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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.

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.

Concept of social responsibility: Social responsibility of business refers to its obligation to take those decisions and perform those actions which are desirable in terms of the objectives and values of our society.

Meaning and significance of business finance: Finance required by business to establish and run its operations is known as business finance. No business can function without adequate amount of funds for undertaking various activities.

Role of small business in India: Small Scale Industries play a very important role in the socio-economic development of the country. These industries account for 95 per cent of industrial units, contributing up to 40 per cent of the gross industrial value added and 45 per cent of the total exports.

Trade refers to buying and selling of goods and services with the objective of earning profit on the basis of geographical location of buyers and sellers. It can be classified into two categories (i) internal trade; and (ii) external trade.

International business refers to business activities that take place across national frontiers. Though many people use the terms international business and international trade synonymously, the former is a much broader term.

In every society, people undertake various activities to satisfy their needs. These activities may be broadly classified into two groups - economic and non-economic.

Forms of business organisation refers to the types of organisations which differ in terms of ownership and management. The major forms of organisation include proprietorship, partnership, joint Hindu family business, cooperative society and company.

Organising is the process of defining and grouping activities and establishing authority relationships among them.

Staffing has been described as the managerial function of filling and keeping filled, the positions in an organisation structure. This is achieved by, first of all, identifying requirement of work force, followed by recruitment, selection, placement, promotion, appraisal and development of personnel, to fill the roles designed into the organisation structure.

Directing is a complex managerial function consisting of all the activities that are designed to encourage subordinates to work effectively. It includes supervision, motivation, communication and leading. The principles which guide effective directing may be classified as principles related to the purpose of directing and principles related to direction process.