1
2
  -2022
..  
  II
  :    : 100
:               
-
1.              
      
2.        ?     ?  
     
-
3. (i)   ?       
(ii)     ,      :

0-10
10-20
20-30
30-40
40-50
50-60

8
15
22
20
10
5
2
2
4. (i)    ?       
(ii)           :


 

0-10
10-20
20-30
30-40
40-50
50-60
 

12
17
23
39
16
3
-
5. (i)    ,    ?
(ii)         :
X 



52
53
42
60
45
41
37
38
25
27
Y 



65
68
43
38
77
48
35
30
25
50
6.(i)       ?     
(ii)          :
6
2
10
4
8
9
11
5
8
6
3
2
-
7.(i)           ?
(ii)          :


2010

2010

2011

2011

20
12
30
14

13
13
15
20

12
10
20
15

8
6
10
4

5
8
5
6
8. (i)             
(ii)         :

2001
2002
2003
2004
  
38
45
65
68

2005
2006
2007
2008
  
75
87
60
95
4
2
  -2022
..  
  II
  :    : 100
:               
-
1.              
      
:   ?
                 
    ,     ,  ,  ,
                
  -                
     ,             
          :
    
    
     
  
                
  :
       ,         
        :
5
2
1.  :            
               
   ,            
             ,   
    
2.  :             
              
           ,  
             

     :
1.   :
     
    
   
           
              , 
          
2.    :
     
     
     
    -19   ,  
    ,          
       
3. :
  
 
  
           
               
  
6
2
4.  :
 
 
  
          
              
   
5. :
  
  
   
   :       
      ()   
6.  :
  
  
  
           
           
  :
1.  :          
     ,          
    :         
             
2.         ,   
               
 :          
             
3.   :          
        :     
              
7
2
4.  :          
   :         
         
5.   :            
    :          
       
6.            
  :          
       
7.  :          
    :       
     
   :
                  :
            
           
               
 
        
        
  :
   ,       :
         
         
        
  -     
 ,      ,         
              
        ,    , -   
         ,     , 
     ,  , -    
  
8
2
2.        ?     ?  
     
:    ?        
                -,
       -          
              ,
        
  :
1.   
    ,           
 
       : ? ? ?
  : "   , - 2023"  " "   
2.   
     ,    
         (  , " ( )", " (₹
)")
            
3.    
         (  ,   ,
   )
       (   )
        
4.   

       
         
   
          
5.   
         
9
2
               
(  
   )
       
  :
1.         ,       
:
      (2023)
   
 150
 175
325
2.          :
:
        (2023)
   
 85% 82% 78%
 83% 86% 82%
 84% 84% 80%
10
2
3.           
:
   2023
       
   
 250 300 50 50 100 50
. 300 275 60 60 120 40
 200 225 45 45 90 30
4.               
:
      
   
0-20 5 5%
21-40 15 15%
41-60 45 45%
61-80 25 25%
81-100 10 10%
100 100%
5.            
:
   (2021-2023)
   
2021 300 250 200 750
2022 325 275 225 825
2023 350 300 250 900
11
2
   :
1. 
       
          
          
2.  
         
         
       
3.  
      
         
-      
    :
1. 
         
   ,      
2.  
     
       
     
3. 
     
     
      
12
2
  :
1.  
      
   
    
2.  
         
     
       
3.   
      
     
       100%  
4.  
         
     -  
    
 ,     :
  
 
  
  
    
  ,             
                
      
13
2
-
3. (i)   ?       
(ii)     ,      :

0-10
10-20
20-30
30-40
40-50
50-60

8
15
22
20
10
5
:  1:   
                
  ""  " "             
           , "    ?" 
            
       :
1.   ( )
  ""        
               
  :       85, 90, 88, 92  95    
 : (85 + 90 + 88 + 92 + 95) ÷ 5 = 90
:
o     
o        
o       
:
o      ()
o              
2. 
        
     ,     
    ,        
14
2
:  2, 3, 4, 7, 9     : 2, 3, 4, 7, 9  = 4 (
)
:
o     
o          
:
o        
o        
3. 
          
      ,           
:  2, 3, 3, 4, 4, 4, 5, 6  = 4 (   )
:
o          
o   
o        
:
o      
o     
o       
4.  
               
             
:    : (40% ): 85  (30% ): 92 
 (30% ): 88   = (85 × 0.4) + (92 × 0.3) + (88 × 0.3) = 88
5.  
,          
     n     
15
2
      10%, 20%  30%      =
((1.1 × 1.2 × 1.3)) - 1
        
 2:     
(ii)     ,      :

0-10
10-20
20-30
30-40
40-50
50-60

8
15
22
20
10
5
: ,     
              :
  (X)
 (f)
0-10
8
10-20
15
20-30
22
30-40
20
40-50
10
50-60
5
 ,          
1. 
 ( )          :
:
    
     
-- :
1.      (x)  :        
  :
16
2
     
  :
o 0-10:
o 10-20:
o 20-30:
o 30-40:
o 40-50:
o 50-60:
2. f, x,       :
  (X)
 (f)

(x)
0-10
8
5
40
10-20
15
15
225
20-30
22
25
550
30-40
20
35
700
40-50
10
45
450
50-60
5
55
275
3.   :
4.    :
2. 
               , 
     :
:
=     
=   
=      
=    
17
2
=  
-- :
1.  :
2.  :
3.             40   
   
o 0-10:   = 8
o 10-20:   = 8 + 15 = 23
o 20-30:   = 23 + 22 = 45
  20-30        40   
4.   :
o (  20-30)
o (20-30    )
o ( 20-30)
o ( )
3. 
             ,   
      :
:
=     
=    
=       
=       
=  
-- :
1.     :      
o   22 ,    20-30
18
2
2.   :
o (  20-30)
o ( 20-30)
o ( 10-20)
o ( 30-40)
o ( )

:28
:27.73
:27.78
               ,  
         
4. (i)    ?       
(ii)           :


 

0-10
10-20
20-30
30-40
40-50
50-60
 

12
17
23
39
16
3
:   :   
  ?
                  
            ()      
         ""  ""  
  ,        10    ,  
    50       ,  10  90      
      ,          
    
19
2
       :
1. :              
2. :            
  
          ,       
 :
1.    
2.    
1.    
                  
    :
)
                
:
=   -  
:
      : 15, 20, 25, 30, 35,   : 35 - 15 = 20
 :
               , 
       
)   ( )
                 
      
:

= (     ) / (  )
:
   5, 10, 15  (5 + 10 + 15) ÷ 3 = 10
  :|5 - 10|, |10 - 10|, |15 - 10| = 5, 0, 5.   = (5 + 0 + 5) ÷ 3 =
3.33.
:
       ,         
20
2
)    
          ,       
   
:
            
:
 = (     ) / (  )
  ():
             
        ,       
:
   :2, 4, 6.  = (2 + 4 + 6) ÷ 3 = 4.  = [(2 - 4)² + (4 - 4)² + (6 - 4)²] ÷
3 = (4 + 0 + 4) ÷ 3 = 2.67.   = √2.67 ≈ 1.63.
  :
           ,      
 ,  ()  
2.    
            ,     
          
)  
          
:
  = (  -  ) ÷ (  +  )
:
  10  90 ,   = (90 - 10) ÷ (90 + 10) = 80 ÷ 100 = 0.8.
)   ()
CV       ,         
          
:
CV = (  ÷ ) × 100
:
 A: = 50, .. = 5. .. = (5 ÷ 50) × 100 = 10%.
 : = 100, .. = 20. .. = (20 ÷ 100) × 100 = 20%.
: B    .
21
2
   ?
1.   :
 ,          
2.   :
           ,   
   - 
3.   :
  ,  ,          
 
1. :
      (   )  ,    
  ,        ,     
2.  :
 , A  B,   A  85-90     ,  B 
 50-95          , B      
3.  :
               , 
                
,       

             
,  , ,           
 ,              ,   
   ,              
    
22
2
(ii)           :


 

0-10
10-20
20-30
30-40
40-50
50-60
 

12
17
23
39
16
3
:2:     
 
           
1:   
: = 110 ΣfX = 3,140 ΣfX² = 107,750
 : (X
)    X
= ΣfX/N = 3,140/110 = 28.55  
 3:    
(σ) : σ = √[fX²/N) - (X
)²]
σ = √[(107,750/110) - (28.55)²]
σ = √(979.55 - 814.90) σ = √164.65
σ = 12.83  
 4:   (CV)   
 = (σ/X
) × 100
.. = (12.83/28.55) × 100
.. = 44.94%
23
2
:
1. 12.83                
         28.55  
2. 44.94%              
  :
o CV < 25%    
o 25-75%   CV    
o CV > 75%     
  ,      ,         
 
        :
     
    
  
-  

        :
  ,         
 ,       
  ,       
 ,               
 ,        ,          
                
      ,          

24
2
-
5. (i)    ,    ?
(ii)         :
X 



52
53
42
60
45
41
37
38
25
27
Y 



65
68
43
38
77
48
35
30
25
50
:  1:   
                   
             1  10   
            (      
 ),             
  (         ),    
 
     :
  ?             
      ()           
  :
1.      
2.    (             )
3.        
4.   ()          
  :
1.
        
         
     
25
2
2. 
      
           
            
3.
  (,  ,  )       

         
         
4. 
         
       
       
5. 
          
         
    
  :
1. 
    
      
          
2.
          
-         
    
3.  
              
26
2
       
            
 :
1.  
           
       
2.   
   ,       
    
3.   
       
        
  :
1.  
   
-1  +1  
     
2.   
 
      
    
 2:    
 ,            ,   
         :
:            :
: 85, 90, 75, 95, 70
: 88, 85, 80, 92, 75
27
2
  :
1.       : 3, 2, 4, 1, 5 : 2, 3, 4, 1, 5
2. ()    
3.      (d²)
4.    
5.   : rs = 1 - (6Σd²)/(n(n²-1))  n    
    :
+1.0       
-1.0     
0    
      -   
  :
0.00  0.19:   
0.20  0.39:  
0.40  0.59:  
0.60  0.79:  
0.80  1.00:   
    , , ,       
             
         
28
2
(ii)         :
X 



52
53
42
60
45
41
37
38
25
27
Y 



65
68
43
38
77
48
35
30
25
50
:   :
              
                   
                
  ,     -    ,        
       -- 
1.   
         :
X    :52, 53, 42, 60, 45, 41, 37, 38, 25, 27
Y    :65, 68, 43, 38, 77, 48, 35, 30, 25, 50
  -  (X  Y)           
2.   
       ,   X  Y     
      1         
X   :
o    :25, 27, 37, 38, 41, 42, 45, 52, 53, 60
o  :
25 = 1
27 = 2
37 = 3
38 = 4
41 = 5
42 = 6
29
2
45 = 7
52 = 8
53 = 9
60 = 10
 , X   :8, 9, 6, 10, 7, 5, 3, 4, 1, 2
Y   :
o    :25, 30, 35, 38, 43, 48, 50, 65, 68, 77
o  :
25 = 1
30 = 2
35 = 3
38 = 4
43 = 5
48 = 6
50 = 7
65 = 8
68 = 9
77 = 10
 , Y   :8, 9, 5, 4, 10, 6, 3, 2, 1, 7
3.       
 ,                 



 ()
^2
8
8
0
0
9
9
0
0
6
5
1
1
10
4
6
36
30
2
7
10
-3
9
5
6
-1
1
3
3
0
0
4
2
2
4
1
1
0
0
2
7
-5
25
,       (∑d^2):
∑d^2 = 0 + 0 + 1 + 36 + 9 + 1 + 0 + 4 + 0 + 25 = 76
4.     
   (ρ)   :
ρ = 1 - [(6 ∑d^2) / (n(n^2 - 1))]
:
∑d^2 =    
n =    (  , 10)
  :
ρ = 1 - [(6 * 76) / (10(10^2 - 1))]
ρ = 1 - [(456) / (10 * 99)]
ρ = 1 - [456/990] ρ = 1 - 0.46 ρ = 0.54
5.   
    (ρ) 0.54   X  Y       
      ,     - X    
 , Y       ,     
       
      ,   ,        
                
               
      ,    (1  )       
 ,    (-1  )       ,   0  

31
2
 
             ,   
              ( 
     )    ,       , 
    ,    -   
         ,        
,     
     ?
          :
       ()
     -    
          

              
      ,             
              
 ,               
 
6.(i)       ?     
(ii)          :

6
2
10
4
8
9
11
5
8
6
:    :          
                  
                 
,         ,      
                
32
2
     
     "- "  
       :
          
          
        
2.  
   :
1. X  Y    (X    Y    )
2. Y  X    (Y    X    )
    :
Y = a + bX (Y  X)
X = a + bY (X  Y)
:
'a' Y-  (  Y-   )
'b'   ( X  1      Y     )
3.     
             :
: 6, 2, 10, 4, 8
: 9, 11, 5, 8, 6
1: ()   
X   (X
) = (6 + 2 + 10 + 4 + 8) ÷ 5 = 30 ÷ 5 = 6
Y   (Ȳ) = (9 + 11 + 5 + 8 + 6) ÷ 5 = 39 ÷ 5 = 7.8
 2:   
33
2
∑=30 ∑=39 ∑=220 ∑=327 ∑=206
 3:     :
 = 30
∑Y = 39
∑X² = 220
∑Y² = 327
∑XY = 206
n = 5 (  )
 4:     
X  Y   (byx):
byx = (n∑XY - ∑X∑Y) ÷ (n∑X² - (∑X)²)
= (5(206) - 30×39) ÷ (5(220) - 30²)
= (1030 - 1170) ÷ (1100 - 900)
= -140 ÷ 200 = -0.7
Y  X   (bxy):
bxy = (n∑XY - ∑X∑Y) ÷ (n∑Y² - (∑Y)²)
= (5(206) - 30×39) ÷ (5(327) - 39²)
= -140 ÷ (1635 - 1521) = -140 ÷ 114
= -1.228
 5:     (a)
X  Y  :
a = Ȳ - byx(X)
= 7.8 - (-0.7)(6)
= 7.8 + 4.2 = 12
Y  X  :
=  - (Ȳ)
= 6 - (-1.228)(7.8)
= 6 + 9.578
= 15.578
34
2
 6:  
1. X  Y   : Y = 12 - 0.7X
2. Y  X   : X = 15.578 - 1.228Y
:
     X  Y      
X      , Y 0.7    ( )
Y      , X 1.228    ( )
4.     
          :
1.   X    Y     (Y = 12 - 0.7X    )
2.   Y    X     (X = 15.578 - 1.228Y    )
  :
 X = 7,  Y = 12 - 0.7(7) = 7.1
 Y = 10,  X = 15.578 - 1.228(10) = 3.298
5.     
6.    -         
7.     (X
, Ȳ)    
8.                 
  
9.               
 
10.  
     
     :
        
      
         
         
         
35
2
                
        , ,      
              
-
7.(i)           ?
(ii)          :


2010

2010

2011

2011

20
12
30
14

13
13
15
20

12
10
20
15

8
6
10
4

5
8
5
6
:  :   
   ?
                 
          ,     
    -           
               (   
)        ,         
              
    ,           
     ,       -     
 ,             
    
1.  :        ,      ,
       
36
2
2.  :           ,   
        
3.  :   (,   )     (,  
 )      
4.  :            , 
       
   
1.  :          (,  
)
2.  :,           
3.  :       
   
   ,           
        
1.     ( )
               
                 ,    

:   (CPI)         
        CPI  ,      
    
   :
o    ,          
o               
   
:                   
         ,        
2.   
         ,        
      - ,    
37
2
:    6% ,          
  6%       (WPI)    
     
   :
o             
 
o            
:          -    
       (  )  
3.    
              
 ,     ,           
    
:   ()           
      
   :
o       
o           
4.     
   -         ,    
           
   5%    ,       
       5%     
   :
o        
o       
:              
              "" 
5.    
      ,         
        
38
2
:             
         
   :
o         
o        
:            
6.   
   
              , 
           
             
  
   :
o         
o 
    
7.    
   ()         
 ,             
      
   :
o           
o        
8.     
           ,    
      
     -    ,  
      

              
       ,        
  ,             
              ,  
39
2
      ,  ,       
 
(ii)          :


2010

2010

2011

2011

20
12
30
14

13
13
15
20

12
10
20
15

8
6
10
4

5
8
5
6
:     
              
                
  -    -       
                 
    
  (  )
        :
1:  
             :
40
2
  ()
   (2010)            :
:
P1 = 2011  
P0 = 2010  
Q0 = 2010  
  ()
    (2011)            
:
:
Q1 = 2011  
 2:  (L)   
     
41
2
42
2
 
          138.87 
      ?
1.  :          
2.             
 ,  138.87       2010  2011     38.87% 
 
43
2
8. (i)             
(ii)         :

2001
2002
2003
2004
  
38
45
65
68

2005
2006
2007
2008
  
75
87
60
95
:    
             ,   
  ,   ,         , 
   ,            
     
                   
   ,             
   
        ,        :
,  ,             
       
1.  ( )
      ,          , 
         :
   :         
 :           

 :            
           -   

44
2
:
    10              
   -   ,          
    
2.  
  -     -     ,   
      ,          
          
:
             
            
    ,            
           
:
                  
 -              
   ,      
3.  
       -   ,      
              
      
:
              
    -,            
 
   ,        -   
  ,           
:
             ( ),   
    ()  -   -    , 
        
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4.  
             
               

:
        
COVID-19         
            
      ,     ""    
          
:
              
               
     
         :
1.  :  =  +  +  + 
o                
2.  :   =  ×  ×  × 
o               
  ,           ,     
 :
            
           -  
        
           
  
 :   

            :
1. :                
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2
2.  :              
 ,      
3.            ,    
     ,         
4.   :        ,    
 
     ,            
,       
    
        :
1. :     ,        

2.  :,         
3.   :       

                
  - ,   ,      - 
   ,            
     ,           
  ,              
    
47
2
(ii)         :

2001
2002
2003
2004
  
38
45
65
68

2005
2006
2007
2008
  
75
87
60
95
:         
    ,              
                  
        ,       
       ,            
     
  ,   2001  2008       ( )   :
                 
         :
=+
:
Y    (   ),
X   () ,
a   (     X=0X = 0X=0),
b   (         )
     
 1:     (X)
     ,              
  2001, 2002,       ,  X=−3,−2,−1,0,1,2,3,4 
48
2
    0    (2004)         
 
 2: a  b     
a  b   :
:
ΣXY: X  Y    ,
ΣX2: X    ,
ΣY:  Y   ,
n:    .
49
2
       
                 -  ,
 -              
      ,          
  
12.27   ()         ₹12.27      66.63
  ()       X=0X = 0X=0 (  )
50
2

                    
       -    -    (
2007,     ),        ,    
               
 ,   ,        !
:             ( )      
         ,              