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## PHYSICS

#### JEE MAIN AND JEE ADVANCED SYLLABUS OF GENERAL PHYSICS

In order to be able to answer all scientific inquiries into a universally intelligible format, one has to develop a commonly accepted language in which to converse. It was this need which led to the development of units and dimensions. It is an effort to do away with subjectivity of forms and personal prejudices and introduce a common objectivity. If we are to report the result of a measurement to someone who wishes to reproduce this measurement, a standard must be defined. Therefore, in order to reduce and eliminate such and other discrepancies, an international committee set up in 1960, established a set of standards for measuring the fundamental quantities.

From the IIT JEE point of view Measurement, Dimensions, Vectors and Scalars do not hold lot of significance as we cannot expect a number of questions directly based on this. However, we cannot completely ignore this chapter as this forms the basis of all chapters to follow

Topics covered under IIT JEE General Physics are :

• Dimensions
• Applications of Dimensions
• Scalars and Vectors
• Addition and Subtraction of Vectors
• Multiplication of Vectors
• Vector Components

#### JEE MAIN AND JEE ADVANCED SYLLABUS OF MECHANICS

Mechanics is one of the basic units in the preparation of IIT JEE, AIEEE and other engineering examinations. A beginner mostly starts with this unit in Physics. Mechanics is not only a lengthy portion in Physics but it forms the basis of entire Physics. It will not be an exaggeration to say that it is the most important unit in Physics from the point of view of preparation of IIT JEE, AIEEE and other engineering examination.

Mechanics begins with Kinematics which deals with the motion of particle in one and two dimension. This portion along with the next topic Newton's Laws of Motion fetches 2-3 questions in the IIT JEE, AIEEE and other engineering examinations every year. Work Power and Energy, Conservation of Momentum and Collision, Centre of Mass and Rotational Dynamics becomes very important portion from the viewpoint of Examination. These topics together form the heart of Mechanics.IIT JEE, AIEEE and other engineering examination fetches about 15 % of the total Physics questions. Those who get good IIT JEE rank always do well in this section. Simple harmonic Motion and Hydrostatics are also important as this fetches question in the IIT JEE, AIEEE and other engineering examination almost every year.

It is a point to note that this unit of Mechanics can be easily dealt with the proper understanding of the concept which is strengthened with practice of numerical problems.

Following topics are covered under mechanics:

• Kinematics
• Newtons Laws of Motion
• Conservation of Momentum
• Work Energy and Power
• Gravitation
• Fluid Mechanics
• Thermal Physics

#### JEE MAIN AND JEE ADVANCED SYLLABUS OF WAVE MOTION

Waves are present everywhere. Whether we recognize it or not, we encounter waves on a daily basis. We experience a variety of waves on daily basis including sound waves, radio waves, microwaves, water waves, visible light waves, sine waves, stadium waves, earthquake waves, cosine waves and waves on a string. Besides these waves we also experience various other motions which are similar to those of waves and are better referred as wavelike. These phenomena include the motion of a pendulum, the motion of a mass suspended by a spring and the motion of a child on a swing. Wave phenomena emerge in unexpected contexts. The flow of traffic along a road can support a variety of wave-like disturbances as anybody who has experienced a slowly moving traffic will know. The beat of your heart is regulated by spiral waves of chemical activity that swirl across its surface. You control the movement of your body through the action of electrochemical waves in your nervous system. Finally, quantum physics has revealed that, on a small enough scale, everything around us can only be described in terms of waves. The universe isn't really mechanical in nature. It's made of fields of force. When a radio antenna makes a disturbance in the electric and magnetic fields, those disturbances travel outward like ripples of water in a pond. In other words, waves are fundamental to the way the universe works.

Wave Motion Definition:
A waves motion can be defined as a disturbance that travels through a medium from one place to another. We consider the case of a slinky wave. When the slinky is stretched from end to end and is held at rest, it assumes an equilibrium position which is the position of rest. In order to induce a wave in slinky we first displace a particle of slinky from its position of rest. Wherever we move the coil whether upward or downward, forward or backward, it returns to its original position. But this movement creates a disturbance. If the slinky was moved in a back and forth direction then the disturbance observed in the slinky is called a slinky pulse. A pulse is a single disturbance that moves through a medium form one place to another. However, if the first coil of the slinky is continuously and periodically vibrated in a back-and-forth manner, it induces a repeating disturbance that continues for a longer duration. This disturbance is termed as a wave.

Frequency and Period of Wave:
The frequency of a wave refers to how often the particles of the medium vibrate when a wave passes through the medium. In mathematical terms, the frequency is the number of complete vibrational cycles of a medium per a given amount of time. The unit of frequency is the Hertz (abbreviated Hz) where 1 Hz is equivalent to 1 cycle/second. If a coil of slinky makes 2 vibrational cycles in one second, then the frequency is 2 Hz. If a coil of slinky makes 3 vibrational cycles in one second, then the frequency is 3 Hz. The period of a wave is the time for a particle on a medium to make one complete vibrational cycle. Period, being a time, is measured in units of time such as seconds, hours, days or years. The period of orbit for the Earth around the Sun is approximately 365 days; it takes 365 days for the Earth to complete a cycle.

Types of Wave Motion:
Waves come in various shapes and forms. Though the basic characteristics of wave motion are same and present in all waves but they can be distinguished on the basis of some distinguishing features.

Transverse Wave Motion:
The wave in which particles of the medium move in a direction perpendicular to the direction of the wave is called a transverse wave. Now again if we consider the case of a slinky then if it is stretched in a horizontal direction and a movement is produced in the first coil by moving it up and down, energy is transported from left to right. Since the movement of particles is perpendicular to the direction of movemnt of wave so it is an example of traneverse wave.

Longitudinal Wave Motion:
The wave in which the particles move in a direction parallel to the direction of the movement of the wave is called a longitudinal wave. As discussed in the last case, in a slinky, once a disturbance is produced, the energy is transported from left to right. The particles of the medium move in a direction parallel to that of the pulse. Hence, such waves are longitudinal waves.

Waves can also be categorized on the basis of their capability of transferring energy through a vacuum.

Electromagnetic wave motion:
Waves which are capable of transmitting energy through a vacuum and are produced by the vibration of charged particles are called electromagnetic waves. Light waves are an example of these waves. Electromagnetic waves are produced on Sun and travel to Earth through vacuum. These waves are responsible for the existence of life on Earth.

Mechanical Wave Motion:
Those waves which cannot transmit their energy through a vacuum and require a medium for same are called mechanical waves. Various examples include sound waves, water waves, slinky waves etc.

Equation of Wave Motion:
The wave motion equation can be expressed as
Speed = Wavelength • Frequency
It states the mathematical relationship between the speed (v) of a wave and its wavelength () and frequency (f). Using the symbols v, λ , and f, the equation can be rewritten as v = f • λ
Wave Motion is an important topic in the Physics syllabus of the IIT JEE. The topic usually fetches around 5-6 questions on an average.

#### JEE MAIN AND JEE ADVANCED SYLLABUS OF ELECTROSTATICS

Electrostatics is a vital branch of Physics. It is an interesting branch and questions are often asked from it in the JEE. It is important to have a strong grip on the topics of electrostatics in order to remain competitive in the JEE..

Electrostatics is the branch of Physics which is concerned with the study of those electric charges which are at rest or are stationary. It goes way beyond the study and exploration of the various properties of electricity and its applications. We illustrate this concept with the help of an example. When a rod of plastic is rubbed with fur or a glass rod is rubbed against silk, then it is generally observed that the rods start attracting some pieces of paper and seem to be electrically charged. While the charge on plastic is defined to be negative, that on silk is considered positive. The vast amount of charge in an everyday object is usually hidden, comprising equal amount of two kinds – positive and negative. The imbalance is always small compared to the total amounts of positive charge and negative charge contained in the object.

Electrostatics involves the building up of charge on the surface of the objects as a result of contact with other surfaces. The charge exchange occurs only when two surfaces come in contact and then finally separate. The effects as a result of this exchange can be noticed only when at least one of the two surfaces who came in contact is highly resistive to the electrical flow. The reason behind this is that these kinds of charges which are transmitted to or from highly resistive surface remain trapped there for a sufficient duration so that their effects can be observed. These charges stay there and ultimately they either ooze out in the form of bleeding to the ground or need to be immediately neutralized by a discharge.

The electromagnetic forces are vital forces which interact with particles in a variety of ways and this includes the minimal reactions of electrostatics.
Types of charges and their nature….

We have already discussed that there are two types of charges- positive and negative. It is a fundamental fact that while like charges repel, the unlike charges attract each other. This law forms the base of electrostatics. If two unlike charges are brought close to one another, they are pulled towards each other whereas if two like charges are brought close they are pulled away from each other

Electric charge

We discuss some of the characteristic features of an electrically charged object:

Firstly, the law which has already been discussed applies i.e. like charges repel and unlike charges attract. Main point to be note here is that the charge is conserved. As discussed in the earlier example of rod and fur, the net negative charge on the rod is same as the net positive charge on the fur.

Concepts of conductor and insulator...
As the name suggests, a conductor is the material which conducts electric charges or through which the charges can easily pass. Contrary to this is the concept of insulators. Insulators do not allow electric charges to pass through them easily. The charges on a conductor always assemble at the sharp points as a result of properties of electric fields. Let us consider the example of a metal cone. Obviously, due to the result explained just now, the charged cone will have maximum charge on its sharp edge and less charge elsewhere.

Remarks:

1. Human body acts like a conductor and lets current to easily flow through it. It is for this reason that one should be careful with the useful of electrical appliances.
2. Each type of material has a different arrangement of atoms, electrons and protons. This arrangement makes it a conductor or an insulator.
3. Metals are considered to be very good conductors while rubber is an insulator as it does not allow electric current to pass through it easily.
4. We list some of the items which are considered to be positively charged:

• Humanhair
• Wool
• Silk
• Nylon
• Cotton
• Teflon
• Wood

Applications of Electrostatics

Electrostatics deals with electromagnets and is used in numerous fields. Its major application is in paint spraying. Other uses include:

• Smokestacks
• Air fresheners
• Xerography
• Painting cars
• Insecticide spraying
• Inkjet printers
• Photocopiers

#### JEE MAIN AND JEE ADVANCED SYLLABUS OF MAGNETISM

Magnetism is a class of physical phenomena that includes forces exerted by magnets on other magnets. It has its origin in electric currents and the fundamental magnetic moments of elementary particles.

Now we answer the question what is magnetism? Magnetism is the force by which the objects are attracted or repelled by one another. Usually these objects are metals such as iron. Besides iron, other materials that are easily magnetized when placed in a magnetic field include nickel and cobalt. Magnetism is a force of attraction or repulsion that acts at a distance. It is due to a magnetic field, which is caused by moving electrically charged particles. Iron is not the only material that is easily magnetized when placed in a magnetic field; others include nickel and cobalt.

Magnets can also be formed and such magnets are called electromagnets. A simple electromagnet is formed with a battery and copper wire coiled around a metal rod such as a nail. There is evidence that there is an electrical basis for magnetism.

Pierre de Mari court checked angles pointed out by an iron rod placed at various points of a natural magnet. He found that the directions were in such a way that they rounded the sphere and passed through two points diagonally opposite, which he called the ends or poles of the magnet. Later experiments showed that every magnet, regardless of its shape, has two poles, called north and south poles, that exert repulsive as well as attractive forces on other magnetic poles just as electric charges exert forces on one another.

Later William Gilbert extended de Mari court's experiments to a variety of materials. Using the fact that a iron rod orients in some preferred directions, as per his hypothesis Earth itself is a large permanent magnet. Further a Torsion Balance was used for experiments and it was postulated that the force exerted varies inversely with the square of distance between them.

Magnetic Poles, Forces, and Fields

Magnetic Poles:
Every magnet has two poles. The poles are the points in a magnet where the magnetic strength is at its peak. These poles are called the north and the south poles or the north seeking and south seeking poles. When a magnet is suspended or hung somewhere the magnet lines up in a north - south direction. As we know that the like charges repel and the unlike charges attract, so when the North Pole of one magnet is kept close to the north pole of another magnet, the poles are repelled. When the south poles of two magnets are placed near one another, they also are repelled from one another. When the north and south poles of two magnets are placed near one another, they attract each other.

The attraction or repulsion of two magnets towards one another depends on how close they are to each other and how strong the magnetic force is within the magnet. The further apart of the magnets are the less they are attracted or repelled to one another.

Note : Even when a magnet is broken into pieces each broken piece has its own north and South Pole.

Magnetic Field:
The imaginary lines of flux originating from moving or spinning electrically charged particles constitute the magnetic field. For example the spin of a proton and the motion of electrons through a wire in an electric circuit constitute magnetic fields. It is undoubtedly a special property of space. All materials are influenced to some extent by a magnetic field. Most materials do not have permanent moments. Some are attracted to a magnetic field, others are repulsed by a magnetic field while others have a much more complex relationship with an applied magnetic field. Substances that are negligibly affected by magnetic fields are known as non-magnetic substances. They include copper, aluminium, gases and plastic. Even pure oxygen exhibits magnetic properties when cooled to a liquid state.

Magnetic Force:
The magnetic field of an object can create a magnetic force on other objects with magnetic fields. When a magnetic field is applied to a moving electric charge, such as a moving proton or the electrical current in a wire, the force on the charge is called a Lorentz force.

Attraction
When two magnets or magnetic objects are close to each other, there is a force that attracts the poles together. Attraction always occurs between unlike poles. Except iron, magnets also attract nickel and cobalt.

Repulsion
When two magnetic objects have like poles facing each other, the magnetic force pushes them apart. Magnets might also repel diamagnetic materials.

Magnetic and electric fields
There is a close relation between the magnetic and electric fields. They are similar as well as different.

Electric charges and magnetism similar
The poles of a magnet behave in the same way as the electric charges. The positive (+) and the negative (-) electrical charges attract each other and so do the North and the South Poles in a magnet. In electricity like charges repel, and in magnetism like poles repel.

Electric charges and magnetism different
The magnetic field is a dipole field as every magnet has exactly two poles. But in electricity, a positive (+) or a negative (-) charge can exist independently. Electrical charges are called monopoles, since they can exist without the opposite charge but a single magnetic pole can never be isolated.

#### JEE MAIN AND JEE ADVANCED SYLLABUS OF CURRENT ELECTRICITY

Electric current in simple words refers to the rate at which the electric charge flows in an electric field or an electric circuit.

Basic concept of current:
Electric current can also be defined as the rate of flow of charge through a particular area of cross section of a conductor. The current always flows in a direction which is from a region of higher potential to a region of lower potential. The direction of flow of electrons is opposite to direction of current because they carry negative charge and will move from a region of higher potential. Electric current is a scalar quantity. If we consider the case of water pipes, the water current flowing through the pipe can be assumed to be the electric current.

The unit of measurement of electric current is ampere (amp). If we consider the case of electric charges, then the rate at which electric charges pass through a conductor is also defined to be electric current. These charged particles may have any charge either positive or negative. Generally, some kind of force or a push is required by a charge to flow and this force is provided by either voltage or potential difference. The word 'current' is indeed an abbreviation for electric current. When we discuss this topic, the context of the situation is such that it automatically implies the adjective 'electrical'.

Current in gases and liquids includes flow of positive ions in one direction and of negative ions in the direction opposite to the first direction. If there exists a current of negative charge which is moving in opposite direction, then even that is included in the total current as it is assumed to be equal to positive charge of the same magnitude moving in the usual direction.

Electric current also leads to the formation of magnetic fields similar to the case of electromagnetics. Any kind of heat loss or loss of energy that occurs in a conductor by electric current is proportional to the square of the current.

CURRENT DENSITY
Current density, as it follows from the word itself refers to the density of the current. Mathematically, current density is the ratio of the electric current that flows in a conductor at a particular point to the cross-sectional area of the conductor. Hence, it denotes the amount of current flowing across a particular area. It is denoted by the symbol 'J' and its unit of measurement is amperes per square metre.

The mathematical formula for the calculation of current density is given by
J = I/A, where
'I' is the current in amperes that flows through the conductor.
'A' is the cross sectional area in m2.
Current density is a vector quantity and has the same direction as that of current. If we consider 'I' to denote the total electric current then the relationship between 'I' to the current density can be represented as
I = ∫ J. dS, where the integral runs over the area where current is flowing
This shows that the total current (I) equals the summation of current density over the area where charge is flowing.

Illustration :
A copper wire of area 5 mm2 has a current of 5 mA of current flowing through it. Calculate the current density?

Solution:
Given: Total Current I = 5 mA,
Total Area A = 5 mm2
The Current density is given by J = IA
= 5×10−3A5×10−3m
= 1 A/m2.

Drift Velocity:
It is a known fact that charged particles don't travel in straight lines in a conductor due to the obvious reason that they often collide with other particles present in the material. Therefore, the average speed at which the particle travels along the conductor is called the drift velocity. Inn other terms, the drift velocity may be defined as the average or the mean velocity attained by a particle as a result of electric field. It can also be called as the axial drift velocity since the particles are assumed to be moving in the plane .

Drift velocity formula:

The mathematical formula for calculation of drift velocity in a material exhibiting constant cross-sectional area is given by :
v = I/ nAq, where
v is the drift velocity of electrons
I is the current flowing through the conductor
n is the density of the charge-carrier
q is the charge on carrier
A is the area of the cross-section

#### JEE MAIN AND JEE ADVANCED SYLLABUS OF ELECTROMAGNETIC INDUCTION

Electromagnetic induction refers to the production of a voltage or a potential difference across a conductor when it is exposed to a changing magnetic field. The name of Faraday is generally acknowledged with the discovery of induction.

We first discuss the terms which will be used frequently in this topic and later move on to the Faraday Law of electromagnetic Induction .

Flux: Flux is defined as the rate of flow of a property per unit area. For example, the magnitude of a rivers current which gives the quantity of water flowing through a cross-section of the river each second is a kind of flux.

Emf: Emf is an abbreviation of electromagnetic force which is the voltage developed by any source of electrical energy such as a battery. It is denoted by and is measured in volts. The EMF is also given by the rate of change of the magnetic flux:

= - dφB/ dt, where is the electromagnetic force Emf in volts and ΦB is the magnetic flux.

Faradays laws of electromagnetic induction is a basic law of electromagnetism which describes the interaction of a magnetic field with an electric circuit to produce an emf I. It is the prime operating principle of various kinds of motors and generators.

Christian Oersted's discovery of magnetic field around a current carrying conductor was quite accidental. If a flow of electric current can produce a magnetic field then why cant a Magnetic field produce an electric current? While searching for an answer to this Michel Faraday ended up inventing generators.

We now discuss in detail the Michael Faraday Law:

Relationship between Induced Emf and Flux
In this experiment Faraday took a magnet and a coil and connected a galvanometer across the coil. In the beginning the magnet is at rest so there is no deflection in the galvanometer and hence the needle of galvanometer is at center or zero position. When the magnet is moved toward the coil, the needle of galvanometer deflects in one direction. When the magnet is held stationary at that position, the needle of galvanometer returns back to zero position. Now when the magnet is moved away from the coil , there is some deflection in the needle but in opposite direction and again when the magnet becomes stationary at that point with respect to coil , the needle of galvanometer return back to zero position.

Also if the magnet is held stationary and the coil is moved away and towards the magnet, the galvanometer shows deflection in a similar manner. It is also observed that faster the change in the magnetic field, the greater will be the induced emf or voltage in the coil.

The induced electromotive force in any closed circuit is equal to the negative of the time rate of change of the magnetic flux through the circuit. This version of the Faraday's law is valid only when the closed circuit is a loop of indefinitely thin wire.

Application of Electromagnetism in Physics (including the Faraday's Law):
Faraday's Law is a fundamental law of electromagnetism. This law has widespread applications in various fields including industries, electrical machines etc. some of the major ones are listed below:

Electrical Transformers
This is a static device which is used for increasing or decreasing thhe voltage or current. It has its applications in generating station, transmission and distribution system. The transformer works on Faradays law.

Electrical Generators
The basic working principle of electrical generator is Faradays law of mutual induction .Electric generator is used to convert mechanical energy into electrical energy.

Induction Cookers
The Induction cooker also works on principle of mutual induction. When current flows through the coil of copper wire placed below a cooking container, it produces a changing magnetic field. This alternating or changing magnetic field induces an emf and hence the current in the conductive container.

Electromagnetic Flow Meters
It is used to measure velocity of blood and certain fluids. When a magnetic field is applied to electrically insulating pipe in which conducting fluids are flowing then according to Faradays law an electromotive force is induced in it. This induced emf is proportional to velocity of fluid flowing.

Musical Instruments
It is also used in musical instruments like electric guitar, electric violin etc.

#### JEE MAIN AND JEE ADVANCED SYLLABUS OF RAY OPTICS

There are several ways to think about light in physics. One very useful way is to think of it in terms of rays. That is, to imagine light to be traveling in very narrow beams. When you do that, we say that you are modeling light as rays. This method allows one to develop an understanding of several light phenomena including common reflections and refractions.

An image in which the rays of light radiating from a point on the object converge to a physical point in space is called a real image. Parallel to this is the concept of a virtual image. When the rays from an object point never actually converge towards a different physical point in space but simply appear to the eye as if they were radiating from an image point, we call the image a virtual image.

Ray Optics covers a large variety of topics like the refractive index, mirror formula, lens formula, refraction at spherical surfaces, refraction through prism etc. We shall give a brief outline of these topics as they have been discussed in detail in the coming sections:

Reflection
Reflection is a simple yet important concept. Almost all objects in the world reflect a certain amount of light falling on them. When light falls on the object, it gets reflected and this is the color that is visible to the human eye. The ray that falls on the surface is termed as the incident ray and the angle that it makes with the normal is called the incident angle. The normal is the imaginary line that is perpendicular to the surface at the point where it is intersected by the incident ray. This ray again springs back as the reflected ray and it has the same angle of reflection as the angle of incidence from the normal.

Hence, from this discussion, we obtain the law of reflection which states that Angle of incidence = Angle of reflection

One point to be noted here is that all these lines including the incident ray, the reflected ray and the normal to the surface lie in the same plan. Reflection can broadly be categorized as Specular Reflection and Diffuse Reflection.

Specular Reflection
When light rays which are in the form of parallel lines strike against a smooth or a plane surface and then get reflected again in the form of parallel lines, then this form of reflection is called as the specular reflection. The following figure will prove useful in furthering clearing the concept of this kind of reflection:

Diffuse Reflection: When light rays fall in the form of parallel lines on a rough surface and as result they get reflected in all directions, such type of distortion is termed as diffuse reflection.

Refraction
The concept of refraction and the index of refraction are of immense importance. If light travels from point A to point B, then its speed will be highest if it travels in a straight line. But it has to pass through various materials then it will pass through them at different speeds and the motion will not be in a straight line. Hence, when it enters a new medium, it gets bent a bit and this bending is termed as refraction. The human eye always assumes light to be traveling in a straight line and hence when an object appears to be bent slightly due to refraction, we assume that the object is bent and not the light.

The index of refraction depends on the medium through which light passes. The speed of light is more in medium which are less optically dense and less in more optically dense mediums. The mathematical formula for calculation of index of refraction is:

n = Speed of light in vacuum/ Speed of light in the medium = c/v

The index of refraction is represented by the letter 'n' and denotes the angle at which the light bends.

When light travels from a medium with refractive index n1 to the other with refractive index n2, the relation between the angle of incidence θ1 and the angle of refraction θ2 is given by n1sin θ1 = n2 sin θ2

Since the speed of light in air is almost the same as the speed of light in a vacuum, so in most of the cases a value of one for the index of refraction of air. We have listed the indices of refraction for various substances in the table given below:

Substance (at 20° C) Index of refraction, n
Sodium Chloride 1.544
Water 1.333
Benzene 1.501

Refraction through a glass prism
When the visible white light passes through an equilateral prism, it experiences dispersion. It was proposed by Newton that it was possible to divide the white light into its various component colors with the help of an isosceles prism having equal sides and angles. As soon as the light ray falls on the surface of a dispersing prism, on entering it gets refracted and then passes through the glass unless it reaches the second boundary. Again the light gets refracted then it follows a new path on exiting. When the waves pass through prism, they get deviated by a certain angle which can be calculated. We obtain the angle of minimum deviation when the angle at which the light wave enters the prism permits the beam to pass through the glass in a parallel direction to the base.

As the values of the refractive index of a prism are increased, it also leads to an increase in the angle of deviation of light passing through prism. Refractive index is also affected by wavelength of light. The shorter wavelengths are refracted at greater angles while the longer wavelengths like red light are refracted at small angles. This variation in the angle of deviation in prism is termed as dispersion.

Ray optics discusses topics like refraction of light through prism or the angle of prism and it is quite different form Wave Optics. Ray theory does not describe phenomena such as interference and diffraction, which require wave theory (involving the phase of the wave).

Difference between Deviation and Refraction
Deviation and refraction are two concepts which are quite inter-related and often confused with each other. But there is a difference between the two. When light enters a different medium, the change that occurs in its path like bending or turning is termed as refraction. Deviation refers to the amount of this deflection in the path of light when it enters some other medium.

#### JEE MAIN AND JEE ADVANCED SYLLABUS OF WAVE OPTICS

Wave optics, also termed as Physic optics is the branch of optics which deals with various phenomenon including diffraction, polarization of light, interference pattern in optics and solved problems, Young's double slit experiment etc. It is an eminent branch which assumes great importance as a large number of questions in various competitions are generally asked from it. We shall discuss some of the chief topics of wave optics for IIT JEE here in brief as they have been discussed in detail in the coming sections:

Huygens' Wave Theory of Light
This is one of the most important principles of wave analysis which was introduced by an eminent physicist Huygens. The crux of the principle is that every point of a wave front can be treated as the source of secondary wavelets that spread out in all directions with the same speed as that of the speed of propagation of waves. This just implies that an edge of a wave can actually be viewed as creating a series of circular waves. Usually, these waves combine to carry on the propagation, but sometimes there are noteworthy evident results. The wave front which is defined as the surface, on which the wave disturbance is in phase, actually appears to be tangent to all the circular waves. Though these results can be easily derived from Maxwell's equations as well, but Huygens' Principle is better suited for performing calculations on waves. Maxwell was the one who provided solid theoretical basis to what Huygens' had already anticipated around two centuries back. According to this principle, a plane light wave passes through free space at the speed of light. The below figure demonstrates the motion of light rays associated with the propagation of wave front. They move in straight lines as shown here:

A critical description of the Huygens' Principle is:

(i) Each point on a wave front acts as a source of a new disturbance and therefore emits its own set of spherical waves which are called secondary wavelets. The secondary wavelets travel in all directions with the velocity of light as long as they move in the same medium.

(ii) The locus or the envelope of these wavelets in the forward direction indicates the position of new wave front at any subsequent time.

Diffraction
Diffraction basically refers to the bending of light around hindrances. This basically means that it creates some sort of interference in the passage of light. Another associated concept is of a diffraction grating. A diffraction grating refers to the screen with a bunch of parallel slits which are placed at a distance'd' from each other. Diffraction is in fact a special case of interference. It takes place when a wave hits against the barrier of an edge. The passing of light through some edge or gap is involved in almost all optical phenomena which clearly imply that diffraction takes place in almost all of them, though the impact might be negligible. A wave tends to bend around the hindrance as a result of diffraction. Diffraction can also be used for studying the structure of particular objects. It is even possible to reverse and move backward from the diffraction pattern to know about the nature of the object.

Young's Double –Slit Experiment
As a result of double slit experiment by Thomas Young in 1801, the wave theory of light came into the limelight. The double-slit experiment is based on the doctrines of constructive interference and destructive interference and hence proves that light resembles some of the properties of waves.

The experiment involves throwing up of light on a screen containing two narrow slits separated by a distance 'd'. At a distance 'L' from the first screen, a second screen is placed and the light which passes through the two slits shines on falling on this screen.

It is apparent form the figure that the light of single wavelength λ falls on the first screen and since the slits are narrower than λ, so the light passes and spreads all over the second screen. As depicted in the figure above the point P on the back screen is the point which receives light from two different sources.

#### JEE MAIN AND JEE ADVANCED SYLLABUS OF MODERN PHYSICS

Modern Physics is a very important constituent of Physics portion of the IIT JEE. It is interesting as well as usually fetches many questions in the JEE. It includes topics like Nuclear Fission and Fusion which are easy to master. These topics are quite fascinating but involve concepts which must be understood properly.

Modern Physics for IIT JEE refers to the Physics based on the two major branches: relativity and quantum mechanics. Classical Physics refers to the traditional Physics which was based on the concepts before coming up of Modern Physics. There were various theoretical and experimental paradoxes that forced thinking out of the traditional path. Modern physics is generally encountered when dealing with extreme conditions. Quantum mechanical effects appear in circumstances dealing with "lows" (low temperatures, small distances), while relativistic effects tend to appear when dealing with "highs" (high velocities, large distances), the "middles" being classical behavior. The Classical Physics was indeed in accord with common sense. Modern Physics has in fact come over that and imparts a better understanding of nature. Modern Physics in IIT JEE syllabus is the most scoring part.

Nuclear Fission and Fusion
Nuclear Fission and Fusion are two different kinds of energy releasing reactions. In these reactions, the energy is released from high- powered atomic bonds between the particles present in the nucleus. The two processes are quite opposite in nature. While Fission involves the splitting of an atom into two or more atoms, in Fusion, two or more smaller atoms combine to form a larger atom. We discuss both the processes one by one.

Nuclear Fission
Nuclear Fission is the process of splitting atoms. It is a process in nuclear physics in which the nucleus of the atom into smaller nuclei as fission products along with some by-produce particles. Fission hence may be termed as a form of elemental transmutation.

The by-products comprise free neutrons and photons which are generally in the form of gamma rays in addition to other nuclear fragments such as beta particles and alpha particles. Fission is an exothermic reaction which means there is a release of huge amount of energy when it takes place. Fission of heavy elements releases considerable amount of useful energy either in the form of gamma rays or as kinetic energy of the fragments. This energy may be used for nuclear power or for the explosion of nuclear weapons.

The sum of the masses of these fragments is less than the original mass. This gap in the mass which is around 0.1 percent of the original total mass has been converted into energy according to Einsteins equation.

Nuclear Fusion
In simple words, Fission refers to the process in which two or more atoms combine to form a larger atom. Nuclear energy can also be released by fusion of two light elements (elements with low atomic numbers). The power that fuels the sun and the stars is nuclear fusion. In a hydrogen bomb, two isotopes of hydrogen, deuterium and tritium are fused to form a nucleus of helium and a neutron. It may also be defined as the process in which multiple nuclei join together to form a heavier nucleus. It is accompanied by the release or absorption of energy depending on the masses of the nuclei involved.

Radioactive decay, as the word suggests refers to the decay to attain stability. It refers to the loss of particles from an unstable atom in order to attain more stability. The unstable elements emit some particles from their nucleus to gain stability and this process is termed as radio-activity. For elements, uniformity is produced by having an equal number of neutrons and protons which determines and henceforth directs the nuclear forces to keep the nuclear particles inside the nucleus. There may be cases when a particle becomes more frequent than another and hence creates an unstable nucleus. The unstable nucleus then releases radiation in order to gain stability. This radio-active decay can occur in five forms:

• Alpha emission
• Beta emission
• Positron emission
• Electron capture
• Gamma emission

As stated above, each decay emits some specific particle which also changes the type of product produces. The nuclei produced from the decay are called the daughter nuclei. The type of decay also determines the number of neutrons and protons found in the daughter nuclei. Let us consider an example of a radioactive substance.

The stable Beryllium contains 4 protons and 5 neutrons in its nucleus. A lighter isotope of beryllium is also available which contains 4 protons and only 3 neutrons, which gives a total mass of 7 amu. This isotope decays into Lithium-7 through electron capture. A proton from Beryllium-7 captures a single electron and becomes a neutron. This reaction produces a new isotope (Lithium-7) that has the same atomic mass unit as Beryllium-7 but one less proton which stabilizes the element.

Beta decay occurs when the neutron to proton ratio is too great in the nucleus and causes instability. In basic beta decay, a neutron is turned into a proton and an electron. The electron is then emitted. Heres a diagram of beta decay with hydrogen-3:

Half-Life
Half-Life is a very common term associated with radio-active decay. It cannot be easily detected when a single radioactive atom will decay. But, we can get an idea about the time required for half a large number of identical radioactive atoms to decay. This time is called the half-life.

Structure of Atom and Nucleus
An atom is made up of three subatomic particles: protons, neutrons and electrons. The protons and neutrons are placed inside the nucleus. The nucleus is at the center of the atom. The electrons keep on moving in orbits around the nucleus. The nature of the atom is determined by the number of protons. The protons carry positive charge, while electrons are negatively charged. Neutrons, as the name suggests are neutral and do not carry any charge. If the nucleus contains 17 protons, then the atom is chlorine. An atom of oxygen contains 8 protons in its nucleus.

​The nucleus is the dense central core of the atom which contains both the protons as well as the neutrons. Electrons are outside the nucleus in energy levels. Protons have a positive charge, neutrons have no charge, and electrons have a negative charge. An atom is said to be neutral of it has the same number of protons and electrons. The neutrons can vary in number in the atom of a particular element. Atoms of the same element that have differing numbers of neutrons are called isotopes.

## CHEMISTRY

### JEE Main and JEE Advanced Syllabus of Chemical Bonding

Chemical bonding is the major part of chemistry which is an interaction between two or more atoms that holds them together by reducing the potential energy of their electrons. In other words Bonds are the like chemists "glue" - which hold atoms together in molecules or ions. Valence electrons are the outer shell electrons of an atom which take part in chemical bonding.

Atoms gain or lose electrons to attain a more stable noble gas - like electron configuration (octet rule). There are two ways in which atoms can share electrons to satisfy the octet rule:

Ionic Bonding - occurs when two or more ions combine to form an electrically-neutral compound

The positive cation "loses" an electron (or 2 or 3)

The negative anion "gains" the electron (or 2 or 3)

The anion steals the electrons from the cation.

Covalent Bonding - occurs when two or more atoms combine to form an electrically-neutral compound. The electrons are shared between the two atoms.

Both atoms dont have charge in the beginning and the compound remains with zero charge.

The chemical activity of an atom is determined by the number of electrons in its valence shell. With the help of concept of chemical bonding one can define the structure of a compound and is used in many industries for manufacturing products.

### JEE Main and JEE Advanced Syllabus of Ionic Equilibrium

Chemical reactions mostly take place in solutions. Solution chemistry plays a very significant role in chemistry. All chemical substances are made up of either polar units (called ions) or non-polar units. The activity of these entities is more evident and pronounced in solution. The behaviour of these substances depends upon their nature and conditions of the medium in which they are added. It is therefore necessary to understand the principles that govern their behaviour in solution.

This type of equilibrium is observed in substances that undergo ionization easily, or in polar substances in which ionization can be induced. Ionic and polar substances are more easily soluble in polar solvents because of the ease of ionization taking place in the solvent medium. With the dissolution of ionic and polar substances in the solvent, these solutions become rich in mobile charge carriers (ions) and thus can conduct electricity. Substances, which are capable of conducting electricity are called as electrolytes while those substances which are non-conducting are called as non-electrolytes

### JEE Main and JEE Advanced Syllabus of States of Matter

States of matter is of prime importance to physicists. Everyday elements and compounds form three states of matter, however there are many other states, less common but equally important.

The liquid-crystal state of certain compounds has the properties of solids as well as liquids, and is the basis of electronic displays. More states are obtained when the particles are lighter.

The electrons in metals and ceramics undergo a change due to which electricity is conducted without dissipation.

Study States of matter for Preparation of IIT JEE AIEE at askiitians, with several underlined topics and important examples. All provided to you by ex IITians. States of matter as explained above is important both from exam as well as research point of view for future reference.

Solutions

This chapter besides being simple is very important too especially the section dealing with Raoult's Law , Colligative properties and Van't Hoff Factor.Colligative properties are those properties of solutions that depend on the number of dissolved particles in solution, but not on the identities of the solutes. For example, the freezing point of salt water is lower than that of pure water, due to the presence of the salt dissolved in the water.

To a good approximation, it does not matter whether the salt dissolved in water is sodium chloride or potassium nitrate; if the molar amounts of solute are the same and the number of ions are the same, the freezing points will be the same. For example, AlCl3 and K3PO4 would exhibit essentially the same colligative properties, since each compound dissolves to produce four ions per formula unit. The four commonly studied colligative properties are freezing point depression, boiling point elevation, vapor pressure lowering, and osmotic pressure. Since these properties yield information on the number of solute particles in solution, one can use them to obtain the molecular weight of the solute.

### JEE Main and JEE Advanced Syllabus of Redox Reactions

Oxidation can be defined as the loss of electrons by an atom or ion, while Reduction can be called as the gain of electrons by an atom or ion. Neither reduction nor oxidation occurs alone. Both of them occur simultaneously. Since both these reactions must occur at the same time they are often termed as "redox reactions".

The oxidation or reduction portion of a redox reaction, including the electrons gained or lost can be determined by means of a Half-Reaction:

• Reduction:
Fe3+(aq.) + 3e- → Fe(s)
• Oxidation:
Fe(s) → Fe3+(aq.) + 3e-

The substance that causes the oxidation of a metal is called the oxidizing agent. Or in other words, the Oxidizing agent is the substance being reduced. The substance that causes the reduction of the metal is called the reducing agent. Or in other words reducing agent gets oxidized (undergoing oxidation).

Redox reactions are mainly of two types:

• Inter Molecular Redox Reactions
• Intra Molecular Redox Reactions

It is very important to master these concepts at early stage as this forms the basis of your preparation for IIT JEE, AIEEE, DCE, EAMCET and other engineering entrance examinations.

### JEE Main and JEE Advanced Syllabus of Surface Chemistry

Surface chemistry is the study of chemical phenomena that occurs at the interface of two phases, usually between a gas and a solid or between a liquid and a solid. It can be roughly defined as the study of chemical reactions at interfaces. One of the important aspects of surface chemistry study is to determine whether a molecule attaches itself to a surface by chemisorption or by physisorption. Surface chemistry is of particular importance to the field of heterogeneous catalysis. It is the surface chemistry which deals with Colloidal states which finds a lot of uses

From the IIT JEE point of view it is really easy to score marks in the questions based on this topic as the questions are direct and simple.

### JEE Main and JEE Advanced Syllabus of Electrochemistry

Electrochemistry encompasses chemical and physical processes that involve the transfer of charge. It deals with the interactions of electrical energy with chemical species. It is broadly divided into two categories, namely (i) production of chemical change by electrical energy (phenomenon of electrolysis) and (ii) conversation of chemical energy into electrical energy, i.e., generation of electricity by spontaneous redox reactions. In this section, we will discuss Faraday's Laws of Electrolysis, Applications of Electrolysis, Electrochemical cell, Daniell Cell, Electrode Potential, Emf of a Galvanic Cell, Solved examples on Electrochemistry etc.

### JEE Main and JEE Advanced Syllabus of Chemical Kinetics

Chemical kinetics, also known as reaction kinetics, is the study of rates of chemical processes. Chemical kinetics includes investigations of how different experimental conditions can influence the speed of a chemical reaction and yield information about the reactions mechanism and transition states, as well as the construction of mathematical models that can describe the characteristics of a chemical reaction. In 1864, Peter Waage and Cato Guldberg pioneered the development of chemical kinetics by formulating the law of mass action, which states that the speed of a chemical reaction is proportional to the quantity of the reacting substances. Concept of order of reaction and how to determine order of reaction along with integrated rate laws are the most important topics of this chapter. Repeatedly questions appear in IITJEE and AIEEE from these topics.

### JEE Main and JEE Advanced Syllabus of Isomerism

What is Isomerism
In the study of organic chemistry we come across many cases when two or more compounds are made of equal number of like atoms. A molecular formula does not tell the nature of organic compound; sometimes several organic compounds may have same molecular formula. These compounds possess the same molecular formula but differ from each other in physical or chemical properties, are called isomers and the phenomenon is termed isomerism (Greek, isos = equal; meros = parts).

Since isomers have the same molecular formula, the difference in their properties must be due to different modes of the combination or arrangement of atoms within the molecule. Broadly speaking,there are two types of isomerism:

1. Structural Isomerism

2. Stereo Isomerism

Structural isomerism
When the isomerism is simply due to difference in the arrangement of atoms within the molecule without any reference to space, the phenomenon is termed structural isomerism. In other words, while they have same molecular formulas they possess different structural formulas. This type of isomerism which arises from difference in the structure of molecules, includes:

1. Chain or Nuclear Isomerism;

2. Positional Isomerism

3. Functional Isomerism

4. Metamerism and

5. Tautomerism

Stereoisomerism
When isomerism is caused by the different arrangements of atoms or groups in space, the phenomenon is called Stereoisomerism (Greek, Stereos = occupying space). The stereoisomers have the same structural formulas but differ in the spatial arrangement of atoms or groups in the molecule. In other words, stereoisomerism is exhibited by such compounds which have identical molecular structure but different configurations.Stereoisomerism is of two types:

1. Geometrical or cis-trans isomerism

2. Optical Isomerism.

### JEE Main and JEE Advanced Syllabus of Purification Classification And Nomenclature of Organic Compounds

Carbon atoms have a versatile nature to attach themselves to one another to an extent not possible for any other element. Carbon atoms can form long chains and rings containing thousand of atoms. The chains and rings can be branched and cross-linked. This versitile nature of carbon is the reason why there are millions of compounds of carbon present around us. Organic molecules are everywere around us.These are the part of not our body but also food & medicins. Organic chemistry is vey important for technology also as it is chemistry of ink, papaer, dyes, paint, galsoline, rubber and plastic. As it is mentioned above, there are millions of organic compounds around us, it becomes very important to classify them in groups in order to study them properly. Further as organic compounds play a very important role in our life, it is also important to identify and name them all. In this chapter we will study about the classification of organic compounds and their nomenclature and also about the techniques used for their purification.

### JEE Main and JEE Advanced Syllabus of Alkyl Halides

The haloalkanes (also known as halogenoalkanes or alkyl halides) are a group of chemical compounds, consisting of alkanes, such as methane or ethane, with one or more halogens linked, such as chlorine or fluorine, making them a type of organic halide. They are a subset of the halocarbons, similar to haloalkenes and haloaromatics. They are known under many chemical and commercial names. As flame retardants, fire extinguishants, refrigerants, propellants and solvents they have or had wide use. Some haloalkanes (those containing chlorine or bromine) have been shown to have negative effects on the environment such as ozone depletion. The most widely known family within this group is the chlorofluorocarbons (CFCs).

### JEE Main and JEE Advanced Syllabus of Alcohols and Ethers

In chemistry, an alcohol is any organic compound in which a hydroxyl group (-OH) is bound to a carbon atom of an alkyl or substituted alkyl group. The general formula for a simple acyclic alcohol is CnH2n+1OH. In common terms, the wordalcohol refers to ethanol, the type of alcohol found in alcoholic beverages.

Ethanol is a colorless, volatile liquid with a mild odor which can be obtained by the fermentation of sugars. (Industrially, it is more commonly obtained by ethylene hydration-the reaction of ethylene with water in the presence of phosphoric acid.) Ethanol is the most widely used depressant in the world, and has been for thousands of years. This sense underlies the term alcoholism (addiction to alcohol).

Other alcohols are usually described with a clarifying adjective, as in isopropyl alcohol (propan-2-ol) or wood alcohol (methyl alcohol, or methanol). The suffix -ol appears in the "official" IUPAC chemical name of all alcohols.

There are three major subsets of alcohols: primary (1°), secondary (2°) andtertiary (3°), based upon the number of carbon atoms the C-OH groups carbon is bonded to. Ethanol is a simple primary alcohol. The simplest secondary alcohol is isopropyl alcohol (propan-2-ol), and a simple tertiary alcohol is tert-butyl alcohol (2-methylpropan-2-ol).

### JEE Main and JEE Advanced Syllabus of Carbohydrates Amino Acids and Peptides

In The later part of 1950 resulted in classic advance in the knowledge of how living cells engage themselves with molecules such as carbohydrates additionally the metabolism of carbohydrates also became clarified. In Biochemistry Carbohydrates belong to basic category of chemical compounds. They are biological means of consuming energy or storing energy; other forms being via fat and protein. Complex carbohydrates are known as polysaccharides.

Amino acids and Peptides link in a head-to-tail style, i.e. the molecules are bounded by ionic interactions and H-bonds involving α-amino and α-carboxylate groups. Further, from several crystal structures it was confirmed that head to tail arrangement is unaffected by presence of water molecules. However, in the hydrated cases the case is different.

Askiitians provides free Study Material - Course Material for IIT JEE AIEEE preparation in Carbohydrates Amino Acids and Peptides. From the examination point of view Biomolecules and carbohydrates are very important in better understanding of many graduate courses specifically medical.

### JEE Main and JEE Advanced Syllabus of Hydrocarbons

What are hydrocarbons?
The term 'hydrocarbon' is self –explanatory which refers to the compounds formed by combination of carbon and hydrogen only. Hydrocarbons have very important role in our daily life. The Vegetable oil which is part of our food and also the gasoline which we use to run our vehicles is hydrocarbons. Hydrocarbons are the source of energy. The oil, ghee & butter are the hydrocarbon which are part of our diet and provide our body with energy required to perform various physical and biological functions.

LPG, CNG, LNG , Petrol, Diesel and Kerosene oil are the hydrocarbons which are used as fuels for automobiles and domestic uses. LPG is the abbreviated form of liquified petroleum gas, LNG is the abbreviated form of liquified natural gas whereas CNG stands for compressed natural gas. Petrol, diesel and kerosene oil are obtained by the fractional distillation of petroleum found under the earth's crust while coal gas is obtained by the destructive distillation of coal.

Hydrocarbons are also used for manufacture of polymers like polythene, polypropene, polystyrene etc and also as solvents for paints. They are also used as the starting materials for manufacture of many dyes and drugs.

Classification of Hydrocarbons
Hydrocarbons are broadly divided into aliphatic hydrocarbons and aromatic hydrocarbons. Aromatic hydrocarbons or arenes are the cyclic hydrocarbons with alternating double and single bonds betweeen carbon atoms. Benzene is an example of aromatic hydrocarbon.

Aliphatic hydrocarbons can further be divided into saturated hydraocarbons or alkane and unsaturated hydrocarbons. Saturated hydrocarbons cotain only single bonds throughout the length of carbon chain.

Unsaturated hydrocarbons contain one or more multiple bonds i.e double or triple bond, between the carbon atoms anywhere throughout the carbon chain. Unsaturated hydrocarbons with double bonds between the carbon atoms are called alkenes while those with triple bonds are called alkynes.

There is another class of aliphatic hydrocarbons called alicyclic hydrocarbons. These hydrocarbons are present in the form of rings.

### JEE Main and JEE Advanced Syllabus of Hydrogen and Its Compounds

Hydrogen is the most abundant element in universe and third most abundant element on the surface of earth. It is the simplest element with only one electron in its orbit around the nucleus congaing only one proton. It exist as a diatomic molecule i.e. H2 in its elemental form. The global concern related to clean energy makes it so important to study hydrogen separately from the other elements. This concern can be overcome to a greater extent by the use of hydrogen as a source of energy.

Position of Hydrogen in Periodic Table:
Hydrogen is the first element of periodic table but its position in the periodic table has been a subject of discussion for the past few years due to its similarities with both halogens and alkali metals. A proper position could not be assigned to hydrogen either in the Mendeleev's periodic table or Modern periodic table because of the following reason: In some properties, it resembles alkali metals and in some properties it resembles halogens. So hydrogen can be placed both in group 1 and group 17 with alkali metals and halogen respectively.

Isotopes of Hydrogen:
Hydrogen has three isotopes: Protium 1H1, or H , Deuterium 2H1 or D & Tritium 3H1 or T. These all differ from each other in respect of number of neutrons present in the nucleus. Protium does not contain any neutron, Deutrium ( also known as heavy hydrogen) contains one neutron while the number of neutrons in the nucleus of tritium is 2. Tritium is radioactive and emits low energy β- particles.

Name of Isotope Symbol Atomic Number Mass Number Relative Abundance
Protium 1H1 or H 1 1 99.99%
Deuterium 2H1 or D 1 2 0.015%
Tritium 3H1 1 3 10-18%

Resemblance with alkali metals

1. Electronic configuration:
Hydrogen contains one electron in the valence shell like alkali metals

Element Electronic Configuration
H 1s1
Li [He]2s1
Na [Ne]3s1
K [Ar]4s1
Rb [Kr]5s1

2.  Electropositive Character:
Like alkali metal, hydrogen also loses its only electron to form hydrogen ion, i.e, H-

3.  Oxidation state:
Like alkali metals, hydrogen exhibits an oxidation state of +1 in its compounds.

4.  Reducing agent :
Alkali metals act as reducing agents because of their tendency to lose valence electron. Hydrogen is also a very good reducing agent as evident from the following reactions:

5. Combination with electronegative elements :
Just like alkali metals hydrogen combines with electronegative elements such as halogen, oxygen, sulphur, etc to form compounds with similar formulae

Difference from Alkali Metals

1. Ionization enthalpy :
Ionization enthalpy of hydrogen (1312 kJ mol-1) is very high in comparison with the ionization enthalpy of alkali metals.

2. Existence of H- :
It has been established that H- ion does not exist freely in a aqueous solution. This is because of the fact that has a very small size as compared to normal atomic and ionic size (which range from 50 to 220 pm). Thus it exists in aqueous solution in the form of hydrated proton with a formula,H9O4-. However, for the sake of simplicity hydrated proton is represented by hydronium ion,H3O+.

On the other hand, the alkali metal ions mostly exist as hexahydrated ions

3. Difference in halides :
Hydrogen halides are different from the halides of alkali metals although they have similar molecular formulae. For example

(i) Pure HCl is a covalent compound while NaCl is an ionic compound.

(i) HCl is a gaseous compound while NaCl is a solid at ordinary temperature.

Resemblance With Halogens

1. Electronic configuration.
Just like halogens, hydrogen needs one electron to attain the configuration of nearest noble gas.

2. Atomicity.
Like halogens, hydrogen also exists in a diatomic state. The atomicity of hydrogen as well as halogens is two.

3. Electrochemical nature.
During electrolysis of LiH, CaH2, etc, in molten state hydrogen is evolved at the anode indicating its electronegative nature. In this respect, hydrogen shows resemblance with halogens which are also liberated at the anode during electrolysis.

4. Oxidation state.
Just like halogens, hydrogen also exhibit state of -1 in some of its compounds such as metal hydrides.

5. Combination with alkali metals.
Just like halogen, hydrogen also combines with alkali metals to form salts with similar formulae

6. Combination with non-metals.
Just like halogens hydrogen also react with non-metals such as carbon, silicon, germanium, etc, to form covalent compounds.

7. Ionization energy.
Ionization energy of hydrogen is comparable to the ionization energies of halogens as shown below:

Element H F CI Br
Ionization energies (Kj mol-1) 1312 1681 1255 1121

### JEE Main and JEE Advanced Syllabus of S and P Block Elements

The outermost orbital of s block elements consists of one or two electrons. Next to the outer most penultimate shell has either 2 or 8 electrons. s-block elements show a fixed valency which depends on the number of electrons present in the outermost shell.

Except hydrogen , all s-block elements have low values of ionisation potential decreases in the case of alkali metals and alkaline earth metals as the atomic number increases. Ionisation potential increases on moving horizontally from IA to IIA. On account of low values of ionisation potentials , these elements are highly electropositve , i.e., easily lose valency electrons and form cations.

The elements of groups of p block elements exhibit a range of properties. Many of the trends observed in this group can be understood from considerations of their electron configurations and their respective positions in the periodic table. For example, the inert pair of valence-shell electrons retained in the Ti+ ion is also found in several other cations following a transition series.

### JEE Main and JEE Advanced Syllabus of D and F Block Elements

The elements that lie in between S-block and P-block are the d-block elements. These elements are called transition elements as they show transitional properties between s and p-block elements. These elements contain partially filled d-orbitals and hence they are called as d-block elements.

The f-block of the periodic table of the elements consists of those elements whose atoms or ions have valence electrons in f-orbitals. Actual electronic configurations may be slightly different from what is predicted by the aufbau principle. The elements are also known as inner transition elements.

This section includes the important topics like screening effect, periodic table, hydration energy, magnetic and periodic properties etc.

### JEE Main and JEE Advanced Syllabus of Co-ordination Compounds

Co-ordination compounds are the compounds in which the central metal atom is linked to ions or neutral molecules by co-ordinate bonds, e.g. [Cr(H2O)5Cl]2+.

Coordination Compounds includes topics like Ligands, IUPAC Nomenclature, Isomerism, Valence Bond Theory and Organometallic Compounds etc which are very important from the point of view that these are prerequisite to Inorganic chemistry Sections. It is very important to master these concepts at early stage as this forms the basis of your preparation for IIT JEE, AIEEE, DCE, EAMCET and other engineering entrance examinations.

## MATHEMATICS

### JEE MAIN AND JEE ADVANCED SYLLABUS OF ALGEBRA

Algebra is one of the building blocks of Mathematics in IIT JEE examination. In fact, in the preparation of JEE this is the starting point. Algebra is a very scoring and an easy portion in the Mathematics syllabus of JEE. Though Algebra begins with Sets and Relations but we seldom get any direct question from this portion. Functions can be said to be a prerequisite to Calculus and hence it is critical in IIT JEE preparation. Sequence and series is another section which is mixed with other concepts and then asked in the examination. Quadratic equations fetch direct questions too and are also easy to grasp. Binomial Theorem is also a marks fetching topic as the questions on this topic are quite easy. Permutations and Combinations along with Probability is the most important section in Algebra. IIT JEE exam fetches a lot of questions on them. Those who get good IIT JEE rank always do well in this section. Complex Numbers are also important as this fetches question in the IIT JEE exam almost every year. Matrices and Determinant mostly give direct question and there are no twist and turns in the questions based on them.

We cover some of these contents here in brief as they have been discussed in detail in the coming sections:

Sets
A set is a well-defined collection of distinct objects. The different members of a set are called the elements of the set. Moreover all the elements are unique. Example: {1, 2, 3, 6} represents a set of numbers less than 10. Note that there is no repetition of elements in a set.

Relations and Functions
A relation is a set of ordered pairs. A function is a relation for which each value from the set the first components of the ordered pairs is associated with exactly one value from the set of second components of the ordered pair.

The first elements in the ordered pairs i.e. the x values form the domain while the second elements i.e. the y-values form the range. But, only the elements used by the relation are counted in the range.

The figure given below describes a mapping which shows a relation from set A to set B. As is clear from the figure, the relation consists of ordered pairs (1, 2), (3, 2), (5, 7), and (9, 8). The domain is the set {1, 3, 5, 9} and the range which is the dependent variable is the set {2, 7, 8}. Note that 3, 5, 6 are not a part of the range as they are not associated to any member of the first set.

Example: f(x) = x/2 is a function because for every value of x we get another value x/2, so f(2) = 1 f (3) = 3/2f (-6) = -3

Quadratic Equations: An equation of the form ax2 + bx +c = 0
where x represents the variable and a,b,c are constants with a not equal to zero.
If a = 0, the equation becomes linear and is no more a quadratic equation. An equation must have a second degree term in order to be a quadratic equation.
We give the formula of solving the quadratic equation
The general form of quadratic equation is ax2+bx+ c
Dividing by a, we obtain
x2 + b x/a = -c /a
(x+ b/2a)2 = -c/a + b2 / 4a2 = (b2-4ac) / 4a2
x+ b/2a = ±√b2-4ac / 2a
Solving this for x we get
x = (-b±√b2-4ac) / 2a
The above equation is called the quadratic formula

Binomial Theorem
The binomial theorem is used for expanding binomial expressions (a+b) raised to any given power without direct multiplication. Mathematically a binomial theorem can be defined as the theorem that gives the expansion of any binomial raised to a positive integral power say n. Such an expansion contains (n+1) terms.
The general expression for it is
( x + a)n = xn + nxn-1 a + [n(n-1)/2] xn-2a2 +…+ (nk) xn-kak + … + an, where (nk) = n!/(n-k)!k!, the number of combinations of kitems selected from n.

Permutation
Permutation is an ordered arrangement of the numbers, terms, etc., of a set into specified groups. The number of permutations of n objects taken r at a time is given by n! / (n-r)! The permutations of a, b, and c, taken two at a time, are ab, ba, ac, ca, bc, cb.

Combinations
A combination is also a way selecting certain things out of a larger group. But here order does not matter unlike permutation.

For example: If we need to form combination of two out f given three balls as in the previous case, only three cases are possible. As in the previous figure the first two constitute the same case in combination. Similarly the middle two and the last two also represent the same case.

The number of combinations of n objects taken r at a time is given by n! / r! (n-r)!

Complex Numbers
A complex number is a combination of a real and a imaginary number. They are written as a+ib, where a and b are real and I is an imaginary number with value √-1. Even 0 is also a complex number as 0 = 0 + 0i. Examples: 1+i, 2-3i, 6i, 3.

### JEE MAIN AND JEE ADVANCED SYLLABUS OF TRIGONOMETRY

Trigonometry is a vital constituent of Mathematics in IIT JEE examination. Trigonometric functions and trigonometry ratios are some of the most imperative areas of trigonometry. Since the IIT JEE exam asks a good amount of questions on this topic, so getting the knack of the basic trigonometry can surely help an IIT JEE aspirant to smooth his way through the exam.

Trigonometry is the branch of Mathematics that deals with triangles and the relationships between the sides and angles. The trigonometry functions are universal in parts of pure mathematics and applied mathematics which also lay the groundwork for many branches of science and technology. The IIT JEE Trigonometry problems range from the trigonometry basics to the applications of trigonometry.

Some of the trigonometry questions are simply based on trigonometry formulae and are quite easy to crack while others may demand some trigonometry tricks. Thus IIT JEE trigonometry syllabus is a perfect blend of questions of all levels

Given below is the trigonometry table that illustrates the values of the functions at different angles:

It would be an added advantage if the aspirants could memorize all above trigonometry formulas but if not; they must at least grasp the major ones like

The graphs also constitute a vital component in Trigonometry. A student who is well versed with the graphs of the major functions is able to tackle questions with ease. Some of the fundamental graphs are sketched below:

Enlisted below are some of the prime heads that come under trigonometry and are covered in the coming sections:

• Multiple and Sub-multiple Angles
• Trigonometric Equations
• Inverse circular Functions
• Trigonometric Inequality
• Trigonometric Functions
• Trigonometric Identities and Equations

For more in depth knowledge refer
Illustration 1
Find the angles and sides indicated by the letters in the diagram. Give each answer correct to the nearest whole number.
Since in general, it is simpler to use the value of 30°, so we will have to consider the triangle with 30°. Here,
r/60 = tan 30°.
Hence, r = 60tan (30°) = 34.64101615...
So answer correct to the nearest whole number is r= 35.
Now, that we have obtained the value of r, we use the first triangle to evaluate s.
r/s = tan(55°)
so, 35/s = tan(55°)
so, 35/tan(55°) = s = 24.50726384...
r = 35, s = 25

Illustration 2
Evaluate the value of (1 + cos π/8) (1 + cos 3π/8) (1 + cos 5π/8) (1 + cos 7π/8).
Solution: The given expression is
(1 + cos π/8) (1 + cos 3π/8) (1 + cos 5π/8) (1 + cos 7π/8)
= (1 + cos π/8) (1 + cos 3π/8) (1 - cos 3π/8) (1 - cos π/8)
= (1 – cos2π/8) (1 - cos23π/8)
= ¼ [2 sin π/8 sin 3π/8]2
= ¼ [2 sin π/8 cos π/8]2
= ¼ [sin π/4]2
= 1/4. 1/2 = 1/8

Illustration 3
If cos (α – β) = 1 and cos (α + β) = 1/e, where α, β ∈ [-π, π], then the values of α and β satisfying both the equations is/are
Solution: It is given in the question that cos (α – β) = 1 and cos (α + β) = 1/e, where α, β ∈ [-π, π].
Now, cos (α – β) = 1
α – β = 0, 2π, -2π
Hence, α – β = 0 (since α, β ∈ [-π, π])
So, α = β
Hence, cos 2α = 1/e
So, the number of solutions of above will be number of points of intersection of the curves y = cos 2α and y = 1/e
where α, β ∈ [-π, π]
It is quite clear that there are four solutions corresponding to four points of intersection P1, P2, P3 and P4.

Illustration 4
If k = sin (π/18) sin (5π/18) sin (7π/18), then what is the numerical value of k?
Solution: The value of k is given to be k = sin (π/18) sin (5π/18) sin (7π/18).
Solution: The value of k is given to be k = sin (π/18) sin (5π/18) sin (7π/18).
Hence, k = sin 10° sin 50° sin 70°
= sin 10° sin (60° - 10°). sin (60° + 10°)
= sin 10° [sin260° - sin210°]
= sin 10° [(√3/2)2 - sin210°]
= sin 10° [3/4 - sin210°]
= 1/4 [3sin 10° - 4sin310°]
= 1/4 x sin (3 x 10) (since, sin 3θ = 3sinθ- 4sin3 θ)
= 1/4 sin 30° = 1/8
Hence, the numerical value of k is 1/8

### JEE MAIN AND JEE ADVANCED SYLLABUS OF CO-ORDINATE GEOMETRY

Co-ordinate Geometry is a method of analyzing geometrical shapes. Co-ordinate Geometry is one of the most scoring topics of the mathematics syllabus of IIT JEE and other engineering exams. Besides calculus, this is the only topic that can fetch you maximum marks. It is a vast topic and can further be divided into various parts like:

• Circle
• Parabola
• Ellipse
• Hyperbola
• Straight Lines

All these topics hold great importance from examination point of view but the Straight Line and the Circle are the most important. These topics together fetch maximum questions in the JEE and moreover they are a pre requisite to conic sections as well.

The Coordinate Plane
In Coordinate geometry, points are placed on the coordinate plane. The horizontal line is the x-axis while the vertical line is the y-axis. The point where they cross each other is called the origin. A points location on the plane is given by two numbers, the first tells where it is on the x-axis and the second tells where it is on the y-axis. Together, they define a single, unique position on the plane. In the figure below, we have plotted the point (20, 15). Note here that the order is important as the first in the pair always stands for the x coordinate. Sometimes these are also referred to as the rectangular coordinates.

We have listed some of the important facts here, but the rest have been covered in detail in the later sections:

The Midpoint of a Line Joining Two Points
The midpoint of the line joining the points (x1, y1) and (x2, y2) is:
[½(x1 + x2), ½(y1 + y2)]
Illustration:
Find the coordinates of the midpoint of the line joining (11, 2) and (3, 4).
Midpoint = [½(11+ 3), ½(2 + 4)] = (7, 3)
You may refer the maths past papers to get an idea about the type of questions asked.

The Gradient of a Line Joining Two Points
The gradient of a line joining two points is given by
y2-y1 ∕ x2-x1

Parallel and Perpendicular Lines
Two parallel lines have the same gradient while if two lines are perpendicular then the product of their gradient is -1.Example

a) y = 4x + 1

b) y = -1/4 x + 12

c) ½y = 2x – 3

The gradients of the lines are 4, -1/4 and 4 respectively. Hence, as stated above lines (a) and (b) are perpendicular, (b) and (c) are perpendicular and (a) and (c) are parallel.

The figure below shows the various topics included in the conic sections:

Generally, conic section includes ellipse, parabola and hyperbola but sometimes circle is also included in conic sections. Circle can actually be considered as a type of ellipse. When a cone and a plane intersect and their intersection is in the form of a closed curve, it leads to the formation of circle and ellipse. As it is visible from the figure above, the circle is obtained when the cutting plane is parallel to the plane of generating circle of the cone. Similarly, in case the cutting plane is parallel to one generating line of the cone, the resulting conic is unbounded and is called parabola. The last case in which both the halves of the cone are intersected by the plane, produce two distinct unbounded curves called hyperbola.

Conic section is a vital organ of coordinate geometry. It is easy to gain marks in this section as there are some standard questions asked from this section so they can be easily dealt with. Some topics no doubt are difficult but can be mastered with continuous practice. The importance of Co-ordinate Geometry lies in the fact that almost all the students who aspire to get high All India rank in IIT JEE, AIEEE, DCE, EAMCET and other engineering entrance examinations give a good emphasis on Co-ordinate Geometry.

### JEE MAIN AND JEE ADVANCED SYLLABUS OF DIFFERENTIAL CALCULUS

Differential Calculus is one of the most important topics in the preparation of IIT JEE. This is the easiest part of Calculus and there is no doubt in the fact that it is scoring too. It is also important to attain proficiency in Differential Calculus as it is a prerequisite to the learning of Integral Calculus too.

Differential Calculus is a branch of mathematical analysis which deals with the problem of finding the rate of change of a function with respect to the variable on which it depends. So, differential calculus is basically concerned with the calculation of derivatives for using them in problems involving non constant rates of change. Applications also include computation of maximum and minimum values of a function.

The study of Differential Calculus includes Functions, Sets and Relations though they are considered to be a part of Algebra. Limits, Continuity and Differentiability are the favorite topics of those who have a bent towards Differential Calculus. It is not only an easy topic but also fetches direct question in the examination. A person who has already done a good practice of this chapter is also likely to do well in the next topic of Differentiation. The lifeline of Differential Calculus is basically the topics which include the application of Derivatives i.e.Tangent and Normal and Maxima and Minima. Differential calculus is closely related to integral calculus. In fact, differentiation is the reverse process of integration.

Here we shall discuss the main heads that are counted under differential calculus. These topics have been discussed in detail in the coming sections: Relation: A relation is a set of ordered pairs and is usually defined by a rule. The domain of a relation is the set of all first elements (usually x values) of its ordered pairs. The range of a relation is the set of all second elements (usually y values) of its ordered pairs.

Function
A function is a relation for which each value from the set of the first components of the ordered pairs is associated with exactly one value from the set of second components of the ordered pair.

Here is an example of a function:

It is an important example as students are likely to get confused as to whether it is a function or not. As stated in the definition, since each value of x is associated with exactly one value of y (though in this case that y is same for every x) hence it is a function.

Example: Consider f to be defined by
{(0,0),(1,1),(2,2),(3,3),(1,2),(2,3),(3,1),(2,1),(3,2),(1,3)}
This is a relation (not a function) since we can observe that 1 maps to 2 and 3, for instance
Limit of a function
Suppose f : R → R is defined on the real line and p,LR. Then we say that the limit of the function f is l if
For every real ε > 0, there exists a real δ > 0 such that for all real x,
0 < | x − p | < δ implies | f(x) − L | < ε.
Mathematically it is represented as
Example: Find the limit of the function f(x) = (x2-6x + 8) / x-4, as x→5.
Solution
The limit is 3, because f (5) = 3 and this function is continuous at x = 5.
Such easy questions are not asked in the exam so it was just meant to clear the concept of limit. We now move on to a bit difficult question:
Example: Find the limit of the function g(x) = √(x-4) -3 / (x-13) as x approaches 13.
Solution: In such a question you first need to reduce the function in simple form so that the computation of limit becomes simple.
First rationalize the numerator and denominator by multiplying by its conjugate
So, [√(x-4) -3] / (x-13) x [√(x-4) +3]/ [√(x-4) +3]
On multiplying and solving we get the result,
1/ √( x-4) +3
Now since the terms have been simplified the limit can now be calculated by substituting the value of x as 13.
Hence putting x=13 in the last equation we get the limit as 1/6.
For more on limits, you may refer the video
Continuity of a function A function y = f(x) is continuous at point x=a if the following three conditions are satisfied:
(1) f(a) is defined ,
(2) exists (i.e., is finite)
A function is continuous when its graph is a single unbroken curve. This definition proves to be useful when it is possible to draw the graph of a function so that just by the graph the continuity of the function can be judged.

Differentiation: Differentiation is an operation that allows us to find a function that outputs the rate of change of one variable with respect to another variable. The derivative of a function at a chosen input value describes the rate of change of the function near that input value. The process of finding a derivative is called differentiation. Geometrically, the derivative at a point is the slope of the tangent line to the graph of the function at that point, provided that the derivative exists and is defined at that point.

Since Differential Calculus is new to the students as they do not study it in their 10th standard examination, so they are advised to master the topic by practicing questions on Limits, Continuity and Differentiability. The preparation of Differential Calculus also gives another opportunity to prepare and revise the chapter on Functions, Sets and Relations. To get an idea about the types of questions asked you may also consult the Previous Year Papers.

### JEE MAIN AND JEE ADVANCED SYLLABUS OF INTEGRAL CALCULUS

Integral Calculus is the part of calculus that deals with integration and its application in the solution of differential equations and in determining areas or volumes etc.

It involves problems like solution of area and volume. In the figure given above, integral calculus may be used to find out the area between the two curves F1 and F2.

It would not be an exaggeration if we call Calculus to be the castle of Mathematics and its most important part is the integral calculus. The topic is quite scoring and has a good weightage in exams like IIT JEE, AIEEE etc. but it can be mastered only through constant practice.

Definite integrals and Indefinite integrals are the major tools which are used for applications under the topic of Area. Calculation of area is considered a hard topic but since it fetches many questions in JEE so it must be given due importance.

Indefinite Integral:

An integral of the form i.e. without upper and lower limits is called an indefinite integral. Indefinite integrals are often written as

where c is an arbitrary constant. Indefinite integrals are also called Antiderivatives.

Note : he constant of integration is very important. In fact, the graphs of Antiderivatives of a function are vertical translations of each other, the location of each graph depending on the value of c. Consider the following example:

The function F(x) = x3/3 is an antiderivative of f(x) = x2. As the derivative of constant is zero, x2 will have an infinite number of antiderivatives; such as (x3/3) + 0, (x3/3) + 7, (x3/3) − 42, (x3/3) + 293 etc. Thus, all the antiderivatives of x2 can be obtained by changing the value of C in F(x) = (x3/3) +C
Definite Integral : Given a function that is continuous on the interval [a,b] we divide the interval into n subintervals of equal width and from each interval choose a point x1* Then the definite integral of f(x) from a to b is

Properties of Definite Integral:

We list below various important rules that form the basis of solving numerical in definite integral:

(1) The limits on any definite integral can be interchanged. You just need to add a minus sign on the integral while doing so.

(2) When the upper and lower limits coincide, the value of the integral is zero.

(3) where c is any number.
A constant can be taken out of the integral sign in case of both definite and indefinite integral.

(4) A definite integral can be broken into parts across a sum or difference.

(5) where c is any number.
This property tells us how to integrate a function over adjacent intervals, [a,c] and [c,b]. Note that it is not necessary for c to be between a and b.

(6) This property shows that as long as the function and limits are same, the variable used for integration does not make any difference.

Illustration 1 :
Let f(x) = x/(1+xn)1/n for n ≥ 2 and g(x) = (fofofofof… n times) (x). Then find the value of ∫xn-2 g(x) dx.
Solution: The value of f(x) is given to be f(x) = x/(1+xn)1/n
Then ff(x) = f(x)/(1 + f(x)n)1/n
= x/(1+2xn)1/n
So, fff(x) = x/(1+3xn)1/n
Hence, g(x) = (fofof…..n times)(x) = x/(1+nxn)1/n
Hence, I = ∫xn-2 g(x) dx
= ∫ xn-1dx/(1 + nxn)1/n
= 1/n2 ∫ n2xn-1dx/(1 + nxn)1/n
= 1/n2 ∫ [d/dx (1 + nxn)] / (1 + nxn)1/n . dx
Hence, I = [(1+ nxn)1-1/n ]/ n(n-1) + K.

Illustration 2 :
Evaluate the given expression
Solution: Let I = ∫ dx/ [x2(x4 + 1)]3/4
= ∫ dx/ x2x3(1+x-4)3/4
= ∫ dx/ x5(1+x-4)
Put 1+x-4 = z
Then -4x-5 dx = dz
Hence, I = ∫ -dz/4z3/4
= -1/4 ∫ dz/z3/4
= -z1/4 + c

Illustration 3 :
Evaluate ∫ (x +1)/ x(1+xex)2 dx
Solution: Let I = ∫ (x +1)/ x(1+xex)2 dx
= ∫ ex (x +1)/ xex (1+xex)2 dx
Put (1+xex) = t, then (ex+xex) dx = dt
Hence, I = ∫ dt/ (t-1)t2 = ∫[1/(t-1) – 1/t – 1/t2] dt
= log |t-1| - log |t| +1/t + c
= log | [(t-1)/t] | +1/t + c
= log | xex/ (1+xex)| + 1/ (1+xex) + c

Most of the students are weak in Integral calculus but this weakness must be converted into strength with regular practice. All the high rank holders in JEE have their expertise in Integral calculus.

### JEE MAIN AND JEE ADVANCED SYLLABUS OF VECTOR

Vectors constitute an important topic in the Mathematics syllabus of JEE. It is important to master this topic to remain competitive in the IIT JEE. It often fetches some direct questions too. The topic of Vectors is quite simple and it also forms the basis of several other topics.

Various topics that have been covered in this chapter include:

• Introduction
• Fundamental Theorem of Vectors
• Orthogonal System of Vectors

We now discuss some of these topics in brief as they have been covered in detail in the coming sections:

Length / Magnitude of a vector: The length or magnitude of the vector w = (a, b, c) is defined as |w|= w= √a2+b2+c2

Unit vectors :
A unit vector is a vector of unit length. A unit vector is sometimes denoted by replacing the arrow on a vector with a "^" or just adding a "^" on a boldfaced character (i.e., ȇ or ȇ). Therefore,
|ȇ| = 1.
Any vector can be made into a unit vector by dividing it by its length.
ȇ = u / |u|
Any vector can be fully represented by providing its magnitude and a unit vector along its direction. A vector can be written as u = uȇ

Base vectors and vector components :
Base vectors represent those vectors which are selected as a base to represent all other vectors. For example the vector in the figure can be written as the sum of the three vectors u1, u2, and u3, each along the direction of one of the base vectors e1, e2, and e3, so that u= u1+u2+u3

It is clear from the figure that each of the vectors u1, u2 and u3 is parallel to one of the base vectors and can be written as a scalar multiple of that base. Let u1, u2, and u3 denote these scalar multipliers such that one has.
u1= u1e1
u2=u2e2
u3=u3e3

The original vector u can now be written as
u= u1e1+u2e2+u3e3
Negative of a vector:
A negative vector is a vector that has the opposite direction to the reference positive direction.

A vector connecting two points:

The vector connecting point A to point B is given by
r= (xB-xA) i+ (yB – yA) j + (zB – zA)k , here i, j and k denote the unit vectors along x, y and z axis respectively.

Some Key Points :

• The magnitude of a vector is a scalar and scalars are denoted by normal letters
• Vertical bars surrounding a boldface letter denote the magnitude of a vector. Since the magnitude is a scalar, it can also be denoted by a normal letter; |w| = w denotes the magnitude of a vector
• The vectors are denoted by either drawing a arrow above the letters or by boldfaced letters.
• Vectors can be multiplied by a scalar. The result is another vector.

Suppose c is a scalar and v = (a, b) is a vector, then the scalar multiplication is defined by cv= c (a,b)= (ca,cb). Hence each component of a vector is multiplied by the scalar.

• If two vectors are of the same dimension then they can be added or subtracted from each other. The result is gain a vector

Then the sum of these two vectors is defined by
v + u = (a + e, b + f, c + g).

• We can also subtract two vectors of the same direction. The result is again a vector. As in the previous case subtracting vector u from v yields
v - u = (a - e, b - f, c - g). the difference of these vectors is actually the vector v - u = v + (-1)u.

​Some Basic Rules of Vectors
If u, v and w are three vectors and c, d are scalars then the following hold true:

• u + v = v + u (the commutative law of addition)
• u + 0 = u
• u + (-u) = 0 (existence of additive inverses)
• c (du) = (cd)u
• (c + d)u = cu +d u
• c(u + v) = cu + cv
• 1u = u
• u + (v + w) = (u + v) + w (the associative law of addition)