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2023
/
( , ++ )
: 3 :75
: ,
-
1. ? ?
f(x) = 5x⁴ + 7x + 3, 0(x¹)
2. (a) -, - -
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() ,
-
Easy2Siksha.com
3. (a)
b) (EAS*Y*QUE**ST*IO*N*) ,
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, ( E)
()
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-
5. 15, 10, 13, 9, 12, 7
6. ?
-
7. (a) ++ ,
(b)
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8. (a)
?
(b) ++ - ++
2023
/
( , ++ )
: 3 :75
: ,
-
1. ? ?
f(x) = 5x⁴ + 7x + 3, 0(x¹)
: ?
:
Easy2Siksha.com
1.
2.
3.
4.
5.
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-- ,
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3.
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6.
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Easy2Siksha.com
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Easy2Siksha.com
:
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f(x) = 5x⁴ + 7x + 3 O(x⁴)
- , :
f(x) ≤ C × g(x)
x ≥ k , g(x) x⁴ , O(x⁴)
Easy2Siksha.com
:
f(x) = 5x⁴ + 7x + 3
:
x , x⁴, x
:
• 7
• 3
• 5x⁴
:
5x⁴ + 7x + 3 ≤ 5x⁴ + x⁴ + x⁴ ( x )
≤ 7x⁴
, x ≥ 1 , f(x) ≤ 7x⁴
, C = 7k = 1
-
,
f(x) = O(x⁴)
x⁴
( )
--
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:
• - ()→
• (Ω)→
• (Θ)→ /
Easy2Siksha.com
,
, 5x⁴ + 7x + 3 O(x⁴) x⁴
2. (a) -, - -
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() ,
: () :
?
() ,
1. -
• :
int arr[5] = {10, 20, 30, 40, 50};
, arr 5
• :
1000 4 , :
o arr[0] → 1000
o arr[1] → 1004
o arr[2] → 1008
o arr[3] → 1012
o arr[4] → 1016
,
Easy2Siksha.com
2. -
• :
int matrix[2][3] = {
{1, 2, 3},
{4, 5, 6}
};
2x3
• : - (C )
,
o [0][0] → 1000
o [0][1] → 1004
o [0][2] → 1008
o [1][0] → 1012
o [1][1] → 1016
o [1][2] → 1020
,
3. -
• :
int cube[2][2][3] = {
{
{1, 2, 3},
{4, 5, 6}
},
{
{7, 8, 9},
{10, 11, 12}
}
};
2x2x3
• : - , cube[0][0] ,
cube[0][1], cube[1][0],
Easy2Siksha.com
,
( )
#include <stdio.h>
() {
int arr[5] = {10, 20, 30, 40, 50};
int matrix[2][3] = {{1, 2, 3}, {4, 5, 6}};
int cube[2][2][3] = {
{{1, 2, 3}, {4, 5, 6}},
{{7, 8, 9}, {10, 11, 12}}
};
printf("1D : %d\n", arr[2]); // : 30
printf("2D : %d\n", matrix[1][2]); // : 6
printf("3D : %d\n", cube[1][0][2]); // : 9
0 ;
}
()
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Easy2Siksha.com
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2. -> = NULL
3.
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1.
2. last->prev->next = NULL
3.
• :
1.
2.
3.
()
Easy2Siksha.com
CreateDLL():
newNode ←
newNode.data ← data
newNode.next ← NULL
newNode.prev ← NULL
= NULL
←
←
while temp.next ≠ NULL do
← .
. ←
newNode.prev ← temp
InsertAtBeginning(data):
newNode ←
newNode.data ← data
newNode.next ← head
newNode.prev ← NULL
≠ NULL
. ←
←
DeleteAtEnd():
= NULL
" "
←
while temp.next ≠ NULL do
← .
.. ←
()
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• :
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/
Easy2Siksha.com
,
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3. (a)
b) (EAS*Y*QUE**ST*IO*N*) ,
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()
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:
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Easy2Siksha.com
( A + B ) * C
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+
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+
/
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•
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/
(Ctrl + Z) ?
,
Easy2Siksha.com
:
→ →
,
: INFIX POSTFIX
:
• : : A + B
• : : AB+
:
•
•
•
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(, , ...)
→2 → , :
• ( → )
→ ( 5 →
Easy2Siksha.com
:
( + ) *
:
• ( →
• A → : A
• + →
• B → : AB
• ) → ( → : AB+
• * → *
• C → : AB+CEnd → *
:
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():
:
* * * * * * *
:
• =
• * = POP POP
--
Easy2Siksha.com
( POP )
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Easy2Siksha.com
4. ?
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Easy2Siksha.com
: ,
• ,
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• ,
1. ( )
:
( , ):
1. ( ( + 1) ) = :
" " ( )
2. ( = -1 = -1):
← 0
← 0
[] ←
3. :
← ( + 1)
[] ←
:
• ,
• , 0
• ,
2. ()
Easy2Siksha.com
:
( ):
1. ( = -1):
" " ( )
2. ( = ):
← []
← -1
← -1
3. :
← []
← ( + 1)
:
• = -1 ,
• = , ; ,
• ,
5
• : = -1, = -1.
• 10 → = 0, = 0.
• 20 → = 1.
• 30 → = 2.
• → 10 , = 1.
• 40 → = 3.
• 50 → = 4.
• 60 → 0 ( )
:
• : —
,
• : —
Easy2Siksha.com
,
,
, ,
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-
5. 15, 10, 13, 9, 12, 7
:
( ),
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Easy2Siksha.com
1.
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2.
:
o
o
3.
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o
4. (
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:
15, 10, 13, 9, 12, 7
1
= 15
:
• 15 → : 10, 13, 9, 12, 7
• 15
:
7, 10, 13, 9, 12 | 15
( , ) 15
:
10, 13, 9, 12, 7
Easy2Siksha.com
: 10, 13, 9, 12, 7 = 10
:
• 10 → 9, 7
• 10 → 13, 12
:
7, 9 | 10 | 13, 12
10 - :
: 7, 9 : 13, 12
3
7, 9 = 7
? → 9
:
7 | 9
4
13, 12 = 13
→ 12 →
:
12 | 13
Easy2Siksha.com
:
7, 9, 10, 12, 13, 15
:
7, 9, 10, 12, 13, 15
, -
– O(n log n)
, :
| |
:
•
•
• ,
= log n = n log n
,
= O(n log n)
– O(n²)
Easy2Siksha.com
:
:
1, 2, 3, 4, 5
= 1
,
:
n + (n-1) + (n-2) + ... + 1 = n²
,
= O(n²)
:
•
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– O(n log n)
,
:
= O(n log n)
,
?
,
:
✔
✔
✔
✔ ( C, C++, Java, Python
)
Easy2Siksha.com
(
2
-
:
15, 10, 13, 9, 12, 7
: 7, 9, 10, 12, 13, 15
:
• → O(n log n)
• → O(n log n)
• → O(n²)
6. ?
:
-
-
:
[5, 12, 18, 23, 45, 67, 89]
Easy2Siksha.com
45 :
• 5 →
• 12 →
• 18 →
• 23 →
• 45 → !
5
, - :
1.
2. →
3.
4. →
; , ,
:
[5, 12, 18, 23, 45, 67, 89]
45 :
• = 23 → 45 > 23 → .
• - = [45, 67, 89]. = 67 → 45 < 67 → .
• - = [45]. = 45 → !
5 3
?
• : , n
, O(n)
Easy2Siksha.com
• : n , O(log n)
:
• 1,000 :
o → 1,000
o → 10 ( log₂(1000) ≈ 10)
,
(arr, low, high, key):
1. > :
-1 //
2. ← ( + ) / 2
3. arr[mid] = key:
4. < arr[mid]:
(arr, low, mid - 1, key)
5. :
(arr, mid + 1, high, key)
( )
#include <stdio.h>
int binarySearch(int arr[], int low, int high, int key) {
( > ) -1 ;
int mid = (low + high) / 2;
(arr[mid] == key) mid ;
else if (key < arr[mid]) return binarySearch(arr, low, mid - 1, key);
(arr, mid + 1, high, key) ;
}
() {
int arr[] = {5, 12, 18, 23, 45, 67, 89};
int n = sizeof(arr)/sizeof(arr[0]);
int key = 45;
Easy2Siksha.com
int = (arr, 0, n-1, );
( != -1)
printf(" %d \n", result);
printf(" \n");
0 ;
}
1.
• :O(1) ( ).
• :O(n) ( )
2.
• :O(1) ( ).
• :O(log n) ( )
O(log n) ? , :
• 1 → n
• 2 → n/2
• 3 → n/4 … 1 = log₂(n)
• : -
• : ,
,
,
(1)
(1)
)
O( n)
Easy2Siksha.com
,
,
, ,
-
7. (a) ++ ,
(b)
:(a) C++
, C++ ?
C++ C
, - (OOP)
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:
C++
C++ :
•
•
•
•
Easy2Siksha.com
•
C , C++
,
C++ : , ,
, ,
C++ ( )
OOPS
( OOP) , C++ :
• ( C)
• ,
, :
,
Easy2Siksha.com
C++
?
, (
) ,
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- (
)
:
{
:
;
:
( ) {
= ;
}
() {
cout << " : " << roll;
}
};
:
• ,
• setRoll() display()
Easy2Siksha.com
.
;
s.setRoll(10);
s.display();
->
*;
= &;
->();
:
&ref = s;
.();
, ( ),
-
()
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C++ ... ,
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:
Easy2Siksha.com
•
• ( )
•
🏷
, C++
{
:
() {
cout << " ";
}
};
{
:
x;
:
( ) {
x = a;
}
};
( &obj) {
x = obj.x;
}
Easy2Siksha.com
-
#include <iostream>
;
{
;
;
:
//
() {
= 0;
= "";
cout << "\n ";
}
//
( r, n) {
= ;
= ;
cout << "\n ";
}
//
( &) {
= .;
= .;
cout << "\n ";
}
() {
cout << "\n: " << << " : " << ;
}
};
Easy2Siksha.com
() {
s1; //
s1.display();
s2(101, ""); //
s2.display();
s3(s2); //
s3.display();
0 ;
}
?
s1 → s2 →
3 s2 s3 →
C++
• -
•
•
• C OOP
• :
• , ,
• :
o
o
o
Easy2Siksha.com
•
• :
o
o
o
o
8. (a)
?
(b) ++ - ++
: ()
?
( +, -, *,
) - ()
Complex
, + ,
+
1.
o
2.
o
3. -
o -
4.
o ,
5. . (), :: ( ), , ?: (),
.
o
6.
Easy2Siksha.com
o , '+' ,
• :: ( )
• ( )
• .* ( )
• ?: ( )
•
•
: +
Complex +
#include <iostream>
;
{
, ;
:
( r = 0, i = 0) {
= r;
= i;
}
// +
+ ( & ) {
;
. = + .;
. = + .;
;
}
() {
cout << real << " + " << image << "i" << endl;
}
};
() {
c1(3, 4), c2(1, 2);
Complex c3 = c1 + c2; // +
Easy2Siksha.com
cout << " : ";
c3.display();
0 ;
}
:
: 4 + 6i
()
?
: - , -
- -
C++
1. :
2. :
3. : ,
( )
4. :
-
:
• (- → - → )
•
:
• (-)
• ,
• ,
Easy2Siksha.com
#include <iostream>
;
//
{
:
() {
cout << " " << endl;
}
};
// Person ()
: {
:
() {
cout << " " << endl;
}
};
//
{
:
() {
cout << " " << endl;
}
};
// () ()
: , {
:
() {
cout << " " << endl;
}
};
() {
;
m.showPerson(); // Person
m.showEmployee(); //
m.showDepartment(); //
m.showManager(); //
0 ;
}
Easy2Siksha.com
:
(
)
?
• : -
,
• : : ,
• :
:
• , ( )
• , ( )
• ( , )
• , , ,
C++ -
-
+
, ::, .,
sizeof
-
,
→ →
+
Easy2Siksha.com
+
,
C++
,
: - -
, C++
“
, ”
