1
2
-2023
..
( - I)
: : 100
: - ,
-
(1) :
(ii) : (x+1) (x+2)(x+3)(x+4) = 120
(iii) D = , ,
2. (i) ( AP 12 , n 5n + 2
(ii) 12
(iii) a, b, c, d GP ..,
:
2
2
-
3. (i) -
(ii) (3,5) (4, 7)
(iii) :
()
1
100
2
50
3
0
4. (i) -
(ii) , ,
-
5. (i)
(ii) , -
(iii)
6. (i) f(x) = x = 0
(ii) cX
3
2
-
7. (i) wrtx
(ii) : y= log
(iii) (1+x)x wrtx
8. (1) , q=9 q=0
(ii) p = (50 - x)/5 , = x ' MR MR
x = 0 x = 25
4
2
-2023
..
( - I)
: : 100
: - ,
-
(1) :
(ii) : (x+1) (x+2)(x+3)(x+4) = 120
(iii) D = , ,
:
:
,
;
(1) :
:
:
5
2
:
- x
:
:
x , x :
x !
x = 5/18 :
x :
, :
:
y :
6
2
:
:
(ii) :
:
:
( 4)
:
:
:
:
:
:
7
2
:
:
LHS 120 :
u=z2:
:
:
= 6.25+477.75=484
:
x=z−2.5x = z - 2.5x=z−2.5
8
2
(iii) :
:
:
:
:
:
2 :
:
,
, :
:
9
2
:
! = = 42 = 4
:
: p=4p = 4p=4
: q=42q = 42q=42
:
?
,
:
--
-
:
- -
2. (i) (12 n AP , 5n + 2
(ii) 12
(iii) a, b, c, d GP ..,
:
: , ,
'
(AP) (GP) ,
10
2
, - -
- , ,
, ,
(i): AP 12 n n 5n + 2
1:
AP :
ₙ = 5n + 2
:
n = 1: T₁ = 5(1) + 2 = 7
n = 2: T₂ = 5(2) + 2 = 12
n = 3: T₃ = 5(3) + 2 = 17
…
, AP :
7, 12, 17, 22, 27, 32, 37, 42, 47, 52, 57, 62
AP 12
2: AP
AP , n (Sₙ) :
Sₙ = (n / 2) × [2a + (n - 1)d]
:
,
,
:
a = 7
d = 12 - 7 = 5
n = 12
,
11
2
S₁₂ = (12 / 2) × [2×7 + (12 - 1)×5]S₁₂ = 6 × [14 + 55]S₁₂ = 6 × 69 = 414
:
AP 12 n 5n + 2 , 414
(ii): 1 - 2/3 + 4/9 - 8/27 + … 12
AP
1:
:
T₁ = 1T₂ = -2/3T₃ = 4/9T₄ = -8/27T₅ = 16/81T₆ = -32/243...
: 1, 2, 4, 8, 16, 32 → GP = 1, = 2
: 1, 3, 9, 27, 81, 243 → GP = 1, = 3
:
Tₙ = (2ⁿ⁻¹)/(3ⁿ⁻¹) × (-1)ⁿ⁺¹ = (-1)ⁿ⁺¹ × (2/3)ⁿ⁻¹
:
S = ∑[n=1 12] (-1)ⁿ⁺¹ × (2/3)ⁿ⁻¹
:
(a)= 1
(r)= -2/3
2: GP
:
Sₙ = a × [(1 - rⁿ) / (1 - r)], |r| < 1
:
a = 1
= -2/3
n = 12
:
12
2
S₁₂ = 1 × [(1 - (-2/3)¹²) / (1 - (-2/3))]= [(1 - (4096 / 531441)) / (1 + 2/3)]= [(1 - 4096/531441) /
(5/3)]
, :
1 531441/531441 :
= [(531441 - 4096)/531441] ÷ (5/3)= (527345 / 531441) × (3/5)
:
= (527345 × 3) / (531441 × 5) = 1582035 / 2657205
:
15 :
= 105469 / 177147
:
1 - 2/3 + 4/9 - 8/27 + … 12 :≈ 0.5955 (,
)
(iii): a, b, c, d GP , :
1/(a² + b²), 1/(b² + c²), 1/(c² + d²) GP
1: GP
a, b, c, d GPSo ,
b = = ar²d = ar³
( a r )
:
T₁ = 1 / (a² + b²) = 1 / (a² + a²r²) = 1 / (a²(1 + r²))
T₂ = 1 / (b² + c²) = 1 / (a²r² + a²r⁴) = 1 / (a²r²(1 + r²))
T₃ = 1 / (c² + d²) = 1 / (a²r⁴ + a²r⁶) = 1 / (a²r⁴(1 + r²))
2:
K = 1 / (a²(1 + r²))
13
2
:
₁ =
T₂ = K / r²
₃ = / ⁴
:
₁ = = × ⁰
T₂ = K × r⁻²
₃ = × ⁻⁴
', GP
:
₂² = ₁ × ₃
:
(K / r²)² = K × (K / r⁴)(K² / r⁴) = (K² / r⁴)
,
:
a, b, c, d GP , 1 / (a² + b²), 1 / (b² + c²), 1 / (c² + d²) GP
,
:
()
: Tₙ = a + (n - 1)d
n : Sₙ = (n/2)[2a + (n - 1)d]
(GP)
: Tₙ = arⁿ⁻¹
n : Sₙ = a(1 - rⁿ) / (1 - r), |r| < 1
2 3 (-1)ⁿ
-
14
2
-
3. (i) -
(ii) (3,5) (4, 7)
(iii) :
()
1
100
2
50
3
0
: 3(i)
-
3
- , - 5
:
5 3 -
?
?
- ,
-
:
=
=
15
2
!= ( , 4!=4×3×2×1=24)
:
5 ,
3 (, , ):
, 60
?
:
:
5 3 ,
, ,
:
,
5 3 10
-
?
'
-
-
, ,
16
2
?
:
-
3(ii) (3,5) (4,7)
,
1:
:
A=(3,5)A = (3, 5)A=(3,5)
B=(4,7)B = (4, 7)B=(4,7)
2: -
, (m) :
:
- :
17
2
(3, 5) m=2m = 2m=2:
:
:
(3,5) (4,7) :
3(iii)
- :
:
(/)
()
1
100
2
50
3
0
,
:
:
?
18
2
,
1:
:
- , → ,
' :
2:
:
1:(P1=1, Q1=100)(P_1 = 1, Q_1 = 100)(P1=1, Q1=100)
2:(P2=2, Q2=50)(P_2 = 2, Q_2 = 50)(P2=2, Q2=50)
-
:
, a=−50a = -50a=−50
3: -
:
:
:
19
2
:
1 ', Q=−50(1)+150=100Q = -50(1) + 150 = 100Q=−50(1)+150=100
2 ', Q=−50(2)+150=50Q = -50(2) + 150 = 50Q=−50(2)+150=50
3 ', Q=−50(3)+150=0Q = -50(3) + 150 = 0Q=−50(3)+150=0
(i)
(ii)
(3,5) (4,7)
y=2x−1y = 2x - 1y=2x−1
(iii)
Q=−50P+150Q = -50P + 150Q=−50P+150
-
- :
, , , , ,
:
, ( ), (/ ),
:
, ,
,
:
-
- ,
-
20
2
-
, ,
, , ,
4. (i) -
(ii) , ,
:(i) -
?
, "
?"
, ", , " ,
,
:
- ,
"-"
?
' A, B, C ,
{ }
:
= {a, e, i, o, u}
5 = {1, 2, 3, 4}
,
:
a A ( a A )
,
:
b A ( b A
)
21
2
-
1. ( )
,
: {}
:
1 =
2 3 =
2.
,
:
A = {2, 4, 6, 8}
= {, , }
3.
,
:
= {1, 2, 3, 4, …}
=
4.
,
:
A = {1, 2, 3}
= {3, 2, 1}
, A = B
22
2
5.
,
:
A = {1, 2, 3}
= {, , }
A B
6.
A B A B
l: A B
:
= {2, 4}
= {1, 2, 3, 4, 5}
A, B B
,
7.
,
: ' U
:
U = {1, 2, 3, 4, 5, 6}, A = {2, 4}, A U
8.
- ,
:
= {1, 2}
23
2
A = {, {1}, {2}, {1, 2}}
9.
,
:
= {5}
10.
:
A = {1, 2}, B = {3, 4}, A ∩ B =
(ii) , ,
'
1.
A B A B
:
:
A = {1, 2, 3}, B = {3, 4, 5}
A B = {1, 2, 3, 4, 5}
:
24
2
2.
A B A B
: ∩
:
A = {1, 2, 3}, B = {2, 3, 4}
A ∩ B = {2, 3}
:
3.
A B (A − B)
A B
: -
:
A = {1, 2, 3, 4}, B = {3, 4, 5}
− = {1, 2}
:
A B
25
2
, B − A B A
4.
A B
:A B A Δ B
:
A = {1, 2, 3}, B = {3, 4, 5}
A Δ B = {1, 2, 4, 5}
:
-
?
' :
,
,
,
,
,
,
26
2
,
:
,
,
, , , , ,
' , , ,
' ,
, ,
-
5. (i)
(ii) , -
(iii)
: (i)
, " '
, ?"
, ", ', ' "
, "
"
"
"
?
,
27
2
:
2, -7, 3.14 (π), 0
y = 5x + 3 , 3 x
- :
(24)
π ( 3.1416)
' (100°C)
?
:
y = 5x + 3 , x y
x = 1, y = 8; x = 2, y = 13
- :
( )
( )
' ( )
:
?
(ii) -
?
28
2
,
:
' :
f(x) =
:
f
x
f(x) , x '
:
:
f(x) = x²
x = 2 , f(2) = 4 x = 3 , f(3) = 9
,
:
1.
,
:
f(x) = 5
x = 1, f(x) = 5 x = 100, f(x) = 5
,
!
2.
29
2
:
f(x) = x
x = 2, f(x) = 2 x = -5, f(x) = -5
-
3.
:
f(x) = 2x + 3
x = 1, f(x) = 5 x = 0, f(x) = 3
4.
x² U- ( )
:
f(x) = x² + 2x + 1
( , )
5.
x³ S-
:
f(x) = x³
6.
x
:
30
2
f(x) = 3x⁴ + 2x² + 1
:
f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀
7.
:
f(x) = (x² + 1)/(x + 1)
8.
:
f(x) = 2^x
, ,
9.
:
f(x) = (x)
10.
x - -
:
:
31
2
f(x) = 5
x
f(x) = x
=
f(x) = 2x + 3
f(x) = x²
U-
f(x) = x³
S-
f(x) = x⁴ + 2x²
f(x) = (x²+1)/(x+1)
f(x) = 2^x
f(x) = (x)
x -
(iii) :
x → (x³ - 1) / (x² - 1) 1
:
(x → 1) (x³ - 1)/(x² - 1)
, :
(1³ - 1)/(1² - 1) = (1 - 1)/(1 - 1) = 0/0
!
1:
:
x³ - 1 = (x - 1)(x² + x + 1)
x² - 1 = (x - 1)(x + 1)
:
32
2
[(x - 1)(x² + x + 1)] / [(x - 1)(x + 1)]
(x - 1) :
(x² + x + 1) / (x + 1)
2: x = 1
(1² + 1 + 1)/(1 + 1) = (1 + 1 + 1)/2 = 3/2
:
(x → 1) (x³ - 1)/(x² - 1) = 3/2
:
:
( 3, 5)
( x, y)
:
: , , , , ,
:
0/0 () ,
3/2
6. (i) f(x) = x = 0
(ii) cX
:
,
- ',
f(x)f(x)f(x) ' x=ax = ax=a :
33
2
:
x=ax = ax=a '
x→ax \to ax→a
'
,
:
:
x=0 '
, f(0)
,
:
x=0 ' , - (LHL) - (RHL) x→0
1: \ 0^+ 0 )
:
:
,
2: ( 0 )
34
2
:
:
,
(i) :
:
,
(ii) f(x) = cx
x ' f(x) :
35
2
'
f(x)=cx
f′(x)
1:
2:
,
:
:
cx :
36
2
: c=e
:
(i): x = 0 '
x=0x = 0x=0 ' - -
, x=0x = 0x=0 '
(ii): cx
:
:
- , ,
, , , ,
37
2
,
-
-
7. (i) wrtx
(ii) : y= log
(iii) (1+x)x wrtx
: :
-
'
,
'
:
7(i):
x
-- :
:
38
2
:
:
:
:
:
:
:
:
:
39
2
(i) :
7(ii):
:
-- :
:
:
,
:
,
40
2
:
,
,
(ii) :
7(iii):
x (1+x)x
-- :
:
1:
:
41
2
2:
' :
:
:
:
?
:
y :
, y=(1+x)x
(iii) :
42
2
:
- :
:
:
:
:-
: ( )
:
;
:
-- ,
-
!
8. (1) , q=9 q=0
43
2
(ii) p = (50 - x)/5 , = x ' MR MR
x = 0 x = 25
:
,
-
(1):
?
, ?
:
:
(Ed) :
q=9q = 9q=9
q=0q = 0q=0
1: dp/dq
44
2
2:
q = 9
q = 0
45
2
(1) :
q=9q = 9q=9 ', -0.75 , (
)
q=0q = 0q=0 ', 0 , ( '
)
(2): (MR)
?
:
TR = = ×
x = /
:
:
(MR)
MR x=0x = 0x=0
MR x=25x = 25x=25
1: (TR)
:
46
2
,
2: MR TR
:
MR :
' MR
x = 0:
x = 25:
47
2
(2) :
MR 50−2x5\frac{50 - 2x}{5}550−2x
x=0x = 0x=0, MR = 10
x=25x = 25x=25, MR = 0
:
,
( x = 25), MR ,
-
:
,
MR , -
"
, "
