recursal/Fanatic-Fandom huggingface dataset:
Limits

The value of a function in the neighbourhood of a point , where the function is discontinuous is called the limiting value .

## Limits

There are three ways in which a function can approach the limiting value

1) LHL and RHL approach the same value . Limit exists .

2) LHL and RHL approach two different limiting values . Limit does not exist , but we can find the individual limiting values for both .

3) The Graph has a weird behavior or trajectory .

The limitting Value can be infinity e.g. 1/|x| .

## Reasons for Non - Existence of Limits

1) Function is not defined in the neighbourhood of x . (even if one of the limits is not defined )

2) LHL and RHL approach different values .

3) The Graph has a weird behaviour .

## L-Hospital's Rule

If while , evaluating the limits , we get infinity or an undefined value , then we can differentiate the numerator and denominator separately , till the infinity factor is removed .

For MCQs with limits of the form 0/0 , L-Hospital's Rule is crucial .

## Evaluation of Limits

**Types:-**

1) Algebraic Evaluation

2) Trigonometric Evaluation

3) Logarithmic Evaluation

4) Exponential Evaluation

5) Miscellaneous

## Hints

1) For evaluating RHL or LHL with a variable a , put x tends to (a + h) or (a - h).
2) While dealing with trigonometric functions , if x does not tend to 0 , use a variable h such that h tends to 0 .
Carboxylic Acids

## Preparation

1. Oxidation
	* Alcohols
	* Aldehydes and Ketones
	* Alkyl Benzene (Oxidation of aliphatic chain)
2. Hydrolysis
	* Acyl Chloride
	* Esters
3. From Dry Ice (Using Grignard Reagent

## Reactions

1. Substitution of H in O-H bond
	* Na
		+ Na
		+ NaOH
		+ NaHCO3 or Na2CO3
	* Alcohol (Formation of Esters)
2. Substitution of -OH bond
	* NH3
	* Chlorination using PCl3 , PCl5 , SOCl2
3. Condensation
	* Formation of Acid Anhydride
4. Reduction
	* to Alcohols
5. Coupling
	* Formation of Alkane
6. Halognation of α Carbon atom
The World Is Mine

Length
 Genre
 Year

**The World Is Mine** is a song by MainStreet. It was sung for the Argentinian teen-telenovela *Violetta* and is also the Dutch title song.

## Lyrics

Yeah the world is mine

De dagen gaan zo snel

Er gaat zoveel aan mij voorbij

Maar één ding weet ik wil

Ik kan niet terug

Terug in de tijd, no

I don't know which way to go

Blijf zoeken naar de juiste weg

This time I will run the show

No I won't wait again

Yeah the world is mine

This time I know things will go my way

Yeah this is my time

And I will make it

No matter what they say

And the ups and downs

Zal ik overleven

What doesn't kill you makes you strong

Yeah the world is mine

Niemand houd me tegen

I'm gonna live the dream

I'll always keep singing my song

Soms wordt het iets te veel

En zeggen ze dat ik het op moet geven

But that's not how I feel

Want ik zal altijd overleven

I don't know which way to go

Blijf zoeken naar de juiste weg

This time I will run the show

No I won't wait again

Yeah the world is mine

This time I know things will go my way

Yeah this time is mine

And I will make it

No matter what they say

And the ups and downs

Zal ik overleven

What doesn't kill you makes you strong

Yeah the world is mine

Niemand houd me tegen

I'm gonna live the dream

I'll always keep singing my song

Things will go my way

No matter what they say

Yeah the world is mine

This time I know things will go my way

Yeah this time is mine

And I will make it

No matter what they say

And the ups and downs

Zal ik overleven

What doesn't kill you makes you strong

Yeah the world is mine

Niemand houd me tegen

I'm gonna live the dream

I'll always keep singing my song.
Tips & Tricks

Steps of Solving MCQs :

1. Scan the Question
	1. Keep known variables in Mind
	2. Keep the variables that can be found using the question ready
	3. Keep the final variable needed in Mind
2. Evaluate Concepts
	1. Try to pin point the concept .
	2. You may get wrong in first attempt . Try with another related concept .
	3. Recall the entire concept in an instant !
3. Calculations
4. Option Marking
	1. This step has to be done carefully .
	2. Sometimes , the option can give an idea about the concept to be used in step 2 .

Physics

Chemistry

Maths
Quadratic Equations

An equation whose highest power is 2 is an quadratic equation . The graph of an quadratic is always a parabola .

## Solution of a Quadratic Equation

A Quadratic Equation cannot have more than two roots .

### **Substitution Method**

Form aZ +b/Z + c

Form (x + a)(x + b)(x + c)(x + d) + k = 0

Step 1 :De-factorize (2 at a time)

Step 2 : Substitute [ if a + b = c+ d ]

Step 3 : Solve and Re-substitute

## Complex & Irrational Quadratic Equations

If any coefficient is a non-real complex and p + iq is a complex root of equation , then p - iq need not b the other root .

If any coefficient is an irrational number and p + √q is a complex root of equation , then p - √q need not b the other root .

The above holds true for even higher degree polynomials .

## Relation between Roots and Coefficients

α + β = - b/a

α β = c/a

## Nature of Roots

b2 - 4ac > 0 Roots are **Real** and **Unequal**

b2 - 4ac = 0 Roots are **Real** and **Equal . The equation is a perfect square**

b2 - 4ac < 0 Roots are **Complex**

b2 - 4ac is not a perfect square

Roots are **Real** and **Irrational**

## Sum and Product of Roots

α + β = - b/a

αβ = c/a

## Formation of Quadratic Equation

x2 - (Sum of Roots)x + (Product of Roots) = 0

## Sign of Roots

{Based on laws of inquations}

**Positive ,**  if both sum and product of roots are positive .

**Negative ,**  if both sum and product of roots are negative .

**Opposite Signs ,**  if product of roots is negative .

**Equal Magnitude and Opposite Signs ,**  if Sum of Roots is 0 .

## Graph

Parabolic Graph

b2  = 4ca y2 = 4ax

## Quadratic Inequalities

Quadratic Inequalities are solved in the same way as quadratic equations (Interpretation is important) .

## Quadratic & Higher Order Polynomials

A polynomial of n order , has n roots . Note the similarities in quadratic equations and higher order polynomials .

Consider α β γ to be roots of third order polynomial ax3 + bx2 + cx + d  ; then

α+β+γ = -b/a

αβ + βγ + αγ = c/a

αβγ = -d / a

Consider α β γ δ to be the roots of a fourth order polynomial ax4 + bx3 + cx2 + dx + e ; then

α+β+γ+δ = -b/a

αβ + βγ + γδ + αγ + αδ + βδ = c/a

αβγ+ βγδ + αγδ + αβδ = -d/a

αβγδ = e/a

## Common Roots in two Quadratic Equations

α is a common root in a1x2+b1x+c1 = 0 and a1x2+b1x+c1 = 0; thus α **satisfies** both equations .

If α β are the roots of first equation ; α γ are the roots of second equation ; then

Sum and Products of roots of both equations can be equated due to the common root α . (This helps in solving such equations)
IIT JEE Wiki

## Welcome to the IIT JEE Wiki

Boards Strategy      Mock Test Strategy      Exam Day Strategy      Last Minute Strategy      Exam Strategy Stress Management JEE : Important Skills

For Boards preparation , we highly recommend the site : hscboards.wikia.com (HSC Boards Wiki)

For Better preparation for any Entrance Exam, we highly recommend the site : www.meritstore.in (MeritStore.in)

For JEE Online Preparation, Crash Course, go to http://www.meritstore.in/jee/

For NEET Online Preparation, Crash Course, go to http://www.meritstore.in/neet/

## JEE Main

Revise & Practice these chapters : (January)

* Physics
* Chemistry
* Maths

Kinematics

Force

Heat

Ray Optics

Electrostatics

Organic Chemistry (All together)

Solutions & Colligative Properties

Equilibrium

Surface Chemistry

Theory Chapters of inorganic Chemistry

Biomolecules & Chemistry in action

Circle & Conics

Permutations & Probability

Integration & Differentiation

Differential Equations
1. Solve Higher level MCQs from Marvel and mark those chapters that were difficult to handle . Also solve Easier chapters from HC Verma (January - February)
2. Study important chapters from NCERT thoroughly (January - February - March)
	1. Study these chapters :

* Physics
* Chemistry
* Maths

Force

Current Electricity

Electrostatics & Magnetism
Ionic Equilibrium

Circle & Conics

1. Solve 5 - 10 Numericals from books like HC Verma from all chapters and note the chapters that were difficult to handle . (March 14-March 31)
	1. Study & Practice these chapters intensively . Take chapter wise tests of these chapters on Embibe .

* Physics
* Chemistry
* Maths

1. Solve Practice Papers and Note down your problems : (March 14 - April 9)
	1. Making decision amongst options
	2. Getting Stuck in questions

## JEE Advanced

1. Order these Books : Arihant JEE Advanced Past year's IIT JEE Questions ; Guidebook by Arpit Agarwal
2. Do Complete Revision of JEE Advanced . Read in-depth abbout chapters in Physics & Chemistry on Wikipedia (April 10 - April 15)
3. Solve 10 problems from each book daily ( April 11 - May 20 )
4. Solve JEE Advanced Practice Tests and note down the problems ( May 1 - May 20 )
	1. Above the head questions

## Bucket List

Transisters

Optics

Equilibrium

Organic Chemistry
Ray Optics

## Spherical Mirror

A spherical mirror is a curved reflecting surface . It is painted with silver coating on one side , so that it reflects light that is incident on it . The reflected ray always passes through the focus , because the mirror is curved .

F = R/2 ; where F is the focus & R is the radius of Curvature .

**Concave Mirror**

**Convex Mirror**

## Mirror Formula

1/f = 1/u + 1/v

Sign Conventions :

Follow the cartesian system ; Pole at the Origin ; Distances to the left and bottom will be negative and distances to the right and top will be positive .

## Lenses

Lenses are curved Refracting Surfaces . The refraction takes place such that the refracted ray always passes through the primary focus , since the lens is curved .

**Concave Lens**

**Convex Lens**

Plano Convex & Concave Lenses : These lenses have only one hemispherical Surface .

## Lens Formula

1/f = 1/v - 1/u

Sign Conventions :

Follow the cartesian system ; optical center at the Origin ; Distances to the left and bottom will be negative and distances to the right and top will be positive .

## Lens Maker's Equation

1/f = (μ2 - 1)[ 1/R1 - 1/R2 ]

Where R1 and R2 are radius of curvatures of the two surfaces .

For Plano Convex lens , one surface is straight , thus Radius of curvature for on surface is infinity .

## Power of Lens

Power of Lens = 1 / f

Power of lens is measured in Dioptres , if the focal length is in metres .

## Vision Correction

**Myopia**

Myopia is near Sightedness . A concave lens of suitable focal length is used for vision correction .

**Hypermetropia**

Hypermetropia is far sightedness . A convex lens of suitable focal length is used for vision correction .

**Presbyopia**

Presbyopia is the condition of weak eye sight for distant as well as for nearby objects . A bifocal lens is used for vision correction .

## Optical Instruments

Optical instruments are used to either magnify objects nearby or to view distant objects . The arrangement of lenses matters in these instruments . The Distance of Distinct Vision (DDV) is the minimum distance upto which an object can b seen clearly . The DDV is said to be 25 cm .

Magnifying Power = Angle subtended at eye by image / angle subtended at eye by object at DDV

## Simple Microscope

Magnifying Power = β/α

Magnifying Power = D/v + D/f

Thus , magnifying power of Simple Microscope could be anything from D/f to 1 + D/f

## Compound Microscope

 Magnifying Power = β/α

Magnifying Power = - {vo / uo}{1 + D/fe}

## Tips And Tricks

1. The number of images formed by two mirrors inclined at an angle x is [360/x] - 1.
Integral Calculus

## Concept of Integration

Integration is the inverse function of a derivative . Hence it is also called as anti-derivative .

## Indefinite Integration

Integration without limits or intervals is called as indefinite integration . A constant is always generated in this case . The main challenge of integration is to simplify the given expression (integrand) and bring it into a suitable form , so that it can b integrated using standard formulae .

## Substitution Method

If the derivative of one part of equation exists in another part of equation (numerator or denominator ) then we use the substitution method .

## Integration Factor

For a differential equation of the form dy/dx + Py = Q

Integration Factor = e∫pdx

## Definite Integration

Integration with limits or intervals is called as definite integration .

NOTE : If substitution method is to be used for definite integration , then the limits have to be changed according to the substitution made .

## Application of Definite Integration

Integration can be used to multiply two quantities if the relation between them is known .

## Tips and Tricks

1) u.v. shortcut :- Coming Soon

2) You can differentiate the options one by one to reach the qustion .
Oscillations

### **Harmonic Oscillations**

Harmonic oscillations are those , which can be represented as a sine or cosine function of an angle .

## **Linear Simple Harmonic Motion**

Th motion of an object moving to and fro along a straight line is called linear simple harmonic motion .

F = -kx where F is the restoring Force ; k is the spring constant of force ; x is the displacement from equilibrium .

a = -kx / m

k/m = ω2

## Spring Constant

a) Parallel Connection : k = k1 + k2

b) Series Connection : 1/k = 1/k1 + 1/k2

## Kinetics of SHM

**Displacement**

x = asin (ωt + α)

☀α is a constant of integration .

**Velocity**

v = ω sqrt[a2 - x2]

**Acceleration**

d2x/dt2 = - ω2x

**Period**

ω = 2π / T

... T = 2π / ω

but , ω = sqrt [k/m]

... T = 2π / sqrt [k/m]

T = 2π / sqrt [a/x]

T = 2π / sqrt [acceleration per unit distance]

**Frequency**

ω = 2πn

... n = ω / 2π

n = 1/T

**Phase**

Displacement is given by :- x = a sin(ωt + α)

... Phase Angle is given as :- ωt + α

**Epoch**

Epoch is the phase angle of a SHM when t = 0

## S.H.M. and U.C.M.

x = R sin (wt + δ)

R = sqrt [a12 + a22 + 2a1a2cos(α1-α2)]

δ = tan-1[a1sinα1 + a2sinα2 / a1cosα1 + a2cosα2]

## Energy of a Particle performing S.H.M.

Kinetic Energy = 1/2 k (a2 - x2)

Potential Energy = 1/2 kx2

Total Energy = 1/2 mω2a2

## Resultant S.H.M.

During the superposition of two oscillations , their displacements at the point of superposition are added up to give the resultant Amplitude .

x = Rsin

## Simple Pendulum

F = -mqsinθ

θ = x/L

acceleration = -gx/L = ω2x

T = 2π√(L/g)

f = 1/T

## Laws of Simple Pendulum

1) Law of length : The period of a simple pendulum is directly proportional to square root of it's length .

2) Law of Acceleration due to gravity : The period of a simple pendulum is inversely proportional to square root of acceleration due to gravity .

3) Law of Mass : The period of a simple pendulum does not depend upon it's mass .

4) Law of isochronous : The period of a simple pendulum does not dpend upon its amplitude .

## Damped Oscillations

Periodic oscillations of gradually decreasing amplitude are called damped harmonic oscillations .

A' = Ae(-BT/2m)
Determinants

## Expansion

Only square determinants can be solved .

For determinants having more than three rows and columns , Det = aA - bB + cC - dD ........

where a,b,c,d are the first row elements and A,B,C,D are their corresponding minors

A determinant having all elements of a row or column zero , has value zero .

## Reduction

A determinant evaluation can be made easy by reducing the determinant into a unit matrix .

## Properties

1) Det = aA - bB + cC - dD ........

where a,b,c,d are the first row elements or column elements and A,B,C,D are their corresponding minors

2) One row or column can be subtracted or added by another row or column , respectively ; the value remains same .

3) An entire row or column can be multiplied by a scalar ; the value remains same .

## Sum of Determinants

Corresponding elements of determinants can be added directly . Thus , both the determinants should be of the same order .

## Product of Determinants

Multiplication of Determinants is not commutative .

## Derivative of Determinant

Der = Sum of Determinants with first row derivative then second row derivative .... n row derivative

## Cramer's Rule

Determinants can be used to find the solution of linear equations

## Number of Solutions of a Linear Equation

Δ =/ 0 ; then one unique solution

Δ = 0 & Δx , Δy , Δz is equal to zero ; then infinite solutions

Δ=0 , but at least one of Δx , Δy , Δz is not equal to zero ; then equation has no solutions .

## Consistency of Equation

Consider a determinant of co-efficients and constants of three linear equations .

if Det = 0 , then the equations are consistent ; i.. at least one set of solution exists .

## Area

Area of Quadrilateral : Divide quadrilateral into two triangles and use the above formula .
Stop The Time

Genre
 Year

**Stop The Time** is a song by MainStreet. It's featured on Breaking the Rules.

## Lyrics

Ja ze lacht mooi, en als ik met haar praat

Is het net alsof de hele wereld stil staat

Al zegt ze enkel 'hoi'

Weet ik mij geen raad

Ben opeens helemaal van de kaart

So many words to say

Maar mijn lippen werken nu niet mee

De manier waarop ze naar me kijkt oh

Ja mijn hart klopt in m'n keel

Hoop dat dit nooit meer verdwijnt no

Zij is alles wat ik wil

Ze ziet het niet maar zij is beautiful

She's always on my mind

En wanneer ze naar me kijkt oh

It will almost stop the time

Ze is zo uniek

Er is niemand zo cool

Ja ik hoop dat ze ziet hoe ik me voel

So many words to say

Maar mijn lippen werken nu niet mee

De manier waarop ze naar me kijkt oh

Ja mijn hart klopt in m'n keel

Hoop dat dit nooit meer verdwijnt no

Zij is alles wat ik wil

De manier waarop ze naar me kijkt oh

Ja mijn hart klopt in m'n keel

Hoop dat dit nooit meer verdwijnt no

Zij is alles wat ik wil

Ze ziet het niet maar zij is beautiful

She's always on my mind

En wanneer ze naar me kijkt oh

It will almost stop the time

Stop the time

It will almost stop the time
Progress of Technology

## Applications of Electrical Components

1. Resistors
2. Capacitors
3. Inductors
4. Diodes
5. Transistors
6. Triodes

## Applications of Devices

1. Van de Graffe Generator
2. Cyclotron
3. Galvanometer
4. Ammeter
5. Voltmeter
6. Potentiometer
7. Multimeter
8. Wheatstone's Meterbridge
9. Transformers
10. LC Oscillator

## Role of Transisters

Transisters have helped us to develop logic circuits that can perform certain logic operations (i.e. they can virtually think !)

## Important Establishments

1. Hadron Collider (CERN Laboratory)
2. International Space Station
3. Observation of Stars through Radio technology

There are stars in the universe that cannot be seen through our eyes . The reason is that these stars do not radiate EM waves in the visible region , instead they radiate Radiowaves . But , today we can detect these stars by using Radio technology . Huge Observatories are set up with dish antennae that can detect radio waves from these stars . One such observatory is in New Mexico , called as VLA (Very Large Array) , it has 27 huge dish antennae and it's range can move through miles into the space .
Molecular Orbital Theory

Molecular Orbitals are formed by the overlapping of two atomic orbitals of two different atoms . Bonding Molecular Orbitals have lower energy than the individual atomic orbitals , i.e. stable . The opposite is applicable for Anti-bonding Molecular Orbitals . The number of Molecular Orbitals is the same as the number of Atomic Orbitals .

For Bonding Molecular Orbitals ; b = A + B

For Anti-bonding Molecular Orbitals ; a = A - B

1) For every bonding molecular orbital , there is a corresponding anti-bonding molecular orbital .

2) Anti-Bonding Molecular Orbital will be opposite to Bonding Orbital .

3) Bonding Molecular Orbitals will participate in bonding .

4) Generally Anti-bonding Molecular Orbitals are empty .

5) Bonding and Anti-bonding Molecular orbitals can together hold 2 electrons .

**Bond Order = (Bonding - Anti-bonding electron )/ 2**
Alcohols & Phenols

# Alcohols

Alcohols can be prepared by hydration of Alkenes , Hydrolysis of Alkyl halides , Oxidation of Ketones / Aldehydes , From Grignard's Reagent , Hydroboration - Oxidation .

Alcohols undergo Oxidation , Esterification , Reactions with Metals , HCl , HBr , HI , PCl3 , PCl5 , SOCl2 .

# Phenols

Phenol is Carbolic Acid .

Phenols are prepared by Dow Process , Raschig Process , Diazotization and from Cumene .

Phenols undergo Halogenation , Sulphonation , Nitration , Kolbe's Reaction , Riemer - Tiemann Reaction , Zinc dust reaction , Oxidation .
Organic Chemistry

Organic Chemistry is the branch of Chemistry which deals with the study of Compounds containing Carbon . This branch is vast because Carbon can form millions of compounds with itself as well as other functional groups and elements .

## Carbon

Carbon is the 6th element of the periodic table having atomic number 6 and atomic mass number 12 .Carbon has three isotopes - C12 , C13 , C14 .

## Catenation

Carbon shows the remarkable property of Catenation , due to which it can form long chains of bonds with itself . This is possible due to Tetravalent Character of Carbon .

## Physical Properties

The MP & BP increases with increase in number of Carbon atoms but decreases as branching increases .

## Carbon Ions

### Carbanions

A Carbanion is unstable because :-

1) Carbon atom is small in atomic size .

2) Carbon Atom has low value of electronegativity .

## IUPAC Nomenclature

(only hydrocarbons , mono and bifunctional compounds)

## Resonance

A higher charge density , decreases the stability of a compound . Thus , the phenomenon of Resonance spreads the charges and decreases the electron charge density , thus making it more stable

## Optical Isomerism

(upto two optical centres) (RS EZ Nomenclature excluded)

## Conformation

(Ethane & Butane) Newmann Projections

The isomerism which arises due to rotation about C-C single bond is called conformational isomerism .

## Commercial Uses

1. During fractional Distillation of Petroleum , a variety of products can be recovered .
	1. Gasoline : C6 - C10
	2. Kerosene : C10 - C13
	3. Oil : C13 - C18
2. Polyurethane and Ammonium perchloride are used as solid fuel propellants .

## Tips & Tricks

1. For Stability , consider Resnance . If the compound has double/triple bond or lone pair of electrons in conjugation , then it is stabilized by resonance .
2. For Organic Chemistry Reactions , for predicting the product , use the concept of Nucleophiles/Electrophils . Negative part goes to positive part of carbon chain and vice versa . The positive/negative part of carbon chain may shift according to stability .
Practical Physics

## Measuring Instruments

1. Meter Scale
2. Vernier Calipers
3. Micrometer Screw Gauge
4. Spherometer
5. Thermometer
6. Digital Multimeter
7. Ammeter
8. Voltmeter

## Methods of Measurements

(For the following Experiments in JEE )

1. Vernier Calipers
2. Micrometer Screw Gauge
3. Determination of g using simple pendulum
4. Young's Modulus by Searle's Method
5. Specific Heat of a Calorimeter
6. Focal Length of a Concave Mirror and a Convex lens (uv method)
7. Speed of Sound using Resonance Column
8. Verification of Ohm's Law (Using Voltmeter and Ammeter)
9. Specific Resistance of the material of a wire using Wheatstone's meterbridge and post office box
Permutations and Combinations

A permutation is an arrangement and combination is a selection . Obviously , there would be more permutations possible than combinations for a given set of elements .

In a permutation the order of arrangement is important , where as in a combination , order is not important since we are concerned only with what elements we are selecting .

## Fundamental Principle of Counting

If a work can be done in either m **or** n ways , the total number of ways of doing it is **m + n .**

If a work can has to be done by both m **and** n ways , the total ways of doing it is mn ways .

If a work has to be done under a restriction , then No. of Ways = Total ways without restriction - no. of ways under opposite restriction

### **Permutations of repeated objects**

if an object is repeated p times and another object is repeated q times , then

No. of Permutations = nPr / p!!q!

### **Circular Permutations**

No. of Permutations = (n-1)!  ; If clockwise and anticlockwise are different

No. of Permutations = 1/2 (n-1) !  ; if clockwise and anti clockwise are identical

## Combinations

**No. of Combinations** = **n****Cr = n! / r!(n-r)!**

Number of selections from n objects of zero or more objects = **2n**

without zero = **2n - 1**

No. of selections from n identical objects without selecting zero = **n**

with zero = **n + 1**

No. of selections from p identical , q identical and r distinct objects :

if all and none can be selected = **(p+1)(q+1)2r .**

If none cannot be selected = **(p+1)(q+1)2r - 1**

if all and none cannot be selected = **(p+1)(q+1)2r - 2**

Number of Selections of r things from n different things when each can be repeated unlimited times

= **n+r-1Cr**

## Distributions

No. of ways of distributing n different things between 2 people , one reciving p things and other receiving q things , where p + q = n (all distributed)

= **nCpxn-pCq == n! / p!q!**

similarly for three people , = **n! / p!q!r!  ;** p+q+r = n

Number of ways to distribute combination of two objects (mxn) among n people equally is

= (**mn)!/ (m!)n**

No. of ways to divide (mxn) objects into n bundles

= (**mn)!/ (m!)nn!**

 No. of ways to divide n identical objects among r people

= **n+r-1Cr-1**

## Tips And Tricks

1) nCr = nCn-r

2) nCr + nCr-1 = n+1Cr
Circle and Conics

## Circle

**Equation of Circle**

**Standard Form**

(x-h)2 + (y-k)2 = 0

**Diameter Form**

(x-x1)(x-x2) + (y-y1)(y-y2) = 0

**General Form**

x2 + y2 + 2gx + 2 fy + c = 0 (Note : There is no term of xy in this equation !)

Centre co-ordinates = (-g , -f)

Radius = sqrt [g2 + f2 - c]

**Parametric Form**

(x,y)= (asinθ + acosθ )

**Equation of a circle passing through intersection of two other circles**

**Tangents**

The lengths of tangents from a point (x1,y1) is :

sqrt [x12 + y12 + 2gx1 + 2fy1 + c]

**Normals**

**Intersection**

## Parabola

**Equation**

y2 = 4ax

**Directrix**

x = - a

**Focus**

(a,0)

**Latus Rectum**

LR = 4a

**Parametric Form**

x = at2 ; y = 2at

**Tangents and Normals at (x1,y1)**

**Tangent :** yy1 = 2a (x + x1)  ; t = 1/m

**Normal :** [x - x1] / [-2a] = [y - y1] / y1

## Ellipse

**Equation**

x2/a2 + y2/b2 = 1

**Directrix**

**Focus**

(+ae,0) ; (-ae,0)

**Axes**

Minor Axis = Major Axis =

**Parametric Form**

(x,y) = ( asinθ , bcosθ )

**Tangents and Normals**

**Tangent :** xx1 / a2 + yy1 / b2 = 1  ; y = mx + sqrt (a2m2 - b2)

**Normals :**

## Hyperbola

**Equation**

x2/a2 - y2/b = 1

**Directrix**

**Focus**

**Axes**

**Parametric Form**

x = (asecθ , btanθ)

**Tangents and Normals**

**Tangent :** xx1 / a2 - yy1 / b2 = 1  ; y = mx - sqrt (a2m2 - b2)

**Normals :**

## Tips and Tricks

1) The properties of radius of circle are useful for solving circle based questions .
Surface Chemistry

# Adsorption

Adsorption is the process of accumulation of absorbate particles on the adsorbent surface .

## Physisorption & Chemisorption

Physisorption is the process of adsorption of particles on a surface by weak Vaan der Waals Forces .

Chemisorption is the process of adsorption of particles on a surface by stronger chemical bonds .

## Adsorption of Gases on Solids

x/m = kPn (where n is 0 for high pressure and n is 1 for low pressure)

# Catalysis

Catalysis is the process of speeding up a reaction by adding certain specific materials . The phenomenon of Adsorption is useful for Catalysis .

## Catalytic Activity

Catalyst makes the chemical reaction easier or faster by reducing it's activation energy or providing a surface for attack of reagent .

## Catalytic Selectivity

Catalytic Selectivity refers to the idea that each chemical reaction needs a certain , specific catalyst and that catalysts are not universal .

# Colloids

## Types of Solutions

**True Solutions**

Particle Size = less than 10-10 m

**Colloidal Solutions**

Particle Size = 10-7 to 10-9 m

**Suspension**

Particle Size = More than 10-6 m

## Classification

 Liquid in Solid
 Gas in Solid
 Solid in Liquid
 Liquid in Liquid
 Gas in Liquid
 Solid in Gas
 Liquid in Gas

## Sols

**Lyophilic Sols :** Liquid loving sols . They can be prepared directly by mixing solid with liquid . Relatively stable as they have affinity for liquid .

**Lyophobic Sols :** Liquid hating sols . Special methods are employed to prepare lyophobic sols and additional stabilizers are required during their preparation. They are less stable .

**Electrophoresis :** It is similar to electrolysis .

## Surfactants and Micelles

(Only definition and examples)

**Surfactants** are compounds that lower the surface tension (or interfacial tension) between two liquids or between a liquid and a solid. Surfactants may act as detergents, wetting agents, emulsifiers, foaming agents, and dispersants. are compounds that lower the surface tension (or interfacial tension) between two liquids or between a liquid and a solid. Surfactants may act as detergents, wetting agents, emulsifiers, foaming agents, and dispersants.

A **micelle** (/maɪˈsɛl/) or **micella** (/maɪˈsɛlə/) (plural **micelles** or **micellae**, respectively) is an aggregate of surfactant molecules dispersed in a liquid colloid. A typical micelle in aqueous solution forms an aggregate with the hydrophilic "head" regions in contact with surrounding solvent, sequestering the hydrophobic single-tail regions in the micelle centre.
D-Block Elements

(3d series to be learnt precisely)

d- block elements are intermediate between highly electropositive s-block elements and highly electronegative p-block elements . The differentiating electron enters the d-orbital of the penultimate shell . d-block elements have partly filled (n-1) d-orbital . ... Zn, Cd , Hg & Uub should be excluded from d-block . But since their chemical properties are similar , they are considered as d-block elements .

## Periods

3d Series : Sc(21) to Zn (30)

4d Series : Y (39) to Cd (Z=48)

5dSeries : La(57) to Hf(72) to Hg (80)

6d Series : Ac(89) to Rf(104) to Uub(112)
Basic Mathematics

## Co-ordinate Geometry

Distance Formula

Section Formula

Mid Point Formula

Centroid Formula

Incentre Formula

Area Formulas

## Ratios

If a/b = c/d THEN ;

**Alternendo**

d/b = c/a

**Invertendo**

b/a = d/c

**Componendo**

(a+b)/b = (c+d)/d

**Dividendo**

(a-b)/b = (c-d)/d

**Componendo - Dividendo**

(a+b)/(a-b) = (c+d)/(c-d)

**Theorem on Equal Ratios**

(a+c)/(b+d) = (a-c)/(b-d) = k

## Expansions

(a+b)2 = a2 + b2 + 2ab

(a-b)2 = a2 + b2 - 2ab

a2 - b2 = (a-b)(a+b)

(a+b)3 = a3 + 3a2b +3ab2 +b3

(a-b)3 = a3 - 3a2b + 3ab2 - b3

a3 + b3 = (a+b)(a2-ab+b2)

a3 - b3 = (a - b)(a2+ab+b2)

(a+b+c)2 = a2 + b2 +c2 + 2(ab+bc+ac)

a2 + b2 = (a + b)2 - 2ab

a2 + b2 = (a - b)2 + 2ab

(a - b)2 = (a + b)2 - 4ab

## Identity

f(x) = g(x) is an identity if both have the same value for all real values of x .

## Polynomials

1) First order polynomial : f(x) = 0 ;

e.g. ax + b = 0

x = -b /a (only one solution)

2) Second Order Polynomial : ax2 + bx + c = 0

x = [-b ± √(b2-4ac)] / 2a (two solutions)

3) Third Degree Polynomial :

Can be solved if the equation can be factorized into linear and quadratic sub-equations .

4) nth Degree Polynomial : has n solutions

## Exponential Equations

e.g. af(x) = ag(x)  , where a =/ 1

then f(x) = g(x)

Equate the exponents to find the solution .

If it cannot be put in the form , substitute another variable y so as to get a polynomial equation .

## Logarithmic Equations

if logaf(x) = logag(x) ; then f(x) = g(x) . f(x) and g(x) should be positive .

if the above form cannot be achieved , put a variable y , so as to get a polynomial equation .

## Functions

Every function can be plotted on graph . Based on the behavior of graphs , we have an important section of Mathematics , called the Calculus , which studies the following :-

1. x and corresponding y co-ordinates
2. Continuity of graph of a function
3. Rate of change of y co-ordinate with respect to x co-ordinates
4. Revival of function from rate of change of y w.r.t. x .

If a function is defined as max{ 2x + 5 , 4x + 7} ; it means that our function is the outer boundary amongst the to given functions .

## Some Important Functions

A Function is an expression which defines the behaviour of numbers .

**Modulus Function : f(x) = |x-a|**

= x-a x>a

= a-x x<a

Not continuous at a

**Greatest Integer Function : f(x) = [x]**

f(x) = [x+n] = [x] + n

Not continuous at all integral values of x

## Inequations

An expression which is defined by using ">" , "<" instead of "=" is called an inequation .

Laws of inequations

1) a + b > a + c ; then b > c

2) if a > b  ; then ca > cb

3) if ab > ac ; then b > c  ; if a > 0

4) if a > 0 , b > 0 ; then a+b > 0 , ab > 0

if a < 0 , b < 0 ; then a+b < 0 , ab > 0

if a<0 , b >0 ; then ab > 0

5) ax > ay  ; then x > y if a > 1

6) logax > logay  ; then x > y if a > 1

x < y if 0 < a < 1

for evaluating roots of an inequations , use basic rules discussed in the chapter earlier . The answer always lies as an inequation .
Kinematics

## Rectilinear Motion

Rectilinear Motion is the motion which takes place along a straight line . There is no angular component related to it .

## Displacement

Displacement is the change in position of an object with respect to a certain coordinate system or frame of reference .

## Speed

Average Speed : The average speed in a time interval

Or Average Speed = 3xyz / [xy + yz + xz]

Instantaneous Speed : The speed of a particle at a certain point in time .

## Velocity

Velocity is the rate of change of position of an object with respect to a specific co-ordinate system or frame of reference .

Average Velocity : The average velocity in a time interval .

Instantaneous Velocity : The velocity of a particle at a certain point in time .

## Acceleration

The rate of change of velocity with respect to time is called as acceleration . Zero acceleration means uniform velocity . Positive acceleration means the velocity is increasing . Negative acceleration means velocity is decreasing .

Average acceleration : The average acceleration in a time interval

Instantaneous acceleration : The acceleration of a particle at a certain point in time .

## Momentum

Momentum is the Product of Mass and Velocity .

p = mv

## Force

The rate of change of momentum with respect to time is called Force . It can also be formulated by Mass x Acceleration .

F = ma

## Kinematical Equations

v = u + at

s = ut + 1/2 at2

v2 = u2 + 2as

sn = u + (a/2)(2n - 1)

## Motion of a Freely Falling Body

v = gt

h = 1/2 gt2

v2 = 2gh

hn = (g/2)(2n - 1)

Maximum height = u2 / 2g

Time of Flight = 2u / g

v = sqrt (2gh)

## Relative Velocity

1) In the same direction ; a - b

2) In opposite directions ; a + b

3) If inclined at an angle θ , then

Relative Velocity = sqrt ( a2 + b2 - 2abcosθ)

## Motion

Motion in Straight Line : Moves along one co-ordinate axis

Motion in a Plane : Moves in a plane with 2 d co-ordinate axis .

Motion in Space : Moves in Space with 3 d co-ordinate axis .

Time is always uni-directional .

## Change of Frame

If there are 2 different frames of reference w.r.t. which measurement is to be done , we use shift of origin concept of Maths .

## Tips and Tricks

1. Preferably make use of Kinematical Equations only .
Scalars and Vectors

Mathematics is the language of Physics .

Scalar Quantities can be denoted by a number and a unit , i.e. Magnitude . Vector Quantities are denoted by magnitude and direction . Tensor Quantities

## Properties of Vectors

**Equal Vectors**

Vectors having same direction and magnitude .

**Parallel Vectors**

Vectors having same direction .

**Anti-Parallel Vectors**

Vectors having opposite directions .

**Position Vector**

Vector drawn from the origin to a point .

**Zero Vector**

Vector of magnitude zero .

**Free Vector**

Vector which can be moved through space , without changing it's direction .

## Triangle Law

If two vectors exist such that their directions are taken in order , then their resultant is equal to the third of the triangle formed ; & the direction is from head of first to tail of second .

## Parallelogram Law

The parallelogram law of Vectors states that if two vectors originate or meet at a common point , then their resultant is given by the diagonal of parallelogram formed .

## Scalar Multiplication

Scalar multiplication refers to the multiplication of a vector with a number having no direction .

In Scalar Multiplication , the number is multiplied to all components of the vector . Thus , scalar multiplication does not change the direction of the vector , it only affects the magnitude .

## Resolution of Vectors

for co-ordinate system , we know that **:** x = rcosθ y = rsinθ

for 3-d coordinate system , we resolve using direction cosines : r = acosα + acosβ + acosγ

## Scalar Product

a.b = abcosθ

Scalar Product gives us a Scalar result .

## Vector Product

axb = absinθ

Vector product gives us a vector result .

## Unit Vector

A unit vector has unit magnitude and any direction . If we divide a vector by it's magnitude , we get an unit vector .

## Mathematics in Physics

### Differential Calculus

Rate Measurer : Using graphs in physics , we can find the rate of change of quantity on Y-Axis w.r.t. change in quantity on x-Axis by differentiating the function and applying the values at that point . or by finding the slope of tangent at that point .

Maxima & Minima : We can interpret data from the graph by finding it's maximum & minimum value by second order derivative test .

## Tips & Tricks

1) If in any question , based on any chapter , direction is being asked , then always work with vectors (xi + yj + zk)
Vectors

## Vector

A vector has a magnitude and direction . Vectors can also be called as directed numbers .

## Triangle Law

If two vectors are taken in order , then their resultant would be the third side of triangle formed by joining the tail of first and head of second .

## Parallelogram Law

If two vectors have a common source of origin or intersection , then their resultant is given by the diagonal of the parallelogram formed by extending parallel sides .

## Unit Vector

A vector having magnitude one (unity) is called a Unit Vector . So Unit vector is used to specify the direction of another vector .

## Scalar Product

Scalar Product is commutative

a.b = abcosθ

## Vector Product

Vector Product is not commutative .

axb = abcosθ

## Area

Area of Triangle = 1/2 AB x AC

Area of parallelogram = 1/2 axb

## Collinearity of Vectors

If a vector can be expressed as a scalar multiple of the other , then the vectors are collinear

## Linear Combination of Vectors

If two vectors are co-planar , then they can be expressed int he linear form , e.g. r = px + qy .

Any two vectors are always co-planar . for three co planar vectors , we need their linear combination .

## Section Formula

Internal Division : Position Vector = (mb + na) / (m+n)

External Division : Position Vector = (mb - na) / (m-n)

Mid Point = ( a + b ) / 2

## Scalar Triple Product

if a , b , c are three vectors , then

[a b c] = a.(bxc) =

a1 a2 a3

b1  b2 b3

c1 c2 c3

(Multiplied by [a b c] if non-coplanar vectors)

If any 1 is zero vector , [a b c ] = 0

If any two are collinear or equal, [a b c ] = 0

The vectors are coplanar if , [a b c] = 0

If all three are interchanged , the scalar triple product remains same .

If any two are interchanged , the sign changes .

## Parallelopiped & Tetrahedron

Volume of Parallelopiped = [a b c]

Volume of Tetrahedron = A - BCD = 1/6 [AB AC AD]

A,B,C,D are coplanar if [AB AC AD] = 0

## Vector Geometry

The sum of all vectors taken in order of a closed figure , is Zero .
JEE : Chemistry

When there is inadequate information given regarding any numerical , always begin with assumptions ; for e.g. You can consider 100 mL Volume and and make calculations accordingly .

Make Substitutions for temperature only in Kelvin (Depends) .

Electronic Configurations up to Element 36 to be known .

For Solid state , study fcc , bcc , hcp and simple ionic compound structures

Redox Reaction Functional Groups

Important Compounds

Molar Mass
Mole Concept

## 1 Gram Atom

1 Gram Atom is the number of atoms present in 1 formula gram of that atom .

## 1 Gram Mole

1 Gram Molecule is the number of molecules present in 1 formula gram of that atom .

## Mole Concept

### 1 **Mole**

1 Mole is the amount of substance which contains 6.022 x 1023 fundamental particles .

Mass of one mole = Molar Mass

Volume of 1 mole liquid = (Molar Mass) ml

Volume of 1mole gas = 22.4 L

### **Formulae**

1) Number of Moles = Mass / Molar Mass

2) Number of Moles = Volume / Molar Volume

3) Number of Moles = Number of Particles / Avogadro's Number

4) For compound MxNy , x moles of N = y moles of M

## Principle of Atom Conservation

1) No. of moles of an atom in Reactants = No. of moles of the atom in Products .

2) In reaction , H2 + O2 -------> H2O

No. of moles of O2  = No. of moles of H2O

2 x No. of moles of O2 = No. of moles of O in H2O
Gaseous State

## Absolute Scale of Temperature

P = dRT / M

P1V1 / T1 = P2V2 / T2

PT = P1 + P2 + P3 + ........ + Pn

Where , P1 = x1P1 ; x1 = n1 / nT  ( x = mole fraction )

### **Graham's Law of Diffusion**

Rate of diffusion (r) = Volume of Gases Diffused                                                                                                                                 ------------------------------------------TIme

### r = V/t

r1 / r2 = V1T2 / V2T1
r α 1/√d

... r1/ r2 = √ (d2/d1)

... r1/ r2 = √ (M2/ M1)

**Amagat's Law of Equal Volumes**

The total volume of a gas mixture is equal to the sum of partial volumes of individual gases .

## **Van der Wahl's Equation or Real Gas Equation**

(for real gases)

(P+(a/V2))(V-b) = RT {for 1mole }

(P+(an/V2))(V-nb) = nRT

Unit of a = atm lit2 / mol2

Unit of b = lit / mol

Note : For Low Pressure , ignore b and for low volme ignore a .

## Properties of Gases

### **Standard Temperature Pressure**

Temperature : 0o C = 273 K

Pressure = 1 atm = 1.0132 x 105 N/m2 (Pa)

### **Liquefaction**

#### Mean Square Velocity

Average Velocity

Most Probable Velocity

###
Sterownia

**Sterownia** to lokacja w grze Five Nights at Wario's, połączona jest ona z Biurem.

## Rozgrywka

W tym miejscu można zobaczyć jedynie Luigi'ego i Peach. Stoją oni obok maszyn.

W tym pomieszczeniu, z prawej strony, jest tak zwany ślepy punkt.

## Wygląd

W tym miejscu jest dużo maszyn kontrolujących całą fabrykę. Po lewej stronie można zobaczyć jakieś biurko z krzesłem.
Preparation Life Management Strategy

Whenever there is a problem or you feel stressed , simply SMILE !

Drink a lot of water to stay hydrated

Exercise everyday to stay fit.

Whenever the going becomes tough , give yourself a push - Self Motivation

Whenever you are being compared or feel offended when someone is performing better than you , just decide that you have to study that chapter more and become better than the person . Work for 12 hours !
Rotational Motion

## Rigid Body

A Rigid body is defined as a body whose particles move in the same way as the body moves .

## Rotational Motion

Rotational Motion is the motion of a body which rotates about its own axis . The body performs rotational motion while , the individual particles perform circular motion . The axis of rotation about the centre of mass of a body , is called as its natural axis .

## Moment of Inertia

Moment of Inertia of a body is defined as the product of Mass and the square of radius of gyration .

I = MR2

I = MK2

## Radius of Gyration

Radius of Gyration is defined as the distance of the axis from its natural axis .

K = sqrt (I/m)

## Rolling Motion

(Rings , Cylinders & Spheres for JEE)

Kinetic Energy = 1/2 MV2 [1+K2/R2]

Velocity = sqrt {2gh / [1+K2/R2]}

Acceleration = gsinθ / [1+K2/R2]

## Theorem of Parallel and Perpendicular Axes

Ip = Ic + Mh2

Iz = Ix + Iy

## Applications

 Disc
about it's height
 Solid Sphere
through natural Axis
about diameter
about tangent

## Tips and Tricks

1. For a system of bodies , add up their individual momnts of inertia .
Refraction of Light

Refraction is the phenomenon of change in direction of light when it travels from a rarer medium to a dnser medium and vice versa .

## Snell's Law

μ = sin i / cos r

Ratio of width = cos i / cos r

## Dispersion of Light

The separation of white light into it's constituent colours is called as Dispersion .

δ is the angle of deviation .

δv > δy > δr

## Prism Formula

μ = sin[A + δm / 2] / sin[a/2]

δ = A(μ - 1)

## Angular Dispersion

Angular DIspersion = δv - δr = A(μv - μr)

## Dispersive Power

Dispersive Power =  ω = (δv - δr) / δy

```
 ω = (μv - μr) / (μy - 1)

```

## Scattering of Light

The phenomenon of deflection of light in all possible directions by small particles is called as Scattering .

## Rayleigh Scattering

The Intensity of scattered light is inversely proportional to the fourth power of wavelength .

I α 1/λ4
Gravitation

## Universal Law of Gravitation

F = G m1m2 / r2

Gravitational Force is independent of the medium . The gravitational force inside a body is zero .

## Acceleration due to Gravity

### **Variation with Height**

g = GM / (R+h)2 gh = g (R/(R+h)2) gh = g (1 - 2h/R)

### **Variation with Depth**

g = gh(1 - d/R)

for every percentage decrease in radius , with mass constant ; there is 2 % increase in acceleration due to gravity

### **Relation between Variation of height and depth**

If acceleration due to gravity is same at a certain height and depth ;

then d = 2h

## Gravitational Field Intensity

Gravitational field is the field of attraction around a mass . Gravitational Field Intensity is the amount of Force experienced by an object of unit mass in a gravitational Field .

... Intensity = Force / mass

## Gravitational Potential Energy

Gravitational Potential Energy is the work done in bringing a charge of unit mass from infinity to a point against the gravitational field .

Potential Energy (U) = Force . Displacement

= - GMm /r

... U = Vm

## Gravitational Potential

Gravitational potential is the amount of work done per unit mass in a gravitational field .

Potential(V) = - GM / r

... V = U / m

Gravitational Potential is always directed opposite to Gravitational Field .

**Cases of Potential :**

1. **Due to point mass :** V = - GM/r ... at r = 0 , Z is negative infinite . The graph will be negative , curved and parallel to V and r axes .
2. **Inside a Solid Sphere :** V = - GM (1.5R2 - 0.5r2) / R3 . At r=R ,V = - GM**/**R . At r = 0 ,V = -1.5GM**/**R The graph is negative , parabolic .
3. **Outside a Solid Sphere :** V = -GM/r , at r = infinity , V = 0 ; On surface , V = -GM/R
4. **Due to a Spherical Shell :** Outside V = -GM/r . Inside V = - GM/r
5. **On the Axis of a Ring :** V = - GM / sqrt(R2+x2)

## Kepler's Laws

### **Law of Orbits**

Kepler's Law of Orbits states that all planets revolve around the sun in elliptical orbits with sun as the focus .

### **Law of Areas**

Kepler's Law of Areas states that a planet sweeps equal Areas in equal intervals of time around the sun .

### **Law of Periods**

Kepler's Law of Periods states that the square of Time Period is directly proportional to the cube of Distance .

... T2 α r3

## Rocket Science

### **Critical Velocity**

vc = sqrt (GM/r)

vc = sqrt (ghr)

### **Binding Energy**

B.E. = - T.E.

B.E. = GMm / r {At rest}

B.E. = (GMm) / 2r

### **Escape Velocity**

ve = sqrt (2ghr)

ve = sqrt (2GM/r)

The escape velocity remains independent of angle of projection .

Escape Velocity for an object on earth is 11.2 km/s