2.
2023
/
(
)
: 3 :100
: - ,
-
1.(a) , : (i) (ii) (iii)
()
(c) :
2.(a) :
(i)
(ii)
(b) :
-
3.(a) :
2.
20–29
30–39
40–49
50–59
5
14
24
22
60-69
70-79
80-89
16
6
3
(b) ,
2,400 :
(₹100 )
0-10
10-20
20-30
30–40
40-50
5
25
?
18
7
-
4.(a) 9 43 5
63 , : (i) (ii) 10
() :
0-10
10-20
20-30
30–40
40-50
5
7
15
25
8
5. :
1
2
3
4
5
6
7
10
18
30
25
12
3
2
6. , :
23
26
33
30
29
34
27
36
32
30
41
44
47
39
34
30
29
28
31
37
2.
-
7.
:
6.5
500
10.8
560
2.8
124
8.2
148
4.7
69
13.4
78
10.9
38
10.8
25
8.6
49
13.4
20
8. , :
1
2
3
4
5
6
('000 )
200
220
260
?
350
430
2.
2023
/
(
)
: 3 :100
: - ,
-
1.(a) , : (i) (ii) (iii)
()
(c) :
:1.(a) , ,
:
•
•
•
:
2.
(i)
-
" " :
'
:
• →
• →
• 10 →
' { }
:
• A = {1, 2, 3, 4}
• = {, , }
:
• 32
=
:
2.
• = {a, e, i, o, u}
:
• = {1, 2, 3, 4, …}
:
• A = {1,2,3}
• = {3,2,1}
A = B (
)
A B , A, B -
:
• = {1,2}
• = {1,2,3,4}
.
-
= =
(ii)
2.
:
A = {1,2,3}B = {4,5,6}
A B
A B (a,b) , a .
: A = {1,2,3}B = {2,4,6}
: “ ”
:
• (1,2)
• (2,4)
• (3,6)
A B
: A = {1,2,3}
: (1,1), (2,2), (3,3)
(a,b) , (b,a)
2.
: 1, 2 , 2, 1
(a,b) (b,c) , (a,c)
: A B , B C A C
:
✔
✔
✔
: " "
-
:
•
→
• →
()
(iii)
A B
:
→
2.
A = {1,2,3}B = {2,4,6}
: f(x) = 2x
:
• (1,2)
• (2,4)
• (3,6)
→ ✔
:(1,2) (1,3)
1 →
-
- → -
: f(x)=2x
-
- →
: f(x)=x²
(-2)² = 42² = 4
'
2.
: A={1,2,3}B={2,4,6}f(x)=2x
B ✔
B
-
:
→
1.(b)
'
• →
•
→ -
y = f(x) :
• → -
• - → -
2.
:
-:
f′(x) = 0
= 0 ( )
f′(x)=0 , f″(x)
1: f″(x) < 0
2: f″(x) > 0
-
/- :
f(x) = x²
1: f′(x) = 2x
= 0:2x = 0x = 0
2:
f″(x) = 2
2 > 0
- x=0
:
f(0)=0
2.
- = (0,0)
-
:
• →
• - → -
/ :
•
•
1.(c)
:
lim(x→3) (x² − 9)/(x − 3)
1:
x=3 :
: 3² − 9 = 9 − 9 = 0
: 3 − 3 = 0
:
0/0
2:
2.
x² − 9 = (x−3)(x+3)
:
(x−3)(x+3)/(x−3)
(x−3):
= x + 3
3:
:
lim(x→3) (x+3)
x=3 :
= 3+3= 6
:
lim(x→3) (x² − 9)/(x − 3) = 6
2.(a) :
(i)
(ii)
(b) :
: (a):
' ,
2.
(i)
, , :
:
,
:
,
• : . .something
something
• : .
:
"2" "1/2" :
:
! , ? : ,
(ii)
, ,
• : . .
something
something
• : . .
:
2.
:
!
,
"-5" ,
:
():
- ,
: ,
,
()
1:
:
'
2:
= 0:
:
:
2.
•
•
3:
(
) ( )
:
• ':
→ -
• ':
→
4:
-values :
• ':
-
• ':
2.
(27) :
: " ,
?" ,
:
• /
• (
)
• —
'
/- ,
-
-
3.(a) :
20–29
30–39
40–49
50–59
5
14
24
22
2.
60-69
70-79
80-89
16
6
3
:
( )
20–29
5
30–39
14
40–49
24
50–59
22
60–69
16
70–79
6
80–89
3
:
• 5 20-29
• 14 30-39
• 24 40-49
• …
,
,
1: ( )
( 20-29),
:
Midpoint
Lower limit + Upper limit
:
(x)
20–29 → (20+29)/2 = 24.5
30–39 → 34.5
40–49 → 44.5
50–59 → 54.5
60–69 → 64.5
70–79 → 74.5
80–89 → 84.5
2.
2:
,
:
• (f)
• (x)
•
• x²
• fx²
x²
fx²
20–29
5
24.5
122.5
600.25
3001.25
30–39
14
34.5
483
1190.25
16663.5
40–49
24
44.5
1068
1980.25
47526
50–59
22
54.5
1199
2970.25
65345.5
60–69
16
64.5
1032
4160.25
66564
70–79
6
74.5
447
5550.25
33301.5
80–89
3
84.5
253.5
7140.25
21420.75
3:
:
4:
:
✔ ≈ 51.17
2.
?
51 -
40-
59 - ,
5:
:
:
:
:
✔ ≈ 14.21
= 51.17
= 14.21
( )
2.
:
• ≈ 51
• ≈ 14
:
:
and
— 40-69
-
:
" "
:
" "
SD → SD →
SD ≈ 14 →
:
2 → = / SD
:
2.
:
:
, ,
51 , - 14
-
- ,
40 69
(b) ,
2,400 :
(₹100 )
0-10
10-20
20-30
30–40
40-50
5
25
?
18
7
: (
0-10, 10-20, ) ,
2.
(20-30 )
,
₹2,400
:
(₹100 )
0-10
10-20
20-30
30–40
40-50
5
25
?
18
7
1:
,
-
:
Median
:
• =
• = ( )
• =
• =
• = ( )
2:
₹2,400 "₹100 "
:
• 0–10 ₹0–₹1,000
• 10–20 ₹1,000–₹2,000
• 20–30 ₹2,000–₹3,000
• 30–40 ₹3,000–₹4,000
• 40–50 ₹4,000–₹5,000
₹2,400 20-30
3:
2.
:
• (20-30 , )
• ( ,
20–30 ₹2000–₹3000)
• ( , )
• ( )
• ( )
2400 :
4:
2000 :
1000 :
5:
, . :
:
:
2.
6:
:
:
:
7:
25
:
(₹100 )
0-10
10-20
20-30
30–40
40-50
5
25
25
18
7
₹1,000
, ₹1,000–₹2,000 ,
, (₹2,000–₹3,000)
- - ₹2,400
₹2,000–₹3,000
₹2,400
: ,
₹2,400 '
:
•
2.
•
' ,
"" ,
• ,
-
: , , ,
" "
,
-
4.(a) 9 43 5
63 , : (i) (ii) 10
: 1 —
, :
Mean
Sum of observations
Number of observations
,
Sum Mean
( )
2.
• = ..
• =
• =
•
=
2 — 9
: = 43 = 9
,
Sum of 9 items
, = 387
3 —
= 63
:
= 10
4 —
New mean
= 45
2.
(63) 43 → 45
:
5 — 9
SD :
: SD = 5 → = = 43n = 9
,
:
,
1849 :
9 :
9 = 16866
2.
6 —
= 63
:
10 :
7 —
:
: n = 10 = 45 = 20835
,
SD ≈ 7.65
2.
(i) = 45
(ii) ≈ 7.65
:
•
•
:
• 43 → 45
• SD 5 → 7.65
() :
0-10
10-20
20-30
30–40
40-50
5
7
15
25
8
: 1:
, :
GM antilog
:
• = ( )
• = -
• = ( )
:
1.
2.
3.
4.
2.
5.
6. GM
2:
, -
• 0–10 → = 5
• 10–20 → = 15
• 20–30 → = 25
• 30–40 → = 35
• 40–50 → = 45
: 5, 15, 25, 35, 45
3:
:
(x)
(f)
(x) ()
(x)
0-10
5
5
0.6990
.
10-20
15
7
1.1761
8.2327
20-30
25
15
1.3979
20.9685
30–40
35
25
1.5441
38.6025
40-50
45
8
1.6532
13.2256
4: (x)
•
•
5:
:
GM antilog
:
GM
6:
2.
:
GM
5. :
1
2
3
4
5
6
7
10
18
30
25
12
3
2
:
(x)
1
2
3
4
5
6
7
(f)
10
18
30
25
12
3
2
:
Sk
Mean Mode
Standard Deviation
:
1.
2.
3.
- '
1:
:
2.
:
1
10
10
2
18
36
3
30
90
4
25
100
5
12
60
6
3
18
7
2
14
:
:
= 3.28
2:
=
:
10, 18, 30, 25, 12, 3, 2
x = 3 = 30
= 3
3:
:
2.
= 3.28
:
x−3.28
(x−3.28)²
f(x−3.28)²
1
10
−2.28
5.1984
51.984
2
18
−1.28
1.6384
29.491
3
30
−0.28
0.0784
2.352
4
25
0.72
0.5184
12.96
5
12
1.72
2.9584
35.501
6
3
2.72
.
22.195
7
2
.
13.8384
27.677
:
(approx)
100 :
(approx)
= 1.35
4: 's
:
Sk
Mean Mode
:
Sk
(approx)
2.
Karl Pearson’s coefficient of skewness
( )
:
'
>
(4-7 )
100 1 7
3 4 6 7
3 → 3.28
6. , :
23
26
33
30
29
34
27
36
32
30
41
44
47
39
34
30
29
28
31
37
: 1:
, ( ) ( )
:
1. '' :
2.
2. '' :
2:
, , , ,
X²
²
XYName
23
41
529
1681
943
26
44
676
1936
1144
33
47
1089
2209
1551
30
39
900
1521
1170
29
34
841
1156
986
34
30
1156
900
1020
27
29
729
841
783
36
28
1296
784
1008
32
31
1024
961
992
30
37
900
1369
1110
:
•
•
•
•
•
3: X Y
, .
4:
:
:
2.
5: X Y
:
:
on :
6:
:
7: Y X
:
:
2.
on :
8:
:
1. '' :
2. '' :
-
7.
:
6.5
500
10.8
560
2.8
124
8.2
148
4.7
69
13.4
78
10.9
38
10.8
25
8.6
49
13.4
20
: ?
,
:
• (L)→ -
2.
• (P)→ -
:
Fisher Price Index
,
1: -
₀
₀
₁
₁
6.5
500
10.8
560
2.8
124
8.2
148
4.7
69
13.4
78
10.9
38
10.8
25
8.6
49
13.4
20
• p₀ =
• q₀ =
• p₁ =
• q₁ =
2:
:
• p₀q₀ = 6.5×500 = 3250
• p₁q₀ = 10.8×500 = 5400
• p₀q₁ = 6.5×560 = 3640
• p₁q₁ = 10.8×560 = 6048
2.
• p₀q₀ = 2.8×124 = 347.2
• p₁q₀ = 8.2×124 = 1016.8
• p₀q₁ = 2.8×148 = 414.4
• p₁q₁ = 8.2×148 = 1213.6
• p₀q₀ = 4.7×69 = 324.3
• p₁q₀ = 13.4×69 = 924.6
• p₀q₁ = 4.7×78 = 366.6
• p₁q₁ = 13.4×78 = 1045.2
• p₀q₀ = 10.9×38 = 414.2
• p₁q₀ = 10.8×38 = 410.4
• p₀q₁ = 10.9×25 = 272.5
• p₁q₁ = 10.8×25 = 270
• p₀q₀ = 8.6×49 = 421.4
• p₁q₀ = 13.4×49 = 656.6
• p₀q₁ = 8.6×20 = 172
• p₁q₁ = 13.4×20 = 268
3:
4:
2.
5: '
' = 179.24
:
79.24%
6:
' :
,
:
:
2.
✔
7:
:
Price Index Quantity Index Value Index
:
:
✔
= 179.24
:
•
→
2.
•
→
,
8. , :
1
2
3
4
5
6
('000 )
200
220
260
?
350
430
:
1
2
3
4
5
6
('000 )
200
220
260
?
350
430
1:
:
,
,
( 1, 2, 3, …)
2:
:
:
•
= ( 1 ')
•
• =
( 4)
•
= ( 1)
• = (, 1 )
•
=
2.
3:
--
(y)
Δy
Δ²y
Δ³y
Δ⁴y
1
200
20
20
-10
70
2
220
40
10
60
3
260
?
?
4
?
?
5
350
80
6
430
, :
• 1 2 : Δy = 220 - 200 = 20
• 2 3 : Δy = 260 - 220 = 40
• 3 5 (
4 ,
): Δy = 350 -
260 = 90
• 5 6 : Δy = 430 - 350 = 80
: 20, 40, 90, 80
:
• 40 - 20 = 20
• 90 - 40 = 50
• 80 - 90 = -10
: 20, 50, -10
:
• 50 - 20 = 30
• -10 - 50 = -60
: 30, -60
:
• -60 - 30 = -90
4:
:
• , ,
2.
•
•
:
:
• : 200
• : 3 × 20 = 60
• : (3 × 2 / 2) × 20 = 3 × 20 = 60
• : (3 × 2 × 1 / 6) × 30 = 1 × 30 = 30
• (×0 )
:
5:
4 :
Production ‘
"
,
"
