GENERAL SOLUTION OF SOME SIMPLE EQUATION

sin Θ = 0                                   →  Θ = nΠ

cos Θ = 0                                  → Θ = (2n + 1) Π/2

tan Θ = 0                                  → Θ = nΠ

cot Θ = 0                                 → Θ = (2n + 1)Π/2

sin Θ = 1                                  → Θ = (4n + 1) Π/2

sin Θ = -1                                → Θ = (4n + 3) Π/2

cos Θ = 1                                → Θ = 2n Π

cos Θ = -1                              → Θ = (2n + 1) Π

tan Θ = not defined           → Θ = (2n + 1) Π/2

cot Θ = Not defined          → Θ = nΠ

cosec Θ = Not defined     → Θ = nΠ

sec Θ = not defined            → Θ= (2n + 1) Π/2

ü      There are some cautions to be taken while solving some Trigonometric Equations. They are listed down here.

Þ   Check the validity of the given equation

e.g., 2sin Θ – cos Θ = 4 can never be true for any Θ as the value (2 sin Θ – cos Θ) can never exceed √(2²+(-1)² = √5. So there is no solution to this equation.

Þ   Equation involving sec Θ and / or tan Θ can never have a solution of the form Θ = (2n + 1) Π/2. similarly, equation involving cosec Θ and / or cot Θ can never have solution of the form Θ = nΠ. The corresponding function are undefined at these values of Θ.

Þ   Avoid squaring the equations as far as possible because it leads to extraneous solution. If it has to be squared, check whether the solution (s) arrived at satisfy the original unsquared equation or not. e.g., given that x = 4 → x² = 16 → x = ± 4. But originally x = 4 only.

Þ   Do not cancel common factor involving the unknown angle on L.H.S. and R.H.S because it may delete some solution. e.g. In the equation sin Θ (2cos Θ – 1) = sin Θ cos²Θ if we cancel sin Θ on both sides we get cos² Θ – 2cos Θ + 1 = 0 → (cos Θ – 1)² = 0 → cos Θ = 1 → Θ = 2n Π.

Þ   But Θ = nΠ also satisfies the equation because it makes sin Θ = 0. So, the complete solution is Θ = nΠ, n Î Z.

Þ   Denominator terms of the equation if present should never become zero at any stage while solving for any value of Θ contained in the answer.

Þ        Something the equation has some limitation also. e.g., cot² Θ cosec² Θ = 1 can be true only if cot² Θ = 0 and cosec² Θ = 1 simultaneously as                 cosec² Θ ≥ 1. Hence the solution is Θ= (2n + 1) Π/2.

Illustrations

1. Obtain the general solution of secx + tanx =√3

Sol. Since secx and tanx. Both are undefined for x = (2n + 1)Π/2, the final solution should not include any odd multiple of Π/2.

Now,√3 cosx – sin x = 1 => cos(Θ + Π/6) = ½ = cosΠ/3

Þ Θ-+ Π/6 = 2nΠ + Π/3.

With ‘+’ sign, it given Θ = 2nΠ + Π/6

With ‘-‘ sign, it given Θ = 2nΠ – Π/2 = (4n – 1)Π/2

Since (4n -1) is always odd therefore we should delete this solution.

Thus, Θ= (12n + 1)Π/6, n Î Z.

Significant figures in the measured value of a physical quantity tells us the number of digits in which we have confidence. Larger the number of significant figures obtained in a measurement, greater is the accuracy of the measurement. The reverse is also true.

COMMON RULES FOR COUNTING

SIGNIFICANT FIGURES

Following are some of the common rules for counting significant figures in a given expression :

P                 All zeros occurring between two non zero digits are significant. For example: x = 5008 has four significant figures. Again x = 7.0102 has five significant figures.

P                 All non zero digits are significant.

For example: x = 7284 has four significant figures. Again x = 457 has only three significant digits.

P                 All zeros on the right of the last non zero digit in the decimal part are significant. For example                      x = 0.00400 has three significant figures.

P                 e.g., x = 0.00800, x=1.00;  The zeros before 8 are not significant.  1.00 has three significant figures.

P                 In a number less than one, all zeros to the right of decimal point and to the left of a non zero digit are NOT significant.

P                 For example: x = 0.0088 has only two significant digits. Again x = 1.0088 has five significant figures.

P                 All zeros on the right of the last non-zero digit become significant, when they come from a measurement.

For example, suppose distance between two stations is measured to be 3850 m. It has four significant figures. The same distance can be expressed as 3.850×10⁵ cm. In all these expressions, number of significant figures continues to be four.

P                 All zeros on the right of non-zero digit are NOT significant.

For example, x = 7000 has only one significant figure. Again x = 848000 has three significant figures.

ROUNDING OFF

While rounding off measurements, we use the following rules by convention:

P     If the digit to be dropped is more than 5, then the preceding digit is raised by one.

For example, x = 6.87 is rounded off to 6.9. Again x = 12.78 is rounded off to 12.8 .

P     If the digit to be dropped is less than 5, then the preceding digit is left unchanged.

For example, x = 7.82 is rounded off to 7.8 . Again x = 3.94 is rounded off to 3.9 .

P     If the digit to be dropped is 5 followed by digits other than zero, then the preceding digit is raised by one.

For example, x = 16.351 is rounded off to 16.4. Again x = 6.758 is rounded off to 6.8 .

P     If the digit to be dropped is 5 or 5 followed by zeros, then the preceding digit is raised by one, if it is odd.

For example, x = 3.750 is rounded off to 3.8. Again x = 16.150 is rounded off to 16.2 .

If the digit to be dropped is 5 or 5 followed by zeros, then the preceding digit is left unchanged, if it is even.

For example, x = 3.250 becomes 3.2 on rounding off, Again x = 12.650 becomes 12.6 on rounding off.

ARITHMENTICAL OPERATIONS WITH SIGNIFICANT FIGURES

(i) Addition and subtraction.

In addition or subtraction, the number of decimal places in the result should equal the smallest number of decimal places of terms in the operation. Suppose, in the measured values to be added or subtracted, the least number of significant digits after the decimal is N. Then in the sum or difference also, the number of significant digits after the decimal should be N.

For example, the sum of three measurements of length; 2.1 m, 1.78 m and 2.046 m is 5.926m, which is rounded off to 5.9 m (upto smallest number of decimal places).

In the subtraction of quantities of nearly equal magnitudes, accuracy is  almost destroyed. For example, if x = 42.87m and y = 12.86m, then

x – y = 12.87 – 12.86 = 0.01 m. The difference has only one significant figure, whereas x and y have four significant digits each.

(ii) Multiplication and Division

In multiplication and division, the number of significant figures in the product or in the quotient is the same as the smallest number of significant figures in any of the factors.

For example, suppose x = 3.8 and y = 0.125.  Therefore, xy = (3.8) (0.125) = 0.475. As least number of significant figures is 2 (in x = 3.8). Therefore, xy = 0.475 = 0.48 is rounded off to two significant figures.

Example

(i)      3.24 + 4.200018 + 5.0

= 12.440018≈  12.4

Here least number of significant digits after the decimal is one in 5.0. Same is the case with the sum

(ii)      6.21192 – 3.10   = 3.11192  ≈3.11

Here least number of significant digits after the decimal is two in 3.10. Same is the case with the difference.

Multiplication and division of measured values

ü     Suppose, in the measured values to be multiplied or divided, the least number of significant digits be N. Then in the product or quotient, the number of significant digits should be N.

Example

(i)   3.224 * 2.3 = 7.4152 ≈7.4

(ii)  46.64/2.3 =20.3=≈20

Here least number of significant digits is two in 2.3 and same should be the case with product or quotient

Þ  Change in the position of decimal point does not change the number of significant digits in the measured value.  For example, the number of significant digits both in 12.340 * 10² as well as 1234.0 is 5.

Þ The change in the units of measured value does not change the significant digits.

Rounding off a Digit

ü     We round off the number to obtain its value with a definite number of significant digits. Following are the rules for rounding off.

If the number lying to the right of cut off digits be

less than 5, then the cut off digit is retained as

such. However if it is more than 5, then the cut off

digit is increased by 1.

Example

Consider the number 324.1283. To round it off to 4 significant digits, we can write:

324.1283  ≈ 324.1. Again consider the number 324.1823. To round it off to 4 significant digits, we can write: 324.1823 ≈  324.2

Þ If the number to the right of cut off digit be 5, then we proceed as follows:

(a) Increase the cut off digit by 1 if it is odd.

(b) Retain the cut off digit as such if it is even

Example

324.1532  ≈324.2 and 324.2532  ≈324.2

Important

ü     While rounding off, the process should, in fact, be carried out from the last digit to the right. For example to round off 324.14821 to 4 significant digits, we should proceed as follows:

324.14821  ≈324.1482 ≈324.148 ≈324.15 ≈324.2

Þ In general no finally calculated value should have more significant figures than the least significant figures in the given data to be multiplied or divided. However, if multiple steps are involved, in the intermediate steps it is better to retain one significant figure more than the least number of significant figures in the given data.

The process of finding out the concentration of a solution by reacting it with another solution of known concentration is called volumetric analysis.

Volumetric analysis is done with the help of titrations. Suppose, we have a solution of unknown strength of a strong acid. We know that strong acids react with strong bases to give salts. We can prepare a standard solution (i.e. a solution of known strength) of base. Now a fix volume of solution is taken and base is slowly added to acid, in presence on an indicator (Phenolphthalein). After addition of a specific amount of base we find that pink colour appears in the reaction mixture which indicates that solution is completely neutralised.

In terms of m-equivalents same no. of (m-eq.) of reactants react and same no. of m-equivalent of product are formed. This is basic principle involved in volumetric analysis.

For the above reaction

Important

Þ    Acidic salts react with acid as well as base.

Þ      Neutral salts react with neither acid nor base.

Þ    Salts of strong acid and strong base do not react  with base.

Þ    Metal Oxide normally reacts with acid & non metal oxide reacts with base.

Þ    Metal normally reacts with acid and not with base at normal temperature.

• Imagine flying birds, moving planets, gushing air, flowing water and so on, all these phenomena are happening around us continuously. All the above mentioned phenomena can be summarized in one word “motion”.
• When something moves, there are several factors, which can be observed. If it moves, it will cover some distance, the distance covered may be different in different time periods. It may be moving slow or fast. Further its speed or velocity may be changing with time. All these factors depend on each other and we need to study their relationships.             Various kinds of motion can be systematically grouped under few broad categories. The point of distinction is made on grounds of the velocity vector. On these grounds motion can be:
• one dimensional, where only one dimension is required to describe the motion
• two-dimensional, where we require two dimensions to describe the motion of the particle
• three-dimensional, where three dimensions are required.

Pre-requisite

State of Rest and Motion

• If the position of a particle is changing with respect to its surroundings in a given time interval, we say the particle is in motion and on the other hand if the position particle is not changing with respect to its surroundings, we can say the particle is at rest.

Position

The position of a particle refers to its location in the space at a certain moment. A vector joining the moving particle with the origin can describe the position of that particle at a particular moment.

Displacement

It is the vector joining the initial position of the particle to its final position in a given time interval.

Velocity

The rate of change of position of a moving particle is known as its velocity.

Acceleration

The rate of change of velocity with time is called the acceleration.

Distance and Displacement

• To understand the difference between distance and displacement, we study the motion of vertical throw of a ball with respect to point O to height h.
• After some time it will come again to the same point O. The displacement of ball is zero but there is some distance traversed by the ball. It’s because distance is a scalar quantity but displacement is a vector quantity.

Speed and Velocity

• Speed is the rate of change of distance without regard to directions. Velocity is the rate at which the position vector of a particle changes with time. Velocity is a vector quantity whereas speed is scalar quantity but both are measured in the same unit m/sec.

• The mathematical definition of a function from set A to set B is that to each element a ε A there exists a unique element b ε B. As we know that in direct trigonometric functions, we are given the angle and we calculate the trigonometric ratio or the value at that angle. Also for many values of the angle, the values of trigonometric ratio is same. For example for sin Θ = 1/√2 , we have

Θ =Π/4 ,5Π/4 ,  9Π/4 etc. Now, direct trigonometric functions follow the definition of a function. But in inverse trigonometry, if we say that to a certain value of the trigonometric ratio there correspond many values of the angle, it violates the definition of function as it becomes a one – many relation. That is why, some restrictions have been imposed on the angles, and these are based on the principle values of the angle.

PRE-REQUISITE

ü        If A and B are two non–empty sets then a function from A to B associated to each element x in A, a unique element f(x) in B.

f : A → B

Þ            The set A is called Domain of f.

Þ            The set B is called the co-domain of f.

Þ      The range of f is the set consisting of all the images of the element of the domain A.

Þ      If          Range of f = {f(x) : x ε A}  then the function is onto.

Þ      One-One function: If x₁, x₂ ε A then f(x₁) = f(x₂) → x₁ = x₂

Þ            Contra positively x₁ ≠ x₂ ε f(x₁) ≠ f(x₂)

Þ      Many one function: f(x₁) = f(x₂)
where x₁ ≠ x₂.

Þ      Onto function: If f(A) = B i.e.
Range = co-domain then the function is onto.

Þ      Into function: If f(A) ς B the function is into

Þ      A function is invertible iff it is a one-one onto function. The inverse of a function is defined as if y = f(x),       x ε A, y ε B and f(x) is one-one and onto in A then            x = f ^–1(y) y ε B, x ε A

Þ      If A and B are domain and range of f(x) then B and A are those
of f^–1(x).

• The unit of any derived quantity depends upon one or more fundamental units. This dependence can be expressed with the help of dimensions of that derived quantity. In other words, the dimensions of a physical quantity show how its unit is related to the fundamental units.
• To express dimensions, each fundamental unit is represented by a capital letter. Thus the unit of length is denoted by L, unit of mass by M, unit of time by T, unit of electric current by I, unit of temperature by K and unit of luminous intensity by C.
• Remember that speed will always remain distance covered per unit of time, whatever be the system of units, so the complex quantity speed can be expressed in terms of length L and time T. Now, we say that dimensional formula of speed is LT-1. We can relate the physical quantities to each other (usually we express complex quantities in terms of base quantities) by a system of dimensions.
• Dimension of a physical quantity are the powers to which the fundamental quantities must be raised to represent the given physical quantity.

Illustrations

1. Find the dimension of density.

Sol. Density of a substance is defined to be the mass contained in unit volume of the substance.
Hence, [density] =[Mass]/[volume]

= M/L³

So, the dimensions of density are 1 in mass, – 3 in length and 0 in time.
Hence the dimensional formula of density is written as

[ρ]=ML^-3T^0

Important

• Constants such as ½, Π, or trigonometric functions such as “sin Αt” have no units or dimensions because they are numbers, or ratios which are also numbers.
• Two physical quantities can be equated, added (or subtracted) if and only if they have the same dimension. Why? [Verify if two quantities which have different dimensions can be multiplied (or divided) or not.

Broadly speaking, dimension is the nature of a Physical quantity. Understanding of this nature helps us in many ways.

Cathode rays and electrons

Electrical discharge through partially evacuated tubes produced radiation. This radiation originated from near the negative electrode, known as the cathode (thus, these rays were termed cathode rays).

• The “rays” traveled towards, or were attracted to the positive electrode (anode)
• Not directly visible but could be detected by their ability to cause other materials to glow, or fluoresce
• Traveled in straight line
• Their path could be “bent” by the influence of magnetic or electrical fields
• A metal plate in the path of the “cathode rays” aquired a negative charge
• The “cathode rays” produced by cathodes of different materials appeared to have the same properties

These observations indicated that the cathode ray were composed of negatively charged particles (now known as electrons).

J.J. Thompson (1897) measured the charge to mass ratio for a stream of electrons (using a cathode ray tube apparatus) at 1.76 x 10⁸ coulombs/gram.

• Charged particle stream can be deflected by both an electric and by a magnetic field
• An electric field can be used to compensate for the magnetic deflection – the resulting beam thus behaves as if it were neutral
• The field needed to “neutralize” the magnetic field indicates the charge of the beam

Thompson determined the charge to mass ratio for the electron, but was not able to determine the mass of the electron.

However, from his data, if the charge of a single electron could be determined, then the mass of a single electron could also be determined.

Robert Millikan (1909) was able to successfully measure the charge on a single electron (the “Milliken oil drop experiment”). This value was determined to be 1.60 x 10^-19 coulombs.

Thus, the mass of a single electron was determined to be:

(1 gram/1.76 x 10⁸ coulombs)*(1.60 x 10^-19 coulombs) = 9.10 x 10^-28 grams

Note: the currently accepted value for the mass of the electron is 9.10939 x 10^{- 28} grams.

• Trigonometric equations are natural sequel to the trigonometric ratios and identities which constitute basis of many problems in Mathematics.

• The trigonometric equations have a large number of concepts associated with relevant applications.

OBJECTIVE

• This chapter focuses on the solutions of different trigonometric equations. After studying this chapter we will learn how to find different possible solutions of a given trigonometric equation or how to find the general solution of a trigonometric equation. We will also be able to distinguish a trigonometric identity from a trigonometric equation.

PRE-REQUISITE

=>        sin (A + B) = sin A cos B + cos A sin B

=>        sin (A – B) = sin A cos B – cos A sin B

=>        cos (A + B) = cos A cos B – sin A sin B

=>        cos (A – B) = cos A cos B + sin A sin B

=>        tan (A + B) = (tanA+tanB)/ 1-tanA tanB

where A ≠ nΠ +Π/2 , B ≠ nΠ +Π/2

=>   tan (A – B) =(tanA-tanB)/1+ tanA tanB and A ± B ≠ mΠ +Π/2

=>   cot (A + B) =(cotA cot B – 1)/(cotA + cot B),

where A ≠ nΠ, B ≠n Π

=>        cot (A – B) = and A ± B≠ np

=>   sin (A + B) sin (A – B) = sin² A – sin² B

= cos² B – cos² A

=>        cos (A + B) cos (A – B) = cos² A – sin² B

= cos² B – sin² A

=>        sin2 Θ = 2 sinΘ cos Θ =2tanΘ/(1+tan 2Θ)

=>   cos 2 Θ =cos²Θ – sin² Θ =2 cos²Θ -1 = 1 – 2sin² Θ =(1-tan²Θ)/(1+tan²Θ)

=>        1 + cos 2 Θ = 2 cos²Θ,  1 – cos 2 Θ = 2 sin² Θ

or         (1+cosΘ)/2= cos²Θ, (1-cosΘ)/2 = sin² Θ

=>       tan2 Θ = 2tanΘ/(1-tan²Θ), where Θ≠ (2n + 1)Π/4

=>      (1-cosΘ)/sinΘ  = tan Θ/2, where Θ ≠ (2n + 1)Π

=>        (1+cosΘ)/ sinΘ= cot Θ/2, where Θ≠ 2n Π

=>         (1-cosΘ)/(1+cosΘ)= tan² , where Θ ≠ (2n + 1)Π

=>       (1+cosΘ)/(1-cosΘ) = cot^{2 Θ/2}, where Θ ≠ 2n Π

=>        sin 3Θ= 3sinΘ -sin³Θ

=>       cos3 Θ = 4cos³ Θ – 3cos Θ

=>           cos A cos2 A cos2² A … cos2^n-1 A =sin2″A/2″sinA

CORE CONCEPTS

• An equation involving one or more trigonometric ratios of unknown angle is called a trigonometric equation. A trigonometric equation can be written as

Q₁ (sinΘ, cosΘ, tanΘ, cotΘ, secΘ, cosecΘ)

= Q₂ (sinΘ, cosΘ, tanΘ, cotΘ, secΘ, cosecΘ)

where Q₁ and Q₂ are rational functions.

cos² x – 4 sin x = 1.

• All possible values of unknown which satisfy the given equation are called solution of the given equation.

• For complete solution “all possible values” satisfying the equation must be obtained.

• This is trigonometric equation as it is not satisfied for all values of x e.g.,  does not satisfy the given equation.

Identify whether the following are trigonometric equations or trigonometric identities.

1. sin³ A = 3sin A – 4 sin³ A

Sol. Trigonometric Identity

2. cos⁷ x + sin⁴ x = 1

Sol. Trigonometric Equation

The tiny constituent of an element is an atom. The word atom is a Greek word meaning indivisible, i.e., an ultimate particle which cannot be further subdivided. The idea that all matter ultimately consists of extremely small particles was conceived by ancient Indian and Greek philosophers. The old concept was put on a firm footing by John Dalton in the form of atomic theory which he developed in the years 1803-1808. This theory was a landmark in the history of chemistry.

OBJECTIVE

Towards the end of the nineteenth century, it began to appear that the atom itself might be composed of even smaller particles. In 1833, Michael Faraday showed that there is a relationship between matter and electricity. This was the first major breakthrough to suggest that an atom was not a simple indivisible particle yet smaller but was made up of  particles. On the basis of Faraday’s work, Stoney proposed that units of electrical charge are associated with atoms. In 1891, he suggested that these units be called electrons. Electron is a Greek word meaning amber, a material which becomes electrically charged when rubbed with wool or silk.

It is now believed that the atom consists of several particles called sub-atomic particles like electron, proton, neutron, etc. the electron, the proton and the neutron are called Fundamental particles and are building blocks of the atoms about which we shall deal with in this chapter.

PRE-REQUISITE

Law of conservation of Mass

In a chemical reaction the weight of products is equal to the weight of reactants.

Law of definite proportions

If a compound is analysed from various sources, its elemental composition remains the same i.e., analysis of water from a river or ditch or pond either in India or in USA would always give H:O ratio as 2:1. (atom ratio)

Law of Multiple Proportions

Elements combine in simple whole number ratios to form various types of compounds e.g. The ratio of N:O is 1:1, 1:2 and 2:1 in NO, NO₂ and N₂O, respectively.

Atomic Number (Z)

The total number of protons present in the nucleus of an atom is called as atomic number of that atom.

Mass Number (A)

Total number of nucleons (protons + neutrons) in the nucleus of an atom is called it mass number.

Isotopes

Atoms with the same atomic number but different mass numbers are called isotopes of each other. For example the isotopes of hydrogen atom are: ₁H¹, ₁H², ₁H³.

Isobars

Atoms with same mass number but different atomic number are called as isobars of each other. For example ₁₅P³² and ₁₆S³² are isobars of each other.

Isotones

Atoms having the same number of neutrons but different number of protons are called isotones. For example ₆C¹⁴, ₈O¹⁶, ₇N¹⁵ are isotones as they all have 8 neutrons.

Isodiaphers

Atoms having the same value of (A – 2Z) but different value of A or Z are called isodiapheres. For example ₁₁Na²³, ₉F¹⁹, ₇N¹⁵ are isodiapheres as (A – 2Z) for all these three atoms is 1.

Nuclear Isomers

Nuclear isomers (isomeric nuclei) are the atoms with the same atomic number and same mass number but different radioactive properties are called nuclear isomers. This type of isomerism is due to the different energy states of the two isomeric nuclei. For example ₃₀Zn⁶⁹ and ₃₀Zn⁶⁹ are two atoms with their half life periods 13.8 hours and 57 minutes respectively.

Isoelectronic

Atoms, molecules or ions with same number of electrons are called isoelectronic. For example N₂, CO, CN^- are isoelctronic.

Do you know how difficult is IIT JEE and what makes it more difficult? Do you know which of the colleges in India that admits students based on IIT JEE? Do you know what mark a student should secure in IIT JEE to get into IIT? Do you know what rank an IIT aspirant should get in IIT JEE to get into IIT? Would you like to know more about new IIT’s that admitted students in 2008? What are the top branches of IIT which admits students every year? To know more about IIT JEE go ahead….

How difficult is the JEE? And what makes it difficult?

The difficulty in getting through lies in obtaining a merit rank within the top 8,200 or so, to qualify for admission to IIT’s. The performance of a candidate is assessed comparatively among the numerous competitors.

This year about 3 lakh students took this examination and the seats offered were about 8200. Compared to 2007, 70,000 more students took this exam in 2008. The acceptance level is about 2 %.

Indian Institute Of Technology Joint Entrance Examination (IIT JEE)

A total of fifteen colleges use JEE (Joint Entrance Examination) as a sole criterion for admission to their undergraduate programs. The fifteen colleges include the seven old and six new (2008) Indian Institutes of Technology (IIT), IT-BHU Varanasi, and ISMU Dhanbad. Starting in 2007, newly established institutions such as Indian Institutes of Science Education and Research (IISERs) at Kolkata, Pune, Mohali, Bhopal & Thiruvananthapuram, Indian Institute of Space Science and Technology (IIST), Thiruvananthapuram, Kerala, Indian Institute of Maritime Studies, Mumbai and Rajiv Gandhi Institute of Petroleum Technology (RGIPT), rae bareilly Uttar Pradesh are also admitting students through the JEE (Extended Merit List). The exam is conducted by the various IITs by a policy of rotation. IIT JEE is one of the toughest engineering entrance exams in the world with a success rate of around 1 in 45. Candidates who qualify in the IIT-JEE can apply for admission to the B Arch (Bachelor of Architecture), B Des (Bachelor of Design), B Tech (Bachelor of Technology), Dual Degree (Integrated Bachelor of Technology and Master of Technology) and Integrated MSc (Master of Sciences) courses in the various institutes. Achieving entrance into an IIT is often considered the pinnacle of achievement for a student of the sciences, and the IITs/IT-BHU/ISM attracts most of the brightest students of the nation.

Of the 384,977 candidates who appeared in the examination conducted on April 12, 2009, 10,035 candidates have been declared qualified to seek admission, giving a selectivity of 1 in 38 overall, 1 in 46 for the 8,295 seats in IITs,IT-BHU and ISMU and 1 in 59 for the IITs only.

VITAL STATISTICS

Aggregate Total And Subject wise Marks For The First And Last Admitted Candidates

The New IIT’s That Admitted Students In 2008

The TOP 5 Branches at Various IITs
The tables given below give the top 5 branches in all IITs. The branches have been ranked based on student preferences . That means the most preferred branch by students counseling has been ranked No. 1. The tables will help you in choosing the branch in IIT of the time of counseling

IMPORTANT DATES FOR ENGINEERING ENTRANCE EXAMS

OPENING AND CLOSING RANKS DURING AIEEE COUNSELLING

(TOP 3 NITs)