡
󰇢
󰇛󰇜 
󰇛󰇜 
 ,   '         0.041 
 4:     A  B   
2.
   :
󰇛 or 󰇜
  :
"      ,      A  B   ?"
   
󰇛 󰇜
󰇛󰇜󰇛󰇜
󰇛󰇜
    :
󰇛󰇜
 
󰇛󰇜

 
 :
󰇛 󰇜


󰇛 󰇜


󰇛 󰇜 
4.           
:    ?
 ,        ,      
        
   
 
      :

       -   
 
    ,  
       
  
2.
 ""   
  ,  -     
,        ,    -   
  ',              ,
         
󰇛󰇜 󰇛󰇛󰇜󰇜
:
=  
󰇛󰇜=    
󰇛󰇜=  
  
  
     :
 
6   ,  
₹60  
 
    ,  
 

    ?
6     = 
6     = ₹60
  = .

 ,  
  ₹60         
, ,  
₹10         -       " "  
  
            
 
1.   
          :
󰇛󰇜 󰇛󰇜󰇛󰇜
  :  
   (    )  ,   
        
:       3.5 ,       3.5 ,  
      
2.
2.    
         (  5),     
 
󰇛󰇜 
   :    ₹100  ,    ,     
  ' ₹100 
3.   
 
        ,     
    
󰇛󰇜 󰇛󰇜
:       3.5 ,  
   
     ,  
 
4.   
          ,       
    
            -   
 ,        
5. - (-  )
     -      (     
   ),     - 
6.    
   '      '     
 ,      
 ,    6 '      ,     
 3.5   
  
 ?
       :
2.
        
              
 
           

        
         
   ,     
   
  
     " "          ,
        
    
-      -     
 
 
      
 
     
 -         
    ,       ,  , 
        
-
5.    ?       
:   
        
  
  :
      
    
    
 
    
 ,         
 󹺩󹺪󹺫              
 
,   :
2.
            ()  -
  ,      '     ,   
   
 
         , 
   ,  
       
  
  
     ,     ,     ' 
:
  
      ,      
    
 
   IQ   ,        IQ   
 
  
       
 ,   ,  
     '       
     
 :
󷷑󷷒󷷓󷷔       ,      
  ( )
          :
1.  (μ)- 
2.
2.   (σ)-      
   
    ,       
    
          :
    
  
     ' 
     
 (   
    
)
         
 
  
                
  
1.     
        
 -  ,  '  ,   -  
       -    
 
    
       -   , 
      
2.   
     
    =    
   ,          
2.
 :  10%         ,  
10%         
3.  =  = 
     
   :
()
( )
(    )
󷷑󷷒󷷓󷷔        
       ,       

4.  '    
       ' 
  :
       
     
- 
       ,      
5.       
 :
     
   X- 

  :
   
      
2.
 ,       ,      
6.       1   
    
      1 ( 100%)    
       
 
   '  :
    (50%)
    (50%)
7.   (68–95–99.7 )    
            
   :
68%    ±1     
95%  ±2     
99.7%  ±3     
              
8.        
   
 :
 (μ)
  (σ)

  :
 
   
   
    

2.
9.   
( )
   :
 
     
        ,          
  
10.       '   
       ,       
 :
 (  )
 ( )
 ( )
  ( )
   
     
 
      
?
          :
     
   
  
       
-       
    "    "   
  
2.
      ,   
     
 =  = 
 6895–99.7     
     100% 
          
 󷊆󷊇
   
    ,          
, ,          - 
   
       '    ,      
 
6. (a)            
󰇛 󰇜 󰇛 󰇜 
(b)  X    :
󰇛 󰇜 󰇛 󰇜
     
: (a)   
   :
󰇛 󰇜 󰇛 󰇜

,              
 
 1:    
       :
󰇛 󰇜 󰇡
󰇢
󰇛󰇜

 2:   
:
2.
󰇛 󰇜

󰇛󰇜
:
󰇛 󰇜

󰇛󰇜
 3:      
   :

󰇛󰇜


󰇛󰇜
,    .       



  :
󰇛󰇜

󰇛󰇜
    :
󰇛󰇜
󰇛
󰇜

  :
 :
 
 4:    
  :
 =
 =󰇛󰇜
  =
󰇛󰇜
:
 =


2.
 =


  =


    3.33      1.49 
 ()   
   :
󰇛 󰇜 󰇛 󰇜
         
 1:    
󰇛 󰇜

 2:   
:
󰇛 󰇜



:
󰇛 󰇜



 3:      
   :


  :

  2   :

2.
 :
 󰇛󰇜
 ,  . 
        (  
)     
:
 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     ,
            
 
2.
        
     
               ,
     " "  
-
7. ()      
()       ?
:-: 
   
 ?
              ,  
              
       -     ,    ,  
  
         
    ',   :
1.   ()
2.  
    -         
  -  
   
    -   
(a)     
1.    ( )
   ?
               
      ,      

  100 ,   100           
2,000  ,        
         
2.
  
             ,  
  , , , ,         
 ;
    
      |
1.     
 ,        
2.   

     
,      

3.  
         
4.  
        
5.  
 ,      
   
1.   

       ,       
2.      
""      

3.    
     ,       
   
1.  
       
2.     

       

3.  
        ,     
4.     
 ,       
2.
   
?
    
       
       
      
2.   
    ?
             ()   
 ,           
   , 
       
  
      1,000         ,
 -       100      
 100       1,000     
  
   
1.          
2.   
3.  
4.   
5.     '   
   
1.  
       
2.
2.   
        
3.   
   
4.     
     '     
   
1.    
       ,      
2.    
   
,      
3.    
       
     ?
    
     
    
     
       (  )
 
 
     
     
 
   


 
  
    
       
()     
      
 :
 
           ,  
 '
    ?
  :     
2.
         
      
  
1.   
         
   , , , , ,    -     
        
󷷑󷷒󷷓󷷔    
 ,    
2.  
            
        
          
        '     
3.   
           
      :


      
 ,           
4.  
           
   
    
 
2.
:  ,  , 
5.    
          
        
6.   
       :
   
       
    
 -      
7.  
        
   ,     
8.   
        :


    

 ,              
  
2.
              
           
             ,   
    ,             
8.   -           
 
:   ?
          -      
 


 - 
     ,     ,     

        :   (   
 )     , 
   ()       

 
  
1.  
                  
 
:   
:
o      
o     
      
:
o      
o           ,    
 
2.   
,      '  (    )   
,        
:  ,      ' -  
   ,         
:
2.
o        
 
o            
:
o      
o     
3.  
                

:   1,000      
100   , 

 10  
:
o   
o       
:
o      ,       
o       
4.  
    , 
  ()    
: 
        ,  

   - ()           

:
o        -  
o   '   
:
o     
     
o       
5.  
           '   
:         
  
  
:
o    
o    
:
o  
2.
o    
   

6.  
,    
    -        
:        50   50  
 
:
o      
 
o        
:
o          
o   
7.  
      ,     ,  
       
:   ,     ,    

:
o       
o    
 
:
o    
o     '    
   


 
, 
    


    

 
,   



, 
   
 
-, 
 ,  
 
, 
 ,  
 
    
 
  
, 



   , 
,   
 
2.
 
           -         
 
          ,   
'   :
 
  ,    
    ,        
   ,          
   :              
  
"              
 
       ,       
 "