### 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.

*Find the Integrating Factor*

**(ii)*** (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:*

** (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 $\vec{b}$**

**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

## Probability:-

**(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**