Author: Azhar

What is Conic Sections
Conic sections or simply conics are the curves obtained by cutting a (double) right circular cone by a plane. Let be a line through the center of a given circle and perpendicular to its plane. Let be a fixed point on . All lines through and points on the circle generate a right circular cone. […]

Product Rule of Derivatives, Definition and Examples
Product Rule of Derivatives:If and are differentiable at , then is also differentiable at and . . . Subtracting . . Subtracting and Adding . . . . Taking. . . . Example 1 {Product rule of derivatives} Let and Now we use product Rule Example 2 [Product rule of derivatives] Let and Now we […]

Order of a Group, Cosets, definition,Examples
Order of a Group: The order of group is the number of elements present in that group , also say it’s cardinality. It is denoted by or . Examples: Dihedral group has order Symmetric group has order ={} modulo has order . Order of an element: Order of element is the smallest positive integer such that , where […]

Lagrange Theorem Statment, Proof
Lagrange Theorem Statement: Both the order and Index of a subgroup of a finite group divide the order of a group. Lagrange Theorem Proof: Let be a subgroup of order of a finite group of order and let be the index of in . since is finite , the set {} all of distinct left […]

Quadratic Formula Derivation, Examples:
Quadratic Formula Derivation: Again there are some quadratic polynomials that are not factorable at all using integral coefficients. In such a case we can always find the solution of a quadratic equation by the quadratic formula. by applying a formula known as the quadratic formula. This formula is applicable for every quadratic equation. Derivation of […]

Quadratic Equation definition, Examples
Quadratic Equation definition: A quadratic equation in is an equation that can be written in the form of are real numbers and .Another name for a quadratic equation in is 2nd Degree Polynomial in . Quadratic Equation Examples: The following equations are the quadratic equations: Solution of Quadratic Equations: There are three basic techniques for […]

Monoid, Definition and Examples :
Definition: A non empty set G having binary operation “*” (say) is called Monoid if it satisfies the following axioms: Closure Property w.r.t “” i.e, Associative Law w.r.t “” Identity element exist. There is identity element e in G such that If the above three properties hold then the set is called Monoid. We can […]

Semi Group, Definition and Examples:
Definition: A non empty set G having binary operation “*” (say) is called semi group in math if it satisfies the following two axioms: Closure Property w.r.t “” i.e, Associative Law w.r.t “” Examples of Semi Group: The set of W {0,1} with respect to “Addition” and “Multiplication”. The set of Natural Number with respect to “Addition”. The […]

Angle in Math, Concept of an Angle,
Angle in Math: Two rays with a common starting point form an angle. One of the rays of angle is called the initial side and the other is the terminal side. The angle is identified by showing the direction of rotation from the initial side to the terminal side.An angle is said to be positive/negative […]

Rotation of Axis, Concept, and Examples:
Rotation of Axis: Let coordinate system be given. We rotate and and about the origin through an angle so that the new axes are and as shown in the figure. Let a point have coordinates referred to the system of coordinates. Suppose has coordinates referred to the coordinate system. We have to find in terms […]