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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.

The origin of life on earth can be understood only against the background of origin of universe especially earth. Most scientists believe chemical evolution, i.e., formation of biomolecules preceded the appearance of the first cellular forms of life.

Health is not just the absence of disease. It is a state of complete physical, mental, social and psychological well-being. Diseases like typhoid, cholera, pneumonia, fungal infections of skin, malaria and many others are a major cause of distress to human beings.

Microbes are a very important component of life on earth. Not all microbes are pathogenic. Many microbes are very useful to human beings. We use microbes and microbially derived products almost every day.

Biotechnology deals with large scale production and marketing of products and processes using live organisms, cells or enzymes.

Biotechnology has given to humans several useful products by using microbes, plant, animals and their metabolic machinery. Recombinant DNA technology has made it possible to engineer microbes, plants and animals such that they have novel capabilities.

As a branch of biology, Ecology is the study of the relationships of living organisms with the abiotic (physico-chemical factors) and biotic components (other species) of their environment. It is concerned with four levels of biological organisation-organisms, populations, communities and biomes.

An ecosystem is a structural and functional unit of nature and it comprises abiotic and biotic components. Abiotic components are inorganic materials- air, water and soil, whereas biotic components are producers, consumers and decomposers.

Since life originated on earth nearly 3.8 billion years ago, there had been enormous diversification of life forms on earth. Biodiversity refers to the sum total of diversity that exists at all levels of biological organisation. Of particular importance is the diversity at genetic, species and ecosystem levels and conservation efforts are aimed at protecting diversity at all these levels.

Genetics is a branch of biology which deals with principles of inheritance and its practices. Progeny resembling the parents in morphological and physiological features has attracted the attention of many biologists.

Nucleic acids are long polymers of nucleotides. While DNA stores genetic information, RNA mostly helps in transfer and expression of information. Though DNA and RNA both function as genetic material, but DNA being chemically and structurally more stable is a better genetic material. However, RNA is the first to evolve and DNA was derived from RNA.

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.

Bank Reconciliation Statement: A statement prepared to reconcile the bank balance as per cash book with the balance as per passbook or bank statement, by showing the items of difference between the two accounts.

Meaning of trial balance: A statement showing the abstract of the balance (debit/credit) of various accounts in the ledger.

Meaning of depreciation: Depreciation is decline in the value of a tangible fixed asset. In accounting, depreciation is the process of allocating depreciable cost over useful life of a fixed asset.

Meaning, usefulness and types of financial statements: After the agreement of the trial balance, a business enterprise proceeds to prepare financial statements. Financial statements are the statements, which present periodic reports on the process of business enterprises and the results achieved during a given period.

Need for adjustments: For the preparation of financial statements, it is necessary that all the adjustments arising out of the accrual basis of accounting are made at the end of the accounting period. Another important consideration in the preparation of final accounts with adjustments, is the distinction between capital and revenue items. Entries which are recorded to give effect to these adjustments are known as adjusting entries.

New Profit Sharing Ratio: New profit sharing ratio is the ratio in which the remaining partner will share future profits after the retirement or death of any partner.

Dissolution of Partnership Firm: The dissolution of a firm implies the discontinuance of partnership business and separation of economic relations between the partners.

Company: An organisation consisting of individuals called shareholders by virtue of their holding the shares of a company, who can act as legal person as regards its business through board of directors.

Debenture: Debenture is the acknowledgements of debt. It is a loan capital raised by the company from general public. A person or holder of such a written acknowledgement is called debenture holder.

Financial Statements: Financial statements are the end products of accounting process, which reveal the financial results of a specified period and financial position as on a particular date. Financial Statements are prepared and published by corporate undertakings for the benefit of various stakeholders. These statements include Statement of profit and loss and balance sheet.

Major Parts of an Annual Report: An annual report contains basic financial statements - Balance Sheet, Statement of Profit and Loss and Cash Flow Statement. It also carries management’s discussion of corporate performance of the year under review for futuristic prospects.

Ratio Analysis: An important tool of financial statement analysis is ratio analysis. Accounting ratios represent relationship between two accounting numbers.

Cash Flow Statement: The Cash Flow Statement helps in ascertaining the liquidity of an enterprise. Cash Flow Statement is to be prepared and reported by Indian companies according to AS-3 issued by The Institute of Chartered Accountants of India. The cash flows are categorised into flows from operating, investing and financing activities.

Definition of partnership and its essential features: Partnership is defined as Relation between persons who have agreed to share the profits of a business carried on by all or any one of them acting for all.

Matters requiring adjustments at the time of admission of a partner: Various matters which need adjustments in the books of firm at the time of admission of a new partner are: goodwill, revaluation of assets and liabilities, reserves and other accumulated profits and losses and the capitals of the old partners (if agreed).

There are all kinds of business organisations - small or large, industrial or trading, privately owned or government owned existing in our country. These organisations affect our daily economic life and therefore, become part of the Indian economy.

Services are those separately identifiable, essentially intangible activities that provide satisfaction of wants, and are not necessarily linked to the sale of a product or another service. There are five basic features of services.

The world of business is changing. e-business and outsourcing are the two most obvious expressions of this change. The trigger for the change owes its origin to both internal and external forces.