Compendium

AIEEE

FORMULAE BOOK

IIT

BITS

PHYSICS General

1. If x = ambnco, 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. m2= mass of the cold liquid

T1=temperature of cold liquid. T2=temperature of hot liquid.

T = final temperature of the system c1= specific heat of the material of

calorimeter and stirrer.

c2= specific heat of material of solid m1 = mass of the calorimeter and stirrer

Mechanics

Vectors

1. = , where is angle between the vectors.

And direction of from ,

2. Two vectors (a1 + a2 + a3 ) and (b1 + b2 + b3 ) are equal if:

a1 = b1 a2= b2 and a3 = b3

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

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

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

Kinematics

t1 = initial time t2 = final time u= initial velocity v = final velocity v av = average velocity a = acceleration s = net displacement

1.Average speed, Vav =

2.Instantaneous speed,

V = =

3. Displacement =

4. Total distance =

5. a= =v =

6. When acceleration is constant

v= u +at

s= ut + at2 = vt - at2

v2 = u2+2as

Projectile motion

1. Time of flight , t0 =

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 =

atotal=

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 : frictional

forces, viscous force..

Center of Mass, Linear

Momentum, 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 constant

U = Gravitational potential energy

V = Gravitational potential

E = Gravitational field

F = 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 Mechanics

P= pressure =density V=volume of solid v=volume immersed D= density of solid

d=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 = vd 4) Equation of continuity, A1v1= A2v2

5) Bernoulli’s equation, P+ p + gh = constant

Elasticity

Y= Young’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 tension

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

η=coefficient of viscosity; F=force V=velocity; ζ = density of liquid ; A = Cross

section area

1. F = Stoke’s Law, F = 6πrηv Terminal Velocity, v0 =

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 = y1 + y2

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 Physics

P=pressure V=volume n= no. of moles T=temperature R= universal gas constant

co-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 of

state 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 )

Optics :

u=dist. Of the object from the lens/mirror v=dist. Of the image from the lens/mirror

m= magnification i= angle of incidence r=angle of reflection/ refraction

n=refractive index θc=critical angle δ=angle of deviation R= radius of

curvature

P=power

1. Spherical Mirrors, m = -

2. Refraction at plane surfaces n= = real depth / apparent depth

Θc= 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 =

Electricity and Magnetism :

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 charges

3. 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 distributions

Charged ring – V = q/( 2 + x2))

Spherical shell -

(h) Capacitance of a parallel plate capacitor, C = ε0A/d

(i) Ohm’s Law, E = ρJ

(j) Force between the plates of a capacitor – Q2/2ε0A

(k) Capacitance of a spherical capacitor, C = ab/(b-a)

(l) Wheatstone Bridge : R1/R2 = R3/R4 , where R1 and R3, R2 and R4 are part of the same

bridges respectively.

Charge on a capacitor in an RC circuit Q(t)= Q0(1-e-t/(RC) ) where Q0 is the charge on the

capacitor 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=qvBsinθ

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 Bc= where n=N/l (No.of turns per unit length )

7) For a very long solenoid Bc=µ0nI 8) At the end of a long solenoid, B= µ0nI/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=µ0 n2A 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 LI2

AC –Circuits

R.M.S current , Irms=I0 /

In RC Circuit Peak current , I0= 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 = VrmsIrmscos 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, cosѲ = 0.

Modern Physics :

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 equate

them.

(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) mn - mZA] where mp, mn and mZA be the

masses of proton, neutron and nucleus respectively. ‘Z’ is number of protons, (A-Z) is

number of neutrons.

When a radioactive material decays by simultaneous ‘ and ‘ ’ emission, then decay

constant ‘λ’ is given by λ = λ1 + λ2

CHEMISTRY Organic Chemistry

Certain important named reactions

1. Beckmann rearrangement This 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 conjugated diene to form a ring compound

3. Michael reaction It’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 .Sn1 mechanism

Rate α [R3CX]

First order reaction and rate of hydrolysis of alkyl halides- allyl>benzyl>3⁰>2⁰>1⁰>CH3

2. Sn2 mechanism

Rate α [RX][Nu-] > Rate of hydrolysis-CH3>1⁰>2⁰>3⁰

Elimination Reaction mechanisms 1.E1 mechanism(first order)

2. E2 mechanism(second order)

3. E1cB mechanism(elimination, unimolecular)

Comparison between E2 and Sn2 recations

Halogenation

Order of substitution- 3⁰ hydrogen>2⁰ hydrogen>1⁰ hydrogen

RH+X2 -- RX+HX (in presence of UV light or heat)

Reactivity of X2: F2>Cl2>Br2>I2

Polymers

Polymer classification 1.based on origin

Natural. Eg-silk, wool, starch etc

Semi-synthetic. Eg-nitrocellulose, cellulose xanthate etc

Synthetic. Eg-teflon, polythene

2. based upon synthesis

Addition polymers. Eg-ethene, polyvinyl chloride

Condensation polymers. Eg- proteins, starch etc

3.based upon molecular forces

Elastomers- they are polymers with very weak intermolecular forces. Eg-vulcanised rubber

Fibres- used for making long thread like fibres. Eg-nylon-66

Thermoplastics- can be moulded by heating. Eg-polyethylene

Thermosetting polymers- becomes hard on heating. Eg-bakelite

Inorganic Chemistry

1. PERIODIC CLASSIFICATION OF

ELEMENTS A. Atomic radius

radius order-van der waal’s>metallic>covalent

in case of isoelectronic species, when proton number increases, radii decreases

B. Ionisation potential (i)It decreases when

Atomic size increases.

Screening effect increases.

Moving from top to bottom in a group.

(ii)It increases when

Nuclear charge increases.

Element has half filled or fully filled subshells.

Moving from left to right in a period. (iii)Order of ionization potential is

I1<I2<….<In

C. Electron Affinity(E.A)

2nd E.A is always negative.

E.A of a neutral atom is equal to the ionization potential of its anion

For inert atoms and atoms with fully filled orbitals,E.A is zero.

D. Oxidation state

Oxidation state of s-block elements is equal to its group number.

P-block elements show multivalency.

Common oxidation state of d-block elements is +2 though they also show variable oxidation states.

The common oxidation state of f-block elements is +3.

No element exceeds its group number in the oxidation state.

Ru and Os show maximum oxidation state of +8 and F shows only -1 state.

E. Diagonal Relationship

It occurs due to similar electronegativites and sizes of participating elements.

Disappears after IV group

2. TYPES OF COMPOUNDS A.Fajan’s rules

A compound is more ionic(less covalent) if it contains larger cation than anion and has an inert gas configuration.

A compound is more covalent if it has small cation and has pseudo inert gas configuration(18 e- configuration).

B.VSEPR Theory Non-metallic compounds

B4C3 is hardest artificial substance

Acidic nature of hydrides of halogens-HI>HBr>HCl>HF

Acidic nature of oxy-acids of halogens-HClO<HClO2<HClO3<HCIO4

Acidic character decreases with decrease in electronegativity of central halogen atom

(CN)2, (SCN)2 etc are called pseudohalogens

Compounds containing C,Cl, Br, F elements are called halons.

Nature of compounds

Non-metallic oxides are generally acidic in nature and metallic oxides are generally basic in nature.

Al2O3, SiO2 etc are amphoteric in nature and CO,NO etc are neutral.

Silicones are polymeric organosilicon compounds containing Si-O-Si bonds.

3.EXTRACTIVE METALLURGY

Mineral is naturally occurring compound with definite structure and ore is a compound from which an element can be extracted economically.

The worthless impurities sticking to the ore is called gangue and flux is a chemical compound used o remove non-fusible impurities from the ore.

Roasting is the process of heating a mineral in the presence of air.

Calcination is the process of heating an ore in the absence of air

Smelting is the process by which a metal is extracted from its ore in a fused state

4. TRANSITION ELEMENTS

The effective magnetic moment is √n(n+2) B.M(bohr magneton)

Color of the compound of transition metals is related to the existence of incomplete d-shell and corresponding d-d transition

Catalytic behavior is due to variable oxidation states

They form complexes due to their small size of ions and high ionic charges

5.CO-ORDINATION CHEMISTRY AND

ORGANO-METALLICS

Compounds containing complex cations are cationic complexes and anionic complexes.

The compounds which dont ionize in aqueous solutions are neutral complexes

Monodentate ligands that are capable of coordinating with metal atom by 2 different sites are called ambidentate ligands like nitro etc.

Ligands containing π bonds are capable of accepting electron density from metal atom into empty antibonding orbital π* of their own are called π acid ligands.

Coordination number of a metal in complex C.N=1 x no.of monodentate ligands C.N=2 x no.of bidentate ligands C.N=3 x no.of tridentate ligands

Charge on complex= O.N of metal atom+O.N of various ligands

Physical Chemistry

1. BASIC CONCEPTS

Average atomic mass =

Moles =

Atomicity (γ) =

Cp-Cv = R

n = [Molecular Formula Weight]/[Empirical formula Weight]

2. STATES OF MATTER

Density=

Partial pressure= Total pressure x Mole fraction

Vmp:Vavg:Vrms=1 : 1.128 : 1.224

Van der waal’s equation:

Avogadro’s Law - V

For 1 molecule, the K.E = 1.5

= 1.5KT

Critical Pressure :

Critical Volume : 3b

Critical Temperature :

Graham’s law of diffusion (effusion) :

Rate of diffusion

3. ATOMIC STRUCTURE

Specific charge = e/m = 1.76 x 108 c/g

Charge on the electron = 1.602 x 10-19 coulomb

Mass of electron = 9.1096 x 10-31 kg

Radius of nucleus, r = (1.3 x 10-

13) A1/3 cm, where A = mass number of the element

Energy of electron in the nth orbit

=

Photoelectric effect : hν = hν0 + 0.5mv2

Heisenberg’s Uncertainty Principle : ∆x.∆p = h/4π

Spin Magnetic Moment =

BM ; 1 BM = 9.27 x 10-

24 J/T

The maximum number of

emission lines =

Radioactivity : Decay Constant

4. SOLUTIONS

% by weight of solute = ,

where W = weight of the solution in g

Molarity = , where w2 =

weight in g of solute whose molecular weight is M2, V = volume of solution in ml

Raoult’s Law (for ideal solutions of non-volatile solutes): p = p0X1, where p =Vapour pressure of the solution, p0 = Vapour pressure of the solvent and X1 = Mole Fraction of the solvent

Van’t Hoff factor :

=

5. CHEMICAL EQUILIBRIUM

In any system of dynamic equilibrium, free energy change, ∆G = 0

The free energy change, ∆G and equilibrium constant, K are related as ∆G = -RT lnK.

Common Ion Effect : By addition of X mole/L of a common ion to a weak acid (or weak base), becomes equal to Ka/X

Solubility Product : The Ionic product (IP) in a saturated solution of the sparingly soluble salt = solubility product(SP)

IP > SP Precipitation occurs

IP = SP Solution is saturated

IP < SP Solution is unsaturated

6. THERMOCHEMISTRY

Heat of reaction = Heat of formation of products – Heat of formation of reactants = Heat of combustion of reactants – Heat of combustion of products = Bond Energy of the reactants – Bond energy of the products

∆H = ∆E + ∆n RT

∆G = ∆H - T∆S (Gibb’s Helmoltz Equation)

∆E = ∆q + w

Order of a reaction = The sum of the indices of the concentration terms in the rate equation. It is an experimental value. It can be zero, fractional or whole number.

Molecularity = the number of molecules involved in the rate determining step of the reaction. It is a theoretical value, always a whole number.

7. ELECTROCHEMISTRY

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

E = electrochemical equivalent of the substance, I = current strength in amperes and t = time in seconds.

Faradays 2nd Law – The amount of substances liberated at the electrodes are proportional to their chemical equivalent, when the same quantity of current is passed through different electrolytes. 1g eq. wt. of the element will be liberated by passing 96500 coulombs of electricity.

8. NUCLEAR CHEMISTRY

1 amu = 1.66 x 10-24 g

1 eV = 1.6 x 10-19 J

1 cal = 4.184 J

Decay Constant λ =

Half life Period t1/2=

Amount N of Substance left after ‘n’

half lives =

9. SOLID STATE CRYSTAL SYSTEM

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 is

a material that loses abruptly its

resistance to the electric current when

cooled to a specific characteristic

temperature. Superconductors are non

– stoichiometric compounds consisting

of rare earthen silicates.

10. SURFACE CHEMISTRY

Isothermal variation of extent of

adsorption with pressure is

Where x is mass of gas adsorbed by

the mass m of adsorbent at pressure P.

K and n are constant for a given pair of

adsorbant and adsorbate.

Hardy Schultz Rule

1. The ion having opposite charge to sol

particles cause coagulation

2. Coagulating power of an electrolyte

depends on the valency of the ion, i.e.

greater the valency more is the

coagulating.

Gold Number: The no. of milligrams of

protective colloid required to just

prevent the coagulation of 10ml of red

gold sol when 1 ml of 10% solution of

NaCl is added to it.

Maths Complex Numbers

i denotes a quantity such that i2=-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 +sinnq

Euler’s Theorem : eiq=cosq + isinq And e-iq=cosq-isinq

Therefore cosq=(eiq+e-iq)/2 And sin q = (eiq-eiq)/2

Conjugate complex numbers

and

and

Rotational approach If z1, 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 Theorem

If n is a positive or negative integer then (cosA + isinA)n = cos nA + isin nA

Quadratic Equations

A quadratic expression is like ax2+bx+c=0 and its roots are (–b+(b2-4ac)1/2)/2a

and (-b-(b2-4ac)/2a.

If p and q are the two roots of the equation then ax2+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 equal

b) If D<0,then D is imaginary. Therefore the roots are imaginary and unequal.

Conditions for Common Roots

a1x2+b1x+c1=0 a2x

2+b2x+c2=0 condition is

a1/a2 = b1/b2= c1/c2

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)/2

4. 12+22+32+.....+n2=n(n+1)(2n+1)/6 and 13+23+33.......+n3=n2(n+1)2/4.

Geometric Progressions:

1. A general form of GP is a,ar,ar2,ar3,......

2. The nth term is arn-1 , and the sum of n terms is a(1-rn)/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. G2=HA Also A>=G>=H.

Series of Natural Numbers

Logarithms

(i) loga(mn) = logam + logan (ii) loga(m/n) = logam - logan

(iii) loga(mp) = plogam

(vi) logba logcb = logca

(vii) logba = 1/logab (a≠1.b≠1,a>0,b>0) (viii) logba logcb logac = 1

Permutations :

1. Multiplication Principle : There are m ways doing one work and n ways doing another

work then ways of doing both work together = m.n

2. Addition Principle: There are m ways doing one work and n ways doing another work

then ways doing either m ways or n ways = m+n.

3. npr=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,q

are 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 where

any object may be repeated any number of times is n-r

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

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

9. npr = n-1pr +

n-1pr-1

10. The number of permutations of n different objects taken r at a time is nr.

11. n unlike bjects can be arranged in a circle in n-1pr .

Combinations :

1. A selection of r objects out n different objects without reference to the order of placing

is given by nCr.

* nCr = npr/r! *

nCr= nCn-r . *

nCr= n-1Cr-1 +

n-1Cr

Some important results

1)nC0= nCn = 1. nC1=n 2) nCr + nCr-1 = n+1Cr

3)2n+1C0+ 2n+1C1+.....+ 2n+1Cn = 22n 4)nCr = n/r . n-1Cr-1

5) nCr/ nCr-1 = n-r+1/r 6)nCn + n+1Cn+

n+2Cn+......+ 2n-1Cn= 2nCn+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-cosAsinB

cos(A+B)=cosAcosB – sinAsinB cos(A-B)=cosAcosB – sinAsinB

tan(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 and

s is semi-perimeter of a triangle)

a/sinA =b/sinB=c/sinC=2R(Sine Formula) cosC=(a2+b2- c2)/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/2

tan-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-1x+sin-1y=sin-1[x (1-y2)1/2 + y(1-x2)1/2] sin-1x-sin-1y=sin-1[x (1-y2)1/2 - y(1-x2)1/2]

cos-1x+cos-1y=cos-1[xy- (1-x2)1/2(1-y2)1/2] cos-1x-cos-1y=cos-1[xy+(1-x2)1/2(1-y2)1/2]

Analytical Geometry :

Points : let A,B,C be respectively the points (x1,y1),(x2,y2) and (x3,y3).

The centroid of the triangle is [x1+x2+x3/3 , y1+y2+y3/3]

In centre of triangle [ax1+bx2+cx3/a+b+c , ay1+by2+cy3/a+b+c ]

The Area is given by ½[y1(x2-x3)+y2(x3-x1)+y3(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 line

makes 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=p

Line passing through the points (x1,y1) and (x2,y2) is (x-x1)/(x1-x2)= (y-y1)/(y1-y2)

Length of a perpendicular from a point (x1,y1) to the line ax+by+c=0 is |(ax1+by1+c)/(a2+b2)(1/2) |

Circle :

1. Equation of a circle: A circle having centre(h,k) and radius r (x-h)2+(y-k)2=r2.

2. General equation of a circle : x2+y2+2gx + 2fy+c=0 here the centre is (-g,-f) and radius is

(g2+f2-c)

3. Equation of circle described by joining the points (x1,y1) and (x2,y2) as diameter is:

(x-x1)(x-x2)+(y-y1)(y-y3) 4. Length of tangent : From (x1,y1) to the circle x2+y2+2gx + 2fy+c=0 is

[x12+y1

2+2gx1 + 2fy1+c]1/2

5. Equation of tangent: the equation of a tangent to x2+y2+2gx + 2fy+c=0 at(x1,y1) is

xx1+yy1+g(x+x1)+f(y+y1)+c=0.

6. Condition for the line y=mx + c to touch a circle is x2+y2=a2 is c2=a2(1+m2).

7. Condition for orthogonal intersection of two circles S= x2+y2+2gx + 2fy+c=0 and

S1=x12+y1

2+2gx1 + 2fy1+c = 0 is given by 2gg1+2ff1=c+c1

Parabola: 1. The standard equation is y2=4ax where x-axis is axis of parabola and y-axsi is tangent at

the 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 y2=4ax is P(at2,2at). At P the slope of tangent is 1/t. 3. Equation of tangent is x-yt+at2=0. Equation of normal is y+tx-2at-at3=0. 4. If P(at1

2,2at1) and Q(at22,2at2) then the slope of chord PQ is 2x-y(t1+t2)+2at1t2=0

Ellipse: 1. Standard equation is x2/a2+ y2/b2=1 ; x-axis is major axis length 2a y-axis is minor axis

length 2b And b2= a2 (1-e2) [ 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 on ellipse then i) SP + S’P=2a ii)SP. SP’=CD2 where CD is semi-diameter

parallel to the tangent at P. 4. Parametric Form of a point P on x2/a2+ y2/b2=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=a2-b2 5. The locus of points of intersection of perpendicular tangents of the ellipse x2/a2+ y2/b2-1=0 is

called the director circle and is given by x2+y2=a2+b2 .

Hyperbola:

1. Standard equation of Hyperbola is x2/a2- y2/b2=1 . x-axis- transverse axis length -2a ,

y-axis conjugate axis, length 2b where e2=1+ b2/a2.

2. Parametric equation of a point on x2/a2- y2/b2=1 are x=asecq and y=b tanq where q is the

parameter.

3. Auxiliary circle : The circle described on the transverse axis of the hyperbola as diameter is

called auxiliary circle and is given by x2+y2=a2.

4. Condition for tangency : A line y=mx+c is a tangent to x2/a2- y2/b2=1 iff c2=a2m2-b2 and the

equation is xx1/a2- yy1/b

2 =1

5. Asymptotes of a Hyperbola: Asymptotes of hyperbola x2/a2- y2/b2=1 are given by x2/a2 –

y2/b2 = 0.

6. Conjugate hyperbola : x2/a2- y2/b2=1 is the conjugate hyperbola of y2/b2-x2/a2 = 1. If e1 and e2

are their eccentricities then 1/e12 + 1/e2

2 =1.

7.Rectangular Hyperbola: It is denoted by xy=c2.A point on xy=c2 is represented in the

parametric form by( ct, c/t ). At P(ct, c/t) , the slope of the tangent is -1/t2 . Equation of tangent is

x + yt2-2ct =0. Slope of normal is xt3-yt+c-ct4=0

Coordinate Geometry : Let α, β and γ be the angles made by the plane with the X, Y and z axes respectively. Then cosα , cosβ and cosγ and are denoted by l, m and n respectively and are called direction cosines of the plane or line. If P(x, y, z) is the point and if Op=r, then x/r = cosα, y/r= cosβ and z/r= cosγ . Also cos2α + 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 the

equation of the plane is lx+my+nz=Φ . 2) The equation of the plane parallel to ax+by+cy+d=0 and passing through (x1, y1, z1 ) is given

by a(x-x1) + b(y-y1) + c(z-z1) +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=0 5) The perpendicular distance between point P(x1,y1,z1) on the plane ax+by+cz+d =0 is given

by (ax1+by1+cz1+d)/ .

Differential Calculus:

A polynomial of x is given as a0xn+a1x

n-1 +......... +an-1x + an . Here a0,a 1,a2...... are constants .

Laws of limits :

1) = 2) =

3) = 4) =

5) = nan-1 6) = 7)

Differentiation:

1) f’(x) =

α

γ

β

Some derivatives of common functions :

Function Derivative Function Derivative

C 0

Cu C* du/dx

u + v du/dx + dv/dx uv u dv/dx + vdu/dx

u/v ( v du/dx - udv/dx) /v2 xn nxn-1

ex ex log x 1/x

sin x cos x cos x - sin x

tan x sec2 x cosec x -cosec x cot x

sec x sec x tan x cot x - cosec2 x

sin-1 x 1/ cos-1 x - 1/

tan -1 x 1 / 1+ x2

Geometric Meaning of Derivative : If the tangent at x=a to the curve y=f(x) makes an angle of θ of with the x-axis then tan θ = the value of dy/dx at at x=a i.e. f’(a).

Maxima and Minima : f(x) attains a maximum at x=a if f’(a) = 0 and f’’(a) is negative . Also f(x) attains a minimum at x=a if f’(a) = 0 and f’’(a) is positive .

Rolle’s Theorem : If a function f(x) is differentiable in the interval (a,b) then there exists at least one value of x1 of x in the interval (a,b) such that f’(x1) = 0.

Le’ Hospitals Rule : 0/0 form:

= and so on

Integral Calculus :

If

= F(b) – F(a) this is known as Definite integral .

Methods of Intigeration :

a) By Substitution

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)

ALL THE BEST