Infinite Geometric Series Sum of n terms of a G.P: The sum of n terms of a G.P. with first term a and common ratio r is given by or . Proof: Let Sn denote the sum of n terms of the G.P. with first term a and common ratio r Then, Sn = a Read more about Infinite Geometric Series[…]
Arithmetic Progression Sequence: A sequence is a function whose domain is the set N of natural numbers. Real Sequence: A sequence whose range is a subset of R is called a real sequence. In other words, a real sequence is a function with domain N and the range a subset of the set R of Read more about Arithmetic Progression[…]
General Equation of Conic If focus, directrix and eccentricity of conic are given, then Let S (α, β) be focus, ax + by + c = 0 be directrix. Let e be the eccentricity of conic. Let P (h, k) be any point on the conic such that PM is perpendicular from P on the Read more about General Equation of Conic[…]
Rank of a Matrix: Rank is defined for any matrix Amxn (need not be square) Submatrix of a Matrix: Suppose A is any matrix of the type m ´ n. then a matrix obtained by leaving some rows and some columns from A is called sub-matrix of A. Rank of a Matrix: A number r Read more about Rank of a Matrix[…]
Graphs of Simple Functions Constant Function: Let k be a fixed real number. Then a function f(x) given by f(x) = k for all x ϵ R is called a constant function. Sometimes we also call it the constant function k.Identity Function: The function defined by I(x) = x for all x ϵ R is Read more about Graphs of Simple Functions[…]
Pole and Polar Polar of a Point with Respect to a Circle: If through a point P(x₁, y₁) (within or without a circle) there be drawn any straight line to meet the given circle at Q and R the locus of the point of intersection of the tangents at Q and R is called the polar Read more about Pole and Polar[…]
Various forms of Circle Equation Circle passing through 3 points: 1. Let x² + y² + 2gx + 2fy + c = 0 be circle passing through 3 points (non-collinear) then P (x₁, y₁), Q (x₂, y₂), R (x₃, y₃). x₁² + y₁² + 2gx₁ + 2fy₁ + c = 0 x₂² + y₂² + Read more about Various forms of Circle Equation[…]
Collinear and Coplanar Vectors Collinear Vectors: They can have equal or unequal magnitudes and their directions may be same or opposite. Two vectors are collinear if they have the same direction or are parallel or anti-parallel. They can be expressed in the form a = k b where a and b are vectors and ‘ k ‘ is a scalar Read more about Collinear and Coplanar Vectors[…]
Condition for Parallel or Perpendicular Lines When the two lines are perpendicular, the angle between the lines is 90° which gives the condition of perpendicularity as l₁ l₂ + m₁ m₂ + n₁ n₂ = 0 or this implies a₁ a₂ + b₁ b₂ + c₁ c₂ = 0. Similarly, when the two lines are parallel, the angle between them i.e. θ = Read more about Condition for Parallel or Perpendicular Lines[…]
Direction Cosines: the cosines of the angles made by a directed line segment with the coordinate axes are called as the direction cosines of that line.if α, β, and γ are the angles made by the line segment with the coordinate axis then these angles are termed to be the direction angles and the cosines of these angles Read more about Relation between the Direction Cosines[…]