compendium book for iit -mains

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AIEEE

FORMULAE BOOK

IIT

BITS



PHYSICS

General

1. If x = a mbnco, then

2. For vernier calipers, least count = s-v(s=length of one division on main scale, v=length of one division on vernier scale)

3. Length measured by vernier caliper = reading of main scale + reading of vernier scale ×least count.

4. For screw gauge, least count =

where, Pitch =

5. Length measured by screw gauge = Reading of main scale + Reading of circular scale ×least count.

6. Time period for a simple pendulum =

Where, l is the length of simple pendulum and g is gravitational acceleration.7. Young’s modulus by Searle’s method, ,

where, L= initial length of the wire, r=radius of the wire and ∆l= change in length. 8. Specific heat of the liquid,

where, m=mass of the solid. m 2= mass of the cold liquidT1=temperature of cold liquid. T 2=temperature of hot liquid.T = final temperature of the system c 1= specific heat of the material of

calorimeter and stirrer.c2= specific heat of material of solid m 1 = mass of the calorimeter and stirrer

Mechan ics

Vectors

1. = , where is angle between the vectors.

And direction of from ,

2. Two vectors (a 1 + a 2 + a 3 ) and (b 1 + b 2 + b 3 ) are equal if:

a 1 = b1 a 2= b2 and a 3 = b3

3. If angle between two vectors and i s ‘ ’ , = ab

x = (ab ) ( is a unit vector perpendicular to both and )

4. Velocity of ‘B’ with respect to ‘A’ , = -



Kinematics

t1 = initial time t 2 = final time u= initial velocity v = final velocityv av = average velocity a = acceleration s = net displacement

1.Average speed, V av =

2.Instantaneous speed,

V = =

3. Displacement =

4. Total distance =

5. a= =v =

6. When acceleration is constant

v= u +at

s= ut + at 2 = vt - at 2

v2 = u 2+2as

Projectile motion

1. Time of flight , t 0 =

Range , R=

2. Maximum height H=

3. (x,y)=(ucos )

4. Equation of Projectile ,y =xtan

Forces

1. F - R=ma

2. Frictional force =f, force applied =F

3. = F



Circular Motion

1. = =

Α = = =

2. =

AN =

a total=

3. R at (x,y) =

4. Banking of roads, tan =

5. Centripetal force = = mr

Work and Energy

1. Total Energy = Kinetic energy =potential energy

2. =

3. Conservative forces : spring force,electrostatic forces…

4. Non- conservative forces : frictionalforces, viscous force..

Center of Mass, LinearMomentum, Collision

1.

2.

3.4.

5. For perfectly inelastic collision,6. .If coefficient of restitution is e

(0<e<1),Velocity of separation = e (Velocity

of approach)V =

7. Impulse = =

Rotational Mechanics

η=torque, F=force I=moment of inertia α=angular acceleration L=angular momentum

1. Moment of inertia,

I = = dm

2. Angular momentum,

L=Iω

3.

4. (i) Pure translation(ii) Rotation 0(iii) Pure rotation 0(iv) Translation(v) Rolling(vi) Sliding or sliding



Gravitation

G = Universal gravitational constantU = Gravitational potential energyV = Gravitational potential

E = Gravitational fieldF = Gravitational force

1. F = G (attraction force)

2. Gravitational potential energy, U = -G

3.

4. Gravitational potential, V =

5. Gravitational field, E =

6. Escape velocity, u

Planets and satellites

1. v = T = 2 K.E. = , P.E =- => E =-

Simple Harmonic Motion

angular frequency I = moment of inertia T = time period = length of pendulum

1. + = 0

2.

3. Angular simple harmonic motion,

T= 2 , =

4. Physical Pendulum

T= =

5. Simple Pendulum,T= 2 , =

Fluid Mechanic s

P= pressure =density V=volume of solid v=volume immersed D= density of solidd=density of liquid A=cross section area U=upthrust

1) p=

2) Variation of pressure with height, dP = - dh

3) Archimedes Principle mg = v dg (or) VD = vd4) Equation of continuity, A 1v1= A 2v2 5) Bernoulli’s equation, P+ p + gh = constant



Elas t ic i ty

Y= Youn g’s modulus = stress = strain B=Bulk modulus

F= force A=cross section area =initial length = change in length

1. Y= = 2) B = - 3) Elastic potential energy = × stress × volume

Surface tens ion

T = surface tension Θ = angle of contact R = radius of the bubble/drop R = radius of the tube

1. Excess pressure inside a drop, ΔP = Excess pressure inside a soap bubble, ΔP =

2. Rise of liquid inside a capillary tube, h =

Viscosity

=coefficient of viscosity; F=force V=velocity; ζ = density of liquid ; A = Crosssection area

1. F = Stoke’s Law, F = 6πr v Terminal Velocity, v 0 =

Wave Motion

A = Amplitude y = Displacement ΔФ = Phase Difference γ = Frequency λ = wavelength

L = Length of the wire Μ = Mass per unit length ω = Angular frequency Δx = Path difference

1. Equation of a wave, y =

2. Velocity of a wave on a string, V =

3.

4.Resultant Wave, y = y 1 + y 2

Constructive interference, Δθ = 2nπ Or Δx = nλ Destructive interference, Δθ = (2n-1)π Or Δx = (n-1/2)λ

Fundamental Frequency, γ0 =



Sound Waves

1. Speed of sound in :

1)Fluids = (b: Bulk Modulus, ρ: Density 2)Solids = (Y: Young’s Modulus, ρ:

Density 3) Gas =

1. Closed organ Pipe,

2. Open organ Pipe,

3. Freq. of beats = |ν1 - ν2|4. Doppler Effect, ν =

Thermal Phys ic s

P=pressure V=volume n= no. of moles T=temperature R= universal gas constantco-efficient of linear thermal expansion β=co-efficient of superficial themal expansion

=coefficient of volume thermal expansion

1. Ideal gas equation , PV=nRT . 2.Thermal expansion , α= ;

β= ; =

Kinetic Theory of Gases

1. = Translational kinetic energy , K=

= p; =

2. Vander Waal’s Equation: ( p+ )(v-b)=nRT

Calorimetry

Q=heat taken /supplied ; s=specific heat; m=mass; =change in temperature; L=latent heat ofstate change per mass

1. Q=ms 2.Q=mL

Laws of Thermodynamics

W=work done by gas , U=internal energy =initial volume =final pressure =initial pressure=final pressure

1. dq = dW + dU 2.W = 3. Work done on an ideal gas:



4.Isothermal process , W = nRT ln( ) Isobaric process W = P( )

Isochoric Process, W = 0

Adiabatic process, W =

5)Entropy, 6). = 7)

8) = R, = 9) Adiabatic Process ,P =constant

Heat Transfer

e = coefficient of emission =stefan’s constant1 K=coefficient of thermal conductivity

1. =K =-KA 2)Thermal resistance ,R =

2) Heat current , I= ) 4)Series connection ,R=

5) Parallel connection , 6)U=e A (U=energy emitted per second)

2. Newton's law of cooling , f =-K( )( taken in Celsius scale )

Opt ic s :u=dist. Of the object from the lens/mirror v=dist. Of the image from the lens/mirrorm= magnification i= angle of incidence r=angle of reflection/ refraction

n=refractive index c=critical angle δ=angle of deviation R= radius ofcurvatureP=power

1. Spherical Mirrors, m = -

2. Refraction at plane surfaces n= = real depth / apparent depthΘc= sin -1(1/n)

3. Refraction at spherical surface m=

4. Refraction through thin lenses , , m=

5. Prism r + r’ =A , δ = i + i’ – A n =



Electr ic i ty and Magnet ism :

Coloumb’s law: The force between 2 point charges at rest: =

Electric field: =q

Electric potential: ∆V=-E∆r cosѲ V=-

Electrical potential energy: U=

Fields for a particular point: E= . V= .

Gauss law: Net electric flux through any closed surface is equal to the net charge enclosed

by the surface divided by ε0. =

Electric dipole: It is a combination of equal and opposite charges. Dipole moment

,where d is the separation between the 2 point charges.

Electric field due to various charge distribution:

(a) Linear charge distribution: E= , λ is the linear charge density

(b) Plane sheet of charge E=ζ/2ε0 where ζ is the surface charge density (c) Near a charged conducting surface: E=ζ/ε0 (d) Charged conducting spherical shell:

= , r>R = , r=R

(e) Non conducting charged solid sphere:

= , r>R = , r=R

(f) Facts: 1. In an isolated capacitor, charge does not change.2. Capacitors in series have equal amt of charges3. The voltage across 2 capacitors connected in parallel is same.4. In steady state, no current flows through a capacitor.5. Sum of currents into a node is zero.

6. Sum of voltages around a closed loop is zero.7. The temperature coefficient of resistivity is negative for semiconductor.

(g) Electric potential due to various charge distributionsCharged ring – V = q/( 2 + x 2))

Spherical shell -

(h) Capacitance of a parallel plate capacitor, C = ε0 A/d(i) Ohm’s Law, E = ρJ



(j) Force between the plates of a capacitor – Q 2/2ε0 A(k) Capacitance of a spherical capacitor, C = ab/(b-a)

(l) Wheatstone Bridge : R 1/R 2 = R 3/R 4 , where R 1 and R 3, R 2 and R 4 are part of the samebridges respectively.

Charge on a capacitor in an RC circuit Q(t)= Q 0(1-e-t/(RC)

) where Q 0 is the charge on thecapacitor at t=0.

Capacitance of a capacitor partially filled with a dielectric of thickness t,

Force between two plates of a capacitor:

Capacitance of a spherical capacitor

Capacitance of a cylindrical capacitor: C=

Grouping of cells:

a) Series combination

If polarity of m cells is reversed,

b) Parallel combination

c) Mixed combination

Current will be maximum when

Heat produced in a resistor=



MAGNETISM

Facts:

1.Force on a moving charge in magnetic field is perpendicular to both and

2.Net magnetic force acting on any closed current loop in a uniform magnetic field is zero.

3.Magnetic field of long straight wire circles around wire.

4.Parallel wires carrying current in the same direction attract each other.

Formulae : Force in a charge particle, F=q( ) thus F=qvBsin

When a particle enters into a perpendicular magnetic field,it describes a circle.Radius of the

circular path, r= = Time period, T=

Magnetic force on a segment of wire, F=I( )

Force between parallel current carrying wires, =

A current carrying loop behaves as a magnetic dipole of magnetic dipole moment, .

Torque on a current loop: .

Magnetic field on due to a current carrying wire ,

Ampere’s Law : =

Magnetic Field Due to various current disributions :

1)Current in a straight wire :- B=

2)For an infinitely long straight wire :- a=b=π/2. Thus B=

3)On the axis of a circular coil :-

4) At the centre of the circular coil :- B=

5) For a circular arc, B=

6) Along the axis of asolenoid B c= where n=N/l (No.of turns per unit length )

7) For a very long solenoid B c=µ0nI 8) At the end of a long solenoid, B= µ 0nI/2



Electromagnetic – Induction

Induced emf ε=- N where A is the area of the loop . Induced current ,

I=ε/R Induced electric field = - Self –Inducatnce, N =LI ε=-L

Inducatance of a solenoid L=µ 0 n2 A l where ‘n’ is the number of turns per unit length

Mutual Inducatance, NΦ = MI and ε= - M

Growth of curent in an L-R circuit, I= where = L/R

Decay of current in an L-R circuit I= | Energy stored in a conductor , U = ½ LI 2

AC –Circuits

R.M.S current , I rms =I0 /

In RC Circuit Peak current , I 0= 0 /Z = 0/

In LCR Circuits – If 1/ c > ωL, current leads the voltage *If 1/ωc < ωL, current lags behind the voltage. If 1 = ω2LC, current is in phase with the voltage.Power in A.C circuit, P = V rms Irms cos Ѳ, where co Ѳ is the power factor.For a purely resistive circuit, Ѳ = 0 * For a purely reactive circuit, Ѳ = 90 or 270. Thus, cos Ѳ = 0.

Modern Phys ic s :

Problem solving technique ( for nuclear physics)

(a) Balance atomic number and mass number on both the sides. (b) Calculate the total energy of the reactants and products individually and equatethem. (c) Finally equate the momenta of reactants and products. If a particle of mass ‘m’ and charge ‘q’ is accelerated through a potential difference ‘v’,then wavelength associated with it is given by λ = (h/√(2mq)) x (1/√v)) The de-Broglie wavelength of a gas molecule of mass ‘m’ at temperature ‘T’(in Kelvin) is

given by λ = h/√(3mkt), where k = Blotzmann constant Mass defect is given by ∆m = [Zmp + (A – Z) m n - m ZA] where m p, m n and m ZA be themasses of proton, neutron and nucleus respectively. ‘Z’ is number of protons, (A-Z) isnumber of neutrons. When a radioactive material decays by simultaneous ‘ and ‘ ’ emission, then decayconstant ‘λ’ is given by λ = λ1 + λ2



CHEMISTRY

Organ ic Chemis t ry

Certain important named reactions

1. Beckmann rearrangementThis reaction results in the formation of an amide(rearrangement product)

2. Diels alder reaction

This reaction involves the addition of 1,4-addition of an alkene to a conjugateddiene to form a ring compound

3. Michael reactionIt’s a base catalysed addition of compounds having active methylene group to an activated olefinic bond.

Addition reactions

1. Electrophilic addition



Markonikov’s rule

Addition to alkynes

2. Nucleophilic addition

3. Bisulphite addition

4. Carbanion addition

Substitution reaction mechanism

1 . S n1 mechanismRate α [R3CX]First order reaction and rate of hydrolysis of alkyl halides- allyl>benzyl>3 >2 >1 >CH 3

2. S n2 mechanismRate α [RX][Nu-] > Rate of hydrolysis-CH 3>1 >2 >3



Elimination Reaction mechanisms

1. E1 mechanism(first order)

2. E2 mechanism(second order)

3. E1cB mechanism(elimination, unimolecular)

Comparison between E2 and S n2 recations

Halogenation

Order of substitution- 3 hydrogen>2 hydrogen>1 hydrogenRH+X 2 -- RX+HX (in presence of UV light or heat)Reactivity of X 2: F 2>Cl 2>Br 2>I2



Polymers

Polymer classification

1.based on originNatural. Eg-silk, wool, starch etcSemi-synthetic. Eg-nitrocellulose, cellulose xanthate etcSynthetic. Eg-teflon, polythene

2. based upon synthesis Addition polymers. Eg-ethene, polyvinyl chlorideCondensation polymers. Eg- proteins, starch etc

3.based upon molecular forces Elastomers- they are polymers with very weak intermolecular forces. Eg-vulcanised

rubberFibres- used for making long thread like fibres. Eg-nylon-66Thermoplastics- can be moulded by heating. Eg-polyethyleneThermosetting polymers- becomes hard on heating. Eg-bakelite



Inorganic Chemistry

1. PERIODIC CLASSIFICATION OFELEMENTS

A. Atomic radiusradius order-van derwaal’s>metallic>covalentin case of isoelectronic species,when proton number increases, radiidecreases

B. Ionisation potential(i)It decreases when

Atomic size increases.Screening effect increases.

Moving from top to bottom in agroup.(ii)It increases when

Nuclear charge increases.Element has half filled or fully filledsubshells.Moving from left to right in a period.

(iii)Order of ionization potential isI1<I2<….<In

C. Electron Affinity(E.A)2nd E.A is always negative.

E.A of a neutral atom is equal to theionization potential of its anionFor inert atoms and atoms with fullyfilled orbitals,E.A is zero.

D. Oxidation stateOxidation state of s-block elementsis equal to its group number.P-block elements show multivalency.Common oxidation state of d-blockelements is +2 though they alsoshow variable oxidation states.The common oxidation state of f-block elements is +3.No element exceeds its groupnumber in the oxidation state.Ru and Os show maximum oxidationstate of +8 and F shows only -1state.

E. Diagonal RelationshipIt occurs due to similarelectronegativites and sizes of

participating elements.Disappears after IV group

2. TYPES OF COMPOUNDS

A.Fajan’s rules A compound is more ionic(lesscovalent) if it contains larger cationthan anion and has an inert gasconfiguration.

A compound is more covalent if ithas small cation and has pseudo

inert gas configuration(18 e-

configuration).

B.VSEPR TheoryNon-metallic compounds

B4C3 is hardest artificial substance Acidic nature of hydrides ofhalogens-HI>HBr>HCl>HF

Acidic nature of oxy-acids ofhalogens-HClO<HClO 2<HClO 3<HCIO 4

Acidic character decreases withdecrease in electronegativity ofcentral halogen atom(CN) 2, (SCN) 2 etc are calledpseudohalogensCompounds containing C,Cl, Br, Felements are called halons.Nature of compounds

Non-metallic oxides are generallyacidic in nature and metallic oxidesare generally basic in nature.

Al2O 3, SiO 2 etc are amphoteric innature and CO,NO etc are neutral.Silicones are polymericorganosilicon compounds containingSi-O-Si bonds.



3.EXTRACTIVE METALLURGYMineral is naturally occurringcompound with definite structure andore is a compound from which anelement can be extractedeconomically.The worthless impurities sticking tothe ore is called gangue and flux is achemical compound used o removenon-fusible impurities from the ore.Roasting is the process of heating amineral in the presence of air.Calcination is the process of heatingan ore in the absence of airSmelting is the process by which ametal is extracted from its ore in afused state

4. TRANSITION ELEMENTSThe effective magnetic moment is√n(n+2) B.M(bohr magneton) Color of the compound of transitionmetals is related to the existence ofincomplete d-shell andcorresponding d-d transitionCatalytic behavior is due to variableoxidation statesThey form complexes due to theirsmall size of ions and high ioniccharges

5.CO-ORDINATION CHEMISTRY ANDORGANO-METALLICS

Compounds containing complexcations are cationic complexes andanionic complexes.The compounds which dont ionize inaqueous solutions are neutralcomplexesMonodentate ligands that arecapable of coordinating with metalatom by 2 different sites are calledambidentate ligands like nitro etc.Ligands containing π bonds arecapable of accepting electrondensity from metal atom into emptyantibonding orbital π* of their ownare called π acid ligands. Coordination number of a metal incomplexC.N=1 x no.of monodentate ligandsC.N=2 x no.of bidentate ligandsC.N=3 x no.of tridentate ligandsCharge on complex= O.N of metalatom+O.N of various ligands

Physic a l Chemis try

1. BASIC CONCEPTS

Average atomic mass =

Moles =

Atomicity ( γ) =

Cp-C v = Rn = [Molecular FormulaWeight]/[Empirical formula Weight]

2. STATES OF MATTERDensity=Partial pressure= Total

pressure x Mole fraction

Vmp :Vavg :Vrms =1 : 1.128 : 1.224Van der waal’s equation:

Avogadro’s Law- VFor 1 molecule, the K.E = 1.5

= 1.5KTCritical Pressure :Critical Volume : 3bCritical Temperature :Graham’s law of diffusion(effusion) :

Rate of diffusion



3. ATOMIC STRUCTURE

Specific charge = e/m = 1.76 x10 8 c/gCharge on the electron = 1.602 x10 -19 coulombMass of electron = 9.1096 x 10 -31 kgRadius of nucleus, r = (1.3 x 10 -

13) A1/3 cm, where A = massnumber of the elementEnergy of electron in the nth orbit=Photoelectric effect : hν = hν0 +0.5mv 2 Heisenberg’s UncertaintyPrinciple : ∆x.∆p = h/4π

Spin Magnetic Moment =BM ; 1 BM = 9.27 x 10 -

24 J/TThe maximum number ofemission lines =Radioactivity : Decay Constant

4. SOLUTIONS

% by weight of solute = ,where W = weight of the solutionin gMolarity = , where w 2 =

weight in g of solute whosemolecular weight is M 2, V =volume of solution in mlRaoult’s Law (for ideal solutionsof non-volatile solutes):

p = p 0X1, wherep =Vapour pressure of thesolution, p 0 = Vapour pressure of

the solvent and X 1 = MoleFraction of the solventVan’t Hoff factor :

=

5. CHEMICAL EQUILIBRIUM

In any system of dynamicequilibrium, free energy change,∆G = 0 The free energy change, ∆G andequilibrium constant, K arerelated as ∆G = -RT lnK.Common Ion Effect : By additionof X mole/L of a common ion to aweak acid (or weak base),becomes equal to K a /XSolubility Product : The Ionicproduct (IP) in a saturatedsolution of the sparingly solublesalt = solubility product(SP)IP > SP Precipitation occursIP = SP Solution is saturatedIP < SP Solution is unsaturated

6. THERMOCHEMISTRY

Heat of reaction = Heat offormation of products – Heat offormation of reactants = Heat ofcombustion of reactants – Heatof combustion of products =Bond Energy of the reactants – Bond energy of the products∆H = ∆E + ∆n RT ∆G = ∆H- T∆S (Gibb’s HelmoltzEquation)∆E = ∆q + w Order of a reaction = The sum ofthe indices of the concentrationterms in the rate equation. It isan experimental value. It can bezero, fractional or whole number.Molecularity = the number ofmolecules involved in the ratedetermining step of the reaction.

It is a theoretical value, always awhole number.

7. ELECTROCHEMISTRY

Faraday’s 1st Law – W = Elt,where W= amount of substanceliberated



E = electrochemical equivalent ofthe substance, I = currentstrength in amperes and t = timein seconds.Faradays 2 nd Law – The amountof substances liberated at theelectrodes are proportional totheir chemical equivalent, whenthe same quantity of current ispassed through differentelectrolytes. 1g eq. wt. of theelement will be liberated bypassing 96500 coulombs ofelectricity.

8. NUCLEAR CHEMISTRY

1 amu = 1.66 x 10 -24 g1 eV = 1.6 x 10 -19 J1 cal = 4.184 JDecay Constant λ =

Half life Period t 1/2=

Amount N of Substance left after ‘n’half lives =

9. SOLID STATE

CRYSTALSYSTEM

INTERCEPTS

CRYSTAL ANGLES

Cubic a = b = c

Ortho-rhombic a b

Tetragonal a = b

Monoclinic a

Triclinic a

Hexagonal a = b

Rhombohedral a = b = c

Superconductivity : A superconductor isa material that loses abruptly itsresistance to the electric current whencooled to a specific characteristictemperature. Superconductors are non

– stoichiometric compounds consistingof rare earthen silicates.

10. SURFACE CHEMISTRYIsothermal variation of extent of

adsorption with pressure is

Where x is mass of gas adsorbed bythe mass m of adsorbent at pressure P.K and n are constant for a given pair ofadsorbant and adsorbate.Hardy Schultz Rule

1. The ion having opposite charge to solparticles cause coagulation

2. Coagulating power of an electrolyte

depends on the valency of the ion, i.e.greater the valency more is thecoagulating.Gold Number: The no. of milligrams ofprotective colloid required to justprevent the coagulation of 10ml of redgold sol when 1 ml of 10% solution ofNaCl is added to it.



Maths

Complex Numbersi denotes a quantity such that i 2=-1 .

A complex number is represented as a+ib , If a+ib=0 then a=0 and b=0. Also a+ib=c+id

then a=c and b=d.De Moivre’s Theorem : (cosq +isinq) n=cosnq +sinnqEuler’s Theorem : e iq=cosq + isinq And e -iq=cosq-isinq

Therefore cosq=(e iq+e -iq)/2 And sin q = (e iq-e iq)/2Conjugate complex numbers

and

and

Rotational approach If z 1, z2, z3 be vertices of a triangle ABC described in counter-clockwise sense, then:

or

Properties of Modulus

1) 2) 3) 4)

5) 7)

DeMoivre’s TheoremIf n is a positive or negative integer then (cosA + isinA) n = cos nA + isin nA

Quadratic Equations

A quadratic expression is like ax 2+bx+c=0 and its roots are ( –b+(b 2-4ac) 1/2)/2aand (-b-(b 2-4ac)/2a.If p and q are the two roots of the equation then ax 2+bx+c=a(x-p)(x-q)

Also p + q=-b/a. Product of the roots p *q=c/a.Nature of the roots

D=b2-4ac where D is called discriminant

a) If D>0, roots are real and unequal b) If D=0,roots are real and equalb) If D<0,then D is imaginary. Therefore the roots are imaginary and unequal.

Conditions for Common Roots

a 1x2+b 1x+c 1=0 a 2x2+b 2x+c 2=0 condition is

a 1/a 2 = b 1/b 2= c 1/c 2



Progressions :

Arithmetic Progressions:1. A general form of AP is a,a+ad,a+2d,.......

2. The nth

term of the AP is a+(n-1)d. The sum of n terms are n/2(2a+(n-1)d).3. Some other sums 1+2+3+4......+n=n(n+1)/24. 1 2+2 2+3 2+.....+n 2=n(n+1)(2n+1)/6 and 1 3+2 3+3 3.......+n 3=n 2(n+1) 2/4.

Geometric Progressions:1. A general form of GP is a,ar,ar 2,ar 3,......2. The nth term is ar n-1 , and the sum of n terms is a(1-r n)/1-r where r>1 And a/(1-r)

where (|r|<1) .

Harmonic Progressions:1. A general forms of a HP is 1/a, 1/(a+d),1/(a+2d),......Means : Let Arithmetic mean be A, Geometric mean be G and Harmonic Mean be H,between two positive numbers a and b, then

A=a+b/2 , G=(ab) 2, H=2ab/(a+b). A,G,H are in GP i.e. G 2=HA Also A>=G>=H.

Series of Natural Numbers

Logarithms

(i) log a(mn) = log am + log an (ii) log a(m/n) = log am - log an

(iii) log a(m p) = plog am

(vi) log ba log cb = log ca

(vii) log ba = 1/log ab (a≠1.b≠1,a>0,b>0) (viii) logba log cb log ac = 1

Permutations :

1. Multiplication Principle : There are m ways doing one work and n ways doing anotherwork then ways of doing both work together = m.n

2. Addition Principle: There are m ways doing one work and n ways doing another workthen ways doing either m ways or n ways = m+n.



3. np r =n(n-1)(n-2)(n-3).......(n-r+1) =

4. The number of ways of arranging n distinct objects along a circle is (n-1)!

5. The number of permutations of n things taken all at a time,p are alike of one kind,qare alike of another kind and r are alike of a third kind and the rest n-(p+q+r) are all

different is

6. The number of ways of arranging n distinct objects taking r of them at a time whereany object may be repeated any number of times is n -r

7. The coefficient of x r in the expansion of (1-x) -n = n+r-1 C r

8. The number of ways of selecting at least one object out of ‘n’ distinct objects = 2n-1

9. np r = n-1 p r + n-1 p r-1

10. The number of permutations of n different objects taken r at a time is n r .11. n unlike bjects can be arranged in a circle in n-1 p r .

Combinations :

1. A selection of r objects out n different objects without reference to the order of placingis given by nC r .

* nC r = np r /r! * nC r = nC n-r . * nC r = n-1 C r-1 + n-1 C r

Some important results

1) nC 0= nCn = 1. nC1=n 2) nC r + nC r-1 = n+1 C r 3) 2n+1 C 0+ 2n+1 C 1+.....+ 2n+1 C n = 2 2n 4) nC r = n/r . n-1C r-1 5) nC r / nC r-1 = n-r+1/r 6) nCn + n+1 Cn+ n+2 Cn+......+ 2n-1 C n= 2n C n+1

Probability :

Let A and B be any two events. Then A or B happening is said to be A union B(A+B) and A and B happening at the same time is said to be A intersection B(AB).

1.P(A)=P(AB) + P(AB’) 2. P(B)=P(AB)+P(A’B) 3. P(A+B)=P(AB)+P(AB’)+P(A’B) 4. P(A+B)=P(A)+P(B)-P(AB)5.P(AB)=1- P(A’+B’) 6.P(A+B)=1- P(A’B’)



Trigonometry :

sin(A+B)=sinAcosB + cosAsinB sin(A-B)=sinAcosB-cosAsinBcos(A+B)=cosAcosB – sinAsinB cos(A-B)=cosAcosB – sinAsinBtan(A+B)=(tanA + tanB)/(1-tanAtanB) tan(A-B)=(tanA-tanB)/(1+tanAtanB)

Transformations:

sinA +sinB = 2 sin(A+B/2)cos(A-B/2) sinA – sinB=2 cos(A+B/2)sin(A-B/2)cosA + cosB=2cos(A+B/2)cos(A-B/2) cosA – cosB=2sin(A+B/2)sin(A-B/2)

Relations between the sides and angles of a triangle

(Here a,b,c are three sides of a triangle , A,B,C are the angles , R is the circumradius ands is semi-perimeter of a triangle)

a/sinA =b/sinB=c/sinC=2R(Sine Formula) cosC=(a 2+b 2- c 2)/2ab(Cosine Formula)a=c cosB + b cosC(Projection Formula)

Half Angle Formulas

sinA/2={(s-b)*(s-c)/bc} (1/2) cosB/2={s(s-b)/ca} 1/2

tanA/2={(s-b)(s-c)/s(s-a)} 1/2

Inverse Functions :

sin -1x+cos -1x=π/2 tan -1x+cot -1x=π/2tan -1x + tan -1y= tan -1(x+y/1-xy) if xy<1 tan -1x + tan -1y= π- tan -1(x+y/1-xy) if xy>1

sin-1

x+sin-1

y=sin-1

[x (1-y2

)1/2

+ y(1-x2

)1/2

] sin-1

x-sin-1

y=sin-1

[x (1-y2

)1/2

- y(1-x2

)1/2

]

cos -1x+cos -1y=cos -1[xy- (1-x 2)1/2(1-y 2)1/2] cos -1x-cos -1y=cos -1[xy+(1-x 2)1/2 (1-y 2)1/2 ]

Analytical Geometry :

Points : let A,B,C be respectively the points (x 1,y1),(x 2,y2) and (x 3,y3).The centroid of the triangle is [x 1+x2+x3/3 , y 1+y2+y3/3]In centre of triangle [ax 1+bx 2+cx 3/a+b+c , ay 1+by 2+cy 3/a+b+c ]The Area is given by ½[y 1(x2-x3)+y 2(x3-x1)+y 3(x1-x2)]

Locus: When a point moves in accordance with a geometric law, its path is called a locus.

Line: Standard form : ax+by+c=0 Slope form :y=(tanq)x+c where q is the angle the linemakes with the x- axis and ‘c’ is the intercept on y-axis.Intercept form : x/a + y/b =1 where a and b are intercepts on x and y axes.Normal form : xcosq +ysinq=pLine passing through the points (x 1,y1) and (x 2,y2) is (x-x 1)/(x1-x2)= (y-y 1)/(y1-y2)Length of a perpendicular from a point (x 1,y1) to the line ax+by+c=0 is |(ax 1+by 1+c)/(a 2+b 2)(1/2) |



Circle :

1. Equation of a circle: A circle having centre(h,k) and radius r (x-h) 2+(y-k) 2=r 2.2. General equation of a circle : x 2+y2+2gx + 2fy+c=0 here the centre is (-g,-f) and radius is

(g 2+f 2-c)3. Equation of circle described by joining the points (x

1,y

1) and (x

2,y

2) as diameter is:

(x-x1)(x-x2)+(y-y 1)(y-y 3)4. Length of tangent : From (x 1,y1) to the circle x 2+y2+2gx + 2fy+c=0 is

[x12+y 1

2+2gx 1 + 2fy 1+c] 1/2

5. Equation of tangent: the equation of a tangent to x 2+y2+2gx + 2fy+c=0 at(x 1,y1) isxx1+yy 1+g(x+x 1)+f(y+y 1)+c=0.

6. Condition for the line y=mx + c to touch a circle is x 2+y2=a 2 is c 2=a 2(1+m 2).7. Condition for orthogonal intersection of two circles S= x 2+y 2+2gx + 2fy+c=0 and

S 1=x12+y1

2+2gx 1 + 2fy 1+c = 0 is given by 2gg 1+2ff 1=c+c 1

Parabola:

1. The standard equation is y 2=4ax where x-axis is axis of parabola and y-axsi is tangent atthe vertex. Vertex is A(0,0) and Focus is S(a,0) and Directrix is x+a=0

2. Parametric Form of a point on y 2=4ax is P(at 2,2at). At P the slope of tangent is 1/t.3. Equation of tangent is x-yt+at 2=0. Equation of normal is y+tx-2at-at 3=0.4. If P(at 1

2,2at 1) and Q(at 22,2at 2) then the slope of chord PQ is 2x-y(t 1+t2)+2at 1t2=0

Ellipse:

1. Standard equation is x 2/a 2+ y 2/b 2=1 ; x-axis is major axis length 2a y-axis is minor axislength 2b And b 2= a 2 (1-e 2) [ e is eccentricity and e < 1]

2. There are two foci S(ae,0) and S’(-ae,0). And the two directrices are x=a/e and x=-a/e.

3. If P is any point o n ellipse then i) SP + S’P=2a ii)SP. SP’=CD2

where CD is semi-diameterparallel to the tangent at P.4. Parametric Form of a point P on x 2/a 2+ y 2/b 2=1 is P(acosq, bsinq) . The equation of the

tangent is x/a cosq + y/b sinq -1=0 . Equation of normal is ax/cosq - by/sinq=a 2-b 2 5. The locus of points of intersection of perpendicular tangents of the ellipse x 2/a 2+ y 2/b 2-1=0 is

called the director circle and is given by x 2+y2=a 2+b 2 .

Hyperbola:

1. Standard equation of Hyperbola is x 2/a 2- y2/b 2=1 . x-axis- transverse axis length -2a ,y-axis conjugate axis, length 2b where e 2=1+ b 2/a 2.

2. Parametric equation of a point on x 2/a 2- y2/b 2=1 are x=asecq and y=b tanq where q is theparameter.

3. Auxiliary circle : The circle described on the transverse axis of the hyperbola as diameter iscalled auxiliary circle and is given by x 2+y 2=a 2.

4. Condition for tangency : A line y=mx+c is a tangent to x 2/a 2- y2/b 2=1 iff c 2=a 2m 2-b 2 and theequation is xx 1/a 2- yy 1/b 2 =1

5. Asymptotes of a Hyperbola : Asymptotes of hyperbola x 2/a 2- y 2/b 2=1 are given by x 2/a 2 – y2/b 2 = 0.



6. Conjugate hyperbola : x2/a 2- y2/b 2=1 is the conjugate hyperbola of y 2/b 2-x2/a 2 = 1. If e 1 and e 2 are their eccentricities then 1/e 1

2 + 1/e 22 =1.

7. Rectangular Hyperbola: It is denoted by xy=c 2.A point on xy=c 2 is represented in theparametric form by( ct, c/t ). At P(ct, c/t) , the slope of the tangent is -1/t 2 . Equation of tangent isx + yt 2-2ct =0. Slope of normal is xt 3-yt+c-ct 4=0

Coordinate Geometry :

Let α, β and γ be the angles made by the plane with the X, Y and z axes respectively. Thencosα , cosβ and cosγ and are denoted by l, m and n respectively andare called direction cosines of the plane or line.

If P(x, y, z) is the point and if Op=r, th en x/r = cosα, y/r= cosβ andz/r= cosγ . Also cos 2α + cos2β + cos2γ =1

Standard Form of the equation of a plane:

1) If p is the length of the normal from the origin on the plane then theequation of the plane is lx+my+nz=Φ .

2) The equation of the plane parallel to ax+by+cy+d=0 and passing through (x 1, y 1, z 1 ) is givenby a(x-x 1) + b(y-y 1) + c(z-z 1) +d =0

3) The equation of a plane parallel to the z-axis is ax + by + d= 0 etc.4) a,b,c are direction ratios of the normal to plane ax+by+cz+d=05) The perpendicular distance between point P(x 1,y1,z 1) on the plane ax+by+cz+d =0 is given

by (ax 1+by 1+cz 1+d)/ .

Differential Calculus:

A polynomial of x is given as a 0xn+a 1xn-1 +......... +a n-1 x + a n . Here a 0,a 1,a 2...... are constants .

Laws of limits :

1) = 2) =

3) = 4) =

5) = na n-1 6) = 7)

Differentiation:

1) f’(x) =

α

γ



Some Standard Substitution

Integral Substitutions

(i) t=ax+b

(ii) t= xn

(iii) t=f(x)

(iv) f(x)(v)

(b) Integration By Parts :

(c) Integration of

(i) If n be odd and m be even put t=cos x(ii)If n be even and m be odd, put t=sinx(iii) If m and n are both odd then put t=cos x or sin x

(d) Properties of Definite Integrals :

(a) = -

(c) function

and =0 if

(2a-x)

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