There are Three terms
(1) Undefined
(2) Infinity
(3) Indeterminate

(1) Undefined
Let varify division rule, let we are deviding M by n and it will give x ( M is multiple of n)
M/n = x
that means if we will multiply n with x ..it will give M
as n×x = M
Let suppose we are dividing H by 0 and it will give T
H/0 = T
so
T×0 = H ( so here T is not defined that can give H when multiplied by Zero(0)
So any number / 0 = Undefined ( remember not infinity)

(2) Infinity
it can be defined as ny large Quantity with respect to very small quantity
if we compare 1kg sugar with respect to
.00000000000000000000000000000000000000000000000000000000000000001 gram sugar
so 1kg sugar is treated as infinity ..
Lim x->0(1/x) = infinity

(3) Indeterminate
let x/y = m
so x = y×m
if x = 0 and y =0
0 = 0×m
m may be 1, 2, 3 , 4 , 5 2000, 40000, and so on ...
so 0/0 called indeterminate form
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Solve this Problem

The Difference between Expression, Equation, Inequality and Identity is given below
(1) Definition of Equations:
Equation, statement of equality between two expressions consisting of variables and/or numbers.
or
The definition of an equation is a mathematical statement that shows that two mathematical expressions are equal.
or
In other word if we use equality between two expression, it become equation
Example:
Quadratic Equation
ax² + bx + c = 0

Linear Equation
4x − 8 = −5 + 3x

Cubic Equation
ax³ + 4x² +4 =0

Quartic equation
ax⁴ +f = 0
Differential Equation
(d2y/dx2)+ 2 (dy/dx)+y = 0
(2) Definition of Expression:
An expression in math is a sentence with a minimum of two numbers or variables and at least one math operation. This math operation can be addition, subtraction, multiplication, or division.
The structure of an expression is:
3x + 4
x+y
4a+7b+4c
(3) Definition of Inequality:
This is also called inequations
If we use these inequality symbols, less than (<), greater than (>), less than or equal (≤), greater than or equal (≥) and the not equal symbol (≠) between two expression or between expression and number so it become inequality
Example of Linear Inequality
4x - 1 > 9
10 ≥ 77x - 2
-448 > 56x + 12

Example of modulus Inequality
|x+5| > 4

Example of Quadratic Inequality
X² +4x -1>0

(4) Definition of Identity
An identity is an Equation that is true for all value of variable,
One variable identity
(x +2)² = x² + 4x + 4
Two variable identity
(x + 2y)² = x² + 4y² + 4xy
Function :
F(x) = x² + 4x +1
F(x) = Sin(x)
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Q1. Solve for x ?
{x²} + {x} = 1
Where { } denotes fractional part of x function.

Q2. Solve for x ?
[x²] + [x] = 1
Where [ ] denotes  greatest Integer less than or equal to x

Q3. Solve for x
If x[x] = 15
[ ] Denotes greatest integer less than or equal to x.

Q If 4 {x} = x + [x], where {x} and [x] denotes fractional part of x function and greatest integer function less than or equal to x respectively , then x=? [IIT-JEE 1994]

Sequences and series
(A) Arithmetic Progression :-
A list of numbers in a specific order is known as a sequence or progression.
Let
f(n) = 4n + 1
For n = 1, f(n) = 5
For n = 2, f(n) = 9
For n = 3, f(n) = 13
For n = 4, f(n) = 17
So sequence or progression will be
5, 9, 13, 17......
An Arithmetic Progression or Arithmetic sequence (AP) is a sequence of numbers  such that the difference(Common Difference of AP) between any two consecutive terms is constant.

Given progression is in AP
A1, A2, A3, A4, A5, A6, A7, A8
If and only if
A2-A1= A3-A2 = A4-A3 = A5-A4 = A6-A5 = A7-A6 = A8 - A7
Note : Each number in the sequence is known as term.
Example 1
2, 4, 6, 8, 10, 12, 14, ....................40
Here
4-2 = 6-4 = 8- 6 = 10-8 = 2
(B) Formula For Common Difference(d) of AP
Let any AP
A1, A2, A3, A4, A5, ........An-1, An, An+1 ......

d = A2 - A1 = A3 - A4 = An - An-1
So Common difference d of an AP is calculated by
d = (n)th term - (n -1)th term
Or
d = (n+1)th term - (n)th term

Example 2
Let first term of an AP = a
Let Common Difference ( Difference between two consecutive terms of an AP) = d
So,
First term = a = a + (1-1)d
Second term = a + d = a + (2-1)d
Third term = a + d + d = a + (3-1)d
Fourth term = a +2d + d = a + (4-1)d
Fifth term = a + 3d + d = a + (5-1)d
....... ......
........ .........
........ ........
nTh term , Tn = a + (n - 1)d
So Arithmetic Progression can be written as
Or General form of an AP written as
a, a +d, a + 2d, a + 3d, a + 4d, a + 5d, .........
.......... a + (n - 1)d , a + n d , a +(n +1)d ... .. infinity
Here,
a = Fist term of AP
d = Common Difference between two consecutive terms of AP
n = Number of terms of AP
Tn = nth term of AP = a + (n - 1)d
(C) nth Term of Arithmetic Progression :-
Tn = a + (n - 1)d
Here,
a = Fist term of AP
d = Common Difference between two consecutive terms of AP
n = Number of terms of AP

If the initial term of an arithmetic progression is "a" and the common difference of successive members is "d' then the nth term (Tn) of the sequence is given by:
Tn = a + (n - 1)d
If there are m terms in the AP, if first term is "a" and common difference of two consecutive term is "d" then last term( Tm) is given by:
Tm = a + (m - 1)d
Note :
(1) If d is Positive, AP terms will grow towards infinity
(2) If d is Negative, AP terms will grow towards negative infinity

(D) Types of Arithmetic Progression:-
There are two type of Arithmetic Progression
(1) Finite Arithmetic Progression: An Arithmetic Progression containing a finite number of terms is called finite Arithmetic Progression. A finite AP has a last term.
For example: 4, 8, 12, 16, 20, .... 36, 40

(2) Infinite Arithmetic Progression : An Arithmetic Progression which does not have a finite number of terms is called infinite Arithmetic Progression. Such APs do not have a last term.
1, 2, 3, 4, 5, 6, 7, 8, 9, ....................
(E) mth term of an AP from the end :-
Let an AP have total numbers of term is "n"
So mth term of AP from end = (n-m +1)th term of AP from the beginning

Let me verify the above the above concept
Let an AP has total number of term = 10 like as below
A1, A2, A3, A4, A5, A6, A7, A8, A9, A10
So,
2nd term from end = A9 = 9th term from beginning = (10-2+1)th term from beginning
3rd term from end = A8 = 8th term from beginning = (10-3+1)th term from beginning
6nd term from end = A5 = 5th term from beginning = (10-6+1)th term from beginning
8th term from end = A3 = 3rd term from beginning = (10-8+1)th term from beginning

(F) Arithmetic Series:-
The arithmetic series is the sum of all the terms of the arithmetic sequence.
The arithmetic series is in the form of
{a + (a + d) + (a + 2d) + (a + 3d) + .................}

(G) Sum of n terms of an Arithmetic Series:
Let an AP has n terms
first term = a
Last term = l = a + (n -1)d
Sum, Sn = n/2( a + l)
Sn = n/2( 2a+(n-1)d)
(H) Important Note :
1. The sum of the infinite arithmetic sequence does not exist.
2. Sum of first n natural numbers
= n(n+1)/2
3. Sum of first n odd natural number = n^2
4. Sum of first n even natural numbers = n(n+1)
5. Sum of Square of first n natural numbers = n(n+1)(2n+1)/6
6. Sum of Cubes of first n natural numbers = [ n(n+1)/2]^2
7. sum of the fourth power of n natural numbers
= n(n +1)(2n +1)( 3n^2 +3n -1)/30
8. If rth term of an AP is
Tr = Ar^3 + Br^2 + Cr + D
then
Sum of n terms of that AP will be
Sn = Summation r =1 to r =n (Tr)
9. If in an AP pth term is q and qth term is p , then rth term will be = p + q - r
10. If in an AP sum of p terms is q and sum of q terms is p so, sum of (p+q) terms will be = -(p+q)
11. If in an AP sum of p terms is equal to sum of q terms then sum of (p +q) term will be Zero (0)
12. For an Arithmetic Series if sum of n terms given as Sn so we can calculate nth term Tn as Tn = Sn - Sn-1
13. Common Difference(d) of an AP can be calculated by d = S2 - 2S1
where S2 = Sum of first 2 terms of that AP
S1 = Sum of First term of that AP

14. Sum of infinite Arithmetic progression is infinity if d is > 0 and Sum of Infinite Arithmetic progression - Infinity if d is <0

15. If sum of n terms of an AP is in the form
Sn = An^2 + Bn +C ( Quadratic form )
So Common Difference(d) will be = 2 × Coefficient of n ^2
d = 2A

16. If nth term Tn of an AP is in the form of
Tn = Pn + Q ( Linear Expression )
Common difference(d) = Coefficient of n = P
17. Let there are two AP, AP1 and AP2
Let sum of n terms of AP1 = S1n = f1(n)
Let sum of n terms of AP2 = S2n = f2(n)
And if
S1n/S2n = f1(n)/f2(n)
Then
T1(n)/T2(n) = f1(2n-1)/f2(2n-1)
Here
T1n = nth term of AP1
T2n = nth term of AP2

18. Let if T1n/T2n = f1(n)/f2(n)
Then S1n / S2n = f1[(n+1)/2]/f2[(n+1)/2]

(I) Arithmetic Mean
(1) Arithmetic mean between two numbers p and q will be AM = (p +q)/2
p , AM, q .... are in AP
AM - p = q - AM
2AM = p + q
AM = (p +q)/2
(2) Arithmetic mean of n numbers a1, a2, a3, .....an
AM = (a1 + a2 + a3 + .... an)/n

(J) n Arithmetic Mean between two numbers p and q
(1)
p, A1, A2, A3, ........ An, q are in AP
First term(a)= p
Total number of term(N) = n +2
Last term(Tn) = q
So,
Tn = a + (N-1)d
d = Common Difference
q = p + (n +2 -1)d
q - p = (n+1)d
d = (q-p)/(n+1)

A1 = p + d = p + (q-p)/(n+1)
A2 = p + 2d = p + 2(q-p)/(n+1)

.........
...
..
An = p + nd = p + n(q-p)/(n+1) = q - d
(2) Sum of n AM between p and q
A1 + A2 + A3 .....+ An = n(p+q)/2
(3) Sum of r AM between p and q
A1 + A2 + A3 ...... + Ar = r(p+q)/2
(4) Sum of n AM between p and q / Sum of r AM between p and q = [n(p+q)/2]/[r(p+q)/2] = n/r
(5) Select 3 terms in AP
a - d , a , a + d
(6) Select 5 terms in AP
a - 2d, a - d, a, a +d, a + 2d
(7) Select 7 terms in AP
a - 3d, a - 2d, a - d, a, a +d, a + 2d, a + 3d
(8) Select 4 terms in AP
a - 3d, a - d, a + d, a + 3d

(9) Select 6 terms in AP
a - 5d, a - 3d, a - d, a + d, a + 3d, a + 5d
(10) Select 8 terms in AP
a - 7d, a - 5d, a - 3d, a - d, a + d, a + 3d, a + 5d, a + 7d
(11) For a series, if mth term in the form of T(m) = Cm + D, Then this Series is in Arithmetic Progression
(12) For a series if sum of m terms in the form of S(m) = Cm^2 + Dm + E, then this Series will be in AP
(13) T(m) = Cm + D, = mth term of a Series ( AP )
Then T(m/k), T(mk), T(m+k), T(m-k) ..Will be in AP k is a non zero constant
(14) If each term of an AP will multiplied, divided, Increased or Decreased by a non zero constant k .. then resulting Sequence will also be in AP
(15 ) For a Series if sum of m terms is S(m) = Cm^2 + Dm + E,

Then S(m/k), S(mk), S(m + k ), S(m - k) will be in AP, k is a non zero constant
(16) If A1, A2, A3, A4, Are in AP
Then 1/A1, 1/A2, 1/A3, 1/A4 are in HP (Harmonic Progression)
(17) The sum of any two terms equidistant from the beginning and end of an Arithmetic progression is always constant. Furthermore, their sum is always equal to the sum of the first and last terms.

Sequences and series
(A) Arithmetic Progression
A list of numbers in a specific order is known as a sequence or progression.
Let
f(n) = 4n + 1
For n = 1, f(n) = 5
For n = 2, f(n) = 9
For n = 3, f(n) = 13
For n = 4, f(n) = 17
So sequence or progression will be
5, 9, 13, 17......
An Arithmetic Progression or Arithmetic sequence (AP) is a sequence of numbers  such that the difference(Common Difference of AP) between any two consecutive terms is constant.

Given progression is in AP
A1, A2, A3, A4, A5, A6, A7, A8
If and only if
A2-A1= A3-A2 = A4-A3 = A5-A4 = A6-A5 = A7-A6 = A8 - A7
Note : Each number in the sequence is known as term.
Example 1
2, 4, 6, 8, 10, 12, 14, ....................40
Here
4-2 = 6-4 = 8- 6 = 10-8 = 2
(B) Formula For Common Difference(d) of AP
Let any AP
A1, A2, A3, A4, A5, ........An-1, An, An+1 ......

d = A2 - A1 = A3 - A4 = An - An-1
So Common difference d of an AP is calculated by
d = (n)th term - (n -1)th term
Or
d = (n+1)th term - (n)th term

Example 2
Let first term of an AP = a
Let Common Difference ( Difference between two consecutive terms of an AP) = d
So,
First term = a = a + (1-1)d
Second term = a + d = a + (2-1)d
Third term = a + d + d = a + (3-1)d
Fourth term = a +2d + d = a + (4-1)d
Fifth term = a + 3d + d = a + (5-1)d
....... ......
........ .........
........ ........
nTh term , Tn = a + (n - 1)d
So Arithmetic Progression can be written as
Or General form of an AP written as
a, a +d, a + 2d, a + 3d, a + 4d, a + 5d, .........
.......... a + (n - 1)d , a + n d , a +(n +1)d ... .. infinity
Here,
a = Fist term of AP
d = Common Difference between two consecutive terms of AP
n = Number of terms of AP
Tn = nth term of AP = a + (n - 1)d
(C) nth Term of Arithmetic Progression
Tn = a + (n - 1)d
Here,
a = Fist term of AP
d = Common Difference between two consecutive terms of AP
n = Number of terms of AP

If the initial term of an arithmetic progression is "a" and the common difference of successive members is "d' then the nth term (Tn) of the sequence is given by:
Tn = a + (n - 1)d
If there are m terms in the AP, if first term is "a" and common difference of two consecutive term is "d" then last term( Tm) is given by:
Tm = a + (m - 1)d
Note :
(1) If d is Positive, AP terms will grow towards infinity
(2) If d is Negative, AP terms will grow towards negative infinity

(D) Types of Arithmetic Progression
There are two type of Arithmetic Progression
(1) Finite Arithmetic Progression: An Arithmetic Progression containing a finite number of terms is called finite Arithmetic Progression. A finite AP has a last term.
For example: 4, 8, 12, 16, 20, .... 36, 40

(2) Infinite Arithmetic Progression : An Arithmetic Progression which does not have a finite number of terms is called infinite Arithmetic Progression. Such APs do not have a last term.
1, 2, 3, 4, 5, 6, 7, 8, 9, ....................
(E) mth term of an AP from the end
Let an AP have total numbers of term is "n"
So
mth term of AP from end = (n-m +1)th term of AP from the beginning

Let me verify the above the above concept
Let an AP has total number of term = 10 like as below
A1, A2, A3, A4, A5, A6, A7, A8, A9, A10
So,
2nd term from end = A9 = 9th term from beginning = (10-2+1)th term from beginning
3rd term from end = A8 = 8th term from beginning = (10-3+1)th term from beginning
6nd term from end = A5 = 5th term from beginning = (10-6+1)th term from beginning
8th term from end = A3 = 3rd term from beginning = (10-8+1)th term from beginning

(F) Arithmetic Series
The arithmetic series is the sum of all the terms of the arithmetic sequence.

The arithmetic series is in the form of
{a + (a + d) + (a + 2d) + (a + 3d) + .................}
(G) Sum of n terms of an Arithmetic Series
Let an AP has n terms
first term = a
Last term = l = a + (n -1)d
Sum, Sn = n/2( a + l)
Sn = n/2( 2a+(n-1)d)
(H) Important Note
1. The sum of the infinite arithmetic sequence does not exist.
2. Sum of first n natural numbers
= n(n+1)/2
3. Sum of first n odd natural number = n^2
4. Sum of first n even natural numbers = n(n+1)
5. Sum of Square of first n natural numbers = n(n+1)(2n+1)/6
6. Sum of Cubes of first n natural numbers = [ n(n+1)/2]^2
7. sum of the fourth power of n natural numbers
= n(n +1)(2n +1)( 3n^2 +3n -1)/30
8. If rth term of an AP is
Tr = Ar^3 + Br^2 + Cr + D
then
Sum of n terms of that AP will be
Sn = Summation r =1 to r =n (Tr)
9. If in an AP pth term is q and qth term is p , then rth term will be = p + q - r
10. If in an AP sum of p terms is q and sum of q terms is p so, sum of (p+q) terms will be = -(p+q)
11. If in an AP sum of p terms is equal to sum of q terms then sum of (p +q) term will be Zero (0)
12. For an Arithmetic Series if sum of n terms given as Sn so we can calculate nth term Tn as Tn = Sn - Sn-1
13. Common Difference(d) of an AP can be calculated by d = S2 - 2S1
where S2 = Sum of first 2 terms of that AP
S1 = Sum of First term of that AP

14. Sum of infinite Arithmetic progression is infinity if d is > 0 and Sum of Infinite Arithmetic progression - Infinity if d is <0

15. If sum of n terms of an AP is in the form
Sn = An^2 + Bn +C ( Quadratic form )
So Common Difference(d) will be = 2 × Coefficient of n ^2
d = 2A

16. If nth term Tn of an AP is in the form of
Tn = Pn + Q ( Linear Expression )
Common difference(d) = Coefficient of n = P
17. Let there are two AP, AP1 and AP2
Let sum of n terms of AP1 = S1n = f1(n)
Let sum of n terms of AP2 = S2n = f2(n)
And if
S1n/S2n = f1(n)/f2(n)
Then
T1(n)/T2(n) = f1(2n-1)/f2(2n-1)
Here
T1n = nth term of AP1
T2n = nth term of AP2

18. Let if T1n/T2n = f1(n)/f2(n)
Then S1n / S2n = f1[(n+1)/2]/f2[(n+1)/2]

(I) Arithmetic Mean
(1) Arithmetic mean between two numbers p and q will be AM = (p +q)/2
p , AM, q .... are in AP
AM - p = q - AM
2AM = p + q
AM = (p +q)/2
(2) Arithmetic mean of n numbers a1, a2, a3, .....an
AM = (a1 + a2 + a3 + .... an)/n

(J) n Arithmetic Mean between two numbers p and q
(1)
p, A1, A2, A3, ........ An, q are in AP
First term(a)= p
Total number of term(N) = n +2
Last term(Tn) = q
So,
Tn = a + (N-1)d
d = Common Difference
q = p + (n +2 -1)d
q - p = (n+1)d
d = (q-p)/(n+1)

A1 = p + d = p + (q-p)/(n+1)
A2 = p + 2d = p + 2(q-p)/(n+1)

.........
...
..
An = p + nd = p + n(q-p)/(n+1) = q - d
(2) Sum of n AM between p and q
A1 + A2 + A3 .....+ An = n(p+q)/2
(3) Sum of r AM between p and q
A1 + A2 + A3 ...... + Ar = r(p+q)/2
(4) Sum of n AM between p and q / Sum of r AM between p and q = [n(p+q)/2]/[r(p+q)/2] = n/r
(5) Select 3 terms in AP
a - d , a , a + d
(6) Select 5 terms in AP
a - 2d, a - d, a, a +d, a + 2d
(7) Select 7 terms in AP
a - 3d, a - 2d, a - d, a, a +d, a + 2d, a + 3d
(8) Select 4 terms in AP
a - 3d, a - d, a + d, a + 3d

(9) Select 6 terms in AP
a - 5d, a - 3d, a - d, a + d, a + 3d, a + 5d
(10) Select 8 terms in AP
a - 7d, a - 5d, a - 3d, a - d, a + d, a + 3d, a + 5d, a + 7d
(11) For a series, if mth term in the form of T(m) = Cm + D, Then this Series is in Arithmetic Progression
(12) For a series if sum of m terms in the form of S(m) = Cm^2 + Dm + E, then this Series will be in AP
(13) T(m) = Cm + D, = mth term of a Series ( AP )
Then T(m/k), T(mk), T(m+k), T(m-k) ..Will be in AP k is a non zero constant

(14) If each term of an AP will multiplied, divided, Increased or Decreased by a non zero constant k .. then resulting Sequence will also be in AP
(15 ) For a Series if sum of m terms is S(m) = Cm^2 + Dm + E,
Then S(m/k), S(mk), S(m + k ), S(m - k) will be in AP, k is a non zero constant
(16) If A1, A2, A3, A4, Are in AP
Then 1/A1, 1/A2, 1/A3, 1/A4 are in HP (Harmonic Progression)
(17) The sum of any two terms equidistant from the beginning and end of an Arithmetic progression is always constant. Furthermore, their sum is always equal to the sum of the first and last terms.

JEE Main 2023 Schedule

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