# Fast revision(Math)

class 12 revision of cbse math part-I 2022-2023

Class 12 revision of cbse math part-II 2022-2023

## Integration: Chapter 7

### Exercise 7.1

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

Methods of Integration:-
1. Integration by Substitution
2. Integration using Partial Fractions
3. Integration by Parts

### Exercise 7.2

Integration by substitution

⇒ Some other formulae:

(1)

(2)

(3)

(4)

### Exercise 7.3

Integration using trigonometric identities

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)(a)

(b)

(c)

(d)

(11) (a)

(b)

(c)

(d)

### Exercise 7.4

Integrals of Some Particular Functions

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

From (7) to (10) types form using completing the square

for completing the square add and subtract by

### Exercise 7.5

Integration by Partial Fractions:

⇒ Partial fraction expression:

(1)

(2)

(3)

(4)

(5)

### Exercise 7.6

Integration by Parts

Formula

(1)

ILATE

I= inverse tigonometry()

L= logrithmic function()

A= Algebraic function

T = trigonometric function

E= Exponential function

(2)

### Exercise 7.7

Integrals of some more types

(1)

(2)

(3)

### Exercise 7.9 & 7.10

Evaluation of Definite Integrals by Substitution

### Exercise 7.11

Some Properties of Definite Integrals

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

## Chapter 8

Area under Simple Curves

Area of region when strip is perpendicular to x-axis

Area of region when strip is perpendicular to x-axis

## Exercise 8.2

Area between Two Curves when

## Chapter 9(Differential equation): Exercise 9.1:

Order of a differential equation:-Order of a differential equation is defined as the order of the highest order derivative of the dependent variable with respect to the independent variable involved in the given differential equation.

Ex:-

Order of d.e. = 2

Degree of a differential equation:-By the degree of a differential equation, when it is a polynomial equation in
derivatives, we mean the highest power (positive integral index) of the highest order derivative involved in the given differential equation.

EX:-

Degree of d.e. = 3

NOTE:- Order and degree (if defined) of a differential equation are always positive integers.

### Exercise 9.2

General solution:The solution which contains arbitrary constants is called the general solution(primitive) of the differential equation.

particular solution:-The solution free from arbitrary constants i.e., the solution obtained from the general solution by giving particular values to the arbitrary constants is called a particular solution of the differential equation.

Methods of Solving First Order, First Degree Differential Equations

### Exercise 9.4

Differential equations with variables separable

### Exercise: 9.5

Homogeneous differential equations:- In homogeneous differential equation substitute or

### Exercise 9.6

Linear differential equations:-

(1) (i) Write the given differential equation in the form where P, Q are constants or functions of x only.
(ii) Find the Integrating Factor

(iii) Solution of differential equation of this form is

(2) (i) Write the given differential equation in the form where P, Q are constants or functions of x only.
(ii) Find the Integrating Factor

(iii) Solution of differential equation of this form is

## Vector algebra:

### Exercise 10.1

Let The point P(x, y, z)

Position vector of P

dr’s of line OP = (x, y, z)

dc’s of the line OP

Relation between dc’s

### Types of Vectors:

(1) Zero Vector: A vector whose initial and terminal points coincide, is called a zero vector (or null vector)

Ex:-

(2) Unit Vector: A vector whose magnitude is unity (i.e., 1 unit) is called a unit vector. The unit vector in the direction of given vector is denoted by .

(3) Coinitial Vectors: Two or more vectors having the same initial point are called coinitial vectors.

(4) Collinear Vectors :Two or more vectors are said to be collinear if they are parallel to the same line, irrespective of their magnitudes and directions.

(5) Equal Vectors:  Two vectors and are said to be equal, if they have the same magnitude and direction regardless of the positions of their initial points, and written
as .

(6) Negative of a Vector :-A vector whose magnitude is the same as that of a given vector(say, ), but direction is opposite to that of it, is called negative of the given vector.
For example, vector is negative of the vector , and written as .

### Exercise 10.2

Let

(i) Modulus of

(ii) Unit vector:-

Components of a vector:

Let

Scalar component of

Vector component of

Remarks:
and

(i) Let two vectors and are collinear vector

(ii)  If , then are also called direction ratios of .

(iii) In case if it is given that l, m, n are direction cosines of a vector, then is the unit vector in the direction of that vector, where and are the angles which the vector makes with x, y and z axes respectively.

Vector joining two points:

If and are any two points, then the vector joining and is the
vector

Then,

Magnitude of is

Section formula: P and q are two points on line and point R divide PQ in m:n Then

And the mid point

Product of Two Vectors

Scalar (or dot) product of two vectors: The scalar product of two nonzero vectors and , denoted by

Observation: (1) Let and be two nonzero vectors, then if and only if and are perpendicular to each other. i.e.

(2) If then,

In particular, in this case

(3) If then,

(4)

(5)

(6) The scalar product is commutative, i.e.

(7) If and

Then

Projection of a vector on a line

Projection of a vector on other vector , is given by

or or

Remarks:

Vector (or cross) product of two vectors

(1) is vector.

(2) Let and be two non zero vectors, then if and only if and are parallel(or collinear)to each other i.e.

In Particular

(3). If and .

(4)

(5)

(6)

(7)

(8)

(9) If and are the adjacent sides of a triangle then its area is given as

(10) If and are the adjacent sides of a parallelogram then its area is given as

(11) If and

## Three dimension geometry

Direction cosine: If makes angles with x, y, and z-axis respectively

Relation between direction cosine of a line

Direction cosines of a line passing through two points:

Let and

Then

Equation of a Line in Space:

(i) it passes through a given point and has given direction,

(ii) it passes through two given points.

Equation of a line through a given point and parallel to a given vector

vector form:

Cartesian form:

If point and direction ratio =

Note: –If l, m, n are the direction cosines of the line, the equation of the line is

Equation of a line passing through two given points:

Two points are and then equation of line

In cartesian form:

Two points and

Equation of line

Angle between Two Lines:

and angle between them is

Or

(i) If the lines are perpendicular,

or

(ii) If two lines are parallel or

Shortest Distance between Two Lines

(i) Distance between two skew lines

Shortest distance=

In cartesian form:

(ii) Distance between parallel lines

Distance

## Linear Programming Problems

Constraints

Objective function;

are called decision variables

Max or min value of Z is called optimal value

(i)

(ii)

(iii)

(iv)

(v)

(vi)

## Exercise 13.1

### Conditional Probability:

Properties of conditional probability:- Let E and F be events of a sample space S of an experiment, then we have

(a)

(b)

(c)

### Exercise 13.2

Multiplication Theorem on Probability:-

(i)

(ii)

Independent Events:

Bayes’ Theorem:- and are three events and A be a non zero events

Theorem of total probability:-

Mean

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