Important Theory, Formula and Derivation of Optional Mathematics

Table of Contents:

1. Algebra
• Function
• Polynomial
• Sequence and Series
• Equation and Graph
2. Matrix
3. Co-Ordinate Geometry
• Angle Between two Straight Lines
• Pair of Straight Lines
• Circle

UNIT 1: ALGEBRA

Lesson One: Function

• Function: If  A and B are non-empty sets and each member of A is related to any member of B then relation from A to B is function. It is denoted as f: A→B

• Composite Function: If A, B, and C are three non-empty sets, f: A → B and g: B→C are two functions. Then a function defined from A to C is the composite function.

• Inverse Function: Function obtained by interchanging domain and range of each ordered pair of the function is inverse function. For Example; the inverse of function f={(1,2), (4,5)} is f^-1={(2,1), (5,4)}

• Algebraic Function: The function which describes the correspondence between two variables x & y which are obtained by the finite rules is called algebraic function. For examples: y = x + 2

• Linear Function: A function is called linear if it can be defined by an equation of the form. F(x) = mx + c

• Constant Function: A function of the form f(x) = mx + c where m = 0 then it is called constant function.

• Identity Function: A function of the form f(x) = mx + c where c + o and m = 1 then it is called identity function.

• Quadratic Function: A function is said to be quadratic if it can be defined by an equation of the form f(x) ax^2 + bx + c; a is not 0.

• Trigonometric Function: The function defined as below are trigonometric functions: f(θ) = Sinθ, f(θ) = Cos θ, f(θ) = tanθ

Lesson Two: Polynomial

• Polynomial: A polynomial is a rational expression each of whose terms consists of a constant multiplied by a positive power if a variable.

• Remainder Theorem: If P(x) is a polynomial of degree n and (x-a) is a divisor of P(x) then P(a) is a reminder, where the degree of the quotient will be (n-1).

• Factor Theorem: If P(x) be a polynomial of degree greater than 0 and p(a) = 0 then (x-a) is a factor of p(x). Conversely if (x-a) is a factor of P(x), then p(a) = 0.

• Division Algorithm: If a polynomial P(x) is divided by D(x) so that the quotient is Q(x) and remainder R then, P(x) = D(x) * Q(x) + R

Lesson Three: Sequence and Series

Definition:

• Sequence: Sequence is a number pattern separated by commas that follow the unique rule.
  

• Series: Series is a Sequence that is connected by addition “+” or subtraction “-“ sign.

• Arithmetic Sequence: A Sequence having the same difference between the successive terms is called the arithmetic sequence.

• Arithmetic Mean: In an arithmetic sequence, the term between the first term and the last term is called arithmetic mean.

• Geometric Sequence: The sequence is called gs in which the ratio of any term to the proceeding term is constant. The constant is called the common ratio.

• Geometric Mean: In GS, the term between the first and last term is gm.

Derivation:

RELATION BETWEEN AM AND GM OF TWO POSITIVE NUMBERS:

Let “a” and “b” be two positive numbers. Then;

I.e. AM – Gm ≥ 0

Therefore, AM-GM

Hence, AM between two positive numbers is greater than or equal to their GM.

Formula:

Arithmetic Series:

Note: Use red one if “n” is from between “a” and “b” (a…….b) & Use Pink one if “n” is whole(a…….b).

Lesson Five: Equation and Graph

Formula:

y = f(x) = ax^2 + bx + c; a ≠ 0   or,  a(x-h)^2 + k

        Vertex = (-b/2a,(4ac-b ̇^2)/4a

UNIT 2: MATRIX

• Idempotent means; A^2=A
• Pre – Multiply means(अगाडी मल्टिप्ल्य गर्ने); matrix of red color; 
  

  

Condition to possess inverse:

• The matrix should be non-singular.
• The matrix must be a square matrix.

Law of Matrix Multiplication:

• Associative law: If matrix (AB)C and A(BC) are defined then; (AB)C =A(BC)
• Distributive law of Matrix multiplication over matrix addition: In Matrix A, B, C: A(B + C) = AB + AC or, (B+C)A = BA + CA
• Identity Property: If A is a square matrix and It is an identity matrix of same order then; AI = IA = A

Condition for Multiplication of Matrices:

• The matrix of multiplication AB of matrices A and B is possible when the number of columns of matrix a is equal to the number of a row of matrix B.  [2 x 3] [3x2]

Types of Matrix:

• Null Matrix: A Matrix which has all element zero is called a null matrix or a zero matrix.

• Diagonal Matrix: A matrix is said to be diagonal matrix if all its diagonal elements are not zero and the remaining elements are zero.

• Scalar Matrix: If all its diagonal elements are the same and the remaining elements are zero.

• Identity matrix: If the main diagonal elements are one and the remaining elements are zero.

• Equal Matrix: Matrix of the same size and whose corresponding elements are equal is an equal matrix.

• Transpose matrix: Matrix obtained by interchanging the row and column is the transpose matrix.

• Inverse matrix: If A and B are two square matrixes of the same order, I is an identity matrix of the same order and AB = BA = I then A and B are said to the inverse of each other. The inverse of A is denoted by A^(-1)

• Singular Matrix: A square matrix A is called singular if |A| = 0

• Non-Singular Matrix: If |A| ≠ 0 then A is a non-singular matrix.

UNIT 3: CO-ORDINATE GEOMETRY

Lesson One: Angle Between Two Straight Lines

Lesson two: Pair Of Straight Lines

Find the general equation of the 2nd degree of pair of lines:

Solution;

Consider two general equations of straight lines are:

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