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What happens to a rubber band when you stretch it and let go? It deforms but regains its original nature when you stop applying a force. But say, you take an aluminium rod and try to bend it using your arm strength. You somehow do manage to bend it a little and then stop applying force. Does the rod regain its original shape? Of course not.

Gravitation or gravity is a naturally occurring phenomenon or a force which exists among all material objects in the universe.

Imagine a rectangular block placed on the slanting edge of a right-angled triangle. If the block is assumed to slide down this edge without any side movement, every point in the rectangular block experiences the same displacement and more importantly, the distance between the points is also maintained.

Work is the measurement of energy transfer that takes place when an object is displaced by an external force applied in the direction of the displacement.

Vertical division of power among different levels of government is one of the major forms of power-sharing in modern democracies. It is most commonly referred to as federalism.

Thermodynamics is concerned with the relationship between other forms of energy and heat. It explains how thermal energy is converted from other forms of energy. The energy coming from heat is termed thermal energy. When the tiny particles within an object move, they generate heat, more amount of heat is generated, when these particles move at a faster pace.

In a democracy, all power does not rest with any one organ of the government. An intelligent sharing of power among legislature, executive and judiciary is very important to the design of a democracy.

We face a dilemma: democracy is seen to be good in principle, but felt to be not so good in its practice. This dilemma invites us to think hard about the outcomes of democracy.

A political party is a group of people who come together to contest elections and hold power in the government. They agree on some policies and programmes for the society with a view to promote the collective good. Parties reflect fundamental political divisions in a society.

The existence of social diversity does not threaten democracy. Political expression of social differences is possible and sometimes quite desirable in a democratic system. Three kinds of social differences that can take the form of social divisions and inequalities are social differences based on gender, religion and caste.

Three-fourth of the earth's surface is covered with water, but only a small proportion of it accounts for freshwater that can be put to use. This freshwater is mainly obtained from surface run off and ground water that is continually being renewed and recharged through the hydrological cycle.

Biodiversity or Biological Diversity is immensely rich in wildlife and cultivated species, diverse in form and function but closely integrated in a system through multiple network of interdependencies.

We use different materials and services in our daily life. Some of these are available in our immediate surroundings, while other requirements are met by bringing things from other places.

Production of goods in large quantities after processing from raw materials to more valuable products is called manufacturing.

We use different things in our daily life made from metal. Various metals are extracted from minerals after proper refinement. Minerals are an indispensable part of our lives.

It is difficult for us to imagine a world without printed matter. We find evidence of print everywhere around us – in books, journals, newspapers, prints of famous paintings, and also in everyday things like theatre programmes, official circulars, calendars, diaries, advertisements, cinema posters at street corners.

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.

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.

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.