1
Easy2Siksha
GNDU -2023
-III
: : 100
: - ,
-
1. (a) /
(b)
u = x4 - 5xy3+ 6x2+ 2xz2- xyz
2. (a) y = x3- 27x + 108 - 108
(b) : y = x3 - 27x + 108
-
3. () : .
() :
2
Easy2Siksha
4. p = sqrt(9 + x) 7
-
5. () ?
(b) :
2z + 4y - z = 9
3x + y + 2z = 7
x + 3y - 3z = 4
6. () ?
(b) : 2 4 -1
3 1 2
1 3 -3
-
7. () -
(b) :
0.4
0.1
50
0.7
0.6
100
3
Easy2Siksha
5
2
:
(i)
(ii)
(iii) , 10 .
8. () ? LPP '
?
() :
Z = 2x1 - 3x2
:
4
Easy2Siksha
GNDU -2023
-III
: : 100
: - ,
-
1. (a) /
(b)
u = x4 - 5xy3+ 6x2+ 2xz2- xyz
: A. /
( )
,
:
1. : '
- (
, f(x)) - ( , f(x,y,z))
2. : '
- :
a) - :
• : f'(x)
• x : f'(x) = 0
• '
( )
5
Easy2Siksha
b) - :
•
• '
• '
( )
3. :
( , , )
a) - :
• : f''(x)
• ' :
o f''(x) > 0 ,
o f''(x) <0,
o f''(x) = 0, ( )
b) - :
• ( )
• ' :
o det(H) > 0 f_xx > 0, -
o det(H) > 0 f_xx <0,
o det(H) <0,
o det(H) = 0, ( )
4. :
, '
'
5. :
6. :
:
[-2, 4] f(x) = x^3 - 3x^2 - 9x + 5
6
Easy2Siksha
1: f(x) = x^3 - 3x^2 - 9x + 5
2: f'(x) = 3x^2 - 6x - 9 f'(x) = 0: 3x^2 - 6x - 9 = 0 x^2 - 2x - 3 = 0 (x - 3)(
x + 1) = 0 x = 3 x = -1
3: f''(x) = 6x - 6 x = 3: f''(3) = 12 > 0, x = 3
x = -1: f''(-1) = -12 <0, x = -1
4: f(-2) = -8 - 12 + 18 + 5 = 3 f(4) = 64 - 48 - 36 + 5 = -15
5: f(-1) = -1 + 3 + 9 + 5 = 16 ( ) f(3) = 27 - 27 - 27 + 5
= -22 f(4) = -15 ( )
6: 16 x = -1
-15 x = 4
- ,
B.
, - -
:
u = x^4 - 5xy^3 + 6x^2 + 2xz^2 - xyz
- :
1. x (∂u/∂x) : ,
y z
x
∂u/∂x = 4x^3 - 5y^3 + 12x + 2z^2 - yz
2. y (∂u/∂y) : ,
x z y
∂u/∂y = -15xy^2 - xz
3. z (∂u/∂z) : x y ,
z
∂u/∂z = 4xz - xy
- :
,
- -
4. x (∂^2u/∂x^2) : x ∂u/∂x :
∂^2u/∂x^2 = 12x^2 + 12
7
Easy2Siksha
5. y (∂^2u/∂y^2) : y ∂u/∂y :
∂^2u/∂y^2 = -30xy
6. z (∂^2u/∂z^2) : z ∂u/∂z :
∂^2u/∂z^2 = 4x
7. x y (∂^2u/∂x∂y) : y ( x ∂u/∂y)
∂u/∂x :
∂^2u/∂x∂y = -15y^2 - z
8. x z (∂^2u/∂x∂z) : z ( x ∂u/∂z)
∂u/∂x :
∂^2u/∂x∂z = 4z - y
9. y z (∂^2u/∂y∂z) : z ( y ∂u/∂z)
∂u/∂y :
∂^2u/∂y∂z = -x
u
:
- :
1. ∂u/∂x = 4x^3 - 5y^3 + 12x + 2z^2 - yz y z x
' u x, y, z '
2. ∂u/∂y = -15xy^2 - xz
x z , y u
x, y, z
3. ∂u/∂z = 4xz - xy
x y , z u
, x, y, z '
- :
4. ∂^2u/∂x^2 = 12x^2 + 12
x (∂u/∂x) u
x ,
x
5. ∂^2u/∂y^2 = -30xy y (∂u/∂y) u y
x y '
6. ∂^2u/∂z^2 = 4x
z (∂u/∂z) u z
x '
8
Easy2Siksha
7. ∂^2u/∂x∂y = -15y^2 - z
x u
y ( ) u
x y
8. ∂^2u/∂x∂z = 4z - y
x z
, x u z (
)
9. ∂^2u/∂y∂z = -x
y z
, y
u z ( )
u = x^4 - 5xy^3 + 6x^2 + 2xz^2 - xyz
:
1. '
2. - (
)
3. -
4. x, y, z
,
x, y, z
- u , ,
-
,
:
1. ( )
2.
3. ,
, - , x,
y, z, u x^4 y^3
-
, xyz
9
Easy2Siksha
,
, -
, , ,
,
2. (a) y = x3- 27x + 108 - 108
(b) : y = x3 - 27x + 108
:(a) y = x³ - 27x + 108 - 108
, :
1. : y = x³ - 27x + 108
' S-
2. : ,
' -
3. :
,
:
1:
,
y = x³ - 27x + 108 y' = 3x² - 27
2:
,
y' = 0 x :
3x² - 27 = 0 3x² = 27 x² = 9 x = ±3
, x = 3 x = -3
3:
-
y' = 3x² - 27 y'' = 6x
10
Easy2Siksha
4: ' x = 3: y'' = 6(3) = 18 (,
- ) x = -3: y'' = 6(-3) = ' -18 (, )
5: x = 3 (- ) ' ' y- : y = 3³ - 27(3) + 108 = 27 -
81 + 108 = 54
x = -3 ( ) ': y = (-3)³ - 27(-3) + 108 = -27 + 81 + 108 = 162
6: : 162 :
54 : 162 - 54 = 108
,
- 108
(b) y = x³ - 27x + 108
()
:
: 162 (x = -3 ' ) : 54 (x = 3 ' )
, :
: , y = x³ - 27x + 108,
x³ x ( )
, (-27x) x = 0
( x
), , ,
x S-
, y = x³ -27x x = 0
, +108
:
(x = 3 x = -3)
- ,
x = -3 ', -3 -3
x- , y- 162
"" ""
x = 3 ', "" -3 3 x-
, y- 54 , 162
:
, y' = 3x² - 27, '
11
Easy2Siksha
(
'), - "" -
, y'' = 6x, (x >
0 ), , (x <0 ),
, ( x = 0)
: x = 3 x = -3
' , x = 0 ' (x³ x) -
- ( 108)
: ,
-
, (
),
, x
(
), y
( ),
:
• 3 -3
(
)
• -3 , 162
• 3 , 54 -
• -3 3 54 162
:
' ,
y = x³ - 27x + 108 ,
:
• ( x )
• x = -3, y = 162 '
• x = 3, y = 54 '
• x
12
Easy2Siksha
"108 " : - 108 ,
108 y =
ax³ + bx + c
2c
: : 162 : 54 : 162 - 54 = 108
: 108 2 * 108 = 216
' :
' ,
' ' :
1. y- : x = 0, y = 108 y-
2. x-: y = 0 ,
: x³ - 27x + 108 = 0
3. : y'' = 6x ,
x = 0 ' ',
:
y =
ax³ + bx + c ' :
1.
2.
3.
4. '
,
: y = x³ - 27x + 108 ,
'
,
, ,
S- ,
13
Easy2Siksha
, -
, ,
,
- - -
,
-
3. () : .
() :
: (a) : ∫x^n log(x) dx
x^n log(x) ,
'
:
∫u dv = uv - ∫v du
u v x
:
1. , u dv : Let u = log(x) Let dv = x^n dx
2. du v: du = (1/x) dx ( log(x) ) v = (x^(n+1))/(n+1) (
x^n )
3. : ∫x^n log(x) dx = log(x) * (x^(n+1))/(n+1)
- ∫(x^(n+1))/( n+1) * (1/x) dx
4. : ∫x^n log(x) dx = log(x) * (x^(n+1))/(n+1) - (1/(n+1)) ∫x^n
dx
5. : ∫x^n dx = (x^(n+1))/(n+1)
6. : ∫x^n log(x) dx = log(x) * (x^(n+1))/(n+1) - (1/(n+1)) * (x^(n) +1))/(n+1)
14
Easy2Siksha
7. : ∫x^n log(x) dx = (x^(n+1)/(n+1)) * ((x) - 1/(n+1)) + C
C
()
:
• x^(n+1)/(n+1) x^n
• (x)
log(x)
• -1/(n+1)
• C
n , n = -1 ( n = -1,
)
, ()
(b) : ∫log/(1+log)^2 dx
--
:
1. , u = log(x) , du
= (1/x) dx x = e^u (
e^(log(x)) = x)
2. : ∫log/(1+log)^2 dx = ∫u/(1+u)^2 * e^u du
3.
'
: ∫u/(1+u)^2 * e^u du
4. .
v = u/(1+u)^2 dw = e^u du
5. dv w : dv = ((1+u)^2 - 2u(1+u)) / (1+u)^4 du = (1-u) / (1+u)^3
du w = e^u
6. : ∫u/(1+u)^2 * e^u du = u/(1+u)^2 * e^u - ∫e^u *
(1-u) / (1+u) )^3 du
7. I : I = ∫e^u * (1-u) /
(1+u)^3 du
8.
: I = ∫e^u / (1+u)^3 du - ∫u*e^u / (1+u)^3 du
9. ' : ∫e^u / (1+u)^3 du
v = 1+u dv = du
u = v-1 : ∫e^( v-1) /
v^3 dv = (1/e) * ∫e^v / v^3 dv
15
Easy2Siksha
10. : (1/e) * (-e^v / (2v^2) - e^v / (2v) + 1/2 * e^v *
ln|v| + C )
11. u = v-1: (-e^u / (2(1+u)^2) - e^u / (2(1+u)) + 1/2 * e^u * ln|1+ u| + C)
12. : -∫u*e^u / (1+u)^3 du ,
u
13. (
), : ∫log/(1+log)^2 dx = x / (1+log(x)) + ln|1+log(x)| +
()
:
• x / (1+log(x))
• ln|1+log(x)|
• , C
, ' (), :
1.
2.
3.
4.
5.
, :
•
•
•
•
, ' ,
() ,
-
,
,
- :
1. : ,
16
Easy2Siksha
2. : ,
3. : , '
,
4. : , '
,
5. : ,
,
,
-
:
1.
2. - '
3. ,
4. ,
5.
, -
, , -
, , ,
-
-
17
Easy2Siksha
4. p = sqrt(9 + x) 7
: ,
:
-
,
' -
, ' '
-
'
, ,
, -- :
:
1. : p = sqrt(9 + x) p x
2. : 7
1:
, (7
) :
p = sqrt(9 + x) p = sqrt(9 + 7) p = sqrt(16) p = 4
, 4 ( )
2: '
, (p = 4) (p =
sqrt(9 + x)), x = 0 x = 7
', y = 4
( ) ' ,
3:
, , '
18
Easy2Siksha
= × = 4 × 7 = 28
= ∫[0 7] (9 + x) dx
= -
4:
, 0 7 sqrt(9 + x)
, :
∫ sqrt(9 + x) dx
,
: Let u = 9 + x du/dx = 1 dx = du
: ∫ sqrt(u) du = (2/3)u^(3/2) + C
, x = u - 9: (2/3)(9 + x)^(3/2) + C
0 7 :
[(2/3)(9 + 7)^(3/2)] - [(2/3)(9 + 0)^(3/2)] = (2/3)(16)^(3/ 2) - (2/3)(9)^(3/2) = (2/3) × 64 - (2/3)
× 27 = 42.67 - 18 = 24.67
5:
= - = 28 - 24.67 = 3.33
, 3.33 ( $3.33
)
, :
1. : $4 , 7
$28
2. : p = sqrt(9 + x) -
' :
o : sqrt(9 + 1) = sqrt(10) ≈ 3.16
o 7
: sqrt(9 + 7) = sqrt(16) = 4
3. : $4
, $3.16 '
4. : ($4 )
( $3.16 )
19
Easy2Siksha
5. :
7
, $3.33
6. : $3.33 ""
-
- ,
:
• - ( ,
)
• $4 '
• ,
$3.16
• , ,
$4
•
7 $4 ,
'
• - $4 '
$3.33
:
1. :
2. :
3. : '
,
4. :
( '
)
20
Easy2Siksha
5. :
'
, ' :
1. $5 :
o 5 × 7 = 35
o ,
2. $4 ' 10 :
o x = 10
o
3. p = sqrt(4 + x) :
o
'
o ' ,
, $3.33
- $4
' ,
, -
, , - ,
, ,
,
,
, !
21
Easy2Siksha
-
5. () ?
(b) :
2z + 4y - z = 9
3x + y + 2z = 7
x + 3y - 3z = 4
():
?
,
' , "det(A)"
A
, 2x2 :
|ab| |cd|
: ad - bc
3x3 :
|abc| |def| ||
: a(ei-fh) - b(di-fg) + c(dh-eg)
, ,
-
:
1. :
2. 1x1 : [a] ,
'a'
3. :
,
22
Easy2Siksha
4.
:
k
, k
5. ( ) ( )
: '
6. , :
,
7. , :
8. ( ) :
9. : det(AB) =
det(A) * det(B)
10. : det(A^(-1)) =
1 / det(A)
11. : det(A^T)
= det(A)
12. (
), :
- , :
• (
b )
•
•
• '
•
():
, :
2x + 4y - z = 9 3x + y + 2z = 7 x + 3y - 3z = 4
23
Easy2Siksha
:
1:
AX = B , :
= | 2 4 -1 | X = |x| = | 9 | | 3 1 2 | |y| | 7 | | 1 3 -3 | |z| | 4 |
2: A (
D )
= | 2 4 -1 | | 3 1 2 | | 1 3 -3 |
D = 2(1(-3)-2(3)) - 4(3(-3)-2(1)) + (-1)(3(3)-1(1)) = 2(-3 -6) - 4(-9-2) + (-1)(9-1) = 2(-9) - 4(-11) +
(-8) = -18 + 44 - 8 = 18
3:A B Dx, Dy Dz
Dx = | 9 4 -1 | | 7 1 2 | | 4 3 -3 |
Dy = | 2 9 -1 | | 3 7 2 | | 1 4 -3 |
Dz = | 2 4 9 | | 3 1 7 | | 1 3 4 |
4:Dx, Dy, Dz
Dx = 9(1(-3)-2(3)) - 4(7(-3)-2(4)) + (-1)(7(3)-4(1)) = 9(-3 -6) - 4(-21-8) + (-1)(21-4) = 9(-9) - 4(-
29) + (-17) = -81 + 116 - 17 = 18
Dy = 2(7(-3)-2(4)) - 9(3(-3)-2(1)) + (-1)(3(4)-1(7)) = 2(-21) -8) - 9(-9-2) + (-1)(12-7) = 2(-29) - 9(-
11) + (-5) = -58 + 99 - 5 = 36
= 2(1(4)-3(7)) - 4(3(4)-1(7)) + 9(3(3)-1(1)) = 2(4-21) - 4( 12-7) + 9(9-1) = 2(-17) - 4(5) +
9(8) = -34 - 20 + 72 = 18
5:x, y, z
: x = Dx / D y = Dy / D z = Dz / D
: x = 18 / 18 = 1 y = 36 / 18 = 2 z = 18 / 18 = 1
, : x = 1, y = 2, z = 1
:
1. 2x + 4y - z = 9 2(1) + 4(2) - 1 = 9 2 + 8 - 1 = 9 9 = 9 ✓
2. 3x + y + 2z = 7 3(1) + 2 + 2(1) = 7 3 + 2 + 2 = 7 7 = 7 ✓
3. x + 3y - 3z = 4 1 + 3(2) - 3(1) = 4 1 + 6 - 3 = 4 4 = 4 ✓
,
24
Easy2Siksha
:
'
'
:
1. D " "
D = 0,
2. Dx, Dy, Dz
(B)
3. Dx, Dy, Dz D ,
' A
"" , x, y, z
:
1. ,
2.
3.
:
1. ' (3 4 )
2. '
3.
( D = 0)
,
,
:
-
,
( ' )
, ,
,
25
Easy2Siksha
, -
,
'
,
6. () ?
(b) : 2 4 -1
3 1 2
1 3 -3
: A:
'
', '
, :
1. ? ,
:
[2 3 1] [4 1 -2] [0 2 5]
3x3 (3 3 )
2. ?
:
• 1
• 2
• 3 -
26
Easy2Siksha
.
3. ? '
:
1: 3x3 ' :
[1 2 3] [2 4 6] [3 6 9]
', 3x3 , ' 3
, :
•
•
-
, '
3x3 1
2: 3x3 ' :
[1 2 3] [0 1 4] [5 6 7]
:
•
•
•
' , 3
4.
?
:
• ,
• ,
• ,
27
Easy2Siksha
• ,
5. ,
-
, :
[1 2 3] [2 4 6] → [1 2 3] → [1 2 3] [3 6 9] [0 0 0] [0 0 0] [0 0 0] [0 0 0]
- , 1
B:
, :
, , ,
',
;
( )
,
:
[2 4 -1] [3 1 2] [1 3 -3]
,
:
1.
2.
3.
1:
3x3 ,
:
det(A) = a(ei - fh) - b(di - fg) + c(dh - )
: [abc] [def] [ghi]
: a = 2, b = 4, c = -1 d = 3, e = 1, f = 2 g = 1, h = 3, i = -3
det(A) = 2[(1)(-3) - (2)(3)] - 4[(3)(-3) - (2)(1)] + (-1)[(3)( 3) - (1)(1)] = 2(-3 - 6) - 4(-9 - 2) + (-1)(9
- 1) = 2(-9) - 4(-11) + ( -1)(8) = -18 + 44 - 8 = 18
,
28
Easy2Siksha
2:
:
a b c
: C11 = +(1*-3 - 23) = -9
C12 = -(3-3 - 21) = +11
C13 = +(33 - 11) = +8 C21 = -(4-3 - (-1)3) = +15 C22 = +(2-3 - (-1)1) = -5 C23 = -(23 - 41) = -2
C31 = +(42 - (-1)1) = +9 C32 = -(22 - (-1)3) = -7 C33 = +(21 - 4*3) = -10
:
[-9 11 8] [15 -5 -2] [9 -7 -10]
3:
= (1/18) *
[(-9/18) (11/18) (8/18) ] [(15/18) (-5/18) (-2/18)] [(9/18) (-7/18) ( -10/18)]
:
[-1/2 11/18 4/9] [5/6 -5/18 -1/9] [1/2 -7/18 -5/9]
,
:
[2 4 -1] [-1/2 11/18 4/9] [3 1 2] * [5/6 -5/18 -1/9] [1 3 -3] [1/2 -7/ 18 -5/9]
i
[1 0 0] [0 1 0] [0 0 1]
1. ? A , A^(-1)
, A ', :
A * A^(-1) = A^(-1) * A = I
2.
•
•
( "" "-"
)
29
Easy2Siksha
• ,
• (A^(-1))^(-1) =
• (AB)^(-1) = B^(-1)A^(-1) ( A B )
3.
:
•
•
• -
• -
•
4.
,
:
•
• LU
•
, '
:
(
)
, 3x3 , 3
( ) '
:
1. :,
2. : () ,
30
Easy2Siksha
3. : - ,
4. : -
,
5. : ,
6. :
,
7. : ,
,
-
" " ,
"" , ,
,
, ,
' ,
, -
,
' , -
31
Easy2Siksha
-
7. () -
(b) :
0.4
0.1
50
0.7
0.6
100
5
2
:
(i)
(ii)
(iii) , 10 .
: -
- 1930
- -
- :
a) :-
'
, , -
, ,
b) : , , -
' ,
-
c) : -
- ,
32
Easy2Siksha
d) : -
'
e) : -
,
f) : ,
-
( CO2 ) ,
g) :
' - '
h) :
-
2.
,
--
: : 0.4 0.1 50 0.7
0.6 100 5 2
:(i) (ii)
(iii)
, 10
1:
-
'
:
• 1 , 0.4 , 0.7 5
• 1 , 0.1 , 0.6 2
2:
33
Easy2Siksha
,
'S' 'C'
: S = 0.4S + 0.1C + 50 ( ) C = 0.7S + 0.6C + 100 ( )
: 0.6S - 0.1C = 50 -0.7S + 0.4C = 100
:
7 6 : 4.2S - 0.7C = 350 -4.2S + 2.4C = 600
: 1.7C = 950 C = 950 / 1.7 = 558.82
C : S = 0.4S + 0.1(558.82) + 50 0.6S = 105.882 S =
176.47
, : : 176.47 : 558.82
3:
,
:
= 5 * 176.47 = 882.35 = 2 * 558.82 = 1117.64
= 882.35 + 1117.64 = 2000
4:
,
'Ps' 'Pc'
10
.
• : Ps = 0.4Ps + 0.7Pc + 5 * 10 Pc = 0.1Ps + 0.6Pc + 2 * 10
• : 0.6Ps - 0.7Pc = 50 -0.1Ps + 0.4Pc = 20
• 4 6 : 2.4Ps - 2.8Pc = 200 -0.6Ps +
2.4Pc = 120
• : 1.8Ps - 0.4Pc = 320
• : 0.6(320 + 0.4Pc)/1.8 - 0.7Pc = 50 106.67 +
0.13Pc - 0.7Pc = 50 106.67 - 0.57Pc = 50 -0.57Pc = 50.57Pc = 749Pc = 50.57Pc.
• Ps: 0.6Ps - 0.7(99.42) = 50 0.6Ps = 119.59 Ps = 199.32
• , : : 199.32
: . 99.42
:
34
Easy2Siksha
1. : ( )
,
176.47 558.82
( 50 100 )
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35
Easy2Siksha
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36
Easy2Siksha
8. () ? LPP '
() :
Z = 2x1 - 3x2
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37
Easy2Siksha
,
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,
2. LPP
, LPP :
Z = 2x₁ - 3x₂ : 4x₁ + 5x₂ ≤ 40 x₁ + 3x₂ ≤ 12 x₁ - x₂ ≥ 2 x₁ ≥ 4 x₁, x₁ ≥ 4 x₁
1:
:
1. 4x₁ + 5x₂ = 40
2. x₁ + 3x₂ = 12
3. x₁ - x₂ = 2
4. x₁ = 4
5. x₂ = 0 (x-)
6. x₁ = 0 (y-)
,
:
1. 4x₁ + 5x₂ = 40 x₁-: (10, 0) x₂-: (0, 8)
2. x₁ + 3x₂ = 12 x₁-: (12, 0) x₂-: (0, 4)
3. x₁ - x₂ = 2 x₁-: (2, 0) x₂-: (0, -2)
4. x₁ = 4 x₁ = 4 '
2:
,
3:
'
:
A. x₁ = 4 x₁ + 3x₂ = 12 4 + 3x₂ = 12 3x₂ = 8 x₂ = 8/3 A: (4, 8/3)
B. x₁ = 4 4x₁ + 5x₂ = 40 4(4) + 5x₂ = 40 16 + 5x₂ = 40 5x₂ = 24 x₂ = 24/5
: (4, 24/5)
38
Easy2Siksha
C. of x₁ - x₂ = 2 x₁ + 3x₂ = 12 x₁ = x₂ + 2 x₁ + 3x₂ = 12 : (x₂ + 2) +
3x₂ = 12 4x₂ + 2₂5 = x 12₁ = x 5 = 12₁ /2 + 2 = 9/2 C: (9/2, 5/2)
D. of x₁ - x₂ = 2 4x₁ + 5x₂ = 40 x₁ = x₂ + 2 4x₁ + 5x₂ = 40 : 4(x₂ +
2) + 5x₂ = 40 4x₂ = 4x₂ 40 = x25 x = 32/9 x₁ = 32/9 + 2 = 50/9 D: (50/9, 32/9)
4: '
Z = 2x₁ - 3x₂
• A (4, 8/3): Z = 2(4) - 3(8/3) = 8 - 8 = 0
• B (4, 24/5) ': Z = 2(4) - 3(24/5) = 8 - 14.4 = -6.4
• C (9/2, 5/2) ': Z = 2(9/2) - 3(5/2) = 9 - 7.5 = 1.5
• D (50/9, 32/9) ': Z = 2(50/9) - 3(32/9) = 100/9 - 96/9 = 4/9 ≈ 0.44
5:
Z , Z = 1.5 C
(9/2, 5/2)
, : x₁ = 9/2 = 4.5 x₂ = 5/2 = 2.5 Z = 1.5
3.
, :
(i) : Z = 2x₁ - 3x₂
', Z = 1.5
( , $1,500
$1,000 )
(ii)
:
,
,
:
:
• x₁ 2
• x₂ 3
,
: = 2x₁ + 3x₂ = 2(4.5) + 3(2.5) = 9 + 7.5 = 16.5
(iii) : , LPP
'
39
Easy2Siksha
,
,
:
1. ( y₁, y₂, y₃, y₄ )
2. ',
: 4y₁ + y₂ + y₃ + y₄ ≥ 2 (x₁
) 5y₁ + 3y₂ - y₃ ≥ -3 (x₂ )
3. . 10
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5.
6. : '
:
o x₁ - x₂ ≥ 2 (x₁ - x₂ = 2 ')
o x₁ + 3x₂ ≤ 12 (4.5 + 3(2.5) = 12)
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x₂
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11. : ' ( C D )
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40
Easy2Siksha
Pareto ,
12. : 4x₁ + 5x₂ ≤ 40 (4(4.5) + 5(2.5) = 30.5 < 40) '
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16. - : x₁, x₂ ≥ 0
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18. : ' x₁ - x₂ = 2 1 2
1
2 , ,
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19. :
,
( )
, x₁ + 3x₂ ≤ 12
41
Easy2Siksha
20. : (1.5) '
21. :
x₂
,
-
22. :
, 4x₁ + 5x₂ ≤ 40 , (4(4.5) + 5(2.5)) / 40 = 30.5 / 40 =
76.25%
23. Breakeven :
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