Articles

In 1900, a popular music publisher E.T. Paull produced a music book that had a picture on the cover page announcing the ‘Dawn of the Century’. At the centre of the picture is a goddess-like figure, the angel of progress, bearing the flag of the new century. She is gently perched on a wheel with wings, symbolising time.

In 1848, Frédéric Sorrieu, a French artist, prepared a series of four prints visualising his dream of a world made up of ‘democratic and social Republics’, as he called them. The first print of the series, shows the peoples of Europe and America - men and women of all ages and social classes - marching in a long train, and offering homage to the statue of Liberty as they pass by it.

The French Revolution opened up the possibility of creating a dramatic change in the way in which society was structured. Before the eighteenth century society was broadly divided into estates and orders and it was the aristocracy and church which controlled economic and social power.

A lot of this diversity is fast disappearing. Between 1700 and 1995, the period of industrialisation, 13.9 million sq km of forest or 9.3 per cent of the world’s total area was cleared for industrial uses, cultivation, pastures and fuelwood.

An event for an experiment is the collection of some outcomes of the experiment. The Probability of an event lies between 0 and 1 (0 and 1 inclusive).

An arithmetic progression is a list of numbers in which each term is obtained by adding a fixed number to the preceding term except the first term.

Facts or figures, collected with a definite purpose, are called data. Statistics is the area of study dealing with the presentation, analysis and interpretation of data. How data can be presented graphically in the form of bar graphs, histograms and frequency polygons.

Two figures are congruent, if they are of the same shape and of the same size. Two circles of the same radii are congruent. Two squares of the same sides are congruent.

When we talk of ‘globalisation’ we often refer to an economic system that has emerged since the last 50 years or so. But as you will see in this chapter, the making of the global world has a long history - of trade, of migration, of people in search of work, the movement of capital, and much else.

As you have seen, modern nationalism in Europe came to be associated with the formation of nation-states. It also meant a change in people’s understanding of who they were, and what defined their identity and sense of belonging.

In this chapter you will read about nomadic pastoralists. Nomads are people who do not live in one place but move from one area to another to earn their living. In many parts of India we can see nomadic pastoralists on the move with their herds of goats and sheep, or camels and cattle.

In the spring of 1945, a little eleven-year-old German boy called Helmuth was lying in bed when he overheard his parents discussing something in serious tones.

One type of polynomial is the quadratic polynomial of the form ax² + bx + c, a ≠ 0. When you equate this polynomial to zero, you get a quadratic equation. Quadratic equations come up when you deal with many real-life situations.

Euclid’s division algorithm, as the name suggests, has to do with divisibility of integers. Stated simply, it says any positive integer a can be divided by another positive integer b in such a way that it leaves a remainder r that is smaller than b.

Trigonometry is one of the most ancient subjects studied by scholars all over the world. Trigonometry was invented because its need arose in astronomy. Since then the astronomers have used it, for instance, to calculate distances from the Earth to the planets and stars.

The word ‘trigonometry’ is derived from the Greek words ‘tri’ (meaning three), ‘gon’ (meaning sides) and ‘metron’ (meaning measure). trigonometry is the study of relationships between the sides and angles of a triangle.

A polynomial p(x) in one variable x is an algebraic expression in x of the form

An equation of the form ax + by + c = 0, where a, b and c are real numbers, such that a and b are not both zero, is called a linear equation in two variables. A linear equation in two variables has infinitely many solutions.

So far, you have been dealing with figures that can be easily drawn on our notebooks or blackboards. These are called plane figures.

Much of mathematics is about finding a pattern - a recognisable link between quantities that change. In our daily life, we come across many patterns that characterise relations such as brother and sister, father and son, teacher and student.

The word ‘trigonometry’ is derived from the Greek words ‘trigon’ and ‘metron’ and it means ‘measuring the sides of a triangle’. The subject was originally developed to solve geometric problems involving triangles.

The equation x² + 1 = 0 has no real solution as x² + 1 = 0 gives x² = -1 and square of every real number is non-negative.

If we interchange any two rows (or columns), then sign of determinant changes. If any two rows or any two columns are identical or proportional, then value of determinant is zero.

A real valued function is continuous at a point in its domain if the limit of the function at that point equals the value of the function at that point. A function is continuous if it is continuous on the whole of its domain.

If a quantity y varies with another quantity x, satisfying some rule y = f(x), then f'(x) represents the rate of change of y with respect to x.

Differential Calculus is centred on the concept of the derivative. The original motivation for the derivative was the problem of defining tangent lines to the graphs of functions and calculating the slope of such lines.

In geometry, we have learnt formulae to calculate areas of various geometrical figures including triangles, rectangles, trapezias and circles. Such formulae are fundamental in the applications of mathematics to many real life problems.

In Class XI and in Chapter 5 of the present book, we discussed how to differentiate a given function f with respect to an independent variable, i.e., how to find f'(x) for a given function f at each x in its domain of definition.

In our day to day life, we come across many queries such as - What is your height? How should a football player hit the ball to give a pass to another player of his team? Observe that a possible answer to the first query may be 1.6 meters, a quantity that involves only one value (magnitude) which is a real number. Such quantities are called scalars.

In Class XI, while studying Analytical Geometry in two dimensions, and the introduction to three dimensional geometry, we confined to the Cartesian methods only. In the previous chapter of this book, we have studied some basic concepts of vectors.

In earlier classes, we have discussed systems of linear equations and their applications in day to day problems. In Class XI, we have studied linear inequalities and systems of linear inequalities in two variables and their solutions by graphical method. Many applications in mathematics involve systems of inequalities or equations.

In earlier Classes, we have studied the probability as a measure of uncertainty of events in a random experiment. We discussed the axiomatic approach formulated by Russian Mathematician, A.N. Kolmogorov (1903-1987) and treated probability as a function of outcomes of the experiment.

In this chapter, you study different types of relations and equivalence relation, composition of functions, invertible functions and binary operations.

In Chapter 1, we have studied that the inverse of a function f, denoted by f^–1, exists if f is one-one and onto. There are many functions which are not one-one, onto or both and hence we can not talk of their inverses.

A matrix is an ordered rectangular array of numbers or functions. A matrix having m rows and n columns is called a matrix of order m × n.

All organisms are made of cells or aggregates of cells. Cells vary in their shape, size and activities/functions. Based on the presence or absence of a membrane bound nucleus and other organelles, cells and hence organisms can be named as eukaryotic or prokaryotic.

Although there is a bewildering diversity of living organisms, their chemical composition and metabolic reactions appear to be remarkably similar. The elemental composition of living tissues and non-living matter appear also to be similar when analysed qualitatively. However, a closer examination reveals that the relative abundance of carbon, hydrogen and oxygen is higher in living systems when compared to inanimate matter.

Green plants make their own food by photosynthesis. During this process carbon dioxide from the atmosphere is taken in by leaves through stomata and used for making carbohydrates, principally glucose and starch.

Plants unlike animals have no special systems for breathing or gaseous exchange. Stomata and lenticels allow gaseous exchange by diffusion. Almost all living cells in a plant have their surfaces exposed to air.

Cells utilise oxygen for metabolism and produce energy along with substances like carbon dioxide which is harmful. Animals have evolved different mechanisms for the transport of oxygen to the cells and for the removal of carbon dioxide from there. We have a well developed respiratory system comprising two lungs and associated air passages to perform this function.

Vertebrates circulate blood, a fluid connective tissue, in their body, to transport essential substances to the cells and to carry waste substances from there. Another fluid, lymph (tissue fluid) is also used for the transport of certain substances. Blood comprises of a fluid matrix, plasma and formed elements. Red blood cells (RBCs, erythrocytes), white blood cells (WBCs, leucocytes) and platelets (thrombocytes) constitute the formed elements.

Movement is an essential feature of all living beings. Protoplasmic streaming, ciliary movements, movements of fins, limbs, wings, etc., are some forms exhibited by animals. A voluntary movement which causes the animal to change its place, is called locomotion. Animals move generally in search of food, shelter, mate, breeding ground, better climate or to protect themselves.

The neural system coordinates and integrates functions as well as metabolic and homeostatic activities of all the organs. Neurons, the functional units of neural system are excitable cells due to a differential concentration gradient of ions across the membrane.

The living world is rich in variety. Millions of plants and animals have been identified and described but a large number still remains unknown. The very range of organisms in terms of size, colour, habitat, physiological and morphological features make us seek the defining characteristics of living organisms.

Biological classification of plants and animals was first proposed by Aristotle on the basis of simple morphological characters. Linnaeus later classified all living organisms into two kingdoms - Plantae and Animalia. Whittaker proposed an elaborate five kingdom classification - Monera, Protista, Fungi, Plantae and Animalia.

Flowering plants exhibit enormous variation in shape, size, structure, mode of nutrition, life span, habit and habitat. They have well developed root and shoot systems. Root system is either tap root or fibrous. Generally, dicotyledonous plants have tap roots while monocotyledonous plants have fibrous roots. The roots in some plants get modified for storage of food, mechanical support and respiration.

Anatomically, a plant is made of different kinds of tissues. The plant tissues are broadly classified into meristematic (apical, lateral and intercalary) and permanent (simple and complex). Assimilation of food and its storage, transportation of water, minerals and photosynthates, and mechanical support are the main functions of tissues.

Meaning of Accounting: Accounting is a process of identifying, measuring, recording the business transactions and communicating thereof the required information to the interested users.

Generally Accepted Accounting Principles (GAAP): Generally Accepted Accounting principles refer to the rules or guidelines adopted for recording and reporting of business transactions in order to bring uniformity in the preparation and presentation of financial statements. These principles are also referred to as concepts and conventions.

Meaning of source documents: Various business documents such as invoice, bills, cash memos, vouchers, which form the basis and evidence of a business transaction recorded in the books of account, are called source documents.