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APEX INSTITUTE was conceptualized in May 2008, keeping in view the dreams of young students by the vision & toil of Er. Shahid Iqbal. We had a very humble beginning as an institute for IIT-JEE / Medical, with a vision to provide an ideal launch pad for serious JEE students . We actually started to make a difference in the way students think and approach problems. We started to develop ways to enhance students IQ. We started to leave an indelible mark on the students who have undergone APEX training. That is why APEX INSTITUTE is very well known of its quality of education

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  2. 2. PHYSICSGeneral 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 stirrerMechanics  Vectors1. = , 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 = b33. 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’ , = -
  3. 3.  Kinematics t1 = initial time t2 = final time u= initial velocity v = final velocity v av = average velocity a = acceleration s = net displacement1.Average speed, Vav = 4. Total distance =2.Instantaneous speed, 5. a= =v =V= = 6. When acceleration is constant v= u +at3. Displacement = s= ut + at2 = vt - at2 v2 = u2+2as  Projectile motion1. Time of flight , t0 = Range , R=2. Maximum height H=3. (x,y)=(ucos )4. Equation of Projectile ,y =xtan  Forces1. F - R=ma2. Frictional force =f, force applied =F3. =F
  4. 4.  Circular Motion1. = = 4. Non- conservative forces : frictional forces, viscous force.. Α= ==  Center of Mass, Linear Momentum, Collision2. = 1. AN = 2. 3. atotal= 4.3. R at (x,y) =4. Banking of roads, tan = 5. For perfectly inelastic collision, 6. .If coefficient of restitution is e (0<e<1),5. Centripetal force = = mr Velocity of separation = e (Velocity of approach)  Work and Energy V= 1. Total Energy = Kinetic energy = potential energy 7. Impulse = = 2. = 3. Conservative forces : spring force, electrostatic forces…  Rotational Mechanics η=torque, F=force I=moment of inertia α=angular acceleration L=angular momentum 3. 4. (i) Pure translation 1. Moment of inertia, (ii) Rotation 0 I= = dm (iii) Pure rotation 0 (iv) Translation 2. Angular momentum, (v) Rolling (vi) Sliding or sliding L=Iω
  5. 5.  Gravitation G = Universal gravitational constant E = Gravitational field U = Gravitational potential energy F = Gravitational force V = Gravitational potential 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 4. Physical Pendulum 2. T= = 3. Angular simple harmonic motion, T= 2 , = 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
  6. 6. 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 × volumeSurface tensionT = 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 =  Viscosityη=coefficient of viscosity; F=force V=velocity; ζ = density of liquid ; A = Crosssection area 1. F = Stoke’s Law, F = 6πrηv Terminal Velocity, v0 =Wave MotionA = Amplitude y = Displacement ΔФ = Phase Difference γ = Frequency λ = wavelengthL = Length of the wire Μ = Mass per unit length ω = Angular frequency Δx = Path difference1. 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 =
  7. 7.  Sound Waves1. 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, 3. Freq. of beats = |ν1 - ν2| 2. Open organ Pipe, 4. Doppler Effect, ν =Thermal PhysicsP=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  CalorimetryQ=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 ThermodynamicsW=work done by gas , U=internal energy =initial volume =final pressure =initial pressure =final pressure1. dq = dW + dU 2.W = 3. Work done on an ideal gas:
  8. 8. 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 Transfere = coefficient of emission =stefan’s constant1 K=coefficient of thermal conductivity1. =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. Newtons 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 -1 Θc= sin (1/n) 3. Refraction at spherical surface m= 4. Refraction through thin lenses , , m= 5. Prism r + r’ =A , δ = i + i’ – A n=
  9. 9. Electricity and Magnetism :Coloumb’s law: The force between 2 point charges at rest: =Electric field: =qElectric 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 enclosedby 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 2 Charged ring – V = q/( + x2)) Spherical shell -(h) Capacitance of a parallel plate capacitor, C = ε0A/d(i) Ohm’s Law, E = ρJ
  10. 10. (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 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 capacitorCapacitance of a cylindrical capacitor: C=Grouping of cells: a) Series combinationIf polarity of m cells is reversed, b) Parallel combination c) Mixed combinationCurrent will be maximum whenHeat produced in a resistor=
  11. 11.  MAGNETISMFacts:1.Force on a moving charge in magnetic field is perpendicular to both and2.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=qvBsinθWhen a particle enters into a perpendicular magnetic field,it describes a circle.Radius of thecircular 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
  12. 12.  Electromagnetic – InductionInduced emf ε=- N where A is the area of the loop . Induced current ,I=ε/R Induced electric field =- Self –Inducatnce, N =LI ε=-LInducatance of a solenoid L=µ0 n2A l where ‘n’ is the number of turns per unit lengthMutual Inducatance, NΦ = MI and ε= - MGrowth of curent in an L-R circuit, I= where = L/RDecay of current in an L-R circuit I= | Energy stored in a conductor , U = ½ LI2  AC –CircuitsR.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. 2If 1 = ω LC, 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
  13. 13. CHEMISTRYOrganic ChemistryCertain 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
  14. 14. Markonikov’s rule Addition to alkynes2. Nucleophilic addition3. Bisulphite addition4. Carbanion additionSubstitution reaction mechanism1 .Sn1 mechanism Rate α [R3CX] First order reaction and rate of hydrolysis of alkyl halides- allyl>benzyl>3⁰>2⁰>1⁰>CH32. Sn2 mechanism Rate α [RX][Nu-] > Rate of hydrolysis-CH3>1⁰>2⁰>3⁰
  15. 15. Elimination Reaction mechanisms1.E1 mechanism(first order) 2. E2 mechanism(second order)3. E1cB mechanism(elimination, unimolecular)Comparison between E2 and Sn2 recationsHalogenation 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
  16. 16.  PolymersPolymer classification1.based on origin Natural. Eg-silk, wool, starch etc Semi-synthetic. Eg-nitrocellulose, cellulose xanthate etc Synthetic. Eg-teflon, polythene2. based upon synthesis Addition polymers. Eg-ethene, polyvinyl chloride Condensation polymers. Eg- proteins, starch etc3.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
  17. 17. Inorganic Chemistry E. Diagonal Relationship1. PERIODIC CLASSIFICATION OF It occurs due to similar ELEMENTS electronegativites and sizes of participating elements.A. Atomic radius Disappears after IV group radius order-van der waal’s>metallic>covalent 2. TYPES OF COMPOUNDS in case of isoelectronic species, when proton number increases, radii A.Fajan’s rules decreases A compound is more ionic(less covalent) if it contains larger cationB. Ionisation potential than anion and has an inert gas(i)It decreases when configuration. Atomic size increases. A compound is more covalent if it Screening effect increases. has small cation and has pseudo Moving from top to bottom in a inert gas configuration(18 e- group. configuration).(ii)It increases when Nuclear charge increases. B.VSEPR Theory Element has half filled or fully filled Non-metallic compounds subshells. Moving from left to right in a period. B4C3 is hardest artificial substance(iii)Order of ionization potential is Acidic nature of hydrides of I1<I2<….<In halogens-HI>HBr>HCl>HF Acidic nature of oxy-acids ofC. Electron Affinity(E.A) halogens- 2nd E.A is always negative. HClO<HClO2<HClO3<HCIO4 E.A of a neutral atom is equal to the Acidic character decreases with ionization potential of its anion decrease in electronegativity of For inert atoms and atoms with fully central halogen atom filled orbitals,E.A is zero. (CN)2, (SCN)2 etc are called pseudohalogensD. Oxidation state Compounds containing C,Cl, Br, F Oxidation state of s-block elements elements are called halons. is equal to its group number. Nature of compounds P-block elements show multivalency. Common oxidation state of d-block Non-metallic oxides are generally elements is +2 though they also acidic in nature and metallic oxides show variable oxidation states. are generally basic in nature. The common oxidation state of f- Al2O3, SiO2 etc are amphoteric in block elements is +3. nature and CO,NO etc are neutral. No element exceeds its group Silicones are polymeric number in the oxidation state. organosilicon compounds containing Ru and Os show maximum oxidation Si-O-Si bonds. state of +8 and F shows only -1 state.
  18. 18. 3.EXTRACTIVE METALLURGY Mineral is naturally occurring 5.CO-ORDINATION CHEMISTRY AND compound with definite structure and ORGANO-METALLICS ore is a compound from which an Compounds containing complex element can be extracted cations are cationic complexes and economically. anionic complexes. The worthless impurities sticking to The compounds which dont ionize in the ore is called gangue and flux is a aqueous solutions are neutral chemical compound used o remove complexes non-fusible impurities from the ore. Monodentate ligands that are Roasting is the process of heating a capable of coordinating with metal mineral in the presence of air. atom by 2 different sites are called Calcination is the process of heating ambidentate ligands like nitro etc. an ore in the absence of air Ligands containing π bonds are Smelting is the process by which a capable of accepting electron metal is extracted from its ore in a density from metal atom into empty fused state antibonding orbital π* of their own are called π acid ligands. 4. TRANSITION ELEMENTS Coordination number of a metal in The effective magnetic moment is complex √n(n+2) B.M(bohr magneton) C.N=1 x no.of monodentate ligands Color of the compound of transition C.N=2 x no.of bidentate ligands metals is related to the existence of C.N=3 x no.of tridentate ligands incomplete d-shell and Charge on complex= O.N of metal corresponding d-d transition atom+O.N of various ligands Catalytic behavior is due to variable oxidation states They form complexes due to their small size of ions and high ionic chargesPhysical Chemistry1. BASIC CONCEPTS Vmp:Vavg:Vrms=1 : 1.128 : 1.224 Van der waal’s equation: Average atomic mass = Avogadro’s Law - V Moles = For 1 molecule, the K.E = 1.5 Atomicity (γ) = = 1.5KT Cp-Cv = R Critical Pressure : n = [Molecular Formula Critical Volume : 3b Weight]/[Empirical formula Weight] Critical Temperature : Graham’s law of diffusion2. STATES OF MATTER (effusion) : Density= Rate of diffusion Partial pressure= Total pressure x Mole fraction
  19. 19. 3. ATOMIC STRUCTURE 5. CHEMICAL EQUILIBRIUM Specific charge = e/m = 1.76 x In any system of dynamic 108 c/g equilibrium, free energy change, Charge on the electron = 1.602 x ∆G = 0 10-19 coulomb The free energy change, ∆G and Mass of electron = 9.1096 x 10-31 equilibrium constant, K are kg related as ∆G = -RT lnK. Radius of nucleus, r = (1.3 x 10- Common Ion Effect : By addition 13 ) A1/3 cm, where A = mass of X mole/L of a common ion to a number of the element weak acid (or weak base), Energy of electron in the nth orbit becomes equal to Ka/X = Solubility Product : The Ionic product (IP) in a saturated Photoelectric effect : hν = hν0 + solution of the sparingly soluble 0.5mv2 salt = solubility product(SP) Heisenberg’s Uncertainty IP > SP Precipitation occurs Principle : ∆x.∆p = h/4π IP = SP Solution is saturated Spin Magnetic Moment = IP < SP Solution is unsaturated BM ; 1 BM = 9.27 x 10- 24 J/T 6. THERMOCHEMISTRY The maximum number of emission lines = Heat of reaction = Heat of Radioactivity : Decay Constant formation of products – Heat of formation of reactants = Heat of combustion of reactants – Heat of combustion of products =4. SOLUTIONS Bond Energy of the reactants – Bond energy of the products % by weight of solute = , ∆H = ∆E + ∆n RT where W = weight of the solution ∆G = ∆H - T∆S (Gibb’s Helmoltz in g Equation) Molarity = , where w2 = ∆E = ∆q + w weight in g of solute whose Order of a reaction = The sum of molecular weight is M2, V = the indices of the concentration volume of solution in ml terms in the rate equation. It is Raoult’s Law (for ideal solutions an experimental value. It can be of non-volatile solutes): zero, fractional or whole number. p = p0X1, where Molecularity = the number of p =Vapour pressure of the molecules involved in the rate solution, p0 = Vapour pressure of determining step of the reaction. the solvent and X1 = Mole It is a theoretical value, always a Fraction of the solvent whole number. Van’t Hoff factor : 7. ELECTROCHEMISTRY = Faraday’s 1st Law – W = Elt, where W= amount of substance liberated
  20. 20. E = electrochemical equivalent of Triclinic a the substance, I = current Hexagonal a=b strength in amperes and t = time in seconds. Rhombohedral a=b=c Faradays 2nd Law – The amount of substances liberated at the electrodes are proportional to Superconductivity : A superconductor is their chemical equivalent, when a material that loses abruptly its the same quantity of current is passed through different resistance to the electric current when electrolytes. 1g eq. wt. of the cooled to a specific characteristic element will be liberated by temperature. Superconductors are non passing 96500 coulombs of – stoichiometric compounds consisting electricity. of rare earthen silicates.8. NUCLEAR CHEMISTRY 10. SURFACE CHEMISTRY Isothermal variation of extent of 1 amu = 1.66 x 10-24 g adsorption with pressure is 1 eV = 1.6 x 10-19 J 1 cal = 4.184 J Decay Constant λ = 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 Half life Period t1/2= adsorbant and adsorbate. Amount N of Substance left after ‘n’ Hardy Schultz Rule 1. The ion having opposite charge to solhalf lives = particles cause coagulation 2. Coagulating power of an electrolyte9. SOLID STATE depends on the valency of the ion, i.e.CRYSTAL INTERCEP CRYSTAL ANGLES greater the valency more is theSYSTEM TS coagulating.Cubic a=b=c Gold Number: The no. of milligrams of protective colloid required to justOrtho-rhombic a b prevent the coagulation of 10ml of redTetragonal a=b gold sol when 1 ml of 10% solution ofMonoclinic a NaCl is added to it.
  21. 21. 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 : e =cosq + isinq And e-iq=cosq-isinq iq 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 a2x2+b2x+c2=0 condition is a1/a2 = b1/b2= c1/c2
  22. 22.  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 p (iii) loga(m ) = 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.
  23. 23. n 3. pr=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’)
  24. 24.  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 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 (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) |
  25. 25. 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+y12+2gx1 + 2fy1+c]1/25. 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+y12+2gx1 + 2fy1+c = 0 is given by 2gg1+2ff1=c+c1Parabola: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=02. 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. 2 24. If P(at1 ,2at1) and Q(at2 ,2at2) then the slope of chord PQ is 2x-y(t1+t2)+2at1t2=0Ellipse: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-b25. 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 e =1+ b2/a2. 22. 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/b2 =15. Asymptotes of a Hyperbola: Asymptotes of hyperbola x2/a2- y2/b2=1 are given by x2/a2 – y2/b2 = 0.
  26. 26. 6. Conjugate hyperbola : x2/a2- y2/b2=1 is the conjugate hyperbola of y2/b2-x2/a2 = 1. If e1 and e2are their eccentricities then 1/e12 + 1/e22 =1.7.Rectangular Hyperbola: It is denoted by xy=c2.A point on xy=c2 is represented in theparametric form by( ct, c/t ). At P(ct, c/t) , the slope of the tangent is -1/t2 . Equation of tangent isx + 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. 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, then x/r = cosα, y/r= cosβ and γz/r= cosγ . Also cos2α + cos2β + cos2γ =1 βStandard Form of the equation of a plane: α1) 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 =03) 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(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+a1xn-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) =
  27. 27. Some derivatives of common functions :Function Derivative Function Derivative C 0 Cu C* du/dxu+v du/dx + dv/dx uv u dv/dx + vdu/dxu/v ( v du/dx - udv/dx) /v2 xn nxn-1ex ex log x 1/xsin x cos x cos x - sin xtan x sec2 x cosec x -cosec x cot xsec x sec x tan x cot x - cosec2 xsin-1 x 1/ cos-1 x - 1/ -1 2tan x 1 / 1+ x Geometric Meaning of Derivative : If the tangent at x=a to the curve y=f(x) makes an angle ofθ 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 leastone 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
  28. 28. 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