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1 Formulae Handbook for ICSE Class 9 & 10 Table of Contents PHYSICS - IX .......................................................................................................................... 4 ESTIMATION AND UNITS ....................................................................................................... 4 EQUATIONS OF MOTION ........................................................................................................ 4 NEWTON S LAWS OF MOTION ............................................................................................... 6 PRESSURE IN FLUIDS AND ATMOSPHERIC PRESSURE .................................................... 6 DENSITY AND RELATIVE DENSITY ....................................................................................... 7 SOUND ......................................................................................................................................8 Electricity ................................................................................................................................. 12 CHEMISTRY IX ................................................................................................................ 13 MATTER AND ITS COMPOSITION........................................................................................ 13 STUDY OF GAS LAWS ............................................................................................................ 19 LANGUAGE OF CHEMISTRY ................................................................................................. 21 WATER .................................................................................................................................... 23 ATOMIC STRUCTURE ............................................................................................................28 THE PERIODIC TABLE .......................................................................................................... 33 STUDY OF FIRST ELEMENT - HYDROGEN ......................................................................... 34 MATHEMATICS IX .......................................................................................................... 36 RATIONAL AND IRRATIONAL NUMBERS........................................................................... 36 PROFIT, LOSS AND DISCOUNT ........................................................................................... 40 COMPOUND INTEREST......................................................................................................... 41 EXPANSIONS .......................................................................................................................... 42 FACTORISATION .................................................................................................................... 43 CHANGE THE SUBJECT OF A FORMULA ............................................................................ 43 GRAPHICAL SOLUTION ........................................................................................................ 43 INDICES (EXPONENTS) ........................................................................................................ 44 LOGARITHMS ......................................................................................................................... 44 RECTILINEAR FIGURES ........................................................................................................ 45 AREA (PLANE GEOMETRIC FIGURES) ................................................................................ 45 MEAN AND MEDIAN ............................................................................................................. 45 PERIMETER AND AREA ........................................................................................................ 46 SOLIDS [AREAS AND VOLUMES] ......................................................................................... 49 TRIGONOMETRICAL RATIOS ............................................................................................... 53 TRIGONOMATRIC RATIOS OF STANDARD ANGLES ......................................................... 54 ICSE Previous Papers 2 Formulae Handbook for ICSE Class 9 & 10 SOLUTION OF RIGHT TRIANGLE ........................................................................................ 54 CO-ORDINATE GEOMETRY .................................................................................................. 54 PHYSICS - X.......................................................................................................................... 57 FORCE ..................................................................................................................................... 57 MACHINES.............................................................................................................................. 58 WORK, POWER AND ENERGY .............................................................................................. 59 REFRACTION THROUGH A LENS ....................................................................................... 60 SOUND ....................................................................................................................................68 CURRENT ELECTRICITY ....................................................................................................... 69 ELECTRIC POWER AND HOUSEHOLD ELECTRICITY ....................................................... 75 ELECTROMAGNETISM .......................................................................................................... 76 CALORIMETRY ...................................................................................................................... 80 MODERN PHYSICS................................................................................................................. 81 CHEMISTRY -X .................................................................................................................... 81 ACIDS, BASES & SALTS .......................................................................................................... 81 ANALYTICAL CHEMISTRY .................................................................................................... 91 METALLURGY ........................................................................................................................ 94 STUDY OF COMPOUNDS HYDROCHLORIC ACID ...........................................................98 STUDY OF COMPOUNDS AMMONIA .............................................................................. 100 STUDY OF COMPOUNDS NITRIC ACID .......................................................................... 102 STUDY OF COMPOUNDS SULPHURIC ACID.................................................................. 105 ORGANIC CHEMISTRY ........................................................................................................ 107 MATHEMATICS X .......................................................................................................... 113 COMPOUND INTEREST........................................................................................................ 113 SHARES AND DIVIDEND ..................................................................................................... 115 BANKING ............................................................................................................................... 115 SALES TAX AND VALUE ADDED TAX (VAT) ......................................................................116 INEQUATIONS.......................................................................................................................116 QUADRATIC EQUATIONS .................................................................................................... 117 REMAINDER AND FACTOR THEOREM ............................................................................. 118 RATIO AND PROPORTION .................................................................................................. 118 MATRICES..............................................................................................................................119 CO-ORDINATE GEOMETRY / EQUATION OF A LINE ...................................................... 120 ICSE Previous Papers 3 Formulae Handbook for ICSE Class 9 & 10 SYMMETRY ........................................................................................................................... 124 SIMILARITY .......................................................................................................................... 124 LOCUS ................................................................................................................................... 127 CIRCLE .................................................................................................................................. 127 ANGLE PROPERTIES ........................................................................................................... 129 AREA OF CIRCLE................................................................................................................... 131 MENSURATION ..................................................................................................................... 131 TRIGONOMETRY ................................................................................................................. 135 STATISTICS ........................................................................................................................... 140 MISCELLANEOUS ................................................................................................................ 144 LOGARITHMS ....................................................................................................................... 145 PROFIT LOSS AND DISCOUNT ........................................................................................... 145 MATHEMATICAL FORMULAE ............................................................................................ 146 ICSE Previous Papers 4 Formulae Handbook for ICSE Class 9 & 10 Physics - Ix Estimation And Units mass volume velocity (ii) Acceleration time distance (iii) Speed= time displacement (iv) Velocity time (v) Momentum mass x velocity (i) Density = Equations Of Motion (i) for accelerating bodies (a) v u at (b) v 2 u 2 2as 1 (c) s = ut + at 2 2 (ii) for deaccelerating bodies (a) v u at (b) v 2 u 2 2as 1 (c) s ut at 2 2 ICSE Previous Papers 5 Formulae Handbook for ICSE Class 9 & 10 (iii ) for free falling bodies (a) a g (b) v u gt (c) v 2 - u 2 =2gs 1 (d) s ut gt 2 2 0 (always) v gt v 2 2gs 1 s gt 2 2 (iv) for body thrown upwards (a) a = - g (b) v = u - gt (c) v 2 u 2 2gs 1 (d) s = ut - gt 2 2 v = 0 (always) u gt u 2 2gs 1 s = ut - gt 2 2 where u = initial velocity; v = final velocity; a = acceleration; t = time and s = displacement. ICSE Previous Papers 6 Formulae Handbook for ICSE Class 9 & 10 Newton s Laws Of Motion (i) Force (F) = mass (m) x accelerration (a) F = ma (ii) Momentum (p) = mass (m ) x velocity (v) (iii) p = (mv) = m v p m v (iv) ma t t p -p mv mu m(v u) (v) f i ma t t t p (vi) F = ma t p = momentum; pi initial momentum and p f final momentum or (delta): used in calculus used to indicate a small change in a given quantity. F = force m = mass a = acceleration v = velocity Pressure In Fluids And Atmospheric Pressure force area (A) S.I. unit of pressure is pascal (pa) (ii) p = h g P or p = pressure h = depth of liquid or gas (rho) = density of liquid or gas g = acceleration due to gravity (i) Pressure (p) = ICSE Previous Papers 7 Formulae Handbook for ICSE Class 9 & 10 Density And Relative Density mass (m) volume (v) = area of cross section height V=a h (i) Density (d) = (ii) Volume (iii) Archimedes' Principle: density of solid density of water at 4 C weight of solid (w1 ) = weight of an equal volume of water weight of solid in air = loss of weight of body in water (weight of body in air - weight of body in water) w1 = w1 w 2 Relative Density of a solid = density of liquid density of water at 4 C loss of weight of solid in liquid = loss of weight of solid in water w1 weight of body in air Relative Density of a liquid = w 2 = weight of body in water w 3 = weight of body in liquid X (iv) Law of Flotation: w1 weight of solid w 2 weight of fluid displaced as per the law: w1 w 2 but: w = mg m1g = m 2g but m = V d (mass = density volume) V1d1 = V2 d 2 or V1 1 V2 2 i.e. 1 V1 2 V2 ICSE Previous Papers 8 Formulae Handbook for ICSE Class 9 & 10 Sound v = where V is velocity; (nu) is frequency and (lambda) is the wave length. Key Points and Concepts Sound is a wave motion, produced by a vibrating source. The audible range of hearing for average human beings is in the frequency range of 20 Hz 20 KHz. A medium is necessary for the propagation of sound waves. Sound is a longitudinal wave in which the particles of medium move along the direction of motion of wave. Speed of Sound : The speed of sound is the rate at which sound travels from the sound producing body of our ears. The speed of sound depends on the (i) Nature of Material (ii) Temperature (iii) Humidity of Air. The point of maximum positive displacement on a transverse wave is known as crest. The point of maximum negative displacement on a transverse wave is known as trough. Sound travels faster in solids than in air. The speed of sound in solids is much more than the speed of sound in liquids or gases. Characteristic of Sound: Loudness: Loudness of a sound depends on the amplitude of the vibration producing that sound. Greater is the amplitude of vibration, louder is the sound produced by it. Pitch: The shrillness of a sound is called its pitch. The pitch of a sound depends upon its frequency. Higher the frequency of a sound, higher is its pitch. Quality: Quality of a sound is also called timbre. The quality of sound is the characteristic which enable us to distinguish between the sounds produced by different sources. Characteristics of wave: wavelength, frequency and time period. Wave Length: The distance between two nearest points in a wave which are in the same phase of vibration is called the wave length. In simple words it is the length of one complete wave. It is denoted by lambda, . Y A Crest Distance Amplitude: The amplitude of a wave is the magnitude of maximum displacement of the vibrating particles on the either side of their mean position. It is denoted by the letter A and its SI unit is metre (m). Time-Period: The time required to produce one complete wave (or cycle) is called time-period of the wave. ICSE Previous Papers 9 Formulae Handbook for ICSE Class 9 & 10 Frequency : The frequency of an oscillating particle is the number of oscillations completed in one second. The unit of frequency is hertz (or Hz). The frequency of the wave is 1/T. It is generally represented by (nu) Wave Velocity: The distance travelled by the wave in one second is called the wave velocity. It is represented by 'v' and its unit is ms 1. Phase: All the points on a wave which are in the same state of vibration are said to be in the same phase. Thus, in the wave shown in fig. B J F A K C v I H Direction of Propagation E D G The distance between two consecutive compressions or two consecutive rarefactions is called the wavelength. Frequency is defined as the number of oscillations per second. The time taken by the wave to complete one oscillation is called the time period, T. When a body repeats its motion continuously on a definite path in a definite interval of time then its motion is called periodic motion. The constant interval of time after which the motion is repeated is called the 'Time period of motion.' (T) If a body in periodic motion moves along the same path to and fro about a definite point (equilibrium position), then the motion of the body is a vibratory motion or oscillatory motion. Wave Motion: The movement of a disturbance produced in one part of a medium to another involving the transfer of energy but not the transfer of matter is called wave motion. Wave which do not require any material medium for their propagation are called electromagnetic waves. Example : Light waves, Radio waves, Television waves, and X-rays are electromagnetic waves. Thus, Light waves, Radio and Television waves, and X-rays can also travel through vaccum. The wave which propagates only in a material medium are called elastic or mechanical waves. Example: Sound waves, Water waves (ripples), Waves on stretched strings, Earthquake waves and the Shock waves produced by a supersonic aircraft are mechanical (or elastic) waves. Sound of frequency less than 20 Hz is known as infrasound and greater than 20 kHz is known as ultrasound. Ultrasound has many medical and industrial applications. The sound returning back towards the source after suffering reflection from a distance obstacle (a wall, a row of building etc.) is called an echo. The repeated reflection that results in the persistence of sound in a large hall is called reverberation. The human ear is sensitive to sound waves of frequency between 20 Hz to 20 kHz. This range is known as audible range and these waves are known as audible waves. Ex. Waves produced by vibrating sitar, guitar, organ pipes, flutes, shehnai etc. Ultrasonic waves: A longitudinal wave whose frequency is above the upper limit of audible range i.e. 20 kHz, is called ultrasonic wave. It is generated by very small ICSE Previous Papers 10 Formulae Handbook for ICSE Class 9 & 10 sources. Ex. Quartz crystal Infrasonic wave: A longitudinal elastic wave whose frequency is below the audible range i.e. 20 Hz, is called an infrasonic wave. It is generally generated by a large source. Ex. Earthquake. SONAR stands for Sound Navigation and Ranging and it works on the principle of reflection of sound waves. The SONAR technique is used to determine the depth of the sea and to locate under water hills, valleys, submarines, icebergs, sunken ships, etc. When a body moves with a speed which is greater than the speed of sound in air, it is said to be travelling at supersonic speed jet fighters, bullets, etc, often travel at supersonic speed, and when they do so, they produce a sharp, loud sound called a sonic boom. Quantities and Units: S.No. Quantities SI Units 1. Velocity m/s 2. Time s (sec) 3. Frequency Hz (hertz) i.e., s-1 4. Wavelength M Important Formulae: Frequency and time period are related as follows: 1 T Distance v= Time = Speed, The wave velocity (v), frequency of the wave (f) and its wavelength (R) are related by the formula, v = Longitudinal and Transverse Waves: S.No. Longitudinal Transverse waves 1. In a transverse wave, the particles of the In a longitudinal wave the particles medium oscillate in a direction of the medium oscillate along the perpendicular to the direction of direction of propagation of the wave. propagation of the wave 2. Longitudinal waves can propagate through solids, liquids, as well as gases. Transverse waves can propagate through solids, and over the surface of liquids, but not through gases. 3. Longitudinal waves consist of compression and rarefactions. Transverse waves consist of crests and troughs. ICSE Previous Papers 11 Formulae Handbook for ICSE Class 9 & 10 Important Graphs and Diagrams: Soft Sound and Louder Sound Low pitched and High pitched Sound The Human Ear Structure of human ear: The ear consists of three compartments : outer ear, middle ear and inner ear. The part of ear which we see outside the head is called outer ear. The outer ears consists of broad part called pinna and about 2 to 3 centimeters long passage called ear canal. At the end of ear canal there is a thin, elastic and circular membrane called ear-drum. The ear-drum is also called tympanum. The outer ear contains air. The middle ear contains three small and delicate bones called hammer, anvil and stirrup. The inner ear has a coiled tube called cochlea. Three tiny ear bones Pinna Sound waves enter here Stirrup Hammar Anvil Auditory nerve (Goes to brain) Ear Ear canal drum Eustachian tube (Goes to threat) Outer ear Oval Cochlea window Middle ear inner ear Working of human ear: Sound waves from outside are collected by the outer ear (called pinna) and reach the eardrum through the auditory canal. When the sound waves strike the eardrum, (tympanic membrane) it starts vibrating. These vibrations are passed on to the oval window by three bones (called the hammer, anvil and stirrup) which act as a lever with the pivot at point P. They magnify the force of the vibrations. ICSE Previous Papers 12 Formulae Handbook for ICSE Class 9 & 10 Electricity q t Where I = current q = charge t = time (i ) I = work done in joule (W) charge in coulomb (q) W V = q (ii) Potential differnce in volts (V) = (iii) Ohm's Law: V R I V = RI Where, V = Potential difference R = Resistance I = Current SI Unit volt ohm ampere ICSE Previous Papers 13 Formulae Handbook for ICSE Class 9 & 10 Chemistry Ix Matter And Its Composition (a) Interconversion of states (b) Kinetic Theory of Matter : (i) All matter is made up of very small particles - such as atoms or molecules. (ii) Particles are moving all the times (higher the temperature, higher the kinetic energy of the particles) Higher particles move more slowly than lighter particles at the same temperature. ICSE Previous Papers 14 Formulae Handbook for ICSE Class 9 & 10 The Kinetic Theory of matter Arrangement State of Matter of Matter SOLID 1. Has a fixed volume and shape. 2. Cannot be compressed Arrangement and Movement of Particles Matter 1.The particles are packed closely together in an orderly manner. (Inter molecular space minimum) 2. There are strong forces between the particles. (Inter molecular attraction) 3. The particles can only vibrate and rotate about their fixed positions. (Brownian movement) LIQUID 1. The particles ard packed closely logother but not in orderly arrangement. 1. Has fixed volume but does 2. Held together by strong forces but not have fixed shape (takes shape weaker than the forces in a solid. of the container). 4. The particles can vibrate, rotate and move throughout the liquid. 2. Cannot be compressed easily 5. They collide against each other. GAS 1. The particles are very far apart from each other and in a random motion. 1. Does not have a fixed shape or 2. Weak forces between the particles. volume. 3. The particles can vibrate, rotate and 2. Can be move freely. The rate of collision is compressed easily. greater than the rate of collision in a liquid. (c) Law of conservcation of Mass : Sum of weight of all reactants = sum of weight of all products For Reaction: A+B C+D Mass of A + Mass of B = Mass of C + Mass of D Key Points and Concepts All the matter around us is not pure. The matter around us is of two types. A mixture contains more than one substance (element and/or compound) mixed in any proportion. A pure substance consists of a single type of particles. ICSE Previous Papers 15 Formulae Handbook for ICSE Class 9 & 10 Pure substances are always homogeneous. Pure substances can be divided into two types. (A) Elements (B) Compounds MATTER PURE SUBSTANCES Only one substance present : no impurities MIXTURE More than one substance present ELEMENTS COMPOUNDS HOMOGENEOUS HETEROGENEOUS Cannot be divided into simpler substances Made from elements chemically bonded together (solutions) substance completely mixed. Mixture exists in one state (or one phase) (Suspensions and collids) Substances separate. One state (or one phase) spread throughout the other Element Symbol Aluminium Al Arsenic As Barium Ba Bromine Br Cadmium Cd Calcium Ca Chlorine Cl Chromium Cr Cobalt Co Fluorine F Hydrogen H Iodine I Magnesium Mg Manganese Mn Nitrogen N Oxygen O Phosphorus P Sulphur S Uranium U Zinc Zn (symbols from latin names) Antimony (stibium) Sb Copper (Cuprum) Cu Gold (Aurum) Au Iron (Ferrum) Fe ICSE Previous Papers 16 Formulae Handbook for ICSE Class 9 & 10 Lead (Plumbum) Pb Mercury (Hydrogyrum) Hg Potassium (Kalium) K Silver (Argentum) Ag Sodium (Natrium) Na Tin (Stannum) Sn Mixtures and Compounds: S.No. Mixtures Compounds 1. Two or more elements or compounds are mixed in any proportion. Two or more elements combine and react. 2. A new substance is formed. A new substance is formed. 3. The new substance shows the properties of its constituents. The new substance has totally different properties. 4. The composition of the new substance is variable. The composition of the new substance is always fixed. 5. The constituents of a mixture can be separated only by physical methods. The constituents of a compound can be separated only by chemical reactions. Homogeneous and Heterogeneous Mixture: S. No. Homogeneous Mixture Heterogeneous Mixture 1. Uniform Composition. Non-Uniform Composition. 2. No distinct boundaries of separation. e.g., sugar + water. Distinct boundaries of separation. e.g., sand + water. Physical Change and Chemical Change: S. No. Physical change Chemical change 1. These are reversible changes and their chemical composition do not change. These are irrreversible changes and the chemical composition also changes. 2. No new substance is formed. e.g. Tearing of paper. New substance is formed. e.g. Burning a match-stick. ICSE Previous Papers 17 Formulae Handbook for ICSE Class 9 & 10 Metals and Non-Metals: Metals Non-Metals 1. Metals are malleable and ductile. That is, metals can be hammered into thin sheets and drawn into thin wires. 1. Non-metals are brittle. They are neither malleable nor ductile. 2. Non-metals are bad conductors of heat and electricity (except diamond which is a good conductor of heat, and graphite which is a good conductor of electricity) 2. Metals are good conductors of heat and electricity. 3. Metals are lustrous (shiny) and can be polished. 3. Non-metals are non-lustrous (dull) and cannot be polished (except iodine which is a lustrous non-metals) 4. Metals are solids at room temperature (except mercury which is a liquid metal). 4. Non-metals may be solid liquid or gases at the room temperature 5. Metals are strong and tough. They have high tensile strength. 5. Non-metals are not strong. They have low tensile strength. 6. Metals are sonorous. They make a ringing sound when struck. 6.Non-metals are not sonorous. Properties of non-metals: Non-metals are not malleable. Non-metals are brittle. Non-metals are not ductile. This means that non-metals cannot be drawn into wires. They are easily snapped on stretching. Non-metals are bad conductors of heat and electricity. Non-metals are not lustrous (not shiny). They are dull in appearance. Non-metals are generally soft Non-metals are not strong. They have low tensile strength. Non-metals may be solid, liquid or gases at the room temperature. Non-metals have comparatively low melting points and boiling points Non-metals have low densities. Non-metals are not sonorous. Non-metals have many different colours. Understanding of true solution, colloidal solution and suspension: S. No. True Solution Colloidal Solution 1. A true solution is a homogenous mixture of solute and solvent. A colloidal solution appears to be a homogenous but actually it is a heterogenous mixture of solute and solvent. It is a heterogenous mixture. 2. It is transparent. It is translucent. It is opaque. ICSE Previous Papers Suspension 18 Formulae Handbook for ICSE Class 9 & 10 3. The solute particles are very small, i.e.,less than 1 nm. The solute particles are between 1-100 nm. The solute particles are quite large i.e. more than 100 nm. 4. The particles are not visible even with a powerful microscope. The particles are visible with the help of microscope. The particles are visible with naked eye. 5. The entire solution passes through filter paper as well as semipermeable membrane. The particles can pass through ordinary filter paper but not through a semi-permeable membrane. The particles cannot pass through either a filter paper or through a semipermeable membrane. 6. The solute particles do not show Tyndall effect. The particles show Tyndall effect. The particles may or may not show Tyndall effect. 7. The particles do not settle due to gravity e.g., salt in water solution. The particles do not settle due to gravity e.g., blood. The particles may settle due to gravity e.g., chalk powder in water. Mass by mass percentage of a solution = No. of moles Mass by volume percentage of a solution = m Mass N Molar mass Mass of Solute Volume of Solution ICSE Previous Papers 19 Formulae Handbook for ICSE Class 9 & 10 Study Of Gas Laws (a) Conversions : Volume : 1 lt = 1 dm3 1000 cc = 1000 mlt. Pressure: 1 atm = 76 cm of Hg = 760 mm of Hg Temperature: K = C + 273 (b) Boyle's law : 1 V at constant temperature p P1V1 P2 V2 where V1 Volume of a given gas at pressure P1 V2 Volume of that gas at pressure P2 (c) Charle's law : V1 V2 = at constant pressure T1 T2 where V1 Volume of a given gas at temperature T1 V2 Volume of a given gas at temperature T2 (d) Gas Equation : P1V1 P2 V2 = T1 T2 Where, V1 volume of a given gas at pressure P1 & temperature T1 V2 volume of a given gas at pressure P2 and temperature T2 ICSE Previous Papers 20 Formulae Handbook for ICSE Class 9 & 10 (e) S.T.P = standard temperature and pressure (f) N.T.P = normal temperature and pressure (g) Standard temperature = 0 = 273 K (h) Standard pressure = 1 atmosphere = 760 mm Hg (i) Absolute Zero or Kelvin zero = - 273 C or 0 K (j) No. of moles = mass of the gas molecular mass ICSE Previous Papers 21 Formulae Handbook for ICSE Class 9 & 10 Language Of Chemistry (a) Chemical Formula of Compounds : Name of Compound Formula Magnesium chloride MgCl2 Aluminium phosphate AlPO 4 Sodium Fluoride NaF Calcium Phosphate Ca 3 PO 4 2 Potassium carbonate K 2 CO3 Barium Chlorate Ba C1O3 2 Sodium Silicate Na 2SiO 3 Barium Sulphate BaSO 4 Calcium bromide CaBr2 Chromium sulphate Cr2 SO 4 2 Potassium Ferrocyanide K 4 Fe CN 6 Stannic Oxide SnO 2 Sodium Zincate Na 2 Zn OH 4 Sodium Thiosulphate Na 2S2O 3 Nickel bisulphate Ni HSO 4 2 Potassium Manganate K 2 MnO 4 Sodium hypochlorite NaClO Sodium Chlorite NaC1O 2 Sodium Chlorate NaC1O3 ICSE Previous Papers 22 Formulae Handbook for ICSE Class 9 & 10 b Formula of common Acids : Acid Name HC1 Hydrochloric acid HNO3 Nitric acid H 3 PO 4 Phosphoric acid H 2SO3 Sulphurous acid HCOOH Acetic acid H 2SO 4 Sulphuric acid H 2 CO 3 Carbonic acid CH 3COOH Acetic acid H 2C2 O4 Oxalic acid H 3 PO3 Phosphorous acid (c) Formula of Common Bases : NaOH Sodium hydroxide NH 4 OH Ammonium hydroxide KOH Potassium hydroxide Ca OH 2 Calcium hydroxide Ba OH 2 Barium hydroxide Al OH 3 Aluminium hydroxide LiOH Lithium hydroxide (d) Steps to Balance Chemical Equations Step 1: Convert the word equation into formula equation. Step 2: Note down the number of different atoms present on left hand side and right hand side of the equation. Step 3: Add coefficients to balance the equation, so that we have the equal number of atoms of each element on both the reactant (left hand side) and product (right hand side) of the equation. ICSE Previous Papers 23 Formulae Handbook for ICSE Class 9 & 10 Example: Balance the chemical equation: Mg + 2HCl > MgCl2 + H2 Step 1: Given chemical equation: Mg + 2HCl Mgcl2 > + H2 Step 2: Atoms on Reactance side Products side 1--------Mg-------1 1--------Cl--------2 1--------H---------2 Step 3: Reactance side Products side 1--------Mg-------1 2 1-----Cl--------2 2 1------H---------2 Mg + 2HCL MgCl2 + H 2 Water (a) Test for Water: (i) It turns anhydrous copper sulphate blue CuSO4 + 5H20 CuSO4.5H20 (ii) It turns anhydrous cobalt chloride pink CoCl2 + 6H20 CoCl2. 6H20 (b) Chemical Properties: (i) Reaction with sodium Na, K react with cold water to form hydroxide and liberate hydrogen. Na melts into a silver globule1 ICSE Previous Papers 24 Formulae Handbook for ICSE Class 9 & 10 2Na + 2H20 2NaOH + H2+ heat This reaction is vigorous and exothermic and the evolved hydrogen burns with golden yellow flame and bubbles of hydrogen are evolved. (ii) Reaction with Calcium Ca + 2H20 Ca(OH)2 + H2 The solution becomes turbid and alkaline. (iii) Reaction With IVlagnesium Mg + H20 MgO + H2 Magnesium bums with intense white light in steam or boiling water liberating hydrogen gas and white ash of magnesium oxide. (iv) Reaction with Iron: 3Fe + 4H2o Fe3o4 4H 2 Red hot iron reacts with steam to give iron triferric tetroxide and hydrogen gas. This reaction is reversible when carried out in closed vessel. (v) Reaction with Carbon: C + H2O CO+ H2 Red hot coke reacts with water to form water gas. (vi) Reaction with Chlorine: sunlight Cl2 + H 2O diffused HCl+ HClO (Hypochlorous acid.) Direct sunlight 2Cl2 2H 2O 4HCl + O2 (vii) Reaction with Carbon dioxide: CO2 + H2O H2CO3 (viii) Reaction with calcium carbide: CaC2 + 2H2 0 Ca(OH)2 + C2 H2 It gives calcium hydroxide and acetylene (ix) Reaction with Nitrogen Dioxide: 2NO2 + H2 0 HNO2 + HNO3 It gives nitrous acid and nitric acid. (x) Reaction with Aluminium: 2Al + 3H2 0 A12O3 + 3H2 Red hot A1 reacts with steam, above 800 C to give aluminium oxide. ICSE Previous Papers 25 Formulae Handbook for ICSE Class 9 & 10 (xi) Reaction with Zinc: Zn + H2O ZnO + H2 Zinc reacts with steam to give zinc oxide and hydrogen. weight of solute 100 weight of solvent weight of solute (ii) Concentration = 100 weight of solution (iii) Formula of salts Salt Formula Blue Vitriol CuSO 4 .5H 2 O (i) Solubility = Quick Lime Oil of Vitriol CaO H 2SO 4 Washing Soda Crystals Na 2CO3 .10H 2 O Epsom Salt MgSO 4 .7H 2 O Green Vitriol FeSO 4 .7H 2 O Borax Na 2 B4 O7 .10H 2 O Glaubers salt Na 2SO 4 .10H 2 O (c) Activity series of elements Activity of metals K Na Ca Can react with cold water and hydrogen. acids, replacing Mg A1 Zn Fe Can react with steam and acids, hydrogen replacing ICSE Previous Papers 26 Formulae Handbook for ICSE Class 9 & 10 Activity of metals Pb H2 Sb Bi Cu Hg2 Ag Pt Au Activity of Halogens Can react with acids, replacing hydrogen. Forms insoluble PbCl2, PbSO4 on top of Pb. Reaction stops. Very very slow reaction. Elements below hydrogen (Cu, Hg, Ag, Pt, Au) and lead do not displace hydrogen from metals. React with oxygen, forming oxides Fairly unreactive, form oxides only indirectly F2 Cl2 Br2 li2 Key Points and Concepts The materials present in natural environment & useful to living organism are called natural resources. Natural Resources can be classified into two groups. Physical resources: E.g, Air, water, soil, minerals, coal etc. Biological resources: E.g. Microorganisms, plants & animals. Life on earth is responsible for the present atmosphere of earth, consisting of gases like nitrogen, oxygen, carbon dioxide and water vapour. The top surface layer of this exposed, solid part of crust containing weathered minerals and humus and capable of supporting plant growth is called soil. Air is a mixture of gases which is odourless, tasteless & invisible. Air also holds water vapour & dust particles. An undesirable change in the physical, chemical or biological characteristics of the air making it harmful for the living organisms (including man) is termed air pollution. Some common harmful effects of air pollution are Respiratory problems, e.g., sneezing, allergy, bronchitis, asthma,, tuberculosis and lung cancer. Carbon monoxide poisoning. Acid rain. Depletion of ozone layer. Global warming (green hourse effect). Serious ailments produced by certain metals and pesticides. ICSE Previous Papers 27 Formulae Handbook for ICSE Class 9 & 10 Smog. Biosphere comprises of biotic and abiotic factors, which interact with each other and maintains a balance. The wind patterns in a particular region direct the rainfall patterns of that region. Burning of fossil fuels releases harmful oxides of sulphur and nitrogen, which gives rise to acid. An undesirable change in the physical, biological or chemical qualities of water (due to addition of foreign organic, inorganic, biological or radioactive substances) that adversely affects the aquatic life, and make the water less fit or unfit for use, is called water pollution. Water is important to living organisms because: 1. All cellular processes require an aqueous medium. 2. Dissolved substances are needed for body reactions as well as for transportation. Water pollution is caused by addition of following to water bodies: 1. Fertilizers and pesticides 2. Sewage 3. Waste from factories 4. Heated water from factories 5. Cold water from dams. Freezing of water in cracks of rocks causes widening of the cracks. Soil is a mixture of rock particles, humus and microscopic and small organisms. Ozone is present in the upper atmospheric strata and contains three atoms of oxygen. The ozone layer absorbs the sun's harmful ultraviolet radiations, thus preventing them from reaching the Earth's surface and damaging life. CFCs and other man-made compounds react with the ozone molecules and causes ozone layer depletion. Green house effect: The process in which green house gases like carbon dioxide, cause thermal radiation emitted by the Earth's surface to be reflected back down, therefore causing the increase in worldwide average temperatures. Global warming: An increase in the average temperature of the earth's atmosphere, brought about by the enhanced greenhouse effect. Important Graphs and Diagrams Water cycle: Biogenic elements (macro-, micro- & other elements) flow from the environment into and out of the plant in a cyclic manner. This flow of nutrients from abiotic to biotic components of the ecosystem and vice-versa constitute the biogeochemical cycles. Water on earth is cycled by two processes, evaporation and precipitation. The atmospheric precipitation occurs in the form of snow, hail or sleet etc. The run off water is finally collected in ocean through rivers. Some water remains solid in the form of snow which gradually melts and reaches the sea. Soil water is used by plants and most of it again reaches the atmosphere through transpiration. Animals consume water directly from water bodies & also the gravitational water. ICSE Previous Papers 28 Formulae Handbook for ICSE Class 9 & 10 By evaporation, the water returns to atmosphere and cycle is repeated. Clouds Snow, Hail, Sleet, Rain Evaporation Transpiration Evaporation Local Rain Run off Rivers Ponds & Lakes Plants Sea Animals Ocean Soil GRAVITATIONAL WATER The Water Cycle Atomic Structure (a) Atomic number (Z) = number of electrons = number of protons (b) Mass number (A) = = number of protons + number of neutrons atomic number + number of neutrons (c) Number of protons = = atomic number mass number - number of neutrons (d) Number of electrons = = atomic number mass number - number of neutrons (e) Number of neutrons = mass number - atomic number (f) Bohr and Bury Scheme - Important Rules (i) Maximum number of electrons that can be accommodated in a shell is given by 2n2 where n = shell number (ii) For 1st energy level, n = 1 Maximum number of electrons in 1st energy level = 2n2 = 2 x (1)2 = 2 (iii) For 2nd energy level n = 2 Maximum number of electrons in the 2nd energy level = 2n2 = 2 x (2)2 = 2 x 4 = 8 ICSE Previous Papers 29 Formulae Handbook for ICSE Class 9 & 10 (iv) For 3rd energy level n = 3 Maximum number of electrons in the 3rd energy level = 2n2 = 2 x (3)2= 2 x 9 = 18 (v) For 4th energy level n = 4 Maximum number of electrons in the 4th energy level = 2n2 = 2 x (4)2 = 2 x 16 = 32 Sl.No Electron Shell Maximum Capacity 1 K Shell 2 electrons 2 L Shell 8 electrons 3 M Shell 18 electrons 4 N Shell 32 electrons The outermost shell of an atom cannot accommodate more than 8 electrons, even if it has a capacity to accommodate more electrons. This is a very important rule and is also called the Octet rule. The presence of 8 electrons in the outermost shell makes the atom very stable. (g) ISOTOPES - are the atoms of same element having same atomic number but different mass number. The isotopes have different number of neutrons Eg. H1 Protium 1 Deutrium 1 Tritium 1 H2 H3 Key Points and Concepts Atoms are made up of three subatomic particles electrons, Protons and neutrons. Electron has negative charge, proton has positive charge, whereas neutron has no charge, it is neutral. Cathode rays always travel in straight line. Cathode rays consist of material particles and possess energy, hence they can produce mechanical effects. Cathode rays consist of negatively charged particles. Anode rays travel in straight lines. Anode rays consist of material particles. Anode rays are deflected by electric field towards negatively charged plate. This indicates that they are positively charged. Electron: Cathode rays consist of small, negatively charged particles called electrons. The electron is a negatively charged particle found in the atoms of all the elements. The electrons are located outside the nucleus in an atom. An electron is usually represented by the symbol e . Charge of an electron: The absolute charge on an electron is 1.6 10-19 coulomb of negative charge. The relative charge of an electron is, 1. Mass of proton: The mass of proton has been found to be equal to 1.67 10-27 kg ICSE Previous Papers 30 Formulae Handbook for ICSE Class 9 & 10 Charge of proton: The charge of proton is equal and opposite to the charge of an electron. The value of charge on proton is 1.602 10 19. Mass of a neutron: The mass of neutron is equal to the mass of a proton. The relative mass of a neutron is 1 u. The absolute mass of a neutron is 1.6 10 27 kg. Valency of an element is the combining capacity of the atoms of the element with atoms of the same or different elements. The outermost orbit of an atom is called its valence shell. The electrons present in the outermost orbit are called valence electrons. Atomic Number: The number of protons present in the nucleus of an atom of an element is known as its atomic number. Atomic number of an element (Z) = Number of protons (p) = Number of positive charges carried by the nucleus of the atom or = Number of electrons (e) Mass Number: The sum of the number of protons and neutrons in an atom of the element. Mass number (A) = No. of protons (p) + No. of neutrons (n) A X Z A = Mass number Z = Atomic number X = Symbol of element Electrovalency: In the formation of an electrovalent compound (or ionic compound), the number of electrons lost or gained by one atom of an element to achieve the nearest inert gas electron configuration is known as its electrovalency. Covalency : In the formation of a covalent compound the number of electrons shared by one atom of an element to achieve the nearest inert gas electron configuration is known as its covalency. If an atom shares 1 electron, its covalency will be 1. Isotopes are atoms Of the same element, which have different mass numbers (i) Carbon, 12 6 (ii) Chlorine, 35 17 14 6 C Cl and 37 17 Cl etc. Isobars are atoms having the same mass number but different atomic numbers. 76 32 C and Ge and 76 34 Se are isobars. Isotones: The atoms having same number of Neutrons but diffrent mass number are called Isotones. The atoms have different number of protons of atomic number. (A Z) is same A & Z are different . ICSE Previous Papers 31 Formulae Handbook for ICSE Class 9 & 10 Comparision between Proton, Neutron and Electron: First 18 elements: Important Graphs and Diagrams Bohr's Model of an atom: An atom is made up of three particles: electons, protons and neutrons. Electrons have negative charge, protons have positive charge whereas neutrons have no charge, they are neutral. Due to the presence of equal number negative electrons and positive protons, the atom on the whole is electrically neutral. The protons and neutrons are located in a small nucleus at the centre of the atom. Due to the presence of protons, nucleus is positively charged. The electrons revolve rapidly around the nucleus in fixed circular paths called energy levels or shells. The energy levels or shells are represented in two ways: either by the ICSE Previous Papers 32 Formulae Handbook for ICSE Class 9 & 10 numbers 1, 2, 3, 4, 5 and 6 or by the letters K, L, M, N, O and P. The energy levels are counted from the centre outwards. There is a limit to the number of electrons which each energy level can hold. Each energy level (or shell) is associated with a fixed amount of energy, the shell nearest to the nucleus having minimum energy and the shell farthest from the nucleus having the maximum energy. Alpha particle scattering experiment by Rutherford: Rutherford model of an atom: Most of the fast moving -particles passed straight through the gold foil without any deflection from their original path. Some of the -particles were deflected from their path through small angles. Very few (about 1 in 12,000) did not pass through the foil at all but suffered large deflections (more than 90 ) or even came back in the direction from which they came. Most of the space inside the atom is empty. Therefore, most of the -particles went through the gold foil without deflecting from their path. The deflections of the -particles to large angles indicate that the -particles has direct collision with the positively charged nucleus. Rutherford s Atomic model: The main features of Rutherford s model of an atom are : The atom consists of a positively charged centre called the nucleus. Most of the mass of the atom is concentrated in the nucleus. The volume of the nucleus is very small as compared to the total volume of the atom. The nucleus is surrounded by the negatively charged electrons. the electrons balance the positive charge of the nucleus. Therefore, the number of electrons in an atom is equal to the number of protons in it. The magnitude of the positive charge on the nucleus is different for different atoms. ICSE Previous Papers 33 Formulae Handbook for ICSE Class 9 & 10 The Periodic Table (a) Dobereiner's Triads arranged elements in an increasing order of atomic mass, in groups of three. The atomic mass of the middle element was the arithmetic mean of the other two elements of the triad. (b) Newland's law of octaves states that on arranging elements in increasing order of their atomic mass, the eighth element resembles the first in physical and chemical properties, just like the eighth node on a musical scale resembles the first note. (c) The law could only be applied upto calcium. No vacant spaces were left for undiscovered elements. (d) According to Mendeleev's periodic law, the physical and chemical properties of elements are periodic functions of their atomic mass. (e) Mendeleev was able to predict the existence of undiscovered elements like Ga, Sc and Ge. (f) Mendeleev corrected the atomic masses of a few elements on the basis of their positions in the periodic table. (g) Mendeleev's table could not assign a proper position to hydrogen or to the lanthanides and actinides and isotopes. (h) Modern periodic law states that the properties of elements are periodic functions of their atomic numbers. It is based on electronic configuration of the elements. (i) Mendeleev - had arranged elements in the periodic table on the basis of increasing atomic weight. ICSE Previous Papers 34 Formulae Handbook for ICSE Class 9 & 10 (j) Modern periodic table - elements are arranged on the basis of increasing atomic number. Salient Feature of Modern Periodic Table (i) The vertical columns are called groups, while the horizontal rows are called periods. There are 7 periods and 8 groups subdivided into 18 sub groups. (ii) Group number is number of electrons in the valence shell. Elements having the same valence number are grouped together. (iii) The number of shells present in the atom gives period number. (iv) Group 1, 2 & 13 to 17 are called normal elements. (v) Group 3 to 12 are called transition elements. (vi) Group 18 (zero) at extreme right contains noble or inert gases. (vii) Reactive metals are placed in group 1 & 2. (viii) Transition elements are placed in the middle. (ix) Non-metals are placed in the upper right corner of the periodic table and metals to the left. Study Of First Element - Hydrogen (a) Preparation Of Hydrogen (i) By the Action of Metals The alkali metals, lithium, sodium and potassium react violently with water at the ordinary temperature, yielding hydrogen. 2Na 2H 2O 2NaOH + H 2 2K 2H 2O 2KOH H 2 Calcium reacts with water more slowly unless the water is hot, when the action is more vigorous. Ca 2H2O H2 Ca(OH)2 2H2 (ii) Preparation of Hydrogen by Decomposition of Water Steam is decomposed when passed over heated magnesium, zinc and iron to form oxides and liberate hydrogen. Zn + H 2SO4 ZnSO4 + H 2 Zn 2HCl ZnCl2 H 2 ICSE Previous Papers 35 Formulae Handbook for ICSE Class 9 & 10 The reaction, (i.e. with iron and steam) is reversible. (iii) Preparation of Hydrogen from Action of Acids Hydrogen is prepared in the laboratory by the action of acids on active metals. Dilute sulphuric acid or dilute hydrochloric acid used is added to granulated zinc. Zinc sulphate or zinc chloride is formed in solution and the hydrogen that is evolved is collected by the downward displacement of water. Zn + H 2SO4 ZnSO4 + H 2 Zn 2HCl ZnCl2 H 2 (iv) BOSCH s Process By passing steam over red hot coke (carbon) at about 1000 C a mixture of carbon monoxide and hydrogen known as water gas is produced. Carbon monoxide is separated from the mixture by converting it to carbon dioxide. For conversion of carbon monoxide into carbon dioxide, excess steam is mixed with the water gas and passed over a catalyst. Fe2 03 + Cr2 O3 at 450 C. H 2 0 + C CO + H 2 water gas CO + H 2 + H 2O C02 + 2H 2 Mixture of carbon dioxide and hydrogen gas is passed through water at a pressure. CO2 is absorbed in water while H2 does not dissolve in water and is collected. (b) Chemical Properties (i) Reaction with Non-Metals Hydrogen readily combines with fluorine and chlorine, less readily with bromine, iodine, sulphur, phosphorous, nitrogen and carbon. H2 + F2 2HF Hydrogen burns in chlorine gas and a mixture of hydrogen and chlorine are exposed to diffused sunlight in the pressure of moisture which act as a catalyst. H2 + Cl2 2HC1 Hydrogen combines with nitrogen on sparking or in presence of a catalyst, forming ammonia. N2 + 3H2 > 2NH3 ICSE Previous Papers 36 Formulae Handbook for ICSE Class 9 & 10 450 - 500 C 200 - 900 atmosphere pressure. Finely divided iron catalyst Molybdenum promoter. (ii) Reaction with Metals Hydrogen forms hydrides, (e.g. NaH) with a number of metals, including lithium, sodium and calcium. H2 + 2Na 2NaH These hydrides when pure are white salt-like compounds rapidly decomposed by water. NaH + H 2 O NaOH + H 2 Ca + H 2 CaH 2 CaH 2 + 2H 2 O Ca OH 2 + 2H 2 (iii) Reducing Properties When hydrogen is passed over many heated metallic oxides (e.g. copper oxide, iron oxide, or lead oxide), they are reduced to the metals. CuO + H 2 Cu + H 2O PbO + H 2 Pb + H 2O Fe2O3 + 3H 2 2Fe + 3H 2 O It is a strong reducing agent and used in metallurgy. Mathematics IX Rational And Irrational Numbers 1. For any two rational numbers a c and b d a c (i) ad = bc b d a c (ii) ad >bc b d a c (iii) ad <bc b d 2. Between any two rational numbers a and b, there exists another rational number ICSE Previous Papers 37 Formulae Handbook for ICSE Class 9 & 10 a+k and 2 a+b >b 2 a+b (ii) If a<b, a < <b 2 (i) If a > b, a > If 3. a and b are irrationals (i) a > b a>b (ii) a < b a<b (iii) b > a b 2 >a (iv) a > b a 2 >b Rationalising factor of: 4. a + b is a -b FUNDAMENTALS Natural Numbers: N = {1, 2, 3, 4, . } Whole Numbers: W = {0, 1, 2, 3, 4, .} Integers: Z = {.. . 3, 2, 1, 0, 1, 2, 3, .} Rational Numbers: A number that can be written in the form p , where p and q q are co-prime integers and q 0. Irrational Numbers: A number that cannot be written in the form p , where p and q q are co-prime integers and q 0. Real Numbers: R = All rational and all irrational numbers taken together. If x and y are any two rational numbers, then : i. x + y is a rational number ii. x y is a rational number iii. x y is a rational number iv. x r y is a rational number, (y 0) ICSE Previous Papers 38 Formulae Handbook for ICSE Class 9 & 10 Real Numbers Rational no. If remainder = 0 If remainder 0 & rem. = devidend If remainder 0 & rem. any devidend Terminating & Non repeating Non terminating & repeating (recurring) Non terminating non repeating Eg: 18 Eg: 3.6 5 Irrational no. 1 3 0.33.... .. If a is a real number, modulus a is written as |a|; |a| is always positive or zero. All natural number which cannot be divided by any number other than 1 and itself is called a prime number. A non-negative integer 'p' is said to be divided by an integer 'q' if there exists an integer 'd' such that: p = qd 1 divides every non-zero integer. 0 does not divide any integer. Identities related to square roots: 1. ab 2. Eg: 0.671234 .. a b a a b b a b a b a b 4. a b a b a b 5. a b c d ac 6. a b a 2 ab 3. 2 ad bc bd 2 ICSE Previous Papers 39 Formulae Handbook for ICSE Class 9 & 10 Important Rationalising factors: Laws of Exponents: 1. a r .a s a r s 2. a r a rs s ar 3. s a r s , r s a 4. a r b r ab 5. a r r 1 ar r s 6. a a r 1 s 1s a r r r a a 7. r b b a 8. b m b a m ICSE Previous Papers 40 Formulae Handbook for ICSE Class 9 & 10 Laws of Radicals: 1. 2. n n a m n a a n m a mn a 3. n a n b n ab , (a, b > 0, be real number) n 4. a na n b b p 5. p an am a n m p 6. a n a m a n m p 7. p p a n m a n m p Profit, Loss And Discount 1. 2. Profit = Selling Price Cost price Profit % = Profit 100 Cost Price 3. Loss = Cost Price Selling Price 4. Loss % = 5. In case of PROFIT: Loss 100 Cost Price 100 profit % Cost Price 100 100 (ii) Cost Price = Selling Price 100 profit % (i) Selling Price = 6. In case of LOSS: 100 loss % Cost Price 100 100 (ii) Cost Price = Selling Price 100 loss % (i) Selling Price = ICSE Previous Papers 41 Formulae Handbook for ICSE Class 9 & 10 7. When discount = d % Selling price = 8. 100 d Market price 100 In case of two successive discount d1 % and d2% 100 d1 100 d 2 Market Price 100 100 Selling Price = 9. In case of three successive discount d1 % and d2% and d3% 100 d1 100 d 2 100 d3 (i) Selling Price = Market Price 100 100 100 Compound Interest Principal Rate Time 100 1. Simple Interest = 2. Amount = Principal + Interest 3. Compound Interest = Final Amount Original Price 4. r A = P 1+ , 100 n 4. a + b = a 2 + 2ab + b2 2 a - b a 2 - 2ab + b2 a 2 - b 2 = a + b a - b 2 2 a + b a - b 2 a 2 + b2 5. a + b a - b 1. 2. 3. 5. 2 2 2 = 4ab Compound Interest = Amount Principal ICSE Previous Papers 42 Formulae Handbook for ICSE Class 9 & 10 Expansions 1. 2. 3. 4. 5. 6. a + b = a 2 + 2ab + b 2 2 a - b a 2 - 2ab + b 2 a 2 - b 2 = a + b a - b 2 2 a + b a - b 2 a 2 + b2 2 a + b a - b = 4ab 2 a + b + c a 2 b2 2 2 c 2 2ab + 2bc + 2ca 2 1 1 7. a + = a 2 2 2 a a 2 1 1 8. a 2 a + 2 a a 2 2 1 1 9. a - = a 2 + 2 - 2 a a 2 1 1 10. a + 2 = a - + 2 a a 2 2 2 1 1 1 11. a + + a - 2 a 2 - 2 a a a 2 2 1 1 12. a + - a - 4 a a 13. a + b a 3 + 3a 2 b + 3ab 2 + b 2 3 14. a 3 + b3 = a + b - 3ab a + b 3 15. a - b 3 = a 3 - 3a 2 b + 3ab 2 - b 2 16. a 3 - b3 = a - b +3ab a - b 3 3 1 1 1 17. a + = a 3 + 3 + 3 a + a a a 3 1 1 1 18. a 3 a + 3 a + a a a 3 3 1 1 1 19. a - = a 3 - 3 - 3 a - a a a 3 1 1 1 20. a - 3 = a - + 3 a - a a a 3 ICSE Previous Papers 43 Formulae Handbook for ICSE Class 9 & 10 1. a - b = a + b a - b 2 Factorisation 2 2. ab + bc + ax + cx = a + c b + x 3. a 3 + b3 = a + b a 2 ab + b 2 4. a 3 - b3 = a - b a 2 +ab + b 2 Change The Subject Of A Formula P R T 100 I 100 P= R T I 100 R= P T I 100 T= P R For eg. I = I = Interest R = Rate per annum T = Time P = Principal Amt. Graphical Solution 1. A linear equation in two variables is given as ax + by + c = 0 where, x and y are variables and a, b, c are constants 2. Equation of x-axis is y = 0 3. Equation of y-axis is x = 0 4. The graph x = a [a is a constant] is a straight line parallel to y-axis and at a distance of 'a' units from the y-axis 5. The graph y = b [b is a constant] is a straight line parallel to x-axis and at a distance of V units from the x-axis ICSE Previous Papers 44 Formulae Handbook for ICSE Class 9 & 10 Indices (Exponents) 1. a m a n = a m + n am 2. n = a m - n a 3. (a m ) n = a mn 4. (a b) m = a m b m n an a 5. n b b 6. If a 0 and n is positive integer 1 n n a= a 7. For a 0 m n a n am 8. For a 0 1 1 a n -n and a -n n a a 9. For a 0 a 1 Logarithms 1. For a b c, log c c = b 2. log a (m n) = log a m + log a n m log a m log a n n 4. log a (m) n n log a m 3. log a 5. log101 = 0 and log1010 1 6. log10100 = 2 ICSE Previous Papers 45 Formulae Handbook for ICSE Class 9 & 10 Rectilinear Figures 1. For a n-sided polygon Sum of interior angles = (2n - 4) x 90 2. For a regular polygon (2n - 4) 90o n Each interior angle 3. For a regular polygon of 'n' sides Each exterior angle 360o n Area (Plane Geometric Figures) 1 base height 2 1. Area of a triangle = 2. Area of a square = (side)2 3. Area of a Rectangle = length breadth 4. Area of a Parallelogram = base height Mean And Median 1. Mean 1 n x xi n i=1 2. Median (i) If n is odd, th n +1 Median = term 2 (ii) If n is even, th th 1 n n +1 Median = [value of term + value of term] 2 2 2 ICSE Previous Papers 46 Formulae Handbook for ICSE Class 9 & 10 Perimeter And Area FUNDAMENTALS: ICSE Previous Papers 47 Formulae Handbook for ICSE Class 9 & 10 Triangle 1 base height 2 1. Area = 2. When a, b, c are sides of a triangle and S is semi perimeter Area = 3. s(s-a) (s-b) (s-c) Area of an equilateral triangle 1 Area = Product of diagonals 4 4. 3 (side) 2 4 Area of an isosceles triangle where a = length of equal sides and b=length of base ICSE Previous Papers 48 Formulae Handbook for ICSE Class 9 & 10 Area = 1 b 4a 2 b 2 4 Rectangle 1. Area = (l b) 2. Perimeter = 2(l b) 3. Length of diagonal d = l 2 b2 Square 1. Area =(side)2 2. Perimeter =4 side 3. Diagonal d = Or Diagonal = 2 area 2 side Parallelogram 1. Area = Base Height Rhombus 1. Perimeter = 4 side 2. Area = 1 Product of diagonals 4 Trapezium 1. Area = 1 (sum of Parallel sides) (Distance between Parallel sides) 2 Circle 2 r or C= d 1. Circumference C= 2. Area = r 2 3. Distance travelled by a wheel or roller in one revolution = its circumference 4. No. of revolutions = Total distance circumference ICSE Previous Papers 49 Formulae Handbook for ICSE Class 9 & 10 Solids [Areas And Volumes] 1. Cuboid l b h (i) Volume = (ii) Total Surface Area = (iii) Diagonal = (iv) Area of four walls of a rectangular box or room = (v) If box is closed, International Dimensions Length = (l 2 x) 2(l b bh h l ) l 2 +b2 +h 2 2(l b) h Breadth = (b 2 x) Height = (h 2 x) If box is open, Length = (l 2 x) Breadth = (b 2 x) Height = (h x) Where x is the thickness of sides of a rectangular box. 2. 3. 4. Cube (i) Volume = (edge) a where a is the edge (ii) Total Surface Area = 6a 2 (iii) Length of diagonal = 3 3 3a Cylinder 2 rh (i) Curved Surface Area = (ii) Total Surface Area = 2 r(h + r) (iii) Volume = r 2 h Hollow Cylinder (i) Total Surface Area = 2 Rh + 2 rh + 2 (R r ) (ii) Volume of material = (R r )h 2 2 2 ICSE Previous Papers 2 50 Formulae Handbook for ICSE Class 9 & 10 FUNDAMENTALS: ICSE Previous Papers 51 Formulae Handbook for ICSE Class 9 & 10 If , b and h denote respectively the length, breadth and height of a cuboid, then (i) Total surface area of the cuboid = 2 ( b+bh+ h) square units. (ii) Volume of the cuboid = Area of the base height = bh cubic units. (iii) Diagonal of the cuboid or longest rod = 2 b 2 h 2 units. (iv) Area of four walls of a room = 2 ( + b) h sq. units. 2. If the length of each edge of a cube is a units, then(i) Total surface area of the cube = 6a2 sq. units. (ii) Volume of the cube = a3 cubic units (iii) Diagonal of the cube = 3 a units. 3. If r and h denote respectively the radius of the base and height of a right circular cylinder, then (i) Area of each end = r2 (ii) Curved surface area = 2 rh = (circumference) height (iii) Total surface area = 2 r (h + r) sq. units. (iv) Volume = r2h = Area of the base height ICSE Previous Papers 52 Formulae Handbook for ICSE Class 9 & 10 4. If R and r (R > r) denote respectively the external and internal radii of a hollow right circular cylinder, then (i) Area of each end = (R2 r2) (ii) Curved surface area of hollow cylinder = 2 (R + r) h (iii) Total surface area = 2 (R + r) (R + h r) (iv) Volume of material = h (R2 r2) 5. If r, h and denote respectively the radius of base, height and slant height of a right circular cone, then(i) 2 = r2 + h2 (ii) Curved surface area = r (iii) Total surface area = r2 + r (iv) Volume = 1 2 r h 3 6. For a sphere of radius r, we have (i) Surface area = 4 r2 (ii) Volume = 4 3 r 3 7. If h is the height, the slant height and r1 and r2 the radii of the circular bases of a frustum of a cone then (i) Volume of the frustum = (r 2 + r1r2 + r22) h 3 1 (ii) Lateral surface area = (r1 + r2) (iii) Total surface area = {(r1 + r2) + r12 + r22} h 2 (r1 r2 ) 2 (iv) Slant height of the frustum = (v) Height of the cone of which the frustum is a part = hr1 r1 r2 (vi) Slant height of the cone of which the frustum is a part = (vii) Volume of the frustum = r1 r1 r2 h A1 A 2 A1A 2 , where A1 and A2 denote the areas of 3 circular bases of the frustum. Tips: ICSE Previous Papers 53 Formulae Handbook for ICSE Class 9 & 10 Trigonometrical Ratios For any right angled triangle, 1. sin = Perpendicular Hypotenuse 2. cos = Base Hypotenuse 3. tan = Perpendicular Base 4. cot = Base Perpendicular 5. sec = Hypotenuse Base ICSE Previous Papers 54 Formulae Handbook for ICSE Class 9 & 10 Hypotenuse Perpendicular 6. cosec = 7. sin 2 + cos2 = 1 8. sec2 - tan 2 = 1 9. cosec2 - cot 2 = 1 Trigonomatric Ratios Of Standard Angles Solution Of Right Triangle 1. cos (90 ) = sin 2. sin (90 ) = cos Co-Ordinate Geometry 1. For x-axis and every line parallel to x-axis; inclination = 0 Slope (m) = tan = tan 0 = 0 2. For y-axis and every line parallel to y-axis; inclination = 90 Slope (m) = tan = tan 90 = (infinity) 3. Equation of line in the form y = mx + c Slope = m y-intercept = c ICSE Previous Papers 55 Formulae Handbook for ICSE Class 9 & 10 FUNDAMENTALS Two perpendicular lines (one horizontal and other vertical) are required to locate the position of a point or an object. The plane is called the Cartesian or coordinate plane and the lines are called coordinate axes or rectangular axes. Horizontal line is called the x-axis and vertical line is called the y-axis. The coordinate axes divide the plane into four parts called quadrants. About Cartesian Plane: 1. 2. 3. 4. 5. 6. 7. 8. Distance of any point from the y-axis is called x co-ordinate or abscissa. Distance of any point from the x-axis is called y co-ordinate or ordinate. Origin: (0, 0). Point on x-axis: (x, 0). Equation of x-axis is y = 0. Equation of y-axis is x = 0. Point on y-axis: (0, y) There are four quadrants in a co-ordinate plane: In the figure OX and OY are called as x-axis and y-axis respectively and both together are known as axes of co-ordinates. It has zero distance from both the axes so that its abscissa and ordinate are both zero. Therefore, the coordinates of origin are (0, 0). The axes divide the plane into four parts. These four parts are called quadrants. So, the plane consists of axes and quadrants. The plane is called the cartesian plane or the coordinate plane or the xy-plane. These axes are called the co-ordinate axes. A quadrant is 1 part of a plane divided by co-ordinate axes. 4 Rules of Signs of Co-ordinates: (i) In the first quadrant, both co-ordiantes i.e., abscissa and ordinate of a point are positive. (ii) In the second quadrant, for a point, abscissa is negative and ordinate is positive. (iii) In the third quadrant, for a point, both abscissa and ordinate are negative. (iv) In the fourth quadrant, for a point, the abscissa is positive and the ordinate is negative. ICSE Previous Papers 56 Formulae Handbook for ICSE Class 9 & 10 Y I II ( ,+) (+,+) X' III O IV ( , ) (+, ) X Y' Quadrant x-co-ordinate y-co-ordinate Point First quadrant + + (+,+) Second quadrant + ( ,+) Third quadrant ( , ) Fourth quadrant + (+, ) Distance Formula: Distance between two points A ( x1 , y1 ) and B ( x2 , y2 ) is: d ( x2 x1 )2 ( y2 y1 )2 Corollary: Distance of point A ( x, y ) from origin is: x2 y 2 Tips: Coordinates will form 1. 2. 3. 4. 5. 6. Rhombus, if all the four sides are equal. Square, if all the four sides and diagonals are equal. Parallelogram, if opposite sides are equal. Rectangle, if opposite sides and diagonals are equal. Right triangle, if it follows Pythagoras theorem. Colinearity condition. (Sum of two distances = Third distance.) Section Formula: Co-ordinates of the point P ( x, y ), dividing the line segment joining the points A ( x1 , y1 ) and B ( x2 , y2 ) internally in the ratio m : n are given by: x mx2 nx1 my ny1 ,y 2 m n m n How to remember the section formula? Corollary: If P ( x, y ) is the midpoint, therefore m : n = 1 : 1 ICSE Previous Papers 57 Formulae Handbook for ICSE Class 9 & 10 x x2 x1 y y ,y 2 1 2 2 Tips: If the ratio in which P divides AB is not given the then we take assumed ratio as k : 1. Centroid Formula: Co-ordinates of the centroid G ( x, y ) of triangle having vertices ( x1 , y1 ), ( x2 , y2 ) and ( x3 , y3 ) is given by x x1 x2 x3 y y y ,y 1 2 3 3 3 Area of Triangle: Area of triangle having vertices ( x1 , y1 ), ( x2 , y2 ) and ( x3 , y3 ) is given by: A 1 x1 ( y2 y3 ) x1 ( y2 y3 ) x3 ( y1 y2 ) 2 Tips: 1. To prove three points to be collinear, Area of Triangle = 0 i.e. x1 ( y2 y3 ) x1 ( y2 y3 ) x3 ( y1 y2 ) 0 2. Area of Quadrilateral ABCD = Area of ABC + Area of ACD. Or If A (x1, y1), B(x2, y2), C(x3, y3) and D(x4, y4) be the vertices of quadrilateral ABCD then area of the quadrilateral ABCD = 1 [x1y2 x2y1 + x2y3 x3y2 + x3y4 x4y3 + x4y1 x1y4] 2 Physics - X Force (i) F = ma ICSE Previous Papers 58 Formulae Handbook for ICSE Class 9 & 10 Where m is mass of a body and a is acceleration. (ii) W = F = mg mg is also considered as the weight of the body. (iii) Momentum = (p) = mv (iv) SI Units : (F) newton, CGS Units : F dyne, (p) Kg m/s P gm cm/s Machines (i) For an ideal machine; Input = Effort x Displacement of effort Output = Load x Displacement of load (ii) = Work output 100 Work input is efficiency of a machine (iii) M.A. = iv V.R= Load ; where M.A is mechanical advantage. Effort vE d E ; where V.R is velocity ratio. vL dL vE is velocity of effort vL is velocity of load. (v) For levers: Load Load arm = Effort Effort arm (at equilibrium) (vi) M.A for levers = Effort Load arm (vii) M.A = V.R. If = 100%, M.A = V.R. (viii) For an inclined plane: Load (W) W 1 s Effort (E) W sin sin h where s is length of slope and h is height of slope. d s V.R.= E dL h M.A. = ICSE Previous Papers 59 Formulae Handbook for ICSE Class 9 & 10 (ix) V.R=M.A + x E x is the weight due to movable parts of machine and E is effort required. Work, Power And Energy (i) W (work done) = f S (both acting in the same direction) (ii) W (work) = f x Scos is the angle of between f & S (iii) Work done = Energy W= mgh P.E.= mgh K.E.= 1 mv2 2 p2 P= 2m p is momentum Heat Energy = mc( ) p2 1 2 mgh = mv = = mc 2 2m (delta ( ) means change in temperature) (iv) Power(p) = = (v) Workdone Time f s t Equation of motion for accelerating bodies (a) v = u + at (b) v 2 - u 2 = 2as 1 (c) s = ut+ at 2 for deaccelerating bodies ICSE Previous Papers 60 Formulae Handbook for ICSE Class 9 & 10 (a) v = u - at (b) v2 - u2 = - 2as (c) s = ut - 1 2 at 2 for free falling bodies (a) a=g (b) v = u + gt (c) v2 - u2 = 2gs (d) s = ut + 1 2 gt 2 u = 0 (always) v = gt v2 = 2gs s= 1 2 gt 2 for a body thrown upwards: (a) a=-g. (b) v = u - gt (c) v2 - u2= - 2gs (d) s = ut - 1 2 gt 2 v = 0 (always) u = gt u2 = 2gs s = ut - 1 2 gt 2 where u = initial velocity; v = final velocity; a = acceleration; t = time and s = displacement. Refraction Through A Lens Key Points and Concepts: 1 1 1 Lens Formula : v u f ICSE Previous Papers 61 Formulae Handbook for ICSE Class 9 & 10 Linear Magnification: m= v h or (in lens) u h1 m=- v h or - (in mirror) u h1 h = Height of object h1 = Height of image v = Image distance u = Object distance Law of refraction of light: First Law: The incident ray, the normal to the transparent surface at the point of incidence and the refracted ray, all lie in one and the same plane. Second Law: The ratio of sine of angle of incidence to the sine of the angle of refraction is constant and is called refractive index of the second medium with respect to the first medium. sin i = sin r Sign Convention: Description: It is a convention which fixes the signs of different distances measured. The sign convention to be followed is the New Cartesian sign convention. It gives the following rules : 1. All distances are measured from the pole of the mirror. 2. The distances measured in the same direction as the direction of incident light from pole are taken as positive. 3. The distances measured in the direction opposite to the direction on incident light from pole are taken as negative. 4. Distances measured upward and perpendicular to the principal axis, are taken as positive. 5. Distances measured downward and perpendicular to the principal axis, are taken as negative. Sign Convention in Lens: Power of a lens P = 1 f (m) ICSE Previous Papers 62 Formulae Handbook for ICSE Class 9 & 10 Characteristics of the images formed in lenses: (i) Convex Lens Position of Nature of object image Type Position Size at F highly diminished At infinity real inverted Beyond 2F real inverted between F & 2F diminished At 2F real inverted at 2F same size as object Between F&2F real inverted beyond 2F magnified - - at infinity highly magnified virtual erect At F Between F&O (ii) on the same side of the lens as the magnified object Concave Lens Position of Nature of object image Type Position At infinity virtual inverted at F At F virtual inverted Within the focal length Size highly diminished diminished Diagrams for Convex Lens: (converging lens) (i) Object at infinity: Characteristics of image: (a) It is real. (b) It is inverted. (c) It is at F. ICSE Previous Papers 63 Formulae Handbook for ICSE Class 9 & 10 (d) It is highly diminished. (ii) Object beyond 2F: Characteristics of image: (a) It is real. (b) It is inverted. (c) It is between F and 2F. (d) It is diminished. (iii) Object at 2F Characteristics of image: (a) It is real. (b) It is inverted. (c) It is at 2F. (d) It is of the same size as the object. (iv) Object between F and 2F ICSE Previous Papers 64 Formulae Handbook for ICSE Class 9 & 10 Characteristics of image: (a) It is real. (b) It is inverted. (c) It is at 2F. (d) It is magnified (v) Object at F The refracted rays are parallel and hence it is assumed that the image is formed at infinity. In case the characteristics are asked: The image is real, inverted and highly magnified. (vi) Objects between F and O, the optic centre Characteristics of image: ICSE Previous Papers 65 Formulae Handbook for ICSE Class 9 & 10 (a) The image is virtual. (b) The image is erect. (c) The image is magnified (d) The image is on the same side of the lens as the object. Diagrams for concave lenses (diverging lens): (i) Object in front of the lens but not at infinity. Characteristics of image: (a) It is virtual. (b) It is erect. (c) It is within the focal length. (d) It is diminished. (ii) Objects at infinity: Characteristics of image: (a) It is virtual. ICSE Previous Papers 66 Formulae Handbook for ICSE Class 9 & 10 (b) It is erect. (c) It is at the focus. (d) It is highly diminished. Image Formation in Convex Lens Using Ray Diagrams ICSE Previous Papers 67 Formulae Handbook for ICSE Class 9 & 10 Image Formation in Concave Lens Using Ray Diagrams ICSE Previous Papers 68 Formulae Handbook for ICSE Class 9 & 10 Sound (i) v=f Where f is frequency; v is wave velocity and is wave length of the wave. (ii) fA B = fB A If velocity / medium is the same for two waves. (iii) v= d t ICSE Previous Papers 69 Formulae Handbook for ICSE Class 9 & 10 d= vt for echo. 2 (a) distance between the sound source & the observer must be atleast 17 m. (b) For hearing echo, sound after reliction in the medium should reach 1 after th of a second. 10 (c) The original energy must have sufficient energy. Current Electricity (i) R= 1 A Where is resistivity or specific resistance. R is resistance (R in ohm) l = Length of a wire (m) a = area of cross section of wire (m2) ohm metre (ii) Resistors in series combination ( n N) R = R1 + R2 + R3 + + Rn In series, current remains same, but potential difference is different. (iii) Resistors in parallel combination 1 1 1 1 1 = + + +.....+ R R1 R 2 R 2 Rn ( n N) In parallel, potential difference remains same, but current is different. (iv) V=IR (ohm s Law) where V=Potential difference (V in volt) I = current (I in ampere) R = resistance (ohm - ) (v) E = I(R+r) where E is electromotive force (emf) r is internal resistance R is the external circuit resistance. (vi) Q=I t ICSE Previous Papers 70 Formulae Handbook for ICSE Class 9 & 10 where Q is charge and t is time taken (Q in coulomb) V2 2 =I R (vii) Power (in elec.) = VI = R (viii) Electricity Energy [P is measured in watt(w)] = Heat energy =V Q = IR It = I2Rt V2t = R = VIt (ix) When a number of cells (of equal emf ) are connected in series; Total emf of all cells = number of cells emf of each cell (x) When a number of cells (of equal emf) are connected in parallel; Total emf of all cells = emf of one cell Key Points and Concepts Coulomb s Law: The force of attraction or repulsion between two point charges is (j) directly proportional to the product (q1 q2) of the two charges and (ii) inversely proportional to the square of distance (r) between them. Mathematically, The value of K depends on the nature of the medium between the two charges and the system of units chosen. For charges in vacuum K = 9 x 109Nm2/C2. The quantity of electric charge flowing through cross section of a given conductor in one second is called current. Thus, if Q is the charge which flows through a conductor in time t, then the current (I) is given by Current (I) = Charge (Q) Time( t ) Ohm s law: This law states that the current passing through a conductor is directly proportional to the potential difference across its ends, provided the physical conditions like temperature, density etc., remain unchanged. ICSE Previous Papers 71 Formulae Handbook for ICSE Class 9 & 10 Factors on which resistance of a conductor depends: The resistance R of a conductor depends on its length L, area of cross-section A and the nature of its material. It is given by The proportionality constant is called resistivity of the conductor. Joule s law of Heating: It states that the heat produced in a conductor is directly proportional to (i) the square of the current I through it, (ii) its resistance R and (iii) the time t, for which current is passed. Mathematically, it can be expressed as Quantities and Units: Quantities Charge S. I. Units Coulomb Electric Potential Difference Resistance Resistivity Current Ampere Volt Ohm Ohm metre Heat Electric Power Joule Watt ICSE Previous Papers 72 Formulae Handbook for ICSE Class 9 & 10 Schematic Diagram of an electric circuit: Flow of Current in Metal Metals show a very different kind of bonding called metallic bonding. According to this bonding, the outermost electrons are not bound to any particular atom, and move freely inside the metal randomly as shown in fig. So, these electrons are free electrons. These free electrons move freely in all the directions. Different electrons move in different directions and with different speeds. So there is no net movement of the electrons in any particular direction. As a result, there is no net flow of current in any particular direction. Metallic wire Electrons move randomly, so no flow of current Fig. Flow of electrons inside a metal wire when no potential is applied across its ends Electrons move from low potential to high potential +Battery High potential Low potential Fig. Flow of electrons inside a metal wire when the two ends of a wire are connected to the two terminals of a battery Classification of Material on Basis of Resistivity: Substances showing very low resistivity: The substances which show very low resistivities allow the flow of electric current through them. These type of substances are called conductors. For example, copper, gold, silver, aluminium and electrolytic solutions are conductors. Substances having moderate resistivity: The substances which have moderate resistivity offer appreciable resistance to the flow of electric current through them. Therefore, such substances are called resistors. For example, alloys such as nichrome, manganin, constantanand carbon are typical resistors. Substances having very high resistivity: The substances which have very high resistivities do not allow electricity to flow through them. The substances which do not allow ICSE Previous Papers 73 Formulae Handbook for ICSE Class 9 & 10 electricity to pass through them are called insulators. For example, rubber, plastics, dry wood, etc. are insulators. Combination of Resistances: When two or more resistances are joined end-to-end so that the same current flows through each of them, they are said to be connected in series. When a series combination of resistances is connected to a battery, the same current (I) flows through each of them. I R1 R2 R3 I Key + Battery When a series combination of resistances is connected to a battery, the same current (I) flows through each of them. Law of combination of resistances in series: The law of combination of resistances in series states that when a number of resistances are connected in series, their equivalent resistance is equal to the sum of the individual resistances. Thus, if R1, R2, R3 ..., etc. are combined in series, then the equivalent resistance (R) is given by, R = R1 + R2 + R3 + ... .... (i) Derivation of mathematical expression of resistances in series combination: Let, R1, R2 and R3 be the resistances connected in series, I be the current flowing through the circuit, i.e., passing through each resistance, and V1, V2 and V3 be the potential difference across R1, R2 and R3, respectively. Then, from Ohm s law, V1 = IR1, V2 = IR2 and V3 = IR3 ... (ii) If, V is the potential difference across the combination of resistances then, V = V1 + V2 + V3 If, R is the equivalent resistance of the circuit, then V = IR ... (iii) ... (iv) Using Equations (i) to (iv) we can write, IR = V = V1 + V2 + V3 = IR1 + IR2 + IR3 or, or, IR = I (R1 + R2 + R3) R = R1 + R2 + R3 Therefore, when resistances are combined in series, the equivalent resistance is higher than each individual resistance. Parallel Combination: When two or more resistances are connected between two common points so that the same potential difference is applied across each of them, they are said to be connected is parallel. ICSE Previous Papers 74 Formulae Handbook for ICSE Class 9 & 10 R1 I I1 I2 I R2 When such a combination of resistance is connected to a battery, all the resistances have the same potential difference across their ends. Derivation of mathematical expression of parallel combination: Let, V be the potential difference across the two common points A and B. Then, from Ohm s law Current passing through R1, I1 = V/R1 ... (i) Current passing through R2, I2 = V/R2 ... (ii) Current passing through R3, I3 = V/R3 ... (iii) If R is the equivalent resistance, then from Ohm s law, the total current flowing through the circuit is given by, and I = V/R ... (iv) I = I1 + I2 + I3 ... (v) Substituting the values of I, I1, I2 and I3 in Eq. (v), V V V V = + + R3 R2 R1 R ... (vi) Cancelling common V term, one gets 1 1 1 1 = + + R3 R2 R1 R The equivalent resistance of a parallel combination of resistance is less than each of all the individual resistances. Components of electric circuit: S.No Components 1 An electric cell 2 A battery or a combination of cells 3 Plug key or switch (open) 4 Plug key or switch (closed) ICSE Previous Papers Symbols 75 Formulae Handbook for ICSE Class 9 & 10 5 A wire joint 6 Wires crossing without joining 7 Electric bulb 8 A resistor of resistance 9 Variable resistance or rheostat 10 Ammeter 11 Voltmeter Electric Power And Household Electricity (i) Electrical Energy = Heat Energy = I2Rt = V2t R = VIt = Power Time SI unit of electrical energy is joule. (ii) Power at home is in Kw (iii) 1kW = 1000 watts (iv) Energy consumed = kWh = 1kWh is the energy used when 1kW of power is used for 1 hour. (v) To find the energy in kWh Power of bulb in watt time for which it is used in hour 1000 ICSE Previous Papers 76 Formulae Handbook for ICSE Class 9 & 10 This gives the no. of units consumed. To calculate the cost; Cost = Number of units cost/unit Electromagnetism For transformers; Output Potential differnce (Vs) o.of turns in secondary Input Potential differnce (Vp) o.of turns in primary current in primary = current in secondasry Vs Ns Ip Vp Np Is Key Points and Concepts Magnetism: The property due to which a substance attracts iron pieces towards it, is called magnetism. The substance having property of magnetism, is called magnet. Right Hand Thumb Rule: Hold the wire carrying current in your right hand, such that the thumb indicates the direction of current, then the folded fingers will indicate the presence of magnetic field (lines) surrounding the wire. Fleming s Left Hand Rule: Fleming s left hand rule is used to find out the direction of motion of a current carrying conductor when placed in a magnetic field. This rule states as follows. ICSE Previous Papers 77 Formulae Handbook for ICSE Class 9 & 10 Stretch out the thumb, the forefinger, and the second (middle) finger of the left hand so that these are at right angles to each other. If the forefinger gives the direction of the magnetic field (N to S), the second (middle) finger the direction of current (+ to ), then the thumb gives the direction of the force acting on the conductor. Since the conductor will move in the direction of the force acting on it hence the thumb gives the direction of motion of the conductor. Faraday s Law: The rate at which the magnetic flux linked with a coil changes, produces the induced emf or current. More the rate, more the current and vice-versa. Properties of Magnetic Lines of Force: N S 1. 2. 3. 4. They start from a north (positive) pole and end at a south (negative) pole. Two lines of force do not intersect each other. They tend to contract longitudinally (longitudinal contraction). They tend to expand laterally (lateral repulsion) so as to exert lateral pressure on neighboring lines. 5. (The above two properties are similar to that of a stretched rubber band). 6. The number of magnetic lines of force passing normally per unit area about a point, gives the intensity of the magnetic field at the point. Electric Motor and Generator: S.No 1 2 Electric Motor Motor converts electrical energy into mechanical energy. Generator Converts mechanical energy to electrical energy. Works on the principle of Fleming s left hand rule. Works on the principle of Fleming s right hand rule. Uniform Magnetic Field: ICSE Previous Papers 78 Formulae Handbook for ICSE Class 9 & 10 Magnetic Lines around a bar magnet: Magnetic Lines due to a current carrying loop: Magnetic field in a solenoid: Magnetic Lines around two magnets: ICSE Previous Papers 79 Formulae Handbook for ICSE Class 9 & 10 Magnetic Lines Produced Around a current carrying conductor: Common Domestic Circuit: Points to Be Remember: A freely suspended magnet always stays north south. The region around a magnet in which its magnetic force can be detected is called the magnetic field. Magnetic field is represented by magnetic field lines. The tangent at any point of a magnetic field line represents the direction of magnetic field at the point. The number of lines of force passing through a unit area represents the strength of the field. If the lines of force are closer, the magnetic field is stronger. The magnetic field lines around a current carrying straight conductor are concentric circle around the conductor. The direction of magnetic field due to a current carrying straight conductor is given by Fleming s right hand rule. The magnetic field due to a current carrying solenoid is similar to that of a bar magnet. The magnetic field inside a solenoid is nearly uniform and is parallel to the axis of the solenoid. A magnet formed due to the magnetic field of a current is called an electromagnet. An electromagnet essentially consists of a soft iron core wrapped around with an insulated copper wire coil. An electric motor is a device that converts electrical energy into mechanical energy. It is based on the principle that: When a current carrying coil is placed in a magnetic field, a torque acts on it. When a conductor moves perpendicular to a magnetic field, an emf is induced across its ends. The direction of induced emf or the induced current is determined by the Fleming s right hand rule. ICSE Previous Papers 80 Formulae Handbook for ICSE Class 9 & 10 Generator is based on the principle of electromagnetic field, with a continuous change in flux due to which an emf is induced. Power is transmitted from the power station to cities at high voltage and low current to minimize power loss. A fuse is a wire of high resistance and made up of a material of low melting point. Calorimetry (i) Heat energy = mc t c= Q Q or m m t where m is mass of the body (kg) c is specific heat capacity (J/kg K or J/Kg C ) is change in temperature. (ii) Heat energy = mL (L is the specific latent heat J/kg) (when there is change of state at its melting or boiling point of substances) (iii) Heat lost by hot body or bodies = heat gained by cold body or bodies, If there is no exchange of heat within surrounding, i.e. mc (of hot body) = mc (of cold body) Specific heat capacity of water = 4200 J kg 1K 1 (iv) = 4.2 Jg 1K 1 = 1 kcal kg 1K 1 Specific heat capacity of ice = 2100 J kg 1K 1 (v) = 2.1 Jg 1K 1 = 0.5 kcal kg 1K 1 Specific latent heat of fusion ice = 336000 J kg 1K 1 (vi) = 336 Jg 1K 1 = 80 kcal kg 1K 1 Specific latent heat of vaporisation of steam = 2268 J kg 1 (vii) = 2268 Jg 1 = 540 cal g 1 ICSE Previous Papers 81 Formulae Handbook for ICSE Class 9 & 10 Modern Physics (i) emission: atomic number reduces by 2 and atomic mass number reduces by 4. (ii) emission: mass number is unaffected but atomic number increases by 1. Chemistry -X Acids, Bases & Salts Key Points and Concepts Those substances which turn blue litmus solution red are called acids. Acids are sour in taste. They give H+ ions in aqueous solution. Example: HCI, H,SO4, HNO3, CH3COOH Those acids which dissociates into ions completely are called strong acids, e.g. H2S04, HCl. Those acids which do not dissociates into ions completely are called weak acids, e.g., citric acid, acetic acid. The reaction in which base or basic oxide reacts with acid or acidic oxide is called neutralization reaction. Example: NaOH(aq) + HCI(aq) > NaCl(aq) + H20 Hydrated salts which are white in colour: Reaction of different solutions with different indicators: S.No Name of the solution Colour change (if any) Phenolphthalein Colour change (if any) Blue litmus 1 Sodium carbonate tunis pink no change 2 Hydrochloric acid no change turns red 3 Sodium chloride no change no change Important Equations: Baking Soda The chemical formula of baking soda is NaHCO3. It is prepared by using sodium chloride. ICSE Previous Papers 82 Formulae Handbook for ICSE Class 9 & 10 On heating: Green Vitriol: Plaster of Paris: Washing Soda and its Properties: Chemical name is Sodium carbonate hydrated It is a basic salt because when dissolved in water it gives a strong base NaOH. It is used as a cleaning agent, in paper and glass industry. Bleaching Powder and its Properties: The common name of CaOCI2 is bleaching powder. By passing chlorine into dry slaked lime Ca(OH)2, bleaching powder is obtained. Two uses: (i) Used for bleaching cotton and linen in the textile industry and wood pulp, paper industry etc. (ii) It is used for disinfecting drinking water. Some of the naturally occurring substances that contain acids are given in Table. ICSE Previous Papers 83 Formulae Handbook for ICSE Class 9 & 10 Substance Acid present 1. Orange, lemon Citric acid, ascorbic acid (vitamin C) 2. Apple Malic acid 3. Tamarind (imli), Tartaric acid grape 4. Vinegar Acetic acid 5. Curd Lactic acid 6. Tomato Oxalic acid 7. Gastric juice Hydrochloric acid 8. Tea Tannic acid 9. Red ants Formic acid Examples of a few acids with their basicities are given in the table below. Acid Basicity HCl 1 HNO3 1 H2SO4 2 H3PO4 3 The following table shows the uses of some organic and inorganic acids. Acid Uses Organic acids Citric acid Ascorbic acid (also called vitamin C) Acetic acid Tartaric acid 1. As a preservative for food 2. As a flavouring agent In the treatment of bone marrow and scurvy diseases Added to pickles to make them sour A component of baking power (baking powder is a mixture of sodium hydrogencarbonate and tartaric acid) Inorganic acids 1. Its presence in the gastric juice helps digestion of food we eat. Hydrochloric acid 2. As a bathroom cleaner 3. In the manufacture of polyvinyl chloride (PVC) Nitric acid 1. Nitric acid present in rainwater forms nitrates in the soil which are then used by plants to obtain nitrogen. ICSE Previous Papers 84 Formulae Handbook for ICSE Class 9 & 10 2. In the manufacture of fertilizers like ammonium nitrate 3. In making explosives like TNT and dynamite 1. In storage batteries Sulphuric acid 2. In the manufacture of fertilizers, paints and pigments, detergents and artificial fibre 3. In the manufacture of hydrochloric acid and alum Phosphric acid In fertilizer and detergent industries 1. In the manufacture of glass, glazes and enamels, leather, paper, adhesives and explosives Boric acid 2. Widely used in detergents 3. As a grain preservative Setup which shows acid solution in water conducts electricity: Some of the bases are listed here in Table. Oxides Soluble hydroxides Insoluble hydroxides Sodium monoxide (Na2O) Sodium hydroxide (NaOH) Ferric hydroxide (Fe(OH)3). Calcium oxide (CaO) Potassium hydroxide (KOH) Aluminium hydroxide (Al(OH)3) Cupric oxide (CuO) ZnO Calcium hydroxide (Ca(OH)2) Ammonium hydroxide NH4OH ICSE Previous Papers 85 Formulae Handbook for ICSE Class 9 & 10 Table lists some of the common bases and their uses. Bases Uses Sodium hydroxide Ammonium hydroxide Calcium hydroxide 1. In the manuacture of soaps, textile, paper, medicines 2. In the refining of petroleum 1. As a reagent in the laboratory 2. In making fertilizers, rayon, plastics and 1. 2. 3. 4. dyes In making cement and mortar In making bleaching powder In whitewashing In removing acidity of soils Some of the most commonly used acid-base indicators that change colour as follows. Indicator Acid solution Basic solution Neutral solution Blue litmus solution Red No change in colour No change in colour Red litmus solution No change in colour Blue No change in colour Methyl orange Phenolphthalein Red Colourless Yellow Red Orange Colourless The following table lists uses of some salts. Salts Uses Sodium chloride 1. 2. 3. 4. 5. An essential requirement of our food In the preservation of food In curing fish and meat In making a freezing mixture which is used by icecream vendors In the manufacture of soaps Sodium carbonate 1. 2. 3. 4. As washing soda for cleaning clothes Used in the manufacture of glass, paper, textiles, caustic soda, etc. In the refining of petroleum In fire extinguishers Sodium bicarbonate Potassium nitrate Copper sulphate 1. Used as baking soda 2. In fire extinguishers 3. As an antacid in medicine 1. To make gunpowder, fireworks and glass 2. As a fertilizer in agriculture 1. Commonly called 'blue vitriol', used as a fungicide to kill certain germs 2. In electroplating 3. In dyeing 1. Used to purify water; makes suspended particles in water settle down Potash alum 2. As an antiseptic 3. In dyeing ICSE Previous Papers 86 Formulae Handbook for ICSE Class 9 & 10 pH Indicator: The acidity or basicity (alkalinity) of a solution is usually expressed in terms of a function of the H+ ion concentration. This function is called the pH of a solution. The pH of an aqueous solution is the negative logarithm of its H+ ion concentration. That is, pH = log [H+]. pOH = log [OH ]. The pH of a neutral solution is 7. The pH of an acidic solution is less than 7. The pH of an alkaline solution is more than 7. Rules for pH scale (at 298 K) 1. Acidic solutions have pH less than 7. 2. The lower the pH, the more acidic is the solution. 3. Neutral solutions or pure water has pH equal to 7. 4. Basic solutions have pH greater than 7. 5. The higher the pH, the more basic is the solution. The pH values of some common solutions: Substance pH Gastric juice 1.0 Lemon juice 2.5 Vinegar 3.0 Wine 3.5 Tomato juice 4.1 Acid rain 5.6 Urine 6.0 Milk 6.5 Pure water 7 Blood 7.4 Lime water 11.0 ICSE Previous Papers 87 Formulae Handbook for ICSE Class 9 & 10 Point to Remember: Everything that tastes sour contains an acid. Acetic acid, citric acid, tartaric acid are a few organic acids. Sulphuric acid, nitric acid and hydrochloric acid are examples of inorganic acids. Acids turn blue litmus red, whereas bases turn red litmus blue. When a solution of an acid contains larger amount of the acid, it is said to be concentrated, while that containing smaller amount of the acid, is said to be dilute. Metals like sodium, potassium and calcium react with an acid to liberate hydrogen gas. Acids react with bases to produce salts and water. Acids react with the carbonates and the hydrogencarbonates to give carbon dioxide gas. The hydrogen atoms of an acid which can be partially or completely replaced by an atom or a group of atoms are called replaceable hydrogen atoms. They are also called acidic hydrogen. The number of replaceable hydrogen atoms present in a molecule of the acid is known as the basicity of the acid. A compound that reacts with an acid to form a salt and water is called a base. Bases that are soluble in water are called alkalis. All alkalis are bases, but all bases are not alkalis. The reaction between an acid and a base is called neutralization reaction. In such a reaction, the acid and the base destroy the properties of each other. The number of hydroxyl groups (OH) present in a molecule of the base is called the acidity of the base. A salt is a compound formed by the reaction of an acid with a base. Na2SO4, CaSO4 and Na3PO4 are normal salts, whereas NaHSO4, NaHCO3, Na2HPO4 are acid salts. A strong acid is one which gets almost completely dissociated when dissolved in water to give hydrogen ions, whereas a weak acid gets only partially dissociated in water to give hydrogen ions. A strong base gets almost completely dissociated when dissolved in water to give hydroxide ions (OH ), whereas a weak base, when treated as such, gets only partially dissociated to provide hydroxide ions. PREPARATION OF ACIDS: (a) (b) Direct combination/synthesis: used to prepare hydra acids H 2 Cl2 2Hcl H2 F2 2HF H 2 l2 2HI H 2 S H 2S Starting non metal (C, P, S) Non metals combine with O2 to form acidic oxides (acid anhydride). Oxides dissolves in H2O to form their resp. acids. ICSE Previous Papers 88 Formulae Handbook for ICSE Class 9 & 10 C + O2 CO2 S + O2 SO2 2P2 5O 2 2P2O5 acid solid CO2 H 2O H 2CO3 SO2 H 2O H 2SO3 P2O5 3H 2O 2H3PO4 carbonic acid sulphurous acid phospheric acid Note: NO2 is called a mixed acid anhydride NO2 dissolves in water to produce nitrous acid and nitric acid (2 acids) 2NO2 + H2O HNO2 + HNO3. It is called mixed acid anhydride (c) Action of non volatile acid (conc. H2SO4) on salt of volatic acid (NaCl, KCl) 220 C NaNO3 H 2SO 4 NaHSO 4 HNO3 220 C KNO3 H 2SO 4 KHSO 4 HNO3 220 C KCL H 2SO 4 KHSO 4 HCL 220 C NaCL H 2SO 4 NaHSO 4 HCL (d) salt of non volatile volatile volatile acid Oxidation of non metals (P, S) with cone. HNO3- volatile acid P + 5HNO3 H 3 PO 4 H 2 O + 5NO 2 S + 6HNO3 H 2SO 4 2H 2O + 6NO 2 Note: Acidic oxides are called acid anhydrides. They are oxides of non metals which dissolve in water in to form acids. SO 2 H 2 O H 2SO3 P2 O5 3H 2 O 2H 3PO 4 SO3 H 2 O H 2SO 4 2NO 2 H 2O HNO 2 HNO3 CO 2 H 2 O H 2CO3 Acid anhydrides are acids without water H 2 CO3 H 2 O CO 2 H 2SO3 H 2 O SO 2 H 2SO 4 H 2O SO3 (acid anhydride) 2H3 PO 4 3H 2 O P2 O5 (acidic solid) acidic acid PREPARATION OF BASES: (a) Metals + O2 Metallic oxides (Bases) ICSE Previous Papers 89 Formulae Handbook for ICSE Class 9 & 10 (b) 4Na +O2 2Na 2O 2Cu + O2 2CuO 4K + O2 2K 2O 2Ca + O2 2CaO Basic oxides dissolve in water to form alkalis K 2O + H 2 O 2KOH CaO + H2 O Ca(OH)2 Na 2O + H 2O 2NaOH NH3 + H 2 O NH 4 OH (non metallic base) (c) Thermal Decomposition of carbonates and nitrates (i) Carbonates: Na 2 CO3 , K 2 CO3 do not decompose on heating CaCO3 CaO + CO 2 MgCO3 MgO+CO 2 CuCO3 CuO + CO 2 FeCO3 FeO CO 2 (ii) Nitrate of heavy metals on heating form bases except KNO3 , NaNO3 2Ca(NO3 ) 2 2CaO + 4NO 2 O 2 2Cu(NO3 ) 2 2CuO + 4NO 2 O 2 2Zn(NO3 ) 2 2ZnO + 4NO 2 O 2 (d) Active Metal + H2O alkali + H2 2K + 2H 2O 2KOH + H 2 2Na + 2H 2O 2NaOH + H 2 Ca + 2H 2O Ca(OH)2 + H 2 (e) By precipitation of 2 solutions 3NaOH + FeCl3 3NaCl + Fe(OH)3 2NaOH + CuSO4 Cu(OH) 2 + Na 2SO4 2NaOH + FeSO4 Fe(OH)2 + Na 2SO4 PREPARATION OF SALTS (V.Imp) (a) Direct Combination: Synthesis (i) Chlorides which are soluble but AgCl3, PbCl2 (ii) Sulphides which are insoluble ICSE Previous Papers 90 Formulae Handbook for ICSE Class 9 & 10 Mg + Cl2 MgCl2 2Fe + 3Cl2 2FeCl3 (ferric chloride is a coloured deliquescent (salt) 2Al + 3Cl2 2AlCl3 2Ag + Cl2 2AgCl NnS FeS PbS Zn + S Fe + S Pb + S (b) (insoluble) (insoluble) Simple Displacement (Generally sulphates) All SO4 are soluble but CaSO4, BaSO4, PbSO4 which is insoluble (salt of active metals prepared) Fe + H 2SO 4 FeSO4 H 2 [Green vitriol - FeSO4 .7H 2 O] Zn + H SO 4 ZnSO 4 + H 2 [White vitriol - ZnSO4 .7H 2 O] Mg + H 2SO4 MgSO4 + H 2 [Epson salt - MgSO4 .7H 2 O] Mg + 2HCl MgCl2 + H 2 Mg + 2HNO3 Mg(NO3 )2 H 2 (c) Neutralisation: (Acid + Alkali) Soluble salts - K, Na, NH 4 + (i) Titration: alkali + acid salt + water NaOH + HCl NaCl + H 2 O NH 4 OH + HCl NH 4 Cl + H 2 O 2NaOH + H 2SO4 Na 2SO 4 2H 2O KOH + HCl (ii) KCl + H 2 O Insoluble base + acid soluble salt + H2O CuO + H 2SO 4 CuSO 4 + H 2 O Cu(OH) 2 + H 2SO 4 CuSO 4 + 2H 2O PbO + 2HNO3 Pb(NO3 ) 2 + H 2O Pb(OH) 2 2HNO3 Pb(NO3 ) 2 + 2H 2O Imp: pb salts are only prepared from lead nitrate ICSE Previous Papers 91 Formulae Handbook for ICSE Class 9 & 10 (d) Decomposition of insoluble Carbonates CuCO3 + H 2SO4 CuSO4 + H 2O + CO2 PbCO3 + 2HNO3 Pb(NO3 ) 2 + H 2O + CO2 (e) Insoluble salts by precipitation or double decomposition Two salt solutions taken. Both salts are soluble in water. Product: one insoluble salt - ppt second soluble salt in solution. (CaSO4, Ba SO4, PbSO4, AgCl, ZnCO3) Carbonates are prepared by double decomposition from sodium carbonate solution. ZnSO 4 (NH 4 ) 2 CO3 ZnCO3 + (NH 4 ) 2 SO 4 CaCl2 + K 2SO 4 CaCl2 + Na 2CO3 CaSO 4 + 2KCl CaCO3 + 2NaCl Note: Pb salts from Pb(NO3 ) 2 Note: Carbonates from Na 2CO3 AgNO3 + NaCl AgCl + NaNO3 Imp. To prepare an insoluble salt (PbCO3) from another insoluble salt PbO 1st covert to nitrate. PbCO3 2HNO3 Pb(NO3 )2 H 2O + CO 2 Pb(NO3 )2 Na 2 CO3 2NaNO3 + PbCO3 Pb(NO3 )2 Na 2SO 4 2NaNO3 + PbSO4 To lead carbonate if H 2SO 4 is to added it forms an insoluble sulphate of PbSO4 on PbCO3 . The reaction will stop. Analytical Chemistry To determine cations by: (i) Flame test (ii) Action of alkalies Zn/Al/Pb reacts with alkalis to form corresponding zincate [ZnO2 2 ], Aluminate AlO2 1 and plumbates PbO2 ICSE Previous Papers 92 Formulae Handbook for ICSE Class 9 & 10 Zn + 2NaOH Na 2 ZnO2 + H 2 Pb + 2NaOH Na 2 PbO2 + H 2 2Al + 2NaOH + 2H 2 O 2NaAlO2 + 3H 2 (V .imp) Oxides ZnO + 2NaOH Na 2 ZnO2 + H 2 O PbO + 2NaOH Na 2 PbO2 + H 2 O Al2O3 + 2NaOH 2NaAlO2 + H 2 O Hydroxides Zn(OH)2 + 2NaOH Na 2 ZnO2 + H 2O Pb(OH)2 + 2NaOH Na 2 PbO2 + H 2O Al(OH)3 + NaOH NaAlO2 + 2H 2 O Note: Note: (a) Alkalis + salt solution insoluble coloured precipitates. e.g. NaOH reacts with CuSO4 to form a blue precipitate (incorrect) NaOH solution reacts with CuSO4 solution to form a blue precipitate (correct) If the word solution is not there, the sentence is incorrect. (b) Both observations - drop by drop and explanation are to be given. Note: ppt soluble in NH4OH: Cu(OH)2, Zn(OH)2 ppt insoluble in NH4OH: Fe(OH)2, Fe(OH)3, Pb(OH)2 ICSE Previous Papers 93 Formulae Handbook for ICSE Class 9 & 10 Note: Ammonium salts react with alkalis to liberate ammonia. 2NH4Cl + Ca(OH)2 CaCl2+2H20 + 2NH3 Soluble ppts - Zn(OH)2, Pb(OH)2 Insoluble ppts - Fe(OH)2, Fe(OH)3, Cu(OH)2, Ca(OH)2, ICSE Previous Papers 94 Formulae Handbook for ICSE Class 9 & 10 Metallurgy (a) Reduction of metal oxides K Na Ca Reduced only by electrolysis in molten state Mg Al Zn Zinc oxide ONLY by COKE. Fe Pb Oxides are reduced by CO, C, H2 Cu (b) Metals + dil acid salt + H2 K Na Ca Mg Violent reaction. Liberate H2 Liberate H2 from dilute acid. Mg + 2HCl MgCl2 + H2c Al Zn Zn+ 2HCl > ZnCl2 + H2 Fe Fe + H2SO4 > FeSO4 + H2 Pb Cu H Do not liberate H2 Hg Ag Au Pt ICSE Previous Papers 95 Formulae Handbook for ICSE Class 9 & 10 (c) Action of on oxides All oxides are stable to but higher oxides, eg. Pb3O4, PbO2 2HgO 2Hg + O2 2Ag2O 4AgO + O2 K Combine with 02 at ordinary temp Na 4K + O2 2K2O Ca Heated metal + steam oxide + H2 Mg Al 4Al + 3O2 2Al2O3 Zn 2Zn + O2 2ZnO Fe 3Fe + 2O2 Fe3O4 Pb H 2Cu + o2 2CuO Cu Hg Ag Au Do not form oxides Pt (d) Action of on Hydroxides KOH NaOH Stable to heat Ca(OH)2 Metal oxide + H2O Mg(OH)2 2Al(OH)3 Zn(OH)2 Al2O3 + 3H2O ZnO + H2O ICSE Previous Papers 96 Formulae Handbook for ICSE Class 9 & 10 Fe(OH)2 FeO + H2O Pb(OH)2 PbO + H2O Cu(OH)2 CuO + H2O Hg(OH)2 Metal + O2 + H2O AgOH (e) Action of on Carbonates K2CO3 Stable to heat Na2CO3 CaCO3 Metal Oxide + CO2 MgCO3 ZnCO3 ZnO + CO2 PbCO3 PbO + CO2 CuCO3 CuO + CO2 FeCO3 FeO + CO2 Hg(OH)2 Ag2CO3 (f) Metal + O2 + CO2 Action of A on nitrates 2KNO3 2NaNO3 2KNO2 + O2 2NaNO2 + O2 2Ca(NO3)2 2Mg(NO3)2 2Zn (NO3)2 2ZnO + 4NO2 + O2 2Pb (NO3)2 2PbO + 4NO2 + O2 2Cu(NO3)2 2 metal oxide + 4NO2 + O2 2CuO +4NO2 + O2 ICSE Previous Papers 97 Formulae Handbook for ICSE Class 9 & 10 Extraction of zinc Ore: Zinc Blend ZnS/Calamine ZnCO3 (i) Roasting: 2ZnS + 3O2 2ZnO + 2SO2 (ii) Calcination: ZnCO3 ZnO + CO2 (iii) Reduction: ZnO + C Zn + CO (By coke - reducing agent) Extraction of Iron Blast Furnace: Main ore: Haematite Fe2O3 Lower region: C + O2 CO2 Middle region: (i) CaCO3 CaO + O2 (ii) (flux) CaO + SiO2 CaSiO2 Gangue slag (iii) CO2 + C > 2CO (reducing agent) Upper region: Fe203 + 3CO 2Fe + 3C02 Extraction of Aluminium (V.Imp) (a) Concentration of the ore - Baeyer s Process <200 C Bauxite: Al2O3 .2H2O + 2NaOH 2NaA1O2 + 3H2O Al(OH)3 NaOH + A1 OH 3 (seeding) NaA1O2 + 2H2O 50 to 60 C 1100 C 2A1 OH 3 A12O3 + 3H2O ICSE Previous Papers 98 Formulae Handbook for ICSE Class 9 & 10 (b) Reduction of Alumina - Hall and Heroult s Process Electrolytic dissociation of electrolyte Na 3AlF6 3Na + +Al+++ +6F ca ++ +2F- CaF2 A12 O3 2A1+++ +3O- Anode reaction Cathode reaction O-- 2e (O) Al+++ + 3e Al (pure Al) (O) + (O) O2 C + O2 CO2 Thermite Welding MnO2 + 4HC1 MnCl2 + 2H 2 0 + Cl2 ii Pb02 + 4HC1 PbCl2 + 2H20 + Cl, | Fe2O3 2Al Al2O3 2Fe Study Of Compounds Hydrochloric Acid Preparation of Hydrochloric Acid: (a) Direct Synthesis: Moisture H2 Cl2 2HCl Diffused sunlight (b) Laboratory Preparation: <200 C NaCl + H2SO4 NaHSO4 + HCl Above 200 C not used 2NaCl + H2S04 Na 2SO4 + 2HCI (c) (d) Drying agent not used: (i) Quicklime: CaO + 2HCl CaCl2 + H2O (ii) 2P2 05 + 3HC1 POC13 , + 3HPO3 , P2O3 - acidic solid Thermal dissociation: ICSE Previous Papers 99 Formulae Handbook for ICSE Class 9 & 10 500 C 2HC1 H2 + G12 (e) NH3(g) + HCl g NH4C1 solid (f) Basicity HC1+ H2 0 H3O + Cl Monobatic (g) Acidic Properties - Typical Acid (i) Active Metal + dil acid salt + H2 Mg + 2HC1 MgCl2 + H2 Fe + 2HC1 FeCl2 + H2 (ii) Base + Acid salt + H20 2HC1 + CuO CuCl2 + H 2 0 HC1 + NaOH NaCl + H 2 0 (iii) Carbonates/Bicarbonates + dil acid salt + H2O + CO2 Na 2CO3 + 2HC1 2NaCl + H 2O + CO2 KHC03 + HC1 KC1 + H 2O + CO2 (iv) Metal sulphites/bisulphites + dil acid salt + H2O+ SO2 K 2S03 + 2HC1 2KC1 + H 2O + SO2 NaHSO3 + HC1 NaCl + H 2O + SO2 (v) Metal sulphide + dil HCl salt + H2S ZnS + 2HC1 ZnCl2 + H 2S FeS + 2HC1 FeCl2 + H 2S (h) Preparation of two insoluble Chloride [PbCl2, AgCl) Pb NO3 2 + 2HC1 PbCl2 + 2HNO3 soluble in hot water AgNO3 + HC1 AgCl + HNO3 soluble in NH 4OH (i) Sodium thiosulphate (Imp) Na 2S2O3 + 2HC1 2NaCl + SO2 + S + H2 0 (j) Cone. HCl - Reducing Agent [Oxidising agents liberate Cl2] (i) MnO2 + 4HC1 MnCl2 + 2H2 0 + Cl2 (ii) PbO2 + 4HC1 PbCl2 + 2H2 0 + Cl2 Aqua Regia ICSE Previous Papers 100 Formulae Handbook for ICSE Class 9 & 10 HNO 3 + 3HC1 NOC1 + 2H 2 0 + 2 C1 Au + 3 Cl AuC13 Pt + 4 C1 PtCl4 Tests: (i) NH 3 HCL (conc) NH 4Cl (dense white fumes) (ii) AgNO3 HCl(dil) HNO 3 AgCl (white ppt) AgCl + 2NH 4 Cl Ag(NH3 ) 2 Cl + 2HCl (soluble) diamine silver chloride (iii) MnO2 4HCl(conc) MnCl2 2H 2 O + Cl2 (greenish yellow gas) suffocating odour Study Of Compounds Ammonia (a) Preparation of ammonia: (i) Laboratory preparation (from ammonium salts) Ammonium chloride + alkali (slacked lime + sal ammonia) 2NH4C1 + Ca OH 2 CaCl2 + 2H2O + 2NH3 (ii) Reactions of ammonia with drying agents. 1. Sulphuric acid (conc.) 2NH3 + H2SO4 NH4 2 SO4 2. Phosphorous pentoxide 6NH3 + P2O5 + 3H2O 2 NH4 3 3PO4 3. Calcium chloride 8NH3 + CaCl2 fused CaCl2 .8NH3 (iii) Laboratory preparation (from metal nitrides) 1. Magnesium nitride. Mg3 N2 + 6H2O 3Mg OH 2 + 2NH3 2. Calcium nitride. Ca 3 N2 + 6H2O 3Ca OH 2 + 2NH3 3. Aluminium nitride. ICSE Previous Papers 101 Formulae Handbook for ICSE Class 9 & 10 AlN + 3H2O Al OH 3 + NH3 (b) Manufacture of ammonia (i) Haber s process: N2 + 3H2 2NH3 + Temperature: 450-500 C Pressure: 200 to 900 atmosphere. Catalyst: finely divided iron Promoter: molybdenum (c) Combustibility (i) Burning of ammonia in oxygen, (greenish yellow frame) 4NH3 + 3O2 6H2O + 2N2 (ii) Catalytic oxidation of ammonia (mfg. of nitric acid-ostwalds process) 4NH3 + 5O2 Pt 6H2O + 4NO + Basic nature (d) Reaction of ammonia gas with (i) Hydrochloric acid NH3 + HCl NH4Cl (ii) Nitric acid NH3 + HNO3 NH4 NO3 (iii) Sulphuric acid 2NH3 + H2SO4 NH4 2 SO4 (iv) Water (Dissociation of aq. soln.) Monoacidic base NH4OH (e) NH4++OH Reaction of ammonia (aq. soln) with (i) Hydrochloric acid NH4OH + HC1 NH4C1 + H2O (ii) Nitric acid NH4OH + HNO3 NH4 NO3 + H2O (iii) Sulphuric acid ICSE Previous Papers 102 Formulae Handbook for ICSE Class 9 & 10 2NH4OH + H2SO4 NH4 2 SO4 +2H2O (f) Reaction of ammonia (aq. soln) with metallic salt solutions (i) Iron(II) sulphate FeSO4 + 2NH 4OH Fe OH 2 + NH 4 2 SO 4 dirty green (ii) Iron(III) chloride FeCl3 + 3NH4OH 3NH4C1 + Fe OH 3 (iii) Lead nitrate Pb NO3 2 + NH4OH PbOH + NH4 NO3 2 (iv) Zinc sulphate Zn OH 2 + NH 4 2SO4 + 2NH 4OH CuZn NH3 4 SO4 + 4H 2O colourless solution (v) Copper sulphate Cu OH 2 + NH4 2 SO4 + 2NH4OH Cu NH3 4 SO4 + 4H2O (g) As a reducing agent (ammonia gas) (i) Heated copper oxide (black baric oxide) 3CuO + 2NH3 3Cu + 3H2O + N2 (ii) Heated lead oxide (yellow amphoteric oxide) 3PbO + 2NH3 3Pb + 3H2O + N2 (iii) Chlorine(ammonia in excess) 8NH3 + 3C12 6NH 4 C1 + N2 NH3 + 3C12 NC13 + 3HC1 Study Of Compounds Nitric Acid (a) Preparation of nitric acid (i) Laboratory preparation from nitrates 1. Potassium nitrate (nitre) 200 C KNO3 + H2SO4 K.HSO4 + HNO3 ICSE Previous Papers 103 Formulae Handbook for ICSE Class 9 & 10 2. Sodium nitrate (chile salt petre) 200 C NaNO3 + H2SO4 NaHSO4 + HNO3 (ii) Atmospheric nitrogen to nitric acid 1. Nitrogen to nitric acid 300 C N2 + O2 2NO 2. Nitrogen oxide to nitric dioxide. 50 C 2NO + O2 2NO2 3. Nitrogen dioxide to nitric acid. 4NO2 + O2 + 2H2 0 4HNO3 (iii) Industrial manufacture of nitric acid - Ostwalds Process 1. Step 1 (Catalytic chamber) pt. 4NH3 + 5O2 4NO + 6H2O 700 800 C 2. Step 2 (Oxidation chamber) 50 C 2NO + O2 2NO2 3. Step 3 (Absorption tower) NO2 + O2 + 2H2O 4HNO3 (b) Chemical Properties of nitric acid (i) Stability of nitric acid Decomposition 4HNO3 2H2O + O2 + 4NO2 (ii) Ionization of dilute nitric acid 1. Dissociation HNO3 + H2O 2. H3O+ + NO3- Base (sodium hydroxide) NaOH + HNO3 NaNO3 + H2O 3. Magnesium Hydroxide Mg OH 2 + 2HNO3 Mg NO3 2 2H2O ICSE Previous Papers 104 Formulae Handbook for ICSE Class 9 & 10 4. Carbonate (calcium carbonate) PbCO3 + 2HNO3 Pb NO3 2 + H2O + CO2 5. Bicarbonate (calcium bicarbonate) Ca(HCO3 )2 + 2HNO3 Ca NO3 2 + 2H2O +2CO2 6. Sulphite (potassium sulphite) K 2SO3 + 2HNO3 2KNO3 + H2O + SO2 7. Bisulphite (calcium bisulphite) Ca HSO3 2 + 2HNO3 Ca NO3 2 + 2H2O + SO2 (iii) Non - metals (cone. Nitric acid) 1. Carbon C + 4HNO3 2H2O + CO2 + 4NO2 2. Sulphur S + 6HNO3 H2SO4 + 6NO2 + 2H2O 3. Phosphorous P + 5HNO3 H3PO4 + H2O + 5NO2 (iv) Metals (Imp) 1. dil. Nitric acid Copper 3Cu + 8HNO3 3Cu NO3 2 + 4H2O + 2NO 2. Cone. Nitric acid (hot nitric acid) Copper Cu + 4HNO3 Cu NO3 2 + 2H2O + 2NO2 (v) Aqua regia (1 part conc. HNO3 + 3 parts HCl) HNO3 + 3HCl 2H2O + 2 C1 + NOCl nitrosyl chloride (c) Test for nitric acid Heat on nitric acid 4HNO3 2H2O+ O2 + 4NO2 reddish brown fumes evolved ICSE Previous Papers 105 Formulae Handbook for ICSE Class 9 & 10 Heat on copper and cone. HNO3 Cu + 4HNO3 Cu NO3 2 + 2H2O + 2NO2 reddish brown fumes evolved Brown ring test FeSO4 + NO FeSO4 .NO nitroso ferrous sulphate Study Of Compounds Sulphuric Acid (a) Manufacture - Contact process (i) Step 1 [sulphur or pyrite burner] - roasting S + O2 SO2 4FeS2 +11O2 2Fe2 O3 + 8SO2 (ii) Step 2 [contact chamber] 2SO2 + O2 2SO3 Catalyst V2O5 temp, 450-500 C, pressure 1 -2 atm (iii) Step 3 [absorption tower] SO3 + H2SO4 H2S2O7 The oleum, so obtained is then passed into water to obtain sulphuric acid. This is then diluted to the desired extent by adding it to water. H2S2O7 + H2O 2H2SO4 (b) Chemical properties of sulphuric acid as an acid(dilute) (i) Forms hydronium ions in aq. soln. H2SO4 + 2H2O 2H3O+ + SO4 2 Reaction with (ii) Active metal (zinc/iron/magnesium) Zn + H2SO4 ZnSO4 + H2 (iii) Base(Sodium oxide) Na 2O + H2SO4 Na 2SO4 + H2O (iv) Base(Sodium hydroxide) ICSE Previous Papers 106 Formulae Handbook for ICSE Class 9 & 10 2NaOH + H2SO4 Na 2SO4 + 2H2O (v) Carbonate (Potassium carbonate) K 2CO3 + H2SO4 K 2SO4 + H2O + CO2 (vi) Sulphite (Sodium sulphite) Na 2SO3 + H2SO4 Na 2SO4 + H2O + SO2 (vii) Bisulphite (Sodium bisulphite) 2NaHSO3 + H2SO4 Na 2SO4 + 2H2O + 2SO2 (viii) Sulphide (Iron(II) sulphide) FeS + H2SO4 FeSO4 + H2S (c) As a dibasic acid (i) Basicity is two H2SO4 (ii) Dissociates in two types H 2SO4 HSO4 (iii) 2H+ +SO4 2 H + +HSO4 2 H + +SO4 2 Forms two types of salts - acid salt, normal salt 2NaOH + H 2SO4 Na 2SO4 + 2H 2O NaOH + H 2SO4 NaSO4 + 2H 2O (d) As a non volatile acid (conc.) Displaces volatile acid from salt below 200 C (i) Sodium chloride (Imp) Below 200 C NaCl + H2SO4 NaHSO4 + HC1 (ii) Sodium nitrate (Imp) Below 200 C NaNO3 +H2SO4 NaHSO4 +HNO3 (e) An oxidising agent (conc. acid) (i) Oxidation of Non-Metals 1. Carbon (Imp) ICSE Previous Papers 107 Formulae Handbook for ICSE Class 9 & 10 C + 2H2SO4 CO2 + 2H2O + 2SO2 2. Sulphur (Imp) S +2H2SO4 2H2O + 2SO2 (ii) Oxidation of Metals 1. Copper (V. Imp) Cu + 2H2SO4 CuSO4 + 2H2O + SO2 2. 3. Sucrose(sugar) (V. Imp) hissing sound C12 H22O11 12C+11H2O black porous mass of sugar charcoal Hydrated copper sulphate. blue crystal turn white crystals turn into a white anhydrous power CuSO4 .5H2O CuSO4 + 5H2O (f) Test for dil (i) (ii) H 2SO4 BaCl2 +dil H 2SO4 2HC1 + BaSO4 white ppt. insoluble in HNO3 Pb NO3 2 + dil H 2SO4 PbSO4 + 2HNO3 white ppt. of PbSO 4 (g) Test for conc. H 2SO4 (i) Cu + 2H2SO4 CuSO4 blue soln + 2H2 0 + SO2 (smell of burning matches) (Imp) (ii) NaCl +H2SO4 NaHSO4 + HC1. (HCl forms dense white fumes with ammonia.) Organic Chemistry (I) Paraffins: ALKANES - Cn H2n+2 C - C Saturated compounds undergo substitution reactions. (a) Laboratory Preparation: Decarboxylation: [mix of NaOH3CaO-soda lime-removes moisture] ICSE Previous Papers 108 Formulae Handbook for ICSE Class 9 & 10 (i) Methane CaO CH3COONa + NaOH CH4 + Na 2CO3 300 C sodium ethconate (ii) methane Ethane CaO C2 H5COONa + NaOH C2 H 6 Na 2CO 300 C sodium propionate (b) collection of gas - download displ. of water Reduction of alkyl halide: Zn, Cu couple in CH3I+2[H] CH 4 +HI alcohol (i) Iodo methane Zn, Cu couple in C2 H5 Br+2[H] C2 H 6 +HBr alcohol (ii) Bromo ethane (c) Chemical Properties: (i) Combustible - Burns with a non luminous flame: 1. Limited supply of air 2CH4 + 3O2 2CO + 4H 2 O 2C2 H6 + 7O2 4CO + 6H2 O 2. Plenty of air CH4 + 2O2 CO2 + 2H 2 O 2C2 H6 + 5O2 4CO2 + 6H 2 O (ii) Substitution Reaction: (V. Imp) 1. u.v.light CH 4 + 4C12 CC14 + 4HC1 sunlight tetra chloro methane 2. sunlight C2 H6 +6C12 CC14 + 4HC1 hexachloro ethane (iii) alkane Conversions [A] using copper tube: Cu-tube 300 C Cu-tube 300 C aldehyde acid acidified K2 Cr2 O7 acidified K2 Cr2 O7 ICSE Previous Papers 109 Formulae Handbook for ICSE Class 9 & 10 Cu-tube 2CH 4 O 2 2CH 3OH 300 C methanol Cu-tube 2CH 3 O 2 2HCHO + 2H 2O 300 C methanal Cu-tube 2HCHO O 2 2HCOOH + 2H 2 O 300 C methanoic acid Cu-tube 2C2 H 6 O 2 2C2 H 5OH 300 C ethanol Cu-tube 2C2 H 5 O 2 2CH 3CHO 2H 2O 300 C ethanal 2CH 3CHO O 2 2CH 3COOH Cu-tube 300 C ethanoic acid (iv) Using acidified K2Cr2O7 Cu-tube 2CH 4 O2 2CH3OH 300 C acidified CH3OH (O) HCHO + H 2O K 2 Cr2 O7 acidified HCHO + (O) HCOOH K 2 Cr2 O7 Cu-tube 2C2 H 6 O2 2C2 H5OH 300 C acidified C2 H5OH (O) CH3CHO H 2O K 2 Cr2 O7 acidified CH3CHO (O) CH3COOH K 2 Cr2 O7 (v) Direct Conversion of methane to methanol is 1 step MoO CH 4 O2 HCHO + H 2 O 100 C MoO C2 H6 O2 CH3CHO H 2O 100 C (II) Paraffins: ALKANES - Cn H2n C - C Ethene ICSE Previous Papers 110 Formulae Handbook for ICSE Class 9 & 10 Unsaturated compound undergoes addition reaction. (a) Laboratory preparation: (i) Dehydration of ethanol boil conc.H2SO4 170 C C2 H5OH+2[H] C2H4 +H 2O Al2O3 at 350 C (ii) Dehydrohalogenation boil C2 H5 Br+KOH KBr + H 2O + C2 H 4 alcoholic Bromo ethane (b) Chemical Properties [Addition reaction] (i) Addition Reactions 1. Nickel 300 C C2 H4 +H2 C2 H 6 2. inert solution of C2 H4 + Cl2 C2 H4C12 dichloro ethane CCl 3. inert solution of C2 H4 + Br2 C2 H4 Br2 dibromoethane CCl o 4 4 The brown colour of bromine disappears. Test for unsaturation. 4. 5. 6. 4 C2 H4 + I2 C2 H4 I2 ethyl dibromide inert soln of CCl C2 H 4 + HBr C2 H5 Br monobromo ethane C2 H 4 + H 2SO 4 C2 H5 HSO 4 ethyl hydrogen sulphate (KMnO4 solution - cold and dilute) Baeyers solution: It decolourises KMnO4 solution. It first turns into a green solution which becomes colourless with a brown residue. (ii) Combustion: C2 H4 + 3O2 2CO2 + 2H2O pale blue flame (III) Acetylene Series: ALKYNES Cn H2n-2 [-C = C-] Highly unsaturated compounds - undergo addition reaction Acetylene Ethyne C2H2 ICSE Previous Papers 111 Formulae Handbook for ICSE Class 9 & 10 (a) Laboratory preparation: CaC2 + 2H 2 O Ca OH 2 + C2 H 2 (i) Calcium Carbide (ii) Dehydrohalogenation of 1, 2 dibromoethane warm water boil C2 H 4 Br2 + 2KOH 2KBr + 2H 2 O + C2 H 2 alcoholic (b) Chemical Properties: Undergo Addition reactions Ni (i) C 2 H 2 + H 2 C2 H 4 300 C Ni : C 2 H 4 H 2 C2H 6 300 C CCl4 (ii) C2 H 2 + Br2 C 2 H 2 Br2 : C 2 H 2 Br2 Br2 C 2 H 2 Br4 CCl4 (iii) C 2 H 2 + Cl 2 C 2 H 2 Cl 2 : C 2 H 2Cl 2 Cl 2 C 2 H 2Cl 4 alcohol (iv) C 2 H 2 + I 2 C 2 H 2 I 2 : [Iodine unreactive] : C 2 H 3Cl HCl C 2 H 4Cl 2 (v) C 2 H 2 + HCl C 2 H 3Cl dichloro methane Chloroethene (vi) C2 H 2 + HBr C2 H 3 Br : C 2 H 3Br HBr C2 H 3Br2 bromo ethene dichloro methane Observation: The brown colour of bromine disappears. This is a test for unsaturation Note: (c) alkyne alkene alkane Test for ethene: (i) With ammonical cuprous chloride it forms a red precipitate of copper acetalide. (ii) With ammonical silver nitrate it forms a white precipitate of silver acetalide. (IV) ALCOHOL - R - OH Ethanol - Ethyl alcohol C2H5OH (a) Laboratory Preparation by hydrolysis of alkyl halide boil C2 H5 Br + KOH (aq) KBr + C2 H5OH ethyl bromide aqueous ethanol boil C2 H 5 Cl + KOH (aq) KCl + C2 H 5OH (ethyl alcohol) ICSE Previous Papers 112 Formulae Handbook for ICSE Class 9 & 10 (b) Industrial Preparation by hydration of ethene with cone. H2SO4 80 C C2 H 4 H 2SO 4 C2 H 5 HSO 4 30 atms pressure C2 H5 HSO 4 + H 2O C2 H 5OH + H 2SO 4 (c) Fermentation of sugar [sucrose] yeast C12 H 22O11 H 2O C6 H12O6 + C6 H12 O6 invertese glucose fructose yeast zymase C6 H12O6 C2 H5OH+ 2CO 2 fermentation (d) Chemical Properties of ethanol (i) (ii) Combustion C2 H5OH + 3O2 2CO2 + 3H2O blue flame Dehydration conc. H2SO4 C2 H5OH + C2 H 4 + H 2O Imp 170 C ethene conc. H2SO4 C2H5OH C2H5 -O-C2H5 diethyl ether 140 C (iii) Estrification conc. H2SO4 C2 H5OH + CH3COOH CH3COOC2 H5 + H2 O ethyl acetate (iv) Test for alcohols 2C2 H5OH + 2Na 2C2 H5OONa + H2 sodium ethoxide Effervescence - A gas bums with a pop sound + blue flame - H2 (e) Conversion: (V) CARBOXYLIC ACIDS [R - COOH] (a) Preparation of acetic acid (i) Ethanoic acid acidified K2Cr2 O7 C2 H5OH + O CH3CHO + H 2 0 ethanol acetaldehyde acidified K2Cr2 O7 CH3CHO + O CH3COOH acetic acid ethanoic acid ICSE Previous Papers 113 Formulae Handbook for ICSE Class 9 & 10 (ii) Hydrolysis of ethyl acetate conc.H2SO4 CH3COOC2 H5 + H 2 O CH3COOH + C2 H5OH ethyl acetate (b) Chemical Properties: (i) Acidic Nature: Acid + alkali salt + H2O CH3COOH + NaOH CH3COONa + H 2 O Na-acetate CH3COOH + NH 4 OH CH3COONH 4 + H 2 O Ammonium acetate 2CH3COOH + Ca OH 2 CH3COO 2 Ca + 2H 2 O calcium acetate (ii) Active metal + acetic acid salt [acetate] + H2 2CH3COOH + 2Na 2CH3COONa + H 2 2CH3COOH + Mg CH3COO 2 Mg + H 2 (iii) Carbonates (CO3--)/Bicarbonates + Acid salt + H2O + CO2 NaHCO3 + CH3COOH CH3COONa + H 2 O + CO2 Na 2CO3 + 2CH3COOH 2CH3COONa + H 2O + CO2 Estrification: conc. H2SO4 Alcohol + Acetic acid Ethyl etheonate + H 2O ethanol ester with a fruity smell Mathematics X Compound Interest 1. The interest calculated every year on the amount due at the end of the year is known as compound interest. 2. The amount at the end of every year becomes the principal for next year and so on. 3. For the first year, Simple Interest and Compound Interest is the same. 4. If some amount of money is paid back at the end of every year, we subtract the money paid back from the amount due at the end of every year to get the amount balance. This amount balance becomes the principal for the next year and so on. ICSE Previous Papers 114 Formulae Handbook for ICSE Class 9 & 10 5. If the interest is calculated semi annually then take time as calculate the interest for every 6. 1 year and 2 1 year. 2 For consecutive years: To find rate percent. (i) If interests are given: CI 2 - CI1 = CI1 R l 100 (ii) If amounts are given: A 2 A1 A1 R l 100 7. If value of a machine or population after x years is asked take the given value of the machine or population as the principal and calculate the amount. 8. If value of a machine or population before x years is asked take the given value of the machine or population as the amount and calculate the principal. 9. If the depreciation for two successive years are given in order to get rate percent use: D1 D2 D1 R l 100 Formula method: 10. A = P(1 + R n ) 100 ......Compounded yearly. R 2n ) ...... Compounded half yearly. 100 R R 12 A = P (1 + 1 ) (1 + 2 ) ...... If rates are different for different 100 100 years. R R 13. A = P(1 + ) (1 + 2 ) ......If lime is given as a fraction, n 100 100 is the whole number 11. A = P (1 + 14. For non-consecutive years: To-get rate percent. (i) R n ) where CI x and CI y are the 100 interest for the two non-consecutive years and n is the time difference between the two non-consecutive years. If interests are given CI x =CI y (1+ ICSE Previous Papers 115 Formulae Handbook for ICSE Class 9 & 10 (ii) If amounts arc given: A x = A y (1+ R n ) where A x and A y are the 100 interest for the two non-consecutive years and n is the time difference between the two non-consecutive years. 15. Depreciation A = P (1 - R n ) where A is the latter value or the depreciated 100 value d and P is the former value. 16. R n ) where A is the latter value and P is the former 100 value, if the population is increasing. Population : A = P (1 + Shares And Dividend Face value is also called the nominal value of the shares. investment market value 1. No. of shares (n) = 2. Sale proceeds = no. of shares selling price of each share. 3. Annual income or dividend = 4. Percentage income on investment or yield % = 5. Yield % market value = dividend % face value. 6. Market value = face value + premium (shares available above par) 7. Market value = face value - discount (shares available below par) 8. Shares available at par means market value = face value. dividend % no. of shares face value 100 income 100 investment Banking To fill the pass book: Add the credit, subtract the debit. Qualifying balance is the minimum balance between 10th and the last day of the month. The month in which the account is opened after the 10th is not counted. The month in which the account is closed is not counted Recurring Deposit: ICSE Previous Papers 116 Formulae Handbook for ICSE Class 9 & 10 If the table of recurring deposit amount is not given, then Step 1: Find the equivalent principal using the formula: P = n(n+1)x 2 Where n = the number of equal monthly instalments and x = the amount deposited every month. Step 2: Find the interest on this equivalent principal for one month using PTR 1 where T = 100 12 PR i.e. I 1200 I= Step 3: The amount payable at the end of the term i.e maturity value will be A = nx +1. Sales Tax And Value Added Tax (Vat) (a) Sales Tax 1. Rate of sales tax = 2. Nett selling price = selling price + sales tax 3. Rebate means discount which is always calculated on the list price 4. Selling price = (list price - rebate) + sales tax. (b) Value Added Tax (VAT) 1. Calculate the tax on the selling price of the first seller. 2. Calculate the tax on the selling price of the second seller. 3. VAT paid by the second seller = tax obtained in step 2 - tax obtained in step 1. 4. Calculate the tax on the selling price of third seller. 5. VAT paid by the third seller = tax obtained in step 5 - tax obtained in step 2. sales tax 100 selling price And so on Inequations ICSE Previous Papers 117 Formulae Handbook for ICSE Class 9 & 10 R or Q = {all negative numbers, all positive numbers, 0, including fractions} A B : common elements between A and B A B : all elements that belong to A and B Compliment of A written as A1 element other than belonging to A A - B: all elements in A which do not belong to B. Quadratic Equations For quadratic equation of the form ax2 + bx + c = 0; a 0 1. The roots of the quadratic are x 2. b2 - 4ac is called the discriminant. b b 2 4ac 2a (i) if b2 - 4ac < 0, roots are complex or imaginary or unreal (ii) if b2 - 4ac = 0, roots are real and equal. (iii) if b2 - 4ac > 0, roots are real and unequal. (iv) if b2 - 4ac is a perfect square, roots are rational. (v) if b2 - 4ac is not a perfect square, roots are irrational. -b c : product of roots = a a 3. Sum of roots = 4. 2 Equation: x - sum of roots x + product of roots = 0. ICSE Previous Papers 118 Formulae Handbook for ICSE Class 9 & 10 Remainder And Factor Theorem Remainder Theorem: When a polynomial f(x) is divided by (x - a), then remainder = f(a) Factor Theorem: If x - a is a factor of f(x), then f(a) = 0. Ratio And Proportion 1. If a : b is the given ratio then a is called antecedent and b is called the Consequent. 2. duplicate ratio a2 : b2 triplicate ratio a3 : b3 sub-duplicate ratio a: b sub-triplicate ratio 3 3 b 1 1 : or b : a a a reciprocal ratio 3. a: If the given ratios a : b and c : d; then compound ratio is a c : b d or Simply multiply the ratios. 4. In a : b = c : d, a and d are called extremes (end - terms) and b and c are called means (middle terms). a : b= c : d a c = a d = b c b d product of extremes = product of means. 5. If a, b and c are in continued proportion a : b = b : c or b2 = ac 6. Fourth proportion of a, b, c is 7. Third proportion of a, b is a : b = b : x where x is third proportion. 8. Mean proportion of a, b is a : x = x : b where x is the mean proportion. 9. Given that a : b = c : x where x is fourth proportion. a c = b d ICSE Previous Papers 119 Formulae Handbook for ICSE Class 9 & 10 b d = a c a b = c d Invertendo a+b c+d = b d a-b c-d = b d componendo a+b c+d = a-b c-d componendo dividendo altcrnendo dividendo a+c = value of each ratio theorem on equal ratios b+d ratio Matrices Unit or identity Matrix: A diagonal matrix, in which each element of the leading diagonal is unity. It is denoted by I. 1 0 e.g. I = 0 1 Transpose of a matrix: Transpose of a matrix obtained on interchanging its row and columns. If A is matrix, then its transpose is denoted by At. 3 0 3 1 2 1 6 e.g. I = t , then its transpose A = 0 6 4 2 4 Addition of two matrices: Possible only if both matrices have the same order. If A, B and C are the matrices of same order, then 1. A + B = B + A, the addition of matrices is commutative. 2. A + (B + C) = (A + B) + C, the addition of matrices is associative. 3. A+ X = B X= B - A ICSE Previous Papers 120 Formulae Handbook for ICSE Class 9 & 10 4. 0 0 Null matrix = 0 0 Multiplication of two matrices: Two matrices are compatible for multiplication if and only if the number of columns of the first matrix is equal to the number of rows of the second. e.g. Am x n X Bn x p Cm x p Remember: 1. For any matrices A, B and C which are compatible for multiplication the number of column of first should be equal to number of rows of the second. 2. Number of rows of first is equal to number of rows of product matrix. 3. Number of columns of second matrix is equal to number of columns of product matrix. 4. In general AB BA i.e product of matrices is not commutative. 5. (AB) C = A(BC) i.e product of matrices is associative. 6. If A 0 and AB = AC, then it is not necessary that B = C. 7. If AB = 0, then it is not necessary that A = 0 or B =0. 8. If A = 0 or B = 0, then AB = 0 = BA. 9. A (B + C) = AB + AC i.e multiplication of matrices is distributive with respect to matrix addition. 10. I - the unit matrix is called the identity matrix for multiplication, i.e. A I = I A = A. 11. a b If M = c d e f then order of M is 2 1 a b = e f then order of M is 1 2 c d If M Co-Ordinate Geometry / Equation Of A Line 1. Reflection: M x x , y = x, - y M y x, y = - x, y M 0 x, y = - x - y ICSE Previous Papers 121 Formulae Handbook for ICSE Class 9 & 10 2. Invariant points: (i) Do not change on being reflected. (ii) They lie on the reflecting line. 3. Equation of a line parallel to x -axis and a units away is y = a Equation of a line parallel to y-axis and b units away is x = b 4. Equation of the x -axis is y = 0. Equation of y-axis is x = 0 5. The mirror line is the perpendicular bisector of the segment joining the point and its image. 6. x co-ordinate is also known as abscissa and y co-ordinate is also known as ordinate 7. If a point lies on the x -axis its y co-ordinate is zero. It is assumed as ( x , 0) 8. If a point lies on the y-axis its x -co-ordinate is zero. It is assumed as (0, y) AB = x2 x1 y2 y1 2 9. Distance formula: 10. Section formula: 11. Mid point formula: 12. Slope (gradient): m = tan . ( -inclination - line makes with the positive direction of x -axis) 14. y = m x + c (m - slope, c y - intercept) 15. If two lines are parallel m1= m2 : perpendicular m1 m2 = -1 16. Equation of line: To find the equation, 17. mx nx1 my2 ny1 (x, y) = 2 , m+n m+n x x y y2 (x, y) = 1 2 , 1 2 2 (i) Slope point form: y y1 = m ( x - x 1) (ii) slope intercept form: y = m x + c : m slope, c = y intercept. Equidistant: PA = PB. Use distance formula. i. Isosceles triangle Two sides equal ii. Right angled triangle Pythagoras satisfied iii. Equilateral triangle All sides equal iv. Parallelogram Opposite sides are equal ICSE Previous Papers 122 Formulae Handbook for ICSE Class 9 & 10 18. V. Rectangle Opposite sides and diagonals equal vi. Rhombus All sides equal vii. Square All sides and diagonals equal 19. Point of Trisection : A divides PQ in ratio 1 : 2 and B divides PQ in ratio 2 : 1. 20. Circumcentre: Point where perpendicular bisectors of sides meet, it is equidistant from the vertices. OA = OB = OC OA2 = OB2 gives equation 1 OB2 = OC2 gives equation 2 Solve simultaneously. 21. Orthocentre: Find the equation of ALTITIDES of two sides of the given triangle. Solve the above equations simultaneously. 22. Centroid: co-ordinates of centroid x1 x2 x3 y1 y2 y2 , 3 3 where ( x 1, y1), ( x 2, y2) and ( x 3, y3) are the vertices of triangle. 23. To find the fourth vertex of parallelogram, rectangle, square, rhombus. Suppose A, B, D are given to find C. (i) Find M - mid point of BD (ii) Find C - using M as mid point of AC. 24. If three points A, B and C are collinear - slope of AB = slope of BC. 25. To find the x - intercept of a line substitute y = 0 in the equation. ICSE Previous Papers 123 Formulae Handbook for ICSE Class 9 & 10 26. To find the y-intercept of a line substitute x = 0 in the equation. 27. To find the point of intersection of two lines - solve the equation simultaneously. 28. If a point lies on a line it satisfies the equation of the line. 29. To find equation of ALTITUDE: (i) Find slope of BC. (ii) Slope of altitude is negative reciprocal B, (iii) Use slope point form, given A is a point on the line. 30. To find equation of MEDIAN: (i) Use mid point formula to find M. (ii) A and M two points are known, get slope and use slope point form. 31. To find the equation of PERPENDICULAR BISECTOR: (i) Mid point of AB (ii) Slope of AB (iii) Negative reciprocal of AB ~ (iv) Use slope point form. ICSE Previous Papers 124 Formulae Handbook for ICSE Class 9 & 10 Symmetry Figure No. of lines of symmetry Whether has point symmetry Scalene triangle no line of symmetry no point symmetry Isosceles triangle one line no point symmetry Equilateral triangle Three lines no point symmetry Parallelogram no line of symmetry One point symmetry Rectangle Two lines One point symmetry Rhombus Two lines-diagonal One point symmetry Square Four lines One point symmetry Kite One line no Isosceles trapezium One line no Regular pentagon Five lines no Regular hexagon Six lines One point symmetry Circle Infinite One Similarity Postulates of similarity: (i) A - A or A - A - A (ii) S A-S (iii) S-S-S Properties of similar triangles: (i) If two triangles are similar, their corresponding sides are proportional. (ii) Their corresponding angles are equal. (iii) The ratio of their areas is square of their corresponding sides. If two triangles have a common vertex then the ratio of their areas is equal to ratio of their bases. ICSE Previous Papers 125 Formulae Handbook for ICSE Class 9 & 10 ABD 2 ABC 5 ABD 2 ; ; ADC 3 ADC 3 ABC 5 If angle is bisected by segment AD in triangle ABC then, BD AB DC AC The length of the perpendicular from the right angle of a right angled triangle to the hypotenuse is equal to the product of the parts of the hypotenuse i.e. BD2 = AD x DC. Mid-point Theorem: The segment joining the midpoints of two sides of a triangle is parallel to the third side and is half of the third side. LM = 1 BC and LM // BC 2 Basic proportionality Theorem: A line drawn parallel to the base of the triangle divides the other two sides proportionally. ICSE Previous Papers 126 Formulae Handbook for ICSE Class 9 & 10 If PQ // BC then AP AQ = PB QC CENTRIOD : Point intersection of medians in a triangle. A centriod divides a median in ratio 2:1. INCENTRE: Point of intersection of angle bisectors of triangle. It is equidistant from the sides of the triangle. CIRCUMCENTRE: Point of intersection of perpendicular bisector of sides. It is equidistant from the vertices of the triangle. ORTHOCENTRE: Point of intersection of the altitudes of triangle. ICSE Previous Papers 127 Formulae Handbook for ICSE Class 9 & 10 Locus Points to remember for construction: 1. Equidistant from two fixed points (AP = BP) draw perpendicular bisector. 2. Equidistant from two sides (P is equidistant from AB and BC) draw angle bisector. 3. At a fixed distance from a fixed point draw CIRCLE 4. At a fixed distance from a fixed line draw a line parallel, equal to the given distance on both sides. Circle 1. The segment joining the centre of the circle to mid point of chord is perpendicular to the chord. 2. CONVERSE: Perpendicular segment from the centre bisects the chord. 3. Equal chords in a circle or equal circles are equidistant from the centre. 4. CONVERSE: Equidistant chords in a circle or equal circles are equal. 5. The tangent and the radius at any point on a circle are perpendicular to each other. ICSE Previous Papers 128 Formulae Handbook for ICSE Class 9 & 10 6. 7. Tangent segment from an exterior points is equal in length and the segment joining the centre to the exterior point, bisects the angle between the tangents, i.e PT = PQ DCT = d 2 - R - r 2 DCT - Direct Common Tangent 8. TCT= d 2 - R - r 2 TCT - Transverse Common Tangent 9. If two circle touch each other externally their centres and the point of contact lie on a straight line, the distance between their centres is (R + r) 10. If two circle touch each other internally, their centres and the point of contact lie on a straight line the distance between their centres is (R - r). ICSE Previous Papers 129 Formulae Handbook for ICSE Class 9 & 10 11. The angle subtended by an arc at the centre is twice the angle subtended on the circumference. Angle Properties 1. The angle subtended by an arc at the centre is twice the angle subtended on the circumference. 2. Angles in the same segment of a circle are equal. 3. The angle inscribed in a semicircle is a right angle. 4. The opposite angles of a cyclic quadrilateral are supplementary. A + C = 180 B + D = 180 5. The exterior angle of a cyclic quadrilateral is equal to interior opposite angle ICSE Previous Papers 130 Formulae Handbook for ICSE Class 9 & 10 6. If two chords are equal, the corresponding arcs are equal and converse. 7. If two chords are equal, the angle inscribed on the circumference and the centre are also equal. 8. If two chords of a circle intersect internally or externally then the product of the lengths of their segments are equal. AP BP = CP DP 9. The angle between a tangent and a chord through the point of contact is equal to an angle in the alternate segment. 10. If a chord and a tangent intersect externally, then the product of the lengths of the segments of the chord is equal to the square of the length of the tangent from the point of contact to the point of intersection. PT2 = PA x PB 11. Base angles of isosceles triangle are congruent. 12. Sum of angles of triangle = 180 13. Sum of angles of quadrilateral = 360 ICSE Previous Papers 131 Formulae Handbook for ICSE Class 9 & 10 14. Opposite angles of parallelogram are equal. Area Of Circle Parallelogram Rectangle Area l b base x height Area 1 2 dA 2 1 d1 d 2 2 1 a + b h 2 4l Semicircle Quadrant Ring Sector r 2 1 2 r 2 1 2 r 4 R+r R-r r 2 360 Perimeter 2 r or 7 d only curved = r Area of a scalene triangle A = Area of Rhombus Trapezium Circle r + 2r s= l 2 or 2 1 b Perimeter where Square r + 2r 2 2 r 360 s s - a s - b s - c a+b+c 2 1 (b h) 2 3 4 Area of equilateral = No. ot revolution = side 2 distance covered distance covered = circumference 2 r Mensuration CSA - Curved Surface Area TSA - Total Surface Area Cylinder VOLUME CSA TSA r 2 h 2 rh 2 r r + h ICSE Previous Papers 132 Formulae Handbook for ICSE Class 9 & 10 r r + l where l = 2 4 3 r 3 Sphere Hollow rl 1 2 r h 3 Cone h R + r R - r r2 +h2 4 r 2 4 r 2 2 rh + 2 RH 2 Rh + 2 rh + 2 R 2 - r 2 t = (R - r) Cylinder t = thickness VOLUME CSA TSA Hemisphere 2 3 r 3 2 r 2 3 r 2 Hollow Sphere 4 R 3 - r3 3 CUBE: Volume = CUBOID: l3 ; diagonal = 1 3 Volume = l b h ; diagonal = TSA = 6l 2 l 2 + b2 + h 2 ; TSA = 2 l b + b h + l h 1. MELTING AND RECASTING Volume of larger = n volume of smaller. 2. DENSITY Weight = density volume (use corresponding formula) 3. TIME Volume of water discharged per second = r 2 speed (use r = radius of pipe) Time = 4. Volume of tank r 2 speed DISPLACEMENT n volume of each solid submereged = r 2 h ICSE Previous Papers 133 Formulae Handbook for ICSE Class 9 & 10 r - radius of cylinder vessel, h - rise in level 5. TENT Area of each = rl rl 2 rh length width of canvass = Area of canvas = rl + length width - area of canvas 6. ROLLER distance travelled in n revolutions = area covered in 'n' revolutions = 7. 2 rn 2 rhn (i) Volume of well = volume of platform r 2 h = l b h (ii) EMBANKMENT Volume of well = volume of hollow cylinder ar2h = (R + r) (R - r)h 8. SECTOR FOLDED TO FORM CONE ICSE Previous Papers 134 Formulae Handbook for ICSE Class 9 & 10 9. (i) area of the sector = curved surface of the cone (ii) the radius r of the sector is the slant height of the cone. (i) Volume of remaining solid = R H - r h 2 1 3 TSA of remaining solid = 2 RH + R 2 + R + r R - r + rL (ii) Volume = TSA = (iii) 2 3 1 r + r 2 h + r 3 3 3 2 r 2 + 2 rh + rl BUOY: volume = TSA = rl 2 r 1 3 2 r h + r 3 3 3 2 ICSE Previous Papers 2 135 Formulae Handbook for ICSE Class 9 & 10 Trigonometry Trigonometric Identities: 1. sin tan cos 2. cos cot sin 3. 1 cosec sin 4. 1 sec cos 5. sin 2 + cos2 = 1 6. 1 + tan 2 = sec2 7. 1 + cot 2 = cosec2 8. sin = cos 90 - 9. cos = sin 90 - 10. tan = cot 90 - 11. cot = tan 90 - 12. sec = cosec 90 - 13. cosec = sec 90 - ICSE Previous Papers 136 Formulae Handbook for ICSE Class 9 & 10 Fundamentals: Trigonometry is the branch of mathematics dealing with the relations of the sides and angles of triangles and with the relevant functions of any angles. Trigonometric Ratios: The values of the ratios of the sides of any right triangle with respect to any angle (other than 90 ) are called trigonometric ratios of that angle. For example: In right triangle ABC, the ratios of the sides of the triangle with respect to A are called trigonometric ratios of A. There are six different trigonometric ratios as follows: 1. Sine A = 2. Cosine A = 3. Tangent A = 4. Cosecant A = 5. Secant A = 6. Cotangent A = Opposite side to angle A Hypotenuse Adjacent side to angle A Hypotenuse Opposite side to angle A Adjacent side to angle A Hypotenuse Opposite side to angle A Hypotenuse Adjacent side to angle A Adjacent side to angle A Opposite side to angle A = = = = = = Tips: 1. sin A is written for sine A. 2. cos A is written for cosine A. 3. tan A is written for tangent A. 4. cosec A is written for cosecant A. 5. sec A is written for secant A. 6. cot A is written for cotangent A. ICSE Previous Papers BC AC AB AC BC AB AC BC AC AB AB BC = = = = = = Perpendicular Hypotenuse Base Hypotenuse Perpendicular Base Hypotenuse Perpendicular Hypotenuse Base Base Perpendicular 137 Formulae Handbook for ICSE Class 9 & 10 Short way to learn above ratios: Just learn first three, because if you see other three are reciprocals of first three respectively. Let P denotes perpendicular, B for base and H for hypotenuse. sin cos tan PANDIT (P) BADRI (B) PRASAD HAR (H) HAR (H) BOLE (P) (B) Similarly: cosec sec cot HAR (H) HAR (H) BOLE PADNIT (P) BADRI (B) PRASAD (B) (P) Relation between Trigonometric Ratios: sin = 1 cosec OR cosec = 1 sin cos = 1 sec OR sec = 1 cos tan = 1 cot OR cot = 1 tan tan = sin cot OR cot = cos sin Trigonometric Ratios of Some Specific Angles: In this part, we will put values of angles as 0 , 30 , 45 , 60 and 90 , hence we will find ratios. 0 30 45 Sin 0 1/2 1/ 2 3/2 1 Cos 1 1/ 2 1/2 0 Tan 0 1/ 3 3 Cosec 2 2 2/ 3 1 Sec 1 2/ 3 2 2 Cot 1/ 3 0 3/2 3 1 1 60 90 How to remember trigonometric ratios of some specific angles? 1. First of all learn only sin row. If you can't learn then follow the step to find sin row ICSE Previous Papers 138 Formulae Handbook for ICSE Class 9 & 10 0 30 Sin 0 4 1 4 0 1/2 45 2 4 1/ 2 60 90 3 4 4 4 3/2 1 2. For cos row, write all the values of sin row in inverse order, i.e. from right to left. 3. For 4. For 5. For 6. For sin cot 1 cosec = sin 1 sec = cos cos cot = sin tan = Trigonometric Ratios of Complementary Angles: In ABC, B = 90 , Let A = , hence C = 90 - . Thus angles and (90 - ) are complementary angles. ICSE Previous Papers 139 Formulae Handbook for ICSE Class 9 & 10 Fundamental Trigonometric Identities: ICSE Previous Papers 140 Formulae Handbook for ICSE Class 9 & 10 Statistics MEAN Ungrouped Data mean = MEDIAN x i) arrange in ascending order n ii) Median rank n + 1 = item 2 mean = Grouped data without C. I. Grouped data with C. I. fx f i) find cumulative frequency ii) median rank N + 1 = 2 [x, c, f] th Direct method fx mean = f Short cut Method fd mean = A + f i) find CF ii) PLOT OGIVE iii) median Step deviation method v) upper quartile mean = A + fu f i MODE number that appears the most x with maximum frequency i) PLOT HISTOGRAM N item 4 = iv) lower quartile N item 4 = 3N item 4 = Interquartile range = Q3 Q1 semi interquartile range = Q3 Q1 2 Fundamentals: 1. The word statistics is used in both singular as well as plural. 2. In singular, it means "science of collection, presentation, analysis and interpretation of numerical data". 3. In plural, it means "numerical facts collected with definite purpose". 4. The number of times an observation occurs in the given data is called the frequency. NCERT Solutions CBSE Sample Papers RD Sharma Solutions 141 Formulae Handbook for ICSE Class 9 & 10 5. Frequency distribution is of two types : i. Discrete Frequency distribution ii. Continuous or Grouped Frequency distribution 6. Classes/class intervals are the groups in which all the observations are divided. 7. Suppose class-interval is 10-20, then 10 is called lower limit and 20 is called upper limit of the class. 8. Mid-value of class-interval is called Class-mark Class-mark = lower limit upper limit 2 Class-mark = lower limit + 1 2 9. If the frequency of first class interval is added to the frequency of second class and this sum is added to third class and so on then frequencies so obtained are known as Cumulative Frequency (c.f.). 10. The commonly used measures of central tendency are as follows : Arithmetic Mean (MEAN), Geometric Mean, Harmonic Mean, Median and Mode. Acc. to course, we will study mean, median & mode Relation between mean, median and mode: 3 Median = Mode + 2 Mean Mean Of Grouped Data: If x1 , x2 , x3 , ....... xn , are observations with respective frequency f 1 , f 2 , f3 , ....... fn , it means observation x1 occurs f 1 times, observation x 2 occurs f 2 times and so on. Mean is denoted by x . There are three different ways to find the mean of a grouped data which are: 1. Direct Method. 2. Assumed Mean Method. 3. Shortcut Method (Step-Deviation Method). 1. Direct Method: Mean (X) = x Sum of all the observations No. of observations f1 x1 f 2 x2 ............ f n xn f1 f 2 ........... f n n x fx i 1 n i i f i 1 i 2. Assumed Mean Method: The formula used in the assumed mean method is given as below: NCERT Solutions CBSE Sample Papers RD Sharma Solutions 142 Formulae Handbook for ICSE Class 9 & 10 n fd Mean (x ) = a+ i i 1 n f i 1 i i Where a is any arbitrary value, chosen as assumed mean (somewhere in the middle of xi ), and di xi a . 3. Step-Deviation Method (Shortcut Method): The formula used is given below: fi ui (x ) = a + f i h , Where a is any arbitrary value, chosen as assumed mean (somewhere in the middle of xi ). h = class-size ui and xi a h Combined Mean: If x1 and x2 are the means of two groups having same unit of measurement computed from n1 and n2 values. Mean (x) n1 x1 n2 x2 n1 n2 Arithmetic mean of raw data (when frequency is not given): The arithmetic mean of a raw data is obtained by adding all the values of the variables and dividing the sum by total number of values that are added. Arithmetic mean (x)= x1 x 2 ... x n 1 = n n n x i i 1 Median of Grouped Data: Condition I: When the data is discrete. Step 1: Arrange data in ascending order. Step 2: If the total frequency n is odd: n 1 Then, , observation is the median. 2 th Step 3: If the total frequency n is even: th n th n Then, mean of and 1 , observations the median. 2 2 Condition II: When the data is continuous and in the form of frequency distribution: NCERT Solutions CBSE Sample Papers RD Sharma Solutions 143 Formulae Handbook for ICSE Class 9 & 10 n 2 c Then, Median = l h f Median class = the class whose cumulative frequency is greater than (nearest to) Where, N . 2 l = lower limit of median class f = frequency of median class h = class-size n = number of observations c = cumulative frequency of class preceding the median class. Mode of Grouped Data: The class with maximum frequency is called the modal class. f1 f 0 Mode l h 2 f1 f 0 f 2 Where, l = lower limit of the modal class h = class-size f1 = frequency of the modal class f 0 = frequency of the class preceding the modal class f 2 = frequency of the class succeeding the modal class OGIVE or Cumulative Frequency Curve: The term ogive is derived from the word ogee. An ogee is a shape consisting of concave arc flowing into a convex arc. An OGIVE of less than type: It is drawn for less than type cumulative frequency distribution. Here we mark upper limit of class interval on horizontal axis while respective cumulative frequency is marked on vertical axis and plot the corresponding points and join them by a free, hand curve. Cumulative frequency is counted up to down. An OGIVE of more than type: It is drawn for more than type cumulative frequency distribution. Here we mark lower limit of class interval on horizontal axis while respective cumulative frequency is marked on vertical axis and plot the corresponding points and join them by a free hand curve. Cumulative frequency is counted down to up. Note: Intersecting point of less than ogive and more than ogive gives median. Note: The median of the grouped data can be obtained on any one of the ogive by locating N 2 on the y-axis. Locate corresponding point on the ogive, x-coordinate of that point determines the median of the data. NCERT Solutions CBSE Sample Papers RD Sharma Solutions 144 Formulae Handbook for ICSE Class 9 & 10 Miscellaneous EXPANSION x y x 2 y2 2 xy 3 x y x3 y3 3xy x y 2 x y + z x 2 y2 z2 2 xy 2 x + y 2 x - y 4 xy 2 1. 2. 3. 4. 2yz 2 xz FACTORISATION 1. x 2 y2 2. x 3 y3 3. x 3 y3 x x x y x y y x 2 xy y 2 y x 2 xy y 2 INDICES 1. a m a n =a m+n a m m-n 2. n =a a 3. (a m ) n a mn 4. n a a1/n 5. n a 2 a1/n 6. a 0 1 7. a n 1 an n n n b a b 8. n a b a 9. (a b) = a m b m NCERT Solutions CBSE Sample Papers RD Sharma Solutions 145 Formulae Handbook for ICSE Class 9 & 10 Logarithms 1. 2. 3. 4. log a b = log a + log b a log =log a - log b b log a b = bloga If a = bc then log b a = c SPEED = distance Time 5 18 18 To convert from m/s to km/hr multiply by 5 To convert from km/hr to m/s multiply by Profit Loss And Discount 1. Profit = Selling price - Cost price 2. Loss = Cost price - Selling price 3. Profit %= 4. Loss %= 5. Discount = Market price - Selling price 6. Discount % = NCERT Solutions Profit 100 Cost Price Loss 100 Cost Price Discount x 100 Market price CBSE Sample Papers RD Sharma Solutions 146 Formulae Handbook for ICSE Class 9 & 10 Mathematical Formulae Algebra: NCERT Solutions CBSE Sample Papers RD Sharma Solutions 147 Formulae Handbook for ICSE Class 9 & 10 NCERT Solutions CBSE Sample Papers RD Sharma Solutions 148 Formulae Handbook for ICSE Class 9 & 10 Power series with real variables: NCERT Solutions CBSE Sample Papers RD Sharma Solutions 149 Formulae Handbook for ICSE Class 9 & 10 Integer series: Physical Constants: List of perfect Square roots: NCERT Solutions CBSE Sample Papers RD Sharma Solutions 150 Formulae Handbook for ICSE Class 9 & 10 1 1 4 2 9 3 16 4 25 5 36 49 64 6 7 8 81 9 100 10 121 11 144 12 169 13 196 14 225 15 256 16 NCERT Solutions 289 17 1156 34 324 18 1225 35 361 19 1296 36 400 20 1369 37 441 21 1444 38 484 22 1521 39 529 23 1600 40 576 24 1681 41 625 25 1764 42 676 26 1849 43 729 27 1936 44 784 28 2025 45 841 29 2116 46 900 30 2209 47 961 31 2304 48 1024 32 2401 49 1089 33 2500 50 CBSE Sample Papers RD Sharma Solutions 151 Formulae Handbook for ICSE Class 9 & 10 Perfect Cube Roots chart: Surface Area and Volume Formulas: SURFACE AREA (SA) of a: NCERT Solutions CBSE Sample Papers RD Sharma Solutions 152 Formulae Handbook for ICSE Class 9 & 10 PERIMETER (P) of a: VOLUME (V) of a: Equations: * If you find any mistakes or want to add more content to this eBook, please mail us at learncbse.in@gmail.com NCERT Solutions CBSE Sample Papers RD Sharma Solutions
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