# 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) x_{1} , x_{2} ϵ A , x_{1 }= x_{2 }in A _{ }⇒ f(x_{1}) = f(x_{2}) 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(x_{1}) = f(x_{2}) in B ⇒ x_{1} = x_{2} 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}

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 2^{n}

of proper subsets of a set of n elements is 2^{n} – 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.

### 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 a_{11} , a_{22} , a_{33} , ………. a_{nn }are 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^{T}A^{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_{3×3 }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.

- 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.

- 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.

- 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(A
^{n}) = (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

Position of vector: If P (x, y, z) is any point in the space, then

**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

**Parallelogram law:** If two vectors

**Scalar multiplication: **Let

**The angle between two non-zero vectors: **Let^{0} is called the angle between the vectors**( **).

**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 _{1}, x_{2}, x_{3}…. x_{n }be scalars, then the vector

**Components: **Consider the ordered triad (a, b, c) of non-coplanar vectors**r** is any vector then there exist a unique triad (x, y, z) of scalars such that

**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 (^{el} to the vector

**Proof:-**

Then AP, are collinear vector proof: let P (

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

- the vecor equation of the line passing through the points A(
) and B( ) is - Cartesian equation of the line passing through A ( x
_{1}, y_{1}, z_{1}) and B ( x_{2}, y_{2}, z_{2}) 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

**5.PRODUCT OF VECTORS**

**Dot product (Scalar product):** Let

- If
= 0, = 0 ⟹ = 0. - If
≠0, ≠ 0 then - The dot product of two vectors is a scalar
- If
are two vectors, then

⟹ If

⟹ If

⟹ If

**Component and Orthogonal Projection:**

Let

In M, then

- The component (projection) vector of
on is - Length of the projection (component) =
- Component of
perpendicular to =

If