Class 12th

Probability Conditional Probability * If A and B are two events of the same sample space S, then the conditional probability of event A, given that even B has occurred, is given by p ( A | B ) = \frac{P (AՈB)}{P (B)} , P (B) ≠ 0 * If A is an event associated ... Read more

Linear Programming Linear Programming Problem Theorem2: If R is open, then a maximum or a minimum value of the object function may not exist. However, if the exists, then it must occur at a corner point of R. Case (i): P is the maximum value of z, if the open half plane determined by ax ... Read more

Direction Cosines and Direction Ratios * Directional angles: The angles made by a line with the positive directions of the X, Y and Z axes are called directional angles. * Directional Cosines: If α, β and γ are the directional angles of a directed line L, them cos α, cos β, and cos γ , ... Read more

Vectors Algebra Basic Concepts * Quantities that are characterised only by their magnitude are called scalar quantities. * Quantities that are characterised by their magnitude as well as direction are called vector quantities. * A line segment with a given direction is called a directed line segment. * A directed line segment AB is written ... Read more

Differential Equations Basic Concepts of Differential Equations * Differential equation: an equation involving a derivative of the dependent variable with respect to the independent variable is called a differential. * The order of the highest order derivative occurring in a differential equation is called the order of the differential equation. * The degree of a ... Read more

Application of Integrals Areas Under Simple Curves * Area of the region bounded by the curve, y = ƒ(x), the coordinates x = a and x = b, and the X- axis = \int_{a}^{b} ƒ(x) dx. * Area of the region bounded by the curve, x = ɡ(y), the coordinates y = c and y ... Read more

Integrals Integration * Integration is the inverse process of differentiation. If \frac{d}{dx} (F(x)) = ɡ(x), then ∫ g (x) dx  = F(x) + C * Where C is an arbitrary constant known as the constant of integration. * Geometrical interpretation of indefinite integrals: * An indefinite integral is a collection of a family of curves, ... Read more

Rate of Change of Quantities Rate of Change: The change in one quantity with respect to time is known as rate of change of quantity. * If a quantity y varies with another quantity x, satisfying y = ƒ(x), then \frac{dy}{dx} represents the rate of change quantity with respect to x. * If x = ... Read more