## Fast revision(Math)

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

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

## Chapter 1: Relation and function

### EXERCISE 1.1(Relation)

**1. Relation:** If A and B are two non empty sets, then any subset R of is called relation from set A to set B.

i.e.

For example: Let A = {1,2},B={3,4}

Then A×B = {(1,3),(1,4),(2,3),(2,4)}

A subset is called relation from A to B.

Similarly, other subsets of are also relation from A to B.

**Domain:-** Domain of R is the set of all first coordinates of elements of R and it is denoted by Dom(R)

**Range:-** Range of R is the set of all second coordinates of R and it is denoted by range (R).

**NOTE:-** A relation R on set A mens, the relation from A to A i.e.,

**Some standard relation:-**

**Let A be a non empty set. then, a relation R on set A ,Then**

**(a) Reflexive :** If for each element

**(b)Symmetric:** If for all

**(c) Transitive:** If and for all

**Equivalence Relation :-** Any relation R on a set A is said to be an equivalence relation if R is reflexive,symmetric and transitive.

**Equivalence class:** Let R be an equivalence relation on a non-empty set A. For all , the equivalence class of “a” is defined as the set of all such elements of A which are related to “a” under R. It is denoted by [a].

i.e. [a] = equivalence class of ‘a’

For example, Let A = {1, 2, 3} and R be the equivalence relation on A given by

R = {(1,1),(2,2),(3,3),(1,2),(2,1)}

The equivalence class are

[1]=equivalence class of 1 = {1,2}

[2] = {1,2} ,[3] ={3}

Hence [1]=[2]

### Exercise 1.2

**Function:-** A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B.

In other words, a function f is a relation from a non-empty set A to a non-empty

set B such that the domain of f is A and no two distinct ordered pairs in f have the

same first element.

If f is a function from A to B and , then f (a) = b, where b is called the image of a under f and a is called the preimage of b under f.

**Types of functions:**

**One-one(Injective):-** A function is defined to be one-one (or injective), if the images

of distinct elements of X under f are distinct,

Let

.

**Many-one: – **A function is defined to be many-one , if the images of distinct elements of X under f are Same,

Let

.

**Onto(Surjective):-**A function is said to be onto (or surjective), if every element of Y is the image of some element of X under f, i.e., for every , there exists an element x in X such that f (x) = y.

**One-one and Onto(Bijective):-** A function is said to be bijective,if f is both one-one and onto.

**Number of function:**

**(i) **If X and Y two finite sets having m and n elements respectively then the number of functions from X to Y is

**(ii)** If , then number of one-one functions from A to B =

**(iii)** If , then number of onto functions from A to B

## Chapter 2: Inverse trigonometric function

### EXERCISE 2.1

**1.** should not be confused with . In fact and similarly for other trigonometric functions.

**2.** Whenever no branch of an inverse trigonometric functions is mentioned, we mean the principal value branch of that function.

**3.** The value of an inverse trigonometric functions which lies in the range of principal branch is called the principal value of that inverse trigonometric functions.

**The following table gives the inverse trigonometric function (principal value branches) along with their domains and ranges**

### EXERCISE 2.2

**Inverse trigonometric formulae**

**(1) (a)** or

**(b)** or

**(c)**

**(2) (a)**

**(b)**

**(c)** i

**(3) (a)**

**(b)** ;

**(c)**

**(4) (a)**

**(b)**

**(c)** i

**(5) (a)**

**(b)**

**(6) (a)** i

**(b)**

**(c)**

**(d)**

**(d)**

**(e)**

**(7) (a)**

**(b)**

**(c)**

**(d)**

**8. Important substitution to simply trigonometrical expression involving inverse trigonometrical functions.**

## Chapter 3:-

**Exercise 3.1**

**Matrix:-** A matrix is a recangular arrangement of numbers or functionsarranged into a fixed number of row and column.

Each entry in a amtrix is called an element of the matrix

Order of matrix:- Row column

Name of elements in matrix

Let

**Row matrix:** A matrix having one row is called row matrix.

**EX:-**

**Column matrix:-** A matrix having only one column is called column matrix.

**EX:-**

**Square matrix:** A matrix in which the number of rows is equal to the the no of column, called a square matrix.

**Diagonal matrix:-** A square matrix is said to be a diagonal matrix for .

**Ex:**

We can write this matrix as

**Scalar matrix:-** A square matrix is said to be scalar matrix and entries on its principal diagonal are equal.

(i) (ii) where .

**Ex:-**

**Identity matrix:** A sqare matrix in which all non diagonal elements are zero and all diagonal elements are equal to 1 is called identity matrix.

(i) (ii)

**EX:-**

**Null matrix or Zero matrix:-** A matrix whose elements are zero is called a null matrix or a zero matrix

**Ex:-**

**Equality of matrices**

**Definition :** Two matrices and are said to be equal if

**(i)** they are of the same order

**(ii)** each element of A is equal to the corresponding element of B, that is for all i and j..

For example, and are equal matrices

## Exercise 3.2

**NOTE:- (i)**matrix addition follow commutative law

**(ii)** Matrix multiplication does not follow commutative law

**(ii)** We know that, for real numbers a, b if ab = 0, then either a = 0 or b = 0. This need not be true for matrices,

**Matrix multiplication:**

**Possibility of multiplication of two matrix**

Let A and B are two matrices whose orders rae and

multiplication of two matrices is possible if . its mean post multiplier of A should be equal to pre multiplier of B.

### Exercise 3.3

**Transpose of a Matrix**

**Definition :-**If be an m × n matrix, then the matrix obtained by interchanging the rows and columns of A is called the transpose of A. Transpose of the matrix A is

denoted by or . In other words, if , then .

**For example:**

Then

Properties of transpose of the matrices

For any matrices A and B of suitable orders, we have

**(i)** , (ii) (where k is any constant)

**(iii)** (iv)

**(iv)**

**Symmetric and Skew Symmetric Matrices**

**Symmetric Matrices:-** A square matrix is said to be symmetric if , that is, for all possible values of i and j.

**Skew Symmetric Matrices:-** A square matrix is said to be skew symmetric matrix if , that is for all possible values of i and j.

**Some special points:-**

**(i)** All main diagonal elements of a skew- symmetric matrix are zero.

**(ii)** Every square matrix can be uniquely expressed as the sum of a symmetric and skew symmetric matrix.

**(iii)** All positive integral powers of a symmetric matrix are symmetric matrix.

**(iv)** All odd positive integral powers of a skew-symmetric matrix are skew symmetric matrix.

**(v)** Any square matrix can be expressed as the sum of a symmetric and a skew symmetric matrix.

is symmetric matrix

is skew symmetric matrix

**(vi)** If A and B are two square matrices such that , then B is the inverse matrix of A and is denoted by and A is the inverse of B.

**(vii)** Inverse of a square matrix, if it exists, is unique.

## Chapter 4: Determinants

### Exercise 4.1

**Determinant:** Every square matrix can be associated to an expression or a number which is known as its determinant.

If , then determinant of A is written as

**Remarks: (i)** For matrix A, |A| is read as determinant of A and not modulus of A.

**(ii)** Only square matrices have determinants.

**Determinants of order two:-**

**Determinants of order three:-**

**(i)** There are six ways of expanding a determinant of order

3 corresponding to each of three rows and three columns giving the same value.

**(ii)** In general, if where A and B are square matrices of order n, then , where

Some property of determinants:

**(i)** The value of the determinant remains unchanged if its rows and columns are interchanged.

**(ii)** If any two rows (or columns) of a determinant are identical (all corresponding elements are same), then value of determinant is zero.

**(iii)** If all the elements of any row or any coloumn are zero then the value of determinant of the will be zero.

**Area of a Triangle: – **In earlier classes, we have studied that the area of a triangle whose vertices are , is given by the expression

**Now this expression can be written in the form of a determinant as**

**Remarks**

**(i)** Since area is a positive quantity, we always take the absolute value of the determinant

**(ii)** If area is given, use both positive and negative values of the determinant for calculation.

**(iii)** The area of the triangle formed by three collinear points is zero.

**(iv)** If A is a skew symmetric matrix of odd order, then

**(v)** The determinant of a skew symmetric matrix of even order is a perfect square.

**Minors factor:-** Minor of an element of a determinant is the determinant obtained by deleting its row and column in which element lies. Minor of an element is denoted by Mij.

**Remark:** Minor of an element of a determinant of order is a determinant of order

**Cofactor:-** Cofactor of an element , denoted by is defined by , where is minor of .

Let

Let

Then

Hence = sum of the product of elements of any row (or column) with their corresponding cofactors.

If elements of a row (or column) are multiplied with cofactors of any other row (or column), then their sum is zero. For example,

Adjoint of a matrix : The adjoint of a square matrix is defined as the transpose of the matrix , where Aij is the cofactor of the element . Adjoint of the matrix A is denoted by .

Let

**Remarks:-** If A be any given square matrix of order n, then

, where I is the identity matrix of order n

**Singular matrix:** A square matrix A is said to be singular if .

**Some property:**

**1:-** If A and B are nonsingular matrices of the same order, then AB and BA are also nonsingular matrices of the same order.

**2:-** The determinant of the product of matrices is equal to product of their respective determinants, that is, , where A and B are square matrices of the same order

**3:-** If A is a square matrix of order n, then .

**4:-** A square matrix A is invertible if and only if A is nonsingular matrix.

**5:-** A is invertible matrix

**6:-**

**7:-**

**8:-**

**9:-**

**10:-**

**11:- **

**12:- **

**13:-**

**14:- **

**15:-** Every invertible matrix possesses a unique inverse.

**16:- ** A square matrix is invertble, if it is non singular.

**17:- ** The inverse of an invertible symmetric matrix is a symmetric matrix.

## EXERCISE 4.6

**Consistent system:-** A system of equations is said to be consistent if its solution (one or more) exists.

**Inconsistent system:-** A system of equations is said to be inconsistent if its solution does not exist.

System of linear equation:-

——————–

**Case II:–** If A is a singular matrix, then .

In this case, we calculate (adj A) B.

If , (O being zero matrix), then solution does not exist and the

system of equations is called inconsistent.

If , then system may be either consistent or inconsistent according as the system have either infinitely many solutions or no solution.

## Chapter 5: Differentiation:-

## Exercis 5.1

**Condition for Continuity:-** A function is said to be continuaous at a point a of its domain if

Every constant function is continuaous function.

Every polynomial function is continuous function.

Identity function is continuous function.

Every logrithmic and exponential function is a continuous function.

**Some formulae for continuity:-**

**(i)**

**(ii)**

**(iii)**

**(iv)**

**(v)**

**(vi)**

**(v)**

**Formulae for differentiation**

**(1)**

**(2)**

**(3)**

**(4)**

**(5)**

**(6)**

**(7)**

**(8)**

**(9)**

**(10)**

**(11)**

**(12)**

**(13)**

**(14)**

**(15)**

**Product rule**

**(16)**

**Divide rule**

**(17)**

**Formulae for Differentiability**

If then the function will be differentiable

## Exercise 5.3

**Explicit function:**When a relationship between x and y is expressed in a way that it is easy to solve for y and write y = f(x), we say that y is given as an explicit function of x.

**Example:**

**Implicit function:** When a relationship between x and y is expressed in a way that it can not solve for y and write y = f(x), we say that y is given as an implicit function of x.

**Example:**

**Some differentiation formulae for inverse trigonometry function:**

**(1)**

**(2)**

**(3)**

**(4)**

**(5)**

**(6)**

### Exercise 5.4 & 5.5

**Differentiation of logrithmic function:**

**(1)**

**(2)**

**(3)**

**(4)**

**(5)**

**(6)**

### Exercise 5.6

**Derivatives of Functions in Parametric Forms**

### Exercise 5.7

**Second order derivative**

## Chapter 6: Application of derivative

### Exercise 6.1

**Rate of Change of Quantities:**

Rate change with respect to time

Rate change with respect to x

is positive if y increases as x increases and

is negative if y decreases as x increases.

### Exercise 6.2:

**Increasing and Decreasing Functions**

**Definition 1:-** Let I be an interval contained in the domain of a real valued function f.

Then f is said to be

**(i)** increasing on I if in for all

**(ii)** decreasing on I, if in for all

**(iii)** constant on I, if for all x in I, where c is a constant.

**Theorem 1:-** Let f be continuous on and differentiable on the open interval Then

**(a)** f is increasing in if for each

**(b)** f is decreasing in if for each

**(c)** f is a constant function in if for each

### Exercise 6.5: Maxima and Minima

First Derivative Test: Let f be a function defined on an open interval I.

Let f be continuous at a critical point c in I. Then

**(i)** If f’ (x) changes sign from positive to negative as x increases through c, i.e., if f ‘(x) > 0 at every point sufficiently close to and to the left of c, and f ‘(x) < 0 at every point sufficiently close to and to the right of c, then c is a point of local maxima.

**(ii)** If f ‘(x) changes sign from negative to positive as x increases through c, i.e., if f ‘(x) < 0 at every point sufficiently close to and to the left of c, and f ‘(x) > 0 at every point sufficiently close to and to the right of c, then c is a point of local minima.

**(iii)** If f ‘(x) does not change sign as x increases through c, then c is neither a point of local maxima nor a point of local minima. Infact, such a point is called point of inflection

**Note:-** If c is a point of local maxima of f , then f (c) is a local maximum value of f. Similarly, if c is a point of local minima of f , then f(c) is a local minimum value of f.

**Second Derivative Test:-** Let f be a function defined on an interval I and Let f be twice differentiable at c. Then

**(i)** x = c is a point of local maxima if and The value f (c) is local maximum value of f .

**(ii)** x = c is a point of local minima if and In this case, f (c) is local minimum value of f .

**(iii)** The test fails if and In this case, we go back to the first derivative test and find whether c is a point of local maxima, local minima or a point of inflexion.

**Note:-** As f is twice differentiable at c, we mean second order derivative of f exists at c.