Ts Inter Maths IA Concept

maths IA feature image head

This note is designed by ‘Basics in Maths’ team. These notes to do help the TS intermediate first year Math students fall in love with mathematics and overcome the fear.

These notes cover all the topics covered in the TS I.P.E  first year maths 1A syllabus and include plenty of formulae and concept to help you solve all the types of Inter Math problems asked in the I.P.E and entrance examinations.

1.Functions

Set: A collection of well-defined objects is called a set.

Ordered pair: Two elements a and b listed in a specific order form. An ordered pair denoted by (a, b).

Cartesian product: Let A and B are two non- empty sets. The Cartesian product of A and B is denoted by A × B and is defined as a set of all ordered pairs (a, b) where a ϵ A and b ϵB

                           cartesion product              

Relation: Let A and B are two non-empty sets the relation R from A to B is a subset of A×B.

  ⇒ R: A→B is a relation if  R⊂ A × B

Function:

A relation f: A → B is said to be a function if ∀ aϵ A there exists a unique element b such that (a, b) ϵ f.                                                                         

(Or) 

A relation f: A → B is said to be a function if 

(i) x ϵ A ⇒ f(x) ϵ B 

(ii)  x1 , x2 ϵ A , x1 = x2 in A  ⇒ f(x1) = f(x2) in B.

 Note:   If A , B are two finite sets then the no. of   functions that can be defined from A to B is  n(B)n(A)

VARIOUS TYPES OF FUNCTIONS

One– one Function (Injective):- A function f: A→ B is said to be a one-one function or injective if different elements in A have different images in B.

(Or)

A function f: A→ B is said to be one-one function if f(x1) = f(x2) in B ⇒ x1 = x2 in A.

Note: No. of one-one functions that can be defined from A into B is n(B) p n(A)   if  n(A) ≤ n(B)

On to Function (Surjection): – A function f: A→ B is said to be onto function or surjection if for each yϵ B ∃ x ϵ A such that f(x) =y

onto function

Note: if n(A) = m and n(B) = 2 then no. of onto functions = 2m – 2

 Bijection: – A function f: A→ B is said to be Bijection if it is both ‘one-one’ and ‘onto’.

bijection

Constant function:  A function f: A→ B is said to be constant function if f(x) = k ∀ xϵA

Identity function:  Let A be a non-empty set, then the function defined by IA : A → A, I(x)=x is called identity function on A.

Equal function:  Two functions f and g are said to be equal if

(i)   They have same domain (D)

(ii)  f(x) = g(x) ∀ xϵ D

Even function:  A function f: A→ B is said to be even function if f (- x) = f(x) ∀ xϵ A

Odd function:   A function f: A→ B is said to be odd function if f (- x) = – f(x) ∀ xϵ A

Composite function:  If f: A→B, g: B→C are two functions then the composite relation gof is a function from A to C.

composite function

gof: A→C is a composite function and is defined by gof(x) = g(f(x)).

Step function:  A number x = I + F

 I → integral part    = [x]

F → fractional part = {x}

∴ x = [x] + {x}

step functionIf y = [x] then domain = R and

Range = Z                                                 

0 ≤ x ≤ 1, [x] = 0

1≤ x ≤ 2, [x] = 1

-1 ≤ x ≤ 0, [x] = -1

If k is any integer [ x + k] = k + [x]

The value of [x] is lies in x – 1 < [x] ≤ 1.

Inverse function: If f : A → B is bijection then f -1  is exists

f-1: B → A is an inverse function of f.

domain and range

in equations

SOME IMPORTANT POINTS

of subsets of a set of n elements is 2n

of proper subsets of a set of n elements is 2n – 1

Let A and B are two non-empty finite sets and f: A → B is a function. This function will

One-one if n(A) ≤ n(B)

On to if n(A) ≥ n(B)

Bijection   if n(A) = n(B).


2. MATHEMATICAL INDUCTION


3. MATRICES

Matrix: An ordered rectangular array of elements is called a matrix

  • Matrices are generally enclosed by brackets like
  • Matrices are denoted by capital letters A, B, C and so on
  • Elements in a matrix are real or complex numbers; real or complex real-valued functions.

Oder of Matrix: A matrix having rows and ‘n’ columns is said to be of order m x n. Read as m by n.

matrix

Types of Matrices

 Square Matrix: A matrix in which the no. of rows is equal to no. of columns is called     square matrix.

square matrix
 Principal diagonal ( diagonal)  Matrix: If A  = [aij] is a square matrix of order ‘n’ the elements  a11 , a22 , a33 , ………. ann are said to constitute its principal diagonal.

principle diagonal matrix

Trace Matrix: The sum of the elements of the principal diagonal of a square matrix A is called the trace of the matrix. It is denoted by Tr (A).

Ex:-

trace of matrix

Diagonal Matrix: If each non-diagonal element of a square matrix is ‘zero’ then the matrix is called a diagonal matrix.

diagonal of matrix

Scalar Matrix: If each non-diagonal elements of a square matrix are ‘zero’ and all diagonal elements are equal to each other, then it is called a scalar matrix.

scallar matrix

Identity Matrix or Unit Matrix: If each of the non-diagonal elements of a square matrix is ‘zero’ and all diagonal elements are equal to ‘1’ , then that matrix is called a unit matrix.

identity matrix

Null Matrix or Zero Matrix: If each element of a matrix is zero, then it is called a null matrix.

null matrix

Row matrix & column Matrix: A matrix with only one row s called a row matrix and a matrix with only one column is called a column matrix.

row and column matrices

Triangular matrices:

A square matrix A = [aij] is said to be upper triangular if aij = 0   ∀ i > j

A square matrix A = [aij] is said to be lower triangular matrix aij = 0  ∀ i < j

triangular matrices

Equality of matrices: matrices A and B are said to be equal if A and B of the same order and the corresponding elements of A and B are equal.

equality of matrices

Product of Matrices:

 Let A = [aik]mxn and B = [bkj]nxp be two matrices ,then the matrix C = [cij]mxp  where

product of matrices

  • Note: Matrix multiplication of two matrices is possible when no. of columns of the first matrix is equal to no. of rows of the second matrix.

Transpose of Matrix: If A = [aij] is an m x n matrix, then the matrix obtained by interchanging the rows and columns is called the transpose of A. It is denoted by AI or AT.  

transpose of matrix

  • Note: (i) (AI)I = A (ii) (k AI) = k . AI    (iii)  (A + B )T = AT + BT  (iv)  (AB)T = BTAT

Symmetric Matrix: A square matrix A is said to be symmetric if AT =A

If A is a symmetric matrix, then A + AT is symmetric.

Skew-Symmetric Matrix: A square matrix A is said to be skew-symmetric if AT = -A

If A is a skew-symmetric matrix, then A – AT is skew-symmetric

Minor of an element: Consider a square matrix   

minor of an elemen

the minor an element in this matrix is defined as the determinant of the 2×2 matrix obtained after deleting the rows and the columns in which the element is present.

minor of an element example

Cofactor of an element: The cofactor of an element in i th row and j th column of A3×3 matrix is defined as it’s minor multiplied by (- 1 ) i+j .

Properties of determinants:

  • If each element of a row (column) of a square matrix is zero, then the determinant of that matrix is zero.

det-1

  • If A is a square matrix of order 3 and k is scalar then.
  • If two rows (columns) of a square matrix are identical (same), then Det. Of that matrix is zero.

det-2

  • If each element in a row (column) of a square matrix is the sum of two numbers then its determinant can be expressed as the sum of the determinants.

         det-3

  • If each element of a square matrix are polynomials in x and its determinant is zero when x = a, then (x-a) is a factor of that matrix.
  • For any square matrix A  Det(A) =  Det (AI).
  • Det(AB) = Det(A) . Det(B).
  • For any positive integer n Det(An) = (DetA)n.

Singular and non-singular matrices: A Square matrix is said to be singular if its determinant is zero, otherwise it is said to be the non-singular matrix.

singular and non-singular matrices

Ad joint of a matrix: The transpose of the matrix formed by replacing the elements of a square matrix A with the corresponding cofactors is called the adjoint of A.

adjoint of matrix 2

 Invertible matrix: Let A be a square matrix, we say that A is invertible if there exists a matrix B such that AB =BA = I, where I is the unit matrix of the same order as A and B.

invertible matrix

Augmented matrix: The coefficient matrix (A) augmented with the constant column matrix (D) is called the augmented matrix. It is denoted by [AD].

augmented matrix

Sub matrix: A matrix obtained by deleting some rows and columns (or both) of a matrix is called the submatrix of the given matrix.

sub matrix

Let A be a non-zero matrix. The rank of A is defined as the maximum of the order of the non-singular submatrices of A.

  • Note: If A is a non-zero matrix of order 3 then the rank of A is:
  • 1, if every 2×2 submatrix is singular
  • 2, if A is singular and at least one of its 2×2 sub-matrices is non-singular

   (iii)  3, if A is non – singular.

Consistent and Inconsistent: The system of linear equations is consistent if it has a solution, in-consistent if it has no solution.

  • Note: The system of three equations in three unknowns AX = D has
  • A unique solution if rank(A) = rank ([AD]) = 3
  • Infinitely many solutions if rank (A) = ([AD]) < 3
  • No solution if rank (A) ≠ rank ([AD])

Solutions of a homogeneous system of linear equations:

hogenious equations

The system of equations AX = 0 has

  • The trivial solution only if rank(A) = 3
  • An infinite no. of solutions if rank(A) < 3

4.ADDITION OF VECTORS

Directed line: If A and B are two distinct points in the space, the ordered pair (A, B) denoted by AB is called a directed line segment with initial point A and terminal point B.

directed line

⇒ A directed line passes through three characteristics: (i) length (ii) support (iii) direction

Scalar: A quantity having magnitude only is called a scalar. We identify real numbers as a scalar.

Ex: – mass, length, temperature, etc.

Vector: A quantity having length and direction is called a vector.

Ex: – velocity, acceleration, force, etc.

⇒ If TS inter addition of vectors 4 is a vector then its length is denoted by TS inter addition of vectors 28

Position of vector: If P (x, y, z) is any point in the space, then TS inter addition of vectors 1  is called the position vector of the point P with respect to origin (O). This is denoted by TS inter addition of vectors 2

Like and unlike vectors:  If two vectors are parallel and having the same direction then they are called like vectors.

like vectors

 

If two vectors are parallel and having opposite direction then they are called, unlike vectors.


un like vectors
Coplanar vectors:
Vectors whose supports are in the same plane or parallel to the same plane are called coplanar vectors.

VECTOR ADDITION                                       

Triangle law: If TS inter 1A product of vectors 2 are two vectors, there exist three points A, B, and C in a space such that   defined by TS inter addition of vectors 7

triangle law

Parallelogram law: If two vectorsTS inter 1A vector a and TS inter addition of vectors 5 represented by two adjacent sides of a parallelogram in magnitude and direction then their sum is represented in magnitude and direction by the diagonal of the parallelogram through their common point.

parallelogram law 2                                                                    parallelogram law

Scalar multiplication: LetTS inter 1A vector a be a vector and λ be a scalar then we define vector λTS inter 1A vector a  to be the vectorTS inter addition of vectors 29 if eitherTS inter 1A vector a is zero vector or λ is the scalar zero; otherwise λTS inter 1A vector a is the vector in the direction of TS inter 1A vector awith the magnitude TS inter addition of vectors 9if λ>0 and λTS inter 1A vector a  = (−λ)(−TS inter 1A vector a ) if λ<0.

add. vectors notes

The angle between two non-zero vectors:   LetTS inter 1A product of vectors 2 be two non-zero vectors, let TS inter addition of vectors 10  then ∠AOB has two values. The value of ∠AOB, which does not exceed 1800 is called the angle between the vectorsTS inter 1A vector a and TS inter addition of vectors 5  , it is denoted by (TS inter 1A product of vectors 2 ).

TS inter addition of vectors 12

Section formula: LetTS inter 1A product of vectors 2 be two position vectors of the points A and B with respect to the origin if a point P divides the line segment AB in the ratio m:n then

section formula

Linear combination of vectors:  let TS inter addition of vectors 13 be vectors x1, x2, x3…. xn be scalars, then the vectorTS inter addition of vectors 14 is called the linear combination of vectors.

Components: Consider the ordered triad (a, b, c) of non-coplanar vectorsTS inter addition of vectors 15 If r is any vector then there exist a unique triad (x, y, z) of scalars such that TS inter addition of vectors 16 . These scalars x, y, z are called the components of TS inter addition of vectors 2with respect to the ordered triad   (a, b, c).

  • i, j, k are unit vectors along the X, Y and Z axes respectively and P(x, y, z) is any point in the space thenTS inter addition of vectors 1 = r = x i + y j +z k   andTS inter addition of vectors 17

Regular polygon: A polygon is said to be regular if all the sides, as well as all the interior angles, are equal.

  • If a polygon has sides then the no. of diagonals of a polygon is TS inter addition of vectors 18 
  • The unit vector bisecting the angle between  is  TS inter addition of vectors 19

Vector equation of a line and plane

⇒The vector equation of the line passing through the point A (TS inter 1A vector a) and ∥el to the vector TS inter addition of vectors 5 is

vector equation of a line

Proof:-

vector equation opf a line 2

 Then AP,  are collinear vector proof: let P (TS inter addition of vectors 2 ) be any point on the line a

TS inter addition of vectors 20      

   the equation of the line passing through origin and parallel to the vectorTS inter addition of vectors 5isTS inter addition of vectors 21      

  • the  vecor equation of the line passing through the points A(TS inter 1A vector a )  and B( TS inter addition of vectors 5 )  is TS inter addition of vectors 23
  • Cartesian equation of the line passing through A ( x1, y1, z1) and  B ( x2, y2, z2) is TS inter addition of vectors 22
  • The vector equation of the plane passing through the point A(TS inter 1A vector a ) and parallel to the vectors TS inter addition of vectors 5andTS inter 1A vector c is  TS inter addition of vectors 24
  • The vector equation of the plane passing through the point A(TS inter 1A vector a ), B(TS inter addition of vectors 5 ) and parallel to the vector TS inter 1A vector c is TS inter addition of vectors 25
  • The vector equation of the plane passing through the points A(TS inter 1A vector a ), B(TS inter addition of vectors 5 ) and C( TS inter 1A vector c) isTS inter addition of vectors 26

large bar{r}= (1-t)bar{a} + t bar{b}

5.PRODUCT OF VECTORS

TS inter 1A vectors dotproduct title

Dot product (Scalar product): LetTS inter 1A product of vectors 2 are two vectors. The dot product or direct product of TS inter 1A vector a and TS inter 1A vector b  is denoted byTS inter 1A product of vectors 3and is defined as 

  • IfTS inter 1A vector a = 0, TS inter 1A vector b = 0 ⟹ TS inter 1A product of vectors 3  = 0.
  • If TS inter 1A vector a≠0,TS inter 1A vector b ≠ 0 thenTS inter 1A product of vectors 4
  • The dot product of two vectors is a scalar
  • If TS inter 1A product of vectors 2 are two vectors, then

     TS inter 1A product of vectors 1

  • If θ is the angle between the vectorsTS inter 1A product of vectors 2 then. TS inter 1A product of vectors 4

         ⟹    TS inter 1A product of vectors 5

         ⟹ IfTS inter 1A product of vectors 3   > 0, then θ is acute angle

         ⟹ If  TS inter 1A product of vectors 3  < 0, then θ is obtuse angle 0

          ⟹ If  TS inter 1A product of vectors 3  = 0, thenTS inter 1A vector a  is perpendicular toTS inter 1A vector b

  • IfTS inter 1A vector a is any vector then  TS inter 1A product of vectors 6

Component and Orthogonal Projection:

LetTS inter 1A vector a=TS inter 1A vector OA,TS inter 1A vector b=TS inter 1A vector OB  be two non-zero vectors. Let the plane passing through B (TS inter 1A vector b ) and perpendicular to TS inter 1A vector a   intersectsTS inter 1A vector OA

TS inter 1A product of vectors 7

In M, then TS inter 1A vector OM is called the component of TS inter 1A vector b on TS inter 1A vector a

  • The component (projection) vector of TS inter 1A vector b on TS inter 1A vector a is TS inter 1A product of vectors 8
  • Length of the projection (component) =TS inter 1A product of vectors 9
  • Component ofTS inter 1A vector b perpendicular toTS inter 1A vector a = TS inter multiplication of vectors 1

If TS inter 1A vector i,TS inter 1A vector j, TS inter 1A vector k   form a right-handed system of an orthonormal triad, then 

TS inter 1A product of vectors 10

  • If TS inter 1A product of vectors 11