1
Easy2Siksha
GNDU -2022
( )
: : 50
: - ,
-
1. - ?
2. - - x3 - x -
1
-
3. () ? ?
() - :
x + 2y + z = 8
2x + 3y + 4z = 20
4x + 3y + 2z = 16
2
Easy2Siksha
4. ? - ?
:
1 2 3
3 -2 1
4 1 1
-
5. () ,
(b) , f(7.5)
1
2
3
4
5
6
7
8
F(x)
1
8
27
64
125
216
343
512
6. (a) n = 4
(b) n = 10 1/3
-
7. () ? - ?
(b) :
25
27
31
35
36
3
Easy2Siksha
3
2
4
1
2
8. ' :
(a)
()
(c)
4
Easy2Siksha
GNDU -2022
( )
: : 50
: - ,
-
1. - ?
: :
,
:
1.
-
2.
3.
:
1.
- :
-
,
,
5
Easy2Siksha
:
1/3 0.333333
... ( ) , 0.33333333 (8 )
,
: ' 0.1 + 0.2 '
0.3 , -
,
0.30000000000000004
-
- , '
2. :
e: '
:
sin(x) = x - x^3/3! + x^5/5! - x^7/7! +...
(x - x^3/3!) (x) ,
, ,
(
)
3. :
, ,
:
,
5.23
5.22
6
Easy2Siksha
0.1
,
:
, ,
:
•
,
:
• , A B,
ea eb
C = A * B
• C , ec , :
• ec ≈ ea + eb
•
:
1% (ea = 0.01)
10 , 2% (eb = 0.02) 5
•
: = * = 10 * 5 =
50 ²
• : ec ≈ ea + eb = 0.01 + 0.02 = 0.03 3%
• , 3% 50 cm²
:
• ,
C = A / B , C
:
• ec ≈ ea + eb
• :
,
0.5% (ea = 0.005) 100 g , 1% (eb =
0.01) 50 cm³
= / = 100 g / 50 cm³ = 2 g/cm³
: ec ≈ ea + eb = 0.005 + 0.01 = 0.015 1.5%
,
1.5% 2 g/cm³
:
7
Easy2Siksha
, ,
:
1. ,
- , -
2. ,
,
,
3. ,
4.
:
:
1. : -
2. : '
,
,
-
3. : ,
4. :
,
8
Easy2Siksha
5. :3D - ,
-
:
,
:
1. : ,
-
2. :
3. :
4. : -
,
5. : , (
)
-
6. : ,
, -
:
- -
-, , -
- :
, ,
, ' ,
,
, '
,
, ,
'
, , ,
9
Easy2Siksha
, '
,
,
,
-
2. - - x3 - x -
1
: - :
- , , -
( ) , x
' '
:
1.
2. '
3. x-
4.
5. 2-4
, :
1. :
-
,
,
x₀
2. : ',
'
10
Easy2Siksha
3. - :
x-
4. :
x- ,
x₁
5. :
- ,
'
,
:
, .
:
x_{n+1} = x_n - f(x_n) / f'(x_n)
:
• x_n
• x_{n+1}
• f(x_n) '
• f'(x_n) '
-
!
:
-
'
" " ,
(
)
:
1. : , -
,
2. :
3. : ' ,
:
11
Easy2Siksha
1. : ,
2.
:
3. : ' ,
4. : ,
, x^3 - x - 1 = 0
1:
f(x) = x^3 - x - 1 f'(x) = 3x^2 - 1
2:
,
f(1) = 1 - 1 - 1 = -1 f(2) = 8 - 2 - 1 = 5. 1 2
.
x₀ = 1.5
3: -
x_{n+1} = x_n - f(x_n) / f'(x_n)
1: x₁= 1.5 - (1.5^3 - 1.5 - 1) / (3(1.5)^2 - 1) = 1.5 - (0.875) / (5.75) = 1.5 - 0.152173913
= 1.347826087
2: x₂= 1.347826087 - (1.347826087^3 - 1.347826087 - 1) / (3(1.347826087)^2 - 1) =
1.347826087 - 0.022411859 / - 741859 / 4740 = 4750 = 4750 005036087 = 1.342790000
3: x₃= 1.342790000 - (1.342790000^3 - 1.342790000 - 1) / (3(1.342790000)^2 - 1) =
1.342790000 - 0.000069624 / - 400069624 =
4000000000000000000000000000000000000000 000015760 = 1.324774240
4: x₄= 1.324774240 - (1.324774240^3 - 1.324774240 - 1) / (3(1.324774240)^2 - 1) =
1.324774240 - 0.000000007 / 4312407 = 432407. 000000002 = 1.324774238
, 1.325
: :
f(1.325) = 1.325^3 - 1.325 - 1 = 2.325390625 - 1.325 - 1 = 0.000390625
12
Easy2Siksha
, '
:
•
x ≈ 1.325 x^3 - x - 1 = 0 x
1.325 ,
• ,
y = x^3 - x - 1 ,
x- x- x-
• :
(1 + √5) / 2 , ( 1.618034)
!
:
- , ' -, :
1. : , ,
2. :
3. : 3D
4. :
5. :
6. :- ,
,- -
, : , '
,
x^3 - x - 1 = 0
, ,
-
,
, ,
13
Easy2Siksha
, , ,
-
3. () ? ?
() - :
x + 2y + z = 8
2x + 3y + 4z = 20
4x + 3y + 2z = 16
: a) :
,
, - '
:
,
:
1. 2 3 $7
2. 3 2 $8
:
x y
1: 2x + 3y = 7 2: 3x + 2y = 8
,
:
1. :
,
,
14
Easy2Siksha
2. - : , - -
1
,
3. :
: 2x + 3y = 0 4x - y = 0
4. - : -
( )
5. : -
6. :
7. : , -
8. :
,
b) - :
: x + 2y + z = 8 2x + 3y + 4z = 20 4x + 3y + 2z = 16
-
-- :
1:
(x, y, z)
:
| 1 2 1 | 8 | | 2 3 4 | 20 | | 4 3 2 | 16 |
2:
1 , 0s.
1 ,
2 2 1 : | 1 2 1 | 8 | | 0 -1 2 | 4 | | 4 3 2 | 16 |
3 4 1 : | 1 2 1 | 8 | | 0 -1 2 | 4 | | 0 -5 -2 | -16 |
3:
15
Easy2Siksha
1 ,
0s.
2 -1 : | 1 2 1 | 8 | | 0 1 -2 | -4 | | 0 -5 -2 | -16 |
1 2 2 : | 1 0 5 | 16 | | 0 1 -2 | -4 | | 0 -5 -2 | -16 |
3 5 2 : | 1 0 5 | 16 | | 0 1 -2 | -4 | | 0 0 -12 | -36 |
4:
1 , 0s.
3 -12 : | 1 0 5 | 16 | | 0 1 -2 | -4 | | 0 0 1 | 3 |
1 5 3 : | 1 0 0 | 1 | | 0 1 -2 | -4 | | 0 0 1 | 3 |
2 2 3 : | 1 0 0 | 1 | | 0 1 0 | 2 | | 0 0 1 | 3 |
5:
:
x = 1 y = 2 z = 3
:
1: x + 2y + z = 1 + 2(2) + 3 = 1 + 4 + 3 = 8 ✓ 2: 2x + 3y + 4z = 2(1) + 3(2) +
4(3) = 2 + 6 + 12 = 20 ✓ 3: 4x + 3y + 2z = 4(1) + 3(2) + 2(3) = 4 + 6 + 6 = 16 ✓
,
, -
:
- ' '
:
1. -
2.
3.
,
(
)
16
Easy2Siksha
'
:
1. :
,
2. : , -
3. :
, ,
4. ' :-
,
, - ( )
:
1. : 3D , ,
-
2. :
, ,
, -
3. : , (
) -
4. :
5. : - ,
- ,
6. :
,
, -
17
Easy2Siksha
, -
- ,
,
, -
,
- ,
,
, '
,
- -
4. ? - ?
:
1 2 3
3 -2 1
4 1 1
: ?
, ',
- 2 * 1/2 = 1,
A ,
A^(-1)
A
, :
A * A^(-1) = A^(-1) * A = I
I ( ' 1s 0s)
(
) , , "-"
""
18
Easy2Siksha
?
:
1. : Ax = b
, A , x , b ,
x
A^(-1) : x = A^ (-1) .
2. :
3. :- ,
4. : ,
5. :
:
1. :
2. ( ):
3. - :
4. LU :
5. : , -
,
:
1.
2.
3.
' :
A = [1 2 3] [3 -2 1] [4 1 1]
19
Easy2Siksha
1:
3x3 ,
:
det(A) = a(ei - fh) - b(di - fg) + c(dh - )
: a = A[0][0], b = A[0][1], c = A[0][2] d = A[1][0], e = A[1][1] , f = A[1][2] g = A[2][0], h =
A[2][1], i = A[2][2]
:
det(A) = 1[(-21) - (11)] - 2[(31) - (14)] + 3[(31) - (4-2)] = 1(-3) - 2(- 1) + 3(11) = -3 + 2 + 33 = 32
, 32
2:
,
3x3 , 2x2
, (-1)^(i+j) i j
.
:
C11 = (-1)^(1+1) * [(-21) - (11)] = 1 * (-3) = -3
C12 = (-1)^(1+2) * [(31) - (14)] = -1 * (-1) = 1
C13 = (-1)^(1+3) * [(31) - (4-2)] = 1 * 11 = 11
C21 = (-1)^(2+1) * [(21) - (31)] = -1 * (-1) = 1
C22 = (-1)^(2+2) * [(11) - (33)] = 1 * (-8) = -8
C23 = (-1)^(2+3) * [(14) - (31)] = -1 * 1 = -1
C31 = (-1)^(3+1) * [(21) - (3-2)] = 1 * 8 = 8
C32 = (-1)^(3+2) * [(11) - (32)] = -1 * (-5) = 5
C33 = (-1)^(3+3) * [(1*-2) - (2*3)] = 1 * (-8) = -8
,
:
= [-3 1 11] [ 1 -8 -1] [ 8 5 -8]
= [-3 1 8] [ 1 -8 5] [11 -1 -8]
3:
,
:
20
Easy2Siksha
A^(-1) = (1/32) * [-3 1 8] [ 1 -8 5] [11 -1 -8]
:
A^(-1) = [-3/32 1/32 1/4 ] [ 1/32 -1/4 5/32] [11/32 -1/32 -1/4]
:
,
, :
[1 2 3] [-3/32 1/32 1/4] [3 -2 1] * [ 1/32 -1/4 5/32] [4 1 1] [11/32 -1/32 - 1/4]
, :
[1 0 0] [0 1 0] [0 0 1]
,
-
:
1.
, ,
- (
)
2. ,
3. : (AB)^(-1) = B^(-1)A^(-
1)
4. : (A^(-1))^(-1) = A
5. : (A^T)^(-1) = (A^(-1))^T
6. 2x2 , : A = [ab] [cd] A^(-1) =
(1 / (ad-bc)) * [ d -b] [-ca]
,
7. ' ,
, '
- :
21
Easy2Siksha
1. :3D ,
,
3D ""
,
,
2. : , -
: θ = (X^TX)^(-1) X^T y
3. : - ,
4. : ,
-
5. : -
,
6. : ,
7. : , ,
( )
,
,
LU '
, ,
,
x Ax = b ,
' A^(-1) b '
,
,
-
- , ,
- 3x3
:
22
Easy2Siksha
[1 2 3] [3 -2 1] [4 1 1]
:
[-3/32 1/32 1/4] [ 1/32 -1/4 5/32] [11/32 -1/32 -1/4]
, , ,
,
-
-
5. () ,
(b) , f(7.5)
1
2
3
4
5
6
7
8
F(x)
1
8
27
64
125
216
343
512
:(a) :
, ,
:
-
' ,
.
23
Easy2Siksha
:
9:00 20°C
11:00 24°C 10:00 ,
22 -
-
:
1. :
2. :
3. :
4. : -
:
- -
-
:
3 5
,
5 ,
:
1.
2.
3. :
4. :
:
1. :
o : -
o : -
2. :
o : '
-
'
o : , '
-
,
24
Easy2Siksha
3. :
o :
o : '
4. :
o :
o :
5. :
o : -
o : -
,
() f(7.5) :
:
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 F(x) | 1 | 8 | 27 | 64 | | 216 | 343 | 512
f(7.5) ,
7.5
: 7 8
-- :
1: - - x = 7 x = 8
,
7.5
x = 7 , f(x) = 343 x = 8 , f(x) = 512
2: ,
,
3: :
f(x) = f(x1) + (x - x1) * (f(x2) - f(x1)) / (x2 - x1)
: x
( 7.5) x1 x-
(7) x2 x- (8) f(x1) x1 (343) ' ) f(x2) x2 (512) '
25
Easy2Siksha
:
f(7.5) = 343 + (7.5 - 7) * (512 - 343) / (8 - 7) = 343 + 0.5 * (169) / 1 = 343 + 0.5 * 169 = 343 +
84.5 = 427.5
, f(7.5) 427.5
:
, x = 7.5 427.5 y-
( f(x) ) .
:
: ,
f(x)
:
1 8: 7 8 27: 19 27 64: 37 64 125: 61 125 216: 91 216 343: 127 343 512: 169
, :
7, 19, 37, 61, 91, 127, 169
: 12, 18, 24, 30, 36, 42
: 6
f(x) = x^3 (x cubed) :
1^3 = 1 2^3 = 8 3^3 = 27 4^3 = 64 5^3 = 125 6^3 = 216 7^3 = 343 8^3 = 512
, f(x) = x^3 !
,
f(7.5) :
f(7.5) = 7.5^3 = 421.875
427.5 ,
?
,
( , )
, ,
:
x^3 , (7, 343) (8, 512)
'
' x = 7.5 ' ,
, (427.5) (421.875)
26
Easy2Siksha
:
' :
1. ( )
2. - ( ' )
3. (, , , )
- ,
,
-
' :
1. : ,
,
,
2. :
,
, ' -
3. :
, ,
' , '
4. : ,
5. : ' '
'
' :
•
- ,
'
• ,
, ,
27
Easy2Siksha
, f(9) ,
, f(9) 9^3
= 729 , ,
- .
:
1. :
2. : -
'
3. Lagrange :
' '
4. :
, '
,
,
'
,
- -
,
- -
,
' , '
' ,
-
, -
,
f(7.5) 427.5
f(x) = x^3 , f(7.5)
421.875 -
28
Easy2Siksha
, -
,
6. (a) n = 4
:,
:
∫_0^5▒x/(1+x^4 )
1/(1+x^4) 0 5
'
' :
1. n -
2. - '
3. -
4.
Trapezoidal :
∫_a^bf(x)dx ≈ (h/2)[f(x₀) + 2f(x₁) + 2f(x₂) + ... + 2f(xₙ₋₁) + f(xₙ)]
:
• a
• b
• n -
• h = (ba)/n -
• x₀, x₁, ..., xₙ [a,b] n
29
Easy2Siksha
, ' :
0 5 , a = 0 b = 5 n = 4 -
1:hh = (ba)/n = (5-0)/4 = 5/4 = 1.25
2:x [0, 5] 4 : x₀ = 0 x₁
= 1.25 x₂ = 2.5 x₃ = 3.75 x₄ = 5
3: x f(x) f(x) = 1/(1+x^4)
f(x₀) = f(0) = 1/(1+0^4) = 1 f(x₁) = f(1.25) = 1/(1+1.25^4) ≈ 0.6097561 f(x₂) = f(2.5 ) =
1/(1+2.5^4) ≈ 0.0609756 f(x₃) = f(3.75) = 1/(1+3.75^4) ≈ 0.0125313 f(x₄) = f(5) = 1/(1+5) ^4) ≈
0.0015625
4: ∫_0^5▒x/(1+x^4 ) ≈ (h/2)[f(x₀) + 2f(x₁) + 2f(x₂) + 2f(x₃) +
f(x₄) ]
:
≈ (1.25/2)[1 + 2(0.6097561) + 2(0.0609756) + 2(0.0125313) + 0.0015625]
≈ 0.625[1 + 1.2195122 + 0.1219512 + 0.0250626 + 0.0015625]
≈ 0.625[2.3680885]
≈ 1.4800553125
, 4 -
1.4801
,
:
1.
? ,
2.
?
- ,
-
' , -
3.
?
' ,
- -
4.
2
? x (
) - ,
30
Easy2Siksha
2
5. ? ' :
a) - (n): ', n b)
: ,
,
, n = 4 ,
,
,
n 8, 16,
6.
?Trapezoidal
: a)
b)
' c)
7. ?
: a) ( ):
' b) :
c)
:
8. ? , : a)
b)
- , '
c)
9.
?
,
1/(1+x^4) x
x = 0 , x '
x = 0
, x
10. ? ( 1.4801) x = 0 x = 5
1/(1+x^4)
,
,
11.
? ,
:
a) ( )
b) n c)
31
Easy2Siksha
12.
?
,
'
13. ?
: a) :
, b) :
c) : n
,
14.
?
,
: a)
'
b)
c)
15. - ?
- : a)
: b) :
c) : -
d) :
,
,
,
-
32
Easy2Siksha
(b) n = 10 1/3
() , 1/3
:
1/3
""
,
1/3
,
:
∫_0^5▒x/(4x+5)
y = 1/(4x+5) x = 0 x = 5
n = 10 1/3 , n -
1/3 , n
' :
1: -
(0 5 ) 10 - ,
:
h = (b - a) / n b (5), a (0), n - (10)
h = (5 - 0) / 10 = 0.5
- 0.5
2: x-
11 ( 10 - )
:
x0 = 0 x1 = 0.5 x2 = 1.0 x3 = 1.5 x4 = 2.0 x5 = 2.5 x6 = 3.0 x7 = 3.5 x8 = 4.0 x9 = 4.5 x10 = 5.0
33
Easy2Siksha
3: y-
x- y = 1/(4x+5) :
y0 = 1/(4(0)+5) = 1/5 = 0.2000000
y1 = 1/(4(0.5)+5) = 1/7 ≈ 0.1428571
y2 = 1/(4(1.0)+5) = 1/9 ≈ 0.1111111
y3 = 1/(4(1.5)+5) = 1/11 ≈ 0.0909091
y4 = 1/(4(2.0)+5) = 1/13 ≈ 0.0769231
y5 = 1/(4(2.5)+5) = 1/15 ≈ 0.0666667
y6 = 1/(4(3.0)+5) = 1/17 ≈ 0.0588235
y7 = 1/(4(3.5)+5) = 1/19 ≈ 0.0526316
y8 = 1/(4(4.0)+5) = 1/21 ≈ 0.0476190
y9 = 1/(4(4.5)+5) = 1/23 ≈ 0.0434783
y10 = 1/(4(5.0)+5) = 1/25 = 0.0400000
4: 1/3
1/3 :
∫_a^bf(x)dx ≈ (h/3) * [y0 + 4(y1 + y3 + y5 + y7 + y9) + 2(y2 + y4 + y6 + y8) + y10]
:
(0.5/3) * [0.2000000 + 4(0.1428571 + 0.0909091 + 0.0666667 + 0.0526316 + 0.0434783) +
2(0.1111111 + 0.07620503 + 0.076503) + + 0.0400000]
5:
:
0.5/3 = 0.1666667
y1, y3, y5, y7, y9 : 0.1428571 + 0.0909091 + 0.0666667 + 0.0526316 + 0.0434783 =
0.3965428 4: 8175
y2, y4, y6, y8 : 0.1111111 + 0.0769231 + 0.0588235 + 0.0476190 = 0.2944767 2
: 0.5889534
: 0.2000000 + 1.5861712 + 0.5889534 + 0.0400000 = 2.4151246
, 0.1666667: 2.4151246 * 0.1666667 ≈ 0.4025208
, 0.4025208
34
Easy2Siksha
,
:
∫_0^5▒x/(4x+5) y = 1/(4x+5) x = 0 x = 5
0.4025208
, y = 1/(4x+5)
x = 0 x = 5 ' , x-
, 0.4025208
. ,
' , , 1/3
' , '
-
( , 10)
,
:
(1/4) * ln((45 + 5)/(40 + 5)) ≈ 0.4054651
(0.4025208) 0.7% ,
1/3
- 10 - ,
,
- ,
n = 10 n = 100 ,
1/3
- ,
'
, (
) ' , .
1/3 :
1. : '
, '
2. : ,
3. :
, :
1. - : -
,
35
Easy2Siksha
2. : ,
,
3. : , ,
'
,
:
1. :
,
2. : - ,
,
3. :
,
4. :
, ,
'
5. :- ,
,
,
'
:
1. ,
2.
3. ' -
4. - , ,
, , -
-
, ,
,
36
Easy2Siksha
,
0.4025208 , ∫_0^5▒x/(4x+5)
1/3 -
, ' ,
x = 0 x = 5 y = 1/(4x+5)
' ,
-
7. () ? - ?
(b) :
25
27
31
35
36
3
2
4
1
2
: a) ? - ?
,
,
( )
,
:
: 85, 87, 90, 88, 85 : 60, 75, 90, 100, 85
() 87 ,
- :
1. :
2. (IQR): 50% 75
(Q3) 25 (Q1)
37
Easy2Siksha
3. :
,
4. :
5. (MAD):
6. : ,
-
-
,
'
(b) :
:
X () | F () 25 | 3 27 | 2 31 | 4 35 | 1 36 | 2
(N) = 3 + 2 + 4 + 1 + 2 = 12
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38
Easy2Siksha
-- :
| F | |X - | | F × |X - | 25 | 3 | |25 - 30| = 5 | 3 × 5 = 15 27 | 2 | |27 - 30| =
3 | 2 × 3 = 6 31 | 4 | |31 - 30| = 1 | 4 × 1 = 4 35 | 1 | |35 - 30| = 5 | 1 × 5 = 5 36 | 2 | |36 -
30| = 6 | 2 × 6 = 12
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, 3.5
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2 | (36 - 30)² = 36 | 2 × 36 = 72
F × (X - )² = 75 + 18 + 4 + 25 + 72 = 194
= 194 ÷ 12 = 16.1667
= √16.1667 ≈ 4.0207
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