IIT JEE Blogs

The order of magnitude of a number is the power of ten closest to the number.

Following table gives us some of the commonly used prefixes for power of ten.

Positive Powers of 10

Negative Powers of 10

Some Derived SI units and their symbols

The following conventions are adopted while writing a unit.

• Even if a unit is named after a person the unit is not written in capital letters. i.e. we write joules not Joules.

• For a unit named after a person the symbol is a capital letter e.g. ‘J’ for joules and the rest of them are in lowercase letters e.g. ’s’ for seconds.

• The symbols of units do not have plural form i.e. 70 m not 70 ms or 10 N not 10 Ns.

• Not more than one solid’s is used i.e. all units of numerator is written together before the ‘ / ‘ sign and all in the denominator are written after that.

i.e. It is 1 ms-2 or 1 m/s2 not 1m/s/s.

• Punctuation marks are not written after the unit e.g. 1 litre = 1000 cc not 1000 c.c.

• Constant function is a periodic function without any period. This happens because of the non-existence of the least positive real number which is due to the continuity of real number system.

• If f(x) has it’s period T then f(ax + b) has its period .

• If f(x) has its period T₁ and g(x) has its period                T₂ then   (af(x) + bg(x))  has  its  period                         £ L.C. M.(T₁, T₂).  Moreover if f(x) and g(x) are basic trigonometric functions then period of                        [af(x) + bg(x)]  =  L.C.M. (T₁, T₂)

Examine whether sin x is a periodic function or not. If so, find its period.

Given f(x) = sin x. Let’s assume sin x to be periodic. So, it must have some positive value independent of x say T such that f(x + T) = f(x)

• sin (x + T) = sin x

• x + T = n p + (–1)ⁿ x where n = 0, ± 1, ± 2 ……

The positive values of T independent of x are given by n p where n = 2, 4, 6……..

Further according to definition for periodic number, it should be least. So, here we have T = 2p.

Thus, it is proved that sin x is periodic function having periodicity 2p.

Prove that f(x) = sin√x  is not a periodic function.

Proof : Let the positive real number T be such that f (x + T) = f(x)

This above relation does not give any positive value of T independent of x because it holds only when T = 0.

f(x) is non-periodic function.

Let f(x) = x – [x] where [x] is the greatest integer less than or equal to x. Find out the periodicity of f(x). Assume f(x + T) = f(x)

• (x + T) – [x + T] = x- [x]

• T = [x + T] – [x] = an integer.

Hence least positive value of T independent of x is 1.

Thus f(x) is a periodic function of period 1.

This can be explained through graphs. As in the case of algebraic function, we can have same idea about the nature of a trigonometric function by its graph.

The variations in the values of the trigonometric ratios generated the concept of graph in trigonometric functions.

From the graph, we observe that:

• The value of sin x repeats itself after an interval of 2p. So sin x is a periodic function with period of 2p. Actually a revolution of 2p is the complete revolution.

• sin x takes value from –1 to 1.

When an atom/ion/molecule loses electrons, oxidation of the species takes place, such a molecule is termed as reductant.

or

When oxidation number of an atom increases in a reaction, it is said to be oxidised.

Oxidation Number / Oxidation State

The real or hypothetical charge present over an element is called oxidation number. Whereas oxidation state defines charge on one atom but oxidation number refers to charge present on all atoms of one element in a compound.

Certain features of oxidation number

• The pure oxidation number is always an integer, but the mathematical average may be in fraction.
• Oxidation number may be positive as well as negative.
• Oxidation number of I(A) group elements is +1, II(A) group element +2, and in III(A) group Al & B have +3 oxidation number and rest are variable.
• The molecules which exist in free state, always have zero oxidation number (NH₃, H₂O etc).
• Oxidation state of hydrogen is always +1, but when hydrogen is directly attached to metal (metal hydride) then its oxidation no. is always –1.
• The oxidation No. of oxygen is –2 but in peroxide compounds its oxidation No. is –1.
• Oxidation No. of oxygen is +ve when it is directly attached to fluorine.
• ^· In superoxide compounds oxidation number of oxygen  –¹/₂.

• The sum of oxidation No. in neutral species is zero.

• In halogens oxidation number of F is ¬–1 because F has maximum electronegative value in periodic table.

Let us now focus our attention on the calculation of the oxidation number

• Break a molecule into its atoms

• Now valencies of all the atom’s are added

• This is than equated to the total formal charge on the molecule

• The total formal charge on a neutral molecule is taken as zero. For a charged molecule, total formal charge is taken equal to charge on cation/anion.

Important
For finding out formal charge on an atom, hypothetically break all bonds to that atom. The e-pair of bond goes to more electronegative atom. After this exercise total charge left on central atom would be called the formal charge on that atom.

Illustrations

The HRD Minister Kapil Sibal on Tuesday, Oct 20 clarified that any change in the eligibility criteria to appear in IIT-JEE exam will have to be taken by an IIT committee, and dubbed the media reports suggesting that he would be fixing the marks to 80 per cent for the entrance to IITs as ‘baseless’.

“The eligibility criteria to appear in JEE is decided by the Indian Institutes of Technology (IITs) themselves, and the only decision that has been taken by the IIT Council is that the IITs will submit a report in January 2010 to rationalize the JEE,” he said.

“The government of India has no role to play and any report which suggests that there is a proposal to allow only those who obtain 80 per cent marks in their class 12 examination to sit for JEE is baseless,” he said.

It was entirely up to the IITs to decide the criteria. It was they who would consider what weightage was to be given to the Class XII examination, and whether marks or percentile should be the basis for admission, he said. “The government has no jurisdiction in the matter, and the Human Resource Development Ministry can in no way, either directly or indirectly, decide or make any proposal for a decision.”

On the lines of the Common Aptitude Test (CAT) for management courses, the All India Engineering Entrance Exam (AIEEE) plans to conduct the exam online. The government has set up a committee of National Institute of Technology (NIT) directors to explore the possibility of making the exam online, in addition to the regular pen and paper test. The committee will submit its report by January 2010.

The committee will also deliberate on giving Class XII board exam marks weightage during the entrance exam. This came up after NIT directors said that the proliferation of coaching institutes was skewing the pattern of entrants into NITs.

Nearly 10 lakh students appeared for AIEEE for roughly 24,000 seats in engineering colleges across the country. This includes NITs, state engineering colleges, the Indian Institutes of Information Technology (IIITs) and deemed and central universities. The number of aspirants is expected to increase significantly over the next few years and the NIT council feels that an online test would facilitate the process.

However, some members of the council felt that an online test would not be feasible considering poor internet connectivity and the non-availability of computers in the rural areas. “All students are not familiar with computers. Hence, written tests should be a part of the system,” director NIT Hamirpur IK Bhatt said.

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Physics

MECHANICS

• Physics by H.C. Verma
• Problems in physics by I.E. Irodov
• Resnick and Halliday

ELECRICITY AND MAGNETISM

• Resnick and Halliday
• Circuits devices and systems by R.J. Smith
• Problems in physics by I.E. Irodov

OPTICS

• Physics by H.C. Verma

MODERN PHYSICS

• Physics by H.C. Verma
• Problems in Physics by I.E. Irodov

HEAT AND WAVES

• Physics by H.C. Verma
• Resnick and Halliday

Chemistry

ORGANIC CHEMISTRY

• Morrison & Boyd
• Solutions to Morrison Boyd
• Reaction mechanism in Organic Chemistry by Parmar ∓ Chawla

INORGANIC CHEMISTRY

• NCERT Inorganic Chemistry
• Concise Inorganic Chemistry by J.D. Lee
• IIT Chemistry by O.P. Aggarwal

GENERAL CHEMISTRY

• J.D. Lee
• O.P. Aggarwal
• R.C.Mukerjee

Maths

ALGEBRA

• High school mathematics by Hall and Knight
• IIT Maths by M.L. Khanna

CALCULUS amp; ANALYTIC GEOMETRY

• G.N.Berman
• Calculus and analytic geometry by Thomas and Finney
• Coordinate geometry by Loney
• IIT Maths by M.L. Khanna
• I.A.Maron

VECTORS

• IIT Maths by M.L. Khanna

Introduction

Physics is that branch which deals with the study of nature and natural phenomenon. The word physics comes from the Greek word ‘fusis’ meaning nature. In this unit we will discuss some of the important aspect of measurement in physics. We will also discuss why we need a unit to measure a physical quantity.

In the measurement of any physical quantity, we require some ‘reference standard’. This reference standard of measurement is called a unit. These are independent quantities i.e. they do not need any other quantity to represent them. Let us consider three physical quantities mass, length and time. These quantities are independent of each other. So, three separate units are required for the measurement of these quantities. Thus, it becomes important to establish a system of units.

Measurement in Physics

Fundamental Units

Measurement of a physical quantity involves:

• The standard or unit in which the quantity is being measured
• The numerical value representing the number of times the quantity contains that unit.

The physical quantities which do not depend upon other quantities are called fundamental quantities. In M.K.S. system the fundamental quantities are mass, length and time, while in more general Standard International (S.I.) system the Fundamental quantities are mass, length, time, temperature, luminous intensity, current and amount of substance. The units of fundamental quantities are called fundamental units and are discussed below.

Derived Units

The units of physical quantities which may be derived from fundamental units are called derived units, for example:

Unit of area:

area = length × breadth

unit of area =  unit of length × unit of breadth

= m × m = m²

Unit of Velocity:

velocity = Displacement/Time

unit of velocity =Unit of Displacement/Unit of Time

= m/s = ms^-1

Hence m2 and ms-1 are derived units.

Systems of Units :

There are following principal system of units:

1. C.G.S System :

length → centimetre (cm),
mass  → gram  (g)
time    → second (s).

2. F.P.S System :

length → foot (ft),
mass  → pound (lb),
time    → second (s).

3. M.K.S. System:

length  → metre (m),
mass   → kilogram (kg),
time     → second (s).

4. S.I. System :

It has SEVEN fundamental units.

Length                                     → metre (m),
Mass                                       → kilogram (kg),
Time                                     → second (s).
Temperature                          → kelvin (K),
Luminous intensity                 → candela (cd),
Electric current                     → ampere (A),
Amount of substance             → mole (mol).

In S.I. system there are two supplementary units.

P        Radian (rad) : Unit of plane angle

P        Steradian (st) : Unit of solid angle

INTRODUCTION

Trigonometry is the subject which deals with the properties of triangles. You might recall from your previous knowledge that in a right angled triangle the ratios between any two sides may be defined in terms of what are called trigonometric ratios, but the scope of trigonometry is not limited to this only. These rules or laws of trigonometry are used as tools for mathematical analysis of various problems in physics and engineering.

Trigonometric functions

A real number Θ can be interpreted as the measure of the angle constructed as follows: wrap a piece of string of length Θ units around the unit circle  x² + y² = 1 (counterclockwise if Θ≥ 0, clockwise if Θ < 0) with initial point P(1, 0) and terminal point Q(x, y). This gives rise to the central angle with vertex O(0, 0) and sides through the points P and Q. All six trigonometric functions of Θ are defined in terms of the coordinates of the point Q(x,y), as follows:

cos Θ = x                                                       sec Θ = 1/x if x ≠ 0

sin Θ = y                                                       cosec Θ = 1/y, if y ≠ 0

tan Θ = y/x if x ≠ 0                                         cot Θ = x/y if y ≠ 0

Since Q(x,y) is a point on the unit circle, we know that x² + y² = 1. This fact and the definitions of the trigonometric functions give rise to the following fundamental identities:

Pythagorean identity           sin² Θ + cos² Θ = 1

Reciprocal identities

sec Θ = 1/cos Θ

cosec Θ = 1/sin Θ

tan Θ = sin Θ/cos Θ

tan Θ = 1/cot Θ

cot Θ = 1/tan Θ

This modern notation for trigonometric functions is due to L. Euler (1748).

More generally, if Q(x,y) is the point where the circle x² + y² = R² of radius R is intersected by the angle Θ, then it follows (from similar triangles) that

cos Θ = x/R                                             sec Θ = R/x, if x ≠ 0

sin Θ= y/R                                               cosec Θ = R/y, if y ≠ 0

tan Θ = y/x, if x ≠ 0                                 cot Θ = y/x if y ≠ 0

Periodic Functions

If an angle Θ corresponds to a point Q(x, y) on the unit circle, it is not hard to see that the angle Θ + 2Π corresponds to the same point Q(x, y), and hence that

cos (Θ + 2Π) = cos Θ,  sin (Θ + 2Π) = sin Θ …(1)

Moreover, 2Π is the smallest positive angle for which Equations (1) are true for any angle Θ. In general, we have for all angles Θ:

cos (Θ + 2nΠ) = cos Θ, sin (Θ + 2nΠ) = sin Θ, n = 0, ±1, ±2 ….                                                                                       …(2)

We call the number 2Πthe period of the trigonometric functions sin and cos, and refer to these functions as being periodic. Both sec and cosec are periodic functions as well, with period 2Π, while tan and cot are periodic with period Π.

Illustrations

1. Find the period of the function
f(x) = – 3 cos (3x).

Sol. The function f(x) = – 3 cos (3x) runs through a full cycle when the angle 3x runs from 0 to 2Π, or equivalently when x goes from 0 to 2Π/3. The period of f(x) is then 2Π/3.

2. Find the period of the function f(t) = 8 sin (7Πt).

Sol. For the function f(t) = 8 sin (7Πt) to run through a full cycle, the angle 7Πt should run from 7Πt = 0 to 7Πt = 2Π, and hence t should run from t = 0 to t = 2/7. The period of f(t) is then 2/7.

Introduction:

The branch of chemistry which deals with mass relationship in chemical reactions is called stoichiometry. Stoichiometry is the quantitative analysis of various types of chemical reactions. Most of these calculations are done on the basis of mole concept. The term ‘mole’ was first introduced by ‘Ostwald’. It is a Latin version of the term ‘heap’ or ‘pile’ or ‘weight’, which refers the amount of a substance containing a fixed number of its elementary particles equivalent to the Avogadro’s number                (6.023 × 10²³). In modern practice, it is easy to express the mole of substance in terms of its weight or its volume. The analysis based on weight is called Gravimetric analysis whereas the analysis based on volume is known as volumetric analysis.
Core Concepts
Equivalent weight
The minimum weight of any chemical species, which reacts (completely) or liberates 1 g hydrogen (11.2 litre), 8 g Oxygen (5.6 litre), 35.5 g Chlorine (11.2 litre), 80 g Bromine (11.2 litre), 127 g Iodine (11.2 litre) is called Equivalent weight of that particular chemical species.

The above definition for equivalent weight is not sufficient. For example if an acid is given then the equivalent weight of an acid is defined as the ratio of Molecular weight of acid to its basicity.
Basicity means number of acidic hydrogens present in the molecule.
Acidic hydrogen means hydrogen atoms directly attached to electronegative element.