[FREE PDF sample] Physics at a glance for class xi & xii, engineering entrance and other competitive

Page 1


Visit to download the full and correct content document: https://ebookmass.com/product/physics-at-a-glance-for-class-xi-xii-engineering-entra nce-and-other-competitive-exams-abhay-kumar/

More products digital (pdf, epub, mobi) instant download maybe you interests ...

Objective Physics for NEET and Other Medical Entrance Examinations Abhay Kumar

https://ebookmass.com/product/objective-physics-for-neet-andother-medical-entrance-examinations-abhay-kumar/

Class 8th Physics book 1st Edition Pawan Kumar

https://ebookmass.com/product/class-8th-physics-book-1st-editionpawan-kumar/

Atmospheric Remote Sensing Abhay Kumar Singh

https://ebookmass.com/product/atmospheric-remote-sensing-abhaykumar-singh/

Independent and Supplementary Prescribing At a Glance (At a Glance (Nursing and Healthcare)) (Nov 14, 2022)_(111983791X)_(Wiley-Blackwell) 1st Edition Barry Hill

https://ebookmass.com/product/independent-and-supplementaryprescribing-at-a-glance-at-a-glance-nursing-and-healthcarenov-14-2022_111983791x_wiley-blackwell-1st-edition-barry-hill/

Clinical Nursing Skills at a Glance (At a Glance (Nursing and Healthcare)) (Nov 8, 2021)_(1119035902)_(Wiley-Blackwell) 1st Edition

Fordham-Clarke

https://ebookmass.com/product/clinical-nursing-skills-at-aglance-at-a-glance-nursing-and-healthcarenov-8-2021_1119035902_wiley-blackwell-1st-edition-fordham-clarke/

Fundamentals of Inorganic Chemistry For Competitive Exams: Ananya Ganguly

https://ebookmass.com/product/fundamentals-of-inorganicchemistry-for-competitive-exams-ananya-ganguly/

Atmospheric Remote Sensing: Principles and Applications

Abhay Kumar Singh

https://ebookmass.com/product/atmospheric-remote-sensingprinciples-and-applications-abhay-kumar-singh/

Wiley's ExamXpert BBA Entrance Exams, 2ed Wiley

Editorial

https://ebookmass.com/product/wileys-examxpert-bba-entranceexams-2ed-wiley-editorial/

Pain Medicine at a Glance 1st Edition Beth B. Hogans

https://ebookmass.com/product/pain-medicine-at-a-glance-1stedition-beth-b-hogans/

Kumar for Class XI & XII, Engineering & Medical Entrance and other Competitive Exams

Abhay

Physics at a Glance

This page is intentionally left blank.

Physics at a Glance

Copyright © 2014 Dorling Kindersley (India) Pvt. Ltd.

Licensees of Pearson Education in South Asia

No part of this eBook may be used or reproduced in any manner whatsoever without the publisher’s prior written consent.

This eBook may or may not include all assets that were part of the print version. The publisher reserves the right to remove any material in this eBook at any time.

ISBN 9789332522053

eISBN 9789332537101

Head Office: A-8(A), Sector 62, Knowledge Boulevard, 7th Floor, NOIDA 201 309, India

Registered Office: 11 Community Centre, Panchsheel Park, New Delhi 110 017, India

Dedicatedtomyparents KaushalandPushpa

This page is intentionally left blank.

5.6.2

5.7 Spring 61

5.8 Non-concurrent Coplanar Forces 62

6. WORK, ENERGY, POWER AND CIRCULAR MOTION 63

6.1 Work Done 63

6.1.1 By a Constant Force 63

6.1.2 By a Variable Force 63

6.1.3 By Area Under F-x Graph 63

6.2 Power of a Force 64

6.2.1

6.3

6.3.1

6.3.2

6.3.3 Work-energy Theorem 65

6.3.4 Types of Equilibrium 66

6.3.5 Circular Motion 66

6.3.6 Turning of a Cyclist Around a Corner on the Road 67

6.3.7 A Car Taking a Turn on a Level Road 67

6.3.8

6.3.9

6.3.10

6.3.11

7.1.1

7.1.2 Position of Centre of Mass of Continuous System of

7.1.3 Position of Centre of Mass of More than Two Rigid Bodies 73

7.1.4 Position of Centre of Mass of a Rigid Body from Which Some Portion

8.5.4 Theorem on Moment of Inertia 89

8.5.5 Rolling of a Body on Horizontal Rough Surface 90

8.5.6 Rolling of a Body on Inclined Rough Surface of Inclination q 90

8.5.7 For Rolling with Forward Slipping 91

8.6 Radius of Gyration 95

8.6.1 Couple 96

8.6.2 Conditions for Equilibrium of a Rigid Body 96

9. GRAVITATION

9.1 Properties of Gravitational Force 99

9.1.1 Inertial Mass and Properties of Inertial Mass 100

9.2 Gravitational Mass 100

9.3 Acceleration Due to Gravity 100

9.4 Gravitational Field Strength 101

9.5 Gravitational Potential 102 9.6 Gravitational Potential Energy 103

9.6.1 Relation Between Field Strength E and Potential V 103

10. SOLIDS AND FLUIDS

10.1 Intermolecular Forces 107 10.2 Types of Bonding

10.3 Four States of Matter

10.4 Elasticity

Interatomic Force

(k) 111

10.4.5 Cantilever and Beam 111

10.4.6 Torsion of a Cylinder and Workdone in Twisting 111 10.4.7

10.5.2 Variation of Pressure in a Fluid with the Height from the Bottom of the Fluid 113

10.5.3 Hydrostatic Force Due to Many Liquid Layers 114 10.5.4 Pascal’s Law 114 10.5.5 Archimede’s Principle 115 10.5.6 Variation of Pressure in a Liquid in a Container If the Container Is to Be Accelerated 116

Excess Pressure Due to Surface Tension

10.7.3 Radius of New Bubble When Two Bubbles Coalesce

Radius of Interface

Capillarity

10.7.6 Zurin’s Law

10.7.7 Poiseuille’s Formula and Liquid Resistance 121 10.7.8 Stoke’s Law and Terminal Velocity 122 10.7.9 Reynold’s Number 123 10.7.10 Bernouilli’s Theorem

10.7.11 Torricelli’s Theorem 124

11. OSCILLATIONS AND WAVES (ACOUSTICS)

11.1 Different Equations in SHM 126

11.2 Graphs Related to SHM 127

11.2.1 Spring Block System 128

11.2.2 Pendulum 130

11.2.3 Physical Pendulum 130

11.3 Some Other Important Points Concerning SHM 130

11.3.1 Wave Equation 132

13.1.2

12.3.1 Entropy

15.

CAPACITORS

15.1 Capacitance 170

15.2 Isolated Conductor 170

15.3 Parallel Plate Capacitor 171

15.4 Spherical Capacitor 174

15.5 Cylindrical Capacitor 174

15.6 Combination of Capacitors 174

15.7 Dielectrics 175

15.7.1 Polarization of Dielectric Medium Placed in an Electric Field 175

16. OHM’S LAW, THERMAL AND CHEMICAL EFFECT OF ELECTRICITY 177

16.1 Electric Current 177

16.1.1 Series Combination 178

16.1.2 Parallel Combination 178

16.2 Resistance of a Conductor 179

16.2.1 Variation of Resistivity 179

16.3 Ohm’s Law: V = IR 180

16.4 How to Find Equivalent Resistance 181

16.4.1 Successive Reduction Method 181

16.4.2 Using Symmetry of the Circuit 184

16.4.3 Using Star-delta Conversion Method 189

16.4.4 Using Infinite Ladder Method 191

16.5 Colour Code for Carbon Resistors 194

16.5.1 Superconductivity 195 16.5.2 Potentiometer 195

16.6 Study About R–C Circuit 196

R–C Discharging Circuit 199

17. MAGNETIC EFFECT OF CURRENT AND

17.1 Magnetic Field Produced by Moving Charge or Current 203

Magnetic Force on a Moving Charge in Uniform Magnetic Field 203

Path of a Charged Particle in Uniform Magnetic Field

List of Formulae in Uniform Circular Motion

To Find Velocity and Position at Time t

Magnetic Force on a Current Carrying Wire in a Uniform Magnetic Field

Magnetic Force on a Curved Wire in Uniform B 206 17.8 Torque on a Current Carrying Coil Placed Inside a Magnetic Field 207

17.9 Magnetic Field at a Point Due to a Current or System of Current 208 17.10 List of Formulae 209

Ampere’s Circuital Law (ACL)

Magnet and Its Characteristics

17.13 Properties of a Magnet 214

17.14 Magnetic Lines of Force and Their Characteristics 216

17.14.1 Intensity of Magnetization 218

17.14.2 Magnetic Permeability 218

17.14.3 Magnetic Susceptibility 218

17.15 Earth’s Magnetism 219 18. ELECTROMAGNETIC INDUCTION AND ALTERNATING CURRENT 222

18.1 Magnetic Flux 222

18.1.1 Faraday and Lenz Law (I from B) 222

18.1.2 Lenz’s Law 223

18.2 Mechanism of Electromagnetic Induction Across a Conductor 223

18.3 How to Solve Problems Related to Motional EMF 225

18.3.1 Self-inductance 226

18.3.2 Mutual Inductance (M) 227

18.3.3 Inductor (Solenoid and Toroid) 227

18.3.4 Current Growth in L–R Circuit 228

18.3.5 Current Decay in L–R Circuit 229

18.3.6 LC-oscillatory Circuit 229

18.3.7 Alternating Current (AC) 231

18.3.8 Choke Coil 234

18.3.9 Transformer 234

19.1 Conduction Current 236

19.2 Displacement Current

19.3.1 Maxwell’s Equations

19.4 Electromagnetic Waves 237 19.5 Electromagnetic Spectrum 238

19.5.1 Radiowaves (Frequency Range: 500 kHz to About 1000 MHz) 238

19.5.2 Microwaves (Frequency Range: 1 GHz to 100 GHz) 238

19.5.3 Infrared (IR) Waves (Frequency Range: 1011 Hz to 5 × 1014 Hz) 238

19.5.4 Visible Light (Frequency Range: 4 × 1014 Hz to About 7 × 1014 Hz) 239

19.5.5 Ultraviolet (UV) Radiation (Frequency Range: 1014 Hz to 1017 Hz) 239

19.5.6 X-rays (Frequency Range: 1017 Hz to 1019 Hz) 239

19.5.7 Gamma Rays (Frequency Range: 1018 Hz to 1022 Hz) 239

Division of Wavefront

Division of Amplitude

Young’s Double Slit Experiment

19.9 Diffraction of Light

19.9.1 Fraunhofer Diffraction Due to a Single Slit

19.9.2 Fraunhofer Diffraction at a Circular Aperture

20. RAY OPTICS AND OPTICAL INSTRUMENTS

20.1 Reflection of Light 250

20.2 Characteristics of Image Due to Reflection by a Plane Mirror 250

20.2.1 Effect of Rotation of Plane Mirror on the Image 251

20.2.2 Number of Images Formed by Two Inclined Plane Mirrors 252

20.2.3 Concept of Velocity of Image in the Plane Mirror 254

20.3 Curved Mirrors 255

20.3.1 Concept of Velocity of Image in Spherical Mirrors 258

20.4 Refraction of Light 259

20.5 Laws of Refraction 259

20.5.1 Refraction at Plane Surface 260

20.5.2 Total Internal Reflection 260

20.5.3 Refractive Index (R.I.) and Critical Angle 260

20.5.4 Spherical Refracting Surfaces 261

20.5.5 Refraction from Spherical Surface 262

20.6 Lens 262

20.6.1 Lens Maker’s Formula 262

20.6.2 Nature of Image Formation by Convex Lens and Concave Lens 263

20.6.3 Concept of Velocity of Image in the Refraction Through Spherical Surface and Plane Surface 265

20.6.4 Concept of Velocity of Image in the Refraction Through Lens 265

20.7 Power of the Lens 267

20.7.1 Combinations of the Lenses 267

20.8 Prism 270

20.9 Defects of Vision of Human Eye 270

20.9.1 Simple Microscope 271

20.9.2 Compound Microscope 271

20.9.3 Astronomical Telescope 272

20.9.4 Terrestrial Telescope 272

21. ATOMS AND NUCLEI 273

21.1 Atoms 273

21.1.1 Dalton’s Atomic Theory 273

21.1.2 Thomson’s Atomic Model 273

21.1.3 Rutherford’s Atomic Model 273

21.1.4 Impact Parameter and Angle of Scattering 274

21.1.5 Bohr’s Atomic Model 275

21.1.6 Bohr’s Formulae 276

21.1.7 Hydrogen Spectrum 277

21.1.8 Kossel Diagram 278

21.1.9 Energy Level Diagram of Hydrogen Atom 279

21.1.10 Wave Model 280

21.1.11 Work Function 280

21.1.12 Electron Emission 281

21.1.13 Photoelectric Effect 281

21.1.14 Properties of Photon 282

21.2 Matter Wave or de Broglie Wave or Wavelength 283

21.3 X-rays 284

21.3.1 Mosley’s Law 284

21.3.2 Isotopes 284

21.3.3 Isobars 285

21.3.4 Isotones 285

21.3.5 Isomers 285

21.3.6 Mass Defect (Δm) 285

21.3.7 Binding Energy (ΔE) 285

22.4

22.5

22.6

Preface

Encouraged by the response to my earlier books, PracticeProblemsinPhysics, Volume I and II. I decided to work on a handbook that gives information to students for preparing basic study material such as concepts, definitions, tips, formulae and equations. In my 10 years of teaching experience, I have observed that students find it difficult to revise complete textbooks due to lack of time just before the exams. Therefore, it gives me immense pleasure to present this book, PhysicsataGlance, in which all essential topics are presented in the form of points. This book would be highly beneficial to students at the +2 level for competitive examinations like engineering and medical entrances. I am confident this book will help students brighten their chances of improving their ranks. I will appreciate comments and criticisms from the readers for further improvements of this book. Students can directly contact me at kumar.abhayk@gmail.com.

Abhay Kumar

Acknowledgements

At the outset, I want to express gratitude to my teachers—Professor M. M. R. Akhtar, Professor S. K. Sinha, Professor H. C. Verma, Professor S. N. Guha, Dr Shankar Kumar and Dr Vijay Kumar for their constant encouragement and appreciation. It is not possible for me to acknowledge everyone individually for their valuable suggestion. I would also like to thank all other individuals who have given their valuable suggestions leading to this book.

I am thankful to my parents, Kaushal and Pushpa, and my brother, Ajeet (Tinku) and Navin (Bittu) for their cooperation.

I am also thankful to my computer operator, Ravindra, for his sincere work and also to my students Ashutosh and Mohit for going through the manuscript.

I owe a special debt to my wife Awani (Reshma) for being supportive, understanding and a constant source of motivation. I am also grateful to my little daughter Meethee (Vanshika) whose cheerful face gives me enough patience to work.

Finally, I extend my sincere thanks to the Pearson team (Jitendra Kumar, Bhupesh Sharma and Satendra Sahay) for their constant support, suggestions and positive criticism.

Abhay Kumar

To the Students

A teacher without a student is lame and a student without a teacher is blind. Since it is not always possible to enjoy the personal presence of a teacher therefore an exhaustive and lucid textbook is needed. However, in a book you have to go through all definitions, equations, formulae of vast theories and concepts of physics before the examinations which becomes confusing. To guide you better, this book will help you to revise all topics in a short duration of time and it is loaded with concise text in the form of points.

I hope you will enjoy reading the book. Readers can directly reach me at kumar. abhayk@gmail.com

Mathematical Tools

1.1 TRIGONOMETRY

The branch of mathematics which deals with measurement of sides and angle of triangle is called trigonometry. There are two methods for measuring angles of triangle.

1. Degree method:

1(rt. ∠) = 90°, 1° = 60′ (min), 1′ = 60″ (sec)

2. Radian method:

(a) p radians = 180°

(b) An arc of length l makes angle q o at the centre of circle whose radius is r, then

3. Degree measure = 180 π × Radian measure

4. Radian measure = 180 π × Degree measure

5. sin2q + cos2q = 1, sin2q = 1 cos2q, cos2q = 1 sin2q

q 1

cosec2q cot2q = 1, cosec2q = 1 + cot2q, cot2q = cosec2q 1 8. tan q = sin cos θ θ , cot q = cos sin θ θ 9. sin q ⋅ cosec q = 1, tan q ⋅ cot q = 1, cos q ⋅ sec q = 1

Table 1.1 Sign of T-function

Quadrant1st2nd3rd4th

sin, cosec ++−−

cos, sec +−−+

tan, cot +−+−

1.1.1 Formulae for Compound Angle

1. sin (A + B) = sin A cos B + cos A sin B

2. sin (AB) = sin A cos B cos A sin B

3. cos (A + B) = cos A ⋅ cos B sin A ⋅ sin B

4. cos (AB) = cos A ⋅ cos B + sin A ⋅ sin B

5. tan (A + B) = + −⋅ tantan 1tantan AB AB

6. tan (AB) = +⋅ tantan 1tantan AB AB

1.1.2 Transformational Formula

1. 2 sin A ⋅ cos B = sin (A + B) + sin (AB)

2. 2 cos A ⋅ sin B = sin (A + B) sin (AB)

3. 2 cos A ⋅ cos B = cos (A + B) + cos (AB)

4. 2 sin A ⋅ sin B = cos (AB) cos (A + B)

5. sin C + sin D = 2 sin +− ⋅ cos 22 CDCD

7. cot (A + B) = ⋅− + cotcot1 cotcot AB AB

8. cot (AB) = ⋅+ cotcot1 cotcot AB BA

9. sin (A + B) ⋅ sin (AB) = sin2A sin2 B

10. cos (A + B) cos (AB) = cos2A sin2B

6. sin C sin D = 2 cos +− ⋅ sin 22 CDCD

7. cos C + cos D = 2 cos +− ⋅ cos 22 CDCD

8. cos C cos D = 2 sin +− sin 22 CDDC

1.1.3 Formulae for Multiple and Sub-multiple Angles

1. sin 2A = 2 sin A cos A = + 2 2tan 1tan A A

2. cos 2A = cos2A sin2 = + 2 2 1tan 1tan A A A

3. tan 2A = 2 2tan 1tan A A

4. sin 3A = 3 sin A 4 sin3A

5. cos 3A = 4 cos3A 3 cos A

6. = 3 2 3tantan tan3 13tan AA A A

9. tan A = 2 2tan 2 1tan 2 A A 10. 1cos 1cos A A + = tan2 2 A (II) sin (+ve) (I) all (+ve) (III) tan (+ve) (IV) cos (+ve)

7. sin A = 2 sin 2 A cos 2 A = 2 2tan 2 1tan 2 A A + 8. cos A = cos2 2 A sin2 2 A = 2 2 1tan 2 1tan 2 A A +

Notes

sin 18° = 51 4

sin 36° = 1026 4

1.1.4

cos 18° = + 1025 4

cos 36° = 51 4 +

Trigonometric Equations

The equation which contain trigonometric function is called T-Equation, e.g., cos x = 2 sin x

1. If sin x = 0 ⇒ x = np

2. If cos x = 0 ⇒ x = (2n + 1) 2 π

3. If tan x = 0 ⇒ x = np

4. If sin x = ± 1 ⇒ x = (4n ± 1) 2 π

5. If cos x = 1 ⇒ x = 2np

6. If cos x = 1 ⇒ x = (2n + 1)p

7. If sin x = sin y ⇒ x = np + ( 1)ny

8. If cos x = cos y ⇒ x = 2np ± y

9. If tan x = tan y ⇒ x = np + y

10. If sin2 x = sin2y ⇒ x = np ± y

11. If cos2 x = cos2 y ⇒ x = np ± y

12. If tan2 x = tan2y ⇒ x = np ± y where n = 0, ± 1, ± 2, ± 3, .....

Table 1.2

Some Trigonometrical Values

( q )(90° q )(90° + q )(180° q )(180° + q ) sin sinq cosq cosq sinq sinq cos cosq sinq sinq cosq cosq tan tanq cotq cotq tanq tanq cot cotq tanq tanq cotq cotq sec secq cosecq cosecq secq secq cosec cosecq secq secq cosecq cosecq

1.1.5 Value

of (2p ± q)

1. sin (2p + q) = sin q

2. cos (2p + q) = cos q

sin (2p q) = sin q

cos (2p q) = cos q

3. tan (2p + q) = tan q tan(2p − q) = tan q

4. cot (2p + q) = cot q

5. sec (2p + q) = sec q

6. cosec (2p + q) = cosec q

cot (2p q) = cot q

sec (2p q) = sec q

cosec (2p q) = cosec q

1.1.6 Value

3 sin cos, 2 π ⎛⎞θθ

2. 3 sin cos, 2 π ⎛⎞θθ

Table 1.3 Value of Some Standard Angles

Note: ∞ means undefined.

1.1.7 Inverse Trigonometric Functions

The value of inverse T-functions lies between the given range.

sin 1x, x ∈ , 22 ππ ⎡⎤ ⎢⎥ ⎣⎦ cot 1x, x ∈ (0, p)

cos 1x, x ∈ [0, p] sec 1x, x ∈ [0, p] 2 π ⎛⎞ ⎜⎟ ⎝⎠

tan 1x, x ∈ , 22 ππ ⎛⎞ ⎜⎟ ⎝⎠ cosec 1x, x ∈ , 22 ππ ⎡⎤ ⎢⎥

1. sin 1x + cos 1x = 2 π ( 1 ≤ x ≤ 1)

2. tan 1x + cot 1x = 2 π , x ∈ R 3. sec 1x + cosec 1x = 2 π x ≥ 1 4. sin x = cosec 1 1 x , x ≤ 1

5. cos 1x = sec 1 1 x , x ≤ 1

6. tan 1x = cot 1 1 x , ∞ < x < ∞

7. sin 1 ( x) = sin 1x

8. cos 1 ( x) = p cos 1x

15. sin 1x ± sin 1y = sin 1 () 1122 xyyx −±−

16. cos 1x ± cos 1y = cos 1 () 1122 xyxy ∓

17. tan 1x ± tan 1y = tan 1 1 xy xy ± ⎛⎞

18. 2 tan 1x = tan 1 2 2 1 x x

19. 2 cos 1x = cos 1(2x2 1)

9. tan 1 ( x) = tan 1x

10. cot 1 ( x) = p cot 1x

11. sec 1 ( x) = p sec 1x

12. cosec 1 ( x) = cosec 1x

13. y = sin 1x ⇒ x = sin y

14. x = sin y ⇒ y = sin 1x

20. 3 sin 1x = sin 1(3x 4x3) 21. 3 cos 1x = cos 1(4x3 3x) 22. 3 tan 1x = tan 1 3 2 3 13 xx x ⎛⎞

1.2.1 Quadratic Equation

An equation of the form ax2 + bx + c = 0, where a, b, c are certain numbers and a ≠ 0, is called a quadratic equation.

1. Discriminant of a quadratic equation: The numbers (b2 4ac) is called discriminant of the quadratic equation ax2 + bx + c = 0 and is denoted by D. i.e., D = b2 4ac.

2. Nature of roots of the quadratic equation: The value of x which satisfy the equation ax2 + bx + c = 0 are called roots of the equation. The roots a and b of the equation ax2 + bx + c = 0 are given by, −−−−−−+−−+ === 2244 , 2222 bbacbDbbacbD aaaa β

Now there are three possibilities :

Case I: When D < 0, i.e., b2 4ac < 0. In this case D will be imaginary, hence a and b will be both imaginary.

Case II: When D = 0, i.e., b2 4ac = 0. In this case D = 0, from the above equation, a = , 2 b a b = 2 b a . Hence both roots a and b will be real and equal.

Case III: When D > 0, i.e., b2 4ac > 0. Then the roots a and b will be real and different (distinct).

1.2.2 Determinants

Let a, b, c, d be any four numbers, the symbol ab cd denotes adbc and is called a determinant of second order. The elements of a determinant are multiplied diagonally, like, ab cd = adbc

For example, 24 32 = 4 12 = 8

The elements which lie in the same horizontal line constitute one row and the elements which lie in the same vertical line constitute one column.

Row1 Row2 ab

Column 1 Column 2

1.2.3 Determinant of Third Order

The determinant of 3rd order has three rows and three columns.

the expansion of the determinant along its first row will be

1.2.4 Progression

If the terms of a sequence are written under specific conditions, then the sequence is called progression. Here we shall study only two types of progressions.

Arithmetic Progression (A.P.)

An arithmetic progression is a sequence of numbers such that the difference between any two successive terms is a constant called common difference.

Examples

1. 1, 4, 7, 10, 13 … are in A.P., whose first term is 1 and common difference (c.d.) is 3.

2. The sequence of numbers 10, 8, 6, 2, 0, 2, 4, … are in A.P., whose first term is 10 and c.d. = 2.

In general, an A.P. is expressed as, a1, a2, a3 an, and the common difference is defined as d = a2 a1 =

Properties

1. The nth term of an A.P. is given by, an = a1 + (n 1)d.

2. The sum of the first n terms of an A.P. is given by

1.2.5 Geometric Progression

A geometric progression is a sequence of numbers such that the ratio of each terms to the immediately preceeding one is a constant called the common ratio.

Examples

1. The numbers 2, 4, 8, 16, 32, 64 … form a G.P. with common ratio = 2.

2. The numbers 1, 0.1, 0.01, 0.001, … constitute a G.P. with ratio 0.1. In general, a G.P. is expressed as, a1, a2, a3, … an; and the common ratio is defined as

Properties

1. The nth term of G.P. is given by an =

1; where a1 is the first term and r is the common ratio.

2. The sum of first n terms of G.P. is given by

when (r < 1)

The sum of infinite terms of G.P. for r < 1 is given by

1.2.6 Some Important Summation of Series

1. The sum of the first n natural number Sn = 1 + 2 + 3 + ……… + n = (1) 2 nn +

2. The sum of the squares of the first n natural numbers i.e., Sn2 = 12 + 22 + 32 + + n2 = (1)(21) 6 nnn++

3. The sum of the cubes of the first n natural numbers

n3 = 13 + 23 + 33 + ……… + n3 = 2 (1) 2 nn + ⎡⎤

1.2.7 Binomial Theorem for Any Index (1 + x)2 = 1 + 2x + x2 (1 + x)3 = 1 + 3x + 3x2 + x3

1. (1 + x)n = 1 + nx ++++ … (1)(1)(2)23 , 2!3! nnnnnn xxx where n is a +ve integer.

Number of terms in (1 + x)n is n + 1.

Meaning of factorial 2!

2. But if n is a ve integer or positive or negative fraction; then (1 + x)n = 1 + nx + (1)(1)(2)23 2!3! nnnnn xx++∞ …

Provided x < i.e., 1 < x < 1

Number of terms in this case will be infinite.

1.2.8 Exponential and Logarithmic Series

1. 111 1to , 1!2!3! e =++++∞ ……… which is 2.71828 …… is read as exponential number.

2. 23 1to , 1!2!3! xxxx e =++++∞ where x is any number.

3. 23 1to , 1!2!3! xxxx e =−+−+∞ ……… where x is any number.

4. loge (1 + x) = −+−+∞−≤≤ …… 234 to (11) 234 xxx xx

5. loge (1 x) = −−−−−∞−≤≤ 234 to (11) 234 xxx xx

1.3 CALCULUS

1.3.1 Limits

Let us consider the function y = f (x) = 2 4 . 3 x x

If we put x = 3, we have y =

94

33 = 5 0 = ∞ , which is meaningless. It means that the function is not defined at x = 3. But still, we want to know the value of the function at a value slightly smaller or greater than 3. If we could define the function at a value slightly smaller or greater than 3, then we say that the limit of function exists as x approaches 3. In mathematics it is represented by the symbol 3 lim x →

The expected value of the function f (x) to the left of a point x = a is called left hand limit. It is denoted by lim(). xa fx →

The expected value of function f (x) to the right of a point x = a is called right hand limit is denoted by lim(). xa fx + →

The limit of a function f (x) at point x = a is the common value of left and right hand limit. It is denoted by lim() xa fx →

A variable whose limit is zero is termed as infinitely small quantity (infinitesimal). Mathematically, it may be written as x → 0. A variable that constantly increases in absolute magnitude is termed as infinitely large quantity. Although infinitely large quantities do not have any limits but it is conventional to say that an infinitely large quantity ‘tends to an infinite limit’. The symbol → reads as ‘tends to’.

1.3.2 Basic Formulae of Limit

If f(x) and g(x) are two function then,

1. {} lim()()lim()lim() xaxaxa fxgxfxgx →→ → +=+

2. {} lim()()lim()lim() xaxaxa fxgxfxgx →→ → −=−

3. →→ == lim{()}{lim()}whereconstant xaxa kfxkfxk

4. →→ → ⋅=⋅ lim{()()}lim()lim() xaxa xa fxgxfxgx

5. lim() () lim ()lim() xa xa xa fxfx gxgx → → → ⎧⎫ = ⎨⎬ ⎩⎭

6. 1 lim nn n xa xa na xa → ⎧⎫ ⎪⎪ = ⎨⎬ ⎪⎪ ⎩⎭

7. 0 1 lim1 x x e x → = 8. 0 1 limlog x e x a a x → =

9. 0 log(1) lim1 x x x → + =

1/ 0 lim(1) x x xe → +=

Chapter 1

1.3.3 Continuity

Function f (x) at point x = a is said to be continuous if, L.H. lim = R.H lim = value of function at a. i.e., +− →→ == lim()lim()() xaxa fxfxfa

Notes

1. Function is discontinuous if f (a) is not defined.

2. lim()() xa fxfa → ≠

3. Constant polynomial identity and modulus function are continuous function.

1.3.4 Differentiability and Differentiate

If y = f (x) then differential coefficient of y with respect to (w.r.t) x is given by

Note

Every differentiable function is continuous but every continuous function is not differentiable.

1.3.5 For Two Functions: u and v

1. () duvdudv dxdxdx + =+

2. () duvdudv dxdxdx =−

3. ⋅ =+ () duvdvdu uv dxdxdx

4. 2 (/) dudv duvvudxdx dxv =

1.3.6 Chain Rule 1and

5. (constant) 0 d dx =

6. {()} {()}dkfxdfx k dxdx = where k = constant

7. 1 / dy dxdxdy =

1.3.7 D.C. of Some Important Functions

1. d dx (xn) = nxn 1

2. d dx (ax) = ax logea 3. d dx (ex) = ex 4. d dx (logex) = 1 x

5. d dx (sin x) = cos x

6. d dx (cos x) = sin x

7. d dx (tan x) = sec2 x

8. d dx (cot x) = cosec2 x

9. d dx (sec x) = sec x ⋅ tan x

10. d dx (cosec x) = cosec x ⋅ cot x

1.3.8 Maxima and Minima

11. d dx (sin 1x) = 2 1 1 x

12. d dx (cos 1x) = 2 1 1 x

13. d dx (tan 1x) = 2 1 1 x +

14. d dx (cot 1x) = 2 1 1 x + 15. d dx (sec 1x) = 2 1 1 xx

17. d dx (cosec 1x) 2 1 1 xx =

If, y = f(x) is a function and f ′(x) = 0 then at point x = a

1. Maximum if =− 2 2 dy ve dx 2. Minimum if =+ 2 2 dy ve dx

1.3.9 Integration

Integration is inverse process of differentiation. It is denoted by ∫ ∴ () d dxfx = g(x) then ∫g(x) dx = f (x) + c where c = Integration constant. There are two type of integration:

1. Definite integration

2. Indefinite integration

1.3.10 Indefinite Integration

1. ∫ [ f(x) + g(x)]dx = ∫ f(x)dx + ∫ g(x)dx

2. ∫ kf(x) = k∫ f(x)dx, where k = constant

3. ∫ 1dx = x

4. ∫ xndx 1 1 n x n + = +

5. ∫ 1 x dx = logex

6. ∫ axdx = log x a e a

7. ∫ sin xdx = cos x 8. ∫ cos xdx = sin x

9. ∫ sec2xdx = tan x

10. ∫ cosec2xdx = cot x

11. ∫ sec x ⋅ tan xdx = sec x

12. ∫ tan xdx = log sec x

13. ∫ cot xdx = log sin x

14. ∫ cosec dx = log (cosec x cot x)

15. ∫ sec xdx = log (sec x + tan x)

Turn static files into dynamic content formats.

Create a flipbook
Issuu converts static files into: digital portfolios, online yearbooks, online catalogs, digital photo albums and more. Sign up and create your flipbook.