Preview prep guide to BITSAT 2020 by arihant publications

Preview prep guide to BITSAT 2020 by arihant publications Preview prep guide to BITSAT 2020 by arihant publications Preview prep guide to BITSAT 2020 by arihant publications Preview prep guide to BITSAT 2020 by arihant publications Preview prep guide to BITSAT 2020 by arihant publications Preview prep guide to BITSAT 2020 by arihant publications Preview prep guide to BITSAT 2020 by arihant publications

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3 Motion in 1, 2 & 3 Dimensions and Projectile Motion 24-44

28 Alternating Current and EM Wave 319-330

29 Cathode Rays, Photoelectric Effect of Light and X-Rays 331-340

20 Coordination Compounds and Organometallics 598-607

22 Purification and Estimation of Organic Compounds 626-631

29 Polymers, Biomolecules and Chemistry in Action 721-738

16 Limits, Continuity and Differentiability 968-986

8 Mechanics of Solids and Fluids

10.3 Superposition of waves, beats

5.5 Conservation of mechanical energy

10.4 Doppler Effect

7.1 Newton’s law of gravitation

6.3 Moment of inertia, Parallel and perpendicular axes theorems, rotational kinetic energy

8.3 Viscosity and Surface Tension

7.3 Motion of planets – Kepler’s laws, satellite motion

9.1 Kinematics of simple harmonic motion

6.2 Rotational motion with constant angular acceleration

5.4 Conservative forces and potential

energy

6.5 Conservation of angular momentum

9.2 Spring mass system, simple and compound pendulum

11.2 Thermal equilibrium and temperature

6.1 Description of rotation (angular displacement, angular velocity and angular acceleration)

8.2 Pressure, density and Archimedes’

principle6.6 Rolling motion

9.3 Forced & damped oscillations, resonance

10 Waves

3.5 Inertial and non-inertial frames

4 Impulse and Momentum

4.1 Definition of impulse and momentum

4.4 Momentum of a system of particles

4.5 Center of mass

2.4 Projectile motion

1.4 Fundamental measurements in Physics

(Vernier calipers, screw gauge,

Physical balance etc.)

3.4 Circular motion – centripetal force

4.3 Collisions

5 Work and Energy

5.1 Work done by a force

1 Units & Measurement

1.3 Precision and significant figures

3.1 Newton’s laws (free body diagram,

resolution

of forces)

3.3 Motion of blocks with pulley systems

1.1 Units (Different systems of units, SI

units, fundamental and derived units)

1.2 Dimensional Analysis

2.2 Position, velocity and acceleration

vectors

2.6 Relative motion

3 Newton’s Laws of Motion

3.2 Motion on an inclined plane

2.3 Motion with constant acceleration

2 Kinematics

2.5 Uniform circular motion

11.4 Work, heat and first law of

thermodynamics

11.3 Specific heat, Heat Transfer -

Conduction, convection and radiation,

thermal conductivity, Newton’s law of

cooling

12 Electrostatics

12.1 Coulomb’s law

11.5 2nd law of thermodynamics, Carnot

engine Efficiency and Coefficient of

performance

12.2 Electric field (discrete and continuous

charge distributions)

12.3 Electrostatic potential and

Electrostatic potential energy

16.4 Interference – Huygen’s principle, Young’s double slit experiment

17.3 Hydrogen atom spectrum17.4 Radioactivity

12.4 Gauss’ law and its applications

16.2 Lenses and mirrors

12.6 Capacitance and dielectrics (parallel

plate capacitor, capacitors in series

and parallel)

15.3 Transformers and generators

17.2 Atomic models – Rutherford’s experiment, Bohr’s atomic model

17.5 Nuclear reactions Fission and fusion, binding energy

18.2 Semiconductor diode – I-V characteristics in forward and reverse bias, diode as a rectifier;

I-V characteristics of LED, photodiode, solar cell, and Zener diode; Zener diode as a voltage regulator

15.2 Self and mutual inductance

18.1 Energy bands in solids (qualitative ideas only), conductors, insulators and semiconductors;

13 Current Electricity

15.5 AC circuits, LCR circuits

16.5 Interference in thin films

15.1 Faraday’s law, Lenz’s law, eddy

currents

13.3 Electrical Resistance (Resistivity,

origin and temperature dependence of

resistivity)

14 Magnetic Effect of Current

14.1 Biot-Savart’s law and its applications

16.1 Laws of reflection and refraction

13.1 Ohm’s law, Joule heating

14.2 Ampere’s law and its applications

16.6 Diffraction due to a single slit16.7 Electromagnetic waves and their characteristics (only qualitative ideas), Electromagnetic spectrum

16.8 Polarization – states of polarization, Malus’ law, Brewster’s law

14.4 Magnetic moment of a current loop,

torque on a current loop,

Galvanometer and its conversion to

voltmeter and ammeter

17 Modern Physics

14.3 Lorentz force, force on current

carrying conductors in a magnetic

field

13.2 D.C circuits – Resistors and cells in

series and parallel, Kirchoff’s laws,

potentiometer and Wheatstone bridge,

18.4 Logic gates (OR, AND, NOT, NAND and NOR) Transistor as a switch

1.3 Three states of matter, intermolecular

interactions, types of bonding, melting

and boiling points Gaseous state: Gas

Laws, ideal behavior, ideal gas equation,

empirical derivation of gas equation,

Avogadro number, Kinetic theory –

Maxwell distribution of velocities,

Average, root mean square and most

probable velocities and relation to

temperature, Diffusion; Deviation from

ideal behaviour – Critical temperature,

Liquefaction of gases, van der Waals’

equation

Crystal Structures Simple AB and AB2

type ionic crystals, covalent crystals –

diamond & graphite, metals Voids,

number of atoms per unit cell in a cubic

unit cell, Imperfections- Point defects,

non-stoichiometric crystals; Electrical,

magnetic and dielectric properties;

Amorphous solids qualitative

description Band theory of metals,

conductors, semiconductors and

insulators, and - and - type n p

semiconductors

1.5 Solid State Classification; Space lattices

& crystal systems; Unit cell in two

dimensional and three dimensional

lattices, calculation of density of unit cell

– Cubic & hexagonal systems; Close

packing;

2 Atomic Structure

1.4 Liquid State Vapour pressure, surface

tension, viscosity

2.1 Introduction Radioactivity, Subatomic

particles; Atomic number, isotopes and

isobars, Thompson’s model and its

limitations, Rutherford’s picture of atom

and its limitations; Hydrogen atom

spectrum and Bohr model and its

limitations

1.1 Measurement Physical quantities and SI

units, Dimensional analysis, Precision,

Significant figures

1.2 Chemical Reactions Laws of chemical

combination, Dalton’s atomic theory;

Mole concept; Atomic, molecular and

molar masses; Percentage composition

empirical & molecular formula; Balanced

chemical equations & stoichiometry

of formation, phase transformation, ionization, electron gain;

2.4 Periodicity Brief history of the development of periodic tables Periodic law and the modern periodic table; Types

of elements: , , , and blocks; Periodic s p d f

trends: ionization energy, atomic, and ionic radii, inter gas radii, electron affinity, electro negativity and valency

Nomenclature of elements with atomic number greater than 100

2.2 Quantum Mechanics Wave-particle duality de-Broglie relation, Uncertainty principle; Hydrogen atom: Quantum numbers and wavefunctions, atomic orbitals and their shapes (s, p, and d), Spin quantum number

Molecular Structure

3.1 Valence Electrons, Ionic Bond Lattice Energy and Born-Haber cycle; Covalent character of ionic bonds and polar character of covalent bond, bond parameters

3.2 Molecular Structure Lewis picture &

resonance structures, VSEPR model &

molecular shapes

3.4 Metallic Bond Qualitative description

3.3 Covalent Bond Valence Bond Theory- Orbital overlap, Directionality of bonds &

hybridization ( , & orbitals only), s p d

Resonance; Molecular orbital theory- Methodology, Orbital energy level diagram, Bond order, Magnetic properties for homonuclear diatomic species (qualitative idea only)

3 Chemical Bonding &

2.3 Many Electron Atoms Pauli exclusion principle; Aufbau principle and the electronic configuration of atoms, Hund’s rule

4.4 Third Law Introduction

7 Chemical Kinetics

5.1 Concentration Units Mole Fraction,

Molarity, and Molality

4.3 Second Law Spontaneous and

reversible processes; entropy; Gibbs

free energy related to spontaneity and

non-spontaneity, non-mechanical work;

Standard free energies of formation,

free energy change and chemical

equilibrium

5.3 Physical Equilibrium Equilibria

involving physical changes

(solid-liquid, liquid-gas, solid-gas), Surface

chemistry, Adsorption, Physical and

Chemical adsorption, Langmuir

Isotherm, Colloids and emulsion,

classification, preparation, uses

6.1 Redox Reactions Oxidation-reduction

reactions (electron transfer concept);

Oxidation number; Balancing of redox

reactions; Electrochemical cells and

cell reactions; Standard electrode potentials; EMF of Galvanic cells;

Nernst equation; Factors affecting the electrode potential; Gibbs energy change and cell potential; Secondary cells; dry cells, Fuel cells; Corrosion and its prevention

7.1 Aspects of Kinetics Rate and Rate expression of a reaction; Rate constant;

Order and molecularity of the reaction;

Integrated rate expressions and half life for zero and first order reactions

Thermochemistry; Hess’s Law,

Enthalpy of bond dissociation,

combustion, atomization, sublimation,

solution and dilution

5.5 Ionic Equilibria Strong and Weak

electrolytes, Acids and Bases

(Arrhenius, Lewis, Lowry and Bronsted)

and their dissociation; degree of

ionization, Ionization of Water;

ionization of polybasic acids, pH; Buffer

solutions; Henderson equation,

Acid-base titrations; Hydrolysis; Solubility

Product of Sparingly Soluble Salts;

Common Ion Effect

5 Physical and Chemical Equilibria

5.6 Factors Affecting Equilibria

Concentration, Temperature, Pressure,

Na, Al, Cl & F 2 2

5.2 Solutions Solubility of solids and gases

in liquids, Vapour Pressure, Raoult’s

law, Relative lowering of vapour

pressure, depression in freezing point;

elevation in boiling point; osmotic

pressure, determination of molecular

mass; solid solutions, abnormal

molecular mass, van’t Hoff factor

Equilibrium: Dynamic nature of

equilibrium, law of mass action

5.4 Chemical Equilibria Equilibrium

constants (K , K ), Factors affecting P C

equilibrium, Le- Chatelier’s principle

8 Hydrogen and s-block Elements

7.3 Mechanism of Reaction Elementary reactions; Complex reactions;

Reactions involving two/three steps only

7.2 Factor Affecting the Rate of the Reactions Concentration of the reactants, catalyst; size of particles, Temperature dependence of rate constant concept of collision theory (elementary idea, no mathematical treatment); Activation energy;

Catalysis, Surface catalysis, enzymes, zeolites; Factors affecting rate of collisions between molecules

7.4 Surface Chemistry Adsorption Physisorption and chemisorption;

factors affecting adsorption of gasses

on solids; catalysis: homogeneous and heterogeneous, activity and selectivity:

enzyme catalysis, colloidal state:

distinction between true solutions, colloids and suspensions; lyophillic, lyophobic multi molecular and macromolecular colloids; properties of colloids; Tyndall effect, Brownian movement, electrophoresis, coagulations; emulsions – types of emulsions

8.1 Hydrogen Element Unique position in periodic table, occurrence, isotopes;

Dihydrogen: preparation, properties,

9 p - , d - and f - block Elements

9.1 General Abundance, distribution,

physical and chemical properties,

isolation and uses of elements; Trends in

chemical reactivity of elements of a

group; electronic configuration, oxidation

states; anomalous properties of first

element of each group

9.8 f - Block Elements Lanthanoids and

actinoids;O xidation states and chemical reactivity of lanthanoids compounds;

Lanthanide contraction and its consequences, Comparison of actinoids and lanthanoids

8.4 Alkaline Earth Metals Magnesium and

calcium: Occurrence, extraction,

reactivity and electrode potentials;

Reactions with O , H O, H and halogens; 2 2 2

Solubility and thermal stability of oxo

salts; Biological importance of Ca and

Mg; Preparation, properties and uses of

important compounds such as CaO,

Ca(OH) , plaster of Paris, MgSO , MgCl , 2 4 2

CaCO , and CaSO ; Lime and limestone, 3 4

cement

9.3 Group 14 Elements Carbon, carbon

catenation, physical & chemical

properties, uses, allotropes (graphite,

diamond, fullerenes), oxides, halides and

sulphides, carbides; Silicon: Silica,

silicates, silicone, silicon tetrachloride,

Zeolites, and their uses

9.6 Group 17 and group 18 Elements Structure and properties of hydrides, oxides, oxoacids of halogens (structures only); preparation, properties & uses of chlorine & HCl; Inter halogen

compounds; Bleaching Powder; Uses of Group 18 elements, Preparation, structure and reactions of xenon fluorides, oxides, and oxoacids

9.2 Group 13 Elements Boron, Properties

and uses of borax, boric acid, boron

hydrides & halides Reaction of

aluminium with acids and alkalis;

8.2 s-block Elements Abundance and

occurrence; Anomalous properties of the

first elements in each group; diagonal

relationships; trends in the variation of

properties (ionization energy, atomic &

ionic radii)

8.3 Alkali Metals Lithium, sodium and

potassium: occurrence, extraction,

reactivity, and electrode potentials;

Biological importance; Reactions with

oxygen, hydrogen, halogens water and

liquid ammonia; Basic nature of oxides

and hydroxides; Halides; Properties and

uses of compounds such as NaCl,

Na CO , NaHCO , NaOH, KCl2 3 3

and KOH

9.5 Group 16 Elements Isolation and chemical reactivity of dioxygen; Acidic, basic and amphoteric oxides;

Preparation, structure and properties of ozone; Allotropes of sulphur;

Preparation/production properties and uses of sulphur dioxide and sulphuric acid; Structure and properties of oxides, oxoacids (structures only), hydrides and halides of sulphur

9.4 Group 15 Elements Dinitrogen;

Preparation, reactivity and uses of

nitrogen; Industrial and biological

nitrogen fixation; Compound of nitrogen;

Ammonia: Haber’s process, properties and reactions; Oxides of nitrogen and their structures; Properties and Ostwald’s process of nitric acid production;

Fertilizers – NPK type; Production of phosphorus; Allotropes of phosphorus;

Preparation, structure and properties of hydrides, oxides, oxoacids (elementary idea only) and halides of phosphorus, phosphine

9.7 d - Block Elements General trends in the

chemistry of first row transition elements;

Metallic character; Oxidation state;

ionization enthalpy; Ionic radii; Color;

Catalytic properties; Magnetic properties;

Interstitial compounds; Occurrence and extraction of iron, copper, silver, zinc, and mercury; Alloy formation; Steel and some important alloys; preparation and properties of CuSO , K Cr O , KMnO , 4 2 2 7 4

Mercury halides; Silver nitrate and silver halides; Photography

reactions, and uses; Molecular, saline,

ionic, covalent, interstitial hydrides;

Water: Properties; Structure and

aggregation of water molecules; Heavy

water; Hydrogen peroxide: preparation,

reaction, structure & use, Hydrogen as a

fuel

9.9 Coordination Compounds Coordination number; Ligands; Werner’s coordination theory; IUPAC nomenclature; Application and importance of coordination

compounds (in qualitative analysis, extraction of metals and biological systems e.g chlorophyll, vitamin B , and 12

hemoglobin); Bonding: Valence-bond approach, Crystal field theory

13.2 Carbohydrates Classification;

Monosaccharides; Structures of pentoses and hexoses; Anomeric carbon; Mutarotation; Simple chemical reactions of glucose, Disaccharides:

reducing and nonreducing sugars – sucrose, maltose and lactose;

Polysaccharides: elementary idea of structures of starch, cellulose and glycogen

11.2 Conformations Ethane conformations;

Newman and Sawhorse projections

12 Organic Compounds with Functional Groups Containing Oxygen and Nitrogen

12.1 General Nomenclature, electronic structure, important methods of preparation, identification, important reactions, physical and chemical properties, uses of alcohols, phenols, ethers, aldehydes, ketones, carboxylic acids, nitro compounds, amines, diazonium salts, cyanides and isocyanides

11.1 Introduction Chiral molecules; optical

activity; polarimetry; R,S and D,L

configurations; Fischer projections;

enantiomerism; racemates;

diastereomerism and meso structures

12.2 Specific Reactivity of a-hydrogen in carbonyl compounds, effect of substituents on alphacarbon on acid strength, comparative reactivity of acid derivatives, mechanism of nucleophilic addition and dehydration, basic character of amines, methods of preparation, and their separation, importance of diazonium salts in synthetic organic chemistry

13 Biological, Industrial and Environmental Chemistry

13.1 The Cell Concept of cell and energy cycle

11.3 Geometrical isomerism in alkenes

13.3 Proteins Amino acids; Peptide bond;

Polypeptides; Primary structure of proteins; Simple idea of secondary , tertiary and quarternary structures of proteins; Denaturation of proteins and enzymes

13.5 Vitamins Classification, structure, functions in biosystems; Hormones

13.4 Nucleic Acids Types of nucleic acids;

Primary building blocks of nucleic acids (chemical composition of DNA &

RNA); Primary structure of DNA and its double helix; Replication;

Transcription and protein synthesis;

Genetic code

10 Principles of Organic Chemistry

and Hydrocarbons

10.1 Classification General Introduction,

classification based on functional

groups, trivial and IUPAC

nomenclature Methods of purification:

qualitative and quantitative

10.2 Electronic Displacement in a

Covalent Bond Inductive, resonance

effects, and hyperconjugation; free

radicals; carbocations, carbanions,

nucleophiles and electrophiles; types

of organic reactions, free radial

halogenations

10.3 Alkanes and Cycloalkanes Structural

isomerism, general properties and

chemical reactions, free redical

helogenation, combustion and

pyrolysis

10.4 Alkenes and Alkynes General

methods of preparation and reactions,

physical properties, electrophilic and

free radical additions, acidic character

of alkynes and (1,2 and 1,4) addition

to dienes

10.6 Haloalkanes and Haloarenes

Physical properties, nomenclature,

optical rotation, chemical reactions

and mechanism of substitution

reaction Uses and environmental

effects; di, tri, tetrachloromethanes,

iodoform, freon and DDT

10.7 Petroleum Composition and refining,

uses of petrochemicals

11 Stereochemistry

10.5 Aromatic Hydrocarbons Sources;

properties; isomerism; resonance

delocalization; aromaticity;

polynuclear hydrocarbons; IUPAC

nomenclature; mechanism of

electrophilic substitution reaction,

directive influence and effect of

substituents on reactivity;

carcinogenicity and toxicity

(qualitative); Stability constants;

Shapes, color and magnetic properties;

Isomerism including stereoisomerisms;

Organometallic compounds

14.4 Purification Methods Filtration, crystallization, sublimation, distillation, differential extraction, and chromatography Principles of melting point and boiling point determination;

principles of paper chromatographic separation – Rf values

Equilibrium studies involving ferric

2+

and thiocyanate ions (ii) [Co(H O) ] 2 6

and chloride ions; Enthalpy determination for strong acid vs

strong base neutralization reaction(ii) hydrogen bonding interaction between acetone and chloroform;

Rates of the reaction between (i) sodium thiosulphate and hydrochloric acid, (ii) potassium iodate and sodium sulphite (iii) iodide vs hydrogen peroxide, concentration and temperature effects in these reactions

14.6 Quantitative Analysis of Organic Compounds Basic principles for the quantitative estimation of carbon, hydrogen, nitrogen, halogen, sulphur and phosphorous; Molecular mass determination by silver salt and chloroplatinate salt methods;

Calculations of empirical and molecular formulae

14.7 Principles of Organic Chemistry Experiments Preparation of iodoform, acetanilide, p-nitro acetanilide, di-benzayl acetone, aniline yellow,b-naphthol; Preparation of acetylene and study of its acidic character

14.8 Basic Laboratory Technique Cutting glass tube and glass rod, bending a glass tube, drawing out a glass jet, boring of cork

14.3 Physical Chemistry Experiments

Preparation and crystallization of

alum, copper sulphate Benzoic acid

ferrous sulphate, double salt of alum

and ferrous sulphate, potassium ferric

sulphate; Temperature vs solubility;

Study of pH charges by common ion

effect in case of weak acids and weak

bases; pH measurements of some solutions obtained from fruit juices, solutions of known and varied concentrations of acids, bases and salts using pH paper or universal indicator; Lyophilic and lyophobic sols;

Dialysis; Role of emulsifying agents in emulsification

14.5 Qualitative Analysis of Organic Compounds Detection of nitrogen, sulphur, phosphorous and halogens;

Detection of carbohydrates, fats and proteins in foodstuff; Detection of alcoholic, phenolic, aldehydic, ketonic, carboxylic, amino groups and

unsaturation

14.1 Volumetric Analysis Principles;

Standard solutions of sodium

carbonate and oxalic acid; Acidbase

titrations; Redox reactions involving

KI, H SO , Na SO , Na S O and H S; 2 4 2 3 2 2 3 2

Potassium permanganate in acidic,

basic and neutral media; Titrations of

oxalic acid, ferrous ammonium

sulphate with KMnO , K 4 2

Cr O /Na S O , Cu(II)/Na S O 2 7 2 2 3 2 2 3

14 Theoretical Principles of

Experimental Chemistry

13.7 Pollution Environmental pollutants;

soil, water and air pollution; Chemical

reactions in atmosphere; Smog; Major

atmospheric pollutants; Acid rain;

Ozone and its reactions; Depletion of

ozone layer and its effects; Industrial

air pollution; Green house effect and

global warming; Green Chemistry,

study for control of environmental

pollution

14.2 Qualitative Analysis of Inorganic

Salts Principles in the determination of

13.6 Polymers Classification of polymers;

General methods of polymerization;

Molecular mass of polymers;

Biopolymers and biodegradable

polymers; methods of polymerization

(free radical, cationic and anionic

addition polymerizations);

Copolymerization: Natural rubber;

Vulcanization of rubber; Synthetic

rubbers Condensation polymers

13.8 Chemicals in medicine, health-care

and food: Analgesics, Tranquilizers,

antiseptics, disinfectants,

anti-microbials, anti-fertility drugs,

antihistamines, antibiotics, antacids;

Preservatives, artificial sweetening

agents, antioxidants, soaps and

detergents

PART III

a English Proficiency, b Logical Reasoning

5.4 Logical Deduction – Reading Passage Here a brief passage is given and based on the passage the candidate is required to identify the correct or incorrect logical conclusions

6 Non-verbal Reasoning

6.1 Pattern Perception Here a certain pattern is given and generally a quarter is left blank

The candidate is required to identify the correct quarter from the given four alternatives

6.3 Paper Cutting It involves the analysis of a pattern that is formed when a folded piece of paper is cut into a definite design

5.5 Chart Logic Here a chart or a table is given that is partially filled in and asks to complete it in accordance with the information given either in the chart / table or

in the question

6.2 Figure Formation and Analysis The candidate is required to analyze and form a figure from various given parts

3.1 Content/ideas

a English Proficiency

2.1 Synonyms,Antonyms,Odd Word,One

Word,Jumbled letters,Homophones,

1.1 Agreement, Time and Tense, Parallel

construction, Relative pronouns

This test is designed to assess the test takers’

general proficiency in the use of English

language as a means of self-expression in real

life situations and specifically to test the test

takers’ knowledge of basic grammar, their

vocabulary, their ability to read fast and

comprehend, and also their ability to apply the

elements of effective writing

1.2 Determiners, Prepositions, Modals,

Adjectives

3.3 Referents

The test is given to the candidates to judge their

power of reasoning spread in verbal and

nonverbal areas The candidates should be able

to think logically so that they perceive the data

accurately, understand the relationships

correctly, figure out the missing numbers or

words, and to apply rules to new and different

contexts These indicators are measured

through performance on such tasks as detecting

missing links, following directions, classifying

words, establishing sequences, and completing

5.2 Classification Classification means to assort the items of a given group on the basis of certain common quality they possess and then spot the odd option out

5.3 Series Completion Here series

of numbers or letters are given and one is asked to either complete the series or find out the wrong part in the series

2 Vocabulary

b Logical Reasoning

2.2 Contextual meaning

6.4 Figure Matrix In this more than one

set of figures is given in the form of a

matrix, all of them following the same

rule The candidate is required to

follow the rule and identify the

missing figure

6.5 Rule Detection Here a particular rule is given and it is required to select from the given sets of figures, a set of figures, which obeys the rule and forms the correct series

1.9 Sets, Relations and Functions, algebra

of sets applications, equivalence

relations, mappings, one-one, into and

onto mappings, composition of

mappings, binary operation, inverse of function, functions of real variables like polynomial, modulus, signum and greatest integer

1.2 Theory of Quadratic equations,

quadratic equations in real and

complex number system and their

solutions, relation between roots and

coefficients, nature of roots, equations

reducible to quadratic equations

2.5 Heights and distances

1.8 Matrices and determinants of order

two or three, properties and evaluation

of determinants, addition and

multiplication of matrices, adjoint and

inverse of matrices, Solutions of

simultaneous linear equations in two

or three variables, elementary row and

column operations of matrices,

1.7 Binomial theorem for a positive

integral index, properties of binomial

coefficients, Pascal’s triangle

3 Two-dimensional

3.1 Cartesian coordinates, distance between two points, section formulae, shift of origin

1 Algebra

1.5 Exponential series

1.3 Arithmetic, geometric and harmonic

progressions, arithmetic, geometric

and harmonic means,

arithmetico-geometric series, sums of finite

arithmetic and geometric progressions,

infinite geometric series, sums of

squares and cubes of the first n natural

numbers

1.6 Permutations and combinations,

Permutations as an arrangement and

combination as selection, simple

applications

1.1 Complex numbers, addition,

multiplication, conjugation, polar

representation, properties of modulus

and principal argument, triangle

inequality, roots of complex numbers,

geometric interpretations;

Fundamental theorem of algebra

2.1 Measurement of angles in radians and degrees, positive and negative angles, trigonometric ratios, functions and identities

1.11 Linear Inequalities, solution of linear inequalities in one and two variables

2.4 Inverse trigonometric functions

1.4 Logarithms and their properties

2 Trigonometry

2.2 Solution of trigonometric equations

1.10 Mathematical Induction

3.2 Straight lines and pair of straight lines:

Equation of straight lines in various forms, angle between two lines, distance of a point from a line, lines through the point of intersection of two given lines, equation of the bisector of the angle between two lines,

concurrent lines

2.3 Properties of triangles and solutions of triangles

3.3 Circles and family of circles : Equation

of circle in various form, equation of tangent, normal & chords, parametric equations of a circle , intersection of a circle with a straight line or a circle, equation of circle through point of intersection of two circles, conditions for two intersecting circles to be orthogonal

Coordinate Geometry

3.4 Conic sections : parabola, ellipse and hyperbola their eccentricity, directrices

7 Ordinary Differential Equations

5.5 Rolle’s Theorem, Mean Value Theorem

and Intermediate Value Theorem

6 Integral Calculus

6.1 Integration as the inverse process of

differentiation, indefinite integrals of

standard functions

6.2 Methods of integration: Integration by

substitution, Integration by parts,

integration by partial fractions, and

integration by trigonometric identities

6.4 Application of definite integrals to the determination of areas of regions bounded by simple curves

6.3 Definite integrals and their properties,

Fundamental Theorem of Integral

Calculus, applications in finding areas

under simple curves

11 Linear Programming

10 Statistics

10.2 Measures of skewness and Central Tendency, Analysis of frequency distributions with equal means but different variances

4.3 Equation of a plane, distance of a point

from a plane, condition for coplanarity

of three lines, angles between two

planes, angle between a line and a

plane

4 Three Dimensional

4.2 Angle between two lines whose

direction ratios are given, shortest

distance between two lines

5 Differential Calculus

5.1 Domain and range of a real valued

function, Limits and Continuity of the

sum, difference, product and quotient

of two functions, Differentiability

5.2 Derivative of different types of

functions (polynomial, rational,

trigonometric, inverse trigonometric,

exponential, logarithmic, implicit

functions), derivative of the sum,

difference, product and quotient of two

functions, chain rule

5.3 Geometric interpretation of derivative,

Tangents and Normals

5.4 Increasing and decreasing functions,

Maxima and minima of a function

4.1 Co-ordinate axes and co-ordinate

planes, distance between two points,

section formula, direction cosines and

direction ratios, equation of a straight

line in space and skew lines

Coordinate Geometry

& foci, parametric forms, equations of

tangent & normal, conditions for y =

mx + c to be a tangent and point of

tangency

7.1 Order and degree of a differential equation, formulation of a differential equation whole general solution is given, variables separable method

7.2 Solution of homogeneous differential equations of first order and first degree7.3 Linear first order differential equations

8 Probability

8.1 Various terminology in probability, axiomatic and other approaches of probability, addition and multiplication rules of probability

8.3 Independent events8.4 Discrete random variables and distributions with mean and variance

9 Vectors

9.1 Direction ratio/cosines of vectors, addition of vectors, scalar multiplication, position vector of a point dividing a line segment in a given ratio

9.3 Scalar triple products and their geometrical interpretations

9.2 Dot and cross products of two vectors, projection of a vector on a line

8.2 Conditional probability, total probability and Baye’s theorem

Science is a systematic attempt to understand natural phenomena in as much detail and depth as

possible and use the knowledge, so gained to predict, modify and control the phenomena

Every natural occurrence around us like the Sun, the wind, the planets, atmosphere, human body etc.,

follows some basic laws To understand these laws, by observing natural occurrence is called Physics.

These laws of physics are related and applicable to every aspect of life, thus understanding them leads

to their applications in several fields for further development of society, which is also known as

technology.

Physical Quantities

All those quantities which can be measured directly or indirectly and in terms of which the laws ofPhysics can be expressed, are called physical quantities For example, length, mass, temperature,speed and force, electric current, etc

Units of Physical Quantities

Unit of any physical quantity is its measurement compared to certain basic, arbitrarily chosen,

internationally accepted reference standard There are several systems of units like CGS (Centimetre,Gram and Second), FPS (Foot, Pound and Second) and MKS (Metre, Kilogram and Second)

Fundamental and Derived Units

The number of physical quantities is quite large Thus, we may define a set of fundamental quantities

and all other quantities may be expressed in terms of these fundamental quantities These all other

quantities are known as derived quantities Units of fundamental and derived quantities are known as the fundamental units and derived units, respectively A complete set of these units, both fundamental and derived units is known as the system of units.

Units, Measurement

and Dimensions

1

System of Units

There are some systems used in units, can be defined as

1 CGS System (Centimetre, Gram, Second) is often used

in scientific work This system measures, length in

centimetre (cm), mass in gram (g) and time in

second (s)

2 FPS System (Foot, Pound, Second) It is also called the

British Unit System This unit measures, length in foot

(foot), mass in gram (pound) and time in second (s)

3 MKS System (Metre, Kilogram, Second) This system

measures length in metre(m), mass in kilogram (kg)and

time in second (s)

4 SI Units (International System of Units) A variety of

system of units (CGS, FPS and MKS) leads to the need

of a unique system of units which is accepted

world-wide So, in 1971, a system of units named SI

(System International in French) was developed and

recommended by general conference on weights and

measures It is an extended version of the MKS system

SI system has seven fundamental units and two

supplementary units, which are as follows

The two supplementary units of SI system are

(i) Radian for Plane Angle Angle subtended by an arc

at the centre of the circle having length equal to

radius of circle has unit radians It is denoted by rad.

(ii) Steradian for Solid Angle It is the solid angle which

has the vertex at the centre of the sphere and cut-off

an area of the surface of sphere equal to that of

square with sides of length equal to radius of sphere

It is expressed in unit steradian and denoted by sr.

Precision of Measuring

Instruments

Measurement is the foundation of all experimental science

and technology The instruments used for measurement in

any experiment is called measuring instruments

Accuracy, Precision and Resolution of

an Instrument

(i) Accuracy An instrument is said to be the

accurate, if the physical quantity measured by a

measuring instrument resembles very close to its

true value

(ii) Precision An instrument is said to have high degree

of precision, if the value measured by it remains

unchanged, however large number of times it may

have been repeated

(iii) Resolution It stands for the minimum reading,

which an instrument can read

Least Count (LC)

The least count of a measuring instrument is the leastvalue, that can be measured using the instrument It isdenoted as LC

Least Count of Certain Measuring Instruments

● Vernier calliper, Least count = 1 mm

10 divisions=0.1 mm

● Screw gauge, Least count

= Value of 1 pitch scale reading

Total number of head scale divisionsLeast count= 1 mm

100 divisions

=0.01 mm

● Travelling microscope,Least count= Value of 1 main scale division

Total number of vernier scale divisions

The uncertainty in results of every measurement by any

measuring instruments, is called error in measurement.

There can be several causes of errors like instrumentalerrors, imperfection in experimental techniques orprocedures, error caused by random changes intemperature, pressure, humidity etc In systematic errors,mean of many separate measurement differs significantly

Calculation of Magnitude of Errors

(i) True Value

It is the mean of observed values

Σ

where, a a1, 2,K,a n are observed values and n is the number

of observations

(ii) Absolute Error

Absolute error of a particular measurement is the

difference between mean of observed value and true value

Absolute error,

∆a1=amean −a1,

∆a2=amean −a2,

M M Mand ∆a n=amean −a n

(iii) Mean Absolute Error

The arithmetic mean of the magnitudes of different values

of absolute errors, is known as the mean absolute error

∴ Mean absolute error,

n

n

mean =| 1| |+ 2|+ +K | |The final result of measurement can be written as

a=amean±∆amean This implies that value of a is likely to

lie as amean+ ∆amean and amean − ∆amean

(iv) Relative or Fractional Error

The ratio of the mean value of absolute error and the true

value, is known as the mean relative error

Mean relative error= Mean absolute error

Mean value of measurement

= ∆a

a

mean mean

(v) Percentage Error

When relative error is expressed in terms of percentage,

then relative error is called the percentage error

Hence,

Percentage error=∆a ×

a

mean mean

100%

Combination of Errors

(i) Sum of errors (Z) of two physical quantities A and B,

where∆Aand∆B are their absolute errors, is

∆Z= ±(∆A+∆B)

(ii) Difference of errors (Z) of two physical quantities A

and B, where∆Aand∆Bare their absolute errors, is

B B

B B

=   +  

Significant Figures

Significant figure in the measured value of a physicalquantity tells the number of digits in which we haveconfidence All accurately known digits in a measurementplus the first (only one uncertain digit together in ameasured value form significant figures) Larger thenumber of significant figures obtained in a measurement,greater is the accuracy of the measurement

Rules for Counting Significant Figures

(i) All the non-zero digits are significant In 2.738, thenumber of significant figures is 4

(ii) All the zeroes between two non-zero digits aresignificant, no matter where the decimal point is, if atall As examples, 209 and 3.002 have 3 and

4 significant figures respectively

(iii) If the measurement of number is less than 1, thezero (es) on the right of decimal point and to the left

of the first non-zero digit are non-significant

In 000807, first three underlined zeroes arenon-significant and the number of significant figures

is only 3

(iv) The terminal or trailing zero (es) in a numberwithout a decimal point are not significant Thus,12.3=1230cm=12300mm has only 3 significantfigures

(v) The trailing zero (es) in number with a decimal pointare significant Thus, 3.800 kg has 4 significantfigures

(vi) A choice of change of units does not change thenumber of significant digits or figures in ameasurement

Rules for Arithmetic Operations with Significant Figures

(i) In addition or subtraction, the final results shouldretain as many decimal places as there are in thenumber with the least decimal place As an examplesum of 423.5 g, 164.92 g and 24.381 g is 612.801 g, but

it should be expressed as 612.8 g only because theleast precise measurement (423.5 g) is correct to onlyone decimal place

(ii) In multiplication or division, the final result shouldretain as many significant figures, as are there in theoriginal number with the least significant figures.For example, suppose an expression is performed like

24.3

676.481522

×1243 =

44 65.Rounding the above result upto three significantfigures, the result would become 676

UNITS, MEASUREMENT AND DIMENSIONS 5

Rules for Rounding off the

Uncertain Digits

Result of arithmetic computation, we get a number having

more digits than the appropriate number of significant

figures, then these uncertain digits are rounded off as per

the rules given ahead

(i) The preceding digit is raised by 1, if the insignificant

digit to be dropped is more than 5 and is left

unchanged, if the latter is less than 5

e.g.18.764 will be rounded off to 18.8 and 18.74 to 18.7

(ii) If the insignificant figure is 5 and the preceding digit

is even, then the insignificant digit is simply dropped

However, if the preceding digit is odd, then it is

raised by one, so as to make it even e.g 17.845 will be

rounded off to 17.84 and 17.875 to 17.88

Dimensions of Physical

Quantities

The dimensions of a physical quantity are the power to

which the base quantities are raised to represent that

quantity The expression which shows how and which base

quantities represent the dimensions of a physical quantity,

is called the dimensional formula e.g for volume,

dimensional formula is [M L T ].0 3 0 An equation, where a

physical quantity is equated with its dimensional formula is

called dimensional equation e.g dimensional equation for

to establish relation among various physical quantities

Dimensional Analysis and Its Applications

Dimensional analysis help us in deducing certain relationsamong different quantities Main applications ofdimensional analysis are as follows:

To check the correctness of a given physical equation

If both sides of a physical relation have same dimensions,then the relation is dimensionally correct Dimensionalanalysis is also used to deduce relation among the physicalquantities, i.e if the dimensions of physical quantities onboth sides is known, then we can deduce relationscorrelating the quantities with these dimensions

To convert a physical quantity from one system to another

Let dimensional formula of a given physical quantity be[M L Ta b c] If a physical quantity is known in one system ofunit ( )n1 Then, we can relate it with another system of unit(n2)as below

n n

1 2

=



MM1 LL  TT 2

1 2 1 2

NOTE Here, a system having base units [M L T1, 1, 1]the numerical value of the given quantity be n1, and the numerical value n2in another unit system having the base units M L T2, 2, 2.

Practice Exercise

1. Which one is not a unit of time?

a Leap year b Year c. Shake d. Light year

2. The height of the building is 50 ft The same in

c. A screw gauge of pitch 1 mm and 100 divisions on

the circular scale

d. None of the above

4. The radius of hydrogen atom in ground state is

5 10× −11 m Find the radius of hydrogen atom in

7. The density of iron is 7.87 g/cm3 If the atoms are

spherical and closely packed The mass of iron atom

is 9 27 10 × −26kg What is the volume of an iron atom?

a. 1.18×10−29m3 b. 2 63 10 × −29m3

c. 173 10. × − 28m3 d. 053 10. × − 29m3

8. In the previous question, what is the distance between

the centres of adjacent atoms?

a. 2 82 10 × − 9m b. 0 282 10 × − 9m

c. 0 63 10 × −9m d. 6 33 10 × −9m

9. The world’s largest cut diamond is the first start of

Africa (mounted in the British Royal Sceptre and kept

in the tower of London) Its volume is 1.84 cubic inch

What is its volume in cubic metre?

a. 30.2 10× − 6 m3 b. 33.28 m2

10. Crane is British unit of volume

(One crane=170 474 litre) Convert crane into SI unit

13. One light year is defined as the distance travelled bylight in one year The speed of light is 3 10× 8 m/s.Find the same in metre

a. 8 ns b. 10

3 ns

16. The time taken by an electron to go from ground state

to excited state is one shake (one shake =10− 8 s).Find this time in nanosecond

19. Assuming the length of the day uniformly increases

by 0.001 second per century Calculate the neteffect on the measure of time over 20 centuries

a. 3.2 hour b 2.1 hour c 2.4 hour d. 5 hour

20. Find the number of molecules of H O2 in 90 g of water

a. 35 6 10. × 23molecules b. 4122 10. × 23molecules

c. 27 2 10. × 23molecules d. 3011 10. × 23molecules

21. The mass of Earth is 5 98 10 × 24 kg The average

atomic weight of atoms that make up Earth is 40 u

How many atoms are there in Earth?

a. 9 10× 51 b. 9 10× 49 c. 9 10× 46 d. 9 10× 55

22. One amu is equivalent to 931 MeV energy The rest

mass of electron is 9 1 10 × − 31kg The mass equivalent

energy is (Here, 1 amu=1.67 10× − 27kg)

23. One atomic mass unit in amu =166 10 × − 27 kg The

atomic weight of oxygen is 16 Find the mass of one

where, m=mass of the body, c=speed of light

Guess the name of physical quantity E

26. One calorie of heat is equivalent to 4.2 J BTU (British

Thermal Unit) is equivalent to 1055 J The value of

one BTU in calorie is

27. It is claimed that the two cesium clocks, if allowed to

run for 100 yr, free from any disturbance, may differ by

only about 0.02s Which of the following is the correct

fractional error?

a. 10−9 b. 10−5 c. 10−13 d. 10−11

28. Which of the following is the average mass density of

sodium atom assuming, its size to be about 2.5 Å

(Use the known values of Avogadro's number and the

atomic mass of sodium)

a. 0 64 10 × 3kg / m3 b. 8 0 10 × 2kg / m3

c. 8 6 10 × 3kg / m3 d. 6 4 10 × 5kg / m3

29. Electron volt is the unit of energy (1 eV 1.6 10= × − 19J)

In H-atom, the binding energy of electron in first orbit

is 13.6 eV The same in joule (J) is

a. 10 10× − 19J b. 21.76×10− 19J

c. 13.6×10− 19J d. None of these

30. 1 mm of Hg pressure is equivalent to one torr and one

torr is equivalent to 133.3 N/m2 The atmospheric

pressure in mm of Hg pressure is

31. One bar is equivalent to 10 N/m5 2 The atmosphere

pressure is 1.013 10 N/m× 5 2The same in bar is

32. 1 revolution is equivalent to 360° The value of

1 revolution per minute is

33. The height of a man is 5.87532 ft But measurement iscorrect upto three significant figures The correctheight is

38. If v=velocity of a body, c=speed of light

Then, the dimension ofv

c. F mv r

r

2

40. The maximum static friction on a body is F = µN

Here, N=normal reaction force on the body,

µ =coefficient of static friction The dimensions ofµis

a. [MLT− 2] b. [M L T0 0 0θ− 1]

c. dimensionless d. None of these

41. What are dimensions of Young’s modulus ofelasticity?

44. If∆H =mL , where m is mass of body.

∆H =total thermal energy supplied to the body

L=latent heat of fusion

Find the dimensions of latent heat of fusion

a. [ML T ]2 − 2 b. [L T ]2 − 2 c. [M L T ]0 0 − 2 d. [ML T ]0 − 1

45. Solar constant is defined as energy received by Earth

per cm2 per minute Find the dimensions of solar

a. [A M L T2 − − 1 3 4] b. [AM L T− − 1 3 4]

c. [A M L T2 − − 1 3 0] d. [A M L T2 0 − 3 4]

47. A physical relation isε ε ε= 0 r

where, ε =electric permittivity of a medium

ε0=electric permittivity of vacuum

εr =relative permittivity of medium

What are dimensions of relative permittivity?

d.None of the above

49. The electric flux is given by scalar product of electric

field strength and area What are the dimensions of

E=electric field strength

Find the dimensions of electric displacement

a. [AML T]−2 b. [AL T ]−2 −1

51. The energy stored in an electric device known as

52. The work done by a battery is W = ε ∆q, where

∆q=charge transferred by battery ε = emf of thebattery What are dimensions of emf of battery?

Here, J=current density,

n=number of electrons per unit volume,

e=16 10 × − 19unit

The unit and dimensions of e are

a. coulomb and [AT]

b. ampere per second and [AT ]− 1

c. no sufficient information

d. None of the above

54. The unit of current element is ampere-metre Find thedimensions of current element

55. The magnetic force on a point moving charge is

F=q(v×B)

Here, q=electric charge

v=velocity of the point charge

58. In the formula, a=3bc2‘a’ and ‘c’ have dimensions

of electric capacitance and magnetic induction,

respectively What are dimensions of ‘b’ in MKS

60. The magnetic energy stored in an inductor is given by

E= 1L I a b

2 Find the value of ‘a’ and ‘b’.

Here, L= self-inductance, I=electric current

a. a=3,b=0 b. a=2,b=1

c. a=0,b=2 d. a=1,b=2

61. In L-R circuit, I I= − e−t

0[1 /λ]

Here, I=electric current in the circuit Then,

a. the dimensions of I0andλare same

b. the dimensions of t andλare same

c. the dimensions of I and I0are not same

d. All of the above

62. A physical quantity u is given by the

relation u= B2

0

2µ .

Here, B=magnetic field strength

µ0=magnetic permeability of vacuum

The name of physical quantity u is

63. The energy of a photon depends upon Planck’s

constant and frequency of light Find the expression

for photon energy

65. The radius of nucleus is r =r A0 1 3/ , where A is mass

number The dimensions of r0is

where f is focal length of

the lens The dimensions of power of lens is

sinθ cosθ, then

a. the dimensions of x and a are same

b. the dimensions of a and b are not same

∫ sin− on the basis of

dimensional analysis, the value of n is

1. For the equation F ∝A v d a b c , where F is the force, A is

the area, v is the velocity and d is the density, the

values of a b , and c are, respectively [2014]

a. 1, 2, 1 b. 2, 1, 1 c. 1, 1, 2 d. 0, 1, 1

2. If edge lengths of a cuboid are measured to be

1.2 cm, 1.5 cm and 1.8 cm, then volume of the cuboid

a. 3.240 cm3 b. 3.24 cm3 c. 3.2 cm3 d. 3.0 cm3

3. If the force is given by F =at+bt2with t as time The

dimensions of a and b are [2012]

7. A resistor of 10 kΩ has a tolerance of 10% and

another resistor of 20 kΩhas a tolerance of 20% The

tolerance of the series combination is nearly [2009]

a. 10% b. 20% c. 15% d. 17%

8. The energy ( )E , angular momentum ( )L and universalgravitational constant ( )G are chosen as fundamentalquantities The dimensions of universal gravitationalconstant in the dimensional formula of Planck'sconstant ( )h is [2008]

β θ, where p is the pressure, z

the distance, k is Boltzmann constant and θ is thetemperature, the dimensional formula ofβwill be

a. [M L T ]0 2 0 b. [ML T]2

[2007]

c. [ML T ]0 − 1 d. [ML T ]2 − 1

10. A physical quantity is given by X=[M L T ] Thea b c

percentage error in measurement of M, L and T are

α, βandγ, respectively Then, the maximum % error in

d. None of the above

11. Which one of the following is not a unit of Young'smodulus? [2006]

1. (d) leap year, year and shake are units of time and light

year is the unit of distance

∴ 90 g of H O2 =6 022 10× ×

23

49

31 27

∴Fractional error=Difference in time (s)

28. (a) Average radius of sodium atom,

r =2.5 Å=2.5×10− 10m

∴Volume of sodium atom= 4

33

10 3 ( ) =65 42 10 × − 30m3Mass of a mole of sodium=23 g=23 10× − 3kgOne mole contains 6 023 10 × 23atoms, hence the mass ofsodium atom,

∴Average mass density of sodium atom

3 82 10

65 42 10

26 30

2 2 2]

46. (a) Unit ofε = C

Nm

2 2Dimensions ofε = [(AT)−

[MLT L

2

2 2

]]=[A M L T2 − −1 3 4]

ε

r =0Relative permittivity is the ratio ofεandε0, hence it is

2

Cm

L [LT

2

3

1 =[AT] amp-second= =coulomb

54. (d) Dimensions of current element are [ampere-metre]

Am

C

AmAs

ms

[ML T ] [M L T A ] [M L T A]2 − 2 = 1 2 − 2 − 2 a 0 0 0 b[MLT ]− 2 =[M L Ta 2a − 2aA'− 2a+b]

2 2

m

Nmm

Jm

=energy per unit volume=energy density

63. (a) E=h a.νb …(i)

where, h=Planck’s constant andν =frequency

UNITS, MEASUREMENT AND DIMENSIONS 13

∴ [Intensity of wave]=[ML T−

[TL

2 2 2

]] =[MT ]− 3

+sinθ cosθ

a n

2 =

or [ ][ ] [ ]

number of significant figures present in measurement

which has least number of significant figures, here all

measurement have 2 significant figures

dt dI

Since length (l) has two significant figure, the volume ( ) V

will also have two significant figure

Therefore, the correct answer isV=1.7×10− 6m3

Physical Quantity

Physical quantity is that which can be measured by available apparatus

Scalar and Vector Quantities

A scalar quantity is one whose specification is completed with its magnitude only Two or more than

two similar scalar quantities can be added according to the ordinary rules of algebra e.g., mass,distance, speed, energy etc

A vector quantity is a quantity that has magnitude as well as direction Not all physical quantities have

a direction Temperature, energy, mass, and time, for example, do not ‘‘point’’ in the spatial sense Wecall such quantities scalars, and we deal with them by the rules of ordinary algebra

Vector quantities can be added according to the law of parallelogram or triangle law.

A vector quantity can be represented by an arrow The front end (arrow head) represents the directionand length of the arrow gives its magnitude

NOTE Orthogonal vectors If two or more vectors are perpendicular to each other, then they are known as orthogonal vectors.

Unit vector A vector of unit magnitude and whose direction is same as the given vector is called unit vector Basically, unit vector represents the direction of the given vector.

Consider a vector A This vector is represented as

Vector=(Magnitude of the vector)×(Direction of the vector)

where, $A is a unit vector drawn in the direction of A.

Unit vector is a dimensionless physical quantity Unit vectors along X Y, and Z-axes are $,$ i j and $k respectively.

Scalar and Vectors

2

Laws of Vector Addition

There are three laws for the addition of vectors

(i) Triangle law of vector addition

(ii) Parallelogram law of vector addition

(iii) Polygon law of vector addition

Triangle Law of Vector Addition

If two vectors are represented

both in magnitude and direction

by the two sides of a triangle

taken in the same order, then

the resultant of these vectors is

represented both in magnitude

and direction by the third side of

the triangle taken in reverse

order as shown below

B

A B

Parallelogram Law of Vector Addition

According to parallelogram law of vector addition, if two

vectors acting on a particle are represented in magnitude

and direction by two adjacent sides of a parallelogram, then

the diagonal of the parallelogram represents the magnitude

and direction of the resultant of the two vectors acting on

the particle

i.e OA + AC = OC; A B = R+

Magnitude of the resultant R is given by

R= A2 +B2+2ABcosθ

Here,θ =Angle between A and B So, the direction of R can

be found by angleαorβof R with A and B.

Here, tan sin

cos

θ

=+

A

Polygon Law of Vector Addition

If a number of non-zerovectors are represented bythe (n −1 sides of an n sided)polygon taken in same orderthen the resultant is given by

the closing side or the nth

side of the polygon taken inopposite order So,

NOTE If the vectors form a closed n sided polygon with all the sides

in the same order, then the resultant is 0.

Vector in three dimension

If r =xi$+ y$j+zk$(a) | |r = x2+y2+z2

(b) Let r makesα β, andγ angles with x-axis, y-axis and

z-axis respectively, thencosα =

A

B C

u v–

v

–v

β α

The Scalar Product or Dot

Product

The scalar product of two vectors a and b in Fig (a) is

written as a b⋅ and is defined to be

a b⋅ =abcosφ …(i)where,φis the angle between the vectors a and b.

Because of the notation, a b⋅ is also known as the dot

product and is spelled as ‘‘a dot b.’’

(i) Dot product of the vectors with itself is equal to the

square of the magnitude of the vector

a a⋅ = ⋅a acos0° ⇒ ⋅ =a a a2 (cos0° =1)

Ifθ =180 , i.e vectors are anti-parallel.°

Then, a b⋅ =ab( )−1 [Qcos 180° = −1]

a b⋅ = −ab

i.e If two vectors are anti-parallel then their dot

product equals the negative product of the

magnitudes of vectors

Ifθ =90 , i.e vectors are perpendicular.°

a b⋅ =abcos 90°=ab( )0 =0

Vectors are perpendicular⇔Dot product=0

(ii) If a=a x^ i+a y^ j+a z^ k and b=b x^ i+b y^ j+b z^ kand θ

is the angle between a and b, then cosθ = ⋅a b

ab where,

a b⋅ =abcosθ

The component of a parallel to b in the vector form is

c a b b b

(a) Angle between−AandBis (180° − θ)

(b) Angle between A and−B is (180° − θ)

(c) Angle between−A and−B isθ

Important Points

● The dot product of forceF and displacement s gives work

(scalar quantity), i.e F s ⋅ = W.

● The dot product of force ( ) F and velocity ( ) v is equal to power (scalar quantity), i.e F v ⋅ = P.

● The dot product of magnetic induction ( ) B and area vector ( ) A is equal to the magnetic flux ( ) φ linked with the surface (scalar quantity) B A ⋅ = φ B

The Vector Product or Cross Prduct

The vector product of a and b, written

as a×b, produces a third vector c

whose magnitude is c=absinφ

where, φ is the smaller of the two

angles between a and b.

Because of the notation, a×b is

also known as the cross product, and

it is spelled as ‘‘a cross b’’.

(i) If two vectors are perpendicular to each other, wehave θ =90° and therefore, sinθ =1 So that,

i , j k

and (eachperpendicular toeach other)

SCALAR AND VECTORS 17

a

b

φ

a b

Plus

j

Minus

j k

1 An insect moves on a circular path of radius 7 m Find

the maximum magnitude of displacement of the

insect

a.7 m b.14πm c.7πm d.14 m

2 In previous problem, if the insect moves with constant

speed 10 m/s Find the minimum time to achieve

maximum magnitude of displacement

3 Two forces of magnitudes 3 N and 4 N are acted on a

body The ratio of magnitude of minimum and

maximum resultant force on the body, is

a.3/4 b.4/3

4 A vector a makes 30° and b makes 120° angle with the

x-axis The magnitude of these vectors are 3 unit and

4 unit, respectively The magnitude of resultant vector

is

a.3 unit b.4 unit c.5 unit d.1 unit

5 If two forces of equal magnitude 4 units acting at a

point and the angle between them is 120°, then find

the magnitude and direction of the sum of the two

8 Three forces are acted on a body Their magnitudes

are 3 N, 4 N and 5 N Then,

a.the acceleration of body must be zero

b.the acceleration of body may be zero

c.the acceleration of the body must not be zero

d.None of the above

9 In the given figure, O is the centre

of regular pentagon ABCDE Five

forces each of magnitude F0 are

acted as shown in figure The

resultant force is

a. 5F0 b.5F0cos72°

c.5F0sin72° d.zero

10 ABCD is a parallelogram, and a b c, , and d are the

position vector of vertices A, B, C and D of a

parallelogram, choose the correct option

11 A man walks 4 km due West, 500 m due South finally

750 m in South-West direction Find the distance andmagnitude of displacement travelled by the man

a 4646.016 m and 5250 m b 5250 m and 4646.016 m

c 4550.016 m and 2300 m d None of these

12 Calculate the resultant force, when four force of 30 N

due East, 20 N due North, 50 N due West and 40 Ndue South, are acted upon a body

a.20 2 N, 60°, South of West

b.20 2 N, 45°, South of West

c.20 2 N, 45°, South of East

d.20 2 N, 45°, South of East

13 A block of 150 kg is placed on an inclined plane with

an angle of 60° Calculate of the weight parallel to theinclined plane

a.1300 N b.1400 N c.1100 N d.750 N

14 A cat is situated at a point A (0, 3, 4) and rat is situated

at point B (5, 0,−8) The cat is free to move but the rat

is always at rest Find the minimum distance travelled

by cat to catch the rat

a.5 unit b.12 unit c.13 unit d.17 unit

15 An insect fly start from one corner of a cubical room

and reaches at diagonally opposite corner Themagnitude or displacement of the insect is 40 3 ft.Find the volume of cube

a.64 3ft3 b.1600ft3

16 In above problem, if the insect does not fly but crawls.

What is the minimum distance travelled by the insect?

a.89 44 ft b.95 44 ft

17 If a particle is moving on an elliptical path given by

r=bcosωti$+asinωt$j, then find its radial

18 Obtain the magnitude and direction cosines of vector

magnitude 2, 3, 2 N are acting at point A along the

lines AB, AC, AD, respectively Find their resultant.

20 A force F=ai$+bj$+ck$ is acted upon a body of

mass m If the body starts from rest and was at the

origin initially, find its new coordinate after time t.

2 ,2 ,2

b. at m

bt m

ct m

2

22

c.zero d.cos−1 2

15

22 The resultant of two vectors P and Q is R If the vector

Q is reversed, then the resultant becomes S, then

choose the correct option

24 Calculate the work done by a force F= +($i 2$j+3k$ )N

to displace a body from position A to position B The

position vector of A is r1= +($i 3$j+k$ ) m and the

c. the direction of c does not change, when the angle

between a and b increases

d.None of the above

26 The unit vector perpendicular to vectors a=3$i+$ andj

b=2i$− −j$ 5k$ is

a.± −($i 3$j+k$ )

11

b.± 3 +11

27 If three vectors along coordinate axes represent the

adjacent sides of a cube of length b, then the unit

vector along its diagonal passing through the originwill be

a.$i+ +$j k$2

(iv) C is perpendicular to ( A×B)

a.Only (i) and (ii) are correct

b.Only (ii) and (iv) are correct

c.(i), (ii) and (iii) are correct

d.All of the above

29 Find the vector area of a triangle whose vertices are

30 If three vectors xa −2b+3 ,c −2a+yb−4c and

−zb+3c are coplanar, where a b, and c are unit

(or any) vectors, then

a. xy +3zx −3z=4 b.2xy −3zx −3z− =4 0

c.4xy −3zx −3z =4 d. xy −2zx −3z− =4 0

31 A force F=( $2i+3j$−k$ ) N is acting on a body at a

position r=( $6i+3$j−2 Calculate the torque aboutk$ )the origin

a.( $3i+ 2$j+12k$ )Nm b.( $9i+2$j+7k$ )Nm

c.($i+2$j+12k$ )Nm d.( $3i+12$j+k$ )Nm

32 Find the values of x and y for which vectors

A=( $6i+x$j−2k$ ) and B=( $5i−6$j−yk$ ) are beparallel

a. x=0, y= 2

5, y=53

c. x= −15

3, y =235

d. x =36

5, y =154

33 Find the area of the parallelogram determined by

34 Choose the correct option.

a. a×(b×c)+ ×b (c×a)+ ×c (a×b)=0

b.a×(c×b)+ ×b (c×a)+ ×c (a×b)=0

c.a×(c×b)+ ×b (c×a)− ×c (a×b)=0

d.None of the above

35 The three conterminous edges of a parallelopiped are

a=2$i−6$j+ 3k $, b=5$, c j = −2$i+k$

Calculate the volume of parallelopiped

a.36 cubic units b.45 cubic units

c.40 cubic units d.54 cubic units

36 If the three vectors are coplanar, then find x.

A= −i$ 2j$+3 , B k$ =x$j+3 , C k$ =7i$+3j$−11k$

a.36 21/ b.−51 32/ c.51 32/ d.−36 21/

37 A particle is moving along a circular path with a

constant speed 30 m/s What is change in velocity of a

particle, when it describe an angle of 90° at the centre

of the circle

a.zero b 30 2 m/s c 60 2 m/s d 30 2 m/s

38 One day in still air, a motor-cyclist riding north at

30 m/s, suddenly the wind starts blowing Westwardwith a velocity 50 m/s, then calculate the apparentvelocity with which the motor-cyclist will move

a.58.3 m/s b.65.4 m/s c.73.2 m/s d.53.8 m/s

39 Calculate the distance travelled by the car, if a car

travels 4 km towards north at an angle of 45° to theeast and then travels a distance of 2 km towards north

at an angle of 135° to the east

a.6 km b.8 km

c.5 km d.2 km

40 On one rainy day a car starts moving with a constant

acceleration of 1.2 m/s2 If a toy monkey is suspendedfrom the ceiling of the car by a string, then find theangle with the vertical with the string be now inclined

2AB (A and B are magnitude of A and B

respectively), the angle between A and B is [2014]

a.30° b.45°

c.60° d.90°

2 A vector F1 acts along positive x-axis If its vector

product with another F2is zero, then F2could be[2009]

5 If a= +i$ 2j$−3k$ and b=3i$− +j$ 2k$, then the angle

between the vectors a + b and a – b is [2005]

According to parallelogram law of vectors,

8 (b) The magnitude of three forces

3N, 4N and 5 N will be zero, if these

vectors from a close polygon will all

the sides in the same order as

shown in figure

Hence, option (b) is correct

9 (d) According to polygon law, resultant force will be zero.

11 (b) The given figure shows the

direction of motion of man

14 (c) The minimum distance

=The magnitude of displacement of cat=|rB −rA|

5

15

25

C

G S

O E N

150

150sin

b a

D C

B A

O