Ts Inter Maths IA Concept

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

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= x₂in 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

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

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

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 I_A: 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.

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}

If 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.

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.

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

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:-

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

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.

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

Triangular matrices:

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

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

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.

Product of Matrices:

Let A = [a_ik]_mxn and B = [b_kj]_nxp be two matrices ,then the matrix C = [c_ij]_mxp where

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 = [a_ij] 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 A^I or A^T.

Note: (i) (A^I)^I = A (ii) (k A^I) = k . A^I (iii) (A + B )^T = A^T + B^T (iv) (AB)^T = B^TA^T

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

If A is a symmetric matrix, then A + A^T is symmetric.

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

If A is a skew-symmetric matrix, then A – A^T is skew-symmetric

Minor of an element: Consider a square matrix

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.

Cofactor of an element: The cofactor of an element in i ^th row and j ^th column of A_3×3matrix 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 (A^I).
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.

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.

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.

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

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

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.

⇒ 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 is a vector then its length is denoted by

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

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

If two vectors are parallel and having opposite direction then they are called, unlike 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 are two vectors, there exist three points A, B, and C in a space such that defined by

Parallelogram law: If two vectors and 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.

Scalar multiplication: Let be a vector and λ be a scalar then we define vector λ to be the vector if either is zero vector or λ is the scalar zero; otherwise λ is the vector in the direction of with the magnitude if λ>0 and λ = (−λ)(− ) if λ<0.

The angle between two non-zero vectors: Let be two non-zero vectors, let then ∠AOB has two values. The value of ∠AOB, which does not exceed 180⁰ is called the angle between the vectors and , it is denoted by ( ).

Section formula: Let 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

Linear combination of vectors: let be vectors x₁, x₂, x₃…. x_nbe scalars, then the vector is called the linear combination of vectors.

Components: Consider the ordered triad (a, b, c) of non-coplanar vectors If r is any vector then there exist a unique triad (x, y, z) of scalars such that . These scalars x, y, z are called the components of with 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 then = r = x i + y j +z k and

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
The unit vector bisecting the angle between is

Vector equation of a line and plane

⇒The vector equation of the line passing through the point A () and ∥^el to the vector is

Proof:-

Then AP, are collinear vector proof: let P ( ) be any point on the line a

the equation of the line passing through origin and parallel to the vectoris

the vecor equation of the line passing through the points A( ) and B( ) is
Cartesian equation of the line passing through A ( x₁, y₁, z₁) and B ( x₂, y₂, z₂) is
The vector equation of the plane passing through the point A( ) and parallel to the vectors and is
The vector equation of the plane passing through the point A( ), B( ) and parallel to the vector is
The vector equation of the plane passing through the points A( ), B( ) and C( ) is